[    {        "title": "1512.09051v1.Magnetization_dynamics_and_spin_pumping_induced_by_standing_elastic_waves.pdf",        "content": "Magnetization dynamics and spin pumping induced by standing elastic waves\nA. V . Azovtsev and N. A. Pertsev\nIoffe Institute, 194021 St. Petersburg, Russia\n(Version 1 – submitted 30.12.2015)\nThe magnetization dynamics induced by standing elastic waves excited in a thin ferromagnetic film is described with the aid of micromagneticsimulations taking into account the magnetoelastic coupling between spins and lattice strains. Our calculations are based on the numerical solution ofthe Landau-Lifshitz-Gilbert equation comprising the damping term and the effective magnetic field with all relevant contributions resulting from theexchange interaction, magnetocrystalline anisotropy, magnetoelastic coupling, and dipole-dipole magnetostatic interactions. The simulations havebeen performed for the 2 nm thick Fe 81Ga19 film dynamically strained by longitudinal and transverse standing waves with various frequencies, which\nspan a wide range around the resonance frequency res of coherent magnetization precession in unstrained Fe 81Ga19  film. It is found that standing\nelastic waves give rise to complex local magnetization dynamics and spatially inhomogeneous dynamic magnetic patterns. The spatio-temporaldistributions of the magnetization oscillations in standing elastic waves have the form of standing spin waves with the same wavelength. Remarkably,the amplitude of magnetization precession does not go to zero at the nodes of these spin waves, which cannot be precisely described by simpleanalytical formulae. In the steady-state regime, the magnetization oscillates with the frequency of elastic wave, except for the case of longitudinal\nwaves with frequencies well below  res, where the magnetization precesses with a  variable frequency strongly exceeding the wave frequency.\nImportantly, the precession amplitude at the antinodes of standing spin waves strongly increases when the frequency of elastic wave becomes close to\nres. The results obtained for the magnetization dynamics driven by elastic waves are used to calculate the spin current pumped from the dynamically\nstrained ferromagnet into adjacent paramagnetic metal. The numerical calculations demonstrate that the transverse charge current in the paramagneticlayer, which is created by the spin current via the inverse spin Hall effect, is high enough to be measured experimentally.\nI. INTRODUCTION\nThe  magnetization  dynamics  in  ferromagnets  is\nusually excited and controlled by magnetic fields or spin-\npolarized electric currents [ 1-4].  However, these methods\nare generally associated with high energy losses, which\nmake  them  unsuitable  for  applications  in  advanced\nspintronic  devices  aimed  at  low  power  consumption.\nTherefore, intensive research efforts are currently focused\non the development of alternative excitation techniques,\nsuch as the exploitation of elastic waves and strain pulses\nto induce the magnetization precession and switching in\nferromagnets  [5-8].  This  “acoustic  spintronics”  [ 6],  a\npromising emerging direction in the modern physics of\nferromagnets,  is  based  on  the  magnetoelastic  coupling\nbetween spins and lattice strains [ 9], which leads to a\nvariety of interesting physical phenomena. In particular,\nrecent  experimental  studies  revealed  dynamic\nmodulations of the magnetization direction by picosecond\nacoustic  pulses  [5],  spin  pumping  via  the  injection  of\nsound waves into a ferromagnetic film [ 6], excitation of a\nferromagnetic resonance by surface acoustic waves in a\nferromagnetic-ferroelectric hybrid [ 7], and generation of\nspin currents at the acoustic resonance [ 8].\nAlthough  the  magnetic  dynamics  driven  by  elastic\nwaves  and  strain  pulses  is  inevitably  spatially\ninhomogeneous, this important feature was either ignored\nin the theoretical studies [ 7, 10] or described for special\nsituations in the approximation of small deviations from\nthe equilibrium magnetization direction [ 11-13]. In this\nwork,  we  employed  micromagnetic  simulations  taking\ninto account both magnetoelastic coupling and exchange\ninteraction to describe the inhomogeneous magnetization\ndynamics excited by standing elastic waves generated in a\nthin ferromagnetic film. Computations were carried out\nfor longitudinal and transverse waves in a galfenol film\nbecause  Fe-Ga  alloys  have  very  high  magnetoelastic\ncoefficients [14]. The results were used to calculate the\nspin  current  pumped  from  the  dynamically  strained\nferromagnet into adjacent paramagnetic metal. It shouldbe  emphasized  that  standing  elastic  waves  can  be\ngenerated in plate-like magnetic crystals by femtosecond\nlaser  pulses  and  were  found  to  induce  unusual\nmagnetization dynamics [15].\nII. MICROMAGNETIC SIMULATIONS OF\nSTRAIN-DRIVEN MAGNETIZATION DYNAMICS\nOur approach is based on the numerical inte gration of\nthe  Landau-Lifshitz-Gilbert  (LLG)  torque  equation\ndescribing  the  temporal  evolution  of  the  local\nmagnetization  M(t).  In  the  considered  case  of  highly\nmagnetostrictive  materials,  the  magnetization  dynamics\ndriven  by  elastic  waves  can  be  described  by  the\nconventional LLG equation [ 16], which may be written\nas  ,\nwhere   is the gyromagnetic ratio,  is the dimensionless\nGilbert  damping parameter,  Ms =  |M| is the  saturation\nmagnetization,  and Heff is  the  effective  magnetic  field\nacting on  M. Since at a fixed temperature much lower\nthan the Curie temperature the saturation magnetization\nmay be regarded as a constant quantity, the LLG equation\ncan  be  reduced  to\n,\nwhere m = M/Ms and * = 1/(1 + 2).\nThe effective field Heff involved in the LLG equation\nis the sum of the external magnetic field  H, field Hdip\ncaused  by  magnetostatic  dipolar  interactions  between\nspins,  and  contributions  resulting  from  the\nmagnetocrystalline  anisotropy  ( Hmca),  magnetoelastic\ncoupling  (Hmel),  and  exchange  interaction  ( Hex).  The\ncalculation  of  Hdip  is  computationally  most  time\nconsuming because it requires the summing of magnetic\nfields created by all spins in the studied ensemble at each\nspin  position.  To  reduce  the  simulation  time  to  a\nreasonable level, we introduce nanoscale computational\ncells with dimensions much larger than the unit cell size\nbut  smaller  than  the  exchange  length.  The  second\ncondition  guarantees  that  the  magnetization  orientation\n1\nFig. 1. Schematic representation of a ferromagnetic film grown\non a nonmagnetic substrate and covered by a thick layer of\nparamagnetic metal. The magnetization oscillations driven by an\nelastic wave create a spin current Js pumped from the film to the\nadjacent paramagnetic layer.\ndoes not change significantly within an individual cell.\nTherefore,  the  introduced  cells  may  be  modeled  by\nuniformly magnetized rectangular prisms, and the dipolar\nfield  Hdip can be calculated as a sum of magnetic fields\ncreated by such prisms. Accordingly, Hdip may be written\nin the general form as \n,              (1)\nwhere the summation is carried out over all computational\ncells,  rn denote the vectors defining spatial positions of\ntheir centers, m(rn) = mn is the magnetization direction in\nthe n-th cell, and the matrix N is described by analytical\nrelations [17,  18]. For the numerical computation of the\nexchange field Hex, it is convenient to use the relation\n,           (2)\nwhere  A is the exchange  stiffness coefficient [ 19],  the\nsummation is carried out over the six nearest neighbors of\nthe n-th cell, dp = |rp  rn|, and the differences between the\nmagnetization  orientations  in  neighboring  cells  are\nassumed  to  be  smaller  than  30°  [ 20].  The  remaining\ncontributions to Heff = H + Hmca + Hmel + Hex + Hdip can be\nfound by differentiating the magnetic energy density  F\nwritten as a  polynomial in terms  of the magnetization\ndirection  cosines  mi in  the  Cartesian  reference  frame\n(x,y,z) [19]. For ferromagnets with a cubic paramagnetic\nphase, the relation  Heff =  F/M  gives (no summation\nover repeated indices i = x,y,z, j ≠ i, and k ≠ i,j)\n,        (3)\n,       (4)\nwhere  K1 and  K2 are the magnetocrystalline anisotropy\nconstants of  fourth  and  sixth order,  B1 and  B2 are  the\nmagnetoelastic coupling constants, and  uij are the lattice\nstrains. \nSince  the  magnetization  reorientations  modifyFig. 2. Schematic of a ferromagnetic film divided into nanoscale\ncomputational  cells  for  simulations  of  the  magnetization\ndynamics induced by standing elastic waves. The axes of the\nrectangular  coordinate  system  ( x,  y,  z)  are  parallel  to  the\ncrystallographic  axes  of  the  ferromagnet,  the  unit  vector  m\nshows the initial orientation of the magnetization, and  is the\nwavelength of the standing wave.\nlattice  strains,  the  LLG  equation  generally  should  be\nsolved  together  with  the  elastodynamic  equation  of\nmotion [11, 12, 16]. However, the magnetostrictive strains\nare very small (~105) so that we can neglect them in\ncomparison with the strains uij induced by an elastic wave\neven at  uij  ~ 103. Therefore, the distribution of lattice\nstrains  in  the  film  was  assumed  independent  of  the\nmagnetic pattern. In our model simulations, two types of\nstanding waves were considered, namely, the longitudinal\nand  transverse  waves  defined  by  the  relations\n and\n,  respectively.\nStrictly speaking, strain waves with such simple structure\ncan exist only in a film sandwiched between two elastic\nhalf-spaces with the same elastic properties as the film.\nHowever, they also represent a reasonable approximation\nfor elastic waves in a relevant material system having the\nform  of  a  thin  ferromagnetic  film  grown  on  a\nnonmagnetic substrate and covered by a thick layer of a\nparamagnetic metal (see Fig.  1).\nThe simulations were performed using a home-made\nsoftware  which  operates  with  a  finite  ensemble  of  N\ncomputational  cells  characterized  by  their  spatial\npositions  rn  and time-dependent unit vectors  mn(t) (n =\n1,2,…N).  First,  the  effective  fields  Heff(mn)  acting  on\nvectors mn(t) at the moment t are calculated with the aid\nof Eqs. (1)-(4). Using the computed fields and the known\nset of  mn(t), we integrate the LLG equation numerically\nand  determine  the  magnetization orientations  mn in all\ncells at the moment t + t. This procedure is repeated until\na steady periodic solution for the strain-induced dynamic\nmagnetic pattern is obtained. Since in our case the strain\ndistribution has the  form  of  a standing wave, periodic\nboundary conditions along the x axis may be introduced\nfor a ferromagnetic film parallel to the  xy plane, which\nenables  us  to  consider  only  cells  situated  within  one\nwavelength  (Fig. 2).\nThe  developed  computational  scheme  employs  the\nLLG equation written in the Cartesian coordinates. As the\nLLG  equation  is  known  to  be  “stiff”,  the  numerical\n2\nFig.  3. Magnetization  dynamics  induced  by  the  transverse\nstanding wave with the frequency   50 GHz much higher than\nthe resonance frequency res  GHz of unstrained Fe81Ga19\nfilm.  Panel  (a)  shows  the  temporal  evolution  of  the\nmagnetization  direction  cosines  my and  mz in  the  whole\nsimulation including the transient regime.  Panels (b) and (c)\npresent  the  enlarged  view  of  the  regime  of  steady-state\nmagnetization  precession.  Phase  shifts  between  the  periodic\nvariations of the direction cosines my and mz and shear strains in\nthe wave amount to  and, respectively.\nintegration  is  performed  using  the  projective  Euler\nscheme with a fixed integration step t = 5 fs, where the\ncondition |m| = 1 is satisfied automatically. To reduce the\ncomputation time, the dipolar field Hdip is calculated with\nthe aid  of fast Fourier transforms  and the convolution\ntheorem.  In  addition,  the  magnetic  fields of  uniformly\nmagnetized  prisms  are  replaced  by  the  fields  of  point\nmagnetic dipoles at distances exceeding the cell sizes by\nmore than a factor of 50. Based on the symmetry of the\nproblem, the orientation of all vectors mn(t) in each chain\nof cells parallel to the in-plane  y axis is taken to be the\nsame  in  each  moment  t.  To  test  the  accuracy  of  our\nsoftware, we used it to solve the NIST Standard Problem\n#4 [21] and found good agreement with the reference.\nThe  simulations  were  performed  for  a  2  nm  thick\nFe81Ga19  film using the following values of the involved\nmaterial parameters: Ms = 1321 emu cm-3 [22],  = 0.017\n[23], A = 1.8×10-6 erg cm-1 [24], K1 = 1.75×105 erg cm-3, K2= 0 [25],  B1 = 0.9×108  erg cm-3, and B2 =  0.8×108  erg\ncm-3 [14]. To stabilize the single-domain initial state in the\nferromagnetic film, an external magnetic field with the\ncomponents Hx = Hz = 500 Oe was introduced. Since the\nexchange length   [26] of Fe81Ga19  is\nabout 4 nm, one computational cell is sufficient in the\nfilm  thickness  direction.  Therefore,  we  employed  cells\nwith the dimensions 2 22 nm3 and considered standing\nelastic waves with wavelengths equal to even numbers\nof the 2 nm cell size only. The frequencies of these waves\nwere determined from the dispersion relation ,\nwhere  the  phase  velocities   and\n of longitudinal and transverse waves were\ncalculated  using  the  elastic  stiffnesses  c11 =  1.621012\ndyne cm-2, c44 = 1.261012 dyne cm-2, and density  = 7.8\ng  cm-3 of  Fe81Ga19  [27]).  For  both  waves,  the  strain\namplitude umax was set equal to 0.510-2.\nIII. MAGNETIZATION DYNAMICS IN\nELASTIC WAVES\nThe  magnetization  dynamics  caused  by\nmagnetoelastic coupling should depend on the frequency\nof elastic wave. Therefore, we carried out simulations for\nstanding waves with various wavelengths, which provide\na wide frequency range spanning frequencies below and\nabove  the  resonance  frequency  res of  coherent\nmagnetization precession in unstrained Fe 81Ga19 film. This\nresonance  frequency  was  determined  by  studying  the\nrelaxation  of  magnetization  vector  to  the  equilibrium\norientation  and  found  to  be  about  9.89 GHz  at  the\nconsidered applied magnetic field ( Hx = Hz = 500 Oe).\nAll  simulations  started  at  the  equilibrium  magnetic\nstate of unstrained ferromagnetic film, where the uniform\nmagnetization is directed in the  xz plane at an angle of\n1.65° with respect to the film surfaces. This initial state\ntransforms into a nonhomogeneous magnetic pattern just\nupon the introduction of a strain wave at  t  = 0. After a\ntransition period of the order of 1 ns (~105 simulation\nsteps),  the  magnetic  dynamics  acquires  the  form  of  a\nsteady-state  magnetization  precession.  The  angular\ndeviations  from  the  initial  magnetization  direction  are\nmaximal at the antinodes of standing waves, where the\ndriving  force  of  magnetoelastic  origin  has  the  largest\nvalue. Remarkably, the magnetization precession does not\nvanish at the nodes, where the driving force goes to zero,\nwhich  is  due  to  cooperative  effects  caused  by\nmagnetostatic and exchange interactions between spins.\nFigure  3 shows  the  temporal  evolution  of  the\nmagnetization orientation at the antinode ( x = /4) of the\ntransverse (shear) standing wave with the frequency  \n50  GHz  well  above  the  resonance  frequency  res.  The\nmost  important  finding  here  is  that  the  steady\nmagnetization precession, which sets in after a transient\nregime  comprising  about  60  oscillations  [Fig.  3(a)],\noccurs with the frequency of elastic wave [Figs. 3(b) and\n3(c)].  This  is  a  nontrivial  result  because  the\nmagnetoelastic  components  of  the  effective  field\n3\nFig. 4. Typical trajectory of the end of magnetization vector at\nthe  antinodes  of  transverse  standing  waves.  The  three-\ndimensional plot presents the full trajectory of the unit vector m\n= M/Ms at the wave frequency   9.88 GHz. The arrow shows\nthe equilibrium magnetization direction.\nFig. 5. Temporal evolution of the magnetization orientation at\nthe antinode of the transverse standing wave with the frequency\n  9.85 GHz. Panel (a) shows variations of the magnetization\ndirection  cosines  mi in  the  whole  simulation  including  the\ntransient regime, while panel (b) presents the enlarged view of\nthe regime of steady magnetization precession.\n  and  \nacting  on  the  magnetization  oscillate  with  higher\nfrequencies owing to periodic variations of the direction\ncosines  mz and  mx.  Such  oscillations  of  Hmel manifest\nthemselves in the appearance of two maxima of mx during\none period of elastic wave [Fig. 3(b)]. This feature results\nfrom  specific  spatial  trajectory  of  the  end  of\nmagnetization vector, which does not have the form of a\nplanar curve (see Fig. 4). \nWhen the frequency of elastic wave is reduced down\nto res, the magnetization oscillations increase drastically\n(Fig. 5). In this case, the transient regime is distinguished\nby  the  presence  of  intermediate  stage,  where  these\noscillations  are  larger  than  in  the  steady-state  regimeFig. 6. Frequency dependences of the solid angle of steady-state\nmagnetization precession induced in the  Fe 81Ga19  film  at  the\nantinodes of transverse and longitudinal standing waves.\nFig. 7. Magnetization dynamics at the antinode of the transverse\nstanding  wave  with  the  frequency    1.25  GHz.  Panel  (a)\nshows  the  temporal  evolution  of  the  magnetization  direction\ncosines my and mz in the whole simulation including the transient\nregime, while panels (b) and (c) present the enlarged view of the\nregime of steady magnetization precession ( m0 = 0.9995 is the\ninitial value of the direction cosine mx).\n(see Fig.  5(a)).   Interestingly,  the  amplitude  of  steady\nmagnetization  precession  reaches  maximum  at  the\nfrequency  max  9.38  GHz  slightly  lower  than  the\nresonance  frequency  of  unstrained  ferromagnetic  film\n(Fig.  6).  At  this  frequency,  the  solid  angle  of\nmagnetization  precession  exceeds  0.5,  but  it  decreases\n4\nFig. 8. Magnetization trajectories at the antinodes of transverse\nstanding waves projected on the  yz plane orthogonal to their\nwave vectors. Panels (a), (b), and (c) show the projections of the\nend  of  the  unit  vector  m =  M/Ms calculated  at  the  wave\nfrequencies of 50, 9.85, and 1.25 GHz, respectively.\nrapidly at smaller frequencies, falling down to 0.1 already\nat  9.15 GHz.\nFigure  7 illustrates  the  magnetization  dynamics\ninduced  by  the  transverse  standing  wave  with  the\nfrequency    1.25  GHz  well  below  the  resonance\nfrequency  res. A novel feature here is the presence of a\ndouble dynamics in the transient regime [see Fig. 7(a)]. In\ncontrast  to  the  case  of  res,  the  magnetization\nprecesses with the frequency    10 GHz   res much\nhigher than the wave frequency. This fast dynamics is\naccompanied  by  slow  variations  of  the  precession\ntrajectory following the evolution of elastic  wave. In the\nsteady-state  regime,  the  frequency  of  magnetization\nprecession drops down to the wave frequency [Fig. 7(b)],\nwhich  is  accompanied  by  a  drastic  change  in  the\nprecession trajectory (see Fig. 8).\nThe magnetization oscillations excited by longitudinal\nstanding waves with three representative frequencies are\nshown in Figs. 9-11. At the high frequency    50 GHz\n(Fig. 9), the magnetic dynamics is qualitatively similar to\nthe one discussed above for the transverse wave with the\nsame frequency. However, angular  deviations from the\nequilibrium magnetization direction are much smaller in\nthe  longitudinal  wave  despite  the  fact  that  theFig.  9. Magnetization  dynamics  at  the  antinode  of  the\nlongitudinal  standing  wave  with  the  frequency   GHz\nexcited  in  the  Fe81Ga19  film.  Panel  (a)  shows  the  temporal\nevolution of the magnetization direction cosines my and mz in the\nwhole simulation including the transient regime, while panels\n(b) and (c) present the enlarged view of the regime of steady-\nstate magnetization precession ( m0 = 0.9995 is the initial value\nof the direction cosine mx).\nmagnetoelastic constants B1 and B2 have almost the same\nvalue in the considered case of Fe 81Ga19  alloy [14]. This\nsmaller efficiency of longitudinal waves originates from\ndissimilar  structure  of  their  contribution\n to the effective field. \nAt frequencies close to the resonance frequency res of\nunstrained  film,  the  amplitude  of  magnetization\nprecession  strongly  increases  (Fig.  10).  This  is\naccompanied  by  significant  distortions  of  the  time\ndependences  of  direction  cosines  mi,  which  remain\nperiodic  (  GHz)  but  cannot  be  described  by\nsimple sine or cosine functions in the steady-state regime.\nInterestingly, the dependence mx(t) is distinguished by the\npresence of two maxima during one wave period, which\nis similar to the behavior of  mx in the transverse wave\nwith   >>  res shown in Fig.  3(b). The solid angle of\nmagnetization precession induced by longitudinal waves\nreaches  maximum  at  the  frequency  max GHz\nslightly higher than the most efficient frequency  max of\ntransverse waves (see Fig. 6). \nWhen the frequency of longitudinal wave is reduced\nbelow  res, the magnetic dynamics changes dramatically\n(Fig.  11). Most importantly, the magnetization precesses\nwith  a  variable  frequency strongly  exceeding\n5\nFig. 10. Temporal evolution of the magne tization orientation at\nthe  antinode  of  the  longitudinal  standing  wave  with  the\nfrequency    9.89  GHz.  Panel  (a)  shows  variations  of  the\nmagnetization  direction  cosines  mi in  the  whole  simulation\nincluding  the  transient  regime,  while  panel  (b)  presents  the\nenlarged view of the regime of steady magnetization precession.\nthe wave frequency. This feature may be attributed to the\ndependence of the resonance frequency  res of coherent\nprecession on lattice strains [ 28]. Indeed, the calculation\nshows that the change of uxx from – 0.005 to +0.005 gives\nres ranging from 3.44 to 14.45 GHz, which agrees with\nthe  frequency  range  4.3 15  GHz  obtained  from\nmicromagnetic  simulations.  These  frequency  variations\nare accompanied by periodic changes of the precession\ntrajectory  following  the  evolution  of  elastic  wave.  In\ncontrast  to  the  case  of  transverse  waves,  this  double\ndynamics does not disappear in the steady-state regime\n(see  Fig.  11).  The  magnetization  trajectories  at  three\nrepresentative  frequencies  of  longitudinal  waves  are\ncompared in Fig. 12. \nIn  conclusion  of  this  section,  we  consider  spatio-\ntemporal distributions of the magnetization oscillations in\nstanding  elastic  waves.  Figures  S1 and  S2  in  the\nSupplemental Material [ 29] demonstrate that such elastic\nexcitations generate  standing spin waves with the same\nwavelength. Remarkably, the amplitude of magnetization\nprecession does not go to zero at the nodes of these spin\nwaves.  Of  course,  the  precession  amplitude  is  always\nmuch larger at the antinodes than at the nodes, but the\nratio of the amplitude at the antinode to that at the node\ndecreases strongly when the frequency of elastic wave\nincreases  from   <<  res to   >>  res.   Graphs\ndemonstrated by Figs.  S1 and  S2  [29] further show that\nthe elastically generated standing spin waves may have\nvery  complex  structure,  especially  when  exited  by  a\nlongitudinal elastic wave. In all cases, these spin waves\ncannot be precisely described by simple relations of the\nform  .   To\ngain  additional  information  on  their  structure,  we\nperformed  the  Fourier  analysis   of   the  spatialFig.  11. Magnetization  dynamics  at  the  antinode  of  the\nlongitudinal standing wave with the frequency    GHz.\nPanel (a) shows the temporal evolution of the magnetization\ndirection cosines my and mz in the whole simulation including the\ntransient regime, while panels (b) and (c) present the enlarged\nview of the regime of steady magnetization precession.\ndistributions   of mi formed at the moment when the strain\nat  the  antinodes  of  elastic  wave  reaches  its  maximum\nvalue. The calculations showed that, in the case of the\nspin wave induced by the transverse elastic wave with the\nfrequency res, a term proportional to  is\nsufficient to describe the direction cosines my(x) and mz(x)\nwith a good accuracy. This term also provides the main\ncontribution  to  my(x)  and  mz(x)  in  the  spin  waves\ngenerated  by  transverse  and  longitudinal  elastic  waves\nwith frequencies  ≥res, but here significant additional\ncontribution (up to 30% of the leading term) is caused by\na term proportional to  . Finally, in the spin\nwave  exited  by  the  longitudinal  elastic  wave  with\nres,  the  distribution  of  the  direction  cosine  my(x)\nmay be approximated by the sum of terms proportional to\n and  ,  while  approximate\ndescription  of  mz(x)  requires  terms  proportional  to\n and . \nIV . SPIN PUMPING DRIVEN BY ELASTIC WAVES\nWhen a ferromagnet is in contact with a paramagnetic\nmetal, the magnetization precession in the former leads to\na  spin  pumping  into  the  latter  [ 30].  The  spin  current\ndensity  Is at  the  interface  can  be  calculated  from  the\n6\nFig.  12. Magnetization  trajectories  at  the  antinodes  of\nlongitudinal  standing  waves  projected  on  the  yz plane\northogonal to their wave vectors. Panels (a), (b), and (c) show\nthe  projections  of  the  end  of  the  unit  vector  m =  M/Ms\ncalculated at the wave frequencies of 50, 9.89, and 1.25 GHz,\nrespectively.\nrelation [30-32]\n, (5)\nwhere s is the unit vector of the spin-current polarization,\nand ,  are the complex reflection and transmission\nspin mixing conductances per unit contact area [ 33, 34].\nAccording  to  Eq.  (5), the  magnetization  precession\ninduced  by  elastic  waves  should  create  a  spin  current\ncomprising  dc  and  ac  components.  The  results  of  our\nmicromagnetic simulations enable us to calculate both dc\nand ac spin currents generated by standing elastic waves.\nSince  the  first-principles  studies  of  spin  mixing\nconductances [34] show that  and the imaginary part of\n should be negligible for the 2 nm thick ferromagnetic\nfilm considered in this work, in our calculations we used\nthe approximate relation  \n.                 (6)\nFigure  13(a) shows the time dependence of the spin\ncurrent created at the antinode of the transverse standing\nwave with the frequency close to res. It can be seen that,\nin  the  steady-state  regime,  all  three  components  si ofFig. 13. Time dependence of the spin current generated by the\nFe81Ga19  film subjected to the transverse elastic wave with the\nfrequency of 9.38 GHz. The components  of the spin-current\ndensity are normalized by the quantity . Panels\n(a) and (b) show  at the antinode of the standing wave, while\npanels (c) and (d) present the current density averaged over one\nwavelength  of  this  elastic  wave.  The  direct  results  of\ncalculations are given in panels (a) and (c), whereas panels (b)\nand (d) show mean current densities obtained after filtering out\nhigh-frequency oscillations.\nthe  spin-current  polarization  s oscillate  with  the  wave\nfrequency.  Interestingly,  the  oscillations  of  the  spin-\ncurrent component  have much larger amplitude than\nthe  oscillations of   and  .  However,  only the\nprojection of the spin current on the x axis has significant\nmean value. To determine mean values  of all three\nspin-current components, we averaged   during the\ntime  period  comprising  several  oscillations  [see  Fig.\n13(b)].  It  was  found  that  in  the  steady-state  regime\n,  \nwhile  is negligible. These results can be explained by\nthe fact that, according to Eq.  (6),  the dc component of\nthe  spin  current  should  be  parallel  to  the  axis  of\nmagnetization  precession.  In  the  considered  case,  the\nlatter  is  close  to  the  equilibrium  magnetization  in  the\nunstrained film, which is almost parallel to the  x axis,\n7\nFig. 14. Schematic representation of magnetization precessions\nin two halves of the standing elastic wave. Despite the fact that\nthe magnetoelastic contribution  Hmel of the effective field  Heff\nhas opposite signs in the first and second halves of the standing\nwave, the magnetization precesses in the same counterclockwise\ndirection in both regions.  \nhaving additional small projection on the z axis only.\nSince  the  quantity  relevant  to  experimental\nmeasurements  is  the  spin  current  produced  by  a\nmacroscopic  section  of  the  film,  we  calculated  the\naverage current density  pumped from the film region\ncorresponding  to  one  wavelength  .  Figure  13(c)\ndemonstrates  that  this  averaging  strongly  reduces\noscillations of the y and z components of the spin current.\nThe  filtering  of  high-frequency  oscillations  (see  Fig.\n13(d)) further  shows  that  the  x component  retains\nsignificant  mean  value  ,\nwhich  is only about two times  smaller  than   at  the\nantinode of transverse standing wave. This feature is due\nto the fact that the magnetization precesses in the counter-\nclock-wise direction everywhere despite opposite signs of\nthe driving field Hmel in two halves of the standing wave\n(see Fig. 14).\nThe results obtained for the spin currents driven by the\nlongitudinal elastic wave with the frequency close to res\nare  shown  in  Fig.  15.  It  can  be  seen  that  they  are\nessentially similar to the results discussed above, but there\nare two interesting distinctions. First, the mean values \nand   of  the  y component are not negligible in the\nsteady-state  regime,  being  close  to  those  of  the  z\ncomponent of the spin current (see Fig. 15(b) and (d)).\nThis  feature  is  caused  by  the  fact  that  the  axis  of\nmagnetization precession driven by the longitudinal wave\nhas a nonzero projection on the y axis. Second, in contrast\nto  the  case  of  transverse  wave,  the  averaging  of  the\npumped spin current  over the wavelength   leaves the\namplitude  of  the   oscillations larger  than  the  \ncomponent  (compare  panels  (c) of  Figs.  13 and  15).\nNevertheless,  the  mean  value\n of  the  averaged  x\ncomponent  remains  much  larger  than  the  mean  value\n of the z one. \nUsing  the  theoretical  result\n obtained  for  the\nreflection spin mixing conductance of the Fe/Au interface\nby  first-principles  calculations  [ 34],  we  estimated\nnumerical values of the spin currents pumped from the\ndynamically strained Fe 81Ga19 film into adjacent Au layer.\nIn  particular,  the  calculations  giveFig. 15. Time dependence of the spin current generated by the\nFe81Ga19 film subjected to the longitudinal elastic wave with the\nfrequency of 9.61 GHz. The components  of the spin-current\ndensity are normalized by the quantity . Panels\n(a) and (b) show  at the antinode of the standing wave, while\npanels (c) and (d) present the current density averaged over one\nwavelength  of  this  elastic  wave.  The  direct  results  of\ncalculations are given in panels (a) and (c), whereas panels (b)\nand (d) show mean current densities obtained after filtering out\nhigh-frequency oscillations.\n for  the  mean  current\ndensity generated by the transverse elastic wave with the\nfrequency   =  9.38  GHz  and\n for  the  longitudinal  wave\nwith   = 9.61 GHz. These estimates render possible to\nevaluate the dc charge current created in the Au layer by\nthe  pumped  spin  current  due  to  the  inverse  spin  Hall\neffect. The density  Ic of charge current is given by the\nrelation , where SH is the spin\nHall angle, e denotes the elementary positive charge, and\nes is the unit vector in the spin current  direction [ 32].\nHence, the charge current is orthogonal to the spin current\nand almost parallel to the y axis in our case. Taking SH ≈\n0.0035 for Au [35], we find the current density  \ngenerated  at  the  interface  by  the  transverse  and\n8\nlongitudinal waves to be about 3.9 106 and 2.7106 A m-2,\nrespectively. Since the injected spin current decays in the\nnormal metal due  to spin relaxation and diffusion, the\ndensity of charge  current falls down with the  distance\nfrom the interface [36]. The total charge current in the\nnormal metal layer of width  wN and thickness  tN can be\nfound from the relation\n,         (7)\nwhere sd  is the spin diffusion length. Taking sd = 35 nm\nfor Au [36] and assuming wN = 10 m, we obtain the total\ncharge current generated in the Au layer with tN > 5 sd by\nthe considered standing elastic waves to be about 1 A,\nwhich can be readily measured experimentally. \nV . CONCLUDING REMARKS \nIn  this  work,  we  carried  out  micromagnetic\nsimulations  of  the  inhomogeneous  magnetization\ndynamics induced in a ferromagnetic material by elastic\nwaves. In contrast to the preceding analytical treatments\nof the problem [11, 13], our calculations do not involve\nthe assumption of small deviations from the equilibrium\nmagnetization  direction. Our  approach  is  based  on  the\nnumerical solution of the LLG equation comprising the\ndamping term and the effective magnetic field with all\nrelevant contributions, such as those resulting from the\nmagnetoelastic  coupling  between  magnetization  and\nlattice  strains,  exchange  interaction,  and\nmagnetocrystalline  anisotropy.  Furthermore,  to describe\ncorrectly  the  practically  important  case  a  thin\nferromagnetic  film,  we  accurately  calculated  the\nmagnetostatic dipolar interactions in a ferromagnetic slab\nof  finite  thickness.  This  enabled  us  to  simulate  the\nmagnetic  dynamics  induced  by  transverse  and\nlongitudinal standing waves generated in the 2 nm thick\nFe81Ga19 film sandwiched between two elastic half-spaces.\nBoth  the  transient  and  steady-state  regimes  of\nmagnetization oscillations are described in detail.\nThe  simulations  showed  that  elastic  waves  induce\nstrongly inhomogeneous magnetization precession, which\nacquires maximum amplitude at the antinodes of standing\nwaves (near the antinodes in the case of transverse wave\nwith  >> res). This amplitude increases drastically near\nthe  resonance  frequency  res of  the  unstrainedferromagnetic film (Fig. 6), which agrees with the general\ntheoretical predictions [ 11, 12]. In the steady-state regime,\nthe frequency of magnetization oscillations equals that of\nthe elastic wave, except for the case of longitudinal waves\nwith frequencies well below res, where the magnetization\nprecesses with a variable frequency strongly exceeding\nthe wave frequency (Fig. 11(c)). \nThe  spatio-temporal  distributions  of  magnetization\noscillations in the considered elastic waves have the form\nof standing spin waves with the same wavelength see\nSupplemental Material [ 29]). Remarkably, the amplitude\nof magnetization precession does not go to zero at the\nnodes of these spin waves, although angular deviations\nfrom  the  equilibrium  magnetization  direction  here  are\nmuch  smaller  than  at  the  antinodes.  As  a  result,  the\nelastically  generated  standing  spin  waves  cannot  be\nprecisely  described  by  simple  relations  of  the  form\n even  when\nthe  magnetization  oscillates  with  the  frequency   of\nelastic wave. The simplest structure has the spin wave\ngenerated by the transverse elastic wave with   <<  res,\nwhereas the longitudinal wave with the frequency well\nbelow  res  creates  spin  wave  with  the  most  complex\nstructure.\nUsing the results obtained for the magnetic dynamics\ninduced  by  elastic  waves,  we  also  calculated  the  spin\ncurrents  that  can  be  pumped  from  the  dynamically\nstrained Fe81Ga19 film into adjacent layer of paramagnetic\nmetal. It was found that both transverse and longitudinal\nstanding waves with the frequency close to res create spin\ncurrents  comprising  significant  dc  and  ac  components.\nInterestingly, the spin polarization of the dc component is\nnot  exactly  parallel  to  the  equilibrium  magnetization\ndirection in the steady-state regime. The calculations of\nthe transverse charge current, which is created by the spin\ncurrent via the inverse spin Hall effect, showed that the\ncharge current has high density at the interface and can be\neasily  measured  experimentally.  Our  theoretical\npredictions may be useful for the development of  spin\ninjectors driven by elastic waves.\nACKNOWLEDGMENT\nThis work was supported by the Government of the\nRussian Federation through  the program  P220 (Project\nNo. 14.B25.31.0025; leading scientist A. K. Tagantsev).\n9\nReferences\n[1] M. Farle, Rep. Prog. Phys. 61, 755 (1998).\n[2] J. C. Slonczewski and J. Z. Sun , J. Magn. Magn. Mater. \n310, 169 (2007).\n[3] A. Brataas, A. D. Kent, and H. Ohno, Nature Materials 11, \n372 (2012).\n[4] W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and T. \nJ. Silva, Phys. Rev. Lett. 92, 027201 (2004).\n[5] A. V . Scherbakov, A. S. Salasyuk, A. V . Akimov, X. Liu, M. \nBombeck, C. Brüggemann, D. R. Yakovlev, V . F. Sapega, J. K. \nFurdyna, and M. Bayer, Phys. Rev. Lett. 105, 117204 (2010).\n[6] K. Uchida, H. Adachi, T. An, T. Ota, M. Toda, B. \nHillebrands, S. Maekawa, and E. Saitoh, Nature Materials 10, \n737 (2011).\n[7] M. Weiler, L. Dreher, C. Heeg, H. Huebl, R. Gross, M. S. \nBrandt, and S. T. B. Goennenwein, Phys. Rev. Lett. 106, \n117601 (2011). \n[8] M. Weiler, H. Huebl, F. S. Goerg, F. D. Czeschka, R. Gross,\nand S. T. B. Goennenwein, Phys. Rev. Lett. 108, 176601 \n(2012).\n[9] N. Akulov, Z. Phys. 52, 389 (1928).\n[10] T. L. Linnik, A. V . Scherbakov, D. R. Yakovlev, X. Liu, J. \nK. Furdyna, and M. Bayer, Phys. Rev. B 84, 214432 (2011).\n[11] A. Akhiezer, V . Bar’iakhtar, and S. Peletminski, J . Exptl. \nTheoret. Phys. (U.S.S.R.)  35, 228 (1958).\n[12] C. Kittel, Phys. Rev. 110, 836 (1958).\n[13] A. Kamra, H. Keshtgar, P. Yan, and G. E. W. Bauer, Phys. \nRev. B 91, 104409 (2015).\n[14] J. B. Restorff, M. Wun-Fogle, K. B. Hathaway, A. E. \nClark, T. A. Lograsso, and G. Petculescu, J. Appl. Phys. 111, \n023905 (2012).\n[15] D. Afanasiev, I. Razdolski, K. M. Skibinsky, D. Bolotin, S.\nV . Yagupov, M. B. Strugatsky, A. Kirilyuk, Th. Rasing, and A. \nV . Kimel, Phys, Rev. Lett. 112, 147403 (2014).\n[16] Y . V . Gulyaev, I. E Dikshtein, and V . G Shavrov, Phys.-\nUsp. 40, 701 (1997).\n[17] A. J. Newell, W. Williams, and D. J. Dunlop, J. Geophys. \nRes.: Solid Earth 98, 9551 (1993).\n[18] M. J. Donahue, Accurate computation of the \ndemagnetization tensor, \nhttp://math.nist.gov/~MDonahue/talks/hmm2007-MBO-03-\naccurate_demag.pdf [19] C. Kittel, Rev. Mod. Phys. 21, 541 (1949).\n[20] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F.\nGarcia-Sanchez, and B. Van Waeyenberge,  AIP Advances 4,\n107133 (2014).\n[21] See http://www.ctcms.nist.gov/~rdm/std4/spec4.html  \n[22] J. Atulasimha and A. B Flatau, Smart Mater. Struct. 20, \n043001(2011).\n[23] J. V . Jager, A. V . Scherbakov, T. L. Linnik, D. R. Yakovlev,\nM. Wang, P. Wadley, V . Holy, S. A. Cavill, A. V . Akimov, A. W.\nRushforth, and M. Bayer, Appl. Phys. Lett. 103, 032409 (2013).\n[24] K. S. Narayan, Modelling of Galfenol nanowires for \nSensor Applications, Master Thesis (University of Minnesota, \n2010), \nhttp://conservancy.umn.edu/bitstream/handle/11299/93191/Kris\nhnan_Shankar_May2010.pdf  \n[25] G. Petculescu, K. B. Hathaway, T. A. Lograsso, M. Wun-\nFogle, and A. E. Clark, J. Appl. Phys. 97, 10M315 (2005).\n[26] C. A. F. Vaz, J. A. C. Bland, and G. Lauhoff, Rep. Prog. \nPhys. 71, 056501 (2008).\n[27] R. R. Basantkumar, B. J. H. Stadler, R. William, and E. \nSummers, IEEE Transactions on Magnetics  42, 3102 (2006).\n[28] N. A. Pertsev, H. Kohlstedt, and R. Knöchel, Phys. Rev. B\n84, 014423 (2011).\n[29]  See  Supplemental  Material  for  graphs  of  spin  waves\nexcited by transverse and longitudinal standing elastic waves. \n[30] Y . Tserkovnyak, A. Brataas, G. E.W. Bauer, and B. \nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n[31] Y . Tserkovnyak, A. Brataas, G. E. W. Bauer, Phys. Rev. B\n66, 224403 (2002).\n[32] H. J. Jiao and G. E. W. Bauer, Phys. Rev. Lett. 110, 217602\n(2013).\n[33] A. Brataas, Yu. V . Nazarov, and G. E. W. Bauer, Phys. Rev.\nLett. 84, 2481 (2000).\n[34] M. Zwierzycki, Y . Tserkovnyak, P. J. Kelly, A. Brataas,\nand G. E. W. Bauer, Phys. Rev. B 71, 064420 ( 2005).\n[35] O. Mosendz, V . Vlaminck, J. E. Pearson, F. Y . Fradin, G.\nE. W. Bauer, S. D. Bader, and A. Hoffmann, Phys. Rev. B 82,\n214403 (2010).\n[36] O. Mosendz, J. E. Pearson, F. Y . Fradin, G. E. W. Bauer, S.\nD.  Bader,  and  A.  Hoffmann,  Phys.  Rev.  Lett. 104,  046601\n(2010).\n10\nSupplemental material\nMagnetization dynamics and spin pumping induced by standing elastic waves\nA. V . Azovtsev and N. A. Pertsev\nIoffe Institute, 194021 St. Petersburg, Russia\nFig. 1. Standing spin waves generated by transverse elastic standing waves in the 2 nm thick\nFe81Ga19 film (four wavelengths are shown). Panels (a), (b), and (c) demonstrate spin waves\nexcited at the frequency of elastic wave    50 GHz,    9.85 GHz, and    1.25 GHz,\nrespectively. \n1\nFig. 2.  Standing spin waves generated by longitudinal standing elastic waves in the 2 nm\nthick Fe81Ga19 film (four wavelengths are shown). Panels (a), (b), and (c) demonstrate spin\nwaves excited at the frequency of elastic wave   50 GHz,   9.85 GHz, and   1.25 GHz,\nrespectively. The resonance frequency of unstrained Fe 81Ga19 film is about GHz. \n2\n"    },    {        "title": "0710.2543v1.Dynamics_of_Magnetized_Spherical_Accretion_Flows.pdf",        "content": "arXiv:0710.2543v1  [astro-ph]  12 Oct 2007Dynamicsof magnetized spherical accretion\nflows\nRomanShcherbakov\nHarvardUniversity, AstronomyDepartment,60,Gardenst, C ambridge,USA\nAbstract. Transonicaccretionflowwithself-consistenttreatmentof themagneticfieldispresented.\nMywebsite http://www.cfa.harvard.edu/~rshcherb/ .\nKeywords: Transonicflows,Magnetohydrodynamics,turbulence,Galac ticcenter,accretion\nPACS:97.10.Gz,98.35.Jk,98.35.Mp,47.40.Hg,52.35.Ra,52.30. Cv,95.30.Qd\nINTRODUCTION\nTheaveraged quantitiescan be obtained in two different way s in magnetohydrodynam-\nics.Thefirstwayistosolve3DMHDequationsandthenaverage theresults.Thesecond\nway is to solve some system of equations on averages. Combina tion of numerical sim-\nulations and averaged theory brings phenomenology that can describe observations or\nexperimentaldata.\nThe problem of spherically symmetric accretion takes its or igin from Bondi’s work\n[2]. He presented idealized hydrodynamic solution with acc retion rate ˙MB.However,\nmagnetic field /vectorBalways exists in the real systems. Even small seed /vectorBamplifies in\nsphericalinfalland becomes dynamicallyimportant[9].\nMagneticfieldinhibitsaccretion[9].Noneofmanytheories hasreasonablycalculated\nthe magnetic field evolutionand how it influences dynamics. T hese theories have some\ncommonpitfalls.Firstofall,thedirectionofmagneticfiel disusuallydefined.Secondly,\nthe magnetic field strength is prescribed by thermal equipar tition assumption. In third,\ndynamical effect of magnetic field is calculated with conven tionalmagneticenergy and\npressure.All theseinaccuracies can beeliminated.\nInSection2Idevelopamodelthatabandonsequipartitionpr escription,calculatesthe\nmagnetic field direction and strength and employs the correc t equations of magnetized\nfluid dynamics.In Section 3 Ishow thisaccretion pattern to b ein qualitativeagreement\nwithSgrA* spectrummodels.IdiscussmyassumptionsinSect ion 4 .\nANALYTICALMETHOD\nReasonable turbulence evolution model is the key differenc e of my method. I build an\naveraged turbulence theory that corresponds to numerical s imulations. I start with the\nmodel of isotropic turbulence that is consistent with simul ations of collisional MHD in\nthree regimes. Those regimes are decaying hydrodynamic tur bulence, decaying MHD\nturbulence and dynamo action. I introduce effective isotro pization of magnetic field in10/MinuΣ510/MinuΣ40.001 0.01 0.1 110/MinuΣ40.0010.010.11\nradius r/Slash1rBdimensionless velocitysound speed inflow velocity\nradial Alfven speed\nturbulent velocity\nperpendicular Alfven speedforce/MinuΣfree\nboundarysonic point\nAlfven point\nIsotropic outer turbulence turns into anisotropic inner turbulence.\nAlfven point near outer boundary /Equal/Greaterno transport of angular momentum.\nFIGURE 1. Normalized to Keplerian speed characteristic velocities o f magnetized flow. Horizontal\nlinescorrespondtoself-similarsolution v∼r−1/2.\n3Dmodel.Isotropizationistakentohaveatimescaleoftheo rderofdissipationtimescale\nthatisafraction γ∼1 oftheAlfvenwavecrossingtime τdiss=γr/vA.\nCommon misconceptionexistsabout thedynamical influence o f magneticfield. Nei-\nther magnetic energy nor magnetic pressure can represent /vectorBin dynamics. Correct aver-\nagedEulerandenergy equationswerederivedin[7]forradia l magneticfield.Magnetic\nforce/vectorFM= [/vectorj×/vectorB]can be averaged over the solid angle with proper combination of\n/vector∇·/vectorB=0.I extend the derivation to random magnetic field without pref erred direction.\nDynamical effect of magnetic helicity [1] is also investiga ted. I neglect radiative and\nmechanicaltransportprocesses.\nThederivedset ofequationsrequires somemodificationsand boundary conditionsto\nbeapplicabletotherealastrophysicalsystems.Iaddexter nalenergyinputtoturbulence\nto balance dissipative processes in the outer flow. The outer turbulence is taken to\nbe isotropic and has magnetization σ∼1.Transonic smooth solution is chosen as\npossessingthehighestaccretion rateas in [2].10/MinuΣ510/MinuΣ40.001 0.01 0.1 10.101.00\n0.50\n0.202.00\n0.303.00\n0.151.50\n0.70\nradius r/Slash1rBmagnetization Σ\nExternally\nsupported\nisotropic\nturbulenceSupported by compression\nanisotropic turbulenceOuter equipartitionMain part withsubequipartitionInnersuperequipartition\nMagnetic helicity conservation does not influence dynamics.Flow is convectively stable on average\ndespite increasing inwards entropy.\nFIGURE2. Plotofmagnetization σ=(EM+EK)/EThwith radius.\nRESULTS& APPLICATIONTOSGR A*\nThe results of my calculations confirm some known facts about spherical magnetized\naccretion, agree with the results of numerical simulations and have some previously\nunidentifiedfeatures.\nInitiallyisotropicmagneticfieldexhibitsstronganisotr opywithlargerradialfield Br.\nPerpendicularmagneticfield B⊥≪Brisdynamicallyunimportantintheinneraccretion\nregionFig1.Becausemagneticfielddissipates,infallonto theblackholecanproceed[9].\nTurbulence is supported by external driving in the outer flow regions, but internal\ndrivingduetofreezing-inamplificationtakesoverinthein nerflowFig2.Magnetization\nof the flow increases in the inner region with decreasing radi us consistently with sim-\nulations [3]. Density profile appears to be ρ∼r−1.25that is different from traditional\nADAFscaling ρ∼r−1.5[6]. Thustheideaofself-similarbehaviorisnotsupported .\nCompared to non-magnetizedaccretion, infall rate is 2-5 ti messmallerdepending on\nouter magnetization. In turn, gas density is 2-5 times small er in the region close to the\nblack hole, where synchrotron radiation emerges [6]. Sgr A* produces relatively weak\nsynchrotron [6]. So, either gas density nor electron temperature Teor magnetic field\nBare small in the inner flow or combination of factors works. Th us low gas density in\nmagnetizedmodelisinqualitativeagreementwiththeresul tsofmodellingthespectrum.Flow is convectively stable on average in the model of moving blobs, where dissipa-\ntion heat is released homogeneouslyin volume. Movingblobs are in radial and perpen-\ndicularpressureequilibriums.Theyaregovernedbythesam eequationsas themedium.\nDISCUSSION & CONCLUSION\nThepresented accretion studyself-consistentlytreats tu rbulencein theaveraged model.\nThismodelintroducesmanyweak assumptionsinsteadoffew s trongones.\nItakedissipationratetobethatofcollisionalMHDsimulat ions.Butflowinquestion\nis rather in collisionless regime. Observations of collisi onless flares in solar corona [5]\ngivesdissipationrate20timessmallerthanincollisional simulations[1].However,flares\nin solar corona may represent a large-scale reconnection ev ent rather than developed\nturbulence.It isunclearwhich dissipationrateis morerea listicforaccretion.\nMagnetic field presents another caveat. Magnetic field lines should close, or /vector∇·\n/vectorB=0 should hold. Radial field is much larger than perpendicular in the inner region.\nTherefore, characteristic radial scale of the flow is much la rger than perpendicular. If\nradialturbulencescaleislargerthanradius,freezing-in conditiondoesnotholdanymore.\nMatter can freely slip along radial field lines into the black hole. If matter slips already\natthesonicpoint,theaccretion rateshouldbehigherthanc alculated.\nSomeotherassumptionsaremorelikelytobevalid.Diffusio nshouldbeweakbecause\nofhighMachnumberthatapproachesunityatlargeradius.Ma gnetichelicitywasfound\nto play very small dynamical role. Only when the initial turb ulence is highly helical,\nmagnetic helicity conservation may lead to smaller accreti on rate. Neglect of radiative\ncooling is justified a posteriori. Line cooling time is about 20 times larger that inflow\ntimefromouterboundary.\nThe study is the extension of basic theory, but realistic ana lytical models should\nincludemorephysics.Thework isunderway.\nACKNOWLEDGMENTS\nIthankmyadvisorProf. Ramesh Narayan forfruitfuldiscuss ions.\nREFERENCES\n1. D.Biskamp, Magnetohydrodynamicturbulence ,CambridgeUniversityPress, Oxford,2003\n2. H.Bondi, Mon.Not. R.Astro. Soc. 112,195(1952).\n3. I.Igumenshchev, Astrophys.J. 649,361(2006)\n4. L. D. Landau , E. M. Lifshitz , L. P. Pitaevskii , Electrodynamics of Continuous Media , Pergamon\nPress, Oxford,1984\n5. J. B.Noglik,R.W. Walsh, J. Ireland, Astron.& Astroph. 441,353(2005)\n6. R. Narayan,I. Yi,R. Mahadevan, Nature374,623(1995)\n7. E.T.Scharlemann, Astrophys.J. 272,279(1983)\n8. A.A. Schekochihinetal. Astrophys.J. 276,612(2004)\n9. V. F. Schwartzman, SovietAstronomer 15,377(1971)\n10. K.R. Sreenivasan, Phys.ofFluids 7,2778(1995)"    },    {        "title": "1903.11052v1.Dynamical_evolution_of_magnetic_fields_in_the_intracluster_medium.pdf",        "content": "Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 27 March 2019 (MN L ATEX style file v2.2)\nDynamical evolution of magnetic fields in the intracluster medium\nP. Dom ´ınguez-Fern ´andez1?, F. Vazza2;3;1, M. Br ¨uggen1and G. Brunetti3\n1Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Germany\n2Dipartimento di Fisica e Astronomia, Universit ´a di Bologna, Via Gobetti 92/3, 40121, Bologna, Italy\n3Istituto di Radio Astronomia, INAF , Via Gobetti 101, 40121 Bologna, Italy\nReceived / Accepted\nABSTRACT\nWe investigate the evolution of magnetic fields in galaxy clusters starting from constant pri-\nmordial fields using highly resolved ( \u00194 kpc ) cosmological MHD simulations. The magnetic\nfields in our sample exhibit amplification via a small-scale dynamo and compression during\nstructure formation. In particular, we study how the spectral properties of magnetic fields are\naffected by mergers, and we relate the measured magnetic energy spectra to the dynamical\nevolution of the intracluster medium. The magnetic energy grows by a factor of \u001840-50 in\na time-span of \u00189Gyr and equipartition between kinetic and magnetic energy occurs on a\nrange of scales ( <160 kpc at all epochs) depending on the turbulence state of the system.\nWe also find that, in general, the outer scale of the magnetic field and the MHD scale are not\nsimply correlated in time. The effect of major mergers is to shift the peak magnetic spectra\ntosmaller scales , whereas the magnetic amplification only starts after .1 Gyr. In contrast,\ncontinuous minor mergers promote the steady growth of the magnetic field. We discuss the\nimplications of these findings in the interpretation of future radio observations of galaxy clus-\nters.\nKey words: galaxy: clusters, general – methods: numerical – intergalactic medium – large-\nscale structure of Universe\n1 INTRODUCTION\nGalaxy clusters assemble through mergers and accretion until they\nreach an approximate virial equilibrium (e.g. Kravtsov & Borgani\n2012; Planelles et al. 2015). These events affect the space between\ngalaxies which is filled with a dilute plasma, known as the intra-\ncluster medium (ICM). In particular, radio observations shed light\non the non-thermal component of the ICM revealing the existence\nof cosmic rays and magnetic fields permeating galaxy clusters (e.g.\nFeretti et al. 2012; Brunetti & Jones 2014; Donnert et al. 2018). Ob-\nservations of synchrotron emission indicate magnetic fields with\nstrengths of a few \u0016G (corresponding to an energy density of\n\u00181\u00002%of the thermal energy of the ICM) and typical coher-\nence scales in the range of \u001810\u000050kpc (e.g. V ogt & Enßlin\n2005). Typically, this coherence scale is derived by a Fourier anal-\nysis of rotation measure (RM) maps, and inferring the maximum\nand minimum scales in the magnetic spectrum (often assuming\na Kolmogorov-like power spectrum) necessary to reproduce the\nobserved properties within uncertainties (e.g. Murgia et al. 2004;\nBonafede et al. 2010, 2013). In order to explain their observed mor-\nphology and strength, it has been suggested that magnetic fields get\ntangled over time by some other process than gas compression (e.g.\nDolag et al. 1999; Br ¨uggen et al. 2005a; Xu et al. 2009).\nWhile the origin of magnetic fields in galaxy clusters is still\n?E-mail: pdominguez@hs.uni-hamburg.desubject to debate, two scenarios have been widely discussed: (i) the\nprimordial scenario, in which magnetic fields have been generated\nin the early Universe possibly during (or after) inflation but prior to\nthe formation of galaxies (e.g. Turner & Widrow 1988; Kobayashi\n2014; Grasso & Rubinstein 2001; Kandus et al. 2011; Subramanian\n2016) and (ii) the astrophysical scenario, in which magnetic fields\nwere produced from stellar winds (e.g. Donnert et al. 2009) or ac-\ntive galactic nuclei (AGN) (e.g. Xu et al. 2011). A lower bound on\nthe strength of the initial seed field of B>3\u000210\u000016G (comoving)\nhas been inferred for voids from the non-observation of secondary\ngamma-rays around blazars (e.g Neronov & V ovk 2010). On the\nother extreme, upper limits of the order of B610\u00009G (comov-\ning), derived from the observed level of cosmic microwave back-\nground (CMB) anisotropies (e.g. Planck Collaboration et al. 2016),\ncan be used to limit the strength of any primordial seed field with\ncoherence scales of \u0018Mpc or larger.\nRegardless of the magnetogenesis scenario, magnetic fields\nmust have been significantly amplified in order to have reached to-\nday’s values. It is generally assumed that the amplification of the\ninitial magnetic fields occurred via the combined effect of adia-\nbatic compression and the presence of a small-scale dynamo, both\nof which are driven by minor or major mergers (e.g. Roettiger et al.\n1999; Br ¨uggen et al. 2005b; Subramanian 2016). The presence of\na small-scale dynamo requires the existence of turbulence in the\nICM, which is supported by cosmological simulations (e.g. Dolag\net al. 2005; Vazza et al. 2009; Iapichino & Niemeyer 2008; Ryu\nc\r0000 RASarXiv:1903.11052v1  [astro-ph.CO]  26 Mar 20192 P . Dom ´ınguez-Fern ´andez et al.\net al. 2008; Lau et al. 2009; Vazza et al. 2011; Marinacci et al.\n2015, 2018b; Donnert et al. 2018) and more recently, also by obser-\nvations (e.g. Hitomi Collaboration et al. 2018). A dynamo process\nconverts kinetic energy into magnetic energy over the typical dy-\nnamical timescales of the turbulent cascade. It is believed that the\namplification of ICM magnetic fields arises from the turbulence\ndeveloping on scales which are a fraction of cluster virial radius\n(60:5\u00001 Mpc ) (e.g. Donnert et al. 2018, and references therein).\nPrevious simulations have shown that only a few percent of the\nincompressible turbulent energy needs to be dissipated to account\nfor the observed field strength (e.g Miniati & Beresnyak 2015, and\nreferences therein).\nWhenever the characteristic scale of the magnetic field is com-\nparable or smaller than the characteristic scale of fluid motions, the\ndynamo is referred to as a small-scale dynamo (also called fluctu-\nation dynamo) (e.g Zeldovich et al. 1983; Kazantzev 1967). Con-\nversely, a large-scale dynamo refers to magnetic fields that are spa-\ntially coherent on scales comparable to the scale of the underly-\ning astrophysical system (e.g Zeldovich et al. 1983; Moffatt 1978).\nSince galaxy clusters do not show substantial rotation, it is likely\nthat the turbulent small-scale dynamo winds up magnetic fields on\nscales smaller than the turbulence injection scale (e.g Subramanian\net al. 2006; Brandenburg et al. 2012; Kazantzev 1967; Kraichnan &\nNagarajan 1967; Kulsrud & Anderson 1992; Schekochihin & Cow-\nley 2007; Beresnyak & Lazarian 2006; Schekochihin et al. 2008).\nIn previous papers (e.g Beresnyak & Miniati 2016; Miniati &\nBeresnyak 2015), driven turbulence in the ICM has been studied\nin a cosmological context. Still, it remains a challenge to push the\nspatial resolution down to the so-called MHD scale (lA) at which\nthe magnetic energy is strong enough to prevent additional bending\nof the magnetic field lines. It is crucial to resolve lAin order to fully\ncapture the development of the small-scale dynamo amplification,\nbutlAcan in principle be extremely small ( \u001ckpc) for arbitrarily\nsmall seed magnetic fields. The Reynolds number achieved in sim-\nulations is also an important factor that directly affects the magnetic\nfield growth. While the Reynolds number based on the full Spitzer\nviscosity in the ICM is believed to be of the order of Re\u0018102\n(e.g Brunetti & Lazarian 2007; Cho 2014a), the reduced proton\nmean free path in the collisionless ICM can result in much larger\nReynolds numbers (Beresnyak & Miniati 2016; Brunetti & Lazar-\nian 2011b). This suggests that the fluid approximation provides a\nsuited model for the properties of the ICM (e.g Santos-Lima et al.\n2017, 2014).\nMore recently, it has been shown that initial magnetic field\nseeds can be amplified via a dynamo up to strengths of \u0018\u0016G in\ncosmological grid simulations (e.g Vazza et al. 2018a) (hereafter\nPaper I). Here, we present a new sample of galaxy clusters to study\nthe spectral properties of each galaxy cluster in our sample. Firstly,\nwe study the characteristic spectral features of the magnetic energy\nin different types of clusters at z= 0. Secondly, we follow the\nspectral evolution of a particular cluster that is merging.\nThe paper is structured as follows: in Section 2 we present\nthe numerical setup and describe the fitting process of the magnetic\nenergy spectra. In Section 3 we present our results in two parts, the\nfirst one dedicated to the properties of our galaxy cluster sample at\nz= 0, and the second one describing the evolution of a merging\ncluster. In Section 4 we discuss numerical aspects and in Section 5,\nwe discuss the implications of our results.2 METHODS\n2.1 The Simulated Dataset\nWe simulated the formation of massive galaxy clusters in a cos-\nmological framework with the ENZO grid code (The Enzo Col-\nlaboration et al. 2013). We used the Dedner formulation of MHD\nequations (Wang & Abel 2009) and used adaptive mesh refine-\nment (AMR) to increase the dynamical resolution within our clus-\nters, as in Paper I. We assumed a \u0003CDM cosmology ( h= 0:72,\n\nM= 0:258,\nb= 0:0441 and\n\u0003= 0:742) as in Vazza et al.\n(2010).\nEach cluster was selected in a comoving volume of (260\nMpc)3, first simulated at coarse resolution (Vazza et al. 2010), and\nthen resimulated with nested initial conditions (Wise & Abel 2007).\nWe employed two levels of static uniform grids with 2563cells\neach and using 2563particles each to sample the dark matter distri-\nbution, with a mass resolution per particle of mDM= 1:3\u00011010M\f\nat the highest level.\nThen, we further refined the innermost \u0018(25 Mpc)3volume,\nwhere each cluster forms, with additional 7 AMR levels (refine-\nment = 27). The refinement was initiated wherever the gas density\nwas>1%higher than its surroundings. This gives us a maximum\nspatial resolution of \u0001xmax= 3:95 kpc per cell.\nWith our setup (see Paper I), for z61the virial volume of\nclusters is refined at least up to the 6th AMR level ( 15:8kpc) at\nz= 0, and most of the central volume within 61Mpc from the\ncluster centre is simulated with 3.95 kpc/cell.1\nIn this work, we will only discuss non-radiative cosmo-\nlogical simulations, meaning that we only included the effect\nof cosmic expansion, gas, Dark Matter self-gravity and (mag-\nneto)hydrodynamics, in order to solely focus on the growth of mag-\nnetic fields by the turbulence induced by structure formation.\nIn order to seed magnetic fields at the beginning of our runs,\nwe mimic a simple primordial origin of magnetic fields, in which\nwe initialized the field to a uniform value B0across the entire com-\nputational domain, along each coordinate axis. The initial magnetic\nseed field of 0:1 nG (comoving) is chosen to be below the upper\nlimits from the analysis of the CMB (e.g. Subramanian 2015). This\nparticular setup is easy to implement, ensures r\u0001~B= 0 by con-\nstruction, and has been already tested in our previous work on the\nsubject (Vazza et al. 2014, 2018a). Moreover, several studies have\nshown that the impact on the initial magnetic field topology within\ngalaxy clusters (provided that the simulated dynamical range is\nlarge enough to enter the dynamo regime) is negligible (e.g. Mari-\nnacci et al. 2015; Vazza et al. 2017; Vazza et al. 2018a), hence our\nresults do not strongly depend on this particular setup.\nWe refer the reader to Appendix A for a short overview of the key\nfindings of Paper I. There, we showed that our numerical setup pro-\nvides enough resolution to resolve the MHD scale, lA, in a large\nfraction of the cluster volume during its late evolution ( z61).\nMoreover, the simulations show features of small-scale dynamo\namplification. However, as we discuss in depth in Sec. 4, some re-\nsults can be affected by the limited spatial resolution.\n1Each cluster simulation used \u001830;000\u000050;000core hours running on\n64 nodes on JUWELS at J ¨ulich Supercomputing Centre.\nc\r0000 RAS, MNRAS 000, 000–000Evolution of magnetic fields in the ICM 3\n2.2 Fitting the magnetic power spectrum\nThe three-dimensional power spectrum is defined as\nPij(k) =1\n(2\u0019)3Z Z Z\ne\u0000ik\u0001xRij(k)dk; (1)\nwhereRij=hui(x0)uj(x0+x)iis the two-point correlation\nfunction between the velocities uianduj(e.g. Batchelor 1951).\nWhen the corresponding fields do not depend on the position and\nonly depend on the distance between two points, i.e. we consider\nhomogeneous and isotropic fields, the total energy is given by\nEtot=1\n2\nu2\ni\u000b\n=1\n2Rii(0) =Z1\n0E(k)dk; (2)\nwhereE(k)is thus the scalar energy distribution per unit mass for\nthe modekrelated to the diagonal components of the tensor Rij,\nand therefore, the relation between this spectral energy and the one-\ndimensional power spectrum is found to be\nE(k) = 2\u0019k2Pii(k): (3)\nThis approximation works well for the rather chaotic and isotropic\nvelocity field always found in cosmological cluster simulations\n(e.g. Dolag et al. 2005; Vazza et al. 2011; Wittor et al. 2017).\nWe computed first the power spectrum by using standard algo-\nrithms for the three-dimensional Fast Fourier Transform (FFT) of\nthe velocity and magnetic fields within the simulation box and\nthen by summing up the contributions over spheres within a radius\nk=p\nk2x+k2y+k2zin Fourier space. Finally, by multiplying by\nthe factor 2\u0019k2, we obtained the energy spectrum of the magnetic\nand velocity field.\nWhile the velocity power spectra can be characterized by a\npower-law and by an injection scale, the magnetic spectra are more\ncomplex. We fit the magnetic spectra by the equation:\nEM(k) =Ak3=2\u0014\n1\u0000erf\u0014\nBln\u0012k\nC\u0013\u0015\u0015\n; (4)\nwhere theAparameter gives the normalization of the mag-\nnetic spectrum, Bis related to the width of the spectra and Cis a\ncharacteristic wavenumber corresponding to the inverse outer scale\nof the magnetic field (see Fig. 1). Eq. (4) is rooted in dynamo the-\nory as a solution for single-scale turbulent flows (Kazantzev (1967),\nKraichnan & Nagarajan (1967), Kulsrud & Anderson (1992)). In\nthe remainder of the paper we propose to use Eq. (4) as a proxy to\ncharacterize our evolving magnetic spectra with a minimal set of\nparameters (A,B,Cas detailed above), even though the equation\nis not valid for the scales and conditions that we are studying. It\nshould be stressed that the aim of the paper is not to connect di-\nrectly these parameters with Kazantzev’s dynamo model since the\ngeneration and evolution of turbulent magnetic fields in the ICM\nare affected by a hierarchy of complex processes. In particular, we\nnote that:\n1) The assumptions under which Eq. (4) is derived, such as\nhaving a single-scale turbulent flow, a Kolmogorov spectrum for the\nvelocity field, neglecting the resistive scale, etc. (see more details\non the assumptions and derivation in Kulsrud & Anderson 1992)\nare not valid since, in our system, laminar gas motions and advec-\ntion at many scales may also affect the topology of the magnetic\nfields in the ICM. Furthermore, the magnetic field is amplified and\nre-shaped by the turbulence generated every time a merger occurs.\n2) The analysis of non-linear effects such as ambipolar dif-\nfusion or magnetic reconnection are far beyond the scope of this\nwork. But we can comment that some of these affects have been\nFigure 1. Variation of A, B and C parameters in Eq. (4). Top panel: change\nin the normalization. Middle panel: change in the width. Bottom panel:\nchange in the position of the outer scale.\nalso studied in Kulsrud & Anderson (1992), where the final mag-\nnetic power spectrum exhibits a similar shape, i.e. a power law\nmultiplied by a Macdonald function (or modified Bessel function\nof second order) of different orders. For small k, they can reduce to\nEq. (4).\n3) As long as the velocity scale responsible for the dynamo\nforcing is larger than the scales where the magnetic energy spec-\ntrum peaks, Eq. (4) is valid. This condition is matched during the\ninitial stage of cluster formation, and is later violated after the mag-\nnetic field has grown to larger scales. It is our intention to quantify\nthe development of magnetic fields as a function of resolution (as\nin Paper I) as well as of the cluster evolution. For this reason, it is\nconvenient to apply Eq. (4), as the dynamo in our runs is expected\nto stay in the kinematic regime for long due to the finite numerical\nresolution (e.g. Beresnyak & Miniati 2016).\n3 RESULTS\n3.1 Magnetic fields in the cluster sample\nIn this section we analyze a sample of seven clusters in different\ndynamical states: clusters with ongoing mergers (ME) at z= 0,\nrelaxed ones (RE) and post major merger ones (PM). In Fig. 2 we\nshow the projected gas density and magnetic field strength for all\nof our clusters, considering the highest resolution of our simulation\n(3.95 kpc).\nA list of the main parameters of our simulated clusters is given\nin Tab. 1. The estimate of the total (gas+DM) mass inside R100, as\nwell as the tentative classification of the dynamical state at z= 0of\neach object follows from Vazza et al. (2010). Our dynamical clas-\nsification is done in two steps: firstly, clusters with a major merger\n(based on the total mass accretion history within R100) forz <1\nc\r0000 RAS, MNRAS 000, 000–0004 P . Dom ´ınguez-Fern ´andez et al.\nFigure 2. Maps of projected gas density and magnetic field strength for all clusters in our sample at z= 0 (we omit cluster E5A since this cluster is analyzed\nin detail in Section 3.3 and 3.4). The main characteristics of these clusters can be found in Tab. 1.\nare classified as post-mergers (PM). In particular, major mergers in\nthe range 06z61are selected considering that the change of the\nmass increment \u0018=M2=M1is\u0018>0:3, whereM1is the mass at a\ntimetandM2is the mass at time t+ 1Gyr (Fakhouri et al. 2010).\nSecondly, if no major merger is found in this time interval, we ad-\nditionally compute the ratio between the total kinetic energy of gas\nmotions inside R100,EK, and the total energy ( Etot=EK+ET)\ninside the same volume. This parameter has been shown to char-\nacterise the dynamical activity of clusters well (e.g. Tormen et al.\n1997). Relaxed (RE) clusters typically have EK=Etot<0:5whilemerging (ME) clusters have EK=Etot>0:5. In Tab. 1 we also list\nthe redshift of the last major merger ( zlast) for post-merger systems,\nwhile for relaxed systems we conventionally consider zlast= 0and\nzlast= 1 for merging systems. For a more detailed discussion of\nthe classification scheme we refer the reader to Vazza et al. (2010)\nand references therein.\nFor each cluster, we computed the radial profile of the av-\nerage magnetic field from the peak of gas density at z= 0\nat the highest resolution, as shown in Fig. 3. Within the sample\nvariance, we find that the magnetic field follows gas density as\nc\r0000 RAS, MNRAS 000, 000–000Evolution of magnetic fields in the ICM 5\nIDM100[M\f]R100[Mpc] Dynamical state B0[\u0016G] A (10\u000017[G2=k]) B ([-]) C (k[1/2 Mpc])\nE14 1:00\u000110152:60 RE 1.726 5.470 \u00060.111 1.090\u00060.009 6.461\u00060.096\nE5A 0:66\u000110152:18 ME 1.050 1.985 \u00060.059 1.054\u00060.012 8.708\u00060.192\nE1 1:12\u000110152:67 PM(zlast= 0:1) 1.308 2.052\u00060.036 1.118\u00060.009 10.052\u00060.131\nE3A 1:38\u000110152:82 PM(zlast= 0:2) 1.672 2.372\u00060.041 1.167\u00060.009 8.936\u00060.110\nE16B 1:90\u000110153:14 PM(zlast= 0:2) 2.474 9.041\u00060.164 1.134\u00060.009 10.437\u00060.138\nE4 1:36\u000110152:80 PM(zlast= 0:4) 1.572 4.521\u00060.074 1.124\u00060.008 10.236\u00060.123\nE18B 1:37\u000110152:80 PM(zlast= 0:5) 1.716 3.396\u00060.049 1.113\u00060.007 9.974\u00060.106\nTable 1. Main parameters at z= 0 of the galaxy clusters analyzed in this work. The 4th column lists the tentative dynamical classification of each object (with\nthe approximate redshift of the last major merger, in the case of post-merger clusters). The value of B0is the mean magnetic field within 200 kpc from the\ncorresponding radial profiles plotted in Fig. 3\nFigure 3. Radial profile of the magnetic field for all the clusters in our\nsample atz= 0 computed at our highest resolution run at the 8th AMR\nlevel.\nB(n)\u0019B0\u0001(n=n0)0:5(wherenis the gas density and n0is the\ncore gas density) as in Paper I. In fact, the radial profiles in Paper\nI appear to be consistent with what can be derived by Faraday Ro-\ntation analysis of the Coma cluster (Bonafede et al. 2013), despite\nthe fact that the distribution of magnetic field components found in\nour simulations deviate significantly from a Gaussian distribution.\nThe central magnetic field value of each cluster, B0, is given for\nreference in Tab. 1. In general, we can see that the most perturbed\ncluster (E5A) does not show the strongest fields. The central value,\nB0(measured as the average within the innermost 6200 kpc ra-\ndius from the cluster centre), is strongly correlated with the mass of\nthe cluster. Indeed, in Fig. 3 we can see that the higher the mass, the\nhigher the central value of the magnetic field. While observations\ndo not show a clear correlation of the mean magnetic field with the\nmass of the host cluster (e.g. Govoni et al. 2017), our normaliza-\ntionA, which is the parameter most closely linked to the Faraday\nRotation, shows little correlation with mass, and has a large scatter.\n3.2 Spectral properties in the cluster sample\nNext, we proceeded to compute the magnetic energy power spec-\ntra for the innermost region of all clusters at z= 0 as described\nin Sec. 2.2 . We computed power spectra only for the innermost\n\u001923Mpc3region of each cluster, where the resolution is approx-\nimately constant and equal to the 8th and maximum AMR level\n(corresponding to a 5123grid). By doing so, we can neglect the\neffect of coarse-mesh effects in our FFT analysis as the majorityof the central cluster volume is refined up to the highest level for\nall our clusters (see discussion in Vazza et al. 2018a). The corre-\nsponding spectra, along with the best-fit curves are plotted in the\ntop panel of Fig. 4 and the best-fit parameters are listed in Tab.\n1. To a good degree of approximation, all spectra are well fitted\nby Eq. (4) regardless of the dynamical state of each cluster. All\nclusters in the sample show similar spectral shapes, with a peak of\nmagnetic energy in the range \u0018200\u0000300 kpc and differences in\nnormalization of a factor 65. As shown in Vazza et al. (2018b),\nthis non-Gaussian distribution of magnetic field strengths may re-\nsult from the superposition of multiple magnetic field components\nthat have been accreted at different times via mergers. For com-\npleteness, we also show the kinetic spectra of all the clusters in the\ncentral panel of Fig. 4. These kinetic spectra are very similar, i.e.\nwe observe a higher normalization for perturbed clusters as there is\nmore turbulence involved in these systems, and the lowest normal-\nization is observed for the relaxed cluster (E14). Comparing this\nto the magnetic spectra shown in the top panel of Fig. 4, we can\nclearly see that a higher level of turbulence does not necessarily\nimply higher values of the magnetic field. This may seem counter-\nintuitive but it is caused by the fact that the amplification of mag-\nnetic fields from small to large spatial scales is a slow process that\ntakes a few eddy turnover times. Therefore, even in the presence of\na large input of turbulent kinetic energy, significant magnetic am-\nplification can only be observed with a delay of \u0018Gyr. While\npart of this delay is caused by numerical effects (e.g. our numeri-\ncal finite growth rate depends on the limited Reynolds number our\nsimulation can resolve), this delay is of the same order as the eddy\nturnover timescale for \u0018500 kpc turbulent eddies being injected\nwith a\u001bv\u0018500 km=svelocity. This is the necessary time span for\nturbulence to cascade down to the scales that can drive a dynamo\ngrowth.\nIn the bottom panel of Fig. 4 we plot the ratio between ki-\nnetic and magnetic energy in order to visualize the scales at which\nequipartition is reached. RE systems reach equipartition at larger\nscales compared to PM systems, which is consistent with the gen-\neral picture of a small-scale dynamo acting according to the amount\nof turbulence in the system. As expected, we also observe that the\nME system is still not in equipartition at larger scales because this\nis the most perturbed cluster and it is mostly dominated by com-\npressive turbulence.\n3.2.1 Parameterization of cluster magnetic spectra\nOur analysis in Paper I supports that the magnetic spectra show\nsigns of a dynamo near saturation (see Appendix A). However, as\nwe shall see in Sec. 3.3, if a small-scale dynamo is acting, it co-\nexists with bulk motions on larger scales that are affecting the evo-\nlution of the magnetic field during the whole assembly history of\nc\r0000 RAS, MNRAS 000, 000–0006 P . Dom ´ınguez-Fern ´andez et al.\nFigure 4. Magnetic energy ( top panel ) and kinetic energy ( middle panel )\nspectra of all of our cluster sample at z= 0 . The kinetic spectra were\nmultiplied bypn, wherenis the gas density, in order for the spectra in both\npanels to have the same units. The solid lines correspond to the data and the\nscatter plots show the best-fit of the corresponding data using Eq. (4). In the\nbottom panel we show the ratio of kinetic to magnetic energy, EK=EM(k),\nthe horizontal black dashed line indicates where we have equipartition.\nFigure 5. Comparison of best-fit parameters of each cluster in our sample\natz= 0 according to their virial mass.\nFigure 6. Comparison of best-fit parameters of each cluster in our sample\nto their last major merger event.\nthe clusters. As a consequence, the magnetic properties in our sam-\nple result from the cumulative (and discontinuous) action of dy-\nnamo during the entire cluster life-time. Therefore, there is no im-\nmediate connection between the spectral magnetic properties and\nthe turbulent properties of the cluster at a given time.\nIn order to study how the best-fit parameters, A,BandC, are\nrelated to the mass, dynamical state and redshift since the last major\nmerger, we produced Figs. 5, 6 and 7. For our limited sample, we\ncan conclude:\n1) The spectrum normalization ( A): We find a dependence of the\nmass of the host cluster, and also a hint of a dependence on the\ndynamical state of the cluster. For a given mass bin we find AME<\nAPM<A RE. On the other hand, we do not find a correlation with\nthe time since the last major merger.\n2) The spectrum width ( B): This parameter is found to depend on\nthe dynamical state of each cluster, i.e. Bis larger in less perturbed\nc\r0000 RAS, MNRAS 000, 000–000Evolution of magnetic fields in the ICM 7\nFigure 7. Combination of best-fit parameters of each cluster (proportional\nto the rotation measure (RM)) in our sample compared to their virial mass.\nsystems (PM) and smaller in highly perturbed systems (ME). This\npresumably means that relaxed systems have had enough time for\nturbulence to cascade to the small scales and amplify the magnetic\nfield during past mergers resulting in a broader spectrum; whereas\nthe merging systems have a more restricted region of magnetic am-\nplification. As will be mentioned in 3.4, mergers shift the spectrum\ntowards smaller scales, thus the combination of this shift and the\nnarrow spectrum means that merging systems contain more small-\nscale eddies. Therefore perturbed systems have a higher magnetic\ngrowth rate than relaxed systems.\n3) The inverse of the outer scale of magnetic field ( C): we find\na hint of a dependence on the mass and the dynamical state of the\ncluster,CRE<C ME<C PM. On the assumption of the existence\nof a small-scale dynamo, this would suggest that in more relaxed\nsystems the dynamo had more time to grow towards larger scales\n(i.e. lower values of C). In the case of an on-going merger (as will\nbe discussed in more detail in Sec. 3.3), large scale gas motions\nmay also affect the magnetic spectra merely by compression. In\nprinciple, it will be possible that advection and buoyancy in a strat-\nified medium such as the ICM also play a role in the outer scale\nbehaviour, but the study of these effects are beyond the scope of\nthis paper.\n4) Dependence on the time since the last major merger: We did\nnot find a correlation between the epoch of the last major merger\n(zlast) and the best-fit parameters ( A,BandC), which suggests\nthat the magnetic energy spectrum at a given epoch does not re-\ntain much information about specific events, as well as that minor\nmergers are also an important player in setting the spectral proper-\nties of the ICM at z= 0 (see discussion in 3.4). Moreover, given\nthe limited sample size we have here, it is difficult to disentangle\neffects connected to the mass of the host cluster and the dynamical\nstate.\n5) Rotation measure dependence on mass: The rotation measure\n(RM) of polarised radio emission from background sources scales\nasjRMj/R\nBjj\u0001nedl, which can be approximated to jRMj/\nB\u0003\u0001\u0003B, whereB\u0003is the magnetic field at the autocorrelation scale\n\u0003B. To a first approximation, the total jRMjfrom a clusters should\nscale with the/A=C . We find that the RM depends weakly on the\nmass(because of A and C dependence on mass), but scatters due to\na dependence on the dynamical state of each cluster. This causes\nclusters with a factor \u00182difference in mass to show a very similar\nA=C value (i.e. E14 and E16B, see Fig.7).\nIn order to link the evolution of magnetic spectra to the dy-\nnamical growth of galaxy clusters, in the next section we will ana-\nlyze the assembly of one particular cluster.3.3 Detailed evolution of cluster E5A\nWe studied the evolution of the cluster E5A by analysing a total of\n\u0019100snapshots in the range from z= 1:379toz= 0. The cluster\nE5A is an interesting object as it forms via several mergers in the\ncourse of nearly 9 Gyr.\nWe work on uniformly gridded data reconstructed at the 6th\nAMR level (15.8 kpc resolution). This is done even if the simula-\ntion has refined down to the 8th AMR level. In Fig. 8 we show snap-\nshots of the density and magnetic fields in a simulation box of 6403\ncells. The maps in Figs. 8-9 show a volume-weighted projection of\nthe magnetic field strength along the line-of-sight in order to em-\nphasize the diffuse magnetic field structure on large scales. Hence,\nthe magnetic field values in the map are biased towards lower val-\nues than the ones measured in the computational box.\nNext, we identified the centre of the main cluster (cyan dot in\nFig. 8) and then followed the evolution of gas and magnetic fields\nwithin a box of 1003cells co-moving with the main cluster centre.\nThe trajectory of the centre was obtained by computing the location\nof the maximum of the thermal energy after smoothing the data\nover a length of\u001820cells at each snapshot, and by applying a\ncubic spline time interpolation.\nThe final stage of the merger at z= 0is shown in Fig. 9, where\nthe volume distribution of the gas density and the magnetic field\nstrength are plotted in x,yandzdirections. The magnetic field dis-\ntribution is asymmetric, showing a tangled structure and its strength\nincreases towards the centre of the major component. At least two\nprominent peaks in the magnetic field distribution (at the \u0018\u0016G\nlevel) near the central region are visible in all lines of sight. These\npeaks correspond to the largest and the second largest components.\nIn Fig. 10 we show the evolution of the magnetic field\nstrength, temperature and velocity within the moving simulation\nbox. Every merger event is found to perturb the system and to in-\ncrease the thermalization of the ICM, shown as peaks on the tem-\nperature evolution in Fig.10. While gas velocity and gas temper-\nature show pronounced peaks close each merger event, the evo-\nlution of the averaged magnetic field strength is smoother. These\nsmoothed peaks are correlated with the merger events, but show a\ndelay of about\u00180:5Gyr with respect to the velocity peaks. In\nFig. 11 we show the evolution of the total energy budget of the\ncluster (top panel) and the corresponding evolution of the energy\nratios (bottom panel). During the whole period of evolution ( \u00189\nGyr), the magnetic energy has grown by a factor \u001840-50, the ki-\nnetic energy by a factor of \u001890-100, and the thermal energy has\ngrown by a factor\u001810-20. By the end of the simulation ( z= 0),\nthe kinetic energy is \u001810-40% of the thermal energy, while the\nmagnetic energy is \u001810\u00003of the thermal energy.\n3.4 Spectral evolution of cluster E5A\nIn this section we focus on the spectral features of the magnetic and\nkinetic energy power spectra, whose entire evolution is given in Fig.\n12. The spectra are shown using comoving units and are computed\nwithin a box of comoving size L= 1:58Mpc, which moves with\nwith the cluster centre identified as described in Section 3.3. The\nfirst thing to notice is that the shape of both spectra change very\nlittle over the period from z= 1:379toz= 0, while the normal-\nization increases whenever a minor merger occurs. In the bottom\npanel of Fig. 12, we can observe that the evolution of the mag-\nnetic energy spectra shows a global increment on the magnetic field\nstrength up to an approximate state of equipartition for k\u001820\u000050\n(corresponding to scales \u001830\u000080kpc). This means that during a\nc\r0000 RAS, MNRAS 000, 000–0008 P . Dom ´ınguez-Fern ´andez et al.\nFigure 8. Top panels: projected z-component density field averaged a long the line-of-sight at various redshifts. Bottom panels: projected z-component of the\nmagnetic field strength averaged a long the line-of-sight at the same redshifts as the top panels. The dots indicate the centre of the most massive component at\neach redshift. The centre position was obtained by taking into account the kinetic energy within a simulation box of 6403cells (see text for more details).\nc\r0000 RAS, MNRAS 000, 000–000Evolution of magnetic fields in the ICM 9\nFigure 9. Maps averaged along the line-of-sight of the simulation box at z=0. The top panels show the projected magnetic field strength and the bottom panels\nshow the projected density.\nFigure 10. Evolution of the average magnetic field strength, temperature\nand velocity of the 1003simulation box\nsignificant fraction of the system evolution the magnetic tension is\nstrong enough to prevent the further bending of the magnetic lines,\nas would be expected from a classic small-scale dynamo. However,\nthe range of scales in which equipartition is reached does not evolve\nmonotonically with time (as expected in a classic dynamo), but it\nfluctuates in time, with features that are non-trivial to isolate. In\nparticular, the epochs where there is no equipartition coincide with\nthe occurrence of mergers, i.e. when the cluster is more perturbed.\nThe various kinetic injections driving turbulence in this system will\ncontinuously change the magnetic field topology on spatial scales\nlarger than the equipartition scale.In order to examine the evolution of E5A, in Fig. 13 we colour\ncoded the amplitude of the magnetic and kinetic spectra as a func-\ntion of time. This spectral time sequence shows the entire evolution\nof the ICM as a function of time and spatial scale. As the system\nevolves, the magnetic power increases and tends to shift towards\nsmaller scales, while the kinetic spectrum is always characterized\nby a maximum at k= 1, which mirrors the fact that the forcing of\nturbulence always occurs on scales >1 Mpc . Merger events can\nbe seen as horizontal stripes in the plot, which correspond to the\ninjection of kinetic energy.\nThe resulting amplification of the magnetic field strength is\nthen a complex interplay between compression and the small-scale\ndynamo. This is best shown by the appearance of dense gas struc-\ntures at a similar time, as shown in the power spectra of gas density\nin Fig. 14, which is consistent with the relation between velocity\nand density fluctuations in the stratified ICM (e.g. Gaspari et al.\n2014). A general trend is that every merger shifts the magnetic\nspectral power towards higher wave numbers, i.e. during most of\nthese events the peak of the magnetic energy spectrum moves to-\nwards smaller spatial scales, unlike what is expected from the stan-\ndard dynamo model, and most likely due to gas compression. As\ncluster mergers generate shocks and bulk flows that enhance the\ngas density and compress the magnetic field lines, this can also in-\ncrease the normalization of the spectrum. Furthermore, it can also\nmove the peak of the spectrum to higher wave numbers because the\nmagnetic field lines get stretched along the merger direction.\nSimultaneously, mergers inject turbulence, and only after the\nlatter has decayed to small scales (where the eddy turnover time is\nthe shortest), the peak magnetic spectra shifts towards lower wave\nc\r0000 RAS, MNRAS 000, 000–00010 P . Dom ´ınguez-Fern ´andez et al.\nFigure 11. Energy evolution of the 1003simulation box. The top panel\nshows the evolution of the thermal energy (red), kinetic energy (green) and\nmagnetic energy (purple). The bottom panel shows the corresponding en-\nergy ratios.\nnumbers and the magnetic field is boosted again. This effect is char-\nacteristic of a small-scale dynamo.\nOur analysis implies that both, compressive and dynamo am-\nplification, tend to be present at the same time in galaxy clus-\nters. This causes a difficult evolutionary pattern in the simulated\nICM, adding complexity to what has been previously obtained by\nmore idealized MHD simulations (e.g. Beresnyak & Miniati 2016;\nMiniati & Beresnyak 2015).\nFor better visualization, Fig. 15 shows the residual between\nmagnetic and kinetic spectral energies also as a spectral time se-\nquence plot. At all epochs, the excess magnetic energy is found\non wave numbers k > 10(corresponding to scales <160 kpc),\nshowing that after merger events the magnetic tension gets strong\nenough to overcome further bending of the magnetic lines only at\nsmall scales. The magnetic amplification starts only after merger\nevents because the turbulence injected takes a few eddy-turnover\ntimes to cascade.\nIn fact, if the kinetic energy injection is high enough, as we\ncan observe around t\u00189:8Gyr in Fig. 15, the amplification is\nslowed down.\nIn order to identify the specific times of kinetic energy injec-\ntion, we plotted in Fig. 16 the difference of the total kinetic energy\nin the simulation box at timestep tiwith respect to the previous\ntimestep,ti\u00001. A peak in this plot can account mainly for either\nFigure 12. Top panel: Evolution of the spectral kinetic and magnetic energy\nin the simulation box of 1003cells. The top spectra correspond to the ki-\nnetic energy and the bottom spectra correspond to the magnetic energy. The\nvelocity power spectrum was multiplied bypn,nbeing the gas density, in\norder for the spectra to have the same units. Bottom panel: Ratio of kinetic\nto magnetic energy as function of the wave number. The horizontal dashed\nline indicates where we have equipartition.\nthe entrance of a clump into the simulation box, a shock travel-\ning across the cluster or a reflected shock. Since we are interested\nin studying the amplification periods identified in Fig. 15, we re-\nstrict ourselves to point out only some of these events confirmed\nby visual inspection with red arrows in Fig. 16. The shaded areas\nin the plot are placed as a reference for the amplification phases\nfound in the spectral time sequence of Fig. 15. We noticed that, the\nmaximum kinetic injection appears to happen either when gas sub-\nstructures cross close to the cluster centre, which typically leads\nto shock waves (M \u0018 2–3 in this case, as we measured with a\nvelocity-based shock finder following Vazza et al. 2017) sweeping\nthrough the cluster; or when there is a continuous injection of tur-\nbulence by minor mergers (period between t\u001812–13 Gyr). In the\nfirst case, the most significant boosts of kinetic energy are followed\nby the compression of the magnetic field spectra. The injection of\nlarge amounts of kinetic energy on large scales impact the magnetic\nfield only after .1Gyr (white areas after first and second red ar-\nrows in Fig. 15), suggesting that a small-scale dynamo is activated\nonly after such amount of time. In the second case, continuous mi-\nc\r0000 RAS, MNRAS 000, 000–000Evolution of magnetic fields in the ICM 11\nFigure 13. Evolution of the spectral energy in the simulation box of 1003\ncells. The top panel shows the corresponding evolution of the magnetic\nenergy and the bottom panel shows the evolution of the kinetic energy.\nFigure 14. Density power spectrum as a sequence of time.\nnor mergers contribute to the magnetic amplification at small scales\nby starting to shift the power towards higher scales (period between\nt\u001812–13 Gyr). This seems to suggest that minor mergers signifi-\ncantly power the small-scale dynamo amplification.\nFinally, we studied the evolution of the MHD scale ( lA) using\nthe result from Brunetti & Lazarian (2007):\nlA\u00183\u0010\nB\n\u0016G\u00113\u0010\nL0\n1Mpc\u0011\u0000\u001bv\n103kms\u00001\u0001\u00003\u0000n\n10\u00003cm\u00003\u0001\u00003=2kpc;\nwhereL0is the reference scale within the Kolmogorov iner-\ntial range and \u001bvis the rms velocity within the scale L0. In this\ncase, we measure the turbulent velocity by filtering the large mo-\ntions on\u0019300kpc. We obtain a distribution of the MHD scale for\nall of our snapshots and select the mean at each time. In Fig. 17 we\nFigure 15. Energy residual evolution corresponding to the energies in\nFig.13. The highest values appear at small scales showing how the amplified\nmagnetic field is able to overcome the kinetic pressure.\nFigure 16. Kinetic energy residual as function of time. The red arrows are\nrelated to the time when an in-falling gas clump crosses the centre of the\ncluster. The shaded areas are identified directly with Fig.15, therefore indi-\ncating the periods of amplification.\nshow the resulting evolution of the corresponding scale ( lA) and\ncompare it to the evolution of the outer scale of the magnetic spec-\ntrum ( 1=C). It has been suggested in former studies (e.g. Beres-\nnyak & Miniati 2016; Miniati & Beresnyak 2015) that lAwill fol-\nlow closely the evolution of the outer scale of the magnetic spec-\ntrum. Our analysis suggests that in reality the evolution of magnetic\nfields during mergers is more complicated than that. The system is\nsignificantly affected by compression and large-scale coherent mo-\ntions, whose energy is larger than the small-scale turbulent energy\non6300 kpc scales. In fact, the injected energy may contribute\nto advect magnetic field lines on large scales ( >100kpc). Over-\nall, this means that our galaxy clusters exhibit cumulative turbu-\nlence cascades with different injection timescales, able to amplify\nthe existing magnetic fields via a dynamo action. Under these con-\nditions, the evolution of the outer scale is mismatched with respect\nto that of the MHD scale. This has important implications for the\nfuture surveys of magnetic fields in galaxy clusters. The interpre-\ntation of magnetic field spectra inferred by Faraday Rotation will\nnot uniquely constrain the magnetic amplification coming from a\nsmall-scale dynamo, but may also be contaminated by compression\namplification coming from large-scale gas flows.\nc\r0000 RAS, MNRAS 000, 000–00012 P . Dom ´ınguez-Fern ´andez et al.\nFigure 17. Evolution of the MHD scale and the outer scale of the magnetic\nspectrum (inverse of the C parameter). Note that the MHD scale is rescaled\nby a factor of 50 for ease of comparison.\n3.4.1 Evolution of best-fit parameters for cluster E5A\nFollowing the same approach of Section 2.2, we proceeded with the\nfitting of all magnetic spectra in the evolution of E5A, which yields\nthe evolutionary tracks shown in Fig. 18. The top panel shows the\nnormalization of the magnetic energy spectrum, where we can see\na clear result: the overall amplification of the magnetic field con-\ntinues to grow but steepens more where mergers occur. In fact, we\nobserve that the normalization almost increases by one factor on\nthe last\u00180:5Gyr where a major merger is about to happen. As a\nconsequence of these events and the other effects previously men-\ntioned, the magnetic growth is not linear. While the total magnetic\nenergy increases by a factor of \u001840-50 (as mentioned in Section\n3.3), the normalization of the spectrum only increased by a factor\nof\u00185in nearly 9 Gyr.\nIn the middle and bottom panels of Fig. 18 we show the evolu-\ntion of the parameters B and C. It is notable that both evolution pat-\nterns seem to be correlated. The evolution of C (wave number cor-\nresponding to the outer scale of the magnetic spectrum) also shows\na correlation with some identified merger events: the red arrows\nover-plotted corresponding to those in Fig. 16. Mergers induce an\nimmediate change of the outer scale of the spectrum by shifting the\npower towards smaller scales. While this pattern is less obvious in\nthe evolution of the parameter B, we can observe that mergers also\ninduce an immediate broadening of the spectrum. These combined\neffects can be directly associated with the action of compression. A\nparticular thing to notice is that, the change on B and C at the last\n(third arrow) merger event is not as large as the previous events.\nThis suggests that at this point, the cluster has had enough turbu-\nlence input (at different injection scales and timescales) to amplify\nthe magnetic field at smaller scales, making it harder for the spec-\ntrum to broaden or shift its power to even smaller scales.\n4 NUMERICAL ASPECTS\nAs in Paper I, we relied on the Dedner cleaning algorithm (Dedner\net al. 2002) to get rid of magnetic monopoles. The main limitation\nof this method is the reduction of the effective dynamical range,\ncompared to Constrained Transport (CT) schemes at the same res-\nolution, due to the intrinsic dissipation of the scheme by r\u0001~B\nFigure 18. Evolution of the best-fit parameters A,BandCobtained by\nmeans of Eq. (4). The 2\u001berror envelopes are shown in lighter shades.\ncleaning waves which keep the numerical divergence under con-\ntrol (Kritsuk et al. 2011). Several groups have tested that the Ded-\nner cleaning method is robust and accurate for most idealized test\nproblems, as long as the resolution is opportunely increased (e.g.\nWang & Abel 2009; Wang et al. 2010; Bryan et al. 2014). Even\nin the test of more realistic astrophysical applications, the Dedner\nmethod has been shown to quickly converge to the right solution,\nunlike different approaches to clean r\u0001~Bpreserving at run time\nc\r0000 RAS, MNRAS 000, 000–000Evolution of magnetic fields in the ICM 13\n(Stasyszyn et al. 2013; Hopkins & Raives 2016; Tricco et al. 2016;\nBarnes et al. 2018).\nDespite the numerical dissipation introduced by the Dedner\ncleaning, all important features discussed in this paper (e.g. the\npeak in the power spectrum of magnetic fields, and the equipartition\nscales) are much larger than the length scales affected by numerical\ndissipation: e.g. the peak of power spectra are typically on scales\n\u001825\u000050larger than the minimum cell size in our the simula-\ntion. While the dissipation in the Dedner scheme can considerably\nslow down the first stage of the dynamo amplification (Beresnyak\n& Miniati 2016), once that magnetic structure becomes sufficiently\nlarge, they are relatively unaffected by numerical dissipation.\nIn Paper I we verified that in the largest part of the simulation\nbox, the numerical divergence of Bis of order\u00182-3%of the local\nmagnetic field strength, i.e. 610\u00004of the magnetic energy on\nlarger scales. We refer the reader to the recent review by Donnert\net al. (2018) for a broader discussion of the resolution and accuracy\nof different MHD schemes in the context of small-scale dynamo\nprocesses in galaxy clusters.\nOur simulations neglect physical processes other than gravity\nand (magneto)hydrodynamics, such as radiative gas cooling,\nchemical evolution, star formation and feedback from active\ngalactic nuclei. In this way, we can more easily isolate the effects\nof compression and dynamo from additional amplification caused\nby feedback and gas overcooling.2Comparisons between the\npredictions of primordial and astrophysical seeding scenarios of\nmagnetic fields with ENZO can be found in Vazza et al. (2017). For\nrecent high-resolution simulation of extragalactic magnetic fields\nwith a moving-mesh algorithm we refer the reader to Marinacci\net al. (2018c) and to the recent review by Donnert et al. (2018).\nWhile the initial topology of possible seed magnetic fields\nis unknown, we tested in Paper I that variations of the assumed\ninitial topology of seed fields do not to affect the strength of\nsimulated magnetic fields in the ICM at low redshift. Variations\nof the assumed initial strength of magnetic seed fields are harder\nto test, as for very small seed fields resolving the Alfvenic scale\nlAbecomes prohibitive and the amplification is stuck in the\nexponential regime for the entire cluster evolution (Beresnyak &\nMiniati 2016). In Paper I, we provided evidence that our simulated\nmagnetic fields are fairly independent on the initial field strength\nonly for >0:03 nG (comoving) fields. Future re-simulations at\neven higher resolution, or with less diffusive MHD schemes will\nbe needed to test the scenario for lower seed fields.\nFinally, as customary in simulations without explicit viscos-\nity and resistivity, the numerical viscosity and resistivity are of the\nsame order, meaning that the magnetic Prandtl number is PM=\nRM=Re=\u0017=\u0011\u00191.This assumption is reasonable enough given\nthe existing uncertainties and the difficulties in the characteriza-\ntion of the magnetised plasma in galaxy clusters (e.g. Schekochi-\nhin et al. 2004; Brunetti & Lazarian 2011a; Beresnyak & Miniati\n2016), and it further allows us to easily compare with the stan-\ndard literature of small-scale dynamo in a box (e.g. Cho 2014b;\nPorter et al. 2015). A few groups have explored the role of non-\nideal MHD effects in cosmological simulations, such as the pres-\nence of a physical resistivity (e.g. Bonafede et al. 2011; Marinacci\n2See however Katz et al. (2018), for a possible way of monitoring the\ngrowth of different magnetic field components within the same simulation.et al. 2018a), whose usefulness to explain observed ICM magnetic\nfields has been recently questioned by new simulations (Barnes\net al. 2018).\n5 SUMMARY AND CONCLUSIONS\nIn this paper, we have presented new high-resolution cosmologi-\ncal MHD simulations of a sample of galaxy clusters, which allow\nus to study the spectral properties of magnetic amplification with\nunprecedented spatial and temporal detail.\nIn agreement with our earlier work, we find that we can re-\nproduce cluster magnetic fields of the order of \u00181\u00003\u0016Gwith\nprimordial fields of 10\u000010G (comoving) at z= 30 .\nWe computed the magnetic energy spectra at z= 0for all the\nclusters in the sample. The spectral shape remains similar across\nclusters, despite of their different dynamical states. We parameter-\nize the magnetic spectra of all the clusters in our sample at z= 0\nand as a function of time for the merging cluster E5A by means of\nEq. (4). The resulting best-fit parameters are used to characterize\nthe magnetic properties of the ICM. In general, we could not find a\nsimple one-to-one relation between the kinetic and magnetic spec-\ntra and the dynamical state of the clusters: this indicates that highly\nperturbed systems, exhibiting more turbulence, do not necessarily\nimply higher values of the magnetic fields, and that the cycle of\namplification of magnetic fields in the realistic ICM is complex.\nThe normalization of the magnetic spectrum ( A), the spec-\ntrum width ( B) and the inverse of the outer scale of the spectrum\n(C) show a positive correlation with the virial mass of each clus-\nter. In addition, Bis correlated with the dynamical state of clusters.\nIn general, we observe that the magnetic growth rate is larger for\nmerging systems, while it is smaller in the relaxed system in our\nsample.\nFinally, the outer scale of the magnetic spectrum ( /1=C) also\ncorrelates with the dynamical state of the cluster: the relaxed sys-\ntem in our sample reaches higher values of the outer scale ( \u0018300\nkpc) compared to merging ( \u0018230kpc) and post-merging ( \u0018200\nkpc) systems, possibly indicating that the dynamo has acted for a\nlonger time in such systems. We caution that the ubiquitous pres-\nence of large-scale bulk motions in the ICM may introduce larger\ncorrelation scales in the magnetic field, so that our best-fit parame-\nters do not show an evident correlation with the last major merger\nof each cluster. This suggests that the history of minor mergers mat-\nters, but larger statistics of simulated clusters would be necessary\nto reach firmer conclusions.\nMoreover, we studied the co-evolution of magnetic fields and\nthe ICM properties in a merging cluster (E5A), which we could\nsample with a high time resolution. Our analysis reveals that the\npeak of the magnetic power spectrum shifts towards smaller spatial\nscales shortly after mergers, while overall it shifts to larger scales.\nIn the cluster E5A, the peak of the magnetic power spectrum ex-\ntends to\u0018280kpc after\u00189Gyr of evolution, with equipartition\nat scales<160kpc. Large amounts of kinetic energy are injected\nby substructures that fall through the cluster which first amplify\nthe magnetic field mainly via compression. These mergers prevent\nequipartition on the smallest scales, i.e. when the cluster is more\nperturbed, equipartition is not reached at scales above our current\nresolution.\nIn the course of a merger, the spectrum broadens and the\nouter scale is shifted towards smaller scales. While we observe that\nthe total magnetic energy is continuously growing, the magnetic\namplification at smaller scales starts only after the mergers.\nc\r0000 RAS, MNRAS 000, 000–00014 P . Dom ´ınguez-Fern ´andez et al.\nThis behaviour is driven by two mechanisms: 1) strong mergers\nintroduce additional turbulence into the system that raises the\nkinetic energy above equipartition with the magnetic field. Never-\ntheless, this new energy will only become available for magnetic\namplification after a few eddy-turnover times when the turbulence\nhas already cascaded down to the smaller scales; Consequently,\nthis changes the growth timescales by slowing down the process of\namplification soon after a merger event. In particular, when there is\na large input of kinetic energy, the magnetic amplification at small\nscales sets in only after \u00181 Gyr since the merger.\nFinally, our work has important implications for the interpre-\ntation of existing or future radio observations of magnetic fields in\ngalaxy clusters. The total rotation measure jRMjfrom clusters is\nexpected to scale/A=C . Therefore, our previous results imply\nthat the RM only weakly depends on the mass of the galaxy clus-\nter. We measure a scatter of up to a \u00184difference in RM between\nclusters of the same mass, while systems with a \u00182difference in\nmass can have the same RM, due to differences in their magnetic\nfield correlation scale. This implies that the RM across the cluster\npopulation probably is not universal, but can significantly be af-\nfected by the complex sequence of amplification events in the past\nlifetime of each cluster, with important consequences in the predic-\ntions of the RM from galaxy clusters which should be observable\nby future radio polarisation surveys (e.g. Govoni et al. 2015; Taylor\net al. 2015). We defer this analysis to future work.\n6 ACKNOWLEDGEMENTS\nWe acknowledge our anonymous reviewer for helpful comments\non the first version of this manuscript. The cosmological simula-\ntions were performed with the ENZO code (http://enzo-project.org),\nwhich is the product of a collaborative effort of scientists at many\nuniversities and national laboratories. We gratefully acknowledge\ntheENZO development group for providing extremely helpful and\nwell-maintained on-line documentation and tutorials. The analy-\nsis presented in this work made use of computational resources on\nthe JURECA cluster at the at the Juelich Supercomputing Centre\n(JSC), under projects no. 11823, 10755 and 9016 and HHH42, and\npartially on the Piz-Daint supercluster at CSCS-ETHZ (Lugano,\nSwitzerland) under project s805. The original simulations on which\nthis work is based have been produced by F.V . as PI on project\nHHH42 on JSC. We also acknowledge the usage of online storage\ntools kindly provided by the Inaf Astronomica Archive (IA2) initi-\nave (http://www.ia2.inaf.it).\nP.D.F and F.V . and acknowledges financial support from the Euro-\npean Union’s Horizon 2020 program under the ERC Starting Grant\n”MAGCOW”, no. 714196. We acknowledge useful scientific dis-\ncussions with K. Dolag, A. Beresnyak, J. Donnert, D. Ryu and T.\nJones.\nREFERENCES\nBarnes D. J., On A. Y . L., Wu K., Kawata D., 2018, MNRAS, 476,\n2890\nBatchelor G. K., 1951, Mathematical Proceedings of the Cam-\nbridge Philosophical Society, 47, 359374\nBeresnyak A., Lazarian A., 2006, The Astrophysical Journal, 640,\nL175\nBeresnyak A., Miniati F., 2016, ApJ, 817, 127Bonafede A., Dolag K., Stasyszyn F., Murante G., Borgani S.,\n2011, ArXiv e-prints\nBonafede A., Feretti L., Murgia M., Govoni F., Giovannini G.,\nDallacasa D., Dolag K., Taylor G. B., 2010, A & A, 513, A30+\nBonafede A., Vazza F., Br ¨uggen M., Murgia M., Govoni F., Feretti\nL., Giovannini G., Ogrean G., 2013, MNRAS, 433, 3208\nBrandenburg A., Sokoloff D., Subramanian K., 2012, Science &\nSpace Review, 169, 123\nBr¨uggen M., Ruszkowski M., Simionescu A., Hoeft M., Dalla\nVecchia C., 2005a, ApJL, 631, L21\nBr¨uggen M., Ruszkowski M., Simionescu A., Hoeft M., Dalla\nVecchia C., 2005b, ApJ, 631, L21\nBrunetti G., Jones T. W., 2014, International Journal of Modern\nPhysics D, 23, 1430007\nBrunetti G., Lazarian A., 2007, Monthly Notices of the Royal As-\ntronomical Society, 378, 245\nBrunetti G., Lazarian A., 2007, MNRAS, 378, 245\nBrunetti G., Lazarian A., 2011a, MNRAS, 410, 127\nBrunetti G., Lazarian A., 2011b, MNRAS, 412, 817\nBryan G. L. et al., 2014, ApJS, 211, 19\nCho J., 2014a, ApJ, 797, 133\nCho J., 2014b, ApJ, 797, 133\nDedner A., Kemm F., Kr ¨oner D., Munz C.-D., Schnitzer T., We-\nsenberg M., 2002, Journal of Computational Physics, 175, 645\nDolag K., Bartelmann M., Lesch H., 1999, A & A, 348, 351\nDolag K., Vazza F., Brunetti G., Tormen G., 2005, MNRAS, 364,\n753\nDonnert J., Dolag K., Lesch H., M ¨uller E., 2009, MNRAS, 392,\n1008\nDonnert J., Vazza F., Br ¨uggen M., ZuHone J., 2018, ArXiv e-\nprints\nFakhouri O., Ma C.-P., Boylan-Kolchin M., 2010, MNRAS, 406,\n2267\nFeretti L., Giovannini G., Govoni F., Murgia M., 2012, The As-\ntronomy and Astrophysics Review, 20, 54\nGaspari M., Churazov E., Nagai D., Lau E. T., Zhuravleva I.,\n2014, A & A, 569, A67\nGovoni F. et al., 2017, A & A, 603, A122\nGovoni F. et al., 2015, Advancing Astrophysics with the Square\nKilometre Array (AASKA14), 105\nGrasso D., Rubinstein H. R., 2001, Physics Reports, 348, 163\nHitomi Collaboration et al., 2018, Publications of the Astronomi-\ncal Society of Japan, 70, 9\nHopkins P. F., Raives M. J., 2016, MNRAS, 455, 51\nIapichino L., Niemeyer J. C., 2008, MNRAS, 388, 1089\nKandus A., Kunze K. E., Tsagas C. G., 2011, Physics reports, 505,\n1\nKatz H., Martin-Alvarez S., Devriendt J., Slyz A., Kimm T., 2018,\nArXiv e-prints\nKazantzev A. P., 1967, Journal of Experimental and Theoretical\nPhysics, 53, 1806\nKobayashi T., 2014, Journal of Cosmology and Astroparticle\nPhysics, 2014, 040\nKraichnan R. H., Nagarajan S., 1967, Physics of Fluids, 10, 859\nKravtsov A. V ., Borgani S., 2012, ARAA, 50, 353\nKritsuk A. G. et al., 2011, ApJ, 737, 13\nKulsrud R. M., Anderson S. W., 1992, Astrophysical Journal, 396,\n606\nLau E. T., Kravtsov A. V ., Nagai D., 2009, ApJ, 705, 1129\nMarinacci F., V ogelsberger M., Kannan R., Mocz P., Pakmor R.,\nSpringel V ., 2018a, MNRAS, 476, 2476\nMarinacci F., V ogelsberger M., Mocz P., Pakmor R., 2015, MN-\nc\r0000 RAS, MNRAS 000, 000–000Evolution of magnetic fields in the ICM 15\nRAS, 453, 3999\nMarinacci F. et al., 2018b, MNRAS, 480, 5113\nMarinacci F. et al., 2018c, MNRAS, 480, 5113\nMiniati F., Beresnyak A., 2015, Nature, 523, 59\nMoffatt H. K., 1978, Magnetic field generation in electrically con-\nducting fluids\nMurgia M., Govoni F., Feretti L., Giovannini G., Dallacasa D.,\nFanti R., Taylor G. B., Dolag K., 2004, A & A, 424, 429\nNeronov A., V ovk I., 2010, Science, 328, 73\nPlanck Collaboration et al., 2016, Astronomy and Astrophysics,\n594, A19\nPlanelles S., Schleicher D. R. G., Bykov A. M., 2015, Science &\nSpace Review, 188, 93\nPorter D. H., Jones T. W., Ryu D., 2015, ApJ, 810, 93\nRieder M., Teyssier R., 2017, ArXiv e-prints\nRoettiger K., Stone J. M., Burns J. O., 1999, ApJ, 518, 594\nRyu D., Kang H., Cho J., Das S., 2008, Science, 320, 909\nSantos-Lima R., de Gouveia Dal Pino E. M., Falceta-Gonc ¸alves\nD. A., Nakwacki M. S., Kowal G., 2017, MNRAS, 465, 4866\nSantos-Lima R., de Gouveia Dal Pino E. M., Kowal G., Falceta-\nGonc ¸alves D., Lazarian A., Nakwacki M. S., 2014, ApJ, 781,\n84\nSchekochihin A. A., Cowley S. C., 2007, Turbulence and Mag-\nnetic Fields in Astrophysical Plasmas, Molokov S., Moreau R.,\nMoffatt H. K., eds., Springer, p. 85\nSchekochihin A. A., Cowley S. C., Kulsrud R. M., Rosin M. S.,\nHeinemann T., 2008, Physical Review Letters, 100, 081301\nSchekochihin A. A., Cowley S. C., Taylor S. F., Maron J. L.,\nMcWilliams J. C., 2004, Astrophysical Journal, 612, 276\nStasyszyn F. A., Dolag K., Beck A. M., 2013, MNRAS, 428, 13\nSubramanian K., 2015, ArXiv e-prints\nSubramanian K., 2016, Reports on Progress in Physics, 79,\n076901\nSubramanian K., Shukurov A., Haugen N. E. L., 2006, MNRAS,\n366, 1437\nTaylor A. R. et al., 2015, ArXiv e-prints\nThe Enzo Collaboration et al., 2013, ArXiv e-prints\nTormen G., Bouchet F., White S., 1997, MNRAS, 286, 865\nTricco T. S., Price D. J., Federrath C., 2016, MNRAS, 461, 1260\nTurner M. S., Widrow L. M., 1988, Phys. Rev. D, 37, 2743\nVazza F., Brueggen M., Gheller C., Hackstein S., Wittor D., Hinz\nP. M., 2017, Classical and Quantum Gravity\nVazza F., Br ¨uggen M., Gheller C., Wang P., 2014, MNRAS, 445,\n3706\nVazza F., Brunetti G., Br ¨uggen M., Bonafede A., 2018a, MNRAS,\n474, 1672\nVazza F., Brunetti G., Br ¨uggen M., Bonafede A., 2018b, MNRAS,\n474, 1672\nVazza F., Brunetti G., Gheller C., Brunino R., 2010, New Astron-\nomy, 15, 695\nVazza F., Brunetti G., Gheller C., Brunino R., Br ¨uggen M., 2011,\nA & A, 529, A17+\nVazza F., Brunetti G., Kritsuk A., Wagner R., Gheller C., Norman\nM., 2009, A & A, 504, 33\nVazza F., Jones T. W., Br ¨uggen M., Brunetti G., Gheller C., Porter\nD., Ryu D., 2017, MNRAS, 464, 210\nV ogt C., Enßlin T. A., 2005, A & A, 434, 67\nWang P., Abel T., 2009, ApJ, 696, 96\nWang P., Abel T., Kaehler R., 2010, New Astronomy, 15, 581\nWise J. H., Abel T., 2007, ApJ, 665, 899\nWittor D., Vazza F., Br ¨uggen M., 2017, MNRAS, 464, 4448Xu H., Li H., Collins D. C., Li S., Norman M. L., 2009, ApJL,\n698, L14\nXu H., Li H., Collins D. C., Li S., Norman M. L., 2011, ApJ, 739,\n77\nZeldovich I. B., Ruzmaikin A. A., Sokolov D. D., eds., 1983,\nMagnetic fields in astrophysics, V ol. 3\nAPPENDIX A: EARLIER RESULTS ON DYNAMO\nAMPLIFICATION IN SIMULATED CLUSTERS.\nWe summarize here the main results of our previous work (Vazza\net al. 2018a, , Paper I in this work), which motivates the analy-\nsis performed in this paper. Using an AMR prescription to refine\nmost of the innermost regions of galaxy clusters with the ENZO\ncode, in Vazza et al. (2018a) we simulated the growth of 0.1 nG\n(comoving) magnetic field seed, assumed of a cosmological ori-\ngin, fromz= 30 toz= 0. We simulated the field growth as a\nfunction of the maximum cell resolution for a Coma-like galaxy\ncluster (\u00181015M\f) an starting from the same initial field, and ob-\nserved the onset of significant small-scale dynamo for resolutions\n616 kpc , with near-equipartition magnetic fields on 6100 kpc\nscales. For the best resolved run ( \u00194 kpc=cell), we measured a fi-\nnal magnetic fields strength of \u00181\u00002\u0016Gin the cluster core, with\na radial profile that scales as B(n)/n0:487(wherenis the gas\ndensity). For lower resolution, the magnetic field gets increasingly\nsmaller, with a flatter radial profiles and a magnetic power spectrum\nof a power-law shape. In summary, the following are the key evi-\ndences that support that our runs do feature a resolved small-scale\ndynamo:\n\u000fthe measured dependence of magnetic field strength and the\neffective resolution of the simulation: only when the numerical\nReynolds number exceeds Re\u0018102the magnetic field reaches\nvalues much larger than what gas compression ( /n2=3) can pro-\nduce;\n\u000fthe onset of the curved magnetic field power spectrum only\nwhen the spatial resolution exceeds a critical value (estimated to\nbe\u001816 kpc=cell, even if this may vary with the adopted nu-\nmerical scheme, e.g. Donnert et al. 2018), indicating that only at\na large enough Reynolds number and high enough resolution we\nhave enough solenoidal turbulence and we can resolve the lAscale\n(Fig.A1);\n\u000fthe slope of the power spectra for low wavenumbers is com-\npatible with the Kasantsev model of dynamo PB/k3=2(e.g.\nSchekochihin et al. 2004), while after the peak the spectrum rapidly\nsteepens from/k\u00005=3to/k\u00002or less, consistent with (e.g.\nPorter et al. 2015; Rieder & Teyssier 2017);\n\u000fthe evolution of magnetic fields in our most resolved simula-\ntion, and its relation with the measured dissipation of kinetic tur-\nbulent energy, which indicate a \u00184%dissipation rate of turbulent\ninto magnetic energy, in line with Miniati & Beresnyak (2015) and\nBeresnyak & Miniati (2016);\n\u000fthe measured anti-correlation between the curvature of mag-\nnetic field lines in our most resolved simulation and the magnetic\nfield strength, as expected in the dynamo regime (e.g. Schekochihin\net al. 2004);\n\u000fthe measurement that the lAscale, estimated following in\nBrunetti & Lazarian (2007), which is well resolved for a good frac-\ntion of our cluster volume;\n\u000fthe independence of the magnetic profile and power spectra at\nc\r0000 RAS, MNRAS 000, 000–00016 P . Dom ´ınguez-Fern ´andez et al.\nFigure A1. Power spectra of velocity (top lines) and magnetic field (lower\ncurved lines) for resimulation of increasing resolution, presented in Paper\nI. The spectra are measured within the innermost 23Mpc3of a simulated\ncluster atz= 0, and clearly shows that the increase of resolution leads to\nan increase of the dynamical range (also marked by the sequence of Nyquist\nfrequencies at the bottom of the panel) and results into a radial change in\nthe magnetic spectrum for 616 kpc resolutions.\nz= 0, for>0:03 nG (comoving), above which our setup ensures\nto resolvelAin a large fraction of the cluster volume.\nMoreover, the topology of the magnetic fields at z= 0 pro-\nduces profiles of Faraday Rotation of background polarised sources\nin good agreement with the real observations of the Coma cluster,\nwhich are the most stringent to date (Bonafede et al. 2010, 2013). A\nsignificant new finding of our first analysis in Paper I is also the de-\ntection of a significant non-Gaussian distribution of magnetic field\ncomponents in the final cluster, which results from the superposi-\ntion of different amplification patches mixing in the ICM.\nAll results obtained from this first study are also confirmed\nwith the larger set of cluster simulations which is object of this\npaper.\nAPPENDIX B: CORRELATING THE BEST-FIT\nPARAMETERS\nWe computed the cross-correlation matrix of the change in time of\nthe best-fit parameters A,B,Cand the kinetic energy Ekand show\nthe result in Fig.B1. Where \u0001of a variable Q, is defined as (Q(t)\u0000\nQ(t-1))=Q(t-1)as in Fig. 16. The Pearson coefficients for all the\ncross-correlations are shown in the upper part of the diagonal in\nFig. B1. In this way, we can better quantify the existing correlations\nand interpret them:\n(i)corr(\u0001A;\u0001B): a positive change in the normalization im-\nplies a negative change in the parameter B. This implies that a\nsudden increment on the normalization narrows down the spec-\ntrum width shortening the magnetic growth timescale. Therefore,\nthe growth rate increases over time.\n(ii)corr(\u0001A;\u0001C): an increment in the normalization implies\nthatC(t)< C(t-1), i.e the power is shifted towards larger scales.\nWe attribute this feature to the presence of dynamo amplification.\nThis conclusion is supported by Section 3.4, where every merger\nFigure B1. Cross-correlation matrix of the best-fit parameters and the ki-\nnetic energy changes in the system. The Pearson correlation coefficients are\nindicated in the upper part of the diagonal and the corresponding scatter\nplots are shown in the lower part of the diagonal.\nevent carrying enough kinetic energy was shown to shift the mag-\nnetic spectrum towards smaller scales (i.e. amplifying via compres-\nsion).\n(iii)corr(\u0001B;\u0001C): a wider spectrum coming along with a\nshift of the outer scale towards smaller scales is directly related\nto the action of compression. This matches our previous interpre-\ntation of Fig. 15, where compression shifts the power to smaller\nscales (i.e.C(t-1)<C(t)) and the new turbulent cascade does not\nplay a role in the amplification instantaneously, but after an eddy-\nturnover time.\n(iv)corr(\u0001C;\u0001Ek): a shift of the outer scale towards small\nscales is weakly correlated with the injection of kinetic energy. In\nthis case, we checked the cross-correlation at each time and iden-\ntified the times corresponding to some merger events (red arrows\nin Figs.16 and 18). The first two arrows corresponding to small\nclumps falling into the cluster show a higher correlation than the\nlast arrow which corresponds to a larger clump. A plausible ex-\nplanation is that the first two events generated sufficient turbulence\nthat allowed the magnetic field to grow also at smaller scales (via\nthe small-scale dynamo), so by the time of the third event, the ef-\nfect of compression is not enough to shift the outer scale towards\nsmaller scales anymore. This can be considered as a momentary\nstate of ”balance” between the dynamo and compression effects\nand it would also explain the period of amplification between t\u0018\n12-13 Gyr.\nc\r0000 RAS, MNRAS 000, 000–000"    },    {        "title": "2111.13533v1.Three_dimensional_Resonant_Magnetization_Dynamics_Unraveled_by_Time_Resolved_Soft_X_ray_Laminography.pdf",        "content": "Three-dimensional Resonant Magnetization Dynamics Unraveled\nby Time-Resolved Soft X-ray Laminography\nSimone Finizio,1,\u0003Claire Donnelly,2, 3Sina Mayr,1, 4Ales Hrabec,1, 4and Jörg Raabe1\n1Paul Scherrer Institut, 5232 Villigen PSI, Switzerland\n2Cavendish Laboratory, University of Cambridge,\nCambridge CB3 0HE, United Kingdom\n3Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany\n4Laboratory for Mesoscopic Systems, Department of Materials,\nETH Zurich, 8093 Zurich, Switzerland\n(Dated: November 29, 2021)\nAbstract\nThe imaging of magneto-dynamical processes has been, sofar, mostly atwo-dimensional business,\ndue to the constraints of the available experimental techniques. In this manuscript, building on the\nrecent developments of soft X-ray magnetic laminography, we present an experimental setup where\nmagneto-dynamical processes can be resolved in all three spatial dimensions and in time, with the\npossibility to freely tune the frequency of the dynamical process. We then employ this setup to\ninvestigate the three-dimensional dynamics of two resonant magneto-dynamical modes in a CoFeB\nmicrostructureoccurringatdifferentfrequencies, namelythefundamentalvortexgyrationmodeand\na magnetic field-induced domain wall excitation mode. This new technique provides much needed\ncapabilities for the experimental investigation of the magnetization dynamics of three-dimensional\nmagnetic systems.\n1arXiv:2111.13533v1  [cond-mat.mes-hall]  26 Nov 2021Magneto-dynamical processes encompass a wide spectrum of phenomena, including vor-\ntex dynamics [1–3], magnonics [4–7], current and field induced domain wall motion [8–\n10], skyrmion dynamics [11–14], and magnetization switching [15]. The understanding of\nmagneto-dynamical processes is vital not only for the design and operation of magnetic\ndevices, but also for the confirmation of theoretical models. Experimental techniques such\nas time-resolved X-ray magnetic microscopy have provided vital contributions towards the\nunraveling of magneto-dynamical processes at the nanoscale.\nAlmost all of the magneto-dynamical processes that have been imaged have been on two-\ndimensional systems. The extension of such investigations onto the third spatial dimension\nis of great interest both fundamentally and for applications, thanks to the rich dynami-\ncal behavior offered by three-dimensional spin configurations [16, 17]. Examples of such\nthree-dimensional dynamical modes include higher order spin wave processes [4] and their\ngeneration from three-dimensional topological features [6]. However, until recently, the only\nexperimentalinvestigationsthatcouldbecarriedoutonthree-dimensionalsystemshavebeen\nlimited to indirect characterizations, performed through the imaging of two-dimensional pro-\njections followed by comparisons with theoretical predictions and micromagnetic simulations\n[6, 18].\nIn recent years, significant progress has been made towards the imaging of magneto-\ndynamical processes resolved in all three spatial dimensions, driven by the development of\nthree-dimensional X-ray imaging techniques such as magnetic tomography [19] and laminog-\nraphy [20, 21]. Of particular interest is the work reported in Ref. [20], where the dynamics of\na1.2\u0016mthickGdComicrostructuredelementwere, forthefirsttime, resolvedinallthreedi-\nmensions. However, this groundbreaking study relied on the intrinsically pulsed structure of\nsynchrotronradiation, investigatingadynamicalprocesslockedtothepulserepetitionrateof\nthe synchrotron light source (500 MHz in the case of Ref. [20]) to perform the time-resolved\nimaging. This is caused by the requirement to utilize a two-dimensional X-ray detector de-\nmanded by the ptychographic imaging technique used to acquire each projection [22]. The\nlimited bandwidth of two-dimensional detectors allows only for the imaging of processes oc-\ncurring at either the master clock frequency or its harmonics. This approach therefore limits\nthe accessible frequencies, directly impacting the ensemble of dynamical processes that can\nbeinvestigated, asmanysuchprocessesareresonantatparticularfrequencies. Asaresult, to\nfully probe magnetization dynamics in three dimensions, the free control over the frequency\n2ofthedynamicalprocessiskey. Inthiswork, weachievefrequency-flexiblethree-dimensional\nmagnetic imaging through the combination of soft X-ray magnetic laminography with time-\nresolved scanning transmission X-ray microscopy (STXM) imaging without constraints on\nthe excitation frequency, and exploiting the stronger XMCD contrast of magnetic elements\nat soft X-ray energies.\nThe three-dimensional imaging of thin film samples exhibits some unique challenges com-\npared to the standard cylindrical geometries utilized for tomographic imaging. These addi-\ntional challenges are caused by the tomography imaging geometry, where the tomographic\nrotation axis is perpendicular to the imaging beam axis, i.e. lies in the surface plane of\nthe thin film sample. For projections at angles close to 90\u000e, the large apparent thickness of\nthe thin film sample hinders the acquisition of the projection, leading to a missing wedge\nartifact in the Fourier space. Furthermore, the change in apparent thickness of the thin\nfilm sample as a function of the tomographic rotation angle has to be accounted for in the\nreconstruction. To overcome these two issues, laminographic imaging has been developed.\nLaminography is a generalization of the tomographic imaging protocol where the sample\nrotation axis is not perpendicular to the imaging beam axis, but is rather oriented at a\ndefined angle (in our case, 45\u000e) with respect to the beam axis, as sketched in Fig. 1(a)\n[20–22]. The laminography geometry probes a larger volume of the Fourier space (missing\ncone artifact [22]), which allows for an improvement of the quality of the three-dimensional\nreconstruction. Furthermore, the rotation geometry is such that no changes in the effective\nthickness of the sample occur when rotating it about the laminography axis, simplifying\nthe measurements [20–22]. Finally, laminography allows for the imaging of extended sample\nsurfaces, as demonstrated in Ref. [22].\nTo be able to visualize the magnetization dynamics of three-dimensional systems, the de-\nvelopment of a technique that is able to perform time-resolved imaging with freely selectable\nfrequencies combined with three-dimensional imaging at soft X-ray energies is therefore cru-\ncial. In this work, we have combined time-resolved imaging with the soft X-ray laminogra-\nphy setup developed at the PolLux endstation of the Swiss Light Source [21], where each\nlaminography projection is acquired through STXM imaging. Compared to the ptycho-\ngraphic imaging used in Ref. [20], STXM has the advantage of being a scanning microscopy\ntechnique requiring a point detector. By using an avalanche photodiode (APD) with a band-\nwidth larger than the repetition rate of the synchrotron light source (for the Swiss Light\n3Source, 500 MHz), two consecutive X-ray pulses generated by the light source (separated by\n2 ns) can be resolved by the APD. This allows for the possibility to sort each recorded pho-\nton count by the APD into separate time channels depending on the relative phase between\nwhen the count occurred and the excitation signal used to trigger the dynamical process,\nwhich is synchronized to a rational multiple of the repetition rate [23]. This is achieved\nthrough a custom-designed field-programmable gate array (FPGA) setup. Thanks to this\ndetection protocol, a larger comb of accessible frequencies, given by rational multiples of the\nmaster clock frequency, is available. Purely arbitrary excitation frequencies are still not ac-\ncessible, but this final limitation can be easily lifted if time-of-arrival detection is performed\n[24]. For the work presented in this manuscript, the time-resolved imaging was performed\nusing the FPGA setup. Further details about the time-resolved imaging setup are given in\nthe Methods section.\nTo demonstrate the possibility of performing three-dimensional time-resolved imaging\nwith freely selectable excitation frequencies, we performed time-resolved laminographic\nimaging of two spin dynamic modes of a 150 nm thick Co 40Fe40B20(from now on referred\nto as CoFeB) 2.5\u00022.5\u0016m2microstructured square, which stabilizes a flux-closure magnetic\nLandau state at equilibrium. Before performing the time-resolved imaging, we acquired a\nstatic magnetic laminogram of the equilibrium configuration of the CoFeB microstructure.\nThe reconstruction of the magnitude and orientation of the local magnetization vectors\nwas performed using the algorithm described in Ref. [25]. The reconstructed magnetic\nlaminogram is shown in Fig. 2 (a) and (c). Here, the arrows depict the normalized magni-\ntude and orientation of the local magnetization vector as reconstructed from the magnetic\nlaminogram, and are colored in a blue-white-red scale according to the xcomponent of the\nmagnetization vector. In addition to the reconstructed local magnetic vectors, we also show\nthe vortex core stabilized in the CoFeB microstructure, depicted as an isosurface of the z\ncomponent ( cz) of the curl of the magnetization vector c=r\u0002m(see the Methods section\nfor additional details).\nFrom the static magnetic laminogram, we can observe that the CoFeB microstructured\nsquare exhibits a magnetic uniaxial anisotropy along the xaxis, which is evidenced by the\nlarger area of the magnetic domains along the xaxis and by the shape of the vortex core. In\nparticular, the vortex core exhibits a S-shaped structure (in contrast to the one-dimensional\ncolumnar structure that would be expected in absence of anisotropy) elongated along the\n4anisotropy axis, with the vortex core meeting the top and bottom surfaces of the CoFeB\nmicrostructured square at two spots separated by a lateral distance of about 300 nm.\nTo verify that the reconstructed laminogram resembles the expected magnetic configu-\nration, three-dimensional micromagnetic simulations of the CoFeB microstructured square\nwith anisotropy, the value of which was determined from magneto-optical Kerr effect mea-\nsurements, were performed with the finite-differences simulation package MuMax3[26] (de-\ntails about the simulations can be found in the Methods section). The simulated three-\ndimensional magnetic configuration is shown in Figs. 2(b) and (d), where its close resem-\nblance to the experimental data can be observed.\nHaving determined the static magnetic configuration of the CoFeB microstructure, we\nnext investigate its dynamics by performing time-resolved magnetic laminography. Specif-\nically, by injecting a microwave signal across a Cu microantenna fabricated on top of the\nCoFeB microstructure, a set of different magneto-dynamical modes can be excited, ranging\nfrom the gyration of the vortex core to the excitation of the magnetic domain walls, and to\nthe emission and propagation of spin waves. For this work, we performed three-dimensional\ntime-resolved imaging of the fundamental vortex gyration mode (frequency of 326 MHz)\nand of a magnetic field-induced domain wall excitation mode (frequency of 913 MHz). The\nexperimental setup employed for the time-resolved measurements is sketched in Fig. 1(b)\nand described in more detail in the Methods section.\nFigure 3 shows snapshots of the reconstructed magnetization profile for the domain wall\nexcitation and vortex core gyration modes. As for the static laminogram presented in Fig. 2,\nwe again show the reconstructed magnitude and orientation of the local magnetic moments\nand the isosurfaces of the zcomponent of the curl of the magnetization, cz. For the domain\nwall excitation mode (Fig. 3(a)), we show a set of isosurfaces of cz, which allow us to delin-\neate the magnetic domain walls of the CoFeB microstructure while, for the vortex gyration\nmode, we show the isosurface delineating the vortex core. In addition, by determining the\nmagnitude of the change of the magnetization across one cycle of excitation for each voxel of\nthe time-resolved laminogram, we show the localization of the two modes. This is depicted\nin Fig. 3(c) for the 913 MHz mode, where it can be observed that the mode is localized in the\ndomain walls of the CoFeB square, allowing it to be identified as a domain wall excitation.\nFor the 326 MHz mode, shown in Fig. 3(d), instead, the mode is primarily localized in the\nvortex core, allowing it to be identified as a vortex gyration mode.\n5One of the main advantages of time-resolved laminographic imaging is that it allows us\nto resolve differences in the dynamics through the depth of a structure, and this can be well\nobserved comparing the two magnetodynamical modes shown in Fig. 3. In particular, the\nvortex core gyration mode exhibits a strong variation of its dynamics across its thickness,\nwhile the domain wall excitation mode exhibits a practically uniform motion along the\nthickness of the CoFeB microstructure. For the domain wall excitation mode (Fig. 3(a)), a\nprecession of the magnetization within the domain wall can be observed. This precession,\nvisible by the changes in the position of the czisosurfaces across one cycle of excitation,\nis predominantly uniform along the thickness of the microstructure. Therefore, while the\npossibility to measure this domain wall excitation mode provides a demonstration of the\nfree selection of the excitation frequency, for the remainder of this work we will concentrate\non the vortex gyration mode (Fig. 3(b)), where the laminographic imaging allows for the\nunraveling of the vortex core dynamics.\nThe injection of the 326 MHz RF signal, close to the eigenfrequency of the fundamental\nvortex core gyration mode, causes the vortex core to start gyrating around its equilibrium\nposition, shown in Fig. 2(a). This leads to a strong variation of its dynamical behavior\nacross the thickness of the CoFeB microstructure, as the equilibrium configuration itself\nexhibits a non trivial three-dimensional structure. The motion of the vortex core can be\nobserved in the plot shown in Fig. 4(a), where the position of the center of mass of the\nvortex core across each slice of the 3D time-resolved image is shown.\nTo obtain more insight into the three-dimensional dynamics of the vortex core, we fitted\nthexandycoordinates of the center of mass with a sinusoidal function, which allowed us\nto determine the amplitude of the core’s gyration in the x\u0000yplane at each slice of the\nthree-dimensional time-resolved image. This result is shown in Fig. 4(b). In particular, it\ncan be observed that the highest amplitude in the x\u0000ycore’s gyration occurs at the top and\nbottom surfaces of the CoFeB square, where the core is perpendicular to the x\u0000yplane.\nThis experimental result can be well-reproduced by micromagnetic simulations, as shown in\nFigs. 4(c) and (d).\nThe gyrotropic motion of the vortex core is also paired with a breathing mode of the core,\ni.e. an expansion and contraction of the vortex core volume across one cycle of excitation.\nThe breathing of the vortex core can be visualized by determining the volume enclosed by\na given isosurface of czfor each time step. The specific isosurface has been selected to be\n6equal to 90% of the maximum of cz, and allows for the determination of the relative change\nin the vortex core volume across one cycle of excitation. Such time dependence is shown\nin Fig. 5(a), and the corresponding micromagnetic simulations qualitatively confirming the\nexperimental data are shown in Fig. 5(b).\nThe coexistence of gyration and breathing, coupled with the three-dimensional structure\nof the vortex core at its equilibrium configuration provides an additional energy reservoir\nif compared to two dimensional vortex cores. For such three-dimensional vortex cores, this\nadditional energy reservoir requires a generalization of the Thiele model, where a magnetic\ninertial mass term has to be included [2, 14, 27]. Through time-resolved laminographic imag-\ning, the three-dimensional dynamical deformation of topological objects such as magnetic\nvortex cores can therefore be directly imaged, providing a direct experimental verification\nof predictions e.g. from micromagnetic simulations. This imaging protocol will also allow\nto reduce the reliance on micromagnetics for insights into three-dimensional dynamics, and\noffer the possibility to directly observe dynamical behaviors not predicted by micromagnetic\nsimulations.\nIn conclusion, we have presented a new time-resolved setup based on X-ray laminography\ncombined with time-resolved STXM imaging that allows for the investigation of magneto-\ndynamical processes resolved in all three spatial dimensions combined with the possibility\nto freely select the frequency of the excitation signal. This will allow for the possibility to\nresolvearangeofresonantmagnetizationdynamicprocessesinthree-dimensions, suchase.g.\nthe generation, propagation, and localization of spin waves, domain wall motion dynamics,\ndynamics of three-dimensional topological objects such as hopfions, and the investigation of\nthree-dimensional magnetic nanostructures. In particular, we have demonstrated here the\nthree-dimensional mapping of two different dynamic modes that can be excited in a thick\nmicrostructured CoFeB square, stabilizing a Landau flux closure state at its equilibrium\nconfiguration, when applying oscillating magnetic fields of different frequencies. The two\nmodes are the gyration of the vortex core, using an excitation frequency of 326 MHz, and\nthe motion of the domain walls of the Landau pattern, obtained at a frequency of 913 MHz.\nThe two investigated modes show a substantially different behavior along the thickness of\nthe CoFeB square, with the domain wall excitation mode displacing uniformly along the\nthickness, and the vortex core gyration mode exhibiting a strongly non-uniform behavior\nalong its thickness, which is further enhanced by the uniaxial anisotropy of the CoFeB\n7microstructure. For this mode, the counterclockwise gyration of the vortex core around its\nequilibrium position is combined with a breathing of the vortex core, indicating that energy\ncan be stored in the deformation of the core itself, following the behavior observed in other\ntopological structures such as the magnetic skyrmion bubble.\nAs a final note, it should be considered that, for time-resolved laminography imaging,\na set of projections has to be acquired in order to reconstruct the three-dimensional mag-\nnetization configuration. Due to the requirement of acquiring three polarizations (circular\npositive, negative, and linear horizontal) to reconstruct the magnetic configuration [21], to\nthe requirement of more than 30-40 projections to obtain a sufficiently good zresolution [22],\nand to the fact that a time-resolved image is composed of N frames (typically on the order of\n10 for RF dynamics), all of which requiring sufficient statistics [28], a time-resolved laminog-\nraphyimagecurrentlyrequiresaconsiderableinvestmentinmeasurementtime. Nonetheless,\nthanks to the planned upgrades of several synchrotrons towards diffraction limited sources,\nthe future of time-resolved laminography imaging is bright: improvements in the intensity,\nbrilliance, and coherence of the X-ray beam produced by diffraction limited source will allow\nfor the reduction of integration times of several factors.\nMETHODS\nSample fabrication\nMicrostructured CoFeB squares (2.5 \u0016m wide, 150 nm thick) were patterned on top of a\n200 nm thick, 1\u00021 mm2wide Si 3N4membrane on a 5 \u00025 mm2intrinsic Si frame (250 \u0016m\nthick, from Silson Ltd.). The patterning was performed by electron beam lithography using\na Vistec EBPG5000, 100 kV electron beam writer. A bilayer resist composed of a layer\nof poly(methyl-methacrylate) (4% in Anisole, 3000 rpm for 1 minute) on top of a layer of\nmethyl methacrylate (6% in Ethyl-Lactate, 3000 rpm for 1 minute) was spincoated on top\nof the Si 3N4membrane, followed by a 1 minute soft-bake at 175\u000eC for both resist layers.\nFollowing the spincoating, the resist was exposed with a dose of 1500 \u0016C cm\u00002. Following\ndevelopmentbyimmersionina1:3(volume)solutionofmethylisobutylketoneandisopropyl\nalcohol (IPA) and rinsing in pure IPA (each for 45 seconds), a 150 nm thick CoFeB film was\ndeposited by DC sputtering using an AJA sputter coater, where a 2 nm thick Ta seed layer\n8was deposited prior to the CoFeB. The unexposed resist, together with the CoFeB film on\ntop of it was then removed by immersion of the sample in pure acetone, leaving only the\npatterned CoFeB square.\nOn top of the CoFeB square, a 5 \u0016m wide, 300 nm thick, Cu stripline designed to allow\nfor the generation of an oscillating in-plane magnetic field along the xaxis was fabricated,\nagain by electron beam lithography followed by liftoff using the same recipe as for the CoFeB\nsquare. The Cu film was deposited by thermal evaporation from a pure Cu pellet, using\na Balzers BAE evaporator. The quality of the fabricated samples was verified by scanning\nelectron microscopy and time-domain reflectometry measurements.\nTime-resolved soft X-ray laminography\nThethree-dimensional imageswereacquiredfollowingthe laminographyimagingprotocol\n[22], where a set of two-dimensional projections at different rotation angles of the sample\nwas acquired. The single projections were acquired by STXM imaging, utilizing diffractive\noptics (Fresnel zone plate) to focus the X-ray beam. Due to geometrical constraints, a\nlaminography angle of 45\u000ewas employed, combined with a Fresnel zone plate with a 50 nm\noutermost zone width, which defines the achievable spatial resolution. The X-ray imaging\nwas performed at the L 2absorption edge of Cobalt, corresponding to an X-ray energy of\nabout 793 eV. The L 2edge was selected rather than the more common L 3edge due to\nthe lower X-ray absorption at the L 2edge, which is a critical factor when considering the\neffective CoFeB thickness of about 210 nm under a laminography angle of 45\u000e.\nA total of 50 projections were acquired for each time-resolved laminography image, re-\nsulting in a theoretical maximum zresolution of 10 nm, calculated according to the following\nequation [22]:\nP\u0015\u0019d\n\u0001rtan(\u0012);\nwhere \u0001ridentifies the z resolution, dthe thickness of the CoFeB structure (150 nm), Pthe\nnumber of projections (50), and \u0012the laminography angle (45\u000e).\nFor each projection, three time-resolved STXM images were acquired, two with circularly\npolarized X-rays of opposite helicities, and the third with linearly polarized X-rays (linear\nhorizontal). This is a different approach to standard time-resolved two-dimensional STXM\nimaging, where only a single image with circularly polarized X-rays is acquired (e.g. [5]),\n9and is necessary to allow for the reconstruction of the three-dimensional orientation of the\nlocal magnetization vectors [20]. The acquisition of an image with linearly polarized X-\nrays is required to allow for the correction of possible discrepancies in the degree of circular\npolarization that can occur when performing XMCD imaging at a bending magnet beamline\nsuch as PolLux [21, 29].\nFor the time-resolved imaging, a set of 23 frames for each projection was acquired. To\nimprove the statistics of the single frames, the 23 raw frames were binned (average) to 7\nframes. The dynamics were excited by the injection of a RF current across the Cu stripline\nfabricated on top of the CoFeB microstructured square, giving rise to an oscillating magnetic\nfield through the Oersted effect. The amplitude of the RF current was selected in order\nto generate an oscillating magnetic field with a 5 mT amplitude. The RF current was\ngenerated by a Keysight M8195A arbitrary waveform generator (64 GSa s\u00001bandwidth)\nfrequency locked to the 500 MHz master clock of the synchrotron light source, combined\nwith a SHF826H broadband amplifier. The electrical circuit is schematically depicted in\nFig. 1(b). The frequency of the excitation signal was selected according to the relation\nf= 500M\nN[MHz], being Nequal to the number of raw frames in the image (23) and M\nan integer multiplier [23]. The frequency closest to the vortex gyration eigenfrequency was\nattained for M= 15(326 MHz), and the domain wall excitation mode was triggered at\nM= 42(913 MHz). These frequencies were found by two-dimensional TR-STXM imaging,\nwhere a set of time-resolved images as a function of the applied excitation frequency were\nacquired. The temporal resolution of the measurements presented here is determined by the\nwidth of the X-ray pulses generated by the synchrotron, which is of 70 ps FWHM for the\nSwiss Light Source operating with a multibunch filling pattern [24].\nTo contact the stripline, a custom-designed printed circuit board (PCB), designed to be\ncompatible with the constraints of the laminography stage while still guaranteeing good\nRF performances, was utilized. A Boa-Flex III flexible RF cable from Huber+Suhner was\nutilized for the electrical connection of the PCB. The high flexibility of the cable enabled\nfor the possibility to perform several rotations of the laminography stage whilst maintaining\nan electrical connection to the PCB, and good RF properties.\nThe reconstruction of the three-dimensional topographic and magnetic configuration of\nthe CoFeB square was then performed for each of the 7 binned frames according to the\nprocess described in [20, 21]. The final three-dimensional time-resolved images were then\n10rendered using the free software Paraview [30]. There, the orientation and magnitude of\nthe magnetic moment in each voxel of the three-dimensional image were rendered, using a\nmask defined by the reconstructed topography of the CoFeB square. To visualize the vortex\ncore, we also plot the magnitude of the zcomponent, cz, of the curl of the magnetization\n(calculated as c=r\u0002 m, being mthe local magnetization vector). The czcomponent\nprovides an easy method for the direct visualization of the vortex core, as shown in Ref. [20]\nand as demonstrated from comparison with micromagnetic simulations (see Fig. 2 and the\nsupplementary information).\nThelocalization of thetwomodesinvestigatedhere, shown inFig. 3(c-d), wasdetermined\nby calculating, for each voxel in the time-resolved laminogram, a fast Fourier transform, and\nplotting the magnitude of the Fourier transform at the frequency of the mode (913 MHz for\nFig. 3(c) and 326 MHz for Fig. 3(d)).\nThe vortex core position along the thickness of the microstructured CoFeB square was\ndetermined by calculating the center of mass of the regions where cz\u00150:9\u0002max(cz)for each\nof thezslices of the reconstructed laminogram. The calculation of the center of mass was\nnot weighted by the value of cz(i.e. all pixels where cz\u00150:9\u0002max(cz)have the same weight\nin the calculation). For the micromagnetic simulations, the center of mass of the vortex core\nwas determined by calculating the center of mass of the region where mz>0:9, beingmzthe\nnormalized zcomponent of the magnetization. To allow for an easier comparison between\nthe simulations and the experimental data, the simulated data was subdivided in 8 slices.\nThe volume of the vortex core was determined by calculating the volume surrounded by\nthecz= 0:9\u0002max(cz)isosurface. For the micromagnetic simulations, this calculation was\nperformed by considering the volume enclosed by the mz= 0:9isosurface.\nMicromagnetic simulations\nMicromagneticsimulationsofthestaticanddynamicalbehavioroftheCoFeBmicrostruc-\ntured square presented here were performed by numerically solving the Landau-Lifshitz-\nGilbert equation using the finite differences MuMax3framework [26]. A three-dimensional\nsimulation grid was considered, consisting of a 512 \u0002512\u000232 px3lattice with 4.88\u00024.88\n\u00024.68 nm3cell. We selected a smaller step size along the zdirection to keep a power of\n2 as the number of cells for all directions, allowing for a more efficient computation. As\n11micromagnetic parameters, we employed a saturation magnetization of M s= 106A m\u00001, an\nexchange stiffness of A ex= 10\u000011J m\u00001, a Gilbert damping constant of \u000b= 0.05. These\nmicromagnetic parameters result in an exchange length of about 4 nm. To verify that the\nsize of the discretization is not affecting the results of the micromagnetic simulations, a\nsubset of the simulations was performed with a 2.44 \u00022.44\u00022.34 nm3discretization, and\nthe results of the simulation with the finer grid closely resemble the simulations performed\nwith the larger grid.\nFor all of the time-resolved simulations, the starting configuration was determined by\nrelaxing a symmetric Landau pattern under an applied uniaxial anisotropy of 5 kJ m\u00003, the\nvalue of which was determined by magneto-optical Kerr effect measurements, and verified\nby comparing a static micromagnetic simulation with the observed magnetic configuration.\nA sinusoidal magnetic field along the direction of the uniaxial anisotropy ( xaxis) was then\napplied, reproducing the experimental configuration. The amplitude of the magnetic field\nwas 5 mT, and a 200 ns period was simulated prior to recording the simulated data to allow\nfor transient effects caused by the sudden application of the magnetic field to dissipate.\nAUTHOR CONTRIBUTIONS\nSF conceived the experiment with input from CD and JR. SF and JR implemented the\ntime-resolved laminography setup. SF designed the sample and performed the lithographical\npatterning. AH deposited the CoFeB films. SF, SM, CD, and AH performed the time-\nresolved laminography experiments and interpreted the resulting data, using code provided\nby CD. SF performed the micromagnetic simulations. SF wrote the manuscript, with input\nfrom all authors.\nACKNOWLEDGMENTS\nThis work was performed at the PolLux (X07DA) endstation of the Swiss Light Source,\nPaul Scherrer Institut, Villigen PSI, Switzerland. The research leading to these results\nhas received funding from the Swiss National Science Foundation under grant agreement\nNo. 172517. The PolLux endstation was financed by the German Bundesministerium für\nBildung und Forschung through contracts 05K16WED and 05K19WE2. C.D. acknowledges\n12support from the Max Planck Society Lise Meitner Excellence Program.\nSUPPORTING INFORMATION\nThe following supporting information is available:\n-DomainWallExcitation.avi - Video displaying the change in the spin configuration\nand the motion of the magnetic domain walls for the domain wall excitation mode\n(913 MHz). The video is shown in the same perspective as for Fig. 3(a).\n-VortexGyration.avi - Video displaying the change in the spin configuration and the\ndeformation of the vortex core for the vortex core gyration mode (326 MHz). The\nvideo is shown in the same perspective as for Fig. 3(b).\n\u0003simone.finizio@psi.ch\n[1] K. Y. Guslienko, Applied Physics Letters 89, 022510 (2006).\n[2] K. Y. Guslienko, G. N. Kakazei, J. Ding, X. M. Liu, and A. O. Adeyeye, Scientific Reports 5,\n13881 (2015).\n[3] B. Krüger, A. Drews, M. Bolte, U. Merkt, D. Pfannkuche, and G. Meier, Physical Review B\n76, 224426 (2007).\n[4] G. Dieterle, J. Foerster, H. Stoll, A. S. Semisalova, S. Finizio, A. Gangwar, M. Weigand,\nM.Noske, M.Faehnle, I.Bykova, J.Graefe, A.Bozhko, H.Y.Musiienko-Shmarova, V.Tiberke-\nvich, A. N. Slavin, C. H. Back, J. Raabe, G. Schuetz, and S. Wintz, Physical Review Letters\n122, 117202 (2019).\n[5] S. Wintz, V. Tiberkevich, M. Weigand, J. Raabe, J. Lindner, A. Erbe, A. Slavin, and J. Fass-\nbender, Nature Nanotechnology 11, 948 (2016).\n[6] S.Mayr, L.Flajsman, S.Finizio, A.Hrabec, M.Weigand, J.Förster, H.Stoll, L.J.Heyderman,\nM. Urbanek, and S. Wintz, Nano Letters 21, 1584 (2021).\n[7] B.Divinskiy, H.Merbouche, V.E.Demidov, K.O.Nikolaev, L.Soumah, D.Gorere, R.Lebrun,\nV. Cros, J. Ben Youssef, P. Bortolotti, A. Anane, and S. O. Demokritov, Nature Communi-\ncations 12, 6541 (2021).\n13[8] E. Jue, A. Thiaville, S. Pizzini, J. Miltat, J. Sampaio, L. Buda-Prejebeanu, S. Rohart, J. Vogel,\nM. Bonfim, O. Boulle, S. Auffret, I. Miron, and G. Gaudin, Physical Review B 93, 014403\n(2016).\n[9] J. Rhensius, L. Heyne, D. Backes, S. Krzyk, L. J. Heyderman, L. Joly, F. Nolting, and\nM. Kläui, Physical Review Letters 104, 067201 (2010).\n[10] K.-S. Ryu, L. Thomas, S.-H. Yang, and S. Parkin, Nature Nanotechnology 8, 527 (2013).\n[11] K. Litzius, I. Lemesh, B. Krüger, P. Bassirian, L. Caretta, K. Richter, F. Büttner, K. Sato,\nO. A. Tretiakov, J. Förster, R. M. Reeve, M. Weigand, I. Bykova, H. Stoll, G. Schütz, G. S. D.\nBeach, and M. Kläui, Nature Nanotechnology 13, 170 (2017).\n[12] S. Woo, K. M. Song, X. Zhang, M. Ezawa, Y. Zhou, X. Liu, M. Weigand, S. Finizio, J. Raabe,\nM.-C. Park, K.-Y. Lee, J. W. Choi, B.-C. Min, H. C. Koo, and J. Chang, Nature Electronics\n1, 288 (2018).\n[13] S. Finizio, K. Zeissler, S. Wintz, S. Mayr, T. Weßels, A. J. Huxtable, G. Burnell, C. H.\nMarrows, and J. Raabe, Nano Letters 19, 7246 (2019).\n[14] F. Büttner, C. Moutafis, M. Schneider, B. Krüger, C. M. Günther, J. Geilhufe, C. v. Ko-\nrff Schmising, J. Mohanty, B. Pfau, S. Schaffert, A. Bisig, M. Foerster, T. Schulz, C. A. F.\nVaz, J. H. Franken, H. J. M. Swagten, M. Kläui, and S. Eisebitt, Nature Physics 11, 225\n(2015).\n[15] M. Baumgartner, K. Garello, J. Mendil, C. O. Avci, E. Grimaldi, C. Murer, J. Feng, M. Gabu-\nreac, C. Stamm, Y. Ackermann, S. Finizio, S. Wintz, J. Raabe, and P. Gambardella, Nature\nNanotechnology 12, 980 (2017).\n[16] A. Fernandez-Pacheco, R. Streubel, O. Fruchart, R. Hertel, P. Fischer, and R. P. Cowburn,\nNature Communications 8, 15756 (2017).\n[17] P. Fischer, D. Sanz-Hernandez, R. Streubel, and A. Fernandez-Pacheco, APL Materials 8,\n010701 (2020).\n[18] A. Wartelle, B. Trapp, M. Stano, C. Thirion, S. Bochmann, J. Bachmann, M. Foerster,\nL. Aballe, T. O. Mentes, A. Locatelli, A. Sala, L. Cagnon, J.-C. Toussaint, and O. Fruchart,\nPhysical Review B 99, 024433 (2019).\n[19] C. Donnelly, M. Guizar-Sicarios, V. Scagnoli, S. Gliga, M. Holler, J. Raabe, and L. J. Hey-\nderman, Nature 547, 328 (2017).\n[20] C. Donnelly, S. Finizio, S. Gliga, M. Holler, A. Hrabec, M. Odstrcil, S. Mayr, V. Scagnoli,\n14L. J. Heyderman, M. Guizar-Sicarios, and J. Raabe, Nature Nanotechnology 15, 356 (2020).\n[21] K. Witte, A. Späth, S. Finizio, C. Donnelly, B. Watts, B. Sarafimov, M. Odstrcil, M. Guizar-\nSicarios, M. Holler, R. H. Fink, and J. Raabe, Nano Letters 20, 1305 (2020).\n[22] M. Holler, M. Odstrcil, M. Guizar-Sicarios, M. Lebugle, E. Müller, S. Finizio, G. Tinti,\nC. David, J. Zusman, W. Unglaub, O. Bunk, J. Raabe, A. F. J. Levi, and G. Aeppli, Nature\nElectronics 2, 464 (2019).\n[23] A. Puzic, T. Korhonen, B. Kalantari, J. Raabe, C. Quitmann, P. Jüllig, L. Bommer, D. Goll,\nG. Schütz, S. Wintz, T. Strache, M. Körner, D. Marko, C. Bunce, and J. Fassbender, Syn-\nchrotron Radiation News 23, 26 (2010).\n[24] S. Finizio, S. Mayr, and J. Raabe, Journal of Synchrotron Radiation 27, 1320 (2020).\n[25] C. Donnelly, S. Gliga, V. Scagnoli, M. Holler, J. Raabe, L. J. Heyderman, and M. Guizar-\nSicarios, New Journal of Physics 20, 083009 (2018).\n[26] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. Van Waeyen-\nberge, AIP Advances 4, 107133 (2014).\n[27] I. Makhfudz, B. Krüger, and O. Tchernyshyov, Physical Review Letters 109, 217201 (2012).\n[28] S. Finizio, B. Watts, and J. Raabe, Journal of Synchrotron Radiation 28, 1146 (2021).\n[29] J. Raabe, G. Tzvetkov, U. Flechsig, M. Böge, A. Jaggi, B. Sarafimov, M. G. C. Vernooij,\nT. Huthwelker, H. Ade, D. Kilcoyne, T. Tyliszczak, R. H. Fink, and C. Quitmann, Review of\nScientific Instruments 79, 113704 (2008).\n[30] J. Ahrens, B. Geveci, and C. Law, ParaView: an end-user tool for large data visualization\n(Elsevier, 2005).\n15(a)\nBdynCu\nCoFeB\nxy\nzOscilloscope\nFPGATrigger\nAWG500\nMHz\nRF\nAmplifier(b)\nLaminography\naxis\nAPD\nZoneplateFigure 1. (a) Sketch depicting the geometry and coordinate system for the time-resolved laminogra-\nphy experiments presented in this work. The sample is mounted under an angle of 45\u000ewith respect\nto the X-ray beam. To acquire the projections necessary for the laminographic reconstruction,\nthe sample is rotated at different angles around the axis perpendicular to its surface ( zaxis). (b)\nSketch of the electrical configuration utilized for the generation of the oscillating magnetic field\nused to excite the dynamical processes presented here. An arbitrary waveform generator (AWG),\nfrequency locked to the 500 MHz master clock of the synchrotron light source, is used to inject a\nRF current across a stripline fabricated above a Co 40Fe40B20microstructured square, giving rise to\nan oscillating magnetic field along the xaxis through the Oersted effect. The red signals indicate\nsynchronization and timing signals (handled by a custom designed field programmable gate array\n- FPGA - setup).\n16xyz(c)\n(d)500 nm\n(a) (b)\n1 μm\nmx-1 1Figure 2. Three-dimensional rendering of the equilibrium configuration of the CoFeB microstruc-\ntured square. (a) Top view of the reconstructed magnetic configuration, and corresponding micro-\nmagnetic simulation (b), where the red arrows mark the orientation of the magnetization in the\ndomains. (c) Section of the measured laminogram along the the sectioning plane depicted below\nthe image, where the orientation of the reconstructed local magnetization vector and the vortex\ncore are shown; (d) Same section shown for a micromagnetic simulation of the CoFeB square, where\nthe correspondence between the experimental and simulated data can be observed.\n171 μm\nxyzΦ = 200°, Δt = 600 psΦ = 300°, Δt = 900 psΦ = 0°, Δt = 0 ps Φ = 100°, Δt = 300 ps(a)\n913 MHz\nΦ = 200°, Δt = 1700 psΦ = 300°, Δt = 2550 psΦ = 0°, Δt = 0 psΦ = 100°, Δt = 850 ps(b)\n150 nm\nxyz\n326 MHz\n(c)\n(d)\n1 μmAmplitude [a.u.]0 1mx-1 1Figure 3. Three-dimensional rendering of single frames of the time-resolved magnetic laminograms\nacquired at an excitation frequency of (a) 913 MHz and (b) 326 MHz. Both snapshot sets show\na section of the reconstructed laminogram along the red plane sketched below each snapshot set,\nzoomed in on the vortex core for (b). \bindicates the phase difference between the excitation\nsignal and the time instant probed by each snapshot. For the snapshots pictured in (a), the red\nisosurfaces of czfollow the domain walls. The dashed cyan lines are a guide for the eye, showing\nthe configuration of the czisosurfaces cut by the sectioning plane for the previous snapshot. For\nthe snapshots pictured in (b), the vortex core is shown by the red isosurface of cz. The dashed cyan\nlines are a guide for the eye, showing the vortex core position at the previous snapshot. (c) and\n(d) show the areas of the CoFeB square where the two modes are localized. (c) shows that the 913\nMHz mode is localized primarily in the domain walls, while (d) shows that the 326 MHz mode is\nlocalized in the vortex core.\n18(a) (b)\n-50\n0\n50\n100\n150\n-40\n-30\n-20\n-10\n0\n10\nHeight:\n0 nm\n40 nm\n80 nm\n120 nm\n160 nm\nCenter of Mass - y [nm]\nCenter of Mass - x [nm]\n-150\n-100\n-50\n0\n50\n100\n150\n200\n-20\n-10\n0\n10\n20\nHeight:\n0 nm\n40 nm\n80 nm\n120 nm\n160 nm\nCenter of Mass - y [nm]\nCenter of Mass - x [nm]\n-20\n0\n20\n40\n60\n80\n100\n120\n140\n160\n180\n0\n10\n20\n30\n40\nX\nY\nGyration radius [nm]\nHeight [nm]\n(c)(d)\n-20\n0\n20\n40\n60\n80\n100\n120\n140\n160\n180\n0\n10\n20\n30\n40\nX\nY\nGyration radius [nm]\nHeight [nm]Experiment\nExperimentSimulation\nSimulationFigure 4. Experimental (a) and simulated (b) x\u0000yposition of the vortex core’s center of mass along\nthe thickness of the CoFeB microstructured square, determined by calculating the center of mass of\nthe region where cz\u00150:9\u0002max(cz)for each instant and zslice of the reconstructed time-resolved\nlaminogram. Error bars from a combination of counting statistics and image resolution. The fitted\namplitude of the x\u0000yvortex core gyration along the thickness of the CoFeB microstructured\nsquare is shown in (c) for the experimental data, and in (d) for the corresponding micromagnetic\nsimulations. The error bars are statistical errors from the fits.\n19(a)\n(b)\n0\n50\n100\n150\n200\n250\n300\n350\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\n1.1\nNormalized core volume [a.u.]\nPhase [deg]\n0\n50\n100\n150\n200\n250\n300\n350\n0.88\n0.90\n0.92\n0.94\n0.96\n0.98\n1.00\nNormalized core volume [a.u.]\nPhase [deg]Experiment\nSimulationFigure 5. Temporal dependence of the vortex core volume, normalized to the maximum volume\nacrossanexcitationcycle. (a)Experimentaldata, determinedfromthevolumeenclosedbythe 0:9\u0002\nmax(cz)isosurface. Error bars from a combination of counting statistics and image resolution. (b)\nCorresponding micromagnetic simulation, showing a qualitative agreement with the experimental\ndata.\n20"    },    {        "title": "2009.01331v2.Magnetized_Decaying_Turbulence_in_the_Weakly_Compressible_Taylor_Green_Vortex.pdf",        "content": "Magnetized Decaying Turbulence in the Weakly Compressible Taylor-Green Vortex\nForrest W. Glines,\u0003Philipp Grete,yand Brian W. O'Sheaz\nDepartment of Physics and Astronomy, Michigan State University\n(Dated: April 28, 2021)\nMagnetohydrodynamic turbulence a\u000bects both terrestrial and astrophysical plasmas. The prop-\nerties of magnetized turbulence must be better understood to more accurately characterize these\nsystems. This work presents ideal MHD simulations of the compressible Taylor-Green vortex under\na range of initial sub-sonic Mach numbers and magnetic \feld strengths. We \fnd that regardless\nof the initial \feld strength, the magnetic energy becomes dominant over the kinetic energy on all\nscales after at most several dynamical times. The spectral indices of the kinetic and magnetic energy\nspectra become shallower than k\u00005=3over time and generally \ructuate. Using a shell-to-shell energy\ntransfer analysis framework, we \fnd that the magnetic \felds facilitate a signi\fcant amount of the\nenergy \rux and that the kinetic energy cascade is suppressed. Moreover, we observe nonlocal en-\nergy transfer from the large scale kinetic energy to intermediate and small scale magnetic energy via\nmagnetic tension. We conclude that even in intermittently or singularly driven weakly magnetized\nsystems, the dynamical e\u000bects of magnetic \felds cannot be neglected.\nI. INTRODUCTION\nMagnetized turbulence is present in many terrestrial\nand astrophysical plasmas. Turbulence in magnetohy-\ndrodynamics (MHD) has been studied extensively over\nrecent decades, from experimental, theoretical, and nu-\nmerical perspectives, as the \feld continues to work to-\nwards a full understanding of magnetized turbulent plas-\nmas. However, much of the theoretical and numerical\nwork focuses on continuously driven plasmas, where a\ncontinuous (although potentially stochastic) force adds\nenergy to the plasma, resulting in stationary turbulence.\nIn many natural systems, the turbulence can be inter-\nmittently driven by infrequently occurring events or ini-\ntialized from the initial conditions. For example, in the\ncircumgalactic medium (CGM), the hot di\u000buse gas sur-\nrounding galaxies, or in the intracluster medium (ICM),\nthe plasma in galaxy cluster that accounts for the ma-\njority of baryonic mass, turbulence can be introduced by\nvarious mechanisms. These include mergers with other\ngalaxies, brief increases in the birth rate of stars, tem-\nporary out\rows from jets driven by gas accreting onto\nsupermassive black holes, supernovae, and many more\ntransient events [1{4]. In pulsed power plasmas such as\nin a z-pinch, the plasma is driven by a single initial event\nand then allowed to decay into turbulence as kinetic and\nmagnetic energy in the plasma dissipate into heat [5, 6].\nTherefore, to bridge the gap between observed, intermit-\ntently driven turbulent systems and theories of stationary\nMHD turbulence, we can study the behavior of decaying\nmagnetized turbulence in an idealized environment.\nIn decaying turbulence, the turbulent \row arises purely\nfrom the initial conditions in the absence of a continuous\n\u0003glinesfo@msu.edu; Department of Computational Mathematics,\nScience, and Engineering, Michigan State University\nygrete@pa.msu.edu\nzoshea@msu.edu; Department of Computational Mathematics,\nScience, and Engineering and National Superconducting Cy-\nclotron Laboratory, Michigan State Universitydriving force that injects energy. Essentially, the driving\nforce is a delta function forcing at the initialization of\nthe \row. The absence of external forces can avoid some\nof the shortfalls of driven turbulence simulations. As an\nexample of these shortfalls, previous studies have shown\nthat seemingly unimportant driving parameters such as\nthe autocorrelation time and normalization of the driv-\ning \feld can bias plasma properties in turbulence simu-\nlations, in some cases a\u000becting the scaling of the energy\nspectra [7]. In addition, the driving forces contaminate\nthe driven scales, making studies of turbulent plasma\nproperties on those scale di\u000ecult to interpret. Simula-\ntions of decaying turbulence with \fxed initial conditions\navoid these issues since there are no driving forces.\nThe Taylor-Green (TG) vortex provides a useful set of\nsmooth initial conditions that devolve into a turbulent\n\row. It was \frst proposed by Taylor and Green [8] as an\nearly mathematical exploration of the development of the\nturbulent cascade in a three dimensional hydrodynamic\n\ruid. In the modern era, it is a canonical transition-to-\nturbulence problem also used for validation and veri\fca-\ntion of numerical schemes [9]. From a physics point of\nview, the TG vortex has been explored from numerous\nangles, including numerical simulations of inviscid and\nviscous incompressible hydrodynamics with an emphasis\non the development of small scale structures through vor-\ntex stretching [10]. Multiple con\fgurations for TG vor-\ntices with magnetic \felds were proposed in Lee et al. [11]\nin order to study decaying turbulence in incompressible\nMHD. The new magnetic \feld con\fgurations maintain\nall of the symmetries of the original hydrodynamic \row\n[11], and later works [12{14] used these symmetries to\nsave computational resources and allow more highly re-\nsolved simulations of the vortex. These simulations pro-\nduced di\u000bering k\u00002,k\u00005=3, andk\u00003=2spectra depending\non the initial magnetic \feld, where the k\u00002spectra was\nspeculated to be due to weak turbulence. Later work\nby Dallas and Alexakis [15, 16] investigated the mecha-\nnism behind the di\u000berent spectra. They concluded that\nthek\u00002spectra produced by one con\fguration of thearXiv:2009.01331v2  [physics.flu-dyn]  26 Apr 20212\nmagnetic \feld was due to magnetic discontinuities in the\nplasma and not weak turbulence as previously thought.\nIn Dallas and Alexakis [17], perturbations added to the\ninitial conditions lead the symmetries of the TG vortex\nto break and the k\u00002spectra to dissipate to shallower\nk\u00005=3spectra. A similar problem using the hydrody-\nnamic initial con\fguration of the TG vortex but with an\nOrszag-Tang magnetic \feld was studied in imcompress-\nible resistive MHD by Vahala et al. [18], where a k\u00005=3\nenergy spectra was found in their simulations.\nAll of these studies are concerned with incompressible\nturbulence, whereas many astrophysical systems (such\nas the interstellar, circumgalactic, intracluster, and in-\ntergalactic media) are comprised of compressible magne-\ntized plasmas. To our knowledge, the formulation of the\nTG vortex from Lee et al. [11] remains unexplored in the\ncompressible MHD regime. Moreover, there have been\nrecent advances in analytical tools to study the trans-\nfer of energy between reservoirs in compressible MHD\n[19, 20]. Energy transfer analysis enables measurement of\nthe \rux of energies between length scales within and be-\ntween the kinetic, magnetic, and thermal energies of the\nplasma. In a compressible ideal MHD plasma, energy\ncan be redistributed within the kinetic and within the\nmagnetic energy budget via advection and compression.\nMoreover, magnetic tension can facilitate energy transfer\nbetween kinetic and magnetic energies as vortical motion\nin the turbulent plasma contributes to magnetic \felds\nand magnetic \felds constrain the motion of the plasma.\nIn turbulent \row, intra-budget energy transfers via ad-\nvection and compression typically manifest from a larger\nscale to a smaller but similar scale (i.e., \\down scale-\nlocal\"), de\fning the turbulent cascade. Inter-budget en-\nergy transfer via, e.g., magnetic tension, complicates the\npicture of a turbulent cascade as it moves energy between\nreservoirs and potentially allows for nonlocal transfer of\nenergy from large scales directly to much smaller scales.\nGiven the transient nature of the TG vortex, we expect\nthe energy transfers to change over time as, e.g., the ra-\ntio of kinetic to magnetic energy evolves over time or\ndue to the development of increasingly small-scale struc-\nture. This is in contrast to stationary turbulence where\nthe dynamics remain constant over time in a statistical\nsense.\nFor these reasons, we focus on a detailed study of the\ndynamics in the magnetized, weakly compressible Taylor-\nGreen vortex. Moreover, to explore magnetized decay-\ning turbulence in di\u000berent regimes we present nine sim-\nulations of the TG vortex probing all combinations of\nthree di\u000berent initial ratios of kinetic to magnetic energy\n(1, 10, and 100, corresponding to initial Alfv\u0013 enic Mach\nnumbers ofMA=f1;3:2;10g) and three di\u000berent initial\n\ruid velocities (initial root mean squared, or RMS, sonic\nMach numbers of Ms;0=f0:1;0:2;0:4g). Thus, we ex-\nplore strongly and weakly magnetized, subsonic plasmas\nin which density perturbations are present but limited.\nTo summarize our results, we \fnd that magnetic \felds\nsigni\fcantly in\ruence the decaying turbulence in theplasma regardless of the initial \feld strength. In all cases,\nwe \fnd that at late times the magnetic dynamics domi-\nnate kinetic dynamics even if the initial magnetic energy\nis 100 times smaller than the kinetic energy. Moreover,\nthe spectral indices of the kinetic and magnetic ener-\ngies are not \fxed in time but evolve from steep 'k\u00002\nspectra at earlier times to shallower 'k\u00004=3spectra at\nlater times. Using the energy transfer analysis, we see\nthat most energy transfer is dominated by magnetic \feld\ndynamics. This includes both energy \rux from kinetic\nto magnetic energy via magnetic tension and the \rux of\nenergy within the magnetic energy budget via compres-\nsion and advection. Overall, the kinetic energy cascade\nis e\u000bectively absent and the initial sonic Mach number\n(Ms;0) only weakly a\u000bects the observed dynamics. We\nalso see several transient phenomena during the transi-\ntion to turbulence, including temporary inverse turbulent\ncascades in both the magnetic and kinetic energies and\nlarge nonlocal energy transfers between scales separated\nby up to two orders of magnitude from the kinetic to the\nmagnetic energy.\nWe organize the paper as follows. In Section II, we\ndescribe the simulation and analysis setup including nu-\nmerical methods, detailed Taylor-Green vortex initial\nconditions, and the energy transfer analysis. In Sec-\ntion III, we present results of the simulations (focusing\nonMs;0= 0:2) such as the bulk properties of the plasma,\nthe evolution of the energy spectra, and the transient be-\nhaviors seen through the energy transfer analysis:In Sec-\ntion IV, we discuss our \fndings in the broader context\nof magnetized turbulence and astrophysical plasmas and\nconclude in Section V with a summary of our key \fnd-\nings. The online supplementary materials for this paper\ncontain detailed plots of the results of all initial Ms;0.\nII. METHOD\nA. MHD Equations and Numerical Method\nThe equations for compressible ideal MHD plasma can\nbe written as a hyperbolic system of conservation laws.\nIn di\u000berential form the ideal MHD equations are\n@t\u001a+r\u0001(\u001au) = 0\n@t\u001au+r\u0001(\u001au\nu\u0000B\nB) +r\u0000\np+B2=2\u0001\n= 0\n@tB\u0000r\u0002 (u\u0002B) = 0\n@tE+r\u0001\u0002\u0000\nE+p+B2=2\u0001\nu\u0000(B\u0001v)B\u0003\n= 0\nwhere\u001ais the density, uis the \row velocity, Bis the\nmagnetic \feld (that includes a factor of 1 =p\n4\u0019),pis the\nthermal pressure, and Eis the total energy density. We\nclose the system of equations with the equation of state\nfor an adiabatic ideal gas with\np=\u001a(\r\u00001)e3\nwhere\ris the ratio of speci\fc heats and eis the internal\nenergy found from\nE=\u001a\u00121\n2u\u0001u+1\n2B\u0001B+e\u0013\n:\nWe use the open source K-Athena [21] astrophysical\nMHD code, which is a performance portable version of\nAthena++ [22] using the Kokkos performance porta-\nbility library [23]. K-Athena uses an unsplit \fnite vol-\nume Godunov scheme to evolve the ideal MHD equa-\ntions originally presented and implemented in Athena\n[24]. The method consists of a second-order Van Leer\npredictor-corrector integrator with piecewise linear re-\nconstruction (PLM) and HLLD Riemann solver, and con-\nstrained transport to preserve a divergence-free magnetic\n\feld.\nB. Magnetized TG Vortex\nThe TG vortex was \frst proposed by Taylor and Green\n[8] as a mathematical exploration of the development\nof hydrodynamic turbulence in 3D. The initial \row was\nmade to be periodic and symmetrical in order to accom-\nmodate simple approximations to a solution. There exist\na number of di\u000berent formulations. We follow the setup\ndescribed in Wang et al. [9] for the hydro variables and\nLeeet al. [11] for the initial magnetic \feld con\fguration.\nThe simplest hydrodynamic setup of a TG vortex be-\ngins with a periodic \feld of \ruid velocity in the xy-plane\nand periodic pressure and density \feld with constant\nsound speed throughout the domain. Using a cubic pe-\nriodic domain with side length 2 \u0019L, the initial \ruid ve-\nlocity is set to\nux=u0sinx\nLcosy\nLcosz\nL\nuy=\u0000u0cosx\nLsiny\nLcosz\nL\nuz= 0\nwhereu0is the maximum initial velocity. Note that in\nthis formulation the initial \row velocity is con\fned to the\nxy-plane. The initial pressure and density are set to\nP=P0+\u001a0u2\n0\n16\u0012\ncos2x\nL+ cos2y\nL\u0013\u0012\ncos2z\nL+ 2\u0013\n\u001a=P\u001a0=P0\nso thatPand\u001aare proportional to each other. This\nmeans that the sound speed\ncs=p\n\rP=\u001a =p\n\rP0=\u001a0\nis initially constant throughout the domain.\nThe root mean square (RMS) of the initial Mach num-\nber is related to u0by\nMs;0=u0\n2cs:For simplicity, we set P0= 1 and\u001a0= 1. We assume\nthe \ruid is a monatomic ideal gas with an adiabatic index\n\r= 5=3. The resulting total initial kinetic energy is\nEU;0=\u001a0u2\n0(\u0019L)3: (1)\nMagnetic \felds were \frst added to the TG vortex in\n[11] with the express constraint of preserving the same\nsymmetries of the hydrodynamic \row. Here, we follow\nthe proposed insulating con\fguration so that currents are\ncon\fned to \u0019Lboxes, e.g., the cube [0 ;\u0019L]3forms an\ninsulating box. The corresponding initial magnetic \felds\nare given by\nBx=B0cosx\nLsiny\nLsinz\nL\nBy=B0sinx\nLcosy\nLsinz\nL\nBz=\u00002B0sinx\nLsiny\nLcosz\nL\nwhereB0is the initial magnetic \feld strength. In prac-\ntice, we initialize the magnetic \feld from the magnetic\nvector potential A\nAx=\u0000B0sin\u0010x\nL\u0011\ncos\u0010y\nL\u0011\ncos\u0010z\nL\u0011\nAy=B0cos\u0010x\nL\u0011\nsin\u0010y\nL\u0011\ncos\u0010z\nL\u0011\nAz= 0\nusing B=r\u0002A. This guarantees r\u0001B= 0 to machine\nprecision in the initial conditions, which is then preserved\nby the constrained transport algorithm throughout the\nsimulation. The total initial magnetic energy is\nEB;0= 3B2\n0(\u0019L)3(2)\nso that the initial ratio of kinetic to magnetic energy is\nEU;0\nEB;0=\u001a0u2\n0\n3B2\n0: (3)\nSince the magnetic \feld is zero is some regions of the\ndomain, the Alfv\u0013 enic Mach number MA=up\u001a=B is\nalso unde\fned in some regions. For this reason, we use a\nproxy based on the mean energies for the Alfv\u0013 enic Mach\nnumber\nMA:=p\nhEUi=hEBi (4)\nthroughout the rest of the paper. We also adopt a similar\nproxy for the plasma \f(ratio of thermal to magnetic\npressure)\n\f:=2\n\rM2\nA\nM2s(5)\nwhereMsis the RMS of the sonic Mach number.\nThe hydrodynamic and magnetic initial conditions\nexhibit a number of symmetries that are maintained4\nthroughout the simulation. In each of the three dimen-\nsions there are two planes across which the \ruid is anti-\nsymmetric. For our setup, these are planes through x= 0\nandx=\u0019L; planes through y= 0 andy=\u0019L; and\nplanes through z= 0 andz=\u0019L. Additionally, the \row\nis rotationally symmetric through a rotation of \u0019around\nthe two axes x=z=\u0019L=2 andx=z=\u0019L=2 and ro-\ntationally symmetric through a rotation of \u0019=2 around\nthe axisx=y=\u0019L=2. These symmetries are more\nthoroughly explored in [11].\nWe explore the transition to magnetized turbulence\nand the following decay in di\u000berent regimes with our\nsimulation suite of TG vortices and focus on two pa-\nrameters: the initial RMS Mach number using Ms;0=\nf0:1;0:2;0:4gand the initial ratio of kinetic to mag-\nnetic energy using EU;0=EB;0=f1;10;100g, or alterna-\ntively, the initial RMS Alfv\u0013 enic Mach number MA;0=\nf1;3:2;10g. We simulate all nine combinations of the\nthree values of these two parameters. Throughout the\nrest of the text, we use MsX to refer to simulations\nwithMs;0=Xand MaYto refer to simulations with\nMA;0=Y.\nThe initial magnetic \feld amplitude B0is obtained\nfrom Equation 3 using given a speci\fc value of Ms;0and\nMA;0. All simulations employ a cubic [ \u00000:5;0:5]3do-\nmain with periodic boundaries, with L=1\n2\u0019to be con-\nsistent with the de\fnition of the initial condition that is\npresented above. We use a uniform Cartesian grid with\n1;0243cells. The characteristic length scale of the initial\nvortices is\u0019L, so that we de\fne\nT=\u0019L\nu0\nas the dynamical time [25] In order to evolve the simula-\ntions for su\u000ecient time to allow a turbulent \row to form\nand evolve, we run each simulation for \u00196 dynamical\ntimes.\nIn our results, we present all measurements of time in\nterms of the dynamical time Tand all measurements of\nwavenumber in terms of 1 =L. Unless otherwise noted, all\nother results are in terms of simulation units.\nC. Energy Transfer Analysis\nIn order to probe the movement of energy between dif-\nferent energy reservoirs, we use the shell-to-shell energy\ntransfer analysis from Grete et al. [20], which extends\nthe framework presented in Alexakis et al. [26] to the\ncompressible regime.\nThe total transfer of energy from some shell Qin en-\nergy reservoir Xto some shell Kin reservoir Yis denoted\nby\nTXY(Q;K )X;Y2[U;B] (6)\nwhere we use UandBto denote the kinetic and magnetic\nenergy reservoirs, respectively.In this work we focus on the energy transfer within the\nkinetic and magnetic energy reservoirs via advection and\ncompression which are respectively\nTUU(Q;K ) =\u0000R\nwK\u0001(u\u0001r)wQdx\n\u00001\n2R\nwK\u0001wQr\u0001udx\nTBB(Q;K ) =\u0000R\nBK\u0001(u\u0001r)BQdx\n\u00001\n2R\nBK\u0001BQr\u0001udx\nand the energy transferred from kinetic energy to mag-\nnetic energy via magnetic tension (and vice versa) given\nby\nTUBT(Q;K ) =Z\nBK\u0001r\u0000\nvA\nwQ\u0001\ndx (7)\nTBUT(Q;K ) =Z\nwK\u0001(vA\u0001r)BQdx: (8)\nHere we use the mass weighted velocity w=p\u001auso that\nthe spectral energy density is positive de\fnite [27], and\nvAis the Alfv\u0013 enic wave speed.\nThe velocity wKand magnetic \feld BKin a shell K\n(or Q) are obtained using a sharp spectral \flter in Fourier\nspace. The shell bounds are logarithmically spaced and\ngiven by 1 and 2n=4+2forn2f\u0000 1;0;1;:::; 32g. Shells\n(uppercase, e.g., K) and wavenumbers (lowercase, e.g.,\nk) obey a direct mapping, i.e., K= 24 corresponds\nto the logarithmic shell that contains k= 24, i.e.,\nk2(22:6;26:9].\nIII. RESULTS\nIn this section we present results of the Taylor-Green\nvortices we simulated, showing bulk properties of the\n\ruid (Section III A), including the evolution of the di\u000ber-\nent energy spectra. These results demonstrate that the\nkinetic, magnetic, and thermal energy reservoirs interact\nwith each other in a manner that depends signi\fcantly\non the initial strength of the magnetic \feld. The energy\nspectra evolves to a turbulent cascade over 1-2 dynamical\ntimes and then stays there for the remainder of the simu-\nlation. In Section III B, we examine in detail the transfer\nof energy between di\u000berent energy reservoirs, including\nthe transient behaviors we observed in the simulations.\nWe see robust transfer of energy at all scales within the\nkinetic and magnetic energy reservoirs when examined\nseparately, as well as complex and time-varying nonlo-\ncal transfer of energy between the kinetic and magnetic\nenergy reservoirs, including evidence for an intermittent\ninverse turbulent cascade. Since the initial Mach number\nhad much less of an e\u000bect on the results compared to the\ninitial ratio of kinetic to magnetic energy, we focus on re-\nsults using only the three Ms0.2 simulations as reference.\nWe provide more complete plots of all nine simulations\nspanning all Mach numbers in the online supplements.\nStarting with a visual demonstration of the TG vor-\ntex, Figure 1 shows the sonic Mach number and magnetic5\nFIG. 1. Slices of sonic Mach number (left) and magnetic\npressure (right) at t= 0:77Tandt= 5:16Tin thexy\u0000plane\nthroughz=\u0019\n2L, with streamlines on the left showing the\ndirection of \row and streamlines on the right showing the\ndirection of the magnetic \felds, plotting only the 1st quadrant\nfrom the Ms0.2 Ma10 simulation, demonstrating the transition\nof the \row into turbulence.\npressure from the Ms0.2 Ma10 simulation after 0 :77 dy-\nnamical times and after 5 :16 dynamical times in a slice\nin thexy\u0000plane through the origin. Only one quadrant\nof thexy-place is shown, as it exhibits symmetry across\n4 quadrants in the xy-plane. From the slice plot, we can\nsee that the TG vortex begins as a smooth vortical \row\nand magnetic \feld. After several dynamical times, the\nsmooth \row devolves into a chaotic magnetized turbu-\nlent \row. Kinetic and magnetic structures at all scales\npersist throughout the simulation, as will be shown in\nenergy spectra later in this work.\nA. Bulk Properties\n1. Evolution of energy reservoirs\nFigure 2 shows the total kinetic, magnetic, and thermal\nenergies and the dimensionless RMS sonic Mach number\nMs, Alv\u0013 enic Mach number MA, and plasma beta \fof\ntheMs0.2 simulations as a function of time. In this \fg-\nure, we can see that in all simulations kinetic and mag-\nnetic energy convert into thermal energy over time. This\ndecay into thermal energy is not immediate; rather, it\nrequires at least one dynamical time to begin (i.e., it isobserved to occur at a minimum of t= 1Tin all simula-\ntions). In the Ma1simulations, due to the initial condi-\ntions there is even a small transient transfer of thermal\nenergy into kinetic and magnetic energies. After t= 2T,\nall simulations dissipate kinetic and magnetic energy into\nthermal energy. The sonic Mach number generally de-\ncreases by less than a factor of 4 over time from its initial\n0:2 value, and \fremains high (from &20 for Ms0.2 Ma1\nto&100 for Ms0.2 Ma10 ) throughout the simulations.\nIn all cases, the \row becomes dominated by magnetic\nenergy (i.e., become sub-Alfv\u0013 enic with MA<1) at dif-\nferent dynamical times depending on the initial ratio of\nkinetic to magnetic energy and mostly independent of\nthe initial Mach number. In other words, even for the\nsimulations with initially 100 times more kinetic than\nmagnetic energy ( Ma10 ), in the \fnal state the magnetic\nenergy dominates over the kinetic energy. This already\nhighlights the importance of kinetic to magnetic energy\ntransfer. The initial growth of magnetic energy is charac-\nteristic of the insulating magnetic \feld con\fguration and\nis seen in other works on the TG vortex [12]. This be-\nhavior of the magnetic \feld is likely due to the magnetic\n\felds and vorticity beginning parallel to each other every-\nwhere. All simulations experience a peak in the magnetic\nenergy evolution before t= 3Tdepending on the initial\nmagnetic energy. At t= 6T, all simulations are still los-\ning total kinetic and magnetic energy to thermal energy,\nalthough the rate of energy dissipation is slowing by the\nsimulation end. The magnetic and kinetic energies also\nbecome similar in magnitude, cf., MA'1.\nThe Ms0.2 Ma1simulation displays notably di\u000berent\nbehavior than those where the kinetic energy initially\ndominates. In particular, we observe periodic exchanges\nof energy between these two reservoirs before the bulk of\nthe energy is converted into heat, rather than a smooth\ntransfer of energy from the kinetic to magnetic reservoir,\nfollowed by a decline of both as the \row thermalizes.\nAt approximately t= 1T, more than \fve times as much\nenergy is stored in the magnetic reservoir as compared to\nthe kinetic reservoir, which is in stark contrast with other\ncalculations. These results suggest that the large initial\nmagnetic \feld facilitates a more rapid transfer of kinetic\nenergy, which will be examined in more detail later in this\npaper. For reference, we also plot the temporal evolution\nof the energies in the incompressible, magnetized Taylor-\nGreen vortex with Ma=1 presented in Pouquet et al. [13]\nin the top left panel of Fig. 2 next to our Ms0.2 Ma1\nresults. The evolution in [13] covers the \frst oscillation\nand is in good agreement with our simulation. Finally,\nthe oscillations observed in the energy reservoirs for the\nMa1simulations in general have a period that depends on\nthe initial Mach number, which can be seen in the \fgures\nthat we leave for the online supplements.6\n0.000.010.020.030.040.050.060.07Energy\nMs0.2_Ma1\n Ms0.2_Ma3.2\n Ms0.2_Ma10\n0 2 4 6101\n100101102103104\ns\nA\n0 2 4 6\n0 2 4 6\nTime t [units of T]EU\nEB\n EU+EB\nES ES,0\n [13] EU\n[13] EB\n[13] EU+EB\nFIG. 2. Mean energies over over time in the top row with kinetic energy (solid blue), magnetic energy (solid orange), the\nsum of kinetic and magnetic energies (solid green), and the change in thermal energy since the simulation start (solid red),\nand dimensionless numbers over time in the bottom row with RMS sonic Mach number Ms(blue), Alv\u0013 enic Mach number\nMA(orange), and plasma beta \f(green) for the Ms0.2 simulations. Energy over time from the simulation from Fig. 3a in\nPouquet et al. [13] (adjusted to the normalization used here), which matches the setup of the Ms0.2 Ma1simulation, is shown\nwith dashed lines in the upper left panel for reference. Energies and mach numbers for all nine simulations are shown in the\nonline supplements.\n2. Energy Spectra\nFigure 3 shows the temporal evolution of the kinetic\nand magnetic energy spectra of the three Ms0.2 simu-\nlations, compensated by k4=3, which demonstrates how\nboth the kinetic and magnetic energy spectra change\nfrom the smooth initial large scale \row to fully developed\nturbulence. The top row shows the three simulations ear-\nlier in the evolution ( t= 0:77T), when the spectra are\nstill steep with large scale structure from the initial con-\nditions. In the case of the strongest initial magnetiza-\ntion ( Ma1), the magnetic energy is larger than the kinetic\nenergy on all scales and their spectral scaling is compa-\nrable. For Ma3.2 and Ma10 the kinetic energy spectrum\nis steeper than the magnetic one. The spectra cross at\nk'7 andk'20, respectively, so that the kinetic en-\nergy is still dominant on large scales. The middle row\nin Figure 3 shows intermediate times with Ms0.2 Ma1at\nt= 1:29T, which is the time that is discussed in Sec-\ntion III B 2 and Ms0.2 Ma3.2 and Ms0.2 Ma10 simula-\ntions att= 1:81T, which is the time is discussed in\nSection III B 1. Note that the spectra are still evolvingat this intermediate stage. In the Ms0.2 Ma10 simulation\natt= 1:81T, the magnetic spectra has reached a k\u00004=3\nspectrum while the kinetic spectra shows a broken power\nlaw with excess energy at larger length scales. In both\nMa1andMa3.2 the magnetic energy is now dominant on\ne\u000bectively all scales (with the exception of the noisy part\nof the spectrum at the largest scales, k.4). The bottom\nrow shows all three Ms0.2 simulations at t= 5:16T. Here,\nthe magnetic energy is e\u000bectively dominant on all scales\nin all simulations and the kinetic and magnetic spectra\nexhibit a scaling close to k\u00004=3. The spectral indices still\n\ructuate, which we explore in Section III A 3.\nIn Figure 4 we show the kinetic and magnetic energy\nat speci\fc wavenumbers and compensated by k4=3plot-\nted over time. At early times (before t= 2T) the large\nscale (k= 8) kinetic energy shows the fastest growth\nrate compared to smaller scales as expected from an ini-\ntial entirely large scale con\fguration. The kinetic energy\natk= 8 peaks between t= 1Tandt= 2Twith larger\ninitial magnetic \feld leading to an earlier peak. The\nmagnetic energy at k= 8 in the Ms0.2 Ma1simulation\noscillates throughout the duration of the simulation, with7\n108\n106\n104\n102\n100\nMs0.2_Ma1 \n t=0.77T\nMs0.2_Ma3.2 \n t=0.77T\nMs0.2_Ma10 \n t=0.77T\n108\n106\n104\n102\n100\nMs0.2_Ma1 \n t=1.29T\nMs0.2_Ma3.2 \n t=1.34T\nMs0.2_Ma10 \n t=1.81T\n101102108\n106\n104\n102\n100\nMs0.2_Ma1 \n t=5.16T\n101102\nMs0.2_Ma3.2 \n t=5.16T\n101102\nMs0.2_Ma10 \n t=5.16T\nWavenumber k [units of 1/L]E(k)k4/3\nFIG. 3. Kinetic energy spectra (in solid blue) and magnetic energy spectra (in solid orange) compensated by k4=3, with black\ndashed lines showing the power law \ft to the spectral to obtain a spectral index. In the left column we show the Ms0.2 Ma1\nsimulation, in the middle column we show the Ms0.2 Ma3.2 simulation, and in the right column we show the Ms0.2 Ma10\nsimulation. In the top row we show all simulations at t= 0:77T, in the middle row we show the three simulations at di\u000berent\ntimes (t= 1:29,t= 1:81T,t= 1:81T) when the simulations are displaying interesting behavior discussed in sections III B 2\nand III B 1, and in the bottom row we show all simulations at t= 5:16Twhen the initial \row has completely decayed into\nturbulence and both energy spectra \ructuate around a k\u00004=3spectrum.8\n0.000.010.020.030.040.05EU(k)k4/3\nMs0.2_Ma1\nk=8\nk=22k=64\nk=128\nMs0.2_Ma3.2\n Ms0.2_Ma10\n0 2 4 60.000.010.020.030.040.05EB(k)k4/3\n0 2 4 6\n 0 2 4 6\nTime t [units of T]\nFIG. 4. The kinetic energy (top) and magnetic energy (bottom) at wavenumbers k= 8;22;64;128 plotted separately in\ndi\u000berent colors versus time, where the energy at each wavenumber has been compensated by k4=3to make them comparable.\nIn the left column we show the Ms0.2 Ma1simulation, in the middle column we show the Ms0.2 Ma3.2 simulation, and in the\nright column we show the Ms0.2 Ma10 simulation. Energy at the smallest length scales in both reservoirs saturates at t'1T,\nt'1:5T, andt'2:5 in the Ms0.2 Ma1,Ms0.2 Ma3.2 , and Ms0.2 Ma10 simulations respectively, showing approximately when\nthe turbulence has developed at all scales.\nthe kinetic energy oscillating once. No oscillatory behav-\nior is observed in Ms0.2 Ma3.2 andMs0.2 Ma10 for these\nquantities. From this plot we can also see that the small\nscale (k= 128) energies saturate at t'1T,t'1:5T,\nandt'2:5T, respectively.\n3. Spectral Index\nWe measured the spectral indices of the kinetic and\nmagnetic energy spectra \u000bby \ftting a power-law E/k\u000b\nto the energy spectra of each reservoir at each time step.\nFor the inertial range of wavenumbers across which we\n\ft the power-law to the spectra, we used wavenumbers\nk= 10 tok= 32. We chose this inertial range because\nvery little large scale structure persists below k= 10 and\nwavenumbers above k= 32 are not entirely free of nu-\nmerical dissipation any more. The kinetic and magnetic\nspectral indices measured across the inertial range are\nnot \fxed in time across the di\u000berent simulations, with\nthe most variation being due to initial magnetic energy.\nFigure 5 shows the spectral indices of the kinetic, mag-netic, and sum of kinetic and magnetic energy spectra\nover time for the Ms0.2 simulations. In all simulations,\nthe spectral index evolves over time, decaying from the\ninitial steep spectral index ( \u000b.\u00002) as energy is trans-\nferred to small scales. The kinetic and magnetic spectral\nindices evolves separately in the calculations until the\nmagnetic energy exceeds the kinetic energy, after which\nthe spectral indices of the separate and combined reser-\nvoirs \ructuate within \u0001 \u000b'0:2. The crossover of ki-\nnetic and magnetic energies happens immediately in the\nMs0.2 Ma1simulation, early in the Ms0.2 Ma3.2 simula-\ntion before t= 2T, and later in the Ms0.2 Ma10 simula-\ntion att'4T. After the kinetic and magnetic spectral\nindices reach rough parity and the magnetic \feld becomes\ndominant, both spectral indices reach comparable values\nand reach a rough constant 1 \u00002 dynamical times later,\nalthough they continue to vary over time. Since the mag-\nnetic \felds in the Ma1simulations immediately become\ndominant, the spectral indices reach a rough constant\natt'2T, while in the Ma3.2 simulations they reach a\nrough constant at t'4Tand in the Ma10 simulations this\nhappens at t'5T. The Ms0.2 Ma3.2 simulation experi-9\n1 2 3 4 5 62.50\n2.25\n2.00\n1.75\n1.50\n1.25\n1.00\n0.75\nMs0.2_Ma1EU\nEB\nEU+EB\n1 2 3 4 5 6\nMs0.2_Ma3.2\n1 2 3 4 5 6\nMs0.2_Ma10\nTime t [units of T]Spectral Index \nFIG. 5. Evolution of the spectral indices of the kinetic (blue), magnetic (orange), and sum of kinetic and magnetic energy\n(green) spectra over time for the Ms0.2 simulations. The slope is computed from a least squares \ft of the energy spectra limited\nto wavenumbers k2[10;32] which is approximately the inertial range. Shaded bands show how the \ftted slope di\u000bers if a\nrangek2[8;34],k2[10;32], ork2[12;30] is used. Note that the spectral index using the range k2[10;32] is not guaranteed\nto be bounded by the spectral indices obtained using k2[8;34],k2[10;32] andk2[12;30], which is especially evident in\ntheMs0.2 Ma3.2 and Ms0.2 Ma10 simulations from t'2Ttot'4T. Horizontal dashed lines show \u00004=3 and\u00005=3 spectral\nindices. The slope is only shown after t= 1Tas the initial \row conditions dominate the spectra at early times, leading to steep\nspectra. We include the spectral indices versus time for all nine simulations in the online supplements.\nences a brief peak in the spectral index around t'1:5T\nwhile the \row is still in transition. This is also re\rected\nin the large uncertainty of the spectral index during that\ntime, e.g., the index of the kinetic energy spectrum varies\nbetween\u00001 and\u00002:25 by choosing slightly di\u000berent \ft-\nting ranges (as indicated by the shaded blue bands in\nFig. 5). Note that in the Ma10 case, the magnetic spec-\ntrum \rattens and the spectral index reaches a roughly\nconstant value much sooner than in the other two cases,\natt'2Twhen the kinetic energy still dominates. Later\non in the Ma10 simulations, the kinetic spectral index be-\ncomes comparable to the magnetic spectral index. For\nthe high initial magnetic \feld simulations, the spectral\nindex levels out at about \u000b'\u00005=3 while the initially\nkinetically dominated simulations level out at \u000b'\u00004=3.\nThe \fnal spectral indices depend on the initial ratio\nof kinetic to magnetic energy, with more magnetic en-\nergy leading to shallower magnetic spectra. The Ma1\nsimulations end with \u000b'\u00001:7 (close to\u00005=3),Ma3.2\nends with\u000b'\u00001:3 (close to\u00004=3), and Ma10 ends with\nslightly lower values of \u000b'\u00001:2. In the presence of the\nstronger magnetic \felds in the Ma1simulations, the \rat-\ntening of the spectra seems to be suppressed. Before the\nkinetic and magnetic spectral indices become compara-\nble in each simulation, there is also greater variance in\nthe spectral slope when measured using di\u000berent inertial\nranges. This indicates that a power-law might be a poor\n\ft for the spectra at those early times, showing that thespectra is not fully developed until the magnetic energy\nis dominant. For example, as seen in Figure 3, the kinetic\nenergy spectra appears as a broken power law at interme-\ndiate times, which is especially evident in the Ms0.2 Ma10\nsimulation at t= 1:81Tto a lesser extent the Ms0.2 Ma1\nsimulation at t= 1:29Tand the Ms0.2 Ma3.2 simulation\natt= 1:81T. Oscillations in the spectral index of the Ma1\nsimulations also appear, whose period seems to be linked\nto the initial Mach number, with larger Mach numbers\nleading to a smaller period of oscillation.\nWe note that between the three values of MA, the sim-\nulations shown here exhibit a wide variety of behaviors,\nhighlighted by the spectral indices in Fig. 5. More simula-\ntions with intermediate values of MAwould be required\nto determine if the transition between these behaviors is\nsmooth or abrupt.\nB. Energy Transfer\nWhile the total energy and spectra of the kinetic and\nmagnetic reservoirs can broadly describe the isolated\nbehavior of the di\u000berent energy reservoirs, examining\nthe energy transfer within and between reservoirs us-\ning the analysis described in Section II C can provide\ndeeper insights into the physical phenomena, including\ndemonstrating the mechanisms that are responsible for\nthe transfer of energy. The shell-to-shell energy trans-10\nfer \ruxes examined in this section demonstrate the \rux\nfrom wavenumber Qto wavenumber Kwithin and be-\ntween energy reservoirs via di\u000berent pathways.\nFigure 6 shows the energy transfer within the kinetic\n(left) and magnetic (right) energy reservoirs via advec-\ntion and compression in the Ms0.2 Ma1simulation at t=\n0:77T(top) and at t= 5:16T(bottom). This plot encap-\nsulates the energy transfer of a turbulent cascade. Near\nthe beginning of the simulation in the top panels, most\nof the energy is in large scale modes, with energy from\nlargerQwavenumbers moving to smaller Kwavenum-\nbers. Note that the energy transfer is constrained to the\ndiagonal because the bulk of the energy transfer is local,\noccurring between comparable scales of QtoK. White\nspace \flls the o\u000b-diagonals because very little nonlocal\nenergy transfer occurs internally within reservoirs. The\nenergy transfer shown in this \fgure is solely within the ki-\nnetic and magnetic reservoirs { there is no energy transfer\nshown between these reservoirs (although it is occurring,\nas will be discussed in the next paragraph). In the simu-\nlation shown here, the magnetic energy transfer is larger\nin magnitude than the kinetic energy transfer. In all sim-\nulations, the magnetic energy transfer extends to higher\nwavenumbers more rapidly than the kinetic energy. Af-\nter the \row has decayed into turbulence (as shown in the\nbottom panels), energy transfer to smaller local scales\nhappens across the resolved modes down to numerical\ndissipation scales. At large wavenumbers ( Q > 16), the\nenergy transfers are scale-local and of comparable mag-\nnitude. This phenomenon continues to at least Q'200\nin both the kinetic and magnetic energy transfer { i.e.,\nto much larger wavenumbers than an inertial range is ob-\nserved (see, e.g., Figure 3). Thus, the e\u000bective (numeri-\ncal) viscosity and resistivity are not a\u000becting the turbu-\nlent cascade encoded by these transfers to a signi\fcant\ndegree.\nFigure 7 shows the energy transfer within the kinetic\n(top) and magnetic (bottom) energy reservoirs in the\nMs0.2 Ma1 simulation at t= 1:29T(just before the\nmagnetic energy peaks). Energy transfer within the ki-\nnetic and magnetic reservoirs brie\ry reverses directions\nand moves energy from smaller local scales to larger lo-\ncal scales (note the purple color indicating energy loss\nabove the diagonal and orange color below the diagonal,\nwhich is in contrast to Fig. 6). This constitutes a tran-\nsient inverse cascade. Additionally, the inverse cascade\nis present throughout most scales of the magnetic energy\n(K;Q.100) but only apparent at large scales in the\nkinetic energy ( K;Q.16). As seen in Figure 4, at this\nearly time the turbulent \row is just beginning to saturate\nthe smallest scales while the large scale energy oscillates,\nso the energy transfer inversion lasts less than a dynami-\ncal time (see Section III B 2 for further exploration of the\nduration).\nFigure 8 shows the energy transfer between the kinetic\nto magnetic energy reservoirs due to magnetic tension\natt= 1:81Tin the Ms0.2 Ma10 simulation. This Fig-\nure displays nonlocal transfer from kinetic to magneticenergy. Unlike the advection- and compression-driven\nmodes within the magnetic and kinetic energy reservoirs,\nenergy transfers from kinetic to magnetic reservoirs via\ntension can support nonlocal energy transfers. The non-\nlocal transfer happens from large kinetic scales to much\nsmaller magnetic scales, spanning more than an order\nof magnitude downward in spatial scale from the largest\nkinetic modes. The nonlocal energy transfer between ki-\nnetic and magnetic energy was signi\fcant in simulations\nwith lower initial magnetic energy, and especially in the\nMa10 simulations where the magnetic \feld is dynamically\nunimportant at early times. Kinetic energy moves signif-\nicant energy to all magnetic scales from early times at\nt'1:5Tto intermediate times at t'4Tin these simu-\nlations, although some energy continues to \row via this\nmechanism at later times. Additionally, since the transfer\nof energy via tension is between two di\u000berent reservoirs,\nthe energy transfer can transfer at equivalent scales from\none reservoir to the other. This is shown as non-zero\ntransfer along the diagonal of the plot.\n1. Nonlocal Energy Transfer\nLike in some driven turbulence simulations [20, 26],\nthese decaying turbulence simulations also demonstrate\nsigni\fcant nonlocal energy transfer between kinetic and\nmagnetic energy reservoirs. Unlike in driven simulations,\nthe energy transfers in this work are solely due to the \ruid\n\row and not due to externally-applied driving forces.\nFigure 9 shows the total local, nonlocal, and equivalent-\nscale energy transfers via magnetic tension in the Ms0.2\nsimulations over time. We obtain these quantities by\nintegrating the transfer functions over di\u000berent sets of\nscales:\nNonlocal lowerX\nQX\nK2[1;2\u0000`Q)TXY(Q;K )\nLocal-LowerX\nQX\nK2[2\u0000`Q;Q)TXY(Q;K )\nEquivalentX\nQX\nK=QTXY(Q;K )\nLocal-HigherX\nQX\nK2(2`Q;Q]TXY(Q;K )\nNonlocal HigherX\nQX\nK2(2`Q;1]TXY(Q;K )\nwhere`is a parameter for di\u000berentiating local versus\nnonlocal separation of wavenumbers in log space. In Fig-\nure 9, we show the analysis using `= 5=4 with a solid\nline, which corresponds to 5 logarithmic bins above or\nbelowQ(see II C for the description of the binning),\nand show the extent of the \ruxes if `= 5=4\u00061=4 is\nused in shaded regions. As seen in this \fgure from the\nred line, the nonlocal energy transfer from large scale ki-\nnetic modes to small scale magnetic modes (\\downscale\"11\n101102\n= 3.39 × 106\nUU\nMs0.2_Ma1\nt= 0.77T\n= 6.94 × 106\nBB\nMs0.2_Ma1\nt= 0.77T\n101102101102\n= 3.60 × 105\nUU\nMs0.2_Ma1\nt= 5.16T\n101102\n= 8.01 × 105\nBB\nMs0.2_Ma1\nt= 5.16T\n100\n101\n102\n0102\n101\n100\nShell-to-Shell Transfer (Q,K) [units of ]\nWavenumberQWavenumberKMagnetic cascade\ndevelops fasterLocal Transfer\nInitial transfer from \nlarge scales\nLarge scale transfers\nhave weakenedBoth reserviors fill \nthe cascade\nNegligible Nonlocal\n transferTransfers in the inertial\n range are approx. constantEntirely local transfersNonlocal TransferNonlocal Transfer\nFIG. 6. Shell-to-shell energy transfer plots for the energy transfer within the kinetic (left) and magnetic (right) energy\nreservoirs via advection and compression at t= 0:77T(top) andt= 5:16T(bottom) from the simulations with Ms0.2 Ma1,\nshowing the development of the kinetic and magnetic turbulent cascades. Annotations on the \fgure highlight key features of\nthe energy transfer that are characteristic of a developing turbulence cascade. Each bin shows the \rux of energy from shell Q\nto shellK, where orange with white circles showing a positive \rux of energy, so that Kis gaining energy, and purple with white\nx's showing a negative \rux, so that Kis losing energy. The energy \rux in each bin is normalized by \"= maxQ;KjTXY(Q;K )j\nso that a higher \"means a higher energy \rux. The solid black line shows equivalent scale transfers. As the turbulent cascade\ndevelops in the magnetic and kinetic energy reservoirs, more energy transfers along the diagonal \fll out the energy spectrum\ndown to numerical dissipation scales.\ntransfer) is present in all simulations but is only domi-\nnant when the initial kinetic energy exceeds the initial\nmagnetic energy { this nonlocal energy transfer is more\nsigni\fcant in the Ma3.2 andMa10 simulations. Nonlocal\nenergy transfer downscale (red line) peaks depending on\nthe initial magnetic \feld and in all cases before the to-\ntal magnetic energy peaks. The nonlocal transfer helps\fll out the magnetic energy spectrum faster than the\nkinetic energy spectrum, especially in the Ma10 simula-\ntions, which is consistent with the spectral index shown in\nFigure 3 and the turbulent cascades shown in the shell-\nto-shell energy transfer in Figure 6. By the time the\nmagnetic energy has exceeded the kinetic energy in the\nMa3.2 and Ma10 simulations, nonlocal energy transfer is12\n101102\n= 1.15 × 106\nUU\nMs0.2_Ma1\nt= 1.29T\n101102101102\n= 3.83 × 106\nBB\nMs0.2_Ma1\nt= 1.29T\n100\n101\n102\n0102\n101\n100\nShell-to-Shell Transfer (Q,K) [units of ]\nWavenumberQWavenumberK\nInverse cascade throughoutLarge scale\ninverse cascade \nFIG. 7. Shell-to-shell energy transfer plots for the energy\ntransfer within the kinetic (top) and magnetic (bottom) en-\nergy reservoirs via advection and compression at t= 1:29T\nfrom the Ms0.2 Ma1simulation, showing a transient inverse\ncascade within the magnetic energy reservoir (on all scales\nK;Q.100) and kinetic energy reservoir (on large scales\nK;Q.16). Annotations show where along the diagonal the\ninverse cascade is present.\nlargely diminished due to the lack of kinetic energy to\nfeed the transfer.\nLocal energy transfer downscale (orange line) depends\nmore strongly on the initial magnetic \feld, with local\ntransfer to smaller scales reaching double the nonlocal\ntransfer in the Ma1simulation and being less than half\nin other cases. Local energy transfer upscale (blue line)\nis positive for some early times in the Ma1and Ma3.2\nsimulations.\nThe Ma1simulations also display two di\u000berent oscilla-\ntory behaviors, with a low frequency oscillation in the\nlocal energy transfer and a high frequency oscillation\nclearly visible in the equivalent energy transfer but also\npresent in local and nonlocal down scale transfer.\n101102101102\n= 1.09 × 105\nUBT\nMs0.2_Ma10\nt= 1.29T\n100\n101\n102\n0102\n101\n100\nShell-to-Shell Transfer (Q,K) [units of ]\nWavenumberQWavenumberKNonlocal\ntransferFIG. 8. Shell-to-shell energy transfer plots for the energy\ntransfer from kinetic to magnetic energy via magnetic ten-\nsion att= 1:81Tfrom the Ms0.2 Ma10 simulation, showing\nthe nonlocal energy transfer from large kinetic scales to many\nsmaller magnetic scales. Annotations show where the nonlo-\ncal transfer is present.\n2. Inverted Turbulent Cascades\nAt early times during the evolution of the Ma1sim-\nulations, a temporary inverse cascade forms within the\nkinetic and magnetic energy reservoirs where small scale\nenergy transfers to larger spatial scales. Figure 10 shows\nthe local and nonlocal energy transfers within the kinetic\nand magnetic energies to both smaller and larger length\nscales. In the Ma1simulations, the local energy transfer\nfrom larger to smaller length scales temporarily reverses\ninto an inverse cascade in both the kinetic and magnetic\nenergy reservoirs shortly after peak magnetic energy is\nreached. The inversion appears with all three sonic Mach\nnumbers simulated, with the longest inversion appear-\ning in the Ms0.1 Ma1simulation for'1Tand shortest\nin the high Ms0.4 Ma1simulation for'0:5T. For the\nMs0.1 Ma1simulation, the kinetic energy reservoir brie\ry\nreverses to the normal con\fguration, moving energy from\nlarge scales to scales while the magnetic energy is in an in-\nverted cascade, before returning to the inverted cascade,\nlingering longer than the magnetic \feld in the inverted\nstate and \fnally transitioning into a turbulent cascade\nfor the rest of the simulation. As seen in Figure 7, the\nmovement of energy to larger scales is not limited to any\nregion of the spectra { it is present at all length scales.\nThe Ma1simulations, which are the only simulations to\nexhibit an inverse cascade, are also the only ones in which\nthe total kinetic energy increases during any period. Af-\nter peak magnetic energy in the Ma1, the magnetic energy\nincreases while the kinetic energy increases for '1T; the\ninverse cascade appears during this same period.13\n0 2 4 60.75\n0.50\n0.25\n0.000.250.500.751.00\n=6.32×106\nMs0.1_Ma1EU=EB\nmax EB\n0 2 4 6\n=2.37×106\nMs0.1_Ma3.2EU=EB\nmax EB\n0 2 4 6\n=1.89×106\nMs0.1_Ma10EU=EBmax EB\nTime t [units of T]UBT Integrated Energy Flux\nNonlocal lower k\nLocal lower k\nEquivalent kLocal higher k\nNonlocal higher k\nFIG. 9. Integrated energy \rux over time from kinetic to magnetic energy via tension from larger wavenumbers to smaller\nnonlocal wavenumbers (purple), from larger wavenumbers to smaller local wavenumbers (blue), between equivalent wavenumbers\n(green), from smaller wavenumbers to larger local wavenumbers (orange), and from smaller wavenumbers to larger nonlocal\nwavenumbers (red) in the Ms0.2 simulations. We normalize the energy \rux in each panel so that the absolute maximum of all\nof the \rux bins is 1 :0, where\"is the normalization factor use in each panel. Comparisons of the relative strength of energy\n\ruxes in di\u000berent simulations must consider \". The inset plot in the lower right panel shows the color coded regions that\nare integrated to calculate each line at a single time for the same shell-to-shell transfer from Figure 8. Solid lines show the\nintegrated \rux if \\local\" wavenumbers as de\fned as 5 logarithmic bins away from the equivalent wavenumber. The shaded\nregions show the integrated \rux if 4 or 6 bins are used, showing that the behavior is robust if the range \\local\" wavenumbers is\nde\fned closer or further away from transfer between equivalent scales. We include the integrated \rux from kinetic to magnetic\nenergy via tension for all nine simulations in the online supplements\n3. Cross-Scale Flux\nWith additional analysis of the shell-to-shell transfer,\nwe can extract more insight into the movement of energy.\nWe can measure the cross-scale \rux of energy from scales\nbelow a wavenumber kto scales above a wave number k\nby integrating the transfer function\n\u0005X<\nY>(k) =X\nQ\u0014kX\nK\u0015kTXY(Q;K ) (9)\nFigure 11 shows the cross-scale \ruxes via di\u000berent trans-\nfer mechanisms for the simulations with Ms0.2 . The top\nrow shows cross-scale \ruxes early in the simulation at\nt= 0:77T, when the large scale \row is still decaying\ninto smaller scales. The magnetic cross-scale \rux at low\nwavenumbers predictably depends on the initial magnetic\nenergy, while the kinetic energy cross-scale \rux is largely\nthe same between simulations at a given sonic Mach num-\nber. For example, for Ma10 the cross-scale \rux is strongly\ndominated by \u0005U<\nU>, whereas for Ma3.2 it is still the most\nsigni\fcant contribution to the cross-scale \rux, but sub-\nstantial contributions are also seen from \u0005U<\nB>('60%\nof \u0005U<\nU>(4)), \u0005B<\nB>('30%), and \u0005B<\nU>('20%). For the\nstrongest initial magnetization ( Ma1) the early cross-scale\n\rux is dominated by magnetic tension-mediated transfersfrom the kinetic-to-magnetic budget (\u0005U<\nB>) on all scales\nhaving a non-zero cross-scale \rux ( k.64), with a simi-\nlar contribution by the magnetic cascade on intermediate\nscales (9 .k.64). The kinetic cascade is suppressed\non all scales, generally contributing less than 10% to the\ntotal cross-scale \rux.\nAt later times ( t= 5:16T, bottom row of Fig. 11),\nmagnetic energy dominates both the energy budget and\ncross-scale energy \rux. Cross-scale energy \rux via ki-\nnetic interactions is near zero across the inertial range of\nthe spectrum, and thus does not signi\fcantly contribute\nto the total cross-scale energy \rux. Only the magnetic\n\felds facilitate down scale cross-scale \rux at intermediate\nscales, both within the magnetic energy and from kinetic\nto magnetic energy. Moreover, the relative contributions\nof the individual transfer \u0005U<\nB>, \u0005B<\nB>, \u0005B<\nU>, and \u0005U<\nU>(in\norder of decreasing contribution) on intermediate scales\n(16.k.64) is the same independent of initial magneti-\nzation. This continuous cross-scale \rux is consistent with\nthe evolving spectral index discussed in Section III A 3.\nCross-scale \rux through large physical scales is irregu-\nlar, variable, and sometimes negative due to the lack of\nstructure and driving forces at large scales.14\n1.0\n0.5\n0.00.51.0\n=3.34×107\nMs0.2_Ma1UU\nmax EB\n0 1 2 3 4 5 61.0\n0.5\n0.00.51.0\n=1.28×108\nMs0.2_Ma1BB\nmax EB\nTime t [units of T]Integrated Energy FluxNonlocal lower k\nLocal lower k\nEquivalent kLocal higher k\nNonlocal higher k\nFIG. 10. Integrated energy \rux over time within the ki-\nnetic energy (top) and within the magnetic energy (bottom)\nfrom larger wavenumbers to smaller nonlocal wavenumbers\n(purple), from larger wavenumbers to smaller local wavenum-\nbers (blue), between equivalent wavenumbers (green), from\nsmaller wavenumbers to larger local wavenumbers (orange),\nand from smaller wavenumbers to larger nonlocal wavenum-\nbers (red) in the Ms0.2 Ma1simulation. The inset plot in the\nlower middle panel demonstrates the color coded regions that\nare integrated to calculate each line at t= 1:29Tfrom the\nshell-to-shell transfer from Figure 7. Solid lines show the in-\ntegrated \rux if \"local\" wavenumbers as de\fned as 5 logarith-\nmic bins away from the equivalent wavenumber. The results\nchange very little if 4 or 6 bins are used. We include the in-\ntegrated \rux within the kinetic energy and magnetic energy\nfor all nine simulations in the online supplements.\nIV. DISCUSSION\nA. Comparison to driven turbulence simulations\nThe Taylor-Green vortex provides an interesting study\nof a freely evolving transition to decaying turbulence. In\nother words, no external force is applied to the simula-\ntion as is the case in driven turbulence simulations. This\nexternal force may introduce unintended dynamics to the\n\row [7]. For example, in a simulation that is mechani-\ncally driven at large scales, energy may still be injected\non intermediate scales both in the incompressible regime[28] as well as in the compressible regime due to density\ncoupling [20]. Moreover, mechanical driving generally re-\nsults in an excess of energy on the excited, kinetic scales\nthat presents a barrier for magnetic \feld ampli\fcation\non those scales in cases without a dynamically relevant\nmean magnetic \feld. This barrier is often expressed in\nthe lack of a clear power law regime in the magnetic spec-\ntrum and resembles an inverse parabolic shape. At the\nsame time, the magnetic energy spectrum drops below\nthe kinetic one on the driving scales (see, e.g., Figure 1\nin [29] and references therein). In the simulations pre-\nsented here no such barrier is observed. Both kinetic and\nmagnetic energy spectra exhibit a (limited) regime where\npower law scaling is observed once a state of developed\nturbulence is reached.\nAnother important question raised from driven turbu-\nlence simulations pertains the locality of energy transfers.\nWhile there is agreement that TUUandTBBmediated\ntransfers, i.e., within a budget, are highly local, the en-\nergy transfers between budgets (here, TUBT) have been\nobserved to be weakly local and/or contain a nonlocal\ncomponent from the driven scales [19, 20, 26]. Here, we\nshow that in the absence of the driving force the energy\ntransfer mediated by magnetic tension contains both a\nlocal component as well as nonlocal component. The lat-\nter directly transfers large-scale kinetic energy to large\nand intermediate scales in the magnetic energy budget.\nThus, the nonlocal component is not an artifact of an\nexternal driving force.\nFinally, we recently showed that the kinetic energy\nspectra in driven turbulence simulations follow a scaling\nclose tok\u00004=3, i.e., shallower than Kolmogorov scaling,\nand explained this by the suppression of the kinetic en-\nergy cascade due to magnetic tension [29]. This is in\nagreement with our \fndings in the work presented here,\nwhere the same dynamics are observed at late times when\nturbulence is fully developed.\nNaturally, this does not demonstrate that the same\nphysical mechanisms are causing the similar slopes. Nev-\nertheless, the late time evolution of the simulations pre-\nsented here is still comparable to a limited degree to\ndriven simulation of stationary turbulence. For exam-\nple, even at late times (see, e.g., t= 5:16Tin Fig. 6),\nenergy is still cascading down from the largest scales\n(k.8) but the cascade is weaker than its initial magni-\ntude. The reduction in strength of the cascade on large\nscale is directly linked to the decay of the large initial\nvortices. Nevertheless, even at late times the overall en-\nergy balance is still dominated by the largest scales, cf.,\nthe spectra shown in Fig. 3 when taking into account\nthek4=3compensation used in the plot. Overall, while\nhere the inertial range shrinks and becomes weaker (to\na limited degree) over time as the large scale modes lose\nenergy, the dynamics within the inertial range is similar\nto driven turbulence simulations.15\n0.0000.0050.0100.0150.0200.0250.0300.035\nMs0.2_Ma1 \n t=0.77T\nMs0.2_Ma3.2 \n t=0.77T\nMs0.2_Ma10 \n t=0.77T\nUU\nBB\nUBT\nBUT\n1011020.001\n0.0000.0010.0020.003\nMs0.2_Ma1 \n t=5.16T\n101102\nMs0.2_Ma3.2 \n t=5.16T\n101102\nMs0.2_Ma10 \n t=5.16T\nWavenumber [units of 1/L]Cross-scale Flux\nFIG. 11. Cross-scale \rux within the kinetic energy (blue line), within the magnetic energy (orange line), and from kinetic to\nmagnetic energy via tension (green line) in the three Ms0.2 simulations across columns and at dynamical time t= 0:77T(top)\nand later at dynamical time t= 5:16T. Note that the cross-scale \ruxes at later times are an order of magnitude less than early\ncross-scale \ruxes. Positive values of this quantity denote energy transfer from larger to smaller scales.\nB. Comparison to previous results\nIn general, our results in the weakly compressible\nMHD regime are in agreement with the \u000b'\u00002 spec-\ntrum reported by previous works on the TG vortex in\n[12, 13, 15, 16] in the imcompressible MHD regime us-\ning the insulating magnetic \feld con\fguration. We see\nthe same\u000b'\u00002 spectrum early in the evolution be-\nforet= 2T, which corresponds to the time period near\nmaximum energy dissipation that these other studies fo-\ncused on. In all cases that we simulated the spectra be-\ncame shallower at later times, independent of the ini-\ntial magnetization (whereas these other works focused\nonEU=EB= 1, i.e.,MA;0= 1, con\fgurations, which\nare in good agreement with the Ms0.2 Ma1.0 simulation\npresented here, see top left panel of Fig. 2). As noted by\n[15], the\u000b'\u00002 spectrum is likely due to discontinuities\nin a small volume of the \row that can be disrupted by\nsymmetry breaking at either large or small scales [17].\nAccording to [17], a simulated Taylor-Green vortex with\nsu\u000eciently high Reynolds number should show symmetry\nbreaking at the small scales at late times in the evolu-\ntion, causing a break from the \u00002 power law at large\nwavenumbers. Since our simulations do not impose sym-\nmetries on the \row, this is a possible explanation for theobserved behavior. However, we see an \u000b'\u00004=3 inertial\nrange scaling at late times, instead of the \u000b'\u00002 and\n\u000b'\u00005=3 broken power law theorized by [17].\nFinally, work done in [12, 14, 16] shows that the be-\nhavior of the magnetic \feld and spectra changes with the\ninitial magnetic \feld con\fgurations. With the insulating\ninitial magnetic \felds that we use, the vorticity begins\nparallel to the magnetic \feld. This facilitates the early\nenergy \rux from kinetic to magnetic energy. The insulat-\ning case tends towards stronger large magnetic \felds com-\npared to the other magnetic \feld con\fgurations. Both of\nthe other initial magnetic \felds result in di\u000berent energy\nspectra, with the conducting magnetic \feld setup lead-\ning to ak\u00003=2spectra and the alternative insulating \feld\nsetup leading to spectra interpreted as either a k\u00005=3or\nk\u00002spectra as argued by [12] and [16] respectively.\nC. Implication of results\nIn all of our simulations, we see magnetic \felds and\ne\u000bects facilitated by the magnetic \felds dominating the\nevolution of the decaying turbulence, even when the ini-\ntial kinetic energy exceeds the magnetic energy by a fac-\ntor of 100 in the Ma10 simulations. Energy transfer from16\nkinetic to magnetic energy via tension and energy trans-\nfer within the magnetic energy far exceed energy \rux\nvia the kinetic turbulent cascade at later times. En-\nergy transfer from kinetic to magnetic energy at earlier\ntimes leads to the magnetic energy dominating over ki-\nnetic energy in all cases in both total magnitude as well\nas in terms of the scale-wise budget, cf., magnetic ver-\nsus kinetic energy spectra. This is similar to what has\nbeen found in incompressible [26] and compressible sim-\nulations [20, 29] of driven turbulence. Thus, even in\nintermittently-driven systems one can expect the mag-\nnetic \feld to signi\fcantly in\ruence the dynamics after a\nfew dynamical times.\nOur simulations exhibit a magnetic energy spectra\nwith a measurable power law after the turbulent \row is\nrealized. The inertial range is short, from approximately\nk= 10 tok= 32, due to the resolution of these simula-\ntions. Nevertheless, within this region we can reasonably\n\ft a power law to both the kinetic and magnetic spectra,\nwhich is often not possible in driven turbulence simula-\ntion without a dynamically relevant mean magnetic \feld,\ncf., Sec. IV A. Thus, freely evolving and driven turbulence\nsimulations complement each other and both are required\nto disentangle environmental from intrinsic e\u000bects.\nFrom an observational point of view, we demonstrated\nthat the spectral indices evolve over time and \ructuate\neven for similar parameters. Therefore, the derived spec-\ntral indices from observation (e.g., velocity maps in as-\ntrophysics), which represent individual snapshots in time,\nneed to be interpreted with care when trying to infer the\n\\nature\" of turbulence (e.g., Kolmogorov or Burgers) in\nthe object of interest.\nFinally, the observed nonlocal energy transfer has im-\nplications on the dynamical development of small scale\nstructures from intermittent or singular energy injection\nevents. Within the context of natural astrophysical and\nterrestrial plasmas, the nonlocal energy transfer from ki-\nnetic to magnetic energies suggests that small magnetic\n\feld structures develop before small scale kinetic struc-\ntures.\nD. Limitations\nWhile our analysis showed that the results are gener-\nally robust (e.g., with respect to varying the \ftting range\nin the spectral indices or varying range in the de\fnition\nof scale-local in the energy transfers), higher resolution\nsimulations are desirable. With higher resolution in an\nimplicit large eddy simulation (ILES) the dynamic range\nis increased and, thus, the e\u000bective Reynolds numbers of\nthe simulated plasma are raised.\nSimilarly, due to the nature of ILES the e\u000bective mag-\nnetic Prandtl number in all simulations is Pm '1. How-\never, in natural systems (both astrophysical and terres-\ntrial/experimental) Pm is either \u001d1 or\u001c1, motivating\nthe exploration of these regimes in the future as well.\nAll of our simulations started with subsonic initialconditions, leaving the supersonic regime unexplored.\nThe additional shocks, discontinuities, and strong den-\nsity variations that may arise in a supersonic \row could\nalter the energy transfer as the \row transitions into tur-\nbulence. In the simulations we present here, the Mach\nnumber generally did not signi\fcantly a\u000bect the growth\nand behavior of the turbulence. In a supersonic \row,\nhowever, the transitory e\u000bects such as the nonlocal en-\nergy transfer and inverse cascade may be altered or sup-\npressed in addition to generally richer dynamics related\nto compressive e\u000bects and e\u000bective space-\flling of turbu-\nlent structures [30].\nFigure 5 indicates that the spectral index of both the\nkinetic and magnetic energy cascades evolves as a func-\ntion of magnetic \feld strength (i.e., initial MA.) It is\nunclear whether there is a threshold of MAabove which\nthe spectra become shallower, or whether there is a con-\ntinuum of behavior as the initial MAis increased. While\nwe would like to engage in a more thorough exploration\nof the dependence of these behaviors on MA, the simu-\nlations in question are computationally expensive and it\nis infeasible to do so at present. Exploration of this tran-\nsition is a promising venue for future work. Finally, the\nshell decomposition used here to study energy transfer\nhas been shown to violate the inviscid criterion for de-\ncomposing scales in the compressible regime [31]. How-\never, this only pertains to \rows with signi\fcant density\nvariations and, thus, is e\u000bectively irrelevant for the sub-\nsonic simulations presented here.\nV. CONCLUSIONS\nWe have presented in this work nine simulations of the\nTaylor-Green vortex using the insulating magnetic \feld\nsetup from [11] to study magnetized decaying turbulence\nin the compressible ideal MHD regime using the \fnite\nvolume code K-Athena . As a \frst for the Taylor-Green\nvortex, we have also presented an energy transfer analy-\nsis to show the movement of energy between scales and\nenergy reservoirs as facilitated via di\u000berent mechanisms.\nOur key results are as follows:\n•Magnetic \felds signi\fcantly a\u000bect the evolution of\nthe decaying turbulence, regardless of initial \feld\nstrength. Energy \rux from kinetic energy to mag-\nnetic energy leads to the magnetic energy dominat-\ning the energy budget, even in simulations where\nthe magnetic energy is initially very small.\n•The Taylor-Green vortex simulations explored here\ndisplay a power law in both the kinetic and mag-\nnetic energy spectra with a measurable spectral in-\ndex, which is in contrast with the lack of a power\nlaw in the magnetic energy spectrum seen in driven\nturbulence calculations without a signi\fcant mean\n\feld.\n•Decaying turbulent \rows do not exhibit a spectral\nindex that is constant in time in either the kinetic17\nnor magnetic energy reservoirs { these spectra con-\ntinually evolve over time. The spectral indices of\nthe kinetic and magnetic energies become compa-\nrable and roughly constant around 1 \u00002 dynamical\ntimes after the magnetic energy has become dom-\ninant. This can happen as early as t= 2Twhen\nthe initial magnetic energy equals initial the kinetic\nenergy, and as late as t= 5Twhen initial kinetic\nenergy exceeds the magnetic by a factor of 100. For\nsimulations with more initial kinetic energy than\nmagnetic energy, the spectral indices reach a rough\nconstant slightly steeper than \u000b'\u00004=3.\n•Before the turbulent \row fully develops, an inverse\ncascade within the kinetic and magnetic energy\nreservoirs is intermittently observed. This inter-\nmittent behavior moves energy from smaller scales\nto larger scales, and is possible when the magnetic\nenergy is comparable to the kinetic energy.\n•Analysis of energy transfer within and between\nreservoirs indicates that within fully-developed tur-\nbulence, the cross-scale \rux of energy in both the\nkinetic and magnetic cascades are dominated by\nenergy transfer mediated by the magnetic \feld.\n•Magnetic tension facilitates nonlocal transfer from\nlarger scales in the kinetic energy to smaller scales\nin the magnetic energy, and is particularly promi-\nnent in simulations where the magnetic \feld is ini-\ntially weak.\nVI. ACKNOWLEDGEMENTS\nThis research is part of the Blue Waters sustained-\npetascale computing project, which is supported by theNational Science Foundation (awards OCI-0725070 and\nACI-1238993) the State of Illinois, and as of Decem-\nber, 2019, the National Geospatial-Intelligence Agency.\nBlue Waters is a joint e\u000bort of the University of Illi-\nnois at Urbana-Champaign and its National Center for\nSupercomputing Applications. Exploratory simulations\nused the Blue Waters Supercomputer [32, 33]. The au-\nthors acknowledge the Texas Advanced Computing Cen-\nter (TACC) at The University of Texas at Austin for\nproviding HPC resources that have contributed to the\nresearch results reported within this paper. The sim-\nulations presented here used the Stampede 2 Super-\ncomputer. FWG acknowledges support from the 2019\nBlue Waters Graduate Fellowship. BWO acknowledges\nsupport from NSF grants no. AST-1517908 and AST-\n1908109, and NASA ATP grant 80NSSC18K1105. PG\nand BWO acknowledge funding by NASA ATP grant\nNNX15AP39G. This work used the Extreme Science and\nEngineering Discovery Environment (XSEDE), which is\nsupported by National Science Foundation grant num-\nber ACI-1548562, through allocation AST190021. It also\nused the resources of the Michigan State University High\nPerformance Computing Center, which is operated by the\nMSU Institute for Cyber-Enabled Research. K-Athena\n[21],Athena++ [22],Kokkos [23], and yt[34] are de-\nveloped by a large number of independent researchers\nfrom numerous institutions around the world. Their com-\nmitment to open science has helped make this work pos-\nsible.\n[1] M. L. Norman and G. L. Bryan, Cluster turbulence, in\nThe Radio Galaxy Messier 87 , Lecture Notes in Physics,\nedited by H.-J. R oser and K. Meisenheimer (Springer,\nBerlin, Heidelberg, 1999) pp. 106{115.\n[2] R. B. Larson, Turbulence and star formation in molec-\nular clouds, Monthly Notices of the Royal Astronomical\nSociety 194, 809 (1981).\n[3] S. Britzen, C. Fendt, A. Eckart, and V. Karas, A new\nview on the M 87 jet origin: Turbulent loading leading to\nlarge-scale episodic wiggling, Astronomy & Astrophysics\n601, A52 (2017).\n[4] M. J. Korpi, A. Brandenburg, A. Shukurov, I. Tuomi-\nnen, and A. Nordlund, A Supernova-regulated Interstel-\nlar Medium: Simulations of the Turbulent Multiphase\nMedium, The Astrophysical Journal 514, L99 (1999).\n[5] L. I. Rudakov and R. N. Sudan, MHD turbulence in radi-\nating intense Z-pinches, Physics Reports Turbulence and\nIntermittency in Plasmas, 283, 253 (1997).\n[6] E. Kroupp, E. Stambulchik, A. Starobinets, D. Osin, V. I.\nFisher, D. Alumot, Y. Maron, S. Davidovits, N. J. Fisch,and A. Fruchtman, Turbulent stagnation in a z-pinch\nplasma, Physical Review E 97, 013202 (2018).\n[7] P. Grete, B. W. O'Shea, and K. Beckwith, As a Mat-\nter of Force|Systematic Biases in Idealized Turbulence\nSimulations, The Astrophysical Journal 858, L19 (2018).\n[8] G. I. Taylor and A. E. Green, Mechanism of the Produc-\ntion of Small Eddies from Large Ones, Proceedings of the\nRoyal Society of London Series A 158, 499 (1937).\n[9] Z. J. Wang, K. Fidkowski, R. Abgrall, F. Bassi,\nD. Caraeni, A. Cary, H. Deconinck, R. Hartmann,\nK. Hillewaert, H. T. Huynh, et al. , High-order cfd meth-\nods: current status and perspective, International Jour-\nnal for Numerical Methods in Fluids 72, 811 (2013).\n[10] M. E. Brachet, D. I. Meiron, S. A. Orszag, B. G. Nickel,\nR. H. Morf, and U. Frisch, Small-scale structure of the\nTaylor{Green vortex, Journal of Fluid Mechanics 130,\n411 (1983).\n[11] E. Lee, M. E. Brachet, A. Pouquet, P. D. Mininni, and\nD. Rosenberg, Paradigmatic \row for small-scale magne-\ntohydrodynamics: Properties of the ideal case and the18\ncollision of current sheets, Physical Review E 78, 066401\n(2008).\n[12] E. Lee, M. E. Brachet, A. Pouquet, P. D. Mininni, and\nD. Rosenberg, On the lack of universality in decaying\nmagnetohydrodynamic turbulence, Physical Review E\n81, 016318 (2010), arXiv:0906.2506.\n[13] A. Pouquet, E. Lee, M. E. Brachet, P. D. Mininni, and\nD. Rosenberg, The dynamics of unforced turbulence at\nhigh Reynolds number for Taylor-Green vortices general-\nized to MHD, Geophysical and Astrophysical Fluid Dy-\nnamics 104, 115 (2010).\n[14] M. E. Brachet, M. D. Bustamante, G. Krstulovic, P. D.\nMininni, A. Pouquet, and D. Rosenberg, Ideal evolu-\ntion of magnetohydrodynamic turbulence when imposing\nTaylor-Green symmetries, Physical Review E 87, 013110\n(2013).\n[15] V. Dallas and A. Alexakis, Origins of the k\u00002spectrum in\ndecaying Taylor-Green magnetohydrodynamic turbulent\n\rows, Physical Review E 88, 053014 (2013).\n[16] V. Dallas and A. Alexakis, Structures and dynam-\nics of small scales in decaying magnetohydrodynamic\nturbulence, Physics of Fluids 25, 105106 (2013),\narXiv:1304.0695.\n[17] V. Dallas and A. Alexakis, Symmetry breaking of decay-\ning magnetohydrodynamic Taylor-Green \rows and con-\nsequences for universality, Physical Review E 88, 063017\n(2013).\n[18] G. Vahala, B. Keating, M. Soe, J. Yepez, L. Vahala,\nJ. Carter, and S. Ziegeler, MHD Turbulence Studies us-\ning Lattice Boltzmann Algorithms, Commun. Comput.\nPhys. , 23 (2008).\n[19] Y. Yang, Y. Shi, M. Wan, W. H. Matthaeus, and S. Chen,\nEnergy cascade and its locality in compressible magne-\ntohydrodynamic turbulence, Phys. Rev. E 93, 061102\n(2016).\n[20] P. Grete, B. W. O'Shea, K. Beckwith, W. Schmidt, and\nA. Christlieb, Energy transfer in compressible magneto-\nhydrodynamic turbulence, Physics of Plasmas 24, 092311\n(2017).\n[21] P. Grete, F. W. Glines, and B. W. O'Shea, K-Athena:\nA Performance Portable Structured Grid Finite Vol-\nume Magnetohydrodynamics Code, IEEE Transactions\non Parallel and Distributed Systems 32, 85 (2021),\narXiv:1905.04341.\n[22] J. M. Stone, K. Tomida, C. J. White, and K. G. Felker,\nThe Athena++ Adaptive Mesh Re\fnement Framework:Design and Magnetohydrodynamic Solvers, The Astro-\nphysical Journal Supplement Series 249, 4 (2020).\n[23] H. Carter Edwards, C. R. Trott, and D. Sunderland,\nKokkos: Enabling manycore performance portability\nthrough polymorphic memory access patterns, Jour-\nnal of Parallel and Distributed Computing Domain-\nSpeci\fc Languages and High-Level Frameworks for High-\nPerformance Computing, 74, 3202 (2014).\n[24] J. M. Stone and T. Gardiner, A simple unsplit godunov\nmethod for multidimensional mhd, New Astronomy 14,\n139 (2009).\n[25] Note that other works such as Wang et al. [9], Pouquet\net al. [13] use a nondimensional time, t\u0003=L=u 0, in con-\ntrast to the dynamical time used here.\n[26] A. Alexakis, P. D. Mininni, and A. Pouquet, Shell-to-shell\nenergy transfer in magnetohydrodynamics. i. steady state\nturbulence, Phys. Rev. E 72, 046301 (2005).\n[27] S. Kida and S. A. Orszag, Energy and spectral dynamics\nin forced compressible turbulence, Journal of Scienti\fc\nComputing 5, 85 (1990).\n[28] J. A. Domaradzki, B. Teaca, and D. Carati, Local-\nity properties of the energy \rux in magnetohydrody-\nnamic turbulence, Physics of Fluids 22, 051702 (2010),\nhttp://dx.doi.org/10.1063/1.3431227.\n[29] P. Grete, B. W. O'Shea, and K. Beckwith, As a matter\nof tension: Kinetic energy spectra in MHD turbulence,\nThe Astrophysical Journal 909, 148 (2021).\n[30] C. Federrath, On the universality of supersonic turbu-\nlence, Mon. Not. R. Astron. Soc. 436, 1245 (2013).\n[31] D. Zhao and H. Aluie, Inviscid criterion for decomposing\nscales, Phys. Rev. Fluids 3, 054603 (2018).\n[32] B. Bode, M. Butler, T. Dunning, T. Hoe\rer, W. Kramer,\nW. Gropp, and W.-m. Hwu, The Blue Waters super-\nsystem for super-science, in Contemporary High Perfor-\nmance Computing , Chapman & Hall/CRC Computa-\ntional Science (Chapman and Hall/CRC, 2013) pp. 339{\n366.\n[33] W. Kramer, M. Butler, G. Bauer, K. Chadalavada,\nand C. Mendes, Blue Waters Parallel I/O Storage Sub-\nsystem, in High Performance Parallel I/O , edited by\nPrabhat and Q. Koziol (CRC Publications, Taylor and\nFrancis Group, 2015) pp. 17{32.\n[34] M. J. Turk, B. D. Smith, J. S. Oishi, S. Skory, S. W.\nSkillman, T. Abel, and M. L. Norman, Yt: A Multi-code\nAnalysis Toolkit for Astrophysical Simulation Data, The\nAstrophysical Journal Supplement Series 192, 9 (2011)."    },    {        "title": "1904.12892v1.Spontaneous_Transport_Barriers_Quench_Turbulent_Resistivity_in_2D_MHD.pdf",        "content": "Spontaneous Transport Barriers Quench Turbulent Resistivity in 2D MHD\nXiang Fan and P. H. Diamond\nUniversity of California at San Diego, La Jolla, California 92093\nL. Chac\u0013 on\nLos Alamos National Laboratory, Los Alamos, New Mexico 87545\n(Dated: May 1, 2019)\nThis Letter identi\fes the physical mechanism for the quench of turbulent resistivity in 2D MHD.\nWithout an imposed, ordered magnetic \feld, a multi-scale, blob-and-barrier structure of magnetic\npotential forms spontaneously. Magnetic energy is concentrated in thin, linear barriers, located at\nthe interstices between blobs. The barriers quench the transport and kinematic decay of magnetic\nenergy. The local transport bifurcation underlying barrier formation is linked to the inverse cascade\nofhA2iand negative resistivity, which induce local bistability. For small scale forcing, spontaneous\nlayering of the magnetic potential occurs, with barriers located at the interstices between layers.\nThis structure is e\u000bectively a magnetic staircase.\nINTRODUCTION\nThe evolution of mean quantities in turbulence is fre-\nquently modelled as a transport process , using ideas from\nthe kinetic theory of gases. A classic example is that\nof Prandtl's theory of turbulent boundary layers, which\n\frst proposed the use of an eddy viscosity - based upon\nmixing length theory - to calculate mean \row pro\fles at\nhigh Reynolds number. Magnetohydrodynamics (MHD)\npresents additional challenges, especially at high mag-\nnetic Reynolds number Rm. There, models based on\ntransport theory concepts are central to our understand-\ning of mean B (hBi) evolution in turbulent \rows. In-\ndeed, the well-known theory of mean \feld electrodynam-\nics (Mo\u000batt [1]) employs transport coe\u000ecients \u000b,\f- re-\nlated to turbulent helicity and energy, respectively - to\ndescribe the growth and transport of a mean magnetic\n\feld. Such models are heavily utilized in dynamo the-\nory - the study of how large scale \felds are formed. The\nturbulent or \\eddy\" resistivity ,\u0011T, is ubiquitous in these\nmodels (and corresponds to \fabove). While \u0011Tis of-\nten taken as kinematic ( \u0011T\u0018\u0011K\u0018P\nkh~v2ik\u001ccwhere\u001cc\nis the self-correlation time) for many applications, non-\nlinear dependence of \u0011Ton magnetic \feld and potential\nhas been observed in numerous simulations [2{21]. Such\nnonlinearity arises from the fact that the magnetic \felds\nalter the turbulent \rows which scatter them. As this\nnonlinearity tends to reduce \u0011Trelative to kinematic ex-\npectations, such trends are referred to as quenching. Rm\ndependent quenching { i.e. when the product Rm hBi2\nenters { is of particular interest, as it signals that for\nthe relevant case of high Rm, relatively weak \felds can\nproduce signi\fcant feedback on transport and evolution\nprocesses. Such Rm-dependent feedback has been asso-\nciated with Alfvenization (i.e. the conversion of hydro-\ndynamics turbulence to Alfven wave turbulence) and/or\nwith the balance of magnetic helicity hA\u0001Bi(i.e. in 3D)\norhA2i(i.e. in 2D). Both arguments ultimately point\ntomemory , due to the freezing-in law, as the origin ofthe quench. The quenching problem is also relevant to\nmodels of fast reconnection and impulsive energy release\nprocesses in MHD, as it constrains the size of (frequently\ninvoked) anomalous dissipation [22, 23]. More generally,\nit is an important paradigm of the transport of an active\nscalar.\nIn a seminal paper [2] which broached the quenching\nquestion, Cattaneo and Vainshtein (CV) presented nu-\nmerical simulations of 2D MHD turbulence which demon-\nstrated that even a weak large scale magnetic \feld is suf-\n\fcient to quench the turbulent transport of the active\nscalarA(the magnetic potential). Based on ideas from\nmean \feld theory, CV suggested { and presented simula-\ntions to support { the idea that \u0011Tis given by\n\u0011T\u0018hv2i1=2l\n1 +1\n\u00160\u001aRmhBi2=hv2i(1)\nThe mean \feldhBihere is estimated using:\njhBij\u0018p\nhA2i=L0 (2)\nwhereL0is system size. For1\n\u00160\u001aRmhBi2=hv2i<1,\u0011T\u0018\n\u0011K\u0018hv2i1=2l. While for1\n\u00160\u001aRmhBi2=hv2i>1,\u0011T\u001c\n\u0011K, so\u0011Tis quenched. It is important to note that, in\nview of Cowling's Theorem, suppression occurs only for\na time of limited duration , without external forcing of A.\nAfter the afore mentioned suppression stage, rapid decay\nof the magnetic \feld occurs, and \u0011Treverts to\u0011K. The\nevolutions of EB,EK(magnetic and kinetic energy) and\nhA2i(mean square potential) are shown in Fig. 1 (a, b).\nEquation (1) was also obtained analytically from sta-\ntistical theory, assuming the presence of an imposed weak\nlarge scale \feld B0(i.e.hBi=B0) [24{27]. (Note the as-\nsumptions thatjhBijis determined by root-mean-square\nAand the system size in CV.) The derivation made use\nofhA2ibalance to constrain the turbulent resistivity [28{\n30]. Rm-dependence of the quench stems from the fact\nthathA2iis conserved up to resistive di\u000busion. This early\nwork on resistivity quenching triggered a tidal wave ofarXiv:1904.12892v1  [physics.flu-dyn]  29 Apr 20192\nFIG. 1. Time evolution of (a) magnetic energy EBand ki-\nnetic energy EK; (b)hA2iin Run1. The suppression stage\nis marked in orange, and the kinematic decay stage in green.\nThe decay of EBis slow in the suppression stage, which is\nconsistent with previous studies. The decay of hA2iis also\nslow in the suppression stage, and is more smooth compared\ntoEB.\nsubsequent studies of nonlinear dynamo evolution and\nquenching.\nIn this Letter, we show that, without an imposed, or-\ndered magnetic \feld, Rm-dependent quenching is intrin-\nsically an intermittency phenomena, and can occur where\na global mean \feld hBisimply does not exist. Rather,\nturbulent resistivity quenching occurs due to intermit-\ntent transport barriers . A transport barrier is a localized\nregion of mixing and transport signi\fcantly lower than\nthe mean thereof, i.e. \u0011T;local<\u0016\u0011T. These barriers are\nextended, thin, linear features, into which strong hB2iis\nconcentrated. The barriers are formed by the hB2ifeed-\nback on scalar transport, speci\fcally by magnetic \rux\ncoalescence. Thus, transport quenching is manifestly not\na mean \feld e\u000bect, as the structure of the \feld is more\nakin to a random network than to a smooth mean \feld.\nThe barriers form in the interstices between blobs of hA2i,\nwhich are formed by the inverse cascade of hA2i. Over-\nall, the magnetic potential and \feld have a structure of\n\\blob-and-barrier\" at large Rm, as shown in Fig. 2. In\ncontrast to the assumptions of CV, the magnetic \feld ex-\nhibits twonon-trivial scales, i.e. the blob size Lbloband\nthe barrier width W, whereW\u001cLblob.Lblobcharac-\nterizes the magnetic potential while Wcharacterizes the\n\feld intensity.\nTheA\feld in the blob-and-barrier structure of 2D\nMHD resembles the concentration contrast \feld  in the\nCahn-Hilliard Navier-Stokes (CHNS) system [31{35].TABLE I. Initial conditions, kand Rm for the suppression\nstage. For all runs, A0= 1:0 andf0= 30.\nRuns Initial Condition \u0011 \u0017 1=(\u00160\u001a)kRm\nRun1 Bimodal 1 \u000310\u000041\u000310\u000040:04 5\u0018500\nRun2 Unimodal 1 \u000310\u000041\u000310\u000040:04 5\u0018500\nRun3 Bimodal 1 \u000310\u000042\u000310\u000030:01 32\u0018150\nANALYSIS: GLOBAL\nIn this Letter, the 2D MHD equations are solved using\ndirect numerical simulation [37, 38] with doubly periodic\nboundary condition:\n@tA+v\u0001rA=\u0011r2A (3)\n@t!+v\u0001r!=1\n\u00160\u001aB\u0001rr2A+\u0017r2!+f (4)\nHere!is vorticity, \u0011is resistivity, \u0017is viscosity, \u00160\u001ais\nmagnetic permeability and density, and fis an isotropic\nhomogeneous external forcing, with wave number kand\nmagnitude f0. The simulation box size is L2\n0= 1:0\u00021:0\nwith 1024\u00021024 resolution. The parameters used are\nsummarized in Table. I. The initial condition for the !\n\feld is!I= 0 everywhere; the initial condition for A\n\feld is a cosine function in Run1: AI(x;y) =A0cos 2\u0019x.\nThe setup of Run1 di\u000bers from that of Ref. [2] only in\nthe range of Rm studied.\nNon-trivial blob-and-barrier structure is observed in\nreal space at large Rm, and this structure forms quickly\nafter a short transition period. Fig. 2 (a1) shows a snap-\nshot of magnetic potential in the suppression stage for\nRun1. It consists of \\blobs\" (regions in red and blue) and\ninterstices (green), and is very di\u000berent from the initial\ncondition, for which a mean \feld is relevant. Fig. 2 (a2)\nshows theB2\feld for the same run. The high B2regions\n(bright color) occur at the interstices of the Ablobs, since\nB\u0011^z\u0002rA. The interstices have a 1-dimensional shape.\nWe call these 1-dimensional, high B2regions \\barriers\",\nbecause these are the regions where transport is strongly\nsuppressed relative to the kinematic case \u0011K, due to lo-\ncally strong B2, as discussed below. One measure of this\nblob-and-barrier structure is the structure of the proba-\nbility density function (PDF) of A. As is shown in Fig. 2\n(a3), the PDF of Afor Run1 during the suppression stage\nhas two peaks, both at A6= 0.\nNotably, such a structure of the PDF also appears in\nthe analogous CHNS system. Some binary \ruid trans-\nfers from miscible phase to immiscible phase when the\ntemperature dropped to below the corresponding crit-\nical temperature, and this second order phase transi-\ntion is called spinodal decomposition. The Cahn-Hilliard\nNavier-Stokes (CHNS) equations describe a binary \ruid\nundergoing spinodal decomposition:3\nFIG. 2. Row 1: A\feld snapshots; Row 2: B2\feld snapshots; Row 3: PDF of A. Column a: Run1 at t= 10 (suppression\nstage). The system exhibits blob-and-barrier feature, and the PDF of Ais bimodal. Column b: Run1 at t= 17 (kinematic\ndecay stage). The distribution of the \felds are trivial. Column c: Run2 at t= 10. Two peaks still arise on the PDF of Aeven\nthough its initial condition is unimodal. Column d: Run3 at t= 4:5. The system exhibits staircases feature, and the PDF of\nAhas multiple peaks.\n@t +v\u0001r =Dr2(\u0000 + 3\u0000\u00182r2 ) (5)\n@t!+v\u0001r!=\u00182\n\u001aB \u0001rr2 +\u0017r2! (6)\nv=^ z\u0002r\u001e; ! =r2\u001e (7)\nB =^ z\u0002r ; j =\u00182r2 (8)\nwhere =\u001aA\u0000\u001aB\n\u001aA+\u001aBis the local relative concentration, and\n\u0018is a coe\u000ecient describing the strength of the surface\ntension interaction.\n2D CHNS and 2D MHD are both active scalar systems.\nThe two systems are analogous, and the correspondence\nof the physics quantities between the two systems are\nsummarized in Table. II. The comparison and contrast\nof some most important features are summarized in Ta-\nble. III and Table. IV. See Ref. [32{34] for more details\nabout turbulence in 2D CHNS.\nIn comparison with the blob-and-barrier structure de-\nscribed above, in the kinematic decay stage of Run1 (i.e.\nat later time, when the magnetic \feld is so weak that \u0011T\nreverts to\u0011K), the \felds are well mixed and nontrivial\nFIG. 3. Some typical screenshots for the  \feld in the 2D\nCHNS system. Reprint from [32].\nreal space structure is absent. No barriers are discernible\nin the decay stage. The corresponding PDF of Ais a\ndistribution for a passive scalar, with one peak atA= 0,\nas shown in Fig. 2 column (b).\nThe time evolution of PDF of Afor Run1 (Fig. 4 (a))4\nTABLE II. The correspondence between 2D MHD and the\n2D CHNS system. Reprint from [32].\n2D MHD 2D CHNS\nMagnetic Potential A  \nMagnetic Field B B  \nCurrent j j  \nDi\u000busivity \u0011 D\nInteraction strength1\n\u00160\u00182\nhas a horizontal \\Y\" shape. The PDF has two peaks\ninitially, and the interval between the peaks decreases as\ntheA\feld decays. The PDF changes from double peak\nto single peak as the system evolves from the suppression\nstage to the kinematic stage.\nTwo quantities which characterize the \feld structure\nin the suppression stage are the packing fraction P, and\nbarrier width W, de\fned below. In order to identify the\nbarriers, we set a threshold on local \feld intensity, and\nde\fne the barriers to be the regions where B(x;y)>p\nhB2i\u00032. The packing fraction Pis de\fned as:\nP\u0011# of grid points in barrier regions\n# of total grid points(9)\nPis the fraction of the space where intensity exceeds\nthe mean square value. The expression for the barrier\nwidth isW\u0018\u0001A=Bb, where \u0001Ais the di\u000berence in\nAbetween adjacent blobs, and Bbis the magnitude of\nthe magnetic \feld in the barrier regions. We usep\nhA2i\nto estimate \u0001 Afor the bimodal PDF, such as for Run1.\nThe narrow barriers contain most of the magnetic energy.\nFor example, in Run1 at t= 10, the barriers occupy only\nP= 9:9% of the system space, but the magnetic \feld in\nthese regions accounts for 80 :7% of the magnetic energy.\nTherefore, we can use the magnetic energy in the barriers\nhB2\nbito approximate the total magnetic energy, i.e.:\nX\nbarriersB2\nb\u0018Z\nd2xB2(10)\nIt follows thathB2\nbi\u0018hB2i=P. We can thus de\fne W\nbased on the arguments above as:\nW2\u0011hA2i=(hB2i=P) (11)\nThis de\fnition of Wcan be justi\fed by measuring the ap-\nproximate barrier widths. The time evolutions of Pand\nWin Run1 are shown in Fig. 6. Pstays at 0:08\u00180:10\nthroughout the suppression stage. Pstarts to decline\nnear the end of the suppression stage, and drops to the\nnoise level in the kinematic decay stage. Wdecreases\nduring the suppression stage, due mainly to the decrease\nin \u0001A. It is important to note that the decline in P,\nwhich begins at t\u001813, slightly leads the decay in mag-\nnetic energy, which begins at t\u001815. This supports thenotion that barriers, the population of which is measured\nbyP, are responsible for the quenching of mixing and de-\ncay in the suppression stage.\nOne may question whether the bimodal PDF is due\nto the initial condition, since the cosine initial condition\nin Run1 is bimodal. The answer is no. In order to show\nthis, a unimodal initial condition is constructed for Run2,\nsuch that the initial PDF of Ahas one peak at A= 0:\nAI(x;y) =A0\u0003(\n\u0000(x\u00000:25)30<=x<1=2\n(x\u00000:75)31=2<=x<1(12)\nSee Fig. 5 for the comparison between bimodal and\nunimodal initial condition. To make Run2 and Run1\nhave the same time duration of the suppression stage,\nthe initial magnitude A0in Run2 is chosen such that the\ninitialhA2i(notEB!) is the same with Run1.\nFig. 2 column (c) shows a snapshot for Run2 at t= 10.\nThe time evolution of the PDF of Afor that case is shown\nin Fig. 4 (b). It is evident that, two non zero peaks\nin the PDF of Astill arise, even if the initial condition\nis unimodal. The blob structure in Aand the barrier\nstructure in B2are also evident.\nANALYSIS: LOCAL\nOne can easily see from the B2\felds plots in Fig. 2\nthat, a large scale hBidoes not exist. Intermittent\nmagnetic intensity, with low P, is a consequence of the\nblob-and-barrier structure. Therefore, the traditional ap-\nproach of mean \feld theory, especially Eqn. (2), is nei-\nther applicable nor relevant. Globally, no theory exists\nforB0= 0. Usual closure approaches appear useful when\nthe averaging window is restricted to a suitable size, cor-\nresponding to a localized region within which a mean B\nexists. In order to derive an expression for the e\u000bective\n\u0011Tfor such a local region from dynamics, we extend the\ntheory by [24{27], and propose:\n\u0011T=hv2i1=2l\n1 + Rm1\n\u00160\u001ahBi2=hv2i+ Rm1\n\u00160\u001ahA2i\nL2\nblob=hv2i(13)\nHereLblobis the size of the large Ablobs, i.e. the char-\nacteristic length scale for hA2i. The derivation is shown\nbelow.\nWe start from:\n1\n2[@thA2i+hr\u0001(vA2)i] =\u0000\u0000A@hAi\n@x\u0000\u0011hB2i(14)\nwhere \u0000A=hvxAiis the spatial \rux of A. In the past,\nonly the \u0000 A@hAi\n@xterm is kept in (14) to balance \u0011hB2i.\nHowever, in the absence of B0, \u0000A@hAi\n@xterm can be small,\nwhile the triplet term hr\u0001(vA2)ican remain, if the aver-\nage is taken over a window smaller than the system size5\nTABLE III. Comparison of 2D MHD and the 2D CHNS system. Reprint from [32].\n2D MHD 2D CHNS\nIdeal Quadratic Conserved Quantities Conservation of E,HAandHCConservation of E,H andHC\nRole of elastic waves Alfven wave couples vwithB CHNS linear elastic wave couples vwithB \nOrigin of elasticity Magnetic \feld induces elasticity Surface tension induces elasticity\nOrigin of the inverse cascades The coalescence of magnetic \rux blobs The coalescence of blobs of the same species\nThe inverse cascades Inverse cascade of HAInverse cascade of H \nPower law of spectra HA\nk\u0018k\u00007=3H \nk\u0018k\u00007=3\nTABLE IV. Contrast of 2D MHD and the 2D CHNS system. Reprint from [32].\n2D MHD 2D CHNS\nDi\u000busion A simple positive di\u000busion term A negative, a self nonlinear, and a hyper-di\u000busion term\nRange of potential No restriction for range of A  2[\u00001;1]\nInterface Packing Fraction Not far from 50% Small\nBack reaction j\u0002Bforce can be signi\fcant Back reaction is apparently limited\nKinetic energy spectrum EK\nk\u0018k\u00003=2EK\nk\u0018k\u00003\nSuggestive cascade by EK\nkSuggestive of direct energy cascade Suggestive of direct enstrophy cascade\nL0. Note the relevant scale lhere is\nld<W <l<L 0 (15)\nwhereldis the dissipation scale. Retaining all contribu-\ntions, we have\n@thA2i=\u0000hvAi\u0001rhAi\u0000r\u0001h vA2i\u0000\u0011hB2i (16)\nNow assume the \ruxes are Fickian. Note that, in princi-\nple, there are two di\u000busion coe\u000ecients:\nhvAi=\u0000\u0011T1rhAi (17)\nhvA2i=\u0000\u0011T2rhA2i (18)\nPlugging them in, we get\n@thA2i=\u0011T1(rhAi)2+r\u0011T2\u0001rhA2i\u0000\u0011hB2i(19)\nThe \frst term on the RHS is turbulent di\u000busion of hAi,\ncorresponding to the large scale magnetic \feld. The sec-\nond term is the turbulent di\u000busion of hA2i, which controls\ndecay in weak magnetic \feld. The third term is the usual\ncollisional dissipation. In principle, \u0011T16=\u0011T2, though\nthese two are related. Both terms are retained. For sim-\nplicity, we assume \u0011T1=\u0011T2=\u0011T. For a stationary\nstate, we have\nhB2i\u0018\u0011T\n\u0011(hBi2+hA2i=L2\nblob) (20)\nwhereLblobis the blob size, the characteristic length scale\nforhA2i. By standard closure methods, one can obtain\nan expression for \u0011T[29, 30]:\n\u0011T=X\nk\u001cc[hv2ik\u00001\n\u00160\u001ahB2ik] (21)Plugging (20) into (21) yields Eqn. (13) proposed above.\nDetailed comparisons of Eqn. (13) with simulation results\nare nontrivial and will be left for a future paper.\nNote that Lblob\u001cL0. In regions where no\nhigh intensity magnetic \feld is present, i.e. inside\nblobs, Rm1\n\u00160\u001ahBi2=hv2iis negligible. Yet transport is\nstill reduced relative to kinematics by hA2i, via the\nRm1\n\u00160\u001ahA2i\nL2\nblob=hv2iterm. In the barrier regions where mag-\nnetic energy is large, Rm1\n\u00160\u001ahBi2=hv2iis dominant, since\nhB2i\u001dhA2i=L2\nblobforP\u001c1. Such regions { barriers\n{ are where turbulent transport of Ais most strongly\nsuppressed.\nA key question concerns how transport barriers form\nspontaneously in turbulent 2D MHD. We argue that\ntransport barriers result from negative resistivity, driven\nby the inverse cascade of hA2i. In Eqn. (21), the positive\ncontribution to \u0011Tis a consequence of turbulent mixing\nby \ruid advection, while the second, negative, term is\na consequence of \rux coalescence. From the above, we\nsee that the turbulent resistivity can go negative locally,\nwherehB2iis strong. Of course, the system-averaged\nresistivity is positive, so the \feld decays, though slowly.\nNote though that a local negative contribution can trig-\nger a feedback loop, i.e.: B2strong in a speci\fc region !\nlocal\u0011Tnegative!localrAincreases!localB2in-\ncreases further. The feedback process saturates after the\nshort transition period, as the inverse cascade of hA2i\nmust ultimately deplete the small scales.\nAnother way to view this evolution is as a local trans-\nport bifurcation { see the spatially local S-shaped \rux-\ngradient curve for A, shown in Fig. 7 for illustration,\nwhich follows from Eqn. (13). The S-curve describes a\nbi-stable system. Note there are two stable ranges with\npositive slope, and one unstable region between, with\na negative slope (as for negative resistivity). This im-6\nFIG. 4. The time evolutions of PDF of A, and the values are in logarithm scale (base 10). (a) For Run1, the PDF is bimodal\nin the suppression stage, and \u0001 Abetween the two peaks reduces in time. The PDF becomes unimodal in the kinematic decay\nstage. (b) For Run2, two peaks at A6= 0 still arise spontaneously given a unimodal initial condition. (c) For Run3, with\nexternal forcing at smaller scale, layering and coarsening can occur. See further explanations in the text.\nFIG. 5. The initial conditions for Aand their PDFs: (a)\n\\Bimodal\" for Run1 and Run3; (b) \\Unimodal\" for Run2.\nplies that barrier formation is a transport bifurcation,\nwhich occurs when local magnetic intensity exceeds the\nthreshold given by (21). This mechanism resembles a\ntransport bifurcation in magnetically con\fned systems\n[39, 40]. Here, feedback via regions of locally intense B2,\nrather than E\u0002Bshear, is the trigger for barrier forma-\ntion.\nLAYERING OF MAGNETIC POTENTIAL\nInhomogeneous mixing and bistability (of which neg-\native viscosity is a symptom) are the key elements in\nthe dynamics of layering (i.e. staircase formation) in\nFIG. 6. Time evolution of (a) packing fraction P; and (b)\nbarrier width Win Run1.\nmany systems [6, 39, 40]. Given that, and the ubiqui-\ntous blob-and-barrier structure here, it is natural to ask\nif spontaneous layering can occur in 2D MHD. We an-\nswer in the a\u000ermative { see Fig. 2 column (d). The\ninitial condition in Run3 is the same bimodal one as for\nRun1. The key di\u000berence in parameters is the forcing\nscale, which is smaller here, i.e. k= 32 in Run3, rather\nthank= 5 for the other runs. As shown in Fig. 4 (c), the\nlayered structure consists of regions of homogenized A,\nwith small transition layers of sharp gradients in A(and\nthusB2) between them. Layering thus induces trans-\nport barriers. The layered structure persists for the du-\nration of the suppression stage, but coarsens, as shown\nin Fig. 4 (c). Coarsening occurs by a sequence of blob\nmergers. Note that by t\u00184 the staircase has coarsened\nto four transition layers. We note that closure theory\nfor the evolution of hA2ikpredict a positive turbulent7\nFIG. 7. A sketch showing the relationship between \rux \u0000 A\nandrA. The total resistivity is \u0011tot=\u000e\u0000A=\u000erA, and is\ncomposed of turbulent and collisional parts \u0011tot=\u0011T+\u0011. In\nthe small B limit, \u0011tot\u0018\u0011K; in the large B limit, the residual\nresistivity is \u0011res\u0018ul\nRm1\n\u00160\u001ahBi2=hv2i+Rm1\n\u00160\u001ahA2i\nL2\nblob=hv2i+\u0011. The\ntransition between the two limits is the transport bifurcation.\nhyper-resistivity along with the negative component of\nthe turbulent resistivity [26, 27]. This implies that evo-\nlution equation for mean hAihas a structure much like\nthe Cahn-Hilliard equation, the solutions of which are\nknown to manifest mergers and coarsening [34].\nCONCLUSIONS\nIn summary, we observe a blob-and-barrier real space\nstructure in the decay of magnetic \felds in turbulent 2D\nMHD. The magnetic \feld and the resulting barriers are\nhighly intermittent, and cannot be treated by mean \feld\ntheory. The turbulent resistivity is suppressed in the bar-\nriers, where the magnetic \felds are strong. The barriers\nform at blob interstices. For small scale forcing, sponta-\nneous layering of magnetic potential occurs due to inho-\nmogeneous mixing. Barriers form between layers. The\nlayered structure coarsens in time.\nThis analysis has implications beyond 2D. One line of\ndevelopment is to the quenching of transport of magnetic\nhelicity and magnetic dynamo processes by spatially in-\ntermittent but locally strong magnetic \felds. The other\nis to anomalous dissipation in anisotropically ordered 3D\nsystems, such as reduced MHD, where the nonlinear dy-\nnamics are e\u000bectively two dimensional. These topics will\nbe pursued in the future.\nThis research was supported by the US Department\nof Energy, O\u000ece of Science, O\u000ece of Fusion Energy\nSciences, under Award No. DE-FG02-04ER54738 and\nCMTFO Award No. de-sc0008378.[1] H. K. Mo\u000batt, Magnetic \feld generation in electrically\nconducting \ruids (Cambridge University Press, 1983).\n[2] F. Cattaneo and S. Vainshtein, The Astrophysical Jour-\nnal (1991).\n[3] S. I. Vainshtein and R. Rosner, The Astrophysical Jour-\nnal376, 199 (1991).\n[4] S. Vainshtein and F. Cattaneo, The Astrophysical Jour-\nnal (1992).\n[5] F. Cattaneo, The Astrophysical Journal 434, 200 (1994).\n[6] N. J. Balmforth, S. G. L. Smith, and W. R. Young,\nJournal of Fluid Mechanics 355, 329 (1998).\n[7] A. Lazarian and E. T. Vishniac, The Astrophysical Jour-\nnal517, 700 (1999).\n[8] D. Biskamp and E. Schwarz, Physics of Plasmas 8, 3282\n(2001).\n[9] G. B. Field and E. G. Blackman, The Astrophysical Jour-\nnal572, 685 (2002).\n[10] P. D. Mininni, D. C. Montgomery, and A. G. Pouquet,\nPhysics of Fluids 17, 035112 (2005).\n[11] E.-j. Kim, Physical Review Letters 96, 084504 (2006).\n[12] L. J. Silvers, Physics Letters A 334, 400 (2005).\n[13] L. J. Silvers, Monthly Notices of the Royal Astronomical\nSociety 367, 1155 (2006).\n[14] N. Kleeorin and I. Rogachevskii, Physical Review E 75,\n066315 (2007).\n[15] S. R. Keating and P. H. Diamond, Physical Review Let-\nters99, 224502 (2007).\n[16] S. R. Keating, L. J. Silvers, and P. H. Diamond, The\nAstrophysical Journal Letters 678, L137 (2008).\n[17] S. R. Keating and P. H. Diamond, Journal of Fluid Me-\nchanics 595, 173 (2008).\n[18] S. M. Tobias, P. H. Diamond, and D. W. Hughes, The\nAstrophysical Journal Letters 667, L113 (2007).\n[19] G. L. Eyink, A. Lazarian, and E. T. Vishniac, The As-\ntrophysical Journal 743, 51 (2011).\n[20] T. Kondi, D. W. Hughes, and S. M. Tobias, The Astro-\nphysical Journal 823, 111 (2016).\n[21] J. Mak, S. D. Gri\u000eths, and D. W. Hughes, Physical\nReview Fluids 2, 113701 (2017).\n[22] P. Xi, X. Xu, and P. Diamond, Physical Review Letters\n112, 085001 (2014).\n[23] P. W. Xi, X. Q. Xu, and P. H. Diamond, Physics of\nPlasmas 21, 056110 (2014).\n[24] A. V. Gruzinov and P. H. Diamond, Physical Review\nLetters 72, 1651 (1994).\n[25] A. V. Gruzinov and P. H. Diamond, Physics of Plasmas\n3, 1853 (1996).\n[26] P. H. Diamond, E. J. Kim, and D. W. Hughes, in Fluid\nDynamics and Dynamos in Astrophysics and Geophysics\n(CRC Press, 2005) p. 145.\n[27] P. H. Diamond, S.-I. Itoh, and K. Itoh, Modern Plasma\nPhysics, Physical Kinetics of Turbulence Plasmas Vol. 1\n(Cambridge University Press, 2010).\n[28] Y. B. Zeldovich, Sov. Phys. JETP 4, 460 (1957).\n[29] A. Pouquet, U. Frisch, and J. Lorat, Journal of Fluid\nMechanics 77, 321 (1976).\n[30] A. Pouquet, Journal of Fluid Mechanics 88, 1 (1978).\n[31] R. Ruiz and D. R. Nelson, Physical Review A 23, 3224\n(1981).\n[32] X. Fan, P. H. Diamond, L. Chacn, and H. Li, Physical\nReview Fluids 1, 054403 (2016).8\n[33] X. Fan, P. H. Diamond, and L. Chacn, Physical Review\nE96, 041101 (2017).\n[34] X. Fan, P. H. Diamond, and L. Chacn, Physics of Plas-\nmas25, 055702 (2018).\n[35] R. Pandit, D. Banerjee, A. Bhatnagar, M. Brachet,\nA. Gupta, D. Mitra, N. Pal, P. Perlekar, S. S. Ray,\nV. Shukla, and D. Vincenzi, Physics of Fluids 29, 111112\n(2017).\n[36] See Supplemental Material at [URL will be inserted by\npublisher] for details about the unimodal initial conditionused in the main text, and the relationship between the\nCHNS system and MHD.\n[37] L. Chacon, D. Knoll, and J. Finn, Journal of Computa-\ntional Physics 178, 15 (2002).\n[38] L. Chacon and D. Knoll, Journal of Computational\nPhysics 188, 573 (2003).\n[39] A. Ashourvan and P. H. Diamond, Physical Review E\n94, 051202 (2016).\n[40] A. Ashourvan and P. H. Diamond, Physics of Plasmas\n24, 012305 (2017)."    },    {        "title": "2403.13481v1.Molecular_Dynamics_Simulations_of_Microscopic_Structural_Transition_and_Macroscopic_Mechanical_Properties_of_Magnetic_Gels.pdf",        "content": "Molecular Dynamics Simulations of Microscopic\nStructural Transition and Macroscopic\nMechanical Properties of Magnetic Gels\nXuefeng Wei,†,‡Gaspard Junot,¶Ramin Golestanian,§,∥Xin Zhou,‡,†Yanting\nWang,⊥,†,‡Pietro Tierno,¶,#,@and Fanlong Meng∗,⊥,†,‡\n†Wenzhou Institute, University of Chinese Academy of Sciences, Wenzhou, Zhejiang\n325000, China\n‡School of Physical Sciences, University of Chinese Academy of Sciences, 19A Yuquan\nRoad, Beijing 100049, China\n¶Departament de F´ ısica de la Mat` eria Condensada, Universitat de Barcelona, 08028\nBarcelona, Spain\n§Max Planck Institute for Dynamics and Self-Organization (MPIDS), D-37077 G¨ ottingen,\nGermany\n∥Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3PU,\nUnited Kingdom\n⊥CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese\nAcademy of Sciences, Beijing 100190, China\n#Universitat de Barcelona Institute of Complex Systems, 08028 Barcelona, Spain\n@Institut de Nanoci` encia i Nanotecnologia, Universitat de Barcelona, 08028 Barcelona,\nSpain\nE-mail: fanlong.meng@itp.ac.cn\n1arXiv:2403.13481v1  [cond-mat.soft]  20 Mar 2024Abstract\nMagnetic gels with embedded micro/nano-sized magnetic particles in crosslinked\npolymer networks can be actuated by external magnetic fields, with changes in their\ninternal microscopic structures and macroscopic mechanical properties. We investi-\ngate the responses of such magnetic gels to an external magnetic field, by means of\ncoarse-grained molecular dynamics simulations. We find that the dynamics of magnetic\nparticles are determined by the interplay of between magnetic dipole-dipole interac-\ntions, polymer elasticity and thermal fluctuations. The corresponding microscopic\nstructures formed by the magnetic particles such as elongated chains can be controlled\nby the external magnetic field. Furthermore, the magnetic gels can exhibit reinforced\nmacroscopic mechanical properties, where the elastic modulus increases algebraically\nwith the magnetic moments of the particles in the form of ∝(m−mc)2when magnetic\nchains are formed. This simulation work can not only serve as a tool for studying\nthe microscopic and the macroscopic responses of the magnetic gels, but also facilitate\nfuture fabrications and practical controls of magnetic composites with desired physical\nproperties.\nIntroduction\nMagnetic gels, with micro/nano-sized magnetic particles embedded in crosslinked polymeric\ngels, belong to the class of smart magnetic materials, whose physical properties can be con-\ntrolled by external magnetic fields.1–3By tuning such fields, e.g., the direction, the strength\nor the frequency, one can change the materials with different mechanical responses, such as\nmaterial stiffness4–7and deformations.8–10Based on such field-controlling features, magnetic\ngels have been fabricated in laboratories and used as magnetic robots,11–13sensors,14–16ac-\ntuators,17,18etc., and also for bio-medical applications19–21such as drug delivery/release due\nto the bio-friendly characteristics of magnetic fields.\nUnder an applied magnetic field, the materials can respond in different ways depending\n2on how the magnetic particles are embedded in the polymer matrix. For magnetic particles\nable to move freely in the polymer gel, they can form microscopic chains when the magnetic\ndipole-dipole interactions between particles dominate the thermal fluctuation as in ordinary\nsolutions,22,23where the polymer gel serves mainly as a viscoelastic medium without much\nchanges in its mechanical properties. For superparamagnetic particles fixed in the polymer\nnetwork,24the particles can still move but now they induce microscopic distortions in the\npolymer network, and they can even form magnetic chains if the magnetic dipole-dipole\ninteractions are strong enough. In the latter case, the responses of the embedded particles to\nthe external magnetic field with simultaneous deformations of the polymer network underpins\nthe physical mechanism for controlling the mechanical properties of the magnetic gels.\n(a) (b)\nwithout magnetic field with magnetic field\nFigure 1: Magnetic gels with (a) uniformly distributed magnetic particles without the mag-\nnetic field and (b) magnetic chains under an external magnetic field. Spheres denote the\nmagnetic particles and grey lines denote partial connections in the polymer network.\nThe connection between the microscopic motion of the magnetic particles and the macro-\nscopic responses of the material, is the key for understanding how external magnetic fields\ncan be utilised to control the material properties. Experimental methods such as small-angle\nX-ray scattering (SAXS),25small angle neutron scattering (SANS),26electron magnetic res-\nonance (EMR),27etc., are utilised for detecting the internal structure of the magnetic gels\nwith macroscopic mechanical properties which are measurable by tensile testing,28,29dynamic\n3mechanical analysis,30,31etc. Theoretical attempts include analytical modelling such as the\nlattice models32,33and continuum models,24,34,35and numerical methods such as finite ele-\nment calculations,36Monte Carlo simulations37,38and molecular dynamics simulations.39–41\nIn this work, we use coarse-grained molecular dynamics simulations to investigate the mi-\ncroscopic motion and structures of the magnetic particles, together with the macroscopic\nmechanical properties especially about the anisotropic elasticity of the materials.\nSimulation methodology\nPolymer network. We follow Ref.42 to construct the 3D packing-derived (PD) polymer net-\nwork. We first randomly place N=W3radially bidisperse spheres with harmonic repulsive\ninteractions within a cubic unit cell of side length W(W= 20 is taken in later discus-\nsions), where half of the spheres are assigned with radius r=r0and the other half with\nr=νr0(ν= 1.4 is chosen to avoid long-range crystal order42–44). By continuously increas-\ning the radius r0from r0= 0, the spheres will reach a jammed state at rJ\n0. Then, from this\ndisordered packing state, we generate a contact network by connecting the centers of the\noverlapping spheres (excluding rattlers) with springs at their rest lengths and angles. For\nsufficiently large systems, this procedure generates contact networks with the connectivity\nz= 2d, where dis the system dimensionality. After generating the underlying network struc-\nture, we remove randomly chosen bonds and any consequent dangling ends repeatedly until\nthe network reaches the desired average connectivity z. The bonds connecting neighbouring\nnodes are modelled as Hookean springs and the bending effects is considered by introducing\nthe bending energy. The stretching and the bending energy of the network can be written\nas:\nUstretching (r) =µ\n2X\ni,j(rij−rij,0)2\nrij,0, (1)\nUbending (r) =κ\n2X\nijk(θijk−θijk,0)2\nlijk,0, (2)\n4respectively, where µandκdenote the Hookean constant and the bending rigidity, respec-\ntively, rij,0andrijdenote the length of spring ( ij) connecting the ith and the jth node\nbefore and after stretching, respectively, θijk,0andθijkdenote the angles between the spring\n(ij) and ( jk) before and after bending, respectively, and lijk,0= (rij,0+rjk,0)/2 denotes the\naverage length of the two connecting springs.\nWCA potential. We choose polymer nodes randomly with the number density n0to\nbe replaced with magnetic particles, i.e., the magnetic particles are uniformly distributed\nin the prepared state. The diameters of the magnetic particles are set as σm= 1.0σand\nother non-magnetic nodes are set as σn= 0.2σ. In our simulations, the truncated Weeks-\nChandler-Andersen (WCA) potential45is used to simulate the non-bonded and non-magnetic\ninteractions between particles, as:\nU(r) =\n\n4ϵh\u0000σij\nr\u000112−\u0000σij\nr\u00016i\n+Cifr≥rcutoff,\n0 if r < r cutoff,\nwhere rcutoff = 21/6σijis the cutoff distance for given pairs, σij= (σi+σj)/2 is the average\ndiameter of particle iandj, and Cis a constant ensuring the continuity of the potential\nenergy at the cutoff distance. In this work we take ϵandσas the unit energy and length,\nrespectively.\nMagnetic dipole-dipole interactions. The superparamagnetic particles can be magnetized\nunder an external magnetic field Bext, and they acquire magnetic moments m=χvpBext/µ0\npointing along the direction of the magnetic field, where χis the magnetic susceptibility, vp\nis the particle volume and µ0is vacuum permeability. These magnetic particles possess mo-\nments aligned along the field direction, and the interactions between the magnetic particles\nare:\nUdd(rij) =µ0\n4π\u00141\nr3\nij(mi·mj)−3\nr5\nij(mi·rij)(mj·rij)\u0015\n. (3)\n5The magnetic dipole-dipole interactions between magnetic particles can be either attractive\nor repulsive, depending on their relative locations with respect to the separation rij.\nMolecular dynamics simulations. We perform the molecular dynamics simulations on\na coarse-grained level based on the above model setup, using LAMMPS software.46The\nsimulations are done in the canonical NVT ensemble using a Langevin thermostat.47,48The\nequation of motion for each particle is:\nm¨r=−γ˙r+F+ξ, (4)\nwhere mdenotes the mass of the particle, γdenotes the friction constant, Fdenotes the forces\nexerted by other particles and the polymer network, and ξdenotes the thermal noise due\nto the random collisions with implicit solvent molecules, which obeys fluctuation-dissipation\ntheorem. Periodic boundary conditions are adopted in the simulations and the detailed\nsimulation procedure including simulation parameters can be found in the supplement.49\nResults and discussions\nInternal microscopic structures\nThe elastic properties, e.g., the shear modulus and the bulk modulus, of the magnetic gels\nbefore applying any external magnetic field can be controlled by changing the connectivity in\nthe polymer networks, as shown in Fig. 2(a). By increasing the connectivity of the polymer\nnetwork, both the shear and the bulk modulus will increase, in an approximate form of\n∝(z−zc)3with zc≃3.2 as the critical connectivity for the polymer network to exhibit finite\nelasticity.42,50\nAs shown in Fig. 2(b), the magnetic particles can form chain-like structures by increas-\ning the strength of the magnetic field (magnetic moments of superparamagnetic particles),\nwhen the magnetic dipole-dipole interactions between the magnetic particles dominate the\n6(a)(b) (c)\nFigure 2: (a) Dependence of shear modulus G(squares), bulk modulus K(triangles) and\nK+ 4G/3 (circles) of the polymer gels on the network connectivity z. (b) Chain ratio as\na function of the magnetic moments of the particles, in polymer networks with different\nconnectivity. Insets denote the microscopic structures of the magnetic particles and the\ndisplacement field in the cross-sectional plane. (c) Chain ratio as a function of the magnetic\nmoments and the network elasticity. Red line denote the physical conditions for the chain\nformation [Equation (5)], and the white diamonds correspond to simulation results of the\ncritical chain ratio, η= 0.15.\nnetwork elasticity and the thermal fluctuations; this phenomenon resembles the clustering\ninstability found in magnetic microswimmers, where the microswimmers can form clusters\nwhen magnetic dipole-dipole interactions dominate thermal fluctuations.51,52Due to the dis-\nplacements of the magnetic particles in events of chain formations, the polymer network will\nbe deformed with the local deformation field, as shown in the insets of Fig. 2(b). The chain\nratio, η, defined as the ratio of the number of the magnetic particles forming the chains to\nthe total number of the magnetic particles in the material, is introduced to characterize the\namount of the micro-structures in the magnetic gel and also the relative strength of the mag-\nnetic dipole-dipole interactions over the network elasticity. We use the following criterion\nto identify if a particle belong to a chain: the distance to its nearest neighboring magnetic\nparticle is less than 21/6σ, and the angle between the direction of the magnetic moment and\nthe position vector connected to its nearest neighboring particle is less than 15◦. With the\ndecrease in the network connectivity, i.e., the elastic moduli of the material, the chain ratio\nin magnetic gels of the particles with the same magnetic moments will increase.\nBy systematically changing the elastic moduli of the network and the magnetic moments\nof the particles, one can obtain the phase diagram showing the physical conditions for the\n7chain formation, as shown in Fig. 2(c). In simulations, η= 0.15 is taken as the value of the\nchain ratio, for denoting the conditions of the chain formation, and one can observe from\nFig. 2(c) that, for inclusions carrying large magnetic moments in soft polymer gels, they\ncan easily form chain-like structures, coinciding with the theoretical conditions for the chain\nformation as derived in a previous work,24\nµ0n0m2\nkBT>vp\nkBT(K+ 4G/3) + 1 . (5)\nOne can also check how the number density of magnetic particles in the prepared state,\nn0, can influence the formation of the micro-structures in the gel. As shown in Fig. 3(a),\nthe chain ratio is larger for systems characterized by a higher number density n0, since the\nmagnetic dipole-dipole interactions become stronger due to decreased distances between the\nmagnetic particles. The physical conditions for the chain formation in the plane of ( n0, m)\nis provided in Fig. 3(b), which also agrees with our theoretical prediction.\n(a) (b)\nFigure 3: (a) Chain ratio ηas a function of the magnetic moments mof the particles with\ndifferent number density. (b) Dependence of the chain ratio on the magnetic moments and\nthe number density of the magnetic particles. Red line denote the physical conditions for the\nchain formation [Equation (5)], and the white diamonds corresponds to simulation results of\nthe critical chain ratio, η= 0.15.\n8Macroscopic mechanical properties\nThe macroscopic mechanical properties of magnetic gels can become reinforced due to the\nformation of microscopic chains.4–7Here we investigate such effects by studying how magnetic\ngels consisting of particles carrying different magnetic moments can respond under shear\ndeformations.\nAs shown in Fig. 4(a), the stress-strain relations of magnetic gels can change if the\nmagnetic moments of the particles are different. The magnetic gels with chain formation\nexhibit obvious mechanical reinforcements (larger stresses compared with those without chain\nformation at same strains) and weak anisotropy, whose mechanical responses depend on the\napplied plane of the shear deformation. By defining the shear modulus of the material as\nGαβ=dσαβ\ndεαβ|εαβ=0, one can obtain the dependence of this modulus on the magnetic moments\nof the particles, as shown in Fig. 4(b). There is a weak effect of the anisotropy in the shear\nmoduli appears at mc≃0.27, corresponding to the chain ratio as η= 0.15 (same criterion\nas used for chain formation). Compared with the material containing inherent chains in its\nrelaxed state (springs not deformed),32the anisotropy in the mechanical response here is\nweaker, since all the elastic springs are already distorted when the chains are formed in the\ncurrent setup. More importantly, for m≤mc, the shear modulus remains almost unchanged\nby increasing the magnetic moments, but for m > m c, the shear modulus increases quickly\nwith the magnetic moments in an approximate form of\nGαβ(m) =Gαβ,∞(m−mc)2\nam+ (m−mc)2, (6)\nwhere Gαβ,∞,amandmcare fitting parameters for specific materials.6,32,53\nBy changing the network connectivity z, we can change the elasticity of the polymer gel\nin the prepared state, based on which one can compare the effects of magnetic interactions\nbetween the embedded particles and polymer elasticity in the mechanical properties of mag-\nnetic gels. As shown in Fig. 4(c), the contributions of magnetic interactions in the shear\n9(b) (c)\n———\n------\nstrain\nstress(a)Figure 4: (a) Stress-strain relations of magnetic gels without (dash lines) and with (solid\nlines) chain formation, which undergoes the shear deformation in three orthogonal planes. (b)\nDependence of the shear modulus Gαβas a function of the magnetic moment of the particles,\nand lines fitted with Equation (6). (c) Comparison between the magnetic interactions and\nthe polymer elasticity in terms of the shear modulus of the magnetic gels. Lines fitted with\nthe function 1 + A/(z−zc)αwith Aandαas fitting parameters.\nmodulus of magnetic gel where magnetic chains are formed, are significant for z≃zc, which\nbecome less important by increasing the network connectivity due to the increased elasticity\nof the polymer network itself.\nSummary\nWe have studied the microscopic structure formation and the macroscopic mechanical re-\nsponses of magnetic gels, by utilising the coarse-grained molecular dynamics simulations.\nBy increasing the strength of the magnetic dipole-dipole interactions between the particles,\nthe microscopic chain structures can form, which induces reinforced mechanical responses of\nthe magnetic gels. This work can not only help to understand the connection between the\nmicroscopic structure and the macroscopic mechanical properties of the magnetic gels, but\ncan also guide industrial fabrications of magnetic gels with desired mechanical responses and\ntheir controls by applying external magnetic fields for practical applications.\n10Acknowledgments\nF. M. acknowledges supports by National Natural Science Foundation of China (Grant No.\n12275332, 12047503, and 12247130), Chinese Academy of Sciences, Max Planck Society (Max\nPlanck Partner Group), Wenzhou Institute (Grant No. WIUCASQD2023009) and Beijing\nNational Laboratory for Condensed Matter Physics (2023BNLCMPKF005). X. W. acknowl-\nedges supports from the China Postdoctoral Science Foundation (Grants No. 2023M743448)\nand the National Natural Science Foundation of China (Grants No. 12347170). P. T. ac-\nknowledges financial support from the European Research Council (Grant No. 811234) and\nfrom the Generalitat de Catalu˜ na via the program “Icrea Academia”. The computations of\nthis work were conducted on the HPC cluster of ITP-CAS.\nReferences\n(1) Brand, H.; Martinoty, P.; Pleiner, H. Cross-Linked Liquid Crystalline Systems: From\nRigid Polymer Networks to Elastomers ; 2011; pp 529–563.\n(2) Th´ evenot, J.; Oliveira, H.; Sandre, O.; Lecommandoux, S. Magnetic responsive polymer\ncomposite materials. Chemical Society Reviews 2013 ,42, 7099.\n(3) Weeber, R.; Hermes, M.; Schmidt, A. M.; Holm, C. Polymer architecture of magnetic\ngels: a review. Journal of Physics: Condensed Matter 2018 ,30, 063002.\n(4) Jolly, M. R.; Carlson, J. D.; Mu˜ noz, B. C. A model of the behaviour of magnetorheo-\nlogical materials. Smart Materials and Structures 1996 ,5, 607–614.\n(5) Collin, D.; Auernhammer, G. K.; Gavat, O.; Martinoty, P.; Brand, H. R. Frozen-\nIn Magnetic Order in Uniaxial Magnetic Gels: Preparation and Physical Properties.\nMacromolecular Rapid Communications 2003 ,24, 737–741.\n11(6) Varga, Z.; Filipcsei, G.; Zr´ ınyi, M. Smart composites with controlled anisotropy. Poly-\nmer2005 ,46, 7779–7787.\n(7) Varga, Z.; Filipcsei, G.; Zr´ ınyi, M. Magnetic field sensitive functional elastomers with\ntuneable elastic modulus. Polymer 2006 ,47, 227–233.\n(8) Raikher, Y. L.; Stolbov, O. Magnetodeformation effect in a ferroelastic material. Tech-\nnical Physics Letters 2000 ,26, 156–158.\n(9) Wang, Z.; Deng, B.; Yao, K. Physical model of plastic deformation on magnetization\nin ferromagnetic materials. Journal of Applied Physics 2011 ,109.\n(10) Huang, S.; Pessot, G.; Cremer, P.; Weeber, R.; Holm, C.; Nowak, J.; Odenbach, S.;\nMenzel, A. M.; Auernhammer, G. K. Buckling of paramagnetic chains in soft gels. Soft\nMatter 2015 ,12, 228–237.\n(11) Kim, Y.; Yuk, H.; Zhao, R.; Chester, S. A.; Zhao, X. Printing ferromagnetic domains\nfor untethered fast-transforming soft materials. Nature 2018 ,558, 274–279.\n(12) Alapan, Y.; Yasa, O.; Yigit, B.; Yasa, I. C.; Erkoc, P.; Sitti, M. Microrobotics and\nMicroorganisms: Biohybrid Autonomous Cellular Robots. Annual Review of Control,\nRobotics, and Autonomous Systems 2019 ,2, 205–230.\n(13) Kim, Y.; Zhao, X. Magnetic Soft Materials and Robots. Chemical Reviews 2022 ,122,\n5317–5364.\n(14) Gao, R.; Jiang, Y.; Zhao, Y. Magnetic field sensor based on anti-resonant reflecting\nguidance in the magnetic gel-coated hollow core fiber. Optics letters 2014 ,39, 6293–\n6296.\n(15) Hankiewicz, J.; Celinski, Z.; Stupic, K.; Anderson, N.; Camley, R. Ferromagnetic parti-\ncles as magnetic resonance imaging temperature sensors. Nature communications 2016 ,\n7, 12415.\n12(16) Gloag, L.; Mehdipour, M.; Chen, D.; Tilley, R. D.; Gooding, J. J. Advances in the ap-\nplication of magnetic nanoparticles for sensing. Advanced Materials 2019 ,31, 1904385.\n(17) Shigetomi, S.; Takahashi, H.; Tsumori, F. Magnetic actuator using double network gel.\nJournal of Photopolymer Science and Technology 2020 ,33, 193–197.\n(18) He, Y.; Tang, J.; Hu, Y.; Yang, S.; Xu, F.; Zr´ ınyi, M.; Chen, Y. M. Magnetic hydrogel-\nbased flexible actuators: A comprehensive review on design, properties, and applica-\ntions. Chemical Engineering Journal 2023 ,462, 142193.\n(19) Li, Y.; Huang, G.; Zhang, X.; Li, B.; Chen, Y.; Lu, T.; Lu, T. J.; Xu, F. Magnetic\nhydrogels and their potential biomedical applications. Advanced Functional Materials\n2013 ,23, 660–672.\n(20) Sung, B.; Kim, M.-H.; Abelmann, L. Magnetic microgels and nanogels: Physical mech-\nanisms and biomedical applications. Bioengineering & translational medicine 2021 ,6,\ne10190.\n(21) Veloso, S. R.; Andrade, R. G.; Castanheira, E. M. Review on the advancements of mag-\nnetic gels: towards multifunctional magnetic liposome-hydrogel composites for biomed-\nical applications. Advances in Colloid and Interface Science 2021 ,288, 102351.\n(22) Auernhammer, G. K.; Collin, D.; Martinoty, P. Viscoelasticity of suspensions of mag-\nnetic particles in a polymer: Effect of confinement and external field. The Journal of\nChemical Physics 2006 ,124, 204907.\n(23) Weeber, R.; Klinkigt, M.; Kantorovich, S.; Holm, C. Microstructure and magnetic\nproperties of magnetic fluids consisting of shifted dipole particles under the influence\nof an external magnetic field. The Journal of Chemical Physics 2013 ,139, 214901.\n(24) Junot, G.; Wei, X.; Ort´ ın, J.; Golestanian, R.; Wang, Y.; Tierno, P.; Meng, F.\n13Elastically-mediated collective organisation of magnetic microparticles. Soft Matter\n2022 ,18, 5171–5176.\n(25) Sunaryono; Taufiq, A.; Putra, E. G. R.; Okazawa, A.; Watanabe, I.; Kojima, N.; Rug-\nmai, S.; Soontaranon, S.; Zainuri, M.; Triwikantoro; others Small-angle X-ray scattering\nstudy on PVA/Fe3O4 magnetic hydrogels. Nano 2016 ,11, 1650027.\n(26) Lawrence, M. B.; Abbas, S.; Aswal, V. Structure of polyvinyl alcohol-borax ferrogels:\nA small angle neutron scattering study. Journal of Polymer Research 2018 ,25, 1–7.\n(27) Sorokina, O. N.; Kovarski, A. L.; Lagutina, M. A.; Dubrovskii, S. A.; Dzheparov, F. S.\nMagnetic nanoparticles aggregation in magnetic gel studied by electron magnetic reso-\nnance (EMR). Applied Sciences 2012 ,2, 342–350.\n(28) Ramanujan, R.; Lao, L. The mechanical behavior of smart magnet–hydrogel compos-\nites. Smart materials and structures 2006 ,15, 952.\n(29) Wu, J.; Gong, X.; Fan, Y.; Xia, H. Physically crosslinked poly (vinyl alcohol) hydrogels\nwith magnetic field controlled modulus. Soft Matter 2011 ,7, 6205–6212.\n(30) Ponton, A.; Bee, A.; Talbot, D.; Perzynski, R. Regeneration of thixotropic magnetic\ngels studied by mechanical spectroscopy: the effect of the pH. Journal of Physics:\nCondensed Matter 2005 ,17, 821.\n(31) Zhang, G.; Zhang, Z.; Sun, M.; Yu, Y.; Wang, J.; Cai, S. The influence of the tem-\nperature on the dynamic behaviors of magnetorheological gel. Advanced Engineering\nMaterials 2022 ,24, 2101680.\n(32) Ivaneyko, D.; Toshchevikov, V.; Borin, D.; Saphiannikova, M.; Heinrich, G. Mechanical\nProperties of Magneto-Sensitive Elastomers in a Homogeneous Magnetic Field: Theory\nand Experiment. Macromolecular Symposia 2014 ,338, 96–107.\n14(33) Pessot, G.; L¨ owen, H.; Menzel, A. M. Dynamic elastic moduli in magnetic gels: Normal\nmodes and linear response. Journal of Chemical Physics 2016 ,145, 104904.\n(34) Jarkova, E.; Pleiner, H.; M¨ uller, H. W.; Brand, H. R. Hydrodynamics of isotropic\nferrogels. Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Inter-\ndisciplinary Topics 2003 ,68, 041706.\n(35) Romeis, D.; Toshchevikov, V.; Saphiannikova, M. Elongated micro-structures in\nmagneto-sensitive elastomers: a dipolar mean field model. Soft Matter 2016 ,12, 9364–\n9376.\n(36) Spieler, C.; K¨ astner, M.; Goldmann, J.; Brummund, J.; Ulbricht, V. XFEM modeling\nand homogenization of magnetoactive composites. Acta Mechanica 2013 ,224, 2453–\n2469.\n(37) Wood, D. S.; Camp, P. J. Modeling the properties of ferrogels in uniform magnetic\nfields. Physical Review E - Statistical, Nonlinear, and Soft Matter Physics 2011 ,83,\n011402.\n(38) Cremer, P.; Heinen, M.; Menzel, A. M.; L¨ owen, H. A density functional approach to\nferrogels. Journal of Physics: Condensed Matter 2017 ,29, 275102.\n(39) Weeber, R.; Kantorovich, S.; Holm, C. Deformation mechanisms in 2D magnetic gels\nstudied by computer simulations. Soft Matter 2012 ,8, 9923–9932.\n(40) Pessot, G.; Weeber, R.; Holm, C.; L¨ owen, H.; Menzel, A. M. Towards a scale-bridging\ndescription of ferrogels and magnetic elastomers. Journal of Physics: Condensed Matter\n2015 ,27, 325105.\n(41) Weeber, R.; Kantorovich, S.; Holm, C. Ferrogels cross-linked by magnetic particles:\nField-driven deformation and elasticity studied using computer simulations. Journal of\nChemical Physics 2015 ,143, 154901.\n15(42) Shivers, J. L.; Feng, J.; Sharma, A.; MacKintosh, F. C. Normal stress anisotropy and\nmarginal stability in athermal elastic networks. Soft Matter 2019 ,15, 1666–1675.\n(43) Dagois-Bohy, S.; Tighe, B. P.; Simon, J.; Henkes, S.; Van Hecke, M. Soft-sphere packings\nat finite pressure but unstable to shear. Physical review letters 2012 ,109, 095703.\n(44) Koeze, D.; V˚ agberg, D.; Tjoa, B.; Tighe, B. Mapping the jamming transition of bidis-\nperse mixtures. Europhysics Letters 2016 ,113, 54001.\n(45) Weeks, J. D.; Chandler, D.; Andersen, H. C. Role of Repulsive Forces in Determining\nthe Equilibrium Structure of Simple Liquids. The Journal of Chemical Physics 1971 ,\n54, 5237–5247.\n(46) Thompson, A. P.; Aktulga, H. M.; Berger, R.; Bolintineanu, D. S.; Brown, W. M.;\nCrozier, P. S.; in ’t Veld, P. J.; Kohlmeyer, A.; Moore, S. G.; Nguyen, T. D.; Shan, R.;\nStevens, M. J.; Tranchida, J.; Trott, C.; Plimpton, S. J. LAMMPS - a flexible simulation\ntool for particle-based materials modeling at the atomic, meso, and continuum scales.\nComp. Phys. Comm. 2022 ,271, 108171.\n(47) Grønbech-Jensen, N.; Farago, O. A simple and effective Verlet-type algorithm for sim-\nulating Langevin dynamics. Molecular Physics 2013 ,111, 983–991.\n(48) Grønbech-Jensen, N. Complete set of stochastic Verlet-type thermostats for correct\nLangevin simulations. Molecular Physics 2020 ,118, 8.\n(49) See supplementary materials.\n(50) Sharma, a.; Licup, a. J.; Jansen, K. a.; Rens, R.; Sheinman, M.; Koenderink, G. H.;\nMacKintosh, F. C. Strain-controlled criticality governs the nonlinear mechanics of fibre\nnetworks. Nature Physics 2016 ,12, 584–587.\n(51) Meng, F.; Matsunaga, D.; Golestanian, R. Clustering of Magnetic Swimmers in a\nPoiseuille Flow. Physical Review Letters 2018 ,120, 188101.\n16(52) Meng, F.; Matsunaga, D.; Mahault, B.; Golestanian, R. Magnetic Microswimmers Ex-\nhibit Bose-Einstein-like Condensation. Physical Review Letters 2021 ,126, 078001.\n(53) Varga, Z.; Filipcsei, G.; Zr´ ınyi, M. Magnetic field sensitive functional elastomers with\ntuneable elastic modulus. Polymer 2006 ,47, 227–233.\n17"    },    {        "title": "1203.5434v1.Dynamics_of_a_dielectric_droplet_suspended_in_a_magnetic_fluid_in_electric_and_magnetic_fields.pdf",        "content": "Dynamics of a dielectric droplet suspe nded in a magnetic fluid in electric and \nmagnetic fields \n \nArthur Zakinyan, Elena Tkacheva, Yury Dikansky \n \nDepartment of Physics, Stavropol State Un iversity, 1 Pushkin St., 355009 Stavropol, Russian \nFederation \n Address correspondence to A. Zakinyan: postal address is presented above, e-mail: zakinyan.a.r@mail.ru \n \nAbstract.  The behavior of a microdrop of dielectri c liquid suspended in a magnetic fluid and \nexposed to the action of electric and magnetic fiel ds is studied experimentally. With increasing \nelectric field, the deformation of droplets into oblate ellipsoid, toroid and curved toroid was observed. At the further increase in the electric field, the bursting of droplets was also revealed. \nThe electrorotation of deformed  droplets was observed and inve stigated. The influence of an \nadditional magnetic field on the droplet dynamics wa s studied. The main feat ures of the droplet \ndynamics were interpreted a nd theoretically examined. \nKeywords:  dielectric droplet, magnetic fluid, shap e dynamics, electrorotation, droplet burst.  \n \n1. Introduction \n \nElectrohydrodynamic deformation and burst of liq uid drops suspended in another immiscible \nliquid are well described and in vestigated experimentally [ 1–5], theoretically [3–8], and \ncomputationally [9–11]. Briefly summarizing the re sults of previous works, it can be concluded \nthat depending on certain system parameters the drop can take the equilibrium shape of oblate or prolate ellipsoid and can burst in to smaller drops under the action of  electric field. The electric \nfield can also bring into rotation the liquid dr ops as it was reported in  [12, 13]. The droplets \nelectrorotation studied in [12, 13] is  quite similar in nature to the electrorotation of solid particles \nsuspended in a liquid (see [14] for a review). While the dynamics and deformation of freely \nsuspended droplets has been well analyzed and observed, the deform ation of droplets confined in \nthe thin film geometry still rema ins relatively unstudied . In addition, some new peculiarities of \nthe droplet behavior can appear in a system wh ere the droplet relaxation time is large. The \ninvestigation of the mentioned problems is the subject of the present work.  \nMagnetic fluid is an artificially synthesized fluid with noticeable magnetic properties. It \nrepresents a colloidal suspension of ultra-fine ferro- or ferri-m agnetic nanoparticles suspended in \na carrier fluid. The action of a magnetic field on magnetic fluids  was studied intensively, both \ntheoretically and experime ntally, in many works. In particul ar, the action of a uniform magnetic \nfield on magnetic fluid drops has b een much studied. The review of these works is presented in \n[15]. The obtained results demonstrate a wide analogy between the behavi or of a liquid drop in \nan electric field and the behavior of a magnetic  fluid drop in a magnetic field. But, unlike an \nelectric field, a uniform magnetic  field always leads to the st retching of both a magnetic fluid \ndrop in a nonmagnetic liquid and a no nmagnetic drop in a magnetic fluid.  \nBecause the droplets of magnetic fluid and th e nonmagnetic droplets suspended in a magnetic \nfluid can be deformed in both magnetic and electric  fields, the simultaneous effect of electric and \nmagnetic fields on a droplet may lead to behavi ors much more complicated and varied. Some \npeculiarities of these behaviors have been studied  experimentally in [16] and theoretically in \n[17]. In the present work we report on the further development of such studies, here we investigate the dynamics of a singl e droplet of a dielectric liquid  suspended in a magnetic fluid \nunder the action of electric and magnetic fields. \n \n2. Experiments \n \nIn our experiments we used a kerosene-based magnetic fluid with dispersed magnetite \nnanoparticles of about 10 nm diam eter stabilized with oleic aci d. The properties of the magnetic \nfluid are: dynamic viscosity is η\ne = 30 mPa·s, conductivity is σe = 1.3·10–6 S/m, dielectric \nconstant is εe = 5.2, magnetite volume fraction is 13 % and saturation magnetization is 55.4 \nkA/m. Liquid caoutchouc immiscible with the ma gnetic fluid was select ed as the dielectric \nliquid. Its density is ρ = 725 kg·m–3, dynamic viscosity is ηi = 1.5 Pa·s, co nductivity is σi = 10–12 \nS/m and dielectric constant is εi = 2.3. The reasons to use liquid caoutchouc are that its viscosity \nis comparatively high and the interfacial tensio n at the interface between it and the magnetic \nfluid is very low ( γ = 6.7 10–6 N/m). The interfacial tension was determined by the retraction of \nthe deformed drop method proposed in [18]. The experimental sample was prepared by \nmechanical mixing of a small volume of liquid caoutchouc with the magnetic fluid. The radii, R, \nof droplets of the emulsion obtained in  this way are varying from 5 to 30 μm. Since the size of \nmagnetic nanoparticles of magnetic fluid is much  smaller than the sizes of nonmagnetic liquid \ncaoutchouc droplets suspended in the magnetic flui d, the fluid can be considered as a continuous \nliquid magnetizable medium. No stabilizing agents were used in the prepar ation of the emulsion. \nThe volume fraction of the suspended droplet s was about 0.01. Owing to the small volume  \n3 \n fraction we could explore the be havior of a single droplet and neglect the effects of droplets \ninteraction. \nDroplet behavior was experimentally studied by observations with an optical microscope, \nwhich was placed between Helmholtz coils genera ting a constant uniform magnetic field in the \nspace where the cell containing a sample wa s located. Fig. 1 shows schematically the \nexperimental setup, which was used for the droplet  behavior investigation. We used two different \ntypes of cells. The first cell (presented at right-bottom in Fig. 1) was assembled of two \nrectangular flat glass plates covered with a transparent c onducting stannic oxide coating. A \nfluoroplastic film 20–30 μm thick with a circular hole in the center was placed between the \nplates to set the distance between electrodes. The hole was filled with the studied emulsion sample. A variable voltage applied to the plates generated the ac electric field between them. The \nmost droplets of the sample were approximately  of the same size as the distance between \nelectrodes, some droplets were smaller and some  large droplets were slightly compressed by the \nplates of electrodes. For detailed observation of the droplet dynamics peculiarities the second \ncell representing a microscope slide with two rect angular metal plates gl ued on the slide surface \nwas also used (presented at left-bottom in Fi g. 1). The distance between edges of metal plates \nwas 1 mm and their thickness was 0.1 mm. The sp ace between edges of metal plates was filled \nwith an emulsion, and a voltage applied to the pl ates generated an electric field between them. \nThe first cell served for viewing the events through a microscope along the electric field \ndirection and for studying the behavior of a dropl et confined in the th in film geometry. The \nsecond cell served for viewing perpendicularl y to the field direction and for studying the \nbehavior of a freely suspended dr oplet. It should be noted that th e main difficulty encountered in \nthe study of the behavior of bodies in a magnetic fluid is that the magnetic fluid is an opaque \nmedium, which complicates observations. However, relatively thin layers of a magnetic fluid (< \n100 μm) are transparent enough to study the behavior of  drops in such fluids. At first we studied \nthe droplet behavior under the action of ac elect ric field and then under the simultaneous action \nof ac electric and dc magnetic fields. Because of low interfacial tension, under the action of \ncomparatively weak electric and magnetic fields  significant deformation of droplets can take \nplace. \nUsing the first cell it was experimentally observed that at relatively low frequencies of electric \nfield, the droplet is flattened taking the shape of oblate ellipso id of revolution; at higher \nfrequencies, the droplet stretches along the for ce lines of the electric field and becomes prolate \nellipsoid. The deformation parameter, D, has been measured as a function of a frequency, f, and \nof a root-mean-square value, \nE, of an alternating electric field ( D=(b–a)/(b+a), here b is the \nsemiaxis parallel to the field and a is the semiaxis perpendicular to it). The obtained  \nexperimental dependencies of the deformation parameter on the electric field strength and \nfrequency are shown in Figs . 2 and 3. It was found that D linearly depends on 2E and rises with \nf increasing. At some cr itical value of frequency, D reverses a sign. It was observed that the \naction of a magnetic field leads to the stretching of droplets, theref ore when a droplet is flattened \nin a low-frequency electric fiel d, the deformation can be compensated by imposing an additional \nmagnetic field directed in parall el with the electric field. \nIn a low-frequency range of el ectric field the oblate ellipsoids arise at comparatively low E \nvalues (Fig. 4 b). With increasing electric field strength th e shape of a confined droplet changes: a \nhole appears in the middle of the oblate dr oplet and it takes a toroidal shape (Fig. 4 c). Upon a \nfurther rise in the electric fiel d strength the droplet shape beco mes distorted and forms a curved \ntoroid (Fig. 4 d). At the final stage, the droplet bursts into several smaller droplets as the electric \nfield increases (Fig. 4 e). These small droplets then begin to rotate. The evolution of the confined \ndroplet shape with increasing electr ic field strength at a constant field frequency is shown in Fig. \n4. The values of E at which the mentioned droplet conf igurations can appear depend on the \nfield frequency. The experimentally obtai ned phase diagram showing the ranges of E and f in \nwhich the different droplet configura tions appear, is presented in Fig. 5 a. The additional action \nof a dc magnetic field aligned with the electr ic field changes the dr oplet shape evolution. \nParticularly sufficiently strong magnetic field can lead to the disappearance of the toroidal \nconfiguration of a droplet and the phase of dropl et shape distortion comes right after the oblate \nellipsoid configuration. The phase diagrams illust rating the influence of an additional magnetic \nfield on the droplet shape dynamics  at two different values of a magnetic field strength ( H = 550 \nand 920 A/m) are presented in Figs. 5 b and 5 c. \nAs it was already mentioned abov e, the electrorotation of so lid particles and liquid drops \nsuspended in a leaky dielectric liquid has be en widely studied. The undertaken explorations \nshow that the rotation of dropl ets under the action of electric field in our experiments is a \nphenomenon quite different from that observed in the previous works. In the above experiments \nwith a confined droplet, the rotation can take place only for small droplets arisen from the \ndisintegration of the initial  drop. For better study of the electrorotation phenomenon the \nobservations of the behavior of freely suspen ded droplets were carrie d out by means of the \nsecond cell. The observations show  that in our case the electror otation passes in the following \nway. It can be observed only in a low-frequency electric field when droplet flattening takes \nplace. At first the droplet flattens as the electri c field increases. When the droplet deformation \nreaches some critical value it begins to turn ove r and tends to be orientated by a major semiaxis \nto the electric field direction (droplet turns through  90°). After orientation the droplet restores to \nits original form of an oblate ellipsoid with a ma jor semiaxis perpendicular to an electric field.  \n5 \n And this events sequence occurs over and ove r again. The rotation direction may be either \nclockwise or counterclockwise. The consecutive snapshots of the droplet showing the described \nprocess are presented in Fig. 6. It should be noted that when a droplet is suspended freely not \nonly small droplets can rotate; the electrorotati on of droplets of any size was observed in this \ncase. Comparatively large droplets flattened in a low-frequency electric field and confined in \nthin film geometry cannot rotate because they are restricted by electrode s plates. The toroidal \nconfiguration was not observed fo r freely suspended droplets. In c onclusion it may be said that \nthe oblate deformation of a droplet is a nece ssary condition for the dr oplet rotation. On the \ncontrary, the rotation of par ticles and droplets observed in the previous works was not \nconditioned by the deformation. \nIt can be supposed that the observed unusual elec trorotation is caused by  a specific properties \nof a studied system such as quite low surface te nsion and high viscosity of liquid of a droplet \n(shape relaxation time is large, ~ 5 s). The dependences of the droplet rotation frequency, ν, on \nstrength and frequency of the electric field have  been measured (Figs. 7 and 8). For the purposes \nof detection of measurement errors three series of measurements were made. The results of all \nthree series of measurements are presented in  the figures. As is seen, the droplet rotation \nfrequency grows with the electri c field strength increasing. In  a low-frequency range of an \nelectric field the droplet rotation frequency gr ows too, but this dependence rapidly comes to \nsaturation with the electric field frequency increasing. It was found that under the additional \naction of a constant magnetic field directed in pa rallel to the electric field the rotation frequency \nof a droplet grows. The obtained experimental dependence of the droplet rotation frequency on \nthe magnetic field strength is presented in Fig.  9. But in the strong magnetic field the droplet \nrotation comes to a  stop when the compensation of a dr oplet deformation by a magnetic field \ntakes place. \nWhen a droplet confined in a thin film bursts into an ensemble of rotating smaller droplets, \nthese droplets repel each other in an electric fi eld and tend to draw up a hexagonal structure. \nDiffraction light scattering by this structure has been studied. For this purpose a laser beam was \ntransmitted through the emulsion layer (Fig. 10). Th e ratio of the radius of the first diffraction \nring to the distance to diffraction image, r/d, has been measured as a function of an electric field \nstrength and frequency; the resu lts are shown in Fig. 10. As is s een, the radius of the diffraction \nring grows with the electric field strength incr easing, and decreases with increasing frequency. \nThese results can be explained in the following  way. The diffraction ring radius is inversely \nproportional to the structure peri od. The action of sufficiently st rong electric field leads to the \nbreak-up of emulsion droplets, as a result the distance between the adjacent droplets diminishes \nwith electric field increasing and the ring radius grows. With refe rence to Fig. 5, it can be seen  \nthat the electric field strength at which the dr oplet break-up takes place gr ows with the electric \nfield frequency increasing. It can lead to the di minishing of the diffract ion ring radius with \nincreasing electric field frequency when the field strength is  fixed. In summary it may be said \nthat the diffraction experiment corroborates the results obtained by direct observations of the \ndroplet dynamics. \n \n3. Analysis and discussion \n \nThe measured droplet deformation has been co mpared with theoretical results of [5]. \nAccording to [5] the deformation paramete r can be calculated by following expression \nREФ De 2 0\n169\n ,        ( 1 )  \n])2( )1 2)[( 1(5)21)( 1( 15)]16 19()1(15[ )14 11(12 22 222 2\n   \nq A Bq A q B BФ\n     \nwhere 0 is the permittivity of free space, i e A0 , f2 , e i q , e i , \ni e B . The results of calculations are presented in Figs. 2 and 3. The comparison shows \nthat there is a qualitative agreement between theory and experiment, but one can see that the \nresults of measurements differ quantitatively from the calculati ons. The difference can be an \neffect of simplifying assumptions made in a first approximation theory [5]. Among these \nassumptions are the consideration of small defo rmations only, neglect of the surface conductance \nand the convection of charge, negl ect of the effects due to diffuse  ionic layers at the interface, \nneglect of the electrocap illarity effects, etc. \nThe appearance of toroidal shape of a droplet (Fig. 4 c) is a quite complicated phenomenon for \ndetailed analysis. As a hypothesis, it may be supposed that the appearance of a hole and the toroidal configuration of a dropl et result from the flow deve lopment inside and outside the \ndroplet. The flow pressure is maximal at the pole of an oblate droplet [5], at the same time the \nsurface pressure is minimal at the pole, as a resu lt the hole appears at th e droplet pole under the \naction of the flow pressure. It se ems that the possibility of rota tion of a freely suspended droplet \nchanges the flow pattern considerab ly, and the toroidal shape of a droplet does not appear in this \ncase. The described distortion of a droplet  shape in an electric field (Fig. 4 d) can take place in \nthin film geometry only and was not observed for freely suspended droplet. It suggests that this \neffect is similar in nature to the well-known in stability of magnetic fluid drops in a thin layer \nunder the action of a perpendicular magne tic field (see [15] for a review).  \n7 \n We shall restrict our consideration to the e ffect of droplet electr orotation. The specific \nelectrorotation described above can occur in a cas e when a droplet relaxation time is greater than \na characteristic time of orientation of an ellipsoid al droplet along the electric field. Therefore, we \nwill find and compare these times. The electric field produces torque acting on a droplet. According to [14], the electric torque, M\ne, can be written as \n    2sin 2sin Re0 2\n0 02\n31\ne yx x y e e M KK En n ba M    ,   (2) \n yx e i e e i yx n K,* * * * *\n31\n,   , \nwhere 0 , ,*\n,  jei ei ei  is the complex dielectric constant, j the imaginary unit, E0 the \nelectric field amplitude, α is the angle between the field direction and the ellipsoid major \nsemiaxis,  nx,y the depolarization factor given by \n e e\neenx atan 1\n32\n , 2 1x y n n , 122\n\nbae . \nWe suppose that the droplet shape and its semi axes are determined by (1) and are changeless \nduring the orientation process. The viscous torque acts in oppos ition to the el ectric torque. \nAccording to [19], the viscous torque, Mη, can be written in the form  \nC M ,        ( 3 )  \n\n nnb nab an n nVС\ny xab\nbaN\nab\nbaN\nx y\ne2 222 2\n21 , \n  ab\nba\nyab\nxba\nab\nba\ny x ab\nbanab n n nnb nanabN\nei \n2 14\n2 2 2\n, \n2 2b an n\nnx y\n\n  \nwhere ba V2\n34  is a droplet volume, dtd .  \nThe equation of rotary motion of a droplet has a form \nM M\ndtdIe22\n     ( 4 )  \nwhere 2 2\n5b a Im  is the moment of inertia  of an oblate ellipsoid, V m  the droplet mass. \nSubstitution of Eqs. (2) and (3) into Eq. (4) gives the differential equation of a droplet orientation \nin the form \n0 2sin 22\n02 2    dtd dt d     (5) \nwhere IC2 , I Me0 2\n0 . The performed calculations show  that the motion of studied \ndroplets is inertialess and the in ertial term (second derivative of an angle) of the equation of  \nmotion (5) can be neglected. In the approximati on of inertialess motion both torques are equal \nand oppositely directed. On this basis we can obtai n the equation of a dropl et orientation in the \nform  \n\n  0\n0\n01c o s 2 1c o s 2ln41 c o s 2 1 c o s 2eCtM \n      ( 6 )  \nwhere α0 is the initial value of α. The results of calculation of th e dependence (6) are presented in \nFig. 11. \nAccording to the small deformatio n theory [20], the evolution of D(t) during the drop \nrelaxation can be written as \nst DD  exp0 ,       ( 7 )  \n\n  16 193 21 40\n \ne i e ie is,  Re  \nwhere D0 is the initial droplet deformation defined by Eq. (1). The dependence D(t) calculated by \nEq. (7) is also presented in Fig. 11. The dependences α(t) and D(t) were calculated at three \ndifferent values of an electric field strength ( E0 = 170, 210 and 250 kV/m). As is seen, at low \nelectric field, the orientation time is greater than  a relaxation time. But with increasing electric \nfield, orientation time becomes less than relaxation time. From this it follows that the rotation of a droplet can take place when electric field stre ngth reaches some critical value in a complete \nconcordance with that observed in experiments.  \nEvidently the increase in an electric field streng th leads to the increasing of turning moment \nand the frequency of droplet rotation grows. The value of α = \n/4 corresponds to the instable \nequilibrium of a droplet, and α = 0 is an asymptote of the function α(t). To estimate the time, t, of \nthe droplet orientation we will assume that α0 = /4 – 10–5 rad and α = 10–5 rad, and substitute \nthese values into Eq. (6). This gives us an oppor tunity to estimate the droplet rotation frequency \nby an approximate expression ν ~ 1/(2 t). Fig. 7 shows the calculat ed dependence of the droplet \nrotation frequency on the el ectric field strength. As is seen, experimental results compare well \nwith theoretical calculations. The electric torque acting on a dr oplet with fluid less conductive \nthan that of the ambient fluid depends very weak ly on the electric field frequency, and so the \ndroplet electrorotation also should not depend  on it. The observed de pendence of a droplet \nrotation on an electric field frequency in a low fr equency range can be caused by the screening of \nreal electric field inside the cel l by the electric double layers which disappear  with electric field \nfrequency increasing.  \n9 \n Additional action of a parallel magnetic field leads to an increase in a droplet rotation \nfrequency. The magnetic field produces torq ue acting on a droplet. The magnetic torque, Mm, \nacting on a nonmagnetic oblate ellips oid immersed in a magnetic medium can be written as [21] \n\n 2 2\n0 1\nsin 2\n21 1yx\nm\nxynn V H\nM\nnn\n\n\n\n     ( 8 )  \nwhere 0=4·10–7 H/m,  is the magnetic permeability of magnetic fluid. The magnetic torque \n(8) is parallel to the electric torque (2). As a consequence under the ad ditional action of magnetic \nfield a total turning moment increases, and the dr oplet rotates faster. However the magnetic field \ndeforms the droplet and its shape cannot be determ ined by Eq. (1) in this case. It leads to the \ndifficulties in the analysis of the droplet rotation in electric and magnetic fields, and we will not \ndiscuss it here at greater length. \n \n4. Conclusions \n \nThus, the presented study demonstrates some new peculiarities of the droplet behavior such as \nthe appearance of toroidal shape and specific rotation under the action of external fields. The \nfreely suspended droplet and droplet confined in thin film geometry  under the simultaneous \naction of electric and magnetic fields have been studied.  \nIt should be noted that the study and design of  new composite material systems based on \nmagnetic fluids currently attracts much attenti on owing to wide potential applications of such \nsystems. One of new materials based on magnetic fluid is a magne tic fluid emulsion, disperse \nsystem composed of two liquid phases one of whic h is a magnetic fluid. In most studied cases \nthe droplets of such emulsions hold the spherica l shape and align in chain-like aggregates under \nthe action of magnetic field, see e.g. [22, 23]. Much le ss attention has been paid to the study of \nmagnetic fluid emulsion with deformable dropl ets. The deformation effect on the emulsion \nproperties has been considered onl y in [24, 25]. In this study we  show that the structure of \nsynthesized magnetic fluid emulsions can cons iderably depend on the magnetic and electric \nfields. It is obvious that the stru ctural organization in such systems can lead to the appearance of \nspecific features in their macros copic properties. The results of  the present study can also be \nused in the design of devices in wh ich magnetic fluid emulsions are used. \n \nAcknowledgments  \nThis work was supported by Russian Foundation for Basic Research (project No. 10-02-90019-\nBel_a) and also by Ministry of education and sc ience of the Russian Federation in scientific \nprogram “Development of Scientific  Potential of Higher School”. \n \nReferences \n [1] W.A. Macky, Some investigations on the deformation and breaking of water drops in \nstrong electric fields, Proc. R. Soc. Lond. A 133 (1931) 565–587. \n[2] J.S. Eow, M. Ghadiri, A. Sharif, Deforma tion and break-up of aque ous drops in dielectric \nliquids in high electric fields, J. Electrostat. 51–52 (2001) 463–469. \n[3] C.G. Garton, Z. Krasucki, Bubbles in insulati ng liquids: stability  in an electric field, Proc. \nR. Soc. Lond. A 280 (1964) 211–226. \n[4] G. Taylor, Studies in electrohydrodynamics . I. The circulation produced in a drop by \nelectrical field, Proc. R.  Soc. Lond. A 291 (1966) 159–166. \n[5] S. Torza, R.G. Cox, S.G. Mason, El ectrohydrodynamic deformation and burst of liquid \ndrops, Phil. Trans. R. Soc. Lond. A 269 (1971) 295–319. \n[6] J.D. Sherwood, The deformation of a flui d drop in an electric  field: a slender-body \nanalysis, J. Phys. A: Math. Gen. 24 (1991) 4047–4053. \n[7] H. Li, T.C. Halsey, A. L obkovsky, Singular shape of a fluid drop in an electric or magnetic \nfield, Europhys. Lett. 27 (1994) 575–580. \n[8] H.A. Stone, J.R. Lister, M.P.  Brenner, Drops with conical ends in electric and magnetic \nfields, Proc. R. Soc. Lond. A 455 (1999) 329–347. \n[9] O.A. Basaran, T.W. Patzek, R.E. Benner Jr ., L.E. Scriven, Nonlinear oscillations and \nbreakup of conducting, inviscid drops in an exte rnally applied electric  field, Ind. Eng. \nChem. Res. 34 (1996) 3454–3465. \n[10] J.Q. Feng, Electrohydrodynamic behaviour of  a drop subjected to a steady uniform electric \nfield at finite electric Reynolds numbe r, Proc. R. Soc. Lond. A 455 (1999) 2245–2269. \n[11] J. Raisin, J.-L. Reboud, P. Atten, Electri cally induced deformations of water–air and \nwater–oil interfaces in relati on with electrocoalescence, J.  Electrostat. 69 (2011) 275–283. \n[12] S. Krause, P. Chandratreya, Electrorota tion of deformable fluid droplets, J. Colloid \nInterface Sci. 206 (1998) 10–18. \n[13] J.-W. Ha, S.-M. Yang, Electrohydrodynamics and electrorotation of a drop with fluid less \nconductive than that of the ambient fluid, Phys. Fluids 12 (2000) 764–772. \n[14] T.B. Jones, Electromechanics of Partic les, Cambridge University Press, London, 1995. \n[15] E. Blums, A. Cebers, M.M. Maiorov, Magnetic Fluids, de Gruyter, New York, 1997.  \n11 \n [16] Yu.I. Dikanskii, O.A. Nechaeva, A.R.  Zakinyan, Deformation of magnetosensitive \nemulsion microdroplets in magnetic and elec tric fields, Colloid J. 68 (2006) 137–141. \n[17] A.N. Tyatyushkin, M.G. Velarde, On the in terfacial deformation of a magnetic liquid drop \nunder the simultaneous action of electric and ma gnetic fields, J. Colloid  Interface Sci. 235 \n(2001) 46–58. \n[18] W. Yu, M. Bousmina, C. Zhou, Determin ation of interfacial te nsion by the retraction \nmethod of highly deformed drop,  Rheol. Acta 43 (2004) 342–349. \n[19] K.I. Morozov, Rotation of a droplet in a vi scous fluid, J. Exp. Theor. Phys. 84 (1997) 728–\n733. \n[20] J.M. Rallison, The deformation of small vi scous drops and bubbles in shear flows, Annu. \nRev. Fluid Mech. 16 (1984) 45–66. \n[21] L.D. Landau, E.M. Lifshitz, Electrodynami cs of continuous media, Pergamon Press, New \nYork, 1984. \n[22] M. Ivey, J. Liu, Y. Zhu, S. Cutillas, Ma gnetic-field-induced structural transitions in a \nferrofluid emulsion, Phys. Rev. E 63 (2001) 011403. \n[23] G.A. Flores, J. Liu, M. Mohebi, N. Jamasbi, Magnetic-field-induced nonequilibrium \nstructures in a ferrofluid emulsi on, Phys. Rev. E 59 (1999) 751–762. \n[24] A. Zakinyan, Y. Dikansky, Drops deformation and magnetic permeability of a ferrofluid \nemulsion, Colloids Surf. A: Physic ochem. Eng. Aspects 380 (2011) 314–318. \n[25] Y.I. Dikansky, A.R. Zakinyan, A.N. Tyatyushkin, Anisotropy of magnetic emulsions \ninduced by magnetic and electric fi elds, Phys. Rev. E 84 (2011) 031402. \n \nFigure Captions \nFig. 1. Sketch of the experimental setup: 1 – sample cell; 2 – Helmholtz coils; 3 – optical \nmicroscope; 4 – digital video camera; 5 – microscope slide; 6 – metal plates; 7 – cover glass; 8 – \ntransparent glass electrodes; 9 – fluoroplastic film. In the e xperiments electric and magnetic \nfields were parallel. Fig. 2. Droplet deformation parameter vs. the elect ric field strength at th ree different values of \nfield frequency: 1 – f = 20 Hz; 2 – f = 1 kHz; 3 – f = 5 kHz ( R = 28 μm). Dots are experiments; \nlines are calculations. Fig. 3. Droplet deformation parameter vs. the elect ric field frequency at th ree different values of \nfield strength: 1 – \nE = 160 kV/m; 2 – E = 175 kV/m; 3 – E = 190 kV/m ( R = 28 μm). Dots are \nexperiments; lines are calculations.  \nFig. 4. Confined dielectric droplet at  different values of electric field: a – E = 0; b – E = 150 \nkV/m; c – E = 200 kV/m; d – E = 315 kV/m; e – E = 420 kV/m ( f = 8 Hz). Electric field is \ndirected perpendicularly to the plane of a figure. \nFig. 5. Experimental phase diagrams showing the ranges of E and f in which the different \ndroplet configurations appear: ( a) – H = 0; ( b) – H = 550 A/m; ( c) – H = 920 A/m. The numbers \ndenote configurations: 1 – oblate ellipsoid; 2 – toroid; 3 – curved toroid; 4 – droplet burst and \nelectrorotation; 5 – intense electrohydrodynamic flows in the entire sample. Lines are the \napproximation of experimental data. Fig. 6. The consecutive snapshots showing the re petitive process of elec trorotation of a freely \nsuspended droplet. Fig. 7. Droplet rotation frequency vs. the electric field strength ( R = 5 μm, f = 8 Hz). Dots are \nexperiments; line is calculations. \nFig. 8. Droplet rotation frequency vs. the electric field frequency ( R = 6 μm, \nE = 520 kV/m). \nFig. 9. Droplet rotation frequency vs. the magnetic field strength ( R = 4 μm, f = 8 Hz, E = 400 \nkV/m). Fig. 10. Measured ratio of the radius of the fi rst diffraction ring to the distance to diffraction \nimage vs. the electric field strength (open symbols, f = 8 Hz) and frequency (filled symbols, \nE = \n300 kV/m). Lines are the approximation of experimental data. Fig. 11. Calculated dependences α(t) (solid lines) and D(t) (dashed lines) at di fferent values of \nelectric field: 1 – E\n0 = 170 kV/m, 2 – E0 = 210 kV/m, 3 – E0 = 250 kV/m ( f = 20 Hz, R = 25 μm).  \n   \n  \n     \nFIG.\n. 1. \n \n \n13  \n \n     \n \n \n \nFIG. 2. \n   0 1 2 3 4 5 6 7-0.4-0.35-0.3-0.25-0.2-0.15-0.1-0.050\nE2(V )1010 2 2/mD\n1\n23/c451\n/c452\n/c453 \n15 \n  \n       \n \n \n \nFIG. 3. \n    0 0.4 0.8 1.2 1.6 2-0.4-0.3-0.2-0.100.10.2\nf(10 Hz)5D\n123\n/c451\n/c452\n/c453 \nFIG.\n. 4. \n \n \n \n17 \n  \n \n \n \nFIG. 5. \n  0100200300400500600\n10110210310412345\n0100200300400500600\n10110210312345\n0100200300400500600\n101102103104f(Hz)E(kV/m)\n1345()cf(Hz)()bE(kV/m)f(Hz)()aE(kV/m) \nFIG.\n. 6. \n \n \n \n19 \n  \n       \n \n \n \nFIG. 7. \n    340 360 380 400 420 440 4600.050.10.150.20.250.3\nE(kV/m)ν(s )/c451 \n \n       \n \n \n \nFIG. 8. \n    00.10.20.30.40.50.6\n0 1 02 03 04 05 06 07 0ν(s )/c451\nf(Hz) \n21 \n  \n          \n \n \n \nFIG. 9. \n    \n00 4.00 9.01 4.01 9.02 4.02 9.03 4.\n05.1 1 5 .2 2 5 .3\nH(kA/m)ν(s )/c451 \n \n      \n \n \n \nFIG. 10. \n    300 350 400 450235 15 25 35 45\n11.52\nE(kV/m)f(Hz)\n()E\n10/c452rd()f\n10/c452rd \n23 \n  \n        \n \n \n \nFIG. 11. \n0 5 10 15 2000.511.5\n00.5\n0.10.20.30.4||D\nt(s)1 2 3\n123α"    },    {        "title": "1702.06309v1.Numerical_simulations_of_magnetic_billiards_in_a_convex_domain_in___mathbb_R__2_.pdf",        "content": "Numerical simulations of magnetic billiards in a convex domain in R2\nPeter Albers Gautam Dilip Banhatti Michael Herrmann\nSeptember 22, 2018\nAbstract\nWe present numerical simulations of magnetic billiards inside a convex domain in the plane.\n1 Introduction\nIn this article we present some numerical simulations of magnetic billiards inside a convex domain in R2.\nClassical Billiards is a simple dynamical system which shows up in various branches of mathematics.\nIt is extremely well studied with deep results and many open questions at the same time.\nRobnik and Berry, [RB85], were the \frst to study numerically magnetic billiards in the plane. It\nseems that this article is still a main source for numerical results of classical magnetic billiards, in\nparticular in ellipses, see for instance page 1 in [BM16]. Newer sources for numerical results are for\ninstance [MBG93, BK96]. The quantum mechanical analogue seems to be much more studied but\ndoesn't concern us here.\nThe purpose of this article is to provide a more detailed numerical study of classical magnetic\nbilliards inside a convex domain (mostly ellipses). We do not claim any originality and consider\nthis purely as a service to the community. We used Matlab for the numerical computations and\nMathematica to illustrate the results.\n2 Magnetic billiards inside a convex domain\nWe brie\ry describe the dynamical system. For that let \u0006 \u001aR2be a curve bounding a strictly\nconvex domain T. For our purposes being a curve means that \u0006 is a smooth embedded compact 1-\nmanifold without boundary. This assumptions are very restrictive and, for instance, exclude billiards\nin polygons. A much more general account and set-up can be found in the book by Tabachnikov,\n[Tab05]. The domain Tbounded by \u0006 is called the table.\nNon-magnetic billiard inside Tdescribes the motion of a free particle which undergoes elastic\nre\rection at the boundary, that is, the point particle moves with constant speed on a straight line until\nit hits the boundary. Then this straight line is re\rected according to the law 'angle of incidence = angle\nof re\rection', i.e., the tangential component of the velocity is kept whereas the normal component is\n\ripped, see Figure 1.\nFor magnetic billiards, a charged particle moves in a constant magnetic \feld which is perpendicular\nto the plane containing the table. Thus the particle moves on a circle instead of a straight line. The\nre\rection law at the boundary is unchanged, see Figure 1. The radius of the circle is determined by\nthe speed of the particle and the strength of the magnetic \feld. We \fx the speed of the particle such\nthat the radius is precisely the inverse of the strength of the magnetic \feld. We call this the Larmor\nradius.\nBoth billiards can be described as a map on \u0006 \u0002(\u0000\u0019\n2;\u0019\n2). The pair describes a point of incidence\nwith outgoing direction. The billiard map sends such a pair to the next point of incidence together\nwith the outgoing direction arising by following a straight line / \fxed-radius circle. Thus we think\nof the billiard map as discrete dynamical system on \u0006 \u0002(\u0000\u0019\n2;\u0019\n2). It preserves the symplectic form\nInstitutes for Applied and Pure Mathematics, University of M unster, Germany\nE-mail: peter.albers/g banh02/michael.herrmann@uni-muenster.de\n1arXiv:1702.06309v1  [math.DS]  21 Feb 2017\u000b \u000b\u0006\n\u000b \u000b\n\u0006\nFigure 1: Billiard re\rection without (left) and with (right) magnetic \feld\n!= cos\u001ed\u001e^dswheresis the arc-length coordinate on \u0006. Moreover, it is well-known that non-\nmagnetic billiards with \u0006 being an ellipse forms an integrable system, see for instance [Tab05] and\nExample 0. Recently, it was (roughly speaking) proved in [BM16] that the only (algebraic) integrable\nmagnetic billiard occurs for \u0006 being a circle, see the article for the precise statement.\nKAM theory asserts that perturbations of the integrable billiard contains many invariant curves.\nWe will numerically demonstrate this for perturbations being a non-zero magnetic \feld and a non-\nelliptical table.\n3 Numerical setup\nWe brie\ry describe the geometric setup and the numerical algorithm. Our table Tis always bounded\nby the curve\n\u0006 =n\n(x;y)2R2:jxjp+1\n1\u0000\"2jyjp= 10po\nwith eccentricity \"and powerpas free parameters, where p= 2 corresponds to an ellipse. We simulate\nmagnetic billiard inside Tchoosing an external magnetic \feld of strength Band assuming that the\nparticle speed is normalized such that the Larmor radius is B\u00001. For parameterizing phase space we\nuse angular coordinates ( \u0012pos;\u0012vel)2(0;2\u0019)\u0002(\u0000\u0019\n2;+\u0019\n2) , where\u0012posdenotes the polar angle of points\nin \u0006 and is hence not identical with the arc-length.\nThe numerical algorithm is illustrated in Figure 2 and easy to implement. For some choices of the\nparameters we chose a large number (= number of orbits ) of random initial data and computed for\neach orbit a large number of particle re\rections (= points per orbit ). The underlying probability\ndistribution was uniform with respect to ( \u0012pos;\u0012vel)2(0;2\u0019)\u0002(\u0000\u0019\n2+\u000e;+\u0019\n2\u0000\u000e), where the small\ncut-o\u000b parameter \u000eexcludes degenerate orbits.\nThe numerical results are displayed in the following \fgures. Each \fgure contains a phase space\nplot in which six orbits are colored. For these we also plot a certain number (= points on table ) of\nre\rections in con\fguration space, i.e., on the table.\ninside the table\noutside the table\non the boundary of the table\nvelocityw=W(q,m) after reflection\nchoose small flight time dt\ncenterm=M(p, v) of circular trajectory\nupdate flight time dt:=12dt\nq=Q(p, m,dt)2R2by circular flight\nupdate pointp:=q\ninput (p, v)\noutput (q, w)\nlocation ofq\nFigure 2: Flow chart for the numerical computation of the billiard map with input and output belonging to\n\u0006\u0002(\u0000\u0019\n2;\u0019\n2). The prescribed numerical accuracy of order 10\u00009enters in determining whether the point qlies\non \u0006.\n2θvel\nθposmagnetic field B= 0.0 number of orbits = 1000\neccentricity \" = 1.5 points per orbit = 1000\npowerp = 2.0 points on table = 1000\nExample 0: Zero magnetic \feld and elliptic table.\n3θvel\nθposmagnetic field B= 0.01 number of orbits = 1000\neccentricity \" = 1.5 points per orbit = 1000\npowerp = 2.0 points on table = 2000\nExample 1: Small magnetic \feld and elliptic table.\n4θvel\nθposmagnetic field B= 0.5 number of orbits = 1000\neccentricity \" = 1.5 points per orbit = 1000\npowerp = 2.0 points on table = 500\nExample 2: Moderate magnetic \feld and elliptic table.\n5θvel\nθposmagnetic field B= 1.0 number of orbits = 1000\neccentricity \" = 1.5 points per orbit = 1000\npowerp = 2.0 points on table = 500\nExample 3: Large magnetic \feld and elliptic table.\n6θvel\nθposmagnetic field B= 2.0 number of orbits = 1000\neccentricity \" = 1.5 points per orbit = 1000\npowerp = 2.0 points on table = 500\nExample 4: Even larger magnetic \feld and elliptic table.\n7θvel\nθposmagnetic field B= 0.0 number of orbits = 1000\neccentricity \" = 1.5 points per orbit = 1000\npowerp = 2.005 points on table = 1000\nExample 5: Zero magnetic \feld and slightly non-elliptic table.\n8θvel\nθposmagnetic field B= 1.0 number of orbits = 2000\neccentricity \" = 1.5 points per orbit = 3000\npowerp = 2.0\nExample 6: Phase portrait of Example 3 with higher resolution and random color for each orbit.\n9References\n[BK96] N. Berglund and H. Kunz, Integrability and ergodicity of classical billiards in a magnetic\n\feld, J. Statist. Phys. 83(1996), no. 1-2, 81{126.\n[BM16] M. Bialy and A.E Mironov, Algebraic non-integrability of magnetic billiards , Journal of\nPhysics A: Mathematical and Theoretical 49(2016), no. 45, 455101 (18pp).\n[MBG93] O. Meplan, F. Brut, and C. Gignoux, Tangent map for classical billiards in magnetic \felds ,\nJ. Phys. A 26(1993), no. 2, 237{246.\n[RB85] M. Robnik and M. V. Berry, Classical billiards in magnetic \felds , J. Phys. A 18(1985),\nno. 9, 1361{1378.\n[Tab05] S. Tabachnikov, Geometry and billiards , Student Mathematical Library, vol. 30, American\nMathematical Society, Providence, RI, 2005.\n10"    },    {        "title": "0906.1753v1.Unified_nonequilibrium_dynamical_theory_for_exchange_bias_and_training_effects.pdf",        "content": "arXiv:0906.1753v1  [cond-mat.mes-hall]  9 Jun 2009Unified nonequilibrium dynamical theory for exchange bias a nd training effects\nKai-Cheng Zhang and Bang-Gui Liu\nInstitute of Physics, Chinese Academy of Sciences, Beijing 100190, China and\nBeijing National Laboratory for Condensed Matter Physics, Beijing 100190, China\n(Dated: December 4, 2018)\nWe investigate the exchange bias and training effects in the F M/AF heterostructures using a\nunified Monte Carlo dynamical approach. This real dynamical method has been proved reliable and\neffective in simulating dynamical magnetization of nanosca le magnetic systems. The magnetization\nof the uncompensated AF layer is still open after the first fiel d cycling is finished. Our simulated\nresults show obvious shift of hysteresis loops (exchange bi as) and cycling dependence of exchange\nbias (training effect) when the temperature is below 45 K. The exchange bias fields decrease with\ndecreasing the cooling rate or increasing the temperature a nd the number of the field cycling. With\nthe simulations, we show the exchange bias can be manipulate d by controlling the cooling rate, the\ndistributive width of the anisotropy energy, or the magneti c coupling constants. Essentially, these\ntwo effects can be explained on the basis of the microscopical coexistence of both reversible and\nirreversible moment reversals of the AF domains. Our simula ted results are useful to really under-\nstand the magnetization dynamics of such magnetic heterost ructures. This unified nonequilibrium\ndynamical method should be applicable to other exchange bia s systems.\nPACS numbers: 75.75.+a.75.20.-g,75.60.-d,05.70.Ln\nI. INTRODUCTION\nUsually, when the heterostructure consisting of cou-\npledferromagnetic(FM)andantiferromagnetic(AF)lay-\ners is cooled in field below the Neel temperature of its\nAF component, it shows the asymmetric magnetization\n[1, 2, 3, 4, 5, 6], which is referred to as the exchange\nbias effect. Furthermore, the exchange bias field, defined\nas the average of the two coercive fields, is observed to\ndecrease with increasing the number of the consecutive\nfield cycling, which is referred to as the training effect[7].\nThe exchange bias and training effects are very interest-\ning and could be used in future spintronics[8, 9, 10] and\ndata storage. Usually, the FM layer is taken as a whole\nand the AF layer consists of many grains. The AF grain\nis small enough to consists of a single domain, and some\nuncompensated domains (or grains) may be formed by\ndefects or impurities[11, 12, 13] and couple with each\nother and with the FM domains. As the heterostruc-\nture is cooled to a low temperature, the uncompensated\nspins in the grains and domains become locked-in and\nprefer to a unidirection in the interface, thus contribute\nto the magnetization shift[3]. Moreover, under the rever-\nsalofFMdomains,theuncompensatedgrainsordomains\nwill be irreversibly reorganized[14, 15, 16, 17, 18] and\nthus cause the training effect. The idea of domain states\nwas corroborated in some Monte Carlo simulations[19].\nOn the other hand, Hoffmann[20] considered the biaxial\nanisotropy of the AF sublattices and solved it by varia-\ntional method. Actual nonequilibrium dynamical prop-\nerties of the magnetization are still waiting to be eluci-\ndated. It is highly desirable and needed to systematically\ninvestigate the two effects in a unified theory.\nIn this article we use a unified Monte Carlo dynam-\nical approach[21] to study the FM/AF heterostructure\nin order to investigate the exchange bias and trainingeffect. Our simulated result shows the obvious shift of\nhysteresis loops and the cycling dependence of exchange\nbias. The magnetization of uncompensated AF layer is\nstill open after the field cycling is finished. The exchange\nbias fields decrease with decreasing the cooling rate or\nincreasing the temperature and the number of the field\ncycling. With the simulations, we shows the exchange\nbias can be manipulated by controlling the cooling rate,\nthe distributive width of the anisotropy energy, or the\nmagnetic coupling constants. Essentially, these two ef-\nfects can be explained on the basis of the microscopically\nirreversible reversal of the AF domains. More detailed\nresults will be presented in the following.\nThe remaining part of this paper is organized as fol-\nlows. In next section we shall define our model and\ndiscuss our simulation method. In section III we shall\npresent our simulated results and analysis. In section\nIV we shall discuss the microscopic mechanism for the\nphenomena in a unified way. Finally, we shall give our\nconclusion in section V.\nII. MODEL AND METHOD\nAccording to experimental observations[22], for both\ncompensatedand uncompensatedAF layersthe easyaxis\ntends to form alongexternal coolingfield direction rather\nthan later rotating field direction. In our model the AF\nlayer consists of many AF domains, and the FM layer\nconsists of one single domain. Assuming the cooling field\nis applied parallel to the AF/FM interface, then all the\neasy axes of AF and FM domains lie in the plane of the\ninterface. The coupled bilayers of AF and FM domains\nare shown in the inset of Fig. 1(a). The rectangles of the\nwhite pattern represent the AF domains and the larger\nrectangleis the single FM domain. The AF domains cou-2\npleto eachotherantiferromagneticallyandthesingleFM\ndomain couples to all the AF domains ferromagnetically.\nWe define the zaxis along the common easy axis which\nlies in the interface plane. We apply the external field to\nsaturate the magnetization of the FM layer along the z\naxis.\nForsimplicity, weconsideralltheuncompensatedspins\nin the AF domains are the same. We use S′/vector sito denote\nthe spin vector of the ith AF domain and S/vector sto denote\nthat of the single FM domain, where S′andSarethe un-\ncompensated spin values and FM spin respectively. Then\nwe write the Hamiltonian of the bilayers in an external\nfield as\nH=−Ku(sz)2−/summationdisplay\nikui(sz\ni)2−/vectorB·(γ′/summationdisplay\ni/vector si+γ/vector s)\n+J1/summationdisplay\ni,j/vector si·/vector sj−J2/summationdisplay\ni/vector si·/vector s (1)\nwhereγ′=gµ0µBS′andγ=gµ0µBS. The first and sec-\nond terms represents the anisotropy of the FM domain\nand the AF ones, and Kuandkuiare the corresponding\nanisotropy constants. The third term represents the Zee-\nman energy of the moments due to the applied external\nfield. The fourth term represents the antiferromagnetic\ncoupling among the AF domains. The last term repre-\nsentsthe ferromagneticcouplingbetweentheFMandAF\ndomains.\nUsingθiandβto describe the angles of the i-th AF\nmomentandtheFMmomentdeviatingfromthecommon\neasy axis, we can express the energies of the FM domain\nand thei-th AF as\nHFM=−(J2/summationdisplay\nisis+Kucosβ+γBs)cosβ(2)\nand\nHAF\ni= (J1si/summationdisplay\njsj−J2sis−kuicosθi−γ′Bsi)cosθi(3)\nwhere both siandsare the scalars taking either 1\nor -1. Thus for the i-th AF domain the energy in-\ncrement is ∆ Ei=kuisin2θi−hi(cosθi−1), where\nhi= (−J1/summationtext\njsj+J2s+γ′B)si, and for the FM domain\nthe energy increment is ∆ E=Kusin2β−hF(cosβ−1),\nwherehF= (J2/summationtext\nisi+γB)s. We can express ∆ Eand\n∆Eias[21]\n∆E=Ku[(1+hF\n2Ku)2−(cosβ+hF\n2Ku)2] (4)\nand\n∆Ei=kui[(1+hi\n2kui)2−(cosθi+hi\n2kui)2] (5)\nAs a result, to reverse its moment, the the FM layer\nmust overcomes a barrier EF\nb=Ku(1 +hF/2Ku)2if\n|hF| ≤2Ku, or 2hFifhF>2Ku; and the i-th AF graina barrier Ei\nb=kui(1+hi/2kui)2if|hi| ≤2kui, or 2hiif\nhi>2kui. If the condition hF<−2Kuorhi<−2kuiis\nsatisfied, there is no barrier for the reversal.\nActually, for the distribution of the AF anisotropy en-\nergy we use a Gauss function, f(kui) = exp[ −(kui−\nku)2/σ2], whose σandkuare set to 30.0 meV and 50.0\nmeV unless stated otherwise. The anisotropy energy of\nthe FM domain is set 200.0 meV without losing main\nphysics. Thus the reversal rate for a spin to reverse is\nR=R0e−Eb/kBT, where Ebis the energy barrier and\nR0is the characteristic frequency. In our simulations,\nR0is set to 1 .0×109/s. We adopt a square lattice for\nthe AF domains and use 20 ×20 as its size. Since we are\nonly interested in the exchangebias and trainingeffect at\nthe nanoscale, the AF lattice is enough to capture main\nphysics. Furthermore, we assume the AF domains have\nuniform moment 4.0 µBand the FM domain 2000 µB.\nThe coupling constant J1is set to 4.0 meV, and J28.0\nmeV. In our simulations the system is quenched from a\nhigh-enoughtemperature such as 610K, at which the AF\nlayer is paramagnetic, to a low-enough temperature such\nas 10 K. The magnetization and exchange bias fields are\ncalculated at the low temperature 10 K unless the tem-\nperature is explicitly stated otherwise. The basic rate of\nchanging temperature is ν0= 50 K/s. The field sweeping\nrate is set to 0.5 T/s with the basic increment 0.1 T for\neach simulation step.\nIII. SIMULATED RESULTS AND ANALYSIS\nAt first, we let the AF/FM bilayers relax under a mag-\nnetic field of 5.0 T at a high temperature 610 K. This\ntemperature is enough to make both the FM layer and\nthe AF layer remain paramagnetic. When the tempera-\nture decreases, the average magnetization values of the\ntwo layers increases. The external field makes the aver-\nage magnetization of the FM layer have a large increase\nbelow 600 K, and reach nearly to the saturated value at\n500 K. When the temperature becomes lower than 60\nK, the average magnetization of the AF layer looks like\nthat of an antiferromagnet under an applied field and is\ndependent on the cooling rate ν. Then, we further cool\nthe bilayers under the same field. After the temperature\nreaches down to 10 K, we start to change the field while\nkeeping the temperature unchanged. The field decreases\nfrom 5.0T to -10.0T and then increasesback to 5.0T for\nthe first hysteresis. Repeating the field cycling, we will\nmake the second hysteresis loop. The simulated results\nare shown in Fig. 1.\nAs shown in Fig. 1(a), the origin of the first hysteresis\nis clearly shifted in the negative field direction and shows\nthe exchange bias. The exchange bias field is defined as\nHE= (Hcl+Hcr)/2, where HclandHcris the coercivity\nof the left and right branches. The left branch of the sec-\nond hysteresis moves towards the positive direction, but\nthe right branches of the first two loops almost coincide\nwith each other. Actually, any further loop almost does3\n-10 -5 0 5 10 -1.0 -0.5 0.0 0.5 1.0 \n-10 -5 0 5 10 0.06 0.09 0.12 0.15 \n  mFM \n 1st loop \n 2nd loop (a) P1 P2 \nP3 P4 \n  \n  mAF \nB (T) (b) \nFIG. 1: The first two hysteresis loops of the FM (a) and AF\n(b) layer at 10 K. The inset in (a) shows the AF/FM bilayers.\nThe hysteresis loop is obtained by changing the field in the\norder of P1-P2-P3-P4-P1.\nno difference in the right branch from the second hys-\nteresis. The shift of the second loop clearly demonstrates\nthat the bilayers magnetization depends on the cycling\nhistory, which is known as training effect. Fig. 1(b)\nshows the magnetization of the AF layer, which drops\nlargely and opens widely due to the irreversible reversal\nof the AF domains after the first field cycling is finished.\nThe subsequent magnetization is more smooth but still\nnot closed, indicating the continuing cycling dependence\nof exchange bias. This is consistent with other Monte\nCarlo simulations[19].\nWe study the effect of the temperature Ton the ex-\nchange bias field, HE, for different loops. Our simulated\nexchange bias fields as functions of Tfor the first two\nloops are shown in Fig. 2. For both of the two curves,\nthe data can be fitted by the simple function\n−µ0HE=a1e−(T/T1)b1(6)\nwherea1,T1, andb1are fitting parameters. For the\nfitting in Fig. 2, the parameters a1,T1, andb1takes\n4.26 T, 17.95 K, and 1.58 for the first loop, and 3.94\nT, 16.66 K, and 1.59 for the second loop. Our results\nare consistent with experimental observationthat the ex-\nchange bias field decreases with increasing temperature\n[12, 23, 24].\nThe exchange bias field is dependent on the field cy-\ncling number n. Our simulated result from n=1 ton=910 20 30 40 0123 \n -µ0HE (T) \nT (K)  1st loop \n 2nd loop \nFIG. 2: Temperature dependence of the exchange bias fields\nfor the first two loops. The exchange bias field is calculated\nat a given temperature after the system is cooled from 610 K\nto the temperature value. The lines are the fitting curves in\nterms of the simple function defined in Eq. (6).\n0 2 4 6 81.5 2.0 2.5 3.0 \n  -µ0HE (T) \nn  σ=30.0 \n  σ=20.0 \nFIG. 3: The loop-number dependence of the exchange bias\nfields for the Gaussian width σ=30.0 and 20.0 meV. The tem-\nperature is 10 K. All the data except for n= 1 can be well\nfitted by a simple function −µ0HE(n) =a2ρn+b2.\nisshownin Fig. 3. Here, the temperatureis keptat 10K,\nandσis set to 20.0 and 30.0 meV. For both of the curves\nin Fig. 3, the data points excepts of n= 1 are well fitted\nby the simple function −µ0HE(n) =a2ρn+b2, where\na2,b2, andρare the fitting parameters, taking 0.23 T,\n2.26 T, and 0.76 for σ= 30.0 meV, and 0.22 T, 1.51 T,\nand 0.74 for σ= 20.0 meV. The value of −µ0HE(1) usu-\nally is substantially above the extrapolation of the other\n−µ0HE(n) (n >1). These simulated results are in good\nagreement with experimental observation[7].\nSince the training effect reflects the non-equilibrium\ndynamical magnetization which is caused by the irre-\nversible reversal of meta-stable domains formed during\nquenching, the quenching rate must affect the exchange\nbias field. By changing the cooling rate ν, we study the4\n0 2 4 6 8 10 2.0 2.5 3.0 3.5 \n  -µ0HE (T) \nν/ν0 1st loop \n 2nd loop \nFIG. 4: The cooling-rate dependence of the exchange bias\nfields for the first two loops. The lines are the fitting curves\nin terms of Eq. (7).\nexchange bias field as the function of quenching rate ν.\nThe result is shown in Fig. 4. For both of the loops, the\ndata be well fitted by\n−µ0HE=a3ln(b3ν\nν0+1) (7)\nwherea3andb3are 0.292 T and 20847 for the first loop,\nand 0.262 T and 16248 for the second loop. The ex-\nchange bias field at 10 K increases logarithmically with\nincreasing the quenching rate.\nIt is interesting to investigate the dependence of the\nexchange bias field on the coupling constants J1andJ2.\nThe simulated results are shown in Fig. 5. As shown in\nFig. 5(a), the exchange bias field decreases with increas-\ningJ1. The training effect is nearly unchanged when J1\nchanges from 0 to 2meV, but diminishes to zero quickly\nwith increasing J1from 2meV. When J1is larger than 6\nmeV, the exchange bias field already becomes very small\nand the training effect is actually zero. In contrast, the\nJ2data points in Fig. 5(b) can be well fitted by the\nsimple function −µ0HE=a4(exp(J2/b4)−1), where a4\nandb4are the fitting parameters, taking 0.44 T and 4.01\nmeV for the first loop, and 0.51 T and 4.49 meV for\nthe second loop. It is clearly shown that the exchange\nbias field increases exponentially as J2increases. Both\nof the the exchange bias field and the training effect can\nbe enhanced by decreasing J1and increasing J2, or by\nincreasing J2/J1. This is consistent with experimental\ntrend[25]. In Fig. 6 we shows how the distributive width\nσof the AF anisotropy affects the exchange bias fields.\nClearly the exchange bias field increases with σ, and so\ndoes the training effect. Experimentally, the width can\nbe increased by the additional nonmagnetic impurities\nand the enhanced roughnessofthe AF crystallinephases.\nThis implies that the rougher the AF crystalline phases\nare, the larger the exchange bias and training effect. Our\nresult reveals that the exchange bias field is determined\nbyboththecouplingconstantsandthedistributivewidth4 6 8 10 12 14 04812 \n  -µ0HE (T) \nJ2 (meV)  1st loop \n 2nd loop (b) 0.0 2.5 5.0 7.5 0510 15 20 \n  -µ0HE (T) \nJ1 (meV)  1st loop \n 2nd loop \n(a) \nFIG. 5: The exchange bias fields as functions of the coupling-\nconstants J1(a) andJ2(b) for the first two loops.\n10 15 20 25 30 35 0123\n  -µ0HE (T) \nσ (meV)  1st loop \n 2nd loop \nFIG. 6: The exchange bias field as a function of the Gaussian\nwidthσfor the first two loops.\nof the AF domain anisotropy. These are useful to com-\npletely understand the phenomenon[25, 26].\nIV. TRENDS AND MICROSCOPIC\nMECHANISM\nAfter being cooled down to the low temperature, the\nAF layer has a non-zero net FM moment MAdue to the\ndriving of both the field and the FM layer. Assuming5\nthere are NAAF domains, on average we have the mo-\nments in part of all the NAAF domains aligning parallel\nalthough they are coupled with AF interactions. The ex-\nchange bias field is determined by the effective moment\nMA, the difference of MAbetween the first two loops\ndetermines the training effect. Naturally, both the ex-\nchange bias field and the training effect increase with\nincreasing J2and with decreasing J1, as shown in Fig. 5.\nActually, small J1doesnotaffecttheeffects, butlarger J1\nthan 2meV is harmful to the effects at 10 K. In addition,\nit is easily understood that MAdecreases with increasing\nthe temperature T. As a result, both the exchange bias\nfield and the training effect decrease with increasing T,\nas shown in Fig. 2. The exponential description in Eq.\n(6) reflects the fact that moment reversals are thermally\nactivated. It is reasonable that both the exchange bias\nfield and the training effect increase with increasing the\ncoolingrate ν, asshownin Fig. 4. Thisismainly because\nthe average magnetization of the AF layer increases with\nνwhen the temperature is below 50 K. When the cooling\nrate approaches to zero, both the exchange bias field and\nthe training effect should be zero. In another word, our\nresults should approach to those of corresponding equi-\nlibrium systems when the cooling rate νapproaches to\nzero.\nAs shown in Fig. 6, both the exchange bias field and\nthe training effect are nearly zero when the distributive\nwidthσof the AF anisotropy energy is smaller than 15\nmeV, but they increase substantially with increasing σ\nforσ >15 meV. This means that the effects are depen-\ndentonawidedistributionoftheanisotropyenergy. This\ncan be understood in terms of the changing of the energy\nbarriers with the external field. From P1 to P2 in Fig.\n1(a), the effective barrier of the FM layer decreases but\nis still high enough to avoid the reversal, but meanwhile,\nmore and more spins of the AF domains are reversed due\nto their lower energy barriers. At the point P2, the FM\nmoment is reversed with the help of the field and the re-\nversing of the AF domains. Anyway, some AF domains\nwithhighenergybarriershavetheirmomentsunchanged,\neven afterthe FM layerhas been reversed, and thus there\nis a net average moment of the AF domains parallel to\nthe moment of the FM layer. This net average moment\nincreaseswith the distributivewidth σ. Thisexplainstheincreasing of the exchange bias field and training effect\nwith increasing σ. The more the field cycling loops, the\nlonger the time. Actually, this is similar to reducing the\ncooling rate νin effect. As a result, the exchange bias\nfield decreases with increasing the number of the field\ncycling. The turning point of the time scale causes the\nlargest drop happens between the first loops.\nV. CONCLUSION\nIn summary, we use a unified Monte Carlo dynami-\ncal approach[21] to study the FM/AF heterostructure in\norder to investigate the exchange bias and training ef-\nfect. The magnetization of uncompensated AF layer is\nstill open after the first field cycling is finished. Our sim-\nulated result shows the obvious shift of hysteresis loops\n(exchange bias) and the cycling dependence of exchange\nbias (training effect). The exchange bias fields decrease\nwith decreasing the cooling rate or increasing the tem-\nperature and the number of the field cycling. With the\nsimulations, we show the exchange bias can be manip-\nulated by controlling the cooling rate, the distributive\nwidth of the anisotropy energy, or the magnetic coupling\nconstants. Essentially, these two effects can be explained\non the basis of the microscopical coexistence of both re-\nversible and irreversible moment reversals of the AF do-\nmains. Our simulated results are useful to really under-\nstand the magnetization dynamics of such magnetic het-\nerostructures which should be important for spintronic\ndevice and magnetic recording media [25, 26, 27]. This\nunified nonequilibrium dynamical method should be ap-\nplicable to other exchange bias systems.\nAcknowledgments\nThis work is supported by Nature Science Founda-\ntion of China (Grant Nos. 10874232, 10774180, and\n60621091), by the Chinese Academy of Sciences (Grant\nNo. KJCX2.YW.W09-5), and by Chinese Department of\nScience and Technology (Grant No. 2005CB623602).\n[1] S. Bruck, G. Schutz, E. Goering, X. S. Ji, and K. M.\nKrishnan, Phys. Rev. Lett. 101, 126402 (2008).\n[2] M. Gruyters and D. Schmitz, Phys. Rev. Lett. 100,\n077205 (2008).\n[3] Y. Ijiri, T. C. Schulthess, J. A. Borchers, P. J. van der\nZaag, and R. W. Erwin, Phys. Rev. Lett. 99, 147201\n(2007).\n[4] J. Eisenmenger, Z. P. Li, W. A. A. Macedo, and I. K.\nSchuller, Phys. Rev. Lett. 94, 057203 (2005).\n[5] S. Brems, D. Buntinx, K. Temst, and C. V.Haesendonck,\nPhys. Rev. Lett. 95, 157202 (2005).[6] J. Nogues and I. K. Schuller, J. Magn. Magn. Mater. 192,\n203 (1999).\n[7] A. Hochstrat, C. Binek, and W. Kleemann, Phys. Rev.\nB 66, 092409 (2002).\n[8] T. Zhao, A. Scholl, F. Zavaliche, K. Lee, M. Barry, A.\nDoran , M. P. Cruz, Y. H. Chu, C. ederer, N. A. Spaldin,\nR. R. Das, D. M. Kim, S. H. Baek, C. B. Eom, and R.\nRamesh, Nat. Mater. 5, 823 (2006); R. Ramesh and N.\nA. Spaldin, Nat. Mater. 6, 21 (2007).\n[9] V. Laukhin, V. Skumryev, X. Mart, D. Hrabovsky, F.\nSanchez, M. V. Cuenca, C. Ferrater, M. Varela, U. Lud-6\ners, J. F. Bobo, and J. Fontcuberta, Phys. Rev. Lett. 97,\n227201 (2006).\n[10] P. Borisov, A. Hochstrat, X. Chen, W. Kleemann, and\nC. Binek, Phys. Rev. Lett. 94, 117203 (2005).\n[11] P. Miltenyi, M. Gierlings, J. Keller, B. Beschoten, and\nG. Guntherodt, Phys. Rev. Lett. 84, 4224 (2000).\n[12] J. I. Hong, T. Leo, D. J. Smith, and A. E. Berkowitz,\nPhys. Rev. Lett. 96, 117204 (2006).\n[13] F. U. Hillebrecht, H. Ohldag, N. B. Weber, C. Bethke,\nand U. Mick, Phys. Rev. Lett. 86, 3419 (2001).\n[14] M. Bode, E. Y. Vedmedenko, K. V. Bergmann, A. Ku-\nbetzka, P. Ferriani, S.Heinze, andR.Wiesendanger, Nat.\nMater. 5, 477 (2006).\n[15] W. Kuch, L. I. Chelaru , F. Offi, J. Wang, M. Kotsugi,\nand J. Kirschner, Nat. Mater. 5, 128 (2006).\n[16] F. Nolting, A. Scholl, J. Stohr, J. W. Seo, J. Fompeyrine ,\nH. Siegwart, J. P. Locquet, S. Anders, J. Luning, E. E.\nFullerton, M. F. Toney, M. R. Scheinfein, and H. A. Pad-\nmore, Nature (London) 405, 767 (2000).\n[17] X. P. Qiu, D. Z. Yang, S. M. Zhou, R. Chantrell, K. O.\nGrady, U. Nowak, J. Du, X. J. Bai, and L. Sun, Phys.\nRev. Lett. 101, 147207 (2008).\n[18] T. Hauet, J. A. Borchers, P. Mangin, Y. Henry, and S.Mangin, Phys. Rev. Lett. 96, 067207 (2006).\n[19] U. Nowak, K. D. Usadel, J. Keller, P. Miltenyi, B.\nBeschoten, and G. Guntherodt, Phys. Rev. B 66, 014430\n(2002).\n[20] A. Hoffmann, Phys. Rev. Lett. 93, 097203 (2004).\n[21] Y. Li and B.-G. Liu, Phys. Rev. B 73, 174418 (2006);\nPhys. Rev. Lett. 96, 217201 (2006); B.-G. Liu, K.-C.\nZhang, and Y. Li, Front. Phys. China 2, 424 (2007).\n[22] W. Zhu, L. Seve, R. Sears, B. Sinkovic, and S. S. P.\nParkin, Phys. Rev. Lett. 86, 5389 (2001).\n[23] M. Gruyters, Phys. Rev. Lett. 95, 077204 (2005).\n[24] V. K. Valev, M. Gruyters, A. Kirilyuk, and T. Rasing,\nPhys. Rev. Lett. 96, 067206 (2006).\n[25] J. Camarero, J. Sort, A. Hoffmann, J. M. G. Martin, B.\nDieny, R. Miranda and J. Nogues, Phys. Rev. Lett. 95,\n057204 (2005).\n[26] S. Brems, K. Temst, and C. V. Haesendonck, Phys. Rev.\nLett. 99, 067201 (2007).\n[27] V. Skumryev, S. Stoyanov, Y. Zhang, G. Hadjipanayis,\nG. Dominique, and J. Nogues, Nature (London). 423, 850\n(2003)."    },    {        "title": "2305.16182v1.Crystallization_dynamics_of_magnetic_skyrmions_in_a_frustrated_itinerant_magnet.pdf",        "content": "Crystallization dynamics of magnetic skyrmions in a frustrated itinerant magnet\nKotaro Shimizu1, 2and Gia-Wei Chern2\n1Department of Applied Physics, The University of Tokyo, Tokyo 113-8656, Japan\n2Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA\n(Dated: May 26, 2023)\nWe investigate the phase ordering kinetics of skyrmion lattice (SkL) in a metallic magnet. The\nSkL can be viewed as a superposition of magnetic stripes whose periods are determined by the\nquasi-nesting wave vectors of the underlying Fermi surface. An effective magnetic Hamiltonian that\ndescribes the electron-mediated spin-spin interaction is obtained for a two-dimensional s-d model\nwith the Rashba spin-orbit coupling. Large-scale Landau-Lifshitz-Gilbert dynamics simulations\nbased on the effective spin Hamiltonian reveal a two-stage phase ordering of the SkL phase after a\nthermal quench. The initial fast crystallization of skyrmions is followed by a slow relaxation domi-\nnated by the annihilation dynamics of dislocations, which are topological defects of the constituent\nmagnetic stripe orders. The late-stage phase ordering also exhibits a dynamical scaling symmetry.\nWe further show that the annihilation of dislocations follows a power-law time dependence with a\nlogarithmic correction that depends on magnetic fields. Implications of our results for SkL phases\nin magnetic materials are also discussed.\nComplex magnetic textures such as vortices and\nskyrmions are not only of great fundamental interest\nin magnetism but also have important implications in\nthe emerging technology of spintronics [1–4]. Both vor-\ntices and skyrmions are nano-sized particle-like spin-\ntextures characterized by nontrivial topological invari-\nants. The presence of such complex patterns in metallic\nmagnetscouldgiverisetointriguingelectronicandtrans-\nport properties due to a nontrivial Berry phase acquired\nby electrons when traversing over closed loops of non-\ncollinear or noncoplanar spins [1, 5–7]. The well-studied\ntopological Hall effects [8–13] and topological Nernst ef-\nfects [14–16] are some of the representative examples.\nAlso importantly, such topological electronic responses\nin metallic magnets can be controlled via the manipula-\ntion of magnetic textures.\nIn magnetic materials, skyrmions are often stabilized\nin the form of a skyrmion lattice (SkL), which is a peri-\nodic array of such particle-like topological spin-textures.\nIndeed, SkL as a spontaneous ground state was al-\nready predicted in the pioneering work of Bogdanov and\nYablonskii that later triggered the enormous interest in\nmagnetic skyrmions [17, 18]. SkLs have since been re-\nported in several chiral magnets such as MnSi [19] and\nother B20 compounds, as well as centrosymmetric ma-\nterials [20]. While the picture of SkL as an array of\nparticle-like objects offers an intuitive framework to un-\nderstand certain structural and dynamical aspects of\nskyrmion phases [21], recent studies have revealed the\nclose connection between SkL and multiple- Qmagnetic\nordering driven by partial nesting of the electron Fermi\nsurface [22–24]. Indeed, this mechanism has been conjec-\ntured to be ubiquitous for itinerant spin systems with a\nwide range of filling fractions.\nDespite the huge interest and extensive research on\nmagnetic SkL over the past decades, the phase-ordering\ndynamics of SkLs remains an open subject. Specifically,here one concerns the dynamical evolution and potential\nuniversal behaviors of skyrmion crystallization when a\nmagnet is quenched into a skyrmion phase. It is known\nthat the kinetics of phase ordering depends crucially\non topological defects of the symmetry-breaking phase.\nSeveral super-universality classes of domain-growth laws\nhave been established over the years [25, 26]. The fact\nthatbothmagneticandtranslationalsymmetriesarebro-\nken in a skyrmion crystal indicates rich relaxational dy-\nnamics of SkL phases which has yet to be systematically\ninvestigated. Understanding the phase ordering of SkL\nis also crucial to the engineering and control of skyrmion\nphases in real materials.\nIn this paper, we make an important step toward this\ngoal by investigating the crystallization dynamics of a\nSkL in a realistic model of chiral metallic magnets. A\nminimum microscopic model for such itinerant spin sys-\ntems is the s-d Hamiltonian Hsd=P\nk,σϵk,σc†\nk,σck,σ−\nJsdP\niSi·c†\niσσσ,σ′ciσ′, where the first term describes\nelectron hopping on a lattice with tijthe transfer in-\ntegrals and the second term represents a local coupling\nbetween the itinerant s-electron and the local moments\nSiof d-electrons; Jsdrepresents the coupling strength.\nSpin-orbit coupling (SOC) within such single-band mod-\nelscanbedescribedbyeitherRashbaorDresselhaushop-\nping terms. Dynamical simulations based on such elec-\ntron models requires solving a disordered electron tight-\nbinding Hamiltonian at every time-step, which could be\nprohibitively expensive for large systems. Instead, here\nwe consider an effective spin Hamiltonian, similar in\nspirit to the Ruderman-Kittel-Kasuya-Yosida (RKKY)\ninteraction [27–29], with SOC properly included [30].\nAt the leading second-order perturbation, the effective\nHamiltonian has the general form\nH=−X\nijSi·J(ri−rj)·Sj−X\niH·Si.(1)arXiv:2305.16182v1  [cond-mat.str-el]  25 May 20232\nHere we have included a Zeeman coupling to an external\nfieldH= (0,0, H), and J(r)represents an effective 3×3\ninteraction matrix between two spins separated by r. Its\nFourier transform is given by\n˜J(q) =J2\nsd\nNX\nkX\nσ,σ′χ0\nσσ′(k,q)h\nI+Fσσ′(k,q)i\n,(2)\nwhere χ0\nσσ′= [f(ϵk,σ)−f(ϵk+q,σ′)]/(ϵk+q,σ′−ϵk,σ)is the\nspin-dependent susceptibility, ϵk,σis the electron band\nenergy, f(ϵ)is the Fermi-Dirac function, Iis the identity\nmatrix, and the dimensionless matrix Fσσ′accounts for\nthe anisotropic spin interaction due to SOC [30, 31].\nThe interaction matrix ˜J(q)is often dominated by a\nfew wave vectors Qηwhen part of the electron Fermi\nsurface is connected by them, i.e. ϵk+Qη,σ′≈ϵk,σ. Such\npartial nesting of the Fermi surface has been shown to be\na primary mechanism for the stabilization of skyrmion\nor vortex lattices in metallic magnets [23]. An effec-\ntive real-space Hamiltonian is then given by the inverse\nFourier transformation of ˜J(q), which likely can only\nbe done numerically for general dispersion relation ϵk,σ.\nAs a first-order approximation, effective spin Hamiltoni-\nans can be obtained by keeping only contribution from\nthe nesting wave vectors: ˜J(q)≈P\nη˜J(Qη)δ(q−Qη).\nThis approach has been employed to investigate com-\nplex spin textures in the ground state of itinerant mag-\nnets [23, 24, 32–34]. However, this approach gives rise to\nunrealisticinfinite-rangespininteractions. Amorerealis-\ntic analytical approach, while preserving the correct form\nof the anisotropic interaction, is to replace the δ-function\nby a peak-shape function h(q)of a finite width, giving\nrise to spin-spin interactions which decay with distance\nin real space [35].\nFor concreteness, here we apply the procedure de-\nscribedabovetoasquare-lattices-dmodelwithaRashba\nSOC. A square array of skyrmions can be stabilized by\npartial nesting with two wave vectors Q1= (Q,0)and\nQ2= (0, Q). The resultant real-space interaction matrix\nis given by\nJ(r) =X\nη=1,2g(r) [ReJηcos(Qη·r)−ImJηsin(Qη·r)],(3)\nwhere\nJ1=\nJ⊥0 0\n0J⊥−iD\n0iD Jzz\n,J2=\nJ⊥0−iD\n0J⊥0\niD 0Jzz\n.(4)\nandJ⊥,Jzzdescribe electron-mediated exchange in-\nteractions, and Drepresents effective long-ranged\nDzyaloshinskii-Moriya interaction (DMI) induced by\nSOC; both JandDare of the order of J2\nsd/W, where\nWis the electron bandwidth. In the following, we set\n2J⊥+Jz= 1to serve as the unit for energy (and inverse\ntime) and D= 0.3. The function g(r)describes the de-\ncay of spin-spin interaction with distance. For simplicity,\nSkLFFM\n1Q\n(a) (b)FIG. 1. (a) The ground state phase diagram includes a\nsquare Skyrmion lattice (SkL), an 1 Qstate, and a forced fer-\nromagnetic state (FFM). (b) The real-space spin texture for\nJzz= 0andH= 0. The color of the arrows represents the\nout-of-plane component of spins.\nwe assume a Lorentzian function for h(q), which leads to\nan exponential decaying g(r) =Ae−γ(|x|+|y|), where A\nis a normalization constant [31]. A hard cutoff rcsuch\nthat J(r) = 0for|x|+|y|> rcis further introduced\nfor large-scale simulations of N= 10002spins. For re-\nsults presented in the following, parameters γ= 0.3and\nrc= 16are used.\nThe dynamical evolution of the magnet is described by\nthe Landau-Lifshitz-Gilbert (LLG) equation\ndSi\ndt=1\n1 +α2\u0014∂H\n∂Si×Si+αSi×\u0012\nSi×∂H\n∂Si\u0013\u0015\n,(5)\nwhere αistheGilbertdampingcoefficient, whichissetto\n0.1 in the following simulations. A fourth-order Runge-\nKuttamethodisusedtointegratetheLLGequationwith\na time-step ∆t= 0.05. The phase diagram, shown in\nFig. 1(a) in the plane of exchange anisotropy Jzzversus\nfield H, includes a 1 Q-cycloidal order, a double- QSkL,\nand a forced ferromagnetic state at high field. It is worth\nnoting that our results are consistent with the phase dia-\ngram obtained from simulated annealing minimization of\nthe infinite-range effective spin model [30]. The double-\nQmagnetic order at H= 0, shown in Fig. 1(b), exhibits\na non-coplanar Néel-type vortex texture.\nImportantly, for small Jzz, the SkL is stable for a wide\nrange of magnetic field, allowing us to study the crys-\ntallization dynamics of skyrmions and the field effects.\nTo this end, the LLG dynamics was employed to simu-\nlate thermal quenches of a spin system with the effec-\ntive RKKY interaction in Eq. (3). An initial state of\nrandom spins, corresponding to equilibrium at high tem-\nperatures, is suddenly quenched to zero temperature at\ntime t= 0. A typical example of the subsequent evolu-\ntion of spins is shown in Fig. 2 for the case of zero-field\nquench. The color shows the scalar chirality which is de-\nfined as χsc\ni=P\n△iS1·S2×S3/2, where the summation\nis taken over four triplets of nearest spins on a trian-\ngle. As skyrmions, including the Néel vortex texture at\nzero field, are characterized by non-coplanar spins that3\n(a) (b)\n(c) (b)\nFIG. 2. Spatial distribution of the scalar spin chirality χsc\ni\nobtained from spin configurations at different times after a\nthermal quench. Parameters Jzz= 0, and H= 0are used\nin the LLG simulation of a 10002system. Shown here are a\nselected 200×200region of the lattice.\nwrap around a sphere, the emergence of a SkL domain\nis indicated by the staggered checkerboard-like arrange-\nment of positive and negative scalar chirality. As shown\nin Figs. 2(a) and (b), small patches of skyrmion arrays\nquickly emerge after the thermal quench. However, long-\nrange coherence is yet to be established between different\npatches, andlargeareasofvanishingscalarchiralitymark\nthe boundaries between skyrmion crystallites.\nAs long-range crystallization order further develops,\nthe incoherent regions quickly contract to particle-like\nobjectsofsimilarsizeasaskyrmionasshowninFigs.2(c)\nand (d). These “particles”, also characterized by a van-\nishing χsc\ni, correspond to dislocation defects, which are\ntopological defects associated with broken translational\nsymmetries. The dislocations here correspond to a start-\ning point of an extra row or column of skyrmion lines\nin the square lattice. These particle-like defects carry a\ntopological charge corresponding to the so-called Burgers\nvector. Their topological nature also manifests itself in\nthe fact that dislocations are created and annihilated in\npairs. The phase-ordering of SkL is now dominated by\nthe dynamics of dislocation defects.\nThe magnetic structure of the double- QSkL can be\napproximated by an equal superposition of two spirals:\nS(ri)∼(cosQ1,cosQ2, b(sinQ1+ sinQ2) +m0),(6)\nwhere Qη=Qη·ri+const, and the coefficients band\nm0depend on model parameters and magnetic field. Im-\nportantly, the x- and y-component of the spin field cor-\nrespond to simple unidirectional stripes along the xand\nydirections, respectively. The fundamental defects of\n(a) (b) (c)\nFIG. 3. Spatial profile of the three components of spin field\nSiat late stage of the phase ordering of SkL. Parameters are\nthe same as those in Fig. 2.\nstripe order are also dislocations, as shown in Figs. 3(a)\nand (b). We can thus further classify the dislocation de-\nfects of SkL according to whether it is associated with\ntheSxorSycomponent. Indeed, our simulations find\nthat pair annihilations are possible only for dislocations\nof the same spin component.\nTo further quantify the phase ordering of SkL, we ex-\namine the time-dependent spin structure factor, defined\nasS(q, t) =1\nN2⟨|P\niSi(t) exp( iq·ri)|2⟩, where ⟨···⟩de-\nnotes the average of independent initial conditions. Ex-\namples of the structure factor at the early and late stages\nof phase ordering are shown in Figs. 4(a) and (b), respec-\ntively. The structure factor exhibits four broad peaks at\n±Q1and±Q2, where Q1= (Q,0),Q2= (0, Q), and\nQ= 0.785, quickly after the quench; see e.g. Fig. 4(a)\nfort= 40. These peaks at the nesting wave vectors\nbecome sharper as the system relaxes toward equilib-\nrium, as shown in Fig. 4(b). Moreover, satellite peaks\natn1Q1+n2Q2with n1+n2= 2n+ 1(nis an inte-\nger) start to emerge at late times, signaling the onset of\nhigher harmonics of the constituent spiral orders.\nAn overall order parameter of the SkL phase can be\ndefined as the sum of peak intensities M(t) =S(Q1, t)+\nS(Q2, t). As shown in Fig. 4(c), this SkL order pa-\nrameter clearly exhibits a two-stage ordering discussed\nabove: the fast development of quasi-long-ranged crys-\ntalline domains ( t<∼80), followed by a slow power-\nlaw growth dominated by the annihilation dynamics of\ndislocations ( t>∼80). To characterize the late-stage\nphase ordering, Fig. 4(c) also shows the growth of the\ncorrelation length defined as the inverse widths of the\nFourier peaks: ξQη= 1/∆Qη, where ∆Qη=P\nq∼Qη|q−\nQη|S(q, t)/P\nq∼QηS(q, t). Furthermore, the power-law\ngrowth of the correlation lengths is intimately related to\nthat of SkL order parameter M. Indeed, we find that\nthe late-stage ordering of SkL exhibits a dynamical sym-\nmetry. The structure factor at different times can be\ndescribed by a universal function Fwith proper rescal-\ning\nS(q, t) =ξ2\nQη(t)F\u0010\n|q−Qη|ξQη(t)\u0011\n. (7)\nThis is illustrated by the excellent data points collapsing\nshown in Fig. 4(d).4\n00(a) (b)\n(c) (d)\nFIG. 4. Spin structure factor S(q, t)with Jzz= 0and\nH= 0at (a) t= 40and (b) t= 400. (c) The SkL order\nparameter M(t) =S(Q1, t) +S(Q2, t)and the correlation\nlength ξQηversus time. Both exhibit a power-law growth\ntαwith exponent αM≈1.046andαξ≈0.476, respectively.\n(d) Scaling plot of the structure factor around the peak at Q1\nalong the qxaxis.\nThe above late-time power-law behaviors are related\nto the dynamics of dislocations, which are topological\ndefects of emergent square skyrmion arrays discussed\nabove. As the long-range coherence of a crystalline or-\nder is disrupted by dislocations, the correlation length of\nSkL can be interpreted as the average distance ℓbetween\ndislocations, which is related to their number density as\nℓ∼ρ−1/2\nd. The power-law growth of correlation length\nξ∼tαthus implies a power-law decrease of dislocation\ndensity ρd∼t−ηwith the exponent η= 2α. This is in-\ndeed confirmed in our large-scale LLG simulations sum-\nmarized in Fig. 5, where a naive power-law fitting gives\nan exponent ηthat depends weakly on magnetic field H.\nFor example, the zero-field exponent η∼1.03is con-\nsistent with the grow exponent of the correlation length\nshown in Fig. 4(c).\nTo understand the power-law annihilation of dislo-\ncations, we note that these topological defects can be\nmapped to vortices of effective 2D-XY models. As dis-\ncussed in Eq. (6), the SkL can be viewed as comprised of\ntwo spiral spin orders. The xandycomponents of the\ncoarse-grained spin field represent two orthogonal stripes\nSx(r)∼cos[Q1·r+θ1(r)]andSy(r)∼cos[Q2·r+θ2(r)].\nThe dislocations associated with the two stripes, shown\nin Figs. 3(a) and (b), correspond to vortex singularities\nof the phase fields θ1,2(r). At the leading-order approxi-\nmation, our system can thus be described by two coupled\nXY models:\nE=AZ\n{[(∇θ1)2+ (∇θ2)2] + 2κ(∇θ1· ∇θ2)}d2r,(8)\nwhere A > 0represents the stiffness of the phase fields,\nFIG. 5. Real-time dependence of the density of dislo-\ncations defects ρd(t)with Jzz= 0for varying magnetic\nfield H, obtained by averaging over 32 independent runs.\nThe positions of defects are detected from the energy den-\nsity [38]. The pale lines represent the fitting by the formula\nρd(t)/(log(ρ0/ρd(t))−1) = a(t−t0)−1.\nandκdenotes their coupling. A large κcould induce a\nbound state of vortices from the two XY fields. Nonethe-\nless, it is straightforward to show that the above action\nis equivalent to two independent XY fields θ±=θ1±θ2\nwith different stiffness: A±=A(1±κ)/2. Importantly,\nit has been shown that the phase ordering of the 2D XY\nmodel is governed by the annihilation of vortices, whose\nnumber density follows a power-law ρv∼t−1time de-\npendence, up to a logarithmic correction [36, 37]. Our\nresults thus strongly indicate that the ordering kinetics\nof the SkL phase belongs to the same dynamical univer-\nsality class of 2D XY model. In fact, while the extracted\nexponent is weakly dependent on H, the fact that all\nηvalues lie in the vicinity of unity suggests a universal\nρd∼t−1behavior, yet with a field-dependent logarith-\nmic correction. We have checked that the various curves\nin Fig. 5 indeed can be well described by the formula\nρd/(log(ρ0/ρd)−1) = a(t−t0)−1, where ρ0,t0, and aare\nfitting parameters [36].\nTo conclude, we have presented a comprehensive study\non the ordering dynamics of SkL phases in chiral metallic\nmagnets, which is important to the design and engineer-\ning of skyrmion-based spintronics devices. Fundamen-\ntally, magnetic skyrmion lattices also provide another\nplatform to study crystallization phenomena in two di-\nmensions, a field which has yet to be systematically in-\nvestigated. An intriguing fast crystallization process has\nrecently been observed in the “square ice” formed by wa-\nter molecules locked between two graphene sheets [39].\nWhile our work here sheds new light on this fundamen-\ntal subject of 2D crystallization dynamics, several im-\nportant issues remain to be addressed. For example, al-\nthough the long-range nature of electron-mediated inter-\nactions might be crucial to the initial fast crystallization,\na detailed study on the effect of interaction range and\nits interplay with other factors is desired. The built-in5\nchirality due to SOC is another crucial component for\nthe fast establishment of coherent lattices. The recent\nexperimental observation of square SkL in centrosym-\nmetric magnets such as GdRu 2Si2[40] and EuAl 4[41]\nthus calls for theoretical studies of skyrmion crystalliza-\ntion dynamics in non-chiral itinerant magnets [23, 42].\nYet another interesting question is the effects of crystal\ngeometry. More complicated annihilation dynamics of\ndislocations might arise in the triangular SkL observed\nin many skyrmion materials.\nThe authors thank Y. Kato and Y. Motome for fruit-\nful discussions. This work was supported by JSPS KAK-\nENHI Grant Number No. JP21J20812. K.S. was sup-\nported by the Program for Leading Graduate Schools\n(MERIT-WINGS). Parts of the numerical calculations\nwere performed in the supercomputing systems in ISSP,\nthe University of Tokyo. G.W.C was partially supported\nby the US Department of Energy Basic Energy Sciences\nunder Contract No. DE-SC0020330.\n[1] N. Nagaosa and Y. Tokura, Topological properties and\ndynamics of magnetic skyrmions, Nat. Nanotechnol. 8,\n899 (2013).\n[2] K.Everschor-Sitte, J.Masell, R.M.Reeve,andM.Kläui,\nPerspective: Magnetic skyrmions—Overview of recent\nprogress in an active research field, Journal of Applied\nPhysics 124, 10.1063/1.5048972 (2018), 240901.\n[3] C. Back, V. Cros, H. Ebert, K. Everschor-Sitte, A. Fert,\nM. Garst, T. Ma, S. Mankovsky, T. L. Monchesky,\nM. Mostovoy, N. Nagaosa, S. S. P. Parkin, C. Pfleiderer,\nN. Reyren, A. Rosch, Y. Taguchi, Y. Tokura, K. von\nBergmann, and J. Zang, The 2020 skyrmionics roadmap,\nJ. Phys. D: Appl. Phys. 53, 363001 (2020).\n[4] B. Göbel, I. Mertig, and O. A. Tretiakov, Beyond\nskyrmions: Review and perspectives of alternative mag-\nnetic quasiparticles, Physics Reports 895, 1 (2021), be-\nyond skyrmions: Review and perspectives of alternative\nmagnetic quasiparticles.\n[5] G. E. Volovik, Linear momentum in ferromagnets, J.\nPhys. C Solid State Phys. 20, L83 (1987).\n[6] M. V. Berry, Quantal phase factors accompanying adia-\nbatic changes, Proc. R. Soc. Lond. A 392, 45 (1984).\n[7] N. Nagaosa and Y. Tokura, Emergent electromagnetism\nin solids, Phys. Scr. T146, 14020 (2012).\n[8] D. Loss and P. M. Goldbart, Persistent currents from\nBerry’s phase in mesoscopic systems, Phys. Rev. B 45,\n13544 (1992).\n[9] J. Ye, Y. B. Kim, A. J. Millis, B. I. Shraiman, P. Majum-\ndar,andZ.Tešanović,BerryPhaseTheoryoftheAnoma-\nlous Hall Effect: Application to Colossal Magnetoresis-\ntance Manganites, Phys. Rev. Lett. 83, 3737 (1999).\n[10] P. Bruno, V. K. Dugaev, and M. Taillefumier, Topologi-\ncal Hall Effect and Berry Phase in Magnetic Nanostruc-\ntures, Phys. Rev. Lett. 93, 096806 (2004).\n[11] M. Onoda, G. Tatara, and N. Nagaosa, Anomalous Hall\nEffect and Skyrmion Number in Real and Momentum\nSpaces, J. Phys. Soc. Jpn. 73, 2624 (2004).\n[12] B. Binz and A. Vishwanath, Chirality inducedanomalous-Hall effect in helical spin crystals, Physica B:\nCondensed Matter 403, 1336 (2008).\n[13] K. Nakazawa and H. Kohno, Weak coupling theory of\ntopological hall effect, Phys. Rev. B 99, 174425 (2019).\n[14] Y. Shiomi, N. Kanazawa, K. Shibata, Y. Onose,\nand Y. Tokura, Topological Nernst effect in a three-\ndimensional skyrmion-lattice phase, Phys. Rev. B 88,\n064409 (2013).\n[15] Y. P. Mizuta and F. Ishii, Large anomalous Nernst effect\nin a skyrmion crystal, Sci. Rep. 6, 28076 (2016).\n[16] M. Hirschberger, L. Spitz, T. Nomoto, T. Kurumaji,\nS. Gao, J. Masell, T. Nakajima, A. Kikkawa, Y. Ya-\nmasaki, H. Sagayama, H. Nakao, Y. Taguchi, R. Arita,\nT.-h. Arima, and Y. Tokura, Topological Nernst Effect of\nthe Two-Dimensional Skyrmion Lattice, Phys. Rev. Lett.\n125, 076602 (2020).\n[17] A. N. Bogdanov and D. Yablonskii, Thermodynamically\nstable “vortices” in magnetically ordered crystals. The\nmixed state of magnets, Zh. Eksp. Teor. Fiz 95, 178\n(1989).\n[18] U. K. Roessler, A. Bogdanov, and C. Pfleiderer, Sponta-\nneousskyrmiongroundstatesinmagneticmetals,Nature\n442, 797 (2006).\n[19] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer,\nA. Rosch, A. Neubauer, R. Georgii, and P. Böni,\nSkyrmion Lattice in a Chiral Magnet, Science 323, 915\n(2009).\n[20] Y. Tokura and N. Kanazawa, Magnetic Skyrmion Mate-\nrials, Chem. Rev. (2020).\n[21] C. Reichhardt, C. J. O. Reichhardt, and M. V. Milošević,\nStatics and dynamics of skyrmions interacting with dis-\norder and nanostructures, Rev. Mod. Phys. 94, 035005\n(2022).\n[22] R. Ozawa, S. Hayami, K. Barros, G. W. Chern, Y. Mo-\ntome, and C. D. Batista, Vortex crystals with chiral\nstripes in itinerant magnets, J. Phys. Soc. Jpn. 85,\n103703 (2016).\n[23] S. Hayami, R. Ozawa, and Y. Motome, Effective bilinear-\nbiquadratic model for noncoplanar ordering in itinerant\nmagnets, Phys. Rev. B 95, 224424 (2017).\n[24] S. Hayami and Y. Motome, Topological spin crystals\nby itinerant frustration, Journal of Physics: Condensed\nMatter 33, 443001 (2021).\n[25] A. J. Bray, Theory of phase-ordering kinetics, Advances\nin Physics 43, 357 (1994).\n[26] S. Puri and V. Wadhawan, Kinetics of phase transitions\n(CRC press, 2009).\n[27] M. A. Ruderman and C. Kittel, Indirect Exchange Cou-\npling of Nuclear Magnetic Moments by Conduction Elec-\ntrons, Phys. Rev. 96, 99 (1954).\n[28] T. Kasuya, A Theory of Metallic Ferro- and Antiferro-\nmagnetism on Zener’s Model, Prog. Theor. Phys. 16, 45\n(1956).\n[29] K. Yosida, Magnetic properties of cu-mn alloys, Phys.\nRev.106, 893 (1957).\n[30] S. Hayami and Y. Motome, Néel- and Bloch-Type Mag-\nnetic Vortices in Rashba Metals, Phys. Rev. Lett. 121,\n137202 (2018).\n[31] A detailed discussion of the effective Hamiltonian for the\nsquare-lattice s-d model with Rashba SOC is presented\nin the supplemental material.\n[32] S. Okumura, S. Hayami, Y. Kato, and Y. Motome, Mag-\nnetic hedgehog lattices in noncentrosymmetric metals,\nPhys. Rev. B 101, 144416 (2020).6\n[33] R. Yambe and S. Hayami, Skyrmion crystals in cen-\ntrosymmetric itinerant magnets without horizontal mir-\nror plane, Sci. Rep. 11, 11184 (2021).\n[34] K. Shimizu, S. Okumura, Y. Kato, and Y. Motome,\nPhase transitions between helices, vortices, and hedge-\nhogsdrivenbyspatialanisotropyinchiralmagnets,Phys.\nRev. B 103, 054427 (2021).\n[35] Y. Kato, S. Hayami, and Y. Motome, Spin excitation\nspectra in helimagnetic states: Proper-screw, cycloid,\nvortex-crystal, and hedgehog lattices, Phys. Rev. B 104,\n224405 (2021).\n[36] B. Yurke, A. N. Pargellis, T. Kovacs, and D. A. Huse,\nCoarsening dynamics of the xy model, Phys. Rev. E 47,\n1525 (1993).\n[37] A. J. Bray, A. J. Briant, and D. K. Jervis, Breakdown\nof scaling in the nonequilibrium critical dynamics of the\ntwo-dimensional XYmodel, Phys. Rev. Lett. 84, 1503\n(2000).\n[38] A detailed discussion of the way to count the dislocations\nand the relationship between dislocations in the presentsystem and vortices in the XY model are presented in the\nsupplemental material.\n[39] G. Algara-Siller, O. Lehtinen, F. C. Wang, R. R. Nair,\nU. Kaiser, H. A. Wu, A. K. Geim, and I. V. Grigorieva,\nSquare ice in graphene nanocapillaries, Nature 519, 443\n(2015).\n[40] N. D. Khanh, T. Nakajima, X. Yu, S. Gao, K. Shibata,\nM. Hirschberger, Y. Yamasaki, H. Sagayama, H. Nakao,\nL. Peng, K. Nakajima, R. Takagi, T. Arima, Y. Tokura,\nand S. Seki, Nanometric square skyrmion lattice in a cen-\ntrosymmetric tetragonal magnet, Nat. Nanotechnol. 15,\n444 (2020).\n[41] R. Takagi, N. Matsuyama, V. Ukleev, L. Yu, J. S. White,\nS. Francoual, J. R. Mardegan, S. Hayami, H. Saito,\nK. Kaneko, et al., Square and rhombic lattices of mag-\nnetic skyrmions in a centrosymmetric binary compound,\nNat. Commun. 13, 1472 (2022).\n[42] S. Hayami and Y. Motome, Square skyrmion crystal in\ncentrosymmetric itinerant magnets, Phys. Rev. B 103,\n024439 (2021)."    },    {        "title": "2002.03535v1.Collisionless_particle_dynamic_in_an_axi_symmetric_diamagnetic_trap.pdf",        "content": "Collisionless particle dynamic in an axi-symmetric\ndiamagnetic trap\nIvan S. Chernoshtanov\n11, akademika Lavrentieva prospect, Novosibirsk, 630090 Russia\nE-mail: I.S.Chernoshtanov@inp.nsk.su\nNovember 2019\nAbstract.\nParticle dynamic in an axi-symmetric mirror machine with an extremely high\nplasma pressure equal to pressure of vacuum magnetic \feld (so-called regime of\ndiamagnetic con\fnement) is investigated. Extrusion of magnetic \feld from central\nregion due to plasma diamagnetism leads to non-conservation of the magnetic moment\nand can result in chaotic movement and fast losses of particles. The following\nmechanisms can provide particle con\fnement for unlimited time: absolute con\fnement\nof particles with high azimuthal velocity and conservation of adiabatic invariant for\nparticle moving in smooth magnetic \feld. The criteria of particle con\fnement and\nestimations of lifetime of uncon\fned particles are obtained and veri\fed in direct\nnumerical simulation. Particle con\fnement time in the diamagnetic trap in regime\nof gas-dynamic out\row is discussed.\nKeywords : mirror machine, high-beta plasma, diamagnetic con\fnement, particle\ndynamic in magnetic \feld\n1. Introduction\nNewly proposed regime of diamagnetic con\fnement of plasma in a mirror machine [1]\nallows us to essentially reduce particles and energy losses from the trap and increase\npower density of thermonuclear reactions. The idea is to con\fne plasma with extremely\nhigh pressure equal to pressure of magnetic \feld of the trap. It leads to formation of\ncentral region with sharp boundary occupied by dense plasma with extruded magnetic\n\feld (so called diamagnetic \\bubble\"). The e\u000bective mirror ratio inside the \\bubble\"\nis extremely high, so in MHD approximation longitudinal losses of plasma from inner\narea of the diamagnetic trap are suppressed. In frame of the MHD approximation the\nplasma and energy losses are driven by di\u000busion of plasma through magnetic \feld on\nthe border of the \\bubble\" [1, 2]. There losses grow linearly with increasing \\bubble\"\nlength and radius and are reduced when plasma conductivity rises.\nThe structure of magnetic \feld in the diamagnetic trap is close to Field Reversal\nCon\fguration (FRC) [4] with zero \feld reversal. So particle dynamic in the diamagnetic\ntrap has a lot in common with dynamic of fast ions in FRCs.arXiv:2002.03535v1  [physics.plasm-ph]  10 Feb 2020Collisionless particle dynamic in an axi-symmetric diamagnetic trap 2\nInvestigation of single-particle dynamic in the diamagnetic trap is needed for\ndevelopment of kinetic models of diamagnetic con\fnement. Small magnitude of\nmagnetic \feld results in non-conservation of magnetic moment mv2\n?=(2B) and can\nlead to chaotic behavior and fast longitudinal losses of particles. From the other\nhand, particle energy and canonical angular momentum are integrals of motion due\nto stationarity and azimuthal symmetry of magnetic \feld. The aim of this work is\ninvestigation of regimes of particle con\fnement and lifetime of uncon\fned particles in\nthe diamagnetic trap.\nThe magnetic \feld is assumed to be fully axisymmetric later. So in\ruence of\npossible instabilities and non-accuracy of magnetic system of the trap are neglected.\nNon-symmetry of magnetic system is seems to result in slow Arnold di\u000busion of energy\nand angular moment. In presence of dense plasma the Coulomb collisions should to\nmasque the slow di\u000busion. Mechanisms of anomalous losses due to plasma instabilities\nin diamagnetic trap are addressed for future investigations.\nThe article is organized as follow. Hamilton function of particle in the trap is\ndiscussed in the second section. Two mechanisms can provide particle con\fnement in\nunlimited time if collision scattering is absent, namely so-called absolute con\fnement\n[3] and conservation of radial adiabatic invariant. There mechanisms are discussed\nin the third section. The estimations for lifetime of uncon\fned particles and plasma\ncon\fnement time in gas-dynamic regume are found in the fourth and \ffth sections.\nSimple numerical example is presented in the sixth section. The results are discussed in\nthe Conclusion.\n2. Hamiltonian\nWe consider particle dynamic in the axisymmertic magnetic \feld without spirality. This\n\feld can be described by only one function, namely magnetic \rux \t( r;z) =rA\u0012(r;z),\nhereA\u0012,randzare azimuthal component of vector potential, radial and longitudinal\ncoordinate. The Hamilton function can be written in the following form\nH(pr;p\u0012;pz;r;\u0012;z ) =p2\nr\n2m+(p\u0012\u0000e\t(r;z)=c)2\n2mr2+p2\nz\n2m+e'(r;z); (1)\nhere'(r;z) is electrostatic ambipolar potential. The particle energy and canonical\nangular momentum p\u0012are integrals of motion due to stationarity and azimuthal\nsymmetry. If structure of magnetic \feld is \\bubble\"-like that magnetic \feld (and\nmagnetic \rux) is small inside region in central part of the trap. Outside this region\nmagnetic \feld is approximately vacuum (examples of structure of magnetic \feld are\nshown in [1, 2], see also \fgures 2 and 3). We assume that the ambipolar potential is\napproximately constant in region with very small magnetic \feld and that longitudinal\ngradient scale of the potential on \\bubble\" boundary coincides with longitudinal gradient\nscale of magnetic \feld.\nIt is convenient in numerical simulations to express all dimension quantities via\nparticle mass, vacuum magnetic \feld in the center of the trap B0and cyclotronCollisionless particle dynamic in an axi-symmetric diamagnetic trap 3\nfrequency calculated by vacuum magnetic \feld \n \u0011eB0=(mc). Only quantities with\ndimension of length remain after this expressing. So integrals of motion with dimension\nof length will be used later together with energy and angular momentum, namely\nLarmor radius calculated by full energy and vacuum magnetic \feld \u001a\u0011(2\"=(m\n2))1=2\nand minimal possible distance between particle and axis in case of zero magnetic \feld\nrmin=p\u0012=p\n2m\".\nAn example of trajectories in the transversal cross-section are shown in \fgure\n1. Radial dependence of the magnetic \feld is chosen in the form B(r) =B0f1 +\ntanh((r\u0000a)=\u0001r)g=2, herea=\u0001r= 10. The trajectories can be divided into three\nclasses in dependence on sign of particle angular momentum. Orbits of the particles\nwith \np\u0012\u00140 and \np\u0012>0 correspond to betatron and drift orbits in FRC [6]. As in\nthe FRC, con\fnement of particles essentially depends on sign and magnitude of angular\nmomentum (see below).\nFigure 1. An example of trajectories of particles with \n p\u0012<0 (left), \np\u0012= 0 (center)\nand \np\u0012>0 (right). Dashed circle bounds region r < a where magnetic \feld is less\nthanB0=2. Arrows indicate direction of mean azimuthal velocity.\n3. Regimes of particle con\fnement\nNon-conservation of magnetic momentum due to smallness of magnitude of magnetic\n\feld inside the diamagnetic \\bubble\" results in changing of regimes of particle\ncon\fnement and essentially modi\fcation of concept of loss cone. Two mechanisms\ncan provide particle con\fnement in the diamagnetic trap for unlimited time (in absence\nof collision scattering and non-axisymmetry): well-known absolute con\fnement [3] and\nconservation of radial adiabatic invariant.\n3.1. Absolute con\fnement\nThe mechanism of the absolute con\fnement is follow. The Hamiltonian (1) described\ntwo-dimensional motion in e\u000bective potential ( p\u0012\u0000e\t(r;z)=c)2=(2mr2) +e'(r;z). This\npotential is potential well for particles with \n p\u0012<0. Such particles con\fnes in the trap\nif their energy is small enough.Collisionless particle dynamic in an axi-symmetric diamagnetic trap 4\nTo \fnd condition of the absolute con\fnement one can note that in region of mirrors\nthe magnetic \feld if quasi-uniform, \t( r)\u0019RvB0r2=2, hereRvis vacuum mirror ratio\nof the trap. Let's found minimal possible energy \"minof particle moving with angular\nmomentum p\u0012in the mirror region. Minimal value of e\u000bective potential is reached at\na point with radial coordinate r\fwhich satis\fes equation ( \u0000p\u0012=r2\n\f\u0000Rv\n=2)(p\u0012=r\f\u0000\nRv\nr\f=2)=m+e'0\nm(r\f) = 0, here 'm(r) is radial distribution of ambipolar potential in\nmirror. Minimal energy is \"min= (p\u0012=r\f\u0000m\nr\f=2)2=(2m) +e'm(r\f). Particle with\nangular momentum p\u0012cannot penetrate in region of the mirror if their energy less than\n\"min.\nIn simplest case of zero electrostatic potential r\f= (\u00002p\u0012=(m\n))1=2and criterion\nof absolute con\fnement can be written in the form [3]\n\">\u0000Rv\np\u0012: (2)\nThis criterion can be re-written in another form: rmin>\u001a=(2Rv) and \np\u0012<0. Particle\nis con\fned absolutely if it rotates quickly in the direction which coincides with direction\nof cyclotron rotation.\nPreferential con\fnement of ions with negative angular momentum can results in\nspontaneous rotation of plasma in the trap which is similar to particle-loss spin-up in\nFRCs [4].\n3.2. Adiabatic con\fnement\nRadial adiabatic invariant\nIr=1\n2\u0019I\nprdr=1\n2\u0019Iq\n2m\"\u00002me'\u0000p2\nz\u0000(p\u0012\u0000e\t=c)2=r2 (3)\nconserves if magnetic \feld changes in longitudinal direction smoothly and particle\nlongitudinal velocity is not too large. In this case frequency of radial oscillations\n\nr= 2\u0019=0\n@Imdrq\n2m\"\u00002me'\u0000p2\nz\u0000(p\u0012\u0000e\t=c)2=r21\nA\u00001\n(4)\ncan be much greater than inverse time of varying of magnetic \feld during particle\nlongitudinal motion. It should be noted that regular motion of ions in oblate FRCs due\nto conservation of adiabatic invariant is observed also in numerical simulations of FRCs\n(see, for example, [5]).\nIf vacuum magnetic \feld has one local minimum (corrugation of \feld is absent)\nthan character time of varying of magnetic \feld during particle longitudinal motion\nis ratio of distance between mirrors to longitudinal velocity L=vkso criterion of\nadiabaticity is \n r>vk=L. Let us consider case of large radius of diamagnetic \\bubble\"\na\u001d\u001a. In this case it is convenient to introduce radial coordinate of magnetic\n\feld line on the \\bubble\" boundary rb(z). To choose this \feld line the condition\nBz(rb(0);0) =B0=2 is used. If rb\u001d\u001aand electrostatic potential is zero one canCollisionless particle dynamic in an axi-symmetric diamagnetic trap 5\ncalculate \n r=\u0019v2\n?=(v2\n?\u0000\n2\u001a2r2\nmin=r2\nb)1=2(herev?= (2\"=m\u0000v2\nk)1=2) and write criterion\nof adiabaticity in the form\nv?\nvk> drb\ndz!\nmaxvuut1\u0000\n2\u001a2\nv2\n?r2\nmin\nr2\nb: (5)\nExpression ( drb=dz)maxdenotes maximal value of derivative of function rb(z). We assume\nthat radial distribution of ambipolar potential is approximately constant inside the\n\\bubble\" and that longitudinal gradient scale of ambipolar potential is of the order of\nr\u00001\nb(drb=dz). In this case taking the electrostatic potential into account does not changes\ncriterion (5) essentially.\nIf motion of particle is regular than particle is con\fned in the trap if real solutions\nPof equation Ir(\";p\u0012;pz=P;z= 0) =Ir(\";p\u0012;pz= 0;z=zm) are absent (here \u0006zm\nare coordinates of mirrors). Magnetic \rux in the mirror is approximately equal to \rux of\nvacuum magnetic \feld \t \u0019RvB0r2=2. So adiabatic invariant (3) in the mirror is equal to\n(\"\u0000p2\nz=(2m)+e'm(q\n2jp\u0012=(m\n)j))=(Rvj\nj)\u0000jp\u0012jH(\u0000\np\u0012), hereH(x) is Heaviside step\nfunction and 'm(r) is radial distribution of ambipolar potential in mirror (we assume\nthat Larmor radius of particle in mirror is small in comparison with radial scale length\nfor potential). Criterion of con\fnement of regularly moving particle can be written in\nthe form\nIr(\";p\u0012;pz;z= 0) +jp\u0012jH(\u0000\np\u0012)>\"+e'm(q\n2jp\u0012=(m\n)j)\nRvj\nj: (6)\nCriterion (6) can be written in analytical form if radius of the \\bubble\" is large,\nrb(z)\u001d\u001a, and ambipolar potential is approximately constant inside the \\bubble\".\nIn this case the invariant (3) is equal approximately to the radial adiabatic invariant for\nparticle moving inside long cylinder surface with radius rb(z):\nIr(\";p\u0012;pz;z)\u00192jp\u0012j(\u0011+ arctan\u0011); \u0011 =q\nr2\nb(z)(2m\"\u0000p2\nz)=p2\n\u0012\u00001:\n3.3. Criterion of adiabaticity in corrugated \feld\nDiscrete structure of magnetic system leads to corrugation of vacuum magnetic \feld\nwhich results in corrugation of margin of diamagnetic bubble. So time of essential\nchanging of magnetic \feld during longitudinal motion essentially decreases and can\nbecome comparable with period of radial oscillations (such e\u000bect for particles moving\nin vacuum magnetic \feld is described in [8]). Resonant interaction between radial\noscillations and longitudinal motion can destroy adiabatic invariant (3) even if criterion\nof adiabaticity (5) is satis\fed. In this section we will estimate magnitude of corrugation\nof vacuum magnetic \feld at which corrugation not in\ruence on movement of particle.\nLet's look a longitudinally-uniform diamagnetic trap with weak corruga-\ntion of vacuum magnetic \feld. Flux of vacuum magnetic \feld is \t v=\nB0fr2=2 + (\u000eb=k)rI1(kr) cos(kz)g, vacuum magnetic \feld at r= 0 isB0f1+\u000ebsin(kz)g.Collisionless particle dynamic in an axi-symmetric diamagnetic trap 6\nFlux of \feld in the trap is the sum \t = \t 0(r)+\t 1(r) cos(kz), unperturbed part satis\fes\nintegral equation [6]\n\t0(r) =B0r2\n2\u00004\u0019\ncZ\ng(r;r0)j'(\t0(r0);r0)dr0; (7)\nherej'(\t;r) is azimuthal component of plasma current (which depends on distribution\nfunction of ions and electrons),\ng(r;r0) =r2\n2 r2\n0\nr2\nw\u00001!\nH(r0\u0000r) +r2\n0\n2 r2\nr2\nw\u00001!\nH(r\u0000r0)\nis Green function (magnetic \rux generated by electric current \rowing on a cylindrical\nsurface with radius r0),rwis radius of conducting shell surrounding plasma, H(x) is\nHeaviside step function.\nPerturbed part \t 1(r) satis\fes following linear integral equation:\n\t1(r) =\u000ebB 0r\nkI1(kr)\u00004\u0019\ncZ\ng1(r;r0)@j'(\t0;r0)\n@\t0\t1(r0)dr0;\nwith the Green function (which is solution of the equation r@r(r\u00001@rg1)\u0000k2g1=\nr\u000e(r\u0000r0))\ng1(r;r0) =rr0I1(kr)\nI1(krw)(I1(kr0)K1(krw)\u0000I1(krw)K1(kr0))H(r0\u0000r) +\n+rr0I1(kr0)K1(krw) I1(kr)\nI1(krw)\u0000K1(kr)\nK1(krw)!\nH(r\u0000r0):\nThe hamiltonian of particle moving in longitudinally-uniform diamagnetic trap with\nweak corrugation is\nH(pr;p\u0012;pz;r;\u0012;z ) =p2\nr\n2m+(p\u0012\u0000e\t0(r)=c)2\n2mr2+p2\nz\n2m\u0000e\t1(r)\nmc(p\u0012\u0000e\t0(r)=c)\nr2cos(kz)\nOne can make canonical transformation to the action-angle variables of non-\nperturbed hamiltonian:\nH(Ir;p\u0012;pz;\u001e;\u0012;z ) =H?(Ir;p\u0012) +p2\nz\n2m\u0000e\t1(r(Ir;\u001e;p\u0012))\nmc(p\u0012\u0000e\t0(r(Ir;\u001e;p\u0012))=c)\nr(Ir;\u001e;p\u0012)2cos(kz);\nhereIr(\"?;p\u0012) = (2\u0019)\u00001Hdr=(mVr(r)) is radial adiabatic invariant, \u001e=\nfRdr=Vr(r)gfHdr=Vr(r)g\u00001,Vr(r) =m\u00001q\n2m\"?\u0000(p\u0012\u0000e\t0(r)=c)2=r2is radial\nvelocity,H?(Ir;p\u0012) is solution of equation Ir(H?;p\u0012) =Ir. Radial coordinate rdepends\nperiodically on new variable \u001e(becauserdepends periodically on time) so perturbation\nof hamiltonian can be expanded in a Fourier series:\nH(Ir;p\u0012;pz;\u001e;\u0012;z ) =H?(Ir;p\u0012) +p2\nz\n2m\u0000cos(kz)X\nn\u000eHn(Ir;p\u0012)ein\u001e:(8)\nCondition of resonance between radial oscillations and longitudinal motion is n_\u001e=\nn\nr(Ir;p\u0012) =kpz=m.\nConcrete form of the distribution functions of ions and electrons are needed to\ncalculate magnetic \rux and coe\u000ecients \u000eHnin (8) and to follow investigation of\nadiabacity. Analytical treatment can be extended in the following cases: if angular\nmomentum of particle is negative and small enough and if radius of the diamagnetic\nbubble exceeds essentially particle Larmor radius \u001a.Collisionless particle dynamic in an axi-symmetric diamagnetic trap 7\n3.3.1. Particles with \"\u001c\u0000 \np\u0012.Now we consider particles with negative and very\nsmall angular momentum. Ions with such momentum can arises in the trap due to\no\u000b-axis NBI. This ions move along betatron orbits with small harmonic oscillations in\nradial direction. The unperturbed part of hamiltonian can be written as [9]\nH?(pr;p\u0012;r;\u0012) =p2\nr\n2m+(p\u0012\u0000e\t0(r\f)=c)2\n2mr2\n\f+m\n2\n\f(r\f)(r\u0000r\f)2\n2;\nherer\fis solution of equation p\u0012\u0000e\t0(r\f)=c+m\nc(r\f)r2\n\f= 0 (mean radius of betatron\norbit), \n \f(r) =f\nc(r)@r(r\nc(r))g1=2is betatron frequency, \n c(r) =r\u00001@r\t0(r) is\nlocal cyclotron frequency. Amplitude of betatron oscillations is assumed to be small,\njr\u0000r\fj\u001cr\f.\nTransition to the angle-momentum variables describes by canonical transformation\npr=q\n2m\n\f(r\f)Ircos\u001e; r =r\f+s\n2Ir\nm\n\f(r\f)sin\u001e:\nAfter transition to the angle-momentum variables the Hamiltonian transforms to\nH(Ir;p\u0012;pz;\u001e;\u0012;z ) = \n\f(r\f)Ir+p2\nz\n2m+(p\u0012\u0000e\t0(r\f)=c)2\n2mr2\n\f+e\t1(r\f)\nc\nc(r\f) cos(kz):(9)\nResonances between radial oscillations and longitudinal motion are absent in the\nhamiltonian (9) so the particles move adiabatically. Condition of applicability of this\napproximationjr\u0000r\fj\u001cr\fcan be written in the form\n2\"\nm\u0000p2\nz\nm2\u0000(p\u0012\u0000e\t0(r\f)=c)2\nm2r2\n\f\u001c\n2\n\fr2\n\f:\nThis condition means that frequency of radial oscillations (which is the betatron\nfrequency) have to be large enough.\n3.3.2. Particles with \u001a\u001ca.If radius of the diamagnetic \\bubble\" ais much greater\nthan width of transition layer and \\Larmor radius\" \u001athat motion of particles can be\ndescribed approximately as motion of particle inside surface rotation r=rb(z). Here\nfunctionrb(z) is coordinate of magnetic \feld line at boundary of the bubble.\nLet's found the criterion of adiabaticity of particle moving with velocity v0inside\ncorrugated cylinder r=a+\u000eacos(kz) when corrugation is small, \u000ea\u001caandk\u000ea\u001c1.\nParticle dynamic is described by twist mapping (see Appendix A)\nvkn+1=vkn\u00001\nk@G(vkn+1;zn)\n@zkn; kzn+1=kzn+ 2k\u0001rn+1In+1+@G(vkn+1;zn)\n@vkn+1;\nG(vkn+1;zn) =\u00002\u000ea\nak\u0001rn+1vkn+1\nIn+1cos(kzn+k\u0001rn+1In+1); (10)\nherevknandznare longitudinal velocity and coordinate at times when radial component\nof velocity is zero, In=vkn=(v2\n0\u0000v2\nkn)1=2is the ratio of longitudinal and transversal\ncomponents of velocity, \u0001 rn= (a2\u0000r2\nminv2\n0=(v2\n0\u0000v2\nkn))1=2is amplitude of radial\noscillations and rmin=jp\u0012j=(mv0).Collisionless particle dynamic in an axi-symmetric diamagnetic trap 8\nAfter linearization near resonances vk=Vj(hereVjare solutions of equation\n2k\u0001rn+1In+1= 2\u0019jwith integer j) and transition to new variable J(j)\nn= (vkn\u0000\nVj)f2k(a2\u0000(2(v2\n0\u0000V2\nj)=V2\nj+ 1)r2\nmin)g(v0=Vj)3\u0001r(j)\nn=v0one can write the twist mapping\n(10) in the form of the Chirikov standard map\nJ(j)\nn+1=J(j)\nn+Ksin(kzn+\u0019j+\u0019); kzn+1=kzn+J(j)\nn+1;\nK= 4\u000ea\nav2\n0\nv2\n0\u0000V2\njk2a2 \n1\u0000r2\nmin\na2v2\n0+V2\nj\nv2\n0\u0000V2\nj!\n;\nhereKis so-called stochasticity parameter. To estimate magnitude of corrugation\nneeded for destroying adiabatic invariant we use the Chirikov criterion of resonances\noverlapping\nK > 1 (11)\nand estimation of magnitude of corrugation of boundary of diamagnetic \\bubble\" found\nin MHD approximation [2]\n\u000ea\na=\u000eb\nkaI0(ka) I1(ka)\nI0(ka)+K1(ka)\nK0(ka)!\n: (12)\nFor small-scale perturbations with ka> 2 estimation (12) can be simpli\fed:\n\u000ea\na\u00192\u000eb\nkaI0(ka): (13)\nWe combine expressions (11) and (13) to found criterion of adiabaticity of motion\nin the diamagnetic trap for particles with \u001a\u001ca:\n8\u000ebkaI 0(ka)v2\n0\nv2\n? \n1\u0000r2\nmin\na22v2\n0\u0000v2\n?\nv2\n?!\n<1; (14)\nherev?= (v2\n0\u0000v2\nk)1=2.\nThis condition is \frst broken for particles with zero angular momentum, rmin=\n0. Most dangerous are small-scale perturbations with ka\u001d1, but magnitude of\nperturbations with very large kis seems to be small due to \fnite Larmor radius\ne\u000bects which is neglected in expression (12). Particles with great value of \u0000p\u0012move\nadiabatically which consistent with results of previous section. Particles with great\nvalue ofp\u0012move in region outside the bubble where magnetic \feld is strong so this\nparticles moves adiabatically also. This behavior consistents with criterion (14).\n4. Lifetime of uncon\fned particles\nLet's us now looks particles which move chaotically and are not con\fned absolutely.\nIf particle moves regularly than particle arrival to mirror at the same longitudinal\nvelocity after each excursion from mirror to mirror. If radial adiabatic invariant not\nconserves than longitudinal velocity changes chaotically with each approach to mirror\nso particle leave the diamagnetic trap after several excursions from mirror to mirror.\nTo estimate lifetime of chaotically moving particles we consider population of particles\nwith same energy \"and angular momentum p\u0012which move inside quasi-cylindricalCollisionless particle dynamic in an axi-symmetric diamagnetic trap 9\ndiamagnetic \\bubble\" with radius aand lengthL. Distribution function of this particles\nisf(v;r) =\u000e(mv2=2+e'\u0000\")\u000e(m(xvy\u0000yvx)+e\t=c\u0000p\u0012)H(v2\nz\u0000Vz0(\";p\u0012;z)2), hereVz0\nis solution of equation Ir(\";p\u0012;Vz0;z) =Ir(\";p\u0012;vz0;z= 0),vz0is value of longitudinal\nvelocity corresponding margin of adiabaticity. Full number of particles in the trap N\nand particle \row through mirrors Jare approximately\nN=\u0019LZ1\n0dr2Z\nd3vf(v;r;z); J = 2\u0019Z1\n0dr2Z1\n0dvzZ\ndvrdv'vzf(v;r;zm);\nherezmis coordinate of the mirror. Particle lifetime is \u001c=N=J.\nIn the simplest case when all particles with energy \"and angular momentum p\u0012\nmove chaotically, Vz0= 0, \\bubble\" radius is large, a\u001d\u001a, and electrostatic potential\nis neglectingly small, estimation of particle con\fnement time is:\n\u001c\u0018Rv\u001cb(\n(a\u0000rmin)=\u001a; \np\u0012\u00150;\n(a\u0000rmin)=(\u001a\u00002Rvrmin);\np\u0012<0;(15)\nhere\u001cb=L=(\n\u001a) is of the order of particle transit time from mirror to mirror (period\nof bounce-oscillations).\n5. Estimation of plasma lifetime in gas-dynamic regime\nCalculation of particle con\fnement time in regime of diamagnetic con\fnement requires\nsophisticated calculations including solving of kinetic equation together with equilibrium\nequation. This calculation can be essentially simpli\fed if plasma is quite dense and\nangular scattering time is less than lifetime of uncon\fned particles. In this case particle\ndistribution functions are locally Maxwellian and one can estimate particle con\fnement\ntime by calculating full number of particles and their \row through mirrors.\nLet's assume that distribution function of particles of type sapproximately\ncoincides with distribution of Maxwellian particles inside cylinder with radius a\nfi(\";p\u0012) =ni0\u0012m\n2\u0019Ti\u00133=2\ne\u0000\"=TiH(2mia2\"\u0000p2\n\u0012): (16)\nThis distribution allows particle density to be uniform inside the diamagnetic \\bubble\"\nwith radius a, like in MHD models (see next section). One can estimate lifetime of\nparticles with distribution sas\n\u001cs\u0018RvL\nvsa\n\u001as; (17)\nherevs= (2Ts=ms)1=2is thermal velocity of particles of type s,\u001as=vs=\nsis mean\nLarmor radius calculated by vacuum magnetic \feld. Estimation (17) gives same lifetime\nfor ions and electrons with same temperatures so plasma out\row in such regime should\nnot be accompanied by appearing essential ambipolar potential.\nCombination RvL=viis of the order of time of gas-dynamic out\row from trap\nwith vacuum magnetic \feld \u001cGDT. So transition to regime of diamagnetic con\fnementCollisionless particle dynamic in an axi-symmetric diamagnetic trap 10\nincreasesa=\u001aitimes particle con\fnement time in gas-dynamic regime. This estimation\nshould be compared with estimation\n\u001c=\u001cGDTa\n\u0015\nof particle con\fnement time in MHD model [1, 2], here \u0015is thickness of boundary\nlayer in MHD approximation. This comparison demonstrates that kinetic e\u000bects are\nimportant when \u001ai>\u0015.\nIt should be noted that e\u000bects of adiabatic con\fnement of part of particles not\ntaken into account in estimation (17). This e\u000bects can be important if plasma \row\nthrough mirror is collisionless (\\short\" mirrors). E\u000bects of adiabatitity of motion are\nseems to decrease particle losses. In this sense the estimation (17) is most pessimistic\nestimation of particle con\fnement time in the gas-dynamic regime.\n6. Numerical example\nTo illustrate in\ruence of structure of magnetic system on collision-less particle dynamic\nsome results of numerical simulation of ions movement in diamagnetic trap are presented\nin this section. Magnetic \rux is calculated similarly article [6]. Namely, magnetic \rux\nsatis\fes Amperes's law ( r\u0002B)\u0012=@rr\u00001@r\t = 4\u0019j\u0012=c(herej\u0012is azimuthal component\nof plasma current) which can be write in the form of integral equation\n\t(r;z) = \tv(r;z) +4\u0019\ncZ\ndr0dz0 G(r;r0;z;z0)j\u0012(r0;z0);\nj\u0012(r0;z0) =X\ns=i;eesZp\u0012\u0000es\t(r0;z0)=c\nr0fsdprdp\u0012dpz\nr0; (18)\nhere \tv(r;z) is \rux of vacuum magnetic \feld and\n G(r;r0;z;z0) =q\n(z\u0000z0)2+ (r+r0)2E(\u0018) + (\u0018=2\u00001)K(\u0018)\n2\u0019;\nis Green function (magnetic \rux of thin coil), E(\u0018) andK(\u0018) are complete elliptic\nintegrals of \frst and second kinds, \u0018= 4rr0=((z\u0000z0)2+ (r+r0)2).\nThe integral equation (18) can be solved numerically by iterations. To calculate\nplasma current j\u0012we assume electrons to be cold and choose distribution function of\nions (16). Dependence of density and azimuthal current of ions on radial coordinate\nand magnetic \rux are given in Appendix B. Example of radial dependence of plasma\ndensity and longitudinal component of magnetic \feld on radius is shown on \fgure 2.\nPlasma density is constant inside the \\bubble\" like in the MHD model [1, 2].\nMagnetic system of trap consists of two mirror coils and of set of equidistant coaxial\ncoils which generate quasi-uniform magnetic \feld (see \fgure 3). In one case corrugation\nof vacuum magnetic \feld does not exceeds tenths of percent (smooth magnetic \feld).\nIn second case distance between the coils is doubled and radius of the coils is reduced\n(currents in coils are changed correspondingly so that value of magnetic \feld in center is\nthe same in both cases). It results in observable corrugation of the \\bubble\" boundary.Collisionless particle dynamic in an axi-symmetric diamagnetic trap 11\nFigure 2. An example of dependence of plasma density n=ni0(red) and magnetic\n\feldBz=B0(blue) on radius in trap center. Parameters: (2 Ti=(mi\n2))1=2= 2,a= 20,\nvacuum mirror ratio Rv= 2.\nFigure 3. An examples of magnet coils (rectangles) and magnetic \feld line on\n\\bubble\" boundary (solid curve) for smooth (left) and corrugated (right) vacuum\nmagnetic \feld. Parameters: (2 Ti=(mi\n2))1=2= 2,a= 20.\nNumerical simulation allows us to found maximal value of longitudinal velocity\nat which ions con\fne in the trap. Ions move regularly in trap with smooth \feld and\nmaximal critical velocity is restricted only by criterion of con\fnement (6). An example\nof dependence of critical velocity on angular momentum for ions moving in trap with\ncorrugated \feld is shown on \fgure 4. Ions with small jp\u0012jscatter due to corrugation\nso their critical velocity is relatively low. This velocity increases when jp\u0012jrises (in\naccording with criterion of adiabaticity (14)). Ions with negative and small p\u0012are\ncon\fned absolutely so their maximal velocity is restricted by full energy (2 \"=mi)1=2.\nCritical velocity of ions with large p\u0012decreases because this ions move in region with\n\fnite magnetic \feld outside the \\bubble\".\nAn example of number of uncon\fned ions in trap with corrugated \feld after\nnbounce-oscillations is shown on \fgure 5. This number decreases approximately\nexponentially with time. Con\fnement time 18 :3\u001cbis consistent with analytical\nestimation (15).\n7. Conclusion\nSmallness of magnetic \feld in central region of the diamagnetic trap results in non-\nconservation of magnetic moment so regimes of particle con\fnement are modi\fed.\nParticle can con\fne inside the diamagnetic \\bubble\" either due to conservation of\nradial adiabatic invariant (this mechanism occurs if vacuum magnetic \feld is smooth)\nor in regime of adiabatic con\fnement (if particle rotates quickly around axis of trapCollisionless particle dynamic in an axi-symmetric diamagnetic trap 12\nFigure 4. An example of value of critical longitudinal velocity for ions with \u001a= 2\nmoving in corrugated magnetic \feld (points) and margin of adiabaticity (14) at\nka= 2:7 and\u000eb= 0:01 (dashed line).\nFigure 5. An example of number of uncon\fned ions after nbounce oscillations\n(points) and function 19 e\u0000x=18:3(solid). Parameters: \u001a= 2,rmin= 0.\nin direction coinciding with direction of cyclotron rotation). Possibility of conservation\nof the adiabatic invariant depends strongly on geometry of magnetic \feld, especially\non small-scale perturbation of vacuum magnetic \feld. Lifetime of uncon\fned particles\nincreases with increasing the \\bubble\" radius and vacuum magnetic \feld in the mirrors\nof the trap and decreasing particle \\Larmor radius\" \u001a. Even in the worst case when all\nparticles move chaotically particle con\fnement time exceeds the gas-dynamic time in\nthe vacuum \feld in ratio of the \\bubble\" radius to mean ion \\Larmor radius\".\nThe author wish to thank all the members of the laboratories 9-0, 9-1 and 10\nof BINP SB RAS who participated in discussion of the results of this work. Author\nespecially grateful to the Dr. Alexei Beklemishev, Dr. Dmitriy Skovorodin and Mikhail\nKhristo for fruitful discussions.Collisionless particle dynamic in an axi-symmetric diamagnetic trap 13\n8. Appendix A. Twist mapping for particle inside corrugated surface\nNow we consider particle moving with velocity v0inside corrugated cylindrical surface\nand re\recting elastically from it. Let znandvknto be longitudinal coordinate and\nlongitudinal component of velocity of particle at time tn. At this point of time\nradial velocity equal to zero and azimuthal component of velocity is ( v2\n0\u0000v2\nkn)1=2.\nRadial coordinate of particle is rn=rminv0=(v2\n0\u0000v2\nkn)1=2, herermin=jp\u0012j=(mv0).\nParticle will collide with surface at point of time twwhich is solution of equation\n(r2\nn+ (v2\n0\u0000v2\nkn)t2\nw)1=2=a+\u000eacos(kzn+kvkntw). Longitudinal and radial coordinate of\nthe particle at the moment of collision are zw=zn+vkntwandrw=a+\u000eacos(kzw).\nRadial, azimuthal and longitudinal components of velocity at the moment of collision\narevr= (v2\n0\u0000v2\n0r2\nmin=r2\nw\u0000v2\nkn)1=2,v\u0012=v0rmin=rwvk=vkn. After collision radial\nand longitudinal components of velocity are vkn+1= (2f0vr+ (1\u0000f02)vkn)=(1 +f02)\nv0\nr= (2f0vkn\u0000(1\u0000f02)vr)=(1 +f02), heref0=\u0000k\u000easin(kzw). Azimuthal component\ndoes not change. Radial coordinate of particle will reach minimal value rn+1=\nrminv0=(v2\n0\u0000v2\nkn+1)1=2through time \u0001 t= (r2\nw\u0000r2\nminv2\n0=(v2\n0\u0000v2\nkn+1))1=2=(v2\n0\u0000v2\nkn+1)1=2after\ncollision. Longitudinal velocity is zn+1=zn+vkntw+vkn+1\u0001twhen radial coordinate\nis minimal.\nIf corrugation is weak \u000ea\u001cathat time before collision is approximately tw\u0019\n(\u0001rnIn=vkn)f1 + (\u000eaa=\u0001r2\nn) cos(kzn+k\u0001rnIn)g, here \u0001rn= (a2\u0000r2\nminv2\n0=(v2\n0\u0000v2\nkn))1=2\nandIn=vkn=(v2\n0\u0000v2\nkn)1=2. When radial coordinate minimal longitudinal component of\nvelocity and longitudinal coordinate of particle are described by expressions (10).\n9. Appendix B. Density and current of ions.\nDensity of ions with distribution function (16) is\nni=ni0=H(rb\u0000r)e\u00002e'=(mw2)+H(r\u0000rb)rb\nr+\n+H(\n2 2\u00002(r2\nb\u0000r2)e'=mi)\n2 rb\nr(\nerf (p+=mi\nrbw)\u0000erf (p\u0000=mi\nrbw))\n\u0000\n\u0000e\u00002e'=(mw2)(\nerf (p+=mi\u0000\n \nrw)\u0000erf (p\u0000=mi\u0000\n \nrw))!\n;\nhere (r;z) = \t(r;z)=B0is normalized magnetic \rux, 'is electrostatic potential,\nw= (2Ti=mi)1=2is thermal velocity and\np\u0006=mi\nr2\nb \u0006rbrq\n2(r2\u0000r2\nb)e'=mi+ \n2 2\nr2\nb\u0000r2\nCurrent of ions is\nji=(eni0) =\u0000H(r\u0000rb)\n rb\nr2+H(\n2 2\u00002(r2\nb\u0000r2)e'=mi)\n2\u0002\n\u0002 \ne\u00002e'=(mw2)wp\u0019(e\u0000(p+=mi\u0000\n )2=(r2w2)\u0000e\u0000(p\u0000=mi\u0000\n )2=(r2w2))+Collisionless particle dynamic in an axi-symmetric diamagnetic trap 14\n+rb\nr2(\n(e\u0000(p+=mi)2=(r2\nbw2)\u0000e\u0000(p\u0000=mi)2=(r2\nbw2))wrbp\u0019+ \n ( erf (p+=mi\nrbw)\u0000erf (p\u0000=mi\nrbw)))!\n:\n[1]A.D. Beklemishev. Diamagnetic \\bubble\" equilibria in linear traps // Physics of Plasmas 23,\n082506 (2016), doi: 10.1063/1.4960129\n[2]A.D. Beklemishev and M.S. Khristo. High-Pressure Limit of Equilibrium in Axisymmetric Open\nTraps // Plasma and Fusion Research 142403007, (2019), doi: 10.1585/pfr.14.2403007\n[3]Ming-Yuan Hsiao and George H. Miley. Velocity-space particle loss in \feld-reversed con\fgurations\n// Physics of Fluids 28, 5, (1985), doi: 10.1063/1.864978\n[4]Loren C. Steinhauer. Review of \feld-reversed con\fgurations // Physics of Plasmas 18070501\n(2011), doi: 10.1063/1.3613680\n[5]Elena V. Belova, Ronald C. Davidson, Hantao Ji, and Masaaki Yamada. Advances in the\nnumerical modeling of \feld-reversed con\fgurations // Physics of Plasmas 13056115 (2006);\ndoi: 10.1063/1.2179426\n[6]Artan Querushi and Norman Rostoker. Equilibrium of \feld reversed con\fgurations with rotation.\nI. One space dimensions and one type of ion // Physics of Plasmas 93057 (2002), doi:\n10.1063/1.1475683\n[7]Artan Querushi and Norman Rostoker. Equilibrium of \feld reversed con\fgurations with rotation.\nIV. Two space dimensions and many ion species // Physics of Plasmas 10737 (2003), doi:\n10.1063/1.1539853\n[8]B.V. Chirikov. Resonance processes in magnetic traps // J. Nucl. Energy, Part C Plasma Phys 1\n253 (1960).\n[9]H. Vernon Wong, H. L. Berk, R. V. Lovelace, and N. Rostoker. Stability of annular equilibrium\nof energetic large orbit ion beam // Physics of Fluids B: Plasma Physics 32973 (1991), doi:\n10.1063/1.859931"    },    {        "title": "1605.00710v2.Electrical_Detection_of_Magnetization_Dynamics_via_Spin_Rectification_Effects.pdf",        "content": "Electrical Detection of Magnetization Dynamics via Spin Rectification Effects\nMichael Harder, Yongsheng Gui, Can-Ming Hu\u0003\nDepartment of Physics and Astronomy, University of Manitoba, Winnipeg, Canada R3T 2N2\nAbstract\nThe purpose of this article is to review the current status of a frontier in dynamic spintronics and contem-\nporary magnetism, in which much progress has been made in the past decade, based on the creation of a\nvariety of micro- and nano-structured devices that enable electrical detection of magnetization dynamics.\nThe primary focus is on the physics of spin rectification effects, which are well suited for studying magneti-\nzation dynamics and spin transport in a variety of magnetic materials and spintronic devices. Intended to be\nintelligible to a broad audience, the paper begins with a pedagogical introduction, comparing the methods\nof electrical detection of charge and spin dynamics in semiconductors and magnetic materials respectively.\nAfter that it provides a comprehensive account of the theoretical study of both the angular dependence\nand line shape of electrically detected ferromagnetic resonance (FMR), which is summarized in a handbook\nformate easy to be used for analyzing experimental data. We then review and examine the similarity and\ndifferences of various spin rectification effects found in ferromagnetic films, magnetic bilayers and magnetic\ntunnel junctions, including a discussion of how to properly distinguish spin rectification from the spin pump-\ning/inverse spin Hall effect generated voltage. After this we review the broad applications of rectification\neffects for studying spin waves, nonlinear dynamics, domain wall dynamics, spin current, and microwave\nimaging. We also discuss spin rectification in ferromagnetic semiconductors. The paper concludes with both\nhistorical and future perspectives, by summarizing and comparing three generations of FMR spectroscopy\nwhich have been developed for studying magnetization dynamics.\nKeywords: spin rectification, ferromagnetic resonance, magnetization dynamics, magnetoresistance, spin\ntorque, spin pumping, dynamic spintronics, contemporary magnetism\nContents\n1 Introduction 2\n2 The Ingredients of Spin Rectification 4\n2.1 Magnetotransport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5\n2.1.1 Two-Current Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n2.1.2 Anisotropic Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n2.1.3 Giant Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8\n2.1.4 Tunnel Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11\n2.1.5 Spin Hall Magnetoresistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n2.2 Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14\n2.2.1 Field Torque Induced Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . . 14\n2.3 The Method of Analyzing Spin Rectification: Basic Examples . . . . . . . . . . . . . . . . . . 16\n\u0003Corresponding author at: Dynamic Spintronics Group, University of Manitoba, http://www.physics.umanitoba.ca/ \u0018hu/\nEmail addresses: michael.harder@umanitoba.ca (Michael Harder), ysgui@physics.umanitoba.ca (Yongsheng Gui),\nhu@physics.umanitoba.ca (Can-Ming Hu)\nPreprint submitted to Physics Reports August 19, 2016arXiv:1605.00710v2  [cond-mat.mtrl-sci]  18 Aug 20162.3.1 In-Plane Angular Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n2.3.2 Out-of-Plane Angular Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n3 Spin Rectification Effects in Magnetic Structures 20\n3.1 Ferromagnetic Films: Early Studies Using Pulsed Microwave Sources . . . . . . . . . . . . . . 20\n3.2 Micro-Structured Monolayers: The Spin Dynamo . . . . . . . . . . . . . . . . . . . . . . . . . 22\n3.2.1 Method and Device Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24\n3.2.2 Lineshape Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25\n3.2.3 Angular Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\n3.3 Magnetic Bilayers: The Spin Battery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\n3.3.1 The Basic Physics of Spin Pumping and Spin Hall Effects . . . . . . . . . . . . . . . . 29\n3.3.2 Spin Battery and \"Longitudinal\" Spin Pumping Voltage . . . . . . . . . . . . . . . . . 32\n3.3.3 Inverse Spin Hall Effect and \"Transverse\" Spin Pumping Voltage . . . . . . . . . . . . 33\n3.3.4 Spin Rectification vs Spin Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35\n3.3.5 Spin-Transfer Torque Rectification in Bilayers . . . . . . . . . . . . . . . . . . . . . . . 36\n3.3.6 Spin Hall Magnetoresistance Rectification . . . . . . . . . . . . . . . . . . . . . . . . . 38\n3.4 Magnetic Tunnel Junctions: The Spin Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39\n3.4.1 Spin-Transfer Torque Rectification in Magnetic Tunnel Junctions . . . . . . . . . . . . 39\n3.4.2 Voltage Torque Rectification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42\n4 Applications of Spin Rectification 43\n4.1 Electrical Detection of Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43\n4.1.1 Review of Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44\n4.1.2 Detection of Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46\n4.2 Electrical Detection of Nonlinear Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . 48\n4.2.1 Review of Nonlinear Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . . . 48\n4.2.2 Detection of Nonlinear Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . 49\n4.3 Electrical Detection of ac Spin Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50\n4.4 Electrical Detection of Magnetization Dynamics in Ferromagnetic Semiconductors . . . . . . 51\n4.5 Electrical Detection of Domain Wall Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 53\n4.6 Spintronic Microwave Sensing and Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55\n4.7 Spintronic Rectifier for Electromagnetic Energy Harvesting . . . . . . . . . . . . . . . . . . . 57\n5 Summary and Outlook: Three Generations of FMR Spectroscopy 58\n1. Introduction\nMagnetization dynamics is a venerable subject in solid state physics, originating with the discovery of\nferromagnetic resonance (FMR) absorption [1–6] in the first half of the 20thcentury. Yet still today, well into\nthe 21stcentury, the study of magnetization dynamics continues to be an active and fast paced research field\n[7–9]. Early understanding of magnetization dynamics was based solely on the phenomenological Landau-\nLifshitz-Gilbert (LLG) equation that does not consider magneto-transport effects, with experimental studies\nemploying microwave absorption measurements. However this has changed recently. The application of\nmodern material growth and nano-lithographic techniques, which was extended from the semiconductor to\nmagnetism community, has revealed a wealth of new and rich physics demonstrating that charge transport\nand spin dynamics are strongly correlated in magnetic materials. On one hand charge transport can be\ncontrolled through magnetization dynamics (or magnetization configurations) through, for example, giant\nmagnetoresistance (GMR) in magnetic multilayer devices [10, 11]. On the other hand, the magnetization in\nnano-structured magnetic multilayers patterned with a sub-micrometer lateral size can be reversed by a dc\ncurrent [12, 13]. Further studies have even found that dc currents and dc voltages can be generated by the\nrectification effect of high-frequency magnetization dynamics [14–48]. This so-called spin rectification effect\nprovides a novel technique for the study of magnetization dynamics – electrical detection.\n2Electrical detection techniques are not new to the study of dynamic material properties, having long\nbeen used in semiconductors and photovoltaics. For example, it is well known that the dynamic interaction\nbetweenphotonsandelectronchargecangeneratenon-thermalelectrondiffusion, thusconvertingsomeofthe\nphoton energy into electric energy when an electromagnetic wave is incident upon a material surface. This\nso-called photovoltaic effect was first observed by French physicist Edmond Becquerel in 1839 [49] using a\npair of platinum electrodes dipped in an electrolyte solution [50]. Nowadays, exploitation of the photovoltaic\neffect in solar cells provides the third most important renewable energy source after hydro and wind power.\nThe key element of a solar cell is a semiconducting device in which electrons in the valence band can absorb\nphoton energy, jump into the conduction band, and then generated electron-hole pairs which are separated\nby a p-n junction and collected by different electrodes [51]. In addition to the photovoltage produced in this\nmanner, the generation of electron-hole pairs also results in photoconductivity [52] – the enhanced electrical\nconductivity of semiconducting materials. Hence this effect is widely used in photodetectors over a broad\nfrequency range, from infrared up to gamma radiation [53–56].\nThe high conversion efficiency from photon energy to an electric signal in semiconductors also makes\nphotovoltageandphotoconductivityahighlysensitivedetectionmethodtostudychargedynamics(especially\nin two-dimensional electron systems). These techniques have provided insight into, for example, inter-\nband transitions [57], inter-subband transitions [58], magneto-plasmonics [59] and spin-flip excitations [60].\nSuch success naturally begs the question, could electrical detection provide insight into spin dynamics in\nmagneticallyorderedmaterials? Onthesurfaceitseemslikethetransitiontospindynamicsinferromagnetic\nmaterials would be difficult. First, the excitation of spin dynamics occurs in d-shell electrons, which have\nless influence on the electric response of a material. Early work on spin-induced current and voltage in\nsemiconductors [61] and ferromagnetic metal films [62] indicated that the conversion efficiency between\nphoton energy and electricity through spin excitations was very poor, often requiring high power (up to\nkW) GHz and THz sources. Second, the conductivity of ferromagnetic metals is at least four orders of\nmagnitude larger than that of semiconductors. Hence the impact of microwave photons on the conductivity\nof ferromagnetic metals was expected to be negligible.\nHowever, starting from the middle 2000s, triggered by the rapid development of micro/nano-structuring\ntechniques applied to ferromagnetic devices, there has been a surge of research interest in the study of spin\ndynamics in ferromagnetic materials by means of electrical detection. A vast number of experimental and\ntheoretical investigations have been carried out, not only on all aspects of traditional spin waves in ferro-\nmagnetic monolayers, but also on novel spin phenomena such as spin pumping (SP) and spin-transfer torque\n(STT) in ferromagnetic multilayers. Through this surge of research, which was sometimes accompanied by\nintense scientific debate, a comprehensive understanding of one of the key mechanisms responsible for dc\nvoltage generation in a variety of magnetic devices, the spin rectification effect (SRE), has been developed.\nThe simplest way to understand the SRE is to highlight the close analogy between spin rectification\nand the well known optical rectification which occurs in nonlinear media with large second-order suscep-\ntibility [63, 64]. In optical rectification the nonlinear optical response of the time-dependent electric field\ne0cos (!t)is governed by the trigonometric relation cos2(!t) = [1 + cos (2 !t)]=2, and hence results in op-\ntical rectification and second harmonic generation. Similarly, spin rectification is the generation of a dc\nvoltage/current due to the nonlinear coupling between an oscillating current and an oscillating resistance in\nmagnetic structures. Despite the similarities, including that both optical and spin rectification are linearly\ndependent on the electromagnetic power, spin rectification has several unique features. First, the dc signal\nis linearly proportionally to the resistance of the sample, which makes it a powerful tool to investigate spin\ndynamics in monolayer nano-structured samples, which are notoriously difficult to study using traditional\ntransmission/reflection measurements since such a signal is proportional to sample volume. Second, the os-\ncillating resistance caused by the dynamic magnetization can be driven by a time dependent magnetic field,\nh0cos (!t+\u001e), where\u001eis the relative phase between microwave eandhfields and plays an important role\nin the curious line shape of electrically detected ferromagnetic resonance. Due to its high sensitivity and fea-\nsibility of implementation, spin rectification has significant impact on the study of spin dynamics. Over the\npast decade, and with the great effort of many research groups, the SRE has been widely employed to study\nferromagnetic resonance (FMR), spin wave resonance, and domain wall resonance in various ferromagnetic\nmaterials including ferromagnetic metals, ferromagnetic semiconductors, ferrimagnetic insulators as well as\n3multilayer structures. Furthermore, properly analyzing the SRE has been found essential for studying novel\nspin dynamics including phase-resolved FMR, spin-transfer torque, spin pumping, and the spin Hall effect.\nParallel to such a path of basic research on spintronics, the SRE has also found its way into novel microwave\napplications, such as spintronic microwave imaging and wireless microwave energy harvesting.\nThis article provides a review of the rectification effects in magnetic structures and their applications. We\nwill examine the physical basis of electrical detection techniques as they apply to magnetization dynamics,\nand examine a variety of applications in which such techniques have been employed. In Sec. 2 we begin by\nexamining the “ingredients\" of spin rectification, namely magnetoresistance, magnetization dynamics and\ntheir nonlinear coupling through the generalized Ohm’s law. In Sec. 3 we turn to various device structures,\nstarting with early pulsed microwave studies followed by an examination of rectification in ferromagnetic\nmonolayers, bilayers and magnetic tunnel junctions. Along the way we discuss spin pumping and spin Hall\neffects and the important role of line shape and angular analyses in distinguishing the many competing\nvoltage producing effects in magnetic devices. Having explained the basic techniques of electrical detection\nof magnetization dynamics, in Sec. 4 we turn to several important applications. On the side of fundamental\nphysics we look at the detection of spin waves and nonlinear magnetization dynamics, dc electrical detection\nof ac spin current, magnetization dynamics in ferromagnetic semiconductors and studies of domain wall\ndynamics. On the applied side, we examine the role of spin rectification in novel microwave sensing, imaging,\nand wireless energy harvesting technologies. The article is concluded in Sec. 5 including both historical\nand future perspectives. By reviewing the historical evolution of three generations of FMR spectroscopy,\nincluding cavity transmission, flip chip absorption, and electrical detection techniques, we also glimpse into\nthe future of FMR spectroscopy, where the venerable science of magnetism is merging with the modern\nphysics of cavity QED, leading us to a new field of cavity spintronics [65].\n2. The Ingredients of Spin Rectification\nSpin rectification (SR) is the production of a dc voltage Vdcin a ferromagnetic structure due to the\nnonlinear coupling between a dynamic resistance R[H(t)]and a dynamic current I(t). To understand the\norigin of this effect, consider a ferromagnetic structure excited by a microwave field at angular frequency\n!. The electric and magnetic fields inside the structure therefore take the form e(t) =e0e\u0000i!tandh(t) =\nh0e\u0000i(!t\u0000\b), where \bis a phase shift associated with losses in the system [66]. The e(t)field will drive a\ncurrent, I(t) =I0e\u0000i!t=\u001be(t)and the h(t)field will drive magnetization precession, m(t) =\u001fh(t), where\n\u001band\u001fare the high frequency material response functions, the conductivity and susceptibility respectively.\nIf the structure is a simple monolayer, h(t)alone will drive the magnetization (Recently it has actually\nbeen found that spin-orbit torques may also be present in single layer devices [67]. We will discuss spin-\norbit torques briefly in Sec. 3.3.5, however for the purpose of this review we will focus on field driven\nmagnetization dynamics in single layer devices). In bilayer devices or magnetic tunnel junctions where a\nspin current js(t)is present, an additional spin torque will contribute to the magnetization precession so that\nm(t) =\u001fh(t)+\u001fsjs(t), where\u001fsis the frequency dependent effective susceptibility due to the spin current.\nDue to the magnetoresistance (MR) of the heterostructure, the effect of the magnetization precession is to\nproduce a dynamic resistance, R(t) =R[H(t)], dependent on the magnetic field H(t) =H0+h(t). As a\nconsequence, a nonzero time averaged voltage can be measured along the current direction,\nV=hRefI(t)gRefR[H(t)]gi=1\n2I0h0\u0001rR(H0) cos \b: (2.1)\nHere we have expanded the resistance in a Taylor series around the dc field H0to first order in the rf field\nh(t),R(H(t)) =R(H0) +h(t)\u0001rR(H0)(rR(H0) =@R=@ HjH0is the field gradient of the resistance\nevaluated at H0), and taken the time average over one period, denoted by hi. Based on this discussion\nwe see that spin rectification requires three ingredients: 1) Some form of magnetoresistance, whereby the\nresistance of the heterostructure depends on the orientation of the magnetization. 2) A torque which drives\nthe rf magnetization, resulting in a time varying magnetoresistance. 3) An rf current which can couple to\nthe rf resistance and produce a dc voltage. These three requirements are summarized in Fig. 1.\n4Figure 1:The ingredients of spin rectification. Magnetoresistance effects result in a magnetization orientation dependence in\nthe resistance of the ferromagnetic heterostructure. The origin of the magnetoresistance will depend on the device structure\nand various forms of magnetoresistance are discussed in Sec. 2.1. By driving the magnetization with microwave frequency\nfields the resulting precession will produce an rf resistance. The magnetization may be driven by a field torque produced by\nan rf magnetic field, hrf, or from a spin torque produced by a spin polarized current js. The rf resistance can then couple\nnonlinearly to an rf current and produce a rectified voltage.\nIn this section we provide a general review of the ingredients of spin rectification: Sec. 2.1 contains a\ndiscussion of the relevant magnetoresistance effects while in Sec. 2.2 we review magnetization dynamics and\nthe Landau-Lifshitz-Gilbert equation. For simplicity we only concern ourselves with field torques in Sec.\n2.2, leaving the treatment of spin torques to Sec. 3.3.5 and Sec. 3.4 where we focus on bilayer devices and\nmagnetic tunnel junctions (MTJs). Finally in Sec. 2.3 we present the simplest example of spin rectification,\nthe voltage produced in a ferromagnetic monolayer, which will serve as an illustrative example which sets\nthe stage for the detailed description of SR in more complex structures presented in Sec. 3.\n2.1. Magnetotransport\nMagnetotransport – the modification of transport properties due to magnetic fields – can depend on\nboth the charge and spin of the carrier and as such is responsible for a myriad of phenomena in metals and\nsemiconductors [68]. However the magnetotransport effects which result in spin rectification are those which\ncausemagnetoresistance, wheretheelectricalresistanceofthematerialdependsonanappliedmagneticfield,\nR=R(H). Although even non-magnetic metals display so called ordinary magnetoresistance (OMR) due to\ncharged-inducedmagnetotransport(theLorentzforce), themaximummagnetoresistanceratioofsucheffects,\nMR = [R(H)\u0000R(0)]=R(0), is too small to be of technological interest. On the other hand in magnetic\nheterostructures large magnetoresistance due to spin-dependent transport dominates and as a result MR\neffects in magnetic materials have a rich history of applications. The anisotropic magnetoresistance (AMR),\nfirst discovered by William Thomson in 1856 [69], was the first MR effect used by IBM to build hard disk\ndrive read heads in 1991. The chief limitation of AMR for memory applications was the low MR ratio of\n\u0018a few %. For this reason the celebrated discovery of giant magnetoresistance (GMR), independently by\nFert [10] and Grünberg [11], with room temperature MR ratios up to \u0018100%, revolutionized the magnetic\nrecording industry. By 1997 IBM had used spin valve sensors based on GMR to replace the AMR sensors in\nmagnetoresistive hard disk drive read heads, providing an increase in areal density of up to 100% per year.\nWhile GMR in multilayer structures has become important for memory applications, it is important to note\nthat from a physics perspective GMR is not limited to multilayer structures but is more generally due to\ninhomogeneities and can be realized e.g. in granular magnetic solids [70].\nEvenfurtherincreasesintheMRratiobecamepossiblewithadvancesinthefabricationoftunnelbarriers\nmade of crystalline magnesium oxide (MgO) which made tunnel magnetoresistance (TMR) an attractive\npossibility for next generation memory devices. TMR, originally discovered in 1975 by Jullière in Fe/Ge-\nO/Co junctions at 4.2 K [71], has recently been used to achieve MR ratios of 600% at room temperature\nand over 1100% at 4.2 K in CoFeB/MgO/CoFeB magnetic tunnel junctions (MTJs)[72]. The large MR\nratio of MTJs allows the creation of non volatile, high endurance memories with fast random access making\nmagnetic random access memory (MRAM) a good candidate for a ‘universal memory’ [73].\nAll of these memory applications we have described rely on the staticproperties of MR, which are sum-\nmarized in Table 1 and which we will discuss further in this section as the first ingredient of spin rectification.\n5Table 1:Summary of static magnetoresistance effects discussed in detail in Sec. 2.1. The magnetoresistance effects for\nferromagnetic monolayers are considered for an in-plane Hfield and in-plane magnetization M.\u0012mdenotes the angle between\nthe magnetization and the current flow, \u0012is the angle between the magnetization of the two ferromagnetic layers, and \u0012sis the\nangle between the magnetization and the spin polarization.\nMR\nEffectOriginDevice\nStructureAngular Dependence MR ratio\nAMR SO CouplingFM mono-\nlayerRAMR (\u0012m) =R(0)\u0000\u0001Rsin2\u0012m\u0018a few %\nPHE SO couplingFM mono-\nlayerRPHE(\u0012m) =1\n2\u0001Rsin 2\u0012m \u0018a few %\nAHE SO couplingFM mono-\nlayerRAHE(\u0012m) = 0 \u0018a few %\nGMRSpin accumula-\ntionFM/NM\nmultilayersGGMR (\u0012) =GP+ \u0001Gsin2(\u0012=2)or, if GMR\nratio is small, RGMR (\u0012) =RP+\u0001Rsin2(\u0012=2)\u0018100 %\nNonaligned FM\ninclusionsGranular\nmagnetic\nsolids\u001810 %\nTMRSpin dependent\ntunnellingFM/Insulator\nmultilayer\n(MTJ)Same as GMR \u0018600 %\nSMRSHE and spin\nrelaxation\nthrough STTFMI/NM\nbilayerRSMR(\u0012s) =R(0) + \u0001Rsin2\u0012s \u00180.01 %\nHowever, when combined with the second ingredient, magnetization dynamics, spin rectification opens the\nbroader possibility of novel applications utilizing the dynamic properties of magnetoresistance.\n2.1.1. Two-Current Model\nThe two-current model, which attributes the resistivity in ferromagnetic materials to two independent\nspin channels, provides the basis for understanding OMR, AMR, GMR and TMR [74]. This model is based\non three key properties of 3d ferromagnets. First, as noted by Mott [75], the low effective mass of the\nvalencespband electrons, compared to the high effective mass of the dband electrons, means that sp\nelectrons carry most of the electric current. Second, because spin-flip scattering in ferromagnetic materials\nis negligible [76] the current can be modelled as two independent spin channels – spin-up (majority) and\nspin-down (minority) electrons. And finally, due to exchange splitting, the density of states (DOS) of d\nband electrons is greater for spin-up than spin-down electrons. Since the scattering rate is proportional to\nthe DOS, this final observation means that the scattering of up spins will be greater, and therefore the up\nspin channel will provide the greatest resistance contribution (hence the term majority electrons). Although\noriginally described for collinear systems, spin rectification often occurs under non-collinear conditions and\nthe two-current model has also been extended to such situations [77, 78].\nIn the case of OMR the electron current corresponds to the spband electrons while the hole current\ncorresponds to the dband holes and it is sufficient to only consider the effect of the Lorentz force on these\ntwo currents, ignoring the spin degree of freedom. However for AMR, GMR and TMR the spin dependence\nof the two currents is necessary. Insight into the origin of this spin dependence can be gained by considering\n6the Drude conductivity per spin for free electrons [74]\n\u001b=e2\nhk2\nF\n3\u0019\u0015: (2.2)\nHerekFis the Fermi momentum and \u0015is the mean free path which depends on the Fermi velocity and the\nrelaxation time \u001c,\u0015=vF\u001c. The relaxation time can be estimated by Fermi’s golden rule\n\u001c\u00001=2\u0019\n\u0016hhV2\nscatiN(\u000fF) (2.3)\nwherehV2\nscatiis the average value of the scattering potential and N(\u000fF)is the DOS at the Fermi energy.\nFrom Eqs. 2.2 and 2.3 we see that the conductivity has an intrinsic spin dependence caused by the spin\ndependence of kF;vFandN(\u000fF)which is due to the band structure of the material, but there is also a\ncontribution from the scattering potential which is not an intrinsic property of the ferromagnetic material\nand can be due to e.g. defects, impurities or lattice vibrations. In the case of AMR the scattering potential,\ndue to the spin-orbit interaction, is fundamental, however in GMR and TMR the effect of the scattering\npotential may be averaged out by interface effects in the multilayer structure, leaving the band structure as\nthe key contribution to the spin dependent conductivity [74].\n2.1.2. Anisotropic Magnetoresistance\nThe earliest discovered magnetoresistance effect in ferromagnetic metals was AMR [69]. The key char-\nacteristic of AMR is the angular dependence,\nR(H) =R(0)\u0000\u0001Rsin2(\u0012M) (2.4)\nwhere\u0012Mis the angle between the current and magnetization direction and \u0001R=Rk\u0000R?is the difference\nin resistance between the current and magnetization aligned ( Rk) and orthogonal ( R?). It should be noted\nthat Eq. 2.4 may not be appropriate for single crystalline films due to the band structure influence on AMR\n[79]. In such samples AMR can exhibit additional four-fold symmetry [80–83], asymmetric behaviour [84],\nand strong dependence on the angle between the current and crystalline axis [80–85]. In both poly and single\ncrystalline films typically \u0001R> 0[86], although there are exceptions [79, 87], and while the effect of AMR\nis small compared to GMR or TMR in multilayer structures, in monolayers AMR is the dominate effect and\nis typically as large as a few percent, although this depends on, amongst other factors, the material, sample\ngeometry and temperature [79], for example as the film thickness decreases, AMR decreases [88].\nExperimentally \u0012Mis controlled by an applied magnetic field Hand depends on the anisotropy fields in\nthe sample [43]. Therefore the magnetic field range over which AMR takes place is determined by the field\nneeded to change the direction of the magnetization, typically on the order of 10 mT. The exact relationship\nFigure 2:(a) AMR measurement configuration. A dc current is sent through the ferromagnetic sample and the voltage is\nmeasured as a function of the angle \u0012between the magnetization and current direction. Typically \u0012is controlled by changing\nthe field strength with \u0012H= 90\u000efixed. With zero applied field the magnetization will lie along the easy axis of the sample,\nin the current direction, and the resistance will be large. As the magnetic field strength is increased toward saturation, the\nmagnetization rotates to align with the field and the resistance decreases. (b) AMR of \u00180:4%in a permalloy (Py) microstrip\nat\u0012H= 90\u000e. The arrows denote the anisotropic field \u00160HA=\u00160Nx0M0= 4:0mT. The open circles are experimental data\nand the solid curve is a fit using R(0) = 112:66 \n;\u0001R= 0:47 \nandHA= 4:0mT.Source :Adapted from Ref. [43].\n7between\u0012MandHcan be determined by minimizing the free energy, F/H\u0001Mwith respect to \u0012M, taking\ninto account the anisotropy fields. For an in-plane magnetization with only shape anisotropy \u0012Msatisfies\nHsin (\u0012H\u0000\u0012M)\u0000Mcos\u0012Msin\u0012M(Nx0\u0000Nz0) = 0:\nHere\u0012His the angle of Hwith respect to the current direction, and Nx0andNz0are the demagnetization\nfactors in the coordinate system defined in Fig. 2 (a), where ( bx0;by;bz0) are fixed with the sample length\nalongbz0and the sample width along bx0and the measurement coordinate system ( bx;by;bz) rotates with the\nHdirection along bz. Typically the dimension of the sample in the bx0direction is much less than in the\nbz0direction, so the Nz0factor can be ignored. Fig. 2 (b) shows the AMR measured in a Py microstrip of\ndimension 300 \u0016m\u000220\u0016m\u000250 nm at an angle \u0012H= 90\u000ein which case sin\u0012M=H=Nx0M. The measured\ndata (symbols) has been fit using R(0) = 112:66 \n;\u0001R= 0:47 \n;\u00160HA= 4:0mT andNx0= 0:004and the\nAMR ratio is found to be \u0001R=R(0)\u00180:4%.\nThe physical origin of AMR is spin-dependent scattering in ferromagnetic metals due to their band\nstructure and the spin-orbit interaction [79, 86, 89]. This means that to understand the magnitude and sign\nof the effect a detailed knowledge of the band structure is required. However the angular dependence can\nbe determined either by symmetry arguments [79] or through the generalized Ohm’s law [90, 91],\nE=\u001a?J+\u0001\u001aO\nH2(J\u0001H)H+\u0001\u001a\nM2(J\u0001M)M\u0000\u001aH\njHjJ\u0002H\u0000\u001aAHE\njMjJ\u0002M (2.5)\nwhich is a phenomenological modification of the usually linear Ohm’s law to include the nonlinear effects\nintroduced by the presence of magnetic fields and a nonzero magnetization. The second and third terms\ndescribe OMR and AMR respectively while the third and fourth terms describe the ordinary Hall effect\n(OHE) and the anomalous Hall effect (AHE). OMR and the OHE are included here for completeness.\nAlthough the angular dependence of OMR is the same as AMR, as we will see, the spin rectified voltage\nis proportional to \u0001Rwhich in ferromagnetic monolayers is much smaller for OMR than it is for AMR.\nAlso, as illustrated in Fig. 2 (b) AMR is measured by sweeping the magnetic field and therefore in such\nmeasurements OMR would just contribute a constant background since \u0012His fixed. Likewise the AHE\ndominates over the OHE. We note that recently additional galvanometric effects have been predicted [92].\nThe resistivity along a general direction bnis defined as\n\u001an=bn\u0001E\njJj: (2.6)\nUsing this definition the generalized Ohm’s law immediately leads to the angular dependence of AMR\ndescribedbyEq. 2.4. Thein-planeresistancemeasuredtransversetothecurrentdirectioncanbedetermined\nby takingbnalong the width of the microstrip so that bn\u0001J= 0. This transverse AMR is know as the planar\nHall effect (PHE) and has an angular dependence\nRPHE (\u0012M) =\u0001R\n2sin (2\u0012M):\nTheAHEwillnotcontributewhenthevoltageismeasuredalongthestripduetothecrossproduct. However,\nalthough there is no static transverse contribution either, the AHE will contribute to the rectified transverse\nvoltage.\n2.1.3. Giant Magnetoresistance\nGMRisaspinaccumulationeffectpresentinmultilayerstructuresconsistingofalternatingferromagnetic\n(FM) and normal metal (NM) layers and as the name suggests results in much larger magnetoresistance\nratios than AMR, up to nearly 100%. The simplest description of GMR is the resistor model [93], based\non the idea of the two band model discussed above, that the current is carried by two non-interacting\nspin channels. If the magnetizations of the alternating FM layers are aligned, the majority electrons will\n8Figure 3:Resistor model of GMR. Due to the small probability of spin flip scattering the majority (up) and minority (down)\nelectrons can be treated as two independent conduction channels. (a) When the magnetization of the FM layers is aligned, the\nmajority electrons with spin along the magnetization direction can move with little scattering and therefore have an overall low\nresistance, while the minority electrons are strongly scattered in each FM layer and have a resulting high resistance. Adding\nthe resistance from the two channels in parallel leads to a low resistance state. (b) When the magnetization of the FM layers\nis antiparallel, the minority and majority electrons are scattered equally after passing through two FM layers. Adding the\nresistance for the two channels in parallel leads to a high resistance state. Source :Adapted from Ref. [74].\nexperience a resistance R\"in each layer, whereas the minority electrons which are more strongly scattered\nwill have resistance R#>R\". Adding these resistances in parallel, the P state has a resistance\nRP=2R\"R#\nR\"+R#: (2.7)\nOntheotherhand, ifthemagnetizationsareantiparallel, theminorityandmajorityelectronswillexperience\nthe same resistance after travelling through the two layers, so that the AP state has resistance\nRAP=R\"+R#\n2(2.8)\nand therefore the GMR ratio is\nGMR =\u0001R\nR=RAP\u0000RP\nRP=\u0000\nR#\u0000R\"\u00012\n4R#R\"=(\u000b\u00001)2\n4\u000b(2.9)\nwhere\u000b=R#=R\"=\u001a#=\u001a\"is the spin-asymmetry parameter. Therefore in order to achieve GMR the\nmultilayer structure must be capable of achieving two resistance states – the parallel, P and antiparallel,\nAP, states where the magnetization of the alternating FM layers is aligned or anti aligned respectively. The\nP state can always be achieved by applying sufficiently large magnetic fields, however the AP state must be\nachieved by sample design. Generally RP> R AP, although there are exceptions for multilayers comprised\nof different FM layers [94]. One limitation of this resistor model is that surface scattering at the FM/NM\ninterface has been ignored [95]. We should note that here we have adopted the most optimistic definition of\nthe GMR ratio. An alternative definition, GMR = (RAP\u0000RP)=(RAP+RP)is also used by many authors.\nInitial studies of GMR achieved the AP state using antiferromagnetic (AFM) coupling between the FM\nlayers [96] at H= 0. Adjusting the thickness of the NM layer produces oscillations in the exchange coupling\nbetween layers [97–100] and therefore it is possible to choose a thickness such that the layers are AFM\ncoupled. The disadvantage of the AFM coupled multilayer structures is that it takes a large Hto switch\nbetween the AP and P states. Fortunately the AP states can be achieved in other ways. Fig. 4 summarizes\nthe different structures that give GMR. One option is to have a [FM/NM] nmultilayer repeating nidentical\nFM and NM layers. The AP state can then be realized by AFM exchange coupling between layers or by\ndipolar coupling, which can be made the dominant coupling effect by increasing the thickness of the NM\nlayer to reduce the exchange interactions. Alternatively a FM/NM/FM multilayer can be used. This so\ncalled pseudo spin valve contains both a soft FM layer with small coercivity field, Hc, and a hard FM\n9Figure 4:Sample structures which display GMR. The resistance state in all structures is controlled with an in-plane magnetic\nfield. (a) A multilayer structure composed of alternating ferromagnetic (FM) and nonmagnetic (NM) layers. When the\nmagnetization of all FM layers are aligned (all solid arrows) the structure will be in the low resistance P-state, while the high\nresistance AP-state will occur when alternating FM layers point in opposite directions (dashed arrows). The FM layers may or\nmay not be exchange locked. Also shown are the CIP and CPP current directions. In both cases the resistance is measured in\nthe current direction. (b) A pseudo spin valve composed of two FM layers of high (hard) and low (soft) coercivities separated\nby a NM layer. (c) A spin valve where a FM layer is pinned by exchange coupling to an antiferromagnetic (AF) layer. The\nthickness of the NM layer is adjusted so that the magnetization of the free FM layer can still be adjusted. Source :Adapted\nfrom Ref. [74].\nlayer [101–103]. An applied field will switch the soft layer first, providing the AP state. A true spin valve\nstructure, AF/FM/NM/FM contains a free FM layer that can be switched in a small field and a FM layer\nwhich is pinned by an antiferromagnetic layer and therefore only reverses under a large field [104].\nAs shown in Fig. 4 (a), GMR experiments are normally performed with the current in the layer plane\n(CIP) or current perpendicular to the layer plane (CPP), with the resistance measured in the current\ndirection. However experiments have been performed with the current at an angle to the layer plane (CAP)\n[105, 106]. For CIP GMR the relevant length scale is the mean free path, \u0015of the NM and FM, however for\nCPP GMR due to spin dependent accumulation effects the longer (compared to \u0015) spin diffusion length is\nimportant [107]. Although CPP GMR >CIP GMR for a sample with fixed lateral dimension and thickness,\nthe CPP resistance will be as much as 8 orders of magnitude less than the CIP resistance [108]. Therefore\ninitiallyGMRexperimentswereperformedusingtheCIPgeometryduetothe0.01-1 \nresistanceswhichcan\neasily be achieved. However improvements in fabrication techniques have lead to CPP becoming standard,\nwith a best MR ratio at room temperature of 80% as reported by Jung et al. [109], which is even higher\nFigure 5:(a) GMR measurement setup, which is similar to that used for AMR measurements. The key difference is that\nfor a multilayer sample the resistance could be measured using a current applied in the sample plane (shown) or the current\ncould be applied perpendicularly. Typically the magnetoresistance ratio for the CPP geometry will be greater than for the CIP\ngeometry, however the absolute resistance measured will be less. (b) GMR curve for Fe/Cr and Co/Cu multilayer structures\n[115]. The MR ratio is up to 2 orders of magnitude greater than AMR. (c) The angular dependence of GMR. While the\nconductance is linear in (1\u0000cos\u0012)the resistance has slight deviations from linearity, however these variations are higher order\nin the GMR ratio [116].\n10than TMR in Al-O based MTJs. For a good review of experimental techniques see Ref. [95].\nThe angular dependence of GMR has been studied in both CIP [104, 110] and CPP geometries [111] and\nwas found to follow\nR(\u0012) =RP+1\n2(RAP\u0000RP) (1\u0000cos\u0012) =RP+ \u0001Rsin2(\u0012=2) (2.10)\nwhere\u0012is the angle between the magnetization of the two layers and \u0001R=RAP\u0000RP. Different approaches\nto a theoretical analysis of the angular dependence predict that the conductance, not the resistance, varies\nascos\u0012[112–114]. Although the deviations of the resistance from the cos\u0012dependence is second order in\nthe GMR ratio they are noticeable in the data shown in Fig. 5 (c). This deviation is more pronounced in\nthe CPP geometry.\n2.1.4. Tunnel Magnetoresistance\nSimilar to GMR, TMR is due to a resistance difference between P and AP configurations in a multilayer\nstructure. However in TMR the FM layers are separated by an insulator and therefore TMR is a result\nof spin polarized tunnelling, rather than spin accumulation effects. A typical schematic structure of a spin\nvalve magnetic tunnel junction (MTJ) in which TMR is observed is shown in Fig. 6. Although a pseudo spin\nvalve may also be used, the spin valve is preferred because the resistance change occurs near zero magnetic\nfield and the spin valve has greater magnetic stability [118]. The ability to control the tunnelling probability\nof the majority and minority electrons through high quality interfaces and a choice of electrode and barrier\nmaterials is what allows the high MR ratios observed in MTJs.\nEarly observations of small TMR ratios ( \u001814%at 4.2 K) were made as early as 1975 [71, 119], however\nas these were not reproducible due to the difficulty of the fabrication process, little research was done on the\neffect until the discovery of GMR provided a renewed interest in the possibility of large MR ratios. The first\nreproducible experiments used a pseudo spin valve MTJ with an amorphous aluminum oxide tunnel barrier\nand achieved MR of 20%at room temperature [120, 121]. Further optimization of fabrication techniques\nled to TMR ratios of \u001870%in Al-O MTJs, however this was not high enough for modern technological\napplications. To increase TMR the amorphous tunnel barrier had to be replaced with a crystalline barrier\nsuch as MgO which is capable of symmetry selection. Improvements in the structure quality of MgO based\nMTJs quickly led to the giant TMR results of up to 200% in 2004 [117, 122] and modern devices are capable\nof achieving MR ratios of over 500% at RT [72, 123].\nThe increased TMR due to an MgO tunnel barrier can be understood, at least qualitatively, by consider-\ning the phenomenological model of Jullière [71] where the TMR is due to spin-dependent electron tunnelling.\nIn Jullière’s model spin is assumed to be conserved during tunnelling so that the two-current model can be\napplied and the tunnelling probability will be proportional to the spin dependent DOS of the two electrodes.\nFigure 6:(a) Schematic cross section of a typical spin valve MTJ. Exchange coupling with the AF layer pins the magnetization\nof the bottom FM electrode and a tunnel barrier separates the free and pinned layers. (b) Typical TMR data taken from a\nFe(001)/MgO(001)/Fe(001) MTJ [117]. The resistance is largest in the antiparallel state and a MR ratio of nearly 200% is\nfound at room temperature.\n11Also the tunnelling probability is assumed to be spin independent, which means that spin dependent tun-\nnelling rates will only be a result of the spin dependent DOS. Since the conductance Gis proportional to\nthe DOS,Ni\nj(EF), wherej=L;R indicates the left and right electrodes and i=\";#indicates the majority\nand minority electrons in the electrode, we have\nGP/N\"\nLN\"\nR+N#\nLN#\nR; G AP/N\"\nLN#\nR+N#\nLN\"\nR\nand so the TMR ratio is\nTMR =GP\u0000GAP\nGAP=RAP\u0000RP\nRP=2PLPR\n1\u0000PLPR(2.11)\nwhere\nPj=N\"\nj\u0000N#\nj\nN\"\nj+N\"\nj; j=L;R:\nTMR ratios estimated from measured spin polarizations using Jullière’s model agree with the measured\nTMR ratios fairly well. Therefore using the experimental results for polarization in 3d ferromagnetic metals\nand their alloys, which typically range from 0< P < 0:6below 4.2 K [73, 124], Eq. 2.11 can be used to\nestimate a maximum TMR of \u0018100%. This value is further reduced at room temperature due to thermal\nspin fluctuations.\nOnewaytoincreasethepolarizationoftheelectrodewouldbetousehalf-metallicferromagneticmaterials\nwhich act as metals for one spin direction and insulators for the other, and therefore theoretically have\npolarizations of 100% [125, 126]. Half metal electrodes have been made using e.g. CrO 2or certain Heusler\nalloys, howeverlargeTMRratioshaveonlybeenachievedatlowtemperatures[127]. Anotherwaytoincrease\nthe polarization of the electrodes would be to exploit another limitation of Jullière’s model – the assumption\nthat the tunnelling is completely incoherent. This assumption is incorrect even for amorphous insulating\nlayers and is clearly incorrect for crystalline tunnel barriers such as MgO where the tunnelling states have\nspecific symmetries. The different symmetry states experience different decay rates determined by their\nsymmetry matching with the lattice, while all states tunnel at the same rate through the amorphous Al-O.\nThischaracteristiccanbeexploitedtoallowonlyhighlyspinpolarizedstatestotunnelthrough, dramatically\nincreasing the TMR ratio.\nThe magnitude of TMR depends on the electrode and barrier materials, the fabrication technique and\nquality of interfaces. These details will not be discussed in detail here, however for more information see\ne.g. [127–129]. It is useful to mention though that for device applications the Fe/MgO/Fe MTJs we have\ndiscussed cannot be used. This is because the pinned layer used in spin valve MTJs uses a synthetic\nferrimagnetic tri-layer that has an fcc(111) orientation on which the bcc(001) Fe/MgO/Fe structure cannot\nbe grown. The reliability of the standard fcc(111) pinned layer is essential and it is therefore easier to use a\nnew MTJ structure. The standard alternative to the Fe/MgO/Fe MTJ is a CoFeB/MgO/CoFeB structure\n[130]. The amorphous CoFeB electrode layer can be grown on the synthetic pinning layer and also allows\nthe growth of a MgO(001) tunnel barrier.\nThe angular dependence for small TMR ratios is the same as Eq. 2.10 which was already discussed for\nGMR [131]. However for larger TMR ratios, the deviations from cos\u0012dependence may become pronounced\n[132]andthecorrectangulardependenceismosteasilyexpressedintermsoftheconductance, Gas[133,134]\nG(\u0012) =R(\u0012)\u00001=GP+GAP\n2+GP\u0000GAP\n2cos\u0012=GP+ \u0001Gsin2(\u0012=2): (2.12)\nHereGP=G(\u0012= 0) =R\u00001\nP,GAP=G(\u0012=\u0019) =R\u00001\nAPand\u0001G=GAP\u0000GP. Eq. 2.12 and Eq. 2.10 agree\ntoO\u0000\nTMR2\u0001\n.\n2.1.5. Spin Hall Magnetoresistance\nThe MR effects discussed so far all require an electrical current to pass through the FM material in which\nthe resistance change is observed, meaning these effects cannot be used to study the magnetic properties of\nferromagnetic insulators (FMIs). However the discovery of spin Hall magnetoresistance (SMR) in FMI/NM\n12hybrid structures [135–148] can be used to study FMI since this effect produces a magnetoresistance in the\nNM layer through the influence of the magnetization in the FMI. This is dramatically different than the\nMR effects already discussed where the magnetoresistance occurs in the FM layer. The key distinguishing\nfeatures of SMR, compared to AMR, is the greater perpendicular resistance, RT\u0018Rk> R?for SMR\nwhileRk> R?\u0018RTfor AMR, where RTis the perpendicular MR with M?IandMperpendicular to\nthe sample plane [135]. Most studies have focussed on Yttrium-Iron-Garnet (YIG)/Pt structures and the\nremoval of Pt or the use of a non magnetic insulator is found to remove the effect.\nFigure 7:(a) - (d) Systematic measurements of the SMR in a Pt jYIG bilayer showing the \u000b;\fand\r(defined in (f)-(h)\nrespectively) dependence of the magnetoresistance [135]. Black circles are experimental data while red and blue curves are\nthe expected curves due to SMR and AMR respectively. (b) Schematic illustration of spin Hall magnetoresistance in a\nFMI(green)/NM(blue) hybrid structure. The charge current in the NM generates a spin current through the SHE. If the\nmagnetization in the FMI is perpendicular to the spin polarization at the FMI/NM interface, a spin-transfer torque will act on\nthe magnetization, reducing the current in the NM and increasing the resistance. When the spin polarization and magnetization\nare aligned, the spins are reflected and the resistance decreases [149]. (a) shows the \u000bdependence of the longitudinal MR, (b)\nshows the\u000bdependence of the transverse MR, (c) shows the \fdependence of the longitudinal MR. For these cases the angular\ndependence of SMR and AMR is the same. (d) shows the \rdependence of the longitudinal MR which shows the key difference\nbetween SMR and AMR. Source: Data from Ref. [135]. Schematic adapted from Ref. [149].\nSMR is due to a combination of the spin Hall effect (SHE) [150–153] and spin dependent scattering at the\nFMI/NM interface. A charge current in the NM will generate a spin current through the SHE. This results\nin an accumulation of spin at the NM/FMI interface. If the spin polarization is parallel to the magnetization,\nthen the spin currents do not enter the FMI and are reflected back at the interface. The inverse spin Hall\neffect(ISHE)thengeneratesanelectriccurrentwhichisparalleltotheoriginalcurrent, increasingthecurrent\nin the NM, resulting in a decreased resistance. On the other hand when the polarization of the spin current is\nperpendicular to the magnetization, spin diffusion into the FMI can occur via spin-transfer torque (discussed\nmore in Secs. 3.3.5 and 3.4). When spin current does cross the interface, spin accumulation is reduced and\nthe charge current in the NM is reduced, increasing the resistance. Therefore the net result of the spin Hall\neffect in the metal, and spin dependent scattering at the metal-insulator interface is that the resistance will\nbe high (low) when the spin current polarization is perpendicular (parallel) to the magnetization of the FMI\nand the resistance change may be written as\nR=R0+ \u0001Rmaxsin2\u0012s\nwhere\u0012sis the angle between the magnetization and the spin polarization [138]. \u0001Rmaxwill depend on the\ndirection of magnetization rotation as shown in Fig. 7 (a) - (d). Since SMR is due in part to spin diffusion\nat the FMI/NM interface, the sample size must be comparable to the spin diffusion length.\n13SMR is much smaller than AMR, \u00180:01%in YIG/Pt [135], though this value depends on both tem-\nperature [144] and the Pt thickness [140]. SMR has been used to measure the spin mixing conductance in\nYIG/Pt [142] and comparative studies of spin pumping, the spin Seebeck effect and SHR have been per-\nformed [141, 147]. Since the initial work on YIG/Pt samples, different FMI/NM hybrid structures have also\nbeen studied using for example Fe 3O4, NiFe 2O4[139] and CoFe 2O4[145] as FMI and Au, Cu [139] and Ta\n[138] as NM. It has been found that SMR is enhanced in trilayer (FMI/NM/FMI) structures and it exhibits\na different angular dependence from AMR [137]. However, it should be noted that for thin ferromagnetic\npolycrystalline films the angular dependent magnetoresistance is much more complex than that generally\naccepted for isotropic bulk samples [79, 154–157]. In particular, single layer polycrystalline Fe thin films\nsandwiched between insulating nonmagnetic layers [158] have recently been found to reproduce all AMR\ncorrelations previously reported in polycrystalline films, including the ones that resembles SMR. These find-\nings suggest that caution should be taken when using angular dependent MR to ascertain the prescence of\nSMR.\nTo rule out the possibility that the MR was caused by known proximity effects, which cause NM layers\nto acquire FM properties when deposited on a FM layer [159, 160], studies have been performed with Cu\ninserted between the NM and FMI. Since the proximity effect is not present in Cu and exchange forces will\nnot be effective over the thick Cu layer used, the observation of MR in Pt/Cu/YIG confirms the explanation\ndue to SMR, since the long diffusion length of copper would still enable SMR. Nevertheless proximity effects\nmay still be important [136] and in fact both effects may be responsible [143].\n2.2. Magnetization Dynamics\nThe second requirement of spin rectification is some form of magnetization dynamics which can be\ncontrolled to produce a dynamic magnetoresistance. Magnetization dynamics may be produced by either\nafield torque , where an applied rf magnetic field interacts with the magnetization through a Zeeman type\ninteraction, or by a spin torque , where a spin polarized current influences the magnetization motion through\nan exchange interaction. Depending on device structure either or both of these torques may be present\nand it is necessary to know the response function of the magnetization motion to the field or spin torque.\nDetermining such a response function in the case of field torques will be the focus of this section, with spin\ntorques discussed in Secs. 3.3.5 and 3.4.\n2.2.1. Field Torque Induced Magnetization Dynamics\nOne way to excite a dynamic magnetization is by applying a microwave field which will cause the mag-\nnetization to precess. This precessional motion is described by the phenomological Landau-Lifshitz-Gilbert\n(LLG) equation. The Landau-Lifshitz (LL) equation [161] without damping follows from the Heisenberg\nequation of motion for the spin operator Sin the presence of a magnetic field Hiwith a Zeeman type\ninteraction [7],\ndM\ndt=\u0000\rM\u0002Hi (2.13)\nwhere\r=\u00160gjqj=2m > 0is the gyromagnetic ratio of the material undergoing FMR. gis the Landé g-\nfactor which is approximately 2 for a free electron and in general depends on the ferromagnetic material\nand the frequency of the applied microwave field, typically ranging from 1.7 to 2.3. Hiis the internal\nmagnetic field which contains contributions from the external applied field, H, the anisotropy fields and the\ndemagnetization fields. Together these magnetic fields produce a torque which causes the magnetization to\nprecess around Hi.\nWithout damping the precession described by the LL equation will continue regardless of the magnitude\nofH. Of course this behaviour is not observed experimentally and instead hysteresis curves show that for\nlarge enough Hthe magnetization should saturate and align with the magnetic field. This behaviour can be\nintroduced by the addition of a damping term that will produce a torque towards the magnetic field. Such\ndamping can be introduced in various ways. When it is necessary to distinguish the spin-lattice relaxation\nfrom the spin-spin relaxation Bloch damping may be used [162]. However the simplest way to introduce\n14Figure 8:(a) Without damping the magnetization experiences a torque which causes it to precess at a constant cone angle\naroundthe Hfielddirection. (b)Dampingproducesaninwardtorquewhichreducestheconeangleandcausesthemagnetization\nto align with H. (c) The Lorentz and dispersive line shape components which both contribute to the susceptibility determined\nby solving the LLG equation. Lis symmetric around the resonance field, while Dis antisymmetric. (d) The spin resonance\nphase, \u0002, which is another key characteristic (along with the line shape components) of the magnetization dynamics. Near\nresonance the spin resonance phase changes from \u0019to 0 within a range on the order of the line width \u0001H.\ndamping is via the Gilbert damping parameter \u000b[163],\ndM\ndt=\u0000\rM\u0002Hi+\u000b\nM\u0012\nM\u0002dM\ndt\u0013\n: (2.14)\nThe Gilbert damping term produces a torque which causes the magnetization to move inward and align with\nthe field and is applicable over a wide power range. Since this torque is perpendicular to the magnetization\ndirection the LLG equation of Eq. 2.14 describes precession and damping that keep the magnitude of M\nconstant. Note that, although the Gilbert damping is introduced phenomenologically, it has a microscopic\norigin in the spin-orbit interaction [164].\nThe internal magnetic field Hihas both a dc and rf contribution, Hi=H0i+hie\u0000i!t. As mentioned\nthe internal field consists of the externally applied field and any anisotropy fields which are present. If the\ndominant effect is the shape anisotropy, then the internal field can be related to the applied field and the\nmagnetization through the demagnetization factors, H0ik=Hk\u0000NkM0kandhik=hk\u0000Nkmk. Here the\nksubscript denotes the kthcomponent of the respective fields, H0iis the internal dc field, His the applied\ndc field,hiis the internal rf field, his the applied rf field, Nkis the demagnetization factor in the kth\ndirection, and M=M0+m(t)whereM0andm(t)are the equilibrium and non-equilibrium magnetizations\nrespectively. The demagnetization factors, which relate the internal magnetic fields to the externally applied\nmagnetic fields, are geometry dependent and obey the sum rule Nx+Ny+Nz= 1[165]. We define our\ncoordinate system so that the external magnetic field is always along the zaxis. By doing so both the dc field\nin and perpendicular to the plane can be described by the same solution to the LLG equation by choosing\nthe correct demagnetization factors. M0will also be along bzand sinceMis constant m(t)will only have\ndynamical components along bxandby(actually for elliptical precession mwill have a 2!dynamic component\nalongbz, however we will not consider this higher order term here). Using the dc and rf components in the\nLLG equation and keeping only linear terms in mandh(motivated by the low microwave power used by\ntypical experiments, but see Sec. 4.2) the Polder tensor \u001f, which relates the rf magnetization to the rf\n15magnetic field, can be determined,\nm=\u001fh=0\n@\u001fxxi\u001fxy0\n\u0000i\u001fxy\u001fyy 0\n0 0 01\nAh=0\n@j\u001fxxj j\u001fxyjei\u0019=20\nj\u001fxyje\u0000i\u0019=2j\u001fyyj 0\n0 0 01\nAhei\u0002(2.15)\nwhere (\u001fxx;\u001fxy;\u001fyy) = (D+iL) (Axx;Axy;Ayy)with [24]\nAxx=\rM0[M0Ny+ (H\u0000NzM0)]\n\u000b![2 (H\u0000NzM0) +M0(Nz+Ny)]\nAxy=\u0000M0\n\u000b[2 (H\u0000NzM0) +M0(Nz+Ny)]\nAyy=\rM0[M0Nx+ (H\u0000NzM0)]\n\u000b![2 (H\u0000NzM0) +M0(Nz+Ny)]\nand\nL=\u0001H2\n(H\u0000Hr)2+ \u0001H2; D =\u0001H(H\u0000Hr)\n(H\u0000Hr)2+ \u0001H2:\nHereH=jHjandM0=jM0j. Also note that the rf magnetic field hcontains a possible phase shift with\nrespect to the electric field which will drive the rf current in the sample. Here this phase shift is kept implicit\nfor simplicity, but we will have to include it explicitly in the next section when we analyze spin rectification.\nHris the resonance field which depends on !and the sample configuration according to the Kittel formula\n!2=\r2[Hr+M0(Ny\u0000Nz)] [Hr+M0(Nx\u0000Nz)]; (2.16)\nand\u0001His a measure of the resonance line width (for the symmetric Lorentz function L,\u0001His the half\nwidth half maximum) which in general depends on !;Hand the sample configuration,\n\u0001H=2 (H\u0000NzM0) +M0(Nx+Ny)\nH+Hr+M0(Nx+Ny\u00002Nz)\u000b!\n\r\nbut reduces to \u0001H\u0019\u000b!=\rnear resonance. The form of the line width which follows directly from the\nLLG equation does not take into account the effect of magnetization inhomogeneities and for comparison to\nexperimental results the expression \u0001H= \u0001Hin+\u000b!=\ris typically used, where \u0001Hinis the inhomogeneous\nline width broadening. \u0002is the spin resonance phase which describes the phase shift between the rf driving\nforce and the dynamic magnetization response. It is defined by tan \u0002 = \u0001H=(H\u0000Hr) =L=Dso that\ncos \u0002 =D=p\nL2+D2andsin \u0002 =L=p\nL2+D2. Near FMR \u0002changes sign over a field range of order \u0001H\nfrom\u0019forH <Hrto 0 forH >Hras shown in Fig. 8 (d).\nAn important feature of \u001fis that each element contains both a Lorentz and dispersive component which\nhave opposite symmetries around HrwithL(Hr+\u000eH) =L(Hr\u0000\u000eH)andD(Hr+\u000eH) =\u0000D(Hr\u0000\u000eH)\nwhere\u000eHis the field detuning. This symmetry which is illustrated in Fig. 8 (c) is strongly dependent on\nthe spin resonance phase, as we can see from the fact that for any element of \u001f,j\u001fijj2/L2+D2=L. In\naddition, both the Lorentz and dispersive components are symmetric under a field rotation of \u0019,H!\u0000H\n(note that under this rotation Hr!\u0000Hr).\n2.3. The Method of Analyzing Spin Rectification: Basic Examples\nHaving discussed magnetoresistance and magnetization dynamics we are now in a position to consider\nanalyzing spin rectification [24, 43, 166]. The dc voltage produced by SR results from the nonlinear coupling\nbetween a time dependent resistance caused by magnetoresistance and a time dependent current driven by\nan rfefield. SR is nonlinear in the sense that only contributions of O\u0000\nh2;j2;hj\u0001\nwill contribute to the time\naverage which is taken over one period. In this section we present the simplest example of spin rectification,\n16whichoccursinferromagneticmonolayerswherethemagnetoresistanceiscausedbymagnetizationprecession\ndrivenbyatorquefromanrf hfield. Thereforewewillnamethiseffect field torque rectification todistinguish\nit from SR dueto spintorque which willbe discussed inSecs. 3.3and 3.4. This simplest caseof SRillustrates\nthe key ideas that can be used to investigate SR in more complex device structures and may be analyzed\nusing Eq. 2.1. However if the voltage is not measured parallel to the rf current in general the planar,\nordinary and extraordinary Hall effects will also contribute to the measured voltage. It is therefore easier\nto determine the voltage line shape directly from the generalized Ohm’s law of Eq. 2.5. This method also\nprovides another way to see the nonlinearity of the SR effect since the rf current generates corrections to\nthe electric field due to AMR and the Hall effects. We will consider only the AMR and AHE contributions\nin Eq. 2.5 which are dominant in ferromagnetic metals (the OMR and OHE could be treated analogously).\nAn important feature of the voltage line shape is its angular dependence, which allows the different\nphysical contributions and the different driving fields to be separated. In FM monolayers both the static\nbehaviour,summarizedinTable1,andtherfbehaviour,characterizedbyacoordinatechange,willcontribute\nto the angular dependence. Even before going through a detailed derivation of the line shape it is possible\nto note the key angular dependencies due to the static behaviour from the magnetoresistance summarized\nin Table 1. For AMR, RAMR (\u0012m) =R(0)\u0000\u0001Rsin2\u0012mso ifM0andHare collinear, VAMR/@R=@\u0012H=\nsin 2\u0012H. Proceeding analogously we expect VPHE/cos 2\u0012Hand thatVAHEwill have no angular dependence\nfrom the static behaviour. Each of these effects will also have an angular contribution which is unique to\neach component of the driving rf field, which must be determined by an analysis of the generalized Ohm’s\nlaw.\n2.3.1. In-Plane Angular Dependence\nWefirstconsiderthecasewhere Hisappliedinthesampleplane. Thecoordinatesystemusedisshownin\nFig. 9 (a). (bx0;by;bz0)are the sample coordinates, with bz0fixed along the long axis of the sample, bx0along the\nwidth andbynormal to the sample plane. (bx;by;bz)rotate with Hso thatH=Hbzmakes an angle \u0012Hwithbz0.\nNote that since we always consider field rotations in the x0\u0000z0plane, thebydirection will remain fixed in all\ncoordinate systems and there is no need to define both sample and field bydirections. The in-plane coordinate\nsystems are related by (bx;by;bz) = (cos\u0012Hbx0\u0000sin\u0012Hbz0;by;sin\u0012Hbx0+ cos\u0012Hbz0), which can be conveniently\nwritten as~x=U~x0withUbeing a rotation about byby\u0012H. If the sample length (along bz0) is much greater\nthan its width (along bx0), the current will flow along bz0. If in addition there is no applied dc current then\nJ=jtbz0=Re(jz0e\u0000i!t)bz0(a dc current will lead to a photoresistance [24] which we do not discuss here).\nSplitting the magnetization into its dc and rf components, M=M0bz+mt=M0bz+Re\u0000\nme\u0000i!t\u0001\n, the time\naverage of the electric field is\nFigure 9:(a) Coordinate systems for an in-plane static magnetic field H. Thez0andx0axes are fixed along the sample’s length\nand width respectively with the yaxis normal to the sample, and current is assumed to flow along z0. Thezaxis is along the\nstatic field Hand static magnetization M0which is rotated by an angle \u0012Hwith respect to the z0axis. (b) Coordinate systems\nfor the first out-of-plane rotation where the field is rotated along the sample length. The x0axis is now along the length of the\nsample, the yaxis is along the width of the sample and the z0axis is normal to the sample, with current along x0. Again the\nstatic field and magnetization are along the zaxis which is rotated by an angle \u001eHwith respect to z0. (c) Coordinate systems\nfor the second out-of-plane rotation where the field is rotated along the sample width. The yaxis is now along the length of\nthe sample, the x0axis is along the width of the sample and the z0axis is normal to the sample and the current is along y.\nAgain the static field and magnetization are along the zaxis which is rotated by an angle  Hwith respect to z0.\n17EMW=\u0000\u0001\u001a\nM2\n0hjt\u0001M0mt+jt\u0001mtM0i+RAHEhjt\u0002mti: (2.17)\nHere we have used the fact that to lowest order jMj=M0is constant and therefore m\u0001M= 0. From\nEq. 2.17 we can see that, due to the cross product, only a magnetization component perpendicular to the\nplane will produce a signal due to the AHE. This is why, when His in-plane, there is no static angular\ndependenceassociatedwiththeAHE,andalsowhytheAHEwillnotcontributewhenthevoltageismeasured\nlongitudinally i.e. along the length of the sample. We also see that only mxwill contribute to longitudinal\nmeasurements. These expectations are realized when we calculate the longitudinal voltage by integrating\nEMWalong the length of the strip,\n(VSR)l\n\u0012=\u0001R\nM0sin (2\u0012H)hIt\nz0mt\nxi=\u0001R\n2M0Iz0sin (2\u0012H)Re(mx): (2.18)\nHere \u0001R=\u001al=AandIz0=jz0AwhereAandlarethecrosssectionalareaandlengthofthestriprespectively.\nAs expected the sin (2\u0012H)dependence is the key characteristic of the AMR induced SR.\nEven though only mxcontributes to the SR voltage, since mxcan be related to the rf hfield using the\nsusceptibility in Eq. 2.15 both hxandhycan drivemx. Furthermore hmust be written in the primed\ncoordinates, m=\u001fh=\u001fUh0which will introduce an additional \u0012Hdependence which is different for each\ncomponent of h0and will also allow hz0to contribute. Using the susceptibility the photovoltage is\n(VSR)l\n\u0012=\u0001R\n2M0Iz0\u0002\nA\u0012x\nLL+A\u0012x\nDD\u0003\n(2.19)\nwhere\nA\u0012x\nL= sin (2\u0012H) [\u0000Axxhx0cos\u0012Hsin \bx0\u0000Axyhycos \by+Axxhz0sin\u0012Hsin \bz0];\nA\u0012x\nD= sin (2\u0012H) [Axxhx0cos\u0012Hcos \bx0\u0000Axyhysin \by\u0000Axxhz0sin\u0012Hcos \bz0]:\nHere we have introduced the notation Aij\nLandAij\nDfor the amplitudes of the Lorentz and dispersive con-\ntributions respectively where i=\u0012;\u001e; corresponding to the \u0012H,\u001eHand Hrotations as defined in Fig.\n9 andj=x;yindicating from which component of the dynamic magnetization, mxormythis amplitude\narises. We will use this same notation when discussing out-of-plane rotations in the next section. We also\ndenote the voltage as (VSR)j\niagain withi=\u0012;\u001e; andj=l;tindicating that it is a voltage measured along\nthe length or width of the sample respectively. Similar notation will be used when we discuss spin pumping.\nThe relative phase \bbetween the rf electric and magnetic fields, which may be different in the bx0;byand\nbz0directions, is now explicit. The amplitudes A\u0012x\nLandA\u0012x\nDof the Lorentz and dispersive line shapes show\nthe key characteristics of the AMR SR: V/sin (2\u0012H) cos\u0012Hfor precession driven by hx0,V/sin (2\u0012H)\nfor precession driven by hyandV/sin (2\u0012H) sin\u0012Hfor precession driven by hz0. The different angular\ndependence allows the contribution from each h0component to be separated, and the ratio of Lorentz and\ndispersive amplitudes for each component, A\u0012x\nL=A\u0012x\nD, allows the relative phase to be determined for each h0\ncomponent.\nFor transverse measurements, across the width win thebx0direction, both the AMR and AHE will\ncontribute,\n(VSR)t\n\u0012=\u0000\u0001R\n2M0Iz0cos (2\u0012H)Re(mx)\u0000wRAHE\n2jz0Re(my): (2.20)\nAgain only mxcontributes to the SR due to AMR. On the other hand only mycontributes to the AHE\nas expected. Again the magnetization is related to hthrough the susceptibility and an additional angular\ndependence is introduced by changing from htoh0,\n(VSR)t\n\u0012=\u0000\u0001R\n2M0Iz0cos (2\u0012H)\nsin (2\u0012H)\u0002\nAtx\nLL+Atx\nDD\u0003\n\u0000wRAHE\n2jz0\u0002\nAty\nLL+Aty\nDD\u0003\n18where\nA\u0012y\nL=Axyhx0cos\u0012Hcos \bx0\u0000Ayyhysin \by\u0000Axyhz0sin\u0012Hcos \bz0;\nA\u0012y\nD=Axyhx0cos\u0012Hsin \bx0+Ayyhycos \by\u0000Axyhz0sin\u0012Hsin \bz0\n\u0001R= \u0001\u001aw=AandIz0=jz0A. The characteristic of the SR due to the PHE is the cos (2\u0012H)dependence,\nwhereas the SR due to AHE has no angular dependence, other than that due to the coordinate change of h.\n2.3.2. Out-of-Plane Angular Dependence\nFor an out-of-plane magnetic field we now have two planes of rotation to consider, as shown in Fig. 9\n(b) and (c), however the treatment is almost identical to the in-plane case. For both out-of-plane rotations\nwe choose a coordinate system where again byis parallel to by0,bz0is normal to the sample plane, and bzis\ndirected along the external magnetic field direction. The latter condition allows us to use the same solution\nof the LLG equation as the in-plane case (though the demagnetization factors will change). This choice of\ncoordinates means that for a rotation along the long axis of the sample, as shown in Fig. 9 (b), bx0is along\nthe long axis of the sample, byis along the width of the sample and we choose \u001eHto denote the angle between\nbzandbz0. For a rotation along the short axis of the sample, as shown in Fig. 9 (c) bx0is now along the width\nof the sample, byis along the length and we choose  Hto denote the angle between bzandbz0. In both cases\nthexandx0coordinates are related by ~x=Uy~x0(with\u0012Hreplaced by \u001eHor Has appropriate).\nProceeding as previously for the in-plane case, the photo voltage for the out-of-plane rotation by \u001eH\nmeasured along the strip in the bx0direction is\n(VSR)l\n\u001e=\u0001R\n2M0Ix0sin (2\u001eH)Re(mx) =\u0001R\n2M0Ix0h\nA\u001ex\nLL+A\u001ex\nDDi\n(2.21)\nwhere\nA\u001ex\nL= sin (2\u001eH) [\u0000Axxhx0cos\u001eHsin \bx0\u0000Axyhycos \by\u0000Axxhz0sin\u001eHsin \bz0];\nA\u001ex\nD= sin (2\u001eH) [Axxhx0cos\u001eHcos \bx0\u0000Axyhysin \by+Axxhz0sin\u001eHcos \bz0]:\nAgain for measurements made along the width of the sample, now in the bydirection, both the AMR and\nAHE will contribute. The form of the photo voltage is\n(VSR)t\n\u001e=\u0001R\n2M0Ix0sin\u001eHRe(my)\u0000wRAHE\n2jx0sin\u001eHRe(mx)\n=\u0001R\n2M0Ix0h\nA\u001ey\nLL+A\u001ey\nDDi\n\u0000wRAHE\n4jx0sec\u001eHh\nA\u001ex\nLL+A\u001ex\nDDi\nwhere\nA\u001ey\nL= sin\u001eH[Axyhx0cos\u001eHcos \bx0\u0000Ayyhysin \by+Axyhz0sin\u001eHcos \bz0];\nA\u001ey\nD= sin\u001eH[Axyhx0cos\u001eHsin \bx0+Ayyhycos \by+Axyhz0sin\u001eHsin \bz0]:\nFor the other out-of-plane rotation by angle  Halong the width of the strip, the longitudinal voltage\nmeasured along the strip length in the bywill vanish, (VSR)l\n = 0, since the static magnetization has no com-\nponent along the measurement direction and therefore the average dynamic magnetization in this direction\nwill be zero. This fact can be used to separate spin pumping and spin rectification as we will discuss in Sec.\n3.3.4. On the other hand, in the traverse direction along bx0the voltage will be\n(VSR)t\n =\u0001R\n2M0Iysin ( H)Re(my) +wRAHE\n2jysin HRe(mx)\n=\u0001R\n2M0Iyh\nA y\nLL+A y\nDDi\n+wRAHE\n2jysec H\u0010\nA x\nLL+A x\nDD\u0011\n(2.22)\n19where\nA x\nL= sin (2 H) [\u0000Axxhx0cos Hsin \bx0\u0000Axyhycos \by\u0000Axxhz0sin Hsin \bz0];\nA x\nD= sin (2 H) [Axxhx0cos Hcos \bx0\u0000Axyhysin \by+Axxhz0sin Hcos \bz0];\nA y\nL= sin H[Axyhx0cos Hcos \bx0\u0000Ayyhysin \by+Axyhz0sin Hcos \bz0];\nA y\nD= sin H[Axyhx0cos Hsin \bx0+Ayyhycos \by+Axyhz0sin Hsin \bz0]:\nThe angular dependence of the rectified voltage in all six measurement/rotation configurations is sum-\nmarized in Table 4 for both AMR and AHE. In general when VSRis measured as a function of angle,\nboth Lorentz and dispersive amplitudes will contribute, but angular fitting will allow the hx0;hyandhz0\ncomponents as well as the AMR and AHE contributions to be separated. The line shape also has distinct\nsymmetries under the rotation H!\u0000Hwhich are summarized in Table 3 in Sec. 3.3 after spin pumping\nhas been discussed.\nIn closing our detailed discussion of field torque spin rectification we should note that our analysis here\nassumes that the static magnetization is fully saturated and aligned with the magnetic field. If the sample\nhas large anisotropy fields, there will be a regime where this is not the case. This is particularly true in the\nout-of-plane configurations, although non collinear dynamics can also be observed for an in-plane static field\n[167]. Such non collinear behaviour can clearly be observed in !\u0000Hdispersion curves as deviations from\nthe normal Kittel-like behaviour. Despite this assumption, any formulas determined above will be exact\nif we simply replace \u0012H!\u0012M. The resulting expression can then be related to experimentally accessible\nparameters by determining the relationship, \u0012M(\u0012H). For a detailed discussion of such an analysis see Ref.\n[167]. It should also be mentioned that the effect of eddy currents has recently been investigated in FM/NM\nbilayers with thin NM layers, and can influence the line shape, as well as induce screening effects in such\nsystems [168, 169].\n3. Spin Rectification Effects in Magnetic Structures\nIn Section 2.1 we discussed the physical origins of various magnetoresistance effects which occur in\nferromagnetic heterostructures. Since SR depends on both the underlying MR effect and the cause of\nmagnetization dynamics, the origin and characteristics of spin rectification will also vary between different\ndevice structures. Spin rectification in ferromagnetic monolayers, bilayer devices and MTJs will be the focus\nof this section. Initial studies of field torque induced SR in ferromagnetic monolayers began as early as 1960\nusing pulsed microwave sources and will be the first topic of discussion. These initial studies only became\nnoticed by the community when advances in device fabrication techniques allowed the direct integration\nof ferromagnetic heterostructures and coplanar wave guides (CPWs). For monolayer devices this led to\nthe so called spin dynamo which will be considered next. Parallel to the rediscovery of SR via the spin\ndynamo, SR became known by the spin pumping (SP) community since both effects will contribute to\nvoltage measurements made on FM/NM bilayer samples. Our next topic of discussion will therefore be the\ngeneration of dc voltages in bilayer samples, due to SR, SP and spin-transfer torque (STT). Finally STT in\nMTJs and the so called spin diode will be considered.\n3.1. Ferromagnetic Films: Early Studies Using Pulsed Microwave Sources\nThe pioneering work in the study of SR was performed by Juretschke in the early 1960’s using pulsed\nmicrowave sources [62, 170] with a theoretical basis similar to the one discussed in Section 2.3 [90] (but\nusing the Bloch equations). A systematic study of the power, field strength and angular dependence of\nthe rectified signal confirmed that the dc voltage accompanying FMR in thin FM films arose from the\nnonlinear coupling between the rf current and field torque induced magnetoresistance – the dc signal was\nlinearly dependent on the microwave power, see Fig. 10 (c), and the angular dependence followed V(\u0012H) =\n(V1cos 2\u0012H+V2) cos\u0012H, see Fig. 10 (d). This angular dependence is consistent with the discussion in Sec.\n2.3 of a transverse measurement driven by a single field component, hx0with \bx0= 0, where both AMR,\n20Figure 10:(a) The experimental setup used for early pulsed microwave studies. A Ni thin film was placed at the end of\na rectangular waveguide. The sample holder was made of the same material as the film to simplify the field profile in the\nwaveguide and a recessed shorting block allowed control over the distance Dbetween the short and the sample. (b) Typical\nvoltage curves measured for several values of D, showing both dispersive and Lorentz contributions to the line shape. (c)\nTypical pulsed microwave power sensitivity of \u00181 nV/mW illustrating the high powers required to obtain a measurable voltage\nsignal. (d) The angular dependence of the transverse rectified voltage, corresponding to magnetization precession driven by a\nsinglehfield component ( hx0) and containing both AMR and AHE contributions. Source :Adapted from Ref. [62].\nV1, and the AHE, V2, contribute and the line shape consists of both Lorentz and dispersive contributions.\nThe typical measurement setup and data for pulsed microwave studies is shown in Fig. 10. This early work\nestablished a novel way to study the GHz frequency properties of thin FM materials – the initial samples\nstudied varied in thickness from \u001810 - 100 nm and had lateral dimensions \u0018cm\u0002cm.\nWhile the experimental design itself was sufficiently simple to facilitate adoption within the community,\nJuretschke’s ideas did not become widespread until some 40 years later, due to the following experimental\nand theoretical limitations:\n1. The experimental configuration was not optimal. In a transverse measurement, both AMR and the\nAHE contribute to the measured voltage, leading to a more complex signal. Recent works have allowed\nfor the independent study of AMR and the AHE using either a longitudinal configuration where only\nAMR contributes [24] or special samples where the AHE dominates and AMR is ignorable in the\ntransverse direction [171]. Also the precise control of the field configuration required to justify a\ntheoretical description with a single hfield component was difficult to achieve experimentally. This\nissue has been resolved through the development of a full theoretical description including all hfield\ncomponents[24]andthedevelopmentofnovelonchipdevicestructureswithintegratedCPWs,allowing\nfor precise field orientation control.\n2. Very high microwave powers were required. Due to the low power sensitivities of \u00181 nV/mW shown\nin Fig. 10 (c), large microwave powers of up to 5 kW were required to produce measurable voltage\nsignals of\u0018mV. This has been greatly improved in recent years due to sample design/fabrication,\nwhich allows the FM material to be directly integrated into a CPW chip, leading to typical power\nsensitivities of\u0018\u0016V/mW. Sensitive detection techniques, such as lock-in amplification, capable of\nmeasuring signals down to the nV scale have also helped the situation.\n3. The line shape was not fully understood. Fig. 10 (b) shows the complex line shape observed in these\nearly experiments. At the time the role of the relative phase in controlling such a line shape (discussed\n21in Sec. 2.3) had not been discovered and therefore the precise meaning of the Lorentz and dispersive\ncontributions to the voltage line shape was unknown.\nDespite these difficulties there was a small amount of early work performed using high power pulsed\nmicrowavesfollowingeitherJuretschke’swaveguidetechnique[172]orananalogousapproachwithmicrowave\ncavities[173](whichallowedmVvoltagesatslightlylowerpower). Thisveryearlyworkisbrieflysummarized\nin Ref. [170]. Pulsed microwave techniques were also used to study spin wave resonances [174] and applied\nto FM semiconductors, including in the study of the AHE [175–177] and also used to study the AHE in\nparamagnetic metals [178].\nHowever, asidefromthefewworksmentioned, Juretschke’sworkwentlargelyunappreciateduntiltherec-\ntificationeffectwasrediscoveredfourdecadeslater[18,23]. Twoverydifferentpathsledtotheseexperiments\n[18, 23]. One path came from the semiconductor community studying electrical detection of spin dynam-\nics, which led to the extension of such powerful experimental methods from semiconductor [60, 61, 179] to\nferromagnetic materials [16, 23]. The other path emerged from theoretical progress in studying magnetism,\nmotivated by the prediction of a dc voltage induced by spin pumping in magnetic bilayers (see section 3.3).\nThe spin pumping community, having observed the voltage produced in bilayer samples at FMR, needed to\nensure the source of such voltage was in fact spin pumping, which required careful analysis of the voltage\nproduced by rectification effects [18]. In 2006, these two paths crossed during the development of dynamic\nspintronic devices which integrated micro-structured ferromagnetic monolayers with microwave CPWs.\n3.2. Micro-Structured Monolayers: The Spin Dynamo\nIn this section we consider spin rectification in ferromagnetic monolayers. In such devices the merger of\nadvanced fabrication techniques and integration with coplanar wave guides allows for enhanced control of\nthe driving magnetic field and increased power sensitivity compared to pulsed microwave studies. The first\nobservation of SR in such devices was made independently by Costache et al. [18] and Gui et al. [23] using\npermalloy (Py) in a longitudinal configuration where the rectification is due to AMR. AMR based SR has\nalso been observed in ferromagnetic semiconductors where it allows for broadband material characterization\n[31]. Although SR in single ferromagnetic devices is most commonly due to field torque driven magnetization\ndynamics and AMR, in certain single layer devices, such as FM nanowires and notched FM rings where large\nspatial variations of the magnetization exist [25, 27], or homogenous ferromagnets with certain crystalline\nsymmetries [67], as it is also possible to observe spin torque induced rectification. Also, in certain FM\nmaterials (such as Co 90Zr10) AMR may be suppressed allowing studies of pure AHE SR [171]. Some of\nthe key experimental work related to SR in single ferromagnetic devices is summarized in Table 2. Our\n\u0002\u0003\u0004\u0005\u0006\n\u0007\u0003\b\u0005\u0006\u0002\u0002\u0003\u0004\u0005\u0002\u0003\u0004\u0005\u0006\n\u0007\u0003\b\u0005\u0006\u0002\u0003\u0004\u0002\u0003\u0006\u0005\n\u0003\u0007\u0005\n\t\n\n\u000b\b\b\n\t\u0003\n\u0005\f\t\f\t\n\t\n\n\u000b\r\u000b\u000e\n\u000e\u0002\nFigure 11:A striking analogy exists between Faraday’s dynamo and the spin dynamo. (a) In Faraday’s dynamo a revolving\ncopper disk converts energy from rotation to an electrical current. (b) Meanwhile a spin dynamo, with a FM strip, converts\nenergy from spin precession to a bipolar current of electricity, which can be rectified into a dc voltage. Source :Adapted from\nRef. 23. (c) A schematic illustration and (d) micrograph of the CPW structure used to provide the driving rf magnetic field\nin the spin dynamo.\n22discussion here will focus on ferromagnetic monolayers where SR is due to AMR and field torque driven\nmagnetization dynamics, with spin torque discussed later in the context of bilayer devices and MTJs.\nTable 2:A summary of key experimental results using single ferromagnetic devices. In Refs. [18, 23, 31] the current in the\nFM material was due to capacitive and/or inductive coupling with the current in the CPW, while in Ref. [171] the rf magnetic\nfield was due to the current in the Py.\nSpin Rectification due to Field Torque\nReference Device Structure Result\nCostache et al. [18]\nMeasurement of dc voltage due to SR\nin Py\nGui et al. [23]\nMeasurement of bipolar dc current due\nto SR in Py\nWirthmann et al. [31]\nMeasurement of SR in ferromagnetic\nsemiconductor GaMnAs\nChen et al. [171]\nMeasurement of SR due to AHE in\nCo90Zr10which has suppressed AMR\nSpin Rectification due to Spin-Transfer Torque\nReference Device Structure Result\nYamaguchi et al. [25]\nMeasurement of dc voltage due to SR\nin Py nanowire\nBedau et al. [27]\nDetection of domain wall resonance\nusing SR\n233.2.1. Method and Device Structure\nA spin dynamo [23, 24] is an on-chip device consisting of a FM microstrip, such as Py, placed in the\nmicrowave field of a ground-signal-ground (GSG) CPW, so named due to its analogy with a Faraday dynamo\nas shown in Fig. 11. A first and second generation spin dynamo are shown in Fig. 12 (a) and (d). In the\nfirst generation device Py microstrips are placed either beside the short of the CPW or between the GS lines.\nAs shown in Fig. 12 (a) two FM strips may be located on one device. The key difference between the first\nand second generation devices is: 1) a different component of the microwave field will drive precession ( hy\nvshx0for a first and second generation device respectively) and 2) the field intensity in the FM layer will\nbe greater in the second generation device since it can be located closer to the CPW – a second generation\nspin dynamo has \u0018200 nm spacing between the CPW and microstrip, compared to \u001840\u0016m in the first\ngeneration structure, which results in an increase of the in-plane hfield by two orders of magnitude.\nFig. 12 (b) shows the field dependent voltage measured using a first generation spin dynamo consisting\nof a Cu/Cr CPW fabricated beside a Py microstrip of dimensions 300 \u0016m\u000220\u0016m\u000250 nm on a SiO 2/Si\nsubstrate. The line shape is measured at an in-plane angle of \u0012H= 120\u000eat a frequency of !=2\u0019= 5:56\nGHz. The line shape follows Eq. 2.19 and since the relative phase is nearly zero it has a dominant Lorentz\ncontribution, following the symmetry expected from Table 3. In comparison, Fig. 12 (e) shows the voltage\nmeasured in a second generation device where the Py is above the CPW short, being electrically isolated\nby a 200 nm thick SiO 2layer. Again the in-plane line shape in Fig. 12 (e) is nearly Lorentz, but as can be\nseen from Eq. 2.19, for an hx0driving field a Lorentz line shape means that the relative phase is nearly 90\u000e.\nFig. 12 (c) shows a typical linear power sensitivity curve for a first generation device. In this device\nthe power sensitivity is 41 \u0016V/mW – a three order of magnitude improvement over the power sensitivity\nof pulsed microwave studies. The magnitude of this improvement is typical of spin dynamos, whose power\nsensitivity typically ranges from 10 - 100 \u0016V/mW [23]. This large power sensitivity has enabled studies\nacross a large range of powers, allowing electrical detection of very weak modes due to spin waves, as well\nas the study of nonlinear effects [32, 180] (see Sec. 4.1 and 4.2 respectively).\nFigure 12:(a) and (d) Schematic diagrams of a first and second generation spin dynamo respectively. Circulating arrows\nindicate the direction of the Oersted field. In the first generation device the ferromagnetic microstrip is placed beside the short\nof the CPW, therefore the dominant rf magnetic field in the FM is in the ydirection. In the second generation spin dynamo\nthe FM is placed above (or below) the CPW (with an electrically insulating layer in-between) and therefore the dominant rf\nmagnetic field in the FM is in the x0direction. (b) and (e) The line shape measured at an in-plane angle of \u0012H= 120\u000eat a\nfrequency of !=2\u0019= 5:56GHz for the first generation and !=2\u0019= 8GHz for the second generation spin dynamo respectively\n[43]. The first generation device used a Py microstrip of dimension 300 \u0016m\u000220\u0016m\u000250 nm while the second generation\ndevice used a Py strip of dimension 300 \u0016m\u00027\u0016m\u0002100 nm with a 200 nm thick SiO 2insulating layer. In both cases\nthe line shape is nearly perfect Lorentz, which means for the first generation dynamo that the relative phase is nearly zero,\nwhile for the second generation dynamo the relative phase is nearly 90\u000e. The fits (black curves) are done using Eq. 2.19 with\n(b)\u00160\u0001H= 3:6mT and\u00160Hr= 39:1mT and (e) \u00160\u0001H= 6:0mT and\u00160Hr= 76:5mT. In both cases the line shapes\nfollow the theoretical expectation summarized in Table 3. (c) Linear dependence of the photo voltage on microwave power in\na first generation spin dynamo. The power sensitivity for this device is 41 \u0016V/mW, typical for microstructured spin dynamos.\nSource :Adapted from Refs. [43] and [166].\n243.2.2. Lineshape Analysis\nAn analysis of spin rectification experiments requires understanding both the line shape and the angular\ndependence of the rectified voltage. Experimentally the line shape is studied by controlling the magnitude\nof the applied static magnetic field, while the angular dependence of course involves changing the direction\nof the field. The importance of a careful line shape analysis is emphasized by considering the subtlety of the\nrelative phase dependence of the line shape derived in Eq. 2.19 and shown experimentally in Fig. 12 (b)\nand (e), where it is clear that even though the line shape is nearly Lorentz, there is still a small dispersive\ncomponent. The fact that the spin rectification may have both Lorentz and dispersive contributions is\nespecially important when other voltage producing effects, such as spin pumping, need to be distinguished.\nIn early studies it was assumed that the rectification line shape was dispersive – a fact that was used to\ndistinguish the effect from spin pumping [37, 181]. This assumption is correct when the rf currents are\ndominated by capacitative coupling, e.g. when the sample is in close proximity to the waveguide and FMR\nis driven by the hx0field [152]. Due to the importance of this special case, the line shape symmetry for\n\bx0= \by= \bz0= 0is summarized in Table 3. The SP line shape is also summarized in this table and will\nbe discussed more in Section 3.3. The key features summarized in Table 3 are the Lorentzian and dispersive\nnature of the voltage line shape as well as the symmetry of the line shape under the change H!\u0000H.\nThis latter symmetry is determined both from the fact that L(H;Hr) =L(\u0000H;\u0000Hr)andD(H;Hr) =\n\u0000D(\u0000H;\u0000Hr)and from the angular dependence of the line shape and its symmetry under the change\n\u0012H!\u0012H+\u0019. Despite the importance of this simplified case, in general both dispersive and Lorentz\ncontributions will be present except in certain carefully designed devices and both should be considered\nwhen performing a line shape analysis [42, 43]. This means that the relative phase, which in general\ndepends in a complicated way on e.g. the waveguide, coaxial cables, bonding wires and sample holder, must\nbe calibrated. This can be done by first separating the field components by fitting the angular dependence\n(see Sec. 3.2.3), and then using these results to determine the LandDfitting.\nThe importance of both Lorentz and dispersive contributions to the spin rectification voltage has been\ndemonstratedbysystematicallycontrollingthe relativephasebetweentherfcurrent andmagneticfieldusing\na phase shifter inserted in one path through the technique of spintronic Michelson interferometry [33, 43].\nFig. 13 shows representative data from this study, which demonstrates the frequency dependence of the\nrelative phase in the first generation spin dynamo shown in Fig. 12 (a). The line shape clearly changes from\nalmost purely Lorentz, at !=2\u0019= 5:5GHz where \bx0= 14\u000eto almost purely dispersive at !=2\u0019= 5:0GHz\nwhere \bx0= 87\u000e. This indicates the device dependent nature of the SR line shape, as the relative phase\nFigure 13:A demonstration of the frequency dependence of the relative phase. The data was collected using the first generation\nspin dynamo shown in Fig. 12 (a). The left panel shows FMR spectra at \u0012H= 120\u000efor several frequencies ranging from 5.0\nto 5.5 GHz. The line shape fits (black lines) are performed using Eq. 2.19. The right panel shows the Lorentz (squares) and\ndispersive (circles) amplitudes of the voltage as a function of \u0012H. The solid curves are a fit to sin 2\u0012H. Both panels clearly\nshow the line shape changing between dominantly Lorentz at 5.5 GHz and dispersive at 5.0 GHz. Source :Adapted from Ref.\n[43].\n25will generally be a complicated function of device material, cabling setup, sample holder geometry etc. The\nfact that the line shape depends on the relative phase has been exploited to perform microwave imaging by\nTable 3:Voltage line shapes of SR and SP. Theoretical voltage curves for SR are calculated under the simplification \bx0=\n\by= \bz0= 0and illustrate whether the line shape is Lorentz or dispersive, as well as the symmetry under a 180\u000erotation\nH!\u0000H.His directed at any angle in the indicated rotation plane, other than integer multiples of 90\u000e; at these high\nsymmetry points the voltage may vanish. For SP the line shape in all rotation planes is identical: Lorentz and symmetric in\nH, however the signal may vanish in certain configurations as summarized in Table 4.\nSP\n26Table 4:The angular dependence of SR for in-plane and both out-of-plane configurations. The only contribution to longitudinal\nmeasurements comes from AMR, however both AMR and the AHE contribute to the transverse voltage.\nAMR sin 2\u0012H sin 2\u0012Hcos\u0012Hsin 2\u0012Hsin\u0012H cos 2\u0012H cos 2\u0012Hcos\u0012Hcos 2\u0012Hsin\u0012H\nAHE 0 0 0 constant cos\u0012H sin\u0012H\nAMR sin 2\u001eHsin\u001eHsin 2\u001eH sin 2\u001eHcos\u001eHsin2\u001eH sin\u001eH sin\u001eHcos\u001eH\nAHE 0 0 0 sin2\u001eH sin\u001eH sin\u001eHcos\u001eH\nAMR 0 0 0 sin2 H sin Hcos H sin H\nAHE 0 0 0 sin2 H sin Hcos H sin H\n27studying the phase shift produced when an object to be imaged is placed between the microwave source and\nthe spin dynamo [182, 183]. This approach has also been used to perform dielectric measurements [184] (see\nSection 4.6).\n3.2.3. Angular Dependence\nIn Sec. 3.2.2 it was pointed out that in order to characterize the relative phase, which may be different\nfor eachhcomponent, the voltage must be separated by driving field. This requirement is most clearly\nemphasized by the experimental data shown in Fig. 12 (b) and (e) which were taken for a first and second\ngeneration spin dynamo respectively. Notice that while the line shape is nearly Lorentz in both cases,\nthe relative phase is very different – for these devices \by=\u00009\u000eand\bx0=\u0000102\u000ehave been measured\nrespectively [43]. This is in agreement with the line shape analysis – from the summary provided in Table 3\nwith zero relative phase the hydriven FMR will be Lorentz, however the hx0driven FMR will be dispersive.\nThis means that the hx0driven FMR must have a relative phase near 90\u000eif the line shape will be Lorentzian.\nThis highlights the fact that for \bto be correctly determined, the hcomponent driving FMR must first be\nidentified, which can be done by performing an analysis of the angular dependence.\nThe key angular dependencies, based on the analysis of Sec. 2.3, are summarized in Table 4. The AMR\nand AHE will have different angular dependencies for each hcomponent in each measurement configuration,\nand therefore by measuring the rectified voltage as a function of the static field orientation, the driving field\ncanbedetermined, whichthenallowsonetoproceedwithalineshapeanalysis. Thereforetocharacterizethe\nspinrectificationbothlineshapeandangularexperimentsarenecessary, whicharecarriedoutbyvarying jHj\nandthefielddirectionrespectively. Table4summarizestheangulardependenceforallmeasurement/angular\nconfigurations discussed in Sec. 2.3 however it should be noted that in single crystalline films, where the\nangular dependence of AMR may differ from Eq. 2.4, the angular dependence of SR may differ from Table\n4 [79–85].\n3.3. Magnetic Bilayers: The Spin Battery\nIn ferromagnetic/normal metal (FM/NM) bilayers a dc voltage may still be generated by field torque\nrectification, however an additional signal due to spin-transfer torque rectification and also spin pumping\nmay be present. SP and STT require a conversion between charge and spin currents within the device\nand therefore rely on the spin Hall effects [153] (although the recently studied spin torques arising from\nRashba/Dresselhaus spin-orbit interactions will not require the spin Hall effect, and will be discussed in\nSec. 3.3.5, for simplicity here we generally exclude spin-orbit torques from our definition of spin torque).\nAs a result, the magnitude of the ST and SP voltage will depend on the strength of the spin-orbit coupling\nin the normal metal layer, which is the physical origin of the spin Hall effect. Therefore to observe large\nspin-transfer torque rectification strong spin-orbit coupling is required, which makes platinum (Pt) a popular\nchoice for the NM layer in such devices. In Sec. 3.3.5 we will discuss recent studies which have shown that\ncurrent driven spin-torque may arise from the Rashba/Dresselhaus spin-orbit interactions.\nThe three voltage producing effects in bilayer devices are shown in Fig. 14. In rectification (both\nfield and spin-transfer torque induced) the role of the magnetization dynamics is to generate a dynamic\nmagnetoresistance which can nonlinearly couple to an rf current and produce a dc voltage. This process\nis shown in Fig. 14 (a) for field induced magnetization dynamics. Note that since the rf hfield directly\nexcites FMR in the FM layer the NM layer is not active in this process. However for both STT and SP\nthe NM is necessary since it is where the spin/charge current conversion takes place. In spin pumping,\nschematically illustrated in Fig. 14 (b), the role of magnetization precession its very different from its\npurpose in rectification. In SP the role of the microwave excited FMR is to set up a non-equilibrium\nspin distribution at the FM/NM interface. Via a diffusive process this spin distribution can equilibrate by\ninjecting a spin current from the FM into the NM. The inverse spin Hall effect will then convert this spin\ncurrent into a charge current and the dc component will then produce a dc voltage. Finally STT rectification\nis shown in Fig. 14 (c). An rf current flowing in the NM layer is converted into an ac spin current via the spin\nHall effect. The ac spin current then flows into the FM layer and produces a torque on the magnetization\ndue to the exchange interaction. This torque drives magnetization precession, which will generate an rf\n28Figure 14:In FM/NM bilayer structures the dc voltage may be due to (a) field induced spin rectification, (b) spin pumping\nor (c) spin-transfer torque induced rectification. (a) In field induced rectification a microwave hfield excites FMR in the\nferromagnetic layer. The magnetization precession generates an rf magnetoresistance which couples nonlinearly to the rf\ncurrent flowing in the FM layer, producing a dc voltage which can have both Lorentz and dispersive components and depends\non both the relative and spin resonance phase. (b) In spin pumping the microwave induced magnetization precession generates\na non-equilibrium spin distribution at the FM/NM interface which equilibrates by sending a spin current into the NM. Via\nthe inverse spin Hall effect this spin current is converted into a charge current, and the dc contribution then generates a dc\nvoltage which is purely Lorentzian. (c) In spin-transfer torque rectification an rf charge current in the NM is converted into an\nac spin current through the spin Hall effect and enters the FM layer. The misalignment between the FM magnetization and\nthe spin current polarization produces a torque on the magnetization, resulting in an rf magnetoresistance which can produce\na dc voltage by coupling to the rf charge current already present in the FM.\nmagnetoresistance which can couple to the rf current and produce a dc voltage. Since the line shape of SP\nis determined by the form of the spin current generation, js\u0018h2, there is no relative phase dependence and\nthe SP line shape will be purely Lorentzian. Similarly, for STT rectification since it is jswhich drives FMR,\nand noth, there is no relative phase dependence. The presence of these three different physical effects in\nbilayer devices makes their study intriguing from both a practical and fundamental physics viewpoint and\nas a result there has been a large number of studies performed on bilayers in the past decade, which we will\ndiscuss in detail in the next five sections. We start our discussion with a brief review of spin pumping and\nspin Hall effects, followed by a summary of experimental observations of spin pumping in both the \"vertical\"\nand \"transverse\" configurations. We then discuss the separation of spin rectification and spin pumping and\nlook at spin-transfer torque rectification. To provide some guidance to the relevant literature a summary of\nseveral key studies is provided in Table 5.\n3.3.1. The Basic Physics of Spin Pumping and Spin Hall Effects\nThe kernel of spintronics is the generation and manipulation of spin currents, making experimental\ntechniques which reliably generate spin currents of the utmost importance. One of the widely used methods\nfor spin current generation is the transport of non-equilibrium magnetization pumped by FMR, known as\nspin pumping (see e.g. Ref. [185, 186] for comprehensive reviews). In a FM/NM bilayer, spin pumping\ncan be understood schematically using Fig. 15: At FMR, the precessional magnetization M(t)in the\nferromagnet differs from the saturation magnetization M0by an amount \u0001M(t), which corresponds to\nthe non-equilibrium magnetization generated by FMR. Inside the FM layer the precessing magnetization\nrelaxes towards its equilibrium position via magnetization damping, leading to the partial dissipation of\nthe non-equilibrium magnetization \u0001M(t). At the interface of the FM/NM bilayer, the non-equilibrium\nFigure 15:A schematic illustration of spin pumping in a FM/NM bilayer. The magnetization, M, of the FM layer precesses\naround the static field direction H. This precession \"pumps\" a spin current into the normal metal layer due to a non-equilibrium\nspin accumulation generated at the FM/NM interface.\n29Table 5:A summary of key experimental results using bilayer (FM/NM) devices. In Refs. [20, 42, 187] a microwave cavity is\nused to provide the rf magnetic field, while in Ref. [188] a CPW is used.\nSpin Pumping and Spin Rectification\nReference Device Structure Result\nCostache et al. [19]\nElectrical detection of SP in \"vertical\"\nconfiguration (without ISHE) using\nPt/Py/Al device\nSaitoh et al. [20]\nElectrical detection of SP in\n\"transverse\" configuration through the\nISHE using Py/Pt bilayer\nMosendz et al. [181]\nSeparation of SP and SR voltage\nsignals using line shape symmetry (L\nvs D respectively) using Py/Pt bilayer\nAzevedo et al. [42]\nSeparation of SP and SR voltage\nsignals using line shape symmetry and\nangular dependence using Py/Pt\nbilayer\nChen et al. [187]\nSeparation of SP and SR voltage\nsignals using line shape symmetry and\nangular dependence in\nGaMnAs/p-GaAs bilayer\nBai et al. [188]\nUniversal method of SP and SR\nseparation based on symmetry\nconsiderations\nspin-transfer Torque\nReference Device Structure Result\nLiu et al. [41]\nElectrical detection of SR due to STT\nin Py/Pt bilayer\n30magnetization diffuses from the FM into the NM as a flow of spin current, which contributes to additional\ndissipation of \u0001M(t). Magnetization precession, magnetization damping, and magnetization diffusion, these\nare the three key ingredients of spin pumping. As shown in Fig. 15, since \u0001M(t)involves both a static and\na precessional component, the spin pumping effect simultaneously generates both dc and ac spin current in\nthe FM/NM bilayer.\nAlthough the concept and full theory of spin pumping [189, 190] were not developed until 2002, histori-\ncally, the fact that the three key ingredients of magnetization precession, damping, and diffusion would lead\nto the generation of a spin current in FM/NM bilayers was first revealed in the transmission-electron-spin\nresonance (TESR) experiments performed in the late 1970’s [191, 192]. In such microwave absorption ex-\nperiments, Silsbee et al. found that the TESR of copper foils was greatly enhanced in the presence of the\nferromagnetic films (such as permalloy, iron, and nickel) deposited on one surface of the copper foil [192]. By\ndeveloping a phenomenological theory using the Bloch equations to describe magnetization precession and\ndamping, coupled by the transport of non-equilibrium magnetization across the FM/NM interface, Silsbee\net al. revealed the key physics of spin pumping, i.e, the generation of a spin current via the interplay of\nmagnetization precession, damping, and diffusion in FM/NM bilayers [192].\nMicroscopically, spin pumping is a consequence of spin dependent reflectivity and transmission param-\neters of NM electrons at the FM/NM interface. The microscopic theory of spin pumping was derived by\nTserkovnyak et al. [189, 190], using the spin mixing conductance as the main parameter to rigorously de-\nscribe the spin current. According to this theory the spin current injection into the normal metal relaxes\nover a characteristic length scale \u0015SD, the spin diffusion length, and the spin current density decays away\nfrom the interface as [189, 190]\njs(y) =js(0)sinh [(tNM\u0000n)=\u0015SD]\nsinh [tNM=\u0015SD];js(0) =\u0016hGr\n4\u0019M2\n0M\u0002dM\ndt: (3.1)\nHereGris the real part of the spin mixing conductance, G\"#=Gr+iGi,\u0016his the reduced Planck constant,\ntNMis the thickness of the NM layer, nis the distance normal to the FM/NM interface (see Fig. 15) and\nMandM0are the full and dc magnetization of the FM layer respectively, just as in Sec. 2.3. Only the spin\npolarization direction b\u001bis determined by the direction of jsin Eq. 3.1, with the spatial direction normal to\nthe interface (since the current diffuses away from the interface). The process of spin pumping is also closely\nrelated to the earlier idea of a voltage generated by spin flip scattering at the FM/NM interface [193] and\ndepends on the spin mixing conductance in the same way as the spin Hall magnetoresistance discussed in\nSec. 2.1.5 [194].\nIn addition to TESR experiments that can measure the spin current, other methods have been demon-\nstrated to detect spin pumping. Comparing the spin current of Eq. 3.1 to the LLG equation in Eq. 2.14,\nthe spin current has the same form as the Gilbert damping term and therefore from the perspective of the\nFM the flow of spin current will result in an increased damping due to the transfer of angular momentum\ninto the NM [189]. Indeed such a signature of spin pumping enhanced magnetization damping was observed\nin Refs. [195–199]. However, owing to the simplicity of charge transport measurements, the most common\nway to detect spin currents is through their conversion into a charge current via the inverse spin Hall effect\n(ISHE).\nSimilartospinrectificationandspinpumping, inwhichthebasicphysicsconceptsweredevelopeddecades\nbefore modern nanotechnologies made them practically useful, spin Hall effects (SHEs) were initially studied\nin the 1970’s [150] but they have only become the subject of intense interest recently after their theoretical\nrediscovery [151]. For an excellent review on spin Hall effects see Ref. [153]. Fig. 16 schematically illustrates\nboth the SHE and ISHE which physically result from spin dependent scattering of charge carriers due to\nthe spin-orbit interaction. In the spin Hall effect, a change current is converted to a spin current, and the\nresulting spin accumulation at the sample boundaries can be detected e.g. via Kerr rotation microscopy\n[200] or polarized electroluminescence [201]. The inverse spin Hall effect is the inverse process, where a spin\ncurrent is converted into a charge current according to\nJc=e\u0012SH\n\u0016hJs\u0002b\u001b: (3.2)\n31Figure 16:Schematic illustrations of (a) the spin Hall effect and (b) inverse spin Hall effect. J cand J sdenote the charge\ncurrent and the spatial direction of the spin current respectively. The spin polarization direction of the spin current is in the\nup-spin direction.\nHere\u0012SHis the spin Hall angle which characterizes the efficiency of spin/charge current conversion and is\ndependent on the strength of the spin-orbit interaction in a material. Due to its ability to convert spin to\ncharge currents, for spin pumping experiments the inverse spin Hall effect can be exploited to convert the\nspin current generated through spin pumping into a charge current which can be electrically measured.\n3.3.2. Spin Battery and \"Longitudinal\" Spin Pumping Voltage\nBefore the dc spin pumping voltage was measured via the inverse spin Hall effect, the first measurement\nof such a voltage [19, 202] used a device called the spin battery [190], which was the spintronics analog\nof solar cells. As was known from the spin injection experiment [203] without microwave irradiation, spin\ndiffusion at the FM/NM interface leads to spin accumulation and depletion near the interface. From the\nviewpoint of the two-channel model for spin transport (see section 2.1), the physics of spin accumulation\nand depletion near the interface of a FM/NM is analogous to the charge accumulation and depletion in a\nsemiconductor P/N junction. Comparing the two, spin pumping at FMR in a FM/NM bilayer is analogous\nto a solar cell made of a P/N junction where, instead of the interband electrical dipole transition that\nconverts energy of sunlight to the electrical energy of a charge current, the magnetic dipole transition of\nFMR converts energy from microwaves to produce spin current.\nFigure 17:(a) A schematic diagram of the device used to observe longitudinal spin pumping. A 25 nm thick Py strip with\nlateral dimensions 0.3 \u0016m\u00023\u0016m is placed at the short end of a CPW with 30 nm thick Pt and Al contacts. Py FMR is\ndriven by an hyfield and a voltage is measured between the Pt and Al contacts using lock-in amplification. (b) The dc voltage\nmeasured due to longitudinal spin pumping. Here flowandfhighare the two frequencies of the rf field used during the lock-in\nmeasurement. The peaks and dips correspond to the resonance at fhighandflowrespectively. Source :Panel (b) from Ref.\n[19].\nIn the spin battery, when the spin injection rate (via spin pumping) is lower than the spin relaxation\nrate, the NM acts as a pure spin sink. However if the spin injection rate exceeds the spin relaxation rate the\nspin accumulation at the interface may allow a back flow of spin current into the FM. Due to spin dependent\n32conductivities of the FM this back flow results in the production of a dc voltage. To observe this effect\nCostache et al. [19] used the device structure shown in Fig. 17 (a). FMR was generated in a Py strip by\nthe Oersted field of a CPW. The Py strip was connected at both ends to normal metals. To measure the dc\nvoltage that was longitudinally generated across the interface by spin pumping, two different contact metals\nwere used. Pt, which is an excellent spin sink, will not generate a voltage, whereas Al, with its small spin\nflip relaxation rate, will produce a dc voltage. Such an asymmetry between the two contacts is expected to\nproduce a net voltage across the Py strip. Typical dc voltage curves measured on such a Pt/Py/Al device\nare plotted in Fig. 17 (b), showing the measured voltage of about \u0018200 nV. In contrast, measurements on\nreference samples made of Pt/Py/Pt structure exhibit only weak signals up to 20 nV. Such an experimental\ncontrast, combined with a detailed theoretical investigation [202], supported the case that the dc voltage\nlongitudinally generated across the Py/Al interface was observed in the Pt/Py/Al device [19]. However, it\nshould be mentioned that in 2006 when such a \"longitudinal\" spin pumping experiment was performed [19],\nthe method for line shape analysis of the rectification voltage was not yet established [24, 33, 42, 43, 181].\nHence, the asymmetric line shape shown in Fig. 17 (b), which indicates a contribution to the voltage from\nspin rectification, was not analyzed. In 2007, the issue of how to distinguish the dc voltage signal caused by\nspin pumping [19] and spin rectification [23] was raised and discussed via private communications among\nthe groups of Refs. [19, 23, 202]. In section 3.3.4 we will address this previously controversial subject and\nreview the methods and solutions developed in the past decade to solve this problem.\n3.3.3. Inverse Spin Hall Effect and \"Transverse\" Spin Pumping Voltage\nCompared to the \"longitudinal\" spin pumping configuration, the use of a \"transverse\" configuration\nusing the inverse spin Hall effect, as introduced by Saitoh et al. [20], has several practical advantages, such\nas the ease of charge current detection and larger dc voltages, typically several \u0016V even at low microwave\npowers. Hence, in contrast to the \"longitudinal\" spin pumping voltage, the \"transverse\" spin pumping\nvoltage induced by the inverse spin Hall effect has been studied by many different groups. Fig. 18 shows\na typical experimental setup which may be used to detect spin pumping via the inverse spin Hall effect.\nNormally a FM/NM bilayer is placed on top of a CPW, allowing the hx0field generated by the current in the\nCPW to drive magnetization precession in the FM. This magnetization precession produces a spin current\nvia the spin pumping mechanism which is then converted into a charge current in the NM via the inverse\nspin Hall effect and is detected as a voltage across the bilayer in the z0direction. Due to its large spin-orbit\nFigure 18:(a) A schematic diagram of a typical bilayer device. The device is similar to the spin dynamo, with the microstrip\nreplaced by the FM/NM bilayer. (b) Experimental data for a Py/Pt bilayer [181]. The lateral dimensions of the bilayers are\n2.92 mm\u000220\u0016m and they are each 15 nm thick. The insulating layer separating the 30 \u0016m wide, 200 nm thick Au CPW\nfrom the bilayer was 100 nm thick MgO. The FMR is driven by an hx0field and due to the capacitative coupling the relative\nphase is zero, which means that the AMR in Py will be purely dispersive, as shown with the black triangles in the data. The\nblue circles show the signal from the Py/Pt which contains both the Lorentz (dotted-dashed line) and dispersive (dotted line)\ncomponents.\ncoupling Pt has a large spin Hall angle, \u0012SH\u00180:05[188] (although the exact value has been the subject\nof some controversy [204], and care must be taken to either separate spin pumping and spin rectification\neffects as discussed in Sec. 3.3.4, or to eliminate AMR induced SR altogether [205].) and therefore Pt is a\ncommon NM used in spin pumping experiments.\n33With the charge current determined by the ISHE in Eq. 3.2, the spin pumping voltage can be found by\nintegrating the time averaged charge current density,\nVSP=RNMZ\nhjc(n;t)\u0001bqidA=RNMe\u0012SH\n\u0016h\u0014Z\nhjs(n;t)idA\u0015\n(bn\u0002b\u001b)\u0001bq\nwherebqis the measurement direction (either along the length or width of the sample – in the configuration\nshown in Fig. 18 bq=bz0) anddA=dndwwithdwalong the width of the sample. To determine VSP\nwe will use the coordinate systems in Fig. 9. In either the in-plane or out-of-plane configurations, taking\nM=\u0000\nmt\nx;mt\ny;M0\u0001\nwill yield the same expression for the spin current density\njs(n;t) =\u0016hGr\n4\u0019M2\n0sinh [(tNM\u0000n)=\u0015SD]\nsinh [tNM=\u0015SD]!Im(m\u0003\nxmy)\nand therefore\nVSP=\u0014Im\u0012m\u0003\nxmy\nM2\n0\u0013\n(bn\u0002b\u001b)\u0001bq (3.3)\nwhere the proportionality constant \u0014=\u0012SH\u0015SD\n\u001bNM\u0000e!\n4\u0019\u0001Gr\ntNMwtanh\u0010\ntNM\n2\u0015SD\u0011\ndepends on material and device\nspecific parameters, such as the normal metal conductivity \u001bNM, but is not important for a line shape\nanalysis. We can immediately see that because the voltage depends on m\u0003\nxmythere will be no dependence\non the spin resonance phase and the line shape will be completely Lorentzian. Using the solution to the\nLLG equation found in Eq. 2.15, VSPfor in-plane magnetic fields is\n(VSP)l\n\u0012=\u0014\u0000\nAxxAxyh2\nx0sin\u0012Hcos2\u0012H+AyyAxyh2\nysin\u0012H+AxxAxyh2\nz0sin3\u0012H\u0001\nL;\n(VSP)t\n\u0012=\u0000\u0014\u0000\nAxxAxyh2\nx0cos3\u0012H+AyyAxyh2\nycos\u0012H+AxxAxyh2\nz0cos\u0012Hsin2\u0012H\u0001\nL:\nFor the two out-of-plane rotations the spin pumping voltage is\n(VSP)l\n\u001e= 0;\n(VSP)t\n\u001e=\u0014\u0000\nAxxAxysin\u001eHcos2\u001eHh2\nx0+AxyAyysin\u001eHh2\ny+AxxAxysin3\u001eHh2\nz0\u0001\nL;\n(VSP)l\n =\u0014\u0000\nAxxAxysin Hcos2 Hh2\nx0+AxyAyysin Hh2\ny+AxxAxysin3 Hh2\nz0\u0001\nL;\n(VSP)t\n = 0:\nThe angular dependence of spin pumping is summarized in Table 6 for each measurement/rotation config-\nuration.\nIn the analysis presented here we have directly used the rf magnetization determined from the LLG\nequation. However it should be noted that is also possible to replace the Im (m\u0003\nxmy)term in Eq. 3.3 with\nsin2\u0012cwhere\u0012cis the cone angle. The resulting expression assumes that the ellipticity of the precession is\nsmall, but provides a convenient way to characterize the behaviour at low microwave powers [181].\nReturning to the typical spin pumping experiment shown in Fig. 18, since the microwave magnetic field\nis in thehx0direction, when the relative phase is zero the line shape due the spin rectification will be purely\ndispersive. This is what is observed in the experimental data shown in Fig. 12 (b). The voltage in a Py\nmonolayer is seen to be dispersive, while the voltage in the Py/Pt bilayer has both a dispersive (calculated\nwith the dotted line) and a Lorentz (calculated with the dashed-dotted line) component. Therefore the line\nshape difference may be used to separate spin rectification from spin pumping in this case. However as we\nsaw in Sec. 2.3, when the relative phase is nonzero the SR voltage will have both dispersive and Lorentz\ncontributions, andthereforedistinguishingSPfromSRbecomesamorecomplex. Sincethefirstexperimental\nobservations of spin pumping [19, 20, 206] were made almost concurrently with the first observations of spin\nrectification in microstructured devices, methods to separate the two effects were not yet fully developed.\nInstead these initial studies attempted to reduce the effect of spin rectification by placing the sample at a\nposition of minimal rf electric field so that there would not be any directly applied rf current. However more\n34Table 6:The angular dependence of spin pumping.\nsin\u0012H sin\u0012Hcos2\u0012H sin3\u0012H cos\u0012H cos3\u0012H cos\u0012Hsin2\u0012H\n0 0 0 sin3\u0012H sin\u0012H sin\u0012Hcos2\u0012H\nsin3 H sin Hcos2 H sin H 0 0 0\nrobust methods that can actually separate SP from SR are desirable so that SP in various device structures\ncan be reliably studied. This is a necessary step in order to exploit spin pumping as a direct source of spin\npolarized current. Accurately measuring spin pumping is also needed to reliably determine the spin Hall\nangle, which is of practical interest in the development of spintronic devices. Therefore shortly after the\nfirst spin pumping measurements it was realized that a robust method to separate spin pumping and spin\nrectification signals was necessary.\n3.3.4. Spin Rectification vs Spin Pumping\nSeveral methods have been developed to separate spin pumping and spin rectification in FM/NM bilayers\n(for a concise review see the supplementary material of Ref. [188]). The first method developed relies on\na line shape analysis [181]. This method is effective when the spin rectification signal is purely dispersive\nso that it can clearly be distinguished from the Lorentzian spin pumping. However this method becomes\nproblematic when there is a relative phase shift between rf electric and magnetic fields. Another method\nusing the angular dependence has also been used [42]. As we see from Tables 4 and 6, for the case of an in-\nplane static field, when FMR is driven by a transverse hfield both spin rectification and spin pumping have\nasin\u0012Hcos2\u0012Hangular dependence. However when a normal hfield produces the voltage, spin pumping\nfollowsa sin\u0012Hwhilespinrectificationfollowsa sin\u0012Hcos\u0012Hangulardependence, andthereforeitispossible\nto distinguish the two effects. Using such a special measurement configuration, a pure spin pumping signal\nhas been detected in the Py/Pt bilayer [44].\nA method combining line shape analysis with the even and odd contributions of the voltage signal with\nrespect to the static magnetic field has also been developed [207]. However this method also cannot be used\nwhen the microwave field is parallel to the sample, which is the same limitation encountered when using\nthe angular dependence to separate the effects. More recently a universal method based on the magnetic\n35field symmetries has been implemented [188]. This method exploits the symmetries of VSPandVSRwhen\na magnetic field is applied nearly perpendicular to the sample plane, enabling a reliable measurement\nconfiguration in which onlyspin pumping contributes to the voltage signal. As a result the separation step\ncan effectively be performed duringthe experiment, reducing the need for complex line shape and angular\nanalyses. From Tables 4 and 6 we can see that for an out-of-plane magnetic field and longitudinal voltage\nmeasurements\nat H= 0\u000ethere isno spin pumping andVSR(\u001eH;H) =VSR(\u001eH;\u0000H) =\u0000VSR(\u0000\u001eH;H)(3.4)\nand\nat\u001eH= 0\u000ethere isno spin rectification andVSP( H;H) =\u0000VSP( H;\u0000H) =\u0000VSP(\u0000 H;H):(3.5)\nTherefore as shown in Fig. 19, VSPandVSRcan be distinguished experimentally based on their Hfield\nsymmetries. This method enables an accurate determination of the spin Hall angle and the characterization\nof a materials spin/current conversion efficiency. Since the work of Bai et al. in 2013 [188] a method for\nthe spin pumping and spin rectification separation based on the difference in symmetry dependence of VSP\nandVSRon the spin diffusion direction was used by Zhang et al. [208], which allows separation of the\neffects without line shape fitting. It is also worth noting that in addition to the methods discussed here, the\nrectification signal in a Schottky diode (due to tunnelling anisotropic magnetoresistance) [209, 210] may be\ndistinguished from spin pumping based on its bias dependence [211].\nFigure 19:(a) - (d) dc voltage measurements on a Py/Pt bilayer as a function of the magnetic field Happlied at angles (a)\n\u001eH=\u00001:5\u000e; H= 0\u000e(b)\u001eH= 0\u000e; H=\u00000:65\u000e(c)\u001eH= 1:5\u000e; H= 0\u000e(d)\u001eH= 0\u000e; H= 0:65\u000e. When H= 0\u000ethe\nsignal is purely spin rectification, while at \u001eH= 0\u000ethe signal is purely spin pumping (e) - (h) dc voltage measurements on a Py\nmonolayer at angles (e) \u001eH=\u00001:3\u000e; H= 0\u000e(f)\u001eH= 0\u000e; H=\u00000:2\u000e(g)\u001eH= 1:3\u000e; H= 0\u000e(h)\u001eH= 0\u000e; H= 0:2\u000e.for\ncomparison. At  H= 0\u000ethe spin rectification is still present. However at \u001eH= 0\u000ethere is no voltage signal since there is no\nspin pumping in a monolayer. Source :Adapted from Ref. [188].\n3.3.5. Spin-Transfer Torque Rectification in Bilayers\nThe final voltage producing effect observed in FM/NM bilayers is spin-transfer torque induced rectifi-\ncation. As shown in Fig. 14 (c) STT induced rectification requires the conversion of an rf charge current\nin the NM into an ac spin current via the spin Hall effect. This ac spin current then produces a torque on\nthe magnetization of the FM via the exchange interaction, subsequently generating a dynamic resistance\nand coupling to the rf charge current to create a dc voltage. The key ingredient in all spin torque induced\neffects is the generation of a spin polarized current, and while the focus of this section will be on spin Hall\ninduced STT, it is important to note that there are other physical mechanism which may generate spin\npolarized currents. First, a spin polarized current may result from the flow of an electric current between\nnon collinear magnetic structures. This form of spin-transfer torque may occur within domain wall struc-\ntures [25, 27] and is the basis of spin rectification in MTJs which will be discussed further in Sec. 3.4.\nSecond, in crystalline structures lacking inversion symmetry, spin-orbit coupling may induce a polarization\n36of the conduction electrons in the presence of an electric current [212]. This so-called spin-orbit torque has\nrecently attracted much attention from the spintronics community [67, 213–225] and results in an effective\nwave vector dependent magnetic field. The spin-orbit torque may arise due to either Rashba (interface)\nor Dresselhaus (bulk) spin-orbit interactions in certain FM/NM devices and may produce a field-like term\nwhich can even oppose the field-like torque [218]. In such situations the spin-torque line shape may contain\na dispersive contribution, unlike the case of pure spin Hall induced spin torque, and care must be taken\nto separate the effects [219, 223]. Here we will not discuss the recent and developing subject of spin-orbit\ntorques in great detail, choosing to focus on the spin Hall induced STT and its distinction from SR and SP.\nA typical measurement setup used to detect the spin-transfer torque rectification via the spin Hall effect\nis shown in Fig. 20 (a). Unlike in the detection of field torque or spin pumping, a typical STT rectification\nmeasurement setup will apply the rf current directly to the bilayer sample, avoiding the use of a CPW or\ncavity to apply an rf field and instead uses a bias tee to allow the direct application of both rf and dc currents\nas well as the measurement of the dc voltage across the sample. As we discuss below this means that the\ncontribution due to spin-transfer torque rectification will be purely symmetric. Typical experimental data,\nmeasured on Pt(15 nm)/Py(15 nm) and Pt(6 nm)/Py(4 nm) bilayers at 8 GHz, is shown in Fig. 20 (b). As\nwould be expected the line shape contains both symmetric and dispersive contributions due to both field\nand spin-transfer torque rectification effects.\nFigure 20:(a) A schematic diagram of a typical spin-transfer torque rectification measurement setup. A bias tee is used to\nprovide both an rf and dc current as well as measure the dc voltage produced in the bilayer. (b) Typical experimental data for\nPy/Pt bilayers of different thicknesses (shown in nm) [41]. The voltage is made of a mixture of dispersive and symmetric line\nshapes due to the field and spin-transfer torques respectively. Source: Adapted from Ref. [41].\nThe idea of current induced magnetization dynamics was proposed in the works of Berger [226] and\nSlonczewski [227] and was first seen in Co/Cu multilayers by observing the magnetoresistance changes when\nlarge current densities ( \u0018108A/cm2) were injected through point contacts [228, 229]. For a detailed review\nof spin-transfer torque see e.g. Ref. [186, 230]. The effect of a spin polarized current on the magnetization\ndynamics can be described by modifying the LLG equation to include an extra torque term [227, 231] which\nmay be written as [41],\ndM\ndt=\u0000\rM\u0002H+\u000b\nM0M\u0002dM\ndt\u0000\u0000jsM\u0002(b\u001b\u0002M) (3.6)\nwhere \u0000 =\u0000(\r\u0016h)=\u0000\n2e\u00160M2\n0tFM\u0001\nwithtFMbeing the thickness of the FM layer. The first two terms are just\nthe unmodified LLG equation describing precession about Hand Gilbert damping respectively, and the last\nterm is the modification due to a the spin-transfer torque resulting from a current with spin polarization\nalongb\u001b. Here we have neglected the field-like term arising from the spin-orbit torque. For a discussion of\nthe LLG modification due to spin-orbit torques, see e.g. Ref. [223].\nThe STT term provides an additional torque in the M\u0000b\u001bplane. Since the magnitude of Mis taken\nto be constant, there is no spin-transfer torque component along M, although in principle there may be a\ncomponent perpendicular to the M\u0000b\u001bplane which would act as a field-like torque. However in FM/NM\nbilayers the out-of-plane spin-transfer torque may be neglected [41, 48] (see also the discussion in [186] and\n[232–235]).\nThe voltage generated along the length of the FM can now be determined by solving the LLG equation\nwith the STT term in analogy with the approach taken in Sec. 2.3. When the field torque is due to hx0the\n37voltage is given by\nVmix=1\n2\u0001R\nM0Iz0sin 2\u0012Hcos\u0012H(Axxhx0D+SjsL) (3.7)\nwhereS=\u0000\n\u0000M2\n0\u0001\n=(2\u0001H(d!r=dH)jH=Hr)andVmixindicates that the voltage is due to both the field\ntorque and STT rectifications. In a FM/NM bilayer AMR provides the magnetoresistance which produces\nthe dc signal, however a more general expression may be obtained by replacing \u0000\u0001Rsin 2\u0012HwithdR=d\u0012\n[41], which is obtained by expanding the voltage as a function of \u0012(\u0012may be\u0012Has it is for AMR, however\nit may also be another relevant angle, depending on the system in question, see Table 1 and Refs. [22, 28]).\nThe first term in Eq. 3.7 is just the field torque rectification and has a dispersive line shape. However the\npresence of a spin-transfer torque also has a residual effect on this term by increasing the line width (for the\nsame reasons as the spin pumping induced line width enhancement). The lack of a symmetric component to\nthe field torque line shape is due to the fact that there is no phase shift between the rf current, jrfand the\nOersted field. This is expected, since the Oersted field is produced by the current flowing in the NM layer,\nwhich is exactly the same current injected into the FM. This situation differs from the measurement of the\nfield torque rectification using a CPW, where the Oersted field is generated by the current flowing in the\nCPW which is electrically isolated from the FM microstrip and may therefore have a phase shift compared\ntojrf. On the other hand the second term, which is the voltage due to spin-transfer torque rectification, is\npurely symmetric as expected, since the spin current, js, generated through the spin Hall effect will be in\nphase with the rf current with which it couples to produce the dc signal. This is the same situation as that\nobserved for spin pumping.\n3.3.6. Spin Hall Magnetoresistance Rectification\nThus far we have examined in detail the phenomena of spin rectification in FM/NM bilayers, taking\ncare to highlight the distinction between spin pumping and field or spin-transfer torque driven rectification.\nIn such systems the magnetoresistance required for rectification results from the current flow in the FM\nlayer, however, as discussed in Sec. 2.1.5, small magnetoresistance effects can also be observed in FMI/NM\nbilayers due to spin dependent scattering at the interface, which means that rectification effects can also play\na role in FMI/NM bilayers. Both theoretical [236, 237] and experimental [238–242] investigations of SMR\nrectification have been performed, revealing new information about the magnetization dynamics of magnetic\ninsulators. The first studies focussed on the direct rectification of the ac spin Hall effect by means of SMR\n[238, 239], with theoretical descriptions including the effect of a relative phase shift between rf microwave\nfield and current as well as multiple driving fields [238]. In these studies the angular dependence was used to\ndistinguish SMR rectification from spin pumping, however due to extremely small SMR magnetoresistance\nratios, the electrical signal due to the ac spin Hall effect was overwhelmed by the dc spin Hall effect signal\ninduced by spin pumping.\nIn order to more easily observe SMR rectification, another approach is to use the spin-torque FMR\nin a FMI/NM bilayer. The theoretical description of such systems was provided by Chiba et al. using a\ndrift-diffusion model with quantum mechanical boundary conditions at the FMI/NM interface. They found\na rectified voltage consisting of both symmetric and asymmetric contributions,\nVSMR=\u0000\u0001R\n2Iz0[C(HSTT+\u000bh)L+C+hD] sin\u0012Hcos2\u0012H: (3.8)\nHere\nC=\r\n\u000bq\n(2\u0019M0\r)2+!2; C+=\r\n\u000b![1 + 2\u0019M0\u000bC];\nHSTTis an effective field accounting for the spin-transfer torque,\nHSTT=\u0016h\u0012SHJc\n2eM0tFMIRe(\u0011); \u0011=2\u0015SD\u001aG\"#tanh\u0010\ntNM\n2\u0015SD\u0011\n1 + 2\u0015SD\u001aG\"#coth\u0010\ntNM\n\u0015SD\u0011;\n38and the magnetoresistance ratio, \u0001R= \u0001\u001al=A =\u001a\u00122\nSH(2\u0015SD=tNM) tanh (tNM=2\u0015SD)Re(\u0011=2), depends on\nthe normal metal thickness, tNM. Note that due to the interface nature of SMR, the ratio of the Lorentz\nand dispersive contributions is strongly dependent on sample thickness, with large dispersive contributions\npresent for small thickness, and also that, unlike other rectification effects we have considered which are\nproportional to the microwave power P, SMR rectification is proportional top\nP.\nThe experimental setup for STT SMR experiments is shown in Fig. 21 (a), with a calculation according\nto Eq. 3.8 including the effect of spin pumping in panel (b) for tFMI= 4nm. For simplicity, the treatment\nleadingtoEq. 3.8includesonlyan x0componentoftherfdrivingfield(wehavedroppedthesubscriptsothat\ncomparing to our previous descriptions of spin pumping and spin rectification h=hx0) and also has assumed\nthat the relative phase \b = 0. From Table 6 we see that the spin pumping voltage in this configuration\nwill also have a sin\u0012Hcos2\u0012Hangular dependence, however the contributions can be distinguished based\non a line shape analysis, assuming the relative phase shift is 0. Based on such line shape differences recent\nexperiments have provided evidence of SMR rectification [240–242], taking into account the effect of non-zero\nrelative phase shifts and finding good agreement with the predicted thickness dependence of the dispersive\ncontribution.\nFigure 21:(a) The experimental setup used to observe SMR rectification via spin-transfer torque driven magnetization dynam-\nics. (b) Calculated SMR and SP contributions to the voltage signal. While the VSPis symmetric, VSMRhas a large dispersive\ncomponent. Source: Adapted from Ref. [236].\n3.4. Magnetic Tunnel Junctions: The Spin Diode\n3.4.1. Spin-Transfer Torque Rectification in Magnetic Tunnel Junctions\nspin-transfer torque is not only relevant in bilayer devices but also plays an important role in the voltage\nproduced in spin valves and magnetic tunnel junctions [17, 21, 22, 28, 230, 243]. When a current flows\nthrough the pinned layer of an MTJ it develops a spin polarization due to the exchange interaction. The\ntransfer of spin angular momentum carried by this spin polarized current will then exert a torque on the\nfree layer magnetization, as we have just discussed for bilayers, leading to magnetization dynamics and\nrectification. This rectification effect can be interpreted as a form of spin diode, where high and low\nresistance states are controlled by the direction of current flow (which changes the spin polarization of the\nspin current and therefore modifies the spin-transfer torque) and high switching rates are standard [17].\nAlthough the initial power sensitivity of such diodes was only 1.4 mV/mW\u00001(still well below the 3 800\nmV/mW\u00001of semiconductor diodes) recent breakthroughs in MTJ design have enabled power sensitivities\nof 25 000 mV/mW\u00001at low temperatures by Cheng et al. [246], and at room temperature 12 000 mV/mW\u00001\nby Miwa et al. [244], and 75 400mV/mW\u00001by Fang et al. [245]. While Fang’s method does not require a\nbias magnetic field, Miwa’s method has a better signal-to noise ratio. Such high power sensitivities opens\nthe door to enhanced microwave imaging and sensing techniques which will be discussed in Sec. 4.6. It\nshould be noted that an analogous diode effect due to GMR has also been observed [247–249].\nTable 7 summarizes some of the key studies in the development of the spin transfer based spin diode and\nMTJ rectification and Fig. 22 shows a typical spin diode experiment in detail. Similar to the spin-transfer\n39Figure 22:(a) A typical MTJ spin-transfer torque measurement configuration. A bias tee allows the application of both an rf\nand dc current as well as the measurement of the voltage across the MTJ. The applied current becomes spin polarized by the\npinned layer and the spin polarized current can then apply a torque to the free layer magnetization. (b) Typical experimental\nvoltage signal as a function of frequency for several magnetic field strengths [17]. The line shape has both dispersive and\nsymmetric contributions (from the \fFTand\fSTTterms in Eq. 3.10 respectively).\nTable 7:A summary of key experimental results using magnetic tunnel junctions (spin diodes).\nReference Device Structure Result\nTulapurkar et al. [17]\nElectrical detection of SR due to STT\nin a CoFeB/MgO/CoFeB MTJ\nSankey et al. [28]\nUse of SR to measure bias and angular\ndependence of STT vector in\nCoFeB/MgO/CoFeB MTJ\nMiwa et al. [244]\nFang et al. [245]\nRoom temperature spin diode with\nlarge power sensitivity due to\noptimization of MTJ structure\n(CoFeB/MgO/FeB and\nCoFe/Ru/CoFeB MTJs).\ntorque measurement in bilayer systems, a bias tee is used to allow the application of both an rf and dc\ncurrent/voltage as well as to measure the voltage across the device.\nThe current becomes spin polarized after passing through the fixed layer and can then act as a torque\non the free layer. Typical experimental voltage curves as a function of the current frequency are shown in\nFig. 22 (b) for several magnetic field strengths. This data was collected from an MTJ with a pinned layer\nmade of CoFeB and CoFe layers antiferromagnetically coupled across a Ru spacing layer. The free CoFeB\nlayer was separated from the pinned layer by an MgO tunnel barrier and a 0.55 mA rf current was applied\nwith no bias current/voltage. The line shape is seen to have both Lorentz and dispersive components as\nwould be expected from Eq. 3.10.\nSince the spin current polarization will be along the direction of the pinned layer, it is convenient to\ndescribe the magnetization dynamics in an MTJ by writing the STT modified LLG equation as\ndM\ndt=\u0000\rM\u0002Hi+\u000b\nM0\u0012\nM\u0002dM\ndt\u0013\n+\r\fSTTI\nM0M\u0002(M\u0002Mp) +\r\fFTIM\u0002Mp:(3.9)\nHereMis the magnetization of the free layer, Mpis the magnetization direction of the pinned layer, which\n40polarizes the current and Iis the amplitude of the rf current applied to the MTJ. Finally \fSTTand\fFTare\nthe spin-transfer torque and field-like torque amplitudes per unit current respectively. The \fSTTterm is the\nsame as the in-plane modification made to the LLG equation in Eq. 3.6 while the \fFTterm is an explicit\nout-of-plane torque which is necessary for MTJs [134, 250].\nIn the bilayer system, where the rectification was due to AMR, the voltage could be determined using\nthe generalized Ohm’s law. In an MTJ where the magnetoresistance is due to TMR, the voltage must be\nfound by solving the modified LLG equation to determine the angle \u0012between the pinned and free layer\nmagnetizations and then expanding the magnetoresistance as a function of this angle. Following the initial\nstudy by Tulapurkar et al. [17], here we will analyze in detail the case of zero bias voltage/current. We take\ntheyaxis perpendicular to the easy axis plane and the magnetizations in the x\u0000zplane with the zaxis\nalong the easy axis of the MTJ. The demagnetization fields will be along the yaxis since the thickness of the\nMTJ is small compared to its lateral dimensions. We take Malongbzwhich is at an angle \u0012to the pinned\nlayer magnetization along bz0. If we assume the magnetization of Mis constant, then when we expand M\naround its equilibrium value M0it can have two components perpendicular to M0. It is convenient to write\nthese two directions as M\u0002MpandM\u0002(M\u0002Mp)and expand Mas\nM=M0+ae\u0000i!tM0\u0002Mp\njM0\u0002Mpj+be\u0000i!tM0\u0002(M0\u0002Mp)\njM0\u0002(M0\u0002Mp)j:\nTakingI=I0e\u0000i!tthe LLG equation can be solved for the coefficients aandbin analogy with how mxand\nmywere found for the field torque rectification. In the case of the spin dynamo, since we were measuring a\nvoltage along the current direction, only mxcontributed to the rectified voltage, but mxhad a contribution\nfrom bothhxandhydue to the matrix form of the susceptibility. By analogy, since the resistance depends\noncM\u0001Mp, onlybwill contribute to the spin diode voltage, but will depend on both the amplitudes \fSTT\nand\fFTby the solution to Eq. 3.9. We can write the resistance given by Eq. 2.10 as\nR=RP+\u0001R\n2\u0010\n1\u0000cM\u0001MP\u0011\n=RP+\u0001R\n2[1\u0000cos\u0012+jsin\u0012j(Re(b) cos (!t) +Im(b) sin (!t))]:\nTherefore the voltage is\nV=\u0001R\n2I0jsin\u0012jRe(b)hsin2!ti=\u0001R\n4\rI2\n0sin2\u0012Re\u0014~!\fFT+i!\fSTT=M0\n!2\u0000!2r\u0000i!\u0001\u0015\n(3.10)\nwhere!r=\rp\nH(H+M0)is the resonance frequency which follows the Kittel formula, \u0001 =\u000b\r(2H+M0)\nand~!=\r(H+M0). For more discussion of the zero bias line shape see [21, 251, 252].\nHere we have assumed that the spin-transfer torque coefficients \fare independent of the angle between\nthe magnetization directions of the pinned and free layers. In general this may not be true [253] with\nangular dependence appearing explicitly in the conductance and also in non-collinear systems which require\na rigorous generalization of the two-current model [252] (which is consistent with a scattering approach\n[77]). However it has been suggested [254] that the angular dependence of the spin transfer efficiency and\nthe electric conductance cancel, leaving the coefficients independent of the angle \u0012. However the same\ncancellation does not occur for the bias voltage dependence since the conductivities of the spin channels\nhave different bias voltage dependencies [134, 255] and therefore the \fcoefficients will generally have a Vbias\ndependence. Kubota et al. [22] have performed a systematic study of the bias voltage dependence of the\nspin-transfer torque rectification in MTJs which have also been studied in Ref. [28]. In the analysis of\nthe biased samples, the torque dependence on the bias current dominates and the angular dependence is\nignored in comparison. Spin-transfer torque in spin valves is analogous [256]. We should note that most\nSTT experiments, including those discussed here have employed amplitude modulation. However in 2013\nGonçalves et al. [257] demonstrated that magnetic field modulation, often used in conventional FMR, could\nimprove the sensitivity of STT-FMR.\nIn addition to the STT signal in MTJs in principle a field torque may also exist if an rf magnetic field\nis applied. The voltage produced by field torque rectification in the free layer would then be the same as\n41we have already discussed in Sec. 2.3. Finally, apart from the spin-torque and field-like torque, a voltage-\ninduced torque can also play an important role in driving FMR in the ultrathin ferromagnetic structures\nwhich may be present in MTJs.\n3.4.2. Voltage Torque Rectification\nThe origin of voltage torque rectification in MTJs is voltage-controlled magnetic anisotropy, and as\nthis effect does not rely on the microwave current, it can dominant in MTJs with thick tunnel barriers.\nIn 2008 and 2009 theoretical work indicated that an electric field can substantially alter the interfacial\nmagnetic anisotropy energy and even induce magnetization reversal in 3d transition ferromagnets [258–260].\nLater, in 2011, both Wang et. al [261] and Shiota et. al [262] experimentally realized the direct resistance\nswitching induced by a dc voltage in MgO-based MTJs. Since the estimated power consumption for single\nswitchingiswellbelowthatrequiredforthespin-current-injectionswitchingprocess, theseresultsopenanew\navenue for exploring voltage-controlled spintronic devices analogous to those which are already ubiquitous\nin semiconductor technologies.\nFigure 23:Magnetization dynamics in MTJs can also be driven by a voltage-torque due to electric field induced interfacial\nmagnetic anisotropy changes. (a) Application of a dc voltage can switch the easy axis from the in-plane to out-of-plane\ndirections. (b) Application of an rf voltage can excite magnetization precession about an external magnetic field direction. The\nyellow arrow indicates the effective rf magnetic field induced by the anisotropy control. Source: Adapted from Ref. 263.\nA consequence of the voltage-controlled magnetic anisotropy is that magnetization oscillations in an\nMTJ device can also be excited by an applied microwave voltage, resulting in resistance oscillations. Fig.\n23 shows the basic concept of voltage-induced FMR in an MTJ with an ultrathin ferromagnetic layer, where\nswitching of the magnetic easy axis between the in-plane and the out-of-plane directions is demonstrated by\ncontrolling the sign of the dc voltage bias. As the tunnelling resistance depends on the relative magnetization\nconfiguration, the excited FMR dynamics (Fig. 23 (b)) alter the configuration and generate an oscillating\nresistance. Mixing with the microwave current, a dc voltage is produced [263, 264]. As expected, the\nobserved rectification voltage is linearly proportional to the square of the input microwave voltage and\nshown in Fig. 24 (b). In MgO-based MTJs the magnitude of high-frequency spin-torque and voltage-torque\ncan be similar, and thus quantitative descriptions of voltage-driven magnetization dynamics in MTJs should\ngenerallyincludebothtorqueterms[264]. Ithasalsobeenshownthatvoltage-controlledmagneticanisotropy\ncan be used in spin diode rf detectors, such as to improve sensitivity [264–266].\nIn this section we have discussed the dc voltage produced in monolayer devices with integrated CPWs,\nknown as spin dynamos, in FM/NM bilayers, where field torque, spin pumping and spin-transfer torque\nall contribute, and in MTJs where the voltage signal is due to spin-transfer torque caused by the pinned\nmagnetization. These effects are summarized in Table 8.\n42Figure 24:(a) Radiofrequency power dependence of the voltage spectra measured under a constant external magnetic field of\n0.05 T at\u001eH= 45\u000e. (b) Intensity of the peak voltage signal, which is proportional to the square of the input rf voltage Vrf, as\nexpected. The FMR precession angle is shown on the right hand axis. Source: Adapted from Ref. 263.\nTable 8:Summary of the dc voltage observed in ferromagnetic thin films, FM/NM bilayers and MTJs.\nDevice Effect Voltage Lineshape\nThin Film Field Torque V=Vh\nSR(hjrfhrfi)Lorentz and dispersive, con-\ntrolled by relative phase and\ndriving field component\nBilayerField Torque,\nSpin-transfer\ntorque, Spin\nPumpingV=Vh\nSR(hjrfhrfi)\n+VSTT\nSR(hjrfjs\nrfi)\n+VSP\u0000\nhh2\nrfi\u0001Lorentz and dispersive from field\ntorque, Lorentz from spin pump-\ning, Lorentz from spin-transfer\ntorque (if field-like spin-transfer\ntorque is negligible)\nMTJSpin-Transfer\nTorque, Field\nTorque, Voltage\nTorqueV=Vh\nSR(hjrfhrfi)\n+VSTT\nSR(hjrfjs\nrfi)Lorentz and dispersive from\nfield torque, Lorentz from spin-\ntransfer torque and dispersive\nfrom field-like spin torque (Volt-\nage torque behaviour is analo-\ngous to spin-transfer torque)\n4. Applications of Spin Rectification\nHaving described the physical mechanisms of spin rectification and its characteristics in various device\nstructures, in this section we turn to the applications of spin rectification, both in the study and under-\nstanding of basic physics and in the development of imaging and material characterization techniques. As\nwe will see, spin rectification has proven to be a versatile technique which is used in a wide variety of studies\ndue to its high sensitivity, phase sensitivity and ease of use.\n4.1. Electrical Detection of Spin Waves\nSpin waves are collective excitations which occur in magnetic lattices due to the short-range exchange\nintegrationand/orthelong-rangedipole-dipoleinteraction. Unlikethe uniformprecessionofFMR,spinwave\nexcitations are characterized by a spatially varying spin precession phase and therefore have a propagation\ndirection and a finite wavelength. From a fundamental physics perspective, since a set of discrete spin\nwaves can easily be excited under microwave radiation (with FMR being the lowest order mode), spin wave\nspectroscopy provides a powerful tool for understanding both the exchange and dipole-diple interactions.\n43On the application side, the generation of spin waves is an important energy loss mechanism in modern\nspintronic devices operated at high frequencies and the inverse of the lowest spin wave frequency determines\nthe switching timescale of devices.\n4.1.1. Review of Spin Waves\nWhen determining the nature of spin waves, the boundary conditions at the sample interface are of\nutmost importance. In a thin film these pinning conditions will, roughly speaking, cause the spin waves\nto form standing waves. Therefore the wavelength of spin waves with wave vector perpendicular to the\nfilm will be small and these so called perpendicular standing spin waves (PSSWs) will be dominated by\nthe exchange interaction. Conversely the comparatively large wavelength in the film plane results in dipole-\ndipoleinteractiondominatedspinwaves, socalledmagnetostaticmodes. Thein-planemagnetostaticmodes\ncan be conveniently classified based on the orientation of the wave vector with respect to the magnetization.\nWhen the wave vector is perpendicular to an in-plane magnetization, Damon-Eshbach (DE) spin waves (also\nknown as surface spin waves) will form, while if the wave vector is parallel to the in-plane magnetization\nback ward volume modes (BVM) will be produced. On the other hand when the magnetization is normal\nto the film plane, due to the in-plane symmetry there is only one type of in-plane spin wave, the forward\nvolume modes (FVM). The different types of spin waves in thin magnetic films are sketched in Fig. 25,\nwith panels (a) and (b) illustrating the various types of spin waves classified by the magnetization and wave\nvector orientation, and panel (c) schematically showing the spin wave dispersions.\nFigure 25:Schematic illustration of spin waves in a thin film in the case that exchange and dipole-dipole interactions can\nbe treated separately. For both in-plane and out-of-plane magnetizations the short-range exchange interaction will produce\nperpendicular standing spin waves normal to the thin film. On the other hand the short-range dipole-dipole interaction is\nresponsible for spin waves with an in-plane wave vector. (a) For the case of in-plane magnetization the spin waves due to the\ndipole-dipole interaction will either be Damon-Eshbach modes, with in-plane wave vector perpendicular to M, or backward\nvolume modes, with in-plane wave vector parallel to M. (b) For the perpendicularly magnetized thin film, due to the in-plane\nsymmetry, there are only one type of in-plane modes, the forward volume spin waves. (c) Schematic spin-wave dispersion.\nIn the long wavelength regime ( k\u0018\u0016m\u00001) the Damon-Eshbach (red) and backward volume modes (blue) are shown, with\nthe grey shaded area indicating arbitrary magnetization orientation. In the nanometer wavelength regime the dispersion is\ncosine-like. The red shaded area indicates a region of heavy spin wave damping. Source: Panel (c) adapted from Ref. [267].\nIgnoring the effects of damping, the dispersion of the resonance frequency of PSSWs can be determined\nfrom the LLG equation by including an additional exchange interaction term [268], leading to the PSSW\ndispersion relations\nPSSW\u0000M0in-plane:!2\nr=\r2\u0000\nH+M0+ 2Ak2\nz=\u00160M0\u0001\u0000\nH+ 2Ak2\nz=\u00160M0\u0001\n;\nPSSW\u0000M0out-of-plane: !r=\r\u0012\nH\u0000M0+2Ak2\nz\n\u00160M0\u0013\n:\nwhereAis the exchange stiffness constant characterizing the strength of the exchange interaction [269].\n44The spin wave dispersion relations for the DE, BVM and FVM modes can be determined using a magneto\nstatic approximation [270], leading to\nBVM:!2\nr=\r2H\u0014\nH+M0\u00121\u0000e\u0000kzd\nkzd\u0013\u0015\n;\nDE:!2\nr=\r2\"\nH(H+M0) +\u0012M0\n2\u00132\u0000\n1\u0000e\u00002kxd\u0001#\n;\nFVM:!2\nr=\r2\u0014\n(H\u0000M0)2+M0(H\u0000M0)\u0012\n1\u0000e\u0000kTd\nkTd\u0013\u0015\n:\nwherekT=q\nk2x+k2yis the in-plane wave vector. Notice that all of the spin wave modes, except for\nBVM modes, are shifted to higher frequencies than the FMR and therefore require more energy to excite.\nInterestingly, due to the nature of the dipole-dipole interaction, the BVM modes actually require less energy\nthan FMR due to their negative group velocity.\nThe spin wave discussed above consider the limits where either exchange or dipolar interactions are\ndominant. Of course in general both interactions will contribute to the formation of spin waves, leading to\nso called dipole exchange spin waves (DESW) [271–273] which follow the dispersion [273],\n!2\nr=\r2\u0012\nHi+2Ak2\n\u00160M0\u0013\u0012\nHi+2Ak2\n\u00160M0+M0Fp\u0013\n(4.1a)\nwhere\nFp=Pp+ sin2\u001eH \n1\u0000Pp+M0Pp(1\u0000Pp)\nHi+2Ak2\n\u00160M0!\n; (4.1b)\nand\nPp=ky\n2Zd\n0Zd\n0 p(z) p(z0) exp (\u0000kyjz\u0000z0j)dzdz0: (4.1c)\nThe geometry used here is that of Fig. 25 (b) with the magnetization tilted an angle \u001eHfrom thezaxis\nand we denote the total internal field, including demagnetization factors, by Hi. Herek2=k2\nz+k2\nywith\nky=n\u0019=w(wbeing the width of the thin film) and kz= (p\u0000\u0001p)\u0019=d. The discrete number pis the\ninteger number of half wavelengths along the zdirection, normal to the sample with \u0001pa correction factor\ndetermined by the pinning condition at the interfaces z= 0andz=d,\n\u0012\n2A@ p\n@z\u0000Ks p\u0013\f\f\f\f\nz=0;d= 0: (4.2)\nFinally pis an eigenfunction of the PSSW with the form  p=Cp(\u000bsinkzz+\fcoskzz)whereCpis\na normalization constant and \u000band\fare constraints determined by the surface anisotropy Ksand the\nexchange stiffness. In the limit \u000b=\f!1the spins at the surface of the material are completely pinned and\n\u0001p= 0, while if\u000b=\f!0, the surface spins are completely free and \u0001p= 1, thereforepis constrained by\n0\u0014p\u00141.\nEq. 4.1a describes DESW that depend on p(\u0001p) andn. However this expression may be used to describe\nmagneto static modes and PSSW as well by taking appropriate limits. Setting the exchange stiffness, A= 0\nthe perpendicular profile of the spin waves becomes trivial and the DESW reduce to magneto static modes\nwith quantization number nand the exact form (BVM, FVM, DE) depending on the measurement geometry.\nAlternatively, setting n= 0the dispersion of Eq. 4.1a reduces to the case of PSSW characterized by p\u0000\u0001p.\nTherefore the FMR ( p\u0000\u0001p= 0;n= 0), DESW ( p\u0000\u0001p6= 0;n6= 0), PSSW (p\u0000\u0001p6= 0;n= 0) and\nmagneto static modes ( p\u0000\u0001p= 0;n6= 0) can all be described by Eq. 4.1a, which makes Eq. 4.1a useful\nin the experimental identification of the spin wave modes across the whole Hrange. For more in depth\ndiscussion of spin waves see e.g. Refs. [7, 274, 275].\n454.1.2. Detection of Spin Waves\nExperimentally there are many techniques which have been adapted to study spin waves. Some of these\ntechniques are summarized in Table 9. Generally these techniques fall into the categories of time-domain\nmeasurements, or field and frequency domain measurements. Time domain techniques measure the temporal\nevolution of the magnetization typically using pulse-inductive microwave magnetometry (PIMM) or time\nand spatially resolved magneto-optic Kerr effect (MOKE) microscopy. PIMM uses a pulsed excitation to\ngenerate spin waves, and the inductance induced voltage is measured in a stripline [276]. On the other\nhand MOKE techniques employ a femtosecond laser to carry out pump probe measurements. As both the\npolarization and intensity of light reflected from a magnetized surface will change based on its magnetic\nproperties, the location of a spin wave resonance can be directly measured using time-resolved scanning Kerr\nmicroscopy as a local spectroscopic probe. More recently x-ray detected magnetic resonance (XDMR) which\nuses x-ray magnetic circular dichroism (XMCD) to probe the local magnetization in a microwave pump field\nhas been used to investigate resonance processes with element specificity [282, 283].\nAlternatively, field and frequency domain techniques have also proven successful in the study of spin\nwaves. Such techniques include both direct measurements of the !(k)spin wave dispersion using Brillouin\nlight scattering (BLS) [297–299, 309, 310] as well as broadband measurements of the microwave reflec-\ntion/transmission/absorption as a function of frequency and external magnetic fields [291–293, 311, 312]. In\nBLS the Stokes and anti-Stokes shifts produced by the creation and absorption of magnons are measured us-\ningasensitivetandemFabry-Pérotinterferometer, whichcanaccessboththespatialandtemporalproperties\nof spin wave packets, and simultaneously various frequencies in a wide frequency range [299]. More recently,\nferromagnetic resonance force microscopy (FMRFM) of confined spin-wave modes with improved, 100 nm\nresolution, has been developed [303]. Ferromagnetic resonance spectra in Py disks (diameters ranging from\nFigure 26:Typical voltage spectra focusing on different Hfield ranges, measured under a nearly perpendicular applied\nmagnetic field. All voltage spectra are normalized and vertically offset. (a) Typical spectra over the whole Hfield range at\nvarious frequencies (4.5 to 5.5 GHz in 0.1 GHz steps). Arrows indicate the positions of FMR and perpendicular standing spin\nwaves forp= 2(S2, S20) andp= 3(S3, S30). (b) Spectra of forward volume modes found near FMR. The inset shows the\ndetection and magnetization field static field configuration. The rf field was provided by a CPW (not shown). (c) Spectra of\nthe dipole-exchange spin waves found near the S2perpendicular standing spin wave. Note that in (b) and (c) the 0 of the\nfield axis is centred at the FMR and S2 spin wave position respectively, with \u0001Hindicating the deviation from this position.\nSource :Adapted from Ref. [26].\n46Table 9:A summary of experimental techniques used for the investigation of linear spin waves and nonlinear magnetization\ndynamics. The schematic setups and non exhaustive list of references given here are meant only as a brief overview and starting\npoint for further reading.\nTechniqueExperimental\nSetupLinear Spin Wave\nStudiesNonlinear Studies\nTime-Domain Techniques\nPulse-Inductive\nMicrowave Magnetometer\nSampling Oscilloscope Pulse \n[276–279] [280, 281]\nTime Resolved XMCD\nX-Rays Microwaves [282–284] [285]\nTime and Spatially\nResolved MOKE\nMicroscopy\n[286–289] [290]\nFrequency and Field Domain\nBroadband Spin Wave\nSpectrometer\nVNA \n[291–293] [294–296]\nBrillouin Light Scattering\n [297–299] [300–302]\nFerromagnetic Resonance\nForce Microscopy\nCantilever [303] [304]\nElectrical Detection via\nSpin Rectification\nV Microwave Generator [26, 305–308] [32]\n47100 to 750 nm) has shown multiple distinguishable edge modes.\nAlso falling into the field and frequency domain category is the use of spin rectification [26, 305–308]. A\nkey difference in SR based spin wave detection is that the device itself acts as the detector. This enables\nsimple implementation, simplified data analysis and, increased sensitivity due to the lock-in techniques used.\nOne of the key results arising from SR detection of spin waves is the unambiguous observation of DESW\n[26, 308]. Although spin waves were experimentally studied as early as the 1950’s, for many years only\nmagneto static modes or PSSWs could be observed [286, 298, 313–316] with the detection of DESW and\nthe related question of the boundary condition dependence of spin dynamics [317–320] remaining elusive.\nThis initial inability to observe the DESW was mainly due to the weak signals compared to FMR and\nPSSW and the fact that the spacing between DESW resonances is only a few mT, which means they are\neasily washed out by Gilbert damping broadening of the resonance peak. The high sensitivity of electrical\ndetection based on SR allowed the DESW to be observed for the first time [26] and subsequent studies have\nfurther elucidated the nature of DESW, e.g. the pinning conditions in nanowires [308].\nFig. 26 nicely illustrates the versatility of spin rectification in the study of spin waves, showing PSSW in\nFig. 26 (a), FVM modes in Fig. 26 (b) and DESW in Fig. 26 (c). This sensitive data can be ready used to\nassign the order of each spin wave mode [26]. For a more detailed review of spin-wave detection techniques\nsee e.g. [267].\n4.2. Electrical Detection of Nonlinear Magnetization Dynamics\nThe onset of nonlinear dynamics produces a rich array of new physics compared to the linear regime, such\nas amplitude dependent resonance frequency, foldover effects and bistability. These nonlinear fingerprints\nare found throughout nature, from mechanical [321] to magnetic systems [322, 323]. In magnetic systems\nthe key effects of nonlinear dynamics are 1) power dependent resonance position shifts of both the FMR\nand spin wave modes and 2) line width broadening. The former is due to a nonlinearity of the driving force\nwhile the later is due to additional nonlinear damping. The onset of the foldover and bistability effects also\noccur in magnetic systems when the resonance shift reaches a critical value which was initially investigated\nby Anderson and Suhl [322]. However in the experimental search for foldover effects in magnetic systems,\nlarge discrepancies from the Anderson-Suhl model have been found [324–336]. These discrepancies were\npartially due to the presence of spin wave instabilities and thermal effects, which made the interpretation\nof early experiments difficult, but the primary reason that the critical frequency shift could not be reached\nis due to the effect of nonlinear damping, which effectively pushes the power and frequency required to\nobserve foldover effects beyond the range initially investigated. This later complication was identified by\nspin rectification studies which were subsequently used to study the foldover FMR [32, 180].\n4.2.1. Review of Nonlinear Magnetization Dynamics\nThe effects of resonance shift and line width broadening can be described by determining the cone\nangle of magnetization precession in the nonlinear regime. Physically the shift in resonance frequency\noccurs due to the decrease in Mzas the magnetization precession is tilted to larger cone angles by a high\ndriving power. However as the resonance frequency is shifted, nonlinear damping becomes more important.\nBased on the analysis in Sec. 2.2 the cone angle \u0012cis determined by \u00122\nc\u0018\u0000\nm2\nx+m2\ny\u0001\n=M2\n0, where we\nhave assumed that M2\n0\u001d\u0000\nm2\nx+m2\ny\u0001\n. The solution for the LLG equation can then be used to relate\nthe cone angle to the driving rf field. For example when His applied nearly perpendicular to the sample\nplane\u00122\nc=h2=h\n(H\u0000Hr)2+ \u0001H2i\n, whereh=hx=hy. However to be consistent with this higher order\nexpansion, the nonlinear effects on the resonance field and the line width must also be taken into account,\nwhich can be done by making the replacements [32, 180]\n\u0001H!\u0001H0+\fM0\u00122\nc= \u0001Hin+\u000b!=\r +\fM0\u00122\nc;\nHr!H0\nr\u00001\n2M0\u00122\nc:\nHere \u0001H0= \u0001Hin+\u000b!=\randH0\nrare the line width and resonance position without any nonlinear\nmodifications, respectively, and \fis the nonlinear damping coefficient.\n48Figure 27:(a) - (c) A discontinuity in \u00122\ncoccurs when the rf driving field exceeds a threshold hth. (d) Theh\u0000!phase diagram.\nIn the region below hc, which is the critical field in the absence of nonlinear damping, no foldover is observed. Even with\nnonlinear damping and high powers, below the threshold hthno foldover is observed. Source :Adapted from Ref. [180].\nFor sufficiently large microwave power P, the microwave field h/p\nPexceeds a critical field strength\nhthwhere the FMR response shows a discontinuity in \u0012c, as shown in Fig. 27 (a) - (c). This discontinuity is\nthe foldover effect. In Fig. 27 (a) - (c) the nonlinear damping term is ignored (i.e. \f!0) in which case the\nabove treatment reduces to that of Anderson and Suhl [322]. However the introduction of nonlinear damping\nwill result in a high power region where the foldover effect is suppressed as shown in Fig. 27 (d). Below\nthe critical field in the absence of nonlinear damping, hcno foldover will be observed. For hth> h > hc\nnonlinear damping will suppress the onset of foldover effects, while for h>hthfoldover can now be observed.\n4.2.2. Detection of Nonlinear Magnetization Dynamics\nTable 9 provides a brief summary of experimental techniques that have been used to study nonlinear\nmagnetization dynamics. The schematics shown here indicate the use of CPWs to deliver a driving rf field,\nwhich enables the application of high powers due the enhanced field confinement and close proximity to\nsamples under investigation. However early studies of nonlinear effects in magnetic materials often used\nmicrowave cavities, such as those in a pulsed microwave reflection cavity spectrometer system [294–296]. In\nsuch systems the peak power levels with a pulse width of about 50 \u0016s can be as high as 1 kW, allowing\nthe measurements of spin wave instabilities in both ferrimagnetic (YIG) [294] and ferromagnetic (Py) films\n[296].\nTo investigate the wave vector dependence of nonlinear effects, which is necessary to fully understand\nFMR at high microwave powers [337], combined BLS and microwave pumping techniques have also been\nused [300–302]. More recently micro focussed BLS has enabled sensitive local probing of micro- and/or\nnano-structured devices. For example, Edwards et al. reported an experimental observation of parametric\nexcitation of magnetization oscillations in a Py micro disk adjacent to a Pt layer [302], demonstrating control\nof nonlinear dynamic magnetic phenomena in microscopic structures via pure spin currents. Additionally\nthe longitudinal magneto-optical Kerr effect has been used to investigate large precession angles of up to\n\u0012c= 20\u000e[290],demonstratingtheSuhlthresholdeffectforparametricspinwavegeneration[337]. Meanwhile,\nferromagnetic resonance force microscopy [304] has also proven to be a useful tool for nonlinear studies,\nallowing simplified sample fabrication of single magnetic nano-structures by removing the requirement of\nelectrical contacts. Finally the phase resolved ability of XMCD measurements has allowed new nonlinear\nphenomena to be observed at low magnetic bias fields resulting in the generalization of spin-wave turbulence\ntheories [285].\n49Electricaldetectiontechniquesbasedonspinrectificationhavealsoprovenusefulinthestudyofnonlinear\nmagnetization dynamics in a variety of magnetic structures [32, 180, 338–340]. For example, nonlinear FMR\nin a micro-structured MTJ has been reported [338] and the large precession angle magnetization dynamics\nin a nanoscale MTJ have been used to measure the electric-field modulation ratio of magnetic anisotropy\nenergy density [339]. Zhou et al. have also studied nonlinear effects in YIG/Pt bilayers where foldover has\nbeen observed and the power dependent cone angle due to spin pumping can be explained by accounting\nfor nonlinear damping [340]. Finally nonlinear phenomena has also been investigated in a Py strip using\na second generation spin dynamo, as shown in Fig. 28. Electrical detection enables several key nonlinear\nbehaviours to be easily observed, such as the foldover effect, shown in Fig. 28 (a) at high powers (25\ndBm) with the blue (red) curves measured while sweeping the field up (down), and the onset of nonlinear\nbehaviour as a function of microwave power, seen in panel (b), where, in addition to FMR (dark blue),\nseveral perpendicular standing spin wave modes can be observed at lower field.\nFigure 28:Nonlinear magnetization dynamics detected using spin rectification. (a) Observation of the foldover effect at high\npower (25 dBm). The green curve shows the rectification signal at low power for comparison. (b) Linear to nonlinear transition\nobserved in the normalized spin rectification voltage over a power range, 1 mW <P < 100mW. FMR (dark blue) and several\nperpendicular standing spin wave modes (at lower field) can be observed. Source :Adapted from Ref. [32].\n4.3. Electrical Detection of ac Spin Current\nSpin pumping is perhaps the most common technique used to generate pure spin currents and has been\nemployedinNM/FMbilayers[20],NM/ferromagneticsemiconductorbilayers[187]andevenNM/ferromagnetic\ninsulator (FMI) bilayers [39]. The use of FMI is especially intriguing since it explicitly demonstrates the\nflow of a pure spin current in the absence of charge current (see Ref. [341] for an in depth discussion of\nspin currents in FMI). In the treatment of spin pumping presented in Sec 3.3.3 we tacitly assumed that the\nspin polarization \u001bwas time independent and directed along M0. This assumption of a dc spin polarization\nresults in a dc spin current which, after being converted to a charge current via the inverse spin Hall effect,\ncan be directly detected as a dc voltage. To date, most work on spin pumping and spin currents has focused\non this dc effect. However the assumption that \u001bis constant neglects the fact that the direction of spin\npolarization is actually determined by the full time dependent magnetization, and therefore \u001bis actually\ntime dependent. This means that in addition to a dc component directed along the external field direction,\nthere is also a transverse ac spin current. Recently spin pumping theory has been extended to include such\nan ac spin current [342] and proposals for an ac spin current source have been put forth [343]. A sketch\nof the dc and ac spin currents is shown in Fig. 29 (a). As we already know, the precessing magnetization\nwill pump a spin current into an adjacent material layer due to a non-equilibrium spin distribution at the\ninterface. Here the dc (yellow) and ac (red) component of this spin current are explicitly shown. In addition\nto the obvious dynamical difference between the dc voltage, Vdc, generated through the ISHE and the ac\nvoltage,Vac, generated by the ac spin current, the other key differences between these two signals are: 1)\nVacis an order of magnitude larger than Vdc[342, 345]. This is due to the fact that, as we saw in Sec. 3.3.3,\nthe dc spin pumping voltage is proportional to sin2\u0012c, while the ac spin pumping voltage is proportional\n50Figure 29:(a) The spin polarization of the pumped spin current has a small dc component (red) along the static field direction\nand a larger transverse ac component (blue). The dc voltage can be measured in-plane transverse to the static field while the ac\nvoltage will be along the field direction. The dc spin pumping has been studied more intensively since it can easily be measured\nthrough the inverse spin Hall effect. (b) The experimental setup used to detect the ac spin current through the simultaneous\nenhancement of spin rectification and spin pumping. (c) By tuning the angle of the external field, the Py and YIG resonance\ncan be made to occur simultaneously, showing enhanced Py rectification and (d) enhanced YIG spin pumping at the mutual\nresonance condition. Source :Panels (c) and (d) adapted from Ref. [344].\ntosin\u0012ccos\u0012c. Therefore the dc signal is maximized at \u0012c= 90\u000ewhile the ac signal is largest at \u0012c= 45\u000e.\n2) The ac voltage scales as the root of the microwave power,p\nP, while the dc spin pumping is linearly\nproportional to P[345]. 3) As shown in Fig. 29 (a), Vdcis measured transverse to the applied static field\ndirection, whereas the Vacwill be generated parallel to the static magnetic field.\nIn general detection of ac spin pumping is more difficult than measuring the dc effect, since the ac\nsignal will have the same high frequency as the microwave field and parasitic effects such as hfield induced\neddy currents and the FMR excitation at the same frequency produce large backgrounds which must be\nseparated. Nevertheless a variety of approaches have succeeded in measuring the ac spin current, using for\nexample rectifying diodes [138, 345], a vector network analysis [346] or more recently direct detection via\nx-ray techniques [347]. However it turns out that the ac spin current can also be measured through electrical\ndetection by exploiting the spin torque exerted by an ac spin current [344]. The schematic setup for such an\napproach is shown in Fig. 29 (b). In a conventional dc spin pumping experiment the spin current flowing\nfrom the FMI (or FM metal) to the NM will increase the FMI (FM) damping [198]. However if the NM is\nreplaced by a FM layer the ac spin current generated in the FMI can act as a spin torque on the FM, causing\nthe precession amplitude to increase and pump an ac spin current back into the FMI, thus compensating\nthe initially increased damping of the FMI (the dc spin current cannot produce this effect since it will be\nparallel to the static field direction, and therefore will not produce a torque on the FM). Therefore when\nboth the FMI and FM are undergoing resonance simultaneously, the damping will be reduced and both\nthe spin rectification signal from the FM and the spin pumping from the FMI will be enhanced. As shown\nin Fig. 29 (c) the simultaneous resonance condition can be achieved by tuning the angle of the external\nmagnetic field as shown. An enhancement of both spin rectification and spin pumping is seen at  H\u001812\u000e.\n(The second, smaller peak, near  H\u001815\u000eresults from an additional weak resonance signal in the YIG\ndue to a spin wave mode). In addition to the ac spin current detection via SP and SR enhancement, this\ntechnique also shows that the ac spin current can be used as a spin torque and could therefore be applied\nto magnetization switching techniques.\n4.4. Electrical Detection of Magnetization Dynamics in Ferromagnetic Semiconductors\nIn a 1931 letter to Rudolf Perierls (who pioneered the “hole\" concept in semiconductors), Wolfgang\nPauli offered the following advice regarding the sensitivity of semiconductor properties to impurities: “One\nshouldn’t work on semiconductors, that is a filthy mess; who knows whether any semiconductors exist\" (En-\nglish translation of the original German: “Über Halbleiter soll man nicht arbeiten, das ist eine Schweinerei,\nwer weiß, ob es überhaupt Halbleiter gibt)? [348]. Today the deliberate introduction of impurities into\nthe crystal structure of a semiconductor (“doping\"), which alters the conducting properties and permits the\ncreation of semiconductor junctions between differently-doped regions of the crystal, plays a fundamental\n51role in modern semiconductor technology and related IT industries. Interestingly, when the paramagnetic\nelement Mn is substituted for Ga in a GaAs lattice, it not only acts as an acceptor, making it a p-type\nmaterial, but also the produced holes mediate a ferromagnetic interaction between the local moments of the\nopendshells in the Mn atoms, resulting in ferromagnetic properties at low temperatures [349] (for a review\nof ferromagnetic semiconductors see e.g. Refs. [350–354]). Therefore ferromagnetic semiconductors, such\nas GaMnAs, offer a unique combination of material properties, which are amenable to both data process-\ning (due to p-type conducting behaviours) and data storage (due to hysteretic magnetization) in a single\nmaterial system.\nFigure 30:(a) SR voltage signal at different frequencies measured on a GaMnAs Hall bar placed between the ground and signal\nlines of a CPW. The inset shows the linear frequency dependence of the FMR line width. (b) . Source :Panel (a) adapted\nfrom Ref. [31]. Panel (b) adapted from Ref. [355].\nSimilartoferromagneticmetals, SRalsoexistsinferromagneticsemiconductorsandjustasinmetalsmay\nhave several origins, e.g. AMR [31] and spin-orbit torques [355]. As in other material systems, SR allows\nsimple, broadband FMR measurements enabling, for example, determination of damping characteristics.\nFig. 30 (a) shows a typical voltage signal to due to AMR induced SR in a GaMnAs Hall bar as shown\nin Table 2. By fitting the line shape to Eq. 2.21 both the resonance position (shown by crosses) and the\nline width (shown in the inset) can be determined just as they are for SR in a ferromagnetic metal. In\ngeneral, a better understanding of damping in ferromagnetic semiconductors is necessary for applications to\ndata storage technologies. An interesting step in this direction was made in 2011 by Fang et al. [355] who\nfabricated a nanostructured GaMnAs device and introduced a form of spin-orbit-driven FMR originating\nfrom the effective magnetic field in the magnetic material induced by an rf electric current. Due to the\nimproved homogeneity of the nano-structured sample they found an exceptionally low extrinsic damping,\nimplying that the GaMnAs could be used for data storage if further homogeneity refinements could be made.\nTypical voltage measurements from such a device are shown in Fig. 30 (b) where the inset displays the\nfrequency dependent line width from which the Gilbert damping can be determined.\nNotonlyareAMRandspin-orbittorqueinducedspinrectificationobservedinferromagneticsemiconduc-\ntors, additionally in 2013, Chen et al. [187] reported the observation of spin pumping in a GaMnAs/p-GaAs\nbilayer structure, again using electrical detection of FMR. Just like the case for the FM/NM bilayers dis-\ncussed in Sec. 3.3 the precession of the magnetization in the ferromagnetic layer, GaMnAs, produces a\npure spin current which is pumped into the adjacent non-magnetic GaAs layer. This spin current is subse-\nquently converted to a change current by the ISHE and hence generates a voltage drop across the device. Of\ncourse spin rectification is also present and therefore the techniques of line shape and angular dependence\nanalysis must be used to separate the two effects. Fig. 31 shows the angular dependent measurements\nmade in this ferromagnetic semiconductor system. The angular dependence shown in panels (d) and (e)\nfollow the expectations based on our discussion in Sec. 2.3 (with deviations from the expected sinusoidal\nbehaviour due to normalization by the microwave absorption, I, which also has an angular dependence).\nMore recently Nakayama et al. [356] investigated the rectified voltage in a Pt/GaMnAs bilayer structure\n52using a similar analysis method. Interestingly they found an exceptionally large spin mixing conductance\nat the Pt/GaMnAs interface, 6.2 \u00021019cm\u00002which is ten times greater than that of the GaMnAs/p-GaAs\ninterface. This result indicates that hybrid structures with ferromagnetic semiconductors and metals may be\nof interest in the investigation of spin-current related phenomena due to their large efficiency of spin-current\ngeneration at the interface.\nFigure 31:Measurements of the microwave absorption and spin rectification/spin pumping voltage in a GaMnAs/p-GaAs\nbilayer. (a) Device structure. Voltage measurements are made in the x0direction. (b) Derivative of the microwave absorption\nand (c) dc voltage for a magnetic field nearly perpendicular to the bilayer. (d) The Lorentz and (e) dispersive amplitude of\nthe voltage signal as a function o \u001eHnormalized to the microwave absorption I. In (d) the dotted and dashed lines show\nthe contribution of spin pumping and spin rectification respectively, with the solid fitted curve the total voltage due to both\neffects. In (e) the solid curve is the calculated voltage due only to spin rectification (as spin pumping does not contribute\nto the dispersive component). The theoretical angular dependence follows from our discussion in Sec. 2.3, with deviations\nfrom the expected sinusoidal dependence the result of normalization by the microwave absorption, which also has an angular\ndependence. Source :Adapted from Ref. [187].\n4.5. Electrical Detection of Domain Wall Dynamics\nA magnetic domain is a spatial transition region between different magnetic moments with an angular\ndisplacement of 90\u000eor180\u000e[357]. The spatial dimensions of such a region depends strongly on the material,\nand can be a few to thousands of atoms in soft materials. Micro magnetic calculations based on the\nLLG equation have revealed two types of domain walls which can be classified by the canting direction of\nmagnetic moments: in a Néel wall the moments rotate in a direction parallel to i.e. within the plane of,\nthe domain wall while for a Bloch wall they rotate in a direction perpendicular to i.e. through the plane of,\nthe domain wall. In thin films without perpendicular magnetic anisotropy, so that the easy axis lies in the\nfilm plane, this means that a Néel wall has moments which rotate in-plane, while a Bloch wall has moments\nwhich rotate out-of-plane. In thin films with low anisotropies Néel walls are energetically favourable [358]\nand vortex structures may form when two Néel walls intersect. From a fundamental physics perspective\nstudying magnetic domains can teach us about the link between the microscopic and macroscopic properties\nof ferromagnetic materials [357]. On the other hand the ability to directly introduce domain walls through\nthe design of pinning sites and to control domain wall motion lays the foundation for magnetic domain wall\nlogic [359–363] which promises a non-volatile memory device with the high performance and reliability of\nconventional solid-state memory, but at the low cost of conventional magnetic disk drive storage.\nIn order to study domain wall properties, e.g. inertia and resonance frequency, and to manipulate\ndomain walls for magnetic logic purposes, it is important to study domain wall motion. For this purpose rf\nexcitations have been found to be exceptionally efficient, moving domain walls at far lower current densities\n53Figure 32:(a) Device used to measure the domain wall resonance frequency and depinning field via spin rectification. A\ndomain wall is pinned near the constriction sight formed by notches in a Py ring. (b) Measurements of the pinning field (blue)\nof a vortex domain wall and the voltage (red) due to AMR rectification. (c) The experimental setup to perform a magnetic\nfingerprinting experiment of a ferromagnetic nanowire. (d) The voltage spectrum as a function of the rf current frequency\nfor Py semicircular (S2, S3, S4) and “boomerang\" shaped (S1) nanowires. The black curves are measured in the absence of\na domain wall, while the red curves are measured after a domain wall has been introduced by saturating the magnetization\nperpendicular to the easy axis along the strip. Source :Adapted from Refs. [27] and [364].\nthan pulsed voltages [365–369]. Since magnetoresistance occurs during the domain wall motion, which is\nefficiently driven by rf currents, spin rectification effects will be present during domain wall motion and have\nbecome a useful tool to study the domain dynamics in nano-structured magnets, such as the determination\nof the nonlinear pining potential of a domain wall [370].\nIn general rectification effects and the study of domain walls intersect in three key ways. First, more\nindirectly related to rectification but directly related to our discussion of magnetoresistance in Sec. 2.1, is\nthe direct use of magnetoresistance effects, such as GMR or AMR, to study domain wall properties. For\nexample, GMR has been used to study the time dependence of the domain wall position [371, 372] and to\ndetermine the domain wall mass [365], while domain wall pinning conditions have been investigated through\nAMR [373], and time resolved AMR measurements have provided insight into the effect of current and\nspin-transfer torque on domain wall motion [374].\nSecond, domain walls display their own magnetoresistance, which can yield rectification effects and can\nbe manipulated by controlling the spin structure at the pinning site [375]. In analogy with GMR, the\nswitching of domains will produce a magnetoresistance due to a mistracking effect, where the orientation\nof the electron spins comprising the current lag behind the orientation of the local magnetization inside\nthe domain wall [376–379] resulting in a spin accumulation effect [379]. Direct measurements of domain\nmagnetoresistance have been used to study e.g. spin current asymmetry [380].\nFinally, and most relevant to our discussion of spin rectification, due to their general effect on magneti-\nzation dynamics, domain walls will influence AMR rectification in ferromagnetic nano structures. Fig. 32\n(a) shows a Py ring structure where constriction sites, introduced by notches in the ring, will lead to domain\nwall formation. A bias T allows the injection of an rf current and detection of the dc voltage produced\nthrough AMR rectification. Fig. 32 (b) shows the measurement results with the depinning field (the field\nrequired to move the domain wall past the pinning site) shown in blue and the rectification voltage shown\nin red. Thus SR measurements can be used to determine the domain wall resonance of both transverse and\n54vortex domain walls and have even been used to observe multiple resonances within a magnetic domain\n[27, 381].\nAnotherinterestinguseofrectificationinnanostructureswithdomainwallsisin“magneticfingerprinting\"\nof a sample. When a current is passed through a ferromagnetic nanowire a voltage will be generated due to\nSTT as discussed in Sec. 3.3.5. In the presence of a domain wall the voltage spectrum will strongly depend\non the internal spin structure of the wall, and therefore can be used as a fingerprint of the sample [25]. An\nexample of such “fingerprints\" is shown in Fig. 32 (d). The voltages are measured using either a semicircular\nnanowire of varying thicknesses (S2, S3, S4) or a “boomerang\" structured nanowire (S1), as shown in Fig.\n32 (a). The red (black) curves show the rectification voltage in the presence (absence) of a domain wall.\n4.6. Spintronic Microwave Sensing and Imaging\nIn addition to applications for magnetic storage and logic devices which we have already discussed,\nspin rectification is playing an increasingly important role in the development of novel microwave sensing\nand imaging techniques. High speed telecommunications require highly sensitive microwave devices that\nare operational at room temperatures. In particular the rectifying behaviour of semiconductor diodes is\nof key importance, making conventional diodes a foundation of modern electronics. However, although\nsemiconductor diodes satisfy the high sensitivity and room temperature requirements, they have poor signal-\nto-noise ratios which means they cannot operate at low input power. Spin rectification on the other hand\nbehaves remarkably different from semiconductor rectification and has been used to develop the spin-transfer\ntorque diode, which can form the basis of nano meter scale radio frequency detectors for telecommunications.\nFig. 33 (a) illustrates a semiconductor p-n junction diode. When current flows from the n to p side the\nspace charge region is enlarged resulting in a high resistance state. On the other hand when the current\nflows from the p to n side the space charge region is reduced, producing a low resistance state. For the MTJ\nbased spin-transfer torque diode shown in Fig. 33 (b) the varying resistance states are produced by the\nTMR of an MTJ. For a positive current the angle between the fixed and free layers is maximized, resulting\nin a high resistance state, while for a negative current the angle is minimized, producing a low resistance\nstate. Due to this oscillating resistance a rectified voltage is produced.\nSince the dc voltage generated, and hence the efficiency of microwave rectification, is proportional to\nthe magnetoresistance ratio of the device, an MTJ is logically the best choice for microwave detection and\nin 2005, using a conventional CoFeB/MGO/CoFeB MTJ, Tulapurkar et al. [17] first demonstrated the\nconcept of a spin-transfer torque diode. However the sensitivity achieved at the time, up to 1.4 mV/mW,\nwas still roughly three orders of magnitude lower than commercially available semiconductor Schottky diode\nFigure 33:(a) The behaviour of a semiconductor p-n junction diode compared to (b) an MTJ based spin-transfer torque diode.\nSource :Adapted from Ref. [17].\n55detectors (3 800 mV/mW). More recently, using an improved understanding of spin dynamics in such devices\nto improve device fabrication, highly sensitive nanoscale spin-transfer torque diodes with sensitivity up to\n75 400 mV/mW have been developed [244–246, 382]. This order of magnitude sensitivity enhancement over\nsemiconductor devices, while still meeting the required functionalities, indicates the exciting potential of\nspintronic and spin rectification based devices for microwave sensing. One downside to MTJ based sensing\nis the complex device fabrication required. In this regard the recently realized GMR based diode effect may\nprove advantageous [247, 248].\nMicrowave detection also has important applications for novel imaging techniques. A key advantage\nof microwave imaging compared to conventional optical imaging is the increased penetration depth of mi-\ncrowaves, enabling imaging behind barriers/obstacles and the potential to “see the invisible\". However unlike\noptical imaging there are no microwave lenses, which necessitates the use of both amplitude andphasein-\nformation in order to produce accurate images. The simplest way to obtain the required phase information\nis through an interferometry based technique, known as spintronic Michelson interferometry [33]. The basis\nof this technique is illustrated in Fig. 34. By using two coherently split microwave paths to drive the\ncurrent and field, respectively, in a spintronic device, any change in path length, due to either a purposely\nintroduced phase shift or changes in dielectric path length introduced by obstacles, can be observed. The\nspintronic device used as a sensor can be either a spin dynamo [33] or an MTJ [382, 383]. Other techniques,\nsuch as XMCD, have also been successful in phase resolved measurements [384–387], however an impor-\ntant advantage of spintronic Michelson interferometry is its simplicity. When compared to other microwave\nimaging techniques spintronic Michelson interferometry is inexpensive and avoids complex reconstruction\nalgorithms.\nAs illustrated in Fig. 34, the rectified voltage in the spintronic Michelson interferometer depends directly\non the phase difference between the CPW and Py signals. Since microwaves have the ability to penetrate\ndielectric objects, and interact with their internal structures and defects, this means that an object inserted\ninto path-2 will change the microwave phase and thus influence the measured voltage. Since microwaves\nare non ionizing and, at the power levels required for imaging, have not been shown to cause any long term\ndamage to human tissue, microwave imaging techniques have great potential potential for use in medical\nimaging technologies, particularly for breast cancer imaging due to the good dielectric contrast of breast\ntissue [388–391]. Additionally the fact that working in the near field allows the Abbe diffraction limit to be\novercome [392], means that microwaves have the capability to detect structures even smaller than their \u0018\ncm wavelength.\nFollowing the proof of concept for such near-field spin rectification based microwave imaging systems\n[33, 182, 183] these techniques have been used for a variety of imaging purposes, including breast cancer\n[393] and ground penetrating radar detection [394]. In addition to the actual imaging of objects, rectification\nFigure 34:The rectification voltage measured by adjusting an rf phase shifter inserted into (a) one of the two microwave paths,\nand (b) only one microwave path which is connected to either the CPW or (c) the Py strip of a spin dynamo. The strong\nsinusoidal dependence of V(\b)is observed in (a) when both paths are coupled in the spintronic device, demonstrating the\ncontrollable influence of the relative electromagnetic phase on the rectified voltage. Source :Adapted from Ref. [33].\n56based sensors have been used in a variety of other contexts. For example, it has been demonstrated that\nspin rectification can be used as an effective method to detect the full hfield vector polarization [29, 395].\nPhase resolved techniques have also enabled the identification of non resonant rectification [396], and have\nbeen used to develop non destructive dielectric characterization techniques [184]. Also, imaging has been\nperformed using Seebeck induced rectification [397] in MTJs [398, 399].\n4.7. Spintronic Rectifier for Electromagnetic Energy Harvesting\nIn the move towards renewable energy a multifaceted approach is necessary to provide both sufficient\nenergy levels and sustainability. Well known green energy techniques such as solar, wind, geothermal and\nocean energy are of course important contributors to this vision, however with the ubiquitous presence of\nWiFi and cell phone signals, the use of energy harvesting from ambient electromagnetic fields is also poised\nto be a key contributor in future green energy technologies. The feasibility of energy harvesting relies on\nthe efficient operation of rectifiers, which use a nonlinear voltage-current relationship to convert microwave\nfrequency fields into a dc power source. While there are a variety of potential devices, such as the Schottky\ndiode, semiconductor tunnel junction, metal-insulator-metal (MIM) diode and spin diode based on a MTJ,\nthe spin diode is believe to have the most long term potential [400].\nUndoubtedly the Schottky diode will play an important role in the next decade, as it is currently the\nmost commonly used device in rectifiers, with the most matured technology. Unfortunately, as summarized\nin Fig. 35, the Schottky diode already reached its maximum responsivity of 19.4 A/W (at 300 K) in\nthe 1970s, with fundamental limitations set by the laws of thermionic emission which govern the electric\ncurrent flow. In contrast, the current flow due to quantum tunnelling in MTJs, created by separating two\nconductors with a very thin insulator, are mainly limited by the barrier thickness, and still have much\nroom for growth. Although the MTJ fabrication process is very complex, it has benefited from the push\nfor magnetoresistive random-access memory (MRAM) motivated by the huge consumer electronics market.\nIn the coming years this push will likely result in a maturation of fabrication techniques analogous to that\nexperienced by Schottky diodes in the mid 20thcentury. As a consequence, spin diode technology represents\nthe most promising device platform in the development of effective ambient energy harvesting technology\n[400].\nFig. 35 shows the evolution of the responsivity in different rectifier devices. Despite the fact that the\nfirst realization of spin diodes in 2005 achieved only 0.01 A/W responsivity at zero current bias [17], by 2014\nthis value had been improved by two orders of magnitude to 2.1 A/W [244]. Just as semiconductor tunnel\njunctions have already surpassed the Schottky diode limitations, the trend in Fig. 35 suggests that in the\nnear future spin diodes may do the same. Although it is still too early to conclusively predict which kind\nof tunnelling device will dominate future recitifier technology, certainly the performance and feasibility of\nmicrowave energy harvesting devices will greatly benefit from this technological competition.\nFigure 35:Improvement of the rectification capabilities of various diode technologies over the years. Source :Adapted from\nRef. [400].\n575. Summary and Outlook: Three Generations of FMR Spectroscopy\nIn this review we have examined the role of electrical detection in the study of magnetization dynamics.\nWe saw that, due to spin rectification, which produces a dc voltage via the nonlinear coupling of a dynamic\nmagnetoresistance with a dynamic microwave current, the magnetization dynamics of magnetic materials,\nincluding metals, semiconductors and even insulators, can be probed through electrical detection. Although\nthe examination of rectification spectra may be complicated by extraneous effects, such as spin pumping,\nwell established techniques such as angular and line shape analyses enable a wealth of information to be\nFigure 36:The three generations of FMR spectroscopy techniques. (a) Early investigations of spin dynamics were based on\nferromagnetic resonance cavities. The microwave signal through the cavity was detected electrically using a rectifying diode.\n(b) The second generation of FMR spectroscopy arose with the invention of the CPW and VNA. The CPW provides an\neasy way to generate localized, on chip microwave fields, while a VNA can directly measure microwave transmission/reflection\nproperties. (c) Finally, the third and most recent generation is based on electrical detection of spin rectification using lock-in\ntechniques.\nextracted from the rectified voltage, making such techniques powerful for both fundamental physics studies\nand novel spintronic applications.\nFrom a historical perspective, electrical detection represent the third generation in a long history of FMR\nspectroscopy, dating back to the early 20thcentury [1, 3]. The first generation of ferromagnetic resonance\nspectroscopy made use of microwave cavities, typically waveguides, as shown in Fig. 36 (a). With the\nferromagnetic sample placed at the position of strongest microwave magnetic field (usually the centre of\nthe cavity) the microwave signal through the cavity can be detected using a rectifying semiconductor diode.\nThis technology was commercially developed by the German company Bruker and is actually most widely\nused today in electron paramagnetic resonance studies, although it is also a powerful tool in studies of mag-\nnetization dynamics in bulk ferromagnetic materials and films [198]. However, since microwave cavities work\n58in a very narrow frequency range, measurements of the dispersion relation are difficult and time consum-\ning. Additionally as the absorption of microwaves is proportional to the volume of ferromagnetic materials,\nthe technology is not suitable for the study of micro- and nano-structured samples, due to extremely weak\nsignals.\nAs a result of advances in microwave transmission and detection technology, the second generation\nof FMR spectroscopy was born, combining the broadband capabilities of vector network analyzers and\ncoplanar waveguides. In 1969, Cheng Wen invented the CPW at RCA’s Sarnoff Laboratories in New\nJersey [401] (interestingly CPW is also the acronym of his English name, Cheng P. Wen). This invention\ncompactified traditional three-dimensional microwave devices into two-dimensions while still maintaining\nhigh performance. This not only greatly reduced the size of microwave devices but also allowed the on chip\nintegration of microwave devices and other electronics. The invention of the CPW, which allowed improved\nmicrowave transmission, was complemented by the development of the scalar network analyzer by Hewlett-\nPackard in 1973, which enabled the direct measurement of microwave amplitudes over a frequency range\nof 15 MHz to 18 GHz. The rapid development of the telecommunications industry at the end of the 20th\ncentury further demanded improved high-frequency measurement systems, leading to the commercialization\nof the modern vector network analyzer, which can not only measure the amplitude but also the phase\nof broadband microwaves. As shown in Fig. 36 (b), the second generation of broadband ferromagnetic\nresonance measurement technology combines the CPW and VNA [402, 403]. The magnetic sample is placed\non the signal (S) strip of a GSG CPW and the microwave field carried by the CPW drives FMR precession.\nBy fixing the magnetic field the S-parameter can be measured by the VNA as a function of microwave\nfrequency producing an FMR spectra. This technique is widely used to study thin magnetic films, especially\nultra-thin films [404]. In addition, a related technique combining a modulated magnetic field and broadband\nstripline with pulsed inductive microwave magnetometry (PIMM) has been developed in some laboratories\nto measure FMR [276, 405].\nThe first two generations of FMR spectroscopy indirectly measure the energy loss due to FMR, and\ncontinuetobeusedaspowerfultoolstostudymagnetizationdynamics. Howeverduetotheinevitableenergy\nloss in a transmission line, which is very sensitive to the microwave frequency as well as the measurement\nenvironment, these techniques require careful calibration. Additionally for many applications the expense of\na VNA may be prohibitive. This lead to the development of the third generation of FMR spectroscopy in the\nbeginning of the 21stcentury, electrical detection, as shown in Fig. 36 (c). Electrical detection has proven\nto be a powerful tool in the study of spin excitations in micro- and nano-structured magnetic samples [14–\n28]. In this third generation of FMR spectroscopy, microwave fields are still produced by a CPW, however\nlock-in amplification is used to measure the spin rectification voltage resulting from the spin dynamics of\nthe sample. Since this technique 1) has a high microwave field density due to the CPW, 2) has a voltage\nsignal which is independent of sample volume, and is instead dependent on sample resistance which can be\ntailored by careful device fabrication, and 3) uses the highly sensitive measurements of lock-in amplification\nwhich can detect voltages down to the several nV scale, electrical detection through spin rectification has\nbeen used to study a variety of nanostructured samples from FM nanowires [25] to MTJs [17]. In addition,\ndue to its high sensitivity and ease of use, spin rectification has been used to study a wide range of materials\n– from FM metals, to FM semiconductors, to FM insulators – a wide range of physics – from magnetization\ndynamics, to spin pumping and the spin Hall effect – and to develop new approaches to microwave sensing\nand imaging.\nWhile at the time of writing, in 2015, the third generation of FMR spectroscopy has become a well\nestablished technique for the study of microstructured magnetic devices, a new generation of FMR spec-\ntroscopy is just forming. Known as cavity spintronics [65], this emerging field incorporates elements of first\nand third generation FMR spectroscopy techniques to perform both microwave transmission and electrical\ndetection of high quality ferromagnetic samples in low loss microwave cavities/resonators, and promises to\nreveal important insights into the physics of strongly coupled spin-photon systems. As predicted by Soykal\nand Flatteé [406, 407] high quality YIG samples, combined with high quality microwave cavities can ex-\nhibit coherent spin-photon interactions with exceptionally long decoherence times. Such systems were first\nstudied experimentally by Huebl et al. [408] using microwave transmission and various foundational studies\nhave confirmed key predictions about these strongly coupled systems [409, 410], which can be accurately\n59Figure 37:(a) A schematic picture of the cavity spintronic system, consisting of a YIG/Pt bilayer inside of a microwave\ncavity. The strong electrodynamic coupling between the magnetization dynamics, governed by the LLG equation, and the\nelectrodynamics, governed by Maxwell’s equations, lead to the creation of a new quasiparticle, the cavity-magnon-polariton.\n(b) The CMP will modify the spin current generated in the YIG/Pt bilayer, allowing characteristic coupling signatures, such\nas a large avoided crossing, to be electrically detected. The horizontal and diagonal dashed lines indicate the uncoupled cavity\nand FMR dispersions respectively. Source :Panel (a) adapted from Ref. [411].\ndescribed by coupling the Maxwell and LLG equations to form a cavity-magnon-polariton (CMP) [411–415].\nAn exciting possibility for future spintronic technologies is the electrical detection of the CMP [411, 416],\nwhich indicates the influence of strong coupling on the spin current in bilayer devices and could enable the\ncoherent control of spin current. With the theoretical and experimental foundation in place, a flurry of\nrecent applications have been proposed e.g. magnon dark mode memories [417], magnon qubit coupling\n[418, 419], and the coupling of optical and microwave frequency regimes through whispering gallery modes\n[420–423]. On this new frontier of strong magnon-photon coupling the electrical detection of magnetization\ndynamics is poised to play a key role due to its versatile capabilities as we have reviewed in this article.\nIn this perspective, the twilight of the fourth generation of FMR spectroscopy, with its focus on cavity\nspintronics and quantum magnetism, is on the horizon.\nAcknowledgements\nThis review article has been based on a decade of collaboration and communication with many colleagues\nas identified in the reference section. We thank all members and alumni of the Dynamic Spintronics Group\nat the University of Manitoba, especially N. Mecking, A. Wirthmann, X.L. Fan, L.H. Bai, Y. Hou, B.M.\nYao, L. Fu, Z.H. Zhang, and P. Hyde for their contributions. We are also thankful for the encouragement\nand/or feedback provided during the public screening of our preprint by W.E. Bailey, G.E.W. Bauer, J.W.\nCai, C.L. Chien, M. Costache, V. Flovik, B. Heinrich, T. Jungwirth, M. Kläui, O. Klein, A. Kovalev, I.\nKrivorotov, G.J. Schmidt, D. Stancil, T. Stobiecki, Y. Tserkovnyak, Y.Z. Wu, J.Q. Xiao, H. Yang, S. Yuasa,\nand W.X. Zhang. Specially, the authors would like to thank the pioneer of this field, Hellmut Juretschke\nat the age of 92, for surprising us by sending an inspiring note. This work was supported by NSERC, CFI,\nCBCF, UofM, and NSFC (No. 11429401) Grants.\n60References\nReferences\n[1] V. K. Arkad’yev, J. Russ. Phys.-Chem. Soc. 44 (1912) 165.\n[2] J. Dorfmann, Some remarks on the knowledge of the mechanism of magnetic phenomena, Zeitschrift für Phys. 17 (1923)\n98.\n[3] Griffiths J. H. E., Anomalous High-frequency Resistance of Ferromagnetic Metals, Nature 158 (1946) 670. doi:10.1038/\n158670a0 .\n[4] C. Kittel, Interpretation of anomalous larmor frequencies in ferromagnetic resonance experiment, Phys. Rev. 71 (1947)\n270. doi:10.1103/PhysRev.71.270.2 .\n[5] C. Kittel, On the theory of ferromagnetic resonance absorption, Phys. Rev. 73 (1948) 155. doi:10.1103/PhysRev.73.155 .\n[6] S. V. Vonsovskii, Magnetism, John Wiley & Sons, Inc., 1974.\n[7] B. Hillebrands, K. Ounadjela (Eds.), Spin Dynamics in Confined Magnetic Structures I, Topics in Applied Physics,\nSpringer, 2002. doi:10.1007/3-540-40907-6 .\n[8] B. Hillebrands, K. Ounadjela (Eds.), Spin Dynamics in Confined Magnetic Structures II, Springer, 2002. doi:10.1007/\n3-540-46097-7 .\n[9] B. Hillebrands, A. Thiaville (Eds.), Spin Dynamics in Confined Magnetic Structures III, Springer, 2006. doi:10.1007/\n10938171_4 .\n[10] M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau, F. Petroff, Giant Magnetoresistance of (001)Fe/(001)Cr Magnetic\nSuperlattices, Phys. Rev. Lett. 61 (1988) 2472. doi:10.1103/PhysRevLett.61.2472 .\n[11] G. Binasch, P. Grünberg, F. Saurenbach, W. Zinn, Enhanced magnetoresistance in layered magnetic structures with\nantiferromagnetic interlayer exchange, Phys. Rev. B 39 (1989) 4828. doi:10.1103/PhysRevB.39.4828 .\n[12] J. Katine, F. Albert, R. Buhrman, E. Myers, D. C. Ralph, Current-Driven Magnetization Reversal and Spin-Wave Exci-\ntations in Co /Cu /Co Pillars, Phys. Rev. Lett. 84 (2000) 3149. arXiv:cond-mat/9908231v1 ,doi:10.1103/PhysRevLett.\n84.3149.\n[13] Y. Huai, F. Albert, P. Nguyen, M. Pakala, T. Valet, Observation of spin-transfer switching in deep submicron-sized and\nlow-resistance magnetic tunnel junctions, Appl. Phys. Lett. 84 (2004) 3118. arXiv:cond-mat/0504486 ,doi:10.1063/1.\n1707228.\n[14] M. Tsoi, A. G. M. Jansen, J. Bass, W. Chiang, V. Tsoi, P. Wyder, Generation and detection of phase-coherent current-\ndriven magnons in magnetic multilayers, Nature 406 (2000) 46.\n[15] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley, R. J. Schoelkopf, R. A. Buhrman, D. C. Ralph, Microwave\noscillations of a nanomagnet driven by a spin-polarized current., Nature 425 (2003) 380. arXiv:cond-mat/0306259 ,\ndoi:10.1038/nature01967 .\n[16] Y. S. Gui, S. Holland, N. Mecking, C. Hu, Resonances in Ferromagnetic Gratings Detected by Microwave Photoconduc-\ntivity, Phys. Rev. Lett. 95 (2005) 056807. doi:10.1103/PhysRevLett.95.056807 .\n[17] A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, N. Watanabe,\nS. Yuasa, Spin-torque diode effect in magnetic tunnel junctions, Nature 438 (2005) 339. doi:10.1038/nature04207 .\n[18] M. V. Costache, S. M. Watts, M. Sladkov, C. H. Van Der Wal, B. J. Van Wees, Large cone angle magnetization\nprecession of an individual nanopatterned ferromagnet with dc electrical detection, Appl. Phys. Lett. 89 (2006) 232115.\narXiv:cond-mat/0609190 ,doi:10.1063/1.2400058 .\n[19] M. V. Costache, M. Sladkov, S. M. Watts, C. H. Van Der Wal, B. J. Van Wees, Electrical detection of spin pumping due\nto the precessing magnetization of a single ferromagnet, Phys. Rev. Lett. 97 (2006) 216603. arXiv:cond-mat/0609089 ,\ndoi:10.1103/PhysRevLett.97.216603 .\n[20] E. Saitoh, M. Ueda, H. Miyajima, G. Tatara, Conversion of spin current into charge current at room temperature: Inverse\nspin-Hall effect, Appl. Phys. Lett. 88 (2006) 182509. doi:10.1063/1.2199473 .\n[21] J. C. Sankey, P. M. Braganca, A. G. F. Garcia, I. N. Krivorotov, R. A. Buhrman, D. C. Ralph, Spin-transfer-driven\nferromagnetic resonance of individual nanomagnets, Phys. Rev. Lett. 96 (2006) 227601. arXiv:cond-mat/0602105 ,doi:\n10.1103/PhysRevLett.96.227601 .\n[22] H. Kubota, A. Fukushima, K. Yakushiji, T. Nagahama, S. Yuasa, K. Ando, H. Maehara, Y. Nagamine, K. Tsunekawa,\nD. D. Djayaprawira, N. Watanabe, Y. Suzuki, Quantitative measurement of voltage dependence of spin-transfer torque\nin MgO-based magnetic tunnel junctions, Nat. Phys. 4 (2008) 37. doi:10.1038/nphys784 .\n[23] Y. S. Gui, N. Mecking, X. Zhou, G. Williams, C.-M. Hu, Realization of a room-temperature spin dynamo: The spin\nrectification effect, Phys. Rev. Lett. 98 (2007) 107602. arXiv:cond-mat/0701013 ,doi:10.1103/PhysRevLett.98.107602 .\n[24] N. Mecking, Y. S. Gui, C.-M. Hu, Microwave photovoltage and photoresistance effects in ferromagnetic microstrips, Phys.\nRev. B 76 (22) (2007) 224430. arXiv:cond-mat/0710.1974 ,doi:10.1103/PhysRevB.76.224430 .\n[25] A. Yamaguchi, H. Miyajima, T. Ono, Y. Suzuki, S. Yuasa, A. Tulapurkar, Y. Nakatani, Rectification of radio frequency\ncurrent in ferromagnetic nanowire, Appl. Phys. Lett. 90 (2007) 182507. arXiv:cond-mat/0606305 ,doi:10.1063/1.\n2737134.\n[26] Y. S. Gui, N. Mecking, C.-M. Hu, Quantized spin excitations in a ferromagnetic microstrip from microwave photovoltage\nmeasurements, Phys. Rev. Lett. 98 (2007) 217603. arXiv:cond-mat/0704.3397 ,doi:10.1103/PhysRevLett.98.217603 .\n[27] D. Bedau, M. Kläui, S. Krzyk, U. Rüdiger, G. Faini, L. Vila, Detection of current-induced resonance of geometrically\nconfined domain walls, Phys. Rev. Lett. 99 (2007) 146601. doi:10.1103/PhysRevLett.99.146601 .\n[28] J. C. Sankey, Y. T. Cui, R. A. Buhrman, D. C. Ralph, J. Z. Sun, J. C. Slonczewski, Measurement of the Spin-Transfer-\nTorque Vector in Magnetic Tunnel Junctions, Nat. Phys. 4 (2007) 67. arXiv:0705.4207 ,doi:10.1038/nphys783 .\n61[29] L. H. Bai, Y. S. Gui, A. Wirthmann, E. Recksiedler, N. Mecking, C.-M. Hu, Z. H. Chen, S. C. Shen, The rf magnetic-field\nvector detector based on the spin rectification effect, Appl. Phys. Lett. 92 (2008) 032504. doi:10.1063/1.2837180 .\n[30] X.Hui, A.Wirthmann, Y.S.Gui, Y.Tian, X.F.Jin, Z.H.Chen, S.C.Shen, C.-M.Hu, Electricdetectionofferromagnetic\nresonance in single crystal iron film, Appl. Phys. Lett. 93 (2008) 232502. doi:10.1063/1.3029744 .\n[31] A. Wirthmann, X. Hui, N. Mecking, Y. S. Gui, T. Chakraborty, C.-M. Hu, M. Reinwald, C. Schüller, W. Wegscheider,\nBroadband electrical detection of spin excitations in Ga 0:98Mn0:02As using a photovoltage technique, Appl. Phys. Lett.\n92 (2008) 232106. arXiv:0806.0785 ,doi:10.1063/1.2943280 .\n[32] Y. S. Gui, A. Wirthmann, N. Mecking, C.-M. Hu, Direct measurement of nonlinear ferromagnetic damping via the\nintrinsic foldover effect, Phys. Rev. B 80 (2009) 060402(R). doi:10.1103/PhysRevB.80.060402 .\n[33] A. Wirthmann, X. Fan, Y. S. Gui, K. Martens, G. Williams, J. Dietrich, G. E. Bridges, C.-M. Hu, Direct phase probing\nand mapping via spintronic michelson interferometry, Phys. Rev. Lett. 105 (2010) 017202. doi:10.1103/PhysRevLett.\n105.017202 .\n[34] C. T. Boone, J. A. Katine, J. R. Childress, V. Tiberkevich, A. Slavin, J. Zhu, X. Cheng, I. N. Krivorotov, Resonant\nNonlinear Damping of Quantized Spin Waves in Ferromagnetic Nanowires: A Spin Torque Ferromagnetic Resonance\nStudy, Phys. Rev. Lett. 103 (2009) 167601. arXiv:0905.4699 ,doi:10.1103/PhysRevLett.103.167601 .\n[35] S. Bonetti, V. Tiberkevich, G. Consolo, G. Finocchio, P. Muduli, F. Mancoff, A. Slavin, J. Åkerman, Experimental\nEvidence of Self-Localized and Propagating Spin Wave Modes in Obliquely Magnetized Current-Driven Nanocontacts,\nPhys. Rev. Lett. 105 (2010) 217204. arXiv:0909.3331 ,doi:10.1103/PhysRevLett.105.217204 .\n[36] V.A.Atsarkin, V.V.Demidov, L.V.Levkin, A.M.Petrzhik, Effectofresonancemicrowaveirradiationonmanganitefilm\nconductivity around the ferromagnetic transition, Phys. Rev. B 82 (2010) 144414. doi:10.1103/PhysRevB.82.144414 .\n[37] O. Mosendz, V. Vlaminck, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, A. Hoffmann, Detection and\nquantification of inverse spin Hall effect from spin pumping in permalloy/normal metal bilayers, Phys. Rev. B 82 (2010)\n214403. arXiv:1009.5089 ,doi:10.1103/PhysRevB.82.214403 .\n[38] P. Saraiva, A. Nogaret, J. C. Portal, H. E. Beere, D. A. Ritchie, Dipolar spin waves of lateral magnetic superlattices,\nPhys. Rev. B 82 (2010) 224417. doi:10.1103/PhysRevB.82.224417 .\n[39] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi,\nS. Maekawa, E. Saitoh, Transmission of electrical signals by spin-wave interconversion in a magnetic insulator, Nature\n464 (2010) 262. doi:10.1038/nature08876 .\n[40] C. W. Sandweg, Y. Kajiwara, K. Ando, E. Saitoh, B. Hillebrands, Enhancement of the spin pumping efficiency by spin\nwave mode selection, Appl. Phys. Lett. 97 (2010) 252504. arXiv:1008.4714v1 ,doi:10.1063/1.3528207 .\n[41] L. Liu, T. Moriyama, D. C. Ralph, R. A. Buhrman, Spin-torque ferromagnetic resonance induced by the spin Hall effect,\nPhys. Rev. Lett. 106 (2011) 036601. arXiv:1011.2788 ,doi:10.1103/PhysRevLett.106.036601 .\n[42] A. Azevedo, L. H. Vilela-Leão, R. L. Rodríguez-Suárez, A. F. Lacerda Santos, S. M. Rezende, Spin pumping and\nanisotropic magnetoresistance voltages in magnetic bilayers: Theory and experiment, Phys. Rev. B 83 (2011) 144402.\ndoi:10.1103/PhysRevB.83.144402 .\n[43] M. Harder, Z. X. Cao, Y. S. Gui, X. L. Fan, C.-M. Hu, Analysis of the line shape of electrically detected ferromagnetic\nresonance, Phys. Rev. B 84 (2011) 054423. arXiv:1105.3236 ,doi:10.1103/PhysRevB.84.054423 .\n[44] Z. Feng, J. Hu, L. Sun, B. You, D. Wu, J. Du, W. Zhang, A. Hu, Y. Yang, D. M. Tang, B. S. Zhang, H. F. Ding,\nSpin Hall angle quantification from spin pumping and microwave photoresistance, Phys. Rev. B 85 (21) (2012) 214423.\ndoi:10.1103/PhysRevB.85.214423 .\n[45] H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, F. Y. Yang, Large spin pumping from epitaxial Y 3Fe5O12thin\nfilms to Pt and W layers, Phys. Rev. B 88 (2013) 100406. doi:10.1103/PhysRevB.88.100406 .\n[46] P. Deorani, J. Son, K. Banerjee, N. Koirala, M. Brahlek, S. Oh, H. Yang, Observation of inverse spin Hall effect in\nbismuth selenide, Phys. Rev. B 90 (2014) 094403. arXiv:1404.1146 ,doi:10.1103/PhysRevB.90.094403 .\n[47] W. T. Soh, B. Peng, C. K. Ong, An angular analysis to separate spin pumping-induced inverse spin Hall effect from spin\nrectification in a Py/Pt bilayer, J. Phys. D. Appl. Phys. 47 (2014) 285001. doi:10.1088/0022-3727/47/28/285001 .\n[48] Y. Wang, P. Deorani, X. Qiu, J. H. Kwon, H. Yang, Determination of intrinsic spin Hall angle in Pt, Appl. Phys. Lett.\n105 (2014) 152412. arXiv:1410.1601 ,doi:10.1063/1.4898593 .\n[49] E. Becquerel, Recherches sur les effets de la radiation chimique de la lumière solaire, au moyen des courant électriques,\nC. R. Acad. Sci. 9 (1839) 145.\n[50] R. Williams, Bequerel photovoltaic effect in binary compounds, J. Chem. Phys. 32 (1960) 1505. doi:10.1063/1.1730950 .\n[51] M. Grätzel, Photoelectochemical cells, Nature 414 (2001) 338. doi:10.1038/35104607 .\n[52] R. L. Petritz, Theory of photoconductivity in semiconductor films, Phys. Rev. 104 (1956) 1508. doi:10.1103/PhysRev.\n104.1508 .\n[53] A. Rogalski, Infrared detectors: An overview, Infrared Phys. Technol. 43 (2002) 187. doi:10.1016/S1350-4495(02)\n00140-8.\n[54] G. Konstantatos, I. Howard, A. Fischer, S. Hoogland, J. Clifford, E. Klem, L. Levina, E. H. Sargent, Ultrasensitive\nsolution-cast quantum dot photodetectors., Nature 442 (2006) 180. doi:10.1038/nature04855 .\n[55] C. Soci, A. Zhang, B. Xiang, S. A. Dayeh, D. P. R. Aplin, J. Park, X. Y. Bao, Y. H. Lo, D. Wang, ZnO nanowire UV\nphotodetectors with high internal gain, Nano Lett. 7 (2007) 1003. doi:10.1021/nl070111x .\n[56] G. F. Knoll, Radiation Detection and Measurement, 4th Edition, John Wiley & Sons, Inc., 2010.\n[57] J. C. Maan, T. Englert, D. C. Tsui, A. C. Gossard, Observation of cyclotron resonance in the photoconductivity of\ntwo-dimensional electrons, Appl. Phys. Lett. 40 (1982) 609. doi:10.1063/1.93197 .\n[58] H. Schneider, F. Fuchs, B. Dischler, J. D. Ralston, P. Koidl, Intersubband absorption and infrared photodetection at 3.5\nand 4.2\u0016m in GaAs quantum wells, Appl. Phys. Lett. 58 (1991) 2234. doi:10.1063/1.104936 .\n62[59] S.Holland,K.Bittkau,C.-M.Hu,C.Heyn,D.Heitmann,Filling-factor-dependentoscillationsofthecoupledintersubband\nplasmon-LOphononmodeinatunnelingcoupled-bilayersystem, Phys.Rev.B66(2002)233302. doi:10.1103/PhysRevB.\n66.233302 .\n[60] C.-M. Hu, C. Zehnder, C. Heyn, D. Heitmann, THz collective spin-flip excitation of a two-dimensional electron system,\nPhys. Rev. B 67 (2003) 201302. doi:10.1103/PhysRevB.67.201302 .\n[61] S. D. Ganichev, E. L. Ivchenko, V. V. Bel’kov, S. A. Tarasenko, M. Sollinger, D. Weiss, W. Wegscheider, W. Prettl,\nSpin-galvanic effect, Nature 417 (2002) 153. doi:10.1038/417153a .\n[62] W. G. Egan, H. J. Juretschke, DC Detection of Ferromagnetic Resonance in Thin Nickel Films, J. Appl. Phys. 34 (1963)\n1477. doi:10.1063/1.1729604 .\n[63] M. Bass, P. A. Franken, F. Ward, G. Weinreich, Optical Rectification, Phys. Rev. Lett. 9 (1962) 446. doi:10.1103/\nPhysRevLett.9.446 .\n[64] Y. R. Shen, The Principles of Nonlinear Optics, John Wiley & Sons, Inc., 2002.\n[65] C.-M. Hu, Dawn of Cavity Spintronics, Phys. in Can. 72 (2016) 76. arXiv:1508.01966 .\n[66] J. D. Jackson, Classical Electrodynamics, 3rd Edition, John Wiley & Sons, Inc., 1999.\n[67] H.Kurebayashi, J.Sinova, D.Fang, A.C.Irvine, T.D.Skinner, J.Wunderlich, V.Novák, R.P.Campion, B.L.Gallagher,\nE. K. Vehstedt, L. P. Zârbo, K. Výborný, A. J. Ferguson, T. Jungwirth, An antidamping spin-orbit torque originating\nfrom the Berry curvature, Nat. Nanotechnol. 9 (2014) 211. arXiv:1306.1893 ,doi:10.1038/nnano.2014.15 .\n[68] J. Stöhr, H. C. Siegmann, Magnetism From Fundamentals to Nanoscale Dynamics, Springer, 2006. doi:10.1007/\n978-3-540-30283-4 .\n[69] W. Thomson, On the Electro-Dynamic Qualities of Metals:–Effects of Magnetization on the Electric Conductivity of\nNickel and of Iron, Proc. R. Soc. London 8 (1856) 546. doi:10.1098/rspl.1856.0144 .\n[70] J. Xiao, J. Jiang, C. Chien, Giant magnetoresistance in nonmultilayer magnetic systems, Phys. Rev. Lett. 68 (1992)\n3749. doi:10.1103/PhysRevLett.68.3749 .\n[71] M. Jullière, Tunneling between ferromagnetic films, Phys. Lett. A 54 (1975) 225. doi:10.1016/0375-9601(75)90174-7 .\n[72] S. Ikeda, J. Hayakawa, Y. Ashizawa, Y. M. Lee, K. Miura, H. Hasegawa, M. Tsunoda, F. Matsukura, H. Ohno, Tunnel\nmagnetoresistance of 604% at 300 K by suppression of Ta diffusion in CoFeB/MgO/CoFeB pseudo-spin-valves annealed\nat high temperature, Appl. Phys. Lett. 93 (2008) 082508. doi:10.1063/1.2976435 .\n[73] S. Parkin, C. Kaiser, A. Panchula, K. Roche, M. Samant, Magnetically engineered spintronic sensors and memory, Proc.\nIEEE 91 (2003) 661. doi:10.1109/JPROC.2003.811807 .\n[74] E. Tsymbal, D. G. Pettifor, Perspectives of giant magnetoresistance, in: H. Ehrenreich, F. Spaepen (Eds.), Solid State\nPhys., Vol. 56, Academic Press, 2001, pp. 113–237. doi:10.1016/s0081-1947(01)80019-9 .\n[75] N. F. Mott, The Resistance and Thermoelectric Properties of the Transition Metals, Proc. R. Soc. A Math. Phys. Eng.\nSci. 156 (888) (1936) 368. doi:10.1098/rspa.1936.0154 .\n[76] A. Fert, Historical Overview: From Electron Transport in Magnetic Materials to Spintronics, in: E. Y. Tsymbal, I. Žutić\n(Eds.), Handb. Spin Transp. Magn., CRC Press, Boca Raton, 2011, pp. 3–17. doi:10.1201/b11086 .\n[77] A. A. Kovalev, A. Brataas, G. E. W. Bauer, Spin transfer in diffusive ferromagnet-normal metal systems with spin-flip\nscattering, Phys. Rev. B 66 (2002) 224424. arXiv:cond-mat/0207458 ,doi:10.1103/PhysRevB.66.224424 .\n[78] J. Barnaś, A. Fert, M. Gmitra, I. Weymann, V. K. Dugaev, From giant magnetoresistance to current-induced switching\nby spin transfer, Phys. Rev. B 72 (2005) 024426. doi:10.1103/PhysRevB.72.024426 .\n[79] T. McGuire, R. Potter, Anisotropic magnetoresistance in ferromagnetic 3d alloys, IEEE Trans. Magn. 11 (1975) 1018.\ndoi:10.1109/TMAG.1975.1058782 .\n[80] R.Ramos, S.K.Arora, I.V.Shvets, AnomalousanisotropicmagnetoresistanceinepitaxialFe 3O4thinfilmsonMgO(001),\nPhys. Rev. B 78 (2008) 214402. doi:10.1103/PhysRevB.78.214402 .\n[81] C. R. Hu, J. Zhu, G. Chen, J. X. Li, Y. Z. Wu, Direct comparison of anisotropic magnetoresistance and planar Hall effect\nin epitaxial Fe 3O4thin films, Phys. Lett. A 376 (2012) 3317. doi:10.1016/j.physleta.2012.08.009 .\n[82] Z. Ding, J. X. Li, J. Zhu, T. P. Ma, C. Won, Y. Z. Wu, Three-dimensional mapping of the anisotropic magnetoresistance\nin Fe 3O4single crystal thin films, J. Appl. Phys. 113 (2013) 17B103. doi:10.1063/1.4796178 .\n[83] X. Xiao, J. X. Li, Z. Ding, Y. Z. Wu, Four-fold symmetric anisotropic magnetoresistance of single-crystalline Ni(001)\nfilm, J. Appl. Phys. 118 (2015) 203905. doi:10.1063/1.4936175 .\n[84] B. Chen, Z. Ding, D. Ma, J. Li, X. Xiao, Y. Wu, Asymmetric anisotropic magnetoresistance in epitaxial Fe 3O4thin\nfilms, Phys. Lett. A 379 (2015) 1780. doi:10.1016/j.physleta.2015.05.006 .\n[85] X. Xiao, J. H. Liang, B. L. Chen, J. X. Li, D. H. Ma, Z. Ding, Y. Z. Wu, Current-direction dependence of the transport\npropertiesinsingle-crystallineface-centered-cubiccobaltfilms,J.Appl.Phys.118(2015)043908. doi:10.1063/1.4927620 .\n[86] R. Potter, Magnetoresistance anisotropy in ferromagnetic NiCu alloys, Phys. Rev. B 10 (1974) 4626. doi:10.1103/\nPhysRevB.10.4626 .\n[87] O. Jaoul, I. Campbell, A. Fert, Spontaneous resistivity anisotropy in Ni alloys, J. Magn. Magn. Mater. 5 (1977) 23.\ndoi:10.1016/0304-8853(77)90193-7 .\n[88] A. F. Mayadas, Resistivity of Permalloy thin films, J. Appl. Phys. 45 (1974) 2780. doi:10.1063/1.1663668 .\n[89] J. Smit, Magnetoresistance of ferromagnetic metals and alloys at low temperatures, Physica 17 (1951) 612. doi:10.1016/\n0031-8914(51)90117-6 .\n[90] H. J. Juretschke, Electromagnetic Theory of dc Effects in Ferromagnetic Resonance, J. Appl. Phys. 31 (1960) 1401.\ndoi:10.1063/1.1735851 .\n[91] J.-P. Jan, Galvanomagnetic and Thermomagnetic Effects in Metals, Solid State Phys. 5 (1956) 1–96. doi:10.1016/\ns0081-1947(08)60101-0 .\n[92] Y. Zhang, X. R. Wang, H. W. Zhang, Extraordinary galvanomagnetic effects in polycrystalline magnetic films, Europhys.\n63Lett. 113 (2016) 47003. doi:10.1209/0295-5075/113/47003 .\n[93] D. Edwards, J. Mathon, R. Muniz, A resistor network theory of the giant magnetoresistance in magnetic superlattices,\nIEEE Trans. Magn. 27 (1991) 3548. doi:10.1109/20.102926 .\n[94] J. George, L. Pereira, A. Barthélémy, F. Petroff, L. Steren, J. Duvail, A. Fert, R. Loloee, P. Holody, P. Schroeder,\nInverse spin-valve-type magnetoresistance in spin engineered multilayered structures, Phys. Rev. Lett. 72 (1994) 408.\ndoi:10.1103/PhysRevLett.72.408 .\n[95] J. Bass, Giant Magnetoresistance: Experiment, in: E. Y. Tsymbal, I. Žutić (Eds.), Handb. Spin Transp. Magn., CRC\nPress, Boca Raton, 2012, pp. 69–93. doi:10.1201/b11086-7 .\n[96] P. Grünberg, R. Schreiber, Y. Pang, Layered Magnetic Structures: Evidence for Antiferromagnetic Coupling of Fe Layers\nacross Cr Interlayers, Phys. Rev. Lett. 57 (1986) 2442. doi:10.1103/PhysRevLett.57.2442 .\n[97] S. S. P. Parkin, N. More, K. P. Roche, Oscillations in exchange coupling and magnetoresistance in metallic superlattice\nstructures: Co/Ru, Co/Cr, and Fe/Cr, Phys. Rev. Lett. 64 (1990) 2304. doi:10.1103/PhysRevLett.64.2304 .\n[98] S. S. P. Parkin, Systematic variation of the strength and oscillation period of indirect magnetic exchange coupling through\nthe 3d, 4d, and 5d transition metals, Phys. Rev. Lett. 67 (1991) 3598. doi:10.1103/PhysRevLett.67.3598 .\n[99] S. S. P. Parkin, R. Bhadra, K. Roche, Oscillatory magnetic exchange coupling through thin copper layers, Phys. Rev.\nLett. 66 (1991) 2152. doi:10.1103/PhysRevLett.66.2152 .\n[100] D. Mosca, F. Petroff, A. Fert, P. Schroeder, W. Pratt, R. Laloee, Oscillatory interlayer coupling and giant magnetoresis-\ntance in Co/Cu multilayers, J. Magn. Magn. Mater. 94 (1991) L1. doi:10.1016/0304-8853(91)90102-G .\n[101] J. Barnaś, A. Fuss, R. Camley, P. Grünberg, W. Zinn, Novel magnetoresistance effect in layered magnetic structures:\nTheory and experiment, Phys. Rev. B 42 (1990) 8110. doi:10.1103/PhysRevB.42.8110 .\n[102] T. Shinjo, H. Yamamoto, Large Magnetoresistance of Field-Induced Giant Ferrimagnetic Multilayers, J. Phys. Soc. Japan\n59 (1990) 3061. doi:10.1143/JPSJ.59.3061 .\n[103] C. Dupas, P. Beauvillain, C. Chappert, J. P. Renard, F. Trigui, P. Veillet, E. Vélu, D. Renard, Very large magne-\ntoresistance effects induced by antiparallel magnetization in two ultrathin cobalt films, J. Appl. Phys. 67 (1990) 5680.\ndoi:10.1063/1.345925 .\n[104] B. Dieny, V. Speriosu, S. Parkin, B. Gurney, D. Wilhoit, D. Mauri, Giant magnetoresistive in soft ferromagnetic multi-\nlayers, Phys. Rev. B 43 (1991) 1297. doi:10.1103/PhysRevB.43.1297 .\n[105] T. Ono, T. Shinjo, Magnetoresistance of Multilayers Prepared on Microstructured Substrates, J. Phys. Soc. Japan 64\n(1995) 363. doi:10.1143/JPSJ.64.363 .\n[106] M. A. M. Gijs, M. T. Johnson, A. Reinders, P. E. Huisman, R. J. M. van de Veerdonk, S. K. J. Lenczowski, R. M. J.\nvan Gansewinkel, Perpendicular giant magnetoresistance of Co/Cu multilayers deposited under an angle on grooved\nsubstrates, Appl. Phys. Lett. 66 (1995) 1839. doi:10.1063/1.113339 .\n[107] T. Valet, A. Fert, Theory of the perpendicular magnetoresistance in magnetic multilayers, Phys. Rev. B 48 (1993) 7099.\ndoi:10.1103/PhysRevB.48.7099 .\n[108] W. Pratt, S.-F. Lee, J. Slaughter, R. Loloee, P. Schroeder, J. Bass, Perpendicular giant magnetoresistances of Ag/Co\nmultilayers, Phys. Rev. Lett. 66 (1991) 3060. doi:10.1103/PhysRevLett.66.3060 .\n[109] J. W. Jung, Y. Sakuraba, T. T. Sasaki, Y. Miura, K. Hono, Enhancement of magnetoresistance by inserting thin NiAl\nlayers at the interfaces in Co 2FeGa 0:5Ge0:5/Ag/Co 2FeGa 0:5Ge0:5current-perpendicular-to-plane pseudo spin valves,\nAppl. Phys. Lett. 108 (2016) 102408. doi:10.1063/1.4943640 .\n[110] A. Chaiken, G. A. Prinz, J. J. Krebs, Magnetotransport study of Fe-Cr-Fe sandwiches grown on ZnSe(100), J. Appl.\nPhys. 67 (1990) 4892. doi:10.1063/1.344722 .\n[111] P. Dauguet, P. Gandit, J. Chaussy, S. Lee, A. Fert, P. Holody, Angular dependence of the perpendicular giant magne-\ntoresistance of multilayers, Phys. Rev. B 54 (1996) 1083. doi:10.1103/PhysRevB.54.1083 .\n[112] A. Vedyayev, B. Dieny, N. Ryzhanova, J. B. Genin, C. Cowache, Angular Dependence of Giant Magnetoresistance in\nMagnetic Multilayered Structures, Europhys. Lett. 25 (1994) 465. doi:10.1209/0295-5075/25/6/012 .\n[113] A. Vedyayev, N. Ryzhanova, B. Dieny, P. Dauguet, P. Gandit, J. Chaussy, Angular variation of giant magnetoresistance\nfor current perpendicular to the plane of the layers, Phys. Rev. B 55 (1997) 3728. doi:10.1103/PhysRevB.55.3728 .\n[114] C. Blaas, P. Weinberger, L. Szunyogh, J. Kudrnovský, V. Drchal, P. M. Levy, C. Sommers, On the orientational depen-\ndence of giant magnetoresistance, Eur. Phys. J. B 9 (1999) 245. doi:10.1007/s100510050763 .\n[115] S. S. P. Parkin, Giant Magnetoresistance in Magnetic Nanostructures, Annu. Rev. Mater. Sci. 25 (1995) 357. doi:\n10.1146/annurev.ms.25.080195.002041 .\n[116] L. B. Steren, R. Morel, A. Barthélémy, F. Petroff, A. Fert, P. Holody, R. Loloee, P. Schroeder, Giant magnetoresistance\nin hybrid magnetic nanostructures, J. Magn. Magn. Mater. 140 (1995) 495. doi:10.1016/0304-8853(94)01529-5 .\n[117] S. Yuasa, T. Nagahama, A. Fukushima, Y. Suzuki, K. Ando, Giant room-temperature magnetoresistance in single-crystal\nFe/MgO/Fe magnetic tunnel junctions, Nat. Mater. 3 (2004) 868. doi:10.1038/nmat1257 .\n[118] S. Gider, B.-U. Runge, A. C. Marley, S. S. P. Parkin, The Magnetic Stability of Spin-Dependent Tunneling Devices,\nScience 281 (1998) 797. doi:10.1126/science.281.5378.797 .\n[119] S. Maekawa, U. Gafvert, Electron tunneling between ferromagnetic films, IEEE Trans. Magn. 18 (1982) 707. doi:\n10.1109/TMAG.1982.1061834 .\n[120] J. S. Moodera, L. R. Kinder, T. M. Wong, R. Meservey, Large Magnetoresistance at Room Temperature in Ferromagnetic\nThin Film Tunnel Junctions, Phys. Rev. Lett. 74 (1995) 3273. doi:10.1103/PhysRevLett.74.3273 .\n[121] T. Miyazaki, Giant magnetic tunneling effect in Fe/Al 2O3/Fe junction, J. Magn. Magn. Mater. 139 (1995) L231. doi:\n10.1016/0304-8853(94)01648-8 .\n[122] S. S. P. Parkin, C. Kaiser, A. Panchula, P. M. Rice, B. Hughes, M. Samant, S.-H. Yang, Giant tunnelling magnetoresis-\ntance at room temperature with MgO (100) tunnel barriers, Nat. Mater. 3 (2004) 862. doi:10.1038/nmat1256 .\n64[123] Y.M.Lee,J.Hayakawa,S.Ikeda,F.Matsukura,H.Ohno,Effectofelectrodecompositiononthetunnelmagnetoresistance\nof pseudo-spin-valve magnetic tunnel junction with a MgO tunnel barrier, Appl. Phys. Lett. 90 (2007) 212507. doi:\n10.1063/1.2742576 .\n[124] R. Meservey, P. Tedrow, Spin-polarized electron tunneling, Phys. Rep. 238 (1994) 173. doi:10.1016/0370-1573(94)\n90105-8.\n[125] M. Bowen, M. Bibes, A. Barthélémy, J. P. Contour, A. Anane, Y. Lemaître, A. Fert, Nearly total spin polarization\nin La 2=3Sr1=3MnO 3from tunneling experiments, Appl. Phys. Lett. 82 (2003) 233. arXiv:cond-mat/0211158 ,doi:\n10.1063/1.1534619 .\n[126] T. Ishikawa, T. Marukame, H. Kijima, K. I. Matsuda, T. Uemura, M. Arita, M. Yamamoto, Spin-dependent tunneling\ncharacteristics of fully epitaxial magnetic tunneling junctions with a full-Heusler alloy Co 2MnSi thin film and a MgO\ntunnel barrier, Appl. Phys. Lett. 89 (2006) 192505. doi:10.1063/1.2378397 .\n[127] S. Yuasa, Tunneling Magnetoresistance: Experiment (MgO Magnetic Tunnel Junctions), in: E. Y. Tsymbal, I. Žutić\n(Eds.), Handb. Spin Transp. Magn., CRC Press, Boca Raton, 2012, pp. 217–231. doi:10.1201/b11086-15 .\n[128] P. R. LeClair, J. S. Moodera, Tunneling Magnetoresistance: Experiment (Non-MgO Magnetic Tunnel Junctions), in:\nE. Y. Tsymbal, I. Žutić (Eds.), Handb. Spin Transp. Magn., CRC Press, Boca Raton, 2012, pp. 197–216. doi:10.1201/\nb11086-14 .\n[129] K. D. Belashchenko, E. Y. Tsymbal, Tunneling Magnetoresistance: Theory, in: E. Y. Tsymbal, I. Žutić (Eds.), Handb.\nSpin Transp. Magn., CRC Press, Boca Raton, 2012, pp. 233–250. doi:10.1201/b11086-16 .\n[130] D. D. Djayaprawira, K. Tsunekawa, M. Nagai, H. Maehara, S. Yamagata, N. Watanabe, S. Yuasa, Y. Suzuki, K. Ando,\n230% room-temperature magnetoresistance in CoFeB/MgO/CoFeB magnetic tunnel junctions, Appl. Phys. Lett. 86\n(2005) 092502. doi:10.1063/1.1871344 .\n[131] J. S. Moodera, L. R. Kinder, Ferromagnetic-insulator-ferromagnetic tunneling: Spin-dependent tunneling and large\nmagnetoresistance in trilayer junctions, J. Appl. Phys. 79 (1996) 4724. doi:10.1063/1.361653 .\n[132] H. Jaffrès, D. Lacour, F. Nguyen Van Dau, J. Briatico, F. Petroff, A. Vaurès, Angular dependence of the tunnel magne-\ntoresistance in transition-metal-based junctions, Phys. Rev. B 64 (2001) 064427. doi:10.1103/PhysRevB.64.064427 .\n[133] J. C. Slonczewski, Conductance and exchange coupling of two ferromagnets separated by a tunneling barrier, Phys. Rev.\nB 39 (1989) 6995. doi:10.1103/PhysRevB.39.6995 .\n[134] Y. Suzuki, H. Kubota, Spin-torque diode effect and its application, J. Phys. Soc. Japan 77 (2008) 031002. doi:10.1143/\nJPSJ.77.031002 .\n[135] H. Nakayama, M. Althammer, Y.-T. Chen, K. Uchida, Y. Kajiwara, D. Kikuchi, T. Ohtani, S. Geprägs, M. Opel,\nS. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. Goennenwein, E. Saitoh, Spin Hall Magnetoresistance Induced by\na Nonequilibrium Proximity Effect, Phys. Rev. Lett. 110 (2013) 206601. arXiv:1211.0098 ,doi:10.1103/PhysRevLett.\n110.206601 .\n[136] Y. M. Lu, J. W. Cai, S. Y. Huang, D. Qu, B. F. Miao, C. L. Chien, Hybrid magnetoresistance in the proximity of a\nferromagnet, Phys. Rev. B 87 (2013) 220409(R). doi:10.1103/PhysRevB.87.220409 .\n[137] Y.-T. Chen, S. Takahashi, H. Nakayama, M. Althammer, S. T. B. Goennenwein, E. Saitoh, G. Bauer, Theory of spin\nHall magnetoresistance, Phys. Rev. B 87 (2013) 144411. arXiv:1302.1352 ,doi:10.1103/PhysRevB.87.144411 .\n[138] C. Hahn, G. de Loubens, O. Klein, M. Viret, V. V. Naletov, J. Ben Youssef, Comparative measurements of inverse\nspin Hall effects and magnetoresistance in YIG/Pt and YIG/Ta, Phys. Rev. B 87 (2013) 174417. arXiv:1302.4416 ,\ndoi:10.1103/PhysRevB.87.174417 .\n[139] M. Althammer, S. Meyer, H. Nakayama, M. Schreier, S. Altmannshofer, M. Weiler, H. Huebl, S. Geprägs, M. Opel,\nR. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M. Schmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen, G. E. W. Bauer,\nE.Saitoh, S.T.B.Goennenwein, QuantitativestudyofthespinHallmagnetoresistanceinferromagneticinsulator/normal\nmetal hybrids, Phys. Rev. B 87 (2013) 224401. arXiv:1304.6151 ,doi:10.1103/PhysRevB.87.224401 .\n[140] N. Vlietstra, J. Shan, V. Castel, B. J. van Wees, J. Ben Youssef, Spin-Hall magnetoresistance in platinum on yttrium\niron garnet: Dependence on platinum thickness and in-plane/out-of-plane magnetization, Phys. Rev. B 87 (2013) 184421.\narXiv:1301.3266 ,doi:10.1103/PhysRevB.87.184421 .\n[141] M. Weiler, M. Althammer, M. Schreier, J. Lotze, M. Pernpeintner, S. Meyer, H. Huebl, R. Gross, A. Kamra, J. Xiao,\nY. T. Chen, H. Jiao, G. E. W. Bauer, S. T. B. Goennenwein, Experimental test of the spin mixing interface conductivity\nconcept, Phys. Rev. Lett. 111 (2013) 176601. arXiv:1306.5012 ,doi:10.1103/PhysRevLett.111.176601 .\n[142] N. Vlietstra, J. Shan, V. Castel, J. Ben Youssef, G. E. W. Bauer, B. J. van Wees, Exchange magnetic field torques in\nYIG/Pt bilayers observed by the spin-Hall magnetoresistance, Appl. Phys. Lett. 103 (2013) 032401. arXiv:1305.3117 ,\ndoi:10.1063/1.4813760 .\n[143] B. F. Miao, S. Y. Huang, D. Qu, C. L. Chien, Physical Origins of the New Magnetoresistance in Pt/YIG, Phys. Rev.\nLett. 112 (2014) 236601. doi:10.1103/PhysRevLett.112.236601 .\n[144] S.R.Marmion,M.Ali,M.McLaren,D.A.Williams,B.J.Hickey,TemperaturedependenceofspinHallmagnetoresistance\nin thin YIG/Pt films, Phys. Rev. B 89 (2014) 220404(R). doi:10.1103/PhysRevB.89.220404 .\n[145] M. Isasa, A. Bedoya-Pinto, S. Vélez, F. Golmar, F. Sánchez, L. E. Hueso, J. Fontcuberta, F. Casanova, Spin Hall\nmagnetoresistance at Pt/CoFe 2O4interfaces and texture effects, Appl. Phys. Lett. 105 (2014) 142402. doi:10.1063/1.\n4897544.\n[146] V. L. Grigoryan, W. Guo, G. E. W. Bauer, J. Xiao, Intrinsic magnetoresistance in metal films on ferromagnetic insulators,\nPhys. Rev. B 90 (2014) 161412. arXiv:1407.3571 ,doi:10.1103/PhysRevB.90.161412 .\n[147] N. Vlietstra, J. Shan, B. J. van Wees, M. Isasa, F. Casanova, J. Ben Youssef, Simultaneous detection of the spin-Hall\nmagnetoresistance and the spin-Seebeck effect in platinum and tantalum on yttrium iron garnet, Phys. Rev. B 90 (2014)\n174436. arXiv:1410.0551 ,doi:10.1103/PhysRevB.90.174436 .\n65[148] T. Lin, C. Tang, H. M. Alyahayaei, J. Shi, Experimental Investigation of the Nature of the Magnetoresistance Effects in\nPd-YIG Hybrid Structures, Phys. Rev. Lett. 113 (2014) 037203. doi:10.1103/PhysRevLett.113.037203 .\n[149] A. Brataas, Insulating Magnets Control Neighbor’s Conduction, Physics (College. Park. Md). 6 (2013) 56. doi:10.1103/\nPhysics.6.56 .\n[150] M. I. D’yakonov, V. I. Perel, Possibility of orienting electron spins with current, JETP Lett. 13 (1971) 467.\n[151] J. E. Hirsch, Spin Hall effect, Phys. Rev. Lett 83 (1999) 1834. arXiv:cond-mat/9906160 ,doi:10.1103/PhysRevLett.83.\n1834.\n[152] A. Hoffmann, Spin Hall Effects in Metals, IEEE Trans. Magn. 49 (2013) 5172. doi:10.1109/TMAG.2013.2262947 .\n[153] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, T. Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87 (2015) 1213.\narXiv:1411.3249v1 ,doi:10.1103/RevModPhys.87.1213 .\n[154] I. A. Campbell, A. Fert, Transport Properties of Ferromagnets, in: E. P. Wohlfarth (Ed.), Handb. Ferromagn. Mater.,\nVol. 3, 1982, Ch. 9, pp. 747–804. doi:10.1016/s1567-2719(82)03012-1 .\n[155] T. Rijks, S. Lenczowski, R. Coehoorn, W. de Jonge, In-plane and out-of-plane anisotropic magnetoresistance in Ni 80Fe20\nthin films, Phys. Rev. B 56 (1997) 362. doi:10.1103/PhysRevB.56.362 .\n[156] W. Gil, D. Görlitz, M. Horisberger, J. Kötzler, Magnetoresistance anisotropy of polycrystalline cobalt films: Geometrical-\nsize and domain effects, Phys. Rev. B 72 (2005) 134401. arXiv:cond-mat/0506232 ,doi:10.1103/PhysRevB.72.134401 .\n[157] A. Kobs, S. Hee, W. Kreuzpaintner, G. Winkler, D. Lott, P. Weinberger, A. Schreyer, H. P. Oepen, Anisotropic Interface\nMagnetoresistance in Pt/Co/Pt Sandwiches, Phys. Rev. Lett. 106 (2011) 217207. doi:10.1103/PhysRevLett.106.217207 .\n[158] L. K. Zou, Y. Zhang, L. Gu, J. W. Cai, L. Sun, Tunable angular-dependent magnetoresistance correlations in magnetic\nfilms and their implications for spin Hall magnetoresistance analysis, Phys. Rev. B 93 (2016) 075309. doi:10.1103/\nPhysRevB.93.075309 .\n[159] Y. M. Lu, Y. Choi, C. M. Ortega, X. M. Cheng, J. W. Cai, S. Y. Huang, L. Sun, C. L. Chien, Pt magnetic polarization\non Y 3Fe5O12and magnetotransport characteristics, Phys. Rev. Lett. 110 (2013) 147207. doi:10.1103/PhysRevLett.\n110.147207 .\n[160] S. Y. Huang, X. Fan, D. Qu, Y. P. Chen, W. G. Wang, J. Wu, T. Y. Chen, J. Q. Xiao, C. L. Chien, Transport magnetic\nproximity effects in platinum, Phys. Rev. Lett. 109 (2012) 107204. doi:10.1103/PhysRevLett.109.107204 .\n[161] L. Landau, E. Lifshitz, On the Theory of the disperison of magnetic permeability in ferromagnetic bodies, Phys. Z.\nSowjetunion 169 (1935) 14.\n[162] F. Bloch, Nuclear induction, Phys. Rev. 70 (1946) 460. doi:10.1103/PhysRev.70.460 .\n[163] T. Gilbert, A Phenomenological Theory of Damping in Ferromagnetic Materials, IEEE Trans. Magn. 40 (2004) 3443.\ndoi:10.1109/TMAG.2004.836740 .\n[164] M. C. Hickey, J. S. Moodera, Origin of intrinsic Gilbert damping, Phys. Rev. Lett. 102 (2009) 137601. arXiv:0812.3184 ,\ndoi:10.1103/PhysRevLett.102.137601 .\n[165] M. S. Sodha, N. C. Srivastava, Microwave Propagation in Ferrimagnetics, Springer, 1981. doi:10.1007/\n978-1-4757-5839-9 .\n[166] Y. S. Gui, L. Bai, C.-M. Hu, The physics of spin rectification and its application, Sci. China Physics, Mech. Astron. 56\n(2012) 124. doi:10.1007/s11433-012-4956-6 .\n[167] Y. Huo, L. H. Bai, P. Hyde, Y. Z. Wu, C.-M. Hu, Spin rectification for collinear and noncollinear magnetization and\nexternal magnetic field configurations, Phys. Rev. B 91 (2015) 174430. arXiv:1502.06859v1 ,doi:10.1103/PhysRevB.91.\n174430.\n[168] V. Flovik, F. Macià, A. D. Kent, E. Wahlström, Eddy current interactions in a ferromagnet-normal metal bilayer\nstructure, and its impact on ferromagnetic resonance lineshapes, J. Appl. Phys. 117 (2015) 143902. arXiv:1412.1385 ,\ndoi:10.1063/1.4917285 .\n[169] V. Flovik, B. H. Pettersen, E. Wahlström, Eddy-current effects on ferromagnetic resonance: Spin wave excitations and\nmicrowave screening effects, J. Appl. Phys. 119 (2016) 163903. arXiv:1602.07463 ,doi:10.1063/1.4948302 .\n[170] H. J. Juretschke, DC detection of spin resonance in thin metallic films, J. Appl. Phys. 34 (1963) 1223. doi:10.1063/1.\n1729445.\n[171] H. Chen, X. Fan, H. Zhou, W. Wang, Y. S. Gui, C.-M. Hu, D. Xue, Spin rectification enabled by anomalous Hall effect,\nJ. Appl. Phys. 113 (2013) 17C732. doi:10.1063/1.4799248 .\n[172] M. H. Seavey, Galvanomagnetic effects in ferromagnetic resonance, J. Appl. Phys. 31 (1960) 216S. doi:10.1063/1.\n1984668.\n[173] W. Heinz, L. Silber, dc Effects in Ferromagnetic Resonance in Thin Ferrite Films, J. Appl. Phys. 33 (1962) 1306.\ndoi:10.1063/1.1728705 .\n[174] W. M. Moller, H. J. Juretschke, Determination of spin-wave boundary conditions by dc effects in spin-wave resonance,\nPhys. Rev. B 2 (1970) 2651. doi:10.1103/PhysRevB.2.2651 .\n[175] M. Toda, Transport Properties of HgCr 2Se4at Ferromagnetic Resonance, Appl. Phys. Lett. 17 (1970) 1. doi:10.1063/\n1.1653231 .\n[176] J. N. Chazalviel, I. Solomon, Experimental evidence of the anomalous Hall effect in a nonmagnetic semiconductor, Phys.\nRev. Lett. 29 (1972) 1676. doi:10.1103/PhysRevLett.29.1676 .\n[177] J. N. Chazalviel, Spin-dependent Hall effect in semiconductors, Phys. Rev. B 11 (1975) 3918. doi:10.1103/PhysRevB.\n11.3918.\n[178] F. J. Wald, H. J. Juretschke, Extraordinary Hall effect in paramagnetic gadolinium, Phys. Rev. B 3 (1971) 2412. doi:\n10.1103/PhysRevB.3.2412 .\n[179] C. L. Yang, H. T. He, L. Ding, L. J. Cui, Y. P. Zeng, J. N. Wang, W. K. Ge, Spectral Dependence of Spin Photocurrent\nand Current-Induced Spin Polarization in an InGaAs/ InAlAs Two-Dimensional Electron Gas, Phys. Rev. Lett. 96 (2006)\n66186605. doi:10.1103/PhysRevLett.96.186605 .\n[180] Y. S. Gui, A. Wirthmann, C.-M. Hu, Foldover ferromagnetic resonance and damping in permalloy microstrips, Phys.\nRev. B 80 (2009) 184422. doi:10.1103/PhysRevB.80.184422 .\n[181] O. Mosendz, J. E. Pearson, F. Y. Fradin, G. E. W. Bauer, S. D. Bader, A. Hoffmann, Quantifying spin Hall angles\nfrom spin pumping: Experiments and theory, Phys. Rev. Lett. 104 (2010) 046601. arXiv:0911.2725 ,doi:10.1103/\nPhysRevLett.104.046601 .\n[182] X. Zhu, M. Harder, A. Wirthmann, B. Zhang, W. Lu, Y. S. Gui, C.-M. Hu, Near-field microwave phase imaging by a\nspintronic sensor, in: J. Chu, Z. Wang (Eds.), Proc. SPIE, Seventh Int. Conf. Thin Film Phys. Appl., Vol. 7995, 2010,\np. 79951X. doi:10.1117/12.888266 .\n[183] Z. X. Cao, M. Harder, L. Fu, B. Zhang, W. Lu, G. E. Bridges, Y. S. Gui, C. M. Hu, Nondestructive two-dimensional phase\nimaging of embedded defects via on-chip spintronic sensor, Appl. Phys. Lett. 100 (2012) 252406. doi:10.1063/1.4729785 .\n[184] X. F. Zhu, M. Harder, A. Wirthmann, B. Zhang, W. Lu, Y. S. Gui, C.-M. Hu, Dielectric measurements via a phase-\nresolved spintronic technique, Phys. Rev. B 83 (2011) 104407. doi:10.1103/PhysRevB.83.104407 .\n[185] Y. Tserkovnyak, A. Brataas, G. Bauer, B. Halperin, Nonlocal magnetization dynamics in ferromagnetic heterostructures,\nRev. Mod. Phys. 77 (2005) 1375. arXiv:0409242 ,doi:10.1103/RevModPhys.77.1375 .\n[186] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, P. J. Kelly, Spin Pumping and Spin Transfer, in: S. Maekawa, S. O.\nValenzuela, E. Saitoh, T. Kimura (Eds.), Spin Curr., Oxford University Press, 2012, Ch. 8, pp. 87–135. arXiv:1108.0385 .\n[187] L. Chen, F. Matsukura, H. Ohno, Direct-current voltages in (Ga,Mn)As structures induced by ferromagnetic resonance.,\nNat. Commun. 4 (2013) 2055. doi:10.1038/ncomms3055 .\n[188] L. Bai, P. Hyde, Y. S. Gui, C.-M. Hu, V. Vlaminck, J. E. Pearson, S. D. Bader, A. Hoffmann, Universal Method for\nSeparating Spin Pumping from Spin Rectification Voltage of Ferromagnetic Resonance, Phys. Rev. Lett. 111 (2013)\n217602. arXiv:1308.4041 ,doi:10.1103/PhysRevLett.111.217602 .\n[189] Y. Tserkovnyak, A. Brataas, G. Bauer, Enhanced Gilbert Damping in Thin Ferromagnetic Films, Phys. Rev. Lett. 88\n(2002) 117601. arXiv:cond-mat/0110247 ,doi:10.1103/PhysRevLett.88.117601 .\n[190] A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, B. I. Halperin, Spin battery operated by ferromagnetic resonance, Phys.\nRev. B. 66 (2002) 060404(R). arXiv:cond-mat/0205028 ,doi:10.1103/PhysRevB.66.060404 .\n[191] A. Janossy, P. Monod, Spin waves for single electrons in paramagnetic metals, Phys. Rev. Lett. 37 (1976) 612. doi:\n10.1103/PhysRevLett.37.612 .\n[192] R. H. Silsbee, A. Janossy, P. Monod, Coupling between ferromagnetic and conduction-spin-resonance modes at a\nferromagnetic-normal-metal interface, Phys. Rev. B 19 (1979) 4382. doi:10.1103/PhysRevB.19.4382 .\n[193] L. Berger, Generation of dc voltages by a magnetic multilayer undergoing ferromagnetic resonance, Phys. Rev. B 59\n(1999) 11465. doi:10.1103/PhysRevB.59.11465 .\n[194] M.Weiler, G.Woltersdorf, M.Althammer, H.Huebl, S.T.B.Goennenwein, SpinPumpingandSpinCurrentsinMagnetic\nInsulators, Solid State Phys. 64 (2013) 123–156. doi:10.1016/B978-0-12-408130-7.00005-8 .\n[195] S. Mizukami, Y. Ando, T. Miyazaki, Ferromagnetic resonance linewidth for NM/80NiFe/NM films (NM=Cu, Ta, Pd and\nPt), J. Magn. Magn. Mater. 226 (2001) 1640. doi:10.1016/S0304-8853(00)01097-0 .\n[196] S. Mizukami, Y. Ando, T. Miyazaki, Effect of spin diffusion on Gilbert damping for a very thin permalloy layer in\nCu/permalloy/Cu/Pt films, Phys. Rev. B 66 (2002) 104413. doi:10.1103/PhysRevB.66.104413 .\n[197] R. Urban, G. Woltersdorf, B. Heinrich, Gilbert damping in single and multilayer ultrathin films: role of interfaces in\nnonlocal spin dynamics, Phys. Rev. Lett. 87 (2001) 217204. doi:10.1103/PhysRevLett.87.217204 .\n[198] B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Urban, G. E. W. Bauer, Dynamic exchange coupling in\nmagnetic bilayers, Phys. Rev. Lett. 90 (2003) 187601. arXiv:cond-mat/0210588 ,doi:10.1103/PhysRevLett.90.187601 .\n[199] K. Lenz, T. Toliński, J. Lindner, E. Kosubek, K. Baberschke, Evidence of spin-pumping effect in the ferromagnetic\nresonance of coupled trilayers, Phys. Rev. B 69 (2004) 144422. doi:10.1103/PhysRevB.69.144422 .\n[200] Y. K. Kato, R. C. Myers, A. C. Gossard, D. D. Awschalom, Observation of the spin Hall effect in semiconductors., Science\n306 (2004) 1910. doi:10.1126/science.1105514 .\n[201] J. Wunderlich, B. Kaestner, J. Sinova, T. Jungwirth, Experimental Observation of the Spin-Hall Effect in a Two-\nDimensional Spin-Orbit Coupled Semiconductor System, Phys. Rev. Lett. 94 (2005) 047204. arXiv:cond-mat/0410295 ,\ndoi:10.1103/PhysRevLett.94.047204 .\n[202] X. Wang, G. E. W. Bauer, B. J. Van Wees, A. Brataas, Y. Tserkovnyak, Voltage generation by ferromagnetic resonance\nat a nonmagnet to ferromagnet contact, Phys. Rev. Lett. 97 (2006) 216602. doi:10.1103/PhysRevLett.97.216602 .\n[203] M. Johnson, R. H. Silsbee, Interfacial charge-spin coupling: Injection and detection of spin magnetization in metals,\nPhys. Rev. Lett. 55 (1985) 1790. doi:10.1103/PhysRevLett.55.1790 .\n[204] L. Liu, R. A. Buhrman, D. C. Ralph, Review and Analysis of Measurements of the Spin Hall Effect in Platinum,\narXiv:1111.3702 .\n[205] M. Obstbaum, M. Härtinger, H. G. Bauer, T. Meier, F. Swientek, C. H. Back, G. Woltersdorf, Inverse spin Hall effect in\nNi81Fe19/normal-metal bilayers, Phys. Rev. B 89 (2014) 060407(R). doi:10.1103/PhysRevB.89.060407 .\n[206] A. Azevedo, L. H. Vilela Leão, R. L. Rodriguez-Suarez, A. B. Oliveira, S. M. Rezende, Dc effect in ferromagnetic\nresonance: Evidence of the spin-pumping effect?, J. Appl. Phys. 97 (2005) 10C715. doi:10.1063/1.1855251 .\n[207] O. Rousseau, M. Viret, Interaction between ferromagnetic resonance and spin currents in nanostructures, Phys. Rev. B\n85 (2012) 144413. doi:10.1103/PhysRevB.85.144413 .\n[208] X. Zhang, C. Zou, L. Jiang, H. X. Tang, Superstrong coupling of thin film magnetostatic waves with microwave cavity,\nJ. Appl. Phys. 119 (2016) 023905. doi:10.1063/1.4939134 .\n[209] C. Gould, C. Rüster, T. Jungwirth, E. Girgis, G. M. Schott, R. Giraud, K. Brunner, G. Schmidt, L. W. Molenkamp,\nTunneling Anisotropic Magnetoresistance: A Spin-Valve-Like Tunnel Magnetoresistance Using a Single Magnetic Layer,\n67Phys. Rev. Lett. 93 (2004) 117203. arXiv:cond-mat/0407735 ,doi:10.1103/PhysRevLett.93.117203 .\n[210] A. Matos-Abiague, J. Fabian, Anisotropic tunneling magnetoresistance and tunneling anisotropic magnetoresistance:\nSpin-orbit coupling in magnetic tunnel junctions, Phys. Rev. B 79 (2009) 155303. arXiv:0807.3564 ,doi:10.1103/\nPhysRevB.79.155303 .\n[211] C. Liu, Y. Boyko, C. C. Geppert, K. D. Christie, G. Stecklein, S. J. Patel, C. J. Palmstrøm, P. A. Crowell, Electrical\ndetectionofferromagneticresonanceinferromagnet/n-GaAsheterostructuresbytunnelinganisotropicmagnetoresistance,\nAppl. Phys. Lett. 105 (2014) 212401. arXiv:1408.5842v1 ,doi:10.1063/1.4902088 .\n[212] P. Gambardella, I. M. Miron, Current-induced spin-orbit torques, Philos. Trans. R. Soc. A 369 (2011) 3175. doi:\n10.1098/rsta.2010.0336 .\n[213] J. Kim, J. Sinha, M. Hayashi, M. Yamanouchi, S. Fukami, T. Suzuki, S. Mitani, H. Ohno, Layer thickness dependence\nof the current-induced effective field vector in Ta|CoFeB|MgO., Nat. Mater. 12 (2013) 240. arXiv:1207.2521 ,doi:\n10.1038/nmat3522 .\n[214] X. Fan, J. Wu, Y. Chen, M. J. Jerry, H. Zhang, J. Q. Xiao, Observation of the nonlocal spin-orbital effective field, Nat.\nCommun. 4 (2013) 1799. doi:10.1038/ncomms2709 .\n[215] K. Garello, I. M. Miron, C. O. Avci, F. Freimuth, Y. Mokrousov, S. Blügel, S. Auffret, O. Boulle, G. Gaudin, P. Gam-\nbardella, Symmetry and magnitude of spin-orbit torques in ferromagnetic heterostructures., Nat. Nanotechnol. 8 (2013)\n587. arXiv:1301.3573 ,doi:10.1038/nnano.2013.145 .\n[216] M. Jamali, K. Narayanapillai, X. Qiu, L. M. Loong, A. Manchon, H. Yang, Spin-Orbit Torques in Co/Pd Multilayer\nNanowires, Phys. Rev. Lett. 111 (2013) 246602. arXiv:1311.3033 ,doi:10.1103/PhysRevLett.111.246602 .\n[217] X. Fan, W. Wang, Y. Wang, H. Zhou, J. Rao, X. Zhao, C. Gao, Y. S. Gui, C.-M. Hu, D. Xue, Rapid characterizing of\nferromagnetic materials using spin rectification, Appl. Phys. Lett. 105 (2014) 262404. doi:10.1063/1.4905358 .\n[218] T. D. Skinner, M. Wang, A. T. Hindmarch, A. W. Rushforth, A. C. Irvine, D. Heiss, H. Kurebayashi, A. J. Ferguson,\nSpin-orbit torque opposing the Oersted torque in ultrathin Co/Pt bilayers, Appl. Phys. Lett. 104 (2014) 062401. arXiv:\n1312.2409 ,doi:10.1063/1.4864399 .\n[219] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab, P. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-A. Kim,\nN. Samarth, D. C. Ralph, Spin-transfer torque generated by a topological insulator, Nature 511 (2014) 449. arXiv:\n1402.1124 ,doi:10.1038/nature13534 .\n[220] C. Ciccarelli, K. M. D. Hals, A. Irvine, V. Novak, Y. Tserkovnyak, H. Kurebayashi, A. Brataas, A. Ferguson, Magnonic\nchargepumpingviaspin-orbitcoupling,Nat.Nanotechnol.10(2014)50. arXiv:1411.2779 ,doi:10.1038/nnano.2014.252 .\n[221] T. D. Skinner, K. Olejník, L. K. Cunningham, H. Kurebayashi, R. P. Campion, B. L. Gallagher, T. Jungwirth, A. J.\nFerguson, Complementary spin-Hall and inverse spin-galvanic effect torques in a ferromagnet/semiconductor bilayer, Nat.\nCommun. 6 (2015) 6730. arXiv:1502.0287 ,doi:10.1038/ncomms7730 .\n[222] Y. Wang, P. Deorani, K. Banerjee, N. Koirala, M. Brahlek, S. Oh, H. Yang, Topological Surface States Originated Spin-\nOrbit Torques in Bi 2Se3, Phys. Rev. Lett. 114 (2015) 257202. arXiv:1505.07937 ,doi:10.1103/PhysRevLett.114.257202 .\n[223] C.-F. Pai, Y. Ou, L. H. Vilela-Leão, D. C. Ralph, R. A. Buhrman, Dependence of the efficiency of spin Hall torque on\nthe transparency of Pt/ferromagnetic layer interfaces, Phys. Rev. B 92 (2015) 064426. arXiv:1411.3379 ,doi:10.1103/\nPhysRevB.92.064426 .\n[224] C. Ciccarelli, L. Anderson, V. Tshitoyan, A. J. Ferguson, F. Gerhard, C. Gould, L. W. Molenkamp, J. Gayles, J. Železný,\nL. Šmejkal, Z. Yuan, J. Sinova, F. Freimuth, T. Jungwirth, Room-temperature spin-orbit torque in NiMnSb, Nat. Phys.\narXiv:1510.03356 ,doi:10.1038/nphys3772 .\n[225] T. Nan, S. Emori, C. T. Boone, X. Wang, T. M. Oxholm, J. G. Jones, B. M. Howe, G. J. Brown, N. X. Sun, Comparison\nof spin-orbit torques and spin pumping across NiFe/Pt and NiFe/Cu/Pt interfaces, Phys. Rev. B 91 (2015) 214416.\narXiv:1503.04104 ,doi:10.1103/PhysRevB.91.214416 .\n[226] L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, Phys. Rev. B 54 (1996) 9353.\ndoi:10.1103/PhysRevB.54.9353 .\n[227] J. Slonczewski, Current-driven excitation of magnetic multilayers, J. Magn. Magn. Mater. 159 (1996) L1. doi:10.1016/\n0304-8853(96)00062-5 .\n[228] M. Tsoi, A. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, P. Wyder, Excitation of a Magnetic Multilayer by an\nElectric Current, Phys. Rev. Lett. 80 (1998) 4281. doi:10.1103/PhysRevLett.80.4281 .\n[229] E. B. Myers, Current-Induced Switching of Domains in Magnetic Multilayer Devices, Science 285 (1999) 867. doi:\n10.1126/science.285.5429.867 .\n[230] D. C. Ralph, M. D. Stiles, Spin transfer torques, J. Magn. Magn. Mater. 320 (2008) 1190. arXiv:0711.4608 ,doi:\n10.1016/j.jmmm.2007.12.019 .\n[231] S. Zhang, P. M. Levy, A. Fert, Mechanisms of spin-polarized current-driven magnetization switching, Phys. Rev. Lett.\n88 (2002) 236601. arXiv:cond-mat/0202363 ,doi:10.1103/PhysRevLett.88.236601 .\n[232] X. G. Zhang, W. H. Butler, Large magnetoresistance in bcc Co/MgO/Co and FeCo/MgO/FeCo tunnel junctions, Phys.\nRev. B 70 (2004) 172407. arXiv:cond-mat/0409155 ,doi:10.1103/PhysRevB.70.172407 .\n[233] Z. Li, S. Zhang, Domain-wall dynamics driven by adiabatic spin-transfer torques, Phys. Rev. B 70 (2004) 024417.\narXiv:cond-mat/0407064 ,doi:10.1103/PhysRevB.70.024417 .\n[234] Z. Li, S. Zhang, Domain-wall dynamics and spin-wave excitations with spin-transfer torques, Phys. Rev. Lett. 92 (2004)\n207203. doi:10.1103/PhysRevLett.92.207203 .\n[235] J. He, Z. Li, S. Zhang, Current-driven vortex domain wall dynamics by micromagnetic simulations, Phys. Rev. B 73\n(2006) 184408. arXiv:cond-mat/0605125 ,doi:10.1103/PhysRevB.73.184408 .\n[236] T. Chiba, G. E. W. Bauer, S. Takahashi, Current-Induced Spin-Torque Resonance of Magnetic Insulators, Phys. Rev.\nAppl. 2 (2014) 034003. arXiv:1404.2360 ,doi:10.1103/PhysRevApplied.2.034003 .\n68[237] T. Chiba, M. Schreier, G. E. W. Bauer, S. Takahashi, Current-induced spin torque resonance of magnetic insulators\naffected by field-like spin-orbit torques and out-of-plane magnetizations, J. Appl. Phys. 117 (2015) 17C715. arXiv:\n1409.8406 ,doi:10.1063/1.4913632 .\n[238] R. Iguchi, K. Sato, D. Hirobe, S. Daimon, E. Saitoh, Effect of spin Hall magnetoresistance on spin pumping measurements\nin insulating magnet/metal systems, Appl. Phys. Express 7 (2014) 013003. doi:10.7567/APEX.7.013003 .\n[239] J. Rao, X. Fan, L. Ma, H. Zhou, X. Zhao, J. Zhao, F. Zhang, S. Zhou, D. Xue, Observation of spin rectification in\nPt/yttrium iron garnet bilayer, J. Appl. Phys. 117 (2015) 17C725. doi:10.1063/1.4914962 .\n[240] M. Schreier, T. Chiba, A. Niedermayr, J. Lotze, H. Huebl, S. Geprägs, S. Takahashi, G. E. W. Bauer, R. Gross,\nS. T. B. Goennenwein, Current-induced spin torque resonance of a magnetic insulator, Phys. Rev. B 92 (2015) 144411.\narXiv:1412.7460 ,doi:10.1103/PhysRevB.92.144411 .\n[241] J. Sklenar, W. Zhang, M. B. Jungfleisch, W. Jiang, H. Chang, J. E. Pearson, M. Wu, J. B. Ketterson, A. Hoffmann,\nDriving and detecting ferromagnetic resonance in insulators with the spin Hall effect, Phys. Rev. B 92 (2015) 174406.\narXiv:1505.07791v1 ,doi:10.1103/PhysRevB.92.174406 .\n[242] M. B. Jungfleisch, W. Zhang, J. Sklenar, J. Ding, W. Jiang, H. Chang, F. Y. Fradin, J. E. Pearson, J. B. Ketterson,\nV. Novosad, M. Wu, A. Hoffmann, Large Spin-Wave Bullet in a Ferrimagnetic Insulator Driven by the Spin Hall Effect,\nPhys. Rev. Lett. 116 (2016) 057601. arXiv:1508.0142 ,doi:10.1103/PhysRevLett.116.057601 .\n[243] T. Moriyama, R. Cao, X. Fan, G. Xuan, B. K. Nikolić, Y. Tserkovnyak, J. Kolodzey, J. Q. Xiao, Tunnel Barrier\nEnhanced Voltage Signal Generated by Magnetization Precession of a Single Ferromagnetic Layer, Phys. Rev. Lett. 100\n(2008) 067602. arXiv:0802.3646 ,doi:10.1103/PhysRevLett.100.067602 .\n[244] S. Miwa, S. Ishibashi, H. Tomita, T. Nozaki, E. Tamura, K. Ando, N. Mizuochi, T. Saruya, H. Kubota, K. Yakushiji,\nT. Taniguchi, H. Imamura, A. Fukushima, S. Yuasa, Y. Suzuki, Highly sensitive nanoscale spin-torque diode., Nat. Mater.\n13 (2014) 50. doi:10.1038/nmat3778 .\n[245] B. Fang, M. Carpentieri, X. Hao, H. Jiang, J. A. Katine, I. N. Krivorotov, B. Ocker, J. Langer, K. L. Wang, B. Zhang,\nB. Azzerboni, P. K. Amiri, G. Finocchio, Z. Zeng, Giant spin-torque diode sensitivity in the absence of bias magnetic\nfield, Nat. Commun. 7 (2016) 11259. arXiv:1410.4958 ,doi:10.1038/ncomms11259 .\n[246] X. Cheng, J. A. Katine, G. E. Rowlands, I. N. Krivorotov, Nonlinear ferromagnetic resonance induced by spin torque in\nnanoscale magnetic tunnel junctions, Appl. Phys. Lett. 103 (2013) 082402. doi:10.1063/1.4819179 .\n[247] J. Kleinlein, B. Ocker, G. Schmidt, Using giant magneto resistance stripes to efficiently generate direct voltage signals\nfrom alternating current excitations, Appl. Phys. Lett. 104 (2014) 153507. doi:10.1063/1.4871383 .\n[248] S. Ziętek, P. Ogrodnik, M. Frankowski, J. Chęciński, P. Wiśniowski, W. Skowroński, J. Wrona, T. Stobiecki, A. Żywczak,\nJ. Barnaś, Rectification of radio-frequency current in a giant-magnetoresistance spin valve, Phys. Rev. B 91 (2015)\n014430. doi:10.1103/PhysRevB.91.014430 .\n[249] S.Zi¸ etek, P.Ogrodnik, W.Skowroński, P.Wiśniowski, M.Czapkiewicz, T.Stobiecki, J.Barnaś, Theinfluenceofinterlayer\nexchange coupling in giant-magnetoresistive devices on spin diode effect in wide frequency range, Appl. Phys. Lett. 107\n(2015) 122410. doi:10.1063/1.4931771 .\n[250] Z. Diao, Z. Li, S. Wang, Y. Ding, A. Panchula, E. Chen, L.-C. Wang, Y. Huai, Spin-transfer torque switching in\nmagnetic tunnel junctions and spin-transfer torque random access memory, J. Phys. Condens. Matter 19 (2007) 165209.\ndoi:10.1088/0953-8984/19/16/165209 .\n[251] J. N. Kupferschmidt, S. Adam, P. W. Brouwer, Theory of the spin-torque-driven ferromagnetic resonance in a\nferromagnet/normal-metal/ferromagnet structure, Phys. Rev. B 74 (2006) 134416. arXiv:cond-mat/0607145 ,doi:\n10.1103/PhysRevB.74.134416 .\n[252] A. A. Kovalev, G. E. W. Bauer, A. Brataas, Current-driven ferromagnetic resonance, mechanical torques, and rotary\nmotion in magnetic nanostructures, Phys. Rev. B 75 (2007) 014430. arXiv:cond-mat/0609258 ,doi:10.1103/PhysRevB.\n75.014430 .\n[253] O. Boulle, V. Cros, J. Grollier, L. G. Pereira, C. Deranlot, F. Petroff, G. Faini, J. Barnas, A. Fert, Shaped angular\ndependence of the spin transfer torque and microwave generation without magnetic field, Nat. Phys. 3 (2007) 492.\narXiv:0704.0370 ,doi:10.1038/nphys618 .\n[254] J. C. Slonczewski, Currents, torques, and polarization factors in magnetic tunnel junctions, Phys. Rev. B 71 (2005)\n024411. arXiv:cond-mat/0404210 ,doi:10.1103/PhysRevB.71.024411 .\n[255] I. Theodonis, N. Kioussis, A. Kalitsov, M. Chshiev, W. H. Butler, Anomalous bias dependence of spin torque in magnetic\ntunnel junctions, Phys. Rev. Lett. 97 (2006) 237205. arXiv:cond-mat/0606080 ,doi:10.1103/PhysRevLett.97.237205 .\n[256] A. Manchon, S. Zhang, Spin Torque Effects in Magnetic Systems: Theory, in: E. Y. Tsymbal, I. Žutić (Eds.), Handb.\nSpin Transp. Magn., CRC Press, Boca Raton, 2012, pp. 157–178. doi:10.1201/b11086-11 .\n[257] A. M. Gonçalves, I. Barsukov, Y.-J. Chen, L. Yang, J. A. Katine, I. N. Krivorotov, Spin torque ferromagnetic resonance\nwith magnetic field modulation, Appl. Phys. Lett. 103 (2013) 172406. arXiv:1310.7996 ,doi:10.1063/1.4826927 .\n[258] C.-G. Duan, J. P. Velev, R. F. Sabirianov, Z. Zhu, J. Chu, S. S. Jaswal, E. Y. Tsymbal, Surface Magnetoelectric Effect\nin Ferromagnetic Metal Films, Phys. Rev. Lett. 101 (2008) 137201. arXiv:0808.0841 ,doi:10.1103/PhysRevLett.101.\n137201.\n[259] K. Nakamura, R. Shimabukuro, Y. Fujiwara, T. Akiyama, T. Ito, A. J. Freeman, Giant Modification of the Magnetocrys-\ntalline Anisotropy in Transition-Metal Monolayers by an External Electric Field, Phys. Rev. Lett. 102 (2009) 187201.\ndoi:10.1103/PhysRevLett.102.187201 .\n[260] M. Tsujikawa, T. Oda, Finite Electric Field Effects in the Large Perpendicular Magnetic Anisotropy Surface\nPt/Fe/Pt(001): A First-Principles Study, Phys. Rev. Lett. 102 (2009) 247203. arXiv:0905.0060 ,doi:10.1103/\nPhysRevLett.102.247203 .\n[261] W.-G. Wang, M. Li, S. Hageman, C. L. Chien, Electric-field-assisted switching in magnetic tunnel junctions, Nat. Mater.\n6911 (2011) 64. doi:10.1038/nmat3171 .\n[262] Y. Shiota, T. Nozaki, F. Bonell, S. Murakami, T. Shinjo, Y. Suzuki, Induction of coherent magnetization switching in a\nfew atomic layers of FeCo using voltage pulses, Nat. Mater. 11 (2011) 39. doi:10.1038/nmat3172 .\n[263] T. Nozaki, Y. Shiota, S. Miwa, S. Murakami, F. Bonell, S. Ishibashi, H. Kubota, K. Yakushiji, T. Saruya, A. Fukushima,\nS. Yuasa, T. Shinjo, Y. Suzuki, Electric-field-induced ferromagnetic resonance excitation in an ultrathin ferromagnetic\nmetal layer, Nat. Phys. 8 (2012) 492. doi:10.1038/nphys2298 .\n[264] J. Zhu, J. A. Katine, G. E. Rowlands, Y.-J. Chen, Z. Duan, J. G. Alzate, P. Upadhyaya, J. Langer, P. K. Amiri, K. L.\nWang, I. N. Krivorotov, Voltage-Induced Ferromagnetic Resonance in Magnetic Tunnel Junctions, Phys. Rev. Lett. 108\n(2012) 197203. arXiv:1205.2835 ,doi:10.1103/PhysRevLett.108.197203 .\n[265] Y. Shiota, S. Miwa, S. Tamaru, T. Nozaki, H. Kubota, A. Fukushima, Y. Suzuki, S. Yuasa, High-output microwave\ndetector using voltage-induced ferromagnetic resonance, Appl. Phys. Lett. 105 (2014) 192408. doi:10.1063/1.4902025 .\n[266] W. Skowroński, T. Nozaki, Y. Shiota, S. Tamaru, K. Yakushiji, H. Kubota, A. Fukushima, S. Yuasa, Y. Suzuki, Per-\npendicular magnetic anisotropy of Ir/CoFeB/MgO trilayer system tuned by electric fields, Appl. Phys. Express 8 (2015)\n053003. doi:10.7567/APEX.8.053003 .\n[267] B. Lenk, H. Ulrichs, F. Garbs, M. Münzenberg, The building blocks of magnonics, Phys. Rep. 507 (2011) 107. arXiv:\n1101.0479 ,doi:10.1016/j.physrep.2011.06.003 .\n[268] C. Herring, C. Kittel, On the theory of spin waves in ferromagnetic media, Phys. Rev. 81 (1951) 869. doi:10.1103/\nPhysRev.81.869 .\n[269] D. S. Schmool, H. Kachkachi, Single-Particle Phenomena in Magnetic Nanostructures, Elsevier, 2015. doi:10.1016/bs.\nssp.2015.06.001 .\n[270] L. R. Walker, Magnetostatic modes in ferromagnetic resonance, Phys. Rev. 105 (1957) 390. doi:10.1103/PhysRev.105.\n390.\n[271] M. Sparks, Ferromagnetic resonance in thin films. I. theory of normal-mode frequencies, Phys. Rev. B 1 (1970) 3831.\ndoi:10.1103/PhysRevB.1.3831 .\n[272] M. Sparks, Effect of exchange on magnetostatic modes, Phys. Rev. Lett. 24 (1970) 1178. doi:10.1103/PhysRevLett.24.\n1178.\n[273] B. A. Kalinkos, A. N. Slavin, Theory of dipole-exchange spin wave spectrum for ferromagnetic films with mixed exchange\nboundary conditions, J. Phys. C Solid State Phys. 19 (1986) 7013. doi:10.1088/0022-3719/19/35/014 .\n[274] S. O. Demokritov, Spin Wave Confinement, Pan Stanford Publishing, 2009. doi:10.1142/9789814241205 .\n[275] D. D. Stancil, A. Prabhakar, Spin Waves: Theory and Applications, Springer, 2009. doi:10.1007/978-0-387-77865-5 .\n[276] T. J. Silva, C. S. Lee, T. M. Crawford, C. T. Rogers, Inductive measurement of ultrafast magnetization dynamics in\nthin-film Permalloy, J. Appl. Phys. 85 (1999) 7849. doi:10.1063/1.370596 .\n[277] M. Covington, T. M. Crawford, G. J. Parker, Time-resolved measurement of propagating spin waves in ferromagnetic\nthin films, Phys. Rev. Lett. 89 (2002) 237202. doi:10.1103/PhysRevLett.89.237202 .\n[278] G. Counil, J. V. Kim, T. Devolder, C. Chappert, K. Shigeto, Y. Otani, Spin wave contributions to the high-frequency\nmagnetic response of thin films obtained with inductive methods, J. Appl. Phys. 95 (2004) 5646. doi:10.1063/1.1697641 .\n[279] R. Bonin, M. L. Schneider, T. J. Silva, J. P. Nibarger, Dependence of magnetization dynamics on magnetostriction in\nNiFe alloys, J. Appl. Phys. 98 (2005) 123904. doi:10.1063/1.2143121 .\n[280] J. P. Nibarger, R. Lopusnik, T. J. Silva, Damping as a function of pulsed field amplitude and bias field in thin film\nPermalloy, Appl. Phys. Lett. 82 (2003) 2112. doi:10.1063/1.1564866 .\n[281] T. Gerrits, M. Schneider, A. Kos, T. Silva, Large-angle magnetization dynamics measured by time-resolved ferromagnetic\nresonance, Phys. Rev. B 73 (2006) 094454. arXiv:cond-mat/0508708 ,doi:10.1103/PhysRevB.73.094454 .\n[282] J. Goulon, A. Rogalev, F. Wilhelm, N. Jaouen, C. Goulon-Ginet, C. Brouder, X-ray detected ferromagnetic resonance in\nthin films, Eur. Phys. J. B 53 (2006) 169. doi:10.1140/epjb/e2006-00367-6 .\n[283] G. Boero, S. Rusponi, J. Kavich, A. L. Rizzini, C. Piamonteze, F. Nolting, C. Tieg, J.-U. Thiele, P. Gambardella,\nLongitudinal detection of ferromagnetic resonance using x-ray transmission measurements, Rev. Sci. Instrum. 80 (2009)\n123902. doi:10.1063/1.3267192 .\n[284] R. Salikhov, R. Abrudan, F. Bruüssing, S. Buschhorn, M. Ewerlin, D. Mishra, F. Radu, I. A. Garifullin, H. Zabel,\nPrecessional dynamics and damping in Co/Cu/Py spin valves, Appl. Phys. Lett. 99 (2011) 092509. doi:10.1063/1.\n3633115.\n[285] H. G. Bauer, P. Majchrak, T. Kachel, C. H. Back, G. Woltersdorf, Nonlinear spin-wave excitations at low magnetic bias\nfields, Nat. Commun. 6 (2015) 8274. doi:10.1038/ncomms9274 .\n[286] J. P. Park, P. Eames, D. M. Engebretson, J. Berezovsky, P. A. Crowell, Spatially resolved dynamics of localized spin-wave\nmodes in ferromagnetic wires, Phys. Rev. Lett. 89 (2002) 277201. arXiv:cond-mat/0207022 ,doi:10.1103/PhysRevLett.\n89.277201 .\n[287] S. Tamaru, J. A. Bain, R. J. M. Van De Veerdonk, T. M. Crawford, M. Covington, M. H. Kryder, Imaging of quantized\nmagnetostatic modes using spatially resolved ferromagnetic resonance, J. Appl. Phys. 91 (2002) 8034. doi:10.1063/1.\n1450815.\n[288] M. J. Pechan, C. Yu, R. L. Compton, J. P. Park, P. A. Crowell, Direct measurement of spatially localized ferromagnetic-\nresonance modes in an antidot lattice (invited), J. Appl. Phys. 97 (2005) 10J903. doi:10.1063/1.1857412 .\n[289] S. Neusser, G. Duerr, H. G. Bauer, S. Tacchi, M. Madami, G. Woltersdorf, G. Gubbiotti, C. H. Back, D. Grundler,\nAnisotropic Propagation and Damping of Spin Waves in a Nanopatterned Antidot Lattice, Phys. Rev. Lett. 105 (2010)\n067208. doi:10.1103/PhysRevLett.105.067208 .\n[290] T.Gerrits, P.Krivosik, M.L.Schneider, C.E.Patton, T.J.Silva, DirectDetectionofNonlinearFerromagneticResonance\nin Thin Films by the Magneto-Optical Kerr Effect, Phys. Rev. Lett. 98 (2007) 207602. doi:10.1103/PhysRevLett.98.\n70207602.\n[291] M. Bailleul, D. Olligs, C. Fermon, S. O. Demokritov, Spin waves propagation and confinement in conducting films at the\nmicrometer scale, Europhys. Lett. 56 (2001) 741. doi:10.1209/epl/i2001-00583-2 .\n[292] I. Neudecker, G. Woltersdorf, B. Heinrich, T. Okuno, G. Gubbiotti, C. H. Back, Comparison of frequency, field, and time\ndomain ferromagnetic resonance methods, J. Magn. Magn. Mater. 307 (2006) 148. doi:10.1016/j.jmmm.2006.03.060 .\n[293] J. Podbielski, F. Giesen, D. Grundler, Spin-Wave Interference in Microscopic Rings, Phys. Rev. Lett. 96 (2006) 167207.\ndoi:10.1103/PhysRevLett.96.167207 .\n[294] Y. T. Zhang, C. E. Patton, G. Srinivasan, G. O. White, C. J. Brower, C. A. Edmondson, Measurement of the spin-wave\ninstability threshold h critin microwave ferrites, Rev. Sci. Instrum. 58 (1987) 620. doi:10.1063/1.1139226 .\n[295] R. G. Cox, C. E. Patton, M. A. Wittenauer, P. Kabos, L. Chen, Spin wave instability in single crystal Zn-Y hexagonal\nferrite at 8.93 GHz, J. Appl. Phys. 89 (2001) 4454. doi:10.1063/1.1352689 .\n[296] S. Y. An, P. Krivosik, M. A. Kraemer, H. M. Olson, A. V. Nazarov, C. E. Patton, High power ferromagnetic resonance\nand spin wave instability processes in Permalloy thin films, J. Appl. Phys. 96 (2004) 1572. doi:10.1063/1.1763996 .\n[297] S. O. Demokritov, B. Hillebrands, A. N. Slavin, Brillouin Light Scattering Studies of Confined Spin Waves: Linear and\nNonlinear Confinement, Phys. Rep. 348 (2001) 441. doi:10.1016/S0370-1573(00)00116-2 .\n[298] J. Jorzick, S. O. Demokritov, B. Hillebrands, M. Bailleul, C. Fermon, K. Y. Guslienko, A. N. Slavin, D. V. Berkov,\nN. L. Gorn, Spin wave wells in nonellipsoidal micrometer size magnetic elements, Phys. Rev. Lett. 88 (2002) 047204.\ndoi:10.1103/PhysRevLett.88.047204 .\n[299] T. Sebastian, K. Schultheiss, B. Obry, B. Hillebrands, H. Schultheiss, Micro-focused Brillouin light scattering: imaging\nspin waves at the nanoscale, Front. Phys. 3 (2015) 1. doi:10.3389/fphy.2015.00035 .\n[300] W. Wettling, W. D. Wilber, P. Kabos, C. E. Patton, Light Scattering from Parallel-Pump Instabilities in Yttrium Iron\nGarnet, Phys. Rev. Lett. 51 (1983) 1680. doi:10.1103/PhysRevLett.51.1680 .\n[301] P. Kabos, C. E. Patton, G. Wiese, A. D. Sullins, E. S. Wright, L. Chen, Butterfly curves and critical modes for second-\norder spin-wave instability processes in yttrium iron garnet films, J. Appl. Phys. 80 (1996) 3962. doi:10.1063/1.363354 .\n[302] E. R. J. Edwards, H. Ulrichs, V. E. Demidov, S. O. Demokritov, S. Urazhdin, Parametric excitation of magnetization\noscillations controlled by pure spin current, Phys. Rev. B 86 (2012) 134420. doi:10.1103/PhysRevB.86.134420 .\n[303] F. Guo, L. M. Belova, R. D. McMichael, Spectroscopy and Imaging of Edge Modes in Permalloy Nanodisks, Phys. Rev.\nLett. 110 (2013) 017601. doi:10.1103/PhysRevLett.110.017601 .\n[304] F. Guo, L. M. Belova, R. D. McMichael, Nonlinear ferromagnetic resonance shift in submicron Permalloy ellipses, Phys.\nRev. B 91 (2015) 064426. doi:10.1103/PhysRevB.91.064426 .\n[305] M. Bailleul, R. Höllinger, K. Perzlmaier, C. Fermon, Microwave spectrum of square permalloy dots: Multidomain state,\nPhys. Rev. B 76 (2007) 224401. doi:10.1103/PhysRevB.76.224401 .\n[306] A. Yamaguchi, K. Motoi, H. Miyajima, Y. Nakatani, Magnetic field dependence of rectification radio frequency current\nflowing through a single layered ferromagnetic wire, J. Appl. Phys. 105 (2009) 07D301. doi:10.1063/1.3056143 .\n[307] A.Yamaguchi, K.Motoi, H.Miyajima, T.Uchiyama, Y.Utsumi, High-orderstandingspinwavemodesinFe 19Ni81micron\nwire observed by homodyne method, J. Phys. Conf. Ser. 266 (2011) 012113. doi:10.1088/1742-6596/266/1/012113 .\n[308] Z. Duan, I. N. Krivorotov, R. E. Arias, N. Reckers, S. Stienen, J. Lindner, Spin wave eigenmodes in transversely\nmagnetized thin film ferromagnetic wires, Phys. Rev. B 92 (2015) 104424. doi:10.1103/PhysRevB.92.104424 .\n[309] A. Ercole, A. Adeyeye, J. Bland, D. Hasko, Size-dependent spin-wave frequency in ferromagnetic wire-array structures,\nPhys. Rev. B 58 (1998) 345. doi:10.1103/PhysRevB.58.345 .\n[310] C. Bayer, J. Jorzick, S. O. Demokritov, A. N. Slavin, K. Y. Guslienko, D. V. Berkov, N. L. Gorn, M. P. Kostylev,\nB. Hillebrands, Spin-wave excitations in finite rectangular elements, Top. Appl. Phys. 101 (2006) 57. doi:10.1007/\n10938171_2 .\n[311] C. Kittel, Excitation of Spin Waves in a Ferromagnet by a Uniform rf Field, Phys. Rev. 110 (1958) 1213. doi:10.1103/\nPhysRev.110.1295 .\n[312] Y. Zhai, J. Shi, X. Y. Zhang, L. Shi, Y. X. Xu, H. B. Huang, Z. H. Lu, H. R. Zhai, A study of the non-uniform effect\non the shape anisotropy in patterned NiFe films of ferromagnetic resonance, J. Phys. Condens. Matter 14 (2002) 7865.\ndoi:10.1088/0953-8984/14/34/306 .\n[313] R. L. White, I. H. Solt, Multiple ferromagnetic resonance in ferrite spheres, Phys. Rev. 104 (1956) 56. doi:10.1103/\nPhysRev.104.56 .\n[314] J. F. Dillon, Magnetostatic modes in ferrimagnetic spheres, Phys. Rev. 112 (1958) 59. doi:10.1103/PhysRev.112.59 .\n[315] J. Jorzick, S. Demokritov, C. Mathieu, B. Hillebrands, B. Bartenlian, C. Chappert, F. Rousseaux, A. Slavin, Brillouin\nlight scattering from quantized spin waves in micron-size magnetic wires, Phys. Rev. B 60 (1999) 15194. doi:10.1103/\nPhysRevB.60.15194 .\n[316] Z. K. Wang, M. H. Kuok, S. C. Ng, D. J. Lockwood, M. G. Cottam, K. Nielsch, R. B. Wehrspohn, U. Gösele, Spin-\nwave quantization in ferromagnetic nickel nanowires., Phys. Rev. Lett. 89 (2002) 027201. doi:10.1103/PhysRevLett.89.\n027201.\n[317] G. Rado, J. Weertman, Spin-wave resonance in a ferromagnetic metal, J. Phys. Chem. Solids 11 (1959) 315. doi:\n10.1016/0022-3697(59)90233-1 .\n[318] P. Pincus, Excitation of spin waves in ferromagnets: eddy current and boundary condition effects, Phys. Rev. 118 (1960)\n658. doi:10.1103/PhysRev.118.658 .\n[319] R. F. Soohoo, Excitation and Boundary Effects in Spin-Wave Resonance, J. Appl. Phys. 32 (1961) 1959. doi:10.1063/\n1.2000381 .\n[320] R. F. Soohoo, General exchange boundary condition and surface anisotropy energy of a ferromagnet, Phys. Rev. 131\n(1963) 594. doi:10.1103/PhysRev.131.594 .\n71[321] L. D. Landau, E. M. Lifshitz, Mechanics, 3rd Edition, Elsevier, 1976.\n[322] P. W. Anderson, H. Suhl, Instability in the motion of ferromagnets at high microwave power levels, Phys. Rev. 100 (1955)\n1788. doi:10.1103/PhysRev.100.1788 .\n[323] G. Bertotti, I. D. Mayergoyz, C. Serpico, Spin-wave instabilities in large-scale nonlinear magnetization dynamics., Phys.\nRev. Lett. 87 (2001) 217203. doi:10.1103/PhysRevLett.87.217203 .\n[324] H. Suhl, “Foldover\" effects caused by spin wave interactions in ferromagnetic resonance, J. Appl. Phys. 31 (1960) 935.\ndoi:10.1063/1.1735723 .\n[325] P. Gottlieb, Nonlinear effects of crystalline anisotropy on ferrimagnetic resonance, J. Appl. Phys. 31 (1960) 2059. doi:\n10.1063/1.1735497 .\n[326] J. I. Masters, Instability of magnetic resonance in single crystal spheres of yttrium iron garnet, J. Appl. Phys. 31 (1960)\n41S. doi:10.1063/1.1984596 .\n[327] I. B. Goldberg, Low-power nonlinear effects in the ferromagnetic resonance of spheres of Ba 3Zn2Fe24O41, J. Appl. Phys.\n51 (1980) 1857. doi:10.1063/1.327760 .\n[328] L. M. Silber, C. E. Patton, H. F. Naqvi, Low power nonlinear effects in the ferromagnetic resonance of Zn 2Y and MnZnY\nhexagonal ferrites, J. Appl. Phys. 54 (1983) 4071. doi:10.1063/1.332537 .\n[329] D. J. Seagle, S. H. Charap, J. O. Artman, Foldover in YIG, J. Appl. Phys. 57 (1985) 3706. doi:10.1063/1.334997 .\n[330] K. D. McKinstry, C. E. Patton, M. Kogekar, Low power nonlinear effects in the ferromagnetic resonance of yttrium iron\ngarnet, J. Appl. Phys. 58 (1985) 925. doi:10.1063/1.336167 .\n[331] Y. Zhang, C. Patton, M. Kogekar, Ferromagentic resonance foldover in single crystal YIG films–Sample heating or Suhl\ninstability, IEEE Trans. Magn. 22 (1986) 993. doi:10.1109/TMAG.1986.1064446 .\n[332] K. Gnatzig, H. Dötsch, M. Ye, A. Brockmeyer, Ferrimagnetic resonance in garnet films at large precession angles, J.\nAppl. Phys. 62 (1987) 4839. doi:10.1063/1.338988 .\n[333] Y. T. Zhang, C. E. Patton, G. Srinivasan, The second-order spin-wave instability threshold in single-crystal yttrium-\niron-garnet films under perpendicular pumping, J. Appl. Phys. 63 (1988) 5433. doi:10.1063/1.340364 .\n[334] M. Chen, C. E. Patton, G. Srinivasan, Y. T. Zhang, Ferromagnetic resonance foldover and spin-wave instability in\nsingle-crystal YIG films, IEEE Trans. Magn. 25 (1989) 3485. doi:10.1109/20.42343 .\n[335] Y. Fetisov, C. Patton, V. Synogach, Nonlinear ferromagnetic resonance and foldover in yttrium iron garnet thin films-\ninadequacy of the classical model, IEEE Trans. Magn. 35 (1999) 4511. doi:10.1109/20.809144 .\n[336] Y. K. Fetisov, C. E. Patton, Thermal microwave foldover and bistability in ferromagnetic resonance, IEEE Trans. Magn.\n40 (2004) 473. doi:10.1109/TMAG.2004.824132 .\n[337] H. Suhl, The theory of ferromagnetic resonance at high signal powers, J. Phys. Chem. Solids 1 (1957) 209. doi:\n10.1016/0022-3697(57)90010-0 .\n[338] C. Bi, X. Fan, L. Pan, X. Kou, J. Wu, Q. Yang, H. Zhang, J. Q. Xiao, Electrical detection of nonlinear ferromagnetic\nresonance in single elliptical permalloy thin film using a magnetic tunnel junction, Appl. Phys. Lett. 99 (2011) 232506.\ndoi:10.1063/1.3670302 .\n[339] E. Hirayama, S. Kanai, J. Ohe, H. Sato, F. Matsukura, H. Ohno, Electric-field induced nonlinear ferromagnetic resonance\nin a CoFeB/MgO magnetic tunnel junction, Appl. Phys. Lett. 107 (2015) 132404. doi:10.1063/1.4932092 .\n[340] H. Zhou, X. Fan, L. Ma, L. Cui, C. Jia, S. Zhou, Y. S. Gui, C.-M. Hu, D. Xue, Spin pumping in the nonlinear dynamic\nregime of a Pt/Y 3Fe5O12heterostructure, Appl. Phys. Lett. 108 (2016) 192408. doi:10.1063/1.4949555 .\n[341] M. W. Wu, A. Hoffmann (Eds.), Recent Advances in Magnetic Insulators - From Spintronics to Microwave Applications,\nVol. 64, Elsevier, 2013.\n[342] H. Jiao, G. E. W. Bauer, Spin backflow and ac voltage generation by spin pumping and the inverse spin hall effect, Phys.\nRev. Lett. 110 (2013) 217602. arXiv:1210.0724 ,doi:10.1103/PhysRevLett.110.217602 .\n[343] P. P. Hofer, H. Aramberri, C. Schenke, P. A. L. Delplace, Proposal for an ac spin current source, Europhys. Lett. 107\n(2014) 27003. arXiv:1403.6674 ,doi:10.1209/0295-5075/107/27003 .\n[344] P. Hyde, L. Bai, D. M. J. Kumar, B. W. Southern, C.-M. Hu, S. Y. Huang, B. F. Miao, C. L. Chien, Electrical detection\nof direct and alternating spin current injected from a ferromagnetic insulator into a ferromagnetic metal, Phys. Rev. B\n89 (2014) 180404(R). arXiv:1310.4840 ,doi:10.1103/PhysRevB.89.180404 .\n[345] D. Wei, M. Obstbaum, M. Ribow, C. H. Back, G. Woltersdorf, Spin Hall voltages from a.c. and d.c. spin currents, Nat\nCommun 5 (2014) 3768. doi:10.1038/ncomms4768 .\n[346] M. Weiler, J. M. Shaw, H. T. Nembach, T. J. Silva, Phase-sensitive detection of spin pumping via the ac inverse spin\nHall effect, Phys. Rev. Lett. 113 (2014) 157204. arXiv:1401.6469 ,doi:10.1103/PhysRevLett.113.157204 .\n[347] J. Li, L. R. Shelford, P. Shafer, A. Tan, J. X. Deng, P. S. Keatley, C. Hwang, E. Arenholz, G. van der Laan, R. J. Hicken,\nZ. Q. Qiu, Direct Detection of Pure ac Spin Current by X-Ray Pump-Probe Measurements, Phys. Rev. Lett. 117 (2016)\n076602. doi:10.1103/PhysRevLett.117.076602 .\n[348] W. Pauli, Wolfgang Pauli Wissenschaftlicher Briefwechsel mit Bohr, Einstein, Heisenberg u.a., Vol. 6 of Sources in the\nHistory of Mathematics and Physical Sciences, Springer, 1985. doi:10.1007/978-3-540-78801-0 .\n[349] H. Ohno, H. Munekata, T. Penney, S. Vonmolnar, L. L. Chang, Magnetotransport Properties of P-Type (In,Mn)As\nDiluted Magnetic III-V Semiconductors, Phys. Rev. Lett. 68 (1992) 2664. doi:10.1103/PhysRevLett.68.2664 .\n[350] H. Ohno, Making Nonmagnetic Semiconductors Ferromagnetic, Science 281 (1998) 951. doi:10.1126/science.281.5379.\n951.\n[351] A. H. MacDonald, P. Schiffer, N. Samarth, Ferromagnetic semiconductors: moving beyond (Ga,Mn)As, Nat. Mater. 4\n(2005) 195.\n[352] T. Jungwirth, J. Sinova, J. Mašek, J. Kučera, A. H. MacDonald, Theory of ferromagnetic (III,Mn)V semiconductors,\nRev. Mod. Phys. 78 (2006) 809. arXiv:cond-mat/0603380 ,doi:10.1103/RevModPhys.78.809 .\n72[353] T. Dietl, H. Ohno, Dilute ferromagnetic semiconductors: Physics and spintronic structures, Rev. Mod. Phys. 86 (2014)\n187. arXiv:1307.3429 ,doi:10.1103/RevModPhys.86.187 .\n[354] T. Jungwirth, J. Wunderlich, V. Novák, K. Olejník, B. L. Gallagher, R. P. Campion, K. W. Edmonds, A. W. Rushforth,\nA. J. Ferguson, P. Němec, Spin-dependent phenomena and device concepts explored in (Ga,Mn)As, Rev. Mod. Phys. 86\n(2014) 855. arXiv:1310.1944 ,doi:10.1103/RevModPhys.86.855 .\n[355] D. Fang, H. Kurebayashi, J. Wunderlich, K. Výborný, L. P. Zârbo, R. P. Campion, A. Casiraghi, B. L. Gallagher,\nT. Jungwirth, A. J. Ferguson, Spin-orbit-driven ferromagnetic resonance, Nat. Nanotechnol. 6 (2011) 413. doi:10.1038/\nnnano.2011.68 .\n[356] H. Nakayama, L. Chen, H. W. Chang, H. Ohno, F. Matsukura, Inverse spin Hall effect in Pt/(Ga,Mn)As, Appl. Phys.\nLett. 106 (2015) 222405. doi:10.1063/1.4922197 .\n[357] A. Hubert, R. Schäfer, Magnetic Domains: The Analysis of Magnetic Microstructures, Springer, 2009. doi:10.1007/\n978-3-540-85054-0 .\n[358] R. McMichael, M. Donahue, Head to head domain wall structures in thin magnetic strips, IEEE Trans. Magn. 33 (1997)\n4167. doi:10.1109/20.619698 .\n[359] D. A. Allwood, Magnetic Domain-Wall Logic, Science 309 (2005) 1688. doi:10.1126/science.1108813 .\n[360] S. S. P. Parkin, M. Hayashi, L. Thomas, Magnetic Domain-Wall Racetrack Memory, Science 320 (2008) 190. doi:\n10.1126/science.1145799 .\n[361] G. Malinowski, O. Boulle, M. Kläui, Current-induced domain wall motion in nanoscale ferromagnetic elements, J. Phys.\nD 44 (2011) 384005. doi:10.1088/0022-3727/44/38/384005 .\n[362] O. Boulle, G. Malinowski, M. Kläui, Current-induced domain wall motion in nanoscale ferromagnetic elements, Mater.\nSci. Eng. R Reports 72 (2011) 159. doi:10.1016/j.mser.2011.04.001 .\n[363] C. T. Boone, J. A. Katine, M. Carey, J. R. Childress, X. Cheng, I. N. Krivorotov, Rapid Domain Wall Motion in\nPermalloy Nanowires Excited by a Spin-Polarized Current Applied Perpendicular to the Nanowire, Phys. Rev. Lett. 104\n(2010) 097203. arXiv:0909.1822 ,doi:10.1103/PhysRevLett.104.097203 .\n[364] A. Yamaguchi, H. Miyajima, T. Ono, Y. Suzuki, S. Yuasa, The rectification of radio-frequency signal by magnetic domain\nwall in a single-layered ferromagnetic nanowire, Appl. Phys. Lett. 91 (2007) 132509. doi:10.1063/1.2793620 .\n[365] E. Saitoh, H. Miyajima, T. Yamaoka, G. Tatara, Current-induced resonance and mass determination of a single magnetic\ndomain wall, Nature 432 (2004) 203. doi:10.1038/nature03119.1.\n[366] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, T. Shinjo, Real-Space Observation of Current-Driven Domain Wall\nMotion in Submicron Magnetic Wires, Phys. Rev. Lett. 92 (2004) 77201. doi:10.1103/PhysRevLett.92.077205 .\n[367] K. S. Buchanan, P. E. Roy, F. Y. Fradin, K. Y. Guslienko, M. Grimsditch, S. D. Bader, V. Novosad, Vortex dynamics in\npatterned ferromagnetic ellipses, J. Appl. Phys. 1 (2006) 172. arXiv:cond-mat/0602509 ,doi:10.1063/1.2175602 .\n[368] I. Neudecker, M. Kläui, K. Perzlmaier, D. Backes, L. J. Heyderman, C. A. F. Vaz, J. A. C. Bland, U. Rüdiger, C. H.\nBack, Spatially resolved dynamic eigenmode spectrum of Co rings, Phys. Rev. Lett. 96 (2006) 057207. doi:10.1103/\nPhysRevLett.96.057207 .\n[369] B. Van Waeyenberge, A. Puzic, H. Stoll, K. W. Chou, T. Tyliszczak, R. Hertel, M. Fähnle, H. Brückl, K. Rott, G. Reiss,\nI. Neudecker, D. Weiss, C. H. Back, G. Schütz, Magnetic vortex core reversal by excitation with short bursts of an\nalternating field., Nature 444 (2006) 461. doi:10.1038/nature05240 .\n[370] D. Bedau, M. Kläui, M. T. Hua, S. Krzyk, U. Rüdiger, G. Faini, L. Vila, Quantitative Determination of the Nonlinear\nPinning Potential for a Magnetic Domain Wall, Phys. Rev. Lett. 101 (2008) 256602. doi:10.1103/PhysRevLett.101.\n256602.\n[371] T. Ono, H. Miyajimia, K. Shigeto, K. Mibu, N. Hosoito, T. Shinjo, Propagation of a Magnetic Domain Wall in a\nSubmicrometer Magnetic Wire, Science 284 (1999) 468. doi:10.1126/science.284.5413.468 .\n[372] T. Ono, Y. Ooka, S. Kasai, H. Miyajima, N. Nakatani, N. Hayashi, K. Shigeto, K. Mibu, T. Shinjo, Magnetization reversal\nand electric transport in ferromagnetic nanowires, Mater. Sci. Eng. B 84 (2001) 126. doi:10.1016/S0921-5107(01)\n00585-2.\n[373] M. Kläui, C. A. F. Vaz, J. Rothman, J. A. C. Bland, W. Wernsdorfer, G. Faini, E. Cambril, Domain wall pinning in\nnarrow ferromagnetic ring structures probed by magnetoresistance measurements, Phys. Rev. Lett. 90 (2003) 097202.\ndoi:10.1103/PhysRevLett.90.097202 .\n[374] M. Hayashi, L. Thomas, Y. B. Bazaliy, C. Rettner, R. Moriya, X. Jiang, S. S. P. Parkin, Influence of current on field-\ndriven domain wall motion in permalloy nanowires from time resolved measurements of anisotropic magnetoresistance,\nPhys. Rev. Lett. 96 (2006) 197207. doi:10.1103/PhysRevLett.96.197207 .\n[375] A. von Bieren, A. K. Patra, S. Krzyk, J. Rhensius, R. M. Reeve, L. J. Heyderman, R. Hoffmann-Vogel, M. Kläui,\nDomain-wall induced large magnetoresistance effects at zero applied field in ballistic nanocontacts, Phys. Rev. Lett. 110\n(2013) 067203. arXiv:1208.4985v1 ,doi:10.1103/PhysRevLett.110.067203 .\n[376] M. Viret, D. Vignoles, D. Cole, J. Coey, W. Allen, D. Daniel, J. Gregg, Spin scattering in ferromagnetic thin films, Phys.\nRev. B 53 (1996) 8464. doi:10.1103/PhysRevB.53.8464 .\n[377] J. F. Gregg, W. Allen, K. Ounadjela, M. Viret, M. Hehn, S. M. Thompson, J. M. D. Coey, Giant magnetoresistive effects\nin a single element magnetic thin film, Phys. Rev. Lett. 77 (1996) 1580. doi:10.1103/PhysRevLett.77.1580 .\n[378] P. M. Levy, S. Zhang, Resistivity due to Domain Wall Scattering, Phys. Rev. Lett. 79 (1997) 5110. doi:10.1103/\nPhysRevLett.79.5110 .\n[379] U. Ebels, A. Radulescu, Y. Henry, L. Piraux, K. Ounadjela, Spin Accumulation and Domain Wall Magnetoresistance in\n35 nm Co Wires, Phys. Rev. Lett. 84 (2000) 983. doi:10.1103/PhysRevLett.84.983 .\n[380] C. H. Marrows, B. C. Dalton, Spin mixing and spin-current asymmetry measured by domain wall magnetoresistance,\nPhys. Rev. Lett. 92 (2004) 097206. doi:10.1103/PhysRevLett.92.097206 .\n73[381] S. Sangiao, M. Viret, Electrical detection of internal dynamical properties of domain walls, Phys. Rev. B 89 (2014)\n104412. doi:10.1103/PhysRevB.89.104412 .\n[382] X. Fan, Yunpeng Chen, Chong Bi, Yunsong Xie, J. Kolodzey, J. D. Wilson, R. N. Simons, H. Zhang, J. Q. Xiao, Magnetic\ntunnel junction-based on-chip microwave phase and spectrum analyzer, in: 2014 IEEE MTT-S Int. Microw. Symp., Vol. 1,\n2014, pp. 1–4. doi:10.1109/MWSYM.2014.6848560 .\n[383] B. M. Yao, L. Fu, X. S. Chen, W. Lu, L. H. Bai, Y. S. Gui, C.-M. Hu, Rapid microwave phase detection based on a solid\nstate spintronic device, Appl. Phys. Lett. 104 (2014) 062408. arXiv:1309.3572 ,doi:10.1063/1.4865374 .\n[384] Y. Guan, W. E. Bailey, E. Vescovo, C. C. Kao, D. A. Arena, Phase and amplitude of element-specific moment precession\nin Ni 81Fe19, J. Magn. Magn. Mater. 312 (2007) 374. doi:10.1016/j.jmmm.2006.10.1111 .\n[385] T.Martin, G.Woltersdorf, C.Stamm, H.A.Duerr, R.Mattheis, C.H.Back, G.Bayreuther, Layerresolvedmagnetization\ndynamics in interlayer exchange coupled Ni 81Fe19/Ru/Co 90Fe10by time resolved x-ray magnetic circular dichroism, J.\nAppl. Phys. 103 (2008) 07B112. doi:10.1063/1.2836340 .\n[386] D. A. Arena, Y. Ding, E. Vescovo, S. Zohar, Y. Guan, W. E. Bailey, A compact apparatus for studies of element and\nphase-resolved ferromagnetic resonance, Rev. Sci. Instrum. 80 (2009) 083903. arXiv:0905.2353 ,doi:10.1063/1.3190402 .\n[387] W. E. Bailey, C. Cheng, R. Knut, O. Karis, S. Auffret, S. Zohar, D. Keavney, P. Warnicke, J.-S. Lee, D. a. Arena,\nDetection of microwave phase variation in nanometre-scale magnetic heterostructures., Nat. Commun. 4 (2013) 2025.\ndoi:10.1038/ncomms3025 .\n[388] E. C. Fear, S. C. Hagness, P. M. Meaney, M. Okoniewski, M. A. Stuchly, Enhancing breast tumor detection with near-field\nimaging, Microw. Mag. IEEE 3 (2002) 48. doi:10.1109/6668.990683 .\n[389] E. C. Fear, P. M. Meaney, M. A. Stuchly, Microwaves for breast cancer detection?, Potentials, IEEE 22 (2003) 12.\ndoi:10.1109/MP.2003.1180933 .\n[390] R. K. Amineh, M. Ravan, A. Trehan, N. K. Nikolova, Near-field microwave imaging based on aperture raster scanning\nwith TEM horn antennas, IEEE Trans. Antennas Propag. 59 (2011) 928. doi:10.1109/TAP.2010.2103009 .\n[391] N. Nikolova, Microwave Imaging for Breast Cancer, IEEE Microw. Mag. 12 (2011) 78. doi:10.1109/MMM.2011.942702 .\n[392] E. A. Ash, G. Nicholls, Super-resolution aperture scanning microscope., Nature 237 (1972) 510. doi:10.1038/237510a0 .\n[393] L. Fu, W. Lu, D. Rodriguez Herrera, D. Flores Tapia, Y. S. Gui, S. Pistorius, C.-M. Hu, Microwave radar imaging using\na solid state spintronic microwave sensor, Appl. Phys. Lett. 105 (2014) 122406. doi:10.1063/1.4896691 .\n[394] B. M. Yao, L. Fu, X. S. Chen, W. Lu, H. Guo, Y. S. Gui, C.-M. Hu, Ground penetrating detection using miniaturized\nradar system based on solid state microwave sensor, Rev. Sci. Instrum. 84 (2013) 124702. doi:10.1063/1.4838662 .\n[395] Z. Wang, X. Zhu, W. Lu, Passive detection of polarization and relative phases for L-band microwaves, in: 2013 Int. Conf.\nElectromagn. Adv. Appl., 2013, p. 1386. doi:10.1109/ICEAA.2013.6632478 .\n[396] X. F. Zhu, M. Harder, J. Tayler, A. Wirthmann, B. Zhang, W. Lu, Y. S. Gui, C.-M. Hu, Nonresonant spin rectification\nin the absence of an external applied magnetic field, Phys. Rev. B 83 (2011) 140402. doi:10.1103/PhysRevB.83.140402 .\n[397] Z. H. Zhang, Y. S. Gui, L. Fu, X. L. Fan, J. W. Cao, D. S. Xue, P. P. Freitas, D. Houssameddine, S. Hemour, K. Wu,\nC.-M. Hu, Seebeck rectification enabled by intrinsic thermoelectrical coupling in magnetic tunneling junctions, Phys.\nRev. Lett. 109 (2012) 037206. doi:10.1103/PhysRevLett.109.037206 .\n[398] L. Fu, Z. X. Cao, S. Hemour, K. Wu, D. Houssameddine, W. Lu, S. Pistorius, Y. S. Gui, C.-M. Hu, Microwave reflection\nimaging using a magnetic tunnel junction based spintronic microwave sensor, Appl. Phys. Lett. 101 (2012) 232406.\ndoi:10.1063/1.4769837 .\n[399] Y. Gui, A. M. Mehrabani, D. Flores-Tapia, L. Fu, L. Bai, S. Pistorius, L. Shafai, C.-M. Hu, New horizons for microwave\napplications using spin caloritronics, Solid State Commun. 198 (2014) 45. doi:10.1016/j.ssc.2014.07.026 .\n[400] S.Hemour, K.Wu, Radio-frequencyrectifierforelectromagneticenergyharvesting: Developmentpathandfutureoutlook,\nProc. IEEE 102 (2014) 1667. doi:10.1109/JPROC.2014.2358691 .\n[401] C. P. Wen, C. P. Wen, A Surface Nonreciprocal Applications, IEEE Trans. Microw. Theory Tech. 17 (1969) 1087.\ndoi:10.1109/TMTT.1969.1127105 .\n[402] F. Giesen, J. Podbielski, B. Botters, D. Grundler, Vortex circulation control in large arrays of asymmetric magnetic rings,\nPhys. Rev. B 75 (2007) 184428. doi:10.1103/PhysRevB.75.184428 .\n[403] N. Kuhlmann, A. Vogel, G. Meier, Magnetization dynamics and cone angle precession in permalloy rectangles, Phys.\nRev. B 85 (2012) 014410. doi:10.1103/PhysRevB.85.014410 .\n[404] N. H. G. Grenda, P. A. Hyde, Y. S. Gui, M. P. Wismayer, J. D. A. Jung, C.-M. Hu, B. W. Southern, K.-W. Lin, Angular\ndependence of ferromagnetic resonance measurements in exchange coupled Ni 80Fe20/NiO bilayers, J. Phys. D 46 (2013)\n205002. doi:10.1088/0022-3727/46/20/205002 .\n[405] S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L. Schneider, P. Kabos, T. J. Silva, J. P. Nibarger, Ferromagnetic\nresonance linewidth in metallic thin films: Comparison of measurement methods, J. Appl. Phys. 99 (2006) 093909.\ndoi:10.1063/1.2197087 .\n[406] Ö. O. Soykal, M. E. Flatté, Strong Field Interactions between a Nanomagnet and a Photonic Cavity, Phys. Rev. Lett.\n104 (2010) 077202. arXiv:0907.3926 ,doi:10.1103/PhysRevLett.104.077202 .\n[407] Ö. O. Soykal, M. E. Flatté, Size dependence of strong coupling between nanomagnets and photonic cavities, Phys. Rev.\nB 82 (10) (2010) 104413. arXiv:1005.3068 ,doi:10.1103/PhysRevB.82.104413 .\n[408] H. Huebl, C. W. Zollitsch, J. Lotze, F. Hocke, M. Greifenstein, A. Marx, R. Gross, S. T. B. Goennenwein, High\ncooperativity in coupled microwave resonator ferrimagnetic insulator hybrids, Phys. Rev. Lett. 111 (2013) 127003. arXiv:\n1207.6039 ,doi:10.1103/PhysRevLett.111.127003 .\n[409] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, Y. Nakamura, Coherent coupling between\nferromagnetic magnon and superconducting qubit arXiv:1410.3781 .\n[410] X. Zhang, C.-L. Zou, L. Jiang, H. X. Tang, Strongly Coupled Magnons and Cavity Microwave Photons, Phys. Rev. Lett.\n74113 (15) (2014) 156401. arXiv:1405.7062 ,doi:10.1103/PhysRevLett.113.156401 .\n[411] L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, C.-M. Hu, Spin Pumping in Electrodynamically Coupled Magnon-\nPhoton Systems, Phys. Rev. Lett. 114 (2015) 227201. arXiv:1504.01335 ,doi:10.1103/physrevlett.114.227201 .\n[412] Y. Cao, P. Yan, H. Huebl, S. T. B. Goennenwein, G. E. W. Bauer, Exchange magnon-polaritons in microwave cavities,\nPhys. Rev. B 91 (2015) 094423. arXiv:1412.5809 ,doi:10.1103/PhysRevB.91.094423 .\n[413] B. Zare Rameshti, Y. Cao, G. E. W. Bauer, Magnetic spheres in microwave cavities, Phys. Rev. B 91 (2015) 214430.\narXiv:1503.02419v1 ,doi:10.1103/PhysRevB.91.214430 .\n[414] B. M. Yao, Y. S. Gui, Y. Xiao, H. Guo, X. S. Chen, W. Lu, C. L. Chien, C.-M. Hu Theory and experiment on\ncavity magnon-polariton in the one-dimensional configuration, Phys. Rev. B 92 (2015) 184407. arXiv:1509.05804 ,\ndoi:10.1103/PhysRevB.92.184407 .\n[415] M. Harder, L. Bai, C. Match, J. Sirker, C.-M. Hu, Study of the cavity-magnon-polariton transmission line shape arXiv:\n1601.06049 .\n[416] H. Maier-Flaig, M. Harder, R. Gross, H. Huebl, S. Goennenwein, Cavity spin pumping in different coupling regimes\narXiv:1601.05681 .\n[417] X. Zhang, C.-L. Zou, N. Zhu, F. Marquardt, L. Jiang, H. X. Tang, Magnon dark modes and gradient memory, Nat.\nCommun. 6 (2015) 8914. arXiv:1507.02791v1 ,doi:10.1038/ncomms9914 .\n[418] Y.Tabuchi, S.Ishino, A.Noguchi, T.Ishikawa, R.Yamazaki, K.Usami, Y.Nakamura, Quantummagnonics: Themagnon\nmeets the superconducting qubit, Comptes Rendus Phys. arXiv:1508.05290 ,doi:10.1016/j.crhy.2016.07.009 .\n[419] Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Yamazaki, K. Usami, Y. Nakamura, Coherent coupling between a\nferromagnetic magnon and a superconducting qubit, Science 349 (2015) 405. arXiv:1410.3781 ,doi:10.1126/science.\naaa3693.\n[420] A. Osada, R. Hisatomi, A. Noguchi, Y. Tabuchi, R. Yamazaki, K. Usami, M. Sadgrove, R. Yalla, M. Nomura, Y. Naka-\nmura, Cavity Optomagnonics with Spin-Orbit Coupled Photons, Phys. Rev. Lett. 116 (2016) 223601. arXiv:1510.01837 ,\ndoi:10.1103/PhysRevLett.116.223601 .\n[421] X. Zhang, N. Zhu, C.-L. Zou, H. X. Tang, Optomagnonic whispering gallery microresonators arXiv:1510.03545 .\n[422] J. A. Haigh, S. Langenfeld, N. J. Lambert, J. J. Baumberg, A. J. Ramsay, A. Nunnenkamp, A. J. Ferguson, Magneto-\noptical coupling in whispering-gallery-mode resonators, Phys. Rev. A 92 (2015) 063845. arXiv:1510.06661 ,doi:10.\n1103/PhysRevA.92.063845 .\n[423] J. Bourhill, N. Kostylev, M. Goryachev, D. L. Creedon, M. E. Tobar, Ultrahigh cooperativity interactions between\nmagnons and resonant photons in a YIG sphere, Phys. Rev. B 93 (2016) 144420. arXiv:1512.07773 ,doi:10.1103/\nPhysRevB.93.144420 .\n75"    },    {        "title": "1601.06213v1.Nonlinear_magnetization_dynamics_of_antiferromagnetic_spin_resonance_induced_by_intense_terahertz_magnetic_field.pdf",        "content": " 1Nonlinear magnetization dyna mics of antiferromagnetic \nspin resonance induced by inte nse terahertz magnetic field \n \nY Mukai1,2,4,6, H Hirori2,3,4,7, T Yamamoto5, H Kageyama2,5, and K Tanaka1,2,4,8 \u001f \n1 Department of Physics, Graduate School of  Science, Kyoto University, Sakyo-ku, Kyoto \n606-8502, Japan  \n2 Institute for Integrated Cell-Material Scien ces (WPI-iCeMS), Kyoto University, Sakyo-ku, \nKyoto 606-8501, Japan \n3 PRESTO, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan \n4 CREST, Japan Science and Technology Agency, Kawaguchi, Saitama 332-0012, Japan \n5 Department of Energy and Hydrocarbon Chemistr y, Graduate School of Engineering, Kyoto \nUniversity, Nishikyo-ku, Kyoto 615-8510, Japan \n \nE-mail: \n6 mukai@scphys.kyoto-u.ac.jp \n7 hirori@icems.kyoto-u.ac.jp \n8 kochan@scphys.kyoto-u.ac.jp \n We report on the nonlinear magnetization dynamics of a HoFeO\n3 crystal induced by a strong \nterahertz magnetic field resonantly enhanced with a split ring resonator and measured with \nmagneto-optical Kerr effect microscopy. The terahertz magnetic field induces a large change (~40%) in the spontaneous magnetization. The frequency of the antife rromagnetic resonance \ndecreases in proportion to the square of the magnetization change. A modified \nLandau-Lifshitz-Gilbert equation with a phenomenological nonlinear damping term \nquantitatively reproduced the nonlinear dynamics. \nPACS: 75.78.Jp, 76.50.+g, 78.47.-p, 78.67.Pt \n    21. Introduction \nUltrafast control of magnetization dynamics by a femtosecond optical laser pulse has \nattracted considerable attention from the persp ective of fundamental physics and technological \napplications of magnetic recording and inform ation processing [1]. The first observation of \nsubpicosecond demagnetization of a fe rromagnetic nickel film demonstrated that a femtosecond \nlaser pulse is a powerful stimulus of ultrafast magnetization dynamics [2], and it has led to numerous theoretical and experimental inves tigations on metallic and semiconducting magnets \n[3-8]. The electronic state created by the laser pulse has a strongly nonequilibrium distribution \nof free electrons, which consequently leads to  demagnetization or even magnetic reversal \n[1,2,9-11]. However, the speed of the magnetizat ion change is limited by the slow thermal \nrelaxation and diffusion, and an alternative t echnique without the limits of such a thermal \ncontrol and without excessive thermal energy would be desirable.  \nIn dielectric magnetic media, carrier heating hardly occurs, since no free electrons are present \n[12]. Consequently, great effort has been devoted  to clarifying the spin dynamics in magnetic \ndielectrics by means of femtosecond laser pulses. A typical method for nonthermal optical \ncontrol of magnetism is the inverse Faraday effect, where circularly polarized intense laser \nirradiation induces an effective magnetic field in  the medium. Recently, new optical excitation \nmethods avoiding the thermal effect such as the ma gneto-acoustic effect is also reported [13,14]. \nIn particular, these techniques have been used in  many studies on antiferromagnetic dielectrics \nbecause compared with ferromagne ts, antiferromagnets have inhe rently higher spin precessional \nfrequencies that extend into the terahertz (THz) regime [12,15]. Additionally, ultrafast \nmanipulation of the antiferromagnetic order parame ter may be exploited in order to control the \nmagnetization of an adjacent ferromagnet through the exchange interaction [16]. The THz wave \ngeneration technique is possibly a new way of optical spin control through direct magnetic \nexcitation without undesirable thermal effects [17-19]. As yet however, no technique has been \nsuccessful in driving magnetic motion excited directly by a magnetic field into a nonlinear \ndynamics regime that would presumably be fo llowed by a magnetization reversal [20-22]. \n \nIn our previous work [23], we demonstrated that the THz magnetic field can be resonantly \nenhanced with a split ring resonator (SRR) and may become a tool for the efficient excitation of \na magnetic resonance mode of antiferromagnetic dielectric HoFeO\n3. We applied a Faraday \nrotation technique to detect the magnetization ch ange but the observed Faraday signal averaged  3the information about inhomogeneous magnetiza tion induced by localized THz magnetic field \nof the SRR over the sample thickness [23]. In th is Letter, we have developed a time-resolved \nmagneto-optical Kerr effect (MOKE) micr oscopy in order to access the extremely \nfield-enhanced region, sample surface near th e SRR structure. As a result, the magnetic \nresponse deviates from the linear response in the strong THz magnetic field regime, remarkably \nshowing a redshift of the antiferromagnetic r esonance frequency that is proportional to the \nsquare of the magnetization change. The observe d nonlinear dynamics could be reproduced with \na modified Landau-Lifshitz-Gilbert (LLG) e quation having an additional phenomenological \nnonlinear damping term.  \n2. Experimental setup \nFigure 1 shows the experimental setup of MOKE microscopy with a THz pump pulse \nexcitation. Intense single-cycle THz pulses were  generated by optical rectification of \nnear-infrared (NIR) pulses in a LiNbO\n3 crystal [24-26]; the maximum peak electric field was \n610 kV/cm at focus. The sample was a HoFeO 3 single crystal polished to a thickness of 145 µm, \nwith a c-cut surface in the Pbnm  setting [27]. (The x-, y-, and z-axes are parallel to the \ncrystallographic a-, b-, and c-axes, respectively. ) Before the THz pump excitation, we applied a \nDC magnetic field to the sample to saturate its magnetization along the crystallographic c-axis. We fabricated an array of SRRs on the crystal surface by using gold with a thickness of 250 nm. \nThe incident THz electric field, parallel to the metallic arm with the SRR gap (the x-axis), drove \na circulating current that resulted in a strong  magnetic near-field normal to the crystal surface \n[23,28,29]. The SRR is essentially subwavelength LC circuit, and the current induces magnetic \nfield B\nnr oscillating with the LC resonance frequency (the Q-factor is around 4). The right side \nof the inset in figure 1 shows the spatial distribut ion of the magnetic field of the SRR at the LC \nresonance frequency as calculated by the fin ite-difference time-domain (FDTD) method. \nAround the corner the current density in the metal is very high, inducing the extremely \nenhanced magnetic field in the HoFeO 3 [29]. \n \nAt room temperature, the two magnetizations mi (i=1,2) of the different iron sublattices in \nHoFeO 3 are almost antiferromagnetically aligned along the x-axis with a slight canting angle \n0(=0.63°) owing to the Dzyaloshinskii fiel d and form a spontaneous magnetization MS along \nthe z-axis [30]. In the THz region, ther e are two antiferromagnetic resonance modes \n(quasiantiferromagnetic (AF) and quasiferroma gnetic (F) mode [31]). The magnetic field Bnr  4generated along the z-axis in our setup causes AF-mode motion; as illustrated in figure 2(a), the \nZeeman torque pulls the spins along the y-ax is, thereby triggering precessional motions of mi \nabout the equilibrium directions. The precessional motions cause the macroscopic magnetization M=m\n1+m2 to oscillate in the z-direction [32,33]. The resultant magnetization \nchange Mz(t) modulates the anti-symmetric off-diagonal element of the dielectric tensor \nεxyaሺൌ െεyxaሻ and induces a MOKE signal (Kerr ellipticity change \u001f [34,35] (see Appendix A \nfor the detection scheme of the MOKE measuremen t). The F-mode oscillation is also excited by \nTHz magnetic field along the x or y-axis. Howe ver, the magnetization deviations associated \nwith the F-mode, Mx and My, do not contribute to the MOKE in our experimental geometry, \nwhere the probe light was incident no rmal to the c-cut surface of HoFeO 3 (the xy-plane) [34,35]. \nIn addition, the amplitude of the F-mode is much smaller than AF-mode because the F-mode \nresonance frequency ( F~0.37 THz) differs from the LC resonance frequency ( LC~0.56 THz). \n-10010x position (µm)\n-10 0 10\ny position (µm)\nTHz pump HoFeO3 \nz y \nx \nSRR  \nObjective lens \nNonpolarized \nbeam splitter Quarter \nwave plateWollaston  \nprism \nLens  Balanced photodiodes \nVisible probe Bin Bnr \nEin \n-10 0 10\ny position (µm)\n120\n80\n40\n0\nFigure 1. Schematic setup of THz pump-visible MOKE measurement. The left side\nof the inset shows the photograph of SRR fabricated on the c-cut surface of the\nHoFeO 3 crystal and the white solid line indicat es the edge of the SRR. The red soli d\nand blue dashed circles indicate the probe spots for the MOKE measurement. The\nright side of the inset shows the spatia l distribution of the enhancement facto r\ncalculated by the FDTD method, i.e., the ratio between the Fourier amplitude at LC\nof the z-component of Bnr (at z=0) and the incident THz pulse Bin.  5To detect the magnetization change induced onl y by the enhanced magnetic field, the MOKE \nsignal just around the corner of the SRR (indicated by the red circle in figure 1’s inset), where \nthe magnetic field is enhanced 50-fold at the LC  resonance frequency, was measured with a 400 \nnm probe pulse focused by an objective lens (spot diameter of ~1.5 µm). Furthermore, although \nthe magnetic field reaches a maximum at the surface and decreases along the z-axis with a \ndecay length of lTHz~5 µm, the MOKE measurement in refl ection geometry, in contrast to the \nFaraday measurement in transmission [23], can evaluate the magnetization change induced only \nby the enhanced magnetic field around the sample  surface since the penetration depth of 400 nm \nprobe light for typical orthoferrites is on the orde r of tens of nm [35]. (The optical refractive \nindices of rare-earth orthoferrites in th e near ultraviolet region including HoFeO 3 are similar to \neach other, regardless of the rare-earth ion speci es, because it is mostly determined by the strong \noptical absorption due to charge transfer and orbital promotion transitions inside the FeO 6 \ntetragonal cluster [35].) All experiments in this  study were performed at room temperature. \n \n3. Results and discussions   \nFigure 2(a) (upper panel) shows the calculated  temporal magnetic waveform together with \nthe incident magnetic field. The maximum peak am plitude is four times that of the incident THz \npulse in the time domain and reaches 0.91 T. Th e magnetic field continues to ring until around \n25 ps after the incident pulse has decayed away. The spectrum of the pulse shown in figure 2(c) \nhas a peak at the LC resonance frequency ( LC=0.56 THz) of the SRR, which is designed to \ncoincide with the resonance frequency of the AF-mode ( νAF0=0.575 THz). Figure 2(a) (lower \npanel) shows the time development of the MOKE signal  for the highest THz excitation \nintensity (pump fluence I of 292 µJ/cm2 and maximum peak magnetic field Bmax of 0.91 T). The \ntemporal evolution of  is similar to that of the Faraday rotation measured in the previous \nstudy and the magnetization oscillates harmonically with a period of ~2 ps [23], implying that \nthe THz magnetic field coherently drives the AF-mode motion.  \n As shown in figure 2(b), as th e incident pump pulse intensity increases, the oscillation period \nbecomes longer. The Fourier transform spectra  of the MOKE signals for different pump \nintensities are plotted in figure 2(c). As the ex citation intensity increases, the spectrum becomes \nasymmetrically broadened on the lower freque ncy side and its peak frequency becomes \nredshifted. Figure 2(d) plots the center-of-mass fre quency (open circles) and the integral (closed \ncircles) of the power spectrum P(\n) as a function of incident pulse fluence. The center  6frequency monotonically redshifts and P() begins to saturate. As shown in figure 2(c), the \nMOKE spectra obtained at the center of the SRR (indicated in the inset of figure 1) does not \nshow a redshift even for the highest intensity excitation, suggesting that the observed redshift \noriginates from the nonlinearity of the precessional spin motion rather than that of the SRR \nresponse. We took the analytic signal approach  (ASA) to obtain the time development of the \ninstantaneous frequency (t) (figure 3(c)) and the envelope amplitude 0(t) (figure 3(d)) from \nthe measured magnetization change (t)=Mz(t)/|MS| (figure 3(b)) (see Appendix B for the \ndetails of the analytic signal approach). As is  described in the Appendix C, the MOKE signal \n6\n4\n2\n0\nIntegral of P( ) \n(arb. units)\n300 200 100 0\nFluence (µJ/cm2)0.575\n0.570\n0.565Frequency (THz)1.0\n0.8\n0.6\n0.4\n0.2\n0.0Intensity P( ) (arb. units)\n0.60 0.58 0.56 0.54\nFrequency (THz)  50%\n   100%\n 100% (x 3.7)\n (center)\n \n  10%\n |Bnr|2\n \nz \nm2 m1 \nM \nx Bnr \ny (a) (c) \n(d) (b) -0.020.000.02∆(degrees)\n40 30 20 10 0\nTime (ps)1.0\n0.5\n0.0\n-0.5B (T)  Bnr\n  Bin (x 3)\n0.08\n0.06\n0.04\n0.02\n0.00∆(degrees)\n24 20 16 12 8\nTime (ps)10%100%\nFigure 2. (a) Upper panel: Incident magnetic field of the THz pump pulse Bin estimated by\nelectro-optic sampling (dashed line) and the THz magnetic near-field Bnr calculated by the\nFDTD method (solid line). The illustration s hows the magnetization motion for the AF-mode.\nLower panel: The MOKE signal for a pump fluence of 292 µJ/cm2 (100%). (b) Comparison o f\ntwo MOKE signals for different pump fluences, vertically offset for clarity. (c) The FFT powe r\nspectrum of the magnetic near-field Bnr (black solid line). The spectra P() of the MOKE\nsignals for a series of pump fluences obtained at th e corner (solid lines) and at the center (blue\ndashed circle in the inset of figure 1) for a pump fluence of 100% (dashed line). Each spectru m\nof the MOKE signal is normalized by the peak amplitude at the corner for a pump fluence o f\n100%. (d) Intensity dependence of the center-of-m ass frequency (open circles) and the integral\n(closed circles) of the P().  7(t) is calibrated to the magnetization change (t) by using a linear relation, i.e., (t)=g(t), \nwhere g (=17.8 degrees−1) is a conversion coefficient. The tim e resolved experiment enables us \nto separate the contributions of the applied magnetic field and magnetization change to the \nfrequency shift in the time domain. A comparison of the temporal profiles between the driving \nmagnetic field (figure 3(a)) and the frequency e volution (figure 3(c)) shows that for the low \npump fluence (10%, closed blue circles), the frequency is redshifted only when the magnetic \nfield persists ( t < 25 ps), and after that, it recovers  to the constant AF mode frequency \n(νAF0=0.575 THz). This result indicates that the signals below t = 25 ps are affected by the \npersisting driving field and the redshift may orig inate from the forced oscillation. As long as the \n0.575\n0.570\n0.565\n0.560\n0.555Frequency (THz)\n50 40 30 20 10 0\nTime (ps)-0.4-0.20.00.20.4Magnetization \nchange 0.5\n0.0\n-0.5Bnr (T)\n0.4\n0.3\n0.2\n0.1\n0.0Amplitude 0  Experiment\n100%\n  10%\n  Experiment\n100%\nSimulation\n100% (1=0)\n100% \n  10% Simulation(a) \n(b) \n(c) \n(d) \nFigure 3. (a) FDTD calculated magnetic field Bnr for pump fluence of 100%. (b) Temporal\nevolution of the magnetization change obtained from the experimental data (gray circles) an d\nthe LLG model (red line). (c) Instantaneous frequencies and (d) envelope amplitudes fo r\npump fluences of 100% and 10% obtained by the analytic signals calculated from the\nexperimental data (circles) and the LLG simulation with nonlinear damping paramete r\n(1=1×10−3, solid lines) and without one ( 1=0, dashed line).  8magnetic response is under the linear regime, the instantaneous frequency is independent on the \npump fluence. However, for the high pump fluenc e (100%) a redshift (a maximum redshift of \n~15 GHz relative to the constant frequency νAF0) appears in the delay time ( t < 25 ps) and even \nafter the driving field decays away ( t > 25 ps) the frequency continues to be redshifted as long \nas the amplitude of the magnetization change is large. These results suggest that the frequency \nredshift in the high intensity case depends on the magnitude of the magnetization change, \nimplying that its origin is a nonlinear precessional spin motion with a large amplitude. \n \nThe temperature increase due to  the THz absorption (for HoFeO 3 T=1.7×10−3 K, for gold \nSRR T=1 K) is very small (see Appendix D). In ad dition, the thermal relaxation of the spin \nsystem, which takes more than a nanosecond [36], is much longer than the frequency \nmodulation decay (~50 ps) in figure 3(c). Therefore, laser heating can be ignored as the origin of the redshift.  \n \nFigure 4 shows a parametric plot of the instantaneous frequency \n(t) and envelope amplitude \n0(t) for the high pump fluence (100%). The instantaneous frequency shift for t > 25 ps has a \nsquare dependence on the amplitude, i.e., νAF=νAF0(1െCζ02). To quantify the relationship \nbetween the redshift and magnetization change, it  would be helpful to have an analytical \nexpression of the AF mode frequency AF as a function of the magnetization change, which is \nderived from the LLG equation based on the two- lattice model [32,33]. The dynamics of the \nsublattice magnetizations mi (i=1,2), as shown in the inset of figure 2(a), are described by \n \n dRi\ndt=െγ\n(1+α2)ቀRi×[B(t)+Beff,i]െαRi×൫Ri×[B(t)+Beff,i]൯ቁ,  (1) \n \nwhere Ri=mi/m0 (m0=|mi|) is the unit directional vector of the sublattice magnetizations, \n=1.76×1011 s−1T−1 is the gyromagnetic ratio, V(Ri) is the free energy of the iron spin system \nnormalized with m0, and Beff,i is the effective magnetic field given by െ∂V/∂Ri (i=1,2) (see \nAppendix E). The second term represents the ma gnetization damping with the Gilbert damping \nconstant \u001f \n \nSince Beff,i depends on the sublattice magnetizations mi and the product of these quantities \nappears on the right side of Eq. (1), the LLG e quation is intrinsically nonlinear. If the angle of  9the sublattice magnetization precession is sufficien tly small, Eq. (1) can be linearized and the \ntwo fixed AF- and F-modes for the weak excitation can be derived. However, as shown in figure \n3(b), the deduced maximum magnetization change  reaches ~0.4, corresponding to precession \nangles of 0.25° in the xz-plane and 15° in th e xy-plane. Thus, the magnetization change might \nbe too large to use the linear approximation. For such a large magnetization motion, assuming \nthe amplitude of the F-mode is zero and =0 in Eq. (1), the AF mode frequency AF in the \nnonlinear regime can be deduced as  \n \n νAF =νAF0ට1ିζ02tan2β0\nK(D),       ( 2 )  \n D =ඨ\t\t\t\t\t\tζ02(rAF2ି1) tan2β0\n1ିζ02tan2β0,      ( 3 )  \n \n0.575\n0.570\n0.565Frequency (THz)\n0.4 0.3 0.2 0.1 0.0\nAmplitude 0Experiment\n t > 25 ps\n t < 25 ps\n \n Analytic Solution\n 2nd order expansion\n \nFigure 4.  Relation between instantaneous frequency  and envelope amplitude 0 obtaine d\nfrom the magnetization change; for t < 25 ps (open circles) and for t > 25 ps (closed circles),\nthe analytic solution (blue line) and second orde r expansion of the analytic solution (gree n\ndashed line). Errors are estim ated from the spatial inhomogeneity of the driving magnetic\nfield (see Appendix H).  10where K(D) is the complete elliptic integral of the first kind, rAF(≈60) is the ellipticity of the \nsublattice magnetization precession trajectory of the AF-mode (see Appendix F), and 0 is the \namplitude of the (t). As shown in figure 4, the analytic solution can be approximated by the \nsecond order expansion νAF≈νAF0(1െtan2β0(rAF2െ1)ζ024⁄) and matches the observed redshift \nfor t > 25 ps, showing that the frequency appr oximately decreases with the square of (t). The \ndiscrepancy of the experimental data from the theoretical curve ( t < 25 ps) may be due to the \nforced oscillation of the AF-mode caused by the driving field.  \n \nTo elaborate the nonlinear damping effects, we compared the measured (t) with that \ncalculated from the LLG equation with the damping term. As shown in figures 3(c) and 3(d), the \nexperiment for the high intensity excitation devi ates from the simulation with a constant Gilbert \ndamping (dashed lines) even in the t > 25 ps time region, suggesting nonlinear damping \nbecomes significant in the large amplitude region. To describe the nonlinear damping \nphenomenologically, we modified the LLG equa tion so as to make the Gilbert damping \nparameter depend on the displacement of th e sublattice magnetization from its equilibrium \nposition, (Ri)=0+1Ri. As shown in figures 3(b)-(d), the magnetization change (t) derived \nwith Eq. (1) (solid line) with the damping parameters ( 0=2.27×10−4 and 1=1×10−3) nicely \nreproduces the experiments for both the high (100%) and low (10%) excitations.1 These results \nsuggest that the nonlinear damping plays a signifi cant role in the large amplitude magnetization \ndynamics. Most plausible mechanism for the nonlinear damping is four magnon scattering \nprocess, which has been introdu ced to quantitatively evaluate the magnon mode instability of \nferromagnet in the nonlinear response regime [37]. \n \n4. Conclusions  \nIn conclusion, we studied the nonlinear magnetization dynamics of a HoFeO 3 crystal excited \nby a THz magnetic field and measured by MOKE microscopy. The intense THz field can induce \nthe large magnetization change (~40%), and the ma gnetization change can be kept large enough \n                                            \n1 The damping parameter 0 (=2.27×10−4) and conversion coefficient  g (=17.8 degrees−1) are \ndetermined from the least-squares fit of the calculated result without the nonlinear damping \nparameter 1 to the experimental MOKE signal for the low pump fluence of 29.2 µJ/cm2. The \nnonlinear damping parameter 1 (=1×10−3) is obtained by fitting the experimental result for the \nhigh intensity case ( I=292 µJ/cm2) with the values of 0 and g obtained for the low excitation \nexperiment. The estimated value of g is consistent with the stat ic MOKE measurement; the Kerr \nellipticity induced by the spontaneous magnetization MS is ~0.05 degrees ( g~20 degrees−1). See \nAppendix G for details on the static Kerr measurement.  11to induce the redshift even after the field has gone , enabling us to separate the contributions of \nthe applied magnetic field and ma gnetization change to the frequency shift in the time domain. \nThe resonance frequency decreases in proportion to  the square of the magnetization change. A \nmodified LLG equation with a phenomenologi cal nonlinear damping term quantitatively \nreproduced the nonlinear dynamics. This suggest s that a nonlinear spin relaxation process \nshould take place in a strongly driven regime. Th is study opens the way to the study of the \npractical limits of the speed and efficiency of magnetization reversal, which is of vital \nimportance for magnetic recording and information processing technologies. \n   12Acknowledgments \nWe are grateful to Shintaro Takayoshi, Masah iro Sato, and Takashi Oka for their discussions \nwith us. This study was supported by a J SPS grants (KAKENHI 26286061 and 26247052) and \nIndustry-Academia Collaborative R&D grant fro m the Japan Science and Technology Agency \n(JST). \n   13Appendix A. Detection sche me of MOKE measurement \nWe show the details of the detection scheme  of the MOKE measurement. A probe pulse for \nthe MOKE measurement propagates along the z direction. By using the Jones vector [38], an electric field E\n0 of the probe pulse polarized linearly along the x-axis is described as  \n \n E0 =ቀ1\n0ቁ.       ( A . 1 )  \n \nThe probe pulse E1 reflected from the HoFeO 3 surface becomes elliptically polarized with a \npolarization rotation angle and a ellipticity angle . It can be written as \n \n E1 =R(െ߶)MR(θ)E0ൌ൬cos θ cos ߶െ\t݅ sin θ sin ߶\ncos θ sin ߶\t݅ sin θ cos ߶൰,  (A.2) \n  \nwhere  M is the Jones matrix describing \u001f\u001f phase retardation of the y component with \nrespect to the x component \n \nM=ቀ10\n0െiቁ,       ( A . 3 )  \n \nand R(ψ) is the rotation matrix \n \nR(ψ)=൬cosψ sinψ\nെsinψcosψ൰.      (A.4)  \n \nThe reflected light passes through the quarter wave plate, which is arranged such that its fast \naxis is tilted by an angle of 45° to the x-axis. The Jones matrix of the wave plate is given by \n \nRቀെπ\n4ቁMRቀπ\n4ቁ.        ( A . 5 )  \n Thus, the probe light E\n2 after the quarter wave plate is described as follows, \n  14E2 = ൬E2,x\nE2,y൰=Rቀെπ\n4ቁMRቀπ\n4ቁE1  \n=1\n2൬cosሺθ߶ሻെsin (θെ߶+)i(െcosሺθെ߶ሻsin (θ߶))\ncosሺθെ߶ሻsin (θϕ)+i(c o sሺθ߶ሻsin (θെ߶))൰.  (A.6)  \n \nThe Wollaston prism after the quarter wave plat e splits the x and y-polarization components of \nthe probe light E2. The spatially separated two pulses are incident to the balanced detector and \nthe detected probe pulse intensity ratio of the di fferential signal to the total corresponds to the \nKerr ellipticity angle as follows,    \n〈หாమ,ೣหమ〉ି〈หாమ,หమ〉\n〈หாమ,ೣหమ〉ା〈หாమ,หమ〉ൌെsin2θ.      ( A . 7 )  \n \nIn the main text, we show the Kerr ellipticity change =w−wo, where the ellipticity angles \n(w and wo) are respectively obtained with and without the THz pump excitation. \n \nAppendix B. Analytic signal approach and short time Fourier transform \nThe Analytic signal approach (ASA) allows the extraction of the time evolution of the \nfrequency and amplitude by a simple procedure and assumes that the signal contains a single \noscillator component. In our study, we measure only the MOKE signal originating from the \nAF-mode and it can be expected that the single oscillator assumption is valid. In the ASA, the \ntime profile of the magnetization change (t) is converted into an analytic signal (t), which is a \ncomplex function defined by using the Hilbert transform [39]; \n \nψ(t)=ζ0(t)exp( i߶(t))=ζ(t)+i ζ෨(t),     (B.1) \nζ෨(t)ൌ1\nπ pζ(t)\ntିτ∞\n-∞ dτ.       ( B . 2 )  \n \nwhere the p is the Cauthy principal value. The real part of (t) corresponds to (t). The real \nfunction 0(t) and (t) represent the envelope amplitude  and instantaneous phase of the \nmagnetization change. The instantaneous frequency (t)(=2(t)) is given by (t)=d(t)/dt. In \nthe analysis, we averaged 0(t) and (t) over a ten picosecond time range. \n \nTo confirm whether the ASA gives appropriate results, as shown in figure B.1 we compare  15them with those obtained by the short time Fourie r transform (STFT). As shown in figure B.1(a), \nthe time-frequency plot shows only one oscillato ry component of the AF-mode. As shown in \nfigures B.1(b) and (c), the instantaneous freque ncies and amplitudes obtained by the ASA and \nthe STFT are very similar. Because the ASA provides us the instantaneous amplitude with a \nsimple procedure, we showed the time evolu tions of frequency and amplitude derived by the \nASA in the main text. \n \nAppendix C. Determination of conversion coefficient g and linear damping parameter 0 \nThe conversion coefficient  g and the linear damping parameter 0(=) in Eq. (1) are \ndetermined by fitting the experimental MOKE signal (t) for the low pump fluence of 29.2 \nµJ/cm2 with the LLG calculation of the magnetization change (t). Figure C.1 shows the MOKE \nsignal (t) (circle) and the calculated magnetization change (t) (solid line). From the \nleast-squares fit of the calculated result to th e experiment by using a linear relation, i.e., \n(t)=g(t), we obtained the parameters g(=17.8 degrees−1) and 0(=2.27×10−4). 0.575\n0.570\n0.565\n0.560\n0.555\n0.550Frequency (THz)\n50 40 30 20 100\nTime (ps)ASA\n 100%\n   10%\nSTFT\n 100%\n   10%\n 1.0\n0.8\n0.6\n0.4\n0.2\n0.0\nFourier am plitude (arb. units)\n50 40 30 20 100\nTime (ps)0.4\n0.3\n0.2\n0.1\n0.0Amplitude 0ASA\n   100%\n     10% \nSTFT\n   100%\n     10%\n (a) (b) (c) \n1.2\n1.0\n0.8\n0.6\n0.4\n0.2\n0.0Frequency (THz)\n5040302010\nTime (ps)(arb. units)\n1.0 0.0\nFigure B.1.  (a) Time-dependence of the power spectrum of the magnetization \noscillation for the highest THz excitation ( I=292 µJ/cm2) obtained by the STFT. \nComparison of (b) instantaneous frequencies and (c) amplitudes obtained by the ASA \nand STFT with a time window with FWHM of 10 ps.   16 \n \nAppendix D. Laser heating effect \nThe details of the calculation of the temperature change are as follows: \n \nFor HoFeO 3:  \nThe absorption coefficient abs of HoFeO 3 at 0.5 THz is ~4.4 cm−1 [40]; the fluence IHFO \nabsorbed by HoFeO 3 can be calculated as IHFO=I(1−exp(−absd)), where d (=145 µm) is the \nsample thickness and I is the THz pump fluence. For the highest pump fluence, I=292 µJ/cm2, \nIHFO is 18.1 µJ/cm2. Since the sample thickness is much smaller than the penetration depth, \nd≪abs−1, we assume that the heating of the sample due to the THz absorption is homogeneous. \nBy using the heat capacity Cp of 100 J mol−1 K−1 [27], and the molar volume v of ~1.4×102 \ncm3/mol [27], the temperature change T can be estimated as\u001f T=IHFOv/Cpd ~1.7×10−3 K. \n \nFor gold resonator (SRR): \nThe split ring resonator has an absorption band (center frequency ~0.56 THz, band width ~50 \nGHz) originated from the LC resonance (figure 2( c)). Assuming the SRR absorbs all incident \nTHz light in this frequency band, the absorbed energy accounts for 3 % of the total pulse energy. \nHence, for the highest THz pump fluence, I=292 µJ/cm2, the fluence absorbed by the SRR is \nIgold=8.76 µJ/cm2. By using the heat capacity Cp of 0.13 J g−1 K−1 [41], the number of the SRRs \nper unit area N of 4×104 cm−2, and the mass of the SRR m of 1.6×10−9 g, the temperature change -10x10-3-50510degrees)\n50 40 30 20 10 0\nTime (ps)-0.10.00.1\nMagnetization \nchange  Experiment\n Simulation\nFigure C.1.  Experimentally observed MOKE signal \u001f(circle) and LLG simulatio n\nresult of the magnetization change \u001f(solid line) for the pump fluence of 29.2\nµJ/cm2.  17T can be estimated as\u001f T=Igold/CpNm ~ 1 K \n \nAppendix E. Free energy of HoFeO 3 \nThe free energy F of the iron spin (Fe3+) system based on the two-lattice model is a function \nof two different iron sublattice magnetizations mi, and composed of the exchange energy and \none-site anisotropy energy [32,33]. The free en ergy normalized by the sublattice magnetization \nmagnitude, V=F/m0 (m0=|mi|), can be expanded as a power series in the unit directional vector of \nthe sublattice magnetizations, Ri=mi/m0=(Xi,Yi,Zi). In the magnetic phase 4 (T > 58K), the \nnormalized free energy is given as follows [32,33]: \n \nV=ER1·R2+D(X1Z2െX2Z1)െAxx(X12+X22)െAzz(Z12+Z22),  (E.1) \n \nwhere E(=6.4×102 T) and D(=1.5×10 T) for HoFeO 3 are respectively the symmetric and \nantisymmetric exchange field [42]. Axx and Azz are the anisotropy constants. As mentioned in \nAppendix F, the temperature dependent values of the anisotropy constants can be determined \nfrom the antiferromagnetic resonance frequencies. The canting angle of Ri to the x-axis β0 \nunder no magnetic field is given by \n \ntan 2β0=D\nE+AxxିAzz.        ( E . 2 )  \n \nAppendix F. Linearized resonance modes and anisotropy constants ( Axx and Azz) \nThe nonlinear LLG equation of Eq. (1) can be  linearized and the two derived eigenmodes \ncorrespond to the AF and F-mode. The sublatti ce magnetization motion for each mode is given \nby the harmonic oscillation of mode coordinates; for the AF-mode ( QAF, \nPAF)=((X1−X2)s i nβ0+(Z1+Z2)c o sβ0, Y1−Y2), and for the F-mode ( QF, \nPF)=((X1+X2)sinβ0−(Z1−Z2)cosβ0, Y1+Y2), \n \nQAF=AAFcosωAFt,       ( F . 1 )  \nPAF=AAFrAFsinωAFt,      ( F . 2 )  \n \nQF=AFcosωFt,       ( F . 3 )  \nPF=AFrAFsinωFt,       ( F . 4 )   18 \nwhere AAF,F represents the amplitude of each mode. AF,F, and rAF,F are the resonance frequencies \nand ellipticities, which are given by    \n \nωAF=γට(b+a)(d-c),        ( F . 5 )  \nωF=γට(b-a)(d+c),        ( F . 6 )  \n rAF=γටሺௗିሻ\n(b+a),        ( F . 7 )  \n rF=γටሺௗାሻ\n(b-a),        ( F . 8 )  \n \nwhere =1.76×1011 s−1T−1 is the gyromagnetic ratio, and \n \n a=െ2Axxcos2β0െ2Azzsin2β0െEcos 2β0െDsin 2β0,  (F.9) \n b=E,         ( F . 1 0 )  \n c=2Axxcos2β0െ2Azzcos2β0+Ecos 2β0+Dsin 2β0,    (F.11) \n d=െEcos 2β0െDsin 2β0.       ( F . 1 2 )  \n \nSubstituting the literature values of the exchange fields ( E=6.4×102 T and D=1.5×10 T [42]) and \nthe resonance frequencies at room temperature ( AF/2=0.575 THz and F/2=0.37 THz) to \nEqs. (F.5) and (F.6), Axx and Azz can be determined to 8.8×10−2 T and 1.9×10−2 T. \n \nAppendix G. MOKE measurement for the spontaneous magnetization \nFigure G.1 shows time-development of the MOKE signals for the different initial condition \nwith oppositely directed magnetization. We applied the static magnetic field (~0.3 T) to saturate \nthe magnetization along the z-axis before the TH z excitation. The spontaneous magnetization of \nsingle crystal HoFeO a can be reversed by the much smaller magnetic field (~0.01 T) because of \nthe domain wall motion [27]. Then, we separately measured the static Kerr ellipticity angle \n\u001f\u001f\u001f\u001f\u001f\u001f  and THz induced ellipticity change  for different initial magnetization Mz=±Ms \nwithout the static magnetic field \u001f In figure G.1 we plot the summation of the time resolved \nMOKE signal \u001fand the static Kerr ellipticity \u001f\u001f\u001f\u001f\u001f\u001f  The sings of the ellipticity offset angle  19\u001f\u001f\u001f\u001f\u001f\u001f  for the different spontaneous magnetization (±M S) are different and their magnitudes \nare ~0.05 degrees. The conversion coefficient g(=1/~\u001f/0.05 degrees) is estimated to be ~20 \ndegrees−1, which is similar to the value dete rmined by the LLG fitting (~17.8 degrees−1). In the \ncase of the AF-mode excitation, the phases of the magnetization oscillations are in-phase \nregardless of the direction of the spontaneous magnetization M=±Ms, whereas they are \nout-of-phase in the case of the F-mode excitation. We can explain this claim as follows: In the \ncase of AF-mode excitation, the external THz magne tic field is directed along the z-direction as \nshown in the inset of figure 2(a), the signs of the torques acting on the sublattice magnetization \nmi (i=1,2) depends on the direction of mi, however, the resultant oscillation of the macroscopic \nmagnetization M= m1+m2 along the z-direction has same phase for the different initial condition \nM=±Ms. In the case of the F-mode excitation with the external THz magnetic field along the x \nor y-direction, the direction of the torques acting on the magnetization M depends on the initial \ndirection and the phase of the F-mode osc illation changes depending on the sign of the \nspontaneous magnetization ±Ms. \n \nAppendix H. Influence of the spatial distri bution of magnetic field on magnetization \nchange \nAs shown in the inset of figure 1, the pump magnetic field  strongly localizes near the metallic \narm of the SRR and the magnetic field strength significantly depends on the spatial position r \nwithin the probe pulse spot area. The intensity distribution of the probe pulse Iprobe(r) has an 0.05\n0.00\n-0.05Kerr ellipticity  (degrees)\n25 20 15 10 5\nTime (ps) +MS\n -MS\n \nFigure G.1. The MOKE signals, the temporal change of the Kerr ellipticity , measured \nfor different initial conditions with oppositely directed magnetizations.       20elongated Gaussian distribution with spatial widt hs of 1.1 µm along the x-axis and 1.4 µm along \nthe y-axis [full width at half maximum (FWHM)  intensity]. The maximum magnetic field is 1.2 \ntimes larger than the minimum one in the spot  diameter, causing the different magnetization \nchange dynamics at different positions. To take into account this spatial inhomogeneity to the \nsimulation, the spatially weighted average of magnetization change ζ̅(t) has to be calculated as \nfollows:  \n \n ζ̅(t)=ζ(r,t)Iprobe(r)dr\nIprobe(r)ௗr ,       ( H . 1 )  \n \nwhere (r,t) is a magnetization change at a position r and time t.  \n \nFigure H.1(a) shows the simulation result of the spatially averaged magnetization change ζ̅(t) \nand the non-averaged (r0,t) without the nonlinear damping term ( 1=0), where r0 denotes the \npeak position of Iprobe(r). For the low excitation intensity (10%), ζ̅(t) is almost the same as \n(r0,t) as shown in figure H.1(a). On the other hand, for the high excitation intensity, the spatial \ninhomogeneity of magnetization change dyna mics induces a discrepancy between the ζ̅(t) and (a) (b) (c) \n-0.100.000.10Magnetization change\n50 40 30 20 100\nTime (ps)-0.6-0.4-0.20.00.20.4 Averaged\n Non-averaged\n Averaged\n Non-averaged100%  10%\n0.575\n0.570\n0.565\n0.560\n0.55550 40 30 20 100\nTime (ps)0.575\n0.570\n0.565\n0.560\n0.555Frequency (THz) Averaged\n Non-averaged\n Experiment\n Averaged\n Non-averaged\nExperiment\n100%  10%\n0.4\n0.3\n0.2\n0.1\n0.0\n50 40 3020 100\nTime (ps)0.12\n0.08\n0.04\n0.00Amplitude   Averaged\n Non-averaged\n Experiment\n Averaged\n Non-averaged\n Experiment100%  10%\nFigure H.1.  Comparison of the spatially averag ed and non-averaged magnetization \nchange for the different pump fluences of 10% and 100%. (a) Temporal evolutions of \nthe magnetization change, (b) instantaneous  frequencies and (c) normalized envelope \namplitudes. Open circles show the experimental results.      21(r0,t). This discrepancy is caused by the quasi-i nterference effect between the magnetization \ndynamics with different frequencies and amplit udes at different positions. Figures H.1(b) and \n(c) show the instantaneous freque ncy and envelope amplitude obt ained from the data shown in \nfigure H.1(a) by using analytic signal approach with the experimental result. For the averaged \nmagnetization change, the frequency redshift is more emphasized (figure H.1(b)) and the decay \ntime becomes shorter (figure H.1(c)). Nonetheless,  neither spatially averaged nor non-averaged \nsimulation reproduces the experimental result of  the instantaneous frequency (figure H.1(b)) \nwithout nonlinear damping term.  \n \n   22References \n[1] Kirilyujk A, Kimel A V and Rasing T 2010 Rev. Mod. Phys.  82 2731 \n[2] Beaurepaire E, Merle J-C, Daunois A and Bigot J-Y 1996 Phys. Rev. Lett.  76 4250 \n[3] Barman A, Wang S, Maas J D, Hawkins A R, Kwon S, Liddle A, Bokor J and Schmidt H \n2006 Nano Lett.  6 2939 \n[4] Bovensiepen U 2007 J. Phys.: Condens. Matter  19 083201 \n[5] Wang J, Cotoros I, Dani K M, Liu X, Furdyna J K and Chemla D S 2007 Phys. Rev. Lett.  \n98 217401 \n[6] Fiebig M, Duong N P, Satoh T, Van Aken B B, Miyano K, Tomioka Y and Tokura Y 2008 \nJ. Phys. D: Appl. Phys.  41 164005 \n[7] Au Y , Dvornik M, Davison T, Ahmad E, Kea tley P S, Vansteenkiste A, Van Waeyenberge \nB and Kruglyak V V 2013 Phys. Rev. Lett.  110 097201 \n[8] Bigot J-Y and V omir M 2013 Ann. Phys. (Berlin)  525 2 \n[9] Koshihara S, Oiwa A, Hirasawa M, Katsumoto S, Iye Y , Urano C, Takagi H and Munekata \nH 1997 Phys. Rev. Lett.  78 4617 \n[10] Vahaplar K, Kalashnikova A, Kimel A, Hinzke D, Nowak U, Chantrell R, Tsukamoto A, \nItoh A, Kirilyuk A and Rasing T 2009 Phys. Rev. Lett.  103 117201 \n[11] Ostler T A, Barker J, Evans R and Chantrell R W 2012 Nat. Commun.  3 666 \n[12] Kimel A V , Kirilyuk A, Usachev P A, Pisarev R V , Balbashov A M and Rasing T 2005 \nNature  435 655 \n[13] Kim J-W, V omir M and Bigot J-Y 2012 Phys. Rev. Lett.  109 166601 \n[14] Kim J-W, V omir M and Bigot J-Y 2015 Sci. Rep.  5 8511 \n[15] Satoh T, Cho S-J, Iida R, Shimura T, Kuroda K, Ueda H, Ueda Y , Ivanov B A, Nori F and \nFiebig M 2010 Phys. Rev. Lett.  105 077402 \n[16] Le Guyader L, Kleibert A, Nolting F, Joly L, Derlet P, Pisarev R, Kirilyuk A, Rasing T and \nKimel A 2013 Phys. Rev. B  87 054437 \n[17] Yamaguchi K, Nakajima M and Suemoto T 2010 Phys. Rev. Lett.  105 237201 \nYamaguchi K, Kurihara T, Minami Y , Nakajima M and Suemoto T 2013 Phys. Rev. Lett.  \n110 137204 \n[18] Kampfrath T, Sell A, Klatt G, Pashkin A and Mährlein S 2011 Nature Photon. 5 31 \n[19] Wienholdt S, Hinzke D and Nowak U 2012 Phys. Rev. Lett.  108 247207 \n[20] Back C H, Weller D, Heidmann J, Mauri D, Guarisco D, Garwin E L and Siegmann H C \n1998 Phys. Rev. Lett.  81 3251  23[21] Tudosa I, Stamm C, Kashuba A B, King F, Siegmann H C, Stöhr J, Ju G, Lu B and Weller \nD 2004 Nature  428 831 \n[22] Afanasiev D, Razdolski I, Skibinsky K M,  Bolotin D, Yagupov S V , Strugatsky M B, \nKirilyuk A, Rasing T and Kimel A V 2014 Phys. Rev. Lett.  112 147403 \n[23] Mukai Y , Hirori H, Yamamoto T, Kageyama H and Tanaka K 2014 Appl. Phys. Lett.  105 \n022410 \n[24] Yeh K-L, Hoffmann M C, Hebling J and Nelson K A 2007 Appl. Phys. Lett.  90 171121 \n[25] Hebling J, Yeh K-L, Hoffmann M C, Bartal B and Nelson K A 2008 J. Opt. Soc. Am. B  25 \nB6 \n[26] Hirori H, Blanchard F and Tanaka K 2011 Appl. Phys. Lett.  98 091106 \n[27] Shao M, Cao S, Wang Y , Yuan S, Ka ng B, Zhang J, Wu A and XU J 2011 J. Cryst. Growth  \n318 947 \n[28] Padilla W J, Taylor A J, Highstrete C, Lee M and Averitt R D 2006 Phys. Rev. Lett.  96 \n107401 \n[29] Kumar N, Strikwerda A C, Fan K, Zhang X, Averitt R D, Planken P and Adam A J 2012 \nOpt. Express  20 11277 \n[30] V orob'ev G P, Kadomtseva A M, Krynetkii I B and Mukhin A A 1989 Sov. Phys. JETP 68 \n604 \n[31] Balbashov A, Kozlov G and Lebedev S 1989 Sov. Phys. JETP 68 629 \n[32] Herrmann G F 1963 J. Phys. Chem. Solids 24 597 \n[33] Perroni C A and Liebsch A 2006 Phys. Rev. B  74 134430 \n[34] Iida R, Satoh T, Shimura T, Kuroda K, Ivanov B, Tokunaga Y and Tokura Y 2011 Phys. \nRev. B  84 064402 \n[35] Kahn F J, Pershan P S and Remeika J P 1969 Phys. Rev.  186 891 \n[36] Kimel A, Ivanov B, Pisarev R, Usachev P, Kirilyuk A and Rasing T 2009 Nat. Phys.  5 727 \n[37] Suhl H 1957 Phys. Chem. Solids  1 209 \n[38] Gerrard M and Burch J 1975 Introduction to Matrix Methods in Optics  (Wiley, London) \n[39] Cohen L 1989 Proc. IEEE  77 941 \n[40] Kozlov G V , Lebedev S P, Mukhin A A, Prokhorov A S, Fedorov I V , Balbashov A M and \nParsegov I Y 1993 IEEE Trans. Magn. 29 3443 \n[41] Takahashi Y and Akiyama H 1986 Thermochimica Acta 109 105 \n[42] Koshizuka N and Hayashi K 1988 J. Phys. Soc. Jpn.  57 4418 "    },    {        "title": "0903.3739v1.Quantum_Spinodal_Phenomena.pdf",        "content": "arXiv:0903.3739v1  [cond-mat.other]  22 Mar 2009,\nQuantum Spinodal Phenomena\nSeiji Miyashita∗\nDepartment of Physics, Graduate School of Science,\nThe University of Tokyo, 7-3-1 Hongo,\nBunkyo-Ku, Tokyo 113-8656, Japan and\nCREST, JST, 4-1-8 Honcho Kawaguchi, Saitama 332-0012, Japa n\nHans De Raedt\nDepartment of Applied Physics, Zernike Institute of Advanc ed Materials,\nUniversity of Groningen, Nijenborgh 4,\nNL-9747 AG Groningen, The Netherlands\nBernard Barbara\nInstitut N´ eel, CNRS, 25 Ave. des martyrs,\nBP 166, 38 042 Grenoble Cedex 09, France\n(Dated: November 5, 2018)\n1Abstract\nWe study the dynamical magnetization process in the ordered ground-state phase of the trans-\nverse Ising model under sweeps of magnetic field with constan t velocities. In the case of very slow\nsweeps and for small systems studied previously (Phys. Rev. B 56, 11761 (1997)), non-adiabatic\ntransitions at avoided level-crossing points give the domi nant contribution to the shape of mag-\nnetization process. In contrast, in the ordered phase of thi s model and for fast sweeps, we find\nsignificant, size-independent jumps in the magnetization p rocess. We study this phenomenon in\nanalogy to the spinodal decomposition in classical ordered state and investigate its properties and\nits dependence on the system parameters. An attempt to under stand the magnetization dynamics\nunder field sweep in terms of the energy-level structure is ma de. We discuss a microscopic mecha-\nnismof magnetization dynamics fromaviewpoint of local clu ster flipsandshow that thisprovides a\npicture that explains the size independence. The magnetiza tion dynamics in the fast-sweep regime\nis studied by perturbation theory and we introduce a perturb ation scheme based on interacting\nLandau-Zener type processes to describe the local cluster fl ip dynamics.\nPACS numbers: 75.10.Jm, 75.50.Xx, 75.45.1j\n2I. INTRODUCTION\nIt is well-known that non-adiabatic transitions among adiabatic eigen states take place\nwhen an external field is swept with finite velocity1,2. In particular, at avoided level-crossing\npoints strong non-adiabatic transitions occur, causing a step-wis e magnetization process3.\nIn so-called single molecular magnets4, the energy level diagram consists of discrete levels\nbecause the molecules contain only small number of magnetic ions and hence the quantum\ndynamics plays important roles. In particular, in the easy-axis large spin molecules such as\nMn12and Fe 8, step-wise magnetization processes have been found and they ar e attributed\nto the adiabatic change, that is the quantum tunneling at the avoide d level-crossing points,\nand are called resonant tunneling phenomena5. The Landau-Zener mechanism also causes\nvarious interesting magnetization loops in field cycling processes6.\nThe amount of the change of the magnetization at a step in the magn etization process is\ngoverned by the Landau-Zener mechanism and depends significant ly on the energy gap at\nthe crossing. This dependence has played an important role in the st udy of single-molecule\nmagnets. Observations of the gap have been done on isolated magn etic molecules7.\nThe quantum dynamics of systems of strongly interacting systems which show quantum\nphase transitions is also of much contemporary interest. As far as static properties are\nconcerned, the action inthe path-integral representation of a d-dimensional quantum system\nmaps onto the partition function of a ( d+1)-dimensional classical model, which is the key\ningredient of the quantum Monte Carlo simulation.8From this mapping, it follows that the\ncritical properties of the ground state of the d-dimensional quantum system are the same as\nthose of equilibrium state of the ( d+1)-dimensional classical model, quantum fluctuations\nplaying the role of the thermal fluctuations at finite temperatures .\nHowever, from a view point of dynamics, the nature of the quantum and thermal fluc-\ntuations are not necessarily the same. Thus, it is of interest to stu dy dynamical aspects of\nquantum critical phenomena. As a typical model showing quantum c ritical phenomena, in\nthe present work we adopt the one-dimensional transverse Ising model9.\nRecently, interesting properties of molecular chains which are mode led by the transverse\nIsing model with large spins have been reported10. However, in this paper, we concentrate\nourselves insystems of S= 1/2. Thedynamics ofthetransverse Ising modelplaysimportant\nroles in the study of the quantum annealing in which the quantum fluct uations due to the\n3transverse field are used to survey the ground state in a complex s ystem11. The dynamics of\ndomain growth under the sweep of the transverse field through th e critical point has been\nstudied related to the Kibble-Zurek mechanism12,13.\nIn this paper, we study the hysteresis behavior as a function of th e external field in\nthe ordered state by performing simulations of pure quantum dyna mics, that is by solving\nthe time-dependent Schr¨ odinger equation.14This gives us numerically exact results of the\ndynamical magnetization process of the transverse Ising model u nder sweeps of magnetic\nfield with constant velocities.\nPreviously we have studied time evolution of magnetization of the tra nsverse Ising model\nfrom a view point of Landau-Zener transition, sweeping the field slow ly and finding transi-\ntions at each avoided level crossing point.3However, for fast sweeps the transition at zero\nfieldHz= 0 disappears and the magnetization does not change even after t he field reverses.\nThe magnetization remains in the direction opposite to the external field for a while, and\nwhen the magnetic field reaches an certain value, the magnetization suddenly changes to\nthe direction of the field. This sudden change is also found for very s low sweeps at the level\ncrossing point. However, thepresent case has the following two diff erences: (1) the switching\nfield does not necessarily corresponds to a level crossing, and (2) in all cases the changes\nare independent of the size Lof the system. This sudden change resembles the change of\nmagnetization at the coercive field in the hysteresis loop of ferroma gnetic systems, where it\nis called spinodal decomposition. Therefore, we will call the phenome non that we observe in\nthe quantum system a ”quantum spinodal decomposition” and the fi eld “quantum spinodal\npoint”HSP. We study the dependence of HSPon the transverse field Hx, and also study the\nsweep-velocity dependence of HSP.\nAs in the cases of the single molecular magnets, it should be possible to understand the\ndynamics of the magnetization in terms of the energy levels as a func tion of field. However,\nbecause the structure of the energy-level diagramstrongly dep ends on the size of the system,\nitisdifficulttoexplainthesize-independent propertyofthequantum spinodaldecomposition\nfrom the energy-level structure only. In the case of much faste r sweeps, we find almost\nperfect size-independent magnetization processes. We also find a peculiar dependence of\nmagnetization on the field in the case of weak transverse fields. The se processes can be\nunderstood from the energy-level diagram for local flips of spins, but not from the energy\ndiagram of the total system.\n4Inthis paper, we attempt to understand the microscopic mechanis m that gives rise to this\nsize independent dynamics. We introduce a perturbation scheme fo r fast sweeps, regarding\nthe fast sweeping field term as the unperturbed system and treat ing the interaction term\nas the perturbation. From this viewpoint, we investigate fundamen tal, spatially local time-\nevolutions which yield the size-independent response to the sweep p rocedure. In particular,\nwe propose a perturbation scheme in terms of independent Landau -Zener systems, each of\nwhich consists of a spin in a transverse and sweeping field. A system c onsisting of locally\ninteracting Landau-Zener systems explains well the magnetization dynamics for fast sweeps.\nII. MODEL\nWe study characteristics of dynamics of the order parameter of t he one-dimensional\ntransverse-Ising modelwithperiodicboundaryconditionunderas weeping field.9TheHamil-\ntonian of the system is given by\nH(t) =−J/summationdisplay\niσz\niσz\ni+1−Hx/summationdisplay\niσx\ni−Hz(t)/summationdisplay\niσz\ni, (1)\nwhereσx\niandσz\niare thexandzcomponents of the Pauli matrix, respectively. Hereafter, we\ntakeJas a unit of the energy. The order parameter is the zcomponent of the magnetization\nMz=/summationdisplay\niσz\ni. (2)\nWe study dynamics of the order parameter of the model, i.e., the time dependence of the\nmagnetization under the time dependent field Hz(t)\n∝angbracketleftMz∝angbracketright=∝angbracketleftΨ(t)|Mz|Ψ(t)∝angbracketright, (3)\nwhere|Ψ(t)∝angbracketrightis a time dependent wavefunction given by the Schr¨ odinger equatio n\ni¯h∂\n∂t|Ψ(t)∝angbracketright=H(t)|Ψ(t)∝angbracketright. (4)\nIn the present paper we study the case of linear sweep of the field\nHz(t) =−H0+ct, (5)\nwhere−H0is an initial magnetic field. In the present paper, we set H0/J= 1, andcis the\nspeed of the sweep. We use a unit where ¯ h= 1.\n5In the case Hz= 0, the model shows an order-disorder phase transition as a func tion of\nHx. The transition point is given by Hc\nx=J. In the ordered phase ( Hx< J), the system\nhas a spontaneous magnetization ms:\nms= lim\nHz→+0lim\nL→∞∝angbracketleftG(0)|Mz|G(0)∝angbracketright, (6)\nwhere|G(0)∝angbracketrightis the ground state of the model with Hz= 0. Therefore, the ground state is\ntwofold degenerate with symmetry-broken magnetization, while th e ground state is unique\nwhenHx>J. Because of these twofold symmetry-broken ground states, th e magnetization\nchanges discontinuously at Hz= 0.\nIn a finite system L<∞, this degeneracy is resolved by the quantum mixing (tunneling\neffect) and a small gap opens at Hz= 0. This gap becomes small exponentially with L\nas shown in Appendix A. Therefore, the change of the magnetizatio n becomes sharper as\nLincreases. Dynamical realization of this change by field sweeping bec omes increasingly\ndifficult with L. This phenomenon corresponds to the existence of metastable st ate.\nThe energy-level diagram becomes complicated when Lincrease. However, as shown\nbelow, when cis large the system shows a size-independent magnetization dynamic s which\nis not easily understood in terms of the energy-level diagram. In th is paper, we focus on the\nregime of moderate to large sweep velocities.\nIII. ENERGY STRUCTURE\nIn Fig. 1(left), we present an energy-level diagram for L= 6 andHx= 0.7. We plot all\nenergy levels as a function of Hz. We find that the energy levels show a linear dependence\nat large fields, where quantum fluctuations due to Hxhave little effect. The levels are mixed\nin the region −3< Hz/J <3 where the energy levels come close and are mixed by the\ntransverse field. The isolated two lowest energy levels are located u nder a densely mixed\narea, which represent the ordered states with M=Land−L, and they cross at Hz= 0 with\na small gap ∆ E1, reflecting the tunneling between the symmetry broken states. T he gap\n∆E1is so small that one cannot see it in Fig. 1(left). After the crossing, these states join\nthe densely mixed area. In Fig. 1(right), we show the energy levels f orL= 16 where we plot\nonly energies of a few low-energy states. In this figure, we also find the above mentioned\ncharacteristic structure of two lowest energy levels.\n6-34-32-30-28-26-24-22-20-18-16\n-1 -0.5 0 0.5 1E/J\nHz/J-50-40-30-20-10010203040\n-6-4-20246E/J\nHz/J\nFIG. 1: (Color online) Typical energy-level diagrams of mod el Eq. (1). Left: full spectrum for\nL= 6,Hx= 0.7. Right: A few low-energy states for L= 16,Hx= 0.7.\nLet us point out a few more characteristic features of the energy -level diagram. A finite\ngap ∆E2exists between the crossing point of the low-lying lines and the dense ly populated\nregion of excited states. The d-dimensional Ising model in a transverse field is closely related\nto the transfer matrix of ( d+1)-dimensional Ising model. From this analogy, we associate\n∆E1to symmetry breaking phenomena. When symmetry breaking takes place, the two\nlargest eigenvalues of the transfer matrix of the model become alm ost degenerate. The\nenergy gap corresponds to the tunneling through the free-ener gy barrier between the two\nordered states and vanishes exponentially with the system size. On the other hand, ∆ E2\nis related to the correlation length of the fluctuation of anti-paralle l domains in the order\nstate. The correlation length is finite at a given temperature in the o rdered state and is\nalmost size-independent. At Hz= 0 we can calculate eigenenergies analytically, and we can\nexplicitly confirm that ∆ E1vanishes exponentially with Land that ∆ E2is almost constant\nas a function of the size. The dependencies of the energy gaps at Hz= 0 are discussed in\nAppendix A.\nFor largeHz, the slopes of the low lying isolated lines are ±Lbecause they represent\nthe states with M=±L. Thus, the field at which the lines merge in the area of densely\npopulated excited states is given by\nHz≃∆E2\nL≡H∗\nz(L). (7)\nAt this point, the magnetization shows a jump when the speed of the sweep is very slow.3\n7However, as we will see in the following sections, the dynamical magne tization does not\nshow any significant change at this field value when the sweeping field is fast. Another type\nof jump will occur that we called quantum spinodal jump or quantum s pinodal transition.\nIV. EVOLUTION OF THE MAGNETIZATION FOR FAST SWEEPS OF THE\nFIELD\nA. Quantum spinodal decomposition\nWhen we sweep the magnetic field from Hz=−1 toHz= 1, the magnetization shows\na rapid increase to a positive value. In Fig. 2, we depict examples of dy namics of the\nmagnetization as a function of time for a sweeping velocity c= 0.001. Because Hz(t) =\n−H0+ct,Hzalso represents time.\nThe magnetization stays at a negative value until a certain field stre ngth is reached. The\nsystem can be regarded as being in a metastable state. Then, the m agnetization changes\nsignificantly towards the direction of the field in a single continuous ju mp, the magnetization\nprocessesMz(t) depending very weakly on the system size. In the classical ordere d state,\nwe know a similar behavior. Namely, at the coercive field (at the edge o f the hysteresis),\nthe magnetization relaxes very fast and the relaxation time does no t depend on the size.\nThus, we may make an analogy to the spinodal decomposition phenom ena. We call the\nphenomenon that we observe in the quantum system “quantum spin odal decomposition”\nand we call the field at which the magnetization changes HSP. It should be noted that the\nspinodal decomposition corresponds to the fact that the size of t he critical nuclei becomes\nof the order one. If the size of the critical nuclei is larger than the size of the particle as\nin the case of nanoparticles, the critical field of the sudden magnet ization reversal, which is\nalso a kind of spinodal decomposition, strongly depends on the size.\nFirst, let us attempt to understand this dynamics from the view poin t of the energy\ndiagram. As we mentioned in the previous section, the low-lying levels o fM=±Lmerge\nwith the continuum at Hz=H∗\nz. Thus, we expect that at this point the magnetization\nbegins to change because the states with M=±Lbegin to cross other states. In fact, in\nearlier work, we found stepwise magnetization processes at avoide d level-crossings in very\nslow sweeps, each of which couldbe analyzed interms ofsuccessive L andau-Zener crossings.3\n8-1-0.500.51\n-1 -0.5 0 0.5 1<Mz>/L\nHz/J-1-0.500.51\n-1 -0.5 0 0.5 1<Mz>/L\nHz/J\nFIG. 2: (color online) Left: Magnetization Mz(t) as a function of Hz(t) forc= 0.001,Hx= 0.5\nand various system sizes. Solid (red) line: L= 12; Dashed (green) line: L= 14; Dotted (magenta)\nline:L= 16; Right: Same as left except that Hx= 0.7.\nFromFig. 2(left), we find that the sharp change of Mz(t) starts atHz= 0.2∼0.25, which\nis much larger than H∗\nz(L). We estimate H∗\nz(12)≃0.18, and for larger lattices H∗\nz(L) is even\nsmaller for larger lattices. Moreover, it should be noted that the ma gnetization processes\ndisplay almost no size-dependence. In Fig. 2(right), which shows Mz(t) forHx= 0.7, we also\nfind that the magnetization processes Mz(t) for all sizes Lare very similar. Here HSP≈0.11\nis again significantly larger than H∗\nz(L) (forL= 16 and ∆ E2≈1.4 in the case Hx= 0.7,\nand hence H∗\nz(14)≃0.09). This observation is in conflict with the picture based on the\nstructure of the energy level diagram given earlier.\nIn Fig. 3, we present an example of sweep-velocity dependence for a system with L= 20\n(results of other sizes are not shown). The magnetization proces ses show strong dependence\non the sweep velocity c, as expected. However, for fixed c, there is little dependence on L\n(results of other sizes are not shown).\nWe have found the characteristic change in the cases of relatively la rge quantum fluc-\ntuations, i.e. Hx= 0.5 and 0.7. The size-independence indicates that the change occurs\nlocally. When Hxis small, the quantum fluctuations are weak and local flips of clusters\nconsisting of small number of spins become dominant. In Fig. 4, Mz(t) forHx= 0.1 is\nshown, where a peculiar sequence of jumps is found. It is almost inde pendent of the system\nsize (except for L= 2). Before the large jump of the magnetization at Hz/J= 1, there is a\n9-1-0.500.51\n-1-0.8-0.6-0.4-0.200.20.40.60.81<Mz>/L\nHz/J\nFIG. 3: Magnetization Mz(t) as a function of Hz(t) forL= 20,Hx= 0.7 and various sweep\nvelocities. Solid (red) line: c= 0.1; Dashed (green) line: c= 0.01; Dotted (magenta) line:\nc= 0.001.\nsmall but non-zero precursor jump around Hz≃2/3. After these jumps, the magnetization\nshows a plateau of Mz(t)/L≈ −1/2 until the smooth crossover to the saturated value takes\nplace around Hz/J= 2. The value Hz/J= 2 corresponds to the spinodal point of the\ncorresponding classical model.\nThepositionsofthesejumpscanbeunderstoodfromtheviewpoint oflocal”cluster”flips.\nLetusconsider asinglespinflip, thatis, aflipfromthestatewithallsp ins|−−−−−−−···∝angbracketright\nto a state |−−−+−−−···∝angbracketright . The diabatic energies of these states are E0=−LJ+LHz\nandE1=−(L−4)J−(L−2)Hz, respectively. Thus, the crossing of these states occurs at\nH(1)\nz= 4J/2 = 2J. The transition probability due to the transverse field Hxat this crossing\nis proportional to H2\nx, because the matrix element for a single flip is proportional to Hx.\nIf we consider a collective flip of a connected cluster of mspins, the diabatic energy of\nthis state is\nEm=−(L−4)J+(L−2m)Hz, (8)\n10-1-0.500.51\n-10123456<Mz>/L\nHz/J\nFIG. 4: Magnetization Mz(t) as a function of Hz(t) forHx= 0.1,c= 0.001, and various system\nsizes. Solid (black) line: L= 2; Solid (red) line: L= 4; Long dashed (green) line: L= 6; Dashed\n(magenta) line: L= 8; Dotted (red) line: L= 10; Dashed dotted (blue) line: L= 12;\nand thus, the crossing of the states occurs at\nH(m)\nz= 4J/2m= 2J/m. (9)\nForm= 2,3,...we haveH(m)\nz= 1,2/3,...respectively. These values do not depend on L.\nIt should be noted that for the system L= 2, the 2-spin cluster (m=2) surrounded by +\nspins can not be realized, and no jump appears at Hz/J= 1.\nThe matrix element for the m-spin cluster flip is proportional to Hm\nx(see Appendix A).\nTherefore, for small Hx, only the flips with small values of mare appreciable. In the case of\nHx/J= 0.1 forc= 0.001, jumps for m≤2 are observed. The change of the magnetization\nof each spin is given by a perturbation series and is independent of Las shown in Appendix\nB. These local flips may correspond to the nucleation in classical dyn amics in metastable\nstate.\nIfcbecomes small or Hxbecomes large, contributions from large values of mbecome\nrelevant. Then, magnetization process consists of many jumps, a nd amount of the change\n11becomes large. But, as long as the perturbation series converges , we have a size-independent\nmagnetization process, as shown in Fig. 2. This sharp and size-indep endent nature is con-\nsistent with the property of the classical spinodal decomposition.\nIn the classical system, the magnetization relaxes to its equilibrium v alue at the spinodal\ndecompositionpoint. Incontrast, forpurequantumdynamics, th emagnetizationofthestate\ndoes not change for adiabatic motion along a particular energy level. Only if we include an\neffect of contact with the thermal bath, relaxation to the ground state takes place15.\n1. Phase diagram\nIn Fig. 5, we give a schematic picture of the order parameter Mas a function of the tem-\nperatureTand the field Hzin the thermal phase transition of a ferromagnetic system. The\noverhanging structure signals the existence of the metastable st ate. The spinodal point is at\nthe edge of the metastable branch. In this figure, the magnetic fie ld is swept from positive\nto negative, and the metastable positive magnetization jump down t o the equilibrium value\natHSP(T).\nIn a mean field theory for the magnetic phase transition at a finite te mperature, the\nspinodal point is given by\nHSP=−Jz/radicaligg\n1−kBT\nJ+kBT\n2ln\n1+/radicalig\n1−kBT\nJ\n1−/radicalig\n1−kBT\nJ\n, (10)\nwherezis the number of nearest neighbor sites. We show the dependence o fHSPas a\nfunction of Tby a dash-dot curve in Fig. 5.\nA similar argument can be made for the classical ground state energ y. Let the z-\ncomponent of spin be denoted by σ. Then, the energy is expressed by\nE=−Jσ2−Hx√\n1−σ2−Hσ. (11)\nWe assume that the energy satisfies the condition\ndE\ndσ= 0, (12)\nwhich gives\n−2Jσ+Hxσ√\n1−σ2−H= 0. (13)\n12M\nH\nT\nFIG. 5: (color online) Schematic picture of the magnetizati onMas a function of field Hbelow\nthe critical temperature. Open circles denotes the spinoda l decomposition points. The (red) dash-\ndotted curve in the H–Tplane shows HSP(T) as given by Eq. (10).\nHere, we consider the metastable state and thus we set H=−|H|forσ >0. At the end\npoint of metastability,\ndσ\ndH=∞ordH\ndσ= 0. (14)\nThis leads to\nσ=/parenleftigg\n1−/parenleftbiggHx\n2J/parenrightbigg2/3/parenrightigg1/2\n. (15)\nThe end point of the metastable state is given by\nHSP= 2J/parenleftigg\n1−/parenleftbiggHx\n2J/parenrightbigg2/3/parenrightigg3/2\n. (16)\nwhich gives HSPas a function of Hxand is shown in Fig. 6 as the long-dashed curve.\nItisinteresting tonotethatexpression Eq. (16)isvery similar toth ewell knowexpression\nof the Stoner-Wohfarth model16for the reversal of a classical magnetic moment under the\napplication of a magnetic field tilted with respect of the easy anisotro py axis. This is not\nsurprising because, with both a longitudinal and transverse field co mponent, this model can\n13be considered as a realization of the classical spinodal transition. O ne might derive the\nStoner-Wohfarth model from equation Eq. (11) by replacing the e xchange energy parameter\nJby the anisotropy energy constant D.\nIt should be noted that the critical Hxis a factor of two larger than that of the correct\nvalueHx\nc=Jfor the one-dimensional quantum model. This difference is due to the presence\nof quantum fluctuations. Therefore, inFig. 6 we plot Eq.(16) with an dwithout renormalized\nvalues of the fields. The long-dashed curve denotes the case of Hxscaled by 1/2, and the\ndashed curve denotes the case where both HxandHzare scaled by 1/2.\nAs we saw in Fig. 2, we find a large change of magnetization at a values o fHzfor each\nvalue ofHx, which we called HSP. In Fig. 6, we plot values of Hzat which (1) M(t) shows\na small but clear jump, (2) M(t)/Lis equal to −1/2, and (3)M(t) saturates as a function\nofHz, for various values of c. The data show a dependence on Hxthat shows a similar\ndependence to the dotted line. If we use other value of c, the values of Hzchange. Although\nthe values of Hzfor (1), (2) and (3) for larger values of care larger than those for c= 0.001,\nthevaluesof Hzforc= 0.0001areclosetothosefor c= 0.001. Theyseemtosaturatearound\nthe value of the dotted line, and we may identify a sudden appearanc e of size independent\nchange as an indication for a quantum spinodal point. If we sweep mu ch faster, the jumps\nof the magnetization becomes less clear, as we now study in more det ail.\nB. Very fast sweeps\nFor a fast sweep c= 0.1, the magnetization processes for different sizes almost overlap\neach other, see Fig. 7. The data for L= 14, 16, and 20 are hard to distinguish. This\nalmost perfect overlap is rather surprising from the viewpoint of th e structure of energy-\nlevel diagram. The data for L= 6 deviates from the others. This fact indicates that for\nthese parameters ( Hx= 0.7,c= 0.1) the relevant size of the cluster ( min Eq. (8)) is larger\nthan 6 but smaller than 14.\nLet us now study the behavior if we sweep much faster. In Fig. 8(lef t), we show the\nmagnetization as a function of t(orHz(t)) forL= 12 withc= 10,20,50,100 and 200. For\nthese parameters, the data for other Lare almost indistinguishable from the L= 12 data\nand are therefore not shown. As Fig. 8(left) shows, the magnetiz ation oscillates about a\nstationary value for large values of Hzwhere the energy levels with different magnetization\n1400.511.52\n0 0.5 1 1.5 2HSP/J\nHx\n /J\nFIG. 6: (color online) Spinodal points HSP\nzas a function of the quantum fluctuation Hxfor various\nsweep velocities c. The horizontal dotted lines correspond to 2 /mfor (m= 2,...,10) (see Eq. (9)).\nPlusses (1), crosses (2), and stars (3): c= 0.0001; Open squares (1), solid squares (2), and solid\ndiamonds (3): c= 0.001; Open circles (1), bullets (2), and open diamonds (3): c= 0.01; Open\ntriangles (1), solid triangles (2), and inverted solid tria ngles (3): c= 0.1. The numbers (1), (2),\nand (3), correspond to the field at which M(t) shows a small but clear jump, M(t)/L=−1/2, and\nM(t) saturates as a function of Hz, respectively.\nMare far separated in the energy-level diagram as we saw in Fig. 1. Le t us study the c-\ndependence of the saturated value MS=MS(c). In Fig. 8(right), we plot the change of the\nmagnetization ∆ M/L= (MS(c)−(−L))/Las a function of 1 /c. As shown in Fig. 8(right),\nthe data can be fitted well by the expression\n∆M\nL≃∆M0\nL+a\nc, (17)\nwhere ∆M0/Landaare constants. These constants, to good approximation, do not depend\n15-1-0.500.51\n-1 -0.5 0 0.5 1<Mz>/L\nHz/J\nFIG. 7: (color online) The magnetization M(t) as a function of the sweeping field HzforHx= 0.7,\nc= 0.1, and various system sizes. Solid (black) line: L= 6; Solid (red) line: L= 14; Long dashed\n(green) line: L= 16; Dashed (magenta) line: L= 18; Dotted (red) line: L= 20; Dashed dotted\n(blue) line: L= 12. Except for L= 6, all other curves overlap, indicating that for sufficientl y large\nsystems, the dependence on Lis very weak.\non the system size.\nIn order to explain the observed 1 /cdependence, we introduce a perturbation scheme for\nfast sweeps (see Appendix B). We regard the sweeping field (Zeema n) term as the zero-th\norder system and treat the interaction among spins as the pertur bation term. The result is\na series expansion in terms of H0/c(see Eq. (B8)), which explains the 1 /cdependence.\nIn Appendix B, we also introduce a perturbation scheme based on ind ependent Landau-\nZener systems each of which is given by a spin in a transverse field with a sweeping field.\nIn Appendix B, we show that this scheme can explain the behavior of t he magnetization\ndynamics in the fast-sweep regime.\n16-1-0.95-0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55-0.5\n01020304050<Mz>/L \nHz/J 0.010.020.030.040.050.060.070.080.090.10.110.12\n00.010.020.030.040.050.060.070.080.090.1∆ Mz /L\n1/c \nFIG. 8: Left: The magnetization Mz(t) as a function HzforL= 12,Hx= 0.7 and various sweep\nvelocities. Solid (red) line: c= 10; Long dashed (green) line: c= 20; Dashed (magenta) line:\nc= 50; Dotted (dark blue) line: c= 100; Dashed dotted (magenta) line: c= 200. Right: ∆ M/L\nas a function of 1 /cforHx= 0.7 and various system sizes. Solid (red) line: L= 4; Long dashed\n(green) line: L= 6; Dashed (dark blue) line: L= 10; Dotted (magenta) line: L= 12.\nV. SUMMARY AND DISCUSSION\nWe have studied time-evolution of the magnetization in the ordered p hase of the trans-\nverse Ising model under sweeping field Hz. We found significant jumps of the magnetization\nat a certain value of the magnetic field which we called quantum spinoda l pointHSP\nz. Al-\nthough the energy-level diagram of the system significantly chang es with the system size,\nwe found size-independent magnetization processes for each pair (Hx,c).\nIn principle, it should be possible to understand the quantum dynamic s of the magneti-\nzation from the energy-level diagram of the total system. Indee d the picture of successive\nLandau-Zener scattering processes works in slow sweeping case3. However, for fast sweeps,\nthe time evolution canbe regarded asanassembly of local processe s, the interaction between\nthe spins being a perturbation. Hence the dynamics of the magnetiz ation does not depend\non the size.\nWhen the quantum fluctuations are weak (small Hx), a series of local spin flips governs\nthe magnetization dynamics. The jumps of magnetization can be und erstood on the basis\nof energy-level crossings of certain spin clusters (Eq. (8)). The energy-level structure cor-\nresponding to the local cluster flips is, of course, present in the en ergy-level diagram of the\n17total system but it is hidden in the complicated structure of the hug e number of energy\nlevels.\nForlargevaluesof Hxandfastsweeps, themagnetizationprocess is alsosize-independe nt.\nTo explain this feature, we have introduced a perturbation scheme in which the small pa-\nrameter isH0/c. In addition, we introduced a new perturbation scheme based of sin gle-spin\nfree Landau-Zener processes, which all together have allowed us to provide an understanding\nof the magnetization dynamics under field sweeps in terms of the ene rgy-level structure.\nAcknowledgments\nThis work was partially supported by a Grant-in-Aid for Scientific Res earch on Priority\nAreas “Physics of new quantum phases in superclean materials” (Gr ant No. 17071011), and\nalso by the Next Generation Super Computer Project, Nanoscienc e Program of MEXT.\nNumerical calculations were done on the supercomputer of ISSP.\nAPPENDIX A: SIZE DEPENDENCE OF THE ENERGY GAP AT Hz= 0\nThe eigenvalues of the model Eq. (1) are given by9\nE=Hx/summationdisplay\nqωq/parenleftig\n2η†η−1/parenrightig\n, (A1)\nwhereηandη†are fermion annihilation and creation operators, respectively, and\nωq= 2/radicalig\n1+2λcosq+λ2, (A2)\nwhereλ=J/Hx. When the number of the fermions is even, qtakes the values\nq=±π\nL,±3π\nL,···,±π(L−1)\nL, (A3)\nand when the number of the fermions is odd\nq= 0,±2π\nL,±4π\nL,···,±π(L−2)\nL,π. (A4)\nBecauseωq>0, the ground state is given by η†η= 0. Thus, in the case of even number of\nfermions, the ground state is given by\nEE1=−2HxL/2/summationdisplay\nm=1/radicaligg\n1+2λcos/parenleftbiggL−2m+1\nL/parenrightbigg\n+λ2, (A5)\n18and the first excited state is given by\nEE2=EE1+4Hx/radicaligg\n1+2λcos/parenleftbiggL−1\nL/parenrightbigg\n+λ2. (A6)\nIn the case of an odd number of fermions, the lowest energy state is\nEO1=Hx√\n1+2λ+λ2+Hx√\n1−2λ+λ2\n−2HxL/2−1/summationdisplay\nm=1/radicaligg\n1+2λcos/parenleftbiggL−2m+1\nL/parenrightbigg\n+λ2, (A7)\nand the first excited state is given by\nEO2=EO1+2Hx√\n1−2λ+λ2+/radicaligg\n1+2λcos/parenleftbiggL−2\nL/parenrightbigg\n+λ2. (A8)\nThe energy gaps are given by\n∆E1=EO1−EE1, (A9)\n∆E2=EE2−EE1, (A10)\nand\n∆E3=EO2−EE1. (A11)\nUsing these formulae, we cancalculate the L-dependence of thegaps. The results areplotted\nin Fig. 9(left). We find that ∆ E1vanishes exponentially with L, that is\n∆E1∝exp(−aL), (A12)\nwhereadepends on λ. We also plot −log∆E1/2 to confirm the exponential dependence.\nOn the other hand, we find that ∆ E2is almost independent of L, and ∆E3is very close\nto ∆E2, reflecting the fact that above the 3rd level the infinite system ha s a continuous\nspectrum.\nIt is also of interest to study the dependence of the energy gap ∆ E1onHxfor severalL.\nIn Fig. 9(right) we show the data on double logarithmic scale. In the r egime of small Hxwe\nfind a linear dependence on Hx, suggesting that\n∆E1∝H2S\nx. (A13)\nIndeed, for small Hxthe slopes of the lines is given by 2 S=L. This dependence on Hxand\nLis to be expected when Lspins flip simultaneously.\n19012345678910\n510152025303540\nL 0510152025\n0 0.5 1 1.5 2-ln ∆E1\n-ln Hx\n \nFIG. 9: (color online) Left: Size-dependence of the energy g aps forHx= 0.7. Bullets: Difference\n500×∆E1/Jbetween the energy of the first excited state and the ground st ate energy. This\ndifference vanishes exponentially with L. Solid squares: −(ln∆E1/J)/2; Stars: ∆ E2/J; Open\nsquares: ∆ E3/J. Right: The energy gap ∆ E1/Jas a function of Hxfor several L. Plusses:\nL= 10. Crosses: L= 20; Stars: L= 30; Open squares: L= 40. Note the double logarithmic\nscale. In both figures, lines are guides to the eyes only.\nAPPENDIX B: PERTURBATION ANALYSIS FOR LANDAU-ZENER TYPE\nSWEEPING PROCESSES\nWhen the sweep velocity cis very large, the duration of the sweep is very short. This\nsuggests that it may be useful to study the magnetization proces ses by a perturbational\nmethod in terms of the small parameter 1 /c.\nLet us consider the following model.\nH=H0+ctV, (B1)\nwhereH0andVare time independent. We will work in the interaction representation with\nrespect toctV, that is, we take the motion of ctVas reference, not H0as is usual done. The\nSchr¨ odinger equation is\ni¯h∂\n∂t|Ψ∝angbracketright= (H0+ctV)|Ψ∝angbracketright. (B2)\nIn the interaction representation we have\n|Ψ∝angbracketright=e−ict2V/2¯h|Φ∝angbracketright, (B3)\n20and the equation of motion is given by\ni¯h∂\n∂t|Ψ∝angbracketright=i¯h(−ictV/¯h)e−ict2V/2¯h|Φ∝angbracketright+i¯he−ict2V/2¯h∂\n∂t|Φ∝angbracketright=e−ict2V/2¯h/parenleftigg\nctV+i¯h∂\n∂t/parenrightigg\n|Φ∝angbracketright,\n(B4)\nand therefore the Schr¨ odinger equation for |Φ∝angbracketrightis given by\ni¯h∂\n∂t|Φ∝angbracketright=eict2V/2¯hH0e−ict2V/2¯h|Φ∝angbracketright. (B5)\nDefining\nW(t)≡eict2V/2¯hH0e−ict2V/2¯h, (B6)\nwe can use the usual perturbation expansion scheme for\ni¯h∂\n∂t|Φ∝angbracketright=W(t)|Φ∝angbracketright, (B7)\nand find\n|Φ(t)∝angbracketright=/bracketleftigg\n1+/parenleftbigg1\ni¯h/parenrightbigg/integraldisplayt\nt0W(t1)dt+/parenleftbigg1\ni¯h/parenrightbigg2/integraldisplayt\nt0/integraldisplayt1\nt0W(t1)W(t2)dt1dt2+···/bracketrightigg\n|Φ(0)∝angbracketright.(B8)\nIn the sweep ( −H0< ct < H 0),t0=−H0/candt=H0/c. Thus, the integral is of order\nH0/c. Therefore we can regard the above expansion is a series expansio n in terms of power\nofH0/c. Of course the series can be also regarded as a power of H0as in the usual sense.\n1. Transverse Ising model under a field sweep\nNow, we consider our problem\nH(t) =−J/summationdisplay\njσz\njσz\nj+1−Hx/summationdisplay\njσx\nj−ct/summationdisplay\njσz\nj. (B9)\nWe set\nH0=−J/summationdisplay\njσz\njσz\nj+1−Hx/summationdisplay\njσx\nj (B10)\nand\nV=−/summationdisplay\njσz\nj. (B11)\nThen,W(t) is given by\nW(t) =e−ict2/2¯h/summationtext\njσz\njH0eict2/2¯h/summationtext\njσz\nj\n=−J/summationdisplay\njσz\njσz\nj+1−Hx/summationdisplay\nj/parenleftig\nσ+\nje−ict2/¯h+σ−\njeict2/¯h/parenrightig\n. (B12)\n21We may include the diagonal term −J/summationtext\njσz\njσz\nj+1inV. Then the expansion is regarded as\nseries ofHx. This expansion corresponds to the series of jumps discussed in Eq . (8).\nWe also note that if J= 0 the above process is an ensemble of independent Landau-Zener\nprocesses. Each of them is independently expressed by\ni¯h∂\n∂t|ΦLZ(t)∝angbracketright=−Hx/parenleftig\nσ+e−ict2/¯h+σ−eict2/¯h/parenrightig\n|ΦLZ(t)∝angbracketright. (B13)\n2. Perturbation theory in terms of independent LZ systems\nNext, we consider the case in which the transverse field is included in V. We sweep the\nfield from −H0toH0. The duration of the sweep is 2 H0/c. We assume that\nH0≫J >H x, (B14)\nsuchthatthemotiondueto Visthatofanensemble ofindependent Landau-Zenerprocesses.\nThus, we consider the ensemble of the LZ systems as the unpertur bed system.\nWe know the properties of each system. Namely, we know that the s cattering becomes\nsmall when cbecomes large. The time evolution of each LZ system is given by\n\n1\n0\n→eiφ(t)\n√p\n√1−peiε(t)\n≡ψ(t), (B15)\nin the adiabatic basis, that is in the representation that uses the eig enstates of the system\nwith given Hz(t). Here,pis the probability for staying the ground state. In the Landau-\nZener theory, pis given by the well-known expression\np= 1−exp/parenleftigg\n−πH2\nx\n¯hc/parenrightigg\n. (B16)\nIn the case of small H0,pmay have a different form. Even in those cases, the expression\nEq. (B15) is still correct and the present formulation works if we em ploy a correct expression\nforp.\nThe unperturbed state is given by\nΦ0(t) =/productdisplay\njψj(t) =eiLφ(t)\n√p\n√1−peiε(t)\n\n1⊗\n√p\n√1−peiε(t)\n\n2⊗···⊗\n√p\n√1−peiε(t)\n\nL.\n(B17)\n22-25-20-15-10-50510152025\n-10 -5 0 5 10E/J\nHz/J-25-20-15-10-505101520\n-10 -5 0 5 10E/J\nHz/J\nFIG. 10: Energy level diagram for a two-spin Landau-Zener mo del Eq. (B18) with Hx= 0.7. Left:\nJ= 0; Right: J= 1.\nThe zero-th order is given a usual Landau-Zener process of which the energy diagram is\ngiven by Fig. 10(left), which shows the energy-level diagram for th e two independent spins.\nThus, in this case there are four states, (++) ,(+−),(−+),and (−−). The states consisting\nof (+−) and (−+) are degenerate with energy zero.\nThe interaction term −J/summationtext\njσz\njσz\nj+1is the perturbation. As far as the expansion Eq. (B8)\nconverges with less than L-th terms, it gives a local effect. To first order in J, only the\nnearest-neighbor spins interact, giving a contribution of the orde rJ. The sweep-velocity\ndependence is taken into account through the zero-th order ter m. If we take a large H0,\nthe integral in Eq. (B8) is no longer small, and we have to regard Eq. ( B8) as a series of\nJ. Therefore, we do not have any small parameter, and the Eq. (B8 ) represents the original\ngeneral dynamics. In the case of fast sweeps, the effective rang e of quantum mixing in which\nthe diabatic energy levels (levels for Hx= 0) cross each other, is of order L×J, and therefore\nthe duration of interaction is of order LJ/c. Hence, the integration gives a contribution of\norderLJ/cwhich now becomes the small parameter. In the case of finite H0, the small\nparameter is the minimum of ( H0/c,LJ/c). In the present study, H0= 1. ThenH0/cis the\nsmall parameter and we cannot use the form of pgiven in Eq. (B16). In any case, the series\nconverges for the fast sweeps and we expect that the perturba tion effect does not depend\nonL.\nThesystem described bythefirst-orderperturbationtheoryco rrespondstoaHamiltonian\nof two spins exhibiting the Landau-Zener scattering process and w hich are coupled by an\n23Ising interaction. The Hamiltonian reads\nHCLZ=−Jσz\n1σz\n2−(Hxσx\n1−ctσz\n1)−(Hxσx\n2−ctσz\n2). (B18)\nThe energy-level diagram of this system is shown in Fig. 10(right). L et us study the effect\nof the interaction on the dynamics in this case. We compare the magn etization processes of\nthe model Eq. (B18) with J= 0 andJ= 1. The results are shown in Fig. 11(left). Note\nthat the sweep starts from Hz=−H0=−1.\nNext, in Fig. 11(right), we show the magnetization processes for c= 100 for the model\nEq. (B18) with that of the same model with Jreplaced by 2 J. If we use a small value of\nH0, the ground states of the models at Hz=−1 differ significantly. Therefore, to compare\nthe results, in this figure, we take H0=−60 such that the ground state of both models is\nclose to the all-spins-down state. The average of the first and the third curves is close to\nthe second curve. This fact indicates that the processes are well described by the first-order\nperturbation theory. Indeed, the deviation from the single Landa u-Zener model is 0, J, and\n2J, respectively.\nWe also compare the magnetization processes of the models Eq. (B1 8) withJreplaced\nby 2Jand that of a model with 3 spins in Fig. 12(a). The difference between the models\nof 4 spins and of 12 spins is also shown in Fig. 12(right). In all these ca ses, we start at\nHz=−1 because the magnetizations per spin are very close in all the cases . We find almost\nno difference, indicating that the processes are well described by t he first-order perturbation\ntheory.\nWhen the sweep velocity becomes small, we may need higher order per turbation terms.\nIf the relevant order of the perturbation is less than the length of the chain, we expect a size-\nindependent magnetization process. The size independent magnet ization in the quantum\nspinodal decomposition can be understood in this way.\nThe local motion of magnetization can be understood from a view poin t of an effective\nfield from the neighboring spins. We may study the magnetization pro cess of a single-spin\nin a dynamical mean-field generated by its neighbors.17Let us describe the situation by the\nfollowing Hamitonian:\nHMF=−(Hz(t)+2J∝angbracketleftσz∝angbracketright)σz−Hxσx. (B19)\nBecause the mean field is almost 2 Jduring the fast sweep, the mean field simply shifts H0\nby a constant 2 J. Thus, we conclude that for fast sweeps, the dynamics is very simila r to\n24-1-0.95-0.9-0.85-0.8-0.75-0.7-0.65-0.6\n01020304050<Mz>/L \nHz/J -1-0.95-0.9-0.85-0.8-0.75-0.7-0.65-0.6\n-40-2002040<Mz>/L \nHz/J \nFIG. 11: Left: Comparison of the magnetization processes of the model Eq. (B18) with J= 0\n(thin line) and J= 1 (thick lines) for Hx= 0.7. Solid (red) line: c= 10; Long dashed (green) line:\nc= 20; Dashed (magenta) line: c= 50; Dotted (dark blue) line: c= 100; Dashed dotted (magenta)\nline:c= 200. For each c, the magnetization of the single LZ process is shifted by an a mount such\nthat atHz=−1 it coincides with the magnetization of the model Eq. (B18) w ithJreplaced by\n2J. Right: Comparison of the magnetization of a single LZ proce ss, that of the model Eq. (B18),\nand that of the model Eq. (B18) with Jreplaced by 2 J.Hx= 0.7 andc= 100. Solid (black) line:\nc= 10, single LZ process; Solid (red) line: c= 10, Eq. (B18); Long dashed (green) line: c= 10,\nEq. (B18) with Jreplaced by 2 J; Dashed (magenta) line: c= 100, single LZ process; Dotted (red)\nline:c= 100, Eq. (B18); Dashed dotted (blue) line: c= 100, Eq. (B18) with Jreplaced by 2 J.\nthat of a single spin, meaning that for the dynamics, the effective fie ld on each spin in the\nlattice is essentially that same as the applied field. This conclusion is con sistent with our\nearlier comparison of the zero-th and first-order perturbation r esults.\n∗Corresponding author: miya@spin.phys.s.u-tokyo.ac.jp\n1C. Zener, Proc. R. Soc. London, Ser A 137, (1932), L. Landau, Phys. Z. Sowjetunion, 246\n(1932); E.C. G. St¨ uckelberg, Helv. Phys. Acta 5, 3207 (1932);\n2S. Miyashita, J. Phys. Soc. Jpn. 64, 3207 (1995); S. Miyashita, J. Phys. Soc. Jpn. 65, 2734\n(1996).\n3H. De Raedt, S. Miyashita, K. Saito, D. Garcia-Pablos and N. G arcia, Phys. Rev. B 56, 11761\n25-1-0.95-0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55-0.5\n01020304050<Mz>/L \nHz/J -1-0.95-0.9-0.85-0.8-0.75-0.7-0.65-0.6-0.55-0.5\n01020304050<Mz>/L \nHz/J \nFIG. 12: Left: Comparison of the magnetization processes of the model Eq. (B18) with Jreplaced\nby 2J(thin lines) and the model with three spins (thick lines) for Hx= 0.7. Solid (red) line:\nc= 10; Long dashed (green) line: c= 20; Dashed (magenta) line: c= 50; Dotted (dark blue) line:\nc= 100; Dashed dotted (magenta) line: c= 200. Right: Same as left except that the comparison\nis between models with L= 4 (thin lines) and L= 12 (thick lines) for Hx= 0.7.\n(1997).\n4D. Gatteschi, R. Sessoli, and J. Villain, Molecular Nanomagnets , (Oxford University Press\n2006).\n5L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, an d B. Barbara, Nature, 383, 145\n(1996) J. R. Friedman, M. P. Sarachik, J. Tejada, and R. Ziolo , Phys. Rev. Lett. 76, 3830 (1996)\nJ. A. A. J. Perenboom, J. S. Brooks, S. Hill, T. Hathaway, and N . S. Dalal, Phys. Rev. B 58,\n330 (1998) T. Kubo, T. Goto, T. Koshiba, K. Takeda, and K. Awag a, Phys. Rev. B 65, 224425\n(2002) C. Sangregorio, T. Ohm, C. Paulsen, R. Sessoli and D. G atteschi, Phys. Rev. Lett. 78,\n4645 (1997).\n6I. Chiorescu, W. Wernsdorfer, A. M¨ uller, H. B¨ ogge, and B. B arbara, Phys. Rev. Lett. 84, 3454\n(2000); Rousochatzakis, Y. Ajiro, H. Mitamura, P. Kogerler and M. Luban, Phys. Rev. Lett.\n94, 147204 (2005); En-Che Yang, W. Wernsdorfer, L. N. Zakharov , Y. Karaki, A. Yamaguchi,\nR. M. Isidro, G.-D. Li, S. A. Willson, A.L. Rheingold, H. Ishi moto, and D. N. Hendrickson,\nInorg. Chem. 45, 529 (2006); K-Y. Choi, Y. H. Matsuda, H. Nojiri, U. Kortz, F. Hussain, A.\nC. Stowe, C. Ramsey, and N. S. Dalal, Phys. Rev. Lett. 96107202, (2006); S. Bertaina, S.\nGambarelli, T. Mitra, B. Tsukerblat, A. M¨ uller, and B. Barb ara, Nature, 453, 203, (2008).\n267W. Wernsdorfer andR. Sessoli, Science 284, 133 (1999); M. Ueda, S.Maegawa and S.Kitagawa,\nPhys. Rev. B 66, 073309 (2002).\n8M. Suzuki, Prog. Theor. Phys. 58, 755 (1977).\n9B. K. Chakrabati, A. Dutta and P. Sen, Quantum Ising Phase and Transverse Ising Models ,\n(Springer, Heidelberg, 1996).\n10Yugo Oshima, Hiroyuki Nojiri, Kaname Asakura, Toru Sakai, M asahiro Yamashita, and Hitoshi\nMiyasaka, Phys. Rev. B 73 , 214435 (2006); Jun-ichiro Kishin e, Tomonari Watanabe, Hiroyuki\nDeguchi, MasakiMito, Tˆ oruSakai, TakayukiTajiri, Masahi roYamashita, andHitoshiMiyasaka,\nPhys. Rev. B 74 , 224419 (2006).\n11T. Kadowaki and H. Nishimori, Phys. Rev. E 58, 5355 (1998). A. Das and B. K. Chakrabarti,\nQuantum Annealing And Related Optimization Methods , (Springer, New York 2005), G. E.\nSantoro and E. Tosatti J. Phys. A 39, R393 (2006), A. Das and B. K. Chakrabarti Rev. Mod.\nPhys.80, 1061 (2008), S. Morita and H. Nishimori, J. Phys A 39, 13903 (2006), S. Morita and\nH. Nishimori, J. Phys. Soc. Jpn. 76, 064002 (2007).\n12J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005).\n13T.W. B. Kibble, J. Phys. A 9, 1387 (1976); Phys. Rep. 67, 183 (1980); W. H. Zurek, Nature\n(London) 317, 505 (1985); Phys. Rep. 276, 177 (1996).\n14V.V. Dobrovitski and H. De Raedt, Phys. Rev. E 67, 056702 (2003).\n15K. Saito, S. Miyashita and H. De Raedt, Phys. Rev. B 60, 14553 (1999).\n16E.C. Stoner and E.P. Wohlfarth, Philos. Trans. Roy. Soc. Lon don A20599 (1948).\n17A. Hams, H. De Raedt, S. Miyashita and K. Saito, Phys. Rev. B 62, 13880 (2000).\n27−2 0 2\n−505\nHE(H)unregisteredM\nH\nT−2 0 2\n−505\nHE(H)unregistered0 1 2012\nHxHzSP0 1 2 30102030−ln(E2−E1)\n−ln Hx0 20 400510(E2−E1)x1000\nE3−E1,E4−E1\nL−ln(E2−E1)/2 0 1000 2000 3000 4000 5000 6000 7000\n-40 -20  0  20  40Counts\nθ [degrees] 0 0.2 0.4 0.6 0.8 1\n 0  1000  2000  3000  4000  5000  6000P2\nM 0 1\n 0  100"    },    {        "title": "0907.4105v1.Dynamic_switching_of_magnetization_in_driven_magnetic_molecules.pdf",        "content": "arXiv:0907.4105v1  [nlin.CD]  23 Jul 2009Dynamic switching of magnetization in driven magnetic mole cules\nL. Chotorlishvili1,3, P. Schwab1, J. Berakdar2\n1 Institut f¨ ur Physik, Universit¨ at Augsburg, 86135 Augsb urg, Germany\n2 Institut f¨ ur Physik, Martin-Luther Universit¨ at Halle- Wittenberg,\nHeinrich-Damerow-Str.4, 06120 Halle, Germany\n3 Physics Department of the Tbilisi State University,\nChavchavadze av.3, 0128, Tbilisi, Georgia\nAbstract\nWe study the magnetization dynamics of a molecular magnet dr iven by static and variable\nmagnetic fields within a semiclassical treatment. The under ling analyzes is valid in a regime,\nwhen the energy is definitely lower than the anisotropy barri er, but still a substantial number\nof states are excited. We find the phase space to contain a sepa ratrix line. Solutions far from\nit are oscillatory whereas the separatrix solution is of a so liton type. States near the separatrix\nare extremely sensitive to small perturbations, a fact whic h we utilize for dynamically induced\nmagnetization switching.\nPACS numbers:\n1I. INTRODUCTION\nMolecular magnets (MM) are molecular structures with a large effect ive spin ( S), e.g.\nfor the prototypical MM Mn12acetates [1] S= 10. MM show a number of interesting\nphenomena that have been in the focus of theoretical and experim ental research [1, 2, 3, 4,\n5, 6, 7, 8, 9, 10, 11]. To name but few, as a result of the strong unia xial anisotropy, MM\nshow a bistable behavior [1]; they also exhibit a resonant tunnelling of m agnetization [2]\nthat shows up as steps in the magnetic hysteresis loops [3, 4, 5]. Of s pecial relevance for\napplications in quantum computing is the large relaxation time of MM [12].\nThe present theoretical work focuses on the dynamics of the mag netization. The\nestablished picture of macroscopic quantum tunnelling of the magne tization is as fol-\nlows: The MM effective spin Hamiltonian ˆH=−DS2\nzpossesses degenerate energy levels\n±MS,−S < M s< Sseparated by the finite barrier EB=DS2. At low temperatures only\nthe lowest levels MS=±Sarepopulated. Those two states areorthogonal to each other a nd\nno tunnelling is possible. An anisotropic perturbation E(S2\nx−S2\ny) does not commute with\nthe Hamiltonian ˆH=−DS2\nzand mixes therefore the states at both sides of the anisotropy\nbarrier leading thus to tunnelling [8]. Reversal of the magnetization d ue to macroscopic\nquantum tunnelling has a maximum for the states close to the top of t he barrier. This case\ncorresponds to the high temperature limit.\nIn this work we consider the magnetization dynamics induced by cons tant and harmonic\nexternal magnetic fields: The influence of a variable magnetic field on MM at low tem-\nperatures, i.e. when only 2-3 levels are excited was considered in [10 , 11]. It was shown\nthat in this case the problem is reduced to a three level Jaynes-Cum mings model, the so\ncalled Lambda configuration. Therefore, it is analytically solvable in pr inciple. The low\ntemperature assumption is however quite restrictive [12, 13]: If on ly the levels E0,E1,E2\nare involved the low temperature approximation is applicable for temp eratures obeying [11]\nkBT < E1−E0, wherekBis the Boltzmann constant. For Mn12this leads to the esti-\nmateT <0.6K[13]. Obviously, if the temperature exceeds T, an approximation with a\nlarge number of levels participating in the process is more appropriat e. In this case the\nquasi-classical approximation for the spin dynamics becomes applica ble [14, 15]. It is our\naim here to conduct such a study. MM will be modelled as in previous stu dies, e.g. in\nRefs. [10, 11]. We consider the dynamical reversal of the magnetiz ation, caused not by an\n2anisotropic perturbation but by a constant and varying magnetic fi eld. The energy is such\nthat a large number of levels are excited, but still low enough such th at tunnelling induced\nby an anisotropic perturbation is weak. We shall show that under diff erent conditions (de-\npending on the fields parameters), different types of magnetizatio n dynamics are realized.\nFor the time evolution of the magnetization vector, under certain c onditions we obtain a\nsolution ofthe solitontype. The varioustypes of thedynamics will be linked to thestructure\nof the phase space of the system. In particular, the existence of the separatrix in the phase\nspace has a profound influence on the system behavior. We will show that in this case a\nnew type of field-assisted magnetization dynamics emerges, namely a dynamically induced\nswitching. This occurs when the energy of the system (in the prese nce of the field) is still\nlower than the re-scaled anisotropy barrier [16] and coincides with t he separatrix values of\nthe energy. Therefore, the domain close to the separatrix is ident ified as the phase space\narea where the dynamically induced switching takes place.\nII. MODEL\nWe consider a molecular magnet, e.g. Fe8orMn12acetate. The uniaxial anisotropy axis\n(easy axis) sets the z- direction. The MM is subjected to a constant magnetic field directe d\nalong the x-axis and a radio frequency (rf) magnetic field polarized in the x−y-plain. The\nHamiltonian of the single molecular magnet reads [11]\nˆH=ˆH0+ˆHI,\nˆH0=−DˆS2z+gµBH0ˆSx, (1)\nˆHI=−1\n2gµBH1eiω0t(ˆSy+ˆSx)+H.c..\nHereDis the longitudinal anisotropy constant, ˆSx,ˆSy,ˆSzare the projections of the spin\noperators along the x,y,zaxis,gis the Land´ e factor, and µBis the Bohr magneton. H0\nstands for the constant magnetic field amplitude whereas H1, andω0are the amplitude and\nthe frequency of the rf field. The problem when both fields H0, H1are time dependent was\nstudied in [17]. Using quantum-mechanical perturbation theory, th e probability of quantum\ntunnelling of magnetization has been estimated. However, here we a re interested in the\nexact solution of the semi-classical equations of motion. Typical va lues of the parameter D\nare 90 GHz for Mn12andD= 30 GHz for Fe8[13, 15]. Since we are interested in the case\n3when large number of levels are exited, the spin of the magnetic molec ule can be treated as a\nclassical vector ontheBlochsphere. Takingintoaccount that S2=S2\nx+S2\ny+S2\nzisanintegral\nof motion, it is appropriate to switch to the new variables ( Sz, ϕ) via the transformation\n[14]:Sx=/radicalbig\n1−S2\nzcosϕ,Sy=/radicalbig\n1−S2\nzsinϕand rewrite (1) in the compact form:\nH=−λ\n2S2\nz+/radicalbig\n1−S2\nzcosϕ−ε/radicalbig\n1−S2\nz(sinϕ+cosϕ)cos(ω0t). (2)\nHereafter, if not otherwise stated the energy and the time scales are set by the constant\nmagneticfield H/ma√sto→H/gµBH0S, t/ma√sto→2DS\nλt, ω0/ma√sto→λ\n2DSω0. Weintroducedtwo dimensionless\nparameters λ=2DS\ngµBH0,ε=H1\nH0<1. The corresponding Hamilton equations are\n˙Sz=−∂H\n∂ϕ=/radicalbig\n1−S2\nzsinϕ+ε/radicalbig\n1−S2\nz(cosϕ−sinϕ)cos(ω0t),\n˙ϕ=∂H\n∂Sz=−(λ+cosϕ/radicalbig\n1−S2z)Sz+εSz/radicalbig\n1−S2z(sinϕ+cosϕ)cos(ω0t).(3)\nThese equations are nonlinear. Therefore, the solutions to (3) ca n be regular or chaotic,\ndepending on the values of the magnetic fields (parameters λ,ε). From the intuitive point\nof view it is obvious, that for the low energy case, i.e. close to the gro und states Sz≈ ±1,\nthe system Eq.(2) should become linear. However in the language of v ariables action angle\n(Sz, ϕ) that is not so trivial. Therefore, we will discuss this question in more details when\nstudying solutions of the autonomous system.\nIII. AUTONOMOUS SYSTEM: AN EXACT SOLUTION\nWe inspect at first the autonomous system, i.e. when ε= 0. In this case the system can\nbe integrated exactly: Taking into energy conservation H=const=−Σ\nλ\n2S2\nz−/radicalbig\n1−S2zcosϕ= Σ (4)\nand\n˙Sz=/radicalbig\n1−S2\nzsinϕ, (5)\nwe find\n˙S2\nz+/bracketleftbigλS2\nz\n2−Σ/bracketrightbig2= 1−S2\nz. (6)\nConsequently from eq.(6) we infer\nλt\n2=Sz(0)/integraldisplay\nSz(t)dSz/radicalBig/parenleftbig2\nλ/parenrightbig2/parenleftbig\n1−S2\nz/parenrightbig\n−/bracketleftbig\nS2\nz−2Σ\nλ/bracketrightbig2. (7)\n4This relation can be rewritten in the form\nλt\n2=Sz(0)/integraldisplay\nSz(t)dSz/radicalBig/parenleftbig\na2+S2z/parenrightbig/parenleftbig\nb2−S2z/parenrightbig, (8)\nwherea2=2\nλ2/bracketleftbig\nθ2/2−(Σλ−1)/bracketrightbig\n,b2=2\nλ2/bracketleftbig\nθ2/2 + (Σλ−1)/bracketrightbig\n,θ2(λ) = 2√\nλ2−2Σλ+1.\nPerforming the integration (8) and inverting the result we obtain\nSz(t) =\n\nbcn[(bλ/k)(t−α), k],0< k <1,\nbdn[(bλ/k)(t−α),1/k], k >1.(9)\nHere cn(...) and dn( ...) are the Jacobi periodic functions. The coefficients that enter eq . (9)\nread\nk2=1\n2/parenleftbiggbλ\nθ(λ)/parenrightbigg2\n=1\n2/bracketleftbigg\n1+(Σλ−1)√\nλ2+1−2Σλ/bracketrightbigg\n,\nα= 2/bracketleftbig\nλ√\na2+b2F(arccos[Sz(0)/b],k)/bracketrightbig−1. (10)\nWithF(ϕ,k) =ϕ/integraltext\n0dq(1−k2sin21)−1/2being the incomplete elliptical integral of the first\nkind. From eq.(9) we conclude that, depending on the values of the p arameter k(10),\nthe dynamics of the magnetization is described by different solutions . They are separated\nby the special value k= 1 of the bifurcation parameter kindicating thus the presence\nof topologically distinct solutions. In eq.(9) the Jacobian elliptic funct ions cn(ϕ,k) and\ndn(ϕ,k) are periodic in the argument ϕwith the period 4 K(k) and 2K(k) respectively,\nwhereK(k) =F(π/2,k) is the complete elliptic integral of the first kind [18]. The time\nperiod of the oscillation of the magnetization Sz(t) is given by\nT=\n\n4kK(k)\nbλfor 0 < k <1,\n2kK(1/k)\nbλfork >1.(11)\nIfk−→1, the period becomes infinite because K(k)−→ln(4/√\n1−k2). The evolution in\nthis special case is given by the non-oscillatory soliton solution\nSz(t) =b/cosh[bλ(t−α)]. (12)\nConsidering eq. (10), we infer that the bifurcation value of the par ameterk= 1 is connected\nwith an initial energy of the system via the ratio\nΣS=−HS/gµBH0= 1, H/parenleftbig\nSz(t= 0);ϕ(t= 0)/parenrightbig\n=−gµBH0=HS.(13)\n5If this condition (13) is not fulfilled the dynamics of the magnetization is described by the\nsolutions (9). Finally to conclude this section we consider linear limit of s olutions eq.(9):\ncn(u,k)≈cos(u)+k2sin(u)(u−1\n2sin(2u)), k2≪1,\nand\ndn(u,k)≈1−sin(u)2/k2, k2≫1.\nThe interpretation of those asymptotic solutions is clear. First one corresponds to the\ncase when in the effective magnetic field Heff= (gµBH0,0,−DSz), thex- component is\ndominant. Therefore the magnetization vector performs small os cillations |Sz(t)|<1 trying\nto be aligned along effective magnetic field. While in the second case, co rresponding to the\nground state solution (system is near to the bottom of double pote ntial well) the effective\nmagnetic field is directed along the z- axis.\nIV. TOPOLOGICAL PROPERTIES OF SOLUTIONS\nAs established [19, 20], the existence of a bifurcation parameter ind icates that the solu-\ntions separated by it, have different topological properties. Ther efore, it is instructive to\nconsider the properties of the solutions (9) in the phase plane. The existence of the inte-\ngral of motion (4) in the autonomous case makes it possible to expre ssSzas a function of\nϕ:Sz(ϕ,Σ). The phase portrait of the system is shown in Fig.(1): The differen t phase\ntrajectories correspond to the solutions (9). The phase trajec tories corresponding to the\nsolution Sz(t) = dn(ϕ,k),k >1 are open and they describe a rotational motion of the\nmagnetization. Trajectories corresponding to Sz(t) = cn(ϕ,k),k <1 are closed and they\ndescribe the oscillatory motion of the magnetization. Closed and ope n phase trajectories\nare separated from each other by the special line called separatrix . The existence of a sepa-\nratrix is insofar important as the states in the phase-space area n ear the separatrix are very\nsensitive [20] to external perturbations, which signals the onset o f chaotic behavior. The\nrole of perturbations in our particular case is played by the applied pe riodic magnetic field.\nWe recall that the stochastic layer has finite size and it occupies a sm all part of phase space.\n6    \n \n \n \n         \n \n \n \nFIG. 1: Two types of phase trajectories of the system separat ed by the separatrix k= 1,Σ = Σ S.\nTheopen trajectory (solution Sz(t) = dn(t,1/k),k= 1.52,Σ>ΣS) corresponds to the rotational\nregime of motion. The closed trajectory (solution Sz(t) = cn(t,k),k= 0.89,Σ<ΣS) to the\noscillatory regime. The separatrix crossing point/circlemultiplytextis of special interest: around this point any\nperturbation leads to the formation of homoclinic structur e.\nV. FORMATION OF A STOCHASTIC LAYER\nTo determine the width of the stochastic layer we follow Ref.[20]. For d etails of the\nformation of the stochastic layer and for the general formalism we refer to the monograph\n[20]. Here we only present the main findings. We introduce the canonic al variable of action\nI=1\nπ/contintegraltext\nSz(Σ,ϕ)dϕand rewrite the driven nonlinear system (2) in the following form:\nH=H0+εV(I,ϕ)cos(ω0t). (14)\nHereH0=ω(I)I, ω(I) =/bracketleftbigg\ndI(Σ)\ndΣ/bracketrightbigg−1\n. The trajectories laying far from separatrix of the\nunperturbed Hamiltonian H0are not influenced by perturbation. The motion near the\nhomo-clinic pointsof the separatrix isvery slow [20]. Because the per iodofmotion described\nby (11) is logarithmically divergent, even small perturbations end up with a finite influence\ndue to the large period of motion. Thus, the equations of motion for the canonical variables\n(I,ϕ)\n7˙I=∂I\n∂H0˙H=−ε\nω(I)∂V\n∂Sz˙Szcos(ω0t), (15)\n˙ϕ=∂H\n∂I=ω(I)+ε∂V\n∂Sz˙Szcos(ω0t), (16)\nmay be integrated taking into account the features of the motion n ear to the separatrix.\nNamely, the acceleration ˙Szgives a nonzero contribution in the integral/integraltext\ndt∂V\n∂I˙Szcos(ω0t)\nonly near to the homoclinic points [20] (the particle moves along the ph ase trajectory very\nfast and spends most of the time near the homoclinic points). There fore, the differential\nequations (15),(16) can be reduced to the following recurrence re lations:\nI=I−ε\nω(I)/integraldisplay\n∆tdt∂V\n∂Sz˙Szcos(ω0t), (17)\nϕ=ϕ+πω0\nω(I). (18)\nHereI,ϕ,andI,ϕare the values of the canonical variables just after and before pa ssing the\nhomoclinic point, ∆ tis the interval of the time where ˙Szis different from zero. One can\ndeduce the coefficient of stochasticity by evaluating the maximal Ly apunov exponent for the\nJacobian matrix \n∂I\n∂I∂I\n∂ϕ\n∂ϕ\n∂I∂ϕ\n∂ϕ\n, (19)\nof the recurrence relations (17),(18). All of this subsume to the f ollowing expression for the\nwidth of the stochastic layer\nK0=πεω0\nω2/vextendsingle/vextendsingledω\ndH/vextendsingle/vextendsingle. (20)\nHereε,ω0aretheamplitudeandthefrequency oftheperturbation. Noteth attheexpression\n(20) is general [20] and the only thing one has to do is to calculate the nonlinear frequency\nω(I) and its derivative with respect to the energy for the particular sy stem. Thus, even\nfor small perturbation (in our case it is the magnetic field with the fre quencyω0and the\namplitude ε, see Eq.(2)) the dynamics near the separatrix k= 1,Hc=−gµbH0is chaotic\nand unpredictable. Consequently, the solutions (9) have no meanin g near the separatrix.\nAt the same time far from the separatrix H/ne}ationslash=HS,Σ/ne}ationslash= 1,k/ne}ationslash= 1 they are valid. We note\nthat the expression (20) is valid for a low frequency perturbation ω0≪Dand for a high\n8frequency perturbation ω0/greaterorequalslantDas well. For estimation of the width of the stochastic layer\nK0the variable of action should be determined. Taking into account (4) we find\nI±(Σ) =/contintegraldisplay/bracketleftbigg1\n2λ2/parenleftbigg\n2λΣ−cos2ϕ±2λcosϕ/radicalbigg\n1+1\n4λ2cos2ϕ/parenrightbigg/bracketrightbigg1/2\ndϕ. (21)\nIf the static magnetic field is weak then λ=2DS\ngµBH0≫1 is a large parameter. In this limit,\nwe can simplify expression (21) to obtain\nI(Σ) =I+(Σ>1) =I−(Σ>1) = 2/radicalbigg\nΣ+1\nλE/parenleftbigg2\nΣ+1/parenrightbigg\n, (22)\nwhereE(k) is the complete elliptic integral of the second kind. Taking into accou nt (22)\nthe expression for the width of the stochastic layer acquires the f ollowing form\nK0≈πεω0/radicalbig\nλ(Σ+1)|Σ−1|K/parenleftbigg2\nΣ+1/parenrightbigg\nE/parenleftbigg2\nΣ+1/parenrightbigg\n. (23)\nCondition K0>1 of the emergence of stochasticity imposes certain restrictions o n the\nparameters of the magnetic field ε,ω,H0and on the initial energy Σ of the system. When\nthe energy approaches the separatrix value Σ −→1 the condition K0>1 becomes valid\neven for a very small ε≪1 perturbation. This testifies the fact that the system near the\nseparatrix issensitive to small perturbations. Theemergence ofc haos isproved by numerical\ncalculations as well, see Fig.(2). As one can see from this plot, the dyn amics is not regular.\nThe projection Sz(t) of magnetization changes orientation in a chaotic manner.\nHowever, a chaotic change of orientation is not a reversal to a sta tionary target state.\nUnder dynamical switching we understand here the transition betw een the oscillatory and\nthe rotational types of motion. To be more specific let us discuss th e geometrical aspects of\nthemotionforthetrajectoriesneartheseparatrix. Uponapplyin gastaticmagneticfield, the\nmagnetizationprecessional motioninourcaseismarkedly differentf romthatinthestandard\nNMR set up: The key issue is that the effective magnetic field Heff= (gµBH0,0,−DSz),\ndue to the nonlinearity of the system, depends on the values of Sz. The magnetization\nvector tends to align as dictated by the effective field. However, th e orientation of effective\nfield changes in as much as Szdoes. Only in the special case Sz= 0,ϕ= 0,2πwhich\ncorresponds to the homoclinic points the magnetization vector ten ds parallel to the effective\nfield− →M||− − →Heff. On the other hand, the homoclinic point is an unstable equilibrium point .\nTherefore, theinfluence of thevariablefieldleads toa switching bet ween thetwo types ofthe\nsolutions (9). Hence the following scenario emerges: Suppose at th e initial time the system\n9 \nFIG. 2: Chaotic motion near the separatrix ( k= 1, Σ = Σ S= 1),D= 90GHz. Time independent\nfieldH0, is chosen such that λ=2DS\ngµBH0= 4, and the ratio between the time independent and\nvariable fields is ε=H1/H0= 0.3. The initial energy H=−4.5·103GHzis 8/9 of the re-scaled\nbarrier height E′\nB=DS2/parenleftbig\n1−1\nλ/parenrightbig2. Frequency of the variable field is ω0= 5. One observes that\nthe orientation of the magnetization is changing in time cha otically.\nis prepared in the degenerated ground state Ms=S. We apply a constat magnetic along\nthex−axis and tune its amplitude to realize the separatrix condition Σ s=gµbH0. A small\nperturbation can then lead to the transitions. In particular, switc hing off the perturbation\nwe end up with the transformed state (cf. Fig. (3)).\nVI. DYNAMICS FAR FROM THE SEPARATRIX: THE MEAN HAMILTONIAN\nMETHOD\nTo conclude our study, finally we consider dynamics far from the sep aratrix. The key\npoint is the fact that stochasticity emerges in the small phase-spa ce domain located near\nthe separatrix. Far from the separatrix the dynamics is regular, e ven in the presence of\nsmall perturbations. In this regime, if the frequency of the variab le field is high, analytical\nsolutions are found with the help of the mean Hamiltonian method. The basic idea of\nthe mean Hamiltonian method is the following: for a system having differ ent time scales,\none averages over the fast variables and obtains thus an explicit ex pression for the time\nindependent averaged Hamiltonian [21]. In our case, the following con dition should then\n10 \nFIG.3: Motion neartheseparatrix( k= 1, Σ = Σ S= 1),D= 90GHz,H=−4.5·103GHz,ε= 0.3,\nλ= 4,ω0= 5. The variable field is applied during the finite time interv al between τ1= 100 and\nτ2= 150. Before applying the variable filed, the motion is regul ar and is of an oscillatory nature.\nThe variable field produces a transition into the rotary regi me and then is switched off. During\nthe transition the motion is chaotic.\nhold:\ngµBH0< D < ω 0/parenleftbig2DS\nλ/parenrightbig\n, ε=H1/H0<1. (24)\nThis condition implies that the amplitude of the magnetic fields should be small and the\nfrequency should be high. Provided those conditions hold it is possible to average the dy-\nnamic over the fast frequency ω0. The averaged Hamiltonian is determined by the following\nexpression:\nHav=¯H+1\n2{/an}bracketle{tδH/an}bracketri}ht,H}+1\n3{/an}bracketle{tδH/an}bracketri}ht,{/an}bracketle{tδH/an}bracketri}ht,H+1\n2¯H}}+... (25)\nwhere{A,B}is the Poisson bracket, δH=H−¯H,/an}bracketle{tδH/an}bracketri}ht=/integraltext\nδHdt,(...) means averaging\nover the time. Applying the procedure (25) to the Hamiltonian (1) an d after straightforward\nbut laborious calculations with the accuracy up to the second order terms (1/ω0)2we find\nHav=DS2\nz+gµbH0/radicalbig\n1−S2zcosϕ+1\n2/bracketleftbigg(gµBH1)2\nω2\n0×\n×/parenleftbigg\n−S2\nz(cos(ϕ)+sin(ϕ))2+(1−2S2\nz)(cos(ϕ)−sin(ϕ))2/parenrightbigg/bracketrightbigg\n. (26)\nThe Hamiltonian (26) allows for further simplification: Considering tha t the variable ϕis\nfast in comparison with S2\nz, rotating wave approximation can be used. The Hamiltonian\n11 \nFIG. 4: The dynamics far from the separatrix ( k= 1.6,Σ = 4Σ S) is regular; D= 90GHz,\nH=−0.57·103GHz,ε= 0.3,λ= 100,ω0= 10. The orientation of the magnetization oscillates\nwith time, however without a change of sign. Dynamically ind uced switching is not possible far\nfrom the separatrix. The left plot corresponds to the numeri cal solution of eq.(3). The right side\ncorresponds to the solution (9) Sz(t) = bdn[bλ/k(t−α),1/k], with re-scaled λconstant (27). The\nsolutions are in a good agrement with each other. The only diffe rence is the absence of amplitude\nmodulation in the analytical approximation.\nobtained in this way is completely identical to (4). This means, that th e solutions (9) are\nstill valid. The difference is that, the constant λhas a different form and depends on the\nparameters of the variable field\nλ=/parenleftbigg\n1−/parenleftbiggH1gµB\n2ω0/parenrightbigg2/parenrightbigg2DS\ngµbH0. (27)\nBy comparing the analytical solutions with the results of the numeric al integration of the\nsystem of equations (3) far from the separatrix we verify the valid ity of our approximations.\nFig.(4) is for the parameters of the perturbations that are analog ous to Fig.(2). However,\nunlike Fig.(2), where the system is near the separatrix k≈1, in the case of Fig.(4) k= 1.6\nwhich means that the system is far from the separatrix. That is why the dynamics of\nmagnetization is periodic in time. The difference, between Fig.4. and th e analytical solution\n(9) is that the amplitude of the oscillations is modulated in time. This obs ervation can be\nexplained with the aid of the average Hamiltonian. The point is that the solutions (9) do not\naccount for the existence of multiple angles in the average Hamiltonia n that were ignored\n12by us. They may lead to the appearance of breathing and amplitude m odulations.\nVII. CONCLUSIONS\nWe have considered the spin dynamics of a molecular magnet, when th e number of the\ninvolved levels is large. The dynamics of MM driven by variable filed has be en studied\nbefore [17]. However, in contrast to [17], the applied fields in our case are quite strong,\ni.e. we are in the strongly nonlinear, non-perturbative regime. The u nderlying dynamics is\nthen treated semi-classically. We showed that the phase space of t he system contains two\ndomains separated by a separatrix line. The solutions far from the s eparatrix correspond\nto the rotating and the oscillatory regime, while the separatrix solut ion is non-oscillating\nand is of a soliton type. The existence of the separatrix is important as the states in the\ndomain near to it are extremely sensitive to small perturbations. Th erefore, if a variable\nfield is applied, instead of a soliton type solutions, the spin dynamics tu rns chaotic and\nunpredictable. The control parameter is the initial energy of the s ystem. By a proper choice\nof it each type of the dynamic can be realized. The structure of the system’s phase space\nis directly related to the possible mechanisms of the magnetization re versal. Namely, if\nthe energy is equal to HS=8\n9E′\nBof the re-scaled anisotropy barrier E′\nB=DS2/parenleftbig\n1−1\nλ/parenrightbig2\n[16] (the separatrix condition) an external variable field leads to a c haotic change of the\nmagnetization orientation. The switching process is random and with the equal probability\n1/2, the system may appear in the new state as well as stay in the old one. The information\nabout initial state is lost. This result is different from the case of wea k applied fields [17],\nwhere the dynamics shows a long-term memory of the initial state.\nAcknowledgment: The project is financially supported by theGeorgianNationalFoun-\ndation (grants: GNSF/STO 7/4-197, GNSF/STO 7/4-179). The fin ancial support by the\nDeutsche Forschungsgemeinschaft (DFG) through SFB 672 and t hough SPP 1285 is grate-\nfully acknowledged.\n[1] T. Lis, Acta Crystallogr. B 36, 2042 (1980); A. C. R. Sessoli, D. Gatteschi, and M. A. Novak,\nNature365, 141 (1993).\n13[2] Quantum Tunneling of Magnetization, edited by L. Gunthe r and B. Barbara (Kluwer Aca-\ndemic, Dordrech, 1995).\n[3] J. R. Friedman, M. P. Sarachik, J. Tejada, and R. Ziolo, Ph ys. Rev. Lett. 76, 3830 (1996).\n[4] J. M. Hernandez, X. X. Zhang, F. Luis, J. Bartolom´ e, J. Te jada, and R. Ziolo, Europhys.\nLett.35, 301 (1996).\n[5] L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli , and B. Barbara, Nature 383, 145\n(1996).\n[6] E. M. Chudnovsky and J. Tajida, Macroscopic Quantum Tunn eling of the Magnetic Moment\n(Cambridge University Press, Cambrige, England, 1998).\n[7] L. Thomas et. al, Nature 383, 145 (1996).\n[8] W. Wernsdorfer and R. Sessoli, Science 284, 133 (1999).\n[9] M. Hennion et al, Phys. Rev. B 56, 8819 (1997).\n[10] A. V. Shvetsov, G. A. Vugalter, and A. I. Grebeneva, Phys . Rev. B 74, 054416 (2006).\n[11] Ji-Bing Liu, Xin-You Lu, Na Liu, Men Wang, and Tang-Kun L iu, Phys. Lett. A 373, 413\n(2009).\n[12] A. Ardavan et al, Phys. Rev. Lett. 98, 057201 (2007).\n[13] S. Hill et al, Phys. Rev. Lett. 80, 2453(1998).\n[14] L.Chotorlishvili, Z.Toklikishvili, andJ.Berakdar, Phys.Lett. A. 373, 231(2009), A.Ugulava,\nRadiophysics and Quantum Electronics 30, 748 (1987).\n[15] G. Bellesa et al, Phys. Rev. Lett. 83, 416 (1999), E. Delbarco et al, Phys. Rev. B 62, 3018\n(2000).\n[16] D. A. Garanin and E. M. Chudnovsky, Phys. Rev. B 59, 3671 (1999).\n[17] C. Calero, D. A. Garanin, and E. M. Chudnovsky, Phys. Rev . B72, 024409 (2005).\n[18] Handbook of Mathematical Functions, edited by M. Abram owitz and I. Stegun, (National\nBureau of Standards, Applied Mathematics Series 55, Washin gton 1972), A. Prudnikov, U.\nBrichkov, and O. Marichev, Integrals and Series v.1, (Mosco w, Nauka, 1981).\n[19] A.J. Lichtenberg and M.A. Lieberman, Regular and Stoch astic Motion, (Springer-Verlag, New\nYork, Heidelberg, Berlin, 1983).\n[20] G.M. Zaslavsky, The Physics of Chaos in Hamiltonian Sys tems, 2nd edition, (Imperial College\nLondon, 2007).\n[21] V. M. Volosov and B. I. Morgunov, Averaging Method in the Theory of Nonlinear Oscillations\n14(Nauka, Moscow, 1971).\n15"    },    {        "title": "1009.1671v3.Hydrodynamic_equation_of_a_spinor_dipolar_Bose_Einstein_condensate.pdf",        "content": "arXiv:1009.1671v3  [cond-mat.quant-gas]  10 Nov 2010Hydrodynamic equation of a spinor dipolar Bose-Einstein co ndensate\nKazue Kudo1and Yuki Kawaguchi2\n1Division of Advanced Sciences, Ochadai Academic Productio n,\nOchanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 11 2-8610, Japan\n2Department of Physics, University of Tokyo, 7-3-1 Hongo, Bu nkyo-ku, Tokyo 113-0033, Japan\n(Dated: November 2, 2018)\nWe introduce equations of motion for spin dynamics in a ferro magnetic Bose-Einstein condensate\nwith magnetic dipole-dipole interaction, written usingav ector expressing thesuperfluid velocity and\na complex scalar describing the magnetization. This simple hydrodynamical description extracts\nthe dynamics of spin wave and affords a straightforward appro ach by which to investigate the\nspin dynamics of the condensate. To demonstrate the advanta ges of the description, we illustrate\ndynamical instability and magnetic fluctuation preference , which are expressed in analytical forms.\nPACS numbers: 03.75.Lm, 03.75.Mn,03.75.Kk\nI. INTRODUCTION\nOne of the salient features of a gaseous Bose-Einstein condensat e (BEC) is the internal spin degrees of freedom. In\nspinor BECs, namely, in BECs with internal degrees of freedom, spin and gauge degrees of freedom couple in various\nmanners, leading to nontrivial properties of spin waves and topolog ical excitations. For example, ferromagnetic BECs\nhave continuous spin-gauge symmetry, thus the circulation of the superfluid velocity is not quantized [1–3], whereas\nspin-1 polar BECs and spin-2 cyclic BECs can host fractional vortice s due to the discrete spin-gauge symmetry [4–\n7]. In recent experiments, in situimaging of transverse magnetization has revealed the real-time dyn amics of the\nspontaneous symmetry breaking, spin texture formation, and nu cleation of spin vortices [8–10], opening up a new\nparadigm for studying the static and dynamic properties of spin tex tures.\nOn the other hand, BECs with magnetic dipole-dipole interaction (MDD I) have also attracted much attention both\nexperimentally and theoretically in recent years. The long-range an d anisotropic nature of the MDDI is predicted to\nyield exotic phenomena, such as new equilibrium shapes, roton-maxo nspectra, supersolid states, and two-dimensional\nsolitons[11]. In particular, whenthe BEChasspin degreesoffreedo m, the MDDI ispredicted to developspintextures,\neven when the MDDI is much weaker than the contact interaction [12 –14]. This work is motivated by experiments\ndone by the Berkeley group [9], where small magnetic domains were ob served to develop from a helical spin structure\nin a spin-187Rb BEC. The method presented in this paper simplifies the spin dynamic s in a complicated system of\nspinor dipolar BECs, although we have shown in our previous work tha t mean-field calculations do not reproduce the\nexperimental results [15].\nIn this paper, we propose a new type of hydrodynamic description o f a ferromagnetic BEC with MDDI. The\nhydrodynamic equation of spinor BECs has been discussed for both ferromagnetic phases [16–18] and non-magnetized\nphases [18, 19]. In Ref. [16], Takahashi et al.consider the strong MDDI limit by using the classical spin model, i.e.,\nby neglecting the spin-gauge coupling. On the other hand, Lamacra ft takes into account the spin-gauge coupling by\nintroducing the so-called Mermin-Ho relation, and considers the wea k MDDI [17]. In these papers, the authors use\na unit vector to describe the local magnetization in the ferromagne tic phase. Here we use a single complex scalar\nvariable instead of a unit vector to describe the local magnetization and treat both the spin-gauge coupling and\nMDDI. This simple description allows a straightforward approach to a nalyze the spin dynamics of the condensate.\nIn order to demonstrate the advantages of our description, we a nalyze the dynamical instability and magnetization\nfluctuation preference of the BEC with MDDI.\nThe rest of the paper is organized as follows. In Sec. II, hydrodyn amic equations described using the spin density\nvector are derived from the Gross-Pitaevskii (GP) equation. We t hen rewrite the equations by means of stereographic\nprojection for some simple cases: quasi two-dimensional (2D) sys tems under zero external field and under a strong\nmagnetic field. For both zero-field and strong-field cases, the wav evector dependence of the dynamical instability is\nobtained straightforwardly in an analytical form in Sec. III. In Sec IV, we also illustrate the magnetic fluctuation\npreference for the unstable modes discussed in Sec. III. Conclus ions are given in Sec. V.2\nII. HYDRODYNAMIC DESCRIPTION\nA. Equations of motion of mass and spins\nWe consider a spin- FBEC under a uniform magnetic field Bapplied in the zdirection. The GP equation for the\nspinor dipolar system is given by\ni/planckover2pi1∂\n∂tΨm(r,t) = (H0+pm+qm2)Ψm\n+2F/summationdisplay\nS=0,even4π/planckover2pi12\nMaSS/summationdisplay\nMS=−SF/summationdisplay\nn,m′,n′=−F/angb∇acketleftmn|SMS/angb∇acket∇ight/angb∇acketleftSMS|m′n′/angb∇acket∇ightΨ∗\nnΨm′Ψn′\n+cdd/summationdisplay\nµ=x,y,zF/summationdisplay\nn=−Fbµ(Fµ)mnΨn, (1)\nwhere Ψ m(r,t) is the condensate wavefunction for the atoms in the magnetic sub levelmandH0=−/planckover2pi12∇2/(2M)+\nUtrap(r), withMbeing the atomic mass and Utrapthe spin-independent trapping potential. The linear and quadratic\nZeeman energies per atom are given by p=gFµBBandq= (gFµBB)2/Ehf, respectively, where gFis the hyperfine\ng-factor,µBis the Bohr magneton, and Ehfis the hyperfine energy splitting. The second term on the right-han d side\nof Eq. (1) comes from the short-range part of the two-body inte raction given by\nVs(r,r′) =δ(r−r′)2F/summationdisplay\nS=0,even4π/planckover2pi12\nMaSPS, (2)\nwherePS=/summationtextS\nMS=−S|SMS/angb∇acket∇ight/angb∇acketleftSMS|projects a pair of spin-1 atoms into the state with total spin S, andaSis the\ns-wave scattering length for the corresponding spin channel S. The scattering amplitude for odd Svanishes due to\nBose symmetrization, and /angb∇acketleftmn|SMS/angb∇acket∇ightin Eq. (1) is the Clebsch-Gordan coefficient. The last term on the righ t-hand\nside of Eq. (1) corresponds to the MDDI, where cdd=µ0(gFµB)2/(4π), withµ0being the magnetic permeability of\nthe vacuum. Here, we define the non-local dipole field by\nbµ(r) =/integraldisplay\nd3r′/summationdisplay\nνQµν(r−r′)fν(r′), (3)\nwhereQµν(r) is the dipole kernel, given in Sec. IIC, and\nfµ=/summationdisplay\nmnΨ∗\nm(Fµ)mnΨn (4)\nis the spin density, with Fx,y,zbeing the spin- Fmatrices. Below we omit the summation symbol: Greek indices that\nappear twice are to be summed over x,y, andz, and Roman indices are to be summed over −F,...,F.\nFrom the GP equation, we can immediately derive the mass continuity e quation:\n∂ntot\n∂t+∇·(ntotvmass) = 0, (5)\nwhere\nntot= Ψ∗\nmΨm, (6)\nntotvmass=/planckover2pi1\n2Mi[Ψ∗\nm(∇Ψm)−(∇Ψ∗\nm)Ψm] (7)\nare the number density and superfluid current, respectively. By in troducing a normalized spinor ζmdefined by\nΨm(r,t) =/radicalbig\nntot(r,t)ζm(r,t), the superfluid velocity vmasscan be written as\nvmass=/planckover2pi1\n2Mi[ζ∗\nm(∇ζm)−(∇ζ∗\nm)ζm]. (8)3\nIn the absence of the linear and quadratic Zeeman effects and the M DDI, the continuity equation of the spin density\ncan also be derived from the GP equation as\n∂fµ\n∂t+∇·/parenleftBig\nntotvµ\nspin/parenrightBig\n= 0, (9)\nwherevµ\nspinis the spin superfluid velocity defined by\nvµ\nspin=/planckover2pi1\n2Mi(Fµ)mn[ζ∗\nm(∇ζn)−(∇ζ∗\nm)ζn]. (10)\nThe short-range interaction does not contribute to the equation of motion of spin, since it conserves the total spin of\ntwo colliding atoms. The detailed calculation is given in Appendix A. In the presence of the external magnetic field\nalong thezdirection, the linear Zeeman effect induces a torque term ( p//planckover2pi1)(ˆz×f)µon the right-hand side of Eq. (9),\nwhich causes the precession of spins. In a similar manner, the dipole fi eld also induces a torque term ( cdd//planckover2pi1)(b×f)µ.\nOn the other hand, the quadratic Zeeman term does not conserve the transversemagnetizationand its effect is written\nas (2q//planckover2pi1)ǫµzνntotˆNzν, whereǫijkis the Levi-Civita symbol and\nˆNµν=1\n2ζ∗\nm(FµFν+FνFµ)mnζn (11)\nis a nematic tensor. The derivations of these three terms are given in Appendix A. As a result, we obtain the\nequation of motion of spins in the presence of the MDDI and linear and quadratic Zeeman effects under the external\nfield parallel to the zaxis:\n∂fµ\n∂t+∇·/parenleftBig\nntotvµ\nspin/parenrightBig\n=cdd\n/planckover2pi1(b×f)µ+p\n/planckover2pi1(ˆz×f)µ+2q\n/planckover2pi1ntotǫµzνˆNzν. (12)\nEquations (5) and (12) hold in all phases, independent of scatterin g length.\nB. Ferromagnetic BEC\nIn the following, we consider a ferromagnetic BEC. We assume that t he BEC is fully magnetized, |f|=Fntot, and\nonly the direction of the spin density can vary in space. This assumpt ion is valid when the ferromagnetic interaction\nenergy is large enough compared with the other spinor interaction e nergies, MDDI energy, quadratic Zeeman energy,\nand the kinetic energy arising from the spacial variation of the direc tion off. The linear Zeeman effect is not\nnecessarily weaker than the ferromagnetic interaction, since it me rely induces the Larmor precession. For example,\nthe short-range interaction (2) for a spin-1 BEC can be written as [20]\n/angb∇acketleftmn|Vs(r,r′)|m′n′/angb∇acket∇ight=δ(r−r′)[c0δmnδm′n′+c1(Fµ)mn(Fµ)m′n′], (13)\nwherec0= 4π/planckover2pi12(2a2+a0)/(3M) andc1= 4π/planckover2pi12(a2−a0)/(3M). The ground state is ferromagnetic for c1<0. The\nabove assumption is valid when q≪ |c1|ntot,cdd≪ |c1|, and the length scale of the spatial spin structure is larger\nthan the spin healing length ξsp=/planckover2pi1//radicalbig\n2M|c1|ntot. Moreover in the incompressible limit, namely when the spin\nindependent interaction ( c0ntotfor the case of a spin-1 BEC) is much stronger than the ferromagn etic interaction and\nMDDI, the number density ntotis determined regardless of the spin structure and assumed to be s tationary. This is\nthe case for the spin-187Rb BEC.\nWe then rewrite the equations of motion (5) and (12) in terms of a un it vector ˆf≡f/(Fntot) that describes\nthe direction of the spin density and the superfluid velocity vmassdefined in Eq. (8). The order parameter for the\nspin-polarized state in the zdirection is given by ζ(0)\nm=δmF. The general order parameter is obtained by performing\nthe gauge transformation and Euler rotation as\nζ=eiφe−iFzαe−iFyβe−iFzγζ(0)\n=ei(φ−Fγ)e−iFzαe−iFyβζ(0), (14)\nwhereα,βandγare Euler angles shown in Fig. 1 and φis the overall phase. Due to the spin-gauge symmetry of the\nferromagnetic BEC, i.e., the equivalence between the phase change φand spin rotation γ, distinct configurations of ζ4\nαβγf\nxyz\nFIG. 1: Euler rotation of the unit vector ˆf.\nare characterized with a set of parameters ( α,β,φ′≡φ−Fγ). Here,αandβdenote the direction of ˆfas shown in\nFig. 1. Actually, ˆffor the order parameter (14) is calculated as\nFˆf=ζ∗\nmFmnζn\n=ζ(0)∗\nm/parenleftbig\neiFyβeiFzαFe−iFzαe−iFyβ/parenrightbig\nmnζ(0)\nn\n=Rz(α)Ry(β)/bracketleftBig\nζ(0)∗\nmFmnζ(0)\nn/bracketrightBig\n=F\nsinβcosα\nsinβsinα\ncosβ\n, (15)\nwhereRz(α) andRy(β) are the 3 ×3 matrices describing the rotation about the zaxis byαand about the yaxis by\nβ, respectively. In a similar manner, we obtain the nematic tensor ˆNfor the order parameter (14) as\nˆNµν=Rz(α)Ry(β)ˆN(0)\nµνRT\ny(β)RT\nz(α)\n=F\n2δµν+F(2F−1)\n2ˆfµˆfν, (16)\nwhere T denotes the transpose and ˆN(0)\nµν=1\n2ζ(0)∗\nm(FµFν+FνFµ)mnζ(0)\nnis the nematic tensor for ζ(0), which is given\nby\nˆN(0)=\nF/2 0 0\n0F/2 0\n0 0F2\n\n=F\n2\n1 0 0\n0 1 0\n0 0 1\n+F(2F−1)\n2\n0 0 0\n0 0 0\n0 0 1\n. (17)\nSubstituting Eq. (14) and\n∇ζ=i/bracketleftbig\n∇φ′−(∇α)Fz−(∇β)e−iFzαFyeiFzα/bracketrightbig\nζ (18)\ninto Eq. (8), the superfluid velocity can be written as\nvmass=/planckover2pi1\nM[∇φ′−F(∇α)cosβ], (19)\nwhich satisfies the Mermin-Ho relation [21]:\n∇×vmass=/planckover2pi1F\n2Mǫµνλˆfµ(∇ˆfν×∇ˆfλ). (20)\nAs we mentioned before, ntotis stationary in the incompressible limit. Thus, Eq. (5) becomes\n∇·(ntotvmass) = 0. (21)5\nEquations(20) and(21) areequationsforthesuperfluid velocity. Next, weconsidertheequationforthe spinsuperfluid\nvelocity. Making use of Eq. (16), Eq. (10) can be rewritten in terms ofˆfandvmassas\nvµ\nspin=/planckover2pi1\nM/bracketleftBig\n(∇φ′)Fˆfµ−(∇α)ˆNµz−(∇β)ˆNµycosα+(∇β)ˆNµxsinα/bracketrightBig\n=Fˆfµvmass−/planckover2pi1F\n2M\n(∇α)sinβ\n−cosβcosα\n−cosβsinα\nsinβ\n+(∇β)\n−sinα\ncosα\n0\n\n\nµ\n=Fˆfµvmass−/planckover2pi1F\n2Mǫµνλˆfν∇ˆfλ. (22)\nSubstituting Eqs. (16), (21), and (22) into Eq. (12), we obtain th e hydrodynamic equation in terms of ˆfandvmassas\n∂ˆf\n∂t+(vmass·∇)ˆf=−ˆf×Beff, (23)\nwith\nBeff=−/planckover2pi1\n2M(a·∇)ˆf−/planckover2pi1\n2M∇2ˆf+cdd\n/planckover2pi1b+p\n/planckover2pi1ˆeB+q(2F−1)\n/planckover2pi1(ˆeB·ˆf)ˆeB,\nwherea= (∇ntot)/ntotand ˆeBis the unit vector along the external field (ˆ eB= ˆzin this paper). Here we note that\nEq. (23) has the same form as the extended Landau-Lifshitz-Gilbe rt equation (without damping) which includes the\nadiabatic spin torque term [22].\nC. Quasi-2D system\nWe next consider a quasi-2D system, that is, we consider a BEC confi ned in a quasi-2D trap whose Thomas-Fermi\nradius in the normal direction to the 2D plane is smaller than the spin he aling length. We approximate the wavefunc-\ntion in the normal direction by a Gaussian with width d: Ψm(r⊥,rn) =ψm(r⊥)h(rn), where r⊥is the position vector\nin the 2D plane, rnis the coordinate in the normal direction, and h(rn) = exp[−r2\nn/(4d2)]/(2πd2)1/4. Multiplying the\nwavefunction to Eq. (1) and integrating over rn, we obtain the 2D GP equation. The equation is the same as Eq. (1) if\none replaces Ψ mwithψm,aSwithηaS,cddwithηcdd, andbwith¯b, whereη=/integraltext\ndrnh4(rn)//integraltext\ndrnh2(rn) = 1/√\n4πd2\nand\n¯bµ=/integraldisplay\nd2r′\n⊥Q(2D)\nµν(r⊥−r′\n⊥)[ψ∗\nm(r′\n⊥)(Fν)mnψn(r′\n⊥)], (24)\nwith\nQ(2D)\nµν(r⊥−r′\n⊥) =1\nη/integraldisplay/integraldisplay\ndrndr′\nnh2(rn)h2(r′\nn)Qµν(r−r′). (25)\nStarting from the 2D GP equation and following the above procedure , we derive the 2D hydrodynamic equation:\n∂ˆf\n∂t+(vmass·∇)ˆf=−ˆfׯBeff, (26)\nwith\n¯Beff=−/planckover2pi1\n2M(a·∇)ˆf−/planckover2pi1\n2M∇2ˆf+ηcdd\n/planckover2pi1¯b+p\n/planckover2pi1ˆeB+q(2F−1)\n/planckover2pi1(ˆeB·ˆf)ˆeB,\nwherevmassand∇are the two-dimensional vector and vector operator, respectiv ely. When we consider a quasi-2D\nBEC,ntot,ˆf, andvmassare defined by means of ψminstead of Ψ m.\nD. Dipole kernel\nThis section provides the detailed form of the dipole kernel in 3D and q uasi-2D systems under zero external field\nand under a strong magnetic field ( p≫cddntot). The derivations are given in Ref. [15].6\nThe dipole kernel in the laboratory frame of reference is given by\nQ(lab)\nµν(r) =δµν−3ˆrµˆrν\nr3, (27)\nwithr=|r|andˆr=r/r. The 2D dipole kernel in the laboratory frame is calculated by substit uting Eq. (27) into\nEq. (25). For r⊥= (x,y) andrn=z, the 2D dipole kernel is given by\nQ(2D,lab)\nµν(r⊥) =/summationdisplay\nk⊥eik⊥·r⊥˜Q(2D,lab)\nk⊥µν, (28)\nwhere\n˜Q(2D,lab)\nk⊥=−4π\n3\n1 0 0\n0 1 0\n0 0−2\n+4πG(k⊥d)\nˆk2\nxˆkxˆky0\nˆkxˆkyˆk2\ny0\n0 0 −1\n, (29)\nwithk⊥= (kx,ky),k⊥=|k⊥|,ˆkx,y=kx,y/k⊥, andG(k)≡2kek2/integraltext∞\nke−t2dt=√πkek2erfc(k). It can be shown that\nG(k) is a monotonically increasing function that satisfies G(0) = 0 and G(∞) = 1.\nWhen the linear Zeeman energy is much larger than the MDDI energy, we choose the rotating frame of reference\nin spin space by replacing Ψ mwithe−ipmt//planckover2pi1Ψm, and eliminate the linear Zeeman term from the GP equation. In\nthis case, the contribution of the MDDI is time-averaged due to the Larmor precession, and we use the dipole kernel\nwhich is averaged over the Larmor precession period given by [23]\nQ(rot)\nµν(r) =−1\n21−3ˆr2\nz\nr3(δµν−3δzµδzν). (30)\nSubstituting Eq. (30) into Eq. (25), we obtain the time-averaged 2 D dipole kernel in the rotating frame as\nQ(2D,rot)\nµν(r⊥) = (δµν−3δzµδzν)/summationdisplay\nk⊥eik⊥·r⊥˜Qk⊥, (31)\nwhere\n˜Qk⊥=2π\n3/braceleftBig\n1−3(ˆen·ˆeB)2−3G(k⊥d)/bracketleftBig\n(ˆeB\n⊥·ˆk⊥)2−(ˆen·ˆeB)2/bracketrightBig/bracerightBig\n. (32)\nHere,ˆk⊥=k⊥/k⊥, ˆenis the unit vector normal to the plane, and ˆeB\n⊥is the vector of ˆ eBprojected onto the 2D plane.\nE. Stereographic projection\nThe spin dynamics are now described by Eqs. (20), (21), and (23) o r (26). We rewrite the equations by means of\nstereographic projection [24]: we employ a complex number ϕ= (ˆfx+iˆfy)/(1+ˆfz) to express the spin variables,\nˆfx=ϕ+ϕ∗\n1+ϕϕ∗,ˆfy=−i(ϕ−ϕ∗)\n1+ϕϕ∗,ˆfz=1−ϕϕ∗\n1+ϕϕ∗. (33)\nEquation (20) is rewritten as\n∇×vmass=iF2/planckover2pi1\nM∇ϕ×∇ϕ∗\n(1+ϕϕ∗)2, (34)\nwhile Eq. (21) remains the same. In order to rewrite the equation of spins, we need to specify the dimensionality of\nthe system and the direction and strength of the external field. I n this paper, we consider the following two cases: (i)\na quasi-2D system normal to the zaxis under zero magnetic field, and (ii) a quasi-2D system normal to t heyaxis\nwith a strong magnetic field along the zaxis. Case (ii) corresponds to the situation in the Berkeley experime nt [9].7\nFor case (i), we take ˆ en= ˆzandp=q= 0 and use Eqs. (24), (26), (28), and (29). Using the stereogra phic\nprojection, the equation of motion of spins is given by\n∂ϕ(r,t)\n∂t=−vmass·∇ϕ+i/planckover2pi1\n2Mntot∇ntot·∇ϕ+i/planckover2pi1\n2M∇2ϕ−i/planckover2pi1\nMϕ∗(∇ϕ)2\n1+ϕϕ∗\n−iηcddF\n2/planckover2pi1/integraldisplay\nd2r′ntot(r′)/summationdisplay\nkeik·(r−r′)/bracketleftbigg\n−h1(k)ϕ(r′)\n1+ϕ(r′)ϕ∗(r′)+h2(k)ϕ∗(r′)\n1+ϕ(r′)ϕ∗(r′)/bracketrightbigg\n+iηcddF\n2/planckover2pi1ϕ2/integraldisplay\nd2r′ntot(r′)/summationdisplay\nkeik·(r−r′)/bracketleftbigg\n−h1(k)ϕ∗(r′)\n1+ϕ(r′)ϕ∗(r′)+h∗\n2(k)ϕ(r′)\n1+ϕ(r′)ϕ∗(r′)/bracketrightbigg\n+iηcddF\n/planckover2pi1ϕ/integraldisplay\nd2r′ntot(r′)/summationdisplay\nkeik·(r−r′)h1(k)1−ϕ(r′)ϕ∗(r′)\n1+ϕ(r′)ϕ∗(r′), (35)\nwhere the subscript ⊥was omitted for simplicity and\nh1(k) =8π\n3−4πG(kd), (36)\nh2(k) = 4πG(kd)(ˆkx+iˆky)2. (37)\nFor case (ii), we take ˆ en= ˆy, ˆeB= ˆz, andp= 0, and use Eqs. (24), (26), (31), and (32). Then, the equation of\nmotion of spins is described as\n∂ϕ(r,t)\n∂t=−vmass·∇ϕ+i/planckover2pi1\n2Mntot∇ntot·∇ϕ+i/planckover2pi1\n2M∇2ϕ−i/planckover2pi1\nMϕ∗(∇ϕ)2\n1+ϕϕ∗\n−iηcddF\n/planckover2pi1/integraldisplay\nd2r′ntot(r′)/summationdisplay\nkeik·(r−r′)˜Qkϕ(r′)\n1+ϕ(r′)ϕ∗(r′)\n+iηcddF\n/planckover2pi1ϕ2/integraldisplay\nd2r′ntot(r′)/summationdisplay\nkeik·(r−r′)˜Qkϕ∗(r′)\n1+ϕ(r′)ϕ∗(r′)\n−2iηcddF\n/planckover2pi1ϕ/integraldisplay\nd2r′ntot(r′)/summationdisplay\nkeik·(r−r′)˜Qk1−ϕ(r′)ϕ∗(r′)\n1+ϕ(r′)ϕ∗(r′)\n+iq(2F−1)\n/planckover2pi11−ϕϕ∗\n1+ϕϕ∗ϕ, (38)\nwherer⊥→r= (x,z) andk⊥→k= (kx,kz).\nIII. DYNAMICAL INSTABILITY\nThe hydrodynamic equations derived above give a rather straightf orward approach to the analysis of the spin\ndynamics in a spinor BEC. In this section, we analyze the dynamical ins tability for cases (i) and (ii). Here we\nconsider a uniform quasi-2D system and assume ∇ntot= 0.\nA. Case (i): Instability under zero external field\nHere we analyze the dynamical instability under zero external field f or two initial stationary structures: uniform\nspin structures polarized normal to the x-yplane (ϕ0= 0) and in the x-yplane (ϕ0= 1).\nFirst, we consider the case in which the spins are polarized normal to thex-yplane, i.e., in the zdirection,ϕ0= 0.\nSubstituting ϕ= 0+δϕandvmass=v0+δvinto Eq. (35), we obtain linearized equations of δϕandδϕ∗. Performing\nFourier expansions δϕ=/summationtext\nkδ˜ϕkeik·randδϕ∗=/summationtext\nkδ˜ϕ∗\n−keik·r, we have\nd\ndt/parenleftbigg\nδ˜ϕk\nδ˜ϕ∗\n−k/parenrightbigg\n=i\n/planckover2pi1/parenleftbigg\n−g0−g1−g2\ng∗\n2−g0+g1/parenrightbigg/parenleftbigg\nδ˜ϕk\nδ˜ϕ∗\n−k/parenrightbigg\n, (39)8\nkx/2π (µm   )-1ky/2π (µm   )-1\n s  -1\nFIG. 2: Re λ+(k) of the uniform spin structure polarized in the zdirection under zero external field. The fluctuations in the\nblack region are dynamically unstable and grow exponential ly. Here, ntotis given by√\n2πd2n3Dwithn3D= 2.3×1014cm−3\nandd= 1.0µm. The other parameters are given by the typical values for a s pin-187Rb atom: M= 1.44×10−25Kg,F= 1,\ngF=−1/2, andEhf= 6.835 GHz ×h.\nwhere\ng0(k) =/planckover2pi1v0·k, (40)\ng1(k) =/planckover2pi12k2\n2M−2π˜cdd[2−G(kd)], (41)\ng2(k) = 2π˜cddG(kd)(ˆkx+iˆky)2. (42)\nHere, ˜cdd=ηcddntotFandk= (kx,ky). The eigenvalues of the 2 ×2 matrix in Eq. (39) are\nλ±(k) =−i\n/planckover2pi1g0±1\n/planckover2pi1/radicalBig\n|g2|2−g2\n1. (43)\nThe system becomes dynamically unstable when one of the eigenvalue s has a positive real part; that is when\nReλ+(k)>0 (or|g2|2−g2\n1>0). The wavevector dependence of Re λ+is shown in Fig. 2. When the BEC is polarized\nperpendicular to the 2D plane, the MDDI is repulsive and isotropic in th e 2D plane. Thus, the BEC is unstable\nagainst spin flip, and the unstable modes distribute isotropically in the momentum space. The unstable region in\nthe momentum space has a ring shape. The radius and width of the rin g are estimated as k0= (2//planckover2pi1)√2πM˜cddand\n∆k≃(√π/8)k2\n0d(4−√πk0d), respectively, for kd≪1.\nNext, weconsidertheuniform spinstructurepolarizedinthe xdirection,ϕ0= 1. We obtainthelinearizedequations\nin a similar way to the above. Substituting ϕ= 1+δϕandvmass=v0+δvinto Eq. (35) and performing the Fourier\nexpansion, we obtain the equation of the same form as Eq. (39) with\ng0(k) =/planckover2pi1v0·k, (44)\ng1(k) =/planckover2pi12k2\n2M+2π˜cdd/bracketleftBig\n1−G(kd)(1−ˆk2\ny)/bracketrightBig\n, (45)\ng2(k) = 2π˜cdd/bracketleftBig\n1−G(kd)(1+ˆk2\ny)/bracketrightBig\n. (46)\nIn this case, it can be shown that g2\n2−g2\n1is always negative. Then, the eigenvalues which are given by the same form\nas Eq. (43) are purely imaginary regardless of k. Hence, the spin-polarized state along the 2D plane is stable under\nzero magnetic field.\nB. Case (ii): Instability under a strong magnetic field\nHere we analyze the dynamical instability for the helical spin structu re,ϕ0=ei(κα·r−ωαt), characterized by the\nhelix wavevector καin thex-zplane under a strong magnetic field in the zdirection. Substituting ϕ=ϕ0and\nvmass=v0into Eq. (38) gives ωα=v0·κα. Substituting ϕ=ϕ0(1 +δϕ) andvmass=v0+δvinto Eqs. (21) and\n(34), and holding the terms up to the first order of the fluctuation s, we have ∇·v0= 0,∇·δv= 0,∇×v0= 0, and\n∇×δv=/planckover2pi1\n2M(∇δϕ+∇δϕ∗)×κα. After Fourier expansions of δϕ,δϕ∗andδv=/summationtext\nkδ˜vkeik·r, we have\nδ˜vk=/planckover2pi1\n2M(δ˜ϕk+δ˜ϕ∗\n−k)/bracketleftbigg\nκα−(k·κα)k\nk2/bracketrightbigg\n. (47)9\nkx/2π (µm   )-1(a)   uniform (b)   uniform (c)   uniform\n(d)  helix 120 µm (e)  helix 120 µm (f)  helix 60 µm (g)  helix 60 µmkz/2π (µm   )-1\n120 mG 160 mG 200 mG\n120 mG 160 mG 120 mG 160 mG s  -1\nFIG. 3: Re λ+(k) for (a)–(c) uniform spin structures and (d)–(g) spin helic es in the zdirection with a pitch 2 π/καunder a\nstrong magnetic field. The helical pitch for (d) and (e) is 120 µm, and that for (f) and (g) is 60 µm. The external field Bis\n[(a), (d), and (f)] 120 mG, [(b), (e), and (g)] 160 mG, and (c) 2 00 mG. The other parameters are the same as those in Fig. 2.\nSubstituting ϕ=ϕ0(1+δϕ) into Eq. (38) and applying Eq. (47), we obtain an equation of the sa me form as Eq. (39)\nwith\ng0(k) =/planckover2pi1v0·k, (48)\ng1(k) =/planckover2pi12k2\n2M+/planckover2pi12\n2M/bracketleftbiggκ2\nα\n2−(k·κα)2\nk2/bracketrightbigg\n+˜cdd\n4/parenleftBig\n˜Qk+κα+˜Qk−κα/parenrightBig\n−˜cdd/parenleftBig\n˜Qκα+˜Qk/parenrightBig\n+q(2F−1)\n2,(49)\ng2(k) =/planckover2pi12\n2M/bracketleftbiggκ2\nα\n2−(k·κα)2\nk2/bracketrightbigg\n−˜cdd\n4/parenleftBig\n˜Qk+κα+˜Qk−κα/parenrightBig\n−˜cdd˜Qk+q(2F−1)\n2. (50)\nHere,k= (kx,kz) and the Fourier transform of the dipole kernel is now simply given by ˜Qk= (2π/3)[1−\n3(kz/k)2G(kd)]. The eigenvalues of the 2 ×2 matrix in Eq. (39) are given by Eq. (43) with Eqs. (48)–(50). When\nκα//ˆz,v0= 0, and neither the MDDI nor the quadratic Zeeman effect exists (˜ cdd=q= 0), the eigenvalues coincide\nwith the dispersion relation derived in Ref. [17].\nFigure 3 illustrates the wavenumber dependence of Re λ+(k) for uniform and helical spin structures under various\nmagnetic field strengths, indicating the region of dynamically unstab le modes. The dynamical instability discussed\nhere agrees qualitatively with that obtained by the Bogoliubov analys is [15, 25]. However, there is a quantitative\ndiscrepancy in the magnetic field dependence of unstable modes cau sed by the fact that the local magnetization of\nthe condensate is assumed to be fully polarized in our method. Howev er, whenqis not sufficiently small compared\nwith the ferromagnetic interaction, the amplitude of the magnetiza tion decreases as qincreases. For the parameters\nused in the calculation for Fig. 3, our assumption is valid for B≪480 mG.\nIV. MAGNETIC FLUCTUATION PREFERENCE\nWe also investigate the magnetic fluctuation preference for two ca ses, that of dynamical instability under zero field\nfor the uniform spin structure polarized normal to the x-yplane (ϕ0= 0), which is discussed in Sec. IIIA, and that\nof dynamical instability under a strong magnetic field for the helical s pin structure ( ϕ0=ei(κα·r−ωαt)), which is\ndiscussed in Sec. IIIB.\nFor the case of zero external field for ϕ0= 0, two kinds of magnetic fluctuations are considered: the x-direction\nfluctuation δˆfxand they-direction fluctuation δˆfy. They are described by the first order of δϕ:δˆfx= 2Re(δϕ) and\nδˆfy= 2Im(δϕ). Namely, the x- andy-direction fluctuations are characterized by the real and imaginar y parts ofδϕ,\nrespectively.10\nkx/2π (µm   )-1ky/2π (µm   )-1\n0.1\n-0.050.05\n-0.10\n0 π/2\nxy(a) (b) (c)\nPQ\n0.10.050-0.05-0.1\nFIG. 4: (Color) (a) Magnetic fluctuation preference θ(k) for the unstable mode shown in Fig. 2. The x-direction fluctuation is\ndominant in the red regions ( π/4< θ < π/ 2) and the y-direction fluctuation is dominant in the blue regions (0 < θ < π/ 4).\n(b)(c) Schematic pictures of the magnetic patterns induced by the dynamical instability at points (b) P and (c) Q designa ted\nin (a). The arrows show the local magnetizations projected o nto thex-yplane. The MDDI energy for configuration (c) is lower\nthan that for (b).\nLet us reconsider the Fourier expansion of δϕ,\nδϕ=1\n2/summationdisplay\nk/negationslash=0(δ˜ϕkeik·r+δ˜ϕ−ke−ik·r)+δ˜ϕ0\n=1\n2/summationdisplay\nk/negationslash=0[ARsin(k·r+αR)+iAIsin(k·r+αI)]+δ˜ϕ0, (51)\nwhere\nAR=/radicalbig\n[Re(δ˜ϕk+δ˜ϕ−k)]2+[Im(δ˜ϕk−δ˜ϕ−k)]2, (52)\nAI=/radicalbig\n[Im(δ˜ϕk+δ˜ϕ−k)]2+[Re(δ˜ϕk−δ˜ϕ−k)]2, (53)\nandαR= tan−1[Re(δ˜ϕk+δ˜ϕ−k)/Im(δ˜ϕ−k−δ˜ϕk)] andαI= tan−1[Im(δ˜ϕk+δ˜ϕ−k)/Re(δ˜ϕk−δ˜ϕ−k)]. Asλ+(k=\n0) = 0,δ˜ϕ0= 0. We introduce the quantity θ, which characterizes the magnetic fluctuation preference:\nθ= tan−1(AR/AI). (54)\nHere, we calculate θ(k) for the unstable modes shown in Fig. 2. The eigenvectorwhich corr espondsto the eigenvalue\nλ+of the 2×2 matrix in Eq. (39) is given by\n/parenleftbiggδ˜ϕk\nδ˜ϕ∗\n−k/parenrightbigg\n=/parenleftbigg\n(g1+i/radicalbig\n|g2|2−g2\n1)//radicalbig\n2|g2|2\n−g∗\n2//radicalbig\n2|g2|2/parenrightbigg\n, (55)\nwhereg1andg2are defined by Eqs. (41) and (42). Then, from Eqs. (52)–(55), w e obtain\nθ(k) = tan−1/bracketleftBigg\n|g2|2−g1Re(g2)+Im(g2)/radicalbig\n|g2|2−g2\n1\n|g2|2+g1Re(g2)−Im(g2)/radicalbig\n|g2|2−g2\n1/bracketrightBigg1/2\n. (56)\nWe can consider the x-direction fluctuation to be dominant for π/4<θ <π/ 2 and they-direction fluctuation to be\ndominant for 0 <θ<π/ 4.\nThe wavevectordependence of θ(k) forthe unstablemode shownin Fig. 2isillustratedin Fig.4(a). Inthe red(blue)\nregions, the x(y)-direction fluctuation is more dominant than the y(x)-direction fluctuation. The schematic pictures\nof the magnetic patterns induced by the dynamical instability at poin ts P and Q are illustrated in Figs. 4(b) and\n(c), respectively. At point P in Fig. 4(a), x-direction fluctuation occurs and the wavevector of the magnetic pattern\nis directed in the xdirection as shown in Fig. 4(b). The MDDI energy for this configurat ion is higher than that of\nFig. 4(c), where y-direction fluctuation is induced. In other words, the MDDI energy is reduced by the magnetic\nfluctuation, and the reduction is larger in pattern (c) than in patte rn (b). The reduction in MDDI energy is converted\ninto kinetic energy ( ∼k2), which is higher at point Q than at point P in Fig. 4(a). This explains why the magnetic\nfluctuations change from x-direction to y-direction along the kxaxis.11\nkx/2π (µm   )-1(a)  uniform (b)   uniform\n(d)  helix 60  µm (c)  helix 120  µmkz/2π (µm   )-1\n120 mG 200 mG\n120 mG 120 mG0.04\n0.02\n-0.04-0.020\n0.040.020-0.02-0.04\n0 π/2\nFIG. 5: (Color) Magnetic fluctuation preference θ(k) for uniform and helical spin structures in the zdirection with a pitch\n2π/καµm under a strong magnetic field. The initial structure is unif orm for (a) and (b), a helix with a pitch (c) 120 µm, and\n(d) 60µm. The external field Bis [(a), (c), and (d)] 120 mG and (b) 200 mG. The fluctuations ar e longitudinal in the red\nregions ( π/4< θ < π/ 2) and transverse in the blue regions (0 < θ < π/ 4). The other parameters are the same as those in\nFig. 3.\nNow, we investigate the magnetic fluctuations for the situation disc ussed in Sec. IIIB. The fluctuations are con-\nsidered to be longitudinal or transverse. The longitudinal and tran sverse fluctuations are represented by δˆfzand\nδ(ˆfx+iˆfy), respectively. Substituting ϕ=ϕ0(1+δϕ) and|ϕ0|2= 1 into Eq. (33), we obtain the expressions for the\ntwo types of fluctuations described by the first order of δϕ:δˆfz=−Re(δϕ) andδ(ˆfx+iˆfy) =ϕ0Im(δϕ). Namely, the\nlongitudinal and transverse fluctuations are characterized by th e real and imaginary parts of δϕ, respectively.\nWe can also apply the above method to discuss the magnetic fluctuat ion preference, which is characterized by\nθ= tan−1(AR/AI), in the present case. Since g2=g∗\n2, we can simplify Eq. (56) as\nθ(k) = tan−1/radicalbiggg2−g1\ng2+g1, (57)\nwhereg1andg2are defined by Eqs. (49) and (50), respectively. The magnetic fluc tuation is longitudinal if π/4<\nθ<π/2 and transverse if 0 <θ<π/ 4.\nThe wavevector dependence of θ(k) for the unstable modes shown in Figs. 3 (a), (c), (d), and (f) are demonstrated\nin Fig. 5. When the dynamical instability has a round shape [Figs. 5(a), (c), and (d)], the fluctuations are transverse\nfor smallkand longitudinal for large k. Figure 5(b) looks more complex than the schematics for the other cases:\nfluctuations are transverse for k//ˆzand longitudinal for k//ˆx. The magnetic fluctuation preference is consistent with\nthat obtained by the Bogoliubov analysis [15, 25], and discrepancies a ppear in strong fields for the same reason as\nthat for the dynamical instability.\nV. CONCLUSIONS\nEmploying our hydrodynamic description derived in Sec. II, we have d emonstrated some simple examples of the\nanalysis of dynamical instability and magnetic fluctuation preferenc e in Sec. III and Sec. IV, respectively. Once one\nfinds a stationary solution of the hydrodynamic equations, it is a str aightforward task to obtain the analytical form\nof the dynamical instability: the only necessary step is the diagonaliz ation of a 2 ×2 matrix. The eigenvalues and\neigenvectors of the matrix lead to the dynamical instability and the m agnetic fluctuation preference, respectively.\nAlthough we have discussed just a few types of spin structures an d external fields for simplicity, our method can be\napplied to other conformations.\nIn conclusion, we have introduced the hydrodynamic equations for a ferromagnetic spinor dipolar BEC with an\narbitrary spin by means of stereographic projection. This simple de scription provides a straightforward approach\nby which to investigate spin dynamics, i.e., dynamical instability and mag netization fluctuation preference, which\nare expressed in analytical forms. The description should also be us eful for the study of the exact solutions of\nhydrodynamic equations of a spinor BEC.12\nAcknowledgments\nThe authors thank M. Ueda for his useful comments. This work is su pported by MEXT JSPS KAKENHI (No.\n22103005, 22340114, 22740265), the Photon Frontier Network Program of MEXT, Japan, Hayashi Memorial Foun-\ndation for Female Natural Scientists, and JSPS and FRST under the Japan-New Zealand Research Cooperative\nProgram.\nAppendix A: Contributions from the short-range interactio n, MDDI, and linear and quadratic Zeeman effects\nThe contribution from the short-range interaction to the equatio n of motion of fzis calculated as\n/bracketleftbigg∂fz\n∂t/bracketrightbigg\ns=1\ni/planckover2pi1(Fz)mn/bracketleftbigg/parenleftbigg\ni/planckover2pi1∂Ψ∗\nm\n∂t/parenrightbigg\nΨn+Ψ∗\nm/parenleftbigg\ni/planckover2pi1∂Ψn\n∂t/parenrightbigg/bracketrightbigg\ns\n=2F/summationdisplay\nS=0,even4π/planckover2pi1\niMaSS/summationdisplay\nMS=−S/summationdisplay\nlm′l′(Fz)mn[−/angb∇acketleftml|SMS/angb∇acket∇ight/angb∇acketleftSMS|m′l′/angb∇acket∇ightΨlΨ∗\nm′Ψ∗\nl′Ψn\n+Ψ∗\nm/angb∇acketleftnl|SMS/angb∇acket∇ight/angb∇acketleftSMS|m′l′/angb∇acket∇ightΨ∗\nlΨm′Ψl′]\n=2F/summationdisplay\nS=0,even4π/planckover2pi1\niMaSS/summationdisplay\nMS=−S/summationdisplay\nlm′l′m/angb∇acketleftml|SMS/angb∇acket∇ight/angb∇acketleftSMS|m′l′/angb∇acket∇ight[−ΨmΨlΨ∗\nm′Ψ∗\nl′+Ψ∗\nmΨ∗\nlΨm′Ψl′]\n=2F/summationdisplay\nS=0,even4π/planckover2pi1\niMaSS/summationdisplay\nMS=−S/summationdisplay\nmm′(m−m′)/angb∇acketleftm,MS−m|SMS/angb∇acket∇ight/angb∇acketleftSMS|m′,MS−m′/angb∇acket∇ightΨ∗\nmΨ∗\nMS−mΨm′ΨMS−m′\n=2F/summationdisplay\nS=0,even2π/planckover2pi1\niMaSS/summationdisplay\nMS=−S/bracketleftbigg/summationdisplay\nmm′(m−m′)/angb∇acketleftm,MS−m|SMS/angb∇acket∇ight/angb∇acketleftSMS|m′,MS−m′/angb∇acket∇ightΨ∗\nmΨ∗\nMS−mΨm′ΨMS−m′\n+/summationdisplay\nll′(−l+l′)/angb∇acketleftMS−l,l|SMS/angb∇acket∇ight/angb∇acketleftSMS|MS−l′,l′/angb∇acket∇ightΨ∗\nMS−lΨ∗\nlΨMS−l′Ψl′/bracketrightbigg\n= 0, (A1)\nwhere we have used ( Fz)mn=mδmn. Here, [ ···]sdenotes that only those terms that come from the short-range\ninteraction are extracted. In the following, this notation is applied t o the contributions from the MDDI ([ ···]dd),\nlinear ([···]p), and quadratic ([ ···]q) Zeeman effects. Since the short-range interaction (2) is invarian t under spin\nrotation, [∂fx/∂t]sand [∂fy/∂t]salso vanish, which are shown in a similar way by choosing the spin quantiz ation axis\nalong thexandydirections, respectively.\nThe contribution from the MDDI to the equation of motion of spin is ca lculated as\n/bracketleftbigg∂fµ\n∂t/bracketrightbigg\ndd=1\ni/planckover2pi1(Fµ)mn/bracketleftbigg/parenleftbigg\ni/planckover2pi1∂Ψ∗\nm\n∂t/parenrightbigg\nΨn+Ψ∗\nm/parenleftbigg\ni/planckover2pi1∂Ψn\n∂t/parenrightbigg/bracketrightbigg\ndd\n=cdd\ni/planckover2pi1(Fµ)mn[−b∗\nν(F∗\nν)mlΨ∗\nlΨn+Ψ∗\nmbν(Fν)nlΨl]\n=cdd\ni/planckover2pi1bνΨ∗\nm(FµFν−FνFµ)mnΨn\n=cdd\ni/planckover2pi1bνiǫµνλfλ\n=cdd\n/planckover2pi1(b×f)µ. (A2)\nWe have used the relations F†\nµ=Fµand [Fµ,Fν] =iǫµνλFλ.\nSuppose the magnetic field is applied parallel to the zaxis. Then, the contribution from the linear Zeeman effect13\nis calculated as\n/bracketleftbigg∂fµ\n∂t/bracketrightbigg\np=1\ni/planckover2pi1(Fµ)mn/bracketleftbigg/parenleftbigg\ni/planckover2pi1∂Ψ∗\nm\n∂t/parenrightbigg\nΨn+Ψ∗\nm/parenleftbigg\ni/planckover2pi1∂Ψn\n∂t/parenrightbigg/bracketrightbigg\np\n=p\ni/planckover2pi1(Fµ)mn[−(F∗\nz)mlΨ∗\nlΨn+Ψ∗\nm(Fz)nlΨl]\n=p\ni/planckover2pi1iǫµzνfν\n=p\n/planckover2pi1(ˆz×f)µ. (A3)\nThe contribution from the quadratic Zeeman effect is calculated in th e same way as the that shown above.\n/bracketleftbigg∂fµ\n∂t/bracketrightbigg\nq=1\ni/planckover2pi1(Fµ)mn/bracketleftbigg/parenleftbigg\ni/planckover2pi1∂Ψ∗\nm\n∂t/parenrightbigg\nΨn+Ψ∗\nm/parenleftbigg\ni/planckover2pi1∂Ψn\n∂t/parenrightbigg/bracketrightbigg\nq\n=q\ni/planckover2pi1(Fµ)mn/bracketleftbig\n−(F∗\nz)2\nmlΨ∗\nlΨn+Ψ∗\nm(Fz)2\nnlΨl/bracketrightbig\n=q\ni/planckover2pi1Ψ∗\nm(Fz[Fµ,Fz]+[Fµ,Fz]Fz)mnΨn\n=q\ni/planckover2pi1iǫµzνΨ∗\nm(FνFz+FzFν)mnΨn\n=2q\n/planckover2pi1ntotǫµzνˆNzν, (A4)\nwhereˆNµνis a nematic tensor defined by\nˆNµν=1\n2ζ∗\nm(FµFν+FνFµ)mnζn. (A5)\n[1] T.-L. Ho and V. B. Shenoy, Phys. Rev. Lett. 77, 2595 (1996).\n[2] M. Nakahara, T. Isoshima, K. Machida, S. Ogawa, and T. Ohm i, Physica B: Condensed Matter 284–288 , 17 (2000); T.\nIsoshima, M. Nakahara, T. Ohmi, and K. Machida, Phys. Rev. A 61, 063610 (2000).\n[3] A. E. Leanhardt, Y. Shin, D. Kielpinski, D. E. Pritchard, and W. Ketterle, Phys. Rev. Lett. 90, 140403 (2003).\n[4] H. M¨ akel¨ a, Y. Zhang, and K.-A. Suominen, J. Phys. A: Mat h. Gen.36, 8555 (2003); H. M¨ akel¨ a, J. Phys. A: Math. Gen.\n39, 7423 (2006).\n[5] F. Zhou, Phys. Rev. Lett. 87, 080401 (2001).\n[6] G. W. Semenoff and F. Zhou, Phys. Rev. Lett. 98, 100401 (2007).\n[7] M. Kobayashi, Y. Kawaguchi, M. Nitta, and M. Ueda, Phys. R ev. Lett. 103, 115301 (2009).\n[8] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, Nature 443, 312 (2006).\n[9] M. Vengalattore, S. R. Leslie, J. Guzman, and D. M. Stampe r-Kurn, Phys. Rev. Lett. 100, 170403 (2008).\n[10] M. Vengalattore, J. Guzman, S. R. Leslie, F. Serwane, an d D. M. Stamper-Kurn, Phys. Rev. A 81, 053612 (2010).\n[11] For review of dipolar BECs, see T. Lahaye, C. Menotti, L. Santos, M. Lewenstein and T. Pfau, Rep. Prog. Phys. 72,\n126401 (2009).\n[12] L. Santos and T. Pfau, Phys. Rev. Lett. 96, 190404 (2006).\n[13] Y. Kawaguchi, H. Saito and M. Ueda, Phys. Rev. Lett. 96, 080405 (2006); Phys. Rev. Lett. 97, 130404 (2006).\n[14] S. Yi and H. Pu, Phys. Rev. Lett. 97, 020401 (2006).\n[15] Y. Kawaguchi, H. Saito, K. Kudo, and M. Ueda, Phys. Rev. A 82, 043627 (2010).\n[16] M. Takahashi, Sankalpa Ghosh, T. Mizushima, and K. Mach ida, Phys. Rev. Lett. 98, 260403 (2007); J. A. M. Huhtam¨ aki,\nM. Takahashi, T. P. Simula, T. Mizushima, and K. Machida, Phy s. Rev. A 81, 063623 (2010).\n[17] A. Lamacraft, Phys. Rev. A 77, 063622 (2008).\n[18] R. Barnett, D. Podolsky, and G. Refael, Phys. Rev. B 80, 024420 (2009).\n[19] A. Lamacraft, Phys. Rev. B 81, 184526 (2010).\n[20] T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998).\n[21] N.D. Mermin and T.-L. Ho, Phys. Rev. Lett. 36, 594 (1976).\n[22] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004); A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Europhys.\nLett.69, 990 (2005).\n[23] Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev. Lett. 98, 110406 (2007).\n[24] M. Lakshmanan and K. Nakamura, Phys. Rev. Lett. 53, 2497 (1984).\n[25] R.W. Cherng and E. Demler, Phys. Rev. Lett. 103, 185301 (2009)."    },    {        "title": "1706.05769v1.Origin_and_Structures_of_Solar_Eruptions_II__Magnetic_Modeling__Invited_Review_.pdf",        "content": "Origin and Structures of Solar Eruptions II: Magnetic Modeling\n(Invited Review)\nYang Guo1;2, Xin Cheng1;2, M. D. Ding1;2\n1School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China\nguoyang@nju.edu.cn\n2Key Laboratory for Modern Astronomy and Astrophysics (Nanjing University), Ministry of\nEducation, Nanjing 210023, China\nABSTRACT\nThe topology and dynamics of the three-dimensional magnetic \feld in the solar at-\nmosphere govern various solar eruptive phenomena and activities, such as \rares, coronal\nmass ejections, and \flaments/prominences. We have to observe and model the vector\nmagnetic \feld to understand the structures and physical mechanisms of these solar\nactivities. Vector magnetic \felds on the photosphere are routinely observed via the po-\nlarized light, and inferred with the inversion of Stokes pro\fles. To analyze these vector\nmagnetic \felds, we need \frst to remove the 180\u000eambiguity of the transverse components\nand correct the projection e\u000bect. Then, the vector magnetic \feld can be served as the\nboundary conditions for a force-free \feld modeling after a proper preprocessing. The\nphotospheric velocity \feld can also be derived from a time sequence of vector magnetic\n\felds. Three-dimensional magnetic \feld could be derived and studied with theoretical\nforce-free \feld models, numerical nonlinear force-free \feld models, magnetohydrostatic\nmodels, and magnetohydrodynamic models. Magnetic energy can be computed with\nthree-dimensional magnetic \feld models or a time series of vector magnetic \feld. The\nmagnetic topology is analyzed by pinpointing the positions of magnetic null points, bald\npatches, and quasi-separatrix layers. As a well conserved physical quantity, magnetic\nhelicity can be computed with various methods, such as the \fnite volume method, dis-\ncrete \rux tube method, and helicity \rux integration method. This quantity serves as a\npromising parameter characterizing the activity level of solar active regions.\nSubject headings: Sun: activity | Sun: corona | Sun: coronal mass ejections (CMEs)\n| Sun: \rares | Sun: magnetic \felds | Sun: photosphere\n1. Introduction\nTo understand the origin and structures of solar activities and the related phenomena, such\nas \rares, \flaments/prominencs, and coronal mass ejections (CMEs), we have to know the three-\ndimensional (3D) solar magnetic \feld, since the solar atmosphere is \flled with magnetized plasma.arXiv:1706.05769v1  [astro-ph.SR]  19 Jun 2017{ 2 {\nDue to the high conductivity in the solar atmosphere, the magnetic \feld is frozen to the plasma.\nFrom the upper chromosphere to the transition region and the lower corona, the magnetic pressure\neven dominates over the gas pressure of the plasma. Therefore, the topology and dynamics of\nthe magnetic \feld are critical to controlling the structure and behavior of plasma in the solar\natmosphere.\nMagnetic structures before eruptions could be either a magnetic \rux rope, or sheared magnetic\narcades, or both. A magnetic \rux rope is de\fned as a bundle of magnetic \feld lines twisting around\na common axis. Sheared arcades are regarded as a bundle of magnetic \feld lines that deviates\nfar from the potential state but does not possess an inverse polarity, that is, a magnetic \feld\ncomponent pointing from the negative to the positive polarity. Namely, sheared arcades possess a\nmoderate twist (probably less than one turn) and electric current, but they are not strong enough\nto generate the inverse polarity. A magnetic \rux rope can be formed from sheared arcades by\nmagnetic reconnection and footpoint twisting motion.\nThe central physical mechanisms for solar activities include magnetohydrodynamic (MHD)\ninstabilities and magnetic reconnection. For example, magnetic \rux rope eruptions are explained\nby the loss of equilibrium (D\u0013 emoulin & Priest 1988; Forbes & Isenberg 1991; Lin & Forbes 2000) or\ntorus instability (Bateman 1978; Kliem & T or ok 2006). D\u0013 emoulin & Aulanier (2010) pointed out\nthat these two ideas are two di\u000berent views of the same mechanism, namely, they both resort to the\nLorentz repulsion force. One could increase the electric current in a magnetic \rux rope or decrease\nthe background magnetic \feld strength to build the state that leads to loss of equilibrium or the\ntorus instability, either via an ideal process (e.g., kink instability) or a resistive process (magnetic\nreconnection). The magnetic reconnection could occur either under a \rux rope or sheared arcades\nin the tether cutting model (Moore & Labonte 1980; Moore et al. 2001), or, above sheared arcades\nin the breakout model (Antiochos 1998; Antiochos et al. 1999). Magnetic \feld observations and\nmodeling are critical to understand how these physical mechanisms interplay with each other in\nspeci\fc events.\nStudies on the magnetic \feld can be performed through observations, theoretical models, and\nnumerical models. In this review, we \frst introduce the vector magnetic \feld observations and\nprocessing in Section 2. The observations of Stokes pro\fles from the polarized lights in the solar\natmosphere are mentioned. The magnetic \feld information is extracted from the inversion of Stokes\npro\fles (Section 2.1). For vector magnetic \feld, the transverse components have an intrinsic 180\u000e\nambiguity, which has to be removed before further analysis (Section 2.2). The correction of the\nprojection e\u000bect is also very important for solar vector magnetic \feld, especially when the region\nof interest is close to the solar limb or the \feld of view is large (Section 2.3). Velocities can be\nderived from a time series of vector magnetic \feld by the optical \row techniques (Section 2.4).\nAt present, magnetic \feld in the solar atmosphere is observed routinely and relatively accu-\nrately only on the photosphere. The 3D magnetic \feld above the photosphere could be studied\npurely theoretically, or constructed numerically from observations. In Section 3, we introduce{ 3 {\nsome theoretical force-free \feld models (Section 3.1), numerical nonlinear force-free \feld models\n(Section 3.2), magneto-hydrostatic (MHS) models (Section 3.3), and MHD models (Section 3.4)\nassociated with solar magnetic \rux rope emergence or eruptions. A force-free \feld is de\fned as a\nmagnetic \feld without any Lorentz force. Typical force-free \feld models include the potential \feld,\nlinear force-free \feld, and nonlinear force-free \feld. If the vector magnetic \feld on the photosphere\nis used as the boundary condition for a nonlinear force-free \feld modeling, preprocessing of the vec-\ntor magnetic \feld is also essential for removing the Lorentz force on the boundary (Section 3.2.1).\nSince force-free \feld models are static, they are not suitable for studying dynamic solar eruptive\nphenomena, which have been widely studied by MHD models.\nMagnetic energy computation in a volume and from the boundary is brie\ry discussed in Sec-\ntion 4. To study in detail what are the critical features in a magnetic \feld for MHD instability and\nmagnetic reconnection, we have to know the magnetic topology. The magnetic topology analysis\nincludes searching for the locations of magnetic null points (Section 5.1), bald patches (Section 5.2),\nand quasi-separatrix layers (QSLs; Section 5.3), and analyzing the magnetic structures and evolu-\ntions in light of these speci\fc topologies. We also introduce the applications of magnetic topology\nanalysis to observations in Section 5.4. The helicity computation is discussed in Section 6, where we\nfocus on the \fnite volume method (Section 6.1), discrete \rux tube method (Section 6.2), and helicity\n\rux integration method (Section 6.3). Finally, we give a summary and discussion in Section 7.\nThis paper is focused on the magnetic \feld observations and modelings of various solar eruptive\nactivities. The multi-wavelength observations of the origin and structures of CMEs, \rares, and\nmagnetic \rux ropes are presented in another review by Cheng et al. (2017).\n2. Vector Magnetic Field Observations and Processing\nIn the photosphere with relatively high density and low temperature, the vector magnetic\n\feld is routinely observed by space telescopes, such as the Solar Optical Telescope (SOT; Tsuneta\net al. 2008; Suematsu et al. 2008; Ichimoto et al. 2008; Shimizu et al. 2008) on board Hinode\n(Kosugi et al. 2007) and the Helioseismic and Magnetic Imager (HMI; Scherrer et al. 2012; Schou\net al. 2012) on board the Solar Dynamics Observatory (SDO ). There are also various ground-based\ntelescopes aiming at observing solar magnetic \feld, such as the Solar Magnetic Field Telescope\n(SMFT) of Huairou Solar Observing Station of National Astronomical Observatory of China (Ai\n1987; Zhang et al. 1994; Su & Zhang 2004, 2007), the Multi-Raies (MTR) mode of the T\u0013 elescope\nH\u0013 eliographique pour l'Etude du Magn\u0013 etisme et des Instabilit\u0013 es Solaires (THEMIS; L\u0013 opez Ariste\net al. 2000; Bommier et al. 2007), the New Solar Telescope (NST; Cao et al. 2010; Wang et al.\n2017) at Big Bear Solar Observatory, and so on.\nMagnetic \feld in the photosphere is measured by the polarized light generated by the Zeeman\nsplitting (Hale 1908). Inference of the vector magnetic \feld requires the knowledge of the spectral\npro\fle of the Stokes parameters, I;Q;U; andV. There are basically two types of instruments to{ 4 {\nmeasure the Stokes pro\fles, the \flter type and the spectrograph type. HMI and SMFT belong to\nthe \flter type. They cover a large \feld of view, full solar disk for HMI and 40\u000260for SMFT, have a\nfairly high spatial resolution, 0 :500per pixel for HMI and 0 :400\u00020:700for SMFT, and have a relatively\nhigh cadence, 12 minutes for HMI vector magnetic \feld and 1 minute for SMFT. However, their\nspectral resolution is low. For example, HMI measures the Stokes parameters at six wavelengths\nwith a band width of 76 m \u0017A evenly sampled across the Fe I 6173 \u0017A line covering a tunable range of\n690 m \u0017A, and SMFT acquires data at two wavelengths ( \u000075 m \u0017A and the line center) with a band\nwidth of 125 m \u0017A across the Fe I 5324 \u0017A line.\nThe Spectropolarimeter (SP) of SOT and MTR of THEMIS belong to the spectrograph type.\nThey have very high spectral resolutions. For example, SOT/SP has a spectral sample of 21.5 m \u0017A\nper pixel through the spectral range of 6300.8 \u0017A to 6303.2 \u0017A and THEMIS/MTR has a dispersion\nof 19.5 m \u0017A for the spectral lines 6301.5 and 6302.5 \u0017A. But their \feld of view is restricted to a slit,\n16400\u00020:1600for SOT/SP and 12000\u00020:500for THEMIS/MTR, respectively. SOT/SP spends 4.8\nseconds for observation at one slit position and THEMIS/MTR spends about 3 seconds for each\nscan step. To cover a large \feld of view, they have to take tens of minutes to several hours to scan\nover the region of target by moving perpendicular to the slit direction.\n2.1. Inversion of Stokes Pro\fles\nTo extract the vector magnetic \feld information from observed polarized spectra, we have to\nresort to the inversion of Stokes pro\fles, namely, the inversion of spectral lines of I;Q;U; andV.\nIt faces two problems, namely, the forward modeling problem and the inversion problem. In the\nforward modeling problem, one has to solve the radiative transfer equation of polarized radiation\nto study how the Stokes pro\fles are formed. Pioneered by Unno (1956), this problem has been\nfurther discussed by many other authors by considering the magneto-optic, damping, and other\ne\u000bects (Rachkovsky 1962; Landol\f & Landi Degl'Innocenti 1982; Je\u000beries et al. 1989). This model\nincludes three parameters for the vector magnetic \feld, namely the \feld strength B, inclination\nangle , and the azimuth angle \u001e, seven parameters for the thermodynamics, namely, the line\nstrength\u00110, Doppler width \u0001 \u0015D, damping constant aor equivalently damping parameter \rwith\na=\r=(4\u0019\u0001\u0015D), Doppler velocity vor equivalently line wavelength \u00150, source function constant\nB0, source function gradient B1, and macro-turbulent velocity vm, and \fnally one geometrical\nparameter, the \flling factor \u000b. To simplify the solution of the radiative transfer equations of the\nStokes pro\fles, all the parameters are assumed to be constant as a function of the continuum optical\ndepth\u001c, except the source function B(\u001c), which is assumed to be linearly dependent on \u001cwith\nB(\u001c) =B0+B1\u001c.\nThe inversion problem \fts the modeled Stokes pro\fles to the observations by adjusting the\naforementioned parameters. This is a nonlinear least square problem, which is usually solved by\nthe Levenberg{Marquardt algorithm (Marquardt 1963; Press et al. 1988) to minimize the discrep-\nancies, indicated by an objective function \u001f2, between the modeled and observed Stokes pro\fles.{ 5 {\nMain e\u000borts are devoted to search for the global minimum of \u001f2in the parameter space discussed\nabove. There are already some inversion procedures based on the Unno{Rachkovsky solutions\nand the Levenberg{Marquardt algorithm, such as the Milne{Eddington Line Analysis using an\nInversion Engine (MELANIE; Socas-Navarro 2001), the UNNOFIT inversion code (Landol\f &\nLandi Degl'Innocenti 1982; Bommier et al. 2007), and the Very Fast Inversion of the Stokes Vector\n(VFISV; Borrero et al. 2011; Centeno et al. 2014). These procedures have been applied to the\nobservations of the Stokes pro\fles by di\u000berent instruments, for example, MELANIE to SOT/SP,\nUNNOFIT to THEMIS/MTR, and VFISV to HMI.\nGuo et al. (2010b) presented a detailed comparison of MELANIE and UNNOFIT and found\nthat both procedures provide consistent inversion results of the \feld strength B, inclination angle  ,\nand the azimuth angle \u001e. Teng & Deng (2014) made some tests and improvements to VFISV. They\nfound that using a smooth interpolation for the Voigt function could eliminate spurious inversion\nresults, and using proper initial values for the azimuth angle \u001ecould speed up the code by four\ntimes relative to the original one, but provide accurate results only for the vector magnetic \feld\nand Doppler velocity. Teng (2015) applied a kernel-based machine learning method to the inversion\nof Stokes parameters observed by HMI, and a fast inversion method is further proposed by Teng\n& Deng (2016) based on the quadratic regression.\n2.2. Removing the 180\u000eAmbiguity of the Transverse Components\nThe 180\u000eambiguity of the transverse components of a vector magnetic \feld arises from the\nintrinsic symmetry (or, periodicity) of the radiative transfer of polarized lights (see, e.g., Equa-\ntion (28) of Je\u000beries et al. 1989). If the azimuth of the vector magnetic \feld changes 180\u000e, the\nemergent Stokes pro\fles are identical to those with the original azimuth. Considering the physical\nnature of the Zeeman splitting and the mathematical models of the aforementioned Stokes pro\fle\nformation, there is no known method to resolve this ambiguity. Although it has been proposed\nto measure the vector magnetic \feld at more than one heights to resolve the ambiguity using the\nsolenoidal property, determination of the formation heights of di\u000berent spectral lines turns out to\nbe di\u000ecult (Bommier 2013). Therefore, the ambiguity is usually resolved by additional physical\nassumptions on the magnetic \feld.\nDi\u000berent assumptions have been made to constrain the magnetic \feld. For example, the\npotential \feld model is adopted as an reference model in many algorithms (see Metcalf et al.\n2006), where the ambiguity is resolved by assuming the observed components make an acute angle\nwith the modeled ones. The reference model could also be chosen as a linear force-free \feld\nsuch as that in Wang (1997) and Wang et al. (2001). Moon et al. (2003) proposed a uniform\nshear angle method by assuming that the transverse magnetic \feld makes an acute angle with\nthe azimuth angle of the potential \feld transverse component plus an additional shear angle. The\nmagnetic pressure gradient method (Cuperman et al. 1993) assumes the magnetic \feld to be force-\nfree and the magnetic pressure decreases with height. Georgoulis et al. (2004) introduced a structure{ 6 {\nminimization method by minimizing the electric current density generated by the magnetic \feld\ngradients. Georgoulis (2005) further proposed a nonpotential magnetic \feld calculation method for\nremoving the ambiguity. It employs an iterative method to determine the azimuth by minimizing\nthe discrepancy between the observations and the \feld computed by the nonpotential model. This\nmethod has been improved as described in Metcalf et al. (2006). In the pseudo-current method\n(Gary & D\u0013 emoulin 1995), a multidimensional conjugate gradient method is used to minimize the\nsquare of the vertical current density, by which the azimuth is determined. Another iterative\nmethod developed in University of Hawaii was described in Can\feld et al. (1993), where the azimuth\nis initially determined by the acute angle method and the potential or linear force-free \feld model,\nand then by the minimization of the square of the vertical current density and the divergence of\nthe magnetic \feld. The minimum energy method (Metcalf 1994; Metcalf et al. 2006) employs the\nsimulated annealing method to minimize a function of the magnetic \feld divergence and the total\nelectric current density, which are derived by a linear force-free \feld model. This method has been\nimproved by Leka et al. (2009). Finally, there is an interactive procedure, AZAM, developed in\nHigh Altitude Observatory (HAO), to remove the ambiguity by imposing smoothness and matching\nthe magnetic \feld to expectations of the solar structure (Metcalf et al. 2006).\nMetcalf et al. (2006) provided a comprehensive comparison of all the aforementioned ambiguity-\nremoval methods to test their performances. The authors adopted two reference models to get\nthe vector magnetic \feld, one from an MHD simulation and the other from a theoretical model\ncomputed from multi-point sources buried under a plane. They applied each of the method to\nthe reference model and compare the results with it to determine the area, magnetic \rux, and\nstrong horizontal \feld that have been recovered. It shows that those methods minimizing the\nelectric current density and the magnetic \feld divergence provide the most promising results, such\nas the nonpotential magnetic \feld calculation method, the iterative method developed in University\nof Hawaii, the minimum energy method, and the interactive method developed in High Altitude\nObservatory. Figure 1a shows a vector magnetic \feld observed by SDO /HMI, whose 180\u000eambiguity\nis removed by the minimum energy method. Developing more reliable and e\u000becient methods to\nremove the 180\u000eambiguity is still an ongoing topic. Some new algorithms have been discussed in,\ne.g., Li et al. (2007) and Crouch et al. (2009).\n2.3. Correction of the Projection E\u000bect\nDue to the spherical nature of the solar surface, vector magnetic \feld observed on the pho-\ntosphere is subjected to the projection e\u000bect except at the solar disk center. The projection has\ntwo e\u000bects. On the one hand, it projects the vector magnetic \feld components into an observer-\npreferred coordinate system, while the Sun itself prefers a heliographic coordinate system. On the\nother hand, observations on the image plane distort the geometrical positions of the vector mag-\nnetic \feld on the solar surface. The observer-preferred coordinate system is de\fned by the image\nplane and the line of sight, where the \u0010axis is towards the observer, the \u0018and\u0011axes on the image{ 7 {\nplane, and the three axes are perpendicular to each other. Additionally, the \u0018axis is towards the\nwest and\u0011to the north extremity of the earth rotation axis. The heliographic coordinate system is\nde\fned by the solar longitudinal direction x(towards the west), the latitudinal direction y(towards\nthe solar north), and the radial direction z.\nThe image plane components ( B\u0018;B\u0011;B\u0010) can be transformed to the heliographic components\n(Bx;By;Bz) by the following operation (Gary & Hagyard 1990):\n0\nB@Bx(\u0018;\u0011)\nBy(\u0018;\u0011)\nBz(\u0018;\u0011)1\nCA=R(P;B;B 0;L;L 0)0\nB@B\u0018(\u0018;\u0011)\nB\u0011(\u0018;\u0011)\nB\u0010(\u0018;\u0011)1\nCA: (1)\nThe rotation matrix Rhas 3\u00023 elements and it is a function of P;B;B 0;L, andL0, where the\nsolarPangle is the position angle of the north extremity of the solar rotation axis relative to the\nnorth extremity of the earth rotation axis, BandLare the latitude and longitude of the vector\nmagnetic \feld at the image plane coordinate ( \u0018;\u0011), andB0andL0are the latitude and longitude of\nthe solar disk center. The rotation matrix can be derived by four successive elementary rotations,\nnamely, rotation around the \u0010axis by angle P:\nR\u0010(P)=0\nB@cosP sinP0\n\u0000sinPcosP0\n0 0 11\nCA; (2)\nrotation around the \u0018axis byB0:\nR\u0018(B0)=0\nB@1 0 0\n0 cosB0sinB0\n0\u0000sinB0cosB01\nCA; (3)\nrotation around the \u0011axis byL\u0000L0:\nR\u0011(L\u0000L0)=0\nB@cos(L\u0000L0) 0\u0000sin(L\u0000L0)\n0 1 0\nsin(L\u0000L0) 0 cos( L\u0000L0)1\nCA; (4)\nand rotation around the \u0018axis by\u0000B, where R\u0018(\u0000B) is obtained by substituting B0with\u0000Bin\nEquation (3). The rotation matrix for Equation (1) is then expressed as:\nR(P;B;B 0;L;L 0) =R\u0018(\u0000B)R\u0011(L\u0000L0)R\u0018(B0)R\u0010(P): (5)\nThe full development of Equation (5) can be found in Gary & Hagyard (1990). Equation (5) is\neasier to understand and debug in a code than the fully developed expression in Gary & Hagyard\n(1990).\nFor the geometrical projection correction, Gary & Hagyard (1990) proposed a linear approx-\nimation using a plane tangent to the solar surface at the image center point ( Bc;Lc). This ap-\nproximation omits the curvature of the Sun and assumes a plane as the geometry. Therefore, it{ 8 {\nis suitable for a Cartesian coordinate system but only applies to a small \feld of view. The error\narising from the plane assumption depends on the size of the \feld of view. Gary & Hagyard (1990)\nhas estimated such an error quantitatively. The coordinates ( x;y) on the de-projected plane map to\nthe coordinates ( \u0018;\u0011) on the image plane with the following linear transformation (Gary & Hagyard\n1990): \n\u0018\n\u0011!\n= \nc11c12\nc21c22! \nx\ny!\n: (6)\nBoth coordinates are referred to the center of the image ( Bc;Lc) and the matrix elements are also\nderived by four successive rotations:\nR(P;Bc;B0;Lc;L0) =Rz(\u0000P)Rx(\u0000B0)Ry(L0\u0000Lc)Rx(Bc): (7)\nOnly the upper left four elements in Equation (7) are needed to calculate the coe\u000ecients in Equa-\ntion (6). The full development of Equation (7) can also be found in Gary & Hagyard (1990).\nFigure 1b shows an example where the projection e\u000bect of the originally observed vector\nmagnetic \feld in Figure 1a has been corrected. The vector magnetic \feld components have been\ntransformed to the heliographic coordinate system, and the geometry has been mapped to a plane\ntangent to the solar surface at the center of the \feld of view of Figure 1a, which is (W54 :7\u000e, S15:3\u000e).\nNote that the de-projected vector magnetic \feld in Figure 1b is equivalent to the result if placing\nthe region of interest at the solar disk center. We could also do the reverse operation of the rotations\nin Equations (5) and (7) to project the de-projected magnetic \feld back to the original location.\nThis reverse operation could be done for 3D data as shown in Figure 1c. Figure 1d shows that the\ninverse operation of the de-projected data quantitatively matches the original observation.\nThe geometrical projection correction using Equation (6) is limited to a small \feld of view.\nWhen the \feld of view is large, the plane approximation causes large errors due to the spherical\nsurface of the Sun. This issue has been considered in the HMI data products (Hoeksema et al.\n2014), where the vector magnetic \feld is mapped onto the cylindrical equal area (CEA) coordinate\nsystem (Calabretta & Greisen 2002; Thompson 2006). Note that the CEA coordinate system is\nessentially a spherical coordinate system, and the grids in the latitude are distributed with equal\nspacing of the sine of the latitude. Therefore, the HMI vector magnetic \feld data mapped on the\nCEA coordinate system cannot be used directly as the boundary condition in a problem solved in\na Cartesian coordinate system, especially when the \feld of view is large or the latitude is high.\n2.4. Velocity Derived from Optical Flow Techniques\nThe velocity of magnetized plasma in the photosphere is a critical physical parameter to\ndetermine the evolution of the plasma and magnetic \feld. It can be used as the boundary condition\nof data-driven MHD simulations and can be used to compute the injection of magnetic energy\nand helicity. The ling-of-sight velocity can be observed by the Doppler e\u000bect, while the vector{ 9 {\nvelocity (in some models, only its transverse components) can be computed by the optical \row\ntechniques, which use a series of magnetograms to infer the velocity with an implied physical\nevolution model. Traditionally, the local correlation tracking (LCT) method is adopted using only\nthe normal component of magnetograms to derive the photospheric velocity, uLCT. This velocity\nis regarded as the horizontal plasma velocity, vt, such as in Chae (2001), Moon et al. (2002), and\nNindos & Zhang (2002). However, Schuck (2005) pointed out that the LCT method implies an\nadvection model. It is inconsistent with the magnetic induction equation, which is a continuity\nequation and governs the evolution of magnetic \felds.\nSchuck (2005) showed that the LCT method aims to maximize the correlation coe\u000ecient of\ntwo images in a window. Thus, LCT requires the intensity, I, of an image at a given position and\ntime to satisfy the following equation\nI(x;t2)\u0011I[x\u0000u0(t2\u0000t1);t1]; (8)\nwhich leads to the advection equation\n@I\n@t+u0\u0001rI= 0; (9)\nwhereIis di\u000berentiable. While the evolution of the normal component of the vector magnetic\n\feld,Bn, is governed by the magnetic induction equation. In the photosphere, the resistivity can\nbe omitted such that the plasma is deemed as ideal, and the magnetic induction equation for Bn\nis written as (Kusano et al. 2002; D\u0013 emoulin & Berger 2003; Welsch et al. 2004; Longcope 2004;\nGeorgoulis & LaBonte 2006)\n@Bn\n@t+rt\u0001(Bnvt\u0000vnBt) = 0; (10)\nwherendenotes the normal direction and tthe transverse direction.\nInversion of the velocity from Equation (10) su\u000bers from two ambiguities. On the one hand,\nWelsch et al. (2004) and Longcope (2004) showed that the \rux transport vector can be expressed\nas two scalar functions, namely, the inductive potential, \u001e, and the electrostatic potential,  , as\nuBn=Bnvt\u0000vnBt=\u0000(rt\u001e+rt \u0002^n): (11)\nOnly\u001ecan be determined by Equation (10) while  could be arbitrary. Therefore, the velocity\ncan only be determined with additional assumptions, one of which is the assumption of the a\u000ene\nvelocity pro\fle in a window (Schuck 2005) such that\nu(x) = \nU0\nV0!\n+ \nUxUy\nVxVy! \nx\ny!\n; (12)\nwith (U0;V0) the velocity at the central position, and Ux,Uy,Vx, andVythe \frst order spatial\nderivatives of the velocities. The velocities are derived by a least-square \ftting of the model{ 10 {\nto observations. This method has been implemented as the di\u000berential a\u000ene velocity estimator\n(DAVE) in Schuck (2006).\nOn the other hand, the plasma velocity along a \feld line cannot be determined only by the\nevolution of Bn. Schuck (2008) tackled this problem by extending DAVE for vector magnetograms\n(DAVE4VM), where the velocity is assumed to exhibit a 3D a\u000ene velocity pro\fle with\nu(x) =0\nBB@U0\nV0\nW01\nCCA+0\nBB@UxUy\nVxVy\nWxWy1\nCCA \nx\ny!\n: (13)\nThe vector velocity is derived by \ftting the model to observations of a time sequence of vector\nmagnetic \feld.\n3. Three-Dimensional Magnetic Field Models\nThe magnetic \feld observation is less accessible in the chromosphere and corona than in the\nphotosphere due to a lower density and a higher temperature. A higher temperature widens the\nspectral lines that cover the Zeeman splitting. Moreover, the magnetic \feld is usually weaker\nat higher altitudes. These problems are alleviated by adopting infrared spectral lines (e.g., Lin\net al. 2000, 2004; Kuckein et al. 2009, 2012), because the Zeeman splitting is proportional to the\nwavelength squared and magnetic \feld strength. Nevertheless, there are still some di\u000eculties.\nThe line emission in the chromosphere and corona is weaker than that in the photosphere, which\nlowers the signal-to-noise ratio. Moreover, the optical thin condition in the corona makes the\ninterpretation of the observations more di\u000ecult than that in the photosphere. In addition, the\nplasma in the chromosphere and corona are in a state of non-local thermodynamic equilibrium.\nThe formation of spectral lines there are more di\u000ecult to be interpreted than the photospheric\nspectral lines, where local thermodynamic equilibrium is valid.\nDespite the aforementioned di\u000eculties, observations of the magnetic \feld in the chromosphere\nhave made progress in the past years (Lagg et al. 2015; de la Cruz Rodr\u0013 \u0010guez & van Noort 2016).\nThe way is to measure the polarization signal of chromospheric spectral lines generated by the\nZeeman and Hanle e\u000bect. Some typical magnetic sensitive lines in the chromosphere include the He\ni10830 \u0017A triplet line and the Ca ii8542 \u0017A triplet line. Spectropolarimetric observations and line\nformation theories of the aforementioned spectral lines have been advanced to enable the detection\nof the chromospheric magnetic \feld (Socas-Navarro et al. 2000; Asensio Ramos et al. 2008; Lagg\net al. 2009; de la Cruz Rodr\u0013 \u0010guez et al. 2013). More explanations and applications of the Zeeman and\nHanle e\u000bects to detecting the chromospheric and coronal magnetic \feld can be found in Sten\ro\n(2013) and Schmieder et al. (2014). Magnetic \feld in the corona can also be inferred by radio\nobservations (e.g., White 2002; Wang et al. 2017). However, only the strength can be obtained,\nwhile the full magnetic vector cannot be derived at present.{ 11 {\nAlthough we have the aforementioned methods, reliable measurements of the full 3D magnetic\n\feld is still unavailable. Thus, people resort to theoretical and numerical models to study the solar\nmagnetic \feld, which include theoretical force-free \feld models, numerical nonlinear force-free \feld\nmodels, MHS models, and MHD models. Each model has its own advantage and disadvantage,\nwhile they may complement each other when combined in a study. For example, theoretical force-\nfree \feld models have precise solutions, whose magnetic tension and pressure forces balance each\nother exactly, while they are too simple to describe the complex magnetic structures and dynamical\nevolution revealed in real observations. Numerical nonlinear force-free \feld models have the advan-\ntage of being related with magnetic \feld observations directly. However, they still cannot explain\nthe forced structures in the photosphere and lower chromosphere and their dynamical evolution.\nMHS models partly overcome the disadvantage of the force-free \feld models, while they still belong\nto static models. MHD models may provide the best way to study the observed complex magnetic\nstructures and dynamical evolution though they are the most complicated. At present, it is still a\ndi\u000ecult task to combine MHD models with other observed parameters like magnetic \feld, density,\nand temperature.\n3.1. Theoretical Force-Free Field Models\nThe upper chromosphere and lower corona, especially in an active region, are dominated by\nthe magnetic pressure and tension (Gary 2001). The gradient of the gas pressure and the gravity\nthere are much smaller than the magnetic pressure and tension forces. This implies that, when\nmodeling these atmospheric layers, even a very small Lorentz force can break down any static\nsolutions. Therefore, in static models, one always assumes that the Lorentz force equals zero,\nnamely J\u0002B= 0. The magnetic \feld without any Lorentz force is de\fned as a force-free \feld. The\nsolution requires the electric current density to be parallel to the magnetic \feld. Since \u00160J=r\u0002B,\nwe have\nr\u0002B=\u000bB; (14)\nwhere\u000bis a scalar function of the 3D space. Meanwhile, the magnetic \feld satis\fes the solenoidal\ncondition\nr\u0001B= 0: (15)\nIf\u000bis a constant in the whole space, which represents the linear force-free \feld (where a special\ncase is the potential \feld when \u000b= 0), Equations (14) and (15) are linear, and they can be solved\nby a Green's function method or a Fourier transform method in the Cartesian coordinate system\n(Schmidt 1964; Nakagawa & Raadu 1972; Chiu & Hilton 1977; Teuber et al. 1977; Seehafer 1978).\nFor a potential \feld in the spherical coordinate system, the spherical harmonic transformation\nmethod is often adopted for solving the Laplace's equation, r2\b = 0, for the magnetic \feld\npotential \b, and B=\u0000r\b (Schatten et al. 1969; Altschuler & Newkirk 1969; Altschuler et al.\n1977; Schrijver & De Rosa 2003). Other ways of solving the Laplace's equation include \fnite{ 12 {\ndi\u000berences (T\u0013 oth et al. 2011) and a fast solver by combining spectral and \fnite-di\u000berence methods\n(Jiang & Feng 2012b).\nFor a nonlinear force-free \feld where \u000bvaries in the space, it is di\u000ecult to \fnd analytic\nsolutions for Equations (14) and (15) with given boundary values, because of the nonlinearity of\nthe problem. Some special force-free \feld solutions are discussed as follows, and the numerical\nmethods for solving nonlinear force-free \feld with given boundary conditions are to be discussed\nin Section 3.2.\nA simple but intuitive way to construct potential \felds is the magnetic charge method. Some\nmagnetic point sources are placed below a selected surface at z= 0, and the magnetic \feld in the\nspacez >0 is assumed to be potential. Thus, the magnetic \feld can be derived similarly as the\nelectric \feld derived by the Coulomb's law:\nB=X\nimiri\nr3\ni; (16)\nwhereiis the number of the magnetic charges, miis the strength of the magnetic charge, riis\nthe position vector pointing from the magnetic charge ito the place of magnetic \feld, and riis\nthe scalar distance between them. Potential \felds computed by the magnetic charge method have\nbeen adopted to study various aspects of magnetic structures and solar activities (Seehafer 1986;\nGorbachev & Somov 1988; D\u0013 emoulin et al. 1992; Pontin et al. 2016). Besides the potential \feld,\nthe magnetic charge method can also be used to construct linear force-free \feld (D\u0013 emoulin & Priest\n1992).\nUsing an axis symmetric assumption, Low & Lou (1990) provided a class of nonlinear force-free\n\feld solution in the spherical coordinate system. An axis symmetric and solenoidal magnetic \feld\nBcan be written as\nB=1\nrsin\u0012\u00121\nr@A\n@\u0012er;\u0000@A\n@re\u0012;Qe\u001e\u0013\n; (17)\nwhereAandQare functions of only rand\u0012because of the symmetry. Substituting Equation (17)\ninto Equation (14), one could \fnd that Qis a function of A, andAsatis\fes the Grad-Shafranov\nequation, which is a two-dimensional (2D) nonlinear partial di\u000berential equation (Grad & Rubin\n1958). Low & Lou (1990) further found a class of separable solutions of A. The problem is \fnally\nreduced to a nonlinear ordinary di\u000berential equation of a scalar function, which is solvable with\na numerical method. Thus, the Low and Lou solution is often called semi-analytic. It has been\nwidely adopted as a reference model for testing the numerical nonlinear force-free \feld algorithms.\nFigure 2a shows one of such tests.\nAnother class of semi-analytic nonlinear force-free \feld solution is the \rux rope model proposed\nby Titov & D\u0013 emoulin (1999). The Titov{D\u0013 emoulin model include a nonlinear force-free \feld\nassumption and concentration of the electric current into a partial torus above a selected surface.\nTitov & D\u0013 emoulin (1999) argued that the general magnetic topology is determined by magnetic\n\feld lines, which are a double integration of the electric current distribution. This e\u000bect smoothes{ 13 {\nout the small-scale and weak current distribution and only leaves the large-scale and strong one\nthat is essential to the problem. In this way, a single current channel in a torus shape could\ncharacterize the main feature of an active region with twisted magnetic \feld lines. The hoop force\nof the current torus is balanced by the Lorentz force caused by the interaction between the current\nand the potential \feld, which is generated by two imaginary magnetic charges and a line current\nthat are buried under the selected surface. An example of the Titov{D\u0013 emoulin model is provided\nin Figure 2b.\nSome other analytic solutions of force-free \felds can be found in, e.g., Gold & Hoyle (1960),\nTitov et al. (2011), Kleman & Robbins (2014), and Kleman (2015). All these models provide\nvarious insights into the magnetic \feld topology and stabilities.\n3.2. Numerical Nonlinear Force-Free Field Models\nIn order to extract the full information from vector magnetic \feld observations, one has to re-\nsort to numerical methods that aim at reconstructing nonlinear force-free \feld models from bound-\nary conditions and proper pseudo initial conditions. The latter serve as the initial state for an\niteration process, and the \fnal converged results are usually independent of them. Compared to a\npotential \feld model, a nonlinear force-free \feld involves free magnetic energy and electric current,\nwhich could power solar eruptions. When compared with a linear force-free \feld model, a nonlinear\nforce-free \feld is closer to coronal observations when there are both large scale coronal loops and\nlow-lying \flaments or prominences. Thus, it is closer to the realistic magnetic con\fguration. As a\nresult, a nonlinear force-free \feld provides a better initial condition for further MHD simulations\nthan the potential and linear force-free \feld models. The nonlinear force-free \feld model has also\nbeen adopted to reconstruct magnetic \rux ropes in the solar atmosphere (Canou et al. 2009; Canou\n& Amari 2010; Su et al. 2009; Guo et al. 2010a,c; Cheng et al. 2010; Jing et al. 2010; Jiang et al.\n2014).\nThere are various numerical algorithms to compute nonlinear force-free \feld models, such\nas the Grad{Rubin, vertical integration, MHD relaxation, optimization, and boundary integral\nequation methods. Here, we provide a brief introduction of these methods. Readers are referred to\nWiegelmann & Sakurai (2012) for more details.\nThe Grad-Rubin method was \frst proposed by Grad & Rubin (1958) for computing nonlinear\nforce-free \feld in fusion plasma. The idea is to compute the torsional parameter \u000band magnetic\n\feldBiteratively. Given an initial condition B(0)by the potential magnetic \feld, \u000bat each iteration\nstepnis computed by a hyperbolic equation\nB(n)\u0001r\u000b(n)= 0: (18)\nThe boundary condition of \u000bis provided at either the positive or the negative polarity, which can be\nderived from the vector magnetic \feld. Then, the magnetic \feld at the next step n+1 is computed{ 14 {\nby solving an elliptic equation\nr\u0002B(n+1)=\u000b(n)B(n): (19)\nThe newly computed magnetic \feld B(n+1)is also constrained by the solenoidal condition and\nthe boundary condition for the normal component. The Grad{Rubin method was \frst applied to\ncompute coronal magnetic \felds by Sakurai (1981), and it has been further developed and applied to\nmagnetic \feld reconstruction by Amari et al. (1997, 2006, 2014), Wheatland (2004, 2006), Inhester\n& Wiegelmann (2006), and Gilchrist & Wheatland (2013, 2014). A modi\fed Grad{Rubin method\nhas also been proposed by Malanushenko et al. (2012, 2014) using information of the torsional\nparameter \u000bin the volume, which is derived from the coronal loop geometry. The solenoidal\ncondition of the magnetic \feld is guaranteed by di\u000berent strategies in di\u000berent implementations.\nFor example, Amari et al. (1997, 2006, 2014) used the vector potential to represent the magnetic\n\feld, which could ensure r\u0001B= 0 with the accuracy of round-o\u000b errors.\nThe vertical integration method is straightforward and relatively easy to implement (Nakagawa\n1974). It \frst computes \u000bfrom a vector magnetic \feld on the bottom using the zcomponent of\nthe force-free Equation (14) as\n\u000b=1\nBz\u0012@By\n@x\u0000@Bx\n@y\u0013\n: (20)\nThen, the vector magnetic \feld and \u000bhigher up are derived by integrating the following equations\noverz\n@Bx\n@z=\u000bBy+@Bz\n@x;\n@By\n@z=\u0000\u000bBx+@Bz\n@y;\n@Bz\n@z=\u0000@Bx\n@x\u0000@By\n@y;\n@\u000b\n@z=\u00001\nBz\u0012\nBx@\u000b\n@x+By@\u000b\n@y\u0013\n:(21)\nUnfortunately, Equation (21) is ill-posed in the sense that numerical errors grow fast with height,\nwhich has been discussed in Low & Lou (1990), Wu et al. (1990), and D\u0013 emoulin et al. (1992).\nSome methods have been proposed to regularize the problem by smoothing the physical variables\n(Cuperman et al. 1990; D\u0013 emoulin et al. 1992; Song et al. 2006).\nThe MHD relaxation method solves the MHD momentum equation and magnetic induction\nequation to build an equilibrium state. The momentum equation includes a dissipative term D(v),\nwhich is a function of the velocity v, and the magnetic induction equation considers the resistive\nterm:\n\u001a(@\n@t+v\u0001r)v=J\u0002B\u0000rp+\u001ag+D(v); (22)\n@B\n@t=r\u0002(v\u0002B)\u0000r\u0002 (\u0011J); (23){ 15 {\nwhere\u001ais the density, J=r\u0002B=\u00160is the electric current density, \u00160is the vacuum permeability,\npis the gas pressure, gis the gravitational acceleration vector, and \u0011is the resistivity. Considering\nthat\u0011is uniform in the space, Equation (23) can be recast as\n@B\n@t=r\u0002(v\u0002B) +\u0011mr2B; (24)\nwhere\u0011m= 1=(\u001b\u00160) is the magnetic di\u000busivity, and \u001b= 1=\u0011is the conductivity. The dissipative\nterm D(v) can be either in a friction form such that D(v) =\u0000\u0017v(Yang et al. 1986; Roumeliotis\n1996; Valori et al. 2005, 2007, 2010) or in a viscosity form such that D(v) =r\u0001(\u0017\u001arv) (Miki\u0013 c\n& McClymont 1994; McClymont & Miki\u0013 c 1994; McClymont et al. 1997; Amari et al. 1996, 1997).\nThe free parameter \u0017is used to control the dissipative speed.\nIn its full form, the MHD relaxation method eventually reaches a state of magneto-hydrostatic\nequilibrium (Chodura & Schlueter 1981; Zhu et al. 2013, 2016). When a nonlinear force-free \feld\nmodel is considered, the inertia, pressure gradient, and gravity forces are omitted in Equation (22).\nIf the friction form is adopted, the momentum equation is further reduced to\nv=1\n\u0017J\u0002B: (25)\nA nonlinear force-free \feld model could be reconstructed with Equations (24) and (25), which is\ncalled as the magneto-frictional method. The initial condition could be a potential magnetic \feld,\nand the boundary condition an observed vector magnetic \feld. Some other initial and boundary\nconditions are also possible, such as those adopted in the magnetic \rux rope insertion method (van\nBallegooijen 2004; Su et al. 2009; Savcheva et al. 2015, 2016). The \rux rope insertion method\nonly uses the normal component of the vector magnetic \feld on the bottom boundary, and the\ninitial condition is provided by a potential \feld combined with an arti\fcially inserted magnetic\n\rux rope. Modern MHD codes are recently introduced to nonlinear force-free \feld reconstructions\nbased on the MHD relaxation approach. For example, Jiang et al. (2011, 2012) and Jiang & Feng\n(2012a) introduced a CESE-MHD-NLFFF code with the conservation element/solution element\nscheme (Jiang et al. 2010). The MHD relaxation method has also been developed and applied to a\nseries of researches by Inoue et al. (2011, 2012, 2014). Guo et al. (2016b) has implemented a new\nmagneto-frictional algorithm in the Message Passing Interface Adaptive Mesh Re\fnement Versatile\nAdvection Code1(MPI-AMRVAC; Keppens et al. 2003, 2012; Porth et al. 2014) and tested it with\nanalytic solutions. This method is parallelized with MPI and could be applied to both Cartesian and\nspherical coordinates, with either uniform or adaptive mesh re\fnement (AMR) grids (Figures 2a, 2b,\n2c, and 2d). The solenoidal condition of the magnetic \feld is guaranteed by including a di\u000busive\nterm in the induction equation. Guo et al. (2016a) has applied the magneto-frictional method\nimplemented in MPI-AMRVAC to the vector magnetic \feld observed by SDO /HMI in Cartesian\ncoordinates with uniform or AMR grids and in spherical coordinates with AMR grids (Figures 2c\nand 2d).\n1https://gitlab.com/mpi-amrvac{ 16 {\nThe optimization method constructs a nonlinear force-free \feld model by minimizing an ob-\njective functional\nL=Z\nV\u0002\nB\u00002j(r\u0002B)\u0002Bj2+jr\u0001Bj2\u0003\ndV; (26)\nwhereVis the computational volume. When Lis minimized to a small value by an iteration\nprocess, the force-free and solenoidal conditions are assumed to be ful\flled simultaneously. This\nmethod was proposed by Wheatland et al. (2000) and further developed by Wiegelmann (2004) and\nWiegelmann (2007) in both Cartesian and spherical coordinate systems. Tadesse et al. (2009, 2011)\nfurther tested and applied the optimization method in spherical geometry to reconstruct the global\ncoronal \feld. Jim McTiernan developed another version of this method using IDL2and FORTRAN\nlanguages, in both the Cartesian and spherical coordinate systems. Tests and applications can be\nfound in Schrijver et al. (2006) and Metcalf et al. (2008) for the Cartesian version, and in Guo\net al. (2012a) for the spherical version. Additionally, Inhester & Wiegelmann (2006) proposed a\n\fnite-element scheme for the optimization method, while all the other codes use \fnite-di\u000berence\nschemes.\nThe boundary integration method was proposed by Yan & Sakurai (1997, 2000). Using the\nGreen's theorem (Courant & Hilbert 1962), the magnetic \feld is expressed as a boundary integration\nas\ncB=Z\nS\u0012\nY@B\n@n\u0000@Y\n@nB\u0013\ndS; (27)\nwherec= 1=2 on the boundary Sandc= 1 in the volume above S, andYis a reference function of\na diagonal matrix that can be determined by a volume integration. This method has been improved\nand applied to observations in Yan et al. (2001), Yan & Li (2006), and He & Wang (2006, 2008).\nAnd recently, it has also been implemented with the acceleration of the graphics processor unit\n(GPU; Wang et al. 2013).\n3.2.1. Preprocessing of the Vector Magnetic Field for Nonlinear Force-Free Field Extrapolation\nThe vector magnetic \feld observed on the photosphere is not force-free, since the plasma \f\nis close to one there. Thus, inconsistence arises when the forced photospheric \feld is used as\nboundary conditions of force-free models. For a force-free \feld, its boundary value must satisfy\nthe magnetic force-free and torque-free formulae as derived by Molodenskii (1969) and Aly (1989).\nTherefore, Wiegelmann et al. (2006) proposed a method to remove the magnetic force and torque\nusing an optimization method to minimize a functional Lthat represents the sum of the magnetic\nforce, torque, deviation from observations, and the smoothness of the magnetic \feld. To apply\nthe preprocessing method proposed by Wiegelmann et al. (2006), some prerequisites need to be\n2http://sprg.ssl.berkeley.edu/ ~jimm/fff/optimization_fff.html{ 17 {\nsatis\fed, namely, the vector magnetic \feld on the bottom boundary should be isolated and close to\nthe disk center. Since the force-free and torque-free conditions are asked to be satis\fed on the whole\nboundary of the computation box, while observations are only available on the bottom, the isolated\ncondition is employed that can allow the force and torque on the lateral and top boundaries to be\nneglected. An isolated magnetic \feld means that most of the magnetic \rux is concentrated in the\n\feld of view and the magnetic \rux is balanced. The second prerequisite requires the line-of-sight\ncomponent to be close to the vertical component, as proposed by Wiegelmann et al. (2006). This\nrequirement could be loosened by allowing the vertical component to change in a larger range of\nthe uncertainties.\nFuhrmann et al. (2007) developed another method for preprocessing the vector magnetic \feld\nbased on the principle proposed in Wiegelmann et al. (2006). This method is di\u000berent from that\nof Wiegelmann et al. (2006) in the following three aspects. First, the deviation of the preprocessed\nmagnetic \feld from the original observations is not included in the functional L. Second, the\nmagnetic \feld is smoothed by a windowed median averaging rather than a 2D Laplacian that is\nused in Wiegelmann et al. (2006). Third, a simulated annealing method is adopted for a better\nconvergence to search for the global minimum of the functional Lrather than the Newton{Raphson\nmethod used in Wiegelmann et al. (2006).\nJiang & Feng (2014) also developed a preprocessing method for removing the magnetic force\nand torque of the photospheric vector magnetic \feld. The idea is to split the observed vector\nmagnetic \feld into a potential part determined only by the vertical component and a non-potential\npart. They argued that the potential magnetic \feld at a height of about 400 km, which is approx-\nimately the length of a pixel in the HMI data, can be regarded as the preprocessed potential \feld,\nwhile the non-potential part is preprocessed with the same method of Wiegelmann et al. (2006).\nThe potential \feld is used to guide the preprocessing of the non-potential part, which is required to\npossess the same level of force-freeness and smoothness as that of the potential \feld at the height\nof 400 km.\nAll the aforementioned preprocessing methods are developed in the Cartesian coordinate sys-\ntem. A preprocessing method in the spherical geometry has been developed (Wiegelmann 2007;\nTadesse et al. 2009, 2011), which is also implemented and tested by Guo et al. (2012a) using the for-\nmulae in Tadesse et al. (2009). A series of tests show that preprocessing can improve the nonlinear\nforce-free \feld extrapolation by decreasing the magnetic divergence and Lorentz force (Wiegelmann\net al. 2006; Fuhrmann et al. 2007; Metcalf et al. 2008; Fuhrmann et al. 2011; Guo et al. 2012a;\nJiang & Feng 2013). Note that the preprocessing has to modify the photospheric data to satisfy the\nforce-free condition and the resulted magnetic \feld is assumed to correspond to the chromospheric\n\feld. The validity of this argument is yet to be checked by direct observations of the chromospheric\nmagnetic \feld or more sophisticated models.{ 18 {\n3.3. Magnetohydrostatic Models\nDi\u000berent from the aforementioned nonlinear force-free \feld models, a series of non-force-free\nmodels have also been developed. The models are magnetohydrostatic (MHS) in essence, which\ninclude the physical e\u000bects of pressure gradient and gravity but omit the plasma inertia:\nJ\u0002B\u0000rp+\u001ag= 0: (28)\nThere are two di\u000berent ways to construct an MHS model governed by Equation (28), one being\nanalytic and the other numerical. To use the analytic method, additional assumptions are needed\nto simplify the governing equations of the MHS model to linear problems. For example, assuming\nelectric currents to be perpendicular to gravity everywhere in the computational volume, Low\n(1985) and Bogdan & Low (1986) found a class of analytic solutions for Equation (28). With the\nsame assumption as Low (1985), Zhao & Hoeksema (1993, 1994) showed that the solutions of the\nMHS model can be expressed as the summation of involving spherical harmonics. Neukirch (1995)\nproposed a new mathematical procedure to calculate the analytic solutions of the MHS equations.\nDi\u000berent from Low (1985) and Bogdan & Low (1986), these solutions allow additional \feld-aligned\nelectric currents. The solutions of Neukirch (1995) have been applied to modeling a polar crown\nsoft X-ray arcade (Zhao et al. 2000) and global coronal structures (Ruan et al. 2008). Using the\nprinciple of minimum dissipation rate, Hu & Dasgupta (2008) and Hu et al. (2008) showed that a\ngeneral non-force-free \feld can be expressed as the summation of two linear force-free \felds and\none potential \feld. The two free parameters of constant \u000bare determined by comparing the model\nand the observed transverse magnetic \feld on the photosphere.\nWithout any prerequisite on the solutions of the MHS equation, the problem is fully nonlinear\nand can only be solved by a numerical method. Wiegelmann & Inhester (2003) proposed to use\nthe optimization method, which is similar to that for the nonlinear force-free \feld model but\nincluding the pressure gradient force and gravity, to construct MHS models. Wiegelmann et al.\n(2007) further extended this optimization method for MHS models from Cartesian coordinates to\nspherical geometry. Another numerical method to construct MHS models is the MHD relaxation\nmethod, which is also similar to that for the nonlinear force-free \feld model but including the\npressure gradient and gravity. Chodura & Schlueter (1981) applied the MHD relaxation method to\nreconstruct MHS models. This method has been further implemented and applied to observations\nby Zhu et al. (2013, 2016).\n3.4. Magnetohydrodynamic Models\nThe nonlinear force-free \feld and MHS models mentioned above are all static models. To study\nthe dynamics of the magnetic \feld and its interaction with the plasma, we need MHD numerical\nsimulations. Depending on di\u000berent criteria, MHD simulations are divided into di\u000berent categories.\nFirst of all, MHD simulations can be divided into zero- \f, isothermal, ideal, resistive, and full MHD\nmodels with the order of increasing physical details included.{ 19 {\nThe zero-\fMHD model omits gravity and gas pressure in the momentum conservation equa-\ntion, and omits the energy conservation equation. The magnetic di\u000busion term in the magnetic\ninduction equation could either be neglected (T or ok & Kliem 2003; Kliem et al. 2013) or not\n(Aulanier et al. 2010). An isothermal model considers gravity and gas pressure in the momentum\nconservation equation but the temperature is kept unchanged; thus the energy conservation equa-\ntion is neglected (Xia et al. 2014b). In the ideal MHD model, all the aforementioned physical terms\nand equations are solved except for the omission of the magnetic di\u000busion term in the magnetic\ninduction equation. Meanwhile, the thermal conduction, radiative loss, viscous dissipation, and\nJoule heating are not considered in the energy conservation equation (Fan 2009b). A resistive\nMHD model includes magnetic resistivity in the magnetic induction equation and Joule heating in\nthe energy conservation equation (Leake et al. 2013, 2014). A full MHD model tends to include all\nthe physical e\u000bects, especially the thermal conduction and radiative loss (Xia et al. 2014a). But in\npractice, magnetic \feld di\u000busion in the induction equation, viscous dissipation and Joule heating\nmight be neglected to save computational time.\nSecondly, di\u000berent MHD models have di\u000berent physical domains of simulation. They can be\nrestricted only to the solar corona (T or ok & Kliem 2003; Roussev et al. 2012; Kliem et al. 2013),\nor a more complete domain that includes also the photosphere, chromosphere, and the convection\nzone below the solar surface (Fan 2001, 2009b; Magara & Longcope 2001, 2003; Archontis et al.\n2004; Archontis & T or ok 2008), or an even larger domain extending from the solar atmosphere to\nthe interplanetary space (e.g., Shen et al. 2014; Feng et al. 2015).\nFinally, in terms of the initial and boundary conditions adopted, MHD simulations are either\npurely theoretical (Amari et al. 2000; Zuccarello et al. 2012) or data-driven/data-constrained (Jiang\net al. 2013, 2016a; Kliem et al. 2013; Amari et al. 2014; Inoue et al. 2014, 2015). A recent compre-\nhensive review has been made by Inoue (2016) on this topic. Here, we de\fne a data-driven model\nas the one in which both the initial and boundary conditions are provided by observations. Addi-\ntionally, the boundary condition should be time-varying in correspondence with the data stream\nfrom observations. And a data-constrained model is de\fned as the one in which only the initial\ncondition is provided by observations. Figure 3a shows a data-constrained model (Kliem et al.\n2013), where the initial condition is provided by the nonlinear force-free \feld modeled with the \rux\nrope insertion method. The velocity at the bottom boundary is kept at zero all the time. Another\ndata-constrained model is shown in Figure 3b (Inoue et al. 2014). Two data-driven models are\npresented in Figures 3c (Amari et al. 2014) and 3d (Jiang et al. 2016a), where both the initial\nconditions are provided with a nonlinear force-free \feld model, while the boundary conditions by\ninformation from observations extracted with di\u000berent strategies. We note that the simulation in\nAmari et al. (2014) is not purely data-driven, because their boundary condition for the velocity\n\feld is arti\fcially speci\fed with converging \rows that mimic the \rux cancelation to build up the\n\rux rope until its eruption.\nMHD simulations have been applied to a vast range of topics in the solar and space physics.\nHere, we only focus on a few of the topics closely related to the origin and structure of solar{ 20 {\neruptions. As a key ingredient of solar eruptions, magnetic \rux ropes are present in most MHD\nsimulations related to solar active region formation and solar eruptive activities (e.g., Fan 2001;\nMagara & Longcope 2001; Roussev et al. 2012; Jin et al. 2016). The driving mechanisms for the\neruption of a magnetic \rux rope have been studied by some MHD simulations. For example, T or ok\net al. (2004) found that helical kink instability could drive the initial eruption of a highly twisted\n\rux rope. While there is no evidence showing that the helical kink instability itself could drive the\nfull eruption, T or ok & Kliem (2005) found that the decrease of the overlying magnetic \feld with\nheight should be fast enough to enable the full eruption of a magnetic \rux rope.\nOne possible formation process of magnetic \rux ropes in an active region is through magnetic\n\rux emergence from the convection zone. A series of 3D ideal MHD simulations have been applied\nto study the process of magnetic \rux emergence (e.g., Manchester et al. 2004; Murray et al. 2006;\nGalsgaard et al. 2007; Magara 2008; Archontis et al. 2009; Fan 2009b). It is thought that the\nmagnetic buoyancy instability makes the magnetic \ruxes break through the photosphere into higher\nlayers. Full MHD simulations including radiative transfer have also been applied to study the self-\nconsistent magneto-convection process in the upper convection zone and the photosphere (e.g.,\nStein & Nordlund 2000; Cheung et al. 2008; Mart\u0013 \u0010nez-Sykora et al. 2008; Rempel et al. 2009; Chen\net al. 2017). Comprehensive reviews on the observations and MHD simulations of magnetic \rux\nemergence can be found in Fan (2009a) and Nordlund et al. (2009).\nIn a series of zero- \fMHD simulations, Aulanier et al. (2012, 2013) and Janvier et al. (2013) have\nextended the standard \rare model to 3D. Aulanier et al. (2012) found both direct and return electric\ncurrents in sunspots and faculae. Flare ribbons usually appear as a J-shaped structure as shown by\nmany observations. In the 3D model, the straight part of the J-shaped ribbon corresponds to the\nfootpoints of the reconnected \feld lines, which are formed in the vertical current sheet stretched by\nthe erupting magnetic \rux rope. Only direct electric currents appear in this part. While the curved\npart of the J-shaped ribbon corresponds to the periphery of the legs of the erupting \rux rope (see\nalso Cheng & Ding 2016). Both direct and return currents are present there. The strong-to-weak\nshear transition of the \rare loops are explained by two e\u000bects: one is the transfer of di\u000berential\nmagnetic shear in the pre-eruptive con\fguration to the post-eruptive one, and the other is the\nvertical straightening of the inner legs of the erupting magnetic \rux rope. Combining the zero- \f\nMHD model and historical records of solar active regions, Aulanier et al. (2013) estimated the\nlargest possible \rare on the Sun to be \u00186\u00021033erg. Janvier et al. (2013) analyzed the slipping\nvelocity,vs, and the norm of the magnetic \feld line mapping, N, in this MHD simulation. They\nfound thatvsandNare linearly correlated with each other. A comprehensive review on 3D models\nof solar \rares can be found in Janvier et al. (2015).\nThe formation of solar \flaments/prominenes has been simulated with full 3D MHD models in\nrecent years (Xia et al. 2014a; Xia & Keppens 2016a,b; Keppens et al. 2015). These models focus\non a particular physical mechanism for the prominence formation, namely, the chromospheric evap-\noration and coronal condensation. They are caused by an impulsive heating in the chromosphere\nand a runaway radiative loss in the corona. Therefore, the thermal conduction and radiative loss{ 21 {\nin the energy equation are essential for such a model. Xia et al. (2014a) \frst showed the full 3D\nMHD simulation of prominence formation in which hot plasma could condense to cold material in\nthe corona. Xia & Keppens (2016a) studied the plasma circulation between the chromosphere and\ncorona. They found that when plasmas are heated in the chromosphere, they could be evaporated\ninto the corona. Due to the runaway radiative loss in the corona, hot plasmas are condensed and\ncooled down to form the cold prominence. The dense prominence plasmas in the corona usually\nmove downward, drag magnetic \feld with them, and \fnally fall back to the chromosphere. Kep-\npens et al. (2015) and Xia & Keppens (2016b) further studied the dynamics of the prominence and\nshowed how the magneto-convective motions and Rayleigh-Taylor instability generate the falling\n\fngers and uprising bubbles as observed by Berger et al. (2008) and Berger et al. (2010).\n4. Magnetic Energy Computation\nIn the solar atmosphere, the energy contained in the magnetic \feld is much larger than that\nin other forms, such as the kinematic, thermal, and gravitational potential energy. Therefore, the\nmagnetic energy is thought to be the major reservoir for powering solar \rares and CMEs. The\ntotal magnetic energy in a volume, V, is expressed as\nE=Z\nVB2\n2\u00160dV: (29)\nNot all the magnetic energy can be released in the solar atmosphere. Considering that the evolution\nof the photosphere is much slower than the dynamical eruptions in the corona, only the free energy,\nEf, higher than the potential \feld, Ep, can be released:\nEf=E\u0000Ep; (30)\nwhereEpis determined by the same normal \feld of Bon the boundary of the volume, V. Ad-\nditionally, the free energy is only an upper limit for the available energy to be released, since the\n\fnal state of the magnetic \feld after \rares and/or CMEs is also constrained by the conservation of\nmagnetic helicity (Taylor 1986).\nEquation (30) is valid with another requirement that the magnetic \feld should be solenoidal\n(Valori et al. 2013). This issue matters when we apply Equation (30) to magnetic \feld derived by\nnumerical force-free, MHS, and MHD models, where the solenoidal condition might be violated.\nValori et al. (2013) have shown that, in addition to EpandEf, the magnetic energy are contributed\nby three more terms in non-solenoidal \feld, which come from the non-solenoidal potential \feld, non-\nsolenoidal current carrying \feld, and the mixture of the two \felds. In such cases, the free magnetic\nenergy derived from Equation (30) contains large uncertainties. In some cases, the calculated free\nmagnetic energy is even negative. Such results are caused by the deviations from the solenoidal\ncondition of the magnetic \feld. A good magnetic \feld model should have as small magnetic\ndivergence as possible as demonstrated in Valori et al. (2013).{ 22 {\nThe magnetic energy change in a volume Vis derived by the dot product between the magnetic\n\feld and the magnetic induction equation (Schuck 2006):\n@\n@tB2\n2\u00160=1\n\u00160r\u0001[(v\u0002B)\u0002B]\u00001\n\u00160r\u0001\u0012J\u0002B\n\u001b\u0013\n+ (v\u0002B)\u0001J\u0000J2\n\u001b: (31)\nIntegrating Equation (31) in the volume V, we derive the time evolution of the magnetic energy\nE:\ndE\ndt=1\n\u00160Z\nS\u0014\nB\u0002(v\u0002B) +J\u0002B\n\u001b\u0015\n\u0001^ndS\u0000Z\nV\u0014\nv\u0001(J\u0002B) +J2\n\u001b\u0015\ndV: (32)\nThe normal vector ^ndirects to the inner side of the volume. The energy change is caused by both\na surface integration of energy injection \rux and a volume integration of energy release density.\nThe energy injection \rux includes both the ideal process of the Poynting \rux ( B\u0002(v\u0002B)=\u00160) and\nresistive process of the slippage of magnetic \feld lines in the plasma on the surface ( J\u0002B=(\u001b\u00160)).\nSimilarly, the energy release density includes both the ideal process of the work done by the magnetic\n\feld on the plasma ( v\u0001(J\u0002B)) and the resistive process of the Joule dissipation ( J2=\u001b). If the\nmagnetic \feld is force-free ( J\u0002B= 0) and the plasma is ideal ( \u001bapproaches in\fnity), the energy\nchange can be computed by the Poynting \rux through the photospheric surface Sp:\ndE\ndt=1\n\u00160Z\nSpB\u0002(v\u0002B)\u0001^ndS: (33)\n5. Magnetic Topology Analysis\nThe terminology \\topology\" is de\fned as the unchangeable geometrical properties that are\npreserved under smooth deformations. The topology of a magnetic \feld could be described by the\nmagnetic \feld line linkages, i.e., the \feld line mapping. The singular topology skeletons (Priest\net al. 1997), such as magnetic null points, spines, fans, and bald patches, are places where magnetic\n\feld line linkages are discontinuous. There are also places where the linkages are continuous but\nchange drastically, which are named as quasi-separatrix layers (QSLs; Priest & D\u0013 emoulin 1995;\nD\u0013 emoulin et al. 1996; Titov et al. 2002). Magnetic topology is intimately related to magnetic\nreconnection, because the magnetic topology skeletons (null points, spines, fans, bald patches, and\nQSLs) divide topologically distinct magnetic domains, and electric current sheets are prone to be\nbuilt on the interface between the domains by magnetic shear (Aulanier et al. 2006). Meanwhile,\nthe conductivity in the solar atmosphere is high and the upper chromosphere and lower corona\nare even dominated by magnetic forces. Magnetic \feld lines are then frozen into plasma and they\ngovern single particle motion, thermal conductivity, and Alfv\u0013 en wave propagation (Longcope 2005).\nTherefore, magnetic topology is responsible for the morphology of various manifestations of solar\nactivities, such as location of hard X-ray sources, \flament barb chirality, and \rare ribbon shape\nand motion.{ 23 {\n5.1. Null Point and Spine-Fan Structures\nMagnetic null point is a place where B= 0. To study the structure of a magnetic \feld supposed\nto contain null points, it is necessary to locate their positions. In some special cases, it might be\npossible to solve the equation B= 0 analytically. For general cases such as a magnetic \feld model\nbased on observations, searching for null points must be done by a numerical method. The magnetic\n\feld is discrete in numerical models, so it is usually assumed to be trilinear between neighboring\ngrids to \fll the 3D space. Pascal D\u0013 emoulin developed an FORTRAN code3to search for null points\nusing a modi\fed Powell hybrid method. A Newton{Raphson method has also been employed to\nsearch for null points (Haynes & Parnell 2007; Titov et al. 2011; Sun et al. 2012). An alternative\nmethod to locate the positions of null points is based on the Poincar\u0013 e index (Greene 1992; Zhao\net al. 2005, 2008). Haynes & Parnell (2007) provided a comparison of the Newton{Raphson method\nand the Poincar\u0013 e index method.\nFor the \frst order the magnetic structure at the vicinity of a magnetic null point can be\ndescribed by the linear term of its Taylor expansion:\nB=M\u0001r; (34)\nwhere Mis the Jacobian matrix with Mij=@Bi=@xj(i;j= 1;2;3), and ris the position vector\nof the place of interest related to the null point. The matrix Mcan be diagonalized by solving\nthe eigenfunction equation. There are three eigenvalues \u00151;\u00152, and\u00153corresponding to three\neigenvectors. Due to the solenoidal condition of the magnetic \feld, one can prove that \u00151+\u00152+\u00153=\n0. So there can be one negative eigenvalue (say, \u00153) and two positive ones ( \u00151and\u00152) or one positive\neigenvalue ( \u00153) and two negative ones ( \u00151and\u00152). The former represents a positive null point and\nthe latter a negative null point. The single eigenvector associated with \u00153determines the local\ndirection of a spine, and the other two eigenvectors associated with \u00151and\u00152determine the local\nsurface of a fan.\nThe three eigenvectors are not necessarily to be orthogonal, while they are always linearly\nindependent. If there are electric currents perpendicular to the spine, the fan would be inclined to\nthe spine (Parnell et al. 1996). In addition, Parnell et al. (1996) provided a thorough analysis of\nthe linear structure of a magnetic null point. If the magnetic \feld is potential without any electric\ncurrents, matrix Mis symmetric with three real eigenvalues. The fan is perpendicular to the spine.\nIf there are electric currents in the vicinity of the null, there are four di\u000berent types of spine-fan\nstructures depending on the electric current parallel ( jk) and perpendicular ( j?) to the spine. First,\nwhenjjkjis less than or equal to a threshold of the electric current, jthresh , andj?= 0, there are\nthree distinct or two equal real eigenvalues and the fan is perpendicular to the spine. Second, when\njjkj\u0014jthresh andjj?j>0, the fan is inclined to the spine. Third, when jjkj>jthresh andj?= 0,\nthere are one real and two conjugate complex eigenvalues, and the fan is perpendicular to the spine.\nFinally, whenjjkj>jthresh andjj?j>0, the fan is inclined to the spine.\n3http://www.lesia.obspm.fr/fromage/{ 24 {\nIf there are two null points interconnected with each other, the two fans intersect at a separator,\nwhich is a line connecting the two null points. A separator in 3D space is analogous to an X-point\nin 2D magnetic \feld. The spine of one null point could be the boundary line of a fan of another\nnull point. The end of a spine is a magnetic source or at the in\fnite distance (Priest et al. 1997).\n5.2. Bald Patches and Magnetic Dips\nBald patches are locations on the photosphere and a polarity inversion line, where the magnetic\n\feld line is tangent to them and shaped concave up (Titov et al. 1993; Bungey et al. 1996). Seen\nfrom above, the local magnetic \feld transits from negative polarity to positive one, which forms an\ninverse polarity con\fguration. The magnetic \feld Bat the bald patches satis\fes\n(B\u0001r)Bz>0; (35)\nwhereBz= 0. Similar to bald patches, magnetic dips also satisfy Equation (35). The di\u000berence is\nthat the latter are distributed in the whole space but not restricted to the photosphere.\nTitov et al. (1993) showed that the magnetic \feld strength above bald patches and magnetic\ndips increases with height for force-free magnetic \feld. In general, the Lorentz force can be expanded\nas\nJ\u0002B=1\n\u00160(B\u0001r)B\u0000r\u0012B2\n2\u00160\u0013\n: (36)\nFor a force-free \feld where J\u0002B= 0, the magnetic tension and the gradient of magnetic pressure\nbalance each other as\nB2\nR\u0000@(B2=2)\n@n= 0; (37)\nwhere B=Bsandsis a unit vector along B. The normal unit vector nis de\fned as n=R=ds=ds,\nwhereRis the radius of curvature. Therefore, for a magnetic \feld line that is concave up, one gets\n1\nB@B\n@z=n\u0001ez\nR>0; (38)\nwhich implies that the magnetic strength increases with height.\n5.3. Quasi-Separatrix Layers\nQSLs are 3D thin volumes across which the gradient of the magnetic \feld linkage is large.\nThey divide a magnetic \feld into di\u000berent magnetic domains, between which the \feld line linkage\nchanges drastically. QSLs are a generalization of true separatrix, where the magnetic \feld linkage\nis discontinuous. QSLs have a \fnite thickness, unlike a separatrix that is in\fnitely thin; or, to say,\na QSL is a 3D thin volume, while a separatrix is a 2D surface.{ 25 {\nD\u0013 emoulin et al. (1996) proposed a natural way to compute the locations of QSLs. In a magnetic\n\feld, one could integrate a magnetic \feld line from position ( x;y;z ) to both directions with a\ndistanceson each side. The two end points ( x0;y0;z0) and (x00;y00;z00) de\fne a vector ( X1;X2;X3) =\n(x00\u0000x0;y00\u0000y0;z00\u0000z0). If the magnetic \feld mapping changes fast, a small change in the position\n(x;y;z ) would cause a large change in the vector ( X1;X2;X3). A natural quanti\fcation of this\nchange is the summation of the square of each element in the Jacobian matrix for this vector,\nnamely, the norm\nN(x;y;z;s ) =vuut3X\ni=1\"\u0012@Xi\n@x\u00132\n+\u0012@Xi\n@y\u00132\n+\u0012@Xi\n@z\u00132#\n: (39)\nThe parameter sis free and could be de\fned by a geometrical boundary or a wave propagation\ndistance. The QSLs are places with N\u001d1. Equation (39) can be simpli\fed if z0=z00= 0 and the\nfootpoints ( x0;y0;z0) and (x00;y00;z00) are assumed to be tied on the photosphere:\nN\u0006\u0011N(x\u0006;y\u0006) =s\u0012@X\u0007\n@x\u0006\u00132\n+\u0012@X\u0007\n@y\u0006\u00132\n+\u0012@Y\u0007\n@x\u0006\u00132\n+\u0012@Y\u0007\n@y\u0006\u00132\n; (40)\nwhere (X\u0007;Y\u0007) = (x\u0007\u0000x\u0006;y\u0007\u0000y\u0006) (Priest & D\u0013 emoulin 1995).\nTitov et al. (2002) found that N+andN\u0000usually do not equal each other even along the same\nmagnetic \feld line. They proposed a new parameter, which is uniform along a magnetic \feld line,\nthe squashing degree Q, to measure the mapping of magnetic \feld lines\nQ=N2\n+\nj\u0001+j=N2\n\u0000\nj\u0001\u0000j; (41)\nwhere \u0001 +and \u0001\u0000are the determinants of the following two Jacobian matrices D+andD\u0000:\nD+=0\nBBB@@X\u0000\n@x+@X\u0000\n@y+\n@Y\u0000\n@x+@Y\u0000\n@y+1\nCCCA; (42)\nand\nD\u0000=0\nBBB@@X+\n@x\u0000@X+\n@y\u0000\n@Y+\n@x\u0000@Y+\n@y\u00001\nCCCA: (43)\nDenoting the normal magnetic \feld at the positive and negative polarities by Bn+andBn\u0000, re-\nspectively, Titov et al. (2002) showed that\nQ=N2\n+\njBn+=Bn\u0000j=N2\n\u0000\njBn\u0000=Bn+j: (44){ 26 {\nThe QSLs are places where Q\u001d2. Two QSLs may intersect at a place that is de\fned as a\nhyperbolic \rux tube, where the squashing degree Qis very large and magnetic reconnection is\nprone to occur.\nPariat & D\u0013 emoulin (2012) proposed a numerical scheme to compute the squashing degree Q\nin a 3D volume. Following Pariat & D\u0013 emoulin (2012), this method has been implemented by some\nother authors and applied to 3D magnetic topologies of active regions or eruptions (Savcheva et al.\n2012a,b; Guo et al. 2013b; Zhao et al. 2014, 2016; Yang et al. 2015, 2016; Liu et al. 2016; Tassev &\nSavcheva 2016).\n5.4. Applications to Observations\nMagnetic topology analysis has been applied to interpret various solar activities and phenom-\nena, such as solar \rares, coronal jets, \flament structures, and magnetic \rux rope eruption. First,\nmagnetic null points are closely related with the morphologies of solar \rare ribbons. Flare ribbons\nare found to stop at the border of the closest large-scale QSL. Magnetic null points are also in-\nvolved in magnetic structures responsible for \rares with an EUV late phase. Secondly, coronal jets\ncould occur in a magnetic null con\fguration or QSLs associated with bald patches. Thirdly, solar\n\flaments are supported by magnetic dips. The chirality of \flament barbs (right bearing or left\nbearing) can be determined by the magnetic helicity (positive or negative) and magnetic con\fgu-\nration (magnetic \rux rope or magnetic arcade). Finally, magnetic \rux rope, magnetic null points\nand QSLs may interact each other and lead to solar eruptions and magnetic reconnection.\nCircular ribbon \rares are found to be associated with magnetic null points and spine-fan\nstructures. A typical circular ribbon \rare has three ribbons, a circular, an inner, and a remote\none, which correspond to the traces of the fan separatrix, the inner spine, and the outer spine on\nthe bottom, respectively. Masson et al. (2009) analyzed the magnetic topology (Figure 4a) and\nmagnetic reconnection in such a circular ribbon \rare. Reid et al. (2012) further studied the same\n\rare and found some compact X-ray sources on the circular and inner ribbons and an extended\nsource on the spine. They proposed that a hyperbolic \rux tube embedded in the fan structure\ncould explain the presence of co-spatial X-rays with the strongest UV emission. Note that there\nare other possibilities to explain such phenomena. For example, Yang et al. (2015) found a magnetic\n\rux rope underlying a spine-fan structure. Compact X-ray sources could be explained by magnetic\nreconnection in and around the \rux rope itself. Similar structures with a magnetic \rux rope lying\nunder three null points (Figure 4b) have also been found in Mandrini et al. (2014). A special\npoint is that the three null points in Mandrini et al. (2014) are highly asymmetric, where the fan\nseparatrix is broken into two sections. In this event, the three \rare ribbons correspond to traces of\nthe two sections of the fan and the inner spine, while no \rare ribbon is found at the footpoint of\nthe remote spine.\nAlthough magnetic reconnection is prone to occur in QSLs, when we use a force-free \feld to{ 27 {\nmodel the magnetic structure, we have to be cautious about the explanation. Flare ribbons are\nassociated with the hyperbolic \rux tubes beneath an erupting magnetic \rux rope. However, the\ndynamics of \rare ribbons cannot be fully modeled by a force-free \feld model. A well observed\nfeature for typical two-ribbon \rares is that the two ribbons often separate from each other, while\nQSLs calculated from a force-free \feld model has rarely signi\fcant change before and after a \rare,\nespecially for the traces on the bottom boundary. This is because \rare ribbons are associated\nwith QSLs developed in a dynamical process, which cannot be derived from force-free \feld models.\nMoreover, QSLs derived in a model are only possible places for magnetic reconnection. There exists\nQSLs that do not participate in magnetic reconnection. It has been found that \rare ribbons tend to\nstop at the border of the closest large-scale QSLs, when the size of the QSLs is comparable to that\nof the active region hosting the eruption and the QSLs do not stride over the erupting magnetic\n\rux rope (Chen et al. 2012; Guo et al. 2012b). On the other hand, without using the QSL method,\nJiang et al. (2016b) directly reproduced the location of the \rare ribbons of a major con\fned \rare in\nactive region 12192 by tracing the footpoints of the \feld lines from the reconnection current sheet\nin their data-driven MHD simulation.\nThe EUV late phase was discovered by the EUV Variability Experiment (EVE; Woods et al.\n2011, 2012) on board SDO . The key feature is that there appears a second peak in the relatively\nwarm emission (e.g., Fe XV 28.4 nm and Fe XVI 33.5 nm) after the main GOES soft X-ray peak.\nThere should exist a second set of higher and longer coronal loops to produce the EUV late phase.\nSome typical \rares with an EUV late phase have been analyzed in Hock et al. (2012), Liu et al.\n(2013), Dai et al. (2013), Sun et al. (2013), and Li et al. (2014). Regarding the magnetic topology\nrelated to \rares with an EUV late phase, Sun et al. (2013) found that a hot spine is a viable\nmagnetic structure (Figure 4c). Li et al. (2014) found more cases where the spine is the key\nmagnetic structure for producing the EUV late phase. However, in a speci\fc case of the X2.1 \rare\non 2011 September, both a spine and some large-scale magnetic loops are found (Li et al. 2014).\nDai et al. (2013) showed that the late phase emission is produced by the large-scale magnetic loops\nbut not the spine.\nThere are two categories of 2D models for coronal jets, namely, the emerging \rux model\n(Heyvaerts et al. 1977; Shibata et al. 1992) and the converging \rux model (Priest et al. 1994). In\n3D, both the magnetic null point model (Pariat et al. 2009, 2010; Zhang et al. 2012) and bald patch\nmodel (Guo et al. 2013a; Schmieder et al. 2013) have been proposed. Many coronal jets have a\nperiodicity of about tens of minutes to several hours. Pariat et al. (2010) proposed that photospheric\ntwisting motion drives the null point and spine-fan structure to reconnect periodically. Zhang\net al. (2012) proposed that the modulation of trapped slow-mode waves along the spine \feld lines\n(Figure 4d) might cause such a periodicity. Guo et al. (2013a) found that magnetic reconnection\nin QSLs associated with bald patches could cause the eruption of coronal jets. Schmieder et al.\n(2013) discovered that coronal jets might appear in a bundle of twisted \feld lines, among which at\nleast one of the footpoints is not connected to the bottom. Similar to Guo et al. (2013a), magnetic\nreconnection also occurs in the QSLs associated with bald patches (Figure 5a). Coronal jets could{ 28 {\ngain twists from the \feld lines during the expulsion of the jet plasma.\nKippenhahn & Schl uter (1957) and Kuperus & Raadu (1974) proposed two magnetic \feld\ncon\fgurations, one with a normal polarity (where the horizontal magnetic \feld has a component\npointing from the positive to negative polarity) and the other with an inverse polarity (opposite\nto the normal one), respectively, to explain the magnetic structure for \flaments. Both magnetic\ncon\fgurations contain magnetic dips. Aulanier & D\u0013 emoulin (1998) proposed a series of linear force-\nfree models to explain the presence of \flament barbs, where magnetic dips are assumed to support\n\flament material. Aulanier et al. (1998) found that the shape of a \flament is determined by the\ndistribution of magnetic dips (Figure 5b) in a linear magnetic \feld. Guo et al. (2010c) also found\nthat magnetic dips resemble well the shape of a \flament observed in H \u000busing a nonlinear force-\nfree \feld model (Figure 5c). Additionally, both a magnetic \rux rope and sheared magnetic arcades\nare found in the same \flament. The chirality of \flament barbs are determined by the magnetic\nhelicity and magnetic con\fguration simultaneously. In a magnetic \feld with negative magnetic\nhelicity, a magnetic \rux rope (with inverse polarity) induces right bearing \flament barbs, and\nsheared arcades (with normal polarity) induce left bearing \flament barbs. Similarly, in a magnetic\n\feld with positive magnetic helicity, a magnetic \rux rope induces left bearing \flament barbs, and\nsheared arcades induce right bearing ones. This result has also been discussed in Chen et al. (2014).\nFigure 5d shows a linear force-free \feld model for explaining polar crown prominences with bubbles\nand plumes (Dud\u0013 \u0010k et al. 2012), which are interpreted as a separator magnetic reconnection rather\nthan the Rayleigh{Taylor instability (Hillier et al. 2011, 2012a,b).\nSavcheva et al. (2012b) studied the QSLs of nonlinear force-free \feld models for a long-lasting\ncoronal sigmoid, which is a signature of a magnetic \rux rope. The sigmoid keeps stable for several\ndays although there are bald patches in the constructed magnetic \feld. It erupts as long as the\nmagnetic \feld exhibits hyperbolic \rux tubes. Savcheva et al. (2012a) further compared the QSLs\ncomputed from both the nonlinear force-free \feld model (Figure 6a) and the MHD simulation,\nand suggested that magnetic reconnection at the hyperbolic \rux tube under the \rux rope and\ntorus instability jointly cause the observed CME. Guo et al. (2013b) studied the QSLs, twist\naccumulation, and magnetic helicity injection of a magnetic \rux rope before a major \rare and\nCME. It is found that the \rux rope is surrounded by QSLs (Figure 6b). The twist and magnetic\nhelicity are injected into the \rux rope by continuous magnetic reconnection in the QSLs. The\nmagnetic topology structure, evolution, and stability of a double-decker \rux rope (Figure 6c) have\nalso been studied by Liu et al. (2016). Yang et al. (2015) presented a 3D QSL with both a spine-fan\nseparatrix and a large-scale quadrupolar structure (Figure 6d). The 3D QSL resembles very well\nthe EUV emission in 94 \u0017A observed by SDO /AIA. A magnetic \rux rope is found below the fan.\nThe interplay between MHD instability and magnetic reconnection in the QSL structures could\nexplain the eruption process revealed in multi-wavelength observations. More applications of QSLs\nto the interpretation of observations can be found in Zhao et al. (2014) and Janvier et al. (2016).{ 29 {\n6. Magnetic Helicity Computation\nMagnetic helicity, the volume integration of the product of the vector potential and the mag-\nnetic \feld, is not only conserved in ideal MHD process but also approximately conserved in resistive\nprocess with magnetic reconnection (Taylor 1974; Berger 1984). The conservation of magnetic helic-\nity constrains the \fnal state in the relaxation process of plasma con\fnement (Woltjer 1958; Taylor\n1974). Magnetic helicity is preferentially negative in the northern solar hemisphere and positive in\nthe southern one (Rust & Kumar 1994; Pevtsov et al. 1994, 1995, 2003; Zirker et al. 1997; Zhang &\nBao 1998, 1999; Bao & Zhang 1998; Ouyang et al. 2017) and it seems unchanged with solar cycle,\nalthough this point is under debate (Hagino & Sakurai 2002, 2004; Zhang et al. 2010). Magnetic\nhelicity also plays a critical role in the eruption mechanisms of CMEs (Zhang & Low 2003; Zhang\net al. 2006b), the formation of \flament channels (Antiochos 2013; Knizhnik et al. 2015, 2017), and\nthe physical process in magnetic \feld dynamos (Brandenburg & Subramanian 2005; Zhang et al.\n2006a).\nMagnetic helicity measures the topological complexity of a bundle of magnetic \feld lines. In\na volumeV, magnetic helicity Hmis expressed as\nHm=Z\nVA\u0001BdV; (45)\nwhere Adenotes the vector potential and B=r\u0002A. Note that Equation (45) is gauge invariant\nonly when the magnetic \feld is closed in a volume with no normal component on the boundary.\nIn realistic cases, such as a magnetic \feld rooted on the photosphere, the \feld is open with \ruxes\npassing through the boundaries. Berger & Field (1984) and Finn & Antonsen (1985) proposed a\nrelative magnetic helicity, H, which is gauge invariant for both closed and open con\fgurations:\nH=Z\nV(A+Ap)\u0001(B\u0000Bp) dV; (46)\nwhere Bpis the reference \feld with r\u0002Ap=Bp. The reference \feld is usually selected as the\npotential \feld and it has the same normal component as the magnetic \feld Bon the boundary S\nso that\n(r\u0002Ap)\u0001^n=B\u0001^n; (47)\nand\n(r\u0002A)\u0001^n=B\u0001^n: (48)\nThe unit vector ^nis normal to Sand directs to the inner side of the volume.\nAlthough the relative helicity Equation (46) is widely used in computations of magnetic helic-\nity in open con\fgurations, there are other expressions and interpretations of the magnetic helicity.\nLow (2011) pointed out that the relative magnetic helicities might be not conserved due to the\nevolution of the reference \feld itself. Low (2006) and Low (2011) proposed to compute the \\La-\ngrangian helicity\" and \\absolute helicity\" using a two \rux representation or the representation of{ 30 {\nChandrasekhar & Kendall (1957). Prior & Yeates (2014) argued that although the relative helicity\nEquation (46) is gauge invariant, it is dependent on the choice of the reference \feld. Prior & Yeates\n(2014) proposed to use the classical magnetic helicity Equation (45) but \fx the gauge as the wind-\ning gauge (Equation (18) in Prior & Yeates 2014). Other gauges can be found in Jensen & Chu\n(1984) and Hornig (2006). The value and interpretation of the magnetic helicity are dependent\non the choice of the expressions (classical, relative, Lagrangian, or absolute magnetic helicity) and\ngauges (Coulomb gauge, DeVore gauge, winding gauge, and so on). It still needs further studies to\nclarify these issues.\nIn the following, we introduce three practical methods to compute the magnetic helicity, which\ninclude the \fnite volume method (Rudenko & Myshyakov 2011; Thalmann et al. 2011; Valori et al.\n2012; Yang et al. 2013; Rudenko & An\fnogentov 2014; Moraitis et al. 2014), the discrete \rux tube\nmethod (Guo et al. 2010a, 2013b; Georgoulis et al. 2012), and the helicity \rux integration method\n(Chae 2001; Pariat et al. 2005; Liu & Schuck 2012). A series of papers have been devoted to\nbenchmark, compare, and apply these methods (Valori et al. 2016; Pariat et al. 2017b; Guo et al.\n2017). Besides the above methods, Russell et al. (2015) proposed a \feld-line helicity method, where\nthe helicity density is assigned to each individual \feld line. The helicity density in each \feld line\nis the integration of the vector potential along it. Longcope & Malanushenko (2008) proposed two\ngeneralizations of the relative magnetic helicity, one is the uncon\fned self-helicity and the other is\nthe additive self-helicity, using di\u000berent reference \felds and considering sub-volumes in the corona.\nFurther details can be found in the respective references.\n6.1. Finite Volume Method\nThe \fnite volume method integrates Equation (46) numerically with di\u000berent gauges and\nboundary conditions. It requires an input of the full 3D magnetic \feld information, which is\nprovided by theoretical force-free \feld models, numerical nonlinear force-free \feld models, and\nMHD models as discussed in Section 3. The output is the relative magnetic helicity at a certain\nmoment. Valori et al. (2016) provided a detailed description of six implementations of the \fnite\nvolume method, and made a thorough benchmark and comparison of the six algorithms regarding\ntheir accuracy, mutual consistency, and sensitivity.\nThe six algorithms for the \fnite volume method are divided into two categories according to\nthe employed gauges, one being the Coulomb gauge ( r\u0001A= 0) and the other being the DeVore\ngauge (Az= 0, DeVore 2000). In the Coulomb gauge, the key problem is to solve the Laplace\nproblem for the vector potential Ap:\nr2Ap= 0; (49)\nand the Poisson problem for A:\nr2A=\u0000J; (50)\nwith the boundary conditions of Equations (47) and (48) and the gauges r\u0001A= 0 andr\u0001Ap= 0.{ 31 {\nThe \fnite volume method in the Coulomb gauge has been implemented by Rudenko & Myshyakov\n(2011), Rudenko & An\fnogentov (2014), Thalmann et al. (2011), and Yang et al. (2013).\nValori et al. (2012) derived the formula for the vector potential Ausing the DeVore gauge in\na volume bounded by [ x1;x2], [y1;y2], and [z1;z2]:\nA=b+^z\u0002z2Z\nzBdz0; (51)\nwith\nbx=\u00001\n2yZ\ny1Bz(x;y0;z2)dy0;\nby=1\n2xZ\nx1Bz(x0;y;z 2)dx0;\nbz= 0:(52)\nThe vector potential Apis derived similarly by integrating Bp, which is solved by a scalar Laplace\nequation\nr2\t = 0; (53)\nwhere Bp=r\t.Bphas the same normal component on the boundary as B. The \fnite vol-\nume method in the DeVore gauge has been further implemented by Moraitis et al. (2014) and S.\nAn\fnogentov (Valori et al. 2016) following Valori et al. (2012) with only minor di\u000berences.\n6.2. Discrete Flux Tube Method\nIn the Coulomb gauge, it is found that Equation (45) has a similar form to the Gauss linking\nnumber (Mo\u000batt 1969; Berger & Field 1984; Berger & Prior 2006):\nLk=1\n4\u0019I\nxI\ny^Tx(s)\u0002^Ty(s0)\u0001r\njrj3ds0ds; (54)\nwhere x(s) and y(s0) are the positions of two curves parameterized by sands0,^Tx(s) and ^Ty(s0)\nare the unit tangent vectors to x(s) and y(s0), and r=x(s)\u0000y(s0). Note that Equation (54)\nis valid for closed curves, and Lkis an integer and invariant to continuous deformations without\ntearing or gluing. The Gauss linking number can be expressed as the sum of the writhe Wrand\ntwistTw, namely,\nLk=Wr+Tw; (55){ 32 {\nwhich is known as the C\u0014 alug\u0014 areanu theorem (e.g., Fuller 1978; Mo\u000batt & Ricca 1992; Berger &\nPrior 2006). The writhe measures the non-planarity of an axis curve alone:\nWr=1\n4\u0019I\nxI\nx^T(s)\u0002^T(s0)\u0001r\njrj3ds0ds; (56)\nwithr=x(s)\u0000x(s0). The twist measures the rotation of a secondary curve about the axis:\nTw=1\n2\u0019I\nx^T(s)\u0001^V(s)\u0002d^V(s)\ndsds; (57)\nwhere ^V(s) is a unit vector normal to ^T(s) and pointing from the axis to the secondary curve.\nFuller (1978) pointed out that each of the three quantities, Lk,Wr, andTw, has a unique property\nthat is not possessed by the other two. Speci\fcally, Lkis topologically invariant, Wrdepends only\non the axis, andTwis additive and quanti\fed by a local density.\nWith the above analysis, Berger & Field (1984) assigned the magnetic helicity with a geomet-\nrical (or, topological) meaning under the Coulomb gauge. If a magnetic \feld is divided into a \fnite\nnumber (N) of \rux tubes, the magnetic helicity arises both from the internal structure, namely\ntwist and writhe, of each magnetic \rux tube and the linkage and knotting of di\u000berent \rux tubes.\nThe former is known as the self helicity, and the latter mutual helicity. Quantitatively, Berger &\nField (1984) found that for a closed magnetic con\fguration (also refer to D\u0013 emoulin et al. 2006):\nHm\u0019NX\ni=1Li\b2\ni+NX\ni=1NX\nj=1;j6=iLi;j\bi\bj; (58)\nwhereLiincludes the contributions from both the writhe and twist as Li=Wi+Ti,Li;jis the\nlinking number between \rux tubes iandj, and \biand \bjdenotes the magnetic \ruxes of tubes i\nandj, respectively. Note that the self helicity arises because the number Nof \rux tubes is \fnite\nso that each tube contains a certain \rux. If the number Napproaches in\fnity, the self helicity\nvanishes and only the mutual helicity exists (Berger & Field 1984; D\u0013 emoulin et al. 2006).\nThe above analyses are valid for closed magnetic con\fgurations, where no magnetic \rux pene-\ntrates the boundary. For open con\fgurations, Equations (54) and (56) are not applicable any more.\nHowever, Equation (57) is still applicable with the integration only along an open curve. Further-\nmore, Equation (58) still holds true, but the interpretation has to be changed. In open magnetic\ncon\fgurations, the relative helicity is adopted and the linking number is rede\fned in D\u0013 emoulin\net al. (2006) and the writhe in Berger & Prior (2006). Especially, D\u0013 emoulin et al. (2006) used the\nconcept of helicity injection to de\fne the linking number of open curves, and they proposed an\ninternal angle method to quantify this number.\nIn practice, Equation (58) has been adopted for the computation of magnetic helicity with\ndi\u000berent implementations and assumptions. The twist number method (Guo et al. 2010a, 2013b,\n2017; Xia et al. 2014b; Valori et al. 2016; Yang et al. 2016) only computes the magnetic helicity{ 33 {\ncontributed by the twist. It is applicable for cases where only one major electric current channel\nexists and the writhe is negligible. The twist number method uses QSLs as introduced in Section 5.3\nto determine the boundary of a magnetic \rux rope. QSLs have a property of being a magnetic \rux\nsurface. Therefore, it is meaningful to compute the magnetic helicity for only a magnetic \rux rope.\nThe connectivity-based method (Georgoulis & LaBonte 2007; Georgoulis et al. 2012; Tziotziou\net al. 2012, 2013, 2014; Moraitis et al. 2014) computes the linking number using the internal angle\nmethod. This method can use either a magnetic connectivity provided by force-free, MHS, or MHD\nmodels, or a connectivity inferred by a minimal connection length.\n6.3. Helicity Flux Integration Method\nBoth the \fnite volume method and the discrete \rux tube method require a 3D magnetic \feld\nfor the computation of the magnetic helicity of a volume. However, the accumulated magnetic\nhelicity in a volume can also be estimated from the integration of the helicity \rux through a\nboundary surface, say, the bottom surface that usually has the largest helicity \rux. Berger & Field\n(1984) derived the time variation of H(also refer to Pariat et al. 2015):\ndH\ndt=\u00002Z\nS(Ap\u0002E) dS\u00002Z\nVE\u0001BdV+ 2Z\nS@\t\n@tAp\u0001dS; (59)\nwithBp=r\t. The boundary condition and gauge for the vector potential Aphas a certain\nfreedom. To simplify the expression, a particular boundary condition and gauge are selected,\nnamely, the vector potential Aphas no normal component on the boundary SandApis solenoidal\nin the volume V:\nAp\u0001^n= 0;\nr\u0001Ap= 0:(60)\nNote that ^nis de\fned as directing to the inner side of the volume. In some studies (e.g., Pevtsov\net al. 2014; Pariat et al. 2015), it can also be de\fned as directing to the outer side of the volume,\nwhere the surface integration in Equation (59) should change its sign. Substituting Equation (60)\ninto Equation (59) and using the Ohm's law\nE=\u0000v\u0002B+J\n\u001b; (61)\nEquation (59) is reduced to\ndH\ndt= 2Z\nS[(Ap\u0001B)v\u0000(Ap\u0001v)B]\u0001^ndS\u00002Z\nS1\n\u001b(Ap\u0002J)\u0001^ndS\u00002Z\nV1\n\u001bJ\u0001BdV: (62)\nEquation (62) shows that the helicity in a volume changes owing to helicity \rux across the boundary\nand dissipation in the volume.{ 34 {\nIn the solar atmosphere, the conductivity can be regarded as in\fnitely large. So, the last two\nterms in Equation (62) are omitted. Then we have\ndH\ndt=\u00002Z\nSp(Ap\u0001u)BndS; (63)\nwhere only the \rux across the photosphere Spis considered and\nu=vt\u0000vn\nBnBt: (64)\nD\u0013 emoulin & Berger (2003) argued that horizontal velocities derived from optical \row techniques\nas discussed in Section 2.4 represent uin Equation (64) rather than vt. However, this viewpoint\nhas been questioned by, for example Welsch et al. (2007) and Schuck (2008). The integrand in\nEquation (63) is de\fned as the helicity \rux density,\nGA(x) =\u00002(Ap\u0001u)Bn: (65)\nGA(x) has been adopted to compute the magnetic helicity \rux distribution in active regions (Chae\n2001; Kusano et al. 2002; Moon et al. 2002; Nindos & Zhang 2002).\nPariat et al. (2005) proposed an alternative expression for the magnetic helicity \rux\ndH\ndt=\u00001\n2\u0019Z\nSpZ\nS0pd\u0012(r)\ndtBnB0\nndS0dS; (66)\nwith\nd\u0012(r)\ndt=1\nr2\u0012\nr\u0002dr\ndt\u0013\nn=1\nr2[r\u0002(u\u0000u0)]n; (67)\nwhere r=x\u0000x0is the position vector. Equation (66) shows that the magnetic helicity is injected by\nthe rotation of each pair of elementary \rux tubes weighted by their magnetic \ruxes. Consequently,\nthe helicity \rux density G\u0012(x) is de\fned as\nG\u0012(x) =\u0000Bn\n2\u0019Z\nS0pd\u0012(r)\ndtB0\nndS0: (68)\nNeitherGA(x) norG\u0012(x) itself is meaningful as a measurement of the helicity \rux density. Pariat\net al. (2005) proposed to use a proxy G\b(x), which meaningfully measures the connectivity-based\nhelicity \rux density per elementary magnetic \rux tube. The method has been implemented by\nDalmasse et al. (2014) and applied to observations by Dalmasse et al. (2013).\n6.4. Applications to Models and Observations\nValori et al. (2012) implemented a \fnite volume method using the DeVore gauge and applied\nit to the force-free Titov-D\u0013 emoulin model. They found that this method only needs a small volume{ 35 {\nto derive the full helicity content. Yang et al. (2013) applied another \fnite volume method using\nthe Coulomb gauge to a data-driven MHD simulation, and found that the accumulated magnetic\nhelicity in the volume coincides with the helicity \rux injected through the boundaries. Pariat et al.\n(2015) also made a thorough test on the magnetic helicity conservation by computing the relative\nmagnetic helicity in a \fnite volume and the integration of the helicity \rux through the boundaries.\nIt is found that the dissipation of the magnetic helicity is almost zero in a quasi-ideal MHD process,\nand the dissipation in a resistive process is very low ( <2:2%). By contrast, the dissipation rate of\nmagnetic energy is much higher, or tens of times the dissipation rate of magnetic helicity. Pariat\net al. (2017a) proposed that the relative magnetic helicity can be used as a diagnostic tool for solar\neruptivity. They found that the ratio of the magnetic helicity carried purely by the electric currents\nto the total relative helicity is very high for eruptive simulations. Thus, this parameter can be used\nto distinguish between eruptive and con\fned eruptions.\nGuo et al. (2013b) adopted the twist number method and helicity \rux integration method to\nstudy the twist accumulation and magnetic helicity injection, respectively. It is found that only\na small fraction of the injected helicity is transferred to the internal helicity of a magnetic \rux\nrope. Valori et al. (2016) and Guo et al. (2017) found that the magnetic helicity contributed by\nthe twist matches well the magnetic helicity contributed only by the electric current, which is\ngauge invariant representing part of the relative magnetic helicity. With the connectivity-based\nmethod, Tziotziou et al. (2012) calculated the free magnetic energy and relative magnetic helicity\nfor 162 vector magnetic \felds in 42 active regions, and found a clear linear correlation between the\nmagnetic energy and helicity. Eruptive active regions possess both large free energy and relative\nhelicity exceeding 4 \u00021031erg and 2\u00021042Mx2, respectively. Tziotziou et al. (2013) applied the\nconnectivity-based method to a time series of 600 vector magnetic \felds in active region 11158\nand found that both the free magnetic energy and relative magnetic helicity are accumulated to\nsu\u000ecient amounts to power a series of solar eruptions.\nDalmasse et al. (2013) \frst applied a connectivity-based helicity \rux density to vector magnetic\n\felds observed by SDO /HMI in active region 11158, and con\frmed that the helicity \rux density\nis mixed with both positive and negative signs. Liu et al. (2014) used the helicity \rux integration\nmethod to study the helicity injection in emerging active regions. They found that about 61% of\nthe 28 emerging active regions follow the hemispheric rule, which states that the magnetic helicity is\nnegative in the northern hemisphere and positive in the southern one. The helicity \rux integration\nmethod has been widely used in various studies on the magnetic helicity injection through the\nboundaries (e.g., Tian & Alexander 2008; Yamamoto & Sakurai 2009; Yang et al. 2009; Chandra\net al. 2010; Park et al. 2010; Zuccarello et al. 2011; Jing et al. 2012; Vemareddy et al. 2012; Romano\net al. 2014; Lim et al. 2016).{ 36 {\n7. Summary and Discussion\nIn this review, we introduce the recent progresses in observations, theories, and numerical\nmethods related to magnetic \feld structures and dynamics in the solar atmosphere. The magnetic\n\feld on the photosphere is observed with the polarized light emergent from there, which is rep-\nresented by the Stokes parameters. The magnetic \feld is derived by the inversion of the Stokes\nspectral pro\fles under certain assumptions for the physical environment of the photosphere. The\ntransverse components of a vector magnetic \feld have an intrinsic 180\u000eambiguity, which has to be\nremoved under additional physical assumptions. Correction of the projection e\u000bect is also necessary\nif the \feld of view is large or close to the solar limb. With a time series of vector magnetic \felds,\nthe velocity can be derived by the optical \row techniques.\nThere are di\u000berent ways to obtain the 3D magnetic \feld and its evolution. We introduce\nsome theoretical force-free \feld models, numerical nonlinear force-free \feld models, MHS models,\nand MHD models that are widely used. We also introduce some methods to preprocess the vector\nmagnetic \feld observed on the photosphere as a suitable boundary condition for nonlinear force-\nfree \feld extrapolation. Magnetic energy computation in a volume and from the boundaries is\nbrie\ry discussed. To quantify the structure and stability of a magnetic \feld, the magnetic topology\nanalysis and magnetic helicity computation are essential. The methods to pinpoint null points,\nbald patches, and QSLs are introduced. We also mention some applications of these concepts\nto the interpretation of observations. Some practical methods to compute the magnetic helicity\nare presented. They include the \fnite volume method, discrete \rux tube method, helicity \rux\nintegration method, and other methods. Applications of magnetic helicity to interpreting solar\neruptive activities are also mentioned.\nIt should be pointed out that most of the methods, including those described above and\nsome others not mentioned here, have limitations. Thus, one needs to be cautious when using\nthem for interpreting observations. We should also note that there is still a large gap between\nobservations and models. Therefore, it is still a challenging task to understand the solar magnetic\nstructure and dynamics in a consistent way based on both observations and models. For example,\nobservations of the full Stokes parameters contain much more information than that derived by\nthe Unno{Rachkovsky solution and Milne{Eddington atmosphere model. Without more advanced\ntheoretical models and numerical methods (see, Lagg et al. 2015; de la Cruz Rodr\u0013 \u0010guez & van\nNoort 2016), we cannot extract such information precisely. The MHD simulations of solar eruptions\nshould be improved with more physics included and more realistic boundary and initial conditions\nadopted. Speci\fcally, we should include all necessary physics, such as resistive process, thermal\nconduction, and radiation, in MHD simulations. The computation domain should be broad enough\ncontaining very di\u000berent physical environments, such as the convection zone, solar atmosphere, and\nthe interplanetary space. Meanwhile, more physical parameters, including magnetic \feld, velocity\n\feld, density, and temperature inferred from multi-wavelength observations, should be assimilated\ninto MHD simulations. This is a long way to go. In a word, just like our knowledge on the\nearthquake, we expect a continuous reduction of the gap between observations and models of solar{ 37 {\neruptions though it may never vanish.\nThe authors thank Prof. X.S. Feng and Prof. W.X. Wan for the invitation to write the review\npaper, and thank the three anonymous referees for constructive comments that improve the paper.\nYG, XC, and DMD are supported by NSFC (11533005, 11203014, 11373023, and 11303016) and\nNKBRSF 2014CB744203.\nREFERENCES\nAi, G.-X. 1987, Publications of the Beijing Astronomical Observatory, 9, 27\nAltschuler, M. D., Levine, R. H., Stix, M., & Harvey, J. 1977, Sol. Phys., 51, 345\nAltschuler, M. D. & Newkirk, G. 1969, Sol. Phys., 9, 131\nAly, J. J. 1989, Sol. Phys., 120, 19\nAmari, T., Aly, J. J., Luciani, J. F., Boulmezaoud, T. Z., & Mikic, Z. 1997, Sol. Phys., 174, 129\nAmari, T., Boulmezaoud, T. Z., & Aly, J. J. 2006, A&A, 446, 691\nAmari, T., Canou, A., & Aly, J.-J. 2014, Nature, 514, 465\nAmari, T., Luciani, J. F., Aly, J. J., & Tagger, M. 1996, ApJ, 466, L39\nAmari, T., Luciani, J. F., Mikic, Z., & Linker, J. 2000, ApJ, 529, L49\nAntiochos, S. K. 1998, ApJ, 502, L181\n|. 2013, ApJ, 772, 72\nAntiochos, S. K., DeVore, C. R., & Klimchuk, J. A. 1999, ApJ, 510, 485\nArchontis, V., Hood, A. W., Savcheva, A., Golub, L., & Deluca, E. 2009, ApJ, 691, 1276\nArchontis, V., Moreno-Insertis, F., Galsgaard, K., Hood, A., & O'Shea, E. 2004, A&A, 426, 1047\nArchontis, V. & T or ok, T. 2008, A&A, 492, L35\nAsensio Ramos, A., Trujillo Bueno, J., & Landi Degl'Innocenti, E. 2008, ApJ, 683, 542\nAulanier, G. & D\u0013 emoulin, P. 1998, A&A, 329, 1125\nAulanier, G., D\u0013 emoulin, P., Schrijver, C. J., Janvier, M., Pariat, E., & Schmieder, B. 2013, A&A,\n549, A66\nAulanier, G., D\u0013 emoulin, P., van Driel-Gesztelyi, L., Mein, P., & Deforest, C. 1998, A&A, 335, 309{ 38 {\nAulanier, G., Janvier, M., & Schmieder, B. 2012, A&A, 543, A110\nAulanier, G., Pariat, E., D\u0013 emoulin, P., & DeVore, C. R. 2006, Sol. Phys., 238, 347\nAulanier, G., T or ok, T., D\u0013 emoulin, P., & DeLuca, E. E. 2010, ApJ, 708, 314\nBao, S. & Zhang, H. 1998, ApJ, 496, L43\nBateman, G. 1978, MHD Instabilities (Cambridge, MA: MIT Press)\nBerger, M. A. 1984, Geophysical and Astrophysical Fluid Dynamics, 30, 79\nBerger, M. A. & Field, G. B. 1984, Journal of Fluid Mechanics, 147, 133\nBerger, M. A. & Prior, C. 2006, Journal of Physics A Mathematical General, 39, 8321\nBerger, T. E., Shine, R. A., Slater, G. L., Tarbell, T. D., Title, A. M., Okamoto, T. J., Ichimoto,\nK., Katsukawa, Y., Suematsu, Y., Tsuneta, S., Lites, B. W., & Shimizu, T. 2008, ApJ, 676,\nL89\nBerger, T. E., Slater, G., Hurlburt, N., Shine, R., Tarbell, T., Title, A., Lites, B. W., Okamoto,\nT. J., Ichimoto, K., Katsukawa, Y., Magara, T., Suematsu, Y., & Shimizu, T. 2010, ApJ,\n716, 1288\nBogdan, T. J. & Low, B. C. 1986, ApJ, 306, 271\nBommier, V. 2013, Physics Research International, Volume 2013 (2013), Article ID 195403, 16\npages, 2013\nBommier, V., Landi Degl'Innocenti, E., Landol\f, M., & Molodij, G. 2007, A&A, 464, 323\nBorrero, J. M., Tomczyk, S., Kubo, M., Socas-Navarro, H., Schou, J., Couvidat, S., & Bogart, R.\n2011, Sol. Phys., 273, 267\nBrandenburg, A. & Subramanian, K. 2005, Phys. Rep., 417, 1\nBungey, T. N., Titov, V. S., & Priest, E. R. 1996, A&A, 308, 233\nCalabretta, M. R. & Greisen, E. W. 2002, A&A, 395, 1077\nCan\feld, R. C., de La Beaujardiere, J.-F., Fan, Y., Leka, K. D., McClymont, A. N., Metcalf, T. R.,\nMickey, D. L., Wuelser, J.-P., & Lites, B. W. 1993, ApJ, 411, 362\nCanou, A. & Amari, T. 2010, ApJ, 715, 1566\nCanou, A., Amari, T., Bommier, V., Schmieder, B., Aulanier, G., & Li, H. 2009, ApJ, 693, L27\nCao, W., Gorceix, N., Coulter, R., Ahn, K., Rimmele, T. R., & Goode, P. R. 2010, Astronomische\nNachrichten, 331, 636{ 39 {\nCenteno, R., Schou, J., Hayashi, K., Norton, A., Hoeksema, J. T., Liu, Y., Leka, K. D., & Barnes,\nG. 2014, Sol. Phys., 289, 3531\nChae, J. 2001, ApJ, 560, L95\nChandra, R., Pariat, E., Schmieder, B., Mandrini, C. H., & Uddin, W. 2010, Sol. Phys., 261, 127\nChandrasekhar, S. & Kendall, P. C. 1957, ApJ, 126, 457\nChen, F., Rempel, M., & Fan, Y. 2017, ArXiv e-prints\nChen, P. F., Harra, L. K., & Fang, C. 2014, ApJ, 784, 50\nChen, P. F., Su, J. T., Guo, Y., & Deng, Y. Y. 2012, Chinese Science Bulletin, 57, 1393\nCheng, X. & Ding, M. D. 2016, ApJS, 225, 16\nCheng, X., Ding, M. D., Guo, Y., Zhang, J., Jing, J., & Wiegelmann, T. 2010, ApJ, 716, L68\nCheng, X., Guo, Y., & Ding, M. D. 2017, SCIENCE CHINA Eearth Sciences, submitted\nCheung, M. C. M., Sch ussler, M., Tarbell, T. D., & Title, A. M. 2008, ApJ, 687, 1373\nChiu, Y. T. & Hilton, H. H. 1977, ApJ, 212, 873\nChodura, R. & Schlueter, A. 1981, Journal of Computational Physics , 41, 68\nCourant, R. & Hilbert, D. 1962, Methods of Mathematical Physics, Vol. 2 (New York: Interscience\nPublishers)\nCrouch, A. D., Barnes, G., & Leka, K. D. 2009, Sol. Phys., 260, 271\nCuperman, S., Li, J., & Semel, M. 1993, A&A, 268, 749\nCuperman, S., Ofman, L., & Semel, M. 1990, A&A, 230, 193\nDai, Y., Ding, M. D., & Guo, Y. 2013, ApJ, 773, L21\nDalmasse, K., Pariat, E., D\u0013 emoulin, P., & Aulanier, G. 2014, Sol. Phys., 289, 107\nDalmasse, K., Pariat, E., Valori, G., D\u0013 emoulin, P., & Green, L. M. 2013, A&A, 555, L6\nde la Cruz Rodr\u0013 \u0010guez, J., Rouppe van der Voort, L., Socas-Navarro, H., & van Noort, M. 2013,\nA&A, 556, A115\nde la Cruz Rodr\u0013 \u0010guez, J. & van Noort, M. 2016, Space Sci. Rev.\nD\u0013 emoulin, P. & Aulanier, G. 2010, ApJ, 718, 1388\nD\u0013 emoulin, P. & Berger, M. A. 2003, Sol. Phys., 215, 203{ 40 {\nD\u0013 emoulin, P., Cuperman, S., & Semel, M. 1992, A&A, 263, 351\nD\u0013 emoulin, P., H\u0013 enoux, J. C., & Mandrini, C. H. 1992, Sol. Phys., 139, 105\nD\u0013 emoulin, P., H\u0013 enoux, J. C., Priest, E. R., & Mandrini, C. H. 1996, A&A, 308, 643\nD\u0013 emoulin, P., Pariat, E., & Berger, M. A. 2006, Sol. Phys., 233, 3\nD\u0013 emoulin, P. & Priest, E. R. 1988, A&A, 206, 336\nD\u0013 emoulin, P. & Priest, E. R. 1992, A&A, 258, 535\nDeVore, C. R. 2000, ApJ, 539, 944\nDud\u0013 \u0010k, J., Aulanier, G., Schmieder, B., Zapi\u0013 or, M., & Heinzel, P. 2012, ApJ, 761, 9\nFan, Y. 2001, ApJ, 554, L111\n|. 2009a, Living Reviews in Solar Physics, 6, 4\n|. 2009b, ApJ, 697, 1529\nFeng, X., Ma, X., & Xiang, C. 2015, Journal of Geophysical Research (Space Physics), 120, 10\nFinn, J. H. & Antonsen, T. M. J. 1985, Comments on Plasma Physics and Controlled Fusion, 9,\n111\nForbes, T. G. & Isenberg, P. A. 1991, ApJ, 373, 294\nFuhrmann, M., Seehafer, N., & Valori, G. 2007, A&A, 476, 349\nFuhrmann, M., Seehafer, N., Valori, G., & Wiegelmann, T. 2011, A&A, 526, A70\nFuller, F. B. 1978, Proceedings of the National Academy of Science, 75, 3557\nGalsgaard, K., Archontis, V., Moreno-Insertis, F., & Hood, A. W. 2007, ApJ, 666, 516\nGary, G. A. 2001, Sol. Phys., 203, 71\nGary, G. A. & D\u0013 emoulin, P. 1995, ApJ, 445, 982\nGary, G. A. & Hagyard, M. J. 1990, Sol. Phys., 126, 21\nGeorgoulis, M. K. 2005, ApJ, 629, L69\nGeorgoulis, M. K. & LaBonte, B. J. 2006, ApJ, 636, 475\n|. 2007, ApJ, 671, 1034\nGeorgoulis, M. K., LaBonte, B. J., & Metcalf, T. R. 2004, ApJ, 602, 446{ 41 {\nGeorgoulis, M. K., Tziotziou, K., & Raoua\f, N.-E. 2012, ApJ, 759, 1\nGilchrist, S. A. & Wheatland, M. S. 2013, Sol. Phys., 282, 283\n|. 2014, Sol. Phys., 289, 1153\nGold, T. & Hoyle, F. 1960, MNRAS, 120, 89\nGorbachev, V. S. & Somov, B. V. 1988, Sol. Phys., 117, 77\nGrad, H. & Rubin, H. 1958, in the 2nd International Conference on Peaceful Uses of Atomic Energy,\nInternational Atomic Energy Agency, Geneva\nGreene, J. M. 1992, Journal of Computational Physics, 98, 194\nGuo, Y., D\u0013 emoulin, P., Schmieder, B., Ding, M. D., Vargas Dom\u0013 \u0010nguez, S., & Liu, Y. 2013a, A&A,\n555, A19\nGuo, Y., Ding, M. D., Cheng, X., Zhao, J., & Pariat, E. 2013b, ApJ, 779, 157\nGuo, Y., Ding, M. D., Liu, Y., Sun, X. D., DeRosa, M. L., & Wiegelmann, T. 2012a, ApJ, 760, 47\nGuo, Y., Ding, M. D., Schmieder, B., D\u0013 emoulin, P., & Li, H. 2012b, ApJ, 746, 17\nGuo, Y., Ding, M. D., Schmieder, B., Li, H., T or ok, T., & Wiegelmann, T. 2010a, ApJ, 725, L38\nGuo, Y., Pariat, E., Valori, G., An\fnogentov, S., Chen, F., Georgoulis, M. K., Liu, Y., Moraitis,\nK., Thalmann, J. K., & Yang, S. 2017, ApJ, 840, 40\nGuo, Y., Schmieder, B., Bommier, V., & Gosain, S. 2010b, Sol. Phys., 262, 35\nGuo, Y., Schmieder, B., D\u0013 emoulin, P., Wiegelmann, T., Aulanier, G., T or ok, T., & Bommier, V.\n2010c, ApJ, 714, 343\nGuo, Y., Xia, C., & Keppens, R. 2016a, ApJ, 828, 83\nGuo, Y., Xia, C., Keppens, R., & Valori, G. 2016b, ApJ, 828, 82\nHagino, M. & Sakurai, T. 2002, in Multi-Wavelength Observations of Coronal Structure and Dy-\nnamics, ed. P. C. H. Martens & D. Cau\u000bman, 147\nHagino, M. & Sakurai, T. 2004, PASJ, 56, 831\nHale, G. E. 1908, ApJ, 28, 315\nHaynes, A. L. & Parnell, C. E. 2007, Physics of Plasmas, 14, 082107\nHe, H. & Wang, H. 2006, MNRAS, 369, 207\n|. 2008, Journal of Geophysical Research (Space Physics), 113, 5{ 42 {\nHeyvaerts, J., Priest, E. R., & Rust, D. M. 1977, ApJ, 216, 123\nHillier, A., Berger, T., Isobe, H., & Shibata, K. 2012a, ApJ, 746, 120\nHillier, A., Isobe, H., Shibata, K., & Berger, T. 2011, ApJ, 736, L1\n|. 2012b, ApJ, 756, 110\nHock, R. A., Chamberlin, P. C., Woods, T. N., Crotser, D., Eparvier, F. G., Woodraska, D. L., &\nWoods, E. C. 2012, Sol. Phys., 275, 145\nHoeksema, J. T., Liu, Y., Hayashi, K., Sun, X., Schou, J., Couvidat, S., Norton, A., Bobra, M.,\nCenteno, R., Leka, K. D., Barnes, G., & Turmon, M. 2014, Sol. Phys., 289, 3483\nHornig, G. 2006, ArXiv Astrophysics e-prints\nHu, Q. & Dasgupta, B. 2008, Sol. Phys., 247, 87\nHu, Q., Dasgupta, B., Choudhary, D. P., & B uchner, J. 2008, ApJ, 679, 848\nIchimoto, K., Lites, B., Elmore, D., Suematsu, Y., Tsuneta, S., Katsukawa, Y., Shimizu, T., Shine,\nR., Tarbell, T., Title, A., Kiyohara, J., Shinoda, K., Card, G., Lecinski, A., Streander, K.,\nNakagiri, M., Miyashita, M., Noguchi, M., Ho\u000bmann, C., & Cruz, T. 2008, Sol. Phys., 249,\n233\nInhester, B. & Wiegelmann, T. 2006, Sol. Phys., 235, 201\nInoue, S. 2016, Progress in Earth and Planetary Science, 3, 19\nInoue, S., Hayashi, K., Magara, T., Choe, G. S., & Park, Y. D. 2014, ApJ, 788, 182\n|. 2015, ApJ, 803, 73\nInoue, S., Kusano, K., Magara, T., Shiota, D., & Yamamoto, T. T. 2011, ApJ, 738, 161\nInoue, S., Shiota, D., Yamamoto, T. T., Pandey, V. S., Magara, T., & Choe, G. S. 2012, ApJ, 760,\n17\nJanvier, M., Aulanier, G., & D\u0013 emoulin, P. 2015, Sol. Phys., 290, 3425\nJanvier, M., Aulanier, G., Pariat, E., & D\u0013 emoulin, P. 2013, A&A, 555, A77\nJanvier, M., Savcheva, A., Pariat, E., Tassev, S., Millholland, S., Bommier, V., McCauley, P.,\nMcKillop, S., & Dougan, F. 2016, A&A, 591, A141\nJe\u000beries, J., Lites, B. W., & Skumanich, A. 1989, ApJ, 343, 920\nJensen, T. H. & Chu, M. S. 1984, Physics of Fluids, 27, 2881{ 43 {\nJiang, C. & Feng, X. 2012a, ApJ, 749, 135\n|. 2012b, Sol. Phys., 281, 621\n|. 2013, ApJ, 769, 144\n|. 2014, Sol. Phys., 289, 63\nJiang, C., Feng, X., Fan, Y., & Xiang, C. 2011, ApJ, 727, 101\nJiang, C., Feng, X., Wu, S. T., & Hu, Q. 2013, ApJ, 771, L30\nJiang, C., Feng, X., & Xiang, C. 2012, ApJ, 755, 62\nJiang, C., Feng, X., Zhang, J., & Zhong, D. 2010, Sol. Phys., 267, 463\nJiang, C., Wu, S. T., Feng, X., & Hu, Q. 2014, ApJ, 786, L16\n|. 2016a, Nature Communications, 7, 11522\nJiang, C., Wu, S. T., Yurchyshyn, V., Wang, H., Feng, X., & Hu, Q. 2016b, ApJ, 828, 62\nJin, M., Schrijver, C. J., Cheung, M. C. M., DeRosa, M. L., Nitta, N. V., & Title, A. M. 2016,\nApJ, 820, 16\nJing, J., Park, S.-H., Liu, C., Lee, J., Wiegelmann, T., Xu, Y., Deng, N., & Wang, H. 2012, ApJ,\n752, L9\nJing, J., Yuan, Y., Wiegelmann, T., Xu, Y., Liu, R., & Wang, H. 2010, ApJ, 719, L56\nKeppens, R., Meliani, Z., van Marle, A. J., Delmont, P., Vlasis, A., & van der Holst, B. 2012,\nJournal of Computational Physics, 231, 718\nKeppens, R., Nool, M., T\u0013 oth, G., & Goedbloed, J. P. 2003, Computer Physics Communications,\n153, 317\nKeppens, R., Xia, C., & Porth, O. 2015, ApJ, 806, L13\nKippenhahn, R. & Schl uter, A. 1957, Zeitschrift fur Astrophysik, 43, 36\nKleman, M. 2015, Sol. Phys., 290, 707\nKleman, M. & Robbins, J. M. 2014, Sol. Phys., 289, 1173\nKliem, B., Su, Y. N., van Ballegooijen, A. A., & DeLuca, E. E. 2013, ApJ, 779, 129\nKliem, B. & T or ok, T. 2006, Physical Review Letters, 96, 255002\nKnizhnik, K. J., Antiochos, S. K., & DeVore, C. R. 2015, ApJ, 809, 137{ 44 {\n|. 2017, ApJ, 835, 85\nKosugi, T., Matsuzaki, K., Sakao, T., Shimizu, T., Sone, Y., Tachikawa, S., et al. 2007, Sol. Phys.,\n243, 3\nKuckein, C., Centeno, R., Mart\u0013 \u0010nez Pillet, V., Casini, R., Manso Sainz, R., & Shimizu, T. 2009,\nA&A, 501, 1113\nKuckein, C., Mart\u0013 \u0010nez Pillet, V., & Centeno, R. 2012, A&A, 539, A131\nKuperus, M. & Raadu, M. A. 1974, A&A, 31, 189\nKusano, K., Maeshiro, T., Yokoyama, T., & Sakurai, T. 2002, ApJ, 577, 501\nLagg, A., Ishikawa, R., Merenda, L., Wiegelmann, T., Tsuneta, S., & Solanki, S. K. 2009, in\nAstronomical Society of the Paci\fc Conference Series, Vol. 415, The Second Hinode Science\nMeeting: Beyond Discovery-Toward Understanding, ed. B. Lites, M. Cheung, T. Magara,\nJ. Mariska, & K. Reeves, 327\nLagg, A., Lites, B., Harvey, J., Gosain, S., & Centeno, R. 2015, Space Sci. Rev.\nLandol\f, M. & Landi Degl'Innocenti, E. 1982, Sol. Phys., 78, 355\nLeake, J. E., Linton, M. G., & Antiochos, S. K. 2014, ApJ, 787, 46\nLeake, J. E., Linton, M. G., & T or ok, T. 2013, ApJ, 778, 99\nLeka, K. D., Barnes, G., Crouch, A. D., Metcalf, T. R., Gary, G. A., Jing, J., & Liu, Y. 2009,\nSol. Phys., 260, 83\nLi, J., Amari, T., & Fan, Y. 2007, ApJ, 654, 675\nLi, Y., Ding, M. D., Guo, Y., & Dai, Y. 2014, ApJ, 793, 85\nLim, E.-K., Yurchyshyn, V., Park, S.-H., Kim, S., Cho, K.-S., Kumar, P., Chae, J., Yang, H., Cho,\nK., Song, D., & Kim, Y.-H. 2016, ApJ, 817, 39\nLin, H., Kuhn, J. R., & Coulter, R. 2004, ApJ, 613, L177\nLin, H., Penn, M. J., & Tomczyk, S. 2000, ApJ, 541, L83\nLin, J. & Forbes, T. G. 2000, J. Geophys. Res., 105, 2375\nLiu, K., Zhang, J., Wang, Y., & Cheng, X. 2013, ApJ, 768, 150\nLiu, R., Kliem, B., Titov, V. S., Chen, J., Wang, Y., Wang, H., Liu, C., Xu, Y., & Wiegelmann,\nT. 2016, ApJ, 818, 148\nLiu, Y., Hoeksema, J. T., Bobra, M., Hayashi, K., Schuck, P. W., & Sun, X. 2014, ApJ, 785, 13{ 45 {\nLiu, Y. & Schuck, P. W. 2012, ApJ, 761, 105\nLongcope, D. W. 2004, ApJ, 612, 1181\n|. 2005, Living Reviews in Solar Physics, 2, 7\nLongcope, D. W. & Malanushenko, A. 2008, ApJ, 674, 1130\nL\u0013 opez Ariste, A., Rayrole, J., & Semel, M. 2000, A&AS, 142, 137\nLow, B. C. 1985, ApJ, 293, 31\n|. 2006, ApJ, 646, 1288\n|. 2011, Physics of Plasmas, 18, 052901\nLow, B. C. & Lou, Y. Q. 1990, ApJ, 352, 343\nMagara, T. 2008, PASJ, 60, 809\nMagara, T. & Longcope, D. W. 2001, ApJ, 559, L55\n|. 2003, ApJ, 586, 630\nMalanushenko, A., Schrijver, C. J., DeRosa, M. L., & Wheatland, M. S. 2014, ApJ, 783, 102\nMalanushenko, A., Schrijver, C. J., DeRosa, M. L., Wheatland, M. S., & Gilchrist, S. A. 2012,\nApJ, 756, 153\nManchester, IV, W., Gombosi, T., DeZeeuw, D., & Fan, Y. 2004, ApJ, 610, 588\nMandrini, C. H., Schmieder, B., D\u0013 emoulin, P., Guo, Y., & Cristiani, G. D. 2014, Sol. Phys., 289,\n2041\nMarquardt, D. W. 1963, J. Soc. Ind. Appl. Math. , 11, 431\nMart\u0013 \u0010nez-Sykora, J., Hansteen, V., & Carlsson, M. 2008, ApJ, 679, 871\nMasson, S., Pariat, E., Aulanier, G., & Schrijver, C. J. 2009, ApJ, 700, 559\nMcClymont, A. N., Jiao, L., & Miki\u0013 c, Z. 1997, Sol. Phys., 174, 191\nMcClymont, A. N. & Miki\u0013 c, Z. 1994, ApJ, 422, 899\nMetcalf, T. R. 1994, Sol. Phys., 155, 235\nMetcalf, T. R., De Rosa, M. L., Schrijver, C. J., Barnes, G., van Ballegooijen, A. A., Wiegelmann,\nT., et al. 2008, Sol. Phys., 247, 269{ 46 {\nMetcalf, T. R., Leka, K. D., Barnes, G., Lites, B. W., Georgoulis, M. K., Pevtsov, A. A., et al.\n2006, Sol. Phys., 237, 267\nMiki\u0013 c, Z. & McClymont, A. N. 1994, in ASP Conf. Ser., Vol. 68, Solar Active Region Evolution:\nComparing Models with Observations, ed. K. S. Balasubramaniam & G. W. Simon, 225\nMo\u000batt, H. K. 1969, Journal of Fluid Mechanics, 35, 117\nMo\u000batt, H. K. & Ricca, R. L. 1992, Proceedings of the Royal Society of London Series A, 439, 411\nMolodenskii, M. M. 1969, Soviet Ast., 12, 585\nMoon, Y., Chae, J., Choe, G. S., Wang, H., Park, Y. D., Yun, H. S., Yurchyshyn, V., & Goode,\nP. R. 2002, ApJ, 574, 1066\nMoon, Y.-J., Wang, H., Spirock, T. J., Goode, P. R., & Park, Y. D. 2003, Sol. Phys., 217, 79\nMoore, R. L. & Labonte, B. J. 1980, in IAU Symposium, Vol. 91, Solar and Interplanetary Dynam-\nics, ed. M. Dryer & E. Tandberg-Hanssen, 207\nMoore, R. L., Sterling, A. C., Hudson, H. S., & Lemen, J. R. 2001, ApJ, 552, 833\nMoraitis, K., Tziotziou, K., Georgoulis, M. K., & Archontis, V. 2014, Sol. Phys., 289, 4453\nMurray, M. J., Hood, A. W., Moreno-Insertis, F., Galsgaard, K., & Archontis, V. 2006, A&A, 460,\n909\nNakagawa, Y. 1974, ApJ, 190, 437\nNakagawa, Y. & Raadu, M. A. 1972, Sol. Phys., 25, 127\nNeukirch, T. 1995, A&A, 301, 628\nNindos, A. & Zhang, H. 2002, ApJ, 573, L133\nNordlund, \u0017A., Stein, R. F., & Asplund, M. 2009, Living Reviews in Solar Physics, 6, 2\nOuyang, Y., Zhou, Y. H., Chen, P. F., & Fang, C. 2017, ApJ, 835, 94\nPariat, E., Antiochos, S. K., & DeVore, C. R. 2009, ApJ, 691, 61\n|. 2010, ApJ, 714, 1762\nPariat, E. & D\u0013 emoulin, P. 2012, A&A, 541, A78\nPariat, E., D\u0013 emoulin, P., & Berger, M. A. 2005, A&A, 439, 1191\nPariat, E., Leake, J. E., Valori, G., Linton, M. G., Zuccarello, F. P., & Dalmasse, K. 2017a, A&A,\n601, A125{ 47 {\nPariat, E., Valori, G., An\fnogentov, S., Chen, F., Georgoulis, M. K., Guo, Y., Liu, Y., Moraitis,\nK., Thalmann, J. K., & Yang, S. 2017b, Space Science Reviews, to be submitted\nPariat, E., Valori, G., D\u0013 emoulin, P., & Dalmasse, K. 2015, A&A, 580, A128\nPark, S.-H., Chae, J., Jing, J., Tan, C., & Wang, H. 2010, ApJ, 720, 1102\nParnell, C. E., Smith, J. M., Neukirch, T., & Priest, E. R. 1996, Physics of Plasmas , 3, 759\nPevtsov, A. A., Balasubramaniam, K. S., & Rogers, J. W. 2003, ApJ, 595, 500\nPevtsov, A. A., Berger, M. A., Nindos, A., Norton, A. A., & van Driel-Gesztelyi, L. 2014,\nSpace Sci. Rev., 186, 285\nPevtsov, A. A., Can\feld, R. C., & Metcalf, T. R. 1994, ApJ, 425, L117\n|. 1995, ApJ, 440, L109\nPontin, D., Galsgaard, K., & D\u0013 emoulin, P. 2016, Sol. Phys., 291, 1739\nPorth, O., Xia, C., Hendrix, T., Moschou, S. P., & Keppens, R. 2014, ApJS, 214, 4\nPress, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. 1988, Numerical Recipes in\nC: The Art of Scienti\fc Computing\nPriest, E. R., Bungey, T. N., & Titov, V. S. 1997, Geophysical and Astrophysical Fluid Dynamics,\n84, 127\nPriest, E. R. & D\u0013 emoulin, P. 1995, J. Geophys. Res., 100, 23443\nPriest, E. R., Parnell, C. E., & Martin, S. F. 1994, ApJ, 427, 459\nPrior, C. & Yeates, A. R. 2014, ApJ, 787, 100\nRachkovsky, D. N. 1962, Izv. Krymskoi Astro\fz. Obs., 27, 148\nReid, H. A. S., Vilmer, N., Aulanier, G., & Pariat, E. 2012, A&A, 547, A52\nRempel, M., Sch ussler, M., & Kn olker, M. 2009, ApJ, 691, 640\nRomano, P., Zuccarello, F. P., Guglielmino, S. L., & Zuccarello, F. 2014, ApJ, 794, 118\nRoumeliotis, G. 1996, ApJ, 473, 1095\nRoussev, I. I., Galsgaard, K., Downs, C., Lugaz, N., Sokolov, I. V., Moise, E., & Lin, J. 2012,\nNature Physics, 8, 845\nRuan, P., Wiegelmann, T., Inhester, B., Neukirch, T., Solanki, S. K., & Feng, L. 2008, A&A, 481,\n827{ 48 {\nRudenko, G. V. & An\fnogentov, S. A. 2014, Sol. Phys., 289, 1499\nRudenko, G. V. & Myshyakov, I. I. 2011, Sol. Phys., 270, 165\nRussell, A. J. B., Yeates, A. R., Hornig, G., & Wilmot-Smith, A. L. 2015, Physics of Plasmas, 22,\n032106\nRust, D. M. & Kumar, A. 1994, Sol. Phys., 155, 69\nSakurai, T. 1981, Sol. Phys., 69, 343\nSavcheva, A., Pariat, E., McKillop, S., McCauley, P., Hanson, E., Su, Y., & DeLuca, E. E. 2016,\nApJ, 817, 43\nSavcheva, A., Pariat, E., McKillop, S., McCauley, P., Hanson, E., Su, Y., Werner, E., & DeLuca,\nE. E. 2015, ApJ, 810, 96\nSavcheva, A., Pariat, E., van Ballegooijen, A., Aulanier, G., & DeLuca, E. 2012a, ApJ, 750, 15\nSavcheva, A. S., van Ballegooijen, A. A., & DeLuca, E. E. 2012b, ApJ, 744, 78\nSchatten, K. H., Wilcox, J. M., & Ness, N. F. 1969, Sol. Phys., 6, 442\nScherrer, P. H., Schou, J., Bush, R. I., Kosovichev, A. G., Bogart, R. S., Hoeksema, J. T., et al.\n2012, Sol. Phys., 275, 207\nSchmidt, H. U. 1964, NASA Special Publication , 50, 107\nSchmieder, B., Guo, Y., Moreno-Insertis, F., Aulanier, G., Yelles Chaouche, L., Nishizuka, N.,\nHarra, L. K., Thalmann, J. K., Vargas Dominguez, S., & Liu, Y. 2013, A&A, 559, A1\nSchmieder, B., Tian, H., Kucera, T., L\u0013 opez Ariste, A., Mein, N., Mein, P., Dalmasse, K., & Golub,\nL. 2014, A&A, 569, A85\nSchou, J., Scherrer, P. H., Bush, R. I., Wachter, R., Couvidat, S., Rabello-Soares, M. C., et al.\n2012, Sol. Phys., 275, 229\nSchrijver, C. J. & De Rosa, M. L. 2003, Sol. Phys., 212, 165\nSchrijver, C. J., Derosa, M. L., Metcalf, T. R., Liu, Y., McTiernan, J., R\u0013 egnier, S., Valori, G.,\nWheatland, M. S., & Wiegelmann, T. 2006, Sol. Phys., 235, 161\nSchuck, P. W. 2005, ApJ, 632, L53\n|. 2006, ApJ, 646, 1358\n|. 2008, ApJ, 683, 1134\nSeehafer, N. 1978, Sol. Phys., 58, 215{ 49 {\n|. 1986, Sol. Phys., 105, 223\nShen, F., Shen, C., Zhang, J., Hess, P., Wang, Y., Feng, X., Cheng, H., & Yang, Y. 2014, Journal\nof Geophysical Research (Space Physics), 119, 7128\nShibata, K., Nozawa, S., & Matsumoto, R. 1992, PASJ, 44, 265\nShimizu, T., Nagata, S., Tsuneta, S., Tarbell, T., Edwards, C., Shine, R., et al. 2008, Sol. Phys.,\n249, 221\nSocas-Navarro, H. 2001, in ASP Conf. Ser., Vol. 236, Advanced Solar Polarimetry { Theory, Ob-\nservation, and Instrumentation, ed. M. Sigwarth, 487\nSocas-Navarro, H., Trujillo Bueno, J., & Ruiz Cobo, B. 2000, ApJ, 530, 977\nSong, M. T., Fang, C., Tang, Y. H., Wu, S. T., & Zhang, Y. A. 2006, ApJ, 649, 1084\nStein, R. F. & Nordlund, \u0017A. 2000, Sol. Phys., 192, 91\nSten\ro, J. O. 2013, A&A Rev., 21, 66\nSu, J. T. & Zhang, H. Q. 2004, Chinese J. Astron. Astrophys., 4, 365\n|. 2007, ApJ, 666, 559\nSu, Y., van Ballegooijen, A., Schmieder, B., Berlicki, A., Guo, Y., Golub, L., & Huang, G. 2009,\nApJ, 704, 341\nSuematsu, Y., Tsuneta, S., Ichimoto, K., Shimizu, T., Otsubo, M., Katsukawa, Y., et al. 2008,\nSol. Phys., 249, 197\nSun, X., Hoeksema, J. T., Liu, Y., Aulanier, G., Su, Y., Hannah, I. G., & Hock, R. A. 2013, ApJ,\n778, 139\nSun, X., Hoeksema, J. T., Liu, Y., Chen, Q., & Hayashi, K. 2012, ApJ, 757, 149\nTadesse, T., Wiegelmann, T., & Inhester, B. 2009, A&A, 508, 421\nTadesse, T., Wiegelmann, T., Inhester, B., & Pevtsov, A. 2011, A&A, 527, A30\nTassev, S. & Savcheva, A. 2016, ArXiv e-prints\nTaylor, J. B. 1974, Physical Review Letters, 33, 1139\n|. 1986, Reviews of Modern Physics, 58, 741\nTeng, F. 2015, Sol. Phys., 290, 2693\nTeng, F. & Deng, Y.-Y. 2014, Research in Astronomy and Astrophysics, 14, 1469{ 50 {\n|. 2016, Research in Astronomy and Astrophysics, 16, 71\nTeuber, D., Tandberg-Hanssen, E., & Hagyard, M. J. 1977, Sol. Phys., 53, 97\nThalmann, J. K., Inhester, B., & Wiegelmann, T. 2011, Sol. Phys., 272, 243\nThompson, W. T. 2006, A&A, 449, 791\nTian, L. & Alexander, D. 2008, ApJ, 673, 532\nTitov, V. S. & D\u0013 emoulin, P. 1999, A&A, 351, 707\nTitov, V. S., Hornig, G., & D\u0013 emoulin, P. 2002, Journal of Geophysical Research (Space Physics),\n107, 1164\nTitov, V. S., Miki\u0013 c, Z., Linker, J. A., Lionello, R., & Antiochos, S. K. 2011, ApJ, 731, 111\nTitov, V. S., Priest, E. R., & D\u0013 emoulin, P. 1993, A&A, 276, 564\nT or ok, T. & Kliem, B. 2003, A&A, 406, 1043\n|. 2005, ApJ, 630, L97\nT or ok, T., Kliem, B., & Titov, V. S. 2004, A&A, 413, L27\nT\u0013 oth, G., van der Holst, B., & Huang, Z. 2011, ApJ, 732, 102\nTsuneta, S., Ichimoto, K., Katsukawa, Y., Nagata, S., Otsubo, M., Shimizu, T., et al. 2008,\nSol. Phys., 249, 167\nTziotziou, K., Georgoulis, M. K., & Liu, Y. 2013, ApJ, 772, 115\nTziotziou, K., Georgoulis, M. K., & Raoua\f, N.-E. 2012, ApJ, 759, L4\nTziotziou, K., Moraitis, K., Georgoulis, M. K., & Archontis, V. 2014, A&A, 570, L1\nUnno, W. 1956, PASJ, 8, 108\nValori, G., D\u0013 emoulin, P., & Pariat, E. 2012, Sol. Phys., 278, 347\nValori, G., D\u0013 emoulin, P., Pariat, E., & Masson, S. 2013, A&A, 553, A38\nValori, G., Kliem, B., & Fuhrmann, M. 2007, Sol. Phys., 245, 263\nValori, G., Kliem, B., & Keppens, R. 2005, A&A, 433, 335\nValori, G., Kliem, B., T or ok, T., & Titov, V. S. 2010, A&A, 519, A44\nValori, G., Pariat, E., An\fnogentov, S., Chen, F., Georgoulis, M. K., Guo, Y., Liu, Y., Moraitis,\nK., Thalmann, J. K., & Yang, S. 2016, Space Sci. Rev., 201, 147{ 51 {\nvan Ballegooijen, A. A. 2004, ApJ, 612, 519\nVemareddy, P., Ambastha, A., Maurya, R. A., & Chae, J. 2012, ApJ, 761, 86\nWang, H. 1997, Sol. Phys., 174, 265\nWang, H., Liu, C., Ahn, K., Xu, Y., Jing, J., Deng, N., Huang, N., Liu, R., Kusano, K., Fleishman,\nG. D., Gary, D. E., & Cao, W. 2017, Nature Astronomy, 1, 0085\nWang, H., Yan, Y., & Sakurai, T. 2001, Sol. Phys., 201, 323\nWang, R., Yan, Y., & Tan, B. 2013, Sol. Phys., 288, 507\nWelsch, B. T., Abbett, W. P., De Rosa, M. L., Fisher, G. H., Georgoulis, M. K., Kusano, K.,\nLongcope, D. W., Ravindra, B., & Schuck, P. W. 2007, ApJ, 670, 1434\nWelsch, B. T., Fisher, G. H., Abbett, W. P., & Regnier, S. 2004, ApJ, 610, 1148\nWheatland, M. S. 2004, Sol. Phys., 222, 247\n|. 2006, Sol. Phys., 238, 29\nWheatland, M. S., Sturrock, P. A., & Roumeliotis, G. 2000, ApJ, 540, 1150\nWhite, S. M. 2002, Astronomische Nachrichten, 323, 265\nWiegelmann, T. 2004, Sol. Phys., 219, 87\n|. 2007, Sol. Phys., 240, 227\nWiegelmann, T. & Inhester, B. 2003, Sol. Phys., 214, 287\nWiegelmann, T., Inhester, B., & Sakurai, T. 2006, Sol. Phys., 233, 215\nWiegelmann, T., Neukirch, T., Ruan, P., & Inhester, B. 2007, A&A, 475, 701\nWiegelmann, T. & Sakurai, T. 2012, Living Reviews in Solar Physics, 9\nWoltjer, L. 1958, Proceedings of the National Academy of Science, 44, 489\nWoods, T. N., Eparvier, F. G., Hock, R., Jones, A. R., Woodraska, D., Judge, D., Didkovsky, L.,\nLean, J., Mariska, J., Warren, H., McMullin, D., Chamberlin, P., Berthiaume, G., Bailey,\nS., Fuller-Rowell, T., Sojka, J., Tobiska, W. K., & Viereck, R. 2012, Sol. Phys., 275, 115\nWoods, T. N., Hock, R., Eparvier, F., Jones, A. R., Chamberlin, P. C., Klimchuk, J. A., Didkovsky,\nL., Judge, D., Mariska, J., Warren, H., Schrijver, C. J., Webb, D. F., Bailey, S., & Tobiska,\nW. K. 2011, ApJ, 739, 59\nWu, S. T., Sun, M. T., Chang, H. M., Hagyard, M. J., & Gary, G. A. 1990, ApJ, 362, 698{ 52 {\nXia, C. & Keppens, R. 2016a, ApJ, 823, 22\n|. 2016b, ApJ, 825, L29\nXia, C., Keppens, R., Antolin, P., & Porth, O. 2014a, ApJ, 792, L38\nXia, C., Keppens, R., & Guo, Y. 2014b, ApJ, 780, 130\nYamamoto, T. T. & Sakurai, T. 2009, ApJ, 698, 928\nYan, Y., Deng, Y., Karlick\u0013 y, M., Fu, Q., Wang, S., & Liu, Y. 2001, ApJ, 551, L115\nYan, Y. & Li, Z. 2006, ApJ, 638, 1162\nYan, Y. & Sakurai, T. 1997, Sol. Phys., 174, 65\n|. 2000, Sol. Phys., 195, 89\nYang, K., Guo, Y., & Ding, M. D. 2015, ApJ, 806, 171\n|. 2016, ApJ, 824, 148\nYang, S., B uchner, J., Santos, J. C., & Zhang, H. 2013, Sol. Phys., 283, 369\nYang, S., Zhang, H., & B uchner, J. 2009, A&A, 502, 333\nYang, W. H., Sturrock, P. A., & Antiochos, S. K. 1986, ApJ, 309, 383\nZhang, H., Ai, G., Yan, X., Li, W., & Liu, Y. 1994, ApJ, 423, 828\nZhang, H. & Bao, S. 1998, A&A, 339, 880\n|. 1999, ApJ, 519, 876\nZhang, H., Sakurai, T., Pevtsov, A., Gao, Y., Xu, H., Sokolo\u000b, D. D., & Kuzanyan, K. 2010,\nMNRAS, 402, L30\nZhang, H., Sokolo\u000b, D., Rogachevskii, I., Moss, D., Lamburt, V., Kuzanyan, K., & Kleeorin, N.\n2006a, MNRAS, 365, 276\nZhang, M., Flyer, N., & Low, B. C. 2006b, ApJ, 644, 575\nZhang, M. & Low, B. C. 2003, ApJ, 584, 479\nZhang, Q. M., Chen, P. F., Guo, Y., Fang, C., & Ding, M. D. 2012, ApJ, 746, 19\nZhao, H., Wang, J.-X., Zhang, J., & Xiao, C.-J. 2005, Chinese J. Astron. Astrophys., 5, 443\nZhao, H., Wang, J.-X., Zhang, J., Xiao, C.-J., & Wang, H.-M. 2008, Chinese J. Astron. Astrophys.,\n8, 133{ 53 {\nZhao, J., Gilchrist, S. A., Aulanier, G., Schmieder, B., Pariat, E., & Li, H. 2016, ApJ, 823, 62\nZhao, J., Li, H., Pariat, E., Schmieder, B., Guo, Y., & Wiegelmann, T. 2014, ApJ, 787, 88\nZhao, X. P. & Hoeksema, J. T. 1993, Sol. Phys., 143, 41\n|. 1994, Sol. Phys., 151, 91\nZhao, X. P., Hoeksema, J. T., & Scherrer, P. H. 2000, ApJ, 538, 932\nZhu, X. S., Wang, H. N., Du, Z. L., & Fan, Y. L. 2013, ApJ, 768, 119\nZhu, X. S., Wang, H. N., Du, Z. L., & He, H. 2016, ApJ, 826, 51\nZirker, J. B., Martin, S. F., Harvey, K., & Gaizauskas, V. 1997, Sol. Phys., 175, 27\nZuccarello, F. P., Bemporad, A., Jacobs, C., Mierla, M., Poedts, S., & Zuccarello, F. 2012, ApJ,\n744, 66\nZuccarello, F. P., Romano, P., Zuccarello, F., & Poedts, S. 2011, A&A, 530, A36\nThis preprint was prepared with the AAS L ATEX macros v5.2.{ 54 {\na\nc d\nb\nFig. 1.| A vector magnetic \feld and nonlinear force-free \feld model illustrating how the projection\ne\u000bect is corrected. a.Vector magnetic \feld at 05:24 UT on 2015 August 27 observed by SDO /HMI.\nb.The vector magnetic \feld components have been transformed to the heliographic coordinate\nsystem. Its geometry has been mapped onto a plane tangent to the solar surface at (W54.7\u000e,\nS15.3\u000e).c.The computation box of the nonlinear force-free \feld is rotated back to the place\ntangent to the solar surface at (W54.7\u000e, S15.3\u000e).d.The vector magnetic \feld components are\ntransformed back to the line-of-sight and plane-of-sky coordinate system, where the x-axis points\nto the observer. Solid lines colored with the \feld strength represent the magnetic \feld lines. The\nbackground image shows the line-of-sight magnetic \feld, which is the same as that in panel a.\nWhite lines show the contours of the local vertical component of the magnetic \feld, namely, Bzas\nshown in panel b.{ 55 {\na b\nc d\nFig. 2.| Magnetic \feld models computed by the magneto-frictional method. a.Blue lines show\nthe semi-analytic solution of Low & Lou (1990). White lines are those computed by the magneto-\nfrictional method with all the six boundaries being provided by the original solution. The \fgure is\nfrom Guo et al. (2016b) and reproduced by permission of the AAS. b.Blue lines show the semi-\nanalytic solution of Titov & D\u0013 emoulin (1999). White lines are those computed by the magneto-\nfrictional method with all the six boundaries being provided by the original solution. The \fgure is\nfrom Guo et al. (2016b) and reproduced by permission of the AAS. c.Green and blue lines show\nthe nonlinear force-free \feld model computed by the magneto-frictional method in a Cartesian\ncoordinate system with a uniform grid. It uses a vector magnetic \feld observed by SDO /HMI as\nthe boundary condition. The background image is a 171 \u0017A image by SDO /AIA. The \fgure is from\nGuo et al. (2016a) and reproduced by permission of the AAS. d.Colored solid lines show a twisted\nmagnetic \rux rope modeled by the magneto-frictional method in the spherical coordinate system\nwith an adaptive mesh re\fnement. The grids show the structure of the adaptive mesh re\fnement\non a slice. The background image represents the radial magnetic \feld. The \fgure is adapted from\nGuo et al. (2016a) and reproduced by permission of the AAS.{ 56 {\na b\nc d\nFig. 3.| Some selected data-constrained and data-driven MHD models. a.A snapshot of the\nzero-\fMHD numerical simulation from Kliem et al. (2013) and reproduced by permission of the\nAAS. Green lines show the overlying magnetic \feld lines and other colored lines show a twisted\nmagnetic \rux rope. b.Another zero- \fHMD model from Inoue et al. (2014) and reproduced by\npermission of the AAS. Solid lines with di\u000berent colors show the magnetic \rux rope with di\u000berent\ntwist numbers. the vertical slice shows the distribution of the vertical velocity and the horizontal\nslice shows the distribution of vertical magnetic \feld. c.An erupting twisted magnetic \rux rope\nillustrated by the colored lines from Amari et al. (2014) and reprinted by permission from the\nNature Publishing Group. d.A jet-like structure showing the magnetic reconnection induced by\nthe data-driven MHD simulation. Black/red lines indicate magnetic \feld lines before/after the\nreconnection. The \fgure is adapted from Jiang et al. (2016a), where more explanation of the \fgure\nis provided.{ 57 {\na b\nc d\nFig. 4.| Magnetic null points in di\u000berent magnetic \feld models. a.Red, cyan, and yellow lines\nrepresent the magnetic \feld lines in the vicinity of a null point in a potential \feld model adapted\nfrom Masson et al. (2009) and reproduced by permission of the AAS. The grey-scale image shows\nthe vertical magnetic \feld. b.Blue and red lines represent the magnetic \feld lines in the vicinities\nof three null points in a nonlinear force-free \feld. Yellow lines denote some sheared and twisted\nmagnetic \feld lines. Contours show the vertical magnetic \feld. The \fgure is adapted from Mandrini\net al. (2014) and reproduced by permission of Springer Science+Business Media Dordrecht. c.A\ntoy model showing both the magnetic \feld lines close to the null point and sheared \feld lines under\nit. The \fgure is adapted from Sun et al. (2013) and reproduced by permission of the AAS. d.Blue\nand green lines denote the \feld lines close to two null points. Contours show the vertical magnetic\n\feld and the color image is a soft X-ray image observed by the X-ray Telescope aborad Hinode .\nThe \fgure is from Zhang et al. (2012) and reproduced by permission of the AAS.{ 58 {\na b\nc d\nFig. 5.| Bald patches and magnetic dips in di\u000berent magnetic \feld models. a.Green lines mark\nthe positions of bald patches derived from a nonlinear force-free \feld model. The \fgure is adapted\nfrom Schmieder et al. (2013) and reproduced with permission from Astronomy & Astrophysics,\nc\rESO. b.Dark line sections denote magnetic \feld line sections with magnetic dips that are\ncomputed in a linear force-free \feld model. The \fgure is adapted from Aulanier et al. (1998) and\nreproduced with permission from Astronomy & Astrophysics, c\rESO. c.Green crosses denote\nthe positions of magnetic dips computed in a nonlinear force-free \feld model. Red line sections\nrepresent the magnetic \feld line sections passing the magnetic dips. The \fgure is adapted from\nGuo et al. (2010c) and reproduced by permission of the AAS. d.Red line sections denote magnetic\n\feld line sections with magnetic dips in a linear force-free \feld model. Other lines are magnetic\n\feld lines associated with two null points and an underlying magnetic arcade. The \fgure is adapted\nfrom Dud\u0013 \u0010k et al. (2012) and reproduced by permission of the AAS.{ 59 {\na b\nd\n c\nFig. 6.| QSLs in di\u000berent magnetic \feld models. a.Distribution of the logarithm of the squashing\ndegreeQfor a nonlinear force-free \feld model from Savcheva et al. (2012a) and reproduced by\npermission of the AAS. b.White semitransparent surfaces represent the isosurface of the squashing\ndegree atQ= 104. Solid lines are selected magnetic \feld lines of a nonlinear force-free \feld model.\nYellow surfaces are isosurface of the electric current density. The image on the bottom shows vertical\nmagnetic \feld. The \fgure is adapted from Guo et al. (2013b) and reproduced by permission of\nthe AAS. c.Red and blue lines show the twisted magnetic \feld lines of a nonlinear force-free \feld\nmodel. Black lines denote the axes of the magnetic \rux rope. The lower image shows the isosurface\nof the twist number that equals \u00001. The isosurface of the twist number has a similar distribution\nas that of the squashing degree. The \fgure is from Liu et al. (2016) and reproduced by permission\nof the AAS. d.The semitransparent surfaces represent the 3D distribution of the logarithm of the\nsquashing degree Qcomputed from a potential \feld model. The red lines delineate some \feld lines\nfrom a magnetic null point. The image on the bottom is a SDO /AIA 1600 \u0017A image. The \fgure is\nfrom Yang et al. (2015) and reproduced by permission of the AAS."    },    {        "title": "2206.14152v2.Turbulent_magnetic_helicity_fluxes_in_solar_convective_zone.pdf",        "content": "arXiv:2206.14152v2  [astro-ph.SR]  13 Aug 2022Mon. Not. R. Astron. Soc. 000, 000–000 (0000) Printed 16 August 2022 (MN L ATEX style file v2.2)\nTurbulent magnetic helicity fluxes in solar convective zone\nN. Kleeorin1,2and I. Rogachevskii1,3\n1Department of Mechanical Engineering, Ben-Gurion Univers ity of the Negev, P. O. B. 653, Beer-Sheva 8410530, Israel\n2Institute of Continuous Media Mechanics, Korolyov str. 1, 6 14013 Perm, Russia\n3Nordita, Stockholm University and KTH Royal Institute of Te chnology, 10691 Stockholm, Sweden\n16 August 2022\nABSTRACT\nCombined action of helical motions of plasma (the kinetic αeffect) and non-uniform\n(differential) rotation is a key dynamo mechanism of solar and galactic large-scale\nmagnetic fields. Dynamics of magnetic helicity of small-scale fields is a cr ucial mecha-\nnism in a nonlinear dynamo saturation where turbulent magnetic helicit y fluxes allow\nto avoid catastrophic quenching of the αeffect. The convective zone of the Sun and\nsolar-likestars as well as galactic discs are the source for product ion of turbulent mag-\nnetic helicity fluxes. In the framework of the mean-field approach a nd the spectral τ\napproximation, we derive turbulent magnetic helicity fluxes using the Coulomb gauge\nin a density-stratified turbulence. The turbulent magnetic helicity fl uxes include non-\ngradient and gradient contributions. The non-gradient magnetic h elicity flux is pro-\nportional to a nonlinear effective velocity (which vanishes in the abse nce of the density\nstratification) multiplied by small-scale magnetic helicity, while the grad ient contri-\nbution describes turbulent magnetic diffusion of the small-scale magn etic helicity. In\naddition, the turbulent magnetic helicity fluxes contain source term s proportional to\nthe kinetic αeffect or its gradients, and also contributions caused by the large- scale\nshear (solar differential rotation). We have demonstrated that t he turbulent magnetic\nhelicity fluxes due to the kinetic αeffect and its radial derivative in combination with\nthe nonlinear magnetic diffusion of the small-scale magnetic helicity are dominant in\nthe solar convective zone.\nKey words: dynamo – MHD – Sun: interior — turbulence – activity\n1 INTRODUCTION\nThe large-scale solar and galactic magnetic fields are gen-\nerated by a combined action of helical turbulent motions\nand large-scale differential rotation due to the αΩ dynamo\n(see, e.g., Moffatt 1978; Parker 1979; Krause & R¨ adler 1980;\nZeldovich et al. 1983; Moffatt & Dormy 2019). A non-zero\nkinetichelicity producedbyarotatingdensitystratified c on-\nvective turbulence, causes the kinetic αeffect. The dynamo\ninstability is saturated by nonlinear effects. One of the im-\nportant nonlinear effect is the feedback of the growing large -\nscale magnetic field on the plasma turbulent motions, so\nthat the turbulent transport coefficients (the αeffect, the\neffective pumping velocity and the turbulent magnetic dif-\nfusion) depend on the mean magnetic field B. The simplest\nnonlinear saturation mechanism of the dynamo instability\nis related to the αquenching which prescribes the kinetic\nαeffect to be a decreasing function of the mean magnetic\nfield strength, e.g., α(B) =αK/parenleftBig\n1+B2/B2\neq/parenrightBig−1\n, where\nαK∝ −τ0Huis the kinetic αeffect that is proportional\nto the kinetic helicity Hu=∝angb∇acketleftu·(∇×u)∝angb∇acket∇ight,B2\neq= 4πρ/angbracketleftbig\nu2/angbracketrightbig\nis the squared equipartition mean magnetic field, uis theturbulent velocity field, τ0is the turbulent time and ρis the\nmean density. This implies that the mean magnetic field\nstrength at which quenching becomes significant, is esti-\nmated from the equipartition between the energy density\nof the mean magnetic field and the turbulent kinetic en-\nergy density. When applied to galactic dynamos, this pic-\nture results in robust magnetic field models which are com-\npatible with observations (see, e.g., Ruzmaikin et al. 1988 ;\nShukurov & Subramanian 2021). The above-mention non-\nlinearity is referred as algebraic nonlinearity.\nHoweverthispictureisobviouslyoversimplifiedandvar-\nious attempts to suggest a more advanced version of non-\nlinear dynamo theory have been undertaken (see, e.g., re-\nviews and books by Brandenburg & Subramanian 2005b;\nR¨ udiger et al. 2013; Rincon 2019; Rogachevskii 2021, and\nreferences therein). The quantitative theories of the al-\ngebraic nonlinearities of the αeffect, the turbulent mag-\nnetic diffusion and the effective pumping velocity have been\ndeveloped using the quasi-linear approach for small fluid\nand magnetic Reynolds numbers (R¨ udiger & Kichatinov\n1993; Kitchatinov et al. 1994; R¨ udiger et al. 2013) and the\ntau approach for large fluid and magnetic Reynolds num-\n©0000 RAS2N. Kleeorin and I. Rogachevskii\nbers (Field et al. 1999; Rogachevskii & Kleeorin 2000, 2001,\n2004, 2006).\nIn addition to the algebraic nonlinearity, there is also a\ndynamicnonlinearity caused byanevolution of magnetic he-\nlicity density of small-scale fields during the nonlinear st age\nof the mean-field dynamo. In particular, the αeffect is the\nsum of the kinetic and magnetic parts, α=αK+αm, where\nthe magnetic αeffect,αm∝τ0Hc/(12πρ), is proportional\nto the current helicity Hc=∝angb∇acketleftb·(∇×b)∝angb∇acket∇ightof the small-scale\nmagnetic field b(Pouquet et al. 1976). The dynamics of the\ncurrent helicity Hcis determined by the evolution of the\nsmall-scale magnetic helicity density Hm=∝angb∇acketlefta·b∝angb∇acket∇ight, where\nmagnetic fluctuations b=∇×aandaare fluctuations of\nmagnetic vector potential.\nMagnetic helicity is fundamental quantity in magneto-\nhydrodynamics and plasma physics (see, e.g., Berger 1999).\nIn particular, the total magnetic helicity, i.e., the sum of\nthe magnetic helicity densities of the large-scale and smal l-\nscale magnetic fields, HM+Hm, integrated over the volume,/integraltext\n(HM+Hm)dr3, is conserved for very small microscopic\nmagnetic diffusivity η. HereHM=A·Bis the magnetic he-\nlicity density of the large-scale field B=∇×A. Signature of\nmagnetic helicity has been detected in many solar features,\nincluding solar active regions (see, e.g., Pevtsov et al. 20 14;\nZhang et al. 2006, 2012, and references therein).\nThe governing equation for small-scale magnetic helic-\nity density Hmhas been derived for an isotropic turbu-\nlence by Kleeorin & Ruzmaikin (1982) and for an arbitrary\nanisotropic turbulence by Kleeorin & Rogachevskii (1999).\nThis equation has been used for analytical study of solar dy-\nnamos (Kleeorin et al. 1994, 1995) as well as for mean-field\nnumerical modeling of solar and galactic dynamos (see, e.g. ,\nCovas et al. 1997, 1998; Kleeorin et al. 2000, 2002, 2003b,a,\n2016; Brandenburg & Subramanian 2005b; Sokoloff et al.\n2006; Zhang et al. 2006, 2012; Del Sordo et al. 2013;\nSafiullin et al. 2018).\nAs the dynamo amplifies the large-scale magnetic field,\nthe magnetic helicity density HMof the large-scale field\ngrows in time. In particular, the evolution of the large-sca le\nmagnetic helicity density HMis determined by the following\nequation:\n∂HM\n∂t+∇·F(M)= 2E·B−2ηHC, (1)\nwhereE=∝angb∇acketleftu×b∝angb∇acket∇ightis the turbulent electromotive force\nthat determines generation and dissipation of the large-sc ale\nmagnetic field, 2 E·Bis the source of HMdue to the dynamo\ngenerated large-scale magnetic field, F(M)is the fluxof mag-\nnetic helicity density of the large-scale field that determi nes\nits transport and HC=B·(∇×B) is the current helicity of\nlarge-scale field.\nSince the total magnetic helicity/integraltext\n(HM+Hm)dr3\nis conserved, the magnetic helicity density Hmof the\nsmall-scale field changes during the dynamo action,\nand its evolution is determined by the dynamic equa-\ntion (Kleeorin & Ruzmaikin 1982; Zeldovich et al. 1983;\nKleeorin et al. 1995; Kleeorin & Rogachevskii 1999):\n∂Hm\n∂t+∇·F(m)=−2E·B−2ηHc, (2)\nwhere−2E·Bis the source of Hmdue to the dynamo gener-\nated large-scale magnetic field, F(m)is the flux of magnetichelicity density of the small-scale field that determines it s\ntransport and −2ηHcis the dissipation rate of Hm. The\nsource of the small-scale and large-scale magnetic helicit y\ndensities is only located in turbulent region.\nThe characteristic decay time of the magnetic helic-\nity density Hmof the small-scale field is of the order of\nTm=τ0Rm, while the characteristic time for the decay\nof kinetic helicity is of the order of the turn-over time\nτ0=ℓ0/u0of turbulent eddies in the integral turbulence\nscaleℓ0, where Rm = ℓ0u0/ηis the magnetic Reynoldsnum-\nber. The current helicity Hcof the small-scale field is not a\nconserved quantity, and the characteristic decay time of Hc\nvaries from a short timescale τ0to much larger timescales.\nOn the other hand, the characteristic decay times of the\ncurrent helicity of large-scale field, HC, and of the large-\nscale magnetic helicity HMare of the order of the turbulent\ndiffusion time. For weakly inhomogeneous turbulence the\ncurrent helicity density Hcof the small-scale field is pro-\nportional to the small-scale magnetic helicity density Hm\n(Kleeorin & Rogachevskii 1999).\nUsing the steady-state solution of Eq. (2) with a zero\nturbulent flux F(m)= 0 of magnetic helicity density of\nsmall-scale field and a zero current helicity of large-scale\nfield,HC, it has been concluded that the critical mean mag-\nnetic field strength, Bcr, at which the dynamic αquenching\nbecomes significant, in fact is much lower than the equipar-\ntition value, e.g. Bcr=BeqRm−1/2(Vainshtein & Cattaneo\n1992; Gruzinov & Diamond 1994). In astrophysics, e.g., in\ngalactic disks and in the convective zone of the sun, mag-\nnetic Reynolds numbers are very large. Therefore, for large\nmagnetic Reynolds numbers the dynamo action should sat-\nurate at a magnetic field strength that is much lower than\nthe equipartition value. This effect is referred as to a catas -\ntrophic quenching of the αeffect (Vainshtein & Cattaneo\n1992; Gruzinov & Diamond 1994). On the other hand, the\nobserved large-scale field strengths in spiral galaxies is o f the\norder of the equipartition value (see, e.g., Ruzmaikin et al .\n1988; Shukurov & Subramanian 2021), and the observed so-\nlar and stellar magnetic fields are much larger than Bcr\n(see, e.g., Moffatt 1978; Parker 1979; Krause & R¨ adler 1980;\nZeldovich et al. 1983).\nThe evolution of magnetic helicity appears however\nto be a more complicated process than can simply be\ndescribed by a balance of magnetic helicity in a given\nvolume. It is necessary to take into account fluxes of\nmagnetic helicity (Kleeorin et al. 2000). This implies that\nthe turbulent transport of magnetic helicity through the\nboundaries (the open boundary conditions in simula-\ntions) should be taken into account (Blackman & Field\n2000). Different forms of magnetic helicity fluxes have\nbeen suggested in various studies (Covas et al. 1997, 1998;\nKleeorin & Rogachevskii 1999; Kleeorin et al. 2000, 2002;\nVishniac & Cho 2001; Subramanian & Brandenburg 2004;\nBrandenburg & Subramanian 2005b). Turbulent fluxes of\nsmall-scale magnetic helicity have been measured in nu-\nmerical simulations (K¨ apyl¨ a et al. 2010; Mitra et al. 2010 ;\nHubbard & Brandenburg 2010, 2011, 2012; Del Sordo et al.\n2013), and in solar observations (Chae et al. 2001;\nPariat et al. 2005; Pevtsov et al. 2014; Hawkes & Berger\n2018).\nTaking into account turbulent fluxes of the small-\nscale magnetic helicity, it has been shown by nu-\n©0000 RAS, MNRAS 000, 000–000Turbulent magnetic helicity fluxes in solar convective zone 3\nmerical simulations that a nonlinear galactic dy-\nnamo governed by a dynamic equation for the mag-\nnetic helicity density Hmof small-scale field satu-\nrates at a mean magnetic field comparable with the\nequipartition magnetic field (see, e.g., Kleeorin et al.\n2000, 2002, 2003b,a; Blackman & Brandenburg 2002;\nBrandenburg & Subramanian 2005b; Shukurov et al. 2006;\nDel Sordo et al. 2013). Numerical simulations demonstrate\nthat the dynamics of the small-scale magnetic helicity\nin the presence of the turbulent magnetic helicity fluxes\nplay a crucial role in the solar dynamo as well (see, e.g.,\nKleeorin et al. 2003b, 2016, 2020; Sokoloff et al. 2006;\nZhang et al. 2006, 2012; K¨ apyl¨ a et al. 2010; Guerrero et al.\n2010; Hubbard & Brandenburg 2012; Del Sordo et al. 2013;\nSafiullin et al. 2018; Rincon 2021).\nDue to very important role of the turbulent magnetic\nhelicity fluxes in nonlinear dynamos, in the present study we\nperform a rigorous derivation of these fluxes applying the\nmean-field theory, adopting the Coulomb gauge and consid-\nering a strongly density-stratified turbulence. We show tha t\nthe turbulent magnetic helicity fluxes contain non-gradien t\nand gradient contributions. The non-gradient magnetic he-\nlicity fluxes are product of a nonlinear effective velocity an d\nsmall-scale magnetic helicity. The gradient contribution s de-\ntermine a nonlinear magnetic diffusion of the small-scale\nmagnetic helicity. We also demonstrate that the turbulent\nmagnetic helicity fluxes include source terms proportional\nto the kinetic αeffect or its gradients. In the present study\nwe do not consider an algebraic quenching of the turbulent\nmagnetic helicity fluxes that is a subject of a separate study .\nThis paper is organized as follows. In Section 2, we de-\nrive equation for the magnetic helicity of small-scale field s\nwhich includes divergence of the turbulent magnetic helici ty\nflux. In Section 3 we discuss the results of calculations of th e\nturbulent flux of magnetic helicity of the small-scale fields .\nIn addition, we obtain a general form of turbulent flux of the\nmagnetic helicity using symmetry arguments. In Section 4,\nwe consider the turbulent magnetic helicity flux in the solar\nconvective zone. Finally, in Section 5, we discuss our resul ts\nanddrawconclusions. InAppendixesAandBwe discuss ap-\nproximations and procedure of the derivation of turbulent\nflux of magnetic helicity. In Appendix C we determine the\neffect of large-scale shear on turbulent flux of the magnetic\nhelicity. Applying the method described in Appendixes A–\nC, we determine various contributions to the turbulent flux\nof the small-scale magnetic helicity in Appendix D. In par-\nticular, we present the general form of turbulent transport\ncoefficients entering in the turbulent flux of the small-scale\nmagnetic helicity. For better understanding of the physics\nrelated to various contributions to the turbulent flux of the\nsmall-scale magnetic helicity, in Appendix E we consider a\nmore simple case with a large-scale linear velocity shear an d\npresent turbulent transport coefficients in the Cartesian co -\nordinates.\n2 EQUATION FOR THE MAGNETIC\nHELICITY\nIn this Section, we derive an equation for the small-scale\nmagnetic helicity. The induction equation for fluctuationsof magnetic field breads\n∂b\n∂t=∇×/bracketleftBig\nU×b+u×B+u×b−∝angb∇acketleftu×b∝angb∇acket∇ight\n−η∇×b/bracketrightBig\n, (3)\nwhere in the framework of the mean-field approach, we sep-\narate magnetic and velocity fields into mean and fluctua-\ntions,B=B+bandB=∝angb∇acketleftB∝angb∇acket∇ightis the mean magnetic\nfield,U=U+u, andU=∝angb∇acketleftU∝angb∇acket∇ightis the mean fluid velocity\ndescribing, e.g., the differential rotation, ηis the magnetic\ndiffusion due toelectrical conductivityof fluid. The equati on\nfor magnetic fluctuations is obtained by subtracting induc-\ntion equation for the the mean magnetic field Bfrom that\nfor the total field B(t,x). The equation for fluctuations of\nthe vector potential afollows from induction equation (3)\n∂a\n∂t=U×b+u×B+u×b−∝angb∇acketleftu×b∝angb∇acket∇ight\n−η∇×b+∇φ, (4)\nwhereB=∇×AandA=A+a,andA=∝angb∇acketleftA∝angb∇acket∇ightis\nthe mean vector potential, b=∇×aandφare fluctua-\ntions of the scalar potential. We multiply Eq. (3) by aand\nEq. (4) by b, add them and average over an ensemble of\nturbulent fields. This yields an equation for the magnetic\nhelicityHm=∝angb∇acketlefta(x)·b(x)∝angb∇acket∇ightof the small-scale fields as\n∂Hm\n∂t=−2E·B−2η∝angb∇acketleftb·(∇×b)∝angb∇acket∇ight−∇·F(m), (5)\nwhereE=∝angb∇acketleftu×b∝angb∇acket∇ightis the turbulent electromotive force, and\nthe turbulent flux of magnetic helicity F(m)of the small-\nscale fields is given by\nF(m)=UHm−/angbracketleftbig\nb(a·U)/angbracketrightbig\n+/angbracketleftbig\nu(a·B)/angbracketrightbig\n−B∝angb∇acketlefta·u∝angb∇acket∇ight\n−η∝angb∇acketlefta×(∇×b)∝angb∇acket∇ight+∝angb∇acketlefta×(u×b)∝angb∇acket∇ight−∝angb∇acketleftbφ∝angb∇acket∇ight.(6)\nUsing the Coulomb gauge ∇·a= 0, we obtain that\n∇×b=−∆aanda=−∆−1∇×b. The Coulomb gauge\nalso allows us to find fluctuations of the scalar potential φ.\nIndeed, equation for ∇·awhich follows from Eq. (4), yields\nexpression for fluctuations of the scalar potential φ, so that\nthe correlation function ∝angb∇acketleftbiφ∝angb∇acket∇ightreads\n∝angb∇acketleftbiφ∝angb∇acket∇ight=∝angb∇acketleftbiaj∝angb∇acket∇ightUj−/angbracketleftbig\nbi∆−1(∇×u)j/angbracketrightbig\nBj\n−/angbracketleftbig\nbi∆−1bj/angbracketrightbig\nWj+/angbracketleftbig\nbi∆−1uj/angbracketrightbig\n(∇×B)j\n−/angbracketleftbig\nbi∆−1∇·(u×b)/angbracketrightbig\n. (7)\nwhereW=∇×Uis the mean vorticity and ∝angb∇acketleftbiaj∝angb∇acket∇ight=\n−/angbracketleftbig\nbi∆−1(∇×b)j/angbracketrightbig\n. Equations (6)–(7) yield the turbulent\nflux of magnetic helicity F(m)of the small-scale fields as\nF(m)\ni=UiHm+Wj/angbracketleftbig\nbi∆−1bj/angbracketrightbig\n+Bj∝angb∇acketleftuiaj∝angb∇acket∇ight\n−Bi∝angb∇acketleftujaj∝angb∇acket∇ight+Bj/angbracketleftbig\nbi∆−1(∇×u)j/angbracketrightbig\n+F(η)\ni\n−(∇×B)j/angbracketleftbig\nbi∆−1uj/angbracketrightbig\n+F(III)\ni, (8)\nwhere ∝angb∇acketleftuiaj∝angb∇acket∇ight=−/angbracketleftbig\nui∆−1(∇×b)j/angbracketrightbig\n,F(η)=\n−η∝angb∇acketlefta×(∇×b)∝angb∇acket∇ightis the flux caused by the microscopic\nmagnetic diffusion ηandF(III)is the flux that is determined\nby the third-order moments, and it is given by\nF(III)=/angbracketleftbig\nb∆−1∇·(u×b)/angbracketrightbig\n+∝angb∇acketlefta×(u×b)∝angb∇acket∇ight.(9)\nEquations (5)–(9) are exact equations. Note that only in\nthe Coulomb gauge, the scalar potential φis described by\n©0000 RAS, MNRAS 000, 000–0004N. Kleeorin and I. Rogachevskii\nthe stationary equation. For all other gauge conditions, th e\nscalar potential φis determined by a non-stationary equa-\ntion. Also for the Coulomb gauge the relation between the\nmagnetic αeffect and small-scale magnetic helicity is most\nsimple.\n3 GENERAL FORM OF TURBULENT FLUX\nOF THE MAGNETIC HELICITY\nIn this Section we discuss the results of calculations of the\nturbulent flux of magnetic helicity of the small-scale fields .\nGeneral form of turbulent flux F(m)of the magnetic helic-\nity can be obtained from symmetry reasoning. Indeed, the\nturbulent flux F(m)is the pseudo-vector which should con-\ntain two pseudo-scalars: the magnetic helicity, Hm, and the\nkineticαeffect,αK, and their first spatial derivatives. In\naddition, the contributions F(S0)\nito the turbulent magnetic\nhelicity flux caused by the large-scale shear (differential r o-\ntation)shouldcontainthepseudo-vector W=∇×U,where\nU=δΩ×ris the large-scale velocity describing the differ-\nential rotation δΩ.\nAll turbulent transport coefficients entering in the tur-\nbulent flux F(m)of magnetic helicity of the small-scale fields\nshould be quadratic in the large-scale magnetic field B, i.e.,\ntheyshouldbe proportional to B2orV2\nA=B2/(4πρ), where\nρis the mean plasma density and VAis the mean Alfv´ en\nspeed. On the other hand, the turbulent flux F(m)of the\nmagnetic helicity should vanish in theabsence of turbulenc e.\nThis implies that all turbulent transport coefficients enter -\ning in the turbulent flux F(m)should be proportional to\nturbulent correlation time τ0or turbulent integral scale ℓ0.\nSome of the turbulent transport coefficients are caused by\nthe plasma density stratification, i.e., they are proportio nal\ntoλ=−∇lnρ.\nUsing the theoretical approach based on the spectral τ\napproximation which is valid for large fluid and magnetic\nReynolds numbers, and the multi-scale approach, we obtain\nthe turbulent flux of the small-scale magnetic helicity as\nF(m)\ni=/parenleftBig\nUi+V(H)\ni/parenrightBig\nHm−D(H)\nij∇jHm+N(α)\niαK\n+M(α)\nij∇jαK+F(S0)\ni, (10)\nwhereαK=−τ0Hu/3 is the kinetic αeffect. Details of the\nderivation ofEq. (10)are describedin AppendixesA–C. The\ngeneral form of the turbulent transport coefficients enterin g\nin the turbulent flux (10) of magnetic helicity of the small-\nscale fields is given byEqs. (D2)–(D6) in AppendixD. These\nturbulent transport coefficients of the turbulent magnetic\nhelicity flux in spherical coordinates are given in the next\nsection and in the Cartesian coordinates are discussed in\nAppendix E.\nThe turbulent flux of the small-scale magnetic helic-\nity includes the non-gradient and gradient contributions.\nThe non-gradient contribution to the turbulent flux of mag-\nnetic helicity is proportional to the sum of the mean velocit y\nU=δΩ×rand the turbulent pumping velocity V(H)which\nis multiplied by small-scale magnetic helicity Hm, while the\ngradient contribution −D(H)\nij∇jHmdescribe the turbulent\nmagnetic diffusion of the small-scale magnetic helicity. Th e\neffective pumping velocity of the small-scale magnetic heli c-\nityV(H)vanishes in the absence of the density stratification.Inaddition, theturbulentmagnetic helicity fluxcontains t he\nsource term N(α)αKproportional to the kinetic αeffect,\nand the source term −M(α)\nij∇jαKproportional to the gra-\ndient∇jαKof the kinetic αeffect. The turbulent magnetic\nhelicityfluxalso havecontributions causedbythelarge-sc ale\nshear (differential rotation) in the turbulent flow.\nWe assume that the turbulent flux of the magnetic he-\nlicityF(III)containing the third-order moments [see equa-\ntion (9)], is determined using the turbulent diffusion ap-\nproximation as F(III)=−D(H)\nT∇Hm. The contribution to\ntheturbulentmagnetic helicity flux, −D(H)\nT∇Hm, causedby\nthe turbulent diffusion, has been used in mean-field numer-\nical simulations by Covas et al. (1997, 1998); Kleeorin et al .\n(2002, 2003a).\nThe turbulent diffusion of the small-scale magnetic he-\nlicity can be interpreted as follows. The random flows exist-\ning in the interstellar medium consist of a combination of\nsmall-scale motions, which are affected by magnetic forces\n(tangling turbulence) resulting in a steady-state of the dy -\nnamo, and a background micro-turbulence which is sup-\nported by a strong random driver (e.g., supernovae explo-\nsions which can be considered as independent of the galactic\nmagnetic field). The large-scale magnetic field is smoothed\nover both kinds of turbulent fluctuations, while the small-\nscale magnetic field is smoothed over micro-turbulent fluc-\ntuations only. It is the smoothing over the micro-turbulent\nfluctuations that gives the coefficient D(H)\nT=CDηTwith a\nfree dimensionless constant CD∼0.1. HereηTis the turbu-\nlent diffusion coefficient of the mean magnetic field.\nThe magnetic helicity flux F(η)=−η∝angb∇acketlefta×(∇×b)∝angb∇acket∇ight\ndue to the microscopic magnetic diffusion ηis given by\nF(η)=−1\n3η∇Hm. This flux in astrophysical systems is very\nsmall and neglected here.\n4 TURBULENT MAGNETIC HELICITY FLUX\nIN THE SOLAR CONVECTIVE ZONE\nIn this Section we discuss the results of calculations of the\nturbulent magnetic helicity flux in the solar convective zon e,\nwhere we use spherical coordinates ( r,ϑ,ϕ). The radial tur-\nbulent flux of the small-scale magnetic helicity is given by\nF(m)\nr=V(H)\nrHm−D(H)\nrj∇jHm+N(α)\nrαK\n+M(α)\nrj∇jαK+F(S0)\nr. (11)\nThe general forms of the turbulent transport coefficients en-\ntering in the turbulent flux F(m)of magnetic helicity of\nthe small-scale fields are given by Eqs. (D2)–(D6) in Ap-\npendix D. In view of applications to the solar convective\nzone, the turbulent transport coefficients of the turbulent\nmagnetic helicity flux in spherical coordinates are specifie d\nbelow:\nV(H)\nr=−1\n15τ0V2\nAλ/bracketleftbigg\n1+7β2\nr−173\n14sinϑτ0δΩβrβϕ/bracketrightbigg\n,(12)\nD(H)\nrr=D(H)\nT+1\n30τ0V2\nA/parenleftbig\n5−4β2\nr/parenrightbig\n, (13)\nD(H)\nrϑ=2(80+17 q)\n105τ2\n0V2\nAδΩβrβϕcosϑ, (14)\n©0000 RAS, MNRAS 000, 000–000Turbulent magnetic helicity fluxes in solar convective zone 5\nN(α)\nr=−1\n10ℓ2\n0B2λ/bracketleftbigg\n1+7q−2\nqβ2\nr\n−216(q−1)\n7(3q−1)τ0δΩβrβϕsinϑ/bracketrightbigg\n, (15)\nM(α)\nrr=2q−1\n20qℓ2\n0B2/bracketleftbigg\n1+20q−23\n2q−1β2\nr\n−32q(q−1)\n(2q−1)(3q−1)τ0δΩβrβϕsinϑ/bracketrightbigg\n,(16)\nM(α)\nrϑ=8(q−1)\n3q−1ℓ2\n0B2τ0δΩβrβϕcosϑ, (17)\nF(S0)\nr=−2\n9δΩ cosϑ/braceleftbigg\n4ℓ2\n0B2\nr+/bracketleftbiggV2\nA\n∝angb∇acketleftu2∝angb∇acket∇ight/parenleftBig\n1−3\n11β2\nr/parenrightBig\n+3(q−1)\nq+1/bracketrightbigg\nℓ2\nb/angbracketleftbig\nb2/angbracketrightbig/bracerightbigg\n, (18)\nwhereβ=B/Bis the unit vector along the mean mag-\nnetic field, U=δΩrsinϑeϕis the mean velocity caused\nby the differential rotation δΩ = Ω(r,ϑ)−Ω(r=R⊙,ϑ).\nHere Ω(r=R⊙,ϑ) = Ω 0(1−C2cos2ϑ−C4cos4ϑ) with\nΩ0= 2.83×10−6s−1,C2= 0.121 and C4= 0.173\n(LaBonte & Howard 1982), R⊙is the solar radius, λ=λer,\nℓbis the energy containing scale of magnetic fluctuations\nwith a zero mean magnetic field and qis the exponent in\nthe spectrum of the turbulent kinetic energy (the exponent\nq= 5/3 corresponds to the Kolmogorov spectrum of the\nturbulent kinetic energy).\nIn derivation of Eqs. (12)–(18), we take into account\nthat for weakly inhomogeneous turbulence Hc≈Hm/ℓ2\n0,\nand we neglect small terms ∼O[ℓ2\n0/L2\nm] withLmbeing char-\nacteristic scale of spatial variations of Hm. We neglect also\nsmall contributions proportional to spatial derivatives o f the\nmean magnetic field, and spatial derivatives of/angbracketleftbig\nu2/angbracketrightbig\nandδΩ.\nLet us discuss the obtained results. For illustration, in\nFig. 1 we show the radial profile of the total angular veloc-\nity Ω(r)/Ω⊙in the solar convective zone that includes the\nuniform and differential rotation specified for the latitude\nφ∗= 30◦. The theoretical profile (solid line) of the to-\ntal angular velocity (Rogachevskii & Kleeorin 2018) is com-\npared with the radial profile of the solar angular velocity\n(stars) obtained from the helioseismology observational d ata\n(Kosovichev et al. 1997) specified for the latitude φ= 30◦\nand normalized by the solar rotation frequency Ω ⊙(φ∗= 0)\nat the equator, where Ω /Ω⊙is given by Eq. (3.14) de-\nrived by Rogachevskii & Kleeorin (2018). In Figs. 1–2 we\nalso show the radial profile of the kinetic αeffect,αK/αmax\nwhich is specified for the latitude φ= 30◦and given by\nEq. (22) derived by Kleeorin & Rogachevskii (2003).\nIn the upper part of the solar convective zone for\nthe latitude φ∗>0 (the Northern Hemisphere), the ki-\nneticαeffect is positive, αK>0 (see Fig. 2). On the\nother hand, the magnetic αeffect in this region is nega-\ntive, i.e., αM=τ0Hc/(4πρ)<0. This implies that the\ncurrent helicity Hc<0 as well as the magnetic helicity\nHm<0 are negative the Northern Hemisphere. Here for\nsimplicity, we choose the radial profile of the poloidal and\ntoroidal field as Br=Br0sin[π(r−0.73R⊙)/(0.6R⊙)] and0.8 0.84 0.88 0.92 0.960.940.960.981.00\nFigure 1. The theoretical radial profiles of the total angular ve-\nlocity Ω( r)/Ω⊙(solid) that includes the uniform and differential\nrotation specified for the latitude φ∗= 30◦and the normalized\nkineticαeffect,αK/αmax(dashed). The theoretical profile of the\ntotal angular velocity is compared with the radial profile of the\nsolar angular velocity obtained from the helioseismology o bserva-\ntional data (stars) specified for the latitude φ∗= 30◦and normal-\nized by the solar rotation frequency Ω ⊙(φ∗= 0) at the equator\n(Kosovichev et al. 1997), where R⊙is the solar radius. The profile\nαK(r)≡α(K)\nϕϕis given by Eq. (22) derived by Kleeorin & Ro-\ngachevskii (2003), and Ω( r)/Ω⊙is given by Eq. (3.14) derived\nby Rogachevskii & Kleeorin (2018).\n0.75 0.8 0.85 0.9 0.95-0.20.00.20.40.60.81.0\nFigure 2. The radial profile of the normalized kinetic αeffect,\n˜αK=αK/αmax, specified for the latitude φ∗= 30◦and given by\nEq. (22) derived by Kleeorin & Rogachevskii (2003).\nBϕ=Bϕ0cos[π(r−0.73R⊙)/(0.6R⊙)], where Br0is the\nsurface mean magnetic field measured in Gauss. To avoid\ncatastrophic quenching, the radial component of the turbu-\nlentfluxofthesmall-scale magnetic helicity F(m)\nr<0should\nbe negative for the Northern Hemisphere.\nIn Figs. 3 and 4 we show the radial profiles of the\neffective pumping velocity V(H)\nr(r) and turbulent diffusion\nD(H)\nrr(r) of the small-scale magnetic helicity. In Figs. 5 and 6\nwe plot the radial profiles of the turbulent magnetic helic-\nity fluxes caused by the source terms F(α)\n1(r) =N(α)\nrαK\nandF(α)\n2(r) =M(α)\nrr∇rαK, which are proportional to the\nkineticαeffect and its radial derivative, as well as their\nsumF(α)\nr(r) =N(α)\nrαK+M(α)\nrr∇rαK. In Fig. 6 we also\nshow the contribution F(S0)(r) to the turbulent magnetic\nhelicity flux caused by the large-scale shear (differential\nrotation). Finally, in Fig. 7 we plot the radial profile of\nthe total source flux of the magnetic helicity Ftot(r) =\nN(α)\nrαK+M(α)\nrr∇rαK+F(S0)\nrthat is independent of the\nmagnetic helicity and its radial derivative.\nAs follows from Figs. 3–7 as well as Eqs. (11)–(18), the\nnegative contribution to the turbulent magnetic helicity fl ux\nF(m)\nrin the range of the generation of the mean magnetic\nfield, is due to the source flux F(α)\nr=N(α)\nrαK+M(α)\nrr∇rαK,\nand the contribution F(S0)to the turbulent magnetic he-\nlicity flux caused by the large-scale shear (differential ro-\ntation). Here we take into account that δΩ>0 at 0.8<\n©0000 RAS, MNRAS 000, 000–0006N. Kleeorin and I. Rogachevskii\n0.75 0.80 0.85 0.90 0.95-0.5-1.0-1.5-2.0\nFigure 3. The radial profile of the effective pumping velocity\nV(H)\nrof the small-scale magnetic helicity given by Eq. (12), and\nmeasured in m s−1.\n0.75 0.80 0.85 0.90 0.950.51.01.52.0\nFigure 4. The radial profile of turbulent diffusion D(H)\nrr(r) of the\nsmall-scale magnetic helicity given by Eq. (13) and measure d in\ncm2s−1.\n0.92 0.94 0.96 0.98-15.0-12.5-10.0-7.5-5.0-2.50.02.5\nFigure 5. The radial profile of the turbulent magnetic helicity\nfluxes caused by the source terms F(α)\n1=N(α)\nrαK(dashed) and\nF(α)\n2=M(α)\nrr∇rαK(dashed-dotted) which are proportional to\nthe kinetic αeffect and its radial derivative, as well as their sum\nF(α)\nr=N(α)\nrαK+M(α)\nrr∇rαK(solid), where N(α)\nrandM(α)\nrrare\ngiven by Eqs. (15) and (16), respectively. The fluxes are spec ified\nfor the latitude φ∗= 30◦and measured in G2cm2s−1.\n0.75 0.8 0.85 0.9 0.95-10-7.5-5.0-2.50.02.5\nFigure 6. The radial profiles of the turbulent magnetic helicity\nfluxes caused by the source terms F(α)\n1=N(α)\nrαK(solid),F(α)\n2=\nM(α)\nrr∇rαK(dashed) and the contribution F(S0)\nr(dashed-dotted)\nto the turbulent magnetic helicity flux caused by the large-s cale\nshear (differential rotation) , where N(α)\nr,M(α)\nrrandF(S0)\nrare\ngiven by Eqs. (15), (16) and (18), respectively. The fluxes ar e\nspecified for the latitude φ∗= 30◦and measured in G2cm2s−1.0.75 0.8 0.85 0.9 0.95-7.5-5.0-2.50.02.5\nFigure 7. The radial profile of the total source flux Ftot=\nN(α)\nrαK+M(α)\nrr∇rαK+F(S0)\nrof the magnetic helicity that is in-\ndependent of the magnetic helicity and its radial derivativ e. Here\nthe flux is measured in G2cm2s−1.\n0.75 0.8 0.85 0.9 0.95-7.5-5.0-2.50.0\nFigure 8. Turbulent diffusion flux r2F(D)\nr(solid line) and the\nfluxr2[F(D)\nr(r)+Ftot(r)] (dashed-dotted line) of magnetic helic-\nity per unit solid angle which are measured in Mx2h−1.\nr/R⊙<1 (see Fig. 1), where the differential rotation\nδΩ = Ω(r)−Ω(r=R⊙).\nThe small-scale magnetic helicity is not accumulated\ninside the solar convective zone due to turbulent magnetic\ndiffusionflux, F(D)\nr. InFig. 8we showtheturbulentdiffusion\nfluxr2F(D)\nr(solid line) of magnetic helicity per unit solid\nangle and the flux [ F(D)\nr(r)+Ftot(r)]r2(dashed-dotted line)\nof magnetic helicity per unit solid angle which are measured\nin Mx2h−1. As follows from Fig. 8, the flux [ F(D)\nr(r) +\nFtot(r)]r2(the sum of the turbulent diffusion flux and total\nsource flux of magnetic helicity) of small-scale field per uni t\nsolid angle is independent of r, i.e.,\n[F(D)\nr(r)+Ftot(r)]r2≈Ftot(r= 0.73R⊙)(0.73R⊙)2.(19)\nHere we take into account that the turbulent diffusion flux\nF(D)\nr(r= 0.73R⊙)→0 vanishes at the bottom of the con-\nvective zone, r= 0.73R⊙, where the turbulence intensity\nvanishes (see Fig. 8). Equation (19) implies that there is\nno accumulation of small-scale magnetic helicity inside th e\nsolar convective zone.\nIn Fig. 9 we compare the theoretical predictions for\nflux Φ D≡F(D)\nr(r=R⊙)R2\n⊙δφ∗with the observational\nvalues of Φ Dwhich are taken from Fig. 8a by Chae et al.\n(2001), where time variations of the rates of magnetic helic -\nity change by photospheric motions (which do not include\ndifferential rotation) are shown. Here the flux Φ Dis mea-\nsured in Mx2h−1andδφ∗= 2πsin(π/4) is the solid angle\ncorresponding to the thickness of the Royal sunspot region.\nThe theoretical values for Φ Dare givenfor differentvalues of\nthe mean magnetic field, BbotandBtop, at the bottom and\ntop of the solar convective zone (see the caption of Fig. 9).\nNote that the measurements of the magnetic helicity flux\nare based on the equation ∂Hm/∂t=−2/contintegraltext\n(u·ap)bzdS\n©0000 RAS, MNRAS 000, 000–000Turbulent magnetic helicity fluxes in solar convective zone 7\n16/08 17/08 18/08 19/08-10.0-7.5-5.0-2.50.02.5\nFigure 9. Comparison of the theoretical predictions for Φ D=\nF(D)\nr(r=R⊙)R2\n⊙δφ∗with the observational values ofΦ D(slant-\ning crosses) which are taken from Fig. 8a by Chae et al. (2001) ,\nwhere time variations of the rates of magnetic helicity chan ge\nby photospheric motions (which do not include differential r o-\ntation) are shown. Here the flux Φ Dis measured in Mx2h−1\nandδφ∗= 2πsin(π/4) is the solid angle corresponding to the\nthickness of the Royal sunspot region. The theoretical valu es for\nΦDare given for different values of the mean magnetic field, Bbot\nandBtop, at the bottom and top of the solar convective zone (i.e.,\nthick solid line is for Bbot= 103G andBtop= 8 G; dashed line\nis forBbot= 1.4×103G andBtop= 11 G and dashed-dotted\nline is for Bbot= 2×103G andBtop= 16 G).\n(Chae et al. 2001; Pevtsov et al. 2014), where we use here\nthe lower-case letters for the small-scale fields. This im-\nplies that the measurements by Chae et al. (2001) are based\non the calculation of the third-order moment, ∝angb∇acketleft(u·ap)bz∝angb∇acket∇ight,\nwhich we describe using the turbulent diffusion approxima-\ntion,F(D)\nr=−D(H)\nrr∇rHm. As follows from Fig. 9, the the-\noretical predictions for flux Φ Dare in agreement with the\nobservational values of Φ D.\n5 DISCUSSION AND CONCLUSIONS\nIn the present study, turbulent magnetic helicity fluxes of\nsmall-scale field are derived applying the mean-field ap-\nproach and the spectral τapproximation using the Coulomb\ngauge in a density-stratified turbulence. The turbulent mag -\nnetic helicity fluxes contain non-gradient contribution th at\nis proportional to the effective pumping velocity multiplie d\nby the small-scale magnetic helicity. There is the gradient\ncontribution to the turbulent magnetic helicity flux descri b-\ning the turbulent magnetic diffusion of the small-scale mag-\nnetic helicity. The turbulent magnetic helicity flux includ es\nalso the source term proportional to the kinetic αeffect or\nits radial gradient. Finally, there is a contribution to the\nturbulent magnetic helicity flux due to the solar differentia l\nrotation.\nThe convective zone of the Sun and solar-like stars as\nwell as galactic discs are the source for production of turbu -\nlent magnetic helicity fluxes. The turbulent magnetic helic -\nity flux due to the kinetic αeffect and its radial derivative\nin combination with the turbulent magnetic diffusion of the\nsmall-scale magnetic helicity are dominant in the solar con -\nvective zone. The turbulentmagnetic helicity fluxes result in\nevacuation of small-scale magnetic helicity from the regio ns\nof generation of the solar magnetic field, which allows to\navoid the catastrophic quenching of the αeffect. The small-\nscale magnetic helicity is not accumulated inside the solar\nconvective zone due to turbulent magnetic diffusion flux.\nThe magnetic helicity fluxes are measured in the solarsurface. Most of the measurements of the magnetic helicity\nfluxes are performed in active regions. The contributions to\nthe measured magnetic helicity flux are from both, the solar\nsurface and solar interiors.\nACKNOWLEDGMENTS\nThis work was partially supported by the Russian Science\nFoundation (grant 21-72-20067). We acknowledge the dis-\ncussions with participants of the Nordita Scientific Progra m\non ”Magnetic field evolution in low density or strongly strat -\nified plasmas”, Stockholm (May – June 2022).\nDATA AVAILABILITY\nThere are no new data associated with this article.\nREFERENCES\nBerger M. A., 1999, Plasma Physics and Controlled Fusion,\n41, B167\nBlackman E. G., BrandenburgA., 2002, Astrophys.J., 579,\n359\nBlackman E. G., Field G. B., 2000, Astrophys. J., 534, 984\nBrandenburg A., Subramanian K., 2005a, Astron. Nachr.,\n326, 400\nBrandenburg A., Subramanian K., 2005b, Phys. Rep., 417,\n1\nBrandenburg A., Subramanian K., 2005c, Astron. Astro-\nphys., 439, 835\nBrandenburg A., K¨ apyl¨ a P. J., Mohammed A., 2004, Phys.\nFluids, 16, 1020\nBrandenburg A., R¨ adler K.-H., Rheinhardt M., K¨ apyl¨ a\nP. J., 2008, Astrophys. J., 676, 740\nBrandenburg A., R¨ adler K.-H., Kemel K., 2012, Astron.\nAstroph., 539, A35\nChae J., Wang H., Qiu J., Goode P. R., Strous L., Yun H.,\n2001, Astrophys. J., 560, 476\nCovas E., Tworkowski A., Tavakol R., Brandenburg A.,\n1997, Solar Phys., 172, 3\nCovas E., Tavakol R., Tworkowski A., Brandenburg A.,\n1998, Astron. Astrophys., 329, 350\nDel Sordo F., Guerrero G., Brandenburg A., 2013, Mon.\nNot. Roy. Astron. Soc., 429, 1686\nElperin T., Kleeorin N., Liberman M., Lipatnikov A. N.,\nRogachevskii I., Yu R., 2017, Phys. Rev. E, 96, 053111\nField G. B., Blackman E. G., Chou H., 1999, Astrophys.\nJ., 513, 638\nGruzinov A. V., Diamond P. H., 1994, Phys. Rev. Lett.,\n72, 1651\nGuerrero G., Chatterjee P., Brandenburg A., 2010, Mon.\nNot. Roy. Astron. Soc., 409, 1619\nHaugen N. E. L., Kleeorin N., Rogachevskii I., Branden-\nburg A., 2012, Phys. Fluids, 24, 075106\nHawkes G., Berger M., 2018, Solar Physics, 293, 1\nHubbard A., Brandenburg A., 2010, Geophys. Astrophys.\nFluid Dyn., 104, 577\nHubbard A., Brandenburg A., 2011, Astrophys. J., 727, 11\nHubbard A., Brandenburg A., 2012, Astrophys. J., 748, 51\n©0000 RAS, MNRAS 000, 000–0008N. Kleeorin and I. Rogachevskii\nK¨ apyl¨ a P. J., Korpi M. J., Brandenburg A., 2010, Astron.\nAstrophys., 518, A22\nKitchatinov L. L., Pipin V. V., R¨ udiger G., 1994, Astron.\nNachr., 315, 157\nKleeorin N., Rogachevskii I., 1999, Phys. Rev. E, 59, 6724\nKleeorin N., Rogachevskii I., 2003, Phys. Rev. E, 67,\n026321\nKleeorin N., Ruzmaikin A., 1982, Magnetohydrodynamics,\n18, 116\nKleeorin N., Rogachevskii I., Ruzmaikin A., 1994, Solar\nPhys., 155, 223\nKleeorin N., Rogachevskii I., Ruzmaikin A., 1995, Astron.\nAstrophys., 297, 159\nKleeorin N., Moss D., Rogachevskii I., Sokoloff D., 2000,\nAstron. Astrophys., 361, L5\nKleeorin N., Moss D., Rogachevskii I., Sokoloff D., 2002,\nAstron. Astrophys., 387, 453\nKleeorin N., Moss D., Rogachevskii I., Sokoloff D., 2003a,\nAstron. Astrophys., 400, 9\nKleeorin N., Kuzanyan K., Moss D., Rogachevskii I.,\nSokoloff D., Zhang H., 2003b, Astron. Astrophys., 409,\n1097\nKleeorin Y., Safiullin N., Kleeorin N., Porshnev S., Ro-\ngachevskii I., Sokoloff D., 2016, Mon. Not. Roy. Astron.\nSoc., 460, 3960\nKleeorin N., Safiullin N., Kuzanyan K., Rogachevskii I.,\nTlatov A., Porshnev S., 2020, Mon. Not. Roy. Astron.\nSoc., 495, 238\nKosovichev A., Schou J., Scherrer P. H., Bogart R. S. e. a.,\n1997, Solar Phys., 170, 43\nKrause F., R¨ adler K.-H., 1980, Mean-Field Magnetohydro-\ndynamics and Dynamo Theory. Oxford: Pergamon Press\nLaBonte B. J., Howard R., 1982, Solar Phys., 75, 161\nMitra D., Candelaresi S., Chatterjee P., Tavakol R., Bran-\ndenburg A., 2010, Astron. Nachr., 331, 130\nMoffatt H. K., 1978, Magnetic Field Generation in Electri-\ncally Conducting Fluids. Cambridge: Cambridge Univer-\nsity Press\nMoffatt H. K., Dormy E., 2019, Self-Exciting Fluid Dy-\nnamos. Vol. 59, Cambridge: Cambridge University Press\nPariat E., D´ emoulin P., Berger M., 2005, Astronomy &\nAstrophysics, 439, 1191\nParker E. N., 1979, Cosmical Magnetic Fields: Their Origin\nand their Activity. Oxford: Clarendon Press\nPevtsov A. A., Berger M. A., Nindos A., Norton A. A., van\nDriel-Gesztelyi L., 2014, Space Sci. Rev., 186, 285\nPouquet A., Frisch U., L´ eorat J., 1976, J. Fluid Mech., 77,\n321\nR¨ adler K.-H., Kleeorin N., Rogachevskii I., 2003, Geophys .\nAstrophys. Fluid Dyn., 97, 249\nR¨ adler K.-H., Brandenburg A., Del Sordo F., Rheinhardt\nM., 2011, Phys. Rev. E, 84, 046321\nRincon F., 2019, J. Plasma Phys., 85, 205850401\nRincon F., 2021, Phys. Rev. Fluids, 6, L121701\nRogachevskii I., 2021, Introduction to Turbulent Trans-\nportof Particles, Temperature andMagnetic Fields. Cam-\nbridge: Cambridge University Press\nRogachevskii I., Kleeorin N., 2000, Phys. Rev. E, 61, 5202\nRogachevskii I., Kleeorin N., 2001, Phys. Rev. E, 64,\n056307\nRogachevskii I., Kleeorin N., 2004, Phys. Rev. E, 70,\n046310Rogachevskii I., Kleeorin N., 2006, Geophys. Astrophys.\nFluid Dyn., 100, 243\nRogachevskii I., Kleeorin N., 2018, J. Plasma Phys., 84,\n735840201\nRogachevskii I., Kleeorin N., K¨ apyl¨ a P. J., Brandenburg\nA., 2011, Phys. Rev. E, 84, 056314\nRogachevskii I., Kleeorin N., Brandenburg A., Eichler D.,\n2012, Astrophys. J., 753, 6\nRogachevskii I., Kleeorin N., Brandenburg A., 2018, J.\nPlasma Phys., 84, 735840502\nR¨ udiger G., Kichatinov L. L., 1993, Astron. Astrophys.,\n269, 581\nR¨ udiger G., Hollerbach R., Kitchatinov L. L., 2013, Mag-\nnetic Processes in Astrophysics: Theory, Simulations, Ex-\nperiments. Weinheim: John Wiley & Sons\nRuzmaikin A., Shukurov A. M., Sokoloff D. D., 1988, Mag-\nnetic Fields of Galaxies. Dordrecht: Kluwer Academic\nSafiullinN.,KleeorinN.,PorshnevS.,RogachevskiiI.,Ruz -\nmaikin A., 2018, J. Plasma Phys., 84, 735840306\nShukurov A., Subramanian K., 2021, Astrophysical Mag-\nnetic Fields: From Galaxies to the Early Universe. Cam-\nbridge University Press\nShukurov A., Sokoloff D., Subramanian K., Brandenburg\nA., 2006, Astron. Astrophys., 448, L33\nSokoloff D., Bao S. D., Kleeorin N., Kuzanyan K., Moss\nD., Rogachevskii I., Tomin D., Zhang H., 2006, Astron.\nNachr., 327, 876\nSubramanian K., Brandenburg A., 2004, Phys. Rev. Lett.,\n93, 205001\nVainshtein S. I., Cattaneo F., 1992, Astrophys. J., 393, 165\nVishniac E. T., Cho J., 2001, Astrophys. J., 550, 752\nZeldovich Y. B., Ruzmaikin A. A., Sokoloff D. D., 1983,\nMagnetic Fields in Astrophysics. New-York: Gordon and\nBreach\nZhang H., Sokoloff D., Rogachevskii I., Moss D., Lamburt\nV., Kuzanyan K., Kleeorin N., 2006, Mon. Not. Roy. As-\ntron. Soc., 365, 276\nZhang H., Moss D., Kleeorin N., Kuzanyan K., Ro-\ngachevskii I., Sokoloff D., GaoY., XuH., 2012, Astrophys.\nJ., 751, 47\nAPPENDIX A: DERIVATION OF TURBULENT\nFLUX OF MAGNETIC HELICITY\nIn this Section we derive turbulent flux of the magnetic he-\nlicity. We consider developed turbulence with large fluid an d\nmagnetic Reynolds numbers, so that the Strouhal number\n(the ratio of turbulent time τto turn-over time ℓ0/u0) is\nof the order of unity, and the turbulent correlation time is\nscale-dependent, like in Kolmogorov type turbulence. In th is\ncase, we perform the Fourier transformation only in kspace\nbut not in ωspace, as is usually done in studies of turbulent\ntransport in a fully developed Kolmogorov-type turbulence .\nWetakeintoaccountthenonlinearterms inequationsfor ve-\nlocity and magnetic fluctuations and apply the τapproach.\nTheτapproach is a universal tool in turbulent trans-\nport for strongly nonlinear systems that allows us to ob-\ntain closed results and compare them with the results of\nlaboratory experiments, observations, and numerical simu -\nlations. The τapproximation reproduces many well-known\nphenomena found by other methods in turbulent transport\n©0000 RAS, MNRAS 000, 000–000Turbulent magnetic helicity fluxes in solar convective zone 9\nof particles and magnetic fields, in turbulent convection an d\nstably stratified turbulent flows for large fluid and magnetic\nReynolds and P´ eclet numbers.\nTo derive equations for the turbulent fluxes of the mag-\nnetic helicity, we need expressions in a Fourier space for\nthe cross-helicity tensor gij(k) =∝angb∇acketleftui(t,k)bj(t,−k)∝angb∇acket∇ightand\nthe tensor hij(k) =∝angb∇acketleftbi(t,k)bj(t,−k)∝angb∇acket∇ightfor magnetic fluctua-\ntions. Indeed, as follows from Eq. (8), the turbulent fluxes o f\nthe magnetic helicity depend only on the second moments\ngijandhij(except for the last two terms, η∝angb∇acketlefta×(∇×b)∝angb∇acket∇ight\nandF(III)which are considered separately). Using induction\nequation (3) for magnetic fluctuations band the Navier-\nStokes equation for velocity fluctuations uwritten in a\nFourier space, we derive equations for the cross-helicity t en-\nsorgij(k) and the tensor hij(k) for magnetic fluctuations\nas\n∂gij(k)\n∂t=−/bracketleftbigg\nik·B−1\n2B·∇/bracketrightbigg/bracketleftBig\nfij(k)−hij(k)/bracketrightBig\n+ˆM(b)g(III)\nij(k), (A1)\n∂hij(k)\n∂t= i/parenleftBig\nk·B/parenrightBig/bracketleftBig\ngij(k)−gji(−k)/bracketrightBig\n+1\n2/parenleftBig\nB·∇/parenrightBig/bracketleftBig\ngij(k)+gji(−k)/bracketrightBig\n+ˆM(b)h(III)\nij(k),(A2)\nwhere in Eqs. (A1)–(A2) we neglect terms proportional\nto spatial derivatives of the mean magnetic field [i.e.,\nterms∝O/parenleftbig\n∇iBj/parenrightbig\n]. Here fij(k) =∝angb∇acketleftui(t,k)uj(t,−k)∝angb∇acket∇ight,\nandˆM(b)g(III)\nijandˆM(b)h(III)\nijare the third-order moment\nterms appearing due to the nonlinear terms:\nˆM(b)g(III)\nij(k) =−/angbracketleftBig\nui(t,k)T(b)\nj(t,−k)/angbracketrightBig\n+/angbracketleftbigg∂ui(t,k)\n∂tbj(t,−k)/angbracketrightbigg\n, (A3)\nˆM(b)h(III)\nij(k) =−/angbracketleftBig\nbi(t,k)T(b)\nj(t,−k)/angbracketrightBig\n−/angbracketleftBig\nT(b)\ni(t,k)bj(t,−k)/angbracketrightBig\n, (A4)\nwhere\nT(b)\nj= [∇×(u×b−∝angb∇acketleftu×b∝angb∇acket∇ight)]j. (A5)\nEquations (A1)-(A2) for the second moment includes\nthe first-order spatial differential operators applied to th e\nthird-order moments ˆM(b)g(III)\nij(k) andˆM(b)h(III)\nij(k). A\nproblem arises how to close the system, i.e., how to express\nthe third-order moments through the lower moments, gij\nandhijdenoted as F(II). We use the spectral τapproxi-\nmation which postulates that the deviations of the third-\norder moments, denoted as ˆMF(III)(k), from the contribu-\ntions to these terms afforded by a background turbulence,\nˆMF(III,0)(k), can be expressed through the similar devia-\ntions of the second moments, F(II)(k)−F(II,0)(k) as\nˆMF(III)(k)−ˆMF(III,0)(k) =−1\nτr(k)/bracketleftBig\nF(II)(k)\n−F(II,0)(k)/bracketrightBig\n, (A6)\nwhereτr(k) is the scale-dependent relaxation time, which\ncan be identified with the correlation time τ(k) of the tur-\nbulent velocity field for large fluid and magnetic Reynoldsnumbers. The functions with the superscript (0) corre-\nspond to the background turbulence with a zero mean\nmagnetic field. Validation of the τapproximation for\ndifferent situations has been performed in various nu-\nmerical simulations (Brandenburg et al. 2004, 2008, 2012;\nBrandenburg & Subramanian 2005b,c,a; R¨ adler et al. 2011;\nRogachevskii et al. 2011, 2012, 2018; Haugen et al. 2012;\nElperin et al. 2017). When the mean magnetic field is zero,\nthe turbulent electromotive force vanishes, which implies\nthatg(0)\nij(k) = 0. We also take into account magnetic fluctu-\nations caused by a small-scale dynamo (the dynamo with a\nzero mean magnetic field). Consequently, Eq. (A6) reduces\ntoˆM(b)g(III)\nij(k) =−gij(k)/τ(k) and ˆM(b)h(III)\nij(k) =\n−[hij(k)−h(0)\nij(k)]/τ(k).\nWe assume that the characteristic time of variation\nof the second moments gij(k) andhij(k) are substantially\nlarger than the correlation time τ(k) for all turbulence\nscales. Therefore, in a steady-state Eqs. (A1) and (A2) yiel d\nthe following formulae for the cross-helicity tensor gij(k) =\n∝angb∇acketleftui(k)bj(−k)∝angb∇acket∇ight, and the function hij(k) =∝angb∇acketleftbi(k)bj(−k)∝angb∇acket∇ight:\ngij(k) =−τ(k)/braceleftbigg/bracketleftBig\ni/parenleftBig\nk·B/parenrightBig\n−1\n2/parenleftBig\nB·∇/parenrightBig/bracketrightBig/bracketleftBig\nfij(k)\n−hij(k)/bracketrightBig\n−Bj/parenleftBig\nikn−1\n2∇n/parenrightBig\nfin(k)/bracerightbigg\n, (A7)\nhij(k) =h(0)\nij(k)+τ2(k)/parenleftBig\nk·B/parenrightBig/bracketleftbigg\n2/parenleftBig\nk·B/parenrightBig\nfij(k)\n−kn/parenleftBig\nBjfin(k)+Bifnj(k)/parenrightBig/bracketrightbigg\n. (A8)\nIn Eqs. (A7)–(A8) we neglect small contributions propor-\ntional tospatial derivatives ofthemeanmagnetic field.Sin ce\nwe consider a one way coupling (i.e., we do not consider the\nalgebraic quenching of the turbulent fluxes of the magnetic\nhelicity), the correlation functions fijandhijin the right-\nhand sides of Eqs. (A7)–(A8) should be replaced by f(0)\nijand\nh(0)\nij, respectively.\nWe use the following model for the second moment,\nf(0)\nij(k,R) =∝angb∇acketleftui(k)uj(−k)∝angb∇acket∇ight(0)of velocity fluctuations in\ndensity stratified and helical turbulence in a Fourier space\n(R¨ adler et al. 2003):\nf(0)\nij=Eu(k)\n8πk2/braceleftbigg/bracketleftBig\n(δij−kij)+i\nk2/parenleftbig˜λikj−˜λjki/parenrightbig/bracketrightBig/angbracketleftbig\nu2/angbracketrightbig\n−1\nk2/bracketleftBig\niεijpkp+(εjpmkip+εipmkjp)˜λm/bracketrightBig\nHu/bracerightbigg\n,(A9)\nwhereδijis the Kronecker tensor, kij=kikj/k2and\n˜λm=λm− ∇m/2. The energy spectrum function Eu(k)\nof velocity fluctuations in the inertial range of turbulence\nis given by Eu(k) = (q−1)k−1\n0(k/k0)−q, where the ex-\nponentq= 5/3 corresponds to the Kolmogorov spectrum,\nk0/lessorequalslantk/lessorequalslantkν, the wave number k0= 1/ℓ0, the length ℓ0\nis the maximum scale of random motions, the wave num-\nberkν=ℓ−1\nν, the length ℓν=ℓ0Re−3/4is the Kolmogorov\n(viscous) scale. The expression for the turbulent correlat ion\ntime is given by τ(k) = 2τ0(k/k0)1−q, whereτ0=ℓ0/u0is\nthe characteristic turbulent time. In Eq. (A9) we take into\naccount inhomogeneity of the kinetic helicity.\nThe model for the second moment, h(0)\nij(k,R) =\n∝angb∇acketleftbi(k)bj(−k)∝angb∇acket∇ight(0), of magnetic fluctuations in a Fourier space\n©0000 RAS, MNRAS 000, 000–00010N. Kleeorin and I. Rogachevskii\nis analogous to equation (A9)\nh(0)\nij=1\n8πk2/braceleftbigg\nEb(k)(δij−kij)/angbracketleftbig\nb2/angbracketrightbig\n−1\nk2/bracketleftbigg\niεijpkp\n−1\n2(εjpmkip+εipmkjp)∇m/bracketrightbigg\nHcδ(k−k0)/bracerightbigg\n,(A10)\nwhereHc=∝angb∇acketleftb·(∇×b)∝angb∇acket∇ightis the current helicity, Eb(k) =\n(qm−1)k−1\nb(k/kb)−qmis the magnetic energy spectrum\nfunction in the range kb/lessorequalslantk/lessorequalslantkη, the wave number\nkb= 1/ℓb, the length ℓbis the maximum scale of magnetic\nfluctuations caused by the small-scale dynamo, and the ex-\nponentqm= 5/3 corresponds to the Kolmogorov spectrum\nfor the magnetic energy. In Eq. (A10) we take into account\ninhomogeneity of the current helicity. We also take into ac-\ncount that due to the realizability condition, the current\nhelicity of the small-scale field is located at the integral t ur-\nbulence scale (Kleeorin & Rogachevskii 1999).\nFor the integration over angles in k-space we use the\nfollowing integrals:\n/integraldisplay2π\n0dϕ/integraldisplayπ\n0sinϑdϑk ij=4π\n3δij, (A11)\n/integraldisplay2π\n0dϕ/integraldisplayπ\n0sinϑdϑk ijmn=4π\n15∆ijmn, (A12)\n/integraldisplay2π\n0dϕ/integraldisplayπ\n0sinϑdϑk ijmnpq=4π\n105∆ijmnpq,(A13)\nwhere\n∆ijmn=δijδmn+δimδjn+δinδjm, (A14)\n∆ijmnpq= ∆mnpqδij+∆jmnqδip+∆imnqδjp\n+∆jmnpδiq+∆imnpδjq+∆ijmnδpq−∆ijpqδmn,\n(A15)\nandkij=kikj/k2,kijmn=kikjkmkn/k4and\nkijmnpq=kikjkmknkpkq/k6. We also take into account\nthat ∆ ijmm= 5δijand ∆ ijmnpp= 7∆ijmn.\nFor the integration over kwe use the following integrals\nfor large Reynolds numbers, Re= u0ℓ0/ν≫1:\n/integraldisplaykν\nk0τ(k)Eu(k)dk=τ0, (A16)\n/integraldisplaykν\nk0τ(k)Eu(k)\nk2dk=q−1\nqτ0ℓ2\n0, (A17)\n/integraldisplaykν\nk0τ2(k)Eu(k)\nk2dk=4(q−1)\n3q−1τ2\n0ℓ2\n0, (A18)\n/integraldisplaykν\nk0τ2(k)Eu(k)dk=4\n3τ2\n0. (A19)\nUsing Eqs. (A7)–(A19), and integrating in kspace, we\ndetermine various contributions to the turbulent flux of the\nsmall-scale magnetic helicity, see Eqs. (12)–(17), and Ap-\npendixD. The details ofthe derivations of theeffect of large -\nscale shear on turbulent fluxes of the magnetic helicity are\ndiscussed in Appendix C.APPENDIX B: DERIVATION OF EQUATIONS\nFOR THE SECOND MOMENTS\nIn this Appendix we derive Eqs. (A1)–(A2) for the cross\nhelicity tensor gij(k) =∝angb∇acketleftui(t,k)bj(t,−k)∝angb∇acket∇ightand the tensor\nhij(k) =∝angb∇acketleftbi(t,k)bj(t,−k)∝angb∇acket∇ightfor magnetic fluctuations. To\nthis end, we perform several calculations that are similar\nto the following. We use the equation for magnetic fluctu-\nations obtained by subtracting equation for the mean mag-\nnetic field from the equation for the total field:\n∂b\n∂t−∇×(u×b−∝angb∇acketleftu×b∝angb∇acket∇ight)−η∆b= (B·∇)u−(u·∇)B.\n(B1)\nThe source term, ( B·∇)u, in the right hand side of Eq. (B1)\nin a Fourier space reads:\n/bracketleftbig/parenleftbig\nB·∇/parenrightbig\nuj/bracketrightbig\nk= ikp/integraldisplay\nBp(Q)uj(k−Q)dQ,(B2)\nso that the induction equation for bj(k2) inkspace is given\nby:\n∂bj(k2)\n∂t= ik(2)\np/integraldisplay\nBp(Q)uj(k2−Q)dQ\n−un(k2)∇nBj+N(b)\nj(k2), (B3)\nwherek(2)≡k2=−k+K/2. We use the identity:\n∂\n∂t∝angb∇acketleftui(k1,t)bj(k2,t)∝angb∇acket∇ight=/angbracketleftbigg∂ui(k1,t)\n∂tbj(k2,t)/angbracketrightbigg\n+/angbracketleftbigg\nui(k1,t)∂bj(k2,t)\n∂t/angbracketrightbigg\n. (B4)\nFirst we derive equation for the second term in the right\nhand side of Eq. (B4). To this end, we multiply Eq. (B3)\nbyui(k1) and averaging over ensemble of turbulent velocity\nfield, where k1=k+K/2. This yields:\n/angbracketleftbigg\nui(k1)∂bj(k2)\n∂t/angbracketrightbigg\n= i (−kp+Kp/2)/integraldisplay\ndQBp(Q)\n×∝angb∇acketleftui(k1)uj(k2−Q)∝angb∇acket∇ight−∝angb∇acketleftui(k1)un(k2)∝angb∇acket∇ight ∇nBj\n+/angbracketleftBig\nui(k1)N(b)\nj(k2)/angbracketrightBig\n, (B5)\nwhere for brevity of notations we omit the argument tin the\nvelocity and magnetic fields. Next, we perform in Eq. (B5)\ntheFourier transformation inthe large-scale variable K, i.e.,\nwe use the transformation\nF(R) =/integraldisplay\nF(K)exp(iK·R)dK.\nThe first term Sij(k,R) in the right hand side of the ob-\ntained equation [which originates from the first term in the\nright hand side of Eq. (B3)], is given by:\nSij(k,R) = i/integraldisplay /integraldisplay\nBp(Q) (−kp+Kp/2) exp(iK·R)\n×∝angb∇acketleftui(k+K/2)uj(−k+K/2−Q)∝angb∇acket∇ightdKdQ.(B6)\nNext, we introduce new variables:\n˜k= (˜k1−˜k2)/2 =k+Q/2,\n˜K=˜k1+˜k2=K−Q, (B7)\nwhere\n˜k1=k+K/2,˜k2=−k+K/2−Q. (B8)\n©0000 RAS, MNRAS 000, 000–000Turbulent magnetic helicity fluxes in solar convective zone 11\nTherefore, Eq. (B6) in the new variables reads\nSij(k,R) = i/integraldisplay /integraldisplay\nfij(k+Q/2,K−Q)Bp(Q)\n×(−kp+Kp/2) exp(i K·R)dKdQ. (B9)\nSince|Q| ≪ |k|, we use the Taylor expansion\nfij(k+Q/2,K−Q)≃fij(k,K−Q)\n+1\n2∂fij(k,K−Q)\n∂ksQs+O(Q2), (B10)\nand the following identity:\n∇p[fij(k,R)Bp(R)] = i/integraldisplay\ndKKp[fij(k,R)Bp(R)]K\n×exp(iK·R), (B11)\nwhere\n[fij(k,R)Bp(R)]K=/integraldisplay\nfij(k,K−Q)Bp(Q)dQ.\n(B12)\nTherefore, Eqs. (B9)–(B11) yield\nSij(k,R)≃/bracketleftbigg\n−i(k·B)+1\n2(B·∇)/bracketrightbigg\nfij(k,R)\n−1\n2kp∂fij(k)\n∂ks∇sBp. (B13)\nWe take into account that the terms in gij(k,R) with\nsymmetric tensors with respect to the indexes ”i” and ”j”\ndo not contribute to the turbulent electromotive force be-\ncauseEm=εmij/integraltext\ngij(k,R)dk. Ingij(k,R) we also neglect\nthe second and higher derivatives over R. This procedure\nyields Eq. (A1). Similar calculations are performed to de-\nrive Eq. (A2).\nTodetermine variouscontributions totheturbulentflux\nof small-scale magnetic helicity, we use the following iden ti-\nties:\n/parenleftbig\n∆−1/parenrightbig\nk1=−k−2/bracketleftbigg\n1+i(k·∇)\nk2/bracketrightbigg\n, (B14)\n/parenleftbig\n∆−1/parenrightbig\nk2=−k−2/bracketleftbigg\n1−i(k·∇)\nk2/bracketrightbigg\n. (B15)\nAPPENDIX C: EFFECT OF LARGE-SCALE\nSHEAR\nIn this Appendix we determine the effect of large-scale\nshear on turbulent fluxes of the magnetic helicity. The\ncross-helicity tensor g(S)\nij(k) =∝angb∇acketleftvi(k)bj(−k)∝angb∇acket∇ightin turbulence\nwith large-scale shear is given by (Rogachevskii & Kleeorin\n2004):\ng(S)\nij(k) =−iτ(k·B)/bracketleftbigg\nf(S)\nij(k)−h(S)\nij(k)\n4πρ\n+τ Jijmn(U)/parenleftBigg\nf(0)\nmn(k)−h(0)\nmn(k)\n4πρ/parenrightBigg/bracketrightbigg\n, (C1)\nwhere the effect of large-scale shear on the tensors f(S)\nij(k) =\n∝angb∇acketleftvi(k)vj(−k)∝angb∇acket∇ightandh(S)\nij(k) =∝angb∇acketleftbi(k)bj(−k)∝angb∇acket∇ightis determined\nby\nf(S)\nij(k) =τ Iijmn(U)f(0)\nmn(k), (C2)h(S)\nij(k) =τ Eijmn(U)h(0)\nmn(k), (C3)\nand the tensors Iijmn(U),Eijmn(U) andJijmn(U) are\ngiven by\nIijmn(U) =/braceleftbigg\n2kiqδmpδjn+2kjqδimδpn−δimδjqδnp\n−δiqδjnδmp+4kpqδimδjn+δimδjnkq∂\n∂kp\n−iλr\n2k2/bracketleftbigg/parenleftBig\nkiδjnδpm−kjδimδpn/parenrightBig/parenleftBig\n2krq−δrq/parenrightBig\n+kq/parenleftBig\nδipδjnδrm−δimδjpδrn/parenrightBig\n−2kpq/parenleftBig\nkiδjnδrm\n−kjδimδrn/parenrightBig/bracketrightbigg/bracerightbigg\n∇pUq, (C4)\nEijmn(U) =/bracketleftbigg\nδimδjqδpn+δiqδjnδpm\n+δimδjnkq∂\n∂kp/bracketrightbigg\n∇pUq, (C5)\nJijmn(U) =/braceleftbigg\n2kiqδjnδpm−δiqδjnδpm+δimδjqδpl\n+2kpqδimδjn+δimδjnkq∂\n∂kp−iλr\n2k2/bracketleftbigg\nkiδjnδpm\n×/parenleftBig\n2krq−δrq/parenrightBig\n+δjnδrm/parenleftBig\nkqδip−2kikpq/parenrightBig/bracketrightbigg/bracerightbigg\n∇pUq.\n(C6)\nUsing Eqs. (A9)–(A19), (B14)–(B15), and Eqs. (C1)–(C6),\nand integrating in kspace, we determine various contribu-\ntions to the turbulent flux of the small-scale magnetic he-\nlicity caused by the differential rotation, see Eq. (18) and\nAppendix D.\nAPPENDIX D: GENERAL FORM OF\nTURBULENT TRANSPORT COEFFICIENTS\nApplyingthemethoddescribed inAppendixesA–C,we have\ndetermined various contributions tothe turbulentfluxof th e\nsmall-scale magnetic helicity. In particular, the general form\nof turbulent flux of the small-scale magnetic helicity is giv en\nby\nF(m)\ni=V(H)\niHm−D(H)\nij∇jHm+N(α)\niαK\n+M(α)\nij∇jαK+F(S0)\ni, (D1)\nwhere the turbulent transport coefficients are given below.\nThe turbulent pumping velocity V(H)of the small-scale\nmagnetic helicity is\nV(H)=−1\n15τ0V2\nA/braceleftbigg\nλ+7β(β·λ)+1\n7τ0/bracketleftbigg\n28(W×λ)\n+139\n2(β·λ)(W×β)−2Q(λ)+β/parenleftBig\n17W·(β×λ)\n+58λ·Q(β)/parenrightBig\n−31Q(β)(β·λ)−3λ(β·Q(β))\n−7(β×λ)(β·W)/bracketrightbigg/bracerightbigg\n. (D2)\n©0000 RAS, MNRAS 000, 000–00012N. Kleeorin and I. Rogachevskii\nHereβ=B/Bis the unit vector along the mean magnetic\nfield,VA=B/(4πρ)1/2is the mean Alfv´ en speed, W=\n∇×Uis the mean vorticity, the vectors Q(β)andQ(λ)are\ndefined as Q(β)\ni=βm(∂U)miandQ(λ)\ni=λm(∂U)mi, and\nthe gradient of the mean velocity ∇iUjis decomposed into\nsymmetric, ( ∂U)ij= (∇iUj+∇jUi)/2, and antisymmetric,\nεijpWp/2 parts, i.e., ∇iUj= (∂U)ij+εijpWp/2.\nThe total diffusion tensor D(H)\nijthat describes turbulent\nmagneticdiffusionofthesmall-scale magnetichelicity,re ads:\nD(H)\nij=D(H)\nTδij+1\n30τ0V2\nA/braceleftbigg\n5δij−4βiβj+τ0/bracketleftbigg\n8εijp\n×(W·β)βp+8βi(β×W)j+14βj(β×W)i\n+4εiqmεjpnβmβn(∂U)pq+1\n7/parenleftbigg\n8(q+1)(∂U)ij\n+2(41+34 q)βiQ(β)\nj+2(1−6q)βjQ(β)\ni+(1+8q)δij\n×(β·Q(β))/parenrightbigg/bracketrightbigg/bracerightbigg\n+τ0\n2/bracketleftbigg\nηT+8\n15τ0V2\nA/bracketrightbigg\nεijpWp.(D3)\nIn derivation of Eqs. (D2)–(D3), we take into account that\nHc=Hm/ℓ2\n0, and we neglect small terms ∼O[ℓ2\n0/L2\nm] with\nLmbeing characteristic scale of spatial variations of Hm.\nThe turbulent magnetic helicity flux also includes the sourc e\ntermN(α)αKcausedbythekinetic αeffect with N(α)being\nN(α)=−1\n10ℓ2\n0B2/braceleftbigg\nλ+7q−2\nq(β·λ)β+(q−1)τ0\n(3q−1)\n×/bracketleftbigg\n10(β×W) (β·λ)−37(W·β)(β×λ)−4Q(λ)\n−4(β×Q(β,λ))+2\n7/parenleftbigg\n19β[(β×W)·λ]−4Q(β)\n×(β·λ)−24β(λ·Q(β))+4λ(β·Q(β))/parenrightbigg/bracketrightbigg/bracerightbigg\n,(D4)\nwhereQ(β,λ)\ni= (β×λ)m(∂U)mi. The contribution to the\nturbulentmagnetichelicityflux, ∝ −ℓ2\n0B2λαK[seethefirst\nterm in equation (D4)], caused by the kinetic αeffect, has\nbeen suggested by Kleeorin et al. (2000, 2002, 2003a).\nThe turbulent magnetic helicity flux contains also the\nsource term M(α)\nij∇jαKcaused by the gradient ∇jαKof the\nkineticαeffect with M(α)\nijbeing\nM(α)\nij=1\n20qℓ2\n0B2/braceleftbigg\n(2q−1)δij+(20q−23)βiβj\n+16q(q−1)τ0\n3q−1/bracketleftbigg\nβi(β×W)j+(W·β)εijpβp/bracketrightbigg/bracerightbigg\n.\n(D5)\nThe additional contribution F(S0)to the turbulent magnetic\nhelicity flux caused by the large-scale shear (differential r o-\ntation) is given by\nF(S0)=−q−1\n3(q+1)ℓ2\nb/angbracketleftbig\nb2/angbracketrightbig\nW+2\n45ℓ2\n0B2/bracketleftBig\n11ǫW\n+(3ǫ−10)(β·W)β+(β×Q(β))[8q+35\n+ǫ(8q−20)]/bracketrightBig\n. (D6)\nHereǫ=ℓ2\nb/angbracketleftbig\nb2/angbracketrightbig\n/(ℓ2\n04πρ/angbracketleftbig\nu2/angbracketrightbig\n), andℓbis the energy con-\ntaining scale of magnetic fluctuations with a zero mean mag-\nnetic field. The contribution to the turbulent magnetic he-\nlicity flux, ∝ℓ2\n0B2(β×Q(β)) [see the last term in equa-tion (D6)], caused by the large-scale shear, has been derive d\nby Brandenburg & Subramanian (2005a), using a general\nexpression originally suggested by Vishniac & Cho (2001).\nTo derive equations for the turbulent magnetic helicity\nflux due to the differential rotation in spherical coordinate s,\nwe use the identities given below. The large-scale shear ve-\nlocityU=δΩ×ris caused bythe differential (non-uniform)\nrotation, that is in spherical coordinates ( r,ϑ,ϕ) reads\nδΩ =δΩ(r,ϑ)(cosϑ,−sinϑ,0), (D7)\nand the stress tensor ( ∂U)ijreads\n(∂U)ij=rn\n2(εimn∇j+εjmn∇i)δΩm. (D8)\nThe vectors Q(β)andQ(λ)defined as Q(β)\ni=βm(∂U)mi\nandQ(λ)\ni=λm(∂U)mi, are given by\nQ(β)= (r×β)m(∇δΩm)−r×(β·∇)δΩ,(D9)\nQ(λ)=−r×(λ·∇)δΩ, (D10)\nwhereλ=λerandβ=B/B= (βr,βϑ,βϕ). We also use\nthe identity\nεiqmεjpnβmβn(∂U)pq=1\n2(r·β)/bracketleftBig\n(β×∇)iδΩj\n+(β×∇)jδΩi/bracketrightBig\n−1\n2βm/bracketleftBig\nri(β×∇)j\n+rj(β×∇)i/bracketrightBig\nδΩm. (D11)\nWe have taken into account that/parenleftBig\nβ×Q(β)/parenrightBig\nr= O(∇δΩ),\ni.e it does not contain contributions ∝δΩ, but it includes\ntheir spatial derivatives, ∇δΩ. Using Eqs. (D1)–(D11), we\ndetermine various contributions to the turbulent flux of the\nsmall-scale magnetic helicity in spherical coordinates, s ee\nEqs. (11)–(18).\nAPPENDIX E: TURBULENT TRANSPORT\nCOEFFICIENTS IN THE CARTESIAN\nCOORDINATES\nFor better understanding of the physics related to vari-\nous contributions to the turbulent flux of the small-scale\nmagnetic helicity [see Eqs. (D1)–(D11)], we consider a\nsmall-scale turbulence with large-scale linear velocity s hear\nU= (0,Sx,0) in the Cartesian coordinates. In this case,\nthe large-scale vorticity is W= (0,0,S), the stress ten-\nsor (∂U)ij= (S/2)(ex\niey\nj+ex\njey\ni), the vector λthat\ndescribes the non-uniform mean fluid density, is λ=\nλ(sinϑ,0,cosϑ), the unit vector along the large-scale mag-\nnetic isβ= (cos˜β,sin˜β,0), the vector Q(β)\ni=βm(∂U)mi=\n(S/2)(sin˜β,cos˜β,0) and the vector Q(λ)\ni=λm(∂U)mi=\n(λS/2) sinϑey\ni. We also take into account that\nβ×λ=λ(cosϑsin˜β,−cosϑcos˜β,−sinϑsin˜β),(E1)\n(β×Q(β))i= (S/2) cos(2˜β)ez\ni, (E2)\n(β×Q(λ))i= (Sλ/2) sinϑcos˜βez\ni, (E3)\n©0000 RAS, MNRAS 000, 000–000Turbulent magnetic helicity fluxes in solar convective zone 13\nβ×W=S(sin˜β,−cos˜β,0), (E4)\n(W×λ)i=Sλsinϑey\ni. (E5)\nFirst, we determine various contributions to the turbu-\nlent flux of the magnetic helicity inside the turbulent regio n\nwhere the toroidal mean magnetic field is much larger than\nthe poloidal mean magnetic field, i.e., β= (0,1,0). In this\ncase, the turbulent pumping velocity V(H)of the small-scale\nmagnetic helicity is\nV(H)=−1\n15τ0V2\nAλ/bracketleftbigg/parenleftbigg\n1+3\n14Sτ0/parenrightbigg\neλ+5.6Sτ0ey/bracketrightbigg\n,\n(E6)\nwhereeλ=λ/λ. The turbulent magnetic helicity flux has\nthe source term N(α)αKcaused by the kinetic αeffect with\nN(α)being\nN(α)=−1\n10ℓ2\n0B2λ/bracketleftbigg\n1−4(q−1)\n7(3q−1)Sτ0/bracketrightbigg\n. (E7)\nThe total diffusion tensor D(H)\nijwhich describes the micro-\nscopic and turbulent magnetic diffusion of the small-scale\nmagnetic helicity is given by:\nD(H)\nij=D1δij−D2ey\niey\nj+D3ex\niey\nj−D4ey\niex\nj,(E8)\nwhereD2= (2/15)τ0V2\nA,\nD1=D(H)\nT+1\n3η+1\n6τ0V2\nA/bracketleftbigg\n1−1+8q\n70Sτ0/bracketrightbigg\n,(E9)\nD3=1\n2Sτ0/bracketleftbigg\nηT+159−6q\n105τ0V2\nA/bracketrightbigg\n, (E10)\nD4=1\n2Sτ0/bracketleftbigg\nηT−34q+45\n105τ0V2\nA/bracketrightbigg\n. (E11)\nEquation(E8)impliesthat D(H)\nxx=D(H)\nzz=D1,D(H)\nyy=D1−\nD2,D(H)\nxy=D3,D(H)\nyx=−D4, and other components of the\ntotal diffusion tensor D(H)\nijvanish. The turbulent magnetic\nhelicity flux containing the source term M(α)\nij∇jαKwith\nM(α)\nijbeing\nM(α)\nij=1\n20qℓ2\n0B2/bracketleftbigg\n(2q−1)δij+(20q−23)ey\niey\nj\n+16q(q−1)\n3q−1Sτ0ey\niex\nj/bracketrightbigg\n. (E12)\nThe additional contribution F(S0)to the turbulent magnetic\nhelicity flux caused by the large-scale shear is given by\nF(S0)=−/bracketleftbiggq−1\n3(q+1)−22\n45V2\nA\n∝angb∇acketleftu2∝angb∇acket∇ight/bracketrightbigg\nℓ2\nb/angbracketleftbig\nb2/angbracketrightbig\nSez.(E13)\nNow we determine various contributions to the turbu-\nlent flux of the magnetic helicity at the surface (the up-\nper boundary of the turbulent region), where the toroidal\nmean magnetic field is much smaller than the poloidal mean\nmagnetic field, i.e., β= (1,0,0). In this case, the turbulentpumping velocity V(H)of the small-scale magnetic helicity\nis\nV(H)=−1\n15τ0V2\nAλ/bracketleftbigg\neλ+7 sinϑ/parenleftbigg\nex+81\n49Sτ0ey/parenrightbigg/bracketrightbigg\n.\n(E14)\nThe turbulent magnetic helicity flux has the source term\nN(α)αKcaused by the kinetic αeffect with N(α)being\nN(α)=−1\n10ℓ2\n0B2λ/bracketleftbigg\neλ+7q−2\nqsinϑex\n−2(q−1)\n3q−1Sτ0/parenleftbigg\nez+44\n7sinϑey/parenrightbigg/bracketrightbigg\n. (E15)\nThe total diffusion tensor D(H)\nijwhich describes the micro-\nscopic and turbulent magnetic diffusion of the small-scale\nmagnetic helicity is given by:\nD(H)\nij=D1δij−D2ex\niex\nj+D3ex\niey\nj−D4ey\niex\nj,(E16)\nwhereD2= (2/15)τ0V2\nA,\nD1=D(H)\nT+1\n3η+1\n6τ0V2\nA, (E17)\nD3=1\n2Sτ0/bracketleftbigg\nηT+49+42q\n105τ0V2\nA/bracketrightbigg\n, (E18)\nD4=1\n2Sτ0/bracketleftbigg\nηT+145−2q\n105τ0V2\nA/bracketrightbigg\n. (E19)\nEquation (E16) implies that D(H)\nyy=D(H)\nzz=D1,D(H)\nxx=\nD1−D2,D(H)\nxy=D3,D(H)\nyx=−D4, and other components of\nthe total diffusion tensor D(H)\nijvanish. The turbulent mag-\nnetic helicity flux containing the source term M(α)\nij∇jαK\nwithM(α)\nijbeing\nM(α)\nij=1\n20qℓ2\n0B2/bracketleftbigg\n(2q−1)δij+(20q−23)ex\niex\nj\n−16q(q−1)\n3q−1Sτ0ex\niey\nj/bracketrightbigg\n. (E20)\nThe additional contribution F(S0)to the turbulentmagnetic\nhelicity flux caused by the large-scale shear is given by\nF(S0)=1\n3/bracketleftbigg8q+35\n15ℓ2\n0B2−ℓ2\nb/angbracketleftbig\nb2/angbracketrightbig/parenleftbiggq−1\nq+1\n−2(4q+1)\n15V2\nA\n∝angb∇acketleftu2∝angb∇acket∇ight/parenrightbigg/bracketrightbigg\nSez. (E21)\nThispaperhas beentypesetfrom aT EX/LATEXfileprepared\nby the author.\n©0000 RAS, MNRAS 000, 000–000"    },    {        "title": "0801.0347v2.On_scaling_laws_in_turbulent_magnetohydrodynamic_Rayleigh_Benard_convection.pdf",        "content": "arXiv:0801.0347v2  [physics.flu-dyn]  12 Feb 2008On scaling laws in turbulent magnetohydrodynamic Rayleigh -Benard convection\nSagar Chakraborty∗\nS.N. Bose National Centre for Basic Sciences\nSaltlake, Kolkata 700098, India\n(Dated: November 5, 2018)\nWe invoke the concepts of magnetic boundary layer and magnet ic Rayleigh number and use\nthe magnetic energy dissipation rates in the bulk and the bou ndary layers to derive some scaling\nlaws expressing how Nusselt number depends on magnetic Rayl eigh number, Prandtl number and\nmagnetic Prandtl number for the simple case of turbulent mag netohydrodynamic Rayleigh-Benard\nconvection in the presence of uniform vertical magnetic fiel d.\nPACS numbers: 47.55.P-, 47.65.-d\nTurbulent fluid convectionis an unsolvedproblem hav-\ning very wide applications in the study of convective pro-\ncessesin atmospheres,oceans, metallurgy etc.Ifthe fluid\nis conducting and is acted upon by magnetic field then a\ntheory for this magnetohydrodynamic fluid’s convection\nhopes to explain, in the long run, the convection pro-\ncesses in planetary core, stellar interior and; some other\nimportant astrophysical, industrial and geophysical situ-\nations. Complex nature of these realistic situations have\nprompted the researchers first to try to find a theory\nfor the rather simpler problem of Rayleigh-Benard (RB)\nconvection[1], probablyalsobecausedoingexperimenton\nit is quite feasible.\nBriefly speaking, RB convection in MHD fluid is inves-\ntigated in a set up where a magnetofluid of density ρ,\nkinematic viscosity ν, conductivity σ, thermal diffusiv-\nityκ, magnetic permeability µand isobaric thermal ex-\npansion coefficient αis confined between two horizon-\ntal plates (conducting or non-conducting). Theoretically,\nthe plates are considered to have infinite extent; in real-\nity, this can be realised by making the thickness dof\nmagnetofluid between them very small compared to the\nlateral extent. The plates are maintained at a constant\nrelative temperature difference ∆. The constant acceler-\nation due to gravity /vector gis actingin the downwarddirection\nand the entire set up is acted upon by constant uniform\nvertical magnetic field /vectorB. Experiments[2, 3] on MHD\nRB convection has been done using gallium or mercury\nconfined between two copper plates. However, whether\nthe results of ref.[3] applies to this paper is a question\nbecause this paper is applicable for high Nusselt number\nflows unlike the reference.\nIf/vector u(x,y,z,t),p(x,y,x,t),T(x,y,z,t) and/vectorb(x,y,z,t) be\nthe velocityfield, the kinematic pressurefield (containing\nalso the external divergence-free force terms), the tem-\nperaturefieldandthe magneticfieldperturbationrespec-\ntively, thentheequationsdescribingtheconvectionunder\n∗Electronic address: sagar@bose.res.inthe Boussinesq’s approximation are:\n∂tui+uj∂jui=−∂ip+ν∂j∂jui+αgTδi3\n+1\nµρBj∂jbi (1)\n∂tbi+uj∂jbi=Bj∂jui+bj∂jui+1\nσµ∂j∂jbi(2)\n∂tT+uj∂jT=κ∂j∂jT (3)\nThe second order terms have been purposefully retained\nin these three equations. The boundary conditions at the\ntwo horizontalrigid plates, assumed perfectly conducting\n(electrically), are: a) /vector u= 0 atz= 0 and z=d, b)T=\n±∆/2 onz= 0 and drespectively and, c) hz= 0 on the\nplates. As the convection becomes turbulent, assuming\nthe existence of wind of turbulence i.e., there exists a\nmean large scale velocity Uthat stirs the magnetofluid in\nthe bulk, wecan definea Reynoldsnumber Realongwith\nother important non-dimensional parameters. These are\nlisted below:\nReynolds number: Re=Ud/ν\nRayleigh number: Ra=gα∆d3/κν\nChandrasekhar number: Q=σB2d2/ρν\nPrandtl number: Pr=ν/κ\nMagnetic Prandtl number: Pm=µσν\nNusselt number: Nu= (uzT−κ∂3T)/(κ∆/d)\nwhere the overline denotes average over any horizontal\nplane. By the way, in this paper we shall use angular\nbrackets to denote volume average.\nIt has been conjectured[4] that there are various regimes\nin MHD RB convectionwherein the scaling lawsconnect-\ningNuwithRaandQare different; this fact has been\nverified later by experiments[2]. We shall take the re-\nsults of these experiments at their face values although\nsome deeper meaning behind them must exist; most im-\nportantly we shall make the following two observations\nfrom the results of that paper:\n(i) TheNumostly depends on some power of the ra-\ntioRa/Q. This ratio basically is a sort of Rayleigh\nnumber — which we shall call magnetic Rayleigh2\nnumber Rb— constructed with magnetic viscos-\nityσB2d2/ρthat essentially is the manifestation of\nJoule damping present in the magnetofluid.\n(ii) It is surprising that Nu−1, rather than Nu, seems\nto be proportional to some powers of Rb.\nOf late, a paper[5] on turbulent RB convection in or-\ndinary fluids has presented a unifying theory on how\nNudepends on Raby constructing four regimes each of\nwhich is defined depending on what (the boundary layers\nor the bulk of the fluid) the dominant contributors to the\nkinetic and the thermal dissipation rates are. We shall\nclosely follow this very paper extending the arguments\nand the assumptions to the turbulent RB convection in\nmagnetofluidinthepresenceofuniformverticalmagnetic\nfield keeping in mind the observations (i) and (ii) made\nabove.\nTo begin with, we define εu=ν/angb∇acketleft(∂iuj)(∂iuj)/angb∇acket∇ight,εT=\nκ/angb∇acketleft(∂iT)(∂iT)/angb∇acket∇ightandεb= (µ2σρ)−1/angb∇acketleft(∂ibj)(∂ibj)/angb∇acket∇ight. Using\nequations (1) and (2), and the definition of the Nu, we\narrive at:\nεu+εb=ν3\nd4Ra\nPr2(Nu−1) (4)\nSimilarly, usage of equation (3) yields:\nεT=κ/parenleftbigg∆\nd/parenrightbigg2\nNu (5)\nWe shall now clearly state the main assumptions in-\nvolved. The very first and most important assumption\nof this paper is:\nεu∼ν3\nd4Ra\nPr2(Nu−1) (6)\nεb∼ν3\nd4Ra\nPr2(Nu−1) (7)\nwhere (and henceforth, ‘ ∼’ means ‘scales as’). This ex-\nplains (or is explained by) the observation (ii) above.\nSecondly, we assume that there is a magnetic boundary\nlayer (BL) of thickness δbwhere the perturbation mag-\nnetic field is strongly affected by the magnetic diffusivity\nso that it grows to some definite value at the top of the\nlayer from zero on the surface of the plate. Thirdly, to\ngo with the observation (i) above we shall churn this the-\nory in such a manner that Nudepends on Rband not\nonRaexplicitly. Fourthly, we shall confine ourselves to\nthe laminar boundary layers only though the bulk of the\nmagnetofluid is in turbulent state. So the thickness of\nthe Blasius type kinetic BL δuand the thermal bound-\nary layer δTare given by δu∼d/√\nReandδT∼dNu−1\nrespectively. Lastly, we shall confine our attention to\nthe relevant cases where convection dominates conduc-\ntioni.e.,Nu≫1.\nArmed with these hypotheses, weproposethat theremay\nbe following eight possible different regimes depending\non whether the bulk or the respective boundary layers is\nchief contributor to dissipations:(A):εb(bl),εT(bl),εu(bl).\n(B):εb(bl),εT(bl),εu(bulk).\n(C):εb(bl),εT(bulk),εu(bl).\n(D):εb(bl),εT(bulk),εu(bulk).\n(E):εb(bulk),εT(bl),εu(bl).\n(F):εb(bulk),εT(bl),εu(bulk).\n(G):εb(bulk),εT(bulk),εu(bl).\n(H):εb(bulk),εT(bulk),εu(bulk).\nHere, ‘(bl)’ and ‘(bulk)’ respectively says that the respec-\ntive boundary layerand the bulk is dominant contributor\nto the dissipation concerned. These regimes can be pre-\npared by proper choice of Ra,PrandPm. For example,\nforlarge Raregime(H) ispossible; forsmall Prand large\nPmregime (F) is expected and so on.\nBefore proceeding further, let us estimate the width\nof the magnetic boundary layer. For Pm≪1 (i.e.,\nδu≪δb), we may neglect the first and the second terms\nin the R.H.S. of the equation (2) and under steady con-\nvection one is left to compare the second term of L.H.S.\nto the only unneglected term in the R.H.S: :\n|uj∂jbi| ∼/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nµσ∂j∂jbi/vextendsingle/vextendsingle/vextendsingle/vextendsingle(8)\n⇒UB\nδb∼B\nµσδ2\nb(9)\n⇒δb∼d\nRePm(10)\nNote that while estimating gradients we are using the\nmagnitude of the uniform external magnetic field B\nrather than the typical magnetic field fluctuations. This\nalso is an assumption in analogy with the standard as-\nsumption made in ordinary fluid RB convection where\nwhile estimating the temperature gradients ∆ is used\nfreely. For Pm≫1 (i.e.,δu≫δb), we must balance\nthe first and the third term in the R.H.S. of the equation\n(2):\n|Bj∂jui| ∼/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nµσ∂j∂jbi/vextendsingle/vextendsingle/vextendsingle/vextendsingle(11)\n⇒BU\nδu∼B\nµσδ2\nb(12)\n⇒δb∼d\nRe3\n4Pm1\n2(13)\nwhere the gradient of the velocity field in the kinematic\nboundary layer has been approximated by a linearly in-\ncreasing profile.\nNowunderthe assumptionsmade, followingcanbeeasily3\nderived\nεu(bulk)∼U3\nd=ν3\nd4Re3(14)\nForPr≪1, εT(bulk)∼U∆2\nd=κ/parenleftbigg∆\nd/parenrightbigg2\nPrRe(15)\nForPr≫1, εT(bulk)∼U∆2\ndδT\nδu\n∼κ/parenleftbigg∆\nd/parenrightbigg2PrRe3\n2\nNu(16)\nForPm≪1, εb(bulk)∼B2\n2µρU\nd∼ν3\nd4QRe\nPm(17)\nForPm≫1, εb(bulk)∼B2\n2µρU\ndδT\nδu∼ν3\nd4QRe3\n4\nPm3\n2(18)\nεu(bl)∼ν/parenleftbiggU\nδu/parenrightbigg2δu\nd∼ν3\nd4Re5\n2(19)\nThough we are interested in the limit of large and small\nPrandtl numbers (magnetic and non-magnetic), they can\nnot be too large or too small, for, then either the con-\nvective flow will be suppressed or the thermal and the\nmagnetic diffusivities will be so high that the respective\neffects will be the dominance of thermal conduction and\ntoo quick decay of magnetic fields to have any magnetic\neffect on the fluid. In writing the equation (17), it has\nbeenkeptinmindthatthemagneticenergyperunitmass\nB2/2µρis cascaded down the spatial scales by the turbu-\nlent eddies until it is dissipated at the lowest scale (the\nturbulent condition is taken to be stationarywith contin-\nuous feeding of magnetic energy into the magnetofluid)\nand in the equation (19), the factor δu/dcomes in to as-\nsert that only the kinematic boundary layers’ volume is\ninvolved. Attempttogetsimilarrelationsfor εT(bl)takes\none to the expression (5). Therefore, alternate relations\nmust be sought. Using the equation (3) and the argu-\nments given in the Landau’s text on fluid mechanics[6],\none readily arrives at:\nForPr≪1, Nu∼Re1\n2Pr1\n2 (20)\nForPr≫1, Nu∼Re1\n2Pr1\n3 (21)\nAgain, coming on the scaling expression for εb(bl) we\nwork out as follows: In the magnetic boundary layer if /vectorj\nis the current density, then from the relation /vector∇×/vectorb=µ/vectorj\none can estimate jasB/δbµ. This means that mean rate\nof dissipation of magnetic energy per unit mass j2/σρis\ngiven as:\nεb(bl) =B2\nσρµ2δ2\nb(22)\nThis equation alongwith with relations (10) and (13)\ngivesεb(bl) forPm≪1 andPm≫1 respectively.\nNow we are prepared to derive a range of scaling laws.\nWe shall demonstrate the line of attack for doing so by\nfinding out a illustrative scaling law for the regime (A);TABLE I: For Pm≪1 andPr≪1. (UD=Undetermined)\nRegime Exponent of RbExponent of PrExponent of Pm\nA1\n30 0\nB1\n30 0\nC UD UD UD\nD UD UD UD\nE 1 -1 1\nF 1 -1 1\nG UD UD UD\nH UD UD UD\nTABLE II: For Pm≪1 andPr≫1. (UD=Undetermined)\nRegime Exponent of RbExponent of PrExponent of Pm\nA1\n3−2\n90\nB1\n3−2\n90\nC3\n5−2\n50\nD3\n5−2\n50\nE 1 −4\n31\nF 1 −4\n31\nG 3 -4 3\nH 3 -4 3\nall other seven regimes can be treated similarly. For the\nregime (A) the dominant contributor to the magnetic en-\nergy dissipation are the magnetic BLs. Deciding to work\nforPm≫1 first and as Nu≫1, we equate the corre-\nsponding relation (22) (using the expression (13) for δb)\nto the expression (7) to get:\nν3\nd4RaNuPr−2∼B2\nσρµ2δ2\nb(23)\n⇒Nu∼1\nRb/parenleftbiggPr\nPm/parenrightbigg2/parenleftbiggd\nδb/parenrightbigg2\n(24)\n⇒Nu∼1\nRbPr2\nPmRe3\n2 (25)\nAgain in the regime (A), the dominant contributors to\nthe kinetic energy dissipation are the kinetic BLs so we\nequate relation (19) to relation (6) to get:\nRe5\n2=RaNuPr−2(26)\nAlso, they are the BLs — thermal BLs — that are con-\ntributing to the thermal energy dissipation, so relation\n(20) is of importance (we are considering Pr≪1 to\nbegin with) for this case. For other cases, where the con-\ntribution to the thermal energy dissipation comes from\nthe bulk, thecorresponding εT(bulk)will havetobe com-\npared with equation (5) to obtain the desired relations.\nSubstituting relation (20) into relation (26), we obtain:\nRe∼Ra1\n2Pr−3\n4 (27)\nwhich in turn when put into relation (26), yields:\nNu∼Ra1\n4Pr−1\n8 (28)4\nTABLE III: For Pm≫1 andPr≪1. (UD=Undetermined)\nRegime Exponent of RbExponent of PrExponent of Pm\nA1\n2−1\n81\n2\nB1\n2−1\n81\n2\nC UD UD UD\nD UD UD UD\nE 2 −5\n23\nF 2 −5\n23\nG UD UD UD\nH UD UD UD\nTABLE IV: For Pm≫1 andPr≫1. (UD=Undetermined)\nRegime Exponent of RbExponent of PrExponent of Pm\nA1\n2−1\n21\n2\nB1\n2−1\n21\n2\nC 1 -1 1\nD 1 -1 1\nE 2 -3 3\nF 2 -3 3\nG UD UD UD\nH UD UD UD\nNow, inaccordancewiththe observation(i), weeliminate\nRafrom relations (27) and (28) to write Rein terms of\nNuandPr. This is going to be the general strategy\nthroughout. Thus, for this example we have:\nRe∼Nu2Pr−1(29)\nPutting relation (29) in relation (25), we arrive at the\ndesired scaling law:\nNu∼Rb1\n2Pr−1\n8Pm1\n2 (30)The benefit of this strategy of finding scaling laws is\nthat the exponents of PrandPmare also predicted.\nMoreover, the method of classification of the regimes\nin the present-day experiments[2] differs from what\nhas been done theoretically here. Hence, each of those\nregimes may consist of an overlapping region of some of\nthe eight regimes proposed in this paper. This would\nsuggest that a fit of the form Nu=cRaa/Qb(a,b,care\njust numerical constants) usually done to represent the\nexperimental results can in principle be an equally good\nfit like: Nu=c1Rbd1+c2Rbd2+c3Rbd3+···; this of\ncourse is not mathematically impossible.\nIn closing, we list rest of the all mathematically possible\nscaling results (assuming Nu∼RbaPrbPmc) in four\ntables — table-I to table-IV — without showing explicit\nderivation whose strategy, anyway, has already been\nclearly outlined; and hope that experiments and simula-\ntions would be done in near future to test the conjecture\nproposed in this paper and the exponents of Prandtl\nnumber and magnetic Prandtl number will be concen-\ntrated upon more seriously. Only then one can say if\nthe idea presented herein, which seems to have a flavour\nof being a mere translation of the Grossmann-Lohse\ntheory[5] of thermal flow in Rayleigh-Benard geometry\nfor similar convection processes in magnetohydrody-\nnamic flows, is valid or not. We remind the readers\nthat it has not been possible for us to investigate the\nscaling laws when the laminar BLs turn turbulent: This,\nof course, should draw attention of the theoretical fluid\ndynamists.\nThe author would like to acknowledge his supervi-\nsor — Prof. J. K. Bhattacharjee — for the helpful and\nfruitful discussions. Also, CSIR (India) is gratefully\nacknowledged for awarding fellowship to the author.\n[1] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Sta-\nbility, (Dover Publications, Inc., New York), (1961).\n[2] S. Cioni, S. Chaumat and J. Sommeria, Phys. Rev. E 62,\nR4520 (2000).\n[3] J. M. Aurnou and P. L. Olson, J. Fluid Mech. 430, 283\n(2001).\n[4] J. K. Bhattacharjee, A. Das and K. Banerjee, Phys. Rev.A43, 1097 (1991).\n[5] S. Grossmann and D. Lohse, J. Fluid Mech. 407, 27\n(2000).\n[6] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Second\nEdition: Volume 6 (Course of Theoretical Physics) , (Reed\nEducational and Professional Publishing Ltd), (1987)."    },    {        "title": "1609.02582v1.Influence_of_non_magnetic_impurity_scattering_on_the_spin_dynamics_in_diluted_magnetic_semiconductors.pdf",        "content": "arXiv:1609.02582v1  [cond-mat.mes-hall]  8 Sep 2016Influence of non-magnetic impurity scattering on the spin dy namics in diluted\nmagnetic semiconductors\nM. Cygorek,1F. Ungar,1P. I. Tamborenea,2and V. M. Axt1\n1Theoretische Physik III, Universit¨ at Bayreuth, 95440 Bay reuth, Germany\n2Departamento de F´ ısica and IFIBA, FCEN, Universidad de Bue nos Aires,\nCiudad Universitaria, Pabell´ on I, 1428 Ciudad de Buenos Ai res, Argentina\nThe doping of semiconductors with magnetic impurities give s rise not only to a spin-spin inter-\naction between quasi-free carriers and magnetic impuritie s, but also to a local spin-independent\ndisorder potential for the carriers. Based on a quantum kine tic theory for the carrier and impurity\ndensity matrices as well as the magnetic and non-magnetic ca rrier-impurity correlations, the influ-\nence of the non-magnetic scattering potential on the spin dy namics in DMS after optical excitation\nwith circularly polarized light is investigated using the e xample of Mn-doped CdTe. It is shown that\nnon-Markovian effects, which are predicted in calculations where only the magnetic carrier-impurity\ninteraction is accounted for, can be strongly suppressed in the presence of non-magnetic impurity\nscattering. This effect can be traced back to a significant red istribution of carriers in k-space which\nis enabled by the build-up of large carrier-impurity correl ation energies. A comparison with the\nMarkov limit of the quantum kinetic theory shows that, in the presence of an external magnetic\nfield parallel to the initial carrier polarization, the asym ptotic value of the spin polarization at\nlong times is significantly different in the quantum kinetic a nd the Markovian calculations. This\neffect can also be attributed to the formation of strong corre lations which invalidates the semiclas-\nsical Markovian picture and it is stronger when the non-magn etic carrier-impurity interaction is\naccounted for. In an external magnetic field perpendicular t o the initial carrier spin, the correla-\ntions are also responsible for a renormalization of the carr ier spin precession frequency. Considering\nonly the magnetic carrier-impurity interaction, a signific ant renormalization is predicted for a very\nlimited set of material parameters and excitation conditio ns. Accounting also for the non-magnetic\ninteraction a relevant renormalization of the precession f requency is found to be more ubiquitous.\nPACS numbers: 75.78.Jp, 75.50.Pp, 75.30.Hx, 72.10.Fk\nI. INTRODUCTION\nMost of the devices based on the spintronics paradigm\nthat are commercially available today use the fact that\nspin-up and spin-down carriers exhibit different trans-\nmission and reflection probabilities at interfaces involv-\ning ferromagneticmetals1,2. However,someapplications,\nlike spin transistors3, require the controlnot only of spin-\nup and spin-down occupations, but also of the coher-\nent precession of spins perpendicular to the quantiza-\ntion axis provided by the structure. For this purpose,\nspintronic devices based on semiconductors are prefer-\nable to metallic structures since the dephasing time in\na metal is about three orders of magnitude shorter than\nin a semiconductor4. In the context of semiconductor\nspintronics5–7, a particularly interesting class of materi-\nals for future applications are diluted magnetic semicon-\nductors (DMS)8–22, which are obtained when semicon-\nductors are doped with transition metal elements, such\nas Mn, which act as localized magnetic moments. While\nsome types of DMS, such as Ga 1−xMnxAs, exhibit a fer-\nromagnetic phase8,23, other types of DMS, like the usu-\nally paramagnetic CdMnTe, are especially valued for the\nenhancement of the effective carrier g-factor by the gi-\nant Zeeman effect that can be used, e.g., to facilitate an\ninjection of a spin-polarized current into a light-emitting\ndiode24. Besides causing the giant Zeeman effect, the s-d\nexchange interaction between the quasi-free carriers andlocalized magnetic impurities also leads to other effects,\nsuch as inducing spin-flip scattering and thereby a direct\ntransfer of spins from the carriers to the impurities and\nvice versa25–28.\nTypically, the s-dinteraction is described by a Kondo-\nlike29localizedspin-spin interactionbetween carriersand\nimpurities. However, in real DMS materials, the in-\ntroduction of Mn impurities not only leads to a spin-\ndependent interaction Hamiltonian, but also to a spin-\nindependent local potential for the carriers30. The rea-\nsonfortheappearanceofthisspin-independentpotiential\nis that, in the case of Cd 1−xMnxTe, the semiconductor\nCdTe has a different band structure than MnTe and car-\nriers located at unit cells with Mn impurities experience\na larger local potiential energy than carriers at unit cells\nwith Cd cations. The strength of this local potential can\nbe estimated by the conduction and valence band offsets\nbetween CdTe and MnTe. Note, however, that usually,\nCdTe crystallizes in a zinc-blende structure, while MnTe\nis found in a wurzite structure. Thus, a better estima-\ntion for the strength of the spin-independent local po-\ntential is obtained by studying CdTe/Cd 1−xMnxTe het-\nerostructures where both materials appear in the form\nof a zinc-blende lattice31. From such investigations, the\nstrenght of the local spin-independent potential for car-\nriers at Mn sites of about 1.6 eV can be estimated. In\ncontrast, the spin-dependent local interaction in DMS\nis typically about 220 meV, i.e. one order of magnitude\nlower. Thisconsiderationsuggeststhatthenon-magnetic2\nimpurity scattering caused by the local spin-independent\ninteraction between carriersand impurities should not be\ndisregarded in the study of the spin physics in DMS.\nIt is noteworthy that a theory which takes into ac-\ncount a local magnetic interaction as well as a non-\nmagnetic local potential in a DMS, the V-J tight-binding\nmodel was employed to study the magnetic properties of\nGaMnAs32and it was found that taking into account\nthe non-magnetic interaction is necessary in order to ob-\ntain results in goodquantitative agreementwith ab initio\ncalculations for the Curie temperature and with experi-\nmental data for the optical conductivity.\nFor the spin dynamics, scattering at non-magnetic\nimpurities has already important consequences in non-\nmagnetic semiconductors33in the presence of spin-orbit\nfields, where scattering processes can enhance or reduce\nthe spin relaxation and dephasing significantly, e.g., via\nthe Elliott-Yafet34and D’yakonov-Perel’35mechanisms.\nThe goal of the present article is to investigatehow the\nnon-magneticinteractionbetween carriersandimpurities\naffects the spin dynamics in paramagnetic II-VI DMS.\nTo this end we employ a quantum kinetic theory for car-\nrier and impurity density matrices including the carrier-\nimpuritycorrelationsstartingfromasystemHamiltonian\nthat comprises a kinetic energy term, the magnetic and\nnon-magnetic carrier-impurity interactions as well as the\ncarrier and impurity Zeeman energies. Earlier quantum\nkinetic studies of the spin dynamics in DMS25–27,36,37,\nwhich only considered the spin-dependent s-dinterac-\ntion, predicted that in some cases, such as in narrow\nquantum wells optically excited very close to the band\nedge38, the spin transfer between carriers and impurities\ncannot be well described by rate equations. Rather, the\ntime evolution of the carrier spin is, in these cases, non-\nexponential and it can exhibit non-monotonic features\nsuch as overshoots. These effects are non-Markovian, as\nthey can be tracedback to the finite memory providedby\nthe correlations, since the Markovian assumption of a δ-\nlike memory leads to effective rate equations that predict\nan exponential spin dynamics28.\nHere, we find that these non-Markovian effects pre-\ndicted in the theory of Refs. 25–27, 36, and 37 are\nsuppressed in the case of the conduction band of a\nCd1−xMnxTe quantum well when non-magnetic scatter-\ning of carriers at the impurities is taken into account.\nWhile, in this case, the non-monotonic behavior of the\nspin dynamics disappears, the quantum kinetic theory\npredicts quantitative changes in the effective spin trans-\nfer rate compared with the Fermi’s golden rule value.\nThe suppression of the non-Markovian features is mainly\ncaused by a significant redistribution of carriers away\nfrom the band edge where the non-Markovian effects are\nparticularly strong38. This carrier redistribution is facil-\nitated by the build-up of strong carrier-impurity correla-\ntions providing a correlation energy of the order of a few\nmeV per electron that leads to an increase of the average\nkinetic electron energy by about the same amount. Due\nto the different strengths of the interactions in the con-duction band of Cd 1−xMnxTe, the non-magnetic carrier-\nimpurity correlation energy is also much larger than the\nmagnetic correlation energy studied before in Ref. 39.\nIn other cases, such as in the valence band of\nCd1−xMnxTe, the non-magnetic impurty scattering can\nbe much weaker than the magnetic spin-flip scattering\nand the non-Markovian effects prevail.\nIn the presence of an external magnetic field parallel\nto the initial carrier spin polarization, it was shown40\nthat a quantum kinetic treatment of the magnetic part\nof the carrier-impurity interaction in DMS leads to a sig-\nnificantly different asymptotic value of the carrier spin\npolarization at long times t. Because this is also a conse-\nquence of an energetic redistribution of carriers, includ-\ning non-magnetic scattering increases this effect. If the\ninitial carrier spin polarization is perpendicular to the\nexternal magnetic field, the carrier spins precess about\nthe effective field comprised of the external field and the\nmean field due to the impurity magnetization. As shown\nin Ref. 39, the carrier-impurity correlations built up by\nthe magnetic s-dinteraction renormalize the carrier spin\nprecession frequency. Here, we show that when both, the\nmagnetic and the non-magnetic interactions are taken\ninto account the renormalization of the carrier spin pre-\ncession frequency can be different in sign and magnitude\ncompared with calculations in which only the magnetic\ninteraction is considered.\nThe article is structured as follows: First, quantum\nkinetic equations of motion for the carrier and impurity\ndensity matrices as well as for the magnetic and non-\nmagnetic carrier-impuritycorrelations are formulated for\naDMS with magnetic and non-magneticcarrier-impurity\ninteractions. Then, we derive the Markov limit of the\nquantum kinetic theory which enables a comparison and\nallows us to distinguish the genuine quantum kinetic ef-\nfects from the Markovian behavior. Furthermore, from\nthe Markov limit we can derive analytic expressions for\nthe carrier-impurity correlation energies as well as the\ncorrelation-induced renormalization of the carrier spin\nprecession frequency. After having layed out the theory,\nwe present numerical simulations of the quantum kinetic\nequations for the conduction band of a Cd 1−xMnxTe\nquantum wellincluding magnetic andnon-magneticscat-\ntering at the Mn impurities and discuss the energetic re-\ndistribution of carriersas well as the correlationenergies.\nThen, we estimate the influence of non-magnetic impu-\nrity interaction on the spin dynamics in the valence band\nof Cd1−xMnxTe. Finally, we discuss the effects of the\nnon-magnetic impurity scattering on the spin dynamics\nin DMS in the presence of an external magnetic field\nparallel and perpendicular to an initial non-equilibrium\ncarrier spin polarization.3\nII. THEORY\nA. DMS Hamiltonian\nHere, we consider an intrinsic DMS such as\nCd1−xMnxTe in the presence of an external magnetic\nfield. The total Hamiltonian of this DMS is given by\nH=H0+Hsd+Himp+He\nZ+HMn\nZ, (1a)\nH0=/summationdisplay\nkσ¯hωkc†\nσkcσk, (1b)\nHsd=Jsd\nV/summationdisplay\nkk′σσ′/summationdisplay\nInn′Snn′·sσσ′c†\nσkcσ′k′ei(k′−k)RIˆPI\nnn′,\n(1c)\nHimp=J0\nV/summationdisplay\nkk′σ/summationdisplay\nJc†\nσkcσk′ei(k′−k)RJ, (1d)\nHe\nZ=/summationdisplay\nkσσ′¯hgeµBB·sσσ′c†\nσkcσ′k, (1e)\nHMn\nZ=/summationdisplay\nInn′¯hgMnµBB·Snn′ˆPI\nnn′, (1f)\nwhereH0is the single-electron Hamiltonian due to the\ncrystal potential, Hsddescribes the magnetic s-dex-\nchange interaction between the carriers and the impu-\nrities,Himpdescribes the spin-independent scattering of\ncarriers at impurities and He\nZandHMn\nZare the carrier\nand impurity Zeeman energies.\nIn Eqs. (1), c†\nσkandcσkdenote the creation and an-\nnihilation operators for conduction band electrons with\nwave vector kin the spin subband σ={↑,↓}. The\nmagnetic Mn impurities are described by the operator\nˆPI\nnn′=|I,n/an}bracketri}ht/an}bracketle{tI,n′|where|I,n/an}bracketri}htis then-th spin state\n(n∈ {−5\n2,−3\n2,...5\n2}) of theI-th magnetic impurity lo-\ncated at RI. The band structure of the semiconductor\nis described by ¯ hωk, which we assume to be parabolic\nωk=¯hk2\n2m∗with effective mass m∗.Vdenotes the vol-\nume of the sample. Jsdis thes-dcoupling constant for\nthe spin-spin interaction between carriers and impurities\nandJ0is the non-magnetic coupling constant. Sn1n2and\nsσ1σ2are the vectors with components consisting of spin-\n5\n2and spin-1\n2spin matrices for the impurities and the\nconduction band electrons, respectively, where the unit\n¯hhas been substituted into the definition of Jsdso that\nsσ1σ2=1\n2σσ1σ2, whereσσ1σ2are the Pauli matrices. Fi-\nnally,geandgMnare the g-factors of the electrons and\nthe impurities, respectively, and µBis the Bohr magne-\nton.\nIn order to account for spin-independent scattering\nnot only at Mn impurities but also additional non-\nmagnetic scattering centers, such as in quaternary com-\npound DMSs like HgCdMnTe41, we allow the number of\nscattering centers Nimpin general to be larger than the\nnumber of magnetic impurities NMn. Here, we use the\nnotation that the index Iruns from 1 to NMnwhile the\nindexJruns from 1 to Nimp.B. Quantum kinetic equations of motion\nThe goalofthis artilce is to study the time evolutionof\nthe carrier spin polarization after optical excitation with\ncircularly polarized light which can be extracted from\nthe carrier density matrix. In this section, we derive the\ncorrespondingequationsofmotionstartingfromthetotal\nHamiltonian in Eqs. (1).\nFollowing Ref. 36, where for the conduction band only\nH0andHsdinEqs.(1)wereconsidered,weseektoobtain\na closed set of equations for the reduced carrier and im-\npurity density matrices as well as for the carrier-impurity\ncorrelations:\nMn2\nn1=/an}bracketle{tˆPI\nn1n2/an}bracketri}ht (2a)\nCσ2\nσ1k1=/an}bracketle{tc†\nσ1k1cσ2k1/an}bracketri}ht, (2b)\n¯Cσ2k2\nσ1k1=V/an}bracketle{tc†\nσ1k1cσ2k2ei(k2−k1)RJ/an}bracketri}ht,fork2/ne}ationslash=k1,\n(2c)\nQσ2n2k2\nσ1n1k1=V/an}bracketle{tc†\nσ1k1cσ2k2ei(k2−k1)RIˆPI\nn1n2/an}bracketri}ht,fork2/ne}ationslash=k1.\n(2d)\nMn2n1andCσ2\nσ1k1aretheimpurityandelectrondensityma-\ntrices and ¯Cσ2k2\nσ1k1as well as Qσ2n2k2\nσ1n1k1are the non-magnetic\nand magnetic carrier-impurity correlations, respectively.\nIn Eqs. (2), the brackets denote not only the quantum\nmechanical average of the operators, but also an aver-\nage over a random distribution of impurity positions,\nwhich we assume to be on average homogeneous so that\n/an}bracketle{tei(k2−k1)RJ/an}bracketri}ht=δk1k2.\nThe equations of motion for the variables de-\nfined in Eqs. (2) can be derived using the Heisen-\nberg equations of motion for the corresponding op-\nerators. Note, however, that this procedure leads\nto an infinite hierarchy of variables and equations\nof motion, since, e. g., the equation of motion for\n/an}bracketle{tc†\nσ1k1cσ2k2ei(k2−k1)RIˆPI\nn1n2/an}bracketri}htcontains also terms of the\nform/an}bracketle{tc†\nσ1k1cσkei(k−k1)RIei(k2−k)RI′ˆPI\nn1n2ˆPI′\nnn′/an}bracketri}htforI′/ne}ationslash=I\nwhich cannot be expressed in terms of the variables in\nEqs. (2). Thus, in order to obtain a closed set of equa-\ntions, one has to employ a truncation scheme. Here, we\nfollow the procedure of Ref. 36: we factorize the aver-\nages over products of operators and define the true cor-\nrelations to be the remainder when all combinations of\nfactorizations have been subtracted from the averages.\nFor example, we define (for k2/ne}ationslash=k1)\nδ/an}bracketle{tc†\nσ1k1cσ2k2ei(k2−k1)RIˆPI\nn1n2/an}bracketri}ht:=\n/an}bracketle{tc†\nσ1k1cσ2k2ei(k2−k1)RIˆPI\nn1n2/an}bracketri}ht\n−/parenleftBig\n/an}bracketle{tc†\nσ1k1cσ2k2/an}bracketri}ht/an}bracketle{tei(k2−k1)RI/an}bracketri}ht/an}bracketle{tˆPI\nn1n2/an}bracketri}ht\n+/an}bracketle{tc†\nσ1k1cσ2k2ei(k2−k1)RI/an}bracketri}ht/an}bracketle{tˆPI\nn1n2/an}bracketri}ht\n+/an}bracketle{tei(k2−k1)RI/an}bracketri}ht/an}bracketle{tc†\nσ1k1cσ2k2ˆPI\nn1n2/an}bracketri}ht/parenrightBig\n(3)\nwhereδ/an}bracketle{t.../an}bracketri}htdenotes the true correlations. The basic\nassumption of the truncation scheme of Ref. 36 is that4\nall correlations higher than δ/an}bracketle{tc†\nσ1k1cσ2k2ei(k2−k1)RI/an}bracketri}htand\nδ/an}bracketle{tc†\nσ1k1cσ2k2ei(k2−k1)RIˆPI\nn1n2/an}bracketri}htare negligible. This as-\nsumption resultsin aclosedset ofequationsofmotion for\nthe reduced density matrices and the true correlations.\nHowever, it turns out26that the equations of motion can\nbe written down in a more condensed form when switch-\ning back to the full (non-factorized) higher order densitymatrices as variables, after the higher (true) correlations\nare neglected. For details of this procedure, the reader is\nreferred to Refs. 26 and 36.\nApplying this truncation scheme to the total Hamil-\ntonian (1) including magnetic and non-magnetic carrier-\nimpurity interactions as well as the Zeeman terms for\ncarriers and impurities leads to the equations of motion\nfor the variables defined in Eqs. (2):\n−i¯h∂\n∂tMn2\nn1=/summationdisplay\nn¯hωMn·(Snn1Mn2\nn−Sn2nMn\nn1)+Jsd\nV2/summationdisplay\nn/summationdisplay\nkk′σσ′(Snn1·sσσ′Qσ′n2k′\nσnk−Sn2n·sσσ′Qσ′nk′\nσn1k)/bracketrightbig\n,(4a)\n−i¯h∂\n∂tCσ2\nσ1k1=/summationdisplay\nσ¯hωe·(sσσ1Cσ2\nσk1−sσ2σCσ\nσ1k1)+JsdNMn\nV2/summationdisplay\nnn′/summationdisplay\nkσ(Snn′·sσσ1Qσ2n′k1\nσnk−Snn′·sσ2σQσn′k\nσ1nk1)+\n+J0Nimp\nV2/summationdisplay\nk(¯Cσ2k1\nσ1k−¯Cσ2k\nσ1k1), (4b)\n−i¯h∂\n∂tQσ2n2k2\nσ1n1k1=¯h(ωk1−ωk2)Qσ2n2k2\nσ1n1k1+bσ2n2k2\nσ1n1k1I+bσ2n2k2\nσ1n1k1II+bσ2n2k2\nσ1n1k1III+bσ2n2k2\nσ1n1k1imp, (4c)\n−i¯h∂\n∂t¯Cσ2k2\nσ1k1=¯h(ωk1−ωk2)¯Cσ2k2\nσ1k1+cσ2k2\nσ1k1I+cσ2k2\nσ1k1II+cσ2k2\nσ1k1III+cσ2k2\nσ1k1sd(4d)\nwith\nbσ2n2k2\nσ1n1k1I=/summationdisplay\nnσσ′Jsd[Snn1·sσσ′(δσ1σ′−Cσ′\nσ1k1)Cσ2\nσk2Mn2n−Sn2n·sσσ′(δσσ2−Cσ2\nσk2)Cσ′\nσ1k1Mn\nn1], (4e)\nbσ2n2k2\nσ1n1k1II=/summationdisplay\nσ¯hωe·(sσσ1Qσ2n2k2\nσn1k1−sσ2σQσn2k2\nσ1n1k1)+/summationdisplay\nn¯hωMn·(Snn1Qσ2n2k2\nσ1nk1−Sn2nQσ2nk2\nσ1n1k1), (4f)\nbσ2n2k2\nσ1n1k1III=Jsd\nV/summationdisplay\nn/summationdisplay\nkσ/braceleftBig\n(Snn1·sσσ1Qσ2n2k2\nσnk−Sn2n·sσ2σQσnk\nσ1n1k1)\n−/summationdisplay\nσ′sσσ′·/bracketleftbig\nCσ′\nσ1k1/parenleftbig\nSnn1Qσ2n2k2\nσnk−Sn2nQσ2nk2\nσn1k/parenrightbig\n+Cσ2\nσk2/parenleftbig\nSnn1Qσ′n2k\nσ1nk1−Sn2nQσ′nk\nσ1n1k1/parenrightbig/bracketrightbig/bracerightBig\n,(4g)\nbσ2n2k2\nσ1n1k1imp=J0/bracketleftbig/parenleftbig\nCσ2\nσ1k2−Cσ2\nσ1k1/parenrightbig\nMn2\nn1+1\nV/summationdisplay\nk/parenleftbig\nQσ2n2k2\nσ1n1k−Qσ2n2k\nσ1n1k1/parenrightbig/bracketrightbig\n, (4h)\nand\ncσ2k2\nσ1k1I=J0(Cσ2\nσ1k2−Cσ2\nσ1k1), (4i)\ncσ2k2\nσ1k1II=/summationdisplay\nσ¯hωe·(sσσ1¯Cσ2k2\nσk1−sσ2σ¯Cσk2\nσ1k1), (4j)\ncσ2k2\nσ1k1III=J0\nV/summationdisplay\nk(¯Cσ2k2\nσ1k−¯Cσ2k\nσ1k1), (4k)\ncσ2k2\nσ1k1sd=Jsd/summationdisplay\nnn′/summationdisplay\nσMnn′Snn′·/parenleftbig\nsσσ1Cσ2\nσk2−sσ2σCσ\nσ1k1/parenrightbig\n+Jsd\nVNMn\nNimp/summationdisplay\nnn′/summationdisplay\nkσSnn′·/parenleftbig\nsσσ1Qσ2n′k2\nσnk−sσ2σQσn′k\nσ1nk1/parenrightbig\n,\n(4l)\nwherebσ2n2k2\nσ1n1k1Xare the source terms for the magnetic\ncarrier-impurity correlations, cσ2k2\nσ1k1Xare the sources forthe non-magnetic correlations and\nωMn=gMnµBB+Jsd\n¯h1\nV/summationdisplay\nkσσ′sσσ′Cσ′\nσk,(5a)\nωe=geµBB+Jsd\n¯hNMn\nV/summationdisplay\nnn′Snn′Mnn′(5b)5\nare the mean-field precession frequencies of the impu-\nrity and carrier spins, respectively. The first terms on\nthe right-hand side of Eqs. (4a) and (4b) represent the\nprecession of the impurity and carrier spins in the mean\nfield due to the carrier and impurity magnetization as\nwell as the external magnetic field. The second terms in\nEqs. (4a) and (4b) describe the effects of the magnetic\ncarrier-impurity correlations on the impurity and carrier\ndensity matrices and the last term of Eq. (4b) describes\nthe scattering of carriers at non-magnetic impurities.\nIn analogy to the situation without non-magnetic im-\npurity scattering ( J0= 0) studied in Ref. 26, we label\nthe source terms of the correlations on the right-hand\nside of the Eqs. (4c) and (4d) as follows: The terms\nbσ2n2k2\nσ1n1k1Iare the inhomogeneous driving terms depending\nonly on single-particle quantities. bσ2n2k2\nσ1n1k1IIare homoge-\nneous terms which describe a precession-type motion of\nthe correlations in the effective fields ωeandωMn. The\nsource terms bσ2n2k2\nσ1n1k1IIIcomprise the driving of the mag-\nneticcorrelationsbyothermagneticcorrelationswithdif-\nferent wave vectors and describe a change of the wave\nvectors of the correlations due to the s-dinteraction.\nbσ2n2k2\nσ1n1k1impdenotes the contributions to the equation for\nthe magnetic correlations due to the non-magnetic im-\npurity scattering. The source terms cσ2k2\nσ1k1Xfor the non-\nmagnetic correlations are classified analogously.\nA straightforward but lengthy calculation confirms\nthat Eqs. (4) conserve the particle number as well as\nthe total energy comprised of the single-particle contri-\nbutions and the correlation energies.\nC. Markov limit\nAlthough Eqs. (4) can readily be used to calculate the\nspin dynamics given a set of appropriate initial condi-\ntions, it is instructive also to derive the Markov limit of\nthe quantum kinetic equations26–28. On the one hand,\nthis enables us to distinguish the Markovian behavior\nfromgenuinequantumkineticeffects. Ontheotherhand,\nit allows us to derive analytic expressions for the cor-\nrelation energies and the renormalization of the preces-\nsion frequencies in the presence of an external magnetic\nfield39.\nThe derivation of the Markov limit comprises two\nsteps28: First, the equations of motion for the correla-\ntions are formally integrated yielding explicit expressions\nfor the correlations in the form of a memory integral.\nThis yields integro-differential equations for the single-\nparticle variables, where the values of the single-particle\nvariables at earlier times enter. Second, the memory in-\ntegral is eliminated by assuming a δ-like short memory.\nHowever, the first step, which involves the formal in-\ntegration of the carrier-impurity correlations, can, in\ngeneral, be complicated. Nevertheless, if the source\ntermsbσ2n2k2\nσ1n1k1IIIandcσ2k2\nσ1k1IIIas well as the correlation-dependent part of bσ2n2k2\nσ1n1k1impandcσ2k2\nσ1k1sdare neglected,\nthe formal solution of Eqs. (4c-d) becomes much eas-\nier. In absence of non-magnetic impurity scattering,\nit has been shown that these source terms are indeed\nnumerically insignificant26. Furthermore, a straightfor-\nward calculation shows that neglecting these terms also\nyields a consistent theory with respect to the conserva-\ntion of the total energy. Whether neglecting the terms\nbσ2n2k2\nσ1n1k1III,cσ2k2\nσ1k1IIIand the correlation-dependent parts\nofbσ2n2k2\nσ1n1k1impandcσ2k2\nσ1k1sdis indeed a good approxima-\ntion in the presence of non-magnetic impurity scattering\ncan be tested by comparing the numerical results of the\nquantum kinetic equations with and without accounting\nfor these source terms.\nNeglecting the aforementioned source terms in\nEqs.(4), wefirstformulateasetofquantumkineticequa-\ntions for the new dynamical variables\n/an}bracketle{tSi/an}bracketri}ht=/summationdisplay\nn1n2Si\nn1n2Mn1n2, (6a)\nnk=/summationdisplay\nσCσ\nσk, (6b)\nsi\nk=/summationdisplay\nσ1σ2si\nσ1σ2Cσ2\nσ1k, (6c)\n¯Cαk2\nk1=/summationdisplay\nσ1σ2sα\nσ1σ2¯Cσ2k2\nσ1k1(6d)\nQαk2\nlk1=/summationdisplay\nσ1σ2/summationdisplay\nn1n2sα\nσ1σ2Sl\nn1n2Qσ2n2k2\nσ1n1k1,(6e)\nwhere/an}bracketle{tS/an}bracketri}htis the average impurity spin and nkandsk\nare the occupation density and spin density of the car-\nrier states with wave vector k, respectively. ¯Cαk2\nk1as\nwell asQαk2\nlk1comprise the non-magnetic and magnetic\ncarrier-impurity correlations. In Eqs. (6) we use a nota-\ntion in which the Latin indices are in the range {1,2,3},\nwhile the Greek indices also include the value 0, where\ns0\nσ1σ2=δσ1σ2is the 2x2 identity matrix. The corre-\nsponding equations of motion for the variables defined in\nEqs. (6) are explicitly given in appendix A.\nNote that the source terms bαk2\nlk1Ifor the correlations\nQαk2\nlk1depend on the second moments of the impurity\nspins/an}bracketle{tSiSj/an}bracketri}ht=/summationtext\nn1n2n3Si\nn1n2Sj\nn2n3Mn1n3for which we do\nnot present equations of motions, although such equa-\ntions can, in principle, be derived from Eqs. (4). Here,\nwe use the fact that for typical sample parameters the\noptically induced carrier density is usually much lower\nthan the impurity concentration, so that the average im-\npurity spin only changes marginally over time26. For the\nnumerical calculations we assume that the impurity den-\nsity matrix can be approximately described as being in\nthermal equilibrium at all times where the effective im-\npurity spin temperature TMncan be obtained from the\nvalue of /an}bracketle{tS/an}bracketri}ht. From this thermally occupied density ma-\ntrix, the second moments /an}bracketle{tSiSj/an}bracketri}htconsistent with /an}bracketle{tS/an}bracketri}htcan\nbe calculated in each time step.6\nThe equations of motion for the variables defined in\nEqs. (6) are the starting point for the formal integra-\ntion of the correlations. Note that Eqs. (A1d-g) for the\ncorrelations Qαk2\nlk1and¯Cαk2\nk1can be transformed into the\ngeneral form\n∂\n∂tQk2\nk1=−i(ωk2−ωk1)Qk2\nk1+iχ1ωeQk2\nk1\n+iχ2ωMnQk2\nk1+bk2\nk1I, (7)\nwhereχ1,χ2∈ {−1,0,1}and the terms proportional to\nωe=|ωe|andωMn=|ωMn|originate from the preces-\nsion of the correlations described by the source terms\nbσ2n2k2\nσ1n1k1IIandcσ2k2\nσ1k1II. The term bk2\nk1Ihere denotes the\ncontributions from the source terms bσ2n2k2\nσ1n1k1I,cσ2k2\nσ1k1I,\nbσ2n2k2\nσ1n1k1impandcσ2k2\nσ1k1sdand only depends on the single-\nparticle variables. The formal integration of Eq. (7)\nyields\nQk2\nk1(t) =t/integraldisplay\n0dt′ei[ωk2−(ωk1+χ1ωe+χ2ωMn)](t′−t)bk2\nk1I(t′).\n(8)\nTheMarkovlimitconsistsofassumingashortmemory,\ni.e. the assumptionthat thecorrelationsattime tdepend\nonly significantly on the single-particle variables at the\nsame time t, so that one is inclined to evaluate bk2\nk1I(t′)\nin Eq. (8) at t′=tand to draw the source term out of\nthe integral. However, first, one has to make sure that\nthe source terms are indeed slowly changing variables.For example, the carrier spin can precess rapidly about\nan external magnetic field. Therefore, we first analyze\nthe mean-field precession of the single-particle quantities\nandsplitthesourcetermsintopartsoscillatingwithsome\nfrequencies ωof the form\nbk2\nk1I(t′)MF=/summationdisplay\nω/summationdisplay\nχ∈{−1,0,1}eiχω(t′−t)bk2\nk1ω,χ(t).(9)\nThen, the different oscillating parts bk2\nk1ω,χ(t) can be\ndrawn out of the memory integral and the remaining in-\ntegral can be solved in the limit of large times t28:\nt/integraldisplay\n0dt′ei∆ω(t′−t)t→∞−→πδ(∆ω)−i\n∆ω.(10)\nThis procedure yields particularly transparent results\nin the case where the external magnetic field and the im-\npurity magnetization are collinear, as is usually the case\nwhen the number of impurities exceeds the number of\nquasi-free carriers ( NMn≫Ne), and the impurity den-\nsity matrix is initially occupied thermally. Choosing the\ndirection of ωeas a reference and defining s/bardbl\nk1:=s·ωe\nωe,\nS/bardbl:=ˆS·ωe\nωeandω/bardbl\nMn:=ωMn·ωe\nωe, the Markovian equa-\ntionsobtainedforthespin-upand spin-downoccupations\nand the perpendicular carrier spin component with re-\nspect to the direction of ωe,\nn↑/↓\nk1:=nk1\n2±s/bardbl\nk1, (11a)\ns⊥\nk1:=sk1−ωe\nωes/bardbl\nk1, (11b)\nare given by:\n∂\n∂tn↑/↓\nk1=π\n¯h2V2/summationdisplay\nk2/braceleftbigg\nδ(ωk2−ωk1)/bracketleftbig\nJ2\nsdNMn1\n2/an}bracketle{tS/bardbl2/an}bracketri}ht±JsdJ0(NMn+Nimp)/an}bracketle{tS/bardbl/an}bracketri}ht+2J2\n0Nimp/bracketrightbig\n(n↑/↓\nk2−n↑/↓\nk1)+\n+δ/bracketleftbig\nωk2−/parenleftbig\nωk1±(ωe−ω/bardbl\nMn)/parenrightbig/bracketrightbig\nJ2\nsdNMn/bracketleftbigg/parenleftBig\n/an}bracketle{tS⊥2/an}bracketri}ht±1\n2/an}bracketle{tS/bardbl/an}bracketri}ht/parenrightBig/parenleftbig\n1−n↑/↓\nk1/parenrightbig\nn↓/↑\nk2−/parenleftBig\n/an}bracketle{tS⊥2/an}bracketri}ht∓1\n2/an}bracketle{tS/bardbl/an}bracketri}ht/parenrightBig/parenleftbig\n1−n↓/↑\nk2/parenrightbig\nn↑/↓\nk1/bracketrightbigg/bracerightbigg\n,\n(12a)7\n∂\n∂ts⊥\nk1=−π\n¯h2V2/summationdisplay\nk2/braceleftbigg\nδ(ωk2−ωk1)/bracketleftBig\nJ2\nsdNMn1\n2/an}bracketle{tS/bardbl2/an}bracketri}ht(s⊥\nk2+s⊥\nk1)−2J2\n0Nimp(s⊥\nk2−s⊥\nk1)/bracketrightBig\n+δ/bracketleftbig\nωk2−/parenleftbig\nωk1+(ωe−ω/bardbl\nMn)/parenrightbig/bracketrightbig1\n2/parenleftBig\n/an}bracketle{tS⊥2/an}bracketri}ht−1\n2/an}bracketle{tS/bardbl/an}bracketri}ht(1−2n↓\nk2)/parenrightBig\ns⊥\nk1\n+δ/bracketleftbig\nωk2−/parenleftbig\nωk1−(ωe−ω/bardbl\nMn)/parenrightbig/bracketrightbig1\n2/parenleftBig\n/an}bracketle{tS⊥2/an}bracketri}ht+1\n2/an}bracketle{tS/bardbl/an}bracketri}ht(1−2n↑\nk2)/parenrightBig\ns⊥\nk1/bracerightbigg\n+ωe×s⊥\nk1+1\n¯h2V2/summationdisplay\nk2/braceleftbigg\n−JsdJ0\nωk2−ωk1/an}bracketle{tS/an}bracketri}ht×/bracketleftbig\n(Nimp−NMn)s⊥\nk2+(NMn+Nimp)s⊥\nk1/bracketrightbig\n−J2\nsdNMn\nωk2−/parenleftbig\nωk1+(ωe−ω/bardbl\nMn)/parenrightbig1\n2/parenleftBig\n/an}bracketle{tS⊥2/an}bracketri}ht−1\n2/an}bracketle{tS/bardbl/an}bracketri}ht(1−2n↓\nk2)/parenrightBig/parenleftBigωe\nωe×s⊥\nk1/parenrightBig\n+J2\nsdNMn\nωk2−/parenleftbig\nωk1−(ωe−ω/bardbl\nMn)/parenrightbig1\n2/parenleftBig\n/an}bracketle{tS⊥2/an}bracketri}ht+1\n2/an}bracketle{tS/bardbl/an}bracketri}ht(1−2n↑\nk2)/parenrightBig/parenleftBigωe\nωe×s⊥\nk1/parenrightBig/bracerightbigg\n. (12b)\nThe first line of the right-hand side of Eq. (12a), which\nis proportional to n↑/↓\nk2−n↑/↓\nk1, describes a redistribution\nof occupations of spin-up and spin-down states within\na shell of defined kinetic energy via the term propor-\ntional to δ(ωk2−ωk1). For a parabolic band struc-\nture, this implies a redistribution between states with\nthe same modulus kof the wave vector k, while the to-\ntal carrier spin remains unchanged. If accompanied by\na wave-vector dependent magnetic field like a Rashba or\nthe Dresselhaus field, this term leads to a D’yakonov-\nPerel’-type suppression of the spin dephasing. Here,\nhowever, we do not consider any wave vector dependent\nfield and the system under investigation is isotropic in\nk-space, so that the first line in Eq. (12a) has no in-\nfluence on the dynamics of the total spin. The second\nline in Eq. (12a) describes a spin-flip scattering from\nthe spin-up band to the spin-down band and vice versa.\nSince these bands are energetically split by ¯ hωeand a\nflip of carrier spin involves a corresponding flip of an im-\npurity spin in the opposite direction, which requires a\nmagnetic (Zeeman) energy of ¯ hω/bardbl\nMn, the total magnetic\nenergy released in a spin-flip process is ±¯h(ωe−ω/bardbl\nMn).\nThus,δ/bracketleftbig\nωk2−/parenleftbig\nωk1±(ωe−ω/bardbl\nMn)/parenrightbig/bracketrightbig\nensures a conserva-\ntion of the total single-particle energies in the Markov\nlimit. It is noteworthy that, if the mean-field dynamics\nof the source terms as in Eq. (9) is not correctly taken\ninto account, other energetic shifts are obtained in the δ-\nfunction, which yields equations in the Markovlimit that\nare not consistent with the conservation of the single-\nparticle energies28. Note also that the right-hand side of\nEq. (12a) correctly deals with Pauli blocking effects. Be-\ncause the non-magnetic impurity scattering enters in the\nequations of motion (12a) for the spin-up and spin-down\noccupationonlyviathe firstlinewhich playsnorolein an\nisotropicsystem, it hasno influence on the spin dynamics\nin the Markov limit.\nThe first three lines in Eq. (12b) for the perpendic-\nular carrier spin component, which are proportional to\nδ-functions, indicate an exponential decay of the per-pendicular carrier spin component towards zero. The\nlast three lines describe a precession of the perpendic-\nular carrier spin component. The mean-field precession\nfrequency ωeis renormalized by the carrier-impurity cor-\nrelations. This renormalizationoriginatesfrom the imag-\ninary part of the memory integral in Eq. (10). Besides\nthe terms proportionalto1\nωk2−/parenleftbig\nωk1±(ωe−ω/bardbl\nMn)/parenrightbig, which are\nalso present when only the magnetic s-dinteraction is\ntaken into account39, the non-magnetic impurity scatter-\ningintroducesanothercontributionwhichisacross-term,\ni.e. it is absent when either the magnetic or the non-\nmagneticimpurityscatteringisabsent, whichcanbeseen\nfrom the fact that it is proportional to the product of Jsd\nandJ0. In the quasi-continuous limit, the sum over k2\ncan be replaced by an integral over the spectral density\nof states. In quasi-two-dimensional systems like quan-\ntum wells, the spectral denstiy of states D(ω) =Am∗\n2π¯h\nis constant. Thus, the frequency renormalization can be\nintegrated and yields logarithmic divergences\n/summationdisplay\nk21\nωk2−ω0=ωBZ/integraldisplay\n0dωD(ω)1\nω−ω0\n=Am∗\n2π¯hln/vextendsingle/vextendsingle/vextendsingle/vextendsingleωBZ−ω0\nω0/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (13)\nat the poles ω0=ωk1andω0=ωk1±(ωe−ω/bardbl\nMn). These\nlogarithmic divergences are similar to the ones obtained\nin the discussion of the Kondo-effect in metals with mag-\nnetic impurities29. Despite the formal divergence, the\nsummation over a non-singular carrier distribution al-\nways leads to a finite value of the precession frequency of\nthe total carrier spin, since the logarithm is integrable28.\nFrom Eq. (13), one can see that the cut-off energy ¯ hωBZ,\nwhich corresponds to the width of the conduction band\nand is typically of the order of 1 eV, enters as a new\nmodel parameter in the theory and cannot be eliminated\nby assuming that ωBZ→ ∞, since then the frequency\nrenormalization also diverges. As a consequence, the\nMarkovian expression for the frequency renormalization8\ncan only give an order-of-magnitude estimation and a\nmore detailed treatment of the band structure is neces-\nsary if a quantitatively more accurate description is re-\nquired.\nFor the special case of zero external magnetic field,\nvanishing impurity magnetization and low carrier densi-\nties, Eqs. (12) are equivalent to the simple rate equations\n∂\n∂tsk1=−1\nτsk1, (14)\nwhere the values of the rates coincide with the Fermi’s\ngolden rule value. In two dimensions, one obtains25\n1\nτ2D=35\n12J2\nsdm∗\n¯h3NMn\nV1\nd. (15)\nD. Correlation energy\nIn Eqs. (8) to (10), Markovian expressions for the\ncarrier-impurity correlations are derived as functionals\nof the carrier and impurity variables. Using these ex-\npressions, it is straightforwardto also obtain analytic ex-\npressions for the carrier-impurity correlation energies as\nfunctionals of the carrier spins and occupations28. Split-\nting the averages over the magnetic and non-magnetic\ncarrier-impurity interactions into mean-field and corre-\nlated contributions\n/an}bracketle{tHsd/an}bracketri}ht=/an}bracketle{tHMF\nsd/an}bracketri}ht+/an}bracketle{tHcor\nsd/an}bracketri}ht, (16a)\n/an}bracketle{tHimp/an}bracketri}ht=/an}bracketle{tHMF\nimp/an}bracketri}ht+/an}bracketle{tHcor\nimp/an}bracketri}ht, (16b)\n/an}bracketle{tHMF\nsd/an}bracketri}ht=JsdNMn\nV/summationdisplay\nk/an}bracketle{tS/an}bracketri}ht·sk (16c)\n/an}bracketle{tHcor\nsd/an}bracketri}ht=JsdNMn\nV2/summationdisplay\nk,k′/summationdisplay\niQik′\nik (16d)\n/an}bracketle{tHMF\nimp/an}bracketri}ht=J0Nimp\nV/summationdisplay\nknk, (16e)\n/an}bracketle{tHcor\nimp/an}bracketri}ht=J0Nimp\nV2/summationdisplay\nk,k′¯C0k′\nk, (16f)\none obtains in the Markov limit\n/an}bracketle{tHcor\nsd/an}bracketri}ht=−JsdNMn\nV2/summationdisplay\nk1k2/braceleftbigg1\n2Jsd/an}bracketle{tS/bardbl2/an}bracketri}htnk1+2J0/an}bracketle{tS/bardbl/an}bracketri}hts/bardbl\nk1\nωk2−ωk1\n+Jsd/parenleftbig\n/an}bracketle{tS⊥/an}bracketri}ht−1\n2/an}bracketle{tS/bardbl/an}bracketri}ht/parenrightbig\n(1−n↓\nk2)n↑\nk1\nωk2−/parenleftbig\nωk1+(ωe−ω/bardbl\nMn)/parenrightbig\n+Jsd/parenleftbig\n/an}bracketle{tS⊥/an}bracketri}ht+1\n2/an}bracketle{tS/bardbl/an}bracketri}ht/parenrightbig\n(1−n↑\nk2)n↓\nk1\nωk2−/parenleftbig\nωk1−(ωe−ω/bardbl\nMn)/parenrightbig/bracerightbigg\n,(17a)\n/an}bracketle{tHcor\nimp/an}bracketri}ht=−2J0Nimp\nV2/summationdisplay\nk1k2J0nk1+Jsd/an}bracketle{tS/bardbl/an}bracketri}hts/bardbl\nk1\nωk2−ωk1.(17b)\nEqs. (17) have the same poles as Eq. (12b) for the fre-\nquency renormalization and, thus, also contain formally\nlogarithmic divergences in two-dimensional systems.III. RESULTS\nAfter havingderivedthe quantum kineticequationsfor\nthe description of the spin dynamics in DMS including\nmagnetic and non-magnetic scattering and having ob-\ntained rate-type Markovian equations, we now present\nresults of numerical simulations. Here, we focus on the\ncase of a 4-nm-wide Cd 0.93Mn0.07Te quantum well. For\nthis material, the magnetic coupling constant is Jsd=\n−15 meVnm3(N0Jsd=−220 meV)42, while the non-\nmagnetic coupling constant is approximately J0= 110\nmeVnm3(N0J0= 1.6 eV)31, whereN0is the number of\nunit cells per unit volume. Furthermore, we use a con-\nduction band effective mass of m∗= 0.1m0and assume\nthat the impurity magnetization is described by a ther-\nmal distribution at a temperature of T= 2 K and the\ng-factors of the conduction band carriers and Mn impu-\nrities are ge=−1.77 andgMn= 2, respectively40. If not\nstated otherwise, we choose a value of 40 meV for the\ncut-off energy ¯ hωBZin the numerical calculations and\nwe consider only Mn ions as sources of non-magnetic im-\npurity scattering, i. e. Nimp=NMn. As initial value for\nthe carrier distribution, we use a Gaussian distribution\ncentered at the band edge of the spin-up band with stan-\ndard deviation of Es= 0.4 meV, which corresponds to\nan excitation with a circularly polarized light pulse with\nfull width at half maximum (FWHM) pulse duration of\nabout 350 fs.\nWe first discuss the spin dynamics in the conduction\nband of a Cd 0.93Mn0.07Te quantum well for zero mag-\nnetic field with a focus on the impact of non-magnetic\nimpurity scattering on the spin dynamics and investigate\nthe redistribution of carriers in k-space as well as the\nbuild-up of correlation energy. Then, we study the spin\ndynamics in the valence band in a simplified model. Fi-\nnally, we investigate the spin dynamics in the presence\nof an external magnetic field parallel and perpendicular\nto the carrier spin polarization and discuss, in the latter\ncase, how the non-magnetic impurity scattering affects\nthe carrier spin precession frequencies.\nA. Zero magnetic field\nFigure 1(a) shows the time evolution of an initially\npolarized carrier spin in a Cd 0.93Mn0.07Te quantum well\nfor vanishing magnetic field. The Markovian equations\n(12) predict a simple exponential decay of the carrier\nspin, which is transferred to the impurities. Note that\ndue toNMn≫Ne, the asymptotic value of the carrier\nspin for long times tis close to zero, since the impu-\nrities act as a spin bath. If only the magnetic spin-\nflip scattering is accounted for ( J0= 0), the time evo-\nlution according to the quantum kinetic theory is non-\nmonotonic and shows an overshoot below the asymptotic\nvalue. These non-Markovian effects are strongly sup-\npressed in the calculations including non-magnetic im-\npurity scattering ( J0= 110 meVnm3) and the time evo-9\n-0.200.20.40.60.811.2\n0 5 10 15 20 25 30spin polarization (norm.)\ntime [ps](a)\n-0.0200.020.040.060.080.1\n10 15 20\n00.20.40.60.811.2\n0 5 10carrier distribution\nkin. energy [meV](b) QKT1,J0= 0meVnm3\nQKT2,J0= 0meVnm3\nQKT1,J0= 110 meVnm3\nQKT2,J0= 110 meVnm3\nMarkovt= 0ps\nJ0= 0meVnm3,t= 10ps\nJ0= 110 meVnm3,t= 10ps\nFIG. 1. (a): Time evolution of the carrier spin for zero magne tic field with ( J= 110 meVnm3) and without ( J= 0) non-\nmagnetic impurity scattering. QKT1 (points) denotes the re sults according to the full quantum kinetic equations (4) wh ile\nQKT2 (lines) describes the results of the reduced set of equa tions (A1). The purple dash-dotted line shows the results of the\nMarkovian equations (12), which is independent of non-magn etic impurity scattering. The inset shows a magnification of the\nregion where the quantum kinetic theory for J0= 0 predicts a non-monotonic behavior. (b) Occupation of car rier states at\nt= 0 and t= 10 ps for the calculations with and without non-magnetic im purity scattering.\nlution of the total spin follows the Markovian dynamics\nmore closely. An exponential fit to the dynamics of the\nfull quantum kinetic theory yields an effective spin trans-\nfer rate about 15% smaller than the Markovian rate in\nEq. (15).\nInterestingly, while the full quantum kinetic equations\n(4) yield identical results as the reduced set of equations\n(A1) in the case without non-magnetic impurity scatter-\ning, deviations between both approaches can be clearly\nseen when the non-magnetic impurity scattering is taken\ninto account.\nIn order to understand the suppression of the non-\nmonotonic features in the spin dynamics with non-\nmagnetic impurity scattering, it is useful to recapitu-\nlate the findings of Ref. 38, where the origin of the non-\nMarkovian behavior of the spin dynamics in absence of\nnon-magnetic impurity scattering was discussed: It was\nfound that the depth of the memory induced by the cor-\nrelations is of the order of the inverse energetic distance\nof the carrier state under consideration to the band edge\ntimes ¯h. Memory effects become insignificant if the ki-\nnetic energy of the carrier ¯ hωk1is much higher than the\nenergy scale of the carrier-impurity spin transfer rate\n¯h\nτ. For the parameters used in the simulations, one ob-\ntains from Eq. (15) a value of τ2D= 2.97 ps and there-\nfore¯h\nτ≈0.22 meV. Figure 1(b) shows the redistribu-tion of carriers in the calculations with and without non-\nmagnetic impurity scattering. One can clearly see that,\nwhile without non-magnetic impurity scattering the car-\nrier distribution at t= 10 ps is only slightly broadened,\nincluding the non-magnetic impurity scatterings leads to\na drastic redistribution of carriers to states many meV\naway from the initial distribution. For these states, the\nmemory is very short compared with the spin relaxation\ntime and therefore the Markovian approximation is jus-\ntified.\nThe redistribution of carriers to states several meV\naway from the band edge raises questions about the con-\nservation of energy, since for zero magnetic field the\nmean-field energy of the system is comprised of only the\nkinetic energyofthe carriers. In the quantum kinetic cal-\nculations, however, we also consider the carrier-impurity\ncorrelationswhich introduce correlationenergies that are\nnot captured in a simple single-particle picture. The dif-\nferent contributions to the total energy over the course of\ntime for the simulations presented in Fig. 1 are shown in\nFig. 2. There, it is shown that the averagekinetic energy\nper electron increases from the initial value of the order\nof the width of the initial carrier distribution to a much\nlarger value of about 4 meV on a timescale of about 0 .5\nps. This energy is mostly provided by a decrease of non-\nmagneticcorrelationenergyfromzerotoanegativevalue.10\n-10-8-6-4-2024\n0 0.5 1 1.5 2 2.5 3energy per electron [meV]\ntime [ps]kin. energy\nkin. energy (Markov)\nmagn. cor. energy\nmag. cor. energy (analyt.)\nnon-magn. cor. energy\nnon-mag. cor. energy (analyt.)\ntot. energy\nFIG. 2. Kinetic energy (red line), magnetic correlation ene rgy\n(blue line), non-magnetic correlation energy (purple line ) and\ntotal energy (cyan line) per electron for the quantum kineti c\ncalculation shown in Fig. 1 with J0= 110 meVnm3. The\nred circles show the kinetic energy obtained from the Marko-\nvian calculation in Fig. 1. The pluses and crosses depict the\nresults according to the analytic Markovian expressions fo r\nthe correlation energies in Eqs. (17) evaluated using the ca r-\nrier distribution of the quantum kinetic calculation at sel ected\ntime steps.\nThe magnetic correlation energy is comparatively small\nsince the magnetic coupling constant Jsdis about one or-\nderofmagnitude smallerthan the non-magneticcoupling\nconstant J0. The pluses and crosses in Fig. 2 show the\nresults of the analytic expressions (17) for the correlation\nenergies evaluated using the carrier distributions of the\nfull quantum kinetic theory in the respective time steps.\nThe analytic results are found to coincide with the val-\nues extracted from the quantum kinetic theory after the\nfirst 0.5 ps. Even though the analytic expressions for the\ncorrelation energies are obtained within the Markovian\ndescription, it should be noted that in the Markovian\nequations of motion (12) for the spins and occupations\nonly single-particle energies are considered for evaluating\nthe energy balance. As in our case the single particle en-\nergies comprise only the kinetic energies of the carriers,\nthe latter are constant in the Markovian description in\nsharp contrast to the quantum kinetic treatment.\nNote also that the total energy comprised of single-\nparticle and correlation energies remains constant in the\nquantum kinetic simulations, which provides a further\ntest for the numerics.-0.200.20.40.60.81\n0 0.5 1 1.5 2spin polarization (norm.)\ntime [ps]QKT,J0= 0meVnm3\nQKT,J0= 7meVnm3\nMarkov\nFIG. 3. Spin dynamics in a degenerate valence band of a\nCd0.93Mn0.07Te quantum well with and without accounting\nfor non-magnetic impurity scattering.\nB. Valence band\nThefactthatintheconductionbandofaCd 1−xMnxTe\nquantum well the non-magnetic scattering at the impuri-\nties suppresses the characteristic non-monotonic features\nof genuine quantum kinetic behavior raises the ques-\ntion whether this statement is true in general and non-\nMarkovian effects always only change the spin dynamics\nquantitatively. In this section, we provide an example\nof a situation where the non-Markovian features are not\nsuppressed due to impurity scattering.\nWe consider now the valence band of a Cd 1−xMnxTe\nquantum well. The details of the valence band structure\nin a quantum well are influenced by, e.g., spin-orbit cou-\npling, strain orthe shape ofthe confinement potential. A\nrealistic description of the band structure is beyond the\nscope of this article. Instead, we perform a model study,\nwhere we assume that heavy-hole and light-hole bands\nare degenerate. In this case, we can use the quantum\nkinetic theory derived for the conduction band and take\nthe material parameters for the heavy holes. The mag-\nnetic coupling constant in the valence band is Jpd= 60\nmeVnm342and the heavy-hole mass is mh= 0.7m043.\nThe difference of the band gaps between CdTe and zinc-\nblende MnTe of about 1 .6 eV is split into the conduction\nand valence band offsets by a ratio of 14:144. Thus, one\nobtainsavalue forthe non-magneticcouplingconstantin\nthe valence band of about J0= 7 meVnm3. The results\nof the quantum kinetic simulations for these parameters\nare shown in Fig. 3.\nIn comparison with the conduction band, the 4 times\nlarger magnetic coupling constant in the valence band\nleads to much stronger non-Markovian effects. In par-\nticular, one finds not a single overshoot, but pronounced\noscillations of the spin polarization about its asymptotic11\n-0.200.20.40.60.811.2\n0 5 10 15 20 25 30spin polarization (norm.)\ntime [ps](a)\n00.20.40.60.811.2\n0 5 10carrier distribution\nkin. energy [meV](b) QKT,J0= 0meVnm3\nQKT,J0= 110 meVnm3\nMarkov↑,t= 0ps\nJ0= 0meVnm3,↑,t= 10ps\nJ0= 0meVnm3,↓,t= 10ps\nJ0= 110 meVnm3,↑,t= 10ps\nJ0= 110 meVnm3,↓,t= 10ps\nMarkov, ↑,t= 10ps\nMarkov, ↓,t= 10ps\nFIG. 4. (a): Time evolution of the carrier spin polarization parallel to an external magnetic field ( B= 100 mT). (b): Spin-up\n(↑) and spin-down occupations ( ↓) att= 0 and t= 10 ps.\nvalue. In Fig. 3, the calculations with and without ac-\ncountingfornon-magneticimpurityscatteringyieldprac-\ntically identical results. Thus, due to the fact that in\nthe valence band the non-magnetic coupling constant is\nmuch smaller than the magnetic coupling constant, no\nsuppression of non-Markovian effects in the spin dynam-\nics is observed.\nC. Finite magnetic field: Faraday configuration\nNext, we investigate the effects of non-magnetic im-\npurity scattering on the spin dynamics in DMS in the\npresence of an external magnetic field. In this section,\nwe study the case in which the external field and the\ninitial carrier spins are parallel, which is known as the\nFaraday configuration. This case has also been consid-\nered in Ref. 40, but without accounting for non-magnetic\nimpurity scattering.\nIn Fig. 4(a) the time evolution of the carrier spin po-\nlarization parallel to an external magnetic field B= 100\nmT is shown. Note that the non-monotonic behavior\nthat can be seen in the case without an external mag-\nnetic field is suppressed for finite external fields even if\nthe non-magnetic scattering is disregarded. The most\nstriking feature in the time evolution of the carrier spin\npolarization is that the Markovian result and the quan-\ntum kinetic simulations predict very different asymptotic\nvalues of the spin polarization at long times t.\nAs discussed in Ref. 40, the different stationary values\nare related to a broadening of the distribution of scat-\ntered carriers in the spin-down band, which is shown in\nFig. 4(b). Note that the broadening of the carrier dis-\ntribution is not primarily an effect of energy-time un-certainty, which is commonly found in quantum kinetic\nstudies45,46, since the width of the distribution does not\nshrink significantly over the course of time40. Rather,\nit is a consequence of the build-up of correlation energy\nwhich enables deviations from the conservation of the\nsingle-particle energies in spin-flip scattering processes.\nIn the Markov limit, the stationary value is obtained\nwhen a balance between scattering from the spin-up to\nthe spin-down band and vice versa is reached. In the\nquantumkineticcalculations,thedistributionofthescat-\ntered carriers is broadened, so that also spin-down states\nbelow the threshold ¯ hωe−¯hω/bardbl\nMnare occupied, whose\nback-scattering is suppressed since there are no states\nin the spin-up band with the matching single-particle en-\nergies. If additionally the non-magnetic impurity scat-\ntering is taken into account, the scattered impurity dis-\ntribution is even broaderand more spin-downstates with\nkinetic energies below ¯ hωe−¯hω/bardbl\nMnare occupied, so that\nthe back-scattering is more strongly suppressed and the\ndeviation of the asymptotic value of the spin polarization\nfrom the Markovian value is even larger.\nD. Finite magnetic field: Voigt configuration\nThe situation in which an external magnetic field and\ntheopticallyinducedcarrierspinpolarizationareperpen-\ndicular to each other is usually referred to as the Voigt\nconfiguration and is the subject of this section. In this\nsituation, the carrier spin precesses about the effective\nmagnetic field ωedue to the external field and the im-\npurity magnetization. As shown in Ref. 39, where the\nnon-magnetic impurity scattering was disregarded, the\ncarrier-impurity correlations are responsible for a renor-12\n-1-0.500.51\n0 5 10 15 20 25spin polarization (norm.)\ntime [ps](a)B= 25mT\n0 5 10 15 20 25 30\ntime [ps](b)B= 100 mT\n0.800.850.900.951.001.051.10\n0 1 2 3 4 5 6 7 8 9precession frequency (norm.)\ntime [ps](c)B= 25mT\n0 1 2 3 4 5 6 7 8 9 10\ntime [ps](d)B= 100 mTQKT,J0= 0meVnm3\nQKT,J0= 110 meVnm3\nMarkov (no freq. renorm.)\nFit,J0= 0meVnm3\nanalyt.,J0= 0meVnm3\nFit,J0= 110 meVnm3\nanalyt.,J0= 110 meVnm3\n¯hωBZ= 1eV\nFIG. 5. (a) and (b): Time evolution of the carrier spin polari zation for B= 25 mT (a) and B= 100 mT (b) using the quantum\nkinetic equations (4) and the Markovian equations Eq. (12b) , where the terms responsible for the frequency renormaliza tion in\nthe Markovian equations have been dropped. The precession f requency normalized with respect to its mean-field value ωeis\nshown in (c) and (d) using a fit of an exponentially decaying co sine to the quantum kinetic results and the analytic express ions\nobtained from Eq. (12b) and the occupations from the quantum kinetic calculations. The black dash-dotted lines in (c) an d\n(d) show the analytic results for a cut-off energy of ¯ hωBZ= 1 eV.\nmalization of the precession frequency. There, it was also\nshown that the strength of this renormalization depends\non the details of the carrier distribution and the strength\nof the effective field ωe.\nIn Fig. 5(a), wepresentsimulationsofthe spin dynam-\nics in a DMS in Voigt geometry for an external magnetic\nfieldofB= 25mT,whichcorrespondstoasituationwith\n|/an}bracketle{tS/an}bracketri}ht| ≈0.05, where the magnetic-correlation-induced fre-\nquency renormalization according to Ref. 39 is particu-\nlarly strong. Simulations with ( J0= 110 meVnm3) and\nwithout ( J0= 0) accounting for the non-magnetic impu-\nrity scattering are compared to Markovian calculations\nbased on Eqs. (12). Note that for the Markovian calcula-\ntion shown in Fig. 5 the frequency renormalization was\nnot taken into account. The results of all simulations\nshown in Fig. 5(a) are very similar and follow closely\nthe form of an exponentially damped cosine. Note thatat long times, the phases of the oscillations of the cal-\nculations accounting for non-magnetic impurity scatter-\ningmatchestheMarkoviancalculationwithoutfrequency\nrenormalization,whileaccountingonlyformagneticspin-\nflip scattering leads to oscillations with a slightly higher\nfrequency.\nThe frequency renormalization for the simulations\nshown in Fig. 5(a) is presented in Fig. 5(c), where an\nexponentially decaying cosine is fit to the quantum ki-\nnetic results and, for comparison, the total precession\nfrequency including the correlation-induced renormaliza-\ntion in the Markovian description in Eq. (12b) evaluated\nusingthespin-upandspin-downoccupationsofthequan-\ntum kinetic simulations is depicted. Due to the time evo-\nlution of the occupations, also the renormalization pre-\ndicted by Eq. (12b) becomes a function of time, which,\nhowever, is for all times close to the constant extracted13\nby fitting the quantum kinetic result. The calculations\nwithout non-magnetic impurity scattering predict an in-\ncrease of the carrier spin precession frequency of about\n2−3% with respect to the mean-field value ωe, which\nis consistent with the findings of Ref. 39. On the other\nhand, the contribution from the non-magnetic carrier-\nimpuritycorrelationsleadsto adecreaseofthe precession\nfrequency which partially cancels the contribution from\nthe magnetic correlations.\nIn Figs. 5(b) and 5(d), the time evolution of the car-\nrier spin polarization and the frequency renormalization\nare shown for an external magnetic field of B= 100\nmT. In this case, the envelope of the spin polariza-\ntion decays only exponentially for the calculations with-\nout non-magnetic impurity scattering. For J0= 110\nmeVnm3, the spin polarization follows the exponential\ndecay of the simulation with J0= 0 only up to about\n5 ps. After that, it decays much slower, which is a new\nnon-Markovian effect that is absent if the non-magnetic\nimpurity scattering is disregarded. As can be seen in\nFig. 5(d), the frequency renormalization due to the mag-\nnetic interactionaloneis almostzero. Nevertheless,when\nthe non-magnetic carrier-impurity correlations are taken\ninto account, the precession frequency shows a decrease\nof about 2 −3%. Thus, in contrast to the correlation-\ninducedrenormalizationinabsenceofnon-magneticscat-\ntering where the renormalization is only observable for a\nvery narrowset of initial conditions39, including the non-\nmagnetic carrier-impurity interaction results in a signifi-\ncant renormalization for a much broader set of excitation\nconditions.\nIt is noteworthy that the frequency renormalization in\nthe quantum kinetic calculations is well reproduced by\nthe Markovian expression in Eq. (12b). The numerical\ndemands of the full quantum kinetic equations require a\nrestriction of the conduction band width ¯ hωBZused in\nthe calculations to a few tens of meV. However, in real-\nistic band structures, the band widths are of the order\nof eV. In order to give an order-of-magnitude estimation\nof the frequency renormalization for such band widths,\nwe present in Figs. 5(c) and 5(d) also the results of the\nMarkovian expression for the frequency renormalizations\nusing the value of ¯ hωBZ= 1 eV together with the oc-\ncupations obtained in the quantum kinetic calculations\nfor ¯hωBZ= 40 meV. This estimation yields a renormal-\nization of the precession frequencies due to the combined\neffects of magnetic and non-magnetic scattering of about\n5−7%. A quantitatively more accurate description re-\nquires a more detailed treatment of the band structure,\nwhich is beyond the scope of this article.\nNote also that the frequency renormalization due to\nthe non-magnetic carrier-impurity correlations is domi-\nnated by a cross-term proportional to JsdJ0[cf. fourth\nline in Eq. (12b)]. Thus, the sign of the frequency renor-\nmalization depends on the relative signs of the coupling\nconstants JsdandJ0. In principle, this allows a determi-\nnation of the sign of the magnetic coupling constant Jsd\nwhich cannot be obtained directly, e.g., by measuring thegiant Zeeman splitting of excitons42.\nIV. CONCLUSION\nWehaveinvestigatedtheinfluence ofnon-magneticim-\npurity scattering at Mn impurities on the spin dynamics\nin Cd1−xMnxTe diluted magnetic semiconductors. To\nthis end, we have developed a quantum kinetic theory\ntaking the magnetic and non-magnetic carrier-impurity\ncorrelations into account. The Markov limit of the quan-\ntum kinetic equation is derived in order to distinguish\nthe Markovian dynamics from genuine quantum kinetic\neffects.\nIn contrast to earlier studies25,27,37,40in which only\nthe magnetic contribution to the carrier-impurity inter-\naction has been considered, some non-Markovian effects,\nsuch as a non-monotonic spin transfer between carriers\nand impurities, are strongly suppressed in the case of the\nconduction band of a Cd 1−xMnxTe quantum well, while\nother features stemming from non-Markovian dynamics\nare enhanced, such as the large finite stationary value\nof the spin polarization in a magnetic field reached at\nlong times. The reason for the suppression in the former\ncase is that the non-magnetic impurity scattering leads\ntoastrongredistributionofcarriersin k-spaceawayfrom\nthe states at k= 0. Since memory effects are particu-\nlarly strong for carriers in proximity to the band edge27,\nthis redistribution leads to spin dynamics that are well\ndescribed by Markovian rate equations. The redistribu-\ntion of the carriers implies an increase of their kinetic\nenergies which is provided by a build-up of (negative)\ncarrier-impurity correlation energy and which cannot be\ndescribedbyamean-fieldorsemiclassicalapproximation.\nWe also provide analytic expressions for the correlation\nenergies in the form of functionals of the spin-up and\nspin-down carrier occupations. Numerical calculations\nconfirm that these expressions indeed describe the cor-\nrelation energies obtained from the full quantum kinetic\ntheory very well.\nEventhoughdopingwithmagneticimpuritiesunavoid-\nably also provides a contribution to non-magnetic impu-\nrity scattering, there can still be situations where the\nlatter is too weak to influence the spin dynamics and to\nsuppress otherwise visible non-Markovianeffects. This is\nsubstantiated by a model study of a Cd 1−xMnxTe quan-\ntum well with degenerate valence bands, where the spin\npolarization exhibits a non-monotonic time dependence\nintheformofoscillations,whiletheMarkoviantreatment\npredicts a simple exponential decay. Further investiga-\ntions using a more realistic valence band structure are\nneeded in order to make more precise predictions about\npossible non-Markovian features in the hole spin dynam-\nics in DMS.\nIn the presence of an external magnetic field paral-\nlel to the initial carrier spin (Faraday geometry), earlier\nstudies40that did not consider non-magnetic impurity\nscattering predicted that the asymptotic value ofthe car-14\nrier spins in the conduction band ofa DMS quantum well\nat long times tare significantly different in quantum ki-\nnetic and Markovian calculations. This was attributed\nto a broadening of the distribution of the scattered elec-\ntrons due to the build-up of strong carrier-impurity cor-\nrelations, which, because of the correlation energy, leads\nto a non-conservation of single-particle energies. The\nbroadeningresults in an occupation ofstates by electrons\nwhose back-scattering to the original band is strongly\nsuppresseddue to the lackofstateswith matchingsingle-\nparticle energies. This induces a bias between spin-flip\nscattering from the spin-up to the spin-down subband\nand vice versa. In the presence of a strong non-magnetic\ncarrer-impurity interaction, the correlation energy be-\ncomes much larger and with it also the broadening of the\nscattered carrier distribution and the deviations of the\nasymptotic value of the carrier spin polarization from its\nvalue obtained in Markovian calculations.\nInthe Voigtgeometry, wheretheinitial carrierspinpo-\nlarization is perpendicular to the external field, the car-\nrierspin precessesabout the effective magnetic field com-\nprised of the external field and the mean field due to the\nimpurity magnetization. There, the carrier-impuritycor-\nrelations lead to a renormalization of the spin precession\nfrequencies. An analytic expression for this renormaliza-\ntion is presented and it is found to be of a similar form\nas the expression for the correlation energies. The non-\nmagnetic carrier-impurity interaction influences the fre-\nquencyrenormalizationviaacross-termwhichvanishesif\neither the magnetic or the non-magnetic carrier-impurity\ninteraction is neglected. In the case of the conduction\nband of Cd 1−xMnxTe, the magnetic and non-magneticcontributions to the frequency renormalization have op-\nposite signs. A measurement of the frequency renormal-\nization can therefore indicate the sign of the exchange\ninteraction. For magnetic fields at which the renormal-\nization due to the magnetic correlations is particularly\nstrong, the magnetic and non-magnetic contributions al-\nmost cancel each other. However, in most situations, the\npurely magnetic contributionis relativelyweak39, so that\nthe cross-term dominates the total frequency renormal-\nization. The order of magnitude of the frequency renor-\nmalization for the cases considered here is about a few\npercent of the mean-field precession frequency.\nTosummarize,theinfluenceofthenon-magneticimpu-\nrity scattering on the spin dynamics in DMS is two-fold:\nFirst, it leads to a significant redistribution of carriers in\nk-space, which facilitates the suppression of some non-\nMarkovian effects in certain situations. Second, it causes\na formation of strong many-body correlations between\ncarriers and impurities, which result in large correlation\nenergies and a significant renormalization of the carrier\nspin precession frequency.\nACKNOWLEDGMENTS\nWe gratefully acknowledge the financial support from\nthe UniversidaddeBuenosAires, projectUBACyT2014-\n2017 No. 20020130100514BA, and from CONICET,\nproject PIP 11220110100091.\nAppendix A: Reduced set of equations of motions\nThe equations of motions for the variables defined in\nEq. (6) are:\n∂\n∂t/an}bracketle{tSl/an}bracketri}ht=/summationdisplay\nimǫlimωi\nMn/an}bracketle{tSm/an}bracketri}ht+Jsd\nV/summationdisplay\nkk′/summationdisplay\nimǫlimRe{Qik′\nmk}, (A1a)\n∂\n∂tnk1=JsdNMn\n¯hV2/summationdisplay\nk/summationdisplay\ni2Im{Qik\nik1}+J0Nimp\n¯hV2/summationdisplay\nk2Im{¯C0k\nk1}, (A1b)\n∂\n∂tsl\nk1=/summationdisplay\nijǫlijωi\nesj\nk1+JsdNMn\n¯hV2/summationdisplay\nk/bracketleftbig1\n2Im{Q0k\nlk1}+/summationdisplay\nijǫlijRe{Qjk\nik1}/bracketrightbig\n+J0Nimp\n¯hV2/summationdisplay\nk2Im{¯Clk\nk1},(A1c)\n∂\n∂tQ0k2\nlk1=−i(ωk2−ωk1)Q0k2\nlk1+/summationdisplay\nii′ǫlii′ωi\nMnQ0k2\ni′k1+i\n¯hb0k2\nlk1I+i\n¯hb0k2\nlk1imp, (A1d)\n∂\n∂tQjk2\nlk1=−i(ωk2−ωk1)Qjk2\nlk1+/summationdisplay\nii′ǫjii′ωi\neQi′k2\nlk1+/summationdisplay\nii′ǫlii′ωi\nMnQjk2\ni′k1+i\n¯hbjk2\nlk1I+i\n¯hbjk2\nlk1imp, (A1e)\n∂\n∂t¯C0k2\nk1=−i(ωk2−ωk1)¯C0k2\nk1+i\n¯hc0k2\nk1I+i\n¯hc0k2\nk1sd, (A1f)\n∂\n∂t¯Cjk2\nk1=−i(ωk2−ωk1)¯Cjk2\nk1+/summationdisplay\nii′ǫjii′ωi\ne¯Ci′k2\nk1+i\n¯hcjk2\nk1I+i\n¯hcjk2\nk1sd, (A1g)15\nwith\nb0k2\nlk1I=Jsd/summationdisplay\ni/bracketleftbigg\nRe{/an}bracketle{tSiSl/an}bracketri}ht}(si\nk2−si\nk1)+i/summationdisplay\nmǫilm/an}bracketle{tSm/an}bracketri}ht\n2/parenleftBig\n(1−nk1)si\nk2+(1−nk2)si\nk1/parenrightBig\n+/an}bracketle{tSi/an}bracketri}ht(sl\nk1si\nk2−si\nk1sl\nk2)/bracketrightbigg\n,\n(A1h)\nbjk2\nlk1I=Jsd/summationdisplay\ni/bracketleftbigg\nRe{/an}bracketle{tSiSl/an}bracketri}ht}/bracketleftBig\nδij/parenleftbignk2\n4−nk1\n4/parenrightbig\n+i\n2ǫijk(sk\nk1+sk\nk2)/bracketrightBig\n+i\n2/summationdisplay\nmǫilm/an}bracketle{tSm/an}bracketri}ht/bracketleftBig\nδijnk1+nk2−nk1nk2\n4\n+δijsk1·sk2−(si\nk1sj\nk2+si\nk2sj\nk1)+i\n2ǫijk/parenleftbig\n(1−nk1)sk\nk2−(1−nk2)sk\nk1/parenrightbig/bracketrightBig/bracketrightbigg\n, (A1i)\nb0k2\nlk1imp=J0/an}bracketle{tSl/an}bracketri}ht(nk2−nk1), (A1j)\nbjk2\nlk1imp=J0/an}bracketle{tSl/an}bracketri}ht(sj\nk2−sj\nk1), (A1k)\nc0k2\nk1I=J0(nk2−nk1), (A1l)\ncjk2\nk1I=J0(sj\nk2−sj\nk1), (A1m)\nc0k2\nk1sd=Jsd/an}bracketle{tSi/an}bracketri}ht(si\nk2−si\nk1), (A1n)\ncjk2\nk1sd=Jsd/bracketleftBig1\n4/an}bracketle{tSj/an}bracketri}ht(nk2−nk1)+i\n2ǫijk/an}bracketle{tSi/an}bracketri}ht(sk\nk2+sk\nk1)/bracketrightBig\n. (A1o)\n1M. N. Baibich, J. M. Broto, A. Fert, F. N. Van Dau,\nF. Petroff, P. Etienne, G. Creuzet, A. Friederich, and\nJ. Chazelas, Phys. Rev. Lett. 61, 2472 (1988).\n2S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Moln´ ar, M. L. Roukes, A. Y. Chtchelka-\nnova, and D. M. Treger, Science 294, 1488 (2001).\n3S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990).\n4D. Awschalom and M. Flatt´ e, Nat. Phys. 3, 153 (2007).\n5I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n6T. Dietl, Nat. Mater. 9, 965 (2010).\n7H. Ohno, Nat. Mater. 9, 952 (2010).\n8T. Dietl and H. Ohno, Rev. Mod. Phys. 86, 187 (2014).\n9T. Dietl, H. Ohno, F. Matsukura, J. Cibert, and D. Fer-\nrand, Science 287, 1019 (2000).\n10Z. Ben Cheikh, S. Cronenberger, M. Vladimirova, D. Scal-\nbert, F. Perez, and T. Wojtowicz, Phys. Rev. B 88, 201306\n(2013).\n11Y. S. Chen, M. Wiater, G. Karczewski, T. Wojtowicz, and\nG. Bacher, Phys. Rev. B 87, 155301 (2013).\n12S. Cronenberger, M. Vladimirova, S. V. Andreev, M. B.\nLifshits, and D. Scalbert, Phys. Rev. Lett. 110, 077403\n(2013).\n13J. Debus, V. Y. Ivanov, S. M. Ryabchenko, D. R.\nYakovlev, A. A. Maksimov, Y. G. Semenov, D. Brauk-\nmann, J. Rautert, U. L¨ ow, M. Godlewski, A. Waag, and\nM. Bayer, Phys. Rev. B 93, 195307 (2016).\n14J. Debus, A. A. Maksimov, D. Dunker, D. R. Yakovlev,\nE. V. Filatov, I. I. Tartakovskii, V. Y. Ivanov, A. Waag,\nand M. Bayer, phys. stat. sol. (b) 251, 1694 (2014).\n15A. A. Maksimov, D. R. Yakovlev, J. Debus, I. I. Tar-\ntakovskii, A. Waag, G. Karczewski, T. Wojtowicz, J. Kos-\nsut, and M. Bayer, Phys. Rev. B 82, 035211 (2010).16F. Perez, J. Cibert, M. Vladimirova, and D. Scalbert,\nPhys. Rev. B 83, 075311 (2011).\n17M. D. Kapetanakis, I. E. Perakis, K. J. Wickey, C. Pier-\nmarocchi, and J. Wang, Phys. Rev. Lett. 103, 047404\n(2009).\n18M. D. Kapetanakis and I. E. Perakis, Phys. Rev. Lett. 101,\n097201 (2008).\n19H. Krenn, K. Kaltenegger, T. Dietl, J. Spa/suppress lek, and\nG. Bauer, Phys. Rev. B 39, 10918 (1989).\n20S. A. Crooker, D. D. Awschalom, J. J. Baumberg, F. Flack,\nand N. Samarth, Phys. Rev. B 56, 7574 (1997).\n21P. Barate, S. Cronenberger, M. Vladimirova, D. Scalbert,\nF. Perez, J. G` omez, B. Jusserand, H. Boukari, D. Ferrand,\nH. Mariette, J. Cibert, and M. Nawrocki, Phys. Rev. B\n82, 075306 (2010).\n22J. H. Jiang, Y. Zhou, T. Korn, C. Sch¨ uller, and M. W.\nWu, Phys. Rev. B 79, 155201 (2009).\n23H. Ohno, A. Shen, F. Matsukura, A. Oiwa, A. Endo,\nS. Katsumoto, and Y. Iye, Appl. Phys. Lett. 69, 363\n(1996).\n24R. Fiederling, M. Keim, G. Reuscher, W. Ossau,\nG. Schmidt, A. Waag, and L. W. Molenkamp, Nature\n402, 787 (1999).\n25C. Thurn, M. Cygorek, V. M. Axt, and T. Kuhn, Phys.\nRev. B87, 205301 (2013).\n26M. Cygorek and V. M. Axt, Phys. Rev. B 90, 035206\n(2014).\n27M. Cygorek and V. M. Axt, Semicond. Sci. Technol. 30,\n085011 (2015).\n28M. Cygorek, P. I. Tamborenea, and V. M. Axt, Phys. Rev.\nB93, 205201 (2016).\n29J. Kondo, Prog. Theor. Phys. 32, 37 (1964).\n30J. Kossut, phys. stat. sol. (b) 72, 359 (1975).16\n31A. Twardowski, M. Nawrocki, and J. Ginter, phys. stat.\nsol. (b)96, 497 (1979).\n32R. Bouzerar and G. Bouzerar, EPL (Europhysics Letters)\n92, 47006 (2010).\n33M. W. Wu, J. H. Jiang, and M. Q. Weng, Phys. Rep. 493,\n61 (2010).\n34R. J. Elliott, Phys. Rev. 96, 266 (1954).\n35M. I. D’yakonov and V. I. Perel’, Zh. Eksp. Teor. Fiz. 60,\n1954 (1971), [Sov. Phys. JETP 33, 1053 (1971)].\n36C. Thurn and V. M. Axt, Phys. Rev. B 85, 165203 (2012).\n37C. Thurn, M. Cygorek, V. M. Axt, and T. Kuhn, Phys.\nRev. B88, 161302(R) (2013).\n38M. Cygorek and V. M. Axt, Journal of Physics: Conference\nSeries647, 012042 (2015).\n39M. Cygorek, P. I. Tamborenea, and V. M. Axt, Phys. Rev.\nB93, 035206 (2016); 94, 079906 (2016).40M. Cygorek, F. Ungar, P. I. Tamborenea, and V. M. Axt,\nsubmitted to Proceedings of SPIE, Optics + Photonics:\nNanoscience + Engineering, Spintronics IX (2016).\n41F. Ungar, M. Cygorek, P. I. Tamborenea, and V. M. Axt,\nPhys. Rev. B 91, 195201 (2015).\n42J. Kossut and J. Gaj, eds., Introduction to the Physics of\nDiluted Magnetic Semiconductors , Springer Series in Ma-\nterials Science No. 144 (Springer, Berlin, 2011).\n43F. Long, W. E. Hagston, P. Harrison, and T. Stirner,\nJournal of Applied Physics 82, 3414 (1997).\n44S.-K. Chang, A. V. Nurmikko, J.-W. Wu, L. A.\nKolodziejski, and R. L. Gunshor, Phys. Rev. B 37, 1191\n(1988).\n45J. Schilp, T. Kuhn, and G. Mahler, Phys. Rev. B 50, 5435\n(1994).\n46C. F¨ urst, A. Leitenstorfer, A. Laubereau, and R. Zimmer-\nmann, Phys. Rev. Lett. 78, 3733 (1997)."    },    {        "title": "1807.09059v1.Asymptotic_Dynamics_of_Magnetic_Micro_Swimmers.pdf",        "content": "Asymptotic Dynamics of Magnetic\nMicro-Swimmers\nPauline Rüegg-Reymond and Thomas Lessinnes\nJuly 25, 2018\nAbstract\nMicro-swimmers put into motion by a rotating magnetic field have\nprovided interesting challenges both in engineering and in mathematical\nmodelling. We study here the dynamics of a permanent-magnetic rigid\nbody submitted to a spatially-uniform steadily-rotating magnetic field in\nStokes flow. This system depends on two external parameters: the Ma-\nson number, which is proportional to the angular speed of the magnetic\nfield and inversely proportional to the magnitude of the field, and the\nconical angle between the magnetic field and its axis of rotation. This\nwork focuses on asymptotic dynamics in the limits of low and high Mason\nnumber, and in the limit of low conical angle. Analytical solutions are\nprovided in these three regimes. In the limit of low Mason number, the\ndynamical system admits a periodic solution in which the magnetic mo-\nment of the swimmer tends to align with the magnetic field. In the limit\nof large Mason number, the magnetic moment tends to align with the\naverage magnetic field, which is parallel to the axis of rotation. Asymp-\ntotic dynamics in the limit of low conical angle allow to bridge these two\nregimes. Finally, we use numerical methods to compare these analytical\npredictions with numerical solutions.\n1 Introduction\nUnderstanding motion at low Reynolds number has been an active research\ntopic for decades [1, 2, 3, 4, 5, 6]. Intuitively, this corresponds to the limit of\neither very small bodies moving in water, or bodies in a very viscous fluid. It\nwas initially motivated by the study of micro-organisms [7, 8, 9], and more re-\ncently also found applications in engineering with the advent of artificial micro-\nswimmers [10, 11, 12]. These consist in micrometer to millimetre scale devices\nimmersed in fluid, propelled either by a chemical fuel [13, 14], or by an external\npower source, often an external magnetic field [15, 16]. An example of applica-\ntion is the precise delivery of a microscopic payload to a specific location in a\ncomplicated geometrical environment and through fluids of different rheological\nproperties. In particular, this problem occurs in bio-medical applications, such\nas targeted drug delivery and microsurgery [17].\nIn both [18] and the present study, we focus on the setup where the swim-\nmer is a rigid permanent magnet that experiences a rotating external field, see\nalso [19, 20, 21, 22, 23]. Our theoretical results are valid for rigid swimmers\n1arXiv:1807.09059v1  [math.DS]  24 Jul 2018of arbitrary shapes, while our numerical experiments are performed on helices\nwith circular cross-sections.\nThe swimmer is controlled by adjusting the Mason number aand the angle\n between the magnetic field and its axis of rotation. The Mason number is a\nnon-dimensional number that represents the balance between the drag and the\nmagneticloadontheswimmer; itwillhelptokeepinmindthatitisproportional\nto the angular velocity of the rotating magnetic field, and inversely proportional\nto its magnitude. At any moment, the magnetic moment of the swimmer tries\nto align with the external field but this rotation is slowed by the ensuing Stokes\nflow. In [18], this system is modelled as a first-order nonlinear ODE in SO(3)\n– see also [19, 20, 23]. Depending on aand it has exactly zero, one, or two\nstable relative equilibria. For these motions, the body rotates at the same speed\nas the external field with its magnetic moment lagging behind the external field.\nIf the body is chiral, this rotation generates a controllable translational motion.\nFinally, during that study, we observed that the system can also remain out of\nequilibrium. Experimentally these states would be classified as beyond step-out\nbehaviours. It is the purpose of the present paper to study them analytically.\nMathematically, we study this non-linear equation by a combination of an-\nalytical and numerical methods. We identify a certain scale acsuch that, in\nthe limit of small Mason numbers a\u001cac, an asymptotic expansion shows that\nthe system can be simplified to a single first-order ODE on the circle: all out\nof equilibrium solutions are periodic in this limit. In the limit of large Mason\nnumbersa\u001dac, we use the averaging method [24] to obtain an effective equa-\ntion on SO(3) called the guiding system. Its analysis show that (unsurprisingly)\nthe magnetic moment aligns with the averaged field. But higher order terms\ncoming from the averaging method reveal a secular motion: the swimmer slowly\nrotates – timescale of order ac=a– about the mean field. To bridge between the\nlowaand largearegimes, we study the limit of small sin which corresponds\nto the magnetic field being close to either parallel or antiparallel to its axis of\nrotation. This reveals a continuous change between the low aregime where the\nmagnetic moment of the swimmer is remaining close to the magnetic field, and\nthe largearegime where the magnetic moment is remaining close to the axis of\nrotation of the magnetic field, and allows us to characterise the average position\nof the magnetic moment with respect to the magnetic field, and the amplitude\nof the excursions of the magnetic moment away from its average position.\nDirect numerical integrations for small and large aas well as small sin \ncorroborate the analytical findings for all particular swimmers we tested. Fi-\nnally, we use numerical continuation methods [25] to show that these regimes\ntransform into one another as parameters are varied.\nThe paper is organised as follows. The dynamical system is described in\nsection 2. The limits of low Mason number a, high Mason number and low sin \nare respectively studied in sections 3, 4 and 5. The latter also describes the\ntransitionfromlow atohigha. Finally,section6containsnumericalintegrations\nof the system and discusses how they compare with our predictions from the\nthree previous sections. Note also that sections 3, 4 and 5 open with a short\nsummary of the main results therein.\n22 Governing Equations\nWe consider a neutrally buoyant rigid body immersed in a viscous fluid filling an\ninfinite three-dimensional space. The medium is permeated by a rotating, spa-\ntially homogeneous, magnetic field Bwhich we call the external field for short.\nThe body is assumed to be a permanent magnet with magnetic moment m. The\nshape of the body is taken into account only through its mobility matrix\nM=\u0012M11M12\nMT\n12M22\u0013\n2R6\u00026; (1)\nwhich is the inverse of the drag matrix giving the Stokes flow approximation\nof force and torque due to drag as a linear combination of linear and angular\nvelocities (see [4, 6] for instance).\nIn the Stokes flow limit, the angular velocity !of such a swimmer is ap-\nproximated as\n!=M22[m\u0002]B; (2)\nwhere after non-dimensionalisation and scaling, the magnetic moment mand\nmagnetic field Bare unit vectors. We leave aside the linear velocity, as the dy-\nnamics of the swimmer depend only on its orientation – see [19, 21] for instance;\nfor a detailed derivation, see [18, 26].\n2.1 The Swimmer’s Orientation\nWe assume that the body is equipped with a right-handed and orthonormal\nmaterial frame D=fd1;d2;d3g. We also use a fixed, right-handed and or-\nthonormal lab frame E=fe1;e2;e3g. Throughout, we make the slight abuse\nof notation that involves identifying a vector with its components in the body\nframeD, i.e. mrepresents the triplet (m\u0001d1;m\u0001d2;m\u0001d3)for instance. In\nparticular, the mobility matrix Mand the magnetic moment mare constant in\nthe body frame. This allows to define the constant matrix\nP=M22[m\u0002] (3)\nso that (2) rewrites\n!(t) =PB(t); (4)\nwhere all time dependences are explicit.\nWedenoteby R2SO(3)thematrixrepresentingthechangeofbasisbetween\nthe lab frame and the body frame, i.e. D=ER. By definition of the angular\nvelocity!, this matrix obeys the following ode on SO(3)\n_R=R[!\u0002]; (5)\nwith the notation\n[a\u0002] =0\n@0\u0000a\u0001d3a\u0001d2\na\u0001d3 0\u0000a\u0001d1\n\u0000a\u0001d2a\u0001d1 01\nA: (6)\nTaking the transposed of (5) and working column by column yields\n_ei=\u0000!\u0002ei; (7)\n3where as per our convention, the vectors eiof the lab frame also represent the\ntriplets (ei\u0001dj)j=1;2;3.\nWe define a third frame, the magnetic frame ~E, in which the magnetic field\nBand its axis of rotation are fixed. The axis of rotation is also fixed in the lab\nframe, and without loss of generality we pick it as e3, so that\neE=ER 3(at)whereR3(') :=0\n@cos'\u0000sin'0\nsin' cos' 0\n0 0 11\nA;(8)\nand whereais the Mason number obtained after non-dimensionalisation of the\nangular speed \u000bof the magnetic field:\na=\u000b\u0011`3\nmB; (9)\nwhere\u0011is the dynamic viscosity of the fluid, `is a characteristic length, mis the\nmagnitude of the magnetic moment, and Bis the magnitude of the magnetic\nfield (see [18, 26] for details).\nFinally, we define Qas the matrix representing the change of basis between\nthe magnetic frame and the body frame, i.e. ~E=DQ, which implies\nQ(t) =RT(t)R3(at): (10)\nThus the magnetic field Bcan be written as\nB(t) =Q(t)hsin \n0\ncos i\n; (11)\nwhere > 0is the conical angle between the magnetic field and its axis of\nrotation–notethatthisimpliesthatthelabframeischosensothatthemagnetic\nfield lies in the (e1;e3)-plane and that e1\u0001B>0at timet= 0, which can be\nassumed without loss of generality. In this picture, taking a time derivative\nof (10) and using (5), yields\n_Q=h\u0010\naQh0\n0\n1i\n\u0000!\u0011\n\u0002i\nQ: (12)\nIt is also straightforward to show that if we know e3andB, then e1,e2,\nande3are known (provided sin 6= 0). In this picture, it is advantageous to\nuse (11) to replace (7) by\n(\n_e3=\u0000!\u0002e3;\n_B= (ae3\u0000!)\u0002B:(13)\nThe three systems of equations (7), (12) and (13) are equivalent. Using (4)\nand (11), they can be written in closed form respectively as\n_ei=\u0000(PB)\u0002ei;where B= sin (cos(at)e1+ sin(at)e2) + cos e3;(14)\nand where the eiform a right-handed orthonormal basis;\n_Q=h\u0010\naQh0\n0\n1i\n\u0000PQhsin \n0\ncos i\u0011\n\u0002i\nQ; (15)\n4whereQ2SO(3);\n(\n_e3=\u0000(PB)\u0002e3;\n_B= (ae3\u0000PB)\u0002B;(16)\nwhere e3\u0001B= cos ;e3\u0001e3= 1and1B\u0001B= 1.\nBoth the systems (16) and (15) are autonomous, whereas in (14) Bdepends\nexplicitly on t. When solving analytically, we found it easiest to work with\ne3andBin section 3, and with the eis in sections 4 and 5. When solving\nnumerically in section 6, we found it easiest to work with Q.\nThe behaviour of the swimmer therefore depends on the two parameters a\nand . The aim of this paper is to study its asymptotic dynamics in the limits\nofa!0,a!1, and sin !0. The steady states of these systems are studied\nin detail in [18, 26].\n2.2 The Pmatrix\nWe will use the matrix Pdefined in (3) and its singular value decomposition\nthroughout our analysis. We introduce it here.\nLet\u001b0= 0,\u001b1, and\u001b2be the singular values of Pwith corresponding\nright-singular vectors \f0=m,\f1, and\f2and left-singular vectors \u00110=\nM\u00001\n22m=\r\rM\u00001\n22m\r\r,\u00111, and\u00112. Note that M22is a symmetric and positive defi-\nnite block of the mobility matrix (1) so that its inverse exists [4]. Furthermore,\nthese definitions imply that \f0\u0001\u00110>0so that there exist angles \u00132[0;\u0019=2)\nand\u00102[0;2\u0019)such that\n\u00110= cos\u0013\f0+ sin\u0013(sin\u0010\f1\u0000cos\u0010\f2):\nWe will see in sections 3 and 4 that \u0013is a crucial material parameter for the\nexistence of out of equilibrium solutions of the system.\nBy definition, we have the relations\nP\fi=\u001bi\u0011i; PT\u0011i=\u001bi\fi;\nand each set of singular vectors forms an orthonormal basis that we can assume\nto be right-handed. These bases are also constant in the body frame.\n3 Periodic Orbits at Small a\nLow values of acan be achieved experimentally either by considering strong\nmagnetic effects, slowly rotating external fields, or particularly small bodies –\nsee (9). We show that in this case, the magnetic field and the magnetic moment\nof the swimmer align on a time scale of order 1. Thereafter, the leading order\ndynamic is completely specified by the first order differential equation (25) on\nR. Specifically, we show that\n\u000fif 2(\u0019=2\u0000\u0013;\u0019=2 +\u0013), then there exists a unique stable relative\nequilibrium;\n1This last equation comes from the scaling already mentioned under equation (2).\n5\u000fif =2[\u0019=2\u0000\u0013;\u0019=2+\u0013], then the leading order dynamics exhibits a single\nperiodic orbit of period\n2\u0019cos\u0013\nap\ncos(\u0013+ ) cos(\u0013\u0000 ):\nTo obtain this result, we first note that when a\u001cminf1;\u001b1;\u001b2g, it takes\nanO(1=a)time for the magnetic field to make a full rotation around its axis.\nSince we are interested in the behaviour of the system after many revolutions,\nwe rescale to a longer time scale T=at. The system (16) becomes\n8\n><\n>:ade3\ndT=\u0000(PB)\u0002e3;\nadB\ndT=ae3\u0002B\u0000(PB)\u0002B:(17)\nIn the following, we perform a singular expansion analysis of (17). In the\ninner layer, that is when T=O(1=a)or equivalently when t=O(1), the\ngoverning equation are (16). Expanding\ne3=e[0]\n3+ae[1]\n3+O(a2); B=B[0]+aB[1]+O(a2);(18)\nsubstituting (18) in (16) and matching at each order in awe find that the\nequation for B[0]decouples and reads\n_B[0]=\u0000(PB[0])\u0002B[0]:\nThis equation has two equilibria B[0]=\u0006\f0where B[0]=\f0is stable and\nglobally attracting while B[0]=\u0000\f0is unstable.\nIn the outer layer, that is when T > 1, we substitute an expansion of the\nform (18) in (17) and match at zeroth and first order in ato find\n8\n>>>>>>>><\n>>>>>>>>:(PB[0])\u0002e[0]\n3= 0;\n(PB[0])\u0002B[0]= 0;\nde[0]\n3\ndT=\u0000(PB[1])\u0002e[0]\n3\u0000(PB[0])\u0002e[1]\n3;\ndB[0]\ndT= (e[0]\n3\u0000PB[1])\u0002B[0]\u0000(PB[0])\u0002B[1]:(19)\nBecause Bande3are unit vectors and B\u0001e3= cos is constant, we have\nB[0]\u0001B[0]= 1;e[0]\n3\u0001e[0]\n3= 1; B[1]\u0001B[0]= 0;\ne[1]\n3\u0001e[0]\n3= 0;e[0]\n3\u0001B[0]= cos :(20)\nThe first two equations in (19) imply that PB[0]= 0and hence\nB[0]=\u0006\f0; (21)\nwherewechoosethe‘ +’signbecauseitcorrespondstotheattractingequilibrium\nof the inner layer. Substituting this result in the last equation of (19) yields\nPB[1]\u0002B[0]=e[0]\n3\u0002B[0]: (22)\n6Furthermore, because of (20,21), there exists a function \u0015(T)such that\ne[0]\n3(T) = cos \f0+ sin \u0000\ncos\u0015(T)\f1+ sin\u0015(T)\f2\u0001\n: (23)\nNext, equations (20-23) together with PB[1]?\u00110yield\nPB[1](T) = sin \u0010\ncos\u0015(T)\f1+ sin\u0015(T)\f2+ sin\u0000\n\u0015(T)\u0000\u0010\u0001\ntan\u0013\f0\u0011\n;(24)\nwhere\u00132[0;\u0019=2)and\u00102[0;2\u0019]have been defined in section 2.2.\nFinally, substituting (23,24) in the third equation of (19) yields\nd\u0015\ndT= cos \u0000sin tan\u0013sin(\u0015\u0000\u0010): (25)\nWhen\u0019=2\u0000\u0013< <\u0019= 2 +\u0013, we havejcos j<sin tan\u0013and (25) has two\nequilibria given by\n\u0015=\u0010+\u0019\n2\u0006arccos\u0012cos cos\u0013\nsin sin\u0013\u0013\n:\nThe equilibrium with a minus sign in \u0006is stable and the other one is unstable.\nIfj \u0000\u0019=2j>\u0013, thenjcos j>sin tan\u0013andd\u0015=dTnever changes sign. In\nconsequence, the leading order dynamic exhibits a periodic solution of period\nZ\u0010+\u0019\n\u0010\u0000\u0019d\u0015\ncos \u0000sin tan\u0013sin(\u0015\u0000\u0010)=2\u0019cos\u0013p\ncos(\u0013+ ) cos(\u0013\u0000 ):(26)\nIn the body frame, this periodic dynamic corresponds to the axis of rotation e3\nofthemagneticfielditselfrotatingabout \f0; clockwisewhen  2(0;\u0019=2\u0000\u0013)and\nanti-clockwise when  2(\u0019=2 +\u0013;\u0019). This bifurcation between stable equilibria\nfor 2[\u0013;\u0019\u0000\u0013]and periodic orbit when  =2[\u0013;\u0019\u0000\u0013]occurs through a periodic\nsolution of infinite period.\n4 Asymptotic Dynamics at Large a\nLarge values of the Mason number acorrespond in experiments to either weak\nor rapidly rotating magnetic fields. We show that in this setting, the magnetic\nmoment mtends to align with the average magnetic field, that corresponds\neither to its axis of rotation +e3or to the opposite of its axis of rotation \u0000e3\ndepending on the conical angle  . The mismatch between the magnetic moment\nand\u0006e3is of order 1=a. We also show that there is a slow residual rotation of\nthe swimmer about the average field, with period of order a.\nTo analyse the case of a\u001d1, we work with the version (14) of our system\n_ei=\u0000(PB)\u0002ei; (27)\nand remember that the external field is given as rotating at constant angular\nvelocity with respect to the lab frame so that\nB(t) = sin \u0000\ncos(at)e1(t) + sin(at)e2(t)\u0001\n+ cos e3(t):(28)\n7We will analyse (27) by applying the averaging method described in [24,\n]. The main idea is that because Bchanges much faster than e3, we can ap-\nproximate the effect of Bon the dynamic of the system by averaging Bover\none of its period of revolution2\u0019\na. The method transforms the non-autonomous\nsystem (27,28) into an autonomous averaged differential equation called the\nguiding system. The averaging procedure is carried out in Section 4.1 and the\nresulting guiding system is studied in Section 4.2.\n4.1 Averaged Governing Equations\nWe first rescale time to T=atand define\"= 1=a\u001c1so that (27) becomes\ndei\ndT=\u0000\"(PB)\u0002ei: (29)\nThe averaging operator, noted by an overline, is defined as follows: to any\nfunctionX(e1;e2;e3; T) 2\u0019-periodic in T, it associates the averaged function\nX(e1;e2;e3) =1\n2\u0019Z2\u0019\n0X(e1;e2;e3;T)dT;\nwhere the integration is performed while keeping the eis constant. We also\ndefine the function B?which depends on three vectors and a scalar\nB?(x1;x2;x3;T) = sin \u0000\ncos(T)x1+ sin(T)x2\u0001\n+ cos x3;\nsuchthat B(T) =B?\u0000\ne1(T);e2(T);e3(T);T\u0001\n. Notethat B?(x1;x2;x3) = cos x3.\nGiven two functions u[1]andu[2], the near identity transformation (see ap-\npendix A)\nei=ci+\"\u0010\nu[1]\u0002ci\u0011\n+\"2\u0012\nu[2]\u0002ci+1\n2u[1]\u0002(u[1]\u0002ci)\u0013\n+O(\"3)(30)\ntransformsthedifferentialequation(29)intoadifferentialequationfortheright-\nhanded orthonormal frame of the cis:\ndci\ndT=\"g[1]\u0002ci+\"2g[2]\u0002ci+O(\"3); (31)\nwhere the g[i]are computed hereunder.\nThefunctions u[i](i= 1;2)appearingin(30)aretobeunderstoodasexplicit\nfunctions of the vectors cj(j= 1;2;3) and of time T. We further require that\nthey are 2\u0019-periodic in T. Then they can be chosen such that the functions g[i]\n(i= 1;2) appearing in (31) are independent of time. In that case, solutions of\nthe truncated equation\ndci\ndT=\"g[1]\u0002ci+\"2g[2]\u0002ci (32)\nare guaranteed to remain close to the solutions of (29) up to an order O(\")on a\ntimescale of order T=O(1=\"2)[24]. This means that in the original time scale,\nwe are guaranteed an approximation of order O(1=a)on a time scale of order\nt=O(a)(remember that a\u001d1).\n8Substituting (30) in (29), expanding in \", and matching at each order gives\n@Tu[1]\f\f\f\n(c1;c2;c3;T)=\u0000PB?(c1;c2;c3; T)\u0000g[1](c1;c2;c3); (33)\n@Tu[2]=\u0000P\u0010\nu[1]\u0002B?\u0011\n\u0000@u[1]\n@cj\u0001\u0010\ng[1]\u0002cj\u0011\n\u00001\n2u[1]\u0002@Tu[1]+g[1]\u0002u[1]\u0000g[2];\n(34)\nwith implied summation on repeated indices and where the functions u[1],u[2]\nandB?appearing in (34) are evaluated at (c1;c2;c3; T)and the functions g[1]\nandg[2]are evaluated at (c1;c2;c3).\nRequiring the function u[1]to be periodic implies that the left-hand side\nof (33) vanishes upon averaging over time T. Accordingly, we find that\ng[1](c1;c2;c3) =\u0000PB?(c1;c2;c3) =\u0000cos Pc3: (35)\nAfter substituting (35) in (33), we integrate and find that\nu[1](c1;c2;c3;T) = sin \u0000\n\u0000sinTPc1+ cosTPc2) +A(c1;c2;c3);(36)\nwhere Ais an arbitrary function. Here, we choose A= 0so that u[1]= 0.\nSubstituting (35,36) in (34), and averaging gives\ng[2](c1;c2;c3) =sin2 \n2\u0010\nP(c1\u0002Pc2) +P(Pc1\u0002c2)\u0000(Pc1)\u0002(Pc2)\u0011\n:(37)\nComing back to the time scale t=T=a, and substituting (35,37) in (32),\nthe guiding system becomes\ndci\ndt=\u0000cos (Pc3)\u0002ci\n+\"sin2 \n2\u0010\nP(c1\u0002Pc2) +P(Pc1\u0002c2)\u0000(Pc1)\u0002(Pc2)\u0011\n\u0002ci:(38)\n4.2 Analysis of the Guiding System\nThe argument of the previous section transformed the non-autonomous sys-\ntem (27,28) into the approximate autonomous system (38). In this section we\nshow that the first-order solution of (38) always exhibits stable periodic motion.\nFirst, remark that if we truncate the guiding system (38) to zeroth order in\n\", we obtain\ndci\ndt=\u0000cos (Pc3)\u0002ci: (39)\nThe system (39) has two families of one-parameter equilibria of the form2\n8\n><\n>:c1= cos\u001f\f1+ sin\u001f\f2;\nc2=\u0000sin\u001f\f1+ cos\u001f\f2;\nc3=\u0006\f0:(40)\nItisstraightforwardtoshowthatallequilibriacorrespondingto c3=Sign(cos )\f0\naremeta-stableinthesensethatalleigenvaluesoftheassociatedstabilitymatrix\n2Recall that according to section 2.2 Pc3= 0iffc3=\u0006\f0\n9jcos jP[\f0\u0002]are strictly negative but for one that vanishes and corresponds\nto motion within the continuous family. It is also straightforward to show that\nthe stable manifold is almost globally attracting – that is it is attracting for all\ninitial values that do not lie strictly on the manifold of unstable equilibria.\nHowever as the system gets near this zeroth-order stable manifold, the mag-\nnitude of the zeroth order term in (38) decreases, and the first order term can\nno longer be neglected. For the long term behaviour of the system, we therefore\nexpect the system to be close to but not quite on the equilibrium (40) with the\nsign chosen so as to match that of cos . We therefore define a function\n\u001c(t) =\u001c[0](t) +\"\u001c[1](t) +O(\"2);\nthat specifies elements in the stable manifold via\n8\n><\n>:f1(t) = cos\u001c(t)\f1+ sin\u001c(t)\f2;\nf2(t) =&\u0000\n\u0000sin\u001c(t)\f1+ cos\u001c(t)\f2\u0001\n;\nf3(t) =&\f0;(41)\nwhere&=Sign(cos ). We then expand the cis as follows\nci=fi+\"x\u0002fi+O(\"2); (42)\nwhere x(t) =x1(t)f1(t) +x2(t)f2(t)is to be determined by the differential\nequation (38).\nSubstituting (42) in (38) leads to\n&_\u001c[0]f3\u0002fi+\"\u0010\u0000\n&_\u001c[1]f3+ _x1f1+ _x2f2\u0001\n\u0002fi\n+&_\u001c[0]\u0000\n(f3\u0002x)\u0002fi+x\u0002(f3\u0002fi)\u0001\u0011\n+O(\"2)\n=\"\u0010\n\u0000cos P(x\u0002f3) +g[2](f1;f2;f3)\u0011\n\u0002fi+O(\"2);(43)\nwhere g[2](f1;f2;f3)is given by (37) where substitution according to (41) yields\ng[2](f1;f2;f3) =&sin2 \n2\u0000\nP(\f1\u0002P\f2\u0000\f2\u0002P\f1)\u0000\u001b1\u001b2\u00110\u0001\n;\nwhich is independent from \u001c.\nMatching orders in (43), we find at zeroth order _\u001c[0]= 0and at first order\n8\n><\n>:&_\u001c[1]=h(f1)\u0001f2\n_x1=h(f2)\u0001f3\n_x2=h(f3)\u0001f1;(44)\nwhere\nh(fi) =\u0010\n\u0000cos P(x\u0002f3) +g[2](f1;f2;f3)\u0011\n\u0002fi:\nBecause _\u001c[0]= 0, the function fi(t)evolves on a slow time scale O(1=\"). Ac-\ncordingly we look for equilibria of the equations in (x1; x2)while keeping all fis\nconstant. We find a single equilibrium of the form\n\u0014x1\nx2\u0015\n=1\n\u001b1\u001b2cos cos\u0013\u0014Pf1\u0001\u0000\nf3\u0002g[2](f1;f2;f3)\u0001\nPf2\u0001\u0000\nf3\u0002g[2](f1;f2;f3)\u0001\u0015\n10which leads to\nx=1\n\u001b1\u001b2cos cos\u0013(I\u0000f3fT\n3)PT\u0000\nf3\u0002g[2](f1;f2;f3)\u0001\n:\nSubstituting xaccordingly, the first equation of (44) becomes\n_\u001c[1]=\u0000&\u001b1\u001b2sin2 \n2 cos\u0013;\nso that\n\u001c(t) =\u001c0\u0000\"&\u001b 1\u001b2sin2 \n2 cos\u0013t+O(\"2): (45)\nIn conclusion, we have shown that in the limit of large Mason number a,\nthe magnetic moment mtends to align with the average magnetic field, which\nis\u0006e3depending on the sign of cos . The mismatch between mand\u0006e3is of\norder\". Indeed gathering our findings3we have\ne3=&m+O(\");\nwhere&= sign cos . Furthermore,\ne1= cos\u001c(t)\f1+ sin\u001c(t)\f2+O(\")\ne2=\u0000&sin\u001c(t)\f1+&cos\u001c(t)\f2+O(\");\nwhere\u001c(t)is given by (45), so that viewed from the lab frame, there is a slow\nresidual rotation of the body frame about the average field.\n5 Asymptotic Dynamics at Small sin \nIn this section, we analyse the case of a magnetic field Balmost parallel to its\naxis of rotation e3; this corresponds to sin \u001c1. We show that in the small\nsin regime, the magnetic moment describes a circle in the magnetic frame,\nwhose centre shifts from the time-dependent magnetic field Bto the average\nmagnetic field\u0006e3as the Mason number agoes from asymptotically small to\nasymptotically large, and whose radius goes to zero in both limits a!0and\na!1. This regime thus bridges the small aregime studied in section 3 and\nthe largearegime studied in 4.\nWe analyse the case sin \u001c1by setting \"= sin and performing an\nasymptotic expansion in \". Note that we have sin \u001c1both for \u001c1and for\n\u0019\u0000 \u001c1, that is for Bclose to either of \u0006e3. The equation for the dynamics\nof the lab frame (14) becomes\n_ei=\u0000&(Pe3)\u0002ei\u0000\"(PR3(at)e1)\u0002ei\u0000&\"2\n2(Pe3)\u0002ei+O(\"3);(46)\nwhere&= sign(cos ).\n3Remember that \f0=m.\n115.1 Asymptotic expansion\n5.1.1 Zeroth order\nThe zeroth order dynamics is given by the equation\n_ei=\u0000&(Pe3)\u0002eifori= 1;2;3;\nwhich is in equilibrium for Pe3= 0, i.e. e3=\u0006\f0. This implies that we have\ntwo families of equilibria given by\ne1= cos\u001c\f1\u0006sin\u001c\f2;e2=\u0000sin\u001c\f1\u0006cos\u001c\f2;e3=\u0006\f0;(47)\nwhere\u001cis a parameter. Furthermore, the equilibria for which the \u0006sign is&\nare stable. As in section 4.2, the magnitude of the zeroth order term in (46)\ndecreases as the system approaches this zeroth-order stable manifold. To study\nthe long-term behaviour of the system, we must therefore also take into account\nhigher order terms, as we expect the solutions to (46) to be close to but not\nquite on the equilibrium (47) with the sign matching &. To this end we define a\nfunction\n\u001c(t) =\u001c[0](t) +\"\u001c[1](t) +\"2\u001c[2](t) +O(\"3);\nthat specifies elements of the stable manifolds as\n8\n>><\n>>:e[0]\n1(t) = cos\u001c(t)\f1+&sin\u001c(t)\f2\ne[0]\n2(t) =\u0000sin\u001c(t)\f1+&cos\u001c(t)\f2\ne[0]\n3(t) =&\f0:\nWe then expand the eis as\nei=e[0]\ni+\"e[1]\ni+\"2e[2]\ni+O(\"3); (48)\nwhere\ne[1]\ni=u[1]\u0002e[0]\ni;e[2]\ni=u[2]\u0002e[0]\ni+1\n2u[1]\u0002(u[1]\u0002e[0]\ni);fori= 1;2;3;\nfor some u[1]=u[1](t),u[2]=u[2](t)2R3(see appendix A).\nSubstituting the expansion (48) in (46), we find at zeroth order\n&_\u001c[0]\f0\u0002e[0]\ni= 0;\nimplying that \u001c[0]is constant, and that\ne1= cos\u001c[0]\f1+&sin\u001c[0]\f2+\"(\u0000sin\u001c[1]\f1+&cos\u001c[1]\f2) +O(\"2);\ne2=\u0000sin\u001c[0]\f1+&cos\u001c[0]\f2\u0000\"(cos\u001c[1]\f1+&sin\u001c[1]\f2) +O(\"2):(49)\n5.1.2 First order\nSubstituting (49) in (46), we obtain at first order\n&_\u001c[1]\f0+_u[1]=\u0000&P(u[1]\u0002\f0)\u0000P(cos(at+\u001c[0])\f1+&sin(at+\u001c[0])\f2):(50)\n12We assume that u[1]=u1\f1+u2\f2and solve for u1,u2. A projection of (50)\non\f1and\f2yields\n\u0014\nu1\nu2\u0015\n=\u0000a(a2I+A2)\u00001A\u0014\n&cos(at+\u001c[0])\nsin (at+\u001c[0])\u0015\n\u0000(a2I+A2)\u00001A2\u0014\n\u0000&sin(at+\u001c[0])\ncos(at+\u001c[0])\u0015\n;\n(51)\nwhere\nA=\u0014\u0000\f1\u0001P\f2\f1\u0001P\f1\n\u0000\f2\u0001P\f2\f2\u0001P\f1\u0015\n:\nProjecting on \f0and substituting (51) therein yields\n\u001c[1]= ~\u001c1cos(at+\u001c[0]) + ~\u001c2sin(at+\u001c[0]);\nwhere\n~\u001c1=&\na\u0014P\f2\u0001\f0\n\u0000P\f1\u0001\f0\u0015\n\u0001\u0000\n\u0000a(a2I+A2)\u00001A[0\n1] +&(a2I+A2)\u00001A2[1\n0]\u0001\n\u00001\naP\f2\u0001\f0\n~\u001c2=&\na\u0014P\f2\u0001\f0\n\u0000P\f1\u0001\f0\u0015\n\u0001\u0000\n&a(a2I+A2)\u00001A[1\n0] + (a2I+A2)\u00001A2[0\n1]\u0001\n+&\naP\f1\u0001\f0:\nNote in particular that\nku[1]k \u0018\na!11\na;and (52)\nku[1]k \u0018\na!01; (53)\nwhich is consistent with e3!\na!1\f0ande3!\na!0hsin \n0\ncos i\n.\n5.1.3 Second order\nAt second order, we find that the dynamics is given by\n&_\u001c[2]\f0+_u[2]+&\u001c[1]u[1]\u0002\f0+1\n2u[1]\u0002_u[1]\n=\u0000&P\u0012\nu[2]\u0002\f0+1\n2u[1]\u0002(u[1]\u0002\f0)\u0013\n\u0000P\u0010\ncos(at+\u001c[0])u[1]\u0002\f1+&sin(at+\u001c[0])u[1]\u0002\f2\u0011\n:\nAgain we assume that u[2]?\f0and we find that u[2]is an affine combination of\ncos(2at+ 2\u001c[0])andsin(2at+ 2\u001c[0])with coefficients depending on a.\nWe use this second order solution only to compare with the numerics.\n5.2 Dynamics of the Magnetic Moment\nThe position of the magnetic moment in the magnetic frame is given by\nQTm=RT\n3(at)2\n4e1\u0001\f0\ne2\u0001\f0\ne3\u0001\f03\n5:\n13Substituting the eis by their asymptotic expansion for sin =\"\u001c1, we find\nthat\nQTm=RT\n3(at+\u001c[0])0\n@2\n40\n0\n&3\n5+\"2\n4\u0000u2\n&u1\n03\n5+\"22\n4\u0000u[2]\u0001\f2\n&u[2]\u0001\f1\n\u0000&u2\n1+u2\n2\n23\n51\nA+O(\"3):(54)\nWhen 1=a\u0014\", we use (52) in (54) to obtain\nQTm=h0\n0\n&i\n+O(\"2)\nin agreement with our findings for a\u001d1in section 4. When a\u0014\", we use (53)\nto obtain\nQTm=h0\n0\n&i\n+h\"\n0\n0i\n+O(\"2)\nin agreement with the prediction for a\u001c1in section 3 since in the magnetic\nframe, the magnetic field is [\";0;&(1 +\"2=2) +O(\"3)]T.\nMoreover, the trajectory of QTmat first order is a circle with radius rand\ncentre m0satisfying\nr=\"a\n2 det(a2I+A2)p\nc0+c1a+c2a2+c3a3+c4a4;\nm0=2\n40\n0\n&3\n5+\"\n2 det(a2I+A2)2\n42 (\u001b1\u001b2cos\u0013)2\u0000a2(\f1\u0001P PT\f1+\f2\u0001P PT\f2)\na(a2+\u001b1\u001b2cos\u0013) (\f2\u0001P\f1\u0000\f1\u0001P\f2)\n03\n5;\nwhere\nc0=\u001b2\n1\u001b2\n2cos2\u0013\u0000\n(P\f1\u0001\f1\u0000P\f2\u0001\f2)2+ (P\f1\u0001\f2+P\f2\u0001\f1)2\u0001\n;\nc1=2\u001b1\u001b2cos\u0013\u0000\n\u0000&(P\f1\u0001\f1\u0000P\f2\u0001\f2) (\u0000P\f1\u0001\f2\n0\u0000P\f2\u0001\f2\n0+\u001b2\n1+\u001b2\n2)\n\u00004 (P\f1\u0001\f2+P\f2\u0001\f1) (P\f1\u0001\f0) (P\f2\u0001\f0)\u0001\n;\nc2=\u00002\u001b1\u001b2cos\u0013\u0000\n(P\f1\u0001\f1\u0000P\f2\u0001\f2)2+ (P\f1\u0001\f2+P\f2\u0001\f1)2\u0001\n+ (\u0000P\f1\u0001\f2\n0\u0000P\f2\u0001\f2\n0+\u001b2\n1+\u001b2\n2)2+ 4 (P\f1\u0001\f0)2(P\f2\u0001\f0)2;\nc3=2&(\u0000P\f1\u0001\f2\n0\u0000P\f2\u0001\f2\n0+\u001b2\n1+\u001b2\n2) (P\f1\u0001\f1\u0000P\f2\u0001\f2)\n+ 4 (P\f1\u0001\f0) (P\f2\u0001\f0) (P\f1\u0001\f2+P\f2\u0001\f1);\nc4=(P\f1\u0001\f1\u0000P\f2\u0001\f2)2+ (P\f1\u0001\f2+P\f2\u0001\f1)2;\nanddet(a2I+A2) = (\u001b1\u001b2cos\u0013)2+a2tr(A2) +a4.\nThis behaviour at small sin matches the limits found at low and large\nMasonnumbers ainsections3and4. Thesefindingsaresupportedbynumerical\nexperimentsandcanalsobeobservedatlargervaluesof sin (cffigures5and6).\n6 Numerical Integration\nWe compare analytical predictions of sections 3-5 with solutions obtained by di-\nrect numerical integration with integrator ode45 in MATLAB. Using the MAT-\nLAB package MatCont [27], we also apply numerical continuation starting from\nknown solutions to investigate the existence of periodic solutions to (15) across\nthe whole parameter plane.\n146.1 Adapting the system for numerical treatment using\nquaternions\nWe have already cast our system in the three equivalent forms (14), (16),\nand (15). Numerically, we found it easier to work with (15) as an ode on\nSO(3). However, the Lie group SO(3) of rotation matrices has dimension three\nas opposed to the nine components of a 3 by 3 matrix. Allowing numerical\nintegrators to treat (15) efficiently therefore requires using a parametrisation of\nSO(3). Unit quaternions provide a particularly adapted parametrisation as they\nboth elegantly describe rotations [28] and can be easily processed by numerical\nintegrators as vectors in R4. In this parametrisation, the ode (15) becomes [29,\n]\n_q=1\n2FT(q)u(q;a; ); q(0) =q0; (55)\nwhere\nF(q) =2\n4q4\u0000q3q2\u0000q1\nq3q4\u0000q1\u0000q2\n\u0000q2q1q4\u0000q33\n5\nand\nu(q;a; ) =aQ(q)h0\n0\n1i\n\u0000PQ(q)hsin \n0\ncos i\n;\nwith\nQ(q) =1\njqj22\n4q2\n1\u0000q2\n2\u0000q2\n3+q2\n4 2 (q1q2\u0000q3q4) 2 (q1q3+q2q4)\n2 (q1q2+q3q4)\u0000q2\n1+q2\n2\u0000q2\n3+q2\n4 2 (q2q3\u0000q1q4)\n2 (q1q3\u0000q2q4) 2 (q2q3+q1q4)\u0000q2\n1\u0000q2\n2+q2\n3+q2\n43\n5:\nThis parametrisation of rotations by quaternion is independent of the norm of\nthe quaternion, and in fact we use only unit quaternions. Solutions of (55) are\nanalytically guaranteed to preserve the norm of the initial condition, which is\nconvenient. However, this independence on the norm also causes the Jacobian\nof the RHS of (55) to be singular. As this is an issue for numerical continuation,\ninstead of (55) we integrate the modified system\n_q=1\n2FT(q)u(q;a; )\u00001\n2\u0010\njqj2\u00001\u0011\nq; q (0) =q0: (56)\nSolutions of (56) with unit initial conditions are guaranteed to remain of norm\none. They are also solutions of (55), and the stability of steady states and\nperiodic orbits is kept unchanged.\n6.2 Numerical Solutions\nDirect numerical integration of (56) was performed in MATLAB using the stan-\ndard integrator ode45 [30]. The observed solutions converged either towards\nstable equilibria or towards stable periodic solutions depending on parameters\naand , and on initial conditions. Equilibria of (56) are entirely classified\nand are the object of a separate publication [18]. To explore the existence of\nperiodic solutions for parameters aand for which there are no stable equi-\nlibrium, we used numerical continuation (MatCont [27]) starting from periodic\nsolutions discovered by direct numerical integration. This procedure allowed to\n150\n0.2\n0.4\n0.6\n0.8\n1\n1.2\n1.4\n1.6\n1.8\n2\n-1\n-0.8\n-0.6\n-0.4\n-0.2\n0\n0.2\n0.4\n0.6\n0.8\n1acosψσ12 sinι\nno relative equilibria\nonly unstable relative equilibria\nstable relative equilibria\n×\n×\n×\n×\n×\n×\n×\n×\n×\n×Figure 1: The parameter plane, with regions hatched according to the existence\nof stable relative equilibria as investigated in [18, 26] for a specific swimmer\n(cf appendix B). All the parameters for which steady states exist are shown in\nthis figure. We establish the existence of stable periodic solutions to (16) or\nequivalently (15) for all parameters in the white and single-hatched areas, and\nfor some parameters in the cross-hatched area using numerical methods. The\nred crosses indicate positions of parameter values used in figures 4, 5, and 6.\nfind connected sets of periodic orbits seemingly covering the entire parameter\nplane except for part of the region where stable steady states exist.\nWe choose here to present the example of a specific swimmer that has the\nshape of a helical rod. Unless stated otherwise, the features observed for this\nswimmer are persistent across a range of different helical swimmers, and we\nconjecture that some of them, in particular the existence of either stable steady\nstates or stable periodic orbits for any pair of parameters aand , remain for\nswimmers of any shape. The study of precisely how solutions depend on the\nswimmer’s shape focusing on helical swimmers will be the subject of a separate\npublication.\nFor the example considered (cf appendix B), we actually found two distinct\nsets of periodic orbits, each seemingly covering the entire region of parameter\nplane where there are no stable equilibria, and part of the region where stable\nequilibria exist (cf fig. 1). They are related to each other through the sym-\nmetry of system (15) under transformation Q(t)7!Q(\u0000t)R2(\u0019). These sets\nintersect along a line seemingly close to  =\u0019=2corresponding to a stability\nexchange. One of them contains stable periodic solutions only on one side of\nthe intersection line, and the other one only on the other side. Several other\n16bifurcations corresponding to stability exchanges occur on these sets, notably\nfold bifurcations of periodic orbits resulting in regions where two distinct stable\nperiodic solutions coexist. Loss of stability also occurs along some branches of\nperiodic solutions as they approach a region with stable steady states, although\nthe type of bifurcation could not be identified with certitude. It is noteworthy\nthat not all branches lose stability as they enter this region.\nOther families of periodic orbits that are disconnected from the two sets\nmentioned above also exist. For instance, there are families of periodic orbits\nbifurcating from Hopf bifurcations that are part of the set of equilibria. For the\nswimmer considered here, these families contain stable periodic orbits but this\nis not the case for other helical swimmers. We cannot rule out the possibility\nthat there exist families of periodic that are neither part of the sets spanning\nmostoftheparameterspacenorpartoftheperiodicorbitsbifurcatingfromHopf\nbifurcations. However we are confident that for the specific swimmer considered\nhere, all the stable periodic solutions were found. Indeed a numerical campaign\nwas set up to investigate the existence of other stable solutions: direct numerical\nintegration was performed for various values of the parameters aand and for\n100 randomly chosen initial condition each time. Only solutions corresponding\nto those already discussed here were found.\n6.3 Comparisons between analytical predictions and nu-\nmerical solutions\nFor small Mason number a, we obtain limit cycles both numerically and an-\nalytically for  2[0;\u0019=2\u0000\u0013)[(\u0019=2 +\u0013;\u0019]. Figure 2 displays the periods of\nanalytical and numerical limit cycles for several fixed values of aand varying\n 2[0;\u0019=2\u0000\u0013)[(\u0019=2 +\u0013;\u0019]. Fora= 10\u00003, the numerical solutions were\nobtained by direct integration for different values of  whereas for a= 0:0159\nanda= 0:0208, the solutions were obtained by numerical continuation letting\n vary. Note that to numerically characterise the periods, we must take into\naccount that the quaternion parametrisation of SO(3) is a two to one covering:\n\u0000qandqparametrise the same rotation. Thus there can be symmetric limit\ncycles [25, p. 282] in the quaternion coordinates that actually correspond to\nlimit cycles in SO(3) of half the period. All the limit cycles found numerically\nfor smallafall into this category. Therefore we compare the period obtained\nanalytically in (26) with half the period obtained numerically.\nFora= 10\u00003the relative error between the two is smaller than47:6922\u000110\u00005\nfor < \u0019= 2\u0000\u0013\u00000:1988and > \u0019= 2 +\u0013+ 0:2235. For closer to\u0019=2\u0006\u0013,\nthe relative error becomes large (the maximal computed value is 0:3688). This\nbehaviour is expected when trying to approximate a vertical asymptote.\nNote thata= 0:0159anda= 0:0208don’t fall into the category of asymp-\ntotically small afor this problem. Indeed, the two singular values \u001b1and\u001b2\nprovide characteristic dimensions, and for ato be considered asymptotically\nsmall it must verify a\u001cminf\u001b1;\u001b2g. Here the minimal singular value is\n\u001b2= 0:0497. Fora= 0:0159the relative error is smaller than 0:0198for\n 2[0;\u0019=2\u0000\u0013\u00000:2267)[(\u0019=2 +\u0013+ 0:1002;\u0019]. Fora= 0:0208, the relative\nerror is smaller than 0:0291for 2[0;\u0019=2\u0000\u0013\u00000:2126)[(\u0019=2 +\u0013+ 0:1297;\u0019].\n4The relative error has finite local maxima, and explodes as it approaches \u0019=2\u0000\u0013from the\nleft or\u0019=2 +\u0013from the right. The bounds are chosen taking into account the largest of these\nlocal maxima.\n170100020003000\n a = 0.0208  a = 0.0159 \n  ψ = π/2+ι\nψ = π/2–ι\ncosψperiod\n-1 -0.5 0 0.5 1 -1 -0.5 0 0.5 100.511.522.533.5× 105\nψ = π/2+ι\nψ = π/2–ι\ncosψperiod a = 0.001 Figure 2: Periods of the periodic solutions vs cos for several values of a\nas obtained by numerical computation\nas predicted analytically in section 3.\nOn the left, a= 10\u00003, and the computed period was recovered from solutions\nobtained by direct numerical integration of (56) for various values of  . On the\nright, the computed period was obtained by numerical continuation letting  \nvary starting from a periodic solution found by direct integration.\nThat the analysis and numerics fit so well for these values of aillustrates the\nrobustness of the features captured by the asymptotic expansion described in\nsection 3.\nIn order to visualise the agreement between solutions obtained numerically\nand analytically in both the large aand small sin regimes, we find that it is\nuseful to view the curve described by the magnetic moment min the magnetic\nframe, that is in the frame locked to the rotating magnetic field B. In the large\naregime, figure 3 exhibits a remarkable agreement between the curves obtained\nas a first order expansion as in section 4 and by direct numerical integration\nfora= 100and different values of  . In figure 4 we show the results of direct\nnumerical integrations for a= 2and different values of  . Although a= 2is\nnot large enough to be in the large aregime, we can observe that the magnetic\nmoment stays close to the average magnetic field, especially for small values of\n .\nThe small sin regime is displayed in figure 5 for  = 0:1. Note that\nthe circles predicted as a first order approximation give a good estimate of\nthe trajectories, and that the second order prediction virtually overlaps the\nnumerical solutions. For  =\u0019=4, that is outside the scope of this asymptotic\nexpansion, the curves described by min the magnetic frame have a qualitatively\nsimilar behaviour, with a mean position moving from Btoe3asaincreases,\nand a decreasing amplitude in both the low and large alimits (cf fig. 6).\nNumerical solutions to (56) at large aand at small sin were found to be\n18-8\n-6\n-4\n-2\n0\n2\n4\n6\n8\n-14\n-12\n-10\n-8\n-6\n-4\n-2\n0\n2× 10-3\n× 10-3ψ = π/10\nψ = π/5\nψ = π/3\nψ = 2π/5Figure3: Trajectoriesofthemagneticmomentinthemagneticframefor a= 100\nand several values of parameter  \nas obtained by numerical integration of (56)\nas predicted analytically in section 4\n\u000findicates the position of the axis of rotation of the magnetic field e3.\nThis view is a projection perpendicular to e3(notice the scales).\nperiodic. The corresponding analyses in sections 4 and 5 did not predict peri-\nodicity but are consistent with it. Numerical periodic solutions corresponding\nto the small aand largearegimes of section 3 and 4 are connected numerically\nby branches of periodic orbits that for small sin correspond to the analytical\nregime of section 5.\n7 Conclusion\nUsinganalyticalandnumericalmethods, westudiedout-of-equilibriumsolutions\nfor the motion of a rigid body in Stokes flow submitted to a steadily rotating\nexternalmagneticfielduniforminspace. Theequationsgoverningthisdynamics\nis given in three equivalent forms (14-16) depending on two parameters: the\nMasonnumber aandconicalangle  . Westudyanalyticallytheirsolutionsusing\nasymptotic expansions in three different regimes: in the limit a!0(section 3),\nin the limit a!1(section 4), and in the limit sin !0(section 5).\nIn the limit of small a, which corresponds to either slowly rotating or strong\nmagnetic field, the body aligns its magnetic moment mgiving the dipole axis\nwith the magnetic field. We find a limiting angle \u0013such that the motion in\nthis regime is in relative equilibrium for  2(\u0019=2\u0000\u0013;\u0019=2 +\u0013)and in relative\nperiodic motion elsewhere. This periodic motion arises as the body rotates\nabout its magnetic moment. We give the period of this solution explicitly. The\nchange of regime as  approaches the limit \u0019=2\u0006\u0013occurs through a periodic\n19-0.6\n-0.4\n-0.2\n0\n0.2\n0.4\n-0.8\n-0.6\n-0.4\n-0.2\n0\n0.2\n0.4ψ = π/10\nψ = π/5\nψ = 3π /10\nψ = 2π/5Figure 4: Trajectories of the magnetic moment in the magnetic frame for a= 2\nand several values of parameter  as obtained by numerical integration of (56)\n\u000findicates the position of the axis of rotation of the magnetic field e3.\nThis view is a projection perpendicular to e3.\nThis value of ais out of the scope of the expansion for large ain section 4, but\nthebehaviourweobserveisqualitativelysimilarasthemagneticmomentrotates\naround the average magnetic field. Note that the approximation is better for\nlower values of  .\nsolution of infinite period.\nIn the limit of large a, which corresponds to either rapidly rotating or weak\nmagnetic field, the body aligns its magnetic moment mwith the average mag-\nnetic field. It exhibits a residual rotation about its magnetic moment. The limit\nsin !0gives a continuous change between the small and large aregimes, with\nmagnetic moment mshifting from alignment with the magnetic field Bto align-\nment between the averaged magnetic field as agoes from 0to1.\nWe used numerical integration and continuation to assess the validity of\nour analytical results on a particular example. The predicted features are in\ngood qualitative agreement even for parameter values outside the scope of our\nthree asymptotic expansions. Numerical continuation allowed to show that the\nout-of-equilibrium solutions at large aand small sin are limit cycles of the\ndynamics (15). Therefore at all parameter values for the swimmer tested, we\nfind either stable periodic orbits or stable equilibria.\nComparison of our results with experiments is limited by two main factors:\nwe assume here that the swimmers are permanently magnetic, and that they\nare neutrally buoyant. To our knowledge, these two factors have not arised\nsimultaneously in experiments with magnetic micro-swimmers.\n20Acknowledgements\nWe thank Oscar Gonzalez for providing the code used to compute the drag\nmatrix used in numerical simulations. We also thank Andrew Petruska for the\ninspiring discussions.\nA Asymptotic expansion on SO(3)\nIn sections 4 and 5 we used an asymptotic expansion on SO(3) to obtain (30)\nand (42), and (48) respectively. We show here that an asymptotic expansion on\na right-handed frame fe1;e2;e3gtakes the form\nei=e[0]\ni+\"u[1]\u0002e[0]\ni+\"2\u0012\nu[2]\u0002e[0]+1\n2u[1]\u0002(u[1]\u0002e[0])\u0013\n+O(\"3):\nEverything used in this appendix belongs to classical mathematical knowledge,\nbut to the best of our knowledge, the derivation of generic asymptotic expan-\nsions in SO(3) can’t be found in the literature. We provide it here to ease the\nprogression of the interested reader through sections 4 and 5.\nWe use the matrix form R=\u0000e1e2e3\u0001\n2SO(3) of a right-handed frame\ngiven by vectors eifori= 1;2;3. Suppose we have a curve on SO(3) given by\n\"7!R(\"). Around\"= 0, this curve is approximated by its Taylor expansion\nR(\") =R[0]+\"R[1]+\"2R[2]+O(\"3); (57)\nwith\nR[0]=R(0); R[1]=R0j\"=0;andR[2]=1\n2R00j\"=0;(58)\nwhere0denotes the derivative by \". SinceR(\")2SO(3) for all \", we have the\nidentity\nR(\")RT(\") =I;\nand differentiating it with respect to \"yields\nR0(\")RT(\") +R(\")R0T(\") = 0:\nThis implies that R0RTis a skew-symmetric matrix, so there exists u=u(\")2\nR3such that\n[u\u0002] =R0RT, R0= [u\u0002]R: (59)\nEquivalently, the three columns ei(i= 1;2;3) ofRsatisfy\ne0\ni=u\u0002ei:\nDifferentiating again we obtain\nR00= [u0\u0002]R+ [u\u0002]2Rand e00\ni=u0\u0002ei+u\u0002(u\u0002ei):(60)\nExpanding uin\"asu=u[1]+2\"u[2]+O(\"2), and substituting (59) and (60)\nin (58) we find that\nR[1]=h\nu[1]\u0002i\nR[0];andR[2]=h\nu[2]\u0002i\nR[0]+1\n2h\nu[1]\u0002i2\nR[0]:\n21Thus obtain the expansion of Rin\"(57) takes the form\nR=R[0]+\"h\nu[1]\u0002i\nR[0]+\"2h\nu[2]\u0002i\nR[0]+O(\"3):\nEquivalently the frame vectors ei(i= 1;2;3) satisfy\nei=e[0]\ni+\"u[1]\u0002e[0]\ni+\"2\u0012\nu[2]\u0002e[0]+1\n2u[1]\u0002(u[1]\u0002e[0])\u0013\n+O(\"3):\nB Numerical data for example swimmer\nThe swimmer used as an example for this paper is a body with the shape\nof a helical rod with radius r= 0:1330, pitchp= 1:1076, total arc-length\nL= 4:1628, and rod radius 0.0936 in non-dimensional units chosen so that its\nradius of gyration is 1. The magnetic moment is m= (0;0:1736;0:9848)Tin a\nbody frame that is chosen so that the rod’s centreline is given by\ns7!2\n4rcoss\nrsins\np\n2\u0019s3\n5fors2\"\n0;Lp\nr2+ (p=2\u0019)2#\n:\nThe rod is capped at both ends by half spheres. Assuming uniform density,\nwe compute the drag matrix Dby computing the resultant loads due to Stokes\nflow at the center of mass. To do so, we use a code provided to us by Oscar\nGonzalez based on a Nyström method for exterior Stokes flow [31, 32, 33]. The\ndrag matrix obtained for the example swimmer in the body frame described\nabove is\nD=2\n412:4654 0:0000 0:0000 0:1433 0:0000\u00000:0000\n\u00000:0000 12:4815 0:0582 0:0000 0:0122 0:1178\n\u00000:0000 0:0577 9:2808 0:0000\u00000:5607\u00000:2158\n0:1427\u00000:0000\u00000:0000 20:1070\u00000:0000 0:0000\n\u00000:0000 0:0116\u00000:5610\u00000:0000 20:1725 0:4032\n0:0000 0:1179\u00000:2158 0:0000 0:4031 1:01963\n5:\nSince the result is not exactly symmetric, we correct this by using M=1\n2D\u00001+\n1\n2D\u0000Tin our computations.\nReferences\n[1] H. Brenner. The Stokes resistance of an arbitrary particle. Chemical En-\ngineering Science , 18(1):1–25, 1963.\n[2] Jens Rotne and Stephen Prager. Variational Treatment of Hydrodynamic\nInteraction in Polymers. The Journal of Chemical Physics , 50(11):4831–\n4837, 1969.\n[3] E M Purcell. Life at low Reynolds number. American Journal of Physics ,\n45(1), 1977.\n[4] J. Happel and H. Brenner. Low Reynolds Number Hydrodynamics: With\nSpecial Applications to Particulate Media . Mechanics of Fluids and Trans-\nport Processes. Springer Netherlands, 1983.\n22[5] J. S. Rathore and N. N. Sharma. Engineering Nanorobots: Chronology of\nModeling Flagellar Propulsion. Journal of Nanotechnology in Engineering\nand Medicine , 1(3), 2010.\n[6] Sangtae Kim and Seppo J Karrila. Microhydrodynamics: Principles and\nSelected Applications . Courier Corporation, 2013.\n[7] Howard C Berg and Robert A Anderson. Bacteria Swim by Rotating their\nFlagellar Filaments. Nature, 245(5425):380–382, October 1973.\n[8] James Lighthill. Flagellar Hydrodynamics. SIAM Review , 18(2):161–230,\nApril 1976.\n[9] Eric Lauga and Thomas R Powers. The Hydrodynamics of Swimming\nMicroorganisms. Reports on Progress in Physics , 72(9), September 2009.\n[10] T Honda, K I Arai, and K Ishiyama. Micro Swimming Mechanisms Pro-\npelled by External Magnetic Fields. IEEE Transactions on Magnetics ,\n32(5):5085–5087, 1996.\n[11] Eric E. Keaveny and Martin R. Maxey. Spiral swimming of an artificial\nmicro-swimmer. Journal of Fluid Mechanics , 598:293–319, February 2008.\n[12] Kathrin E Peyer, Soichiro Tottori, Famin Qiu, Li Zhang, and Bradley J\nNelson. Magnetic helical micromachines. Chemistry (Weinheim an der\nBergstrasse, Germany) , 19(1):28–38, January 2013.\n[13] Stephen J. Ebbens and Jonathan R. Howse. In pursuit of propulsion at the\nnanoscale. Soft Matter , 6(4):726–738, 2010.\n[14] Giacomo Gallino, François Gallaire, Eric Lauga, and Sebastien Michelin.\nPhysics of Bubble-Propelled Microrockets. Advanced Functional Materials ,\n28(25):1800686, 2018.\n[15] J. J. Abbott, K. E. Peyer, M. C. Lagomarsino, Li Zhang, Lixin Dong, I. K.\nKaliakatsos, and Bradley J Nelson. How Should Microrobots Swim? In-\nternational Journal of Robotics Research , 28(11-12):1434–1447, July 2009.\n[16] Gilgueng Hwang and Stéphane Régnier. Remotely powered propulsion of\nhelical nanobelts. Encyclopedia of Nanotechnology , pages 2226–2237, 2012.\n[17] Bradley J Nelson, Ioannis K Kaliakatsos, and Jake J Abbott. Microrobots\nfor Minimally Invasive Medicine. Annual Review of Biomedical Engineer-\ning, 12:55–85, August 2010.\n[18] Pauline Rüegg-Reymond, Thomas Lessinnes, and John H Maddocks. Rel-\native Equilibria of Magnetic Micro-Swimmers. In prep., 2018.\n[19] Yi Man and Eric Lauga. The wobbling-to-swimming transition of rotated\nhelices.Physics of Fluids , 25(7):071904, July 2013.\n[20] Arijit Ghosh, Pranay Mandal, Suman Karmakar, and Ambarish Ghosh.\nAnalytical theory and stability analysis of an elongated nanoscale object\nunderexternaltorque. Physical Chemistry Chemical Physics ,15(26):10817–\n10823, 2013.\n23[21] FarshadMeshkatiandHenryChienFu. Modelingrigidmagneticallyrotated\nmicroswimmers: Rotation axes, bistability, and controllability. Physical\nReview E , 90(6):1–11, 2014.\n[22] Henry C. Fu, Mehdi Jabbarzadeh, and Farshad Meshkati. Magnetiza-\ntion directions and geometries of helical microswimmers for linear velocity-\nfrequency response. Physical Review E , 91(4):1–13, 2015.\n[23] Konstantin I. Morozov, Yoni Mirzae, Oded Kenneth, and Alexander M.\nLeshansky. Dynamics of arbitrary shaped propellers driven by a rotating\nmagnetic field. Physical Review Fluids , 2(4):044202, April 2017.\n[24] Jan A. Sanders, Ferdinand Verhulst, and James Murdock. Averaging Meth-\nods in Nonlinear Dynamical Systems . Applied Mathematical Sciences.\nSpringer-Verlag, New York, 2 edition, 2007.\n[25] Yuri A. Kuznetsov. Elements of Applied Bifurcation Theory . Number 112\nin Applied mathematical sciences. Springer, New York, NY, 3. ed edition,\n2004. OCLC: 815949776.\n[26] Pauline Rüegg-Reymond. On the Dynamics of Magnetic Micro-Swimmers .\nPhD thesis, In Prep., 2019.\n[27] A.Dhooge, W.Govaerts, YuA.Kuznetsov, H.G.E.Meijer, andB.Sautois.\nNew features of the software MatCont for bifurcation analysis of dynamical\nsystems. Mathematical and Computer Modelling of Dynamical Systems ,\n14(2):147–175, April 2008.\n[28] Simon L Altmann. Rotations, Quaternions, and Double Groups . Courier\nCorporation, 2005.\n[29] DJDichmann, YLi, andJohnHMaddocks. HamiltonianFormulationsand\nSymmetries in Rod Mechanics. Mathematical Approaches to Biomolecular\nStructure and Dynamics, IMA Volumes in Mathematics and Its Applica-\ntions, 82:71–113, 1996.\n[30] Lawrence F. Shampine and Mark W. Reichelt. The MATLAB ODE Suite.\nSIAM Journal on Scientific Computing , 18(1):1–22, January 1997.\n[31] Oscar Gonzalez. On stable, complete, and singularity-free boundary inte-\ngral formulations of exterior Stokes flow. SIAM Journal on Applied Math-\nematics, 69(4):933–958, 2009.\n[32] J. Li and O. Gonzalez. Convergence and conditioning of a Nyström method\nfor Stokes flow in exterior three-dimensional domains. Advances in Com-\nputational Mathematics , 39(1):143–174, 2013.\n[33] Oscar Gonzalez and Jun Li. A convergence theorem for a class of Nyström\nmethods for weakly singular integral equations on surfaces in R3. Mathe-\nmatics of Computation , 84(292):675–714, 2015.\n24-0.02\n0\n0.02\n0.04\n0.06\n0.08\n0.1\n0.12\n-0.07\n-0.06\n-0.05\n-0.04\n-0.03\n-0.02\n-0.01\n0\n0.01\n0.02a = 10\na = 1a = 0.1a = 0.01\n-0.02\n0\n0.02\n0.04\n0.06\n0.08\n0.1\n0.12a = 10\na = 1a = 0.1a = 0.01\n-0.07\n-0.06\n-0.05\n-0.04\n-0.03\n-0.02\n-0.01\n0\n0.01\n0.02Figure5: Trajectoriesofthemagneticmomentinthemagneticframefor  = 0:1\nand various Mason numbers a\nas obtained by numerical integration of (56)\nas predicted analytically in section 5 (top: first order, bottom: second\norder).\n\u000findicates the position of the axis of rotation of the magnetic field e3.\nFindicates the position of the magnetic field B.\nThis view is a projection perpendicular to e3.\n25-0.2\n-0.1\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n-0.7\n-0.6\n-0.5\n-0.4\n-0.3\n-0.2\n-0.1\n0\n0.1\n0.2\n0.3\na\n = 0.1\na\n = 0.01\na\n = 1\na\n = 10Figure 6: Trajectories described by the magnetic moment in the magnetic frame\nfor =\u0019=4and several values of parameter aas obtained by numerical inte-\ngration of (56)\n\u000findicates the position of the axis of rotation of the magnetic field e3.\nFindicates the position of the magnetic field B.\nThis view is a projection perpendicular to e3.\nThis value of  is out of the scope of the expansion for small sin in section 5,\nbut the behaviour we observe is qualitatively similar.\n26"    },    {        "title": "2211.01356v1.A_study_of_global_magnetic_helicity_in_self_consistent_spherical_dynamos.pdf",        "content": "A study of global magnetic helicity in self-consistent spherical\ndynamos\nP. Gupta\u00031, R.D. Simitev\u00032and D. MacTaggart\u00033y\n\u0003School of Mathematics and Statistics, University of Glasgow, Glasgow, G12 8QQ,\nUK\nARTICLE HISTORY\nCompiled November 3, 2022\nABSTRACT\nMagnetic helicity is a fundamental constraint in both ideal and resistive magnetohy-\ndrodynamics. Measurements of magnetic helicity density on the Sun and other stars\nare used to interpret the internal behaviour of the dynamo generating the global\nmagnetic \feld. In this note, we study the behaviour of the global relative magnetic\nhelicity in three self-consistent spherical dynamo solutions of increasing complexity.\nMagnetic helicity describes the global linkage of the poloidal and toroidal magnetic\n\felds (weighted by magnetic \rux), and our results indicate that there are preferred\nstates of this linkage. This leads us to propose that global magnetic reversals are,\nperhaps, a means of preserving this linkage, since, when only one of the poloidal\nor toroidal \felds reverses, the preferred state of linkage is lost. It is shown that\nmagnetic helicity indicates the onset of reversals and that this signature may be\nobserved at the outer surface.\nKEYWORDS\nmagnetohydrodynamics, magnetic helicity, self-consistent convective dynamo\n1. Introduction\nMagnetic helicity is an invariant of ideal magnetohydrodynamics (MHD) that de-\nscribes \feld line topology weighted by magnetic \rux (Berger and Field, 1984; Mo\u000batt,\n1969). This quantity is important for both laboratory (Taylor, 1986) and astrophysical\nplasmas (MacTaggart and Hillier, 2020; Pevtsov et al. , 2014) since it remains approx-\nimately conserved in the weakly resistive limit (Berger, 1984; Faraco and Lindberg,\n2019; Faraco et al. , 2022). Magnetic helicity has been studied, in detail, in the context\nof mean \feld dynamo models (see Brandenburg, 2018; Brandenburg and Subramanian,\n2005; Cameron et al. , 2017, and references within). In self-consistent convection-driven\ndynamos in spherical shells, however, magnetic helicity has received little attention.\nOn the scales typically considered in such global simulations, it is not possible to reach\nthe high magnetic Reynolds number range needed to maintain an approximately con-\nstant value of the global magnetic helicity. This does not mean, however, that magnetic\n1P. Gupta: orcid.org/0000-0002-2976-5993\n2R.D. Simitev: orcid.org/0000-0002-2207-5789\n3D. MacTaggart: orcid.org/0000-0003-2297-9312\nyCorresponding author. Email: david.mactaggart@glasgow.ac.ukarXiv:2211.01356v1  [astro-ph.SR]  2 Nov 2022helicity is not a useful measure for understanding the nature of self-consistent spher-\nical dynamos. In this work, rather than considering the role that magnetic helicity\nplays in turbulence (such as its in\ruence on the \u000b-e\u000bect in mean-\feld models), we will\nfocus on what the global magnetic helicity can tell us about the global topology of the\nmagnetic \feld, namely how the toroidal and poloidal \felds link with each other and\nhow variations in this linkage manifest themselves, particularly at the outer surface\n(which can be observed in stars).\nSome initial inroads have already been made in this direction in observational studies\nof global magnetic helicity. Recent examples of work in this area, which focus on the\nSun, include Hawkes and Berger (2018); Pipin and Pevtsov (2014); Pipin et al. (2019).\nIn particular, Hawkes and Berger (2018) show that helicity \rux during solar minima\nis a good predictor of the subsequent solar maxima, and can improve upon predictions\nbased on the polar magnetic \feld alone. This result seems to be stronger when the\nhelicity \rux measured in only one hemisphere is considered, a calculation that can\nbe performed due to the equatorial symmetry of helicity density at the solar surface.\nMagnetic helicity has also begun to be measured in stars other than the Sun (e.g.\nLund et al. , 2020, 2021). Again, only the surface density of magnetic helicity can be\nmeasured, and at limited resolution. Despite this, scaling laws relating the strength\nof the surface helicity density to that of the toroidal \feld have been found (Lund\net al. , 2021). Further, the results of the latter study suggest that the \feld topology\nis di\u000berent for stars on di\u000berent parts of their evolutionary track. Thus, it may be\npossible to use magnetic helicity density, in conjunction with other observables, to\ncharacterise di\u000berent stages of stellar, and hence dynamo, evolution.\nThe purpose of this note is to investigate global magnetic helicity in various types\nof self-consistent convective dynamos in spherical shells. Our approach for this pilot\ninvestigation is not to try to model the Sun or a particular star or planet, but rather\nto analyse from a general standpoint some typical solutions to a well-studied model\nfor spherical dynamos (Busse and Simitev, 2011; Simitev and Busse, 2005) that are\nrepresentative of various known dynamo regimes. The three particular solutions that\nwe consider increase in complexity and allow us to assess the usefulness of magnetic\nhelicity as a prediction and analysis tool when confronted with increasingly chaotic\nspatial and temporal \row and \feld structures.\nThe paper is set up as follows. First, the main model equations are introduced,\ntogether with brief remarks on the numerical method used. This is followed by a\ndescription of magnetic helicity in spherical shells. We then consider each particular\ndynamo solution in turn, with a focus on what information is provided by magnetic\nhelicity. The report ends with a summary of the results and a discussion of their\ninterpretation.\n2. Main equations\n2.1. Mathematical model formulation\nFollowing an established model of convection-driven spherical dynamo action (e.g.\nBusse and Simitev, 2006; Simitev and Busse, 2009, 2012) we consider a spherical shell\nthat rotates about a \fxed axis with a constant angular speed \n. A static state is\n2assumed to exist with the temperature distribution\nTS=T0\u0000\fd2r2=2 + \u0001T\u0011r\u00001(1\u0000\u0011)\u00002; (1a)\n\f=q=(3\u0014cp); (1b)\nT0=T1\u0000\u0001T=(1\u0000\u0011); TS(ri) =T1; TS(ro) =T2;\u0001T=T2\u0000T1; (1c)\nwhereT1andT2are constant temperatures at the inner and outer spherical boundaries,\n\u0011=ri=rois the ratio of the inner radius rito the outer radius ro,qis a uniform\nheat source density, \u0014is the thermal di\u000busivity, dis the shell thickness and ris the\nmagnitude of the position vector to a point in the spherical shell.\nWithin the spherical shell, we solve the equations of magnetohydrodynamics (MHD)\nunder the Boussinesq approximation,\nr\u0001u= 0;r\u0001B= 0; (2a)\u0012@\n@t+u\u0001r\u0013\nu=\u0000r\u0019\u0000\u001ck\u0002u+#r+r2u+B\u0001rB; (2b)\nP\u0012@\n@t+u\u0001r\u0013\n#= [Ri+Re\u0011r\u00003(1\u0000\u0011)\u00002]r\u0001u+r2#; (2c)\nPm\u0012@\n@t+u\u0001r\u0013\nB=PmB\u0001ru+r2B: (2d)\nIn the above equations, uis the velocity \feld, Bis the magnetic \feld, #is the deviation\nfrom the static temperature distribution, \u0019is an e\u000bective pressure including all terms\nthat can be represented as a gradient, ris the position vector and kis the unit vector\nin the positive vertical direction about which the shell rotates.\nAbove, the Boussinesq approximation is used, where the density \u001ais constant ev-\nerywhere except in the buoyancy term. There the density varies linearly with respect\nto a constant background density \u001a0and has the form\n\u001a=\u001a0(1\u0000\u000b#); (3)\nwhere\u000bis a constant and represents the speci\fc thermal expansion coe\u000ecient. The\nnon-dimensional parameters in equations (2) are\nRi=\u000b\r\fd6\n\u0017\u0014; Re=\u000b\r\u0001Td4\n\u0017\u0014; \u001c =2\nd2\n\u0017; P =\u0017\n\u0014; Pm=\u0017\n\u0015; (4)\nwhich are the internal and external thermal Rayleigh numbers RiandRe, the Coriolis\nnumber\u001c, the Prandtl number Pand the magnetic Prandtl number Pm, respectively.\nOf the quantities not yet de\fned, \u0015is the magnetic di\u000busivity, \u0017is the viscosity and\n\ris a constant related to the gravitational acceleration g=\u0000d\rr.\nTo solve the above equations, the velocity and magnetic \feld are \frst split into\npoloidal and toroidal parts,\nu=uP+uT=r\u0002(r\u0002rv) +r\u0002rw; (5a)\nB=BP+BT=r\u0002(r\u0002rh) +r\u0002rg; (5b)\n3where r=rer. The scalar quantities are decomposed into spherical harmonics,\nv=1X\nl=0lX\nm=\u0000lVm\nl(r;t)Pm\nl(cos\u0012)eim\u001e; w =1X\nl=0lX\nm=\u0000lWm\nl(r;t)Pm\nl(cos\u0012)eim\u001e;(6a)\ng=1X\nl=0lX\nm=\u0000lGm\nl(r;t)Pm\nl(cos\u0012)eim\u001e; h =1X\nl=0lX\nm=\u0000lHm\nl(r;t)Pm\nl(cos\u0012)eim\u001e;(6b)\nwherePm\nl(\u0001) denotes the associated Legendre polynomials of \frst kind and ( r;\u0012;\u001e )\nare spherical coordinates. Once all the scalars are decomposed as above, the radial\nfunctions are expanded in terms of Chebychev polynomials and the series are trun-\ncated to a chosen resolution ( nr;n\u0012;n\u001e). The MHD equations are solved using a pseu-\ndospectral method and a combination of Crank-Nicolson and Adams-Bashforth time\nintegration schemes. Further details of the numerical method can be found in Tilgner\n(1999) and an open source version of the code is available at Silva and Simitev (2018).\nFor brevity, we do not repeat this description and instead guide the reader to these\nother works. The calculations performed in this work have been run with the resolu-\ntions (nr;n\u0012;n\u001e) = (33;64;129) and (nr;n\u0012;n\u001e) = (41;96;193). Azimuthally averaged\ncomponents of the \felds v,w,handgwill be indicated by an overbar.\nTo complete the speci\fcation of the above mathematical model, we require boundary\nconditions. We assume \fxed temperatures at r=riandr=roand consider both cases\nwith no-slip conditions\nv=@v\n@r=w= 0; (7a)\nand cases with stress-free conditions,\nv=@2v\n@r2=@\n@r\u0010w\nr\u0011\n= 0: (7b)\nFor the magnetic \feld, we assume electrically insulating conditions at r=riand\nr=ro. At these locations, the poloidal function hmatches the function h(e)which\ndescribes the potential \felds outside the spherical shell,\ng=h\u0000h(e)=@\n@r(h\u0000h(e)) = 0: (7c)\n2.2. Magnetic helicity\nThe magnetic \feld is generated and sustained by thermal convective motions inside\nthe spherical shell (e.g. Busse and Simitev, 2005). The \feld emanates outside of the\nspherical \ruid shell where, in the absence of sustaining \ruid motion, it takes the form\nof a freely decaying potential \feld. Thus, the magnetic \feld is not closed and we\nmust consider relative magnetic helicity (Berger and Field, 1984) instead of classical\nhelicity (Mo\u000batt, 1969; Woltjer, 1958). Assuming his continuous at the inner and\nouter boundaries, Berger (1985) showed that the relative helicity in the spherical shell\n4Case 1: Case 2: Case 3:\nSteady dynamo Quasi-periodically Aperiodically-\nreversing dynamo reversing dynamo\n\u0011 0.35 0.4 0.4\nRi 0 3.5\u00021068:5\u0002105\nRe 1050 0\n\u001c 2000 3\u00021043\u0002104\nP 1 0.75 0.1\nPm 5 0.65 1\nBoundary no-slip, stress-free, stress-free,\nconditions \fxed temperatures, \fxed temperatures, \fxed temperatures,\ninsulating outer space insulating outer space insulating outer space\nTable 1. The three dynamo solutions considered in the study.\nvolumeVhas the appealing form\nH= 2Z\nVLh\u0001LgdV= 2Z\nV(curl\u00001BP)\u0001BTdV; (8)\nwhere L=\u0000r\u0002r. Note that the de\fnition of Lis chosen to match the particular\nmagnetic \feld decomposition used in equation (5b). The representation in (8) can be\ninterpreted in terms of the global mutual linkage of the poloidal and toroidal magnetic\n\felds (see also Berger and Hornig, 2018). This is a generalized form of linkage as the\npoloidal and toroidal \felds occupy the same region of space. However, as will be made\nclear in subsequent analysis, the linkage interpretation of magnetic helicity will prove\nvery useful.\nBy simple vector algebra, we have that BT=Lg. There is no simple relation\nbetween BPandLh, but, in spherical coordinates, the latter can be expressed as\nLh=\u0012\n0;1\nsin\u0012@h\n@\u001e;\u0000@h\n@\u0012\u0013\n: (9)\nA test to con\frm that the equation for the helicity Hhas been coded correctly is\ndescribed in the Appendix.\n3. Dynamo solutions\nWe now investigate the behaviour of global magnetic helicity in various dynamo solu-\ntions. Three cases of increasing complexity are considered as summarized in table 1.\n3.1. Case 1: Steady dynamo\nTo begin with, we will consider a laminar dynamo solution that has been used as a\nbenchmark case by the community for validating and testing the accuracy of numerical\ncodes (Christensen et al. , 2001; Matsui et al. , 2016). The non-dimensional parameters\nand boundary conditions are listed in the \frst column of table 1.\n5(a)\n (b)\n (c)\nFigure 1. Plots of the velocity \feld for Case 1 at a particular time (this is a steady \row so other times just\nrepresent rigid rotations of this solution) (a) shows lines of constant u\u001e(left half) and rsin\u0012@\u0012v(right half) in\na meridional plane, visualising the toroidal and poloidal \felds respectively; (b) shows lines of constant urat\nr=ri+ 0:5. There is a distinct pattern of columnar convection at both the equator and near the poles; and\n(c) shows lines of constant r@\u001evin the equatorial plane. Blue is negative, red is positive and green is zero in\nthis \fgure and in all subsequent contour plots. (Colour online)\n(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFigure 2. Plots of the magnetic \feld (top row) and magnetic helicity (bottom row) for Case 1 at a particular\ntime. The top panel (a) shows meridional lines of constant B\u001e(left half) and rsin\u0012@\u0012h(right half) visualising\nthe toroidal and poloidal \felds respectively; (b) shows lines of constant Bratr=ri+ 0:96; and (c) displays\nequatorial streamlines, r@\u001eh= const. The bottom panel (d) shows contour lines of azimuthally averaged helicity\ndensity (left half) and the helicity density in a particular slice \u001e= const (right half); (e) shows the helicity\ndensity atr=ri+ 0:96; and (f) shows the helicity density at the equatorial plane. (Colour online)\nThe main feature of this dynamo is that it develops a steady-state for both the\nmagnetic and velocity \felds. The steady velocity pro\fle exhibits columnar convection,\nas shown in \fgure 1. Focusing now on the magnetic \feld, there is a global dipole with\ninversions at the equator. This situation, together with other details, is displayed in\n\fgure 2.\nIn \fgure 2, the top row reveals the behaviour of the magnetic \feld in (a) the\nmeridional plane, (b) a near-outer spherical surface, and (c) the equatorial plane.\nThe left-most plot shows the toroidal and poloidal components of the magnetic \feld.\nBoth quantities shown in the hemispheres are meridional averages. In the left-hand\nhemisphere, displaying the large-scale behaviour of the toroidal \feld, the magnetic \feld\npattern is perfectly antisymmetric about the equator. In the right-hand hemisphere,\nrevealing the large-scale behaviour of the poloidal \feld, a dominant dipolar \feld is\npresent, except near the surface at the equator. Here, smaller and weaker (compared\nto the polar \felds) positive and negative polarities exist, as can be seen from the\n6(a)\n (b)\nFigure 3. A representation of the linkage of toroidal and poloidal \felds. Colours correspond to those in \fgure\n2(a). The (Gauss) linkage in (a) is \u00001 and in (b) is +1. These signs correspond with those of the helicity\ndensity in \fgure 2(d). (Colour online)\nsurface radial magnetic \feld plot in (b). The radial component of the magnetic \feld\nshows a dipolar symmetry, which means Br= 0 on the equatorial plane. This magnetic\npro\fle, which rotates rigidly with the sphere, is simple enough to provide a complete\ninterpretation of the behaviour of magnetic helicity.\nThe bottom row of \fgure 2 displays the contour lines of (d) azimuthally averaged\nhelicity density (left half) and unaveraged helicity density in a particular slice (right\nhalf), (e) the helicity density at the outer surface, and (f) the helicity density in the\nequatorial plane. Figure 2(d) reveals how the sign of the helicity density in di\u000berent\nparts of the spherical shell depends on the linkage of the toroidal and poloidal \felds.\nIn order to clarify the signs of the helicity density displayed in this \fgure, we sketch an\nidealized linkage of the toroidal and poloidal \felds in \fgure 3. The representation of\nlinkage in \fgure 3 is idealized because the toroidal and poloidal \felds are space-\flling\nin the spherical shell and are not the union of two disjoint regions. However, due to\nthe relative simplicity of the \feld structure (clearly separated regions of positive and\nnegative toroidal \feld), this idealization serves to indicate the correct sign of linkage.\nFigure 3(a) shows the linkage of a negative (blue) part of the toroidal \feld with the\npoloidal \feld and \fgure 3(b) shows the linkage of a positive (red) part of the toroidal\n\feld with the poloidal \feld. The Gauss linking numbers for these links are \u00001 and\n+1 respectively, corresponding to the negative and positive patches of helicity density\nin \fgure 2(d). Just to reiterate, this is only a simple representation of the linkage,\nbut it is e\u000bective in showing how the magnetic helicity reveals information about the\nconnectivity of poloidal and toroidal \felds on large scales.\nFigure 2(d) displays the distribution of helicity density near the surface. This shows\na more detailed morphology compared to the azimuthal average plots, but there is\nstill a clear dominance of positive helicity density in the north and negative helicity\ndensity in the south. As in the other plots, there is perfect antisymmetry about the\nequator. Thanks to this property, together with this being a steady solution, H= 0.\nThis simple dynamo case is a clear example showing that although the total mag-\nnetic helicity is zero, this value is due to the cancellation of positive and negative\nlarge-scale structures. In particular, the signs of these structures depend on the link-\nage of poloidal and toroidal \felds. Magnetic helicity, however, is not only a measure\nof linkage but is weighted by magnetic \rux. This consideration will become impor-\ntant for understanding cases exhibiting more irregular temporal behaviour and spatial\nmorphology as discussed below.\n7(a)\n (b)\n (c)\nFigure 4. Typical structures of the velocity \feld for Case 2. (a) shows lines of constant u\u001ein the left half\nand streamlines rsin\u0012@\u0012v= const in the right half, all in the meridional plane; (b) shows lines of constant ur\natr=ri+ 0:5; and (c) shows streamlines, r@\u001ev= const. in the equatorial plane. These images correspond to\nthe timet= 5:538 in the simulation. (Colour online)\n(a)\n (b)\nFigure 5. Time series of magnetic helicity for Case 2. (a) shows the total helicity as a function of time. (b)\nshows the helicity density integrated in the northern hemisphere (solid) and the southern hemisphere (dashed).\n3.2. Case 2: Quasi-periodically reversing dynamo\nWe now consider a time-dependent quasi-periodic dynamo solution that regularly re-\nverses the signs of both the poloidal and toroidal parts of the magnetic \feld. The\nnon-dimensional parameters for this case, together with the boundary conditions, are\nlisted in the second column of table 1.\nFigure 4 displays the typical structure of the velocity \feld. Although there is colum-\nnar convection like Case 1, the velocity is now evolving chaotically in time and exhibits\nmuch smaller-scale structures.\nUnlike in Case 1, where the \feld structure was \fxed in time and interpreting the\nhelicity density in terms of linkage was straightforward, here the situation is more\ninvolved. Further, the magnetic Prandtl number Pmis an order of magnitude smaller\nfor this case compared to Case 1, so the total magnetic helicity is not expected to be\nstrongly conserved. This is indeed the case upon examination of \fgure 5(a).\nSince magnetic helicity is not conserved in time for this dynamo, we cannot make any\nclear causal conclusions based on helicity conservation. However, this does not mean\nthat magnetic helicity does not reveal useful information. To reveal the structure of\nmagnetic helicity in this dynamo solution, we restrict its calculation to northern and\nsouthern hemispheres separately. The results are shown in \fgure 5(b). There is a clear\nwave solution in both hemispheres, with a complete cycle taking about 0.02 time units.\nThis result indicates that although there can be local changes in the magnetic helicity\ndensity in a hemisphere, there is no complete change in the linkage of hemispheric\n8(a)\n (b)\nFigure 6. Time series related to reversals. (a) displays reversals of the poloidal and toroidal \felds. The\npoloidal and toroidal \felds are represented by their dominant spectral expansion components H0\n1(solid line)\nandG0\n1(dashed line), respectively. Both of these quantities are calculated at r=ri+0:5. (b) displays the total\nmagnetic energy EB, which oscillates at a frequency similar to the hemispheric helicities displayed in \fgure 5.\nFigure 7. A latitude vs time plot of the magnetic helicity density at the surface for Case 2 with a nonlinear\ncolourmap: indigo (-400,-30); maroon (-30, 30) and seismic (blue and red) (30,400). (Colour online)\ntoroidal and poloidal \felds. This global preservation of linkage arises from the nearly\nco-temporal reversal of the global toroidal and poloidal \felds, leaving little time for\na di\u000berent large-scale linkage to develop. The reversals of the poloidal and toroidal\n\felds are indicated in \fgure 6(a). Since the global toroidal and poloidal \felds reverse\ntogether, the linkage, and thus the overall sign, of magnetic helicity in each hemisphere\nremains the same.\nAlthough the integrated magnetic helicity density does not change sign in each\nhemisphere, it does oscillate. Magnetic helicity can change due to variations in \feld\nlinkage and magnetic \feld strength, the latter indicated by the magnetic energy shown\nin \fgure 6(b), and both these e\u000bects cause the oscillation shown in \fgure 5(b). The\nglobal magnetic helicity, in each hemisphere or in the entire spherical shell, is not\nan observable quantity. The magnetic helicity density at the surface is observable,\nhowever, and, for this dynamo solution, provides an indication of when a reversal\ndevelops. Figure 7 displays a time-latitude plot of the magnetic helicity density near\nthe surface.\n9(a)\n (b)\n (c)\nFigure 8. Typical structures of the velocity \feld in the case. (a) shows lines of constant u\u001ein the left half\nand streamlines rsin\u0012@\u0012v= const. in the right half, all in the meridional plane; (b) shows lines of constant ur\natr=ri+ 0:5 and (c) shows streamlines, r@\u001ev= const. in the equatorial plane. These images correspond to\nthe timet= 13:52 in the simulation. (Colour online)\nAtt= 5:54, there is the clear positive/negative split in the north/south hemi-\nspheres. This time corresponds to the peaks of the magnetic helicity magnitudes in\nthe hemispheres (see \fgure 5(b)). As the hemisphere magnitudes decrease to their\nminima, this is represented in \fgure 7 by, \frst, a more mixed pattern of linkage (i.e.\nincreased negative magnetic helicity density in the north and vice versa in the south),\nand then by a decrease in the strength of the magnetic helicity density. These two\nbehaviours can be seen just before and after t= 5:55 in \fgure 7, and repeat for all\nthe cycles shown. The weakening of the density corresponds to the time when the\npoloidal and toroidal \felds reverse and this is followed by the return of these \felds to\ntheir peak values, resulting in a new phase of dominating positive/negative magnetic\nhelicity density in the north/south hemispheres.\n4. Case 3: Aperiodically-reversing dynamo\nWe now consider a dynamo solution in which aperiodic reversals of the global magnetic\n\feld occur. The general behaviour of this dynamo is rather more chaotic compared to\nthe previous two cases considered, and there is no clear precursor pattern for global\nreversals. The non-dimensional parameters and boundary conditions used for this case\nare listed in the third column of table 1. For completeness, we show, in \fgure 8, typical\npro\fles of velocity components for this dynamo solution. Like Case 2, the velocity has\na columnar but chaotic morphology.\nThis particular dynamo solution has been studied in Busse and Simitev (2008).\nTherefore, rather than repeating the description given in that work, we now focus\non the behaviour of magnetic helicity at a reversal. Unlike Case 2, the reversals for\nthis dynamo solution do not occur at regular intervals. That being said, despite the\ndi\u000berences between these two cases, the magnetic helicity can again be used to interpret\nthe behaviour of the global magnetic \feld during a reversal. Figure 9(a) displays\ndominant components of the poloidal and toroidal scalars H0\n1andG0\n1at a reversal\nand \fgure 9(b) displays the integrals of the magnetic helicity density in both the\nnorth and south hemispheres.\nApart from the lack of periodicity, the other striking di\u000berence compared to Case\n2 is that the hemispheric helicities change sign. The blue lines in Figures 9(a) and (b)\nindicate the reversals of H0\n1andG0\n1(both evaluated at r=ri+0:5). The \frst two green\nlines in \fgure 9(b) indicate the reversals of H(VN) andH(VS). The helicity reversals\noccur long before those of H0\n1andG0\n1. The third green line in \fgure 9(b) marks the\ntime at which the signs of both the hemispheric helicities return to their original values\n10(a)\n (b)\nFigure 9. Time series during a reversal. H0\n1(solid) and G0\n1(dashed) are displayed in (a). Both quantities are\nevaluated at r=ri+ 0:5 The blue vertical lines indicate the reversal times of these quantities. The northern\n(solid) and southern (dashed) helicities are shown in (b). Reversal times are indicated by green vertical lines.\nJust before the \fnal green line, there are very small reversals about the H= 0 axis, and these are not shown.\n(Colour online)\n(from the start of the time series). This occurs after the reversal of H0\n1and before that\nofG0\n1. Some care is required in interpreting the reversal times as the helicities are\ndependent on the magnetic \feld throughout an entire hemisphere, whereas the values\nH0\n1andG0\n1presented here are evaluated at a single radius. Despite this, however,\nthe results indicate that an oscillation in the hemispheric helicities, resulting in sign\nchanges, precedes the global reversal. In other words, the linkage of the poloidal and\ntoroidal \felds changes, \ripping and returning to its original state, leading to a global\nreversal. This suggests that the topology of the magnetic \feld changes and reaches an\nunstable state, in the sense of this state not lasting a long time. The way the magnetic\n\feld returns to a stable (longer lasting) con\fguration, and thus a stable topological\nstate, is through a reversal. This pattern of helicity reversing before a global reversal\nis also true for the other reversals (labelled 2 and 3) that occur in the solution (for\nthe time span we have simulated it). These results are displayed in table 2.\nReversals H0\n1G0\n1H(VN)H(VS)H(VN) andH(VS)\n2 26.3232 26.374 26.199 26.3313 26.3504\n3 31.2784 31.3534 31.169 31.2532 31.3047\nTable 2. The helicity and magnetic \feld reversal times for the other observed reversals.\nThe quantities discussed so far in this section are available in simulations but not\nto observers, who only have access to data at the outer surface. Figure 10 displays the\nmagnetic helicity density near the outer surface in a latitude-time plot.\nIn \fgure 10 there is also an indication in the behaviour of the magnetic helicity\ndensity, at the surface, that reversals are taking place. The reversal of H0\n1in \fg-\nure 9, indicating the reversal of the poloidal \feld, occurs at t\u001913:45. From about\nt\u001913:28, there is a signature of stronger patches of positive magnetic helicity density\nnear the equator. At around t\u001913:45, there is an abrupt change with much weaker\nmagnetic helicity density at all latitudes. This then switches to a phase of strong posi-\ntive/negative magnetic helicity density, just south ƒnorth of the equator. This signature\n(a change in sign) is indicative of the reversal of the poloidal \feld but not the toroidal.\n11Figure 10. A latitude vs time plot of the magnetic helicity density at the surface for Case 3 with a nonlinear\ncolourmap: indigo (-500,-30); maroon (-30, 30; shown) and seismic (blue and red) (30,500). (Colour online)\nWhen the toroidal \feld reverses at t\u001913:552, the magnetic helicity density distri-\nbution returns to its original state. Therefore, even with just information about the\nmagnetic helicity density at the surface, it is possible to identify the reversals of both\nthe global poloidal and toroidal \felds.\n5. Discussion\nIn this note, we have analyzed three dynamo solutions in rotating spherical shells\nand have investigated the behaviour of magnetic helicity in each. Our approach has\nbeen to adopt a well-studied model (Boussinesq MHD) for convection-driven dynamos\nin rotating spherical shells and consider dynamo solutions of increasing complexity\n{ steady (Case 1), periodically-reversing (Case 2), and aperiodically-reversing (Case\n3). Although magnetic helicity is not conserved in the \fnal two cases, it does provide\nimportant information about the magnetic \feld in all cases. It has been shown, as\nindicated by its de\fnition, that magnetic helicity can provide clear information about\nthe linkage of toroidal and poloidal magnetic \felds. For the cases with reversals, we\nhave described how magnetic helicity relates to the reversal of both poloidal and\ntoroidal magnetic \felds and that changing states of the magnetic helicity density on\nthe surface (an observable quantity in stars) can indicate the onset of reversals.\nSince magnetic helicity is not strictly conserved in the simulations we have consid-\nered, we cannot claim that reversals are linked causally to magnetic helicity. However,\nwhat our results do suggest is that there are perhaps preferred states of global mag-\nnetic linkage for particular dynamo solutions. When the dynamo moves away from\nsuch states, it can return via a global reversal. For example, if there is a reversal of\nthe large-scale toroidal \feld, this changes the global \feld linkage. To return to the\noriginal linkage, the poloidal \feld reverses, thus completing the global magnetic \feld\nreversal and returning the magnetic helicity distribution to what it was before any\n12reversal took place. Therefore, even if magnetic helicity is not the cause of reversals in\nthese simulations, it is a clear signature of the global poloidal-toroidal linkage, which\nis intimately linked to reversals. Furthermore, maps of the magnetic helicity density\nat the surface, which are measurable quantities in stellar observations, indicate the\nonset of reversals. This result is robust for dynamo solutions with di\u000berent reversal\nmechanisms.\nAs mentioned previously, the main focus on magnetic helicity in dynamos has been\nthe role it plays in the \u000b-e\u000bect, in mean-\feld models. Thus, there is an assumed scale\nseparation of magnetic helicity. Our simulations focus predominantly on the large-scale\nmagnetic helicity, which alone (without the small-scale contribution) is not conserved.\nNevertheless, our results do tie in qualitatively with some results from mean-\feld\nstudies. For example, in a dynamo model based on the Babcock-Leighton \u000b-e\u000bect,\nChoudhuri et al. (2004) found that at the start of cycles, helicities tend to be opposite\nof the preferred hemispheric trends. Temporary changes in the helicity hemisphere\npattern have also been recorded in observations (Zhang et al. , 2010).\nThe interpretation of our results in terms of \\preferred\" states of linkage may also\nprovide a new interpretation of the origin of the hemisphere rule of magnetic helicity.\nOur results also suggest that imbalances in this rule can be caused by changes in the\nlinkage of large-scale \feld (i.e. larger than the small-scale \feld considered in mean-\feld\nmodels). More work is needed to develop this area, but the interpretation of helicity\nin terms of toroidal/poloidal linkage may help to develop existing mean-\feld models\nwhich attempt to explain hemisphere imbalances (e.g. Yang et al. , 2020).\nIn future work, in order to perform a closer comparison with solar models and\nobservations, we will move beyond Boussinesq MHD and study the global magnetic\nhelicity in anelastic models. This will enable us not only to mimic solar parameters\nmore closely, but also the parameters of other speci\fc stars.\nAcknowledgements\nNumerical computations were performed using the DiRAC Extreme Scaling service\nat the University of Edinburgh, operated by the Edinburgh Parallel Computing Cen-\ntre on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). This equipment\nwas funded by BEIS capital funding via STFC capital grant ST/R00238X/1 and\nSTFC DiRAC Operations grant ST/R001006/1. DiRAC is part of the National e-\nInfrastructure. PG acknowledges PhD sponsorship TRAW6PAA8 from the School of\nMathematics and Statistics, the University of Glasgow.\nReferences\nBerger, M.A., Rigorous new limits on magnetic helicity dissipation in the solar corona. Geophys.\nAstrophys. Fluid Dyn. , 1984, 30, 79{104.\nBerger, M.A. and Field, G.B., The topological properties of magnetic helicity. J. Fluid Mech. ,\n1984, 147, 133{148.\nBerger, M.A. and Hornig, G., A generalized poloidal-toroidal decomposition and an absolute\nmeasure of helicity. J. Phys. A Math. , 2018, 51, 495501.\nBerger, M.A., Structure and stability of constant-alpha force-free \felds. Astrophys. J., Suppl.\nSer., 1985, 59, 433{444.\nBrandenburg, A., Advances in mean-\feld dynamo theory and applications to astrophysical\nturbulence. J. Plasma Phys. , 2018, 84, 735840404.\n13Brandenburg, A. and Subramanian, K., Astrophysical magnetic \felds and nonlinear dynamo\ntheory. Phys. Rep. , 2005, 417, 1{209.\nBusse, F. and Simitev, R.D., Dynamos driven by convection in rotating spherical shells. Astron.\nNachr. , 2005, 326, 231{240.\nBusse, F. and Simitev, R.D., Parameter dependences of convection-driven dynamos in rotating\nspherical \ruid shells. Geophys. Astrophys. Fluid Dyn. , 2006, 100, 341{361.\nBusse, F. and Simitev, R., Toroidal \rux oscillation as possible cause of geomagnetic excursions\nand reversals. Phys. Earth Planet. Inter. , 2008, 168, 237{243.\nBusse, F.H. and Simitev, R.D., Remarks on some typical assumptions in dynamo theory.\nGeophys. Astrophys. Fluid Dyn. , 2011, 105, 234{247.\nCameron, R., Dikpati, M. and Brandenburg, A., The global solar dynamo. Space Sci. Rev. ,\n2017, 210, 367{395.\nCantarella, J., DeTurck, D., Gluck, H. and Teytel, M., The spectrum of the curl operator on\nspherically symmetric domains. Phys. Plasma , 2000, 7, 2766{2775.\nChoudhuri, A.R., Chatterjee, P. and Nandy, D., Helicity of Solar Active Regions from a Dy-\nnamo Model. Astrophys. J. Lett. , 2004, 615, L57{L60.\nChristensen, U., Aubert, J., Cardin, P., Dormy, E., Gibbons, S., Glatzmaier, G., Grote, E.,\nHonkura, Y., Jones, C., Kono, M. et al. , A numerical dynamo benchmark. Phys. Earth\nPlanet. Inter. , 2001, 128, 25{34.\nFaraco, D. and Lindberg, S., Proof of Taylor's Conjecture on Magnetic Helicity Conservation.\nComm. Math. Phys. , 2019, 373, 707{738.\nFaraco, D., Lindberg, S., MacTaggart, D. and Valli, A., On the proof of Taylor's conjecture in\nmultiply connected domains. Appl. Math. Lett. , 2022, 124, 107654.\nHawkes, G. and Berger, M., Magnetic helicity as a predictor of the solar cycle. Solar Phys. ,\n2018, 293, 1{25.\nLund, K., Jardine, M., Lehmann, L., Mackay, D., See, V., Vidotto, A., Donati, J., Fares, R.,\nFolsom, C., Je\u000bers, S. et al. , Measuring stellar magnetic helicity density. Mon. Notices Royal\nAstron. Soc. , 2020, 493, 1003{1012.\nLund, K., Jardine, M., Russell, A., Donati, J., Fares, R., Folsom, C., Je\u000bers, S., Marsden, S.,\nMorin, J., Petit, P. et al. , Field linkage and magnetic helicity density. Mon. Notices Royal\nAstron. Soc. , 2021, 502, 4903{4910.\nMacTaggart, D. and Hillier, A., Topics in magnetohydrodynamic topology, reconnection and\nstability theory , CISM International Centre for Mechanical Sciences 2020 (Springer).\nMatsui, H., Heien, E., Aubert, J., Aurnou, J.M., Avery, M., Brown, B., Bu\u000bett, B.A., Busse, F.,\nChristensen, U.R., Davies, C.J., Featherstone, N., Gastine, T., Glatzmaier, G.A., Gubbins,\nD., Guermond, J.L., Hayashi, Y.Y., Hollerbach, R., Hwang, L.J., Jackson, A., Jones, C.A.,\nJiang, W., Kellogg, L.H., Kuang, W., Landeau, M., Marti, P.H., Olson, P., Ribeiro, A.,\nSasaki, Y., Schae\u000ber, N., Simitev, R.D., Sheyko, A., Silva, L., Stanley, S., Takahashi, F.,\nichi Takehiro, S., Wicht, J. and Willis, A.P., Performance benchmarks for a next generation\nnumerical dynamo model. Geochem. Geophys. , 2016, 17, 1586{1607.\nMo\u000batt, H.K., The degree of knottedness of tangled vortex lines. J. Fluid Mech. , 1969, 35,\n117{129.\nPevtsov, A.A., Berger, M.A., Nindos, A., Norton, A.A. and van Driel-Gesztelyi, L., Magnetic\nHelicity, Tilt, and Twist. Space Sci. Rev. , 2014, 186, 285{324.\nPipin, V.V. and Pevtsov, A.A., Magnetic Helicity of the Global Field in Solar Cycles 23 and\n24.Astrophys. J. , 2014, 789, 21.\nPipin, V.V., Pevtsov, A.A., Liu, Y. and Kosovichev, A.G., Evolution of magnetic helicity in\nsolar cycle 24. Astrophys. J. Lett. , 2019, 877, L36.\nSilva, L.A.C. and Simitev, R.D., Pseudo-Spectral Code For Numerical Simulation Of Nonlinear\nThermo-Compositional Convection And Dynamos In Rotating Spherical Shells, zenodo.org\n2018.\nSimitev, R.D. and Busse, F.H., Prandtl-number dependence of convection-driven dynamos in\nrotating spherical \ruid shells. J. Fluid Mech. , 2005, 532, 365{388.\nSimitev, R.D. and Busse, F., Bistability and hysteresis of dipolar dynamos generated by tur-\n14bulent convection in rotating spherical shells. Europhys. Lett. , 2009, 85, 19001.\nSimitev, R.D. and Busse, F., How far can minimal models explain the solar cycle?. Astrophys.\nJ., 2012, 749, 9.\nTaylor, J., Relaxation and magnetic reconnection in plasmas. Rev. Mod. Phys. , 1986, 58, 741.\nTilgner, A., Spectral methods for the simulation of incompressible \rows in spherical shells.\nInt. J. Numer. Meth. Fluids , 1999, 30, 713{724.\nWoltjer, L., A theorem on force-free magnetic \felds. Proc. Natl. Acad. Sci. U.S.A. , 1958, 44,\n489.\nYang, S., Pipin, V.V., Sokolo\u000b, D.D., Kuzanyan, K.M. and Zhang, H., The origin and e\u000bect\nof hemispheric helicity imbalance in solar dynamo. J. Plasma Phys. , 2020, 86, 775860302.\nZhang, H., Sakurai, T., Pevtsov, A., Gao, Y., Xu, H., Sokolo\u000b, D.D. and Kuzanyan, K., A new\ndynamo pattern revealed by solar helical magnetic \felds. Mon. Notices Royal Astron. Soc. ,\n2010, 402, L30{L33.\nAppendix A. Appendix\nIn order to verify the correct implementation of the helicity formula in the code, the\nfollowing test was performed. In a magnetically closed spherical shell, linear force-free\nsolutions exist where\nr\u0002B=\u0015B; (A1)\nwith constant \u0015. The values of \u0015that satisfy equation (A1) in the given domain form a\ndiscrete spectrum of eigenvalues for the curl operator. The corresponding eigenfunction\nof each eigenvalue has the form (in spherical coordinates)\nBr=1\nr3=2\u0002\nc1J3=2(\u0015r) +c2Y3=2(\u0015r)\u0003\ncos\u0012; (A2a)\nB\u0012=\u00001\n2\u00153=2r3(\nsin(\u0019\u0000\u0015r)\"\nc1 r\n2\n\u0019\u00152r2\u0000r\n2\n\u0019!\n+r\n2\n\u0019c2\u0015r#\n+ cos(\u0019\u0000\u0015r)\"\nc2 r\n2\n\u0019\u00152r2\u0000r\n2\n\u0019!\n\u0000r\n2\n\u0019c1\u0015r#)\nsin\u0012; (A2b)\nB\u001e=\u0015\n2pr\u0002\nc1J3=2(\u0015r) +c2Y3=2(\u0015r)\u0003\nsin\u0012; (A2c)\nwhereJ3=2(x) andY3=2(x) are Bessel functions of the \frst and second kind, respectively,\nandc1andc2are constants determined from the boundary conditions (Cantarella\net al. , 2000).\nFor a closed magnetic \feld, we have Br= 0 on the inner and outer boundaries of\nthe spherical shell. The two conditions can be written as the matrix-vector system\n\u0014J3=2(\u0015ri)Y3=2(\u0015ri)\nJ3=2(\u0015ro)Y3=2(\u0015ro)\u0015\u0014\nc1\nc2\u0015\n=\u0014\n0\n0\u0015\n: (A3)\nFor a non-trivial solution, we require that\nJ3=2(\u0015ri)Y3=2(\u0015ro)\u0000Y3=2(\u0015ri)J3=2(\u0015ro) = 0; (A4)\n15(a)\n (b)\n (c)\nFigure A1. (a) shows meridional lines of constant B\u001e(left half) and rsin\u0012@\u0012h(right half); (b) shows averaged\nhelicity density (left half) and unaveraged helicity density (right half); (c) shows the helicity density in the\nequatorial plane. (Colour online)\nand we consider the smallest values of \u0015satisfying equation (A4). With \u0015found,c1\nandc2are readily determined. For ri= 2=3 andro= 5=3,\u0015= 3:383384 and we take\nc1= 1 andc2= 1:974996.\nSince this particular magnetic \feld is force-free, it satis\fes g=\u0015h. Using this\nproperty, the magnetic helicity can be written as\nH= 2\u0015Z\nVjLhj2dV=2\n\u0015Z\nVjLgj2dV: (A5)\nFor this force-free \feld, the above relations represent an alternative way to calculate\nHcompared to the general equation (8), and provide a useful test that the general\nformula has been coded correctly. For the values used in this work, we \fnd, for the\nabove force-free solution, that H= 6:91 using either of equations (8) and (A5).\nIn \fgure A1(b), the meridional plot of the azimuthal average of the helicity density\nis equal to that of a speci\fc slice (shown on the right-had side). This is because the\nmagnetic \feld is symmetric and, also, not dependent on \u001ein this example. However,\nthis result is connected to a more general one regarding the magnetic helicity of any\nmagnetic \feld in a spherical shell. That is,\nZ2\u0019\n0Lh\u0001e\u0012d\u001e=Z2\u0019\n0Lg\u0001e\u0012d\u001e= 0; (A6)\nfor given values of rand\u0012. The proof of this result can be most easily seen by expanding\nthe integrands in terms of their spectral decompositions, e.g.\nLh\u0001e\u0012=1X\nl=0lX\nm=\u0000lim\nsin\u0012Hm\nl(r;t)Pm\nl(cos\u0012)eim\u001e:\nThe azimuthal average is found by setting m= 0, hence con\frming (A6). Thus, plots\nof the azimuthal average of the helicity density depend only on ( Lh\u0001e\u001e)(Lg\u0001e\u001e). This\nconstraint is another way to check that magnetic helicity has been calculated correctly\nin the code.\n16"    },    {        "title": "0806.3519v1.Limiting_Dynamics_for_Spherical_Models_of_Spin_Glasses_with_Magnetic_Field.pdf",        "content": "LIMITING DYNAMICS FOR SPHERICAL MODELS OF SPIN GLASSES\nWITH MAGNETIC FIELD\nMANUEL ZAMFIR\nAbstract. We study the Langevin dynamics for the family of spherical spin glass models of statistical physics,\nin the presence of a magnetic \feld. We prove that in the limit of system size Napproaching in\fnity, the\nempirical state correlation , the response function , the overlap and the magnetization for these N-dimensional\ncoupled di\u000busions converge to the non-random unique strong solution of four explicit non-linear integro-\ndi\u000berential equations, that generalize the system proposed by Cugliandolo and Kurchan in the presence of a\nmagnetic \feld.\nWe then analyze the system and provide a rigorous derivation of the FDT regime in a large area of the\ntemperature-magnetization plane.\n1.Introduction\nMany of the unique properties of magnetic systems with quenched random interactions, namely spin glasses,\nare of dynamical nature (see [15]). Therefore, we would like to understand not only the static properties, but\nalso time dependent features of the spin glass state. This is not an easy task, even for the Sherrington and\nKirkpatrick (SK) model.\nThe extended SK model can be described as follows. Let \u0000 = f\u00001;1gbe the space of spins. Fixing\na positive integer N(denoting the system size), de\fne, for each con\fguration of the spins (i.e. for each\nx= (x1;:::;xN)2\u0000N), a random Hamiltonian HN\nJ(x), as a function of the con\fguration xand of an exterior\nsource of randomness J(i.e. a random variable de\fned on another probability space). For the extended SK\nmodel, the mean \feld random Hamiltonian is de\fned as:\nHN\nJ(x) =\u0000mX\np=1ap\np!X\n1\u0014i1;:::;ip\u0014NJi1:::ipxi1:::xip;\nwherem\u00152, and the disorder parameters Ji1:::ip=Jfi1;:::;ipgare independent (modulo the permutation of\nthe indices) centered Gaussian variables. The variance of Ji1:::ipisc(fi1;:::;ipg)N\u0000p+1, where\n(1.1) c(fi1;:::;ipg) =Y\nklk!;\nand (l1;l2;:::) are the multiplicities of the di\u000berent elements of the set fi1;:::;ipg(for example, c= 1 when\nij6=ij0for anyj6=j0, whilec=p! when allijvalues are the same). Denoting by FN(x) the total magnetization\nof the system:\n(1.2) FN(x) =NX\ni=1xi;\n2000 Mathematics Subject Classi\fcation. 82C44, 82C31, 60H10, 60F15, 60K35.\nKey words and phrases. Interacting random processes, Disordered systems, Statistical mechanics, Langevin dynamics, Aging,\np-spin models.\nResearch supported in part by NSF grants #DMS-0406042 and #DMS-0806211.\nThis work was carried out as a part of my PhD thesis at Stanford University and the author dearly expresses his gratitude to\nhis advisor, Professor Amir Dembo for his helpful support and enlighten discussion.\n1arXiv:0806.3519v1  [math-ph]  21 Jun 20082 MANUEL ZAMFIR\nthe Gibbs measure for \fnitely many spins at inverse temperature \f=T\u00001and intensity of the magnetic \feld\nh>0 is de\fned as:\n(1.3) \u0015N\n\f;h;J(x) =1\nZN\n\f;h;Jexp\u0000\n\u0000\fHN\nJ(x) +hFN(x)\u0001\n1x2\u0000N:\nwhereZ\f;h;Jis a normalizing constant. The propagation of chaos for the dynamics is of much interest. It can\nbe studied from the limit as N!1 of the empirical measure:\n\u0016N=1\nNNX\ni=1\u000exi(t)\nThough the limit was established and characterized in [4] via an implicit non-Markovian stochastic di\u000berential\nequation for the continuous relaxation of the SK model with Langevin dynamics, the complexity of the latter\nequation prevents it from being amenable to a serious understanding.\nSpherical models replace the product structure of the con\fguration space \u0000Nby the sphere SN\u00001(p\nrN) in\nRN, forr= 1, via imposing the hard constraint1\nNPN\ni=1x2\ni=r. The spherical Gibbs measure is then given\nby:\n(1.4) \u0016N\n\f;h;J(dx) =1\nZN\n\f;h;Jexp\u0000\n\u00002\fHN\nJ(x) + 2hFN(x)\u0001\n\u0017N(dx)\nwhere the measure \u0017Nis the uniform measure on the sphere SN\u00001(p\nrN) (the presence of the extra factor of 2\nis just a matter of convenience and is equivalent to the rescaling \f7!2\fandh7!2h). The Langevin dynamics\nfor the normalized spherical mixed spin model (i.e. r= 1) without magnetization (i.e. h= 0), was rigurously\nstudied in [7] and [14]. The authors have shown that the dynamics of the system can be characterized via two\nfunctions, the so called empirical correlation andempirical response and they have derived the pair of coupled\nintegro-di\u000berential equations that characterize them.\nHere, we shall \frst extend their results to allow for a positive magnetic \feld (i.e. h>0) and any radius of\nthe underlying sphere. Due to the extra complexity introduced in the system via the presence of the magnetic\n\feld, that a\u000bects the symmetry of the spins, the dynamics will be characterized via a coupled system of four\nintegro-di\u000berential equations. We rigurously analyze the behavior of the system in the high temperature regime\nand derive equations characterizing the phase transition curve. Along the way, we prove (see Theorem 2.4)\nthat the system simpli\fes dramatically for large radii of the underlying sphere.\nTo work around the complexity induced by the Langevin dynamics on the sphere, we follow [7], by a further\nrelaxation of the hard spherical model, replacing the hard spherical constraint by a softone. Namely, we \frst\nreplace the uniform measure \u0017Non the sphere SN\u00001(p\nrN) by a measure on RN,\ne\u0017N(dx) =1\nZN;fexp \n\u0000Nf \n1\nNNX\ni=1x2\ni!!\ndx\nwherefis a smooth function growing fast enough at in\fnity. The soft spherical Gibbs measure is then given\nby:\n(1.5) de\u0016N\n\f;h;J;f(dx) =1\nZN\n\f;h;J;fexp\u0012\n\u0000Nf\u0012kxk2\n2\nN\u0013\n\u00002\fHN\nJ(x) + 2hFN(x)\u0013N\n\u0005\ni=1dxi:\nThus,e\u0016N\n\f;hJ;fis the invariant measure of the randomly interacting particles described by the (Langevin)\nstochastic di\u000berential system:\n(1.6) dxj\nt=dBj\nt\u0000f0(N\u00001kxtk2)xj\ntdt+\fGj(xt)dt+hdt;\nwhere B= (B1;:::;BN) is an N-dimensional standard Brownian motion, independent of both the initial\ncondition x0and the disorder J, andGi(x) :=\u0000@xi\u0000\nHN\nJ(x)\u0001\n, fori= 1;:::;N . In Proposition 2.2, we\ncharacterize the long term behavior of the Langevin dynamics of this soft spherical model for a general classSPHERICAL SPIN GLASSES 3\nof functions f. We shall then choose an appropriate sequence of functions fn, satisfyinge\u0016N\n\f;h;J;fn!\u0016N\n\f;h;J,\nallowing us to derive, in Theorem 2.3, the limiting behavior of the hard spherical model.\nWe shall \frst prove that, \fxing f, for a.e. disorder J, initial condition x0and Brownian path B, there\nexists a unique strong solution of (1.6) for all t\u00150, whose law we denote by PN\n\f;x0;J.\nWe are interested in the time evolution for large N, of the empirical covariance function :\n(1.7) COVN(s;t) =1\nNNX\ni=1\u0002\nxi\nsxi\nt\u0000EB[xi\ns]EB[xi\nt]\u0003\n;\nwhere EB[\u0001] represents the expectation with respect to the Brownian motion only (and not with respect to\nthe Gaussian law of the couplings), under the quenched law PN\n\f;x0;J, as the system size N!1 . In [7], the\nauthors have formally derived the limiting equations for the empirical state correlation function :\n(1.8) CN(s;t) :=1\nNNX\ni=1xi\nsxi\nt;\nin the absence of a magnetic \feld (i.e. h= 0). The equations characterizing the limit as N!1 ofCN(s;t)\ninvolve the analogous limit for the empirical integrated response function :\n(1.9) \u001fN(s;t) :=1\nNNX\ni=1xi\nsBi\nt;\nand the limits are characterized as the unique solution of a system of two coupled integro-di\u000berential equations.\nThe presence of the magnetic \feld requires us to consider also the empirical averaged magnetization :\n(1.10) MN(s) :=1\nNNX\ni=1xi\ns;\ntheaveraged overlap :\n(1.11) LN(s;t) :=1\nNNX\ni=1EB\u0002\nxi\ns\u0003\nEB\u0002\nxi\nt\u0003\n;\nand the empirical overlap :\n(1.12) QN(s;t) :=1\nNNX\ni=1x1;i\nsx2;i\nt;\nwhere\b\nxk\t\ns,k= 1;2 are two independent replicas, sharing the same couplings J, with the noise given by\ntwo independent Brownian motions fBkgs. With these notations, our primary object of study, the empirical\ncovariance can be written as:\nCOVN(s;t) =CN(s;t)\u0000LN(s;t):\nThe empirical overlap de\fned in (1.12) is the central quantity in the study of the static properties of the\nsystem (see [21] for a comprehensive survey). Its dynamical properties were not rigurously analyzed until now.\nIn the course of our proofs, we show that the limits as N!1 ofLN(i.e. the averaged overlap - that we\nneed to characterize in order to study the empirical covariance) and of QN(i.e. the empirical overlap - that\nis interesting in its own right), coincide. Also, as opposed to the scenario analyzed in [7] (i.e. h= 0), where\nthe authors have characterized the dynamics via a coupled system of two integro-di\u000berential equations, the\npresence of the magnetic \feld will a\u000bect the symmetry of the spins and the dynamics of our system will be\ncharacterized via a coupled system of four integro-di\u000berential equations.\nWe shall analyze the solutions of the latter system in a non-perturbative high temperature region of the\n(\f;h)-plane, rigorously establishing the existence of the so called FDT regime, where the Frequency Dissipation4 MANUEL ZAMFIR\nTheorem in statistical physics holds. We shall see that the phase plane diagram of the system in ( \f;h)\ncoordinates is the one shown in Figure 1 below.\nFigure 1. The Phase Plane Diagram: The hashed\nregion represents the area of applicability of Theorem 2.5,\nwhere we can rigorously prove the FDT regime, the light\nregion represents the expected extend of the FDT regime\nand the red region, past the dynamical phase transition\ncurve, represents the expected extent of the aging regime.\n2.Main Results\nWe shall start by making the same assumptions on the initial conditions as in [7]. Namely, we assume that\nthe initial condition x0is independent of the disorder J, and the limits\n(2.1) lim\nN!1E[CN(0;0)] =C(0;0);\nand\n(2.2) lim\nN!1E[MN(0)] =M(0);\nexists, and are \fnite. Further, we assume that the tail probabilities P(jCN(0;0)\u0000C(0;0)j>x) andP(jMN(0)\u0000\nM(0)j>x) decay exponentially fast in N(so the convergence CN(0;0)!C(0;0) andMN(0)!M(0) holdsSPHERICAL SPIN GLASSES 5\nalmost surely), and that for each k <1, the sequence N7!E[CN(0;0)k] andN7!E[MN(0)k] is uniformly\nbounded. Also, we will assume that each of the two replicas will have the same (random) initial conditions,\nhenceQN(0;0) =CN(0;0).\nFinally, consider the product probability space EN=RN\u0002Rd(N;m)\u0002C([0;T];RN)\u0002C([0;T];RN) (hereT\nis a \fxed time and d(N;m ) is the dimension of the space of the interactions J), equipped with the natural\nEuclidean norms for the \fnite dimensional parts, i.e ( x0;J), and the sup-norm for the Brownian motions Bk,\nk= 1;2. The spaceENis endowed with the product probability measure P=\u0016N\n\rN\nPN\nPN, where\u0016N\ndenotes the distribution of x0,\rNis the (Gaussian) distribution of the coupling constants J, andPNis the\ndistribution of the N-dimensional Brownian motion.\nHypothesis 2.1. For(x0;J;B1;B2)2ENwe introduce the norms\nk(x0;J;B1;B2)k2=NX\ni=1(xi\n0)2+mX\np=1X\n1\u0014i1:::ip\u0014N(Np\u00001\n2Ji1\u0001\u0001\u0001ip)2+2X\nk=1sup\n0\u0014t\u0014TNX\ni=1(Bk;i\nt)2:\nWe shall assume that \u0016Nis such that the following concentration of measure property holds on EN; there exists\ntwo \fnite positive constants Cand\u000b, independent on N, such that, if Vis a Lipschitz function on EN, with\nLipschitz constant K, then for all \u001a>0,\n\u0016N\n\rN\nPN\nPN[jV\u0000E[V]j\u0015\u001a]\u0014C\u00001exp\u0010\n\u0000C\u0010\u001a\nK\u0011\u000b\u0011\n:\nNow, suppose that fis a di\u000berentiable function on R+withf0locally Lipschitz, such that\n(2.3) sup\n\u001a\u00150jf0(\u001a)j(1 +\u001a)\u0000r<1\nfor somer<1, and for some A;\u000e> 0,\n(2.4) inf\n\u001a\u00150ff0(\u001a)\u0000A\u001am=2+\u000e\u00001g>\u00001\n(typically,f(\u001a) =\u0014(\u001a\u00001)rfor somer > m= 2 and\u0014\u001d1). Then the normalization factor Z\f;hJ;f=R\ne\u0000\fHN\nJ(x)\u0000Nf(N\u00001kxk2)+hFN(x)dxis a.s. \fnite (by (2.4)).\nFirst, we shall show that, as N!1 the functions CN(s;t),\u001fN(s;t),MN(s),QN(s;t) andLN(s;t) converge\nto non-random continuous functions C(s;t),\u001f(s;t),M(s) andQ(s;t) =L(s;t) that are characterized as the\nsolution of a system of coupled integro-di\u000berential equations. We denote by \u0000the upper half of the \frst\nquadrant, namely:\n\u0000:=\b\n(s;t)2R2: 0\u0014t\u0014s\t\nAlso, we denote by C1\nsthe class of continuously di\u000berentiable symmetric functions of two variables and by Cs\nthe class of continuous symmetric functions . These notations will be widely used and will appear through\nthis work.\nProposition 2.2. Let (r) =\u00170(r) +r\u001700(r)and\n(2.5) \u0017(r) :=mX\np=1a2\np\np!rp:\nSuppose\u0016Nsatis\fes hypothesis 2.1 and fsatis\fes (2.3) and(2.4) . Fixing any T <1, asN!1 the random\nfunctionsMN,\u001fN,CN,QNandLNconverge uniformly on [0;T]2(or[0;T], whichever applies), almost surely\nand in Lpwith respect to x0,JandBk, fork= 1;2, to non-random functions M(s),\u001f(s;t) =Rt\n0R(s;u)du,\nC(s;t) =C(t;s),Q(s;t) =Q(t;s)andL(s;t) =Q(s;t). Further, R(s;t) = 0 fort > s ,R(s;s) = 1 , and\nfors > t the absolutely continuous functions C,R,M,Q, andK(s) =C(s;s)are the unique solution in\nC1(R+)\u0002C1(\u0000)\u0002C1\ns(R2\n+)\u0002C1\ns(R2\n+)\u0002C1(R+)of the integro-di\u000berential equations:\n@M(s) =\u0000f0(K(s))M(s) +h+\f2Zs\n0M(u)R(s;u)\u001700(C(s;u))du; s \u00150 (2.6)6 MANUEL ZAMFIR\n@1R(s;t) =\u0000f0(K(s))R(s;t) +\f2Zs\ntR(u;t)R(s;u)\u001700(C(s;u))du; s \u0015t\u00150 (2.7)\n@1C(s;t) =\u0000f0(K(s))C(s;t) +\f2Zs\n0C(u;t)R(s;u)\u001700(C(s;u))du (2.8)\n+\f2Zt\n0\u00170(C(s;u))R(t;u)du+hM(t) + 1s<tR(t;s); s;t \u00150\n@1Q(s;t) =\u0000f0(K(s))Q(s;t) +\f2Zs\n0Q(u;t)R(s;u)\u001700(C(s;u))du (2.9)\n+\f2Zt\n0\u00170(Q(s;u))R(t;u)du+hM(t); s;t \u00150\n@K(s) =\u00002f0(K(s))K(s) + 1 + 2\f2Zs\n0 (C(s;u))R(s;u)du+ 2hM(s); s\u00150 (2.10)\nwhere the initial conditions K(0) =C(0;0) =Q(0;0)>0andM(0)are determined by (2.1) and (2.2),\nrespectively. Moreover, C(\u0001;\u0001)andQ(\u0001;\u0001)are non-negative de\fnite kernels, K(s)\u00150,jM(s)j\u0014p\nK(s), for\nalls\u00150and\n(2.11)\f\f\f\fZt2\nt1R(s;u)du\f\f\f\f2\n\u0014K(s)(t2\u0000t1); 0\u0014t1\u0014t2\u0014s<1:\nFor everyr;L> 0, de\fne the function:\nf(x) :=fL;r(x) =L(x\u0000r)2+1\n4k\u0010x\nr\u00112k\n+\u000bhx\nr; k>m= 4; k2Z; L\u00150; (2.12)\nthat is easily seen to satisfy conditions (2.3) and (2.4). We will derive in Section 4 the equations for the hard\nspherical constraint, by taking the limit L!1 . Notice that if there is no magnetic \feld (i.e. h= 0), the\nequations for the correlation C(\u0001;\u0001) and the response R(\u0001;\u0001) will decouple from the magnetization, resulting\nwith the system derived in [14].\nTheorem 2.3. For everyr>0, let(ML;r;RL;r;CL;r;QL;r;KL;r)be the unique solution of the system (2.6) -\n(2.10) with potential fL;r(\u0001)as in (2.12) and initial conditions KL;r(0) =QL;r(0;0) =r>0,ML;r(0) =\u000bpr,\n\u000b2[0;1)andRL;r(t;t) = 1 for everyt\u00150. Then, for any T <1,(ML;r;RL;r;CL;r;QL;r;KL;r)converges\nasL!1 , uniformly in s;t2[0;T], towards (M;R;C;Q;K )that is the unique solution in C1(R+)\u0002C1(\u0000)\u0002\nC1\ns(R2\n+)\u0002C1\ns(R2\n+)\u0002C1(R+)of:\n@M(s) =\u0000\u0016(s)M(s) +hr+\f2Zs\n0M(u)R(s;u)\u001700(C(s;u))du; s \u00150 (2.13)\n@1R(s;t) =\u0000\u0016(s)R(s;t) +\f2Zs\ntR(u;t)R(s;u)\u001700(C(s;u))du; s \u0015t\u00150 (2.14)\n@1C(s;t) =\u0000\u0016(s)C(s;t) +\f2Zs\n0C(u;t)R(s;u)\u001700(C(s;u))du (2.15)\n+\f2Zt\n0\u00170(C(s;u))R(t;u)du+hrM(t); s \u0015t\u00150\n@1Q(s;t) =\u0000\u0016(s)Q(s;t) +\f2Zs\n0Q(u;t)R(s;u)\u001700(C(s;u))du (2.16)\n+\f2Zt\n0\u00170(Q(s;u))R(t;u)du+hrM(t); s;t \u00150SPHERICAL SPIN GLASSES 7\nwherehr=h,k= 1and\n(2.17) \u0016(s) =1\n2r\u0012\nk+ 2\f2Zs\n0 (C(s;u))R(s;u)du+ 2hrM(s)\u0013\nsatisfyingM(0) =\u000bpr,C(t;t) =K(t) =r,R(t;t) = 1 , for allt\u00150. Moreover, C(\u0001;\u0001)andQ(\u0001;\u0001)are\nnon-negative de\fnite kernels, with values in [0;r],M(s)2[0;pr], for alls\u00150,R(s;t)\u00150and\n(2.18)\f\f\f\fZt2\nt1R(s;u)du\f\f\f\f2\n\u0014r(t2\u0000t1); 0\u0014t1\u0014t2\u0014s<1:\nThe predicted structure of the solution is more complicated in the mixed spin case than in the pure spin one.\nHowever, we show in Section 5 that as rincreases, only the highest level interactions will matter, e\u000bectively\nmaking the system behave like a pure spin one. (i.e. \u0017(x) is a monomial). Namely, we prove:\nTheorem 2.4. For\u000b2(0;1)andr > 0, let (Mr;Rr;Cr;Qr)the unique solutions of (2.13) -(2.17) for\nhr=hr(m\u00001)=2, with initial conditions Mr(0) =\u000bpr,Cr(t;t) =Qr(0;0) =r > 0, andRr(t;t) = 1 , for\nallt\u00150. Then for any T <1, the appropriately scaled functions fMr(s) =Mr(sr1\u0000m=2)=pr,eRr(s;t) =\nRr(sr1\u0000m=2;tr1\u0000m=2),eCr(s;t) =Cr(sr1\u0000m=2;tr1\u0000m=2)=randeQr(s;t) =Qr(sr1\u0000m=2;tr1\u0000m=2)=r, converge as\nr!1 , uniformly in s;t2[0;T], towards the solution of the corresponding pure spin system (i.e. towards the\nunique solution of (2.13) -(2.17) withhr=h,k= 0,e\u0017(x) =a2\nm(m!)\u00001xmande (x) =e\u00170(x) +xe\u001700(x), with\ninitial conditions M(0) =\u000b,C(t;t) =Q(0;0) = 1 andR(t;t) = 1 for allt\u00150).\nIn Section 6, we will analyze the solutions of the system (2.13)-(2.17) in the high temperature region of\nthe (\f;h)-plane, formally establishing the existence of the FDT regime. The analysis is done in the absence\nof a random magnetic \feld (i.e. \u00170(0) = 0). In this regime, the correlation, the response and the overlap are\nstationary for large t. Also, both the covariance and the response are decaying exponentially fast to 0. The\nafore-mentioned region is f(\f;h) :\f\u0014\f0;h < h 0gSf(\f;h) :\f\u0014\r0hgfor some non-trivial \r0,\f0and\nh0. The presence of the FDT regime for \fsmall andhsmall region comes as no surprise, in the light of the\nresults proved in [14], where the authors have established similar results for \fsmall andh= 0. However, the\noccurrence of the same regime in the region bounded by\f\nh<\r0as well as the asymptotically linear relation\nbetween the critical inverse temperature and the intensity of the \feld is novel and represents an important\ncontribution to the \feld.\nTheorem 2.5. Suppose\u00170(0) = 0 . Let (M;R;C;Q )be the unique solution of (2.13) -(2.17) , forhr=h,\nk= 1=2andr= 1, with initial conditions R(t;t) =C(t;t) =Q(0;0) = 1 andM(0) =\u000b2(0;1]. Then there\nexist\f0;h0;\r0>0such that if either \r:=\f\nh<\r0or\f <\f 0andh<h 0, then for any \u001c\u00150,\nlim\nt!1Q(t+\u001c;t) =Qfdt; lim\nt!1M(t) =Mfdt= 2h(1\u0000Qfdt)\nlim\nt!1C(t+\u001c;t) =Cfdt(\u001c); lim\nt!1R(t+\u001c;t) =Rfdt(\u001c) =\u00002@Cfdt(\u001c)\nFurthermore, Mfdt;Qfdt;Cfdt(\u001c)2[0;1],Rfdt(\u001c)\u00150,Qfdtis only solution of the equation:\n(2.19) Q= 4(1\u0000Q)2[\f2\u00170(Q) +h2]; Q2[\u0000\n1\u0000(2h)\u00001\u0001\n^0;1]\nandCfdtis the unique [0;1]-valued continuously di\u000berentiable solution of the equation:\n(2.20) C0(s) =\u0000Zs\n0\u001e(C(v))C0(s\u0000v)dv\u00001\n2; C (0) = 1;\nfor\u001e(x) =1\n2(1\u0000Qfdt)+ 2\f2(\u00170(x)\u0000\u00170(Qfdt)). Moreover, Rfdt(\u0001)decays exponentially to zero at in\fnity and\nCfdt(\u0001)converges exponentially fast to Qfdt8 MANUEL ZAMFIR\nEquation (2.20) has been analyzed in detail in Proposition 1.4 of [14]. The authors have shown that, for\nany choice of \u001e(\u0001) such that\n(2.21) sup\nx2[0;1]f\u001e(x)(1\u0000x)g\u00151\n2\nthe equation has an unique solution in [0 ;1], that is decreasing, twice di\u000berentiable and converges as s!1\ntoC1= supfx2[0;1]j\u001e(x)(1\u0000x)\u00151=2g. Furthermore, they show that the condition:\n(2.22) \u001e(C1)>\u001e0(C1)(1\u0000C1);\nis necessary for the exponential convergence of C0(s) to zero as s!1 when\u001e(\u0001) is convex.\nFirst, it is easy to check that for our \u001e(x) of Theorem 2.5, \u001e(Qfdt)(1\u0000Qfdt) = 1=2, hence (2.21) is satis\fed\nand furthermore, C1\u0015Qfdt. Setting\fc(h)2(0;1) via\n(2.23)1\n4\fc(h)2= sup\u001a(\u00170(x)\u0000\u00170(Qfdt))(1\u0000x)(1\u0000Qfdt)\nx\u0000Qfdt:x2(Qfdt;1]\u001b\n;\nit is easy to check that C1=Qfdtif\f <\fc(h) whereasC1>Qfdtfor\f >\fc(h). Further, considering x!0\nin (2.23) we \fnd that\n(2.24)1\n4\fc(h)2\u0015\u001700(Qfdt)(1\u0000Qfdt)2\nso, in particular, the condition (2.22) then holds for any \f <\fc(h) (since in this case, as mentioned C1=Qfdt).\nFurthermore, since Qfdtis a solution of (2.19), from (2.24) we get \fc(h)\u00002(\fc(h)2\u00170(Qfdt)+h2)\u0015Qfdt\u001700(Qfdt),\nso:\n\rc(h)2:=\u0012\fc(h)\nh\u00132\n\u00141\nQfdt\u001700(Qfdt)\u0000\u00170(Qfdt)\u0000!\nh!11\n\u001700(1)\u0000\u00170(1)\nThis indicates that though the values of \f0(h)\u0014\r0hfor which we have formally established the FDT regime\nin Theorem 2.5 are quite small, they should match the predicted dynamical phase transition point \fc(h) of\nour model. Furthermore, 0 <lim inf\nh!1\rc(h)\u0014lim sup\nh!1\rc(h)<1, indicating that \fc(h) will be asymptotically\nlinear inh. Figure 1 in the introduction summarizes all the information above.\n3.Limiting Soft Spherical Dynamics\nThis section is dedicated to proving Proposition 2.2. The line of proof follows closely [7], and references will\nbe given, when appropriate. First, recall that:\n(3.1) Gi(x) :=\u0000@xi\u0010\nHN\nJ(x)\u0011\n=mX\np=1ap\n(p\u00001)!X\n1\u0014i1;:::;ip\u00001\u0014NJii1:::ip\u00001xi1:::xip\u00001;SPHERICAL SPIN GLASSES 9\nWe will start by introducing some notation. For q1;q22f1;2g, de\fne\nCq1;q2\nN(s;t) :=1\nNNX\ni=1xq1;i\nsxq2;i\nt; Kq1;q2\nN(s) :=Cq1;q2\nN(s;s);\n\u001fq1;q2\nN(s;t) :=1\nNNX\ni=1xq1;i\nsBq2;i\nt; Mq1\nN(s) :=1\nNNX\ni=1xq1;i\ns;\nAq1;q2\nN(s;t) :=1\nNNX\ni=1Gi(xq1\ns)xq2;i\nt; Fq1;q2\nN(s;t) :=1\nNNX\ni=1Gi(xq1\ns)Bq2;i\nt;\nRq1\nN(s) :=1\nNNX\ni=1Gi(xq1\ns) Wq1\nN(s) :=1\nNNX\ni=1Bq1;i\ns (3.2)\nand\nDq1;q2\nN(s;t) :=\u0000f0(E(Kq2;q2\nN(t)))Cq1;q2\nN(s;t) +Aq1;q2\nN(t;s)\nEq1;q2\nN(s;t) :=\u0000f0(E(Kq1;q1\nN(s)))\u001fq1;q2\nN(s;t) +Fq1;q2\nN(s;t)\nPq1\nN(s) :=\u0000f0(E(Kq1;q1\nN(t)))Mq1\nN(s) +Rq1\nN(s); (3.3)\nwhere forq= 1;2,fBq\nsgs\u00150=f(Bq;1\ns;:::;Bq;N\ns)gs\u00150, are two iid N-dimensional Brownian motions and\nfxq\nsgs\u00150=f(xq;1\ns;:::;xq;N\ns)gs\u00150are the two replicas sharing the same frustrations J, with the noise given by\nthe realization of the Brownian motions above. Also, when it is clear from the context that there is only one\nreplica, for simplicity of the notation, the superscripts indicating the replica index will be omitted.\nAlso, in order to simplify the (already heavy) notations, we will embed the constant \fintofapgresulting with\n\fGj(\u0001)7!Gj(\u0001) and then having \f= 1 in the stochastic di\u000berential system (1.6). Adopting this convention,\nwe will have from now on \f= 1.\n3.1.Strong Solutions and Self-Averaging. First, by similar arguments to the ones employed in Proposi-\ntion 2.1 of [7], we will show that if f0is locally Lipschitz, satisfying (2.4), then there exist an unique strong\nsolution to (1.6). Namely, we show:\nProposition 3.1. Assume that f0is locally Lipschitz, satisfying (2.4). Then, for any N2Z+, almost any\nJ, initial condition x0and Brownian path B, there exists a unique strong solution to (1.6). This solution is\nalso unique in law for almost any J, and x0, it is a probability measure on C(R+;RN)which we denote PN\nx0;J.\nFurther, with\n(3.4) jjJjjN\n1= max\n1\u0014p\u0014msup\njjuijj\u00141;1\u0014i\u0014p\f\f\fp\nN\u00001X\n1\u0014ik\u0014N;1\u0014k\u0014pNp\u00001\n2Ji1\u0001\u0001\u0001ipu1\ni1\u0001\u0001\u0001up\nip\f\f\f\nwe have for \u000e>0of (2.4),q:=m=(2\u000e) + 1 , some\u0014<1, allN,z>0,J, and x0, that\n(3.5) PN\nx0;J\u0010\nsup\nt2R+KN(t)\u0015KN(0) +\u0014(1 +kJkN\n1+h)q+z\u0011\n\u0014e\u0000zN:\nConsequently, for any L>0, there exists z=z(L)<1such that\n(3.6) P\u0010\nsup\nt2R+KN(t)\u0015z\u0011\n\u0014e\u0000LN:\nProof of Proposition 3.1. The proof follows the same lines as the proof of Proposition 2.1 of [7]. Namely,\nconsidering the truncated drift bM(u) = (bM\n1(u);:::;bM\nN(u)) given by bM\ni(u) =Gi(\u001eM(u))\u0000f0(N\u00001juj2^\nM)ui+h, where\u001eM(x) =xwhenkxk\u0014p\nNM, we see that \u001eMis globally Lipschitz, hence there exist an\nunique square-integrable strong solution u(M)for the SDS\ndui\nt=bM\ni(ut)dt+dBi\nt10 MANUEL ZAMFIR\n(see, for example [20, Theorems 5.2.5, 5.2.9]).\nFixingMand denoting xt=u(M)\nt^\u001cMandZs= 2N\u00001PN\ni=1Rs^\u001cM\n0xi\ntdBi\nt, by applying It^ o's formula for\nCN(t) :=N\u00001jjxtjj2we see that\nCN(s)\u0014CN(0) + 2mX\np=1apjjJjjN\n1\n(p\u00001)!Zs^\u001cM\n0CN(t)p\n2dt+Zs+s^\u001cM (3.7)\n\u00002Zs^\u001cM\n0f0(CN(t))CN(t)dt+ 2hZs^\u001cM\n0CN(t)1\n2dt:\nSincex1\u0000m\n2f0(x)!1 , it follows from (3.7) that there is an almost surely \fnite constant c(jjJjjN\n1;h), inde-\npendent of M, such that\n(3.8) CN(s)\u0014CN(0) +c(jjJjjN\n1;h)s+Zs\nAs the quadratic variation of the martingale Zsis (4=N)Rs^\u001cM\n0CN(t)dt\u00144sN\u00001M, applying Doob's inequal-\nity (c.f. [20, Theorem 3.8, p. 13]) for the exponential martingale L\u0015\ns= exp(\u0015Zs\u00002(\u00152=N)Rs^\u001cM\n0CN(t)dt)\n(with respect to the \fltration fHtgofBt), yields that\n(3.9) P\u0012\nsup\ns\u0014TfZs\u00002Zs\n0CN(t)dtg\u0015z\u0013\n\u0014P\u0012\nsup\ns\u0014TLN\ns\u0015ezN\u0013\n\u0014e\u0000zN;\nfor anyz>0. Therefore, (3.8) shows that with probability greater than 1 \u0000e\u0000zN,\nCN(s^\u001cM)\u0014CN(0) +c(jjJjjN\n1;h)T+z+ 2Zs^\u001cM\n0CN(t)dt;\nfor alls\u0014T, and by Gronwall's lemma then also\n(3.10) sup\nt\u0014TN\u00001ju(M)\nt^\u001cMj2\u0014[CN(0) +c(jjJjjN\n1;h)T+z]e2T:\nSettingz=M=3, for large enough M(depending of N,h,J,x0andTwhich are \fxed here), the right-side of\n(3.10) is at most M=2, resulting with\nP(\u001cM\u0014T)\u0014e\u0000MN= 3;\nwhere\u001cM= infft:jju(M)\ntjj\u0015p\nNMg. and hence that\n(3.11)1X\nM=1P(\u001cM\u0014T)<1:\nso establishing the existence of the solution after an application of the Borel-Cantelli lemma.\nWe also have weak uniqueness of our solutions for almost all Jsince the restriction of any weak solution to\nthe stopped \u001b-\feldH\u001cMfor the \fltrationHtofBtis unique. We denote this unique weak solution of (1.6) by\nPN\nx0;J.\nTurning to the proof of (3.5), by (2.4), for any c>0 there exists \u0014<1such that for all r;x\u00150,\n2\"\nf0(x)x\u0000rmX\np=1apxp\n2\n(p\u00001)!\u0000hx1\n2#\n\u00001\u0015cx\u0000\u0014(1 +r+h)q:\nTakingr=kJkN\n1, we see that by (3.7), for all Nands\u00150,\nCN(s^\u001cM)\u0014CN(0)\u0000Zs^\u001cM\n0\u0002\ncCN(t)\u0000\u0014(1 +kJkN\n1+h)q\u0003\ndt+Zs;\nwhere (Zs)s\u00150is a martingale with bracket (4 N\u00001Rs^\u001cM\n0CN(t)dt;s\u00150).SPHERICAL SPIN GLASSES 11\nBy Doob's inequality (3.9), with probability at least 1 \u0000e\u0000zN,\nsup\nu\u0014s^\u001cMZu\u00142Zs^\u001cM\n0CN(t)dt+z;\nfor alls\u00150. Settingc= 3 we then have that\nCN(s^\u001cM)\u0014CN(0) +z\u0000Zs^\u001cM\n0CN(t)dt+\u0014(1 +kJkN\n1+h)q(s^\u001cM); (3.12)\nso that by Gronwall's lemma,\nCN(s^\u001cM)\u0014e\u0000s^\u001cM(CN(0) +z) +\u0014(1 +kJkN\n1+h)qZs^\u001cM\n0e\u0000tdt\nfrom which the conclusion (3.5) is obtained by considering M!1 .\nIn view of the assumed exponential in Ndecay of the tail probabilities for KN(0) and the bound (B.7) of\n[7] on the corresponding probabilities for kJkN\n1we thus get also the bounds of (3.6). \u0003\nThe next is to extend the arguments in Propositions 2.2 - 2.8 of [7], in order to show that any of the functions\nAq1;q2\nN;Fq1;q2\nN;\u001fq1;q2\nN;Cq1;q2\nN;Wq1\nN;Rq1\nN;Mq1\nNandLNself-averages for Nlarge. More precisely, we show that:\nProposition 3.2. Suppose that \t :R`!Ris locally Lipschitz with j\t(z)j\u0014Mkzkk\nkfor someM;`;k <1,\nandZN2R`is a random vector, where for j= 1;:::;` , thej-th coordinate of ZNis one of the functions\nAq1;q2\nN;Fq1;q2\nN;\u001fq1;q2\nN;Cq1;q2\nN,Wq1\nN;Mq1\nNorLN, evaluated at some (sj;tj)2[0;T]2(or atsj2[0;T], whichever\napplies). Then,\nlim\nN!1sup\nsj;tjjE[\t(ZN)]\u0000\t(E[ZN])j= 0:\nProof of Proposition 3.2. The proof is structured as follows: \frst we show that E\u0002\nsups;t\u0014TjUN(s;t)jk\u0003\nand\nE\u0002\nsups\u0014TjVN(s)jk\u0003\nare bounded uniformly in Nand also that for any \fxed T <1, the sequences UN(s;t)\nandVN(s) are pre-compact almost surely and in expectation with respect to the uniform topology on [0 ;T]2,\nrespectively [0 ;T]. HereUis any of the functions Cq1;q2,Fq1;q2,\u001fq1;q2;Aq1;q2orLandVis one of the functions\nMq1orWq1. The next step is to establish, similarly to Proposition 2.4 of [7], that all the functions UandV\nabove self-averages , namely:\nX\nNP\u0014\nsup\ns;t\u0014TjUN(s;t)\u0000E[UN(s;t)]j\u0015\u001a\u0015\n<1\nX\nNP\u0014\nsup\ns\u0014TjVN(s)\u0000E[VN(s)]j\u0015\u001a\u0015\n<1 (3.13)\nimplying by the uniform moment bounds on kUNk1andkVNk1that we have just established, that:\nlim\nN!1sup\ns;t\u0014TEh\njUN(s;t)\u0000E[UN(s;t)]j2i\n= 0\nlim\nN!1sup\ns\u0014TEh\njVN(s)\u0000E[VN(s)]j2i\n= 0 (3.14)\nThe \fnal step is to establish the claim of the proposition, by using (3.14) and the uniform bounds on the\nmoments that we have just established.\nBy our hypothesis, the mapping N7!E[KN(0)k] is bounded. Since both replicas have the same starting\npointKq;q\nN(0) =KN(0), forq2f1;2g. Also, by the estimate (B.6) of Appendix B, of [7],\n(3.15) sup\nNE\u0002\n(jjJjjN\n1)k\u0003\n<1;12 MANUEL ZAMFIR\nfor anyk <1, for the normjjJjjN\n1of (3.4), the bound (3.5) immediately implies that for each k <1, and\nanyq2f1;2galso\n(3.16) sup\nNE\u0014\nsup\nt2R+[Kq;q\nN(t)]k\u0015\n<1:\nDe\fnekVNk1:= supfVN(t) : 0\u0014t\u0014TgandkUNk1:= supfUN(s;t) : 0\u0014s;t\u0014Tg. Also letBq\nN(t) :=\n1\nNPN\ni=1(Bq;i\nt)2,Gq\nN(t) :=1\nNPN\ni=1(Gi(xq\nt))2andLN(t) :=1\nNPN\ni=1(EB[xi\nt])2. A key result is the bound:\nsup\nNE\u0002\n(jjJjjN\n1)k\u0003\n+ sup\nNE[kLNkk\n1] + sup\nNE[kKNkk\n1] + sup\nNE[kBNkk\n1] + sup\nNE[kGNkk\n1]<1; (3.17)\nfor every \fxed k, where we have dropped the replica index (since we are taking the expected value anyway).\nIndeed, the bounds on kJkN\n1andkKq;q\nNk1are already obtained in (3.15) and (3.16), and by Lemma 2.2 of [7]\nwe have that\n(3.18) ( Gq\nN(t))1\n2\u0014cjjJjjN\n1[1 +Kq;q\nN(t)m\u00001\n2];\nyielding by (3.15) and (3.16) the uniform moment bound on kGq\nNk1. Also, by Jensen's inequality, E[kLNkk\n1]\u0014\nE[kKNkk\n1] and \fnally, the exponential tails of Bq\nN(c.f. [7, (2.16)]), will provide an uniform bound for each\nmoment ofkBq\nNk1, thus concluding the derivation of (3.17).\nSimilarly, by (3.6), (3.18), the exponential tails of Bq\nNmentioned above and the exponential tails of kJkN\n1\n(c.f [7, (B.7)]), we have for each L>0 the bound:\n(3.19) P \nkJkN\n1+kLNk1+2X\nq=1[kKq;q\nNk1+kBq\nNk1+kGq\nNk1]\u0015M!\n\u0014e\u0000LN:\nwill hold for some M=M(L)<1and for all N. Applying Cauchy-Schwartz inequality to UNandVNand\nusing the estimates (3.17) and (3.19), we see that E\u0002\nsups;t\u0014TjUN(s;t)jk\u0003\nandE\u0002\nsups\u0014TjVN(s)jk\u0003\nare bounded\nuniformly in N. The argument is similar to the one employed in Proposition 2.3 of the cited paper.\nWith the previous controls on kUNk1andkVNk1already established, by the Arzela-Ascoli theorem, the\npre-compactness of UN, respectively VNfollows by showing that they are equi-continuous sequences. We notice\nthat suchUN(s;t) andVN(s) are all of the form1\nNPN\ni=1ai\nsbi\nthence,\njUN(s;t)\u0000UN(s0;t0)j \u00141\nNNX\ni=1jai\ns\u0000ai\ns0jjbi\ntj+1\nNNX\ni=1jai\ns0jjbi\nt\u0000bi\nt0j\n\u0014\"\n1\nNNX\ni=1jai\ns\u0000ai\ns0j2#1=2\"\n1\nNNX\ni=1jbi\ntj2#1=2\n+\"\n1\nNNX\ni=1jbi\nt\u0000bi\nt0j2#1=2\"\n1\nNNX\ni=1jai\ns0j2#1=2\n: (3.20)\nand the same is true also for jVN(s)\u0000VN(s0)j, where the functions asandbsare either xq\ns,Bq\ns,G(xq\ns), for\nsomeq2f1;2g,EB[xs] or 1. So, in view of (3.17) and (3.19), it su\u000eces to show that for any \u000f >0, some\nfunctionL(\u000e;\u000f) going to in\fnity as \u000egoes to zero and all N,\nP \nsup\njt\u0000t0j<\u000e\"\n1\nNNX\ni=1jbi\nt\u0000bi\nt0j2#\n>\u000f!\n\u0014e\u0000L(\u000e;\u000f)N\nsup\njt\u0000t0j<\u000eE\"\n1\nNNX\ni=1jbi\nt\u0000bi\nt0j2#\n\u0014L(\u000e;\u000f)\u00001; (3.21)\nforb=xq,Bq,G(xq) and EB[x]. Obviously, this holds for b=Bq. Also, since by (1.6)\njxq;i\nt\u0000xq;i\nt0j\u0014jBq;i\nt\u0000Bq;i\nt0j+kf0(Kq;q\nN)k1Zt0\ntjxq;i\nujdu+Zt0\ntjGi(xq\nu)jdu+h(t0\u0000t):SPHERICAL SPIN GLASSES 13\nwe get, by (2.3), for some universal constant \u001a1<1, allt;t0andN,\n1\nNNX\ni=1jxq;i\nt\u0000xq;i\nt0j2\u00144\nNNX\ni=1jBq;i\nt\u0000Bq;i\nt0j2\n+4jt\u0000t0j2h\n\u001a1(1 +kKq\nNk1)2rkKq\nNk1+kGq\nNk1+h2i\nhence by the bounds established on kGq\nNk1andkKq\nNk1, we establish (3.21) for b=xq. An application of\nJensen's inequality will imply the same result for b=EB[x]. Using the results in Lemma 2.2 of [7], we can\nnow establish (3.21) for b=G(xq), thus concluding the equi-continuity of UNandVN, hence the \fst step of\nthe proof. Note that we have actually shown a stronger result that we will use later, namely that for all \u000f>0\nthere existseL(\u000e;\u000f)!1 for\u000e!0, such that for all N,\nP \nsup\njs\u0000s0j+jt\u0000t0j<\u000ejUN(s;t)\u0000UN(s0;t0)j>\u000f!\n\u0014e\u0000eL(\u000e;\u000f)N\nP \nsup\njs\u0000s0j<\u000ejVN(s)\u0000VN(s0)j>\u000f!\n\u0014e\u0000eL(\u000e;\u000f)N(3.22)\nand also\nsup\njs\u0000s0j+jt\u0000t0j<\u000ejE[UN(s;t)]\u0000E[UN(s0;t0)]j\u0014eL(\u000e;\u000f)\u00001\nsup\njs\u0000s0j<\u000ejE[VN(s)]\u0000E[VN(s0)]j\u0014eL(\u000e;\u000f)\u00001: (3.23)\nThe next step, as mentioned earlier is to establish (3.13) and (3.14). We will use the same approach as in\nthe proof of Proposition 2.4 of [7], by applying the estimate in Lemma 2.5 to UN(s;t) andVN(s), respectively,\nfor any \fxed pair of times s;t. For every M <1and anyN, de\fne the subset:\nLN;M=n\n(x0;J;B1;B2)2EN:jjJjjN\n1+kLNk1+2X\nq=1[kBq\nNk1+kKq;q\nNk1+kGq\nNk1]\u0014Mo\nofEN. ForMsu\u000eciently large, the probability of the complement set Lc\nN;M decays exponentially in Nby\n(3.19). Since the uniform moment bounds for the functions UN(s;t) andUN(s) has been established, as well\nas the stated pointwise bound in LN;M, the only other ingredient that we need to be able to apply the bound\nin Lemma 2.5 in the cited paper is the Lipschitz constant of UNandVNonLN;M.\nTo this end, let xq;fxqbe the two strong solutions of (1.6) constructed from ( x0;J;B1;B2) and (ex0;eJ;eB1;eB2),\nrespectively. If ( x0;J;B1;B2) and (ex0;eJ;eB1;eB2) are both inLN;M, then\nsup\nt\u0014T1\nNX\n1\u0014i\u0014Njxq;i\nt\u0000exq;i\ntj2\u0014Do(M;T )\nNk(x0;J;Bq)\u0000(ex0;eJ;fBq)k2(3.24)\n\u0014Do(M;T )\nNk(x0;J;B1;B2)\u0000(ex0;eJ;fB1;fB2)k2;\nfor someDo(M;T ) independent of N, where the \frst inequality is due to Lemma 2.6 of [7]. Now, equipped\nwith (3.24), we can easily show the desired Lipschitz estimate for all of the functions of interest UN(s;t) and\nVN(s), namely:\n(3.25) sup\ns;t\u0014TjUN(s;t)\u0000fUN(s;t)j\u0014D(M;T )p\nNk(x0;J;B1;B2)\u0000(ex0;eJ;eB1;eB2)k;14 MANUEL ZAMFIR\nand\n(3.26) sup\ns\u0014TjVN(s)\u0000fVN(s)j\u0014D(M;T )p\nNk(x0;J;B1;B2)\u0000(ex0;eJ;eB1;eB2)k;\nwhere the constant D(M;T ) depends only on MandTand not on N. Indeed, since every UN(s;t) and every\nVN(s) is of the form1\nNPN\ni=1ai\nsbi\nt, then (3.20) will hold, with the functions atandbtbeing one of xq\nt,Bq\nt,\nG(xq\nt),EB[x] or 1. By the same proof as the one employed in Lemma 2.7 of [7], we see that:\n\"\n1\nNNX\ni=1jGi(xq\ns)\u0000eGi(exq\ns)j2#1=2\n\u0014C(M;T )p\nNk(x0;J;B1;B2)\u0000(ex0;eJ;eB1;eB2)k:\nand\n\"\n1\nNNX\ni=1jGi(xq\nt)j2#1=2\n\u0014cjjJjjN\n1(1 +Mm\u00001)\u0014C(M):\nAlso, Jensen's inequality applied to (3.24) shows:\nsup\nt\u0014T1\nNX\n1\u0014i\u0014NjEB[xi\nt]\u0000EB[exi\nt]j2\u0014Do(M;T )\nNk(x0;J;B1;B2)\u0000(ex0;eJ;fB1;fB2)k2;\nThe last three bounds, together with the (3.24) plugged into equation (3.20) and is's analogue for V, will\nshow the Lipschitz bounds (3.25) and (3.26), whenever ( x0;J;B1;B2) and (ex0;eJ;eB1;eB2) are both inLN;M.\nAs noticed before, we have all the ingredients for applying Lemma 2.5 of [7] to VN:=UN(s;t) andVN:=\nVN(s), for any \fxed s;t\u0014T, yielding:\nP[jVN\u0000E[VN]j\u0015\u001a]\u0014C\u00001exp\u0012\n\u0000C\u0012\u001a\n2D(M(L))\u0013\u000b\nN\u000b\n2\u0013\n(3.27)\n+4(K+M(L))\u001a\u00001e\u0000LN=2+e\u0000NL:\nfor constants KandD=D(M(L);T) independent of s;t,\u001aandN. Consequently, by the union bound, for any\n\fnite subsetAof [0;T]2andBof [0;T] and any\u001a>0, the sequences N7!P[sup(s;t)2AjUN(s;t)\u0000E[UN(s;t)]j\u0015\n\u001a=3] andN7!P[sups2BjVN(s)\u0000E[VN(s)]j\u0015\u001a=3] are summable. Recalling (3.22) and (3.23), we choose\n\u000e > 0 small enough so that eL(2\u000e;\u001a=3)>3=\u001a > 0, we thus get (3.13) by considering the \fnite subsets\nA=f(i\u000e;j\u000e ) :i;j= 0;1;:::;T=\u000egand respectivelyB=fi\u000e:i= 0;1;:::;T=\u000eg.\nNow, we have all the ingredients needed for \fnalizing the proof. For each r\u0015Rletcrdenote the \fnite\nLipschitz constant of \t( \u0001) (with respect to k\u0001k 2), on the compact set \u0000 r:=fz:kzkk\u0014rg. Then,\njE[\t(ZN)]\u0000\t(E[ZN])j \u0014 Ej\t(ZN)\u0000\t(E[ZN])j 1ZN2\u0000r\n+Ej\t(ZN)j 1ZN=2\u0000r+j\t(E[ZN])jP[ZN=2\u0000r]\n\u0014crE[kZN\u0000E[ZN]k2] + 2`Mr\u0000kEkZNk2k\nk:\nWe have by (3.14) and the uniform moment bounds of UN(s;t) andVN(s) that supsj;tjE[kZN\u0000EZNk2]!0\nasN!1 , whilec0= supsj;tj;NEkZNk2k\nk<1, implying that:\nlim\nN!1sup\nsj;tjjE[\t(ZN)]\u0000\t(E[ZN])j\u00142c0`Mr\u0000k;\nwhich we make arbitrarily small by taking r!1 . \u0003\nNotice that, since LN(s;t) andQN(s;t) have the same \frst moment, for every sandt, the above proposition\nimplies that any limit point of LN(s;t) is also a limit point of QN(s;t).SPHERICAL SPIN GLASSES 15\n3.2.Getting the Limiting Equations. The key step of the proof of Proposition 2.2 is summarized by\nProposition 3.3. Fixing any T <1, any limit point of the sequences E[MN],E[\u001fN],E[CN]andE[QN] =\nE[C1;2\nN]with respect to uniform convergence on [0;T]2, satis\fes the integral equations\nM(s) =M(0) +hs+Zs\n0P(u)du; (3.28)\n\u001f(s;t) =s^t+Zs\n0E(u;t)du; (3.29)\nC(s;t) =C(s;0) +\u001f(s;t) +Zt\n0D(s;u)du+htM(s); (3.30)\nQ(s;t) =Q(s;0) +Zt\n0H(s;u)du+htM(s); (3.31)\nP(t) =\u0000f0(C(t;t))M(t) +\u00170(C(t;t))M(t)\u0000\u00170(C(0;t))M(0) (3.32)\n\u0000Zt\n0\u00170(C(t;u))P(u)du\u0000Zt\n0M(u)\u001700(C(t;u))D(u;t)du\n\u0000h\u0014\nM(t)Zt\n0M(u)\u001700(C(t;u))du+Zt\n0\u00170(C(t;u))du\u0015\nE(s;t) =\u0000f0(C(s;s))\u001f(s;t) +\u001f(s;t)\u00170(C(s;s))\u0000hQ(s)Zs\n0\u001700(C(s;u))\u001f(u;t)du (3.33)\n\u0000Zs\n0\u001f(u;t)\u001700(C(s;u))D(s;u)du\u0000Zt^s\n0\u00170(C(s;u))du\u0000Zs\n0\u00170(C(s;u))E(u;t)du;\nD(s;t) =C(s;t_s)\u00170(C(t_s;t))\u0000C(s;0)\u00170(C(0;t))\u0000f0(C(t;t))C(t;s) (3.34)\n\u0000Zt_s\n0\u00170(C(t;u))D(s;u)du\u0000Zt_s\n0C(s;u)\u001700(C(t;u))D(t;u)du\n\u0000h\u0014\nM(t)Zt_s\n0C(s;u)\u001700(C(t;u))du+M(s)Zt_s\n0\u00170(C(t;u))du\u0015\nH(s;u) =\u0000f0(C(t;t))Q(t;s) +X(s;u) +Y(s;u) (3.35)\nX(s;t) =Q(s;t_s)\u00170(C(t_s;t))\u0000Q(s;0)\u00170(C(0;t))\n\u0000Zt_s\n0\u00170(C(t;u))H(s;u)du\u0000Zt_s\n0Q(s;u)\u001700(C(t;u))D(t;u)du (3.36)\n\u0000h\u0014\nM(t)Zs_t\n0Q(s;u)\u001700(C(t;u))du+M(s)Zs_t\n0\u00170(C(t;u))du\u0015\nandY(s;y)is de\fned similarly to X(s;t), with the roles of CandQand respectively DandHreversed, in\nthe space of bounded continuous functions on [0;T]2, subject to the symmetry conditions C(s;t) =C(t;s)and\nQ(s;t) =Q(t;s)and the boundary conditions E(s;0) = 0 for alls, andE(s;t) =E(s;s)for allt\u0015s.\nWe will then show in Lemma 3.4 that every solution of (3.28)-(3.36) is necessarily a solution of (2.6)-(2.10),\nthus allowing us to conclude the proof of Proposition 2.2, upon showing, in Proposition 3.5, the uniqueness of\nthe solution of (2.6)-(2.10).\nLemma 3.4. FixingT <1, suppose (M;\u001f;C;Q;D;E;P;H )is a solution of the integral equations (3.28){\n(3.36) in the space of continuous functions on [0;T]2subject to the symmetry conditions C(s;t) =C(t;s)\nandQ(s;t) =Q(t;s)and the boundary conditions E(s;0) = 0 for alls, andE(s;t) =E(s;s)for allt\u0015s.\nThen,\u001f(s;t) =Rt\n0R(s;u)duwhereR(s;t) = 0 fort > s ,R(s;s) = 1 and forT\u0015s > t , the bounded16 MANUEL ZAMFIR\nand absolutely continuous functions M;C;R;Q andK(s) =C(s;s)necessarily satisfy the integro-di\u000berential\nequations (2.6){(2.10).\nProposition 3.5. LetT\u00150. There exists at most one solution (M;R;C;Q;K )inC1(R+)\u0002C1(\u0000)\u0002C1\ns(R2\n+)\u0002\nC1\ns(R2\n+)\u0002C1(R+)to(2.6) -(2.10) withR(s;s) = 1 ,C(s;s) =K(s),8s\u00150,C(0;0) =Q(0;0) =K(0)andM(0)\nknown.\nWe will now change the notations in [7], denoting in short bUq1;q2\nN :=E[Uq1;q2\nN], whenever Uis one of the\nfunctions of interest A;C;F;K;\u001f;D;E and respectively, bVq\nN:=E[Vq\nN], whenever Vis one of the functions\nM;P orR. As before, when there is only one replica present, we will drop the index superscript (for example\nbC=bC1;1).\nRecall the integrated form of the equation (1.6), for q= 1;2 andi= 1;:::;N :\n(3.37) xq;i\ns=xq;i\n0+Bq;i\ns\u0000Zs\n0f0(Kq;q\nN(u))xq;i\nudu+Zs\n0Gi(xq\nu)du+hs\nFrom now on, we will write X\u0011Ywhenever the random variables XandYhave the same law and aN'bN\nwhenaN(\u0001;\u0001)\u0000bN(\u0001;\u0001)!0 (oraN(\u0001)\u0000bN(\u0001)!0) asN!1 , uniformly on [0 ;T]2(or [0;T], whichever applies).\nLet us denote by bQN(s;t) :=bC1;2\nN(s;t) =bC2;1\nN(s;t) (sinceC1;2\nN(s;t)\u0011C2;1\nN(s;t)). Applying Proposition 3.2\n(for \t(z) =z1f0(z2) whose polynomial growth is guaranteed by our assumption (2.3)), we deduce that:\nE[f0(Kq1;q2\nN(u))Uq3;q4\nN(u;t)]'f0(bKq1;q2\nN(u))bUq3;q4\nN(u;t)\nand\nE[f0(KN(u))MN(u)]'f0(bKN(u))cMN(u)\nwheneverUis one of the functions Cor\u001f. Hence, upon multiplying (3.37) with xq;i\nt,Bq;i\nt,x3\u0000q;i\nt and 1,\nrespectively, followed by averaging over iand taking the expected value, we get that for any s;t2R+,\ncMN(s)'cMN(0) +hs\u0000Zs\n0f0(bKN(u))cMN(u)du+Zs\n0bRN(u)du (3.38)\nb\u001fN(s;t)'b\u001fN(0;t) +t^s\u0000Zs\n0f0(bKN(u))b\u001fN(u;t)du+Zs\n0bFN(u;t)du (3.39)\nbCN(s;t)'bCN(0;t) +b\u001fN(t;s)\u0000Zs\n0f0(bKN(u))bCN(u;t)du+Zs\n0bAN(u;t)du+hscMN(t) (3.40)\nbQN(s;t)'bQN(0;t)\u0000Zs\n0f0(bKN(u))bQN(u;t)du+Zs\n0bA1;2\nN(u;t)du+hscMN(t): (3.41)\nIn the following proposition, we will approximate the terms bRN,bFN,bRNandbA1;2\nN, in order to compute the\nlimits of (3.38)-(3.41) as N!1 .\nProposition 3.6. We have that\nbAN(t;s)'\u00170(bCN(t;t_s))bCN(s;t_s)\u0000\u00170(bCN(t;0))bCN(s;0) (3.42)\n\u0000Zs_t\n0\u001700(bCN(t;u))bCN(s;u)bDN(t;u)du\u0000Zs_t\n0\u00170(bCN(t;u))bDN(s;u)du\n\u0000h\u0014\ncMN(t)Zs_t\n0bCN(s;u)\u001700(bCN(t;u))du+cMN(s)Zs_t\n0\u00170(bCN(t;u))du\u0015\n;\nbA1;2\nN(t;s)'2X\nr=1h\n\u00170(bCr;1\nN(t;t_s))bC2;r\nN(s;t_s)\u0000\u00170(bCr;1\nN(t;0))bC2;r\nN(s;0)i\n(3.43)\n\u00002X\nr=1\u0014Zs_t\n0\u00170(bCr;1\nN(t;u))bDr;2\nN(s;u)du+Zs_t\n0\u001700(bCr;1\nN(t;u))bC2;r\nN(s;u)bDr;1\nN(t;u)du\u0015SPHERICAL SPIN GLASSES 17\n\u0000h2X\nr=1\u0014\ncMN(t)Zs_t\n0bC2;r\nN(s;u)\u001700(bC2;r\nN(t;u))du+cMN(s)Zs_t\n0\u00170(C2;r\nN(t;u))du\u0015\n;\nbFN(s;t)'b\u001fN(s;t^s)\u00170(bCN(s;s))\u0000Zs\n0\u00170(bCN(s;u))bEN(u;t^u)du (3.44)\n\u0000Zt^s\n0\u00170(bCN(s;u))du\u0000Zs\n0b\u001fN(u;t^u)\u001700(bCN(s;u))bDN(s;u)du\n\u0000hcMN(s)Zs\n0\u001700(bCN(s;u))b\u001fN(u;t^u)du;\nand\nbRN(t)'\u00170(bCN(t;t))cMN(t)\u0000\u00170(bCN(0;t))cMN(0) (3.45)\n\u0000Zt\n0cMN(u)\u001700(bCN(t;u))bDN(u;t)du\u0000Zt\n0\u00170(bCN(t;u))bPN(u)du\n\u0000h\u0014\ncMN(t)Zt\n0cMN(u)\u001700(bCN(t;u))du+Zt\n0\u00170(bCN(t;u))\u0015\ndu:\nIt is clear that using the results in Proposition 3.6 in formulas (3.38)-(3.41), we have proved Proposition\n3.3. We shall start by developing the tools needed to conclude the proof of Proposition 3.6. To begin, we \frst\nprove a slightly more general version of Lemma 3.2 of [7]. The proof is essentially the same, replacing xj\ntby\nxq1;j\ntandxi\nsbyxq2;i\ns, respectively and will not be repeated.\nLemma 3.7. LetEJdenotes the expectation with respect to the Gaussian law PJof the disorder J. Then, for\nthe continuous paths xq2C(R+;RN),q2fq1;q2g, and alls;t2[0;T]andi;j2f1;:::;Ng,\n(3.46) kq1;q2;ij\nts (x) =xq1;j\ntxq2;i\ns\nN\u001700(Cq2;q1\nN(s;t)) + 1i=j\u00170(Cq2;q1\nN(s;t)):\nwherekq1;q2;ij\nts (x) :=EJ[Gi(xq1\nt)Gj(xq2s)].\nFixing continuous paths xq, letkq1;q2\nt denote the operator on L2(f1;\u0001\u0001\u0001Ng\u0002[0;t]) with the kernel k=\nkq1;q2(x) of (3.46). That is, for f2L2(f1;\u0001\u0001\u0001Ng\u0002[0;t]),u\u0014t,i2f1;\u0001\u0001\u0001;Ng\n(3.47) [ kq1;q2\ntf]i\nu=NX\nj=1Zt\n0kq1;q2;ij\nuvfj\nvdv;\nwhich is clearly also in L2(f1;\u0001\u0001\u0001Ng\u0002[0;t]). We next extend the de\fnition (3.47) to the stochastic integrals\nof the form\n[kq1;q2\nt\u000edZ]i\nu=NX\nj=1Zt\n0kq1;q2;ij\nuvdZj\nv;\nwhereZj\nvis a continuous semi-martingale with respect to the \fltration Ft=\u001b(xq1u;xq2u: 0\u0014u\u0014t) and\nis composed for each j, of a squared-integrable continuous martingale and a continuous, adapted, squared-\nintegrable \fnite variation part. In doing so, recall that by (3.46), each kij\nuv(x) is the \fnite sum of terms\nsuch asxq1;i1u\u0001\u0001\u0001xq1;iauxq2;j1v\u0001\u0001\u0001xq2;jbv, where in each term a,bandi1;:::;ia;j1;:::;jbare some non-random\nintegers. Keeping for simplicity the implicit notationRt\n0kq1;q2;ij\nuvdZj\nvwe thus adopt hereafter the convention of\naccordingly decomposing such integral to a \fnite sum, taking for each of its terms the variable xq1;i1u\u0001\u0001\u0001xq1;iau\noutside the integral, resulting with the usual It^ o adapted stochastic integrals. The latter are well de\fned, with\n[kq1;q2\nt\u000edZ]i\nubeing in L2(f1;\u0001\u0001\u0001Ng\u0002[0;t]).\nOur next step is to generalize Proposition C.1 of [7]:18 MANUEL ZAMFIR\nProposition 3.8. Letm2Z+and suppose under the law Pwe have a \fnite collection J=fJ\u000bg\u000bof non-\ndegenerate, independent, centered Gaussian random variables, and Gq;i\ns=P\n\u000bJ\u000bLq;i\ns(\u000b), forq= 1;:::;m , for\nalls2[0;\u001c]andi\u0014N, where for each \u000bthe coe\u000ecients Lq;i\nswhich are independent of Jand also of each other,\nfor di\u000berent q's, are in L2(f1;:::;Ng\u0002[0;\u001c]). Suppose further that Uq;i\nsare continuous semi-martingales,\nindependent of Jand such that for each \u000bandq, the stochastic integral\n\u0016q\n\u000b:=NX\ni=1Z\u001c\n0Lq;i\nu(\u000b)dUq;i\nu;\nis well de\fned and almost surely \fnite. Let P\u0003denote the law of Jsuch that P\u0003=Qm\nq=1\u0003q\n\u001c=E\u0010Qm\nq=1\u0003q\n\u001c\u0011\nP,\nwhere\n(3.48) \u0003q\n\u001c= exp(NX\ni=1Z\u001c\n0Gq;i\nsdUq;i\ns\u00001\n2NX\ni=1Z\u001c\n0(Gq;i\ns)2ds)\n:\nLetVq;i\ns=E\u0003(Gq;i\ns),kq1;q2;ij\nts =E(Gq1;i\ntGq2;j\ns)and\u0000q1;q2;ij\nts =E\u0003[(Gq1;i\nt\u0000Vq1;i\nt)(Gq2;j\ns\u0000Vq2;j\ns)]. Then, for any\ns\u0014\u001c,i\u0014Nandq2f1;:::;mg,\n(3.49) Vq;i\ns+mX\nr=1[kq;r\n\u001cVr]i\ns=mX\nr=1[kq;r\n\u001c\u000edUr]i\ns;\nand for any s;t\u0014\u001c,i;l\u0014Nandq1;q22f1;:::;mg\n(3.50)NX\nr=1NX\nj=1Z\u001c\n0kq1;r;ij\nsu \u0000r;q2;jl\nut + \u0000q1;q2;il\nst =kq1;q2;il\nst:\nProof of Proposition 3.8. Letv\u000b=E(J2\n\u000b)>0 denote the variance of J\u000band\n(3.51) Rq\n\u000b\r:=NX\ni=1Z\u001c\n0Lq;i\nu(\u000b)Lq;i\nu(\r)du;\nobserving that\n\u0003q\n\u001c= expnX\n\u000bJ\u000b\u0016q\n\u000b\u00001\n2X\n\u000b;\rJ\u000bJ\rRq\n\u000b\ro\n:\nWith D= diag(v\u000b) a positive de\fnite matrix and R:=Pm\nq=1Rq=fPm\nq=1Rq\n\u000b\rgpositive semi-de\fnite, it\nfollows from this representation of \u0003q\n\u001c, that under P\u0003the random vector Jhas a Gaussian law with covariance\nmatrix ( D\u00001+Pm\nq=1Rq)\u00001and mean vector w=fw\u000bg= (D\u00001+Pm\nq=1Rq)\u00001(Pm\nq=1\u0016q). Hence, for any \u000b,\n(3.52) w\u000b+v\u000bX\n\r mX\nq=1Rq\n\u000b\r!\nw\r=v\u000bmX\nq=1\u0016q\n\u000b:\nAskq1;q2;ij\nsu =P\n\u000bLq1;i\ns(\u000b)v\u000bLq2;j\nu(\u000b), it is not hard to check that\n[kq1;q2\n\u001c\u000edUq2]i\ns:=NX\nj=1Z\u001c\n0kq1;q2;ij\nsudUq2;j\nu=X\n\u000bLq1;i\ns(\u000b)v\u000bNX\nj=1Z\u001c\n0Lq2;j\nu(\u000b)dUq2;j\nu\n=X\n\u000bLq1;i\ns(\u000b)v\u000b\u0016q2\n\u000b:\nObviously,\nVq;i\ns=X\n\u000bLq;i\ns(\u000b)w\u000bSPHERICAL SPIN GLASSES 19\nand also,\n[kq1;q2\n\u001cVq2]i\ns:=NX\nj=1Z\u001c\n0kq1;q2;ij\nsuVq2;j\nudu=X\n\u000b;\rLq1;i\ns(\u000b)v\u000bw\rNX\nj=1Z\u001c\n0Lq2;j\nu(\u000b)Lq2;j\nu(\r)du\n=X\n\u000b;\rLq1;i\ns(\u000b)v\u000bRq2\n\u000b\rw\r;\nso we get (3.49) out of (3.52), with the last identity due to (3.51). Turning to prove (3.50), since \u0000q1;q2;jl\nut is\nthe covariance of Gq1;j\nuandGq2;l\ntunder the tilted law P\u0003, we have that\n\u0000q1;q2;jl\nut =X\n\u000b;\rLq1;j\nu(\u000b)\"\n(D\u00001+mX\nq=1Rq)\u00001#\n\u000b\rLq2;l\nt(\r);\nand hence by (3.51) we see that\nNX\nj=1Z\u001c\n0kq1;r;ij\nsu \u0000r;q2;jl\nutdu=NX\nj=1Z\u001c\n0kq1;r;ij\nsuX\n\u000b;\rLr;j\nu(\u000b)\"\n(D\u00001+mX\nq=1Rq)\u00001#\n\u000b\rLq2;l\nt(\r)du\n=NX\nj=1Z\u001c\n0X\n\u001bLq1;i\ns(\u001b)v\u001bLr;j\nu(\u001b)X\n\u000b;\rLr;j\nu(\u000b)\"\n(D\u00001+mX\nq=1Rq)\u00001#\n\u000b\rLq2;l\nt(\r)du\n=X\n\u001b;\u000b;\rLq1;i\ns(\u001b)v\u001bRr\n\u001b\u000b\"\n(D\u00001+mX\nq=1Rq)\u00001#\n\u000b\rLq2;l\nt(\r)du\n=X\n\u001b;\rLq1;i\ns(\u001b)v\u001b[Rr\"\n(D\u00001+mX\nq=1Rq)\u00001#\n\u001b\rLq2;l\nt(\r)du\nWith D= diag(v\u000b) we easily get (3.50) out of the matrix identity:\n \nI+D mX\nq=1Rq!! \nD\u00001+mX\nq=1Rq!\u00001\n=D:\n\u0003\nNow, the same proof as in Lemma 3.2 of [7], with \u0003N\n\u001creplaced by:\n\u0003N\n\u001c= exp(mX\nq=1\"NX\ni=1Z\u001c\n0Gi(xq\ns)dUq;i\ns(x)\u00001\n2NX\ni=1Z\u001c\n0(Gi(xq\ns))2ds#)\nand using Proposition 3.8 above instead of Proposition C.1 of [7], will show:\nLemma 3.9. Letm2Z+and consider mreplicasfxqgs, forq= 1;:::;m , sharing the same couplings J,\nwith the noise given by mindependent N-dimensional Brownian motions fBqgs. Fixing\u001c2R+and denoting\nx= (x1;:::;xm), letVq;i\ns(x) =E[Gi(xq\ns)jF\u001c]andZq;i\ns(x) =E[Bq;i\nsjF\u001c]fors2[0;\u001c]. Then, under PJ\nPN\nx0;J\nwe can choose a version of these conditional expectations such that the stochastic processes\nUq;i\ns(x) :=xq;i\ns\u0000xq;i\n0+Zs\n0f0(Kq;q\nN(u))xq;i\nudu\u0000hs (3.53)\nZq;i\ns(x) :=Uq;i\ns(x)\u0000Zs\n0Vq;i\nu(xq)du; (3.54)20 MANUEL ZAMFIR\nare both continuous semi-martingales with respect to the \fltration Ft=\u001b(xk\nu: 0\u0014u\u0014t;1\u0014k\u0014m), composed\nof squared-integrable continuous martingales and \fnite variation parts. Moreover, such choice satis\fes for any\ni;qands2[0;\u001c],\n(3.55) Vq;i\ns+mX\nr=1[kq;r\n\u001cVr]i\ns=mX\nr=1[kq;r\n\u001c\u000edUr]i\ns;\nandVq;i\ns=Pm\nr=1[kr;q\n\u001c\u000edZr]i\nsfor anyi;qand alls\u0014\u001c. Further, for any u;v2[0;\u001c]andi;j\u0014N, let\n(3.56) \u0000q1;q2;ij\nuv (x) :=Eh\n(Gi(xq1\nu)\u0000Vq1;i\nu(x))(Gj(xq2\nv)\u0000Vq2;j\nv(x))jF\u001ci\nFurther, we can choose a version of \u0000q1;q2;il\nuv such that for any s;v\u0014\u001c, anyq1;q22f1;:::;mgand alli;l\u0014N,\n(3.57)mX\nr=1NX\nj=1Z\u001c\n0kq1;r;ij\nsu \u0000r;q2;jl\nut + \u0000q1;q2;il\nst =kq1;q2;il\nst:\nProof of Proposition 3.6. We \frst apply (3.55) to derive (3.43). Fix s;t2[0;T]2, let\u001c=t_sand de\fne:\naq1;q2\nN(t;s) =1\nNNX\ni=1Vq1;i\nt(x)xq2;i\ns;\nSincexq;i\nsis measurable on F\u001c,q= 1;2, we see that:\nbAq1;q2\nN(t;s) =E\"\n1\nNNX\ni=1E[Gi(xq1\nt)xq2;i\nsjF\u001c]#\n=E[aq1;q2\nN(t;s)] =baq1;q2\nN(t;s):\nHence, with t\u0014\u001c, combining (3.55) and (3.53), and suppressing in the notation the dependence of kq1;q2;ij\ntu\nandVq1;j\nuofx, we get:\na1;2\nN(t;s) +2X\nr=12\n41\nNNX\ni;j=1Z\u001c\n0x2;i\nsk1;r;ij\ntuVr;j\nudu+h1\nNNX\ni;j=1Z\u001c\n0x2;i\nsk1;r;ij\ntudu3\n5 (3.58)\n=2X\nr=12\n41\nNNX\ni;j=1Z\u001c\n0f0(Kr;r\nN(u))x2;i\nsk1;r;ij\ntuxr;j\nudu+1\nNNX\ni;j=1Z\u001c\n0x2;i\nsk1;r;ij\ntudxr;j\nu3\n5\nUsing the explicit expression of kq1;q2;ij\ntu from Lemma 3.7, and collecting terms while changing the order of\nsummation and integration, we arrive at the identity:\na1;2\nN(t;s) =\u00002X\nr=1\u0014Z\u001c\n0C2;r\nN(s;u)\u001700(Cr;1\nN(t;u))ar;1\nN(u;t)du+Z\u001c\n0\u00170(Cr;1\nN(t;u))ar;2\nN(u;s)du\u0015\n(3.59)\n\u0000h2X\nr=1\u0014Z\u001c\n0M1\nN(t)C2;r\nN(s;u)\u001700(Cr;1\nN(t;u))du+Z\u001c\n0M2\nN(s)\u00170(Cr;1\nN(t;u))du\u0015\n+2X\nr=1\u0014Z\u001c\n0f0(Kr;r\nN(u))C2;r\nN(s;u)\u001700(Cr;1\nN(t;u))C1;r\nN(t;u)du\u0015\n+2X\nr=1\u0014Z\u001c\n0f0(Kr;r\nN(u))\u00170(Cr;1\nN(t;u))Cr;2\nN(u;s)du\u0015\n+2X\nr=1\u0014Z\u001c\n0C2;r\nN(s;u)\u001700(Cr;1\nN(t;u))duCr;1\nN(u;t) +Z\u001c\n0\u00170(Cr;1\nN(t;u))duCr;2\nN(u;s)\u0015\n:SPHERICAL SPIN GLASSES 21\nApplying Lemma A.1 of [7] for the semi-martingales x=w=xr,y=x1,z=x2and polynomials P(x) =x\nandQ(x) =\u00170(x), the stochastic integrals in the last line of (3.59) can be replaced with:\n2X\nr=1h\n\u00170(Cr;1\nN(\u001c;t))C2;r\nN(\u001c;s)\u0000\u00170(Cr;1\nN(0;t))C2;r\nN(0;s)i\n(3.60)\n\u00002X\nr=1\u00141\n2NC1;1\nN(t;t)Z\u001c\n0\u001700(Cr;1\nN(u;t))C2;r\nN(u;s)du+1\nNC1;r\nN(s;t)Z\u001c\n0\u00170(Cr;1\nN(u;t))du\u0015\n:\nNow, it is easy to see that since E[sups;t\u0014TjAq1;q2\nN(s;t)j] is uniformly bounded in N(see the discussion prior\nto Proposition 3.2), then the same is true for aq1;q2\nN(t;s), hence the terms in the second line (3.60) above will\nconverge almost surely to 0, as N!1 . Furthermore, aq1;q2\nN(t;s) =E[Aq1;q2\nN(t;s)jF\u001c] inherits the self-averaging\nproperty from Aq1;q2\nN, hence, we can apply Corollary 3.2 with possibly aq1;q2\nN as one of the arguments of the\nlocally Lipschitz function \t( z) of at most polynomial growth at in\fnity. Doing so for the functions z1z2\u001700(z3),\nandz1\u00170(z2) and applying Proposition 3.2 also for f0(z1)z2\u001700(z3)z3andf0(z1)\u00170(z2)z3, we deduce from (3.59)\nand (3.60) that\nbA1;2\nN(t;s)'\u00002X\nr=1\u0014Z\u001c\n0bC2;r\nN(s;u)\u001700(bCr;1\nN(t;u))bAr;1\nN(u;t)du+Z\u001c\n0\u00170(bCr;1\nN(t;u))bAr;2\nN(u;s)du\u0015\n\u0000h2X\nr=1\u0014Z\u001c\n0cM1\nN(t)bC2;r\nN(s;u)\u001700(bCr;1\nN(t;u))du+Z\u001c\n0cM2\nN(s)\u00170(bCr;1\nN(t;u))du\u0015\n+2X\nr=1Z\u001c\n0f0(bKr;r\nN(u))bC2;r\nN(s;u)\u001700(bCr;1\nN(t;u))bC1;r\nN(t;u)du\n+2X\nr=1Z\u001c\n0f0(bKr;r\nN(u))\u00170(bCr;1\nN(t;u))bCr;2\nN(u;s)du\n+2X\nr=1h\n\u00170(bCr;1\nN(\u001c;t))bC2;r\nN(\u001c;s)\u0000\u00170(bCr;1\nN(0;t))bC2;r\nN(0;s)i\n:\nFinally, recalling that\nbAq1;q2\nN(t;s) =bDq1;q2\nN(s;t) +f0(bKN(t))bCq1;q2\nN(s;t);\nand noting that bKr;r\nN(t) =bKN(t), for alltandr, setting\u001c=t_s, we indeed arrive at:\nbA1;2\nN(t;s)'\u00002X\nr=1\u0014Zt_s\n0bC2;r\nN(s;u)\u001700(bCr;1\nN(t;u))bDr;1\nN(t;u)du+Zt_s\n0\u00170(bCr;1\nN(t;u))bAr;2\nN(s;u)du\u0015\n\u0000h2X\nr=1\u0014Zt_s\n0cMN(t)bC2;r\nN(s;u)\u001700(bCr;1\nN(t;u))du+Zt_s\n0cMN(s)\u00170(bCr;1\nN(t;u))du\u0015\n+2X\nr=1h\n\u00170(bCr;1\nN(t_s;t))bC2;r\nN(t_s;s)\u0000\u00170(bCr;1\nN(0;t))bC2;r\nN(0;s)i\n:\nthat is (3.43).\nFor deriving (3.42) next, the single-replica equivalent of (3.43), we can apply the same strategy as above.\nNamely, de\fning:\naN(t;s) =1\nNNX\ni=1Vi\nt(x)xi\ns;22 MANUEL ZAMFIR\nwe see that aN(t;s) has the same \frst moment with AN(t;s). Furthermore, since F\u001cis generated only by the\nrealization of one replica up to time \u001c, (3.55) will imply that:\naN(t;s) +1\nNNX\ni;j=1Z\u001c\n0xskij\ntuVj\nudu+h1\nNNX\ni;j=1Z\u001c\n0xi\nskij\ntudu\n=1\nNNX\ni;j=1Z\u001c\n0f0(KN(u))xi\nskij\ntuxj\nudu+1\nNNX\ni;j=1Z\u001c\n0xi\nskij\ntudxj\nu\nNote that the above equation is indeed the one-dimensional version of (3.58) (without the sums and the replica\nindices), so we would expect the results to be similar. Indeed, using the explicit expression of kij\ntufrom Lemma\n3.7, we arrive at the identity:\naN(t;s) =\u0000Z\u001c\n0CN(s;u)\u001700(CN(t;u))aN(u;t)du\u0000Z\u001c\n0\u00170(CN(t;u))aN(u;s)du (3.61)\n\u0000h\u0014Z\u001c\n0MN(t)CN(s;u)\u001700(CN(t;u))du+Z\u001c\n0MN(s)\u00170(CN(t;u))du\u0015\n+Z\u001c\n0f0(KN(u))CN(s;u)\u001700(CN(t;u))CN(t;u)du\n+Z\u001c\n0f0(KN(u))\u00170(CN(t;u))CN(u;s)du\n+Z\u001c\n0CN(s;u)\u001700(CN(t;u))duCN(u;t) +Z\u001c\n0\u00170(CN(t;u))duCN(u;s):\nApplying again Lemma A.1 of [7], this time for the semi-martingales x=y=z=w=xand polynomials\nP(x) =xandQ(x) =\u00170(x), the stochastic integrals in the last line of (3.61) can be replaced with:\n\u00170(CN(\u001c;t))CN(\u001c;s)\u0000\u00170(CN(0;t))CN(0;s)\n\u0000\u00141\n2NCN(t;t)Z\u001c\n0CN(u;s)\u001700(CN(u;t))du+1\nNCN(s;t)Z\u001c\n0\u00170(CN(u;t))du\u0015\n:\nAs before, the terms in the second line above will converge to 0 as N!1 , and,aN(t;s) =E[AN(t;s)jF\u001c]\ninherits the self-averaging property from AN. Hence applying Corollary 3.2 with possibly aNas one of the\narguments of the locally Lipschitz function \t( z), setting\u001c=t_sand recalling that bAN(t;s) =bDN(s;t) +\nf0(bKN(t))bCN(s;t), we arrive at (3.43).\nNow, for (3.45), denoting rN(s) =1\nNPN\ni=1Vi\nt(x) and we easily see that brN(s) =bRN(s), so by (3.55) and\n(3.53) we get that:\nrN(t) +1\nNNX\ni;j=1Z\u001c\n0kij\ntuVj\nudu=1\nNNX\ni;j=1Z\u001c\n0f0(KN(u))kij\ntuxj\nudu+1\nNNX\ni;j=1Z\u001c\n0kij\ntudxj\nu\u0000h1\nNNX\ni;j=1Z\u001c\n0kij\ntudu\nSo, as before, using the explicit expression of kij\ntu, we get to:\nrN(t) =\u0000Z\u001c\n0MN(u)\u001700(CN(t;u))aN(u;t)du\u0000Z\u001c\n0\u00170(CN(t;u))rN(u)du (3.62)\n\u0000h\u0014Z\u001c\n0MN(t)MN(u)\u001700(CN(t;u))du+Z\u001c\n0\u00170(CN(t;u))du\u0015\n+Z\u001c\n0f0(KN(u))MN(u)\u001700(CN(t;u))CN(u;t)duSPHERICAL SPIN GLASSES 23\n+Z\u001c\n0MN(u)\u001700(CN(t;u))duCN(u;t) +Z\u001c\n0\u00170(CN(t;u))duMN(u)\n+Z\u001c\n0f0(KN(u))\u00170(CN(t;u))MN(u)du:\nOnce again, Lemma A.1 of [7], helps, this time for the semi-martingales x=z=w=x,y= 1 and polynomials\nP(x) =xandQ(x) =\u00170(x), hence we replace the stochastic integrals above with:\nMN(\u001c)\u00170(CN(\u001c;t))\u0000MN(0)\u00170(CN(0;t))\n\u00001\n2NCN(t;t)Z\u001c\n0MN(u)\u001700(CN(u;t))du\u00001\nNMN(t)Z\u001c\n0\u00170(CN(u;t))du\nAs before, the terms in the second line above will converge to 0 as N!1 , and,rN(t) =E[RN(t)jF\u001c] inherits\nthe self-averaging property from RN. Hence applying Corollary 3.2 with possibly aNandrNas some of the\narguments of \t( z) and recalling that bPN(t) =bRN(t) +f0(bKN(t))cMN(t), we arrive at (3.45).\nNow the derivation of (3.44) is similar to the derivation of its analogue in the proof of Proposition 3.1 in\n[7]. Namely, since:\n(3.63) E[Gi\nsBi\nt] +E[Zt\n0\u0000ii\nsvdv] =E\u0002\n[ks\u000edZ]i\nsZi\nt\u0003\n:\nthe equation (3.2) implies:\nbFN(s;t) = E\"\n1\nNNX\ni=1E\u0002\n[ks\u000edB]i\nsjFs\u0003\nBi\nt#\n\u0000E\"\n1\nNNX\ni=1Zt\n0\u0000ii\nsvdv#\nhence by (3.37) and (3.46):\nbFN(s;t) +h1\nNNX\ni=1E\u0002\nE\u0002\n[ks]i\nsjFs\u0003\nBi\nt\u0003\n(3.64)\n=1\nNNX\ni=1\u0012\nE\u0002\nE\u0002\n[ks\u000edx]i\ns+ [ksf0(KN)x]i\ns\u0000[ksG]i\nsjFs\u0003\nBi\nt\u0003\n\u0000E\u0014Zt\n0\u0000ii\nsvdv\u0015\u0013\n;\nThe right hand side of (3.64) was evaluated in the proof of Proposition 3.1 of [7]. Using their result into (3.64),\nwe get, for s\u0015t:\nbFN(s;t) +h1\nNNX\ni=1E\u0002\nE\u0002\n[ks]i\nsjFs\u0003\nBi\nt\u0003\n'b\u001fN(s;t)\u00170(bCN(s;s))\u0000Zt^s\n0\u00170(bCN(s;u))du\n\u0000Zs\n0\u00170(bCN(s;u))bEN(u;t^u)du\u0000Zt^s\n0\u00170(bCN(s;u))du\n\u0000Zs\n0b\u001fN(u;t^u)\u001700(bCN(s;u))bDN(s;u)du\nNow, using the explicit formula for ksto compute the remaining term:\nh1\nNNX\ni=1E\u0002\nE\u0002\n[ks]i\nsjFs\u0003\nBi\nt\u0003\n=h1\nNNX\ni=1E\u0014\u0012\nMN(s)Zs\n0\u001700(CN(s;u))xi\nudu+Zs\n0\u00170(CN(s;u))du\u0013\nBi\nt\u0015\n=hE\u0014Zs\n0MN(s)\u001700(CN(s;u))\u001fN(u;t)du+Zs\n0\u00170(CN(s;u))WN(t)du\u0015\n'hcMN(s)Zs\n0\u001700(bCN(s;u))b\u001fN(u;t)du24 MANUEL ZAMFIR\nwhere the last line is obtained by two applications of Proposition 3.2 (eventually with the zero mean random\nvariableWN(t) =1\nNPN\ni=1Bi\ntas one of its arguments), hence concluding the proof of (3.44). \u0003\nProof of Lemma 3.4. We shall show that every solution of (3.28)-(3.36) is necessarily a solution of (2.6)-\n(2.10), where \u001f(s;t) =Rt\n0R(s;u)du.\nFirst, the same argument as in the beginning of Lemma 5.1 of [7] applied to\nh(s;t) :=\u0000f0(C(s;s))\u001f(s;t)\u0000Zs\n0\u001f(u;t)\u001700(C(s;u))D(s;u)du+\u001f(s;t)\u00170(C(s;s))\n\u0000Zt^s\n0\u00170(C(s;u))du\u0000hM(s)Zs\n0\u001700(C(s;u))\u001f(u;t)du\nwill show that t7!\u001f(s;t) is continuously di\u000berentiable on s\u0015t, with\u001f(s;t) =Rt\n0R(s;u)du, whereR(s;s) = 1\nfor allsand\u001f(s;t) =\u001f(s;s) fort>s , implying that R(s;t) = 0, fort>s .\nFrom (3.30) we have that C(s;t)\u0000\u001f(s;t) is di\u000berentiable with respect to its second argument t, hence\n@2C(s;t) =D(s;t) +R(s;t) +hM(s) Further,C(s;t) =C(t;s) implying that @1C(s;t) =@2C(t;s) =D(t;s) +\nR(t;s) +hM(t) on [0;T]2. Thus, combining the identity\nC(s;t_s)\u00170(C(t_s;t))\u0000C(s;0)\u00170(C(0;t)) =Zt_s\n0\u00170(C(t;u))@2C(s;u)du\n+Zt_s\n0C(s;u)\u001700(C(t;u))@2C(t;u)du;\nwith (3.34) we have that for all t;s2[0;T]2,\nD(s;t) =\u0000f0(K(t))C(t;s) +Zt_s\n0\u00170(C(t;u))R(s;u)du+Zt_s\n0C(s;u)\u001700(C(t;u))R(t;u)du: (3.65)\nInterchanging tandsin (3.65) and adding R(t;s) = 0 when s>t , results for s>t with\n@1C(s;t) =\u0000f0(K(s))C(s;t) +Zs\n0\u00170(C(s;u))R(t;u)du+Zs\n0C(t;u)\u001700(C(s;u))R(s;u)du+hM(t);\nwhich is (2.8) for \f= 1.\nNow, from (3.28), M(\u0001) is di\u000berentiable and M0(t) =h+P(t), hence combining the identity\nM(t)\u00170(C(t;t))\u0000M(0)\u00170(C(t;0)) =Zt\n0\u00170(C(t;u))M0(u)du\n+Zt\n0M(u)\u001700(C(t;u))@2C(t;u)du;\nwith (3.32) we have that for all t2[0;T],\n(3.66) P(t) =\u0000f0(K(t))M(t) +Zt\n0M(u)\u001700(C(t;u))R(t;u)du:\nthus showing is (2.9) for \f= 1.\nAlso, since from (3.31), @2Q(s;t) =H(s;t) +hM(s), from the identity\nQ(s;t_s)\u00170(C(t_s;t))\u0000Q(s;0)\u00170(C(0;t)) =Zt_s\n0\u00170(C(t;u))@2Q(s;u)du\n+Zt_s\n0Q(s;u)\u001700(C(t;u))@2C(t;u)du;SPHERICAL SPIN GLASSES 25\nwith (3.36) we have that for all t2[0;T],\n(3.67) X(s;t) =Zt_s\n0\u001700(C(t;u))R(t;u)Q(s;u)du:\nSimilarly,\n(3.68) Y(s;t) =Zt_s\n0\u00170(Q(t;u))R(s;u)du:\nthus showing is (2.9) for \f= 1.\nSinceK(s) =C(s;s), withC(s;t) =C(t;s) and@2C(t;s) =D(t;s) +R(t;s) +hM(t), it follows that for all\nk>0,\nK(s)\u0000K(s\u0000k) =Zs\ns\u0000k(D(s;u) +R(s;u) +hM(s))du\n+Zs\ns\u0000k(D(s\u0000k;u) +R(s\u0000k;u) +hM(s\u0000k))du:\nRecall that R(s;u) = 0 foru > s , hence, dividing by kand taking k#0, we thus get by the continuity\nofDand that of Rfors\u0015tthatK(\u0001) is di\u000berentiable, with @sK(s) = 2D(s;s) +R(s;s) + 2hM(s) =\n2D(s;s) + 1 + 2hM(s), resulting by (3.65) with (2.10) for \f= 1.\nFurther, it follows from (3.29) that @1\u001f(u;t) =E(u;t) + 1u<t. Hence, combining the identity\n\u001f(s;t)\u00170(C(s;s))\u0000\u001f(0;t)\u00170(C(s;0)) =Zs\n0\u00170(C(s;u))@1\u001f(u;t)du+Zs\n0\u001f(u;t)\u001700(C(s;u))@2C(s;u)du;\nwith (3.33) we have that for all T\u0015s\u0015t,\n(3.69) E(s;t) =\u0000f0(K(s))\u001f(s;t) +Zs\n0\u001f(u;t)\u001700(C(s;u))R(s;u)du\n(recall that \u001f(0;t) =\u001f(0;0) = 0). Let\n(3.70) g(s;t) :=\u0000f0(K(s))R(s;t) +Zs\n0R(u;t)\u001700(C(s;u))R(s;u)du;\nfors;t2[0;T]2. Recall that \u001f(s;t) =Rt\n0R(s;v)dv, so by Fubini's theorem, (3.69) amounts to E(s;t) =Rt\n0g(s;v)dvfor alls\u0015t. Further, with E(s;t) =E(s;s) whent>s , it follows that\nE(s;t) =Zt^s\n0g(s;v)dv\nfor alls;t\u0014T. Putting this into (3.29) we have by yet another application of Fubini's theorem that\nZt\n0R(s;u)du=\u001f(s;t) =t+Zs\n0Zt^u\n0g(u;v)dvdu =t+Zt\n0Zs\nvg(u;v)dudv;\nfor anys\u0015t. Consequently, for every t\u0014s,\nR(s;t) = 1 +Zs\ntg(u;t)du;\nimplying that @1R=gfor a.e.s>t , which in view of (3.70) gives (2.7) for \f= 1, thus completing the proof\nof the lemma. \u0003\nProof of Lemma 3.5. We shall show that the system (2.6){(2.10) with initial conditions C(t;t) =K(t),\nR(t;t) = 1,M(0) =\u000bandQ(0;0) =K(0) =C(0;0) =#admits at most one bounded solution ( M;R;C;Q;K )\non [0;T]\u0002[0;T]2\u0002(\u0000\\[0;T]2)\u0002[0;T]2\u0002[0;T]2. First notice that if we denote D(t) :=Q(t;t), by the symmetry of\nQ, we have@D(t) = 2@1Q(t;t). Now consider the di\u000berence between the integrated form of (2.6){(2.9) for two26 MANUEL ZAMFIR\nsuch solutions ( M;R;C;Q;K;D ) and ( \u0016M;\u0016R;\u0016C;\u0016Q;\u0016K;\u0016D) and de\fne the functions \u0001 V(s;t) =jV(s;t)\u0000\u0016V(s;t)j,\nwhenVis one of the functions C;R orQand \u0001U(s) =jU(s)\u0000\u0016U(s)j, whenUisM,DorK. Then, since \u001700\nis uniformly Lipschitz on any compact interval and C;Q; \u0016C;\u0016Qare continuous, hence bounded on [0 ;T]2, we\nhave, for 0\u0014t\u0014s\u0014T,\n\u0001M(t)\u0014\u00141\u0014Zt\n0\u0001M(v)dv+Zt\n0h(v)dv\u0015\n(3.71)\n\u0001R(s;t)\u0014\u00141\u0014Zs\nt\u0001R(v;t)dv+Zs\nth(v)dv\u0015\n(3.72)\n\u0001C(s;t)\u0014\u00141\u0014Zs\nt\u0001C(v;t)dv+Zs\nth(v)dv+ \u0001M(t) + \u0001K(t) +h(t)\u0015\n(3.73)\n\u0001Q(s;t)\u0014\u00141\u0014Zs\nt\u0001Q(v;t)dv+Zs\nth(v)dv+ \u0001M(t) + \u0001D(t) +h(t)\u0015\n(3.74)\n\u0001K(t)\u0014\u00141\u0014Zt\n0\u0001K(v)dv+Zt\n0h(v)dv+ \u0001M(t) +h(t)\u0015\n(3.75)\n\u0001D(t)\u0014\u00141\u0014Zt\n0\u0001D(v)dv+Zt\n0h(v)dv+ \u0001M(t) +h(t)\u0015\n(3.76)\nwhereh(v) :=Rv\n0[\u0001R(v;\u0012) + \u0001C(v;\u0012) + \u0001Q(v;\u0012) + \u0001M(\u0012) + \u0001D(\u0012) + \u0001K(\u0012)]d\u0012and\u00141<1depends on T,\n\f,\u0017(\u0001) and the maximum of jMj,jRj,jCj,jQj,j\u0016Mj,j\u0016Rj,j\u0016Cjandj\u0016Qjon [0;T]2. Integrating (3.71)-(3.76) over\nt2[0;s], since \u0001R(v;u) = 0 foru\u0015v, \u0001C(v;u) = \u0001C(u;v) and \u0001Q(v;u) = \u0001Q(u;v), we \fnd that\nZs\n0\u0001M(t)dt\u0014\u00142Zs\n0h(v)dv;\nZs\n0\u0001R(s;t)dt\u0014\u00142Zs\n0h(v)dv;\nZs\n0\u0001C(s;t)dt\u0014\u00142Zs\n0h(v)dv;\nZs\n0\u0001Q(s;t)dt\u0014\u00142Zs\n0h(v)dv;\nZs\n0\u0001K(t)dt\u0014\u00142Zs\n0h(v)dv;\nZs\n0\u0001D(t)dt\u0014\u00142Zs\n0h(v)dv;\nfor some \fnite constant \u00142(of the same type of dependence as \u00141). Summing the last three inequalities, we\nsee that for all s2[0;T],\n0\u0014h(s)\u0014\u00143Zs\n0h(v)dv:\nwhere\u00143= 6 maxf\u00141;\u00142g. Further,h(0) = 0, so by Gronwall's lemma h(s) = 0 for all s2[0;T]. Plugging this\nresult back into (3.71)-(3.76) and observing that \u0001 R(t;t) = \u0001K(0) = \u0001M(0) = \u0001D(0) = 0, \u0001C(t;t) = \u0001K(t)\nand \u0001Q(t;t) = \u0001D(t), we deduce that \u0001 R(s;t) = \u0001C(s;t) = \u0001M(t) = \u0001Q(s;t) = \u0001D(t) = \u0001K(s) = 0 for\nall 0\u0014t\u0014s\u0014T, hence, by symmetry, the stated uniqueness. \u0003\n4.Limiting Hard Spherical Dynamics\nThrough this section, we will \fx r >0 and, for convenience of notation, suppress the rdependence in the\nsubscripts.SPHERICAL SPIN GLASSES 27\nThe uniform bounds on the moments of KN(s) used to establish Proposition 3.2 (namely equation (3.16)),\nwill show that supt\u00150K(t)<1. Further, as C(s;t) is the limit of CN(s;t) =1\nNPN\ni=1xi\nsxi\nt, it is a non-negative\nde\fnite kernel on R+\u0002R+and in particular, C(s;t)2\u0014K(s)K(t) andC(t;t)\u00150. Also, since Q(s;t) is the\nlimit ofQN(s;t) =1\nNPN\ni=1x1;i\nsx2;i\nt, for two iid replicas x1\ntandx2\nt, by the Cauchy-Schwartz inequality and\nthen taking the limit as N!1 , we haveQ(s;t)2\u0014K(s)K(t).\nTo complete the proof of Theorem 2.3, we \frst prove that any solution ( M;R;C;Q;K ) of (2.6){(2.10)\nconsists of positive functions, a key fact in our forthcoming analysis.\nLemma 4.1. For anyf:R+!Rwhose derivative is bounded above on compact intervals and any K(0)>0,\nM(0)>0, a solution (M;R;C;Q;K )to(2.6) {(2.10) , if it exists, is positive at all times. Furthermore,\neC(s;t) :=C(s;t)\u0000M(s)M(t)is also non-negative.\nProof of Lemma 4.1. By de\fnition K(t)\u00150 for allt2R+. De\fne\nS1= inffu\u00150 :C(u;t)\u00140 for somet\u0014ug:\nand\nS2= inffu\u00150 :M(u)\u00140g:\nand suppose that S= minfS1;S2g<1. By continuity of ( C;K;Q ), sinceK(0)>0 andM(0)>0, also\nS1;S2>0, henceS >0. Set \u0003(s;t) = exp(\u0000Rs\nt\u0016(u)du)>0 for\u0016(u) =f0(K(u)) which is bounded above on\ncompact intervals, and R(s;t) = \u0003(s;t)H(s;t). Then, by [16], for s\u0015t,\n(4.1) H(s;t) = 1 +X\nn\u00151\f2nX\n\u001b2NCnZ\nt\u0014t1\u0001\u0001\u0001\u0014t2n\u0014sY\ni2cr(\u001b)\u001700(C(ti;t\u001b(i)))2nY\nj=1dtj\nwhere NC ndenotes the set of involutions of f1;\u0001\u0001\u0001;2ngwithout \fxed points and without crossings and cr( \u001b)\nis de\fned to be the set of indices 1 \u0014i\u00142nsuch thati<\u001b (i). Consequently,\nR(s;t)\u0015\u0003(s;t)>0 fort\u0014s\u0014S;\nand thus, (2.8) implies that\nC(s;t)\u0015K(t)\u0003(s;t)>0 fort\u0014s\u0014S:\nAlso, (2.6) implies that\nM(s)\u0015M(0)\u0003(s;0)>0 for 0\u0014s\u0014S:\nNote that in the last two estimates we used the fact that \u00170(\u0001) and\u001700(\u0001) are non negative on R+. Similarly,\nfrom the equation (2.10) we see that @[\u0003(s;0)\u00002K(s)]\u0015\u0003(s;0)\u00002for alls\u0014Sresulting with\nK(s)\u0015K(0)\u0003(s;0)\u00002+Zs\n0\u0003(s;v)\u00002dv> 0\nHence, the continuous functions R(s;t);C(s;t) andM(s) are bounded below by a strictly positive constant\nfor 0\u0014t\u0014s\u0014Sin contradiction with the de\fnition of S. We thus deduce that S=1, henceS1=S2=1\nand by the preceding argument and the symmetry of C, the functions R(s;t),C(s;t) andM(s) are positive.\nSimilarly, let S3= inffu\u00150 :Q(u;t)\u00140 for somet\u0014ugand assume S3<1. Then, from the symmetry\nofQ(s;t) =Q(t;s), de\fningD(t) :=Q(t;t), we have@D(t) = 2@1Q(t;t), hence by (2.9) we have:\nD(s)\u0015D(0)\u00032(s;0)>0 for 0\u0014s\u0014S3:\nand hence, using again (2.9):\nQ(s;t)\u0015Q(t;t)\u0003(s;t) =D(t)\u0003(s;t)>0 fort\u0014s\u0014S3:\nHence the continuous function Qis bounded below by a positive constant on 0 \u0014t\u0014s\u0014S3, contradiction to\nthe de\fnition of S3. HenceS3=1and by the symmetry of Q, it is positive on R2\n+. This concludes our proof\nthatM;R;C;Q;K are all positive functions.28 MANUEL ZAMFIR\nFurthermore, from (2.6) and (2.8), we know that eC(s;t) =C(s;t)\u0000M(s)M(t) satis\fes:\n@1eC(s;t) =\u0000f0(K(s))eC(s;t) +\f2Zs\n0eC(u;t)R(s;u)\u001700(C(s;u))du\n+\f2Zt\n0\u00170(C(s;u))R(t;u)du\nhence\neC(s;t)\u0015eC(t;t)\u0003(s;t)\u00150 fort\u0014s\u0014S\nsinceeC(t;t) =K(t)\u0000M2(t)\u00150. \u0003\nWe next show that if ( ML;r;RL;r;CL;r;QL;r;KL;r) are solutions of the system (2.6))-(2.10) with potential\nfL;r(\u0001) as in (2.12), then KL;r(s)!rasL!1 , uniformly over compact intervals. Speci\fcally,\nLemma 4.2. AssumingKL(0) =r, there exist L0>0such thatKL(s)\u0015r\u0000B0L\u00001, for some B0>0,\nfor allL > L 0ands\u00150. Further, for any T\fnite there exists B(T)<1(depending on r), such that\nKL(s)\u0014r+B(T)L\u00001for alls\u0014TandL\u0015maxfB(T);L0g.\nProof of Lemma 4.2. We \frst deal with the lower bound on KL(\u0001). FixL>0 and letgL(x) := 1\u00002xf0\nL(x) =\n1 + 4Lx(r\u0000x)\u0000\u0000x\nr\u00012k\u00002\u000bhx\nr. LetxLbe the largest root of gL(x) smaller than r. It is easy to see that\ngL(r)<0 and also that lim L!1gL(r=2)>0, so there exist L0>0 such that xL> r= 2 whenever L > L 0.\nFurthermore,\nL(r\u0000xL) =\u00001\n4xL+\u0010xL\nr\u00112k1\n4xL+2\u000bhxL\nr\u0014B0\nforB0= 4r\u00001+2\u000bh. By Lemma 4.1, we know that the functions RL(\u0001;\u0001),CL(\u0001;\u0001) andML(\u0001) are non negative,\nas is (x) forx\u00150, so from (2.10) we get the lower bound @KL(s)\u0015gL(KL(s)). SinceKL(0) =r, it follows\nthatKL(s)\u0015xL, for allx\u00150, soKL(s)\u0015r\u0000B0L\u00001, forL\u0015L0.\nTurning now to the complementary upper bound, recall that  (x) is a polynomial of degree m\u00001, hence\nthere exists \u0014<1such that (ab)\u0014\u0014(1 +a2)m=2(1 +b2)m=2for alla;b. Thus, by (2.11), the monotonicity\nof (x) onR+and the non-negative de\fniteness of CL(s;u) we have that for any s;t;u\u00150,\n (CL(s;u))\u0014\u0014(1 +KL(u))m\n2(1 +KL(s))m\n2\nandZt\n0RL(s;u)du\u0014p\ntKL(s); ML(t)\u0014p\nKL(t);\nand from (2.10) we \fnd that\n(4.2) @KL(s)\u0014g(KL(s)) + 2\f2\u0014\u0012\n1 + sup\nu\u0014sKL(u)\u0013mp\nKL(s)ps+ 2hp\nKL(s):\nSetting now B(T) =1\n2r\u0010\n1 + 2pr+ 1\f2\u0014(r+ 2)mp\nT+ 2pr+ 1h\u0011\nand \fxingT <1andL\u0015maxfL0;B(T)g,\nlet\n\u001c:= inffu\u00150 :KL(u)\u0015r+B(T)L\u00001g:\nBy the continuity of KL(\u0001) and the fact that KL(0) =r < r +B(T)L\u00001, we have that \u001c >0 and further, if\n\u001c <1then necessarily\nKL(\u001c) = sup\nu\u0014\u001cKL(u) =r+B(T)L\u00001\u0014r+ 1:\nRecall that gL(x)\u00141 + 4Lx(r\u0000x), whereas from (4.2) we see that if \u001c <1then\n@KL(\u001c)\u00141\u00004KL(\u001c)B(T) + 2p\nr+ 1\f2\u0014(r+ 2)mp\u001c+ 2p\nr+ 1h\n= 2rB(\u001c)\u00004KL(\u001c)B(T)\u00142rB(\u001c)\u00002rB(T):SPHERICAL SPIN GLASSES 29\nwhere the last inequality holds since L\u0015L0impliesKL(s)\u0015r=2, as previously shown. Recall the de\fnition\nof\u001c <1implying that @KL(\u001c)\u00150. Hence the above inequality implies B(\u001c)\u0015B(T), hence\u001c >T , for our\nchoice ofB=B(T). That is, KL(s)\u0014r+B(T)L\u00001for alls\u0014TandL\u0015maxfB(T);B0g, as claimed. \u0003\nLet\u0016L(s) =f0\nL(KL(s)),hL(s) =@KL(s). Fixing hereafter T <1(recallr>0 is \fxed) and denoting eL=\nmaxfL0;B(T)g, we next prove the equi-continuity and uniform boundedness of ( ML;RL;CL;QL;KL;\u0016L;hL),\nen-route to having limit points for ( ML;RL;CL;QL;KL).\nLemma 4.3. The continuous functions ML(s);KL(s);\u0016L(s);hL(s)and their derivatives are bounded uniformly\ninL\u0015eLand0\u0014s\u0014T. The same is true for CL(s;t);QL(s;t)inL\u0015eLand0\u0014s;t\u0014Tand also for\nRL(s;t)inL\u0015eLand0\u0014t\u0014s\u0014T.\nProof of Lemma 4.3. Recall that by Lemma 4.2, for any L\u0015eL,\n(4.3) sup\ns\u0014TjKL(s)\u0000rj\u0014eB\nL:\nwhereeB= maxfB(T);B0g. Consequently, the collections fCL(s;t);0\u0014s;t\u0014T;L\u0015eLgandfQL(s;t);0\u0014\ns;t\u0014T;L\u0015eLgare uniformly bounded (since both jCL(s;t)jandjQL(s;t)jare bounded above byp\nKL(s)KL(t))\nand alsofML(s);0\u0014s\u0014T;L\u0015eLg(sinceML(s)\u0014p\nKL(s)). By (4.3) and our choice of fL(r), we have that\nj\u0016L(s)j\u00142LjKL(s)\u0000rj+\u0012KL(s)\nr\u00132k\u00001\n\u00142eB+\u0012r+ 1\nr\u00132k\u00001\n;8L\u0015eL;s\u0014T:\nBy (4.1), the collection fHL(s;t);0\u0014t\u0014s\u0014T;L\u0015eLgis also uniformly bounded and since RL(s;t) =\nHL(s;t) exp\u0000\n\u0000Rs\nt\u0016L(u)du\u0001\n, the collectionfRL(s;t);0\u0014t\u0014s\u0014T;L\u0015eLgis also uniformly bounded.\nFurther, since by (2.10):\n(4.4) hL(s) = 1\u00002KL(s)\u0016L(s) + 2\f2Zs\n0 (CL(s;u))RL(s;u)du+ 2hML(s);\nit follows from the uniform boundedness of KL,ML,\u0016L,CLandRLthatfhL(s);s2[0;T];L\u0015eLgis\nalso uniformly bounded. By the same reasoning, from (2.6), (2.7), (2.8) and (2.9), we deduce that @ML(s),\n@1CL(s;t),@1RL(s;t),@1QL(s;t) and@DL(s) are bounded uniformly in L\u0015eLands;t2[0;T].\nNext, di\u000berentiating the identity (4.1) with respect to t, we get for f=fLthat\n@2HL(s;t) =X\nn\u00151\f2nX\n\u001b2NCnZ\nt=t1\u0014t2\u0001\u0001\u0001\u0014t2n\u0014sY\ni2cr(\u001b)\u001700(CL(ti;t\u001b(i)))2nY\nj=2dtj;\nwhereNCndenotes the \fnite set of non-crossing involutions of f1;:::; 2ngwithout \fxed points. With the\nCatalan number jNCnjbounded by 4n, and since CL(ti;t\u001b(i))2[0;r+ 1] forti;t\u001b(i)\u0014T,L\u0015eL, we thus\ndeduce by the monotonicity of x7!\u001700(x) that\n0\u0014@2HL(s;t)\u0014X\nn\u00151\f2n\n(2n\u00001)!4n(\u001700(r+ 1))n(s\u0000t)2n\u00001;\nso@2HL(s;t) is \fnite and bounded uniformly in L\u0015eLand 0\u0014t\u0014s\u0014T. Since\n@2RL(s;t) =\u0016L(t)RL(s;t) +e\u0000Rs\nt\u0016L(u)du@2HL(s;t);\nwe thus have that j@2RL(s;t)jis also bounded uniformly in L\u0015B(T) and 0\u0014t\u0014s\u0014T.\nAlso, due to the symmetry of CL,@2CL(s;t) =@1CL(t;s), hence@2CL(s;t) is also bounded uniformly in\nL\u0015eLand 0\u0014s;t\u0014T. The same argument applied to Q, will show that @2QL(s;t) is also bounded uniformly\ninL\u0015eLand 0\u0014s;t\u0014T.30 MANUEL ZAMFIR\nTurning to deal with @hL(s), settinggL(x) := [f0\nL(x)x]0\u00002rL= 4L(x\u0000r) +k\nr\u0000x\nr\u00012k\u00001+\u000bh\nr, we deduce\nfrom (4.3) thatjgL(KL(s))j\u00144eB+k\nr\u0000r+1\nr\u00012k\u00001+\u000bh\nrfor anys\u0014TandL\u0015eL. Di\u000berentiating (4.4) we \fnd\nthat@hL(s) =\u00004LrhL(s) +\u0014L(s) for\n\u0014L(s) =\u00002gL(KL(s))hL(s) + 2\f2@\n@s\u0012Zs\n0 (CL(s;u))RL(s;u)du\u0013\n+ 2h@ML(s);\nwhich is thus bounded uniformly in L\u0015B(T) ands\u0014T(in view of the uniform boundedness of hL,CL,\nRL,@1CL,@1RLand@ML). Further, recall that KL(0) =r, so by (2.10) and our choice of fL(\u0001) we have that\nhL(0) = 1\u00002rf0\nL(r) + 2h\u000b= 0, resulting with\nhL(s) =Zs\n0e\u00004Lr(s\u0000u)\u0014L(u)du:\nhence forL\u0015eL,\n(4.5) sup\ns\u0014TjhL(s)j\u0014supfj\u0014L(u)j:L\u0015eL;u\u0014Tg\n4Lr=A(T)\n4Lr<1;\nwhereA(T) := supfj\u0014L(u)j:L\u0015eL;u\u0014Tg<1and the uniform boundedness of j@hL(s)jfollows.\nFinally, by de\fnition, @\u0016L(s) =f00\nL(KL(s))hL(s), yielding for our choice of fLthat\nj@\u0016L(s)j\u0014 \n2L+2k\u00001\n2r2\u0012r+ 1\nr\u00132k\u00002!\njhL(s)j;8L\u0015eL;s\u0014T;\nwhich by (4.5) provides the uniform boundedness of j@\u0016L(s)j. \u0003\nProof of Theorem 2.3. In Lemma 4.3 we have established that the functions ( ML(s);RL(s;t);CL(s;t);QL(s;t)),\nL\u0015eLare equi-continuous and uniformly bounded on their respective domains for 0 \u0014s;t\u0014T. Further,\n(KL(s);\u0016L(s);hL(s)) are equi-continuous and uniformly bounded on s2[0;T]. By the Arzela-Ascoli theorem,\nthe collection ( ML;RL;CL;QL;KL;\u0016L;hL) has a limit point ( M;R;C;Q;K;\u0016;h ) with respect to uniform\nconvergence on [0 ;T]\u0002(\u0000\\[0;T]2)\u0002[0;T]7.\nBy Lemma 4.2 we know that the limit K(s) =rfor alls\u0014T, whereas by (4.5) we have that h(s) = 0 for all\ns\u0014T. Consequently, considering (4.4) for the subsequence Ln!1 for which (MLn;RLn;CLn;QLn;KLn;\u0016Ln;hLn)\nconverges to ( M;L;R;C;Q;K;\u0016;h ) we \fnd that the latter must satisfy (2.17) for k= 1. Further, re-\ncalling that RL(t;t) = 1,CL(t;t) =KL(t), integrating (2.6), (2.7), (2.8) and (2.9) we \fnd that ML(s) =\nML(0) +Rs\n0fML(\u0012)d\u0012andVL(s;t) =VL(t;t) +Rs\nteVL(\u0012;t)d\u0012, forVany of the functions R,CorQ, where:\nfML(\u0012) =\u0000\u0016L(\u0012)ML(\u0012) +\f2Z\u0012\n0ML(u)RL(\u0012;u)\u001700(CL(\u0012;u))du+h\neRL(\u0012;t) =\u0000\u0016L(\u0012)RL(\u0012;t) +\f2Z\u0012\ntRL(u;t)RL(\u0012;u)\u001700(CL(\u0012;u))du;\neCL(\u0012;t) =\u0000\u0016L(\u0012)CL(\u0012;t) +\f2Z\u0012\n0CL(u;t)RL(\u0012;u)\u001700(CL(\u0012;u))du\n+\f2Zt\n0\u00170(CL(\u0012;u))RL(t;u)du+hML(t);\neQL(\u0012;t) =\u0000\u0016L(\u0012)QL(\u0012;t) +\f2Z\u0012\n0QL(u;t)RL(\u0012;u)\u001700(CL(\u0012;u))du\n+\f2Zt\n0\u00170(QL(\u0012;u))RL(t;u)du+hML(t)SPHERICAL SPIN GLASSES 31\nSincefMLn,eRLn,eCLnandeQLnconverge uniformly on their domains, for 0 \u0014s;t\u0014T, to the right-hand-\nsides of (2.13), (2.14), (2.15) and (2.16), respectively, we deduce that for each limit point ( M;R;C;Q;\u0016 ), the\nfunctionsM(s),R(s;t),C(s;t) andQ(s;t) are di\u000berentiable in sin the region that they are de\fned and all\nlimit points satisfy the equations (2.13){(2.17). Further, since CL(s;t) andQL(s;t) are non-negative de\fnite\nsymmetric kernels, the same properties are inherited by their limits. Similarly, since RL(t;t) = 1 andRL(s;t)\nsatisfy (2.11), the same applies for any limit point R(s;t) and also since CL(t;t)!r, thenC(t;t) =r.\nUsing an argument similar to the one in Lemma 3.5, we show that there exist at most one bounded solution\n(M;R;C;Q ) inC1[0;T]\u0002C1(\u0000\\[0;T]2)\u0002C1\ns([0;T]2)\u0002C1\ns([0;T]2) to the system (2.13){(2.17), with initial\nconditionsC(t;t) =Q(0;0) =r,R(t;t) = 1 andM(0) =\u000bpr,\u000b2[0;1) (actually the uniqueness and the\nresult are true for any choice of starting points, however, it will not be relevant for us).\nIn conclusion, when L! 1 the collection ( ML;r;RL;r;CL;r;QL;r;KL;r) converges towards the unique\nsolution (Mr;Rr;Cr;Qr;Kr\u0011r) of (2.13){(2.17), as claimed. \u0003\n5.Convergence to the Pure Spin Model\nLet (Mr;Rr;Cr;Qr) be the solution of (2.13)-(2.17), for hr=hrm\u00001\n2and the initial conditions Rr(t;t) = 1,\nCr(t;t) =Qr(0;0) =r,Mr(0) =\u000bpr>0,\u000b2(0;1). Set:\ne\u0016r(s) =\u0016(sr1\u0000m=2)\nrm=2\u00001:\nand recall the de\fnitions used in Theorem 2.4:\nfMr(s) =Mr(sr1\u0000m=2)pr; eRr(s;t) =Rr(sr1\u0000m=2;tr1\u0000m=2)\neCr(s;t) =Cr(sr1\u0000m=2;tr1\u0000m=2)\nr; eQr(s;t) =Qr(sr1\u0000m=2;tr1\u0000m=2)\nr\nThe system (2.13)-(2.17) thus becomes:\n@fMr(s) =\u0000e\u0016r(s)fMr(s) +h+\f2Zs\n0fMr(u)eRr(s;u)\u001700(reCr(s;u))\nrm\u00002du; s\u00150 (5.1)\n@1eRr(s;t) =\u0000e\u0016r(s)eRr(s;t) +\f2Zs\nteRr(u;t)eRr(s;u)\u001700(reCr(s;u))\nrm\u00002du; s\u0015t\u00150 (5.2)\n@1eCr(s;t) =\u0000e\u0016r(s)eCr(s;t) +\f2Zs\n0eCr(u;t)eRr(s;u)\u001700(reCr(s;u))\nrm\u00002du (5.3)\n+\f2Zt\n0\u00170(reCr(s;u))\nrm\u00001eRr(t;u)du+hfMr(t); s \u0015t\u00150\n@1eQr(s;t) =\u0000e\u0016r(s)eQr(s;t) +\f2Zs\n0eQr(u;t)eRr(s;u)\u001700(reCr(s;u))\nrm\u00002du (5.4)\n+\f2Zt\n0\u00170(reQr(s;u))\nrm\u00001eRr(t;u)du+hfMr(t); s;t \u00150\nwhere\n(5.5) e\u0016r(s) =1\n2rm=2+\f2Zs\n0 (reCr(s;u))\nrm\u00001eRr(s;u)du+hfMr(s):\nandeCr(t;t) =eRr(t;t) =eQr(0;0) = 1,fMr(0) =\u000b,eCr(t;s) =eCr(s;t) andeQr(t;s) =eQr(s;t).\nFixingT <1, the \frst step of the proof is to establish, in Lemma 5.1, that the function fMr,eRr,eCr,eQr\nande\u0016rare equi-continuous and uniformly bounded. Then we will be able to use Arzela-Ascoli theorem to\nestablish the desired limits.32 MANUEL ZAMFIR\nLemma 5.1. The continuous functions fMr(s);e\u0016r(s);eCr(s;t)andeQr(s;t)and their derivatives are uniformly\nbounded in r\u00151and0\u0014s;t\u0014T. The same is true for eRr(s;t)inr\u00151and0\u0014t\u0014s\u0014T.\nProof of Lemma 5.1. Recall Theorem 2.3 implies that Cr(s;t),Qr(s;t),M2\nr(s)2[0;r], for all 0\u0014s;t\u0014T.\nHence, by construction, eCr(s;t),eQr(s;t) andfM2\nr(s) take values in the interval [0 ;1], for every r >0, thus\nshowing the uniform boundedness of eCr(s;t);eQr(s;t) andfMr(s) on 0\u0014s;t\u0014Tandr\u00151.\nAlso notice that eRr(s;t) =eHr(s;t) exp(\u0000Rs\nte\u0016r(u)du), fors\u0015t, where, by [16], eHr(s;t) satis\fes:\n(5.6) eHr(s;t) = 1 +X\nn\u00151\f2nX\n\u001b2NCnZ\nt\u0014t1\u0001\u0001\u0001\u0014t2n\u0014sY\ni2cr(\u001b)\u001700(reCr(ti;t\u001b(i)))\nrm\u000022nY\nj=1dtj\nSince\u001700(x) is a polynomial of degree m\u00002, there exist an universal constant K1(depending on \u001700) such\nthat, for any r\u00151, andx2[0;1],\u001700(rx)\nrm\u00002<K 1. Hence the collectionn\neHr(s;t);0\u0014t\u0014s\u0014T;r\u00151o\nis also\nuniformly bounded (since eCr(ti;t\u001b(i))2[0;1]). Sincee\u0016r(s)\u00150, for allrands, theneRr(s;t)\u0014eHr(s;t). Since\neRr(s;t)\u00150, for alls;t, the uniform boundedness ofn\neRr(s;t);0\u0014t\u0014s\u0014T;r\u00151o\nis established.\nSince (x) is a polynomial of degree m\u00001, there exist an universal constant K2(depending on  ) such\nthat, for any r\u00151, andx2[0;1], (rx)\nrm\u00001< K 2. Since in addition e\u0016r(s)\u00150, (5.5) implies that the family\nfe\u0016r(s);0\u0014s\u0014T;r\u00151gis uniformly bounded.\nMoving over to the partial derivatives, since by (5.1):\n@fMr(s) =\u0000e\u0016r(s)fMr(s) +\f2Zs\n0fMr(u)eRr(s;u)\u001700(reCr(s;u))\nrm\u00002du+h\nit follows from the uniform boundedness of e\u0016r,fMr,eRrandeCrthat the familyn\n@fMr(s);0\u0014s\u0014T;r\u00151o\nis\nalso uniformly bounded. By similar reasoning, using (5.2), (5.3) and (5.4), we show that @1eRr(s;t),@1eCr(s;t)\nand@1eQr(s;t) are uniformly bounded in r\u00151 and 0\u0014t\u0014s\u0014T(ors;t2[0;T], whichever is relevant).\nNow, di\u000berentiating the identity (5.6) with respect to t, we get\n@2eHr(s;t) =X\nn\u00151\f2nX\n\u001b2NCnZ\nt=t1\u0014t2\u0001\u0001\u0001\u0014t2n\u0014sY\ni2cr(\u001b)\u001700(reCr(ti;t\u001b(i)))\nrm\u000022nY\nj=2dtj;\nwhereNCndenotes the \fnite set of non-crossing involutions of f1;:::; 2ngwithout \fxed points. With the\nCatalan numberjNCnjbounded by 4n, and since 0\u0014\u001700(reCr(ti;t\u001b(i)))\nrm\u00002\u0014K1for 0\u0014ti;t\u001b(i)\u0014Tandr\u00151, we\nthus deduce that\n0\u0014@2eHr(s;t)\u0014X\nn\u00151\f2n\n(2n\u00001)!(4K1)n(s\u0000t)2n\u00001;\nso@2eHr(s;t) is \fnite and bounded uniformly when r\u00151 and 0\u0014t\u0014s\u0014T. Since\n@2eRr(s;t) =e\u0016r(t)eRr(s;t) +e\u0000Rs\nte\u0016r(u)du@2eHr(s;t);\nwe thus have that j@2eRr(s;t)jis also bounded uniformly in r\u00151 and 0\u0014t\u0014s\u0014T. Also, since eCris\nsymmetric, @2eCr(s;t) =@1eCr(t;s), hence@2eCr(s;t) is also bounded uniformly in r\u00151 ands;t2[0;T]. Since\neQris also symmetric, we derive the same conclusion about @2eQr(s;t).\nFinally, by (5.5),\n@e\u0016r(s) =\f2Zs\n0\"\n 0(reCr(s;u))\nrm\u00002@1eCr(s;u)eRr(s;u) + (reCr(s;u))\nrm\u00001@1eRr(s;u)#\ndu+ (r)\nrm\u00001+h@fMr(s):SPHERICAL SPIN GLASSES 33\nand since (x) is a polynomial of order m\u00001, it follows that  (rx)=rm\u00001and 0(rx)=rm\u00002are uniformly\nbounded in r\u00151,x2[0;1]. Since@1eCr,@1eRr,@fMrandeRrare uniformly bounded and eCr(s;u)2[0;1], it\nfollows that the functions @e\u0016r(s) are uniformly bounded on 0 \u0014s\u0014Tandr\u00151, thus concluding the proof.\n\u0003\nProof of Theorem 2.4. In Lemma 5.1 we have established that the functions fMr(s),eRr(s;t),eCr(s;t),\neQr(s;t) ande\u0016r(s) are equi-continuous and uniformly bounded for r\u00151. By the Arzela-Ascoli theorem,\nthe collection ( fMr;eRr;eCr;eQr;e\u0016r) has a limit point ( M;R;C;Q;\u0016 ) with respect to uniform convergence on\nC1[0;T]\u0002C1(\u0000\\[0;T]2)\u0002C1\ns([0;T]2)\u0002C1\ns([0;T]2)\u0002C1[0;T]. Letrnbe an increasing sequence going to in\fnity,\nsuch that (fMrn;eRrn;eCrn;eQrn;e\u0016rn) converges uniformly to ( M;R;C;Q;\u0016 ).\nNow, sinceeCr(\u0012;u)2[0;1], for allr\u00151 and\u0012;u\u00150, the same is true for its limit point C(\u0012;u). Since\n\u0017(\u0001) is a polynomial of degree mwith the dominant coe\u000ecienta2\nm\nm!, (x) =\u00170(x) +x\u001700(x) is a degree m\u00001\npolynomial with dominant coe\u000ecienta2\nm\n(m\u00001)!+a2\nm\n(m\u00002)!. Recalling that e (x) =h\na2\nm\n(m\u00001)!+a2\nm\n(m\u00002)!i\nxm\u00001, we can\neasily see that there exist constant K3(depending only on \u0017(\u0001)), such that\nsup\n0\u0014\u0012;r\u0014T\f\f\f\f (rC(\u0012;u))\nrm\u00001\u0000e (C(\u0012;u))\f\f\f\f\u0014K3\nr\nAlso, sinceeCr;C2[0;1], there exist K4such that\n\f\f\f\f\f (reCr(\u0012;u))\nrm\u00001\u0000 (rC(\u0012;u))\nrm\u00001\f\f\f\f\f\u0014K4jeCr(\u0012;u)\u0000C(\u0012;u)j\u0014K4keCr\u0000Ck1;\nfor everyr. Altogether, we have shown that:\n (rneCrn(\u0012;u))\nrm\u00001nn!1\u0000!e (C(\u0012;u));\nand the convergence is uniform on [0 ;T]2. Using this result, together with the uniform convergence of eCrn;eRrn\nandfMrn, we conclude that e\u0016rn(s), as it is de\fned in (5.5), converges to the right hand side of (2.17) for k= 0\nand the convergence is uniform on [0 ;T].\nFurthermore, since eRr(t;t) = 1,eCr(t;t) = 1,fMr(0) =\u000bandeQr(0;0) = 1 integrating (5.1), (5.2), (5.3) and\n(5.4) we \fnd that fMr(s) =\u000b+Rs\n0\u0016Mr(\u0012)d\u0012andeVr(s;t) =eVr(t;t) +Rs\nt\u0016Vr(\u0012;t)d\u0012, forVany of the functions\nR,CorQ, for:\n\u0016Mr(\u0012) =\u0000e\u0016r(\u0012)fMr(\u0012) +\f2Z\u0012\n0fMr(u)eRr(\u0012;u)\u001700(reCr(\u0012;u))\nrm\u00002du+h\n\u0016Rr(\u0012;t) =\u0000e\u0016r(\u0012)eRr(\u0012;t) +\f2Z\u0012\nteRr(u;t)eRr(\u0012;u)\u001700(reCr(\u0012;u))\nrm\u00002du;\n\u0016Cr(\u0012;t) =\u0000e\u0016r(\u0012)eCr(\u0012;t) +\f2Z\u0012\n0eCr(u;t)eRr(\u0012;u)\u001700(reCr(\u0012;u))\nrm\u00002du\n+\f2Zt\n0\u00170(reCr(\u0012;u))\nrm\u00001eRr(t;u)du+hfMr(t)\n\u0016Qr(\u0012;t) =\u0000e\u0016r(\u0012)eQr(\u0012;t) +\f2Z\u0012\n0eQr(u;t)eRr(\u0012;u)\u001700(reCr(\u0012;u))\nrm\u00002du\n+\f2Zt\n0\u00170(reQr(\u0012;u))\nrm\u00001eRr(t;u)du+hfMr(t)34 MANUEL ZAMFIR\nSimilar arguments as employed earlier will show that:\n\u001700(reCrn(\u0012;u))\nrm\u00002nn!1\u0000!e\u001700(C(\u0012;u));\u00170(reCrn(\u0012;u))\nrm\u00001nn!1\u0000!e\u00170(C(\u0012;u));\nand the same for \u001700(Q(\u0012;u)) and\u00170(Q(\u0012;u)), where the convergence is uniform on [0 ;T]2. Using this re-\nsult, together with the uniform convergence of the quad-uple ( fMrn;eRrn;eCrn;e\u0016rn), we conclude that \u0016Mrn(s)\nconverges to the right hand side of (2.13) and the convergence is uniform on [0 ;T].\nSimilarly we show that \u0016Rrn(s;t),\u0016Crn(s) and \u0016Qrn(s;t) converge uniformly on ( s;t)2[0;T]2to the right\nhand sides of (2.14), (2.15) and (2.16), respectively. Thus, we see that for each limit point ( M;R;C;Q;\u0016 ), the\nfunctionsM(s),R(s;t),C(s;t) andQ(s;t) are di\u000berentiable in son 0\u0014s;t\u0014Tand all limit points satisfy\nthe equations (2.13){(2.17). Since eRrn(t;t) = 1 and the functions eQrnandeCrnare non-negative de\fnite\nsymmetric kernels, the same applies for any limit point R,QorC.\nFinally, using a Gromwell-type argument similar to the one employed in Lemma 3.5, we show that there\nexist at most one bounded solution ( M;R;C;Q ) onC1[0;T]\u0002C1(\u0000\\[0;T]2)\u0002C1\ns([0;T]2)\u0002C1\ns([0;T]2) to the\nsystem (2.13){(2.17), with initial conditions C(t;t) =R(t;t) =Q(0;0) = 1 and M(0) =\u000b2(0;1).\nIn conclusion, when r! 1 the collection ( fMr;eRr;eCr;eQr;e\u0016r) converges towards the unique solution\n(M;R;C;Q;\u0016 ) of (2.13){(2.17), as claimed. \u0003\n6.FDT regime\n6.1.Proof Preliminaries. The arguments that are used for the cases \fandhsmall and, respectively, \r\nsmall, are very similar, and we will be treating them in parallel. On the high level, we will use a perturbation\nargument based on the stability of linear and respectively Ricatti di\u000berential equations. From now on, we will\nrefer to the case when \r=\f\nhis small as the \frst case and when both \fandhare small as the second case .\nFirst, notice that, since r= 1, making the substitution Uh(s;t) =U(s=h;t=h ), forUany ofR;C orQand\nVh(s) =V(s=h) forVany ofMorD, the equations (2.13)-(2.17) are transformed to:\n@Mh(s) =\u0000\u0016h(s)Mh(s) + 1 +\r2Zs\n0Mh(u)Rh(s;u)\u001700(Ch(s;u))du; s\u00150 (6.1)\n@1Rh(s;t) =\u0000\u0016h(s)Rh(s;t) +\r2Zs\ntRh(u;t)Rh(s;u)\u001700(Ch(s;u))du; s\u0015t\u00150 (6.2)\n@1Ch(s;t) =\u0000\u0016h(s)Ch(s;t) +\r2Zs\n0Ch(u;t)Rh(s;u)\u001700(Ch(s;u))du (6.3)\n+\r2Zt\n0\u00170(Ch(s;u))Rh(t;u)du+Mh(t); s \u0015t\u00150\n@1Qh(s;t) =\u0000\u0016h(s)Qh(s;t) +\r2Zs\n0Qh(u;t)Rh(s;u)\u001700(Ch(s;u))du (6.4)\n+\r2Zt\n0\u00170(Qh(s;u))Rh(t;u)du+Mh(t); s;t \u00150\nwith\n(6.5) \u0016h(s) =1\n2h+Mh(s) +\r2Zs\n0 (Ch(s;u))Rh(s;u)du:\nFrom now on, we will be interested in the behavior of the functions C(s;t) andQ(s;t) only fors\u0015t(the\nrest of the plane will be automatically given, by symmetry). We do need, however, to specify initial conditions\nforQ(\u0001;\u0001). De\fning D(s) :=Q(s;s), due to the symmetry of Q, the function Dwill satisfy@D(s) = 2@1Q(s;s),SPHERICAL SPIN GLASSES 35\nhence:\n@D(s)\n2=\u0000\u0016(s)D(s) +hM(s) +\f2Zs\n0Q(u;s)R(s;u)\u001700(C(s;u))du (6.6)\n+\f2Zs\n0\u00170(Q(s;u))R(s;u)du\nhence it's time transform, Dh(s) :=Qh(s;s) solves:\n@Dh(s)\n2=\u0000\u0016h(s)Dh(s) +Mh(s) +\r2Zs\n0Qh(u;s)Rh(s;u)\u001700(Ch(s;u))du (6.7)\n+\r2Zs\n0\u00170(Qh(s;u))Rh(s;u)du;\nIn the course of the proof, we will establish that, when either \ris small or both \fandhare small, the\nlimits:\nMfdt= lim\nt!1M(t); (6.8)\nRfdt(\u001c) = lim\nt!1R(t+\u001c;t); (6.9)\nCfdt(\u001c) = lim\nt!1C(t+\u001c;t); (6.10)\nQfdt(\u001c) = lim\nt!1Q(t+\u001c;t); (6.11)\nare well-de\fned for \u001c\u00150 and, furthermore, that Rfdtdecays to 0 exponentially fast (i.e. 0 \u0014Rfdt(\u001c)\u0014\nK1e\u0000K2\u001c), for some positive constants K1andK2depending on \f,hand\u000b=M(0).\nNotice that if the FDT limits exist for the functions Mh;Rh;ChandQh, the same is true for the functions\nM;R;C andQ. We will establish (6.8)-(6.11) for ( Mh;Rh;Ch;Qh), in the \frst case, and for ( M;R;C;Q ) in\nthe second. Also, until further notice, we will drop the hsubscript in the regime when \ris small (i.e. the \frst\ncase).\nRecalling our notation \u0000=f(s;t) : 0\u0014t\u0014sg\u001aR+\u0002R+, consider the maps \t i: (M;R;C;Q )7!\n(fMi;eRi;eCi;eQi),i= 1;2, on\nA=\b\n(M;R;C;Q )2C1(R+)\u0002C1(\u0000)\u0002C1\ns(R2\n+)\u0002C1\ns(R2\n+)jM(0) =\u000b2(0;1];\nR(t;t) =C(t;t) =Q(0;0) = 1; C(s;t) =C(t;s); Q(s;t) =Q(t;s)\t\n;\nsuch that for s\u00150,\n@fM1(s) =\u0000\u00121\n2h+fM1(s)\u0013\nfM1(s) + 1 (6.12)\n+\r2\u0012Zs\n0M(u)R(s;u)\u001700(C(s;u))du\u0000M(s)Zs\n0 (C(s;u))R(s;u)du\u0013\n@fM2(s) =\u0000\u00162(s)fM2(s) +h+\f2Zs\n0M(u)R(s;u)\u001700(C(s;u)du (6.13)\nand fors\u0015t\u00150:\n@1eRi(s;t) =\u0000\u0016i(s)eRi(s;t) +\u000f2\niZs\nteRi(u;t)eRi(s;u)\u001700(C(s;u))du; (6.14)\n@1eCi(s;t) =\u0000\u0016i(s)eCi(s;t) +kifMi(t) (6.15)\n+\u000f2\ni\u0012Zs\n0C(u;t)R(s;u)\u001700(C(s;u))du+Zt\n0\u00170(C(s;u))R(t;u)du\u0013\n;36 MANUEL ZAMFIR\n@1eQi(s;t) =\u0000\u0016i(s)eQi(s;t) +kifMi(t) (6.16)\n+\u000f2\ni\u0012Zs\n0Q(u;t)R(s;u)\u001700(C(s;u))du+Zt\n0\u00170(Q(s;u))R(t;u)du\u0013\n;\nwith initial conditions eRi(t;t) =eCi(t;t) =eQi(0;0) = 1,eDi(t) :=eQi(t;t) and symmetry conditions eCi(t;s) =\neCi(s;t) andeQi(t;s) =eQi(s;t), whereeDisatis\fes:\n@eDi(s)\n2=\u0000\u0016i(s)eDi(s) +kifMi(t) (6.17)\n+\u000f2\ni\u0012Zs\n0Q(u;s)R(s;u)\u001700(C(s;u))du+Zs\n0\u00170(Q(s;u))R(s;u)du\u0013\nand\n\u00161(s) =!1(s) +\r2Zs\n0 (C(s;u))R(s;u)du =1\n2h+fM1(s) +\r2Zs\n0 (C(s;u))R(s;u)du (6.18)\n\u00162(s) =!2(s) +\f2Zs\n0 (C(s;u))R(s;u)du =1\n2+hM(s) +\f2Zs\n0 (C(s;u))R(s;u)du (6.19)\nandk1= 1,k2=h,\u000f1=\r,\u000f2=\fand the functions !1(s),!2(s) are de\fned implicitly above.\nAssuming ( M;R;C;Q )2A, then both the Ricatti equation, (6.12) and the linear one, (6.13) have unique\nsolutions inC(R+) for the initial conditions fMi(0) =\u000b. Thus,\u0016i(s) are continuous and further, by [16] there\nexists a unique non-negative solution eRi(s;t) of (6.14) which is continuous on \u0000(see for example (4.1) for\nexistence, uniqueness and non-negativity of the solution, and the proof of Lemma 4.3 for the di\u000berentiability,\nhence continuity of eRi(s;t)). WithC,RandfMicontinuous, clearly there is also a unique solution eCi(s;t) to\n(6.15) which is continuous on \u0000and due to the boundary condition eCi(t;t) = 1, its symmetric extension to\nR+\u0002R+remains continuous. By the same reasoning, there exist an unique solution eDi(s) to (6.17), hence\nalso an unique solution eQi(s;t) to (6.16) de\fned on \u0000with boundary condition eQi(t;t) =eDi(t). Furthermore,\nby the boundary conditions, its symmetric extension to R+\u0002R+is di\u000berentiable, hence continuous. Thus, \t i\nis well-de\fned and \t i(A)\u001aA.\nNotice that the solution ( Mh;Rh;Ch;Qh) of (6.1)-(6.5) is a \fxed point of the mapping \t 1and also that\nthe solution ( M;R;C;Q ) of (2.13)-(2.17) is a \fxed point of the mapping \t 2. We will show that, for su\u000eciently\nsmall\r=\f\nh, any \fxed point of \t 1is in the spaceS(\u000e;\u001a;a;d ) and also, for su\u000eciently small \fandh, any \fxed\npoint of \t 2is in the same space, for a suitable choice of constants \u000e;\u001a;a;d , independent of \fandh. Here:\nS(\u000e;\u001a;a;d ) =f(M;R;C;Q )2B(\u000e;\u001a;a;d ) :8\u001c\u00150;9Rfdt(\u001c) = lim\nt!1R(t+\u001c;t);\n9Cfdt(\u0000\u001c) =Cfdt(\u001c) = lim\nt!1C(t+\u001c;t);9Qfdt(\u0000\u001c) =Qfdt(\u001c) = lim\nt!1Q(t+\u001c;t);\n9Mfdt= lim\nt!1M(t);9Q1= lim\nt!1Q(t;0)g;\nand\nB(\u000e;\u001a;a;d ) =f(M;R;C;Q )2A: 0\u0014C(s;t);Q(s;t)\u0014d;0\u0014R(s;t)\u0014\u001ae\u0000\u000e(s\u0000t);\n0\u0014Q(s;s)\u0014d\n2;0\u0014M(s)\u0014a;for alls\u0015tg:SPHERICAL SPIN GLASSES 37\nThis of course will imply that the FDT limits (6.8)-(6.11) exist and are in the space:\nD(\u000e;\u001a;a;d ) =f(M;R;C;Q ) :R;C;Q :R!R; M2R+;\nC(\u001c) =C(\u0000\u001c); Q (\u001c) =Q(\u0000\u001c);0\u0014C(\u001c);Q(\u001c)\u0014d;\n0\u0014R(\u001c)\u0014\u001ae\u0000\u000eh\u001c;0\u0014Q(0)\u0014d\n2;0\u0014M\u0014a; for all\u001c\u00150g:\n6.2.Invariant Spaces. We will begin by \fnding constants ( \u000e;\u001a;a;d ) such thatS(\u000e;\u001a;a;d ) is invariant under\nthe mapping \t i.\nProposition 6.1. There exist \r1,\f1andh1, depending only on \u000b, and a positive, universal constant c1, such\nthat for our choice of constants a=q\n7\n4,b= min\b\n\u000b;1\n2\t\n,\u001a=c1,\u000e=b\n2andd= 2 maxn\n1 +2(a+1)\nb2;a+1\nbo\n, if\n\r:=\f\nh<\r1andi= 1or\f <\f 1,h<h 1andi= 2, then\n(6.20) \t i(B(\u000e;\u001a;a;d ))\u001aB(\u000e;\u001a;a;d );\nand\n(6.21) \t i(S(\u000e;\u001a;a;d ))\u001aS(\u000e;\u001a;a;d ):\nFurthermore, under the same conditions, if (M;R;C;Q )2B(\u000e;\u001a;a;d ), then for every s\u00150:\n(6.22) \u0016i(s)\u0015!i(s)\u0015b>0\nProof of Proposition 6.1: We will start by verifying that (6.20) holds.\nWe will \frst be dealing with the bounds on fMi. Here, due to the di\u000berent nature of the equations (6.12)\nand (6.13) (Ricatti, respectively linear), our analysis will be di\u000berent. Indeed (6.12) is equivalent to:\n@fM1(s) =\u0000\u0010\nfM1(s)\u00112\n\u0000fM1(s)\n2h+ 1 +\r2(I0(s)\u0000I1(s))\nfor\nI0(s) =Zs\n0M(u)R(s;u)\u001700(C(s;u)du (6.23)\nI1(s) =M(s)Zs\n0 (C(s;u))R(s;u)du (6.24)\nSince (M;R;C;Q )2B(\u000e;\u001a;a;d ), then we have the bounds:\n(6.25) \r2jI0(s)\u0000I1(s)j\u0014\r2(jI0(s)j+jI1(s)j)\u0014\r2\u0014a\u001700(d)\u001a\n\u000e+a (d)\u001a\n\u000e\u0015\n\u00143\n4\nfor\rsu\u000eciently small. For k= 1;2, de\fneM1;k(\u0001), to be the unique solutions to the Ricatti di\u000berential\nequations:\n@M1;k(s) =\u0000(M1;k(s))2\u0000M1;k(s)\n2h+ 1 + (\u00001)k3\n4; M 1;k(0) =\u000b\nSince@M1;1(s)\u0014@M(s)\u0014@M1;2(s), for every sand all three functions start at the same point, we can\nsandwichfM1(\u0001) betweenM1;1(\u0001) andM1;2(\u0001), hence:\n(6.26) inf\ns2[0;1)M1;1(s)\u0014M1;1(s)\u0014fM1(s)\u0014M1;2(s)\u0014sup\ns2[0;1)M1;2(s)\nDe\fne the polynomial P2(x) =\u0000\u0000\nx2+x\n2h\u00007\n4\u0001\n. Since its only positive root is x2=\u00001\n4h+q\n1\n(4h)2+7\n4<q\n7\n4\nandM1;2(0) =\u000b2(0;1), a sign analysis of @M1;2will show that M1;2must be monotonic on [0 ;1) and38 MANUEL ZAMFIR\nlim\nt!1M1;2(t) =x2. So\nsup\ns2[0;1)M1;2(s)\u0014maxf\u000b;x2g\u0014max(\n\u000b;r\n7\n4)\n<r\n7\n4\nwhich, together with (6.26) establishes the upper bound on fM1:\n(6.27) fM1(s)\u0014r\n7\n4=a\nDe\fne also the polynomial P1(x) =\u0000\u0000\nx2+x\n2h\u00001\n4\u0001\n. Again, its only positive root is x1=\u00001\n4h+q\n1\n(4h)2+1\n4>\n0 and since M1;1(0) =\u000b2(0;1), analyzing the sign of @M1;1, we conclude that M1;1is monotonic on [0 ;1)\nand lim\nt!1M1;1(t) =x1, so:\ninf\n\u001b2[0;1)M1;1(s)\u0015minf\u000b;x2g>0\nCombining the above inequality with (6.26) and (6.27) we will \fnish establishing the desired bounds on fM1:\n(6.28) 0 \u0014fM1(s)\u0014a\nThe bound on !1will follows suit:\n!1(s) =1\n2h+fM1(s)\u00151\n2h+ inf\ns2[0;1)M1;1(s) (6.29)\n\u0015min(\n\u000b+1\n2h;1\n4h+s\n1\n(4h)2+1\n4)\n>min\u001a\n\u000b;1\n2\u001b\n=b\nFurthermore, since CandRare positive, we are done proving (6.22) for i= 1.\nNow, turning our attention towards M2(s), \frst de\fne, for i= 1;2:\n(6.30) \u0003 i(s;t) =e\u0000Rs\nt\u0016i(u)du\u00150;\nSolving the linear equation (6.13) (recall fM2(0) =\u000b), we obtain:\n(6.31) fM2(s) =\u000b\u00032(s;0) +\f2Zs\n0I0(u)\u00032(s;u)du+hZs\n0\u00032(s;u)du\nwithI0de\fned in (6.23). Since \u000b > 0 andM;R;C are positive, then the RHS above is positive, hence\nfM2(s)\u00150. This implies \u00162(s)\u0015!2(s)\u00151\n2\u0015b, proving (6.22) for i= 2 and consequently \u0003 i(s;t)\u0014\nexp(\u0000b(s\u0000t)). Also, since ( M;R;C;Q )2B(\u000e;\u001a;a;d ),I0(u) is positive and bounded above uniformly by\na\u001700(d)\u001a\n\u000e, hence recalling that \u000b<1, we obtain the desired upper bound on fM2:\nfM2(s)\u00141 +\f2a\u001700(d)\u001a\nb\u000e+h1\nb\u0014r\n7\n4=a\nholding for h;\fsmall enough, as claimed.\nConsidering next the functions eRi, leteRi(s;t) = \u0003i(s;t)eHi(s;t), where \u0003 iis de\fned as in (6.30), with\neHi(t;t) = 1. Further, from [16] we have that for any ( s;t)2\u0000,\n(6.32) eHi(s;t) = 1 +X\nn\u00151\u000f2n\niX\n\u001b2NCnZ\nt\u0014t1\u0001\u0001\u0001\u0014t2n\u0014sY\nk2cr(\u001b)\u001700(C(tk;t\u001bk))2nY\nj=1dtj:SPHERICAL SPIN GLASSES 39\nConsequently, since jNCnj= (2\u0019)\u00001R2\n\u00002x2np\n4\u0000x2dxandC(u;v)2[0;d], by the de\fnition of B(\u000e;\u001a;a;d ),\nwe can bound eHi:\neHi(s;t)\u0014X\nn\u00150\u0000\n\u000f2\ni\u001700(d)\u0001nX\n\u001b2NCnZ\nt\u0014t1\u0014\u0001\u0001\u0001\u0014t2n\u0014s2nY\nj=1dtj (6.33)\n=X\nn\u00150(\u000f2\ni\u001700(d))n(s\u0000t)2n\n(2n!)(2\u0019)\u00001Z2\n\u00002x2np\n4\u0000x2dx\n= (2\u0019)\u00001Z2\n\u00002e\u000fip\n\u001700(d)(s\u0000t)xp\n4\u0000x2dx:\nIt is well known (see for example [5, (3.8)]) that for some universal constant 1 \u0014c1<1and all\u0012,\n(2\u0019)\u00001Z2\n\u00002e\u0012xp\n4\u0000x2dx\u0014c1(1 +j\u0012j)\u00003=2e2j\u0012j;\nfrom which we thus deduce that:\n(6.34) eHi(s;t)\u0014c1\u0010\n1 +\u000fip\n\u001700(d)(s\u0000t)\u0011\u00003=2\ne2\u000fip\n\u001700(d)(s\u0000t)\u0014c1e2\u000fip\n\u001700(d)(s\u0000t):\nFurther, since ( M;R;C;Q )2B(\u000e;\u001a;a;d ) and \u0003i(s;t)\u0014e\u0000b(s\u0000t), then for\u000fi\u0014b\n4p\n\u001700(d)and for our choice of\n\u001a=c1, and\u000e=b\n2\u0014b\u00002\u000fip\n\u001700(d), we can establish the desired upper bound on eRi:\neRi(s;t)\u0014c1e\u0000\u0010\n\u0000b+2\u000fip\n\u001700(d)\u0011\n(s\u0000t)\u0014\u001ae\u0000\u000e(s\u0000t): (6.35)\nFinally, since \u0003 i>0 andeHi>0 (sinceC\u00150), the lower bound on eRifollows:\n(6.36) eRi(s;t)\u00150\nConsidering next the function eCi, recall thateCi(t;t) = 1, hence solving the linear equation (6.15), we get,\nfor (s;t)2\u0000:\neCi(s;t) = \u0003i(s;t) +\u000f2\niZs\nt\u0003i(s;v)I2(v;t)dt+\u000f2\niZs\nt\u0003i(s;v)I3(v;t)dt+kifMi(t)Zs\nt\u0003i(s;v)dv (6.37)\nwhere\nI2(v;t) =Zv\n0C(u;t)R(v;u)\u001700(C(v;u))du (6.38)\nI3(v;t) =Zt\n0\u00170(C(v;u))R(t;u)du (6.39)\nSince \u0003i;C;R andfMiare positive, then I2(v;t);I3(v;t)\u00150 (recall\u0017is a polynomial with positive coe\u000ecients).\nHence the lower bound on eCifollows easily from (6.37):\n(6.40) eCi(s;t)\u00150\nNow, for the upper bound, since ( M;R;C;Q )2B(\u000e;\u001a;a;d ),I2andI3are bounded above, uniformly byd\u001700(d)\u001a\n\u000e\nand\u00170(d)\u001a\n\u000e, respectively, hence, (6.37) implies:\neCi(s;t)\u0014e\u0000b(s\u0000t)+Zs\nte\u0000b(s\u0000v)dv\u0014\n\u000f2\ni\u0012d\u001700(d)\u001a\n\u000e+\u00170(d)\u001a\n\u000e\u0013\n+kia\u0015\n(6.41)\n\u00141 +1\nb\u0014\n\u000f2\ni\u0012d\u001700(d)\u001a\n\u000e+\u00170(d)\u001a\n\u000e\u0013\n+aki\u0015\n\u00141 +a+ 1\nb<d40 MANUEL ZAMFIR\nwhenever\u000f2\ni\u0010\nd\u001700(d)\u001a\n\u000e+\u00170(d)\u001a\n\u000e\u0011\n<1 andki\u00141 (i.e.\ris small enough for i= 1 and\fis small enough and\nh\u00141, fori= 2, respectively).\nNow, foreDi(s) =eQi(s;s), recalling that eDi(0) = 1, solving (6.17) we get:\neDi(s) = \u00032\ni(s;0) + 2\u000f2\niZs\n0\u00032\ni(s;v)I4(v;0)dt+ 2\u000f2\niZs\n0\u00032\ni(s;v)I5(v;0)dt (6.42)\n+2kifMi(0)Zs\n0\u00032\ni(s;v)dv\nwhere\nI4(v;t) =Zv\n0Q(u;t)R(v;u)\u001700(C(v;u))du (6.43)\nI5(v;t) =Zt\n0\u00170(Q(v;u))R(t;u)du (6.44)\nNotice that I4andI5share the same uniform bounds as I2andI3, respectively. Recalling that \u0003 i(s;t)\u0014\nexp(\u0000b(s\u0000t)), we establish the bound:\n(6.45) 0 \u0014eDi(s)\u00141 +2\nb2\u0014\n\u000f2\ni\u0012d\u001700(d)\u001a\n\u000e+\u00170(d)\u001a\n\u000e\u0013\n+aki\u0015\n\u00141 +2(a+ 1)\nb2\u0014d\n2\nfor\rsmall fori= 1 and for \f;hsmall fori= 2.\nMoving over to eQi(s;t), sinceeQi(s;s) =eDi(s), we can solve the linear equation (6.16):\neQi(s;t) =eDi(t)\u0003i(s;t) +\u000f2\niZs\nt\u0003i(s;v)I4(v;t)dt+\u000f2\niZs\nt\u0003i(s;v)I5(v;t)dt+kifMi(t)Zs\nt\u0003i(s;v)dv; (6.46)\nwhereI4andI5are de\fned by (6.43) and (6.44), respectively. Using the same bounds on \u0003 i;I4andI5as\nabove, as well as the controls on eDiprovided by (6.45), we show that:\n0\u0014eQi(s;t)\u0014eDi(t) +1\nb\u0014\n\u000f2\ni\u0012d\u001700(d)\u001a\n\u000e+\u00170(d)\u001a\n\u000e\u0013\n+aki\u0015\n\u00141 +2(a+ 1)\nb2+a+ 1\nb\u0014d\nthus concluding the proof.\nSo, indeed, for our choices of a,\u001a,\u000e,dandb, (fMi;eRi;eCi;eQi)2B(\u000e;\u001a;a;d ), for su\u000eciently small \r=\f\nh\n(i= 1) or su\u000eciently small \fandh(i= 2), thus showing (6.20). Furthermore, \u0016i(s)\u0015!i(s)\u0015b, hence (6.22)\nis true, under the same regime as above, as claimed.\nOur next task is to verify that (6.21), the second statement of the theorem, holds. Namely, assuming\nthat (M;R;C;Q )2S(\u000e;\u001a;a;d ) we are to show that the limits ( fMfdt\ni;eRfdt\ni;eCfdt\ni;eQfdt\ni) exist for the solution\n(fMi;eRi;eCi;eQi) of (6.14){(6.19). The main idea used in this section of the proof is to use the exponential\ndecay ofRand \u0003ito bound all the relevant integrals by L1functions and then apply dominated convergence\ntheorem in order to show the existence of the desired limits. To this end, recall that by (6.12), (6.31), (6.37),SPHERICAL SPIN GLASSES 41\n(6.46), (6.42), (6.30) and (6.32), for any t\u00150 and\u001c\u0015v\u00150,\n@fM1(s) =\u0000\u0010\nfM1(s)\u00112\n\u0000fM1(s)\n2h+ 1 +\r2(I0(s)\u0000I1(s))\nfM2(s) =\u000b\u00032(s;0) +\f2Zs\n0I0(u)\u00032(s;u)du+hZs\n0\u00032(s;u)du\neCi(t+\u001c;t) = \u0003i(t+\u001c;t) +\u000f2\niZ\u001c\n0\u0003i(t+\u001c;t+v)I2(t+v;t)dv\n+\u000f2\niZ\u001c\n0\u0003i(t+\u001c;t+v)I3(t+v;t)dv+kifMi(t)Z\u001c\n0\u0003i(t+\u001c;t+v)dv\neQi(t+\u001c;t) =eDi(t)\u0003i(t+\u001c;t) +\u000f2\niZ\u001c\n0\u0003i(t+\u001c;t+v)I4(t+v;t)dv\n+\u000f2\niZ\u001c\n0\u0003i(t+\u001c;t+v)I5(t+v;t)dv+kifMi(t)Z\u001c\n0\u0003i(t+\u001c;t+v)dv\neDi(t) = \u00032\ni(t;0) + 2\u000f2\niZt\n0\u00032\ni(t;v)I4(v;0)dv+ 2\u000f2\niZt\n0\u00032\ni(t;v)I5(v;0)dv+ 2kifMi(0)Zt\n0\u00032\ni(t;v)dv\neRi(t+\u001c;t) = \u0003i(t+\u001c;t)eHi(t+\u001c;t)\neHi(t+\u001c;t) = 1 +X\nn\u00151\f2nX\n\u001b2NCnZ\n0\u0014\u00121\u0014\u0001\u0001\u0001\u0014\u00122n\u0014\u001cY\ni2cr(\u001b)\u001700(C(t+\u0012i;t+\u0012\u001b(i)))2nY\nj=1d\u0012j\n\u00031(t+\u001c;t+v) = exp\u0012\n\u0000\u001c\u0000v\n2h\u0000I6(t+\u001c;t+v)\u0000\r2Z\u001c\nvI7(t+u;t)du\u0013\n\u00032(t+\u001c;t+v) = exp\u0012\n\u0000\u001c\u0000v\n2\u0000hI8(t+\u001c;t+v)\u0000\f2Z\u001c\nvI7(t+u;t)du\u0013\nwhereI0andI1are given by (6.23) and (6.24), respectively and:\nI2(t+\u001c;t) =Z\u001c\n\u0000tC(t+u;t)R(t+\u001c;t+u)\u001700(C(t+\u001c;t+u))du (6.47)\nI3(t+\u001c;t) =Z0\n\u0000t\u00170(C(t+\u001c;t+u))R(t;t+u)du (6.48)\nI4(t+\u001c;t) =Z\u001c\n\u0000tQ(t+u;t)R(t+\u001c;t+u)\u001700(C(t+\u001c;t+u))du (6.49)\nI5(t+\u001c;t) =Z0\n\u0000t\u00170(Q(t+\u001c;t+u))R(t;t+u)du (6.50)\nI6(t+\u001c;t+v) =Zv\n\u001cfM1(t+u)du (6.51)\nI7(t+\u001c;t) =Z\u001c\n\u0000t (C(t+\u001c;t+u))R(t+\u001c;t+u)du (6.52)\nI8(t+\u001c;t+v) =Zv\n\u001cM(u+t)du (6.53)42 MANUEL ZAMFIR\nWe will show that the limits bIk:= lim\ns!1Ik(s) exist fork= 1;2 and also that bIk(\u001c) := lim\ns!1Ik(t+\u001c;t) exist, for\nk= 3:::;8. ForI0, begin by dividing the integral into two parts:\n(6.54) I0(s) =Zs=2\n0M(u)R(s;u)\u001700(C(s;u))du+Z0\n\u0000s=2M(s+u)R(s;s+u)\u001700(C(s;s+u))du\nSince\u001700(\u0001) is continuous and ( M;R;C;Q )2S(\u000e;\u001a;a;d ), ass!1 the bounded integrand in the second\nintegral above converges pointwise to the corresponding expression for ( Rfdt;Cfdt;Mfdt). Further, by the\nexponential tails of Rthe afore-mentioned integrands are uniformly in sbounded by f(\u0012) :=a\u001a\u001700(d)e\u000e\u0012,\nwhich is integrable on ( \u00001;0]. Thus, by dominated convergence theorem, we deduce that\nlim\ns!1Z0\n\u0000s=2M(s+u)R(s;s+u)\u001700(C(s;s+u))du=Z1\n0MfdtRfdt(u)\u001700(Cfdt(u))du\nThe \frst integral in (6.54) is bounded above by \u001a\u000e\u00001\u001700(s)(e\u0000\u000es=2\u0000e\u0000\u000es) that converges to 0 as s!1 , hence:\n(6.55) bI0:= lim\ns!1I0(s) =MfdtZ1\n0Rfdt(u)\u001700(Cfdt(u))du\nApplying a similar argument to I1, we conclude that:\n(6.56) bI1:= lim\ns!1I1(s) =MfdtZ1\n0Rfdt(u) (Cfdt(u))du\nNow, due to the above limits, for any 0 <\u000f<1\n8\r2\n1there exists\u000f>0 such that if s>s\u000f,\f\f\f[I0(s)\u0000I1(s)]\u0000[bI0\u0000bI1]\f\f\f<\n\u000f. Recalling the Ricatti equation (6.12) that characterizes fM1, we can sandwich fM1between the functions\nM1;3andM1;4that are de\fned for s\u0015s\u000fas the unique solutions of the di\u000berential equations:\n@M1;k(s) =\u0000(M1;k(s))2\u0000M1;k(s)\n2h+ 1 +\r2\u0010\n(bI0\u0000bI1) + (\u00001)k\u000f\u0011\n;\nwhile fors\u0014s\u000f,M1;3(s) =M1;4(s) =fM1(s). Using the joint bound on I0andI1provided by (6.25)\nand observing that our choice of \u000fguarantees \r2\u000f <1\n8, we can conclude that the polynomials Pk(X) =\n\u0000X2\u0000X\n2h+ 1 +\r2\u0010\n(bI0\u0000bI1) + (\u00001)k\u000f\u0011\n, fork= 3;4, have exactly one positive root and one negative root.\nFurthermore, denoting with xk(\u000f) the afore-mentioned positive roots, it is easy to see that:\nlim\nt!1M1;k(t) =xk(\u000f) =\u00001\n4h+s\n1\n(4h)2+ 1 +\r2\u0010\nbI0\u0000bI1+ (\u00001)k\u000f\u0011\nRecalling that fM1is bounded above by M1;4and below by M1;3, we obtain:\nx3(\u000f)\u0014lim inf\nt!1fM1(s)\u0014lim sup\nt!1fM2(s)\u0014x4(\u000f)\nSince lim\n\u000f!0x3(\u000f) = lim\n\u000f!0x4(\u000f) =\u00001\n4h+r\n1\n(4h)2+ 1 +\r2\u0010\nbI1\u0000bI2\u0011\nwe can conclude that:\n(6.57) fMfdt\n1:= lim\nt!1fM1(s) =\u00001\n4h+s\n1\n(4h)2+ 1 +\r2\u0010\nbI0\u0000bI1\u0011\nConsequently, applying again dominated converge theorem, this time to (6.51), we show:\n(6.58) bI6(\u001c;v) := lim\nt!1I6(t+\u001c;t+v) = (\u001c\u0000v)fMfdt\n1\nAlso, since M(s) converges as s!1 :\n(6.59) bI8(\u001c;v) := lim\nt!1I8(t+\u001c;t+v) = (\u001c\u0000v)MfdtSPHERICAL SPIN GLASSES 43\nSince (\u0001),\u001700(\u0001) and\u00170(\u0001) are continuous and ( M;R;C;Q )2 S(\u000e;\u001a;a;d ), ast! 1 the bounded in-\ntegrands in (6.47), (6.48), (6.49), (6.50) and (6.52) converge pointwise to the corresponding expression for\n(Mfdt;Rfdt;Cfdt;Qfdt). Further, by the exponential tail of R, the integrals over [ \u0000t;\u0000m] in afore-mentioned\nformulas, are bounded uniformly in tby\u001a\u000e\u00001 (d)e\u0000\u000em. Thus, applying dominated convergence theorem for\nthe integrals over [ \u0000m;v], then taking m!1 , we deduce that for each \fxed v\u00150,\nbI2(\u001c) := lim\nt!1I2(t+\u001c;t) =Z1\n0Cfdt(\u001c\u0000\u0012)Rfdt(\u0012)\u001700(Cfdt(\u0012))d\u0012; (6.60)\nbI3(\u001c) := lim\nt!1I3(t+\u001c;t) =Z1\n\u001c\u00170(Cfdt(\u0012))Rfdt(\u0012\u0000\u001c)d\u0012; (6.61)\nbI4(\u001c) := lim\nt!1I4(t+\u001c;t) =Z1\n0Qfdt(\u001c\u0000\u0012)Rfdt(\u0012)\u001700(Cfdt(\u0012))d\u0012; (6.62)\nbI5(\u001c) := lim\nt!1I5(t+\u001c;t) =Z1\n\u001c\u00170(Qfdt(\u0012))Rfdt(\u0012\u0000\u001c)d\u0012; (6.63)\nbI7:= lim\nt!1I7(t+\u001c;t) =Z1\n0 (Cfdt(\u0012))Rfdt(\u0012)d\u0012; (6.64)\nhence also:\nb\u00031(\u001c\u0000v) := lim\nt!1\u00031(t+\u001c;t+v) = exp\u0012\n\u0000(\u001c\u0000v)\u00121\n2h+fMfdt\n1+\r2bI7\u0013\u0013\n(6.65)\n= exp(\u0000(\u001c\u0000v)b!1);\nb\u00032(\u001c\u0000v) := lim\nt!1\u00032(t+\u001c;t+v) = exp\u0012\n\u0000(\u001c\u0000v)\u00121\n2+hMfdt+\f2bI7\u0013\u0013\n(6.66)\n= exp(\u0000(\u001c\u0000v)b!2):\nwithb!i=\u0000log \u0003i(t)\nt\u0015b>0.\nMoving over to fM2, we \frst split each integral from the right hand side of (6.31) into [0 ;s=2] and [s=2;s].\nSince the integral over [0 ;s=2] is bounded below by 0 and above by [exp( \u0000bs=2)\u0000exp(\u0000bs)]a\u001a\u001700(d)\u000e\u00001, it con-\nverges to 0 as s!1 . The integrand over [ s=2;s] is dominated by the integrable function exp( \u0000bs)a\u001a\u001700(d)\u000e\u00001\nhence we can and will apply dominated convergence theorem, concluding:\n(6.67) fMfdt\n2:= lim\ns!1fM2(s) =\f2bI0+h\nb!2\nA similar argument will show that\n(6.68) \u0016I4:= lim\ns!1I4(s;0) =Q1Z1\n0Rfdt(u)\u001700(Cfdt(u))du\nSince trivially \u0016I5:= limt!1I5(t;0) = 0, similar arguments applied to the integrals in (6.42) and (6.37) will\nshow:\neDfdt\ni:= lim\nt!1eDi(t) =\u000f2\ni\u0016I4+ki\u000b\nb!i=eQ1\ni:= lim\nt!1eQi(t;0) (6.69)\nBy the preceding discussion we also know that for all v;t\u00150 andi2f2;3;4;5;7g, 0\u0014Ii(t+v;t)\u0014\u001a (d)\u000e\u00001\nand the same bound holds for I0(t), uniformly in t. Since 0\u0014\u0003i(t+\u001c;t+v)\u0014exp(\u0000b(\u001c\u0000v)), we can bound\nall the integrands in the right hand sides of (6.37) and (6.46) by the integrable function \u001a (d)\u000e\u00001exp(\u0000bx)44 MANUEL ZAMFIR\nand then apply dominated convergence theorem, concluding:\neCfdt\ni(\u001c) := lim\nt!1eCi(t+\u001c;t) =b\u0003i(\u001c) +\u000f2\niZ\u001c\n0b\u0003(\u001c\u0000v)bI2(v)dv (6.70)\n+\u000f2\niZ\u001c\n0b\u0003i(\u001c\u0000v)bI3(v)dv+kifMfdt\niZ\u001c\n0b\u0003i(v)dv:\neQfdt\ni(\u001c) := lim\nt!1eQi(t+\u001c;t) =eDfdt\nib\u0003i(\u001c) +\u000f2\niZ\u001c\n0b\u0003(\u001c\u0000v)bI4(v)dv (6.71)\n+\u000f2\niZ\u001c\n0b\u0003i(\u001c\u0000v)bI5(v)dv+kifMfdt\niZ\u001c\n0b\u0003i(v)dv:\nWe also have that for any n2Z+, all\u001b2NCnand each \fxed \u00121;:::;\u0012 2n\u00150,\nlim\nt!1Y\ni2cr(\u001b)\u001700(C(t+\u0012i;t+\u0012\u001b(i))) =Y\ni2cr(\u001b)\u001700(Cfdt(\u0012i\u0000\u0012\u001b(i)));\nBy dominated convergence, the corresponding integrals over 0 \u0014\u00121\u0014\u0001\u0001\u0001\u0014\u00122n\u0014\u001cconverge. Further, the\nnon-negative series (6.32) is dominated in tby a summable series (see (6.33)), so by dominated convergence,\neHfdt\ni(\u001c) := lim\nt!1eHi(t+\u001c;t) (6.72)\n= 1 +X\nn\u00151\u000f2n\niX\n\u001b2NCnZ\n0\u0014\u00121\u0014\u0001\u0001\u0001\u0014\u00122n\u0014\u001cY\ni2cr(\u001b)\u001700(Cfdt(\u0012i\u0000\u0012\u001b(i)))2nY\nj=1d\u0012j:\nIt thus follows that\n(6.73) eRfdt\ni(\u001c) := lim\nt!1eRi(t+\u001c;t) =b\u0003i(\u001c)eHfdt\ni(\u001c);\nexists for each \u001c\u00150, which establishes our claim (6.21) (we have already shown that fMfdt\ni,eCfdt\ni(\u001c),eQfdt\ni(\u001c)\nandeQ1\niexists). \u0003\n6.3.Contraction Mapping. The next step in our proof is to establish that the mappings \t iare contractions\nonS(\u000e;\u001a;a;d ). Thus we will be able to conclude that their unique \fxed point, that coincides with the solution\nof our system, will be stationary in the limit, hence the FDT limits (6.8)-(6.11) are well-de\fned.\nProposition 6.2. For\u000e;\u001a;a;b;d;\r 1;h1;\f1of Proposition 6.1, there exist 0< \r 2\u0014\r1,0< \f 2\u0014\f1and\n0<h2\u0014h1, such that the mappings \tiare contractions on S(\u000e;\u001a;a;d ), equipped with the norm\nk(M;R;C;Q )k:= sup\ns2R+jM(s)j+ sup\ns;t2R+jQ(s;t)j+ sup\ns;t2R+jC(s;t)j+ sup\n(s;t)2\u0000jR(s;t)e\u0018(s\u0000t)j; (6.74)\nwhenever\r2[0;\r2](fori= 1) or\f2[0;\f2]andh2[0;h2](fori= 2), for\u0018=b\n3>0. Also the solution\n(M;R;C;Q )of(2.13) -(2.17) is also the unique \fxed point of \t1inS(\u000eh;\u001a;a;d )and of \t2inS(\u000e;\u001a;a;d ). Con-\nsequently, the functions Mfdt;Rfdt;CfdtandQfdtof(6.8) -(6.11) are then the unique solution in D(\u000eh;\u001a;a;d ),\nrespectivelyD(\u000e;\u001a;a;d )of the FDT equations\n0 =\u0000\u0016M+h+\f2MZ1\n0R(\u0012)\u001700(C(\u0012))d\u0012; (6.75)\nR0(\u001c) =\u0000\u0016R(\u001c) +\f2Z\u001c\n0R(\u001c\u0000\u0012)R(\u0012)\u001700(C(\u0012))d\u0012; (6.76)\nC0(\u001c) =\u0000\u0016C(\u001c) +\f2Z1\n0C(\u001c\u0000\u0012)R(\u0012)\u001700(C(\u0012))d\u0012+\f2Z1\n\u001c\u00170(C(\u0012))R(\u0012\u0000\u001c)d\u0012+hM; (6.77)\nQ0(\u001c) =\u0000\u0016Q(\u001c) +\f2Z1\n0Q(\u001c\u0000\u0012)R(\u0012)\u001700(C(\u0012))d\u0012+\f2Z1\n\u001c\u00170(Q(\u0012))R(\u0012\u0000\u001c)d\u0012+hM; (6.78)SPHERICAL SPIN GLASSES 45\nwhere\n\u0016=1\n2+\f2Z1\n0 (C(\u0012))R(\u0012)d\u0012+hM; (6.79)\nwith initial conditions D(0) =R(0) = 1 andQ0(0) = 0 .\nProof of Proposition 6.2: Keeping\u000e;\u001a;a;b anddas in Proposition 6.1, we will show that \t iis a contraction\nonS(\u000e;\u001a;a;d ) equipped with the uniform norm k((M;R;C;Q )kof (6.74), for any \rsmall enough ( i= 1) or\n\f;hsmall enough ( i= 2). We will \frst recall that in Proposition 6.1, we have shown that if ( M;R;C;Q )2\nS(\u000e;\u001a;a;d ), then!i(s)\u0015b, for alls\u00150, a critical fact that we will use in our upcoming proof. For simplicity\nof notation, we will denote by E(s;t) =R(s;t)e\u0018(s\u0000t)\nConsider a pair of elements in S(\u000e;\u001a;a;d ), (Mk;Rk;Ck;Qk) fork= 1;2 and consider their images through\n\ti, namely (fMi;k;eRi;k;eCi;k;eQi;k) = \ti(Mk;Rk;Ck;Qk) fori= 1;2. We will also use the already established\nnotationDk(s) :=Qk(s;s). We will denote hereafter in short \u0001 f(s;t) =f1(s;t)\u0000f2(s;t) and \u0016\u0001f(s) =\nsup0\u0014u\u0014v\u0014sj\u0001f(v;u)jwhenfis one of the functions of interest to us, such as Q,C,R,E, \u0003 orH. A similar\nnotation will be used for functions fof only one variable, for example MorD, namely \u0001 f(s) =f1(s)\u0000f2(s)\nand \u0016\u0001f(s) = sup0\u0014u\u0014sj\u0001f(u)j\nDenoting by #1=\r2and#2=\f2+h, we shall show that for i= 1;2, there exist \fnite positive constants\nLM;i;LE;i;LC;iandLQ;idepending on \u000e;\u001a;a;b andd, such that for any \fnite s\u00150,\n\u0016\u0001fMi(s)\u0014#iLM;i[\u0016\u0001M(s) +\u0016\u0001E(s) +\u0016\u0001C(s) +\u0016\u0001Q(s)]; (6.80)\n\u0016\u0001eEi(s)\u0014#iLE;i[\u0016\u0001M(s) +\u0016\u0001E(s) +\u0016\u0001C(s) +\u0016\u0001Q(s)]; (6.81)\n\u0016\u0001eCi(s)\u0014#iLC;i[\u0016\u0001M(s) +\u0016\u0001E(s) +\u0016\u0001C(s) +\u0016\u0001Q(s)]; (6.82)\n\u0016\u0001eQi(s)\u0014#iLQ;i[\u0016\u0001M(s) +\u0016\u0001E(s) +\u0016\u0001C(s) +\u0016\u0001Q(s)] (6.83)\nwhenever\r2[0;\r1] andi= 1 orh2[0;h1],\f2[0;\f1] andi= 2. Here\r1,h1,\f1,a,d,\u001a,\u000eandbare the\nones of Proposition 6.1.\nSo, if#iis small enough (i.e. #i\u0014minf1=(5LM);1=(5LE);1=(5LC);1=(5LQ)g,#1\u0014\r2\n1and#2\u0014\f2\n1+h1),\nthen from (6.80)-(6.83) we deduce that\nk(\u0001fMi;\u0001eRi;\u0001eCi;\u0001eQi)k= sup\ns\u00150\u0016\u0001fMi(s) + sup\ns\u00150\u0016\u0001eEi(s) + sup\ns\u00150\u0016\u0001eCi(s) + sup\ns\u00150\u0016\u0001eQi(s)\n\u00144\n5\u0014\nsup\ns\u00150\u0016\u0001M(s) + sup\ns\u00150\u0016\u0001E(s) + sup\ns\u00150\u0016\u0001C(s) + sup\ns\u00150\u0016\u0001Q(s)\u0015\n=4\n5k(\u0001M;\u0001R;\u0001C;\u0001Q)k:\nIn conclusion, the mapping \t iis then a contraction on B(\u000e;\u001a;a;d ), since\n(6.84)k\ti(M1;R1;C1;Q1)\u0000\ti(M2;R2;C2;Q2)k\u00144\n5k(M1;R1;C1;Q1)\u0000(M2;R2;C2;Q2)k;\nwhenever (Mk;Rk;Ck;Qk)2B(\u000e;\u001a;a;d ), fork= 1;2.\nFrom now until the end of the proof, for simplifying the notations, we will denote:\n\u0016\u0001(s) := \u0016\u0001M(s) +\u0016\u0001E(s) +\u0016\u0001C(s) +\u0016\u0001Q(s)\nBefore we start, recall that I0;kis de\fned by (6.23) for ( Mk;Ck;Rk;Qk) andI1;kis de\fned by (6.24). Notice\nthat for every i2f0;1gandk2f1;2g, thatIi;kis of the formRs\n0Rk(s;u)Tk;i(u;s;t)du, whereTk;i(u;s;t)\nare polynomial function depending only on Ck(\u00121;\u00122),Mk(\u00121) andQk(\u00121;\u00122), for\u00121;\u001222fs;t;ug. By the\nde\fnition ofS(\u000e;\u001a;a;d ) the family\bRs\n0Rk(s;u)du\t\ns\u00150is uniformly bounded above by \u001a\u000e\u00001, hence:\n0\u0014Ii;k(s)\u0014KI; i = 0;1 (6.85)46 MANUEL ZAMFIR\nforKI=\u001a\n\u000e\u001e(d) maxfa;dg. Similar arguments will show also that:\n0\u0014Ii;k(s;t)\u0014KI i= 2;3;4;5;7 (6.86)\nwhereI2;k,I3;k,I4;k,I5;kandI7;kare de\fned by (6.38), (6.39), (6.43), (6.44) and (6.52), respectively, for\n(M;R;C;Q ) = (Mk;Rk;Ck;Qk). Now consider the di\u000berence between I0;1andI0;2. SinceRt\n0j\u0001R(t;u)jdu\u0014\n\u0016\u0001E(t)\n\u0018(by the de\fnition of Ek), the di\u000berence between I0;1andI0;2can be controlled, yielding:\n\u0016\u0001I0(s)\u0014LI0[\u0016\u0001M(s) +\u0016\u0001E(s) +\u0016\u0001C(s) +\u0016\u0001Q(s)] =LI0\u0016\u0001(s)\nforLI0= maxn\n\u001a\u001700(d)\n\u000e;a\u001700(d)\n\u0018;a\u001a\u0017000(d)\n\u000eo\n. In a similar manner, we obtain analogous bounds for \u0016\u0001Ii(s), for\ni= 1;2;3;4;5;7, for positive and \fnite constants LIi, depending only on a,d,\u001a,\u000eand\u0018. Hence, de\fning\nLI:= maxi2f0;1;2;3;4;5;7gLIi, we establish an uniform Lipschitz control on Ii:\n\u0016\u0001Ii(s)\u0014LI\u0016\u0001(s); i = 0;1;2;3;4;5;7 (6.87)\n\u000fThe Lipschitz bound (6.80) onfM1.Recall thatfM1;ksatis\fes:\n(6.88) @fM1;k(s) =\u0000\u0010\nfM1;k(s)\u00112\n\u0000fM1;k(s)\n2h+ 1 +\r2(I0;k(s)\u0000I1;k(s))\nLet \u0002(s;t) = exp\u0010\n\u0000Rt\ns\u0010\nfM1;1(\u0012) +fM1;2(\u0012) +1\n2h\u0011\nd\u0012\u0011\n. In Proposition 6.1 we have shown that if ( Mk;Rk;Ck;Qk)2\nB(\u000e;\u001a;a;d ) then bothfM1;k(s)\u00150 and!1;k(s) =fM1;k(s)+1\n2h\u0015bare true, hence 0\u0014\u0002(s;t)\u0014exp(\u0000(t\u0000s)b).\nNow, considering the di\u000berence between the realizations of (6.88) for k= 1 andk= 2, respectively, we get:\n@\u0001fM1(s) =\u0000\u0001fM1(s)\u0012\nfM1;1(\u0012) +fM1;2(\u0012) +1\n2h\u0013\n+\r2(\u0001I0(s)\u0000\u0001I1(s))\nand since \u0001fM1(0) = 0 we get:\n\u0001fM1(s) =\r2Zs\n0(\u0001I0;k(u)\u0000\u0001I1;k(u))\u0002(u;s)du\nhence:\n\u0016\u0001fM1(s)\u0014\r2[\u0016\u0001I0(s) +\u0016\u0001I1(s)]Zs\n0e\u0000(s\u0000u)bdu\u0014LM;1\r2\u0016\u0001(s) (6.89)\nwithLM;1=2LI\nb, where in the last inequality we have used the Lipschitz bound on Ii's established in (6.87).\n\u000fThe Lipschitz bound (6.80) onfM2.We will \frst establish the Lipschitz bounds on \u0016iand \u0003i,i= 1;2, that\nwill be needed later. Namely, for i= 1, from (6.18):\nj\u0001\u00161(v)j\u0014j\u0001fM1(v)j+\r2j\u0001I7(v;0)j\u0014(LI+KM;1)\r2\u0016\u0001(s)\nwhere in the last inequality we have used the bounds in (6.87) and (6.80) for i= 1. Sinceje\u0000x\u0000e\u0000yj\u0014jx\u0000yj\nfor allx;y\u00150 and\u0016k;i(s)\u0015b,i;k= 1;2, denoting K4:=LI+KM;1, we get that\nj\u0001\u00031(s;t)j\u0014e\u0000(s\u0000t)bZs\ntj\u0001!1(v)jdv\u0014h\nK4e\u0000b(s\u0000t)(s\u0000t)i\n\r2\u0016\u0001(s) (6.90)\nSimilarly, for i= 2, we get from (6.19):\nj\u0001\u00162(v)j\u0014hj\u0001M(v)j+\f2j\u0001I7(v;0)j\u0014(LI+ 1)(h+\f2)\u0016\u0001(s)\nand a similar argument as above, for K5:=LI+ 1, will establish:\nj\u0001\u00032(s;t)j\u0014h\nK5e\u0000b(s\u0000t)(s\u0000t)i\n(\f2+h)\u0016\u0001(s) (6.91)SPHERICAL SPIN GLASSES 47\nHence, from (6.90) and (6.91) we establish the Lipschitz bound for \u0003 i:\n\u0016\u0001\u0003i(s)\u0014K6#i\u0016\u0001(s) (6.92)\nwithK6:= maxfK4;K5gsup\u0012\u00150\u0000\n\u0012e\u0000b\u0012\u0001\nand also:\nZs\n0j\u0001\u0003i(s;u)jdu\u0014K7#i\u0016\u0001(s) (6.93)\nwhereK7:= maxfK4;K5gsup\u0012\u00150\u0000\ne\u0000b\u0012\u00122\u0001\n.\nWe can now establish the Lipschitz bound (6.80) for fM2. Recalling that fM2;ksatis\fes (6.31), we get:\nj\u0001fM2(s)j \u0014\u000bj\u0001\u00032(s;0)j+\f2Zs\n0j\u0001(I0(u)\u00032(s;u))jdu+hZs\n0j\u0001\u00032(s;u)jdu\n\u0014\u000bj\u0016\u0001\u00032(s)j+\f2\nb\u0016\u0001I0+ (\f2KI+h)Zs\n0j\u0001\u00032(s;u)jdu\n\u0014K8(\f2+h)\u0016\u0001(s) =K8#2\u0016\u0001(s)\nwithK8:=\u000bK6+LI\nb+K7(\f2\n1KI+h1), where in the last line of the derivation above we have used the bounds\nin (6.85), (6.87), (6.92) and (6.93).\n\u000fThe Lipschitz bound (6.81) oneE.We rely on the formulas (6.32) and eRi;k(s;t) =eHi;k(s;t)\u0003i;k(s;t). Indeed,\nsinceC1andC2are [0;d]-valued symmetric functions, ti2[0;s] and both \u001700(\u0001) and\u0017000(\u0001) are non-negative\nand monotone non-decreasing, it follows that for any n,t2n\u0014sand\u001b2NCn,\n\f\f\f\f\f\fY\ni2cr(\u001b)\u001700(C1(ti;t\u001bi))\u0000Y\ni2cr(\u001b)\u001700(C2(ti;t\u001bi))\f\f\f\f\f\f\u0014n\u001700(d)n\u00001\u0017000(d)\u0016\u0001C(s):\nThus we easily deduce from (6.32) that\nj\u0001eHi(s;t)j \u0014 4\u000f2\ni\u0017000(d)(s\u0000t)2X\nn\u00151n(2n!)\u00001[2r(s\u0000t)]2(n\u00001)\u0016\u0001C(s) (6.94)\n\u0014\u000f2\niK9(s\u0000t)2e2\u000fip\n\u001700(d)(s\u0000t)\u0016\u0001C(s):\nforK9= 2\u0017000(d). Recalling that \u000f2\ni\u0014#iand sinceeEi;k(s;t) =eRi;k(s;t)e\u0018(s\u0000t)=eHi;k(s;t)\u0003i;k(s;t)e\u0018(s\u0000t)we\nnow obtain from (6.34), (6.94), (6.90) and (6.91) that:\n\u0001eEi(s;t)\u0014e\u0018(s\u0000t)h\n\u0003i;1(s;t)\u0001eHi(s;t) +eHi;2(s;t)\u0001\u0003i(s;t)i\n\u0014#ie\u0010\n\u0000b+\u0018+2\u000fip\n\u001700(d)\u0011\n(s\u0000t)[K9(s\u0000t)2+c1(K4+K5)(s\u0000t)]\u0016\u0001(s)\n\u0014#ie\u0000(b=3)(s\u0000t)[K9(s\u0000t)2+c1(K4+K5)(s\u0000t)]\u0016\u0001(s)\n\u0014#iLE;i\u0016\u0001(s)\nfor\u000fi<b\n6p\n\u001700(d)and for the \fnite positive constant\nLE;i:= sup\n\u0012\u00150e\u0000b\u0012=3\u0002\nK9\u00122+c1(K4+K5)\u0012\u0003\n:48 MANUEL ZAMFIR\n\u000fThe Lipschitz bounds (6.82) and (6.83) oneCandeQ, respectively. Recalling the solution (6.37) of Ci;k, we\nhave:\n\u0001eCi(s;t) = \u0001\u0003i(s;t) +\u000f2\niZs\nt\u0001(\u0003i(s;v)I7(v;t))dv\n+\u000f2\niZs\nt\u0001(\u0003i(s;v)I8(v;t))dv+kiZs\nt\u0001(fMi(t)\u0003i(s;v))dv\nUsing the Lipschitz bounds in (6.87), (6.92) and (6.93), the \frst two integrals above are each bounded by:\n(#iK7KI+LI)\u0016\u0001(s)\nwhile by (6.80), the last one is bounded by:\n#i\u0012LM;i\nb+aK6\u0013\n\u0016\u0001(s)\nWrapping all together, we get:\nj\u0016\u0001eCi(s)j \u0014#iLC;i\u0016\u0001(s)\nforLC;i= (K5+K6) + 2((\f1+\r1)K7KI+LI) +\u0010\nLM;i\nb+aK6\u0011\n(h1+ 1) and consequently, (6.82) holds.\nSimilarly, by the solution (6.42) of Di;k(s) :=Qi;k(s;s), we have:\n\u0001eDi(s) = \u0001(\u00032\ni(s;0)) + 2\u000f2\niZs\nt\u0001(\u00032\ni(s;v)I4(v;0))dv+ 2\u000f2\niZs\n0\u0001(\u00032\ni(s;v)I5(v;0))dv\n+ 2\u000bkiZs\n0\u0001(\u00032\ni(s;v))dv\nSince \u0003i(s;t)2[0;1], thenj\u0001L2\ni(s;t)j\u00142j\u0001Li(s;t)j, hence similarly as above, we get:\nj\u0016\u0001eDi(s)j \u0014LD;i#i\u0016\u0001(s)\nforLD;i= 4LCi. Moving over to eQi;k, since:\n\u0001eQi(s;t) = \u0001(eDi(t)\u0003i(s;t)) +\u000f2\niZs\nt\u0001(\u0003i(s;v)I4(v;t))dv\n+\u000f2\niZs\n0\u0001(\u0003i(s;v)I5(v;t))dv+kiZs\nt\u0001(fMi(t)\u0003i(s;v))dv\nusing the Lipschitz bound on eDiand similar reasonings as above, we get:\nj\u0016\u0001eQi(s)j \u0014LQ;i#i\u0016\u0001(s)\nforLQ;i=LD;i+ maxfd;1gLC;i, thus concluding the argument that \t iis a contraction.\nNow suppose that, for a choice of parameters \fandh, the constants d;\u001a;a;b anddare such that \t iis a\ncontraction onB(\u000e;\u001a;a;d ), hence also on its non-empty subset S(\u000e;\u001a;a;d ). Proposition 6.1 shows that both\nB(\u000e;\u001a;a;d ) andS(\u000e;\u001a;a;d ) are invariant under \t i. We start at some Si;0= (M0;R0;C0;Q0)2S(\u000e;\u001a;a;d )\nand construct recursively the sequences Si;k= \ti(Si;k\u00001) fork= 1;2;:::, inS(\u000e;\u001a;a;d ). Fori= 1;2, since\n\tiis a contraction, clearly fSi;kgk2Z+is a Cauchy sequence for the uniform norm k\u0001k of (6.74). Hence,\nSi;k!Si;1= (Mi;1;Ri;1;Ci;1;Qi;1) in the Banach space ( C(R+)\u0002C(\u0000)\u0002Cs(R2\n+)\u0002Cs(R2\n+);k\u0001k). Note\nthatB(\u000e;\u001a;a;d ) is a closed subset of this Banach space, so Si;12B(\u000e;\u001a;a;d ). Further, \fxing \u001c\u00150, since\nSi;k2S(\u000e;\u001a;a;d ) we have that\nlim\nT!1sup\nt;t0\u0015TjCi;1(t+\u001c;t)\u0000Ci;1(t0+\u001c;t0)j\n\u00142kCi;1\u0000Ci;kk1+ lim\nT!1sup\nt;t0\u0015TjCi;k(t+\u001c;t)\u0000Ci;k(t0+\u001c;t0)j= 2kSi;1\u0000Si;kk:SPHERICAL SPIN GLASSES 49\nTakingk!1 we deduce that, for any \u001c\u00150,t7!Ci;1(t+\u001c;t) is a Cauchy function from R+to [0;d], hence\nCi;1(t+\u001c;t) converges as t!1 . A similar bounding procedure as above will show that the same is true for\nEi;1andQi;1and will also show that Mi;1(t) converges as t!1 . Now, since, by de\fnition, Ri;1(s;t) =\nEi;1(s;t)e\u0000\u0018(s\u0000t), thenRi;1will inherit the limiting property from Ei;1. HenceSi;12S(\u000e;\u001a;a;d ) and\nfurtherSi;1is the unique \fxed point of the contraction \t ion the metric space ( S(\u000e;\u001a;a;d );k\u0001k).\nBy our construction of \t i, it follows that ( M1;1;R1;1;C1;1;Q1;1) satis\fes (6.1)-(6.5) and also that\n(M2;1;R2;1;C2;1;Q2;1) satis\fes (2.13)-(2.17). Recalling that any solution of (6.1)-(6.5) is a solution of\n(2.13)-(2.17) that has been time-scaled by a factor of h, we can conclude that the unique solution of (2.13)-(2.17)\nis inS(\u000eh;\u001a;a;d ), for\r2[0;\r2] and inS(\u000e;\u001a;a;d ), respectively, for \f2[0;\f2] andh2[0;h2]. As noted before,\nthis shows that the FDT limits Mfdt,Rfdt(\u001c),Cfdt(\u001c) andQfdt(\u001c) exist, for the unique solution of (2.13)-(2.17)\nand furthermore, ( Mfdt;Rfdt;Cfdt;Qfdt)2D(\u000eh;\u001a;a;d ) if\r\u0014\r2and (Mfdt;Rfdt;Cfdt;Qfdt)2D(\u000e;\u001a;a;d ) if\n\f\u0014\f2andh\u0014h2.\nIn order to conclude the proof, we will show that Mfdt;Rfdt(\u0001);Cfdt(\u0001) andQfdt(\u0001) are the unique solution\ninD(\u000eh;\u001a;a;d ), respectivelyD(\u000e;\u001a;a;d ), of (6.75)-(6.79). While proving Proposition 6.1 we found that on\nS(\u000e;\u001a;a;d ), the mapping \t iinduces a mapping \tfdt\ni: (Mfdt;Rfdt;Cfdt;Qfdt)!(fMfdt\ni;eRfdt\ni;eCfdt\ni;eQfdt\ni) such\nthat\n0 =\u0000\u0010\nfMfdt\n1\u00112\n\u0000fMfdt\n1\n2h+ 1 +\r2\u0010\nbI0\u0000bI1\u0011\n0 =\u0000b!2fMfdt\n2+h+\f2bI0\neRfdt\ni(\u001c) =b\u0003(\u001c)X\nn\u00150\u000f2n\niX\n\u001b2NCnZ\n0\u0014\u00121\u0014\u0001\u0001\u0001\u0014\u00122n\u0014\u001cY\ni2cr(\u001b)\u001700(Cfdt(\u0012i\u0000\u0012\u001b(i)))2nY\nj=1d\u0012j;\neCfdt\ni(\u001c) =b\u0003i(\u001c) +\u000f2\niZ\u001c\n0b\u0003i(\u001c\u0000v)bI2(v)dv+\u000f2\niZ\u001c\n0b\u0003i(\u001c\u0000v)bI3(v)dv+kifMfdt\niZ\u001c\n0b\u0003i(v)dv;\neQfdt\ni(\u001c) =eDfdt\nib\u0003i(\u001c) +\u000f2\niZ\u001c\n0b\u0003i(\u001c\u0000v)bI4(v)dv+\u000f2\niZ\u001c\n0b\u0003i(\u001c\u0000v)bI5(v)dv+kifMfdt\niZ\u001c\n0b\u0003i(v)dv;\n0 =\u0000b!ieDfdt\ni+\u000f2\nibI4(0) +\u000f2\nibI5(0) +kifMfdt\ni;\nwherebI0;bI1;bI2;bI3;bI4andbI5are given by (6.55), (6.56), (6.60), (6.61), (6.62) and (6.63), respectively. In\nparticular,eCfdt,eRfdtandeQfdtare di\u000berentiable on R+, and, for\u001c\u00150,\n0 =\u0000\u0010\nfMfdt\n1\u00112\n\u0000fMfdt\n1\n2h+ 1 +\r2MfdtZ1\n0Rfdt(\u0012)\u001700(Cfdt(\u0012))d\u0012 (6.95)\n\u0000\r2MfdtZ1\n0Rfdt(\u0012) (Cfdt(\u0012))d\u0012\n0 =\u0000b!2fMfdt\n2+\f2MfdtZ1\n0Rfdt(\u0012)\u001700(Cfdt(\u0012))d\u0012+h (6.96)\n@eRfdt\ni(\u001c) =\u0000b!ieRfdt\ni(\u001c) +\u000f2\niZ\u001c\n0eRfdt\ni(\u001c\u0000\u0012)eRfdt\ni(\u0012)\u001700(Cfdt(\u0012))d\u0012; (6.97)\n@eCfdt\ni(\u001c) =\u0000b!ieCfdt\ni(\u001c) +\u000f2\niZ1\n0Cfdt(\u001c\u0000\u0012)Rfdt(\u0012)\u001700(Cfdt(\u0012))d\u0012 (6.98)\n+\u000f2\niZ1\n\u001c\u00170(Cfdt(\u0012))Rfdt(\u0012\u0000\u001c)d\u0012+kifMfdt\ni\n@eQfdt\ni(\u001c) =\u0000b!ieQfdt\ni(\u001c) +\u000f2\niZ1\n0Qfdt(\u001c\u0000\u0012)Rfdt(\u0012)\u001700(Cfdt(\u0012))d\u0012 (6.99)50 MANUEL ZAMFIR\n+\u000f2\niZ1\n\u001c\u00170(Qfdt(\u0012))Rfdt(\u0012\u0000\u001c)d\u0012+kifMfdt\ni\nwitheRfdt\ni(0) = 1,eCfdt\ni(0) = 1,@eQfdt\ni(0) = 0 and\nb!1=1\n2h+\r2Z1\n0 (Cfdt(\u0012))Rfdt(\u0012)d\u0012+fMfdt\n1 (6.100)\nb!2=1\n2+\f2Z1\n0 (Cfdt(\u0012))Rfdt(\u0012)d\u0012+hMfdt(6.101)\nwhere in the derivation of (6.97) we have used the results in [16].\nRecall that if the functions M;R;C andQsolve (2.13)-(2.17), then the functions Mh;Rh;ChandQhare\nthe unique solution of (6.1)-(6.5), hence the unique \fxed point of \t 1. Then, by (6.95){(6.101) the corre-\nsponding quad-uple ( Mfdt\nh;Rfdt\nh;Cfdt\nh;Qfdt\nh) is a \fxed points of \tfdt\n1. ThenMfdt:=Mfdt\nh,Rfdt(\u001c) :=Rfdt\nh(h\u001c),\nCfdt(\u001c) :=Cfdt\nh(h\u001c) andQfdt(\u001c) :=Qfdt\nh(h\u001c) satisfy the FDT equations (6.75)-(6.79). Noticing that the quad-\nuple (Mfdt;Rfdt;Cfdt;Qfdt) that we have just de\fned coincide with the FDT limits of the original ( M;R;C;Q ),\nwe have established that, for \r2[0;\r2], (Mfdt;Rfdt;Cfdt;Qfdt) satisfy (6.75)-(6.79).\nAlso, if (M;R;C;Q ) is the unique solution of (2.13)-(2.17), it is the unique \fxed point of \t 2, hence\n(Mfdt;Rfdt;Cfdt;Qfdt) is a \fxed point of \tfdt\n2, hence it satis\fes (6.75)-(6.79).\nNow, denoting by Efdt(\u001c) =e\u0018\u001cRfdt(\u001c), by the same arguments as in the Lipschitz estimates (6.80)-(6.83)\nof Proposition 6.2, we show that:\n\u0016\u0001fMfdt\ni\u0014#iLM;i[\u0016\u0001Mfdt+\u0016\u0001Efdt(1) +\u0016\u0001Cfdt(1) +\u0016\u0001Qfdt(1)];\n\u0016\u0001eEfdt\ni(\u001c)\u0014#iLE;i[\u0016\u0001Mfdt+\u0016\u0001Efdt(\u001c) +\u0016\u0001Cfdt(\u001c) +\u0016\u0001Qfdt(\u001c)];\n\u0016\u0001eCfdt\ni(\u001c)\u0014#iLC;i[\u0016\u0001Mfdt+\u0016\u0001Efdt(\u001c) +\u0016\u0001Cfdt(\u001c) +\u0016\u0001Qfdt(\u001c)];\n\u0016\u0001eQfdt\ni(\u001c)\u0014#iLQ;i[\u0016\u0001Mfdt+\u0016\u0001Efdt(\u001c) +\u0016\u0001Cfdt(\u001c) +\u0016\u0001Qfdt(\u001c)]\nfor all\u001c <1, where \u0016\u0001f(s) = sup0\u0014u\u0014sjf1(u)\u0000f2(u)jwhenfis one of the function of interest E,CorQ,\nand \u0016\u0001M=jM1\u0000M2j, thus showing that the mappings \tfdt\niare also contractions, they have unique \fxed\npoints inD(\u000eh;\u001a;a;d ) andD(\u000e;\u001a;a;d ), respectively. So (6.76)-(6.79) have an unique solution in D(h\u000e;\u001a;a;d ),\nfor\r2[0;\r2] and inD(\u000e;\u001a;a;d ), for\f2[0;\f2] andh2[0;h2], as claimed. \u0003\n6.4.Exponential Decay of the Covariance. One consequence of Proposition 6.2 is that if either \ris small\nor both\fandhare small, the response function is positive and decays to 0 exponentially fast. In the next\nproposition we will establish an analogous result for the covariance. Namely, we show:\nProposition 6.3. For\r2;\f2;h2>0of Proposition 6.2, if \r2[0;\r2]or\f2[0;\f2]andh2[0;h2]there exist\nM=M(\f;h;\u000b )>0and\u0011=\u0011(\f;h;\u000b )such that for every s\u0015t\u00150:\njC(s;t)\u0000Q(s;t)j \u0014Me\u0000(s\u0000t)\u0011(6.102)\nProof of Proposition 6.3: LetCOV (s;t) :=C(s;t)\u0000Q(s;t) and respectively COVh(s;t) :=Ch(s;t)\u0000\nQh(s;t), withUh(s;t) :=U(s=h;t=h ), whenever Uis one ofCorQ. Subtracting (2.16) from (2.15), we get:\n@1COV (s;t) =\u0000\u0016(s)COV (s;t) +\f2Zs\n0COV (u;t)R(s;u)\u001700(C(s;u))du (6.103)\n+\f2Zt\n0COV (s;u)P(C(s;u);Q(s;u))R(t;u)du; s \u0015t\u00150\nfor the multivariate polynomial P(X;Y ) =\u00170(X)\u0000\u00170(Y)\nX\u0000Y, where\u0016is de\fned by (2.17), hence\n(6.104) COV (s;t) = \u0003(s;t) +\f2Zs\nt\u0003(s;v)I9(v;t)dv+\f2Zs\nt\u0003(s;v)I10(v;t)dvSPHERICAL SPIN GLASSES 51\nwith \u0003(s;v) = exp(\u0000Rs\nv\u0016(u)du),\nI9(v;t) =Zv\n0COV (u;t)R(v;u)\u001700(C(v;u))du; (6.105)\nI10(v;t) =Zt\n0COV (v;u)P(C(v;u);Q(v;u))R(t;u)du: (6.106)\nBy Proposition 6.2 we know that, whenever \f < \f 2andh < h 2,R(s;t)\u0014\u001ae\u0000(s\u0000t)\u000eand\u0016(s)\u0015bimplying\n\u0003(s;v)\u0014e\u0000b(s\u0000v). Also, Theorem 2.3 shows C(s;t);Q(s;t)2[0;1], implying P(C(s;t);Q(s;t))\u0014\u001700(1), since\n\u0017(\u0001) is a polynomial with positive coe\u000ecients. So, we get:\njI9(v;t)j \u0014\u001700(1)Zv\n0jCOV (u;t)j\u001ae\u0000\u000e(v\u0000u)du\u0014\u001700(1)\u001ae\u0000\u000e(v\u0000t);\njI10(v;t)j \u0014\u001700(1)\u001a\u000e\u00001sup\nu\u0014tjCOV (u;v)j:\nand hence, with the symmetric function \u0001( t;s) := supu\u0014t;v\u0014sjCOV (u;v)jwe deduce from (6.104) that for\ns\u0015t\u00150,\n\u0001(t;s)\u0014e\u0000b(s\u0000t)+\f2\u001700(1)\u001aZs\nte\u0000b(s\u0000v)[Zv\n0e\u0000\u000e(v\u0000u)du+\u000e\u00001\u0001(t;v)]dvdu\n\u0014e\u0000b(s\u0000t)+\f2\u001a\u001700(1)Zs\nte\u0000b(s\u0000v)Zt\n0e\u0000\u000e(v\u0000u)dudv\n+\f2\u001a\u001700(1)Zs\nt\u0001(t;v)[\u000e\u00001e\u0000b(s\u0000v)+Zv\nte\u0000b(s\u0000v)\u0000\u000e(v\u0000u)du]dv\nSince for any \u000e2(0;b=2) ands\u0015t,\n(6.107)Zs\nte\u0000b(s\u0000v)\u0000\u000e(v\u0000t)dv\u00142b\u00001e\u0000\u000e(s\u0000t)\nand with\u000e2(0;b) we thus obtain for s\u0015tthe bound\n\u0001(t;s)\u0014M\fe\u0000\u000e\f(s\u0000t)+A\fZs\nt\u0001(t;v)e\u0000\u000e\f(s\u0000v)dv;\nwithM= 1 + 2\f2\u001a\u001700(1)(b\u000e)\u00001andA=\f2\u001a\u001700(1)\u000e\u00001(1 + 2b\u00001). Therefore, \fxing t\u00150, the function\nht(s) =e\u000e(s\u0000t)\u0001(t;s) satis\fes\nht(s)\u0014M+AZs\ntht(v)dv; s\u0015t;\nand so by Gronwall's lemma ht(s)\u0014MeA(s\u0000t). We therefore conclude that for any s\u0015t,\njC(s;t)\u0000Q(s;t)j\u0014Me\u0000(\u000e\u0000A)(s\u0000t);\nwhich proves the lemma in this case, since for \f!0 we have that A=A(\f)!0 (and so\u0011=\u000e\u0000A> 0 for\nany\f >0 small enough).\nSimilarly, from (6.4) from (6.3), we get:\n@1COVh(s;t) =\u0000\u0016h(s)COVh(s;t) +\r2Zs\n0COVh(u;t)Rh(s;u)\u001700(Ch(s;u))du (6.108)\n+\r2Zt\n0COVh(s;u)P(Ch(s;u);Qh(s;u))Rh(t;u)du; s\u0015t\u0015052 MANUEL ZAMFIR\nwhere\u0016his de\fned by (6.5). Recalling that if \r\u0014\r2,\u0016h(s)\u0015b, the same argument as before, with \rin the\nplace of\f, will show that \u0001 h(s;t) := \u0001(s=h;t=h )\u0014Me\u0000(\u000e\u0000A)(s\u0000t), that is equivalent to:\njC(s;t)\u0000Q(s;t)j\u0014Me\u0000h(\u000e\u0000A)(s\u0000t)\nhence concluding out proof. \u0003\n6.5.Simplifying the FDT System. The \fnal step of the proof is to relate the solutions of the limiting\nequations (6.75)-(6.79) to the FDT equations (2.20) and (2.19), hence concluding the proof of Theorem 2.5.\nProposition 6.4. There exist \r3;\f3;h3>0such that whenever \r2[0;\r3]or\f2[0;\f3]andh2[0;h3], the\nequations (2.20) and(2.19) have unique solutions C(\u0001)andQ. Furthermore, the quadruple (M;C;R;Q ), where\nR(\u001c) :=\u00002@C(\u001c)andQ(\u001c) :=Qsolves the system (6.75) -(6.79) with initial conditions C(0) =R(0) = 1 ,\nQ0(0) = 0 . Furthermore, R(\u001c)is positive and decays exponentially fast to 0andC(\u001c)is positive and bounded,\nconverging to Qas\u001c!1 .\nProof of Proposition 6.4: Consider the function f(x) = 4(x\u00001)2[\f2\u00170(x) +h2]\u0000x. Since for any h >\n0,f(1\u0000(2h)\u00001)>0 andf(1)<0 and also f(0)>0, there exist at least a solution to f(x) = 0 in\n[(1\u0000(2h)\u00001)^0;1]. By de\fnition, any of these solutions satis\fes (2.19). Fix Qto be one of them.\nLetCbe the unique [0 ;1]-valued solution of (2.20) for \u001e(x) = 1=2\u00002\f2Q\u00170(Q) + 2h2(1\u0000Q) + 2\f2\u00170(x)\n(see Proposition 1.4 of [14] for existence and uniqueness of the solution). Also, since Q2[(1\u0000(2h)\u00001)^0;1],\nit is easy to see that for small enough \r, the following bound holds:\n2\f2(\u00170(1)\u0000\u00170(Q))\u0015\f\r\u001700(1)\u00152p\n\f2\u00170(1)\nand if\fis small enough, then:\n1\n2\u00152p\n\f2\u00170(1)\nthus concluding that in both scenarios, \u001e(1)>2p\nb\u001e0(1), hence, according to the above-mentioned result, C0\ndecays exponentially to 0 with some positive exponent (it is easy to see that \u001eis convex, so the conditions in\nthe quoted proposition are satis\fed).\nMoreover, by the same result, Cconverges as t!1 to\nC1:= sup\u001a\nx2[0;1] :\u001e(x)(1\u0000x)\u00151\n2\u001b\nNow, from the de\fnition of Q, it is easy to see that \u001e(Q)(1\u0000Q) = 1=2 and since Q2[0;1],C1\u0015Q. Also,\nfor\rsu\u000eciently small, for x2[Q;1],\n2\f2\u0012\u00170(x)\u0000\u00170(Q)\nx\u0000Q\u0013\n\u00142\r2h2\u001700(1)<4h2\u00141\n(1\u0000Q)(1\u0000x)\nhence\u001e(x)(1\u0000x)<1=2 forx2[Q;1], implying C1=Q. Similarly, for \fsmall,\n2\f2\u0012\u00170(x)\u0000\u00170(Q)\nx\u0000Q(1\u0000Q)(1\u0000x)\u0013\n\u00142\f2\u001700(1)<1\nso\u001e(x)(1\u0000x)<1=2 forx2[Q;1], henceC1=Q.\nNow, denoting by R(\u001c) :=\u00002@C(\u001c), andQ(\u001c)\u0011Q, sinceQ= limt!1C(\u001c), some simple algebra will show\nthat (M;R;C;Q ) satisfy (6.75)-(6.79) with initial conditions C(0) = 1,R(0) = 1,Q0(0) = 0, if and only if:\n0 =\u0000\u0016M+h+ 2\f2M(\u00170(1)\u0000\u00170(Q)) (6.109)\n0 =\u0000\u0016Q+ 2\f2(Q\u00170(1)\u00002Q\u00170(Q) +\u00170(Q)) +hM (6.110)\nwith\n\u0016=1\n2+ 2\f2(\u00170(1)\u0000Q\u00170(Q)) +hMSPHERICAL SPIN GLASSES 53\nIt's easy to check that M:= 2h(1\u0000Q) andQare a solution to (6.109)-(6.110), hence ( M;R;C;Q ) satisfy\n(6.75)-(6.79). Furthermore, M;Q2[0;1], as needed.\nNow, for every root of (2.19), we can use the same procedure as above to construct a quad-uple ( M;R;C;Q ),\nthat solves the system. Since Q2[(1\u0000(2h)\u00001)^0;1], the same arguments as above will conclude that\n(M;R;C;Q ) are positive, C(\u0001) is bounded and R(\u0001) decays to 0 exponentially fast. Since according to Proposi-\ntion 6.2, the system (6.75)-(6.79) has an unique solution with these properties, the injectivity of the mapping\nQ7!(M;R;C;Q ) shows that (2.19) has a unique root in [(1 \u0000(2h)\u00001)^0;1], thus concluding the proof. \u0003\nNow we have all the ingredients we need to \fnalize the proof of our theorem:\nProof of Theorem 2.5: Fix\r0= minf\ri:i= 1;2;3g,\f0= minf\fi:i= 1;2;3gandh0= minfhi:i=\n1;2;3g, for\r1;\f1;h1of Proposition 6.1, \r2;\f2;h2of Proposition 6.2 and \r3;\f3;h3of Proposition 6.3. Then,\naccording to Proposition 6.2, the FDT limits (6.8)-(6.11) exist and are the unique solution of (6.75)-(6.79)\nwith initial conditions C(0) =R(0) = 1,Q0(0) = 0, in the space of positive functions such that C(\u0001);Q(\u0001) are\nbounded above and R(\u0001) decays exponentially to 0.\nBy Proposition 6.4, for the same possible values of the parameters \fandh,C(\u001c),R(\u001c) :=\u00002@C(\u001c),\nQ(\u001c) :=QandM:= 2h(1\u0000h) are a solution of (6.75)-(6.79) and furthermore, Rdecays exponentially fast\nto 0 and 0\u0014M;Q (\u001c);C(\u001c)\u00141, so, by the afore-mentioned uniqueness result, they are indeed the solution of\n(6.75)-(6.79), thus concluding the proof. \u0003\nReferences\n[1] AIDA, S. ; STROOCK, D. ; Moment estimates derived from Poincar\u0013 e and logarithmic Sobolev inequalities. Math. Res. Lett.\n1, 75-86 (1994).\n[2] AN \u0013E, C. et altri; Sur les in\u0013 egalit\u0013 es de Sobolev logarithmiques. Panoramas et Syntheses ,10, Soci\u0013 et\u0013 e Math\u0013 ematique de France\n(2000).\n[3] BEN AROUS, G. ; GUIONNET, A. ; Large deviations for Langevin spin glass dynamics. Prob. Th. Rel. Fields 102, 455-509\n(1995).\n[4] BEN AROUS, G. ; GUIONNET, A. ; Symmetric Langevin spin glass dynamics. Ann. Probab. 25, 1367-1422 (1997).\n[5] BEN AROUS, G. ; DEMBO, A. ; GUIONNET, A.; Aging of spherical spin glasses. Probab. Theory Relat. Fields ,120:1{67\n(2001).\n[6] BEN AROUS, G. ; Aging and spin-glass dynamics. Proceedings of the International Congress of Mathematicians , Vol. III ,\n3{14, Higher Ed. Press, Beijing, 2002 (2002).\n[7] BEN AROUS, G. ; DEMBO, A. ; GUIONNET, A. ; Cugliandolo-Kurchan equations for dynamics of Spin-Glasses. Submitted\n(2004).\n[8] BOUCHEAUD J. ; CUGLIANDOLO L. ; KURCHAN J. ; MEZARD M. ; Mode coupling approximations, glass theory and\ndisordered systems. Physica A ,226, 243-273. (1997)\n[9] BOUCHAUD, J.P. ; CUGLIANDOLO, L.F. ; KURCHAN, J. ; MEZARD, M. ; Out of equilibrium dynamics in spin-glasses\nand other glassy systems. Spin glass dynamics and Random Fields , A. P Young Editor (1997).\n[10] CUGLIANDOLO, L.F. ; Dynamics of glassy systems Les Houches (2002)\n[11] CUGLIANDOLO, L.F. ; DEAN, D.S. ; Full dynamical solution for a spherical spin-glass model. J. Phys. A: Math. Gen. 28,\n4213-4234 (1995).\n[12] CUGLIANDOLO, L.F. ; KURCHAN, J. ; Analytical solution of the o\u000b-equilibrium Dynamics of a Long-Range Spin-Glass\nModel. Phys. Rev. Lett. 71, 173-176 (1993)\n[13] CUGLIANDOLO, L.F. ; LE DOUSSAL P.; Large time nonequilibrium dynamics of a particle in a random potential. Phys.\nRev. E ,53, 1525-1552. (1996).\n[14] DEMBO, A. ; GUIONNET, A. ; MAZZA, C.; Limiting dynamics for spherical models of spin glasses at high temperature J.\nof Stat. Physics ,128, No. 4. 847-881. (2007)\n[15] FISCHER, K.H. ; HERTZ, J.A.; Spin Glasses. Cambridge: Cambridge University Press (1991).\n[16] GUIONNET, A. ; MAZZA, C.; Long time behaviour of the solution to non-linear Kraichnan equations Prob. Theory. Rel.\nFields 131:493-518 (2005).\n[17] GUIONNET, A.; Dynamics for spherical models of sping glass and Aging. Proceedings of the Ascona meeting 2004.\n[18] GOTZE W. ; Liquids, Freezing and the Glass Transition Ed. J.P. Hansen, D. Levesque and J Zinn-Justin, Les Houches 1989,\nNorth-Holland. (1991)\n[19] GOTZE W. ; SJOGREN L.; Relaxation processes in supercooled liquids. Rep. Prog. Phys. ,55, 241-376. (1992)\n[20] KARATZAS, I. ; SHREVE, S.E. ; Brownian motion and stochastic calculus (second edition). Springer-Verlag (1991).54 MANUEL ZAMFIR\n[21] M. TALAGRAND ; Spin Glasses: A Challenge for Mathematicians: Cavity and Mean Field Models (\frst edition). Springer\n(2003).\nDepartment of Mathematics, Stanford University, Stanford, CA 94305.\nE-mail address :manuelzamfir@gmail.com"    },    {        "title": "1007.0004v1.Supercurrent_Induced_Magnetization_Dynamics.pdf",        "content": "Supercurrent-Induced Magnetization Dynamics\nJacob Linder1and Takehito Yokoyama2\n1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway\n2Department of Physics, Tokyo Institute of Technology,\n2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan\n(Dated: November 8, 2018)\nWe investigate supercurrent-induced magnetization dynamics in a Josephson junction with two misaligned\nferromagnetic layers, and demonstrate a variety of effects by solving numerically the Landau-Lifshitz-Gilbert\nequation. In particular, we demonstrate the possibility to obtain supercurrent-induced magnetization switch-\ning for an experimentally feasible set of parameters, and clarify the favorable condition for the realization of\nmagnetization reversal. These results constitute a superconducting analogue to conventional current-induced\nmagnetization dynamics and indicate how spin-triplet supercurrents may be utilized for practical purposes in\nspintronics.\nPACS numbers: 74.45.+c\nIntroduction . The interplay between superconducting and\nferromagnetic order is presently generating much interest in\na variety of research communities [1]. Besides the obvious\ninterest from a fundamental physics point of view, a major\npart of the allure of superconductor jferromagnet (SjF) hy-\nbrid structures is the prospect of combining the spin-polarized\ncharge carriers present in ferromagnets with the dissipation-\nless flow of a current offered by the superconducting envi-\nronment. By tailoring the desired properties of a hybrid S jF\nsystem on a nanometer scale, this interplay opens up new per-\nspectives within spin-polarized transport.\nClosely related to the transport of spin is the phenomenon\nof current-induced magnetization dynamics [2]. The gen-\neral principle is that a spin-polarized current injected into a\nferromagnetic layer can act upon the magnetization of that\nlayer via a torque and thus induce magnetization dynamics\n[3]. Previous works in this field have considered mainly non-\nequilibrium spin accumulation via quasiparticle spin-injection\nfrom a ferromagnet into a superconductor [4–8]. The concept\nof supercurrent-induced magnetization dynamics suggests an\ninteresting venue for combining the seemingly disparate fields\nof superconductivity and spintronics. However, magnetiza-\ntion dynamics in a Josephson junction has so far been dis-\ncussed only in a handful of works [9–13]. In particular, it\nwas demonstrated in Ref. [9] how a supercurrent would in-\nduce an equilibrium exchange interaction between two non-\ncollinear ferromagnets in an S jFjNjFjS junction (N stands for\nnormal metal). Taking into account the fact that a Joseph-\nson current flowing through such an inhomogeneous magne-\ntization profile will have a spin-triplet contribution [1], such\nan interaction implies that it should be possible to generate\nsupercurrent-induced magnetization dynamics in this type of\njunction. This would constitute a superconducting analogue\nto magnetization dynamics in a conventional spin-valve setup.\nTo this date, this remains unexplored in the literature.\nMotivated by this, we study in this Letter for the first time\nthe magnetization dynamics of a multilayer ferromagnetic\nJosephson junction when a current bias is applied, based on\nthe Landau-Lifshitz-Gilbert (LLG) equation [14]. Our main\nFIG. 1: (Color online) The proposed experimental structure. Two\nferromagnetic layers separated by a normal spacer are sandwiched\nbetween two s-wave superconductors. The magnetization directions\nmay be non-aligned and tuned by means of an external field, as long\nas the exchange coupling between the ferromagnets is sufficiently re-\nduced by the thickness of the normal spacer. By current-biasing this\nsystem one may generate a supercurrent which induces magnetiza-\ntion dynamics.\nidea is to utilize the spin-triplet nature of the Josephson cur-\nrent, which very recently has been observed experimentally\n[18–20], in order to induce a torque on the magnetic order pa-\nrameter and thus generate magnetization dynamics. The ex-\nperimentally relevant setup is shown in Fig. 1: two ferromag-\nnets with different coersive fields are separated by a normal\nspacer and sandwiched between two conventional s-wave su-\nperconductors. The coersive fields are such that the magnetic\norder parameter is hard in one layer, while soft in the other.arXiv:1007.0004v1  [cond-mat.supr-con]  30 Jun 20102\n05101520253035404550−1−0.8−0.6−0.4−0.200.20.40.60.81\nτMagnetization components(a)\n  \n00.5 1−101\nmy(τ)mz(τ)\n05101520253035404550−1−0.8−0.6−0.4−0.200.20.40.60.81\nτ(b)\n  \n0.80.91−101\n05101520253035404550−1−0.8−0.6−0.4−0.200.20.40.60.81\nτMagnetization components\n1\n(c)\n−1 01−101\n05101520253035404550−1−0.8−0.6−0.4−0.200.20.40.60.81\nτ(d)\n−1 01−101mx(τ)\nmy(τ)\nmz(τ)\nFIG. 2: (Color online) The time-evolution of the normalized magnetization components mj. Here, we have set z=290 with an initial angle\nmisalignment q0=p=0:1. We have considered the weak damping regime a\u001c1 in the upper row, setting (a) w=0:5 and a=0:05 while in\n(b)w=1:5 and a=0:05. In the lower row, we have considered stronger Gilbert damping and set (c) w=0:5 and a=0:5 while in (d) w=1:5\nanda=0:5. The insets display a parametric plot of the ˆyyy- and ˆzzz-components of the magnzetization with the red circles indicating the point\nt=0 (the initial magnetization configuration). The time-span in insets is t2[0;50].\nBy application of an external field HHHext, it is thus possible to\ntune the relative orientation of the local magnetization fields in\nthe two layers. When this junction is current-biased, a super-\ncurrent flows without resistance up to a critical strength. The\nsupercurrent strongly modifies the equilibrium exchange in-\nteraction between the ferromagnetic layers and should thus be\nexpected to result in supercurrent-induced magnetization dy-\nnamics. We investigate this by solving numerically the LLG\nequation, which provides the time-evolution of the magneti-\nzation components in the soft magnetic layer. As we shall\nsee, a number of interesting opportunities arise in terms of the\ntorque exerted on the soft layer by the Josephson current. We\nproceed by first establishing the theoretical framework used in\nthe forthcoming analysis and then present our main results.\nTheory . The magnetization dynamics of the Josephson\njunction is governed by the LLG equation, which in the freemagnetic layer reads:\n¶mmmL\n¶t=\u0000gmmmL\u0002HHHeff+a\u0010\nmmmL\u0002¶mmmL\n¶t\u0011\n; (1)\nwhere gis the gyromagnetic ratio and ais the damping con-\nstant. The effective magnetic field is obtained from the free\nenergy functional via the relation HHHeff=\u0000(dF=dmmmL)=(VM0)\nwhere Vis the unit volume and M0is the magnitude of the\nmagnetization. Note that only transverse magnetization dy-\nnamics are described by the LLG equation due to the cross-\nterms coupling to mmmL, meaning thatjmmmLj=1 is constant in\ntime. To study the magnetization dynamics induced by the\nsupercurrent, we must identify the effective field. To this end,\nwe employ the following phenomenological form [9] stem-\nming from the exchange interaction generated by the Joseph-3\nson current:\nFS=AD0\nl2\nFcosf[J1(mmmL\u0001mmmR)+2J2(mmmL\u0001mmmR)2\u0000J2];(2)\nwhere Ais the unit area, lFis the Fermi wavelength, and D0\nis the gap magnitude. The parameters Jiare analogues to\nthe quadratic and biquadratic coupling constants for a mag-\nnetic exchange interaction. In addition to Eq. (2), one should\nalso include the anisotropy contribution FMto the free energy\nwhich provides an effective field HHHM\neff= (Km y=M0)ˆyyywhere\nKis the anisotropy constant. We have here assumed that the\nanisotropy axiskˆyyy. After some algebra, one arrives at the final\nform of the LLG equation in our system:\n¶mmmL\n¶t=mmmL\u0002\u0010\n\u0000gKm y\nM0ˆyyy+mmmRgD0cosf(t)\ndM0l2\nF[J1+4J2(mmmL\u0001mmmR)]\n+a¶mmmL\n¶t\u0011\n(3)\nwhere f(t) =wJtanddis the thickness of the ferromagnetic\nlayer. Importantly, we note that the anisotropy contribution\nproportional to Kwasnot included in the effective field used\nin previous works [9]. This contribution is nevertheless essen-\ntial since the supercurrent-induced torque must overcome the\nanisotropy contribution in order to switch the magnetization\norientation. In what follows, we will present a full numeri-\ncal solution of this equation to investigate the supercurrent-\ninduced magnetization dynamics. To this end, we first estab-\nlish experimentally relevant values of the parameters in Eq.\n(3). The above equation may be cast into a dimensionless\nform by introducing wF=gK=M0,t=wFt,w=wJ=wF,\nandz=D0=(Kdl2\nF). Here, wFis the ferromagnetic reso-\nnance frequency and wJis the Josephson frequency. Employ-\ning a realistic estimate [16] for transmission probabilities in\nthe FjNjF part of the system, one finds [9] that Eq. (2) ac-\ncounts well for the Josephson current when J1=0:007 and\nJ2=0:025. For a permalloy with weak anisotropy, one may\nestimate [15] K'4\u000210\u00005K˚A\u00003. Moreover, we set D0=1\nmeV , lF=1˚A, and d=10 nm as standard values [17] for\nthe hybrid structure under consideration. The Josephson fre-\nquency wJis typically of order GHz, but may be tuned ex-\nperimentally. We will therefore consider several choices of\nw=wJ=wFand the damping constant ato model a variety of\nexperimentally accessible scenarios. As long as the require-\nment ¯ hwJ\u001cTcis fulfilled, i.e. the time-dependent part is a\nsmall perturbation, one may consider f(t)as a time-dependent\nexternal potential in the static expression for the free energy\n[13]. Since typically wJ\u00181µV , this condition is easily met.\nThe supercurrent-induced magnetization dynamics come into\nplay when the local magnetizations fmmmL;mmmRgare misaligned\nwith an angle q0=q(t=0)to begin with, and vanishes when\nq0=f0;pg. The magnetization in the right (hard) layer is\nfixed at mmmR=ˆyyy.\nResults and Discussion. Using the parameters discussed\nabove, we now proceed to investigate the resulting magneti-\nzation dynamics for both weak damping (a\u001c1)and moreconsiderable damping (a\u00181). Choosing a small initial an-\ngle of misalignment q0=p=0:1, the numerical solution of\nthe LLG-equation is shown in Fig. 2. The upper row shows\nthe weak-damping regime for two different choices of the fre-\nquency ratio ( w<1 and w>1). The qualitative behavior is\nsimilar in (a) and (b): the supercurrent-induced torque induces\noscillations of the magnetization components, but is unable to\nswitch the magnetization direction in the soft layer from its\noriginal configuration mmmL'ˆyyy. The oscillations eventually die\nout and the magnetization orientation of the two layers sat-\nurates in a parallel configuration. This happens on a faster\ntime-scale in (b) due to the larger value of wused compared\nto (a). Consider now the case of stronger Gilbert damping\nshown in the lower row of Fig. 2. The a.c. Josephson current-\ninduced torque now induces more ”violent” oscillations and\nis able to rotate the magnetization orientation by p. When\nthe frequency is large enough, exemplified in (d) by w=1:5,\nfull magnetization reversal is achieved and maintained. On\na larger time-scale, the same behavior is seen for lower fre-\nquencies. In general, we observe in our numerical simula-\ntions that the main difference between the weak and strong\nGilbert damping regime is that the magnetization dynamics is\nperiodic and oscillating in the former case, whereas it tends\nto saturate to a fixed orientation in the latter case. This is\nreasonable as the damping effectively leads to a more rapid\ndecay of the oscillating magnetization dynamics induced by\nthe a.c. current. The results in Fig. 2 suggest that it is pos-\nsible to generate supercurrent-induced magnetization rever-\nsal in the setup shown in Fig. 1. This phenomenon should\nbe intimately linked with the spin-triplet correlations present\nin the non-collinear setup considered here since spin-singlet\ncorrelations cannot carry torque. Recently, controllable long-\nrange triplet supercurrents have been experimentally observed\nin Josephson junctions with strong inhomogeneous ferromag-\nnetic layers [18–20]. In the present setup, torque carried by\nthe supercurrent does not become long-ranged [21] (an addi-\ntional source of triplet correlations would be required for that\npurpose [22]) whereas it is still due to spin-triplet, thus me-\ndiating magnetic correlations between the two ferromagnetic\nlayers.\nWe proceed by investigating the role of the anisotropy pref-\nerence in the soft magnetic layer. This is modelled by the\nterm proportional to Kin Eq. (3). The parameter zeffectively\nmodels the relative weight of the anisotropy energy and the\nJosephson energy related to the presence of superconductiv-\nity. In Fig. 3(a), we focus on the behavior of the magnetiza-\ntion component my. We plot its time-evolution for increasing\nvalues of z. It is seen that supercurrent-induced magnetiza-\ntion reversal occurs for z>zcwith a critical value zc(for\nthe present parameters, we find zc'163). The magnetization\nthen saturates, and the magnitude of the oscillations decreases\naszbecomes larger. In Fig. 3(b)-(d), we give a phase-diagram\nfor the magnetization switching by plotting my(t!¥)in the\na\u0000z,z\u0000w, and a\u0000wplane. In general, the results indi-\ncate that the torque generated by the a.c. Josephson effect\ncan reverse the magnetization orientation when the Gilbert4\n0510152025303540−1−0.500.51\nτmy(τ)(a)\n  \n140\n150\n160\n170\n180\n190\n2003004000.10.20.30.40.5\nζ(b)α\n1 1.50.10.20.30.40.5(d)\nωα\n1 1.5140160180200220240(c)\nωζζ=\nP-regime(switched)AP-regime\nFIG. 3: (Color online) (a) The time-evolution of the normalized mag-\nnetization component my. Here, we have set w=1:1 and a=0:4\nwith an initial angle misalignment q0=p=0:1. Above a critical\nvalue of z, permanent switching occurs from mmmLkˆyyytommmLk(\u0000ˆyyy).\nThe phase-diagram for magnetization reversal [from the parallel (P)\nconfiguration mmmLkmmmRto the anti-parallel (AP) configuration mmmLk\n(\u0000mmmR)] is shown in (b-d). A contour-plot is given for my(t!¥)\nwith (b) w=0:5, (c) a=0:5, and (d) z=200.\ndamping is non-neglible, the anisotropy contribution is suf-\nficiently weak and Josephson frequency is sufficiently small.\nThe phase-diagram in (b) and (d) shows that the magnetization\ncan be trapped in the switched direction when the damping a\nis sufficiently large, in agreement with the results in Fig. 2.\nIt should also be noted that the switching becomes less viable\naswincreases as seen from (c) and (d). When wJ\u001dwF, the\ntorque from the Josephson current oscillates very fast com-\npared to the characteristic motion of the magnetization. In this\nway, the magnetization would experience an averaged torque\nover many oscillations, which results in small effect due to\npartial cancellation of the net torque.Conclusion. In summary, we have investigated\nsupercurrent-induced magnetization dynamics in a Josephson\njunction with two misaligned ferromagnetic layers by solving\nnumerically the Landau-Lifshitz-Gilbert equation. We have\ndemonstrated the possibility to obtain supercurrent-induced\nmagnetization switching for an experimentally realistic\nparameter set. It is clarified that for the realization of mag-\nnetization reversal, large Gilbert damping, small anisotropy\ncontribution or a small Josephson frequency are in general fa-\nvorable. These results constitute a superconducting analogue\nto conventional current-induced magnetization dynamics and\nindicate how spin-triplet supercurrents may be utilized for\npractical purposes in spintronics.\n[1] F. Bergeret et al. , Rev. Mod. Phys. 77, 1321 (2005); A. Buzdin,\nRev. Mod. Phys. 77, 935 (2005).\n[2] Y . Tserkovnyak et al. , Rev. Mod. Phys. 77, 1375 (2005).\n[3] J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[4] J. Clarke, Phys. Rev. Lett. 28, 1363 (1972).\n[5] P. M. Tedrow and R. Meservey, Phys. Rev. B 7, 318 (1973).\n[6] N. C. Yeh et al. , Phys. Rev. B 60, 10522 (1999).\n[7] S. Takahashi et al. , Phys. Rev. Lett. 82, 3911 (1999).\n[8] J. P. Morten et al. , Phys. Rev. B 72, 014510 (2005).\n[9] X. Waintal and P. Brouwer, Phys. Rev. B 65, 054407 (2002).\n[10] S. Takahashi, S. Hikino, M. Mori, J. Martinek, and S. Maekawa,\nPhys. Rev. Lett. 99, 057003 (2007).\n[11] E. Zhao and J. Sauls, Phys. Rev. B 78, 174511 (2008).\n[12] M. Houzet, Phys. Rev. Lett. 101, 057009 (2008).\n[13] F. Konschelle and A. Buzdin, Phys. Rev. Lett. 102, 017001\n(2009).\n[14] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjet. 8, 153 (1935);\nT. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[15] A. Y . Rusanov et al. , Phys. Rev. Lett. 93, 057002 (2004).\n[16] M. D. Stiles, J. Appl. Phys. 79, 5805 (1996); Phys. Rev. B 54,\n14679 (1996).\n[17] V . V . Ryazanov et al. , Phys. Rev. Lett. 86, 2427 (2001); T. Kon-\ntoset al. , Phys. Rev. Lett. 86, 304 (2001).\n[18] T. S. Khaire et al. , Phys. Rev. Lett. 104, 137002 (2010).\n[19] D. Sprungmann et al. , arXiv:1003.2082.\n[20] J. W. A. Robinson et al. , Science, 10 June 2010 (10.1126/sci-\nence.1189246).\n[21] I. B. Sperstad et al. , Phys. Rev. B 78, 104509 (2008)\n[22] M. Houzet and A. I. Buzdin, Phys. Rev. B 76, 060504 (2007)."    },    {        "title": "2111.09441v1.Chaotic_dynamics_of_a_suspended_string_in_a_gravitational_background_with_magnetic_field.pdf",        "content": "BARI-TH/21-730\nChaotic dynamics of a suspended string\nin a gravitational background with magnetic field\nP. Colangeloa, F. Giannuzziaand N. Losaccoa;b\naIstituto Nazionale di Fisica Nucleare, Sezione di Bari, Via Orabona 4, I-70126 Bari, Italy\nbDipartimento Interateneo di Fisica “M. Merlin”, Università e Politecnico di Bari,\nvia Orabona 4, 70126 Bari, Italy\nAbstract\nWe study the effects of a magnetic field on the chaotic dynamics of a string with\nendpoints on the boundary of an asymptotically AdS 5space with black hole. We\nstudy Poincaré sections and compute the Lyapunov exponents for the string per-\nturbed from the static configuration, for two different orientations, with position of\nthe endpoints on the boundary orthogonal and parallel to the magnetic field. We\nfindthatthemagneticfieldstabilizes the stringdynamics, withthelargestLyapunov\nexponent remaining below the Maldacena-Shenker-Stanford bound.\n1 Introduction\nThe aim of this paper is to study the effect of a constant and uniform magnetic field on\nthe chaotic behavior of a suspended string in a gravitational background. Among others,\none purpose is to scrutinize the stabilization role of the magnetic field for different orienta-\ntions of the string. The study follows the tests, carried out using holographic methods, of\nthe Maldacena-Shenker-Stanford (MSS) bound [1]. This conjectures, under general con-\nditions, that for a thermal quantum system at temperature Tsome out-of-time-ordered\ncorrelation functions involving Hermitian operators have an exponential time dependence\nin determined time intervals. The dependence is characterized by the exponent \u0015, and for\nsuch an exponent an upper bound (written in units where ~= 1andkB= 1) holds:\n\u001562\u0019T: (1)\nThe correlation functions are related to thermal expectation values of the squared com-\nmutator of two Hermitian operators at a time separation t, which quantify the effect of\none operator on measurements of the other one at a later time.arXiv:2111.09441v1  [hep-th]  12 Nov 2021The MSS bound has been inspired by the observation that in nature the black holes\n(BH) are the fastest “scramblers”: the time needed for a system near a BH horizon to loose\ninformation depends logarithmically on the number of the system degrees of freedom [2,3].\nConnectionsbetweenchaoticquantumsystemsandgravityhavebeeninvestigatedin[4–7].\nIn a holographic framework, a relation has been worked out between the size of the\noperators of the quantum theory on the boundary, which are involved in the temporal\nevolution of the perturbation, and the momentum of a particle falling in the bulk [8,9].\nHolographic methods have been used to challenge the MSS bound (1). In such studies\nthe quantum system is conjectured to be a 4dboundary theory dual to an AdS 5gravity\ntheory with a black hole [10–12]. Several investigations are described in [13–16]. Some\nstudies concern strings hanging in the bulk with endpoints on the boundary, which are\nthe holographic dual of a static quark-antiquark pair [17–20]. In such systems \u0015is the\nLyapunov exponent characterizing the chaotic behavior of the fluctuations around the\nstatic string configuration [21–23]. The studies include the case of quantum systems\ncharacterized by a global U(1)symmetry, and show that the chemical potential stabilizes\nthe chaotic dynamics [24].\nIt is interesting to consider the role of an external magnetic field on the chaotic\nbehaviour of the string. The magnetic field is relevant in different contexts, including\nheavy-ion collisions or condensed matter problems such as the Quantum Hall Effect and\nsuperconductivity at high temperatures. A general gravity dual for such systems should\ninclude a magnetic field [25–29]. The backreaction of an external magnetic field modi-\nfies the geometry of the 5dspacetime, the metric of which is determined by the Einstein\nequations. As a result, an anisotropy is introduced in the spatial directions. Moreover, in\na finite temperature system the relation between the position of the black-hole horizon,\nthe source of chaos in the 5dgeometry, and temperature, involved in the MSS relation in\nthe boundary theory, is modified by the magnetic field.\nIn this paper we aim at studying how the background magnetic field affects chaos for\nthe hanging string, how this depends on the string orientation, and if the MSS bound is\nsatisfied.\n2 Geometry with a magnetic field\nIn the gauge/gravity duality, a 4dboundary gauge theory at finite temperature is dual\nto a gravity theory in AdS 5with a black hole. A magnetic field is introduced in the\nholographic framework by a U(1)gauge field FMNwhich modifies the 5dgeometry. The\nmetric is determined solving the Einstein equations:\nRMN\u00001\n2gMN(R+ 12)\u0000TMN= 0 (2)\n2with the 5dstress-energy tensor\nTMN= 2 (gABFMAFNB\u00001\n4gMNF2): (3)\nFor a constant magnetic field Bin thex3directionFis given by F=Bdx1^dx2, hence\nthe only nonvanishing components are F12=\u0000F21=B. The Einstein equations have\nbeen solved perturbatively in the low- Band high temperature limits in Refs. [30–32]. The\nresult for the line element, having the general expression\nds2=gttdt2+g11(dx1)2+g22(dx2)2+g33(dx3)2+grrdr2(4)\nwithr>rh, reads:\ngtt=\u0000r2f(r); g 11=g22=r2h(r); g 33=r2q(r); grr=1\nr2f(r): (5)\nThe metric functions are [32]:\nf(r) = 1\u00002B2\n3r4logr+f4\nr4(6)\nq(r) = 1\u00002B2\n3r4logr (7)\nh(r) = 1 +B2\n3r4logr: (8)\nThe magnetic field breaks rotational invariance, hence g226=g33. The geometry has a\nhorizon, the position of which rhis found requiring f(rh) = 0. This gives f4=\u0000r4\nh+\n2\n3B2log(rh)and the blackening function f(r):\nf(r) = 1\u0000r4\nh\nr4\u00002B2\n3r4logr\nrh: (9)\nThe Hawking temperature Tdepends on the magnetic field:\nT=rh\n\u0019\u0012\n1\u0000B2\n6r4\nh\u0013\n: (10)\nThe metric given in terms of the functions (6)-(8) is obtained for large bulk coordinate r\nand lowB, and it is important to reckon the minimum value of rand the largest value of\nBfor which it is a good approximation of Eqs. (2),(3) and (5). In Fig. 1 the differences\nbetween the metric functions f(r),q(r)andh(r)in (6)-(8) and the corresponding ones\ncomputed for low Bin [30,31] and used in [33] are depicted setting r= 1:1and varying\nB. The comparison shows that the largest deviation between the two expressions of\n3Figure 1: ForFone of the metric functions f(r= 1:1)(green line), q(r= 1:1)(red dashed line)\nandh(r= 1:1)(blue dotted line), \u0001Fis the difference between each function in Eqs. (6)-(8) and\nthe corresponding one computed in Ref. [30,31,33], for rh= 1and varying the magnetic field B.\nFigure 2: ForFone of the metric functions f(r= 1:1)(green line), q(r= 1:1)(red dashed\nline) andh(r= 1:1)(blue dotted line), \u0001Fis the difference between each function in (6)-(8) and\nthe one numerically computed from the Einstein equations, for rh= 1and varying the magnetic\nfieldB.\nthe metric, resulting from the different approximations in determining the solution of\nthe Einstein equations, is in the function q. For larger values of the coordinate rthe\ndeviations are smaller. In Fig. 2 the differences between the metric functions in (6)-(8)\nand the ones obtained by a numerical solution of the Einstein equations are depicted\nfor the same value r= 1:1, varying the magnetic field B. The numerical solutions are\nobtained imposing as boundary conditions for r!1the asymptotic expansion in (6)-\n(8), plus additional terms up to O(1=r8). The parameter f4in the asymptotic functions\nis fixed imposing f(rh) = 0. The deviations between (6)-(8) and those in Refs. [30,31,33]\nare larger because in such references different conditions for q(rh)andh(rh)are imposed.\nIn view of this comparison, in our study the magnetic field is increased up to B= 1. For\n4the radial coordinate r, setting the horizon position rh= 1in all our study, we consider\nr>1:1.\n3 String profile in the gravitational background\nWe consider a string described by the functions r(t;`)andxi(t;`), with fixed endpoints\non the AdS boundary r!1. We consider two configurations, i= 1andi= 3. In the\nformer (latter) configuration the string endpoints lie on a line orthogonal (parallel) to the\nmagnetic field. (t;`)are the worldsheet coordinates, with `the proper distance measured\nalong the string. In the probe approximation we ignore the backreaction to the metric\n(4)-(8).\nThe string dynamics is governed by the Nambu-Goto (NG) action:\nS=\u00001\n2\u0019\u000b0Z\ndtd`p\n\u0000h; (11)\nwhere\u000b0is the string tension and his the determinant of the induced metric hij=\ngMN@XM\n@\u0018i@XN\n@\u0018j, with\u0018i;jthe worldsheet coordinates and gthe metric tensor (5). In the\nstatic case the action reads:\nS=\u0000T\n2\u0019\u000b0Z\nd`q\njgttgii\u0013xi2+gttgrr\u0013r2j; (12)\nwhere \u0013xdenotes the derivative with respect to `.xiis a cyclic coordinate, so its conjugate\nmomentum\n@L\n@\u0013xi=\u0000T\n2\u0019\u000b0jgttjgii\u0013xiq\njgttjgii\u0013xi2+jgttjgrr\u0013r2(13)\nis a constant of motion. Denoting with r(`= 0) =r0the position of the tip of the string\nin the bulk, i.e. the point wheredr\ndxi\f\f\f\n`=0= 0, we have:\np\njgttjgii\u0013xip\ngii\u0013xi2+grr\u0013r2=p\njgttjgii\f\f\f\n`=0: (14)\nMoreover, from the condition\nd`2=giidx2\ni+grrdr2(15)\nthe equations determining the string profile can be obtained:\n\u0013x=\u0006p\n\u0000gtt(r0)gii(r0)p\u0000gttgii(16)\n\u0013r=\u0006p\n\u0000gttgii+gtt(r0)gii(r0)p\u0000gttgiigrr: (17)\n5We set as boundary conditions that the string endpoints lie on the AdS 5boundary at\nxi=\u0006L=2. The minimum value r0of the coordinate ris reached at xi= 0(or`= 0).L\nandr0are related, since\nL= 2Z1\nr0dr \ngii(r)\ngrr(r)\u0012gtt(r)gii(r)\ngtt(r0)gii(r0)\u00001\u0013!\u00001\n2\n: (18)\nThe function L(r0)withB= 1is plotted in Fig. 3. It has a maximum separating unstable\nstrings (red dotted line in the figure), corresponding to positive energies, from metastable\n(blue dashed line) and stable strings (black solid line) corresponding to negative energies\n[24]. In the following we focus on the unstable string solutions, varying the magnetic field\nin the range B61.\nx3configurationx1configuration1.0 1.2 1.4 1.6 1.8 2.00.40.50.60.70.80.9\nr0L\nFigure 3: Distance between the string endpoints on the boundary vs the position of the tip of\nthe string, obtained using Eq. (18) with rh= 1andB= 1, for the two string configurations.\n4 Perturbing the static solution\nTo observe the onset of chaos the static solution of the string near the black-hole horizon\nmust be perturbed by a small time-dependent effect. We introduce a perturbation of the\nstring along the orthogonal direction at each point with coordinate `in ther\u0000xplane,\nfor both the x=x1andx=x3configurations [21,24]. The perturbation is depicted in\nFig. 4. Considering the unit vector nM= (0;nx;0;0;nr)orthogonal to tM, we have:\ngrr(r) (nr)2+gxx(r) (nx)2= 1 (19)\n\u0013r(`)grr(r)nr+ \u0013x(`)gxx(r)nx= 0: (20)\n6For an outward perturbation as in Fig. 4 the solution for the components nxandnris\nnx(`) =rgrr\ngxx\u0013r(`),nr(`) =\u0000rgxx\ngrr\u0013x(`): (21)\nThe time-dependent perturbation \u0018(t;`)modifiesrandx:\nr(t;`) =rBG(`) +\u0018(t;`)nr(`);\nx(t;`) =xBG(`) +\u0018(t;`)nx(`);\nwhererBG(`)andxBG(`)are the static solutions obtained integrating Eqs. (16) and (17).\nTo describe the dynamics of the small perturbation, we expand the metric function\naround the static solution rBG(`)to the third order in \u0018(t;`). To this order in \u0018the NG\naction comprises a quadratic and a cubic term. The quadratic term has the form:\nS(2)=1\n2\u0019\u000b0Z\ndtZ1\n\u00001d`\u0010\nCxi\ntt_\u00182+Cxi\n``\u0013\u00182+Cxi\n00\u00182\u0011\n(i= 1;3)(22)\nfor the two chosen string orientations. Cxi\ntt,Cxi\n``andCxi\n00depend on`. For the geometry in\nEq. (4) with metric functions f(r),h(r)andq(r)the coefficients Cxi\ntt,Cxi\n``andCxi\n00read:\nℓ\nrHr0r∞\nt\nnξL\n2\nℓ\nrHr0r∞ L\n2\nFigure 4: Static string profile, and perturbation \u0018(t;`)along the direction orthogonal to the\nstring in each point having coordinate `.\n7Cxi\ntt(`) =1\n2rBGp\nf(rBG);\nCxi\n``(`) =\u00001\n4Cxi\ntt(`);\nCx1\n00(`) =1\n8r3\nBGf(rBG)3=2h(rBG)2\u0010\u0000\nr4\n0f(r0)h(r0)\u0000\n2r2\nBGh(rBG)f0(rBG)2\n+ 2f(rBG)2\u0000\n4h(rBG) +rBGh0(rBG)\u0001\n+rBGf(rBG)\u0000\nrBGf0(rBG)h0(rBG)\n+ 2h(rBG)\u0000\nf0(rBG)\u0000rBGf00(rBG)\u0001\u0001\u0001\n\u0000r4\nBGf(rBG)2\u0000\n2rBGh(rBG)f0(rBG)\n\u0000\n2h(rBG) +rBGh0(rBG)\u0001\n+f(rBG)\u0000\n8h(rBG)2\u0000r2\nBGh0(rBG)2\n+ 2rBGh(rBG)\u0000\n4h0(rBG) +rBGh00(rBG)\u0001\u0001\u0001\u0011\n:(23)\nCx3\n00has the same expression of Cx1\n00, with the metric function h(r)replaced by q(r). The\ncoefficients depend on `throughrBG(`). The metric functions f(r),h(r)andq(r)are\ndefined in Eqs. (6)-(8).\nThe equation of motion from the action (22) is\nCxi\ntt\u0018+@`\u0010\nCxi\n``\u0013\u0018\u0011\n\u0000Cxi\n00\u0018= 0 ( i= 1;3): (24)\nFactorizing \u0018(t;`) =\u0018(`)ei!tit corresponds to the Sturm-Liouville equation\n@`\u0010\nCxi\n``\u0013\u0018\u0011\n\u0000Cxi\n00\u0018=!2Cxi\ntt\u0018 ; (25)\nwithW(`) =\u0000Cxi\ntt(`)the weight function. We solve Eq. (25) for different values of B.\nSince we are interested in unstable configurations we set r0= 1:1near the horizon, and\nimpose the boundary conditions \u0018(`)`!\u00061\u0000\u0000\u0000\u0000! 0. The two lowest lying eigenvalues !2\n0and\n!2\n1, obtained varying Bfor the two different string configurations, are collected in Table 1.\nThe corresponding eigenfunctions \u0018(`) =e0(`)and\u0018(`) =e1(`), for one configuration of\nthe string, are depicted in Fig. 5. Negative eigenvalues corrispond to unstable systems.\nConsidering the results in Table 1, we conclude that the effect of Bis to stabilize the\nsystem, since !2\n0increases with B. The effect of the magnetic field Bis stronger for the\nstring in the x3direction, parallel to the magnetic field, hence the magnetic field stabilizes\nthe system in the x3configuration more than in the x1configuration.\nTo observe the chaotic behaviour we study the contribution of the third order terms\nin\u0018in the action. Up to a surface term, the expression is\nS(3)=1\n2\u0019\u000b0Z\ndtZ1\n\u00001d`\u001a\nD0\u00183+Dxi\n1\u0018\u0013\u00182+Dxi\n2\u0018_\u00182\u001b\n; (26)\n8ω0\nω1\n-2 -1 0 1 2-1.5-1.0-0.50.00.51.01.52.0\nl\nω0\nω1\n-2 -1 0 1 2-1.5-1.0-0.50.00.51.01.52.0\nlFigure 5: Eigenfunctions e0(`)(black line) and e1(`)(red line) of Eq. (25), for r0= 1:1and\nB= 0(left),B= 0:6(right panel).\nwithDxi\n0;1;2functions of `. Expanding the perturbation in terms of the first two eigenfunc-\ntionse0ande1,\n\u0018(t;`) =c0(t)e0(`) +c1(t)e1(`); (27)\nthe time dependence of the perturbation is encoded in the coefficients c0(t)andc1(t).\nWith this form of \u0018(t;`)we have:\nS(3)=\n1\n2\u0019\u000b0Z\ndtZ1\n\u00001d`h\u0010\nDxi\n0e3\n0+Dxi\n1e0\u0013e2\n0\u0011\nc3\n0(t) +\u0010\n3Dxi\n0e0e2\n1+Dxi\n1\u0000\n2\u0013e0e1\u0013e1+e0\u0013e2\n1\u0001\u0011\nc0c2\n1\n+Dxi\n2\u0010\ne0e2\n1c0_c2\n1+e3\n0e2\n1c0_c2\n0+ 2e0e2\n1_c0c1_c1\u0011i\n: (28)\nx1configuration x3configuration\nB !2\n0!2\n1 B !2\n0!2\n1\n0 -1.370 7.638 0 -1.370 7.638\n0.3 -1.327 7.531 0.3 -1.317 7.548\n0.6 -1.202 7.213 0.6 -1.166 7.278\n0.9 -1.004 6.694 0.9 -0.932 6.823\n1 -0.923 6.478 1 -0.841 6.629\nTable 1: Eigenvalues !2\n0and!2\n1of Eq. (25) for r0= 1:1and varying B, for the two string\nconfigurations.\n9The action for c0(t)andc1(t)is obtained by the sum S(2)+S(3), integrating over `:\nS(2)+S(3)=1\n2\u0019\u000b0Z\ndthX\nn=0;1\u0000\n_c2\nn\u0000!2\nnc2\nn\u0001\n+Kxi\n1c3\n0+Kxi\n2c0c2\n1+Kxi\n3c0_c2\n0+Kxi\n4c0_c2\n1+Kxi\n5_c0c1_c1i\n:\n(29)\nIn Eqs. (26)-(29) the index is i= 1ori= 3. The coefficients Kxi\n1;:::;5depend on r0and\nB. They are collected in Tab. 2 for r0= 1:1and different values of B, for the two string\nconfigurations.\nThe potential described by Eq. (29) has a trap for the unstable string configurations.\nWe are interested in the motion of c0andc1in the trap. In some regions of the potential\nthe kinetic term is negative. As shown in [21,24], it is useful to replace c0;1!~c0;1in the\naction, with c0= ~c0+\u000b1~c2\n0+\u000b2~c2\n1andc1= ~c1+\u000b3~c0~c1, neglectingO(~c4\ni)terms, setting\nthe constants \u000biensuring the positivity of the kinetic term. We set \u000b1=\u00002,\u000b2=\u00000:5\nand\u000b3=\u00001. This replacement stretches the potential and stabilizes the time evolution\nof the system. The dynamics is not affected, and a chaotic behaviour shows up also in\nthe transformed system.\nx1configuration\nB K 1K2K3K4K5\n0 11.36 21.72 10.58 3.37 6.73\n0.3 10.89 21.17 10.50 3.36 6.72\n0.6 9.57 19.56 10.24 3.33 6.67\n0.9 7.63 17.05 9.83 3.29 6.58\n1 6.90 16.05 9.65 3.27 6.55\nx3configuration\nB K 1K2K3K4K5\n0 11.36 21.72 10.58 3.37 6.73\n0.3 10.97 21.22 10.52 3.37 6.74\n0.6 9.85 19.76 10.32 3.38 6.77\n0.9 8.16 17.44 9.99 3.41 6.81\n1 7.50 16.50 9.85 3.41 6.83\nTable 2:Kicoefficients in Eq. (29) for the two string configurations, for r0= 1:1and varying\nthe magnetic field B.\n10-0.0010 -0.0008 -0.0006 -0.0004 -0.0002 0.0000-0.0010-0.00050.00000.00050.0010\nc˜\n0c˜\n0\n-0.0010 -0.0008 -0.0006 -0.0004 -0.0002 0.0000-0.0010-0.00050.00000.00050.0010\nc˜\n0c˜\n0\n-0.0008 -0.0006 -0.0004 -0.0002 0.0000-0.0004-0.00020.00000.00020.0004\nc˜\n0c˜\n0\n-0.0008 -0.0006 -0.0004 -0.0002 0.0000-0.0004-0.00020.00000.00020.0004\nc˜\n0c˜\n0\n-0.00020 -0.00015 -0.00010 -0.00005 0.00000-0.00015-0.00010-0.000050.000000.000050.000100.00015\nc˜\n0c˜\n0\n-0.00020 -0.00015 -0.00010 -0.00005 0.00000-0.00015-0.00010-0.000050.000000.000050.000100.00015\nc˜\n0c˜\n0Figure 6: Poincaré sections for a perturbed string in the x1(left column) and x3configurations\n(right column). The initial conditions are changed with fixed energy E= 10\u00005andr0= 1:1.\nThe magnetic field is increased from B= 0:3(top row) to B= 0:6(middle row) and B= 1\n(bottom row). The sections correspond to ~c1= 0and_~c1>0.\n5 Poincaré sections and Lyapunov exponents\nThe onset of chaos is displayed by the Poincaré sections. We construct the sections\ndefined by ~c1(t) = 0and _~c1(t)>0for bounded orbits within the trap in the potential.\nForr0= 1:1and increasing Bthe sections are depicted in Fig. 6. For ~c0near zero\nthe orbits are scattered points which depend on the initial conditions. Increasing Bthe\npoints in the sections form more regular paths, showing that the effect of switching on\nthe magnetic field is to mitigate the chaotic behavior.\nIn Fig. 6 we observe that when the string is along x3we need to go closer to ~c0= 0,\ncloser to the horizon, to observe chaos. This confirms the observation that the strongest\nstabilization effect of the magnetic field is in the x3configuration.\n11Figure 7: Convergency plots of the four Lyapunov exponents for a string along x1, withr0= 1:1\nandB= 0:3(right panel), B= 0:6(left panel). 2\u0002105time steps are shown. For the initial\nconditions, the energy is set to E= 10\u00005together with ~c0=\u00000:0002,_~c0= 0,~c1= 0:0011.\nFor a better understanding of the amount of chaos we evaluate the Lyapunov expo-\nnents. Such exponents in the four dimensional c0,c1phase-space can be computed for\ndifferent values of Busing the numerical method described in [34]. The results are shown\nin Fig. 7. The convergency plot is a damped oscillating function. The value of the largest\nLyapunov exponent can be extrapolated fitting the maximum in each oscillation and con-\nsideringt!+1. The values obtained from the fit decrease as Bincreases, as shown in\nFig. 8: the effect of the magnetic field is to soften the dependence on the initial conditions,\nmaking the string less chaotic. In the numerical procedure it is checked that the sum of\nthe Lyapunov exponents vanishes at large t.\nIn Fig. 8 the results for the two configurations are compared. For the same values of\nFigure 8: Largest Lyapunov exponent \u0015MAXversusBforr0= 1:1. The results for the x1(blue\npoints) and x3string configurations (red points) are shown.\n1200.20.40.60.811.2\n0 1 2 3 4 5 6 7 8λMAX (t)\ntB= 1\nB= 0.6\nB= 0.3\nB= 0Figure 9: Early time convergency plots of the largest Lyapunov exponent, for the x1configu-\nration,r0= 1:1andBindicated in the legend. The energy is set to E= 10\u00005, with the initial\nconditions ~c0=\u00000:0002,_~c0= 0and~c1= 0:0011. The dashed lines show the largest Lyapunov\nexponents at the saddle point discussed in Sec. 6.\nBandr0, so at the same distance from the BH horizon, smaller Lyapunov exponents are\nfound in the x3configuration.\nThePoincaréplotsshowthatchaosisproducedintheproximityoftheBHhorizon,and\nthat the string dynamics is less chaotic if the magnetic field increases. This is confirmed\nby the largest Lyapunov exponent. The effect of the BH horizon can be observed looking\nat the first steps of the Lyapunov convergency plots shown in Fig. 9. In the early times,\nas long as the system is inside a region near ~c0\u00180, the exponents approach a nearly\nconstant value. When the system is far from the origin, they begin to oscillate and drop\nto a lower asymptotic value shown in Fig. 7. This early time behavior is observed when in\ngeneral the trajectory crosses a chaotic region of the phase-space. If initial conditions are\nof a trajectory that does not come close to the origin, such a behavior is not observed. For\nthe time evolution of the convergency plots stopped before the plateaux start decreasing,\nhigher values of the largest Lyapunov exponents with respect to the asymptotic ones in\nFig. 8 would be found. The snapshot of the time evolution near the BH horizon shown in\nFig. 9 further shows that the BH horizon is the source of chaos. The role of the magnetic\nfield to stabilize the system is also displayed by the early time behaviour.\n6 Analysis of the saddle point\nIntheprevioussectionwehavecomputedtheLyapunovexponentsofanextendedbounded\norbit in the phase space, and we have obtained positive values for the largest Lyapunov\nexponents proving that the system is chaotic. In order to complete the analysis of Lya-\npunov exponents and challenge the MSS bound, in this section we compute the Lyapunov\n13Figure 10: Potential obtained from Eq. (29) for B= 1.\nexponents at the unstable fixed point. However, we remark that the sign of the largest\nLyapunov exponent at the fixed point only indicates the stability of that point (it is pos-\nitive for unstable fixed points), but it carries no information about the dynamics of the\nwhole system and cannot be used to argue if the system is chaotic.\nThe evolution of the system governed by the action in Eq. (29) is dictated by the\nequation _~ x=~F, with~ x= (~c0;_~c0;~c1;_~c1). There are two fixed points where ~F= 0: a stable\nfixed point corresponding to the local minimum of the potential obtained from Eq. (29),\nand an unstable fixed point corresponding to the saddle point of the potential, shown in\nFig. 10 for a set of parameters.\nIf an unstable unperturbed string is chosen as solution of Eqs. (16)-(17), for the energy\nE= 0the unstable fixed point is at ~ x= (0;0;0;0). At a fixed point the formula defining\nthe Lyapunov exponents gets simpler [34], and they can be computed analytically as the\nreal part of the eigenvalues of the Jacobian matrix of ~F. At the point (0;0;0;0)this\nJacobian reads:\nJ=0\nBB@0\u0000!2\n00 0\n1 0 0 0\n0 0 0\u0000!2\n1\n0 0 1 01\nCCA; (30)\nand its eigenvalues are (\u0000ip\n!2\n0;ip\n!2\n0;\u0000ip\n!2\n1;ip\n!2\n1). Hence, the Lyapunov exponents\nvanish for!2\ni>0, as for stable solutions. Fig. 11 shows the largest Lyapunov exponents\n\u0015MAX =p\n\u0000!2\n0at the fixed point (0;0;0;0), forB= 1and varying r0. In Fig. 12 the\nexponentsareplottedasafunctionof Bforr0= 1:1. ThevaluesoftheLyapunovexponent\nat this point are large, but they remain below the MSS bound \u0015MSS = 2rh\u0010\n1\u0000B2\n6r4\nh\u0011\n(dashed line in the plots) when the string tip gets close to the black hole horizon, r0!rh.\nAs a final check, in Fig. 13 it is shown that the Lyapunov exponents computed at the\nfixed point (0;0;0;0)by the numerical procedure of Ref. [34] used in the previous section,\n14■■■■\n■\n■\n■\n■●\n●\n●\n●\n●\n●\n1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.170.00.51.01.5\nr0λMAXFigure 11: Largest Lyapunov exponent at the unstable fixed point (0;0;0;0), as a function of\nr0, forB= 1. The blue (red) points refer to the x1(x3) string configuration. The dashed line\nshows the MSS bound in Eq. (1): \u0015MSS = 2rh\u0012\n1\u0000B2\n6r4\nh\u0013\n.\nare equal to the analytical ones (p\n\u0000!2\n0;\u0000p\n\u0000!2\n0;0;0).\n7 Conclusions\nOur investigation of a suspended string in a gravitational background with a black hole,\nthe holographic dual of the heavy quark-antiquark system in a thermal environment,\nconfirms the MSS bounds (1) also in the case of a uniform and constant magnetic field.\n■■■■■■■■■■■●●●●●●●●●●●\n0.0 0.2 0.4 0.6 0.8 1.00.81.01.21.41.61.82.0\nBλMAX\nFigure 12: Largest Lyapunov exponent computed at the unstable fixed point (0;0;0;0)as a\nfunction of Bforr0= 1:1. The symbols are as in Fig. 11.\n150 5 10 15 20 25 30-1.0-0.50.00.51.0\ntλFigure 13: Convergency plot of the Lyapunov exponents for r0= 1:1andB= 1computed\nat the unstable fixed point (0;0;0;0). The horizontal lines correspond to the analytical values\nof the nonvanishing Lyapunov exponentsp\n\u0000!2\n0(top dashed blue line) and \u0000p\n\u0000!2\n0(bottom\ndashed orange line).\nThe system becomes less chaotic increasing B. The anisotropy effect in two different\norientations of the string is found. This conclusion is analogous to the one obtained\nfor different geometries, namely AdS-RN [24], as well as studying the charged particle\nmotion in such a kind of background [35]. Chaos has been observed in the Poincaré\nplots, characterized by scattered points in the region close to the black-hole horizon, and\nquantitatively described computing the Lyapunov exponents, finding that the largest one\nverifies the MSS bound. The stabilization effect of the magnetic field is stronger for the\nstring endpoints lying on a line parallel to the field, keeping the black-hole horizon and\nthe position of the tip of the string fixed. The largest Lyapunov exponents are below\nthe MSS bound also at the fixed unstable point of the potential describing the perturbed\nstring approaching the horizon.\nAcknowledgements. We thank F. De Fazio and S. Nicotri for discussions. This study\nhas been carried out within the INFN project (Iniziativa Specifica) QFT-HEP.\nReferences\n[1] J. Maldacena, S. H. Shenker, and D. Stanford, A bound on chaos ,JHEP 08(2016)\n106, [ arXiv:1503.01409 ].\n[2] Y. Sekino and L. Susskind, Fast Scramblers ,JHEP 10(2008) 065,\n[arXiv:0808.2096 ].\n16[3] L. Susskind, Addendum to Fast Scramblers ,arXiv:1101.6048 .\n[4] S. H. Shenker and D. Stanford, Black holes and the butterfly effect ,JHEP 03(2014)\n067, [ arXiv:1306.0622 ].\n[5] S. H. Shenker and D. Stanford, Stringy effects in scrambling ,JHEP 05(2015) 132,\n[arXiv:1412.6087 ].\n[6] A. Kitaev, Hidden correlations in the Hawking radiation and thermal noise ,\nBreakthrough Prize Fundamental Physics Symposium, 11/10/2014, KITP seminar\n(2014).\n[7] J. Polchinski, Chaos in the black hole S-matrix ,arXiv:1505.08108 .\n[8] L. Susskind, Why do Things Fall? ,arXiv:1802.01198 .\n[9] A. R. Brown, H. Gharibyan, A. Streicher, L. Susskind, L. Thorlacius, and Y. Zhao,\nFalling Toward Charged Black Holes ,Phys. Rev. D 98(2018), no. 12 126016,\n[arXiv:1804.04156 ].\n[10] J. M. Maldacena, Wilson loops in large N field theories ,Phys. Rev. Lett. 80(1998)\n4859–4862, [ hep-th/9803002 ].\n[11] E. Witten, Anti-de Sitter space and holography ,Adv. Theor. Math. Phys. 2(1998)\n253, [ hep-th/9802150 ].\n[12] S. Gubser, I. R. Klebanov, and A. M. Polyakov, Gauge theory correlators from\nnoncritical string theory ,Phys. Lett. B 428(1998) 105–114, [ hep-th/9802109 ].\n[13] J. de Boer, E. Llabrs, J. F. Pedraza, and D. Vegh, Chaotic strings in AdS/CFT ,\nPhys. Rev. Lett. 120(2018), no. 20 201604, [ arXiv:1709.01052 ].\n[14] S. Dalui, B. R. Majhi, and P. Mishra, Presence of horizon makes particle motion\nchaotic,Phys. Lett. B 788(2019) 486–493, [ arXiv:1803.06527 ].\n[15] S. Dalui and B. R. Majhi, Near horizon local instability and quantum thermality ,\nPhys. Rev. D 102(2020), no. 12 124047, [ arXiv:2007.14312 ].\n[16] D. S. Ageev, Butterflies dragging the jets: on the chaotic nature of holographic\nQCD,arXiv:2105.04589 .\n[17] S. D. Avramis, K. Sfetsos, and K. Siampos, Stability of strings dual to flux tubes\nbetween static quarks in N = 4 SYM ,Nucl. Phys. B 769(2007) 44–78,\n[hep-th/0612139 ].\n17[18] R. E. Arias and G. A. Silva, Wilson loops stability in the gauge/string\ncorrespondence ,JHEP 01(2010) 023, [ arXiv:0911.0662 ].\n[19] C. Nunez, M. Piai, and A. Rago, Wilson Loops in string duals of Walking and\nFlavored Systems ,Phys. Rev. D 81(2010) 086001, [ arXiv:0909.0748 ].\n[20] L. Bellantuono, P. Colangelo, F. De Fazio, F. Giannuzzi, and S. Nicotri,\nQuarkonium dissociation in a far-from-equilibrium holographic setup ,Phys. Rev. D\n96(2017), no. 3 034031, [ arXiv:1706.04809 ].\n[21] K. Hashimoto, K. Murata, and N. Tanahashi, Chaos of Wilson Loop from String\nMotion near Black Hole Horizon ,Phys. Rev. D 98(2018), no. 8 086007,\n[arXiv:1803.06756 ].\n[22] T. Ishii, K. Murata, and K. Yoshida, Fate of chaotic strings in a confining\ngeometry ,Phys. Rev. D 95(2017), no. 6 066019, [ arXiv:1610.05833 ].\n[23] T. Akutagawa, K. Hashimoto, K. Murata, and T. Ota, Chaos of QCD string from\nholography ,Phys. Rev. D 100(2019), no. 4 046009, [ arXiv:1903.04718 ].\n[24] P. Colangelo, F. De Fazio, and N. Losacco, Chaos in a Q\u0016Qsystem at finite\ntemperature and baryon density ,Phys. Rev. D 102(2020), no. 7 074016,\n[arXiv:2007.06980 ].\n[25] R. Critelli, R. Rougemont, S. I. Finazzo, and J. Noronha, Polyakov loop and heavy\nquark entropy in strong magnetic fields from holographic black hole engineering ,\nPhys. Rev. D 94(2016), no. 12 125019, [ arXiv:1606.09484 ].\n[26] A. Ballon-Bayona, J. P. Shock, and D. Zoakos, Magnetic catalysis and the chiral\ncondensate in holographic QCD ,JHEP 10(2020) 193, [ arXiv:2005.00500 ].\n[27] I. Y. Aref’eva, K. Rannu, and P. Slepov, Holographic model for heavy quarks in\nanisotropic hot dense QGP with external magnetic field ,JHEP 07(2021) 161,\n[arXiv:2011.07023 ].\n[28] I. Y. Aref’eva, K. Rannu, and P. Slepov, Energy Loss in Holographic Anisotropic\nModel for Heavy Quarks in External Magnetic Field ,arXiv:2012.05758 .\n[29] I. Y. Aref’eva, K. Rannu, and P. S. Slepov, Anisotropic solutions for a holographic\nheavy-quark model with an external magnetic field ,Teor. Mat. Fiz. 207(2021),\nno. 1 44–57.\n[30] E. D’Hoker and P. Kraus, Magnetic Brane Solutions in AdS ,JHEP 10(2009) 088,\n[arXiv:0908.3875 ].\n18[31] E. D’Hoker and P. Kraus, Charged Magnetic Brane Solutions in AdS (5) and the\nfate of the third law of thermodynamics ,JHEP 03(2010) 095, [ arXiv:0911.4518 ].\n[32] D. Li, M. Huang, Y. Yang, and P.-H. Yuan, Inverse Magnetic Catalysis in the\nSoft-Wall Model of AdS/QCD ,JHEP 02(2017) 030, [ arXiv:1610.04618 ].\n[33] N. R. F. Braga, Y. F. Ferreira, and L. F. Ferreira, Configuration entropy and\nstability of bottomonium radial excitations in a plasma with magnetic fields ,\narXiv:2110.04560 .\n[34] M. Sandri, Numerical calculation of lyapunov exponents ,Mathematica Journal 6\n(1996), no. 3 78–84.\n[35] D. S. Ageev and I. Y. Aref’eva, When things stop falling, chaos is suppressed ,\nJHEP 01(2019) 100, [ arXiv:1806.05574 ].\n19"    },    {        "title": "1902.01105v3.Magnetic_eddy_viscosity_of_mean_shear_flows_in_two_dimensional_magnetohydrodynamics.pdf",        "content": "Magnetic eddy viscosity of mean shear \rows in two-dimensional\nmagnetohydrodynamics\nJe\u000brey B. Parker\u0003\nLawrence Livermore National Laboratory, Livermore, CA 94550\nNavid C. Constantinou\nResearch School of Earth Sciences and ARC Centre of Excellence for Climate Extremes,\nAustralian National University, Canberra, ACT, 2601, Australia\nMagnetic induction in magnetohydrodynamic \ruids at magnetic Reynolds number (Rm) less\nthan 1 has long been known to cause magnetic drag. Here, we show that when Rm \u001d1 and the\n\ruid is in a hydrodynamic-dominated regime in which the magnetic energy is much smaller than\nthe kinetic energy, induction due to a mean shear \row leads to a magnetic eddy viscosity. The\nmagnetic viscosity is derived from simple physical arguments, where a coherent response due to\nshear \row builds up in the magnetic \feld until decorrelated by turbulent motion. The dynamic\nviscosity coe\u000ecient is approximately ( B2\np=2\u00160)\u001ccorr, the poloidal magnetic energy density multiplied\nby the correlation time. We con\frm the magnetic eddy viscosity through numerical simulations\nof two-dimensional incompressible magnetohydrodynamics. We also consider the three-dimensional\ncase, and in cylindrical or spherical geometry, theoretical considerations similarly point to a nonzero\nviscosity whenever there is di\u000berential rotation. Hence, these results serve as a dynamical general-\nization of Ferraro's law of isorotation. The magnetic eddy viscosity leads to transport of angular\nmomentum and may be of importance to zonal \rows in astrophysical domains such as the interior\nof some gas giants.\nI. INTRODUCTION\nThe combination of rotation and magnetic \felds is common in astrophysical \ruid dynamics, found in stellar interiors,\ngas giant atmospheres, and astrophysical disks. In the outer atmosphere of planets, magnetic \felds do not play a\nsigni\fcant role; the \row is dominated by rotation, which leads to anisotropic \row in which turbulence, waves, and\ncoherent structures interact. A common feature is the presence of long-lived mean shear \rows, called zonal \row,\nwhich alternate in latitude. In Jupiter's atmosphere, for instance, zonal \rows are remarkably persistent [1]. However,\ndeeper into gas giant interiors, or in exoplanets, magnetic \felds may play an important role and in\ruence the \row.\nIn resistive magnetohydrodynamics (MHD), it is well known that when the magnetic Reynolds number is less than\none, \ruid \row across a magnetic \feld results in magnetic drag [2]. The magnetic Reynolds number Rm = LV=\u0011\nmeasures the relative strength of induction and magnetic di\u000busion, where Lis a characteristic length scale of the\nsystem,Vis a characteristic \ruid velocity, and \u0011is the magnetic di\u000busivity.\nThe regime Rm\u001d1 is more dynamical and complex. Within this regime we examine a subset of parameter space:\nwe assume large Reynolds number and consequently turbulent \row. We also assume a hydrodynamic-dominated\nregime where the in\ruence of the magnetic \feld is small. We characterize this latter restriction with the dimensionless\nparameter A\u001c1, where A=jJ\u0002Bj=j\u001av\u0001rvjis the ratio of the Lorentz force to the inertial force. If the energy-\ncontaining scale lengths of the magnetic \feld and velocity \feld are comparable, then equivalently A= (B2=\u00160)=(\u001av2),\ni.e., the ratio of magnetic energy density to kinetic energy density. Here, Jis the current density, Bthe magnetic\n\feld,vthe \ruid velocity, \u001athe \ruid mass density, and \u00160the permeability of free space.\nIn this paper, we suggest that within the above regime, the Lorentz force on mean shear \row acts not as a\nmagnetic drag but rather as an e\u000bective magnetic eddy viscosity. We demonstrate this result in two-dimensional (2D)\nincompressible MHD simulations. Under conditions where Aapproaches unity, the Lorentz force becomes appreciable.\nIf the mean shear \row is a turbulence-driven zonal \row, the magnetic eddy viscosity can be strong enough to suppress\nit. Figure 1 o\u000bers a schematic diagram showing the parameter space within which the magnetic eddy viscosity is\nexpected in a turbulent \row.\nThis paper is primarily a study of MHD physics in an idealized setup; however, a potential example where the\nregime Rm\u001d1 andA\u001c1 could be realized is in the interiors of gas giants. In the outer atmospheres of Jupiter\nand Saturn, the bulk \ruid is electrically insulating, but the conductivity increases rapidly with depth, giving way to\n\u0003parker68@llnl.govarXiv:1902.01105v3  [physics.plasm-ph]  27 Aug 20192\nmagnetic \ndragmagnetic \neddy \nviscosityzonal flow \nsuppressed\nFIG. 1. Schematic showing the magnetic response to a mean shear \row expected in a turbulent regime. As Rm increases, the\nmagnetic response transitions from a magnetic drag (for \row perpendicular to the magnetic \feld) for Rm <1 to a magnetic\neddy viscosity for Rm >1, assuming also A\u001c1. As Rm increases further, the magnetic energy rises to a value comparable to\nthe kinetic energy, A= 1. If the mean shear \row is turbulence-driven, such as zonal \row, then the mean \row is suppressed.\na conducting \ruid in the interior [3, 4]. Recent spacecraft measurements have shed light on long-standing questions\nregarding the depth of zonal \row in planetary interiors. Gravitometric observations indicate that the zonal \rows\nin Jupiter and Saturn terminate at a depth of approximately 3,000 and 8,500 km, respectively [5{8]. These depths\nare near the transition zone where the electrical conductivity increases rapidly. The greatly enhanced conductivity\nin the interior leads to a stronger in\ruence of the magnetic \feld on bulk \ruid motion. The coincident depths of\nthe suppression of zonal \row and the rise in conductivity in Jupiter and Saturn have led to the hypothesis that the\nmagnetic \feld is responsible for terminating di\u000berential rotation. Moreover, simulations of planetary dynamo models\nprovide some support for this possibility [9{12]. While magnetic drag at Rm <1 has been studied for gas giants\nand hot Jupiters as the mechanism to suppress di\u000berential rotation [13, 14], there remains a great deal of uncertainty\nabout parameter values. Intermediate regimes may exist in which Rm \u001d1 andA\u001c1.\nIdealized settings to study individual e\u000bects prove fruitful. For example, certain aspects of \ruid physics under\nthe in\ruence of rotation and magnetization can be studied in the 2D \fplane or 2D spherical surface [15{19]. A \f\nplane is a Cartesian geometrical simpli\fcation of a rotating sphere that retains the physics associated with rotation\nand the latitudinal variation of the Coriolis e\u000bect [20]. The unmagnetized \fplane is one of the simplest settings\nin which coherent zonal \rows emerge spontaneously from turbulence and is often used for idealized studies [21, 22].\nSuch idealizations eliminate many sources of complexity, including thermal convection, variation in the density and\nelectrical conductivity, and true 3D dynamics. Two-dimensional MHD studies have su\u000eced, however, to demonstrate\nmagnetic suppression of zonal \row [15{17]. Here, our numerical simulations continue with the 2D magnetized \fplane,\nalthough our physical considerations are quite general. The magnetic eddy viscosity also sheds light on earlier results\non the magnetized \fplane and magnetized 2D spherical surface.\nAn outline for the rest of this paper is as follows. In Sec. II, we derive the magnetic eddy viscosity of mean shear\n\row. In Sec. III, we recall the negative eddy viscosity for comparison. In Sec. IV, we describe the setup for our 2D\nsimulations. The simulation results are shown in Sect. V. Section VI provides the 3D generalization of the magnetic\neddy viscosity. Section VII applies these results to consider zonal \rows deep in planetary interiors. We summarize\nour results in Sec. VIII.\nII. MAGNETIC EDDY VISCOSITY\nIn this section we consider a mean shear \row v=U(y)^ xand investigate the zonally averaged magnetic force acting\nback on the \row due to the correlations induced by the \row. The zonal average of a quantity fis de\fned as\nfdef=1\nLxZLx\n0dxf; (1)\nwhereLxis the length of the domain in the xdirection, assumed to be periodic. The deviation from the zonal mean\nis denoted by a prime, i.e.,\nf0def=f\u0000f: (2)3\nThe Lorentz force F=J\u0002Bacting on a \ruid can be expressed as r\u0001T, where the Maxwell stress tensor Tis\nT=BB\n\u00160\u0000B2\n2\u00160I; (3)\nand the stress terms associated with the electric \feld are neglected within the MHD approximation. This formulation\nof the Lorentz force will be most convenient for our purposes, though we return to consideration of the current in\nAppendix A. The force per unit volume in the xdirection is\nFx=@Txx\n@x+@Tyx\n@y+@Tzx\n@z: (4)\nThe \frst term on the right-hand side (RHS) vanishes after a zonal average. In the remainder of this section, we\nassume a quasi-2D system and set Bz= 0 and@z= 0, so\nFx=@Tyx\n@y=@\n@yBxBy\n\u00160: (5)\nWe evaluate the coherent e\u000bect that a shear \row has on inducing correlations in the magnetic \feld and in the\ncorresponding backreaction on the shear \row. We assume a regime Rm \u001d1, in which the magnetic di\u000busivity has\nminimal e\u000bect on energy-containing length scales over short timescales and the magnetic \feld is frozen into the \ruid.\nThe induction equation is then\n@B\n@t=r\u0002(v\u0002B) =B\u0001rv\u0000v\u0001rB; (6)\nwhere the second equality applies for incompressible \row r\u0001v= 0, and r\u0001B= 0 has been used. Then\n@B\n@t=By@U\n@y^ x\u0000U@B\n@x: (7)\nConsider the evolution of the magnetic \feld from some arbitrary initial time t= 0 over a short increment \u0001 t. The\nmagnetic \feld at time \u0001 tis\nB(\u0001t) =\u0012\nBx+ \u0001tBy@U\n@y\u0000\u0001tU@Bx\n@x\u0013\n^ x\n+\u0012\nBy\u0000\u0001tU@By\n@x\u0013\n^ y+O(\u0001t2); (8)\nwhere the magnetic-\feld quantities on the RHS are evaluated at t= 0. At time \u0001 t, we can evaluate the zonally\naveraged Maxwell stress as\nBx(\u0001t)By(\u0001t) =BxBy+ \u0001tB2y@U\n@y\n\u0000\u0001t\u0012\nUBy@Bx\n@x+UBx@By\n@x\u0013\n+O(\u0001t2): (9)\nThe third and fourth terms on the RHS combine to become \u0000\u0001tU@x(BxBy), which is annihilated by the zonal average.\nWe obtain\n\u0001BxBy= \u0001tB2y@U\n@y+O(\u0001t2); (10)\nwhere \u0001BxBydef=Bx(\u0001t)By(\u0001t)\u0000Bx(0)By(0). We have thus derived a relation between the Maxwell stress and the\nshear. The periodicity constraint is essential in producing this exact result; the physical systems in which zonal \row\nis known to occur all have such periodicity. The derivation of Eq. (10) depends on nothing but the standard MHD\ninduction equation at large Rm.\nWe would like our theoretical description to involve statistically observable quantities, but Eq. (10) involves the\nquantities \u0001 BxByandB2yevaluated at arbitrarily selected points in time, and the increment \u0001 t, which is not a\nphysically observable quantity associated with the system. What we do is neglect terms of O(\u0001t2) and then replace4\n\u0001tby the shortest relevant correlation time \u001ccorr. The intuition guiding this choice is that the coherent e\u000bect of\nthe velocity shear builds up in the magnetic \feld and in the Maxwell stress until some process stops it. Various\nprocesses can scramble the coherent e\u000bect of the shear, including the magnetic di\u000busion, turbulent inertial e\u000bects,\nand the shearing itself. In turbulent \row, particularly in hydrodynamic regimes where the magnetic forces are too\nweak to have signi\fcant e\u000bect, the turbulent eddy turnover time and perhaps the shearing time are natural correlation\ntimes. Whether the higher-order terms in \u0001 tare negligible in Eq. (10) depends on the speci\fc regime and the various\ndecorrelation mechanisms.\nIn light of the preceding paragraph, we make a conceptual switch, from interpreting our result in Eq. (10) as a\nliteral initial-value calculation to instead viewing it through the lens of time-averaged statistical observables. With\nthe replacement of \u0001 tby\u001ccorr, we interpret both the left- and right-hand sides of Eq. (10) as statistical, time-averaged\nquantities. The relation between the shear and the magnetic stress becomes\nBxBy=\u000bB2y\u001ccorr@U\n@y; (11)\nwhere we have introduced an order-unity constant \u000b. Hence, the net result is a magnetically originated dynamic\nviscosity of the form\n\u0016m=\u000bB2y\n\u00160\u001ccorr; (12)\nsuch that in the zonally averaged momentum equation, the mean shear results in a coherent contribution to the\nzonally averaged Lorentz force of Fx=@y(\u0016m@yU).\nWe stress that the derivation of Eq. (10) requires no assumptions on the spatial structure of the magnetic \feld.\nThe magnetic \feld could be straight (i.e., a poloidal \feld), but it could also have arbitrary spatial structure. B2y\nincorporates the mean magnetic \feld as well as all magnetic \feld perturbations.\nFigure 2 shows the induction of a magnetic \feld by a shear \row over a short time \u0001 tfor two initial \feld con-\n\fgurations. Also depicted is the decomposition of the xcomponent of the zonally averaged Lorentz force into the\ntension and pressure force J\u0002B=FT+FP, where FT= (B2=\u00160)^b\u0001r^b,FP=\u0000r?B2=2\u00160,^b=B=B, and\nr?= (I\u0000^b^b)\u0001r. When the initial magnetic \feld lines are straight, Fxis due to magnetic tension, akin to the\nbehavior of a shear Alfv\u0013 en wave. However, in the general case with nonstraight \feld lines, as might be expected to\noccur in the tangle of magnetic \felds when induction is important and Rm \u001d1,Fxincludes contributions from both\ntension and pressure. In the alternative de\fnition of pressure and tension, where the pressure is isotropic and the\ntension is given by B\u0001rB=\u00160,Fxis entirely tension. Regardless of the initial \feld con\fguration, Fxstill acts as a\nviscosity, in accordance with Eq. (10).\nIn the derivation of Eq. (10), we have only imposed the e\u000bect of a mean shear \row. Let us consider what would\nhappen if we included the full \ruid velocity \feld in the induction equation, in which the mean shear \row Udef=v\u0001^ x\nis embedded. In a short time increment, there would be additional contributions to \u0001 B. Those contributions not\ncorrelated across the zonal domain would mostly cancel out when the zonal average of the Maxwell stress is taken.\nHence, it is the mean shear \row that gives rise to a coherent zonally averaged e\u000bect.\nIn Sec. V, we compare the prediction Eq. (11) for the Maxwell stress with direct numerical simulations.\nIII. NEGATIVE EDDY VISCOSITY\nFor comparison with the magnetic eddy viscosity, it is worthwhile to recall the so-called negative eddy viscosity.\nThe notion of a hydrodynamic negative eddy viscosity in 2D \row has a long history [23{26]. A negative eddy viscosity\nemerges in a variety of mathematical approaches, which can be seen as a re\rection of the general tendency for energy\nto tend towards large scales in 2D [27]. The early theoretical literature was primarily concerned with the context of\nisotropic turbulent \rows. Recent work, however, has revived the negative eddy viscosity to help explain and interpret\nthe physics of the formation and maintenance of zonal \row.\nFor example, the zonostrophic instability supposes a statistically homogeneous turbulent background and considers\nthe growth of a perturbing zonal \row [28{33]. For a weak zonal \row of wavenumber qand turbulent background with\nspectrum \b( k), the \row reinforces the perturbation at a rate that schematically looks likeR\ndkq2(1\u0000q2=k2)g(k)\b(k),\nwheregcontains anisotropic wave-vector dependence. At small q, this goes like q2. At larger q, there is a cuto\u000b of\nthe form 1\u0000q2=k2, which can be traced back to the v0\u0001r\u0010term, where \u0010is the vorticity [34, 35]. Many treatments\nhave neglected this term compared to v\u0001r\u00100under the assumption of long-wavelength mean \rows. We return to\ndiscussion of this cuto\u000b at the end of the section.5\nU(y) B(t= 0) B(∆t)\nFTxFPx U(y) B(t= 0) B(∆t)\nFIG. 2. Change in the magnetic \feld in a short time increment \u0001 tdue to a mean shear \row for two di\u000berent initial con\fgurations,\nneglecting magnetic di\u000busivity. The modi\fcation of the \feld has been exaggerated for visibility. The resulting xcomponent\nof the zonally averaged Lorentz force Fxin both cases acts as a viscosity to reduce the shear \row. The decomposition into\nmagnetic tension and pressure forces FTxandFPx, respectively, is also shown. The top row shows that from an initially\nstraight \feld line, Fxis due solely to tension (at \frst order in \u0001 t). The bottom row shows that when the magnetic \feld is not\nstraight,Fxwill in general have contributions from both tension and pressure.\nManifestations of a negative eddy viscosity are visible at cloud level in both Jupiter and Saturn. Careful mea-\nsurements have demonstrated that the Reynolds stress of observed cloud-level velocity \ructuations are positively\ncorrelated with the mean shear [36{38]. Hence, the turbulent \ructuations appear to transfer energy to the mean\n\row via Reynolds stress. Measurements of the turbulent Reynolds stress in laboratory plasma devices lead to similar\nconclusions [39, 40].\nIt is instructive to calculate the Reynolds stress in a similar manner as we did the Maxwell stress. The calculation\nalso serves as yet another simple demonstration of a negative eddy viscosity. The details are given in Appendix B.\nWe obtain a prediction for the Reynolds stress,\n\u001av0xv0y=\r\u001av2y\u001ccorr@U\n@y; (13)\nwhere\ris a positive constant of order unity. This corresponds to a negative dynamic eddy viscosity,\n\u0016e=\u0000\r\u001av2y\u001ccorr: (14)\nThe dynamic eddy viscosity has magnitude approximately equal to the kinetic energy density multiplied by a\ncorrelation time. In pure hydrodynamic \row, the eddy turnover time at the energy-containing scale, \u001cturb, is often the\nappropriate correlation time. Using vturb\u0018Lturb=\u001cturbleads to a mixing-length estimate for the kinematic negative\neddy viscosity acting on mean shear \row, \u0017e=\u0000\rL2\nturb=\u001cturb.\nAssuming the relevant correlation times in Eqs. (11) and (13) are equal, the zonally averaged momentum \rux due\nto the turbulent Reynolds and Maxwell stresses is\n@\n@t(\u001avx)\f\f\nturb=@\n@y\" \n\u000bB2y\n\u00160\u0000\r\u001av2y!\n\u001ccorr@U\n@y#\n: (15)\nHence, the magnetic eddy viscosity becomes comparable to the negative eddy viscosity when the \ructuating magnetic\nenergy has reached a level roughly similar to that of the \ructuating kinetic energy. However, in Sec. V we will see\nfrom our 2D simulations that the numerical prefactor \rcan be an order of magnitude smaller than \u000b. Therefore, if the\nturbulent Reynolds stress is the primary driver of zonal \row, the magnetic viscosity can have a signi\fcant in\ruence\neven when the \ructuating magnetic energy is much smaller.\nA re\fnement of Eq. (13) is possible. The aforementioned 1 \u0000q2=k2cuto\u000b in the growth rate motivates a simple\nmodi\fcation of Eq. (13),\n\u001av0xv0y=\r2\u001av2y\u001ccorr^L\u0012@U\n@y\u0013\n: (16)\nHere, ^Lis an operator that multiplies the Fourier amplitude by 1 \u0000q2=k2\ncforq\u0014kc, and suppresses the Fourier\namplitude for q > kc, wherekcis a characteristic eddy wave number. This re\fnement is additionally motivated by\nthe fact that modeling attempts that use only the local shear and neglect higher-order derivatives lead to divergent\ngrowth of small scales, but proper inclusion of the cuto\u000b leads to well-behaved models [35].6\nIV. SIMULATION SETUP\nWe perform a series of numerical simulations to assess our prediction for the Maxwell stress and Reynolds stress.\nWe study the behavior of a 2D incompressible MHD \ruid on a magnetized \fplane [15].\nThe coordinates on the \fplane are xdef= (x;y), wherexis the azimuthal direction (longitude) and ythe meridional\ndirection (latitude). It is convenient to represent both the velocity and magnetic \feld by potentials. The velocity in\nthe rotating frame is written as v=^ z\u0002r , where is the stream function. The vorticity normal to the plane is\n\u0010def=^ z\u0001(r\u0002v) =r2 . The magnetic \feld B=B0+~Bconsists of a constant, uniform background \feld B0and a\ntime-varying component ~B(x;t). We de\fne a corresponding vector potential A=A^ zsuch that B=r\u0002A, where\nA=A0+~AandA0=\u0000B0yx+B0xy. As we have imposed the presence of a background magnetic \feld, this 2D\nsystem is not subject to antidynamo theorems, and magnetic ampli\fcation can result such that B\u001dB0. We perform\nsimulations in which the background magnetic \feld is either purely toroidal (i.e., aligned in the azimuthal direction,\nB0y= 0) or purely poloidal (i.e., aligned in the latitudinal direction, B0x= 0).\nThe evolution of the system can be described by a formulation involving vorticity and magnetic potential,\n@\u0010\n@t+v\u0001r(\u0010+\fy) =\u0000B\u0001rr2A\u0000\u0014\u0010+\u0017r2\u0010+\u0018; (17a)\n@A\n@t+v\u0001rA=\u0011r2A: (17b)\nHere,\fis the latitudinal gradient of the Coriolis parameter, \u0014is the linear drag, \u0017is the viscosity, \u0011is the magnetic\ndi\u000busivity, and \u0018(x;t) is a mechanical forcing to excite hydrodynamic \ructuations. The \frst term on the RHS of\nEq. (17a) stems from the curl of the Lorentz force. Equation (17b) is the magnetic induction equation written for the\npotentialA. For mathematical convenience, we have set the permeability \u00160= 1 and the mass density \u001a= 1.\nWe numerically solve Eq. (17) using the code Dedalus [41]. We employ periodic boundary conditions in both\ndirections. During time stepping, we use a high-wave-number spectral \flter to remove enstrophy from accumulating\nat the grid scale. Unless noted, our domain size is Lx\u0002Ly= 4\u0019\u00024\u0019at a resolution of 5122gridpoints. The forcing \u0018is\nstatistically homogeneous and isotropic white noise. It is also small scale, localized near wave vectors with magnitude\nkf= 12, and injects energy per unit area at a rate 10\u00003(see Ref. [17] for more details).\nThe majority of our simulations use nonzero drag \u0014on the hydrodynamic \row in order to saturate without \flling\nthe largest scales of the box. However, we also show a case with only viscosity, \u0014= 0, to con\frm our results are not\nsensitive to the presence of drag.\nV. SIMULATION RESULTS\nWe perform two series of simulations: one with a background toroidal magnetic \feld B0= 10\u00002, and one with\na background poloidal magnetic \feld B0= 10\u00003. These runs use \u0014= 10\u00002,\u0017= 10\u00004, and\f= 10. For each\nsimulation series, we vary \u0011. It is then convenient to de\fne another magnetic Reynolds number that is referenced to\nthe hydrodynamic limit,\nRm0def=L0V0\n\u0011; (18)\nwhereL0andV0are a characteristic length scale and velocity scale in a regime where the magnetic \feld has no\nin\ruence, i.e., large \u0011. ForL0we use a typical eddy length scale computed from the eddy energy spectrum as\nLturbdef= [R\ndkk2E(k)=R\ndkE(k)]\u00001=2. ForV0we use the rms eddy velocity vturb. Our simulations yield Lturb= 0:14\nandvturb= 0:21, giving Rm 0= 0:03=\u0011. Our reported values of Aare computed as the total magnetic energy divided\nby the total kinetic energy.\nAs we decrease \u0011from large values, we move through the three regimes depicted in Figure 1. First, is the regime\nof large\u0011, Rm 0<1, and magnetic energy much smaller than kinetic energy, A\u001c1. As\u0011is reduced, a second\nregime is entered, where Rm 0>1, and still A\u001c1. Then, as \u0011is decreased even further, Rm 0\u001d1 andA\u00191. The\nintermediate regime is the one where the assumptions behind the magnetic eddy viscosity are valid.\nFigure 3 shows results from three representative simulations. The left column shows space-time diagrams of the\nzonal \rowU(y;t) =vx. The right column shows time series of the di\u000berent contributions to the domain-averaged\nenergy density: zonal-\row kinetic energy (ZKE) hU2=2i, eddy kinetic energy (EKE) h(v02\nx+v02\ny)=2i, zonal magnetic\nenergy (ZME)\n~B2\nx=2\u000b\n, eddy magnetic energy (EME) h(B02\nx+B02\ny)=2i, and the background magnetic energy (BME)\nB2\n0=2. Here,hfidef=R\ndxf=LxLydenotes a domain average of quantity f.7\n0 500 1000 1500 2000 2500 3000\nt04812y(a)\n0 500 1000 1500 2000 2500 3000\nt04812y(c)\n0 500 1000 1500 2000 2500 3000\nt04812y(e)0 500 1000 1500 2000 2500 3000\nt10−710−610−510−410−310−210−1energy(b)\n0 500 1000 1500 2000 2500 3000\nt10−710−610−510−410−310−210−1energy(d)\n0 500 1000 1500 2000 2500 3000\nt10−710−610−510−410−310−210−1energy(f)\nZKE\nEKEEME\nZMEBME\n−0.2 0.0 0.2U(y,t)ToroidalB0= 10−2,η= 10−1\nToroidalB0= 10−2,η= 10−4\nPoloidalB0= 10−3,η= 10−3\nFIG. 3. Results from three representative simulations. The left column shows space-time plots of the zonal \row Uand the\nright column shows time traces of averaged energy densities associated with the zonal \row (ZKE) and the hydrodynamic eddies\n(EKE), the background magnetic \feld (BME), the zonally averaged magnetic \feld (ZME), and the eddy magnetic \feld (EME).\n(a) and (b) Case at Rm 0= 0:3 in which magnetic \ructuations are too weak to have an e\u000bect on the hydrodynamic \row. (c)\nand (d) E\u000bect of magnetic \feld on the zonal \row with toroidal background \feld and Rm 0= 300. (e) and (f) Case with poloidal\nbackground \feld and Rm 0= 30.\nFigures 3(a) and 3(b) show a case with Rm 0= 0:3. Figure 3(a) shows that zonal jets form spontaneously. After an\ninitial transient time of about 5 =\u0014= 500, seven jets equilibrate at \fnite amplitude and remain coherent throughout\nthe rest of the simulation. The eddy magnetic energy is about 4 orders of magnitude smaller than the kinetic energy\nof the \row [Fig. 3(b)]. Figures 3(c) and 3(d) show a case with toroidal background \feld at Rm 0= 300. The magnetic\nenergy is somewhat smaller than the kinetic energy, A= 0:05. The zonal \row is mostly suppressed; it has smaller\namplitude and is less coherent. Figures 3(3) and 3(f) show a case with poloidal background \feld at Rm 0= 30. Here,\nthe zonal kinetic energy is not reduced from the hydrodynamic case as in Fig. 3(a), but the coherence of the zonal\n\row is clearly reduced in Fig. 3(e). There is a continual cycle of jet merging and formation of new jets, and the zonal\n\row never settles down to a steady state. Interestingly, this behavior occurs even when A= 0:006.\nA. Maxwell stress\nWe compare the Maxwell stress predicted by Eq. (11) with the actual stress observed in the simulations. To facilitate\nthe comparison, we perform a least-squares best \ft for the single free parameter \u000b, which serves merely as an overall\nconstant of proportionality and does not a\u000bect the spatial structure. The \ft is performed on the Maxwell stress itself,\nnot its derivative.\nFigure 4 examines a simulation with toroidal background \feld. For this simulation, Rm 0= 30 and A= 0:005.\nThe curves labeled \\predicted B0xB0y\" are computed using correlation time \u001ccorr=Lturb=vturb. Both short-duration\naverages (50 time units, left column) and long-duration averages (500 time units, right column) are shown for this\ncomparison to illustrate that there is no substantial di\u000berence. The proportionality constant \u000b= 1:02 for the short-\ntime average and 1.04 for the long-time average. The predictions for the Maxwell stress from Eq. (11) are in very\ngood agreement with those from the simulations. B02ydoes not vary much with y[Figs. 4(a) and 4(b)], and therefore8\n0.000000.000050.000100.00015(a)averaged for 2000≤t≤2050\n−101(c)\n−0.000050.000000.00005(e)\n−0.250.000.25(g)\n0 2 4 6 8 10 12\ny−0.00050−0.000250.000000.00025(i)0.000000.000050.000100.00015(b)averaged for 2000≤t≤2500\n−101(d)\n−0.000050.000000.00005(f)\n−0.250.000.25(h)\n0 2 4 6 8 10 12\ny−0.00050−0.000250.000000.00025(j)B/prime2x B/prime2y\n∂yU\nB/primexB/primey predictedB/primexB/primey\n−U 0.05∂2\nyU\n∂yB/primexB/primey predicted∂yB/primexB/primeyB/prime2x B/prime2y\n∂yU\nB/primexB/primey predictedB/primexB/primey\n−U 0.05∂2\nyU\n∂yB/primexB/primey predicted∂yB/primexB/primey\nFIG. 4. Results for the simulation with toroidal B0= 10\u00002,\u0011= 10\u00003(Rm 0= 30), and A= 0:005. The left and right columns\nshow short-time and long-time averages, respectively. (a) and (b) Zonally averaged magnetic eddy energy components B02xand\nB02y. (c) and (d) Zonal \row shear @yU. (e) and (f) Maxwell stress B0xB0yand the prediction of Eq. (11). (g) and (h) Structure\nof\u0000Uand@2\nyU. (i) and (j) Divergence of the Maxwell stress @yB0xB0yand the derivative of the prediction of Eq. (11). The\nproportionality constant \u000bis 1:02 for the short-time average and 1 :04 for the long-time average.\nthe Maxwell stress closely resembles the mean \row shear; compare Figs. 4(c) and 4(d) with Figs. 4(e) and 4(f).\nAdditionally, by comparing the spatial dependence of @yB0xB0y[Figs. 4(i) and 4(j)] with \u0000Uand@2\nyU[Figs. 4(g) and\n4(h)], we see that @yB0xB0ymore closely resembles @2\nyU. Thus, the Lorentz force acts like viscosity rather than like\ndrag.\nFigure 5 is much the same, but for a poloidal background \feld, with Rm 0= 30 and A= 0:006. There is an\nadditional subtlety here regarding the calculation of the Maxwell stress. The prediction in Sec. II uses the total\nmagnetic \feld B. For a poloidal background \feld, BxBy=BxB0y+B0xB0y(whereas for a toroidal background \feld,\nBxBy=B0xB0y). However, Eq. (11) can be used as a prediction for B0xB0yif we replace B2ywithB02y. The \fgure shows\nthe prediction for B0xB0yrather than BxBy, though this distinction has little quantitative signi\fcance here because\nB0\u001c~Bfor Rm\u001d1.\nIn Fig. 5, the prediction for the Maxwell stress agrees quite well with the actual values. The proportionality constant\n\u000b= 1:05 for the short-time average and 1.17 for the long-time average. It is interesting to note that unlike the toroidal\ncase,B02yhas signi\fcant variation in y. This dependence leaves a noticeable imprint in the Maxwell stress, showing\nthatB0xB0yis not merely proportional to the \row shear @yU. Figures 5(i) and 5(j) show, in addition to the derivative\nof the \ructuating stresses, the term @y(BxB0y). This term, which is responsible for magnetic drag at Rm \u001c1, is\nmuch smaller than @yB0xB0yhere.\nFigure 6 shows simulations results at di\u000berent values of Rm 0. The left column shows results in the statistically\nsteady state for a toroidal background \feld. The regime of magnetic eddy viscosity is valid for 3 .Rm0.300.\nThe upper bound is determined where the magnetic \feld has a signi\fcant in\ruence on the \row and the condition9\n0.00000.00020.0004averaged for 2000≤t≤2050\n(a)\n−101(c)\n−0.00010.00000.0001(e)\n−0.250.000.25(g)\n0 2 4 6 8 10 12\ny−0.00050−0.000250.000000.00025(i)0.00000.00020.0004averaged for 2000≤t≤2500\n(b)\n−101(d)\n−0.00010.00000.0001(f)\n−0.250.000.25(h)\n0 2 4 6 8 10 12\ny−0.00050−0.000250.000000.00025(j)B/prime2x B/prime2y\n∂yU\nB/primexB/primey predictedB/primexB/primey\n−U 0.05∂2\nyU\n∂y(BxB0y) ∂yB/primexB/primey predicted∂yB/primexB/primeyB/prime2x B/prime2y\n∂yU\nB/primexB/primey predictedB/primexB/primey\n−U 0.05∂2\nyU\n∂y(BxB0y) ∂yB/primexB/primey predicted∂yB/primexB/primey\nFIG. 5. Same as Fig. 4, for the simulation with poloidal B0= 10\u00003,\u0011= 10\u00003(Rm 0= 30), and A= 0:006. In (i) and (j),\nthe term@y(BxB0y) is also shown. The proportionality constant \u000bis 1:05 and 1:17 for the short- and long-time averages,\nrespectively.\nA\u001c1 no longer holds. Figure 6(e) shows the correlation coe\u000ecient between the derivative of the Maxwell stress\nand the prediction, which is close to 1 in the magnetic eddy viscosity regime. Figure 6(g) shows the rms values\nof the Reynolds stress and Maxwell stress, which exhibit a similar dependence as the kinetic energy and magnetic\nenergy. The proportionality constant \u000bfor both the toroidal and poloidal cases is approximately equal to 1, as seen\nin Figs. 6(i) and 6(j).\nThe right column depicts the case with a poloidal background \feld. The details are similar, with the regime of\nmagnetic eddy viscosity existing for 10 .Rm0.100. Figure 6 also demonstrates that the Lorentz force transitions\nfrom a magnetic drag at low Rm 0to magnetic viscosity at high Rm 0. Figure 6(f) shows that at low Rm 0, the\ncorrelation of @y(BxB0y) with\u0000Uis equal to 1, indicating drag. A straightforward calculation shows the amplitude\nof the magnetic drag at low Rm 0is expected to be Fx=\u0000(B2\n0=\u0011\u00160)U(momentarily returning to dimensional\nquantities); this amplitude is quantitatively recovered in the simulation. As Rm 0increases, the Lorentz force due\nto \ructuating magnetic \felds, @yB0xB0y, correlates well with the predicted magnetic viscosity. When Rm 0&10, the\nviscouslike \ructuating force begins to dominate, as seen in Fig. 6(h).\nEven though the kinetic energy is much greater than the magnetic energy for Rm 0<300, Fig. 6(d) shows how a\npoloidal background \feld a\u000bects the coherence of the zonal \row even starting at Rm 0= 3. Figure 6(d) is computed\nusing the time average hUi2\nt=2, which is reduced compared to the zonal kinetic energy when the spatial structure of\nthe zonal \row is not steady in time [see Figs. 3(a) and 3(e)]. The longer the duration of the time average, the more\nhUi2\ntwill be reduced by the decoherence.\nTo check whether the prediction for the Maxwell stress is sensitive to the presence of frictional drag, we also perform\na simulation with \u0014= 0 but maintaining nonzero viscosity. The results are shown in Fig. 7. This simulation uses\ndomain size Lx\u0002Ly= 2\u0019\u00022\u0019, numerical resolution of 2562, and parameter values \f= 2,\u0017= 10\u00004, and\u0011= 10\u00004.\nThe background magnetic \feld is toroidal, with B0= 10\u00002. Without drag, the structures tend to \fll the box size,10\n10−210−110010110210310410−1210−1110−1010−910−810−710−610−510−410−310−210−1\n(a)ToroidalB0= 10−2\n10−210−11001011021031040.00.10.20.3\n(c)\n10−210−1100101102103104−1.0−0.50.00.51.0\n (e)\n10−210−110010110210310410−1010−910−810−710−610−510−410−3\n (g)10−210−110010110210310410−910−810−710−610−510−410−310−210−1\n(b)PoloidalB0= 10−3\nZKE\nEKE\nZME\nEME\nBME\n10−210−11001011021031040.00.10.20.3\n(d)\n1\n2/angbracketleftU/angbracketright2\nt\nZKE + EKE\n10−210−1100101102103104−0.50.00.51.0\n (f)\ncorr(∂yB/primexB/primey,∂yB/primexB/primeypred)\ncorr(∂y(BxB0y),−U)\n10−210−110010110210310410−1010−910−810−710−610−510−410−3\n (h)\nrms∂yv/primexv/primey\nrms∂yB/primexB/primey\nrms∂y(BxB0y)\n10−210−1100101102103104\nRm 00.000.250.500.751.001.25\n(i)\n10−210−1100101102103104\nRm 00.000.250.500.751.001.25\n(j)\nα\nγ\nFIG. 6. Results for several simulations with varying \u0011(Rm 0= 0:03=\u0011). The left column shows the case with background\ntoroidal \feld and the right column with poloidal \feld. (a) and (b) Steady-state energies (see the text for de\fnitions of the\nvarious terms). Ais small for Rm 0.300. (c) and (d) Fraction of kinetic energy in a coherent zonal \row, where h\u0001itis a time\naverage. (e) and (f) Correlations of the magnetic stresses observed in the simulations with the prediction of Eq. (11). For\nthe case of a poloidal \feld, the correlation of @y(BxB0y) with\u0000Uis 1 for Rm 0<1; this is magnetic drag. (g) and (h) The\nrms (overy) of the relevant stresses, showing that the Maxwell stress rises to be su\u000eciently strong to counteract the Reynolds\nstress at large enough Rm. (i) and (j) Best-\ft proportionality constants \u000band\r, where\u000bis shown only for the cases in the\nmagnetic eddy viscosity regime: Rm 0>1, but small enough such that a coherent mean zonal \row still exists. All quantities\nwere averaged over 2000 \u0014t\u00143000.\nand only a single jet exists at late times. Time-averaging around t= 2500, we \fnd Lturb= 0:56,vturb= 0:46, and\nhence Rm\u00192600. The magnetic energy is much smaller than the kinetic energy, with A\u00196\u000210\u00004.\nEquation (11) accurately predicts the Maxwell stress in this simulation as well. This is true both at early times\n200< t < 350, when there are two jets, and at late times, 3000 < t < 3250, after the two jets have merged so that\na single wavelength \flls the domain. Moreover, at late times, the spatial structure of Uand@2\nyUare signi\fcantly11\n0500 1000 1500 2000 2500 3000 3500\nt0.02.55.0y(a)\n0 1000 2000 3000\nt10−710−610−510−410−310−210−1100energy(b)\nZKE\nEKEEME\nZMEBME\n−1 0 1U(y,t)\n−0.50.00.5averaged for 200 <t< 350\n(c)\n−0.000050.000000.00005(e)\n0 2 4 6\ny−0.000250.00000(g)−101averaged for 3000 <t< 3250\n(d)\n−0.000050.000000.00005(f)\n0 2 4 6\ny−0.000250.00000(h)ToroidalB0= 10−2, no drag,ν= 10−4,η= 10−4\nU 0.2∂2\nyU\nB/primexB/primey predictedB/primexB/primey\n∂yB/primexB/primey predicted∂yB/primexB/primey\nFIG. 7. Simulation without frictional drag, with toroidal B0= 10\u00002and parameters \u0017= 10\u00004and\u0011= 10\u00004. In this simulation,\nLturb= 0:56,vturb= 0:46, Rm\u00192600, and A\u00196\u000210\u00004. See the text for more details.\ndi\u000berent, and @yB0xB0yis clearly closer to @2\nyU. We also observe that Eq. (11) holds even though the kinetic energy\nhas not yet saturated under viscous dissipation, because the timescale setting the magnetic quasi-equilibrium is much\nshorter than the viscous time.\nB. Reynolds stress\nWe also examine the negative eddy viscosity in our simulations to determine whether the relation in Eq. (13) holds\nin our simulations. Figure 8 shows results for a simulation with a background toroidal \feld of B0= 10\u00002. The\nsystem has A\u001c1 and so the magnetic \feld has little in\ruence on the \row. Correlations induced in the \ructuating\nReynolds stress by the zonal \row are picked out by zonal averaging. The zonally averaged Reynolds stress and its\ndivergence are in overall good agreement with the predictions from Eq. (13), with a best-\ft \r= 0:08. Observe that\nthe proportionality constant for the Reynolds stress \ris about 10 times smaller than the proportionality constant for\nthe Maxwell stress \u000b[see Figs. 6(i) and 6(j)].\nEquation (13) produces a noticeable over-prediction of the jet forcing \u0000@yv0xv0yat the eastward peaks of the jets.\nFigure 8 also shows the re\fned prediction of Eq. (16). The re\fned prediction removes the over-estimate of \u0000@yv0xv0y\nat the eastward jets and shows much better agreement with the actual value. We have used kc= 1=Lturb= 7:1, so\nthe only free parameter is the overall proportionality factor which, for this case, is \r2= 0:13.12\n−0.0010.0000.001(a)v/primexv/primey predictedv/primexv/primey refined prediction\n0 2 4 6 8 10 12\ny−0.0040.0000.004\n0.0000.008(b)−∂yv/primexv/primey predicted−∂yv/primexv/primey refined prediction\nFIG. 8. Comparison of the zonally averaged Reynolds stress in a simulation with the prediction of Eq. (13) and the re\fned\nprediction of Eq. (16). Shown are (a) the Reynolds stress and (b) its divergence. This simulation used a toroidal background\nmagnetic \feld B0= 10\u00002and\u0011= 10\u00002(Rm 0= 3). The proportionality constants are \r= 0:08 and\r2= 0:13. Quantities\nwere averaged over 2000 \u0014t\u00142500.\nVI. GENERALIZATION TO THREE DIMENSIONS\nThe simple derivation of magnetic eddy viscosity in Sec. II assumed a 2D Cartesian system. Here, we follow a\nsimilar procedure to generalize Eq. (10) to three dimensions in Cartesian, cylindrical, and spherical coordinates. We\n\fnd the relevant components of the Maxwell stress tensor that are required to compute the zonally averaged Lorentz\nforce. As in Sec. II, the following equations are exact to \frst order in \u0001 tand depend only on the large-Rm limit of\nthe MHD induction equation.\nIn Cartesian coordinates, let v=U(y;z)^ x. A zonal average is over x, and periodicity in xassumed. We obtain\n\u0001BxBy= \u0001t\u0014\nB2y@U\n@y+ByBz@U\n@z\u0015\n+O(\u0001t2); (19a)\n\u0001BxBz= \u0001t\u0014\nB2z@U\n@z+ByBz@U\n@y\u0015\n+O(\u0001t2): (19b)\nIn cylindrical coordinates ( s;\u001e;z ), letv=U(s;z)^\u001e, and a zonal average is over the azimuthal angle \u001e. We \fnd\n\u0001BsB\u001e= \u0001t\u0014\nsB2s@\n@s\u0012U\ns\u0013\n+BsBz@U\n@z\u0015\n+O(\u0001t2); (20a)\n\u0001BzB\u001e= \u0001t\u0014\nB2z@U\n@z+sBsBz@\n@s\u0012U\ns\u0013\u0015\n+O(\u0001t2): (20b)\nIn spherical coordinates ( r;\u0012;\u001e ), letv=U(r;\u0012)^\u001e, and again a zonal average is over \u001e. We obtain\n\u0001BrB\u001e=\u0001t\nr\u0014\nr2B2r@\n@r\u0012U\nr\u0013\n+BrB\u0012sin\u0012@\n@\u0012\u0012U\nsin\u0012\u0013\u0015\n+O(\u0001t2); (21a)\n\u0001B\u0012B\u001e=\u0001t\nr\u0014\nB2\n\u0012sin\u0012@\n@\u0012\u0012U\nsin\u0012\u0013\n+r2BrB\u0012@\n@r\u0012U\nr\u0013\u0015\n+O(\u0001t2): (21b)13\nWe make several observations. First, in cylindrical or spherical geometry, the \frst-order coherent e\u000bect in the\nMaxwell stress due to Uvanishes if U\u0018s\u0018rsin\u0012, i.e., solid body rotation. Second, one can straightforwardly\ncompute that in the cross terms ByBz,BsBz, orBrB\u0012, there is no \frst-order coherent e\u000bect due to U. One might\nexpect that the \ructuating \feld's contribution to this term averages to approximately zero. A mean magnetic \feld\ncould lead to nonzero correlation such as BrB\u0012, although in a regime Rm \u001d1 theB2randB2\n\u0012terms likely dominate.\nThird, the computation of the zonally averaged Lorentz force in non-Cartesian geometries, F\u001e=r\u0001T\u0001^\u001e, introduces\nadditional geometric factors in the tensor divergence. Fourth, for rotation about an axis, U=s\n, Eqs. (20) and (21)\nmay be written compactly as\n\u0001Bp\u0001^tB\u001e=s\u0001tBp\u0001^tBp\u0001r\n +O(\u0001t2); (22)\nwhere Bpis the poloidal (nonazimuthal) part of the magnetic \feld and ^tis a unit vector in the plane orthogonal to\n^\u001e, e.g., ^t=^ s;^ z;^ r;^\u0012.\nEquation (22) functions as a dynamical generalization of Ferraro's law of isorotation, which states that for an\naxisymmetric magnetic \feld and axisymmetric azimuthal \row with angular velocity \n( s;z), a steady state is possible\nonly when Bp\u0001r\n = 0 [42]. Hence, Ferraro's law has been cited for the notion that magnetic \felds tend to suppress\ndi\u000berential rotation (although this is not required; the law merely requires that Bplies on surfaces of constant \n). Our\nderivation of a magnetic eddy viscosity generalizes the isorotation law because it (i) allows for turbulent dynamics, (ii)\ndoes not assume axisymmetry of the magnetic \feld, and (iii) provides an estimate for the magnitude of the resulting\nforce that counteracts di\u000berential rotation. The isorotation law is recovered from Eq. (22) if Bpis axisymmetric.\nUnder appropriate physical conditions where second-order e\u000bects can be neglected, one can apply the same argu-\nments as in Sec. II to relate the coherent part of the statistically averaged Maxwell stresses to the mean shear \row. If\nthe \frst-order truncation is applicable, then it is clear that a di\u000berentially rotating body will induce coherent Maxwell\nstresses that act to redistribute angular momentum in a way similar to a viscosity.\nWhile the prediction of the Maxwell stress was veri\fed in 2D MHD \f-plane simulations in Sec. V, comparing\nthe equations of this section with 3D numerical simulations is out of scope of the present work. The expressions\nin Eqs. (19){(21) could be directly compared with the Maxwell stress observed in 3D simulations to determine the\napplicability of the magnetic eddy viscosity in a particular regime.\nVII. ESTIMATES FOR JUPITER AND SATURN\nCould the magnetic eddy viscosity regime apply in some planetary interiors and perhaps be responsible for sup-\npression of di\u000berential rotation? We consider this question along with some Jupiter and Saturn data. We explore\nthe possibility that Rm \u001d1 occurs in regions with signi\fcant mean \row (implicitly with A\u001c1). This analysis is\nrestricted to the upper atmosphere as the deeper planetary dynamo region likely has A\u0018O(1).\nThere is currently a great deal of uncertainty about the dynamics below the cloud-top level in these two gas giants.\nWhile the \row speed is expected to decrease with depth due to the increasing mass density, for this estimate we will\nuse the observed outer-atmosphere values.\nCloud tracking of Jupiter's and Saturn's upper-level atmosphere showed that eddies act to reinforce the zonal jets\nwith an e\u000bective eddy viscosity \u0017e;J\u0019\u0000106m2s\u00001for Jupiter [36] and \u0017e;S\u0019\u00002\u0002105m2s\u00001for Saturn [37, 38].\nWe observe that \u0017e;Jis roughly equal to the mixing-length estimate \u0017e=\u0000\rL2\nturb=\u001cturb, usingLturb= 1000 km and\n\u001cturb= 2:5\u0002105s observed at the Jupiter cloud tops. Supposing the negative eddy viscosity is the sole driving force\nof the zonal \row, we ask: if the negative eddy viscosity is to be counterbalanced solely by magnetic eddy viscosity,\nthen (i) what is the strength of the magnetic \feld needed to do so and (ii) how deep below the cloud tops of the two\ngas giants do we expect such strong magnetic \felds?\nThe mass density increases rapidly beneath the cloud tops. We use a number intermediate within the expected\ndensity range for both Jupiter and Saturn, \u001a\u0019200 kg m\u00003[3, 43, 44]. For the correlation time, we use the turbulent\neddy turnover time \u001cturb= 2:5\u0002105s. This time is not that di\u000berent from the typical shear time associated with\nthe surface-level zonal jets tshear;J\u00192\u0002105s andtshear;S\u00195\u0002104s [1, 37]. We also use typical mean magnetic\n\feld values B0;J= 4\u000210\u00003T [45] and B0;S= 2\u000210\u00005T [46]. With these in hand, the kinematic magnetic eddy\nviscosity\u0017m=\u0016m=\u001acomputed with the mean magnetic \feld values is 2{4 orders of magnitude smaller than the\ncorresponding negative eddy viscosity. Thus, magnetic viscosity can only start becoming important if we take into\naccount the magnetic \ructuations.\nFor convenience, we assume the scaling B2\u0018B2\n0Rm at Rm\u001d1. Accounting for the factor of Rm, we estimate\nfor Jupiter that \u0017m\u0019\u0017erequires Rm J\u001960 and for Saturn requires Rm S\u0019700. Hence these results are consistent\nwith the assumption Rm \u001d1. Then, using the outer-atmosphere typical eddy length scale Lturband speedvturb,\nwe compute that the required Rm values correspond to magnetic di\u000busivities of \u0011J\u00196\u0002104m2s\u00001and\u0011S\u001914\n6\u0002103m2s\u00001. Finally, from Refs. [3, 4, 13] we \fnd that such magnetic di\u000busivity values are achieved at approximately\n0:96RJand 0:8RS. Therefore, under the assumptions we have made, we expect zonal jets would be magnetically\nterminated by these depths. These estimates are not inconsistent with the recent gravitometric estimates of depths\nof 0:96RJand 0:85RS[5{8].\nThe decrease of \row velocity with depth will modify these calculations. Suppose that at the critical depth, the\ncharacteristic velocity and turbulent length scale have decreased by a factor of 10 relative to the observed surface\nvalues. The negative eddy viscosity would be smaller by a factor of 100, using the scaling of Sec. III. Hence, the required\nRmJfalls below 1, making it marginal, while the required Rm Sis about 7, still within the regime of magnetic eddy\nviscosity. The required \u0011S=VL=RmSis unchanged from the previous estimate.\nThis discussion is not intended to assert that magnetic eddy viscosity is the mechanism responsible for terminating\nthe zonal \rows in Jupiter and Saturn, but rather serves as a consideration of whether it is a plausible mechanism\nfor gas giants at all. There are substantial complexities not included in this discussion, such as the variation of\nconductivity and density with depth, meridional \row, thermal convection, and realistic physics of planetary dynamos.\nStill, these estimates suggest that magnetic eddy viscosity is a potential candidate for inhibiting di\u000berential rotation\nin the interior of some gas giants.\nVIII. CONCLUSION\nIn 2D incompressible MHD, under the assumptions of Rm \u001d1 and a turbulent \row regime where the magnetic\nenergy is smaller than the kinetic energy, we have shown that a mean shear \row induces correlations in the magnetic\n\feld that result in a magnetic eddy viscosity. Numerical simulations of a 2D magnetized \fplane, in which zonal \rows\narise naturally, con\frm this result.\nTwo key ingredients lead directly to the prediction of magnetic eddy viscosity, given in Eqs. (11) and (12). First\nis the large-Rm limit of the MHD induction equation. Second is the assumption of turbulent \row, which acts to\nscramble any coherent e\u000bect built up in the magnetic \feld, by enforcing a short correlation time.\nThe expressions for the 3D generalization of the components of the Maxwell stress tensor are given in Sec. VI. They\nshow directly how di\u000berential rotation leads to zonally averaged Maxwell stress that transports angular momentum.\nAdditionally, these expressions function as a dynamical generalization of Ferraro's law of isorotation, because they (i)\nallow for turbulent dynamics, (ii) do not assume axisymmetry of the magnetic \feld, and (iii) provide an estimate for\nthe magnitude of the resulting force that counteracts di\u000berential rotation.\nIn 2D simulations with a mean poloidal magnetic \feld, we observe the Lorentz force transition from magnetic drag\nto magnetic eddy viscosity as Rm increases beyond about 10. Magnetic eddy viscosity also occurs in simulations\nwhere the mean magnetic \feld is toroidal, in which case the !e\u000bect is absent and there is no associated magnetic\ndrag; the magnetic eddy viscosity is due solely to magnetic \ructuations.\nRotating, magnetized \ruids are commonly found in astrophysical domains. The magnetic eddy viscosity may, in\ncertain parameter regimes, provide a simple characterization of the transport of angular momentum and the magnetic\nbraking force due to di\u000berential rotation. Our main numerical example here is a case where the mean shear \row\nconsists of a turbulence-driven zonal \row, with a simple idealized model that could provide insight into the interiors\nof gas giants. Mean shear \row also occurs in astrophysical disks, and magnetic eddy viscosity could be explored in\nthat context as well.\nACKNOWLEDGMENTS\nWe gratefully acknowledge useful discussions with Bill Dorland, Eli Galanti, Tristan Guillot, Yohai Kaspi, Brad\nMarston, Felix Parra, David Stevenson, and Steve Tobias. The work of J.B.P. was performed under the auspices of the\nU.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344.\nN.C.C. thanks Sean Haney for his hospitality in Barber Tract in December 2018, where part of this paper was written.\nNumerical simulations were conducted on the Australian National Computational Infrastructure at ANU, which is\nsupported by the Commonwealth of Australia.\nAppendix A: Calculation of current and force directly from J \u0002B\nAlthough the Maxwell stress tensor is the most direct route to the force and momentum \rux derived in Sec. II,\nwe can also calculate the force by \fnding the current and using J\u0002B. As in Sec. II, we stick to two dimensions for\nsimplicity. The current is given by \u00160J=r\u0002B. For convenience, we use somewhat di\u000berent notation than before;15\nhere we refer to quantities at t= 0 as J0andB0and refer to their \frst-order perturbations at t= \u0001tasJ1andB1.\nFrom the magnetic \feld in Eq. (8), we can compute the current,\n\u00160J0=^ z\u0012@B0y\n@x\u0000@B0x\n@y\u0013\n; (A1)\n\u00160J1=\u0000^ z\u0001t\u0014@\n@x\u0012\nU@B0y\n@x\u0013\n+@\n@y\u0012\nB0y@U\n@y\u0000U@B0x\n@x\u0013\u0015\n: (A2)\nThe Lorentz force per unit volume F=J\u0002Bhas anxcomponent\nFx=\u0000JzBy: (A3)\nThus, we have\nF0x=\u0000J0zB0y; (A4)\nF1x=\u0000(J0zB1y+J1zB0y): (A5)\nWe assume that F0x= 0, which is satis\fed if B0xB0y= 0.\nLet us now consider F1x, and substitute the previous expressions to write everything in terms of UandB0. From\nnow on, we drop the 0 subscript for simplicity. After some algebra, we obtain for the zonally averaged force,\nF1x= \u0001t@\n@y \nB2y\n\u00160@U\n@y!\n+O(\u0001t2): (A6)\nThis expression is equivalent to Eq. (10).\nTo summarize, we have recovered the same net force using J\u0002Bdirectly as we did earlier with the Maxwell stress\ntensor. In the 2D MHD system where Bis in thexyplane, both J0andJ1are along ^ z, implying currents must be\nable to \row in the zdirection.\nAppendix B: Derivation of negative eddy viscosity\nIn this appendix we derive for 2D hydrodynamic \row the negative eddy viscosity discussed in Sec. III using the\nsame technique as we used to derive the magnetic eddy viscosity.\nThe zonally averaged force per unit volume in the xdirection is given by the Reynolds stress\nFx=@\n@y\u001avxvy: (B1)\nWe use the vorticity formulation of the equation of motion. We suppose at t= 0 there is a blob of vorticity\n\u00100(x;y) =z0cos(k\u0001x). One could multiply the spatial dependence by an envelope function to localize the spatial\nextent, but that is unnecessary for the e\u000bect of interest. Consider the evolution of this vorticity blob due to advection\nby an imposed shear v=U(y)^ x=U0sin(qy)^ x. We neglect both the Coriolis and the Lorentz force in this derivation.\nWe have\n@\u0010\n@t=\u0000v\u0001r\u0010=\u0000U0sin(qy)@\u0010\n@x: (B2)\nTo include the high-wave-number cuto\u000b discussed in Sec. III, one should retain the v0\u0001r\u0010term, but we neglect it\nhere for simplicity.\nWe evolve\u0010for a short time increment \u0001 t. The vorticity becomes\n\u0010(\u0001t) =z0\u001a\ncos(k\u0001x)\n+\u0001tkxU0\n2\u0014\ncos(k\u0000\u0001x)\u0000cos(k+\u0001x)\u0015)\n; (B3)16\nwhere k\u0006=k\u0006qandq=q^ y. From\u0010=r2 , we obtain\n (\u0001t) =\u0000z0(\n1\nk2cos(k\u0001x)\n+\u0001tkxU0\n2\u00141\nk2\n\u0000cos(k\u0000\u0001x)\u00001\nk2\n+cos(k+\u0001x)\u0015)\n: (B4)\nCalculating v(\u0001t) =^ z\u0002r (\u0001t) and then constructing vx(\u0001t)vy(\u0001t), we obtain\n\u0001vxvy=z2\n0k2\nx\u0001t\n2k4 \n1\u00004k2\ny\nk2!\nqU0cos(qy) +O(\u0001t2;q3): (B5)\nWe identify qU0cos(qy) =@yU, as well as k2\nxz2\n0=k4=k2\nx 2\n0\u0019v2\nyandv2y=v2\ny=2. Furthermore, we observe that the\nfactor 1\u00004k2\ny=k2is only positive for k2\nx>3k2\ny. This k-dependent prefactor does not arise in the magnetic calculation.\nWe collapse this factor into a k-independent positive constant \r, recognizing that \rmay be small in magnitude\nbecause 1\u00004k2\ny=k2is only positive in a small fraction of kspace. Finally, as in Sec. II, we replace \u0001 twith\u001ccorr, the\nmaximum time the shear can act coherently on a structure before being decorrelated. Again, we interpret our results\nin a statistically averaged sense. We obtain a Reynolds stress\n\u0001\u001avxvy=\r\u001av2y\u001ccorr@U\n@y: (B6)\nThis expression corresponds to a negative dynamic eddy viscosity\n\u0016e=\u0000\r\u001av2y: (B7)\nThis calculation does not directly generalize to 3D. Negative eddy viscosities have in the literature been limited to\n2D or quasi-2D \row, and the quasi-2D nature of \row is a consequence of rotation. In contrast, the positive magnetic\neddy viscosity is derived in 3D in Section VI without such complications.\nWe remark on the relation between the wave-number dependence here and that found in a related Kelvin{Orr\ncalculation. In that calculation with an initial wave shearing in a velocity pro\fle U=Sy^ x, the energy transfer to the\nmean \row is found to have a factor 1 \u00004k2\ny=k2, as in Eq. (B5) [17, 34, 47]. This wave-number dependence arose when\nak-independent friction was used as the decorrelation mechanism. When a viscosity instead acts as the decorrelation\nmechanism, the length-scale dependence of the viscosity modi\fes the factor in the energy transfer from 1 \u00004k2\ny=k2to\n1\u00006k2\ny=k2. Returning to the calculation here, we observe that we did not include any explicit friction or viscosity.\nInstead, we posited that decorrelation occurred due to turbulent inertial motion, and we implicitly assumed it was\nindependent of wave number. Hence, we recovered the same factor 1 \u00004k2\ny=k2as found using a k-independent friction.\n[1] A. R. Vasavada and A. P. Showman, Jovian atmospheric dynamics: an update after Galileo andCassini , Rep. Prog. Phys.\n68, 1935 (2005).\n[2] P. A. Davidson, Magnetic damping of jets and vortices, J. Fluid Mech. 299, 153 (1995).\n[3] M. French, A. Becker, W. Lorenzen, N. Nettelmann, M. Bethkenhagen, J. Wicht, and R. Redmer, Ab initio simulations\nfor material properties along the Jupiter adiabat, Astrophys. J. Suppl. S. 202, 5 (2012).\n[4] J. Liu, P. M. Goldreich, and D. J. Stevenson, Constraints on deep-seated zonal winds inside Jupiter and Saturn, Icarus\n196, 653 (2008).\n[5] Y. Kaspi, E. Galanti, W. B. Hubbard, D. J. Stevenson, S. J. Bolton, L. Iess, T. Guillot, J. Bloxham, J. E. P. Connerney,\nH. Cao, D. Durante, W. M. Folkner, R. Helled, A. P. Ingersoll, S. M. Levin, J. I. Lunine, Y. Miguel, B. Militzer, M. Parisi,\nand S. M. Wahl, Jupiter's atmospheric jet streams extend thousands of kilometres deep, Nature 555, 223 (2018).\n[6] T. Guillot, Y. Miguel, B. Militzer, W. B. Hubbard, Y. Kaspi, E. Galanti, H. Cao, R. Helled, S. M. Wahl, L. Iess, W. M.\nFolkner, D. J. Stevenson, J. I. Lunine, D. R. Reese, A. Biekman, M. Parisi, D. Durante, J. E. P. Connerney, S. M. Levin,\nand S. J. Bolton, A suppression of di\u000berential rotation in Jupiter's deep interior, Nature 555, 227 (2018).\n[7] L. Iess, B. Militzer, Y. Kaspi, P. Nicholson, D. Durante, P. Racioppa, A. Anabtawi, E. Galanti, W. Hubbard, M. J. Mariani,\nP. Tortora, S. M. Wahl, and M. Zanoni, Measurement and implications of Saturn's gravity \feld and ring mass, Science\n10.1126/science.aat2965 (2019).\n[8] E. Galanti, Y. Kaspi, Y. Miguel, T. Guillot, D. Durante, P. Racioppa, and L. Iess, Saturn's deep atmosphere revealed by\nthe Cassini Grand Finale gravity measurements, Geophys. Res. Lett. 46, 10.1029/2018GL078087 (2019).17\n[9] M. Heimpel and N. G\u0013 omez P\u0013 erez, On the relationship between zonal jets and dynamo action in giant planets, Geophys.\nRes. Lett. 38, 10.1029/2011GL047562 (2011).\n[10] L. D. Duarte, T. Gastine, and J. Wicht, Anelastic dynamo models with variable electrical conductivity: An application to\ngas giants, Phys. Earth Planet. Inter. 222, 22 (2013).\n[11] C. Jones, A dynamo model of Jupiter's magnetic \feld, Icarus 241, 148 (2014).\n[12] W. Dietrich and C. A. Jones, Anelastic spherical dynamos with radially variable electrical conductivity, Icarus 305, 15\n(2018).\n[13] H. Cao and D. J. Stevenson, Zonal \row magnetic \feld interaction in the semi-conducting region of giant planets, Icarus\n296, 59 (2017).\n[14] R. Perna, K. Menou, and E. Rauscher, Magnetic drag on hot Jupiter atmospheric winds, Astrophys. J. 719, 1421 (2010).\n[15] S. M. Tobias, P. H. Diamond, and D. W. Hughes, \f-plane magnetohydrodynamic turbulence in the solar tachocline,\nAstrophys. J. Lett. 667, L113 (2007).\n[16] S. M. Tobias, K. Dagon, and J. B. Marston, Astrophysical \ruid dynamics via direct statistical simulation, Astrophys. J.\n727, 127 (2011).\n[17] N. C. Constantinou and J. B. Parker, Magnetic suppression of zonal \rows on a beta plane, Astrophys. J. 863, 46 (2018).\n[18] S. Durston and A. D. Gilbert, Transport and instability in driven two-dimensional magnetohydrodynamic \rows, J. Fluid\nMech. 799, 541578 (2016).\n[19] T. A. Zinyakov and A. S. Petrosyan, Zonal \rows in two-dimensional decaying magnetohydrodynamic turbulence on a\n\f-plane, JETP Lett. 108, 85 (2018).\n[20] J. Pedlosky, Geophysical Fluid Dynamics , 2nd ed. (Springer, 1992).\n[21] P. B. Rhines, Waves and turbulence on a beta-plane, J. Fluid Mech. 69, 417 (1975).\n[22] G. K. Vallis and M. E. Maltrud, Generation of mean \rows and jets on a beta plane and over topography, J. Phys. Oceanogr.\n23, 1346 (1993).\n[23] V. P. Starr, Note concerning the nature of the large-scale eddies in the atmosphere, Tellus 5, 494 (1953).\n[24] R. H. Kraichnan, Eddy viscosity in two and three dimensions, J. Atmos. Sci. 33, 1521 (1976).\n[25] S. Gama, M. Vergassola, and U. Frisch, Negative eddy viscosity in isotropically forced two-dimensional \row: linear and\nnonlinear dynamics, J. Fluid Mech. 260, 95 (1994).\n[26] J. A. Krommes and C.-B. Kim, Interactions of disparate scales in drift-wave turbulence, Phys. Rev. E 62, 8508 (2000).\n[27] R. Fj\u001crtoft, On the changes in the spectral distribution of kinetic energy for twodimensional, nondivergent \row, Tellus 5,\n225 (1953).\n[28] B. F. Farrell and P. J. Ioannou, Structure and spacing of jets in barotropic turbulence, J. Atmos. Sci. 64, 3652 (2007).\n[29] K. Srinivasan and W. R. Young, Zonostrophic instability, J. Atmos. Sci. 69, 1633 (2012).\n[30] J. B. Parker and J. A. Krommes, Zonal \row as pattern formation, Phys. Plasmas 20, 100703 (2013).\n[31] J. B. Parker and J. A. Krommes, Generation of zonal \rows through symmetry breaking of statistical homogeneity, New J.\nPhys. 16, 035006 (2014).\n[32] N. C. Constantinou, B. F. Farrell, and P. J. Ioannou, Emergence and equilibration of jets in beta-plane turbulence:\napplications of Stochastic Structural Stability Theory, J. Atmos. Sci. 71, 1818 (2014).\n[33] J. G. Fitzgerald and B. F. Farrell, Statistical State Dynamics of vertically sheared horizontal \rows in two-dimensional\nstrati\fed turbulence, J. Fluid Mech. 854, 544 (2018).\n[34] N. A. Bakas and P. J. Ioannou, On the mechanism underlying the spontaneous emergence of barotropic zonal jets, J.\nAtmos. Sci. 70, 2251 (2013).\n[35] J. B. Parker, Dynamics of zonal \rows: failure of wave-kinetic theory, and new geometrical optics approximations, J. Plasma\nPhys. 82, 595820602 (2016).\n[36] C. Salyk, A. P. Ingersoll, J. Lorre, A. Vasavada, and A. D. Del Genio, Interaction between eddies and mean \row in Jupiter's\natmosphere: Analysis of Cassini imaging data, Icarus 185, 430 (2006).\n[37] A. D. Del Genio, J. M. Barbara, J. Ferrier, A. Ingersoll, R. West, A. Vasavada, J. Spitale, and C. Porco, Saturn eddy\nmomentum \ruxes and convection: First estimates from Cassini images, Icarus 189, 479 (2007).\n[38] A. D. Del Genio and J. M. Barbara, Constraints on Saturn's tropospheric general circulation from cassini iss images, Icarus\n219, 689 (2012).\n[39] C. Holland, J. H. Yu, A. James, D. Nishijima, M. Shimada, N. Taheri, and G. R. Tynan, Observation of turbulent-driven\nshear \row in a cylindrical laboratory plasma device, Phys. Rev. Lett. 96, 195002 (2006).\n[40] G. R. Tynan, C. Holland, J. H. Yu, A. James, D. Nishijima, M. Shimada, and N. Taheri, Observation of turbulent-driven\nshear \row in a cylindrical laboratory plasma device, Plasma Phys. Control. Fusion 48, S51 (2006).\n[41] K. J. Burns, G. M. Vasil, J. S. Oishi, D. Lecoanet, and B. P. Brown, Dedalus: A \rexible framework for numerical simulations\nwith spectral methods, Submitted. arXiv:1905.10388 (2019), (submitted, arXiv:1905.10388).\n[42] V. C. A. Ferraro, The non-uniform rotation of the sun and its magnetic \feld, Mon. Not. R. Astron. Soc. 97, 458 (1937).\n[43] J. D. Anderson and G. Schubert, Saturn's gravitational \feld, internal rotation, and interior structure, Science 317, 1384\n(2007).\n[44] C. Mankovich, M. S. Marley, J. J. Fortney, and N. Movshovitz, Cassini ring seismology as a probe of Saturn's interior. I.\nRigid rotation, Astrophys. J. 871, 1 (2019).\n[45] K. M. Moore, R. K. Yadav, L. Kulowski, H. Cao, B. J, J. E. P. Connerney, S. Kotsiaros, J. L. J\u001crgensen, J. M. G. Merayo,\nD. J. Stevenson, S. J. Bolton, and S. M. Levin, A complex dynamo inferred from the hemispheric dichotomy of Jupiter's\nmagnetic \feld, Nature 561, 76 (2018).\n[46] T. I. Gombosi, T. P. Armstrong, C. S. Arridge, K. K. Khurana, S. M. Krimigis, N. Krupp, A. M. Persoon, and M. F.18\nThomsen, Saturn's magnetospheric con\fguration, in Saturn from Cassini{Huygens (Springer, 2009) pp. 203{255.\n[47] N. A. Bakas, N. C. Constantinou, and P. J. Ioannou, S3T stability of the homogeneous state of barotropic beta-plane\nturbulence, J. Atmos. Sci. 72, 1689 (2015)."    },    {        "title": "2403.06491v1.Magnetic_vortex_polarity_reversal_induced_gyrotropic_motion_spectrum_splitting_in_a_ferromagnetic_disk.pdf",        "content": "Magnetic vortex polarity reversal induced gyrotropic motion\nspectrum splitting in a ferromagnetic disk\nXiaomin Cui,1,∗Shaojie Hu,1Yohei Hidaka,1Satoshi Yakata,2and Takashi Kimura1,†\n1Department of Physics, Kyushu University,\n744 Motooka, Fukuoka, 819-0395, Japan\n2Department of Information Electronics,\nFukuoka Institute of Technology, 3-30-1 Wajiro-higashi,\nHigashi-ku, Fukuoka, 811-0295 Japan\n(Dated: March 12, 2024)\nAbstract\nWe investigate the gyrotropic motion of the magnetic vortex core in a chain of a few micron-sized\nPermalloy disks by electrical resistance measurement with amplitude-modulated magnetic field. We\nobserve a distinctive splitting of the resistance peak due to the resonant vortex-core motion under\nheightened radio frequency (RF) magnetic field excitation. Our micromagnetic simulation identifies\nthe splitting of the resonant peak as an outcome of vortex polarity reversal under substantial RF\namplitudes. This study enhances our understanding of nonlinear magnetic vortex dynamics amidst\nlarge RF amplitudes and proposes a potential pathway for spintronic neural computing thanks to\ntheir unique and controllable magnetization dynamics.\n∗cui.xiaomin.978@m.kyushu-u.ac.jp\n†t-kimu@phys.kyushu-u.ac.jp\n1arXiv:2403.06491v1  [physics.app-ph]  11 Mar 2024Magnetic vortices are unique topological magnetic structures found in micron or submi-\ncron ferromagnetic elements, characterized by circling in-plane magnetization with a core of\nperpendicular magnetization[1–3]. Dynamics based on magnetic vortex include two types,\none is the gyrotropic precession of the core in the sub-gigahertz range, and the other one\nis the core polarization switching to gigahertz steady-state oscillation[4–7]. The fascinating\nreconfigurable and controllable dynamic properties assure the magnetic vortex of great po-\ntential in numerous spin-related applications such as magnetic sensors, spin wave emission\nsources, and spintronic synapses and neurons for processing, transmitting, and receiving\nradiofrequency signals[8–12]. Correspondingly, a study on vortex dynamics earns of great\nattention. The vortex structure gyrates about its equilibrium position with a characteristic\neigenfrequency[4, 13]. By adjusting the shape, the geometrical ratio or the interval distance\nof the ferromagnetic element containing vortex structure, abundant dynamic properties of\nthe gyration mode could be tailored[5, 14–20]. In addition to other factors, the resonance’s\ngyration sense is intricately linked to the vortex core polarization. To effectively control\nthis resonance, it’s imperative to skillfully manipulate the vortex core polarization[5]. So\nfar, approaches including the utilization of a small AC magnetic field, resonant microwave\npulses, DC spin-polarized current or excitation of spin waves have been introduced to reverse\nthe vortex core polarization[7–9, 21–24]. Studies show the origin of vortex core reversal is\na gyrotropic field, which is proportional to the velocity of the moving vortex[25]. If the\ncore motion speed hits a threshold, the polarity inverts[23, 26]. Several approaches have\nbeen demonstrated to reduce the critical velocity for core switching, such as by introduc-\ning the nanoscale defects, amplifying the perpendicular anisotropy, or leveraging magnetic\ninteractions between units[27–30].\nVortex core reversal coincides with deviations from the linear dynamic properties. Non-\nlinear vortex dynamics often occur at high excitation amplitudes. Investigating novel non-\nlinear vortex core dynamics not only deepens our understanding of vortex physics but also\nadvances the development of vortex-based spintronic devices. This is because the electrical\nmanipulation of nonlinear phenomena provides a significant contribution for advanced func-\ntional devices. Significant theoretical, simulation, and experimental efforts have been made\nto study the dynamical response of the magnetic vortex in the nonlinear regime, revealing\nphenomena like fold-over bifurcation, multiphoton resonance, anti-resonances, and transla-\ntional resonance splitting[31–33]. As for the mechanism of the nonlinear vortex dynamics,\n2some reports claim that the higher-order terms in magnetostatic potential play a greater role\nduring the large amplitude of vortex gyration mode[33–35]. In addition, a large distortion of\nthe vortex core can be effectively excited by high power AC magnetic field with the resonant\nfrequency, leading to nonlinear behaviors [36, 37]. Although the experimental detection of\nnonlinear vortex dynamics is crucial for discerning the relationship between vortex core po-\nlarity and its electrical response, the constraints of injected RF power in the conventional\nmethod based on a monolithic circuit avoid the precise detection of the core dynamics. We\nhave developed a sensitive electrical detection method of the vortex core dynamics by using\nelectrically separated excitation and detection circuits[38]. In this research, we extend our\nsensitive detection technique to investigate the nonlinear vortex dynamics in the chain of the\nmultiple Py disks induced by a high-amplitude AC magnetic field. Our observations high-\nlight a resonant peak splitting at elevated RF power when analyzing the power-dependent\ndynamic response of the magnetic vortex core in gyration mode. Micromagnetic simulations\nwere performed to elucidate this splitting behavior, revealing the role of vortex core polarity\nswitching in inducing resonant peak bifurcation. It proposes one simple way for probing\nthe polarity switching of magnetic vortex, which opens up attractive means for the study\nof nonlinear magnetization dynamics, and applications such as tunable oscillator[8, 39, 40]\nand spintronic neural computing[10, 12].\nUtilizing electron beam lithography and lift-off techniques, We fabricated a chain of\nmicron-sized disks containing 5 Py disks with a thickness of 40 nm. The Scanning Electron\nMicroscopy (SEM) image illustrating the completed device is presented in Fig. 1(a). The\ndiameter and edge-edge interval distance of the adjacent disks are 3 µm and 2 µm, respec-\ntively. The Cu pad with a thickness of 200 nm and a width of 500 nm was prepared to\nconnect the adjacent Py disks. Here, the Cu pad was up-shifted with a distance of 600 nm\nfrom the core of the Py disk. Periodical Cu electrodes with the same thickness as the Cu\npad were also prepared on the top of each Py disk for applying AC current. The Py disk\nand Cu electrodes are electrically isolated through a patterned SiO 2film, precisely tailored\nto a thickness of 100 nm, ensuring insulation of their electrical connection.\nThe dynamic properties of the magnetic vortices were detected using lock-in electric\nmeasurement technique while the vortex core is excited by the amplitude-modulated Oersted\nfield generated accurately by each Cu electrode[38, 41]. As shown in Fig. 1(a), during\nthe measurement, an in-plane external magnetic field was applied along the chain of the\n3FIG. 1: (a) Schematic representation of the electrical measurement setup accompanied by an SEM\nimage displaying the fabricated magnetic vortex device; A static magnetic field is applied along\nthe direction of the disk chain. During the measurement, the amplitude of the AC signal was\nmodulated by a low-frequency sinusoidal wave of 1.73 kHz. A DC current was applied to detect\nthe voltage change of the magnetic vortex device. (b) A representative spectrum showing the\nfrequency dependence of the average resistance change of the device with a diameter of 3 µm\nand edge-to-edge interval of 2 µm measured under RF injection power of -20 dBm. Here, the\n∆Rreswas defined as the difference between the resonant peak and baseline, which represents the\nmagnetoresistance change between the oscillation and the non-oscillation states of the magnetic\nvortices.\nPy disks to create a nonuniform domain structure by displacing the vortex core from the\ncenter. We flow one DC current of 5 mA in the chained disks and detect the responding\nvoltage of the magnetic vortex device by sweeping the RF frequency. Fig. 1(b) shows a\nrepresentative spectrum measured at an RF power of -20 dBm by sweeping frequency from\n50 MHz to 200 MHz under a static magnetic field of 18 mT. A clear resonant peak has been\nobtained at 126.8 MHz due to the gyrotropic motion of the vortex core. The resistance\nchange between the resonant peak and the baseline was defined as the effective resistance\nchange ∆R res, which represents the magnetoresistance change between the oscillation and\nthe non-oscillation states of the magnetic vortex core in this device.\nIn our endeavor to comprehensively analyze the dynamic properties inherent to the\n4chained Py disks, we undertook systematic measurements of the resistance response by\nsweeping the RF frequency under different external magnetic fields. Figures 2(a) and 2(b)\nprovide a visual representation in the form of color images, illustrating the device resis-\ntance as a function of both microwave frequency and the imposed external magnetic field\nunder RF power of -20 and -10 dBm, respectively. A notable observation from Fig. 2(a) is\nthe emergence of a distinct ’M’ shaped profile characterizing the resonance frequency when\nplotted as a function of the external magnetic field at an RF power of -20 dBm. The data\nderived from this figure suggests that the external magnetic field has a relatively subdued\nmodulatory influence on the resonance frequency. In stark contrast, Fig. 2(b) manifests a\ndivergence from this pattern. At an RF power of -10 dBm, the resonance frequency appears\nto adopt a dual ’M’ configuration. This intriguing shift suggests a pronounced splitting in\nthe resonance peak of the spectrum. Furthermore, the frequency disparity between these\ntwo ’M’ shaped configurations appears markedly greater compared to the results influenced\nsolely by field modulation. To delve deeper into these nuanced differences and glean more\ngranular insights, we selected a range of magnetic fields. The resultant frequency-dependent\nspectra derived from these selections are comprehensively illustrated in Fig. 2(c) and 2(d).\nIn our examination of the device’s response, a discernible transformation in the Lorentzian-\nlike spectrum is evident. This transformation progresses from a peak (observed at -20 mT)\nto a dip (registered at 4 mT) and ultimately reverts to a peak (noted at 17 mT). This\nprogression is interspersed with transitions through anti-symmetric Lorentzian-like spectra,\nspecifically observed at magnetic field intensities of 1 mT and 14 mT. The intricate dynamics\nbehind this transition behavior can be ascribed to the atypical resistance alterations contin-\ngent upon the central positioning of the vortex core. This phenomenon has been extensively\ndetailed and rationalized in our recent work[41]. Furthermore, when we modulate the RF\npower, elevating it from -20 dBm to a higher -10 dBm, we observe a nuanced alteration in the\nspectral characteristics. The once singular resonant peak (or dip, as the case may be) within\nthe Lorentzian-like spectrum bifurcates, resulting in the emergence of two distinct resonant\npeaks (or dips). It’s imperative to note, however, that this spectral division predominantly\nmanifests in proximity to the dip of the anti-symmetric Lorentzian-like spectrum, as opposed\nto its peak.\nIn an endeavor to delve deeper into the intricacies of the observed splitting behavior,\nour research team undertook meticulous measurements, examining the power dependence of\n5FIG. 2: (a) Experimental observed dynamic spectra of the chained Py disks at different external\nmagnetic fields and RF power of -20 dBm. (b) Experimental observed dynamic spectra of the\nchained Py disks at different external magnetic fields and RF power of -10 dBm. (c) The resistance\nas a function of the input RF frequency under various external fields marked as the dotted lines\nin (a). (d) The resistance as a function of the input RF frequency under various external fields\nmarked as the dotted lines in (b).\nthe magnetic vortex dynamics. This was systematically conducted across an RF amplitude\nspectrum ranging from -33 dBm to 0 dBm, under the magnetic fields specified earlier in our\ndiscourse. The graphical representation in Fig.3(a), using a color image plot, offers a clear\ndepiction of the resonant frequency’s behavior. Specifically, within magnetic field environ-\nments of -20 mT and 17 mT, a singular resonant peak manifests a discernible bifurcation,\nevolving into two separate resonant peaks as the RF amplitude augments. It’s pivotal to\nhighlight that this augmentation in RF amplitude invariably leads to an expansion in the\nfrequency separation interposed between these two emergent resonant peaks. Furthermore,\nas one continues to elevate the RF amplitude, the extent of this aforementioned separation\n6between the two resonant peaks amplifies correspondingly. Notably, as the power escalates\nto -5 dBm the differential between the two peaks culminates at an impressive 37.2 MHz at\na magnetic field of 17 mT. For perspective, juxtaposing this with the resonant frequency\npinpointed at an RF amplitude of -20 dBm reveals a frequency modulation on the order of\napproximately 29.6%. This substantial figure underscores the profound influence exerted by\nRF power on frequency modulation.\nTo elucidate the power-dependent dynamics intrinsic to the concatenated Py disks, the\nspectral data acquired at a magnetic field strength of 17 mT were chosen for more intricate\nscrutiny, owing to their distinctive power-amplified splitting behavior. The analysis was con-\nducted at an excitation frequency of 125.6 MHz, a resonant frequency empirically determined\nat a power level of -20 dBm and a magnetic field strength of 17 mT. Magnetic resistance\nvalues at the aforementioned excitation frequency (R) and the baseline resistance (R 0) were\nextracted and employed to calculate the difference in resistance ∆R. The collated data is\npresented in Fig.3(b), delineated as a function of radio-frequency (RF) power. Notably, both\nR0and R demonstrate an exponential augmentation as discerned from Figure 3(b), which\nsuggests a potential correlation with the thermal effects engendered by the escalation in RF\npower. Additionally, ∆R exhibits an ascending trajectory, culminating at a maximum value\nat an RF power level of -19 dBm. Subsequently, ∆R begins to exhibit a decremental trend\nconcomitant with the continued escalation of radio-frequency (RF) power. The underlying\nmechanisms contributing to this decline in ∆R at elevated RF power levels remain as yet\nundetermined. To clarify this phenomenon, we propose to employ micromagnetic simula-\ntion techniques. Establishing a correlative framework between the empirical findings and\nthe dynamic processes gleaned from micromagnetic simulations could illuminate the origin\nof both the power-dependent splitting behavior and the attenuation of ∆R.\nWe performed the micromagnetic simulation using MuMax3here. [42] The computational\ndomain was architecturally congruent in both dimensions and thickness with the fabricated\ndevice. Discretization of this domain was executed utilizing mesh dimensions of 4 nm ×4\nnm×40 nm. We employed canonical microstructural parameters pertinent to Permalloy,\nfeaturing an exchange stiffness constant A ex= 1.3×10−11J/m, damping parameter α=\n0.006, saturation magnetization M s= 8×105A/m, and a null magnetocrystalline anisotropy\nconstant. To access the observed unique spectra, the dynamic processes were investigated\nby calculating the corresponding anisotropic magnetoresistance (AMR) in the Py disk. The\n7FIG. 3: (a) Experimentally observed spectra of concatenated Py disks at magnetic fields as a\nfunction of RF power across varying magnetic fields of -20 mT, 1 mT, 4 mT, 14 mT and 17\nmT. (b) Resistance differential ∆R between oscillatory and non-oscillatory states as a function of\nexcitation power, evaluated at a magnetic field strength of 17 mT: pertinent to a magnetic vortex\nexcited at 125.6 MHz, the resonant frequency determined at an RF power level of -20 dBm.\nAMR of Py could be calculated by considering the relation of ρ=ρ0(1 + ηcos2θ). The\nresistivity of Py ( ρ0) is 3 .6×108Ω·m. The AMR ratio ηis about 1.4%. θis the angle\nbetween the magnetization and current density direction in the magnetic unit cell. Assuming\nthe current density is in the x-direction, the resistivity of each unit cell can be determined\nbased on its magnetization direction using the AMR effect. The overall resistance of the\nPy disk can calculated by connecting all the resistor units in parallel and series for each\nmagnetic state.\nTo corroborate our empirical findings, we intend to execute micromagnetic simulations,\nselecting radio-frequency (RF) power levels of -20 dBm and -10 dBm for comparative anal-\nysis. Prior to this, the resonant frequencies at a magnetic field strength of 17 mT were\ncalculated via micromagnetic simulations. Subsequently, dynamic simulations were con-\nducted at an excitation frequency of 123 MHz. Figure 4(a) delineates the core trajectories\nof the singular vortex structure at RF power levels of -20 dBm and -10 dBm, superimposed\non a contour map depicting magnetoresistance within the Permalloy disk. Time-dependent\nanisotropic magnetoresistance, calculated under the constraint of uniform current density,\nis presented in Fig. 4(b) and 4(c)[41, 43]. Remarkably, at an RF power of -20 dBm, the\n8vortex core manifests stable oscillatory behavior. For the core trajectory under -10 dBm,\ndirectional movements are indicated by yellow arrows, elucidating the vortex core’s oscilla-\ntory mechanics. The vortex core commences rotation from an inferior position in a clockwise\ndirection towards point P1, whereupon it transitions into a counterclockwise rotation begin-\nning at point P2, thus illustrating a polarity reversal[25]. To substantiate this observation,\nmagnetization texture snapshots at specific points were captured, as detailed in Figure 4(d).\n(Refer to the supplementary video for a complete overview of the simulation duration.) The\nmagnetization profiles at points P1, P2, and P3 validate the polarity transition from a down-\nward to an upward orientation. Subsequent to this, the vortex core reverts to a clockwise\nrotational direction as it translocates from point P4 to point P5, coinciding with an addi-\ntional polarity inversion. Intriguingly, such polarity reversals are conspicuously absent at an\nRF power of -20 dBm. Collectively, these results insinuate a salient correlation between the\nvortex polarity reversals and variations in ∆R contingent on the applied RF power.\nFurthermore, magnetic resistance values were extracted for the non-oscillatory state at\nt = 0 as well as for the average magnetic resistance during the oscillatory state across\n5-10 periods. Subsequently, the difference in resistance between the non-oscillatory and\noscillatory states, denoted as ∆R was calculated across a range of excitation frequencies.\nThe variation of ∆R as a function of RF frequency is depicted in Figure 5(a). Simulation\nresults reveal a singular resonant peak at 122 MHz under an RF amplitude of -20 dBm. Upon\nincreasing the RF amplitude to -10 dBm, this resonant peak bifurcates into two separate\npeaks with frequencies of 119.5 MHz and 124 MHz. Further amplification to -5 dBm results\nin greater frequency splitting, as corroborated in Figure 5(a). These findings are in close\naccord with the previously mentioned experimental observations. As for ∆R as a function of\nRF power, depicted in Figure 5(b), the behavior of ∆R is consistent with the experimental\noutcomes outlined in Figure 3(b). Notably, the maximum ∆R value is attained at -16 dBm,\nslightly exceeding that of the experimental results. This discrepancy may be attributed to\ndefects induced during the device fabrication process, where the critical value is achieved at\na relatively lower energy level. When the excitation frequency for the magnetic vortex aligns\nwith or is proximal to, the resonant frequency, ∆R commences a decline post the critical\nvalue. Conversely, when the frequency deviates significantly from the resonant frequency, the\nvortex core experiences minimal oscillation, thereby rendering the RF power enhancement\nless impactful on ∆R. In essence, the reduction of ∆R at the resonant frequency, occasioned\n9FIG. 4: (a) The micromagnetic calculated core trajectories of the vortex in the contour map\nof the Py disk, while it was excited under external field of 17 mT, frequency of 123 MHz and\npower of -20 dBm and -10 dBm, respectively. Yellow arrows stand for the moving direction of the\nvortex core during oscillation. The purple dots correspond to the vortex polarity reversal. (b) and\n(c) represent the time-dependent resistance of the Py disk excited under -20 dBm and -10 dBm,\nrespectively. (d) Snapshots of numerical domain structures with the vortex core reversal process\ncorresponding to the highlighted purple points P1 to P6 from (a) and the inset of (c).\nby the reversal of vortex polarity, engenders the observed splitting in the dynamic spectra.\nAdditionally, higher power levels induce a greater incidence of vortex polarity switching,\nfurther contributing to the reduction of ∆R.\nIn conclusion, by employing an amplitude-modulated magnetic field to accurately stim-\nulate the gyroscopic motion of the magnetic vortex core in each disk, we have elucidated\nthe splitting phenomena of the resonant peak in magnetic vortices under elevated RF am-\nplitudes. This investigation was facilitated through the exploration of power-dependent\nresistance spectra. We observed that the frequency separation between the dual resonant\npeaks magnifies concomitantly with increasing RF amplitude. Utilizing micromagnetic simu-\nlations, we successfully replicated the experimental observations and substantiated that this\n10FIG. 5: (a) Calculated ∆R as a function of the excitation frequency under the external field of\n17 mT and excitation power of -20 dBm, -10 dBm and -5 dBm, respectively. (b) Calculated ∆R\nwith respect to the input RF power under the external field of 17 mT and frequency of 123 MHz.\nresonant peak splitting arises due to vortex polarity reversal at high RF amplitudes. These\nempirical findings contribute to an enriched comprehension of nonlinear vortex dynamics\nunder substantial excitation amplitudes and provide a potential avenue for the advancement\nof tunable microwave-assisted spintronic devices or spintronic neural networks.\nAcknowledgement\nThis study is partially supported by JSPS KAKENHI Grant Numbers 17H06227,\n21H05021 and JST CREST (JPMJCR18J1).\nAdditional information\nSupplementary information shows the video of the magnetization texture of the Py disk\nobserved under 17 mT and -10 dBm and excitation frequency of 123 MHz as a function of\nthe simulation time.\n11Data Availability\nThe data that support the findings of this study are available from the corresponding\nauthor upon reasonable request.\n[1] R. P. Cowburn, D. K. Koltsov, a. O. Adeyeye, M. E. Welland and D. M. Tricker, Physical\nReview Letters , 1999, 83, 1042–1045.\n[2] T. Shinjo, Science , 2000, 289, 930–932.\n[3] S.-B. Choe, Science , 2004, 304, 420–422.\n[4] K. Y. Guslienko, B. A. Ivanov, V. Novosad, Y. Otani, H. Shima and K. Fukamichi, Journal\nof Applied Physics , 2002, 91, 8037–8039.\n[5] J. Park and P. Crowell, Physical review letters , 2005, 95, 167201.\n[6] S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno and T. Ono, Physical Review Letters , 2006,\n97, 107204.\n[7] B. Van Waeyenberge, A. Puzic, H. Stoll, K. Chou, T. Tyliszczak, R. Hertel, M. F¨ ahnle,\nH. Br¨ uckl, K. Rott, G. Reiss et al. ,Nature , 2006, 444, 461–464.\n[8] V. S. Pribiag, I. N. Krivorotov, G. D. Fuchs, P. M. Braganca, O. Ozatay, J. C. Sankey, D. C.\nRalph and R. a. Buhrman, Nature Physics , 2007, 3, 498–503.\n[9] M. Kammerer, M. Weigand, M. Curcic, M. Noske, M. Sproll, A. Vansteenkiste,\nB. Van Waeyenberge, H. Stoll, G. Woltersdorf, C. H. Back et al. ,Nature communications ,\n2011, 2, 279.\n[10] L. Martins, A. S. Jenkins, L. S. E. Alvarez, J. Borme, T. B¨ ohnert, J. Ventura, P. P. Freitas\nand R. Ferreira, Scientific reports , 2021, 11, 16094.\n[11] J. Zhou and J. Chen, Advanced Electronic Materials , 2021, 7, 2100465.\n[12] A. Ross, N. Leroux, A. De Riz, D. Markovi´ c, D. Sanz-Hern´ andez, J. Trastoy, P. Bortolotti,\nD. Querlioz, L. Martins, L. Benetti et al. ,Nature Nanotechnology , 2023, 1–8.\n[13] T. Y. Chen, A. T. Galkiewicz and P. A. Crowell, Physical Review B , 2012, 85, 180406(R).\n[14] J. Shibata and Y. Otani, Physical Review B , 2004, 70, 012404.\n[15] A. Barman, S. Barman, T. Kimura, Y. Fukuma and Y. Otani, Journal of Physics D: Applied\nPhysics , 2010, 43, 422001.\n12[16] A. Vogel, A. Corinna Niemann, C. Stenner, A. Drews, M. Y. Im, P. Fischer and G. Meier,\nJournal of Applied Physics , 2012, 112, 063916.\n[17] D.-S. Han, A. Vogel, H. Jung, K.-S. Lee, M. Weigand, H. Stoll, G. Sch¨ utz, P. Fischer, G. Meier\nand S.-K. Kim, Scientific reports , 2013, 3, 2262.\n[18] S. Yakata, T. Tanaka, K. Kiseki, K. Matsuyama and T. Kimura, Scientific reports , 2013, 3,\n3567.\n[19] X. Cui, S. Yakata and T. Kimura, Physica E: Low-Dimensional Systems and Nanostructures ,\n2016, 75, 28–32.\n[20] S. Lend´ ınez, T. Polakovic, J. Ding, M. B. Jungfleisch, J. Pearson, A. Hoffmann and\nV. Novosad, Journal of Applied Physics , 2020, 127, 243904.\n[21] M. Weigand, B. Van Waeyenberge, A. Vansteenkiste, M. Curcic, V. Sackmann, H. Stoll,\nT. Tyliszczak, K. Kaznatcheev, D. Bertwistle, G. Woltersdorf et al. ,Physical review letters ,\n2009, 102, 077201.\n[22] B. Pigeau, G. De Loubens, K. Olivier, A. Riegler, F. Lochner, G. Schmidt and L. W.\nMolenkamp, Nature Physics , 2011, 7, 26–31.\n[23] M. Sushruth, J. P. Fried, A. Anane, S. Xavier, C. Deranlot, M. Kostylev, V. Cros and P. J.\nMetaxas, Physical Review B , 2016, 94, 100402.\n[24] P. Warnicke, P. Wohlh¨ uter, A. K. Suszka, S. E. Stevenson, L. J. Heyderman and J. Raabe,\nPhysical Review B , 2017, 96, 174402.\n[25] K. Y. Guslienko, K.-S. Lee and S.-K. Kim, Physical review letters , 2008, 100, 027203.\n[26] K.-S. Lee, S.-K. Kim, Y.-S. Yu, Y.-S. Choi, K. Y. Guslienko, H. Jung and P. Fischer, Physical\nreview letters , 2008, 101, 267206.\n[27] G. Fior, E. Novais, J. Sinnecker, A. Guimaraes and F. Garcia, Journal of Applied Physics ,\n2016, 119, 093906.\n[28] T. Devolder, D. Rontani, S. Petit-Watelot, K. Bouzehouane, S. Andrieu, J. L´ etang, M.-W.\nYoo, J.-P. Adam, C. Chappert, S. Girod et al. ,Physical review letters , 2019, 123, 147701.\n[29] Y. Luo, T. Zhou, Z. Qian, X. Xiao, Z. Liu, C. Zhou, C. Won and Y. Wu, Journal of Magnetism\nand Magnetic Materials , 2019, 474, 544–550.\n[30] M. Mehrnia, J. Trimble, O. Heinonen and J. Berezovsky, Physical Review Applied , 2021, 16,\n034049.\n[31] K. S. Buchanan, M. Grimsditch, F. Y. Fradin, S. D. Bader and V. Novosad, Physical Review\n13Letters , 2007, 99, 267201.\n[32] Y. S. Gui, A. Wirthmann, N. Mecking and C.-M. Hu, Physical Review B , 2009, 80, 060402.\n[33] K. Y. Guslienko, R. H. Heredero and O. Chubykalo-Fesenko, Physical Review B - Condensed\nMatter and Materials Physics , 2010, 82, 014402.\n[34] O. Sukhostavets, B. Pigeau, S. Sangiao, G. De Loubens, V. Naletov, O. Klein, K. Mitsuzuka,\nS. Andrieu, F. Montaigne and K. Guslienko, Physical Review Letters , 2013, 111, 247601.\n[35] J. Ding, G. N. Kakazei, X. Liu, K. Y. Guslienko and A. O. Adeyeye, Scientific Reports , 2014,\n4, 4796.\n[36] K. Yamada, S. Kasai, Y. Nakatani, K. Kobayashi, H. Kohno, A. Thiaville and T. Ono, Nature\nmaterials , 2007, 6, 270–273.\n[37] A. Vansteenkiste, K. W. Chou, M. Weigand, M. Curcic, V. Sackmann, H. Stoll, T. Tyliszczak,\nG. Woltersdorf, C. H. Back, G. Sch¨ utz and B. Van Waeyenberge, Nature Physics , 2009, 5,\n332–334.\n[38] X. Cui, S. Hu, M. Hidegara, S. Yakata and T. Kimura, Scientific Reports , 2015, 5, 17922.\n[39] G. De Loubens, A. Riegler, B. Pigeau, F. Lochner, F. Boust, K. Y. Guslienko, H. Hurdequint,\nL. Molenkamp, G. Schmidt, A. N. Slavin et al. ,Physical Review Letters , 2009, 102, 177602.\n[40] A. Dussaux, B. Georges, J. Grollier, V. Cros, A. Khvalkovskiy, A. Fukushima, M. Konoto,\nH. Kubota, K. Yakushiji, S. Yuasa et al. ,Nature communications , 2010, 1, 1–6.\n[41] S. Hu, X. Cui, K. Wang, S. Yakata and T. Kimura, Nanomaterials , 2022, 12, 2295.\n[42] A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez and B. Van Waeyen-\nberge, AIP advances , 2014, 4, 107133.\n[43] X. Cui, S. Hu and T. Kimura, Applied Physics Letters , 2014, 105, 082403.\n14"    },    {        "title": "1311.6265v3.Magnetic_catalysis_in_flavored_ABJM.pdf",        "content": "arXiv:1311.6265v3  [hep-th]  2 Jan 2014Magnetic catalysis in flavored ABJM\nNiko Jokela,1∗Alfonso V. Ramallo,1†and Dimitrios Zoakos2‡\n1Departamento de F´ ısica de Part´ ıculas\nUniversidade de Santiago de Compostela\nand\nInstituto Galego de F´ ısica de Altas Enerx´ ıas (IGFAE)\nE-15782 Santiago de Compostela, Spain\n2Centro de F´ ısica do Porto\nand\nDepartamento de F´ ısica e Astronomia\nFaculdade de Ciˆ encias da Universidade do Porto\nRua do Campo Alegre 687, 4169-007 Porto, Portugal\nAbstract\nWe study the magnetic catalysis of chiral symmetry breaking in the ABJM\nChern-Simons matter theory with unquenched flavors in the Ve neziano limit.\nWe consider a magnetized D6-brane probe in the background of a flavored\nblack hole which includes the backreaction of massless smea red flavors in the\nABJM geometry. We find a holographic realization for the runn ing of the\nquark mass due to the dynamical flavors. We compute several th ermodynamic\nquantities of the brane probe and analyze the effects of the dyn amical quarks\non the fundamental condensate and on the phase diagram of the model. The\ndynamical flavors have an interesting effect on the magnetic ca talysis. At zero\ntemperature and fixed magnetic field, the magnetic catalysis is suppressed for\nsmall bare quark masses whereas it is enhanced for large valu es of the mass.\nWhenthetemperatureis non-zerothereis acritical magneti c field, above which\nthe magnetic catalysis takes place. This critical magnetic field decreases with\nthe number of flavors, which we interpret as an enhancement of the catalysis.\n∗niko.jokela@usc.es\n†alfonso@fpaxp1.usc.es\n‡dimitrios.zoakos@fc.up.pt\n11 Introduction\nThe dynamics of gauge theories in external electromagnetic fields h as revealed a rich\nstructure of new phenomena (see [1] for a recent review). One of these effects is\nthe spontaneous symmetry breaking of chiral symmetry induced b y a magnetic field,\nwhich is known as magnetic catalysis [2–5]. It can be understood as d ue to the\nfermionic pairing and the effective dimensional reduction which take p lace in the\nLandau levels. In strongly interacting systems the holographic dua lity [6] can be\nused to study this phenomenon [7] (see [8,9] for reviews and furth er references). The\ngeneral objective of these holographic studies is to uncover new p hysical effects of\nuniversal nature that are difficult to discover by using more conven tional approaches.\nIn the holographic approach, the matter fields transforming in the fundamental\nrepresentation of the gauge group are introduced by adding flavo r D-branes to the\ngravity dual. If these flavor branes are treated as probes and th eir backreaction on\nthe geometry is neglected, we are in the so-called quenched approx imation, which\ncorresponds to discarding quark loops on the field theory side. The magnetic field\nneeded for the catalysis is introduced as a worldvolume gauge field on the D-brane.\nFrom the study of the embeddings of the probe one can extract th e ¯qqcondensate as\na function of the quark mass and verify the breaking of chiral symm etry induced by\nthe magnetic field.\nTo go beyond the probe approximation and to study the effects of q uark loops\nin the holographic approach one has to construct new supergravit y duals which in-\nclude the backreaction of the flavor brane sources on the geomet ry. Finding these\nunquenched backgrounds is a very difficult problem which can be simplifi ed by con-\nsideringacontinuousdistributionofflavorbranes(see[10]forar eviewofthissmearing\ntechnique). In [11,12] the magnetic catalysis for the D3-D7 syste m with unquenched\nsmearedflavorbraneswasstudiedandtheeffectsofdynamical fla vorsonthemagnetic\ncatalysis were analyzed.\nIn this paper we address the problem of the magnetic catalysis with u nquenched\nflavors in the ABJM theory [13]. The unflavored version of the ABJM m odel is a\n(2+1)-dimensional Chern-Simons matter theory with N= 6 supersymmetry, whose\ngauge group is U(N)×U(N), with Chern-Simons levels kand−k. It also contains\nbifundamental matter fields. When the two parameters Nandkare large, the ABJM\ntheory can be holographically described by the ten-dimensional geo metryAdS4×CP3\nwith fluxes. One can naturally add flavor D6-branes extended along theAdS4and\nwrapping an RP3submanifold of the internal CP3[14,15]. The smeared unquenched\nbackground for a large number Nfof massless flavors has been constructed in [16].\nThese results were generalized in [17] to non-zero temperature an d in [18] to mas-\n2sive flavors. The main advantage of the ABJM case as compared to o ther holographic\nsetups is that the corresponding flavored backgrounds have a go odUV behavior with-\nout the pathologies present in other unquenched backgrounds (s uch as, for example,\nthe Landau pole singularity of the D3-D7 case). Moreover, in the ca se of massless\nflavors the geometry is known analytically and is of the form AdSBH4×M6, where\nAdSBH4is a black hole in AdS4andM6is a squashed version of CP3. This simplicity\nwill allow us to obtain a holographic realization of the Callan-Symanzik eq uation for\nthe running of the quark mass due to the anomalous dimension gener ated by the\nunquenched flavors.\nWe will carry out our analysis by considering a magnetized D6-brane p robe in the\ngeometry [16,17] dual to the ABJM theory with unquenched massless flavors (i.e.,\ndynamical seaquarks), corresponding to the backreaction of a large number Nfof\nflavor D6-branes with no magnetic field. We are thus neglecting the in fluence of the\nmagnetic field on the sea quarks. To take this effect into account we would have to\nfind the backreaction to magnetized flavor D6-branes, which is an in volved problem\nbeyond the scope of this work. We will study the system both at zer o and non-zero\ntemperature. In both cases we will be able to study the influence of the dynamical\nsea quarks at fully non-linear order in Nf.\nThe rest of this paper is organized as follows. In Section 2 we introdu ce our\nholographicmodel. Wereviewthebackgroundof[16,17], studythea ctionoftheprobe\nin several coordinate systems and establish the dictionary to relat e the holographic\nparameters to the physical mass and condensate. In Section 3 we obtain the different\nthermodynamic properties of the magnetized brane and we find ana lytic results in\nsome particular limiting cases. Section 4 is devoted to the study of th e phase diagram\nand of the magnetic catalysis of chiral symmetry breaking. Finally, in Section 5 we\nsummarize our results and discuss some possible research direction s for the future.\nAppendix A contains the derivation of the holographic dictionary for the condensate\nat zero temperature.\n2 Holographic model\nIn this section we will recall the background of type IIA supergrav ity dual to un-\nquenched massless flavors in the ABJM Chern-Simons matter theor y at non-zero\ntemperature. This background was obtained [16,17] by including th e backreaction\nofNfflavor D6-branes, which are continuously distributed in the interna l space in\nsuch a way that the system preserves N= 1 supersymmetry at zero temperature.\nThis smearing procedure is a holographic implementation of the so-ca lled Veneziano\nlimit [19], in which both NandNfare large. As the smeared flavor branes are not\n3coincident the flavor symmetry is U(1)Nfrather than U(Nf).\nTo study magnetic catalysis in this gravity dual with unquenched flav ors, we will\nadd an additional flavor D6-braneprobe with a magnetic field in its wor ldvolume. We\nwillobtaintheactionofthisprobeandintroducevarioussystems of coordinateswhich\nare convenient to describe the embeddings of the brane, both at z ero and non-zero\ntemperature.\n2.1 Background metric\nOur model consists of a probe D6-brane in the smeared flavored AB JM background\nof [16,17], oriented such that their intersection is (2+1)-dimension al. We will begin\nby laying out our conventions and reviewing the background geomet ry. The metric\nof the background is [16,17]\nds2\n10=L2/parenleftbigg\n−hr2dt2+r2(dx2+dy2)+dr2\nhr2/parenrightbigg\n+L2\nb2/parenleftbig\nqds2\nS4+(E1)2+(E2)2/parenrightbig\n,(1)\nwhereLis a constant radius and the blackening factor is h(r) = 1−r3\nh\nr3, withrh\nconstant. In our conventions all coordinates are dimensionless an dLhas dimension\nof length. The Bekenstein-Hawking temperature Tof the black hole is related to rh\nasT=3rh\n4π. Notice that Tis dimensionless (the physical temperature is T/√\nα′). The\ninternal metric in (1) is a deformation of the Fubini-Study metric of CP3, represented\nas anS2-bundle over S4. This deformation is generated by the backreaction of the\nmassless flavors and introduces a relative squashing qbetween the S2fiber, corre-\nsponding to the two one-forms E1andE2, and the S4base. We write the metric on\nthe four-sphere in (1) as\nds2\nS4=4\n(1+ξ2)2/bracketleftBig\ndξ2+ξ23/summationdisplay\ni=1(ωi)2/bracketrightBig\n, (2)\nwhere 0≤ξ <∞is a non-compact coordinate and the ωiareSU(2) left-invariant\none-forms satisfying dωi=1\n2ǫijkωj∧ωk. TheS2will be represented by the ordinary\npolar coordinates 0 ≤θ < πand 0≤ϕ <2π, in terms of which E1andE2can be\nwritten as\nE1=dθ+ξ2\n1+ξ2/parenleftbig\nsinϕω1−cosϕω2/parenrightbig\n(3)\nE2= sinθ/parenleftbigg\ndϕ−ξ2\n1+ξ2ω3/parenrightbigg\n+ξ2\n1+ξ2cosθ/parenleftbig\ncosϕω1+sinϕω2/parenrightbig\n.(4)\nThe metric (1) has two parameters qandbwhich deserve pronunciation. The\nparameterqis a constant squashing factor of the internal CP3sub-manifold, whereas\n4brepresents the relative squashing between the internal space an d theAdSBH4part\nof the metric. The explicit expressions for the factors qandbof the smeared solution\nof [16,17] are:\nq= 3+3\n2ˆǫ−2/radicalbigg\n1+ˆǫ+9\n16ˆǫ2 (5)\nb=2q\nq+1, (6)\nwhere ˆǫis the flavor deformation parameter, which depends on the number of flavors\nNfand colorsN, as well as the ’t Hooft coupling λ=N/k, via\nˆǫ=3Nf\n4k=3\n4Nf\nNλ . (7)\nThe radius Lin (1) is also modified by the backreaction of the flavors. Indeed, it c an\nbe written as [16]\nL2=π√\n2λ σα′, (8)\nwhereσis the so-called screening function, which determines the correctio n of the\nradius with respect to the unflavored case and is given by the followin g function of\nthe deformation parameter\nσ≡/radicalBigg\n2−q\nq/bracketleftbig\nq+(1+ˆǫ)(q−1)/bracketrightbigb2=1\n4q3\n2(2−q)1\n2(1+ˆǫ+q)2\n/bracketleftbig\nq+(1+ˆǫ)(q−1)/bracketrightbig5\n2. (9)\nWe note that most of the equations that we will manipulate in this pape r only depend\nonb, in which case it suffices to keep in mind that bis monotonously increasing\nbetween 1 and 5 /4 as one dials ˆ ǫ= 0 to∞. Moreover, σ= 1 for ˆǫ= 0 and it vanishes\nas 1/√\nˆǫwhen ˆǫis large.\nThe type IIA supergravity solution of [16,17] also contains a cons tant dilaton φ,\ngiven by\ne−φ=b\n41+ˆǫ+q\n2−qk\nL√\nα′, (10)\nas well as RR forms F2andF4. In this paper we will only need the seven-form\npotentialC7ofF8=−∗F2. To avoid unnecessary notation, we shall only present its\npullback in the subsection to follow.\n2.2 D6-brane action\nNext we will add a probe D6-branein this background, extended alon g the Minkowski\nand radial coordinates and wrapping a three cycle C3≃RP3inside the internal\n5manifold. The cycle C3extends along two directions of the S4and one direction of\ntheS2fiber. It can be characterized by requiring that the pullbacks of tw o of the\none-formsωivanish (say, ω1andω2) and that the angle θof theS2is a function of\nthe radial variable. By a suitable choice of coordinates,1the induced metric on the\nD6-brane worldvolume can be written as\ndˆs2\n7=−L2r2dt2+L2r2/bracketleftbig\n(dx1)2+(dx2)2/bracketrightbig\n+L2\nr2/bracketleftBig\n1+r2\nb2˙θ2/bracketrightBig\ndr2+\n+L2\nb2/bracketleftBig\nqdα2+qsin2αdβ2+ sin2θ/parenleftbig\ndψ+ cosαdβ/parenrightbig2/bracketrightBig\n,(11)\nwhere˙θ=dθ/drand 0≤α<π, 0≤β,ψ<2π.\nThe D6-brane action has two contributions. As usual there is the D irac-Born-\nInfeld term, but we also have a Chern-Simons term due to the pullbac k of the RR\nseven-form potential\nSD6=−TD6e−φ/integraldisplay\nd7ζ/radicalbig\n−det(g7+F) +TD6/integraldisplay\nˆC7, (12)\nwhere theζ’s are the coordinates of the induced metric and F=dAis the strength\nof the worldvolume gauge field. The explicit form of the the pullback of C7is [17]\nˆC7=L7q\nb3e−φd3x∧/bracketleftbigghr3\nbsinθcosθ˙θ+r2sin2θ+L2(r)/bracketrightbigg\n∧dr∧Ξ3,(13)\nwhere Ξ 3= sinαdα∧dβ∧dψand/integraltext\ndrL2(r) =r3\nh\n4b. To write C7we have chosen a\nparticular gauge which leads to a finite renormalized action with consis tent thermo-\ndynamics. In this paper we will consider a background magnetic field d escribed by a\nspatial component of the D6-brane gauge field:\nAx2=x1L2B . (14)\nIn our conventions, the quantity B, as well as x1, is dimensionless. Notice also that\nthe physical magnetic field is related to Bas\nBphys=L2B\nα′2=π√\n2λσB\nα′. (15)\nA straightforward computation for the full action yields:\nS=−N/integraldisplay\nd3x/braceleftBigg\n4b\nr3\nh/integraldisplay\ndrr2sinθ/parenleftBigg/radicalbigg\n1+B2\nr4/radicalbigg\n1+h/parenleftBigr\nb/parenrightBig2˙θ2−sinθ−hr\nbcosθ˙θ/parenrightBigg\n−1/bracerightBigg\n,\n(16)\n1Let us require the pullbacks ˆ ω1= ˆω2= 0 and parameterize ˆ ω3=dˆψ. Then,α,β, andψare\ndefined as: ξ=: tan/parenleftbigα\n2/parenrightbig\n,β:=ˆψ\n2, andψ:=ϕ−ˆψ\n2.\n6where the prefactor is\nN=2π2r3\nhL7q\nb4TD6e−φ=2√\n2π2(2−b)bσ\n27N√\nλ T3. (17)\nFor later use we also define:\nNr=4b\nr3\nhN=(2−b)b2σ\n4√\n2πN3/2\n√\nk. (18)\nThe equation of motion for the embedding scalar is thus,\n∂r/parenleftbigg\ng/parenleftBigr\nb/parenrightBig2/parenleftbigg\n1+B2\nr4/parenrightbigg\n˙θ/parenrightbigg\n=r2/parenleftbigg3\n2b−1+hr2\n2g/parenrightbigg\nsin2θ , (19)\nwhere we have defined\ng=hr2sinθ/radicalBig\n1+B2\nr4/radicalBig\n1+h/parenleftbigr\nb/parenrightbig2˙θ2. (20)\nThe above equation of motion has generically two kinds of solutions. T he first\nkind are embeddings that penetrate the black hole horizon, those w e shall call black\nhole (BH) embeddings. The other kind are Minkowski (MN) embedding s, which\nterminate smoothly above the horizon at some r0> rh. Examples of both kind are\nthe following. Clearly, the equation of motion is satisfied with trivial co nstant angle\nBH embeddings θ= 0,π/2. The equation of motion possesses a supersymmetric MN\nsolution cos θ(r) =/parenleftbigr0\nr/parenrightbigbatrh= 0 andB= 0. Away from zero temperature and\nvanishing magnetic field, this solution has to be analyzed numerically. O ur focus in\nthis article is to study how these two types of solutions map out the p hase space\nas both the TandBare dialed, and the interesting effects from the variation of\nthe number of background flavors (essentially b). Before we will get absorbed in\nanalyzing several aspects of the system, we wish to introduce new parameterizations\nbetter suited for the analyses.\n2.3 Parameterization at non-zero temperature\nItisusefultointroduceanotherparameterizationasdiscussedin [17]. Letusintroduce\na system with isotropic Cartesian-like coordinates\nR=ucosθ (21)\nρ=usinθ , (22)\nwhere the new radial coordinate uis related to the old one as\nu3\n2b=/parenleftbiggr\nrh/parenrightbigg3\n2\n+/radicalBigg/parenleftbiggr\nrh/parenrightbigg3\n−1. (23)\n7We also define the functions fand˜fas\nf= 1−u−3/b(24)\n˜f= 1+u−3/b. (25)\nWe also rescale the magnetic field as follows:\nˆB= 24/3B\nr2\nh. (26)\nAfter these mappings the action becomes\nS=−N/integraldisplay\nd3x/braceleftBigg/integraldisplay\ndρρf˜fu3/b−2/parenleftBigg/radicalBigg\n1+ˆB2\n˜f8/3u4/b√\n1+R′2−1\n+/parenleftbiggf\n˜f−1/parenrightbiggR\nu2(ρR′−1)/parenrightBigg\n−1/bracerightBigg\n, (27)\nwhere it is understood that u=/radicalbig\nρ2+R2.\nA generic solution to the equation of motion following from the action ( 27), be-\nhaves close to the boundary as:\nR=m+c\nρ3/b−2+... , ρ→ ∞, (28)\nwheremis related to the quark mass and cis proportional to the vacuum expectation\nvalue∝angbracketleft¯ψψ∝angbracketright(see below).\n2.4 Parameterization at zero temperature\nAt zero temperature we also make use of the Cartesian-like coordin ates as in (21) and\n(22), but with\nu=rb. (29)\nThe action (16) maps to\nS=−Nr\nb/integraldisplay\ndρρu3/b−2/braceleftBigg/radicalbigg\n1+B2\nu4/b√\n1+R′2−1/bracerightBigg\n, (30)\nwhere it is understood that u=/radicalbig\nρ2+R2. We can scale out the Bas follows:\nu4/b=B2˜u4/b→u=Bb/2˜u (31)\nR=Bb/2˜R (32)\nρ=Bb/2˜ρ . (33)\n8This leads us to\nS=−B3/2\nbNr/integraldisplay\nd˜ρ˜ρ˜u3/b−2/braceleftBigg/radicalbigg\n1+1\n˜u4/b/radicalbig\n1+˜R′2−1/bracerightBigg\n(34)\nand to the following asymptotic behavior of the embedding function\nR∼m0+c0\nρ3/b−2→˜R∼˜m0+˜c0\n˜ρ3/b−2, (35)\nwhere we defined the dimensionless quantities ˜ m0and ˜c0as:\n˜m0≡B−b/2m0 (36)\n˜c0≡B(b−3)/2c0. (37)\nThe parameters m0andc0can be related to the quark mass mqand the quark\ncondensate at zero temperature (denoted by ∝angbracketleftOq∝angbracketright0). The corresponding relation is\nworked out in the next section and in appendix A.\n2.5 Running mass and condensate\nThe asymptotic value of the embedding function Rshould be related to the quark\nmass. To find the precise relation we will consider a fundamental str ing stretched in\ntheRdirection and ending on the flavor brane. The quark mass is just the Nambu-\nGoto action of the string per unit time. While carrying out this comput ation we\nshould take into account that we are dealing with a theory with unque nched quarks\nin which the quark mass mqacquires an anomalous dimension γmandmqtherefore\nruns with the scale according to the corresponding Callan-Symanzik equation. In our\nholographic setup the value of γmwas found in [16,17] and is simply related to the\nsquashing parameter b:\nγm=b−1. (38)\nIn order to find the scale dependence of mq, we consider a fundamental string located\nat the point ρ=ρ∗. We will start by considering the zero temperature case. Notice\nthatρis the holographic coordinate in our setup and, therefore, it is natu ral to think\nthat the value of ρ∗determines the energy scale. The induced metric on a string\nworldsheet extended in ( t,R) atρ=ρ∗whenT= 0 is given by:\nds2\n2=−L2/bracketleftbig\nR2+ρ2\n∗/bracketrightbig1\nbdt2+L2\nb2dR2\nR2+ρ2\n∗. (39)\nThe running quark mass at zero temperature is then defined as:\nmq=1\n2π(α′)3\n2/integraldisplaym0\n0/radicalbig\n−detg2dR=/radicalbigg\nλ\n2σ\nb√\nα′/integraldisplaym0\n0/bracketleftbig\nR2+ρ2\n∗/bracketrightbig1\n2b−1\n2dR\n=/radicalbigg\nλ\n2σ\nb√\nα′m0ρ1\nb−1\n∗2F1/parenleftBig1\n2,γm\n2b;3\n2;−m2\n0\nρ2\n∗/parenrightBig\n. (40)\n9Notice that, in the unflavored case b= 1,γm= 0 and the effective mass mqis\nindependent of the scale parameter ρ∗, as it should. To determine the precise relation\nbetweenρ∗and the energy scale Λ, let us consider the relation (29) between th e\ncoordinate uand the canonical AdS4radial coordinate r. Taking into account that\nu≈ρin the UV, it is natural to identify rwith the energy scale and define Λ as:\nΛ≡ρ1\nb∗. (41)\nThe dependence of mqon Λ can be straightforwardly inferred from (40). Moreover,\nfrom the integral representation in (40) we can readily obtain an ev olution equation\nformq\n∂mq\n∂logΛ=mq−σ√\nα′/radicalbigg\nλ\n2m0\n(Λ2b+m2\n0)γm\n2b. (42)\nClearly, the second term in (42) incorporates the flavor effects on the running of mq.\nIn the UV regime of large Λ we can just neglect m2\n0in the denominator of (42). The\nsolution of this UV equation can be obtained directly or by taking the la rge Λ limit\nof (40). We get\nmq√\nα′\n√\nλ≈σ√\n2bm0Λ−γm, (43)\nwhich shows that in the UV mqandm0are proportional and that the running of mq\nwith the scale Λ is controlled by the mass anomalous dimension γm. Notice that the\nUV mass (43) satisfies:\n∂mq\n∂logΛ=−γmmq, (44)\nwhich is just the Callan-Symanzik equation for the effective mass.\nThe analysis carried out above for mqis independent of the value of the magnetic\nfieldB. WhenB∝negationslash= 0 it is convenient to write the solution of the evolution equation\nin terms of the reduced mass parameter ˜ m0defined in (36). We get:\nmq√\nα′\n√\nλ=σ√\n2bBb\n2˜m0Λ−γm2F1/parenleftBig1\n2,γm\n2b;3\n2;−Bb˜m2\n0\nΛ2(1+γm)/parenrightBig\n. (45)\nTo find the relation between the parameter c0in (35) and the condensate we have\nto compute the derivative of the free energy with respect to the b are quark mass µ0\nq,\nwhich is the quark mass without the screening effects due to the qua rk loops. These\neffects are encoded inthe functions σandb. By putting σ=b= 1, which corresponds\nto taking ˆǫ= 0, we switch off the dressing due to the dynamical flavors. Accord ingly,\nto getµ0\nqin terms of ˜ m0we just take σ=b= 1 on the right-hand side of (45). We\nget\nµ0\nq=/radicalbigg\nλ\n2√\nB˜m0√\nα′. (46)\n10Notice that the value of ˜ m0does not depend on the magnetic field, which is factorized\nin the action (34). Therefore, the dependence of µ0\nq∼√\nBon the field Bis the same\nas in the unflavored case, as it should.\nTheexplicit calculationofthevacuumexpectationvalue ∝angbracketleftOq∝angbracketright0hasbeenperformed\nin Appendix A, with the result:\n−∝angbracketleftOq∝angbracketright0α′\nN=(3−2b)(2−b)\n4πσBγm\n2c0=(3−2b)(2−b)\n4πσB˜c0.(47)\nEqs. (46) and (47) constitute the basic dictionary in our analysis of the chiral sym-\nmetry breaking at zero temperature.\nFor non-zero temperature we shall proceed as in the T= 0 case. The induced\nmetric for the fundamental string extended in Ratρ=ρ∗is now\nds2\n2=−L2r2\nh\n24\n3/bracketleftbig\nR2+ρ2\n∗/bracketrightbig1\nb/bracketleftbig\nf∗(R)/bracketrightbig2/bracketleftbig˜f∗(R)/bracketrightbig−2\n3dt2+L2\nb2dR2\nR2+ρ2∗,(48)\nwheref∗(R) and˜f∗(R) are the functions defined in (24) and (25) at ρ=ρ∗. Accord-\ningly, the running quark mass at T∝negationslash= 0 is now given by an integral extended from\nthe horizon (for ρ∗<1) toR=m:\nmT\nq=/radicalbigg\nλ\n2σ\nb√\nα′rh\n22\n3/integraldisplaym\nRh/bracketleftbig\nR2+ρ2\n∗/bracketrightbig1\n2b−1\n2f∗(R)/bracketleftbig˜f∗(R)/bracketrightbig−1\n3dR , (49)\nwhereRh=/radicalbig\n1−ρ2∗forρ∗<1 andRh= 0 otherwise. We have not been able to\nintegrate analytically this expression for arbitrary values of ρ∗. In order to relate ρ∗\nwith the scale Λ we recall that, in the UV, ρ1\nb≈u1\nb≈22\n3r/rh. Therefore, identifying\nagainrwith Λ, we have:\nΛ = 2−2\n3rhρ1\nb∗. (50)\nWe readily obtain in the UV domain (Λ ≫1):\nmT\nq√\nα′\n√\nλ≈σ√\n2brb\nh\n22b\n3mΛ−γm. (51)\nThis UV function mT\nqalso satisfies the Callan-Symanzik equation (44). Actually, it is\neasy to relate in the UV the effective mass mT\nqto its zero temperature counterpart. In\norder to establish this connection, let us connect m0andc0with the zero temperature\nlimit ofmandc. To find these relations we recall that these parameters charact erize\nthe leading and subleading UV behaviors of the embedding function. F rom this\nobservation it is easy to prove that\nm1\nb≈22\n3r−1\nhm1\nb\n0, c ≈22−2b\n3rb−3\nhc0, (T→0).(52)\n11By using the relation between mandm0written in (52), we see that mqis just the\nlimit ofmT\nqasT→0:\nmq= lim\nT→0mT\nq. (53)\nThe bare mass at non-zero temperature µqis obtained from the unflavored limit\nof the UV running mass (51). We get [17]:\nµq√\nα′=21\n3π\n3√\n2λTm . (54)\nThe resulting vacuum expectation value at T∝negationslash= 0 has been obtained in appendix D\nof [17] and is given by:\n−∝angbracketleftOq∝angbracketrightα′\nN=22/3π(3−2b)(2−b)\n9σT2c . (55)\nThe relation between the condensates at non-zero and zero temp erature is similar to\nthe one corresponding to the masses. Indeed, by using (52) we ge t that∝angbracketleftOq∝angbracketright0is given\nby the following zero temperature limit:\n∝angbracketleftOq∝angbracketright0= lim\nT→0/bracketleftBig\nˆBγm\n2∝angbracketleftOq∝angbracketright/bracketrightBig\n. (56)\nThe relation (56) is very natural from the point of view of the renor malization\ngroup. Indeed, ∝angbracketleftOq∝angbracketrightand∝angbracketleftOq∝angbracketright0are dimensionful quantities defined at scales de-\ntermined by the temperature and the magnetic field, respectively. The quotient\n∝angbracketleftOq∝angbracketright/∝angbracketleftOq∝angbracketright0should be given by the ratio of these two energy scales (which is ba-\nsically/radicalbig\nˆB) raised to some power which, following the renormalization group logic ,\nshould be the mass anomalous dimension, as in (56).\nIn the rest of this paper, we will use units in which α′= 1. The appropriate power\nofα′can be easily obtained in all expressions by looking at their units.\n3 Some properties of the dual matter\nWe will discuss many of the characteristics of the dual matter as de scribed by the\ngravitational system. The BH phase describes typical metallic beha vior. The phase is\nnongapped to charged and neutral excitations. For example, the former can be easily\nverified by the standard DC conductivity calculation [20] in a simple gen eralization\nof our model by introducing a non-vanishing charge density on the p robe. The MN\nphase, on the other hand, behaves like an insulator: it is gapped to b oth neutral and\ncharged excitations; the latter can be checked by the conductivit y calculation of [21]\n12and the former by fluctuation analysis. The interplay between thes e two phases in\nthe presence of a charge density makes an interesting story, whic h will be addressed\nin a future work.\nIn the absence of the magnetic field, the thermodynamic propertie s of the system\nwere discussed in great detail in [17]. Here we are more interested in t he magnetic\nproperties and on the effects that the magnetic field will bear. We will break this\nnarrative in two parts, so that in this section we will constrain ourse lves to the case\nwhere we have analytic control and in the next section we will confro nt the numerical\nside of the story, most relevantly the magnetic catalysis.\n3.1 Thermodynamic functions\nThe free energy of the system is obtained from evaluating the Wick r otated on-shell\naction (16). As discussed in [16,17], the free energy is finite albeit su btle at non-\nzero temperature; there is no need to invoke holographic renorma lization to get rid\noff infinities. The free energy of the probe is identified with the Euclide an on-shell\nactionSE, through the relation F=TSE. In the calculation of SEwe integrate\nover both the Euclidean time and the non-compact two-dimensional space. Since the\nlatter integration gives rise to an (infinite) two-dimensional volume V2, from now on\nwe divide all the extensive thermodynamic quantities by V2and deal with densities.\nThe free energy density Fcan be written as\nF\nN=G(m,ˆB)−1. (57)\nTheexplicitexpressionforthefunction G(m,ˆB)canbeobtainedfromtheactionofthe\nD6-brane probe. For MN embeddings it is more convenient to use R(ρ) as embedding\nfunction. From the expression (27) of the action in these variables ,G(m,ˆB) is given\nby\nG(m,ˆB) =/integraldisplay∞\n0dρρ/bracketleftbig\nρ2+R2/bracketrightbig3\n2b−1f˜f/bracketleftBigg\n√\n1+R′2/radicalBigg\n1 +ˆB2\n˜f8\n3/bracketleftbig\nρ2+R2/bracketrightbig−2\nb\n−1 +/parenleftbiggf\n˜f−1/parenrightbiggR\nρ2+R2(ρR′−R)/bracketrightBigg\n. (58)\nFor black hole embeddings it is better to use the θ=θ(r) parameterization and\nrepresent G(m,ˆB) as:\nG(m,ˆB) =4b\nr3\nh/integraldisplay∞\nrhdrr2sin2θ/bracketleftBigg/radicalbigg\n1+/parenleftBigrh\n22\n3r/parenrightBig4ˆB2/radicalbigg\n1+h/parenleftBigr\nb/parenrightBig2˙θ2−sinθ−hr\nbcosθ˙θ/bracketrightBig\n.\n(59)\n13The fact that the system under study is defined at a fixed tempera ture and mag-\nnetic field implies that the appropriate thermodynamic potential is\ndF=−sdT− MdB, (60)\nwheresis the entropy density and Mis the magnetization of the system. Following\n(60), the entropy density sis given by the following expression\ns=−/parenleftbigg∂F\n∂T/parenrightbigg\nB=−N\nT/bracketleftBigg\n3F\nN+T/parenleftbigg∂G\n∂m/parenrightbigg\nˆB/parenleftbigg∂m\n∂T/parenrightbigg\nB+T/parenleftbigg∂G\n∂ˆB/parenrightbigg\nm/parenleftBigg\n∂ˆB\n∂T/parenrightBigg\nB/bracketrightBigg\n.\n(61)\nLet us compute the different derivatives on the right-hand side of ( 61). First of all\nwe use that [17]\n∂G\n∂m=2b−3\nbc , (62)\nand thatT∂m/∂T =−bm, as follows from (51) when mT\nqand Λ are fixed. Moreover,\nwe define the function J(m,ˆB) as\nJ(m,ˆB)≡1\nˆB∂G(m,ˆB)\n∂ˆB. (63)\nThen, taking into account the ˆB∝T−2temperature dependence of the rescaled\nmagnetic field in (26), we get:\nTs\nN= 3−3G(m,ˆB) + 2ˆB2J(m,ˆB)−(3−2b)cm . (64)\nFor Minkowski embeddings, J(m,ˆB) is explicitly given by the following integral:\nJ(m,ˆB)≡/integraldisplay∞\n0dρ/bracketleftbig\nρ2+R2/bracketrightbig1\n2b−1f˜f−1\n3√\n1+R′2\n/radicalBig\nˆB2+/parenleftbig\nρ2+R2/parenrightbig2\nb˜f8\n3,(65)\nwhereas for a black hole embedding we have:\nJ(m,ˆB) =brh\n22\n3/integraldisplay∞\nrhdrsin2θ/radicalbigg\n1+h/parenleftBig\nr\nb/parenrightBig2˙θ2\n/radicalbigg\nr4+/parenleftBig\nrh\n22\n3/parenrightBig4ˆB2. (66)\nThe internal energy density Ecan be computed from the relation E=F+Ts,\nwith the result\nE\nN= 2−2G(m,ˆB) + 2ˆB2J(m,ˆB)−(3−2b)cm . (67)\n14The heat capacity density cvis defined as cv=∂E/∂T. Computing explicitly the\nderivative of Ewith respect to the temperature in (67), and using (62), we arrive at\nthe following expression:\nTcv\nN= 2Ts\nN−2ˆB2/parenleftBig\nJ(m,ˆB) + 2ˆB∂J(m,ˆB)\n∂ˆB/parenrightBig\n+\n+(2b−3)/bracketleftBigg/parenleftBig\n3−b−b∂(logc)\n∂(logm)/parenrightBig\ncm−4mˆB∂c\n∂ˆB/bracketrightBigg\n.(68)\nIn order to holographically investigate the joint effect of the prese nce of flavors\nand magnetic field on the speed of sound, we use the following definitio n\nv2\ns=−∂P\n∂E=∂F\n∂T/parenleftbigg∂E\n∂T/parenrightbigg−1\n=s\ncv. (69)\nLet us apply the formula (69) for the background plus probe syste m. Expanding at\nfirst order in the probe functions, we get:\nv2\ns=sback+s\ncv,back+cv≈1\n2−cv−2s\n4sback, (70)\nwhere we have taken into account that cv,back= 2sbackand, therefore, v2\ns= 1/2 for\nthe background, as it corresponds to a conformal system in 2+1 d imensions. Since,\nwe can rewrite the ratio N/Tsbackin the following form [17]\nN\nTsback=1\n4λ\nNq\nb4σ2, (71)\nwe arrive at the following expression for the deviation δv2\ns=v2\ns−1\n2:\nδv2\ns≈λ\nNqσ2\n16b4/bracketleftBigg\n2ˆB2/parenleftBig\nJ(m,ˆB) + 2ˆB∂J(m,ˆB)\n∂ˆB/parenrightBig\n+\n+(3−2b)/parenleftBigg/parenleftBig\n3−b−b∂(logc)\n∂(logm)/parenrightBig\ncm−4mˆB∂c\n∂ˆB/parenrightBigg/bracketrightBigg\n,(72)\nwhere we have used (68) to compute cv−2sfor the probe.\nAccording to (60) the magnetization of the system is given by the fo llowing ex-\npression\nM=−/parenleftbigg∂F\n∂B/parenrightbigg\nT=−24\n3π\n3bNrTˆBJ(m,ˆB). (73)\nThe magnetic susceptibility χis defined as:\nχ≡∂M\n∂B. (74)\n15For generic embeddings, which are numerical, also the thermodynam ic quantities\nneed to be calculated numerically. However, there are two corners were analytic re-\nsultscanbeobtained. Thefirst oneiswhenwe studyembeddings with asymptotically\nlargemand the other when the embeddings are massless. We will consider th ese two\ncases separately in the next two subsections.\n3.2 Massless embeddings\nFor zero mass (and c= 0) the embedding is necessarily a black hole embedding and\nit is more convenient to use the θ=θ(r) parameterization. Actually, the mass-\nless embeddings in these variables are just characterized by the co nditionθ=π/2.\nTherefore, it follows from (59) that the function G(m= 0,ˆB) is given by the following\nintegral:\nG(m= 0,ˆB) =4b\nr3\nh/integraldisplay∞\nrhdrr2/parenleftBig/radicalbigg\n1+/parenleftBigrh\n22\n3r/parenrightBig4ˆB2−1/parenrightBig\n, (75)\nwhich can be explicitly performed:\nG(m= 0,ˆB) =4b\n3/bracketleftBig\n1−2F1/parenleftBig\n−1\n2,−3\n4,1\n4;−ˆB2\n28\n3/parenrightBig/bracketrightBig\n. (76)\nThen, it follows that the free energy is given by\nF\nN=−1 +4b\n3/bracketleftBig\n1−2F1/parenleftBig\n−1\n2,−3\n4,1\n4;−/parenleftBig3\n4π/parenrightBig4B2\nT4/parenrightBig/bracketrightBig\n. (77)\nLet us next compute the entropy density for the massless embedd ings. By taking\nm= 0 in (64), we find\nTs(m= 0,ˆB)\nN=−3G(m= 0,ˆB) + 2ˆB2J(m= 0,ˆB) + 3.(78)\nThe integral J(m= 0,ˆB) can be evaluated explicitly from its definition (66),\nJ(m= 0,ˆB) =b\n22\n32F1/parenleftBig1\n4,1\n2,5\n4;−ˆB2\n28\n3/parenrightBig\n. (79)\nPlugging this result into (78), after some calculation, we arrive at th e following simple\nexpression for the entropy density of the massless embeddings:\nTs(m= 0,B)\nN= 3−4b+4b/radicalbigg\n1 +/parenleftBig3\n4π/parenrightBig4B2\nT4. (80)\nSimilarly, the internal energy for zero mass is obtained from (67):\nE\nN/vextendsingle/vextendsingle/vextendsingle\nm=0= 2+8b\n3/bracketleftBigg/radicalbigg\n1+/parenleftBig3\n4π/parenrightBig4B2\nT4−1+/parenleftBig3\n4π/parenrightBig4B2\nT42F1/parenleftBig1\n2,1\n4,5\n4;−/parenleftBig3\n4π/parenrightBig4B2\nT4/parenrightBig/bracketrightBigg\n.\n(81)\n16We will compute the heat capacity by taking m= 0 in (68) and using the remarkable\nproperty:\nJ(m= 0,ˆB) + 2ˆB∂J(m= 0,ˆB)\n∂ˆB=b\n22\n31/radicalBig\n1+ˆB2\n28\n3, (82)\nwhich combined with (80) leads to the simple result:\nTcv\nN/vextendsingle/vextendsingle/vextendsingle\nm=0= 6+8b/bracketleftBigg\n1/radicalbigg\n1+/parenleftBig\n3\n4π/parenrightBig4B2\nT4−1/bracketrightBigg\n. (83)\nThis result can be confirmed by computing directly the derivative of t he internal\nenergy written in (81). The deviation of the speed of sound with res pect to the\nconformal value v2\ns= 1/2 is readily obtained from (72):\nδv2\ns≈λ\n2Nq σ2\nb3/parenleftBig3\n4π/parenrightBig4B2\nT4/radicalbigg\n1+/parenleftBig\n3\n4π/parenrightBig4B2\nT4, (84)\nand the magnetization of the system at zero mass follows from (73) and (79):\nM(m= 0,B) =−3Nr\n4πB\nT2F1/parenleftBig1\n4,1\n2,5\n4;−/parenleftBig3\n4π/parenrightBig4B2\nT4/parenrightBig\n. (85)\nWe note that the magnetization is always negative and vanishes at ze ro field. It is no\nsurprise that the system is diamagnetic. The magnetic field appears with even power\ninside the DBI action, which implies that the spontaneous magnetizat ion vanishes.\nMoreover, the DBI action has a specific (plus) sign, meaning that th e magnetization\nis always non-positive.2\nTo obtain the magnetic susceptibility we have to compute the derivat ive of the\nright-hand side of (85) with respect to the magnetic field (see (74) ). We get:\nχ(m= 0,B) =−3Nr\n8πT\n1/radicalbigg\n1+/parenleftBig\n3\n4π/parenrightBig4B2\nT4+2F1/parenleftBig1\n4,1\n2,5\n4;−/parenleftBig3\n4π/parenrightBig4B2\nT4/parenrightBig\n.(86)\nIntheequations writtenaboveforthemassless black holeembeddin gs thedependence\non the number of flavors is contained implicitly in the parameter b(see (6)), while the\ndependence on the magnetic field and temperature is manifest. One can take further\n2In other systems, where the gauge fields have Chern-Simons term s, their contribution to the\nmagnetization can be positive thus leading to a competition with the DB I part. As a result one\nmight get a positive overall magnetization leading to paramagnetism ( see [22,23]), or even to ferro-\nmagnetism (as in [23]).\n17limits in some of these functions. For example, the T= 0 values of the free energy\n(77) and entropy (78) are:\nF(m= 0)\nNr/vextendsingle/vextendsingle/vextendsingle\nT=0=B3/2\n6√πΓ/bracketleftbigg1\n4/bracketrightbigg2\n, s (m= 0)/vextendsingle/vextendsingle/vextendsingle\nT=0=4π\n3NrB,(87)\nwhile the magnetization (85) of the massless embeddings at zero tem perature is:\nM(m= 0)/vextendsingle/vextendsingle/vextendsingle\nT=0=−1\n4√πNrΓ/bracketleftbigg1\n4/bracketrightbigg2√\nB. (88)\nLet us pause here for a while. We wish to emphasize, that though we w ere able to\nproduce analytic formulas in the special case of massless embedding , this phase is\nonly relevant for small values of ˆB. In particular, the T= 0 case is never thermody-\nnamically preferred. The phase diagram will be addressed in Section 4 .\n3.2.1 Small magnetic field\nLet us focus on limits of thermodynamic quantities for the massless c ase when the\nmagnetic field is small (actually when B/T2→0). These expressions give the first\ncorrection, due to the magnetic field, to the conformal behavior o f the probe at\nB/T2→0. ForF,s,E, andcvwe find\nF\nN≈ −1+2b/parenleftBig3\n4π/parenrightBig4B2\nT4, Ts\nN≈3+2b/parenleftBig3\n4π/parenrightBig4B2\nT4,\nE\nN≈2 + 4b/parenleftBig3\n4π/parenrightBig4B2\nT4, Tcv\nN≈6−4b/parenleftBig3\n4π/parenrightBig4B2\nT4.(89)\nMoreover, the variation of the speed of sound at leading order in B/T2is:\nδv2\ns≈λ\n2Nq σ2\nb3/parenleftBig3\n4π/parenrightBig4B2\nT4, (B/T2→0), (90)\nand the magnetization becomes:\nM ≈ −3Nr\n4πB\nT=−3(2−b)b2σ\n(4π)2√\n2N√\nλB\nT, (B/T2→0).(91)\nIt follows that the susceptibility at vanishing magnetic field is:\nχ(m=B= 0) =−3(2−b)b2σ\n(4π)2√\n2N√\nλ1\nT. (92)\nThus, the diamagnetic response of the system goes to zero as the temperature ap-\nproaches infinity. The behavior possessed by (92) closely resemble s another (2+1)-\ndimensional construction [25]. In both cases the system behaves a s in Curie’s law\nχ∝1/T, though they are diamagnetic. From a dimensional analysis point of v iew\nthis temperature dependence is the expected one for the magnet ic susceptibility in\n2+1 dimensions, since at high Tconformality is restored.\n183.3 Approximate expressions for large mass\nWhen the D6-brane probe remains far away from the horizon, it is po ssible to obtain\nanalytic results for the free energy and the rest of the thermody namic quantities.\nFollowing the analysis of [17,24], for large mthe embeddings are nearly flat and\ngiven by the following expression\nR(ρ) =R0+δR(ρ), (93)\nwhereR0is a constant and δR(ρ) is much smaller than R0. Before calculating the\nfree energy we want an approximate expression for the condensa te of the theory as a\nfunction of the mass and the magnetic field. For this task we need th e relationship\nbetweenR0andm. A simple calculation yields the following expansion in powers of\nm\nR0=m−a(b)m1−6\nb+1\n4a1(b)ˆB2m1−4\nb+···, (94)\nwhere the function a(b) is given in equation (B.15) of [17], which we record here for\ncompleteness and a1(b) is\na(b) =3\n3+2b/bracketleftBig2b\n3−2b+ψ/parenleftBig3\nb/parenrightBig\n−ψ/parenleftBig3\n2b/parenrightBig/bracketrightBig\n(95)\na1(b) =2b\n3−2b−ψ/parenleftbigg3\n2b/parenrightbigg\n+ψ/parenleftbigg2\nb/parenrightbigg\n, (96)\nwhereψ(x) = Γ′(x)/Γ(x) is the digamma function. Using (94) it is possible to obtain\nanapproximateexpressionforthecondensateasafunctionofth emassforanynumber\nof flavors\nc=6b\n4b2−9m−1−3\nb+1\n2bˆB2\n3−2bm−1−1\nb−4\n3bˆB2\n3−2bm−1−4\nb+···.(97)\nUsing these results in (58) and (65) it is possible to evaluate the func tionsG(m,ˆB)\nandJ(m,ˆB) for large values of the mass parameter m. ForGwe get\nG(m,ˆB) = 1+b\n2ˆB2\nm1\nb−2b\n3 +2b1\nm3\nb−b\n3ˆB2\nm4\nb+···, (98)\nwhileJbehaves for large mas:\nJ(m,ˆB) =b/bracketleftBigg\n1\nm1\nb−2\n31\nm4\nb/bracketrightBigg\n+···. (99)\nIt is now straightforward to compute the different thermodynamic functions in\nthis high mass regime. Indeed, the free energy and the entropy fo llow directly by\n19substituting (98) and (99) into (57) and (64), respectively,\nF\nN=−2b\n3 +2b1\nm3\nb+b\n2ˆB2\nm1\nb−b\n3ˆB2\nm4\nb··· (100)\nTs\nN=12b\n3 +2b1\nm3\nb+bˆB2\nm4\nb+···. (101)\nSimilarly,Eandcvcan be expanded as\nE\nN=b\n2ˆB2\nm1\nb+10b\n3 +2b1\nm3\nb+2b\n3ˆB2\nm4\nb+··· (102)\nTcv\nN=60b\n3 +2b1\nm3\nb+ 2bˆB2\nm4\nb+···, (103)\nand the variation of the speed of sound at large mis given by\nδv2\ns=−9\n4λ\nNσ2\n(3+2b)(2−b)b21\nm3\nb+···. (104)\nFinally, in the high mass regime the magnetization can be expanded as:\nM=−24\n3π\n9NrTˆB/parenleftbigg3\nm1\nb−2\nm4\nb/parenrightbigg\n···. (105)\nThis regime of large mis achieved when the temperature is low. Therefore, it is\ninteresting to compare these results with the ones obtained when T= 0. We will\nperform this analysis in the next subsection.\n3.3.1 Zero temperature limit\nOne can calculate the different thermodynamic functions at zero te mperature by\nworking directly with the parameterization of Section 2.4, in which the embeddings\nare characterized by the two rescaled parameters ˜ m0and ˜c0(see (36) and (37)). For\nlarge values of ˜ m0one can proceed as above and find an approximate expression of\nthe condensate as a function of the mass:\n˜c0=1\n2b\n3−2b˜m−1−1\nb\n0+···, (˜m0→ ∞). (106)\nNotice that, according to our dictionary (46), large ˜ m0corresponds to large µ0\nqor\nsmallB. Actually, we can extract the dependence of the condensate on t heBfield\nby rewriting (106) in terms of the unrescaled parameters c0andm0\nc0=1\n2b\n3−2bB2m−1−1\nb\n0+···. (107)\n20It is worth pointing out that (106) and (107) can also be obtained by using (52) and\nkeeping the leading terms in the T→0 limit. It is instructive to write (107) in terms\nof physical quantities. We can use our dictionary (46) and (47) to t ranslate (107) to\n−∝angbracketleftOq∝angbracketright0\nN=(2−b)b\n8π2Bphys√\n2λ/bracketleftBigg√\nλ\n2√\n2πσBphys\n(µ0q)2/bracketrightBiggb+1\n2b\n, (108)\nwhere we have written the result in terms of Bphys=L2B. Similarly, by direct\ncalculation or by using the limiting expressions (52), one finds that th e free energy\nfor large ˜m0can be approximated as:\nF\nNr=1\n2B2m−1\nb\n0+···, (109)\nwhich, in terms of physical quantities corresponds to\nF=(2−b)b2\n8π2√\n2N√\nλµ0\nqBphys/bracketleftBigg√\nλ\n2√\n2πσBphys\n(µ0q)2/bracketrightBiggb+1\n2b\n+···. (110)\nBy computing the derivatives with respect to the magnetic field of th e free energy\nwritten above one can easily obtain the magnetization and susceptib ility at zero\ntemperature in the regime in which µ0\nq/√\nBis large.\n4 Magnetic catalysis\nIn this section we will address the full phase diagram of the system a t non-zero mag-\nneticfield, zeroandnon-zerotemperature, andinthepresence o fbackground smeared\nflavorsNf∝negationslash= 0. There are excellent reviews [8,9] which discuss some of the inter est-\ning phenomena that occur on probe-brane systems when an exter nal magnetic field\nis turned on. To narrow the scope we focus on a particular effect wh ere the magnetic\nfield induces spontaneous chiral symmetry breaking, known as the magnetic cataly-\nsis [3–5]. Only rather recently have we witnessed attempts in addres sing magnetic\ncatalysis that arise away from the probe limit [11,12]. Current pape r constitutes\nour second step in this direction, the first one being the construct ion of a dual to\nunquenched massive flavors [18]. Ultimately, one wishes to combine th e two and also\nbackreact the magnetic field on the supergravity solution. Our goa l is very ambi-\ntious, but we nevertheless feel that it should not go without seriou s attempt. In the\ncurrent paper, we are more modest and consider the magnetic field only residing on\nthe probe, but as a background we consider the fully backreacted massless flavored\nABJM model.\n21We will begin our discussion with zero temperature case and then mov e on to non-\nzero temperature. Our findings resemble somewhat the results in t he supersymmetric\nD3-D7 probe brane analysis [8,9]. However, surprisingly, the flavo r factors go along\nthe ride and we can thus analyze the background flavor effects exa ctly, in contrast to\nD3-D7 system where the flavors have to be treated perturbative ly [11,12]. We also\nshow that the magnetic catalysis is enhanced (suppressed) with fla vor effects at large\n(small) magnetic field strength when the bare quark mass is non-zer o.\n4.1 Zero temperature\nConsider the equation of motion for the embedding ˜R=˜R(˜ρ) as derived from the\naction (34) at zero temperature T= 0. At non-zero Bthe supersymmetry is broken,\nand thus the D6-branehas a profile (as in (35)) which depends on ˜ ρ. Forlarge ˜m0, the\ncondensate can be obtained analytically (see (106)), but for small values of ˜m0, the\n˜c0needs to be numerically solved for. In Fig. 1 (left panel) we display the parametric\nplot of ˜c0versus ˜m0, which was generated by varying the IR value ˜R(0) =˜R0and\nshooting towards the AdSboundary.3In Fig. 1 (right panel) we plot the free energy\nas a function of ˜ m0. We find several possible solutions for some given small ˜ m0,\nbut immediately infer that the physical solution is the one with lowest f ree energy.\nThis corresponds to the solution with larger condensate, i.e., corresponding to the\nright arms of (˜ m0,˜c0) curves. We also note, that the large ˜ m0tail corresponds to the\nanalytic behavior (106), whereas zooming in toward origin of the plot would probably\nresult in a self-similar behavior of the equation of state, similarly as wa s analyzed\nin [17].\n4.1.1 Zero bare mass\nAn important result is that the ˜ m0= 0 embedding has a non-zero fermion condensate\n˜c0∝ ∝angbracketleftOq∝angbracketright0. In terms of physical quantities, the relation (47) between ∝angbracketleftOq∝angbracketright0and ˜c0\nhas been written in the appendix (eq. (A.7)). The introduction of th e magnetic field\nBhas therefore induced a spontaneous chiral symmetry breaking.\nWe now focus on the flavor effects. While we find in Fig. 1 that the ˜ c0grows\nwith increasing ˆ ǫ, the condensate ∝angbracketleftOq∝angbracketright0given in (A.7) actually has the opposite\nbehavior with more flavor. In Fig. 2 we plot essentially ∝angbracketleftOq∝angbracketright0against ˆǫand see that\nthe condensate actually decreases monotonously with ˆ ǫfor fixedBphys. In a different\nsystem [11] the tendency for the condensate to decrease with fla vor was also observed.\nAt infinite flavor ˆ ǫ→ ∞,∝angbracketleftOq∝angbracketright0reaches a constant non-zero value.\n3Notice that only ˜ m0≥0 embeddings are physical; otherwise the angle θ>π/2.\n22/Minus2 2 4m/OvΕrTiΛdΕ\n00.51.01.5c/OvΕrTiΛdΕ\n0\n/Minus2 2 4m/OvΕrTiΛdΕ\n00.20.40.60.81.01.21.4F\n/ScriptCapNrB/Minus3/Slash12\nFigure1: Plotofthecondensate ˜ c0(left)andthefreeenergy(right)versustherescaled\nmass ˜m0. The solid blue is ˆ ǫ= 0 and the dashed blue is ˆ ǫ=∞. Both of the curves on\nthe right panel start at the value (at ˜ m0= 0)Γ(1/4)2\n6√πas extracted from (87). Notice\nthat there is no phase transition, since for all ˜ m0≥0 we reside on the same solution.\n4.1.2 Non-zero bare mass\nWhile the magnetic catalysis is the main focus of this paper, it is interes ting to study\nthe case with non-vanishing bare mass. In other words, we wish to s tudy the system\nwhen the chiral symmetry is explicitly broken, rather than spontaneously , and ask\nwhat does the condensate care about the magnetic field and backg round flavor.\nWhen the bare mass is non-zero it is convenient to study the value of the con-\ndensate for a fixed value of µ0\nq. From our dictionary (46) we can relate the mass\nparameter ˜m0toµ0\nqand the physical magnetic field. We get:\n˜m0=/radicalBig\n2√\n2πσ/bracketleftBig(µ0\nq)2\n√\nλBphys/bracketrightBig1\n2. (111)\nThe formula (111) enables us to plot the condensate against the ma gnetic field itself,\nin units of µ0\nqrather than the rescaled mass ˜ m0. Indeed, let us now consider the\nquantity:\n−λ\n(µ0q)2∝angbracketleftOq∝angbracketright0\nN=(3−2b)(2−b)\n4π2√\n2√\nλBphys\n(µ0q)2˜c0. (112)\nThe condensate parameter ˜ c0depends non-trivially on ˜ m0(the precise dependence\nmust be found by numerical calculations), which in turn can be writte n as in (111).\nThus, it follows that the left-hand side of (112) depends on√\nλBphys/(µ0\nq)2.\nIn Fig. 3 we show the condensate against the magnetic field and find t hat it in-\ncreases monotonously. Moreover, forsmall Bphys/(µ0\nq)2thecondensate fortheflavored\ntheory is larger than the unflavored one. Thus, for small Bphysor largeµ0\nq, the flavors\nproduce an enhancement of the chiral symmetry breaking. In Fig. 3 we illustrate this\n230 10 20 30 40 50Ε/Hat0.60.70.80.91.0/LAngleBracket2/ScriptCapOq/RAngleBracket20/LParen1Ε/Hat/RParen1\n/LAngleBracket2/ScriptCapOq/RAngleBracket20/LParen1Ε/Hat/EquaΛ0/RParen1\nFigure 2: Plot of the condensate against the deformation paramet er at zero bare\nmass. At infinite flavor the condensate reaches a constant value f or fixed physical\nmagnetic field Bphys. Notice that we normalized the depicted quantity to unity in the\nquenched limit.\nflavor effect by plotting the difference of condensates as a functio n ofBphys/(µ0\nq)2.\nActually, for small values of Bphys/(µ0\nq)2we can use the approximate expression (106)\nto estimate ˜ c0. Plugging (106) and (111) into (112) we get for small Bphys/(µ0\nq)2\n−λ\n(µ0q)2∝angbracketleftOq∝angbracketright0\nN≈(2−b)b\n8√\n2π2(2π√\n2σ)−1\n2−1\n2b/bracketleftBigg√\nλBphys\n(µ0q)2/bracketrightBigg3\n2+1\n2b\n.(113)\nThus, we get a power law behavior with an exponent which depends on the number\nof flavors and matches the numerical results.\nFor large values of Bphys/(µ0\nq)2the flavors suppress the condensate and we have\na behavior similar to the massless case. Curiously, this change of beh avior occurs\nat values of Bphys/(µ0\nq)2which are almost independent of the number of flavors (see\nFig. 3).\n4.2 Non-zero temperature\nHaving understoodthebasicphysics behindintroducing themagnet ic field, let usnow\nheat up the system and study what happens. In addition to the MN e mbeddings, we\nnow also have the BH embeddings at our disposal. At zero magnetic fie ld,B= 0, we\nrecall that there is going to be a phase transition from the MN phase to the BH phase\n245 10 15 20ΛBphys\n/LParen2Μq0/RParen220.000.020.040.060.080.100.120.14/Minus/LAngleBracket2/ScriptCapOq/RAngleBracket20\nNΛ\n/LParen2Μq0/RParen22\n5 10 15 20ΛBphys\n/LParen2Μq0/RParen22\n/Minus0.010/Minus0.0050.0050.010/Minus/LAngleBracket2/ScriptCapOq/RAngleBracket20/LParen1Ε/Hat/RParen1/Minus/LAngleBracket2/ScriptCapOq/RAngleBracket20/LParen10/RParen1\nNΛ\n/LParen2Μq0/RParen22\nFigure3: Plots of thecondensate versus themagnetic field, when t hebare quark mass\nis non-zero. The solid blue is ˆ ǫ= 0 and the dashed blue is ˆ ǫ= 10. In the right panel\nwe have also included the ˆ ǫ= 0.1 (solid red) and the ˆ ǫ= 1 (dotted black) curves.\nas the temperature is increased [17]. The black hole begins to increas ingly attract the\nprobe D6-brane. Turning on Bhas the opposite effect, in some sense the magnetic\nfield makes the D6-braneto repel. We thus have two competing effec ts in play and we\nneed to explore the four-dimensional phase space ( T,B,µ q,ˆǫ), to find out which phase\nis thermodynamically preferred. Recall that at non-zero tempera ture we can form the\ndimensionless ratio (26), which for the physical magnetic field is ˆBphys≡24/3Bphys\nr2\nh\nand that the bare mass µqwas introduced in [17], see below. This narrows down the\nphase space down to three dimensions ( ˆBphys,µq,ˆǫ). Let us begin our journey in the\nsimpler case with vanishing bare quark mass µq= 0.\n4.2.1 Zero bare mass\nWe start exploring the phase space in the case where we set µq= 0. This slice of\nthe full phase diagram is easily obtained. At any given ˆ ǫwe only have two options,\neither the system is in the chirally symmetric BH phase (small ˆBphys) or the system\nis in the MN phase (large ˆBphys) and the chiral symmetry is broken; see Fig. 4. There\nis a first order phase transition at some critical ˆBphys\ncrit, which depends on ˆ ǫ. Above\nthis critical ˆBphys\ncrit, the BH phase is never reached and thus the chiral symmetry is\nspontaneously broken. The phase diagram (ˆ ǫ,ˆB) is presented in Fig. 5 (left panel).\nThe curve plotted ˆBphys\ncrit=ˆBphys\ncrit(ˆǫ) shows that the critical magnetic field decreases\nwith increasing number of flavors. In other words, at fixed temper ature, the more\nflavor there is the smaller magnetic field is needed to realize magnetic c atalysis. As a\nconsequence the critical condensate will also be smaller with more fla vors, as is visible\nin Fig. 5 (right panel).\nWe finish this subsection by presenting the graph Fig. 6, which repre sents the\ncondensate as a function of the magnetic field at selected flavor de formation param-\n25/Minus1 0 1 2 3m24681012c\n/Minus1.0/Minus0.5 0.0 0.5 1.0m5060708090F\n/ScriptCapN\nFigure 4: Plot of the condensate c(left) and the free energy (right) versus mat ˆǫ= 0.\nThe solid curves are for ˆB= 20 and the dashed curves are for ˆB= 15. The blue\ncolor stands for MN embeddings and black for BH embeddings. The ph ase transition\nform= 0 is between these two cases, around ˆBcrit(ˆǫ= 0)∼17.8, above this critical\nvalue the chiral symmetry is always broken.\neters (ˆǫ= 0 and 10) and zero bare quark mass. The swallow-tail structures of the\nfree energy graphs are indications of the first order phase trans ition, and from Fig. 6\nwe conclude that the condensate acts as an order parameter: at criticalˆBphys\ncritthe\ncondensate jumps to a non-zero value and increases thereafter . From the numerics\nwe also infer, that for large ˆB,\n∝angbracketleftOq∝angbracketright\nN∼T2ˆB(3−b)/2∼B·/parenleftbiggT√\nB/parenrightbiggγm\n,ˆB≫1. (114)\nThis behavior conforms with the T= 0 result (47) at m= 0 (recall the relation (56)).\n4.2.2 Non-zero bare mass\nTo complete the investigation of the phase diagram, let us turn on a n on-zero bare\nmass at non-zero temperature. Recall that the bare mass µqis given by [17]\nµq\nT√\nλ=25/6π\n3m . (115)\nGiven the relation (115), instead of directly fixing the bare mass to s ome value, we\ncan fixm(for any flavor deformation parameter ˆ ǫ). We just need to keep in mind\nthat larger mwill then correspond to smaller temperatures, and vice versa.\nWe anticipate that there are essentially two different cases, depen ding on whether\nmis small or large. In Fig. 7 we depict the condensate as a function of ˆBfor various\nmat ˆǫ= 0; the ˆǫ>0 is qualitatively the same with smaller ˆBcrit’s. We find that for\n260 2 4 6 8 10Ε/Hat020406080/LParen1B/Hat\nphys/RParen1crit\nΛ\n0 2 4 6 8 10Ε/Hat0.20.40.60.81.0/LAngleBracket2/ScriptCapOq/RAngleBracket2/LParen1Ε/Hat/RParen1\n/LAngleBracket2/ScriptCapOq/RAngleBracket2/LParen1Ε/Hat/EquaΛ0/RParen1\nFigure 5: On the left we plot the phase diagram for m= 0 in the (ˆ ǫ,ˆB)-plane.\nAbove the curve, the chiral symmetry is spontaneously broken wh ereas below the\ncurve the condensate is zero. We note that the critical magnetic fi eld needed, at fixed\ntemperature, to break the chiral symmetry decreases with the n umber of flavors,\nleading to the decrease of the condensate (and asymptotically van ishing due to the\nscreening function σ), as depicted on the right.\nanygiven ˆǫthereexists alargeenough msuch that thesystem isalways inthe chirally\nbroken MN phase for any ˆB. For small values of m, there can be a phase transition\nfrom the chirally symmetric BH phase to a broken MN phase at some cr iticalˆBcrit.\n5 Conclusions\nLet us shortly recap the main results of our work. We studied the AB JM Chern-\nSimonsmattermodelwithdynamicalflavors, addedassmearedflav orD6-branes. Our\nblack hole geometry includes the backreaction of dynamical massles s flavors at fully\nnon-linear order in the flavor deformation parameter (7). We inves tigated the effect\nof the inclusion of an external magnetic field on the worldvolume of an (additional)\nprobe D6-brane and focused on the flavor effects from the smear ed D6-branes of the\nbackground. We obtained the different thermodynamic functions f or the probe and\nexplored the corresponding phase diagram. In some corners of th is phase space we\nwere able to obtain analytic results.\nAt zero temperature, forany magnetic field strength, the syste m was always in the\nchirally broken MN phase; a phenomenon called magnetic catalysis. At large (small)\nmagnetic field strength, at non-vanishing bare quark mass, the co ndensate was found\ndecreasing (increasing) with the number of flavors. In other word s, for small masses\nthemagneticcatalysisissuppressed whereasforlargevaluesofth emassitisenhanced\ngiven more flavors in the background. This behavior could morally be t hought of as\n270 20 40 60 80 100B/Hat\nphys\nΛ0246810/Minus/LAngleBracket2/ScriptCapOq/RAngleBracket2/LParen1Ε/Hat/RParen1\nNT2\nFigure 6: The condensate versus the magnetic field ˆBatµq= 0. The solid blue\ncurve is for ˆ ǫ= 0 and the dashed blue curve is for ˆ ǫ= 10. On the left of the curves\nthe condensate is zero. Notice that at critical magnetic field ˆBcrit(ˆǫ) there is a first\norder phase transition (from the chirally symmetric BH phase) wher e the condensate\njumps to a non-zero value, thus acting as an order parameter for the transition (to\nthe chirally broken MN phase).\ninverse magnetic catalysis in the sense of [26,27], although is technic ally different.\nAt non-zero temperature there was a critical magnetic field above which the mag-\nnetic catalysis took place. The condensate acted as an order para meter for the first\norder phase transition between the transition from the chirally sym metric BH phase\nto the broken MN phase. We found that the critical magnetic field wa s smaller for\nmore flavors, which we interpret as an enhancement of the magnet ic catalysis.\nLet us finally discuss some possible extensions of our work. First of a ll, we could\nanalyze the effects of having unquenched massiveflavors. The corresponding back-\ngroundfortheABJMtheoryatzerotemperaturehasbeenrecen tlyconstructedin[18].\nIt would be interesting to explore in this setup how the flavor effects on the conden-\nsate are enhanced or suppressed as the mass of the unquenched quarks is varied,\nand to compare with the results found here in Section 4.1.2. Also, one could try to\ninclude the effect of the magnetic field on the unquenched quarks. F or this purpose\na new non-supersymmetric background must be constructed firs t (see [11,12] for a\nsimilar analysis in the D3-D7 setup). To complete the phase structur e of the model\nwe must explore it at non-zero chemical potential. This would require introducing a\nnon-vanishing charge density by exciting additional components of the worldvolume\ngauge field.\n285 10 15 20B/Hat123456/Minus/LAngleBracket2/ScriptCapOq/RAngleBracket2/LParen1Ε/Hat/EquaΛ0/RParen1\nNT2\nFigure 7: The condensate versus the magnetic field ˆBat various fixed bare masses\nat quenched case ˆ ǫ= 0. The critical magnetic field ˆBcrit, whose values correspond to\nthe dotted vertical line segments, decreases as mincreases, and the respective curves\nreadm= 0.1 (blue),m= 0.5 (red),m= 1 (brown), and m= 5 (continuous black).\nAcknowledgments We thank Yago Bea and Johanna Erdmenger for dis-\ncussions and Javier Mas for collaboration at the initial stages of this work. We are\nspecially grateful to Veselin Filev for his comments and help. N. J. and A. V. R. are\nfunded by the Spanish grant FPA2011-22594, by the Consolider-I ngenio 2010 Pro-\ngramme CPAN (CSD2007-00042), by Xunta de Galicia (Conselleria de E ducaci´ on,\ngrant INCITE09-206-121-PR and grant PGIDIT10PXIB206075P R), and by FEDER.\nN. J. is also supported by the Juan de la Cierva program. D. Z. is fund ed by the FCT\nfellowship SFRH/BPD/62888/2009. Centro de F´ ısica do Porto is par tially funded by\nFCT through the projects CERN/FP/116358/2010 and PTDC/FIS /099293/2008.\n29A Zero temperature dictionary\nThe relation between the mass and the parameter ˜ m0has been worked out in detail\nin Section 2.5. In this Appendix we work out the dictionary for the con densate at\nT= 0, which we will denote by ∝angbracketleftOq∝angbracketright0. A similar analysis at non-zero temperature\nwas presented in appendix D of [17]. For simplicity, in this Appendix we us e units in\nwhichα′= 1.\nLetµ0\nqbe the bare quark mass at zero temperature, whose explicit expre ssion in\nterms of ˜m0andBhas been derived in Section 2.5 (Eq. (46)). The expectation value\n∝angbracketleftOq∝angbracketright0is obtained as the derivative with respect to µ0\nqof the zero temperature free\nenergy:\n∝angbracketleftOq∝angbracketright0=∂F\n∂µ0q. (A.1)\nTo compute the derivative in (A.1) we apply the chain rule:\n∂F\n∂µ0\nq=∂F\n∂˜m0∂˜m0\n∂µ0\nq, (A.2)\nand use [17]:\n∂F\n∂˜m0=−3−2b\nb2B3\n2˜c0Nr, (A.3)\nwhere ˜c0=Bb−3\n2c0. We get:\n∝angbracketleftOq∝angbracketright0=−3−2b\nb2B3\n2˜c0˜m0\nµ0\nqNr. (A.4)\nUsing (46) and the expression of Nr, we find:\n˜m0\nµ0qNr=(2−b)b2σ\n4πNB−1\n2. (A.5)\nTherefore, we have the following relation between ∝angbracketleftOq∝angbracketright0and ˜c0:\n−∝angbracketleftOq∝angbracketright0\nN=(3−2b)(2−b)σ\n4πB˜c0, (A.6)\nwhich, after including the appropriate power of α′, coincides with the expression\nwritten in (47). Let us finally write (A.6) in terms of the physical magn etic field\nBphysgiven by (15). We find\n−∝angbracketleftOq∝angbracketright0\nN=(3−2b)(2−b)\n4π2√\n2Bphys√\nλ˜c0. (A.7)\n30References\n[1] D. Kharzeev, K. Landsteiner, A. Schmitt and H. -U. Yee, “Stro ngly Interacting\nMatter in Magnetic Fields,” Lect. Notes Phys. 871(2013) 1.\n[2] K.G.Klimenko, “Three-dimensional Gross-Neveumodelatnonz erotemperature\nand in an external magnetic field,” Theor. Math. Phys. 90(1992) 1 [Teor. Mat.\nFiz.90(1992) 3]; Z. Phys. C 54(1992) 323; “Three-dimensional Gross-Neveu\nmodel in an external magnetic field,” Theor. Math. Phys. 89(1992) 1161 [Teor.\nMat. Fiz. 89(1991) 211].\n[3] V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, “Catalysis of d ynamical\nflavor symmetry breaking by a magnetic field in (2+1)-dimensions,” Ph ys. Rev.\nLett.73(1994) 3499 [Erratum-ibid. 76(1996) 1005] [hep-ph/9405262].\n[4] V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, “Dimensional r eduction and\ndynamical chiral symmetry breaking by a magnetic field in (3+1)-dime nsions,”\nPhys. Lett. B 349(1995) 477 [hep-ph/9412257].\n[5] V. A. Miransky and I. A. Shovkovy, “Magnetic catalysis and aniso tropic confine-\nment in QCD,” Phys. Rev. D 66(2002) 045006 [hep-ph/0205348].\n[6] J. M. Maldacena, “The Large N limit of superconformal field theor ies and su-\npergravity,” Adv. Theor. Math. Phys. 2(1998) 231 [hep-th/9711200].\n[7] V. G. Filev, C. V. Johnson, R. C. Rashkov and K. S. Viswanathan, “Flavoured\nlarge N gauge theory in an external magnetic field,” JHEP 0710(2007) 019\n[hep-th/0701001].\n[8] V. G. Filev and R. C. Raskov, “Magnetic Catalysis of Chiral Symmet ry Break-\ning. A Holographic Prospective,” Adv. High Energy Phys. 2010(2010) 473206\n[arXiv:1010.0444 [hep-th]].\n[9] O. Bergman, J. Erdmenger and G. Lifschytz, “A Review of Magne tic Phenom-\nena in Probe-Brane Holographic Matter,” Lect. Notes Phys. 871(2013) 591\n[arXiv:1207.5953 [hep-th]].\n[10] C. Nunez, A. Paredes, A. V. Ramallo, “Unquenched flavor in the gauge/gravity\ncorrespondence,” Adv. High Energy Phys. 2010, 196714(2010)[arXiv:1002.1088\n[hep-th]].\n[11] V. G. Filev and D. Zoakos, “Towards Unquenched Holographic Ma gnetic Catal-\nysis,” JHEP 1108(2011) 022 [arXiv:1106.1330 [hep-th]].\n31[12] J. Erdmenger, V. G. Filev and D. Zoakos, “Magnetic Catalysis wit h Massive\nDynamical Flavours,” JHEP 1208(2012) 004 [arXiv:1112.4807 [hep-th]].\n[13] O.Aharony, O.Bergman, D.L. JafferisandJ. M. Maldacena, “N =6superconfor-\nmal Chern-Simons-matter theories, M2-branes and their gravity duals,” JHEP\n0810, 091 (2008) [arXiv:0806.1218 [hep-th]].\n[14] S. Hohenegger, I. Kirsch, “A Note on the holography of Chern -Simons matter\ntheories with flavour,” JHEP 0904, 129 (2009) [arXiv:0903.1730 [hep-th]].\n[15] D. Gaiotto, D. L. Jafferis, “Notes on adding D6 branes wrapping RP**3 in\nAdS(4) x CP**3,” [arXiv:0903.2175 [hep-th]].\n[16] E. Conde and A. V. Ramallo, “On the gravity dual of Chern-Simon s-matter\ntheories with unquenched flavor,” JHEP 1107(2011) 099 [arXiv:1105.6045 [hep-\nth]].\n[17] N. Jokela, J. Mas, A. V. Ramallo and D. Zoakos, “Thermodynamic s of the\nbrane in Chern-Simons matter theories with flavor,” JHEP 1302(2013) 144\n[arXiv:1211.0630 [hep-th]].\n[18] Y. Bea, E. Conde, N. Jokela and A. V. Ramallo, “Unquenched mas sive flavors\nand flows in Chern-Simons matter theories,” arXiv:1309.4453 [hep-th ].\n[19] G. Veneziano, “Some Aspects Of A Unified Approach To Gauge, D ual And\nGribov Theories,” Nucl. Phys. B 117, 519 (1976).\n[20] A. Karch and A. O’Bannon, “Metallic AdS/CFT,” JHEP 0709(2007) 024\n[arXiv:0705.3870 [hep-th]].\n[21] O. Bergman, N. Jokela, G. Lifschytz and M. Lippert, “Quantum Hall Effect in\na Holographic Model,” JHEP 1010(2010) 063 [arXiv:1003.4965 [hep-th]].\n[22] O. Bergman, G. Lifschytz and M. Lippert, “Magnetic propertie s of dense holo-\ngraphic QCD,” Phys. Rev. D 79(2009) 105024 [arXiv:0806.0366 [hep-th]].\n[23] N.Jokela, G.Lifschytz andM. Lippert, “Magneticeffects inaho lographicFermi-\nlike liquid,” JHEP 1205(2012) 105 [arXiv:1204.3914 [hep-th]].\n[24] D. Mateos, R. C. Myers and R. M. Thomson, “Thermodynamics o f the brane,”\nJHEP0705(2007) 067 [hep-th/0701132].\n[25] V. G. Filev, “Hot Defect Superconformal Field Theory in an Exte rnal Magnetic\nField,” JHEP 0911(2009) 123 [arXiv:0910.0554 [hep-th]].\n32[26] G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz and A . Schafer,\n“QCD quark condensate in external magnetic fields,” Phys. Rev. D 86(2012)\n071502 [arXiv:1206.4205 [hep-lat]].\n[27] F. Preis, A. Rebhan and A. Schmitt, “Inverse magnetic catalys is in dense holo-\ngraphic matter,” JHEP 1103(2011) 033 [arXiv:1012.4785 [hep-th]].\n33"    },    {        "title": "1506.07864v1.Dynamical_systems_study_in_single_phase_multiferroic_materials.pdf",        "content": "arXiv:1506.07864v1  [cond-mat.mes-hall]  18 Jun 2015epl draft\nDynamical systems study in single-phase multiferroic mate rials\nKuntal Roy(a)\nSchool of Applied and Engineering Physics, Cornell Univers ity, Ithaca, New York 14853, USA(b)\nPACS75.85.+t – Multiferroics\nPACS75.60.Jk – Magnetization reversal\nPACS75.78.-n – Magnetization dynamics\nPACS84.30.Ng – Magnetization oscillation\nAbstract – Electric field induced magnetization switching in single- phase multiferroic materials is\nintriguing for both fundamental studies and potential tech nological applications. Here we develop\na framework to study the switching dynamics of coupled polar ization and magnetization in such\nmultiferroic materials. With the coupling term between the polarization and magnetization as\nan invariant dictated by the Dzyaloshinsky-Moriya vector, the dynamical systems study reveals\nswitching failures and oscillatory mode of magnetization i f the polarization and magnetization\nrelax slowly during switching.\nIntroduction. – Multiferroics usually represent ma-\nterialsthatarebothferroelectricandferromagnetic[1–12].\nSuch materials in single-phase were usually thought to be\nrare [13], and hence multiferroic composites in 2-phase,\ni.e., a ferroelectric layer strain-coupled to a ferromagnet,\nare usually deemed to be the replacement [5,14–20]. How-\never, there have been recent resurgence of interests [21,22]\nandsomemechanismsofcouplingpolarizationandmagne-\ntizationinsingle-phasematerialsarecomingalong[23–25].\nThis can lead to possible technological applications [26] of\nswitching a bit of information (stored in the magnetiza-\ntion direction) by an electric field [27]. This eliminates\nthe need to switch magnetization by a cumbersome mag-\nneticfieldorspin-polarizedcurrent[14], althoughnewcon-\ncepts are being investigated e.g., utilizing giant spin-Hall\neffect [28]. The electric field switches the polarization and\nthe intrinsic coupling between the polarization and mag-\nnetization switches the magnetization between its 180◦\nsymmetry equivalent states. One way to couple polar-\nization and magnetization that has taken attention is due\nto Dzyaloshinsky-Moriya (DM) interaction [29,30], which\narises due to spin-orbit correction to Anderson’s superex-\nchange [31]. This is called ferroelectrically induced weak\nFerromagnetism (wFM), in which two magnetic sublat-\ntices of an antiferromagnet cant in a way to produce a\nresidual magnetization [23,32–35].\n(a)E-mail:royk@purdue.edu\n(b)Present Address: School of Electrical and Computer Enginee r-\ning, Purdue University, West Lafayette, Indiana 47907, USAWhile first-principles calculationsand experiments have\nbeen underway on the search of strongly-coupled multifer-\nroic magnetoelectric materials possibly working at room-\ntemperature, little have been studied on the dynamical\nnature of switching. The study of switching dynamics of\nmagnetization in multiferroic composites, i.e., a piezoelec-\ntric layer strain-coupled to a magnetostrictive nanomag-\nnet, have been very successful to understand the perfor-\nmance metrics, e.g., switching delay, energy dissipation,\nand switching failures [14,15,17]. Here, the switching dy-\nnamics of polarization is studied by forming a Hamilto-\nnian system with Landau-Ginzburg functional [36], while\nthe magnetization dynamics is studied by the Landau-\nLifshitz-Gilbert (LLG) equation of motion [37,38]. We\nfocus on magnetization switching due to electric field in-\nduced polarization switching, i.e., converse magnetoelec-\ntric (ME) effect for technological applications rather than\nthe switching dynamics due to direct ME effect. We par-\nticularly consider the dynamics in single-domains with an\neyetoachievehigh-densityofdevicesratherthanconsider-\ning domain walls in higher dimensions, for which we need\nto consider the competition between the exchange interac-\ntion and dipole coupling among the spins [39]. Note that\nswitching dynamics in BiFeO 3has been studied using a\nfirst-principles-based effective Hamiltonian within molec-\nulardynamicssimulations[40,41]. Hereweperforma com-\nprehensive analysis in emergingstrongly-coupledmultifer-\nroics (ferroelectrically-inducedwFM by design dictated by\nDM interaction [23]) by varying different parameters in\np-1Kuntal Roy\nFig. 1: (a) Energy of the two spins in a predominantly antifer romagnetic configuration with respect to canting angle θc. The\ncanting happens due to Dzyaloshinsky-Moriya (DM) interact ion. The DM vector is proportional to polarization, i.e., D∝P\nand acts in the z-direction. When θcis positive, the net magnetization Mpoints up (+ y-axis), and if θcis negative, the\nnet magnetization Mpoints down ( −y-axis). Without any canting, the net magnetization is zero a s in the case of a perfect\nantiferromagnet. (b) Magnetization initially points alon g the−y-xis. To respect the invariant dictated by the DM interactio n\n[∝P·(L×M)], with the reversal of polarization P, either the magnetization M(case 1) or the AFM vector L(case 2) may\nflip.\nthe LLG dynamics. The analysis of switching dynamics\nreveals very significant motion of magnetization when po-\nlarization is switched by an electric field. It is shown that\nmagnetization may fail to switch or even can go into an\noscillatory state of motion. The phenomenological damp-\ning parameter for both polarization and magnetization\nplays a crucial role in shaping the dynamics of the coupled\npolarization-magnetizationin these multiferroicmaterials.\nModel. – We consider two spins, one representative\ntoeachmagneticsublatticeofanantiferromagnet,tobuild\nup the present model. The dynamics of the two spins S1\nandS2can be described by the Landau-Lifshitz-Gilbert\n(LLG) equation [37,38] as follows:\ndS1\ndt=−|γ′|S1×HS1−α|γ′|\nSS1×(S1×HS1) (1)\ndS2\ndt=−|γ′|S2×HS2−α|γ′|\nSS2×(S2×HS2),(2)\nwhereHS1andHS2are the effective fields on the spins S1\nandS2, respectively, defined as HS1=−(∂H/∂S1) and\nHS2=−(∂H/∂S2),His the potential energy of the two\nspin system, expressed as\nH=−JS1·S2−D·(S1×S2)−KS2\n1,z−KS2\n2,z,(3)\nJdenotes the exchange coupling between the spins, Dis\nthe Dzyaloshinsky-Moriya (DM) vector [29,30] (here, we\nwill consider the case when the vector Dpoints along per-\npendicular to the plane ( x-yplane) on which the spins\nreside, i.e., along the z-direction and proportional to po-\nlarization P, whichisalsointhe z-direction[23])expressed\nasD=D(t)ˆ ezmaking\nD(t)∝P(t), (4)Kis the single-ion anisotropy constant, γ′=γ/(1+α2),γ\nisthegyromagneticratioofelectrons, αisthephenomeno-\nlogical Gilbert damping constant [38], and S=|S1|=\n|S2|. As required, it is possible to include the long-range\ninteraction too in the energy term [42]. The net magneti-\nzationMand the antiferromagnetic (AFM) vector Lfor\nthe two spin system are M=S1+S2andL=S1−S2,\nrespectively. Note that the following two identities hold:\nM·L= 0 and M2+L2= 4S2.\nThe polarization dynamics is based on the Landau-\nGinzburg functional [36]\nG=/bracketleftBig\n−a1\n2P2+a2\n4P4/bracketrightBig\n−E.P, (5)\nwherea1anda2are the ferroelectric coefficients (both are\ngreaterthanzero)and E=Eˆ ezistheappliedelectricfield\nthat switches the polarization Pin thez-direction. We as-\nsume single-domain case [43] and follow the prescription\nin Ref. [36] to trace the trajectory of polarization. Note\nthat polarization is switched by moving ions, which cou-\nples to the magnetization dynamics via the DM term D\n[see Eq. (3)]. On the other hand, rotation of spins does\nnot quite move the heavy ions affecting the polarization.\nFigure 1a depicts how the asymmetric Dzyaloshinsky-\nMoriya (DM) interaction can lead to two anti-parallel\nmagnetization directions, i.e., 180◦symmetry equivalent\nstates. Depending on the sign of the canting angle θcof\nthe spins, the direction of the DM vector changes and\nthe energy expression as in the Equation (3) gives rise to\ntwo magnetization states in opposite directions. The DM\nvectorDis proportional to polarization Pand hence, if\nwe switch the polarization, two cases can happen to re-\nspect the invariant due to DM interaction P·(L×M)\np-2Dynamical systems study in single-phase multiferroic materials\nFig. 2: (a) The potential landscape ofpolarization with ele ctric fieldas aparameter [Equation(5)]. Notethat itrequir es acritical\nelectric field to topple the barrier between polarization’s two 180◦symmetry equivalent states. (b) Switching of polarization\nwith the application of an electric field. Initially, the pol arization was pointing towards −z-axis. With the application of electric\nfield, the polarization does not reach instantly towards the +z-axis, how fast the polarization relaxes to the minimum ener gy\nposition depends on the polarization damping. Note that aft er the withdrawal of electric field, the polarization direct ion is\nmaintained, i.e., the switching is non-volatile. The sligh t increase of polarization over Psis due to the application of electric\nfield [see the potential landscapes in part (a)], which can be followed from the Equation (5) too.\n[∝D·(S1×S2) term in Equation (3)]: (1) The magneti-\nzationMcan change the direction (i.e., switches success-\nfully), and (2) The AFM vector Lmay change the direc-\ntion (i.e., Mfails to switch). The two cases are depicted\nin the Fig. 1b.\nResults and Discussions. – We consider a per-\novskite system NiTiO 3[23,44] in R3c space group [45] as\na prototype to analyzethe switching dynamics. Although,\nNiTiO 3in R3c space group is not yet experimentally re-\nalized, the concept of polarization-magnetization coupling\npredicted from group theory is promising. The parame-\nters are chosen as follows: saturation polarization Ps=\n110µC/cm2, ferroelectric coefficients a1= 1.568×1010\nVm/C, a2= 1.296×1010Vm/C, polarization damp-\ningβ= 0.286 VmSec/C (that switches the polarization\nin realistic time 100 ps [46], see Fig. 2), S= 1.6µB,\nM= 0.25µB,J=−2.2 meV,Ds= 0.35 meV [corre-\nsponding to Ps, i.e.,D(t) = (Ds/Ps)P(t)],K=−0.03\nmeV [23]. We will consider that the electric field switches\nthe polarization from −Psto +Psin thez-direction.\nThe magnetization damping, through which magne-\ntization relaxes to the minimum energy position, can\nhave a wide range of values (10−4– 0.8) by modifying\nthe spin-orbit strength, doping etc. and it can be de-\ntermined by ferromagnetic resonance (FMR), magneto-\noptical Kerr effect, x-ray absorption spectroscopy, and\nspin-current driven rotation with the addition of a spin-\ntorque term [47–49]. Hence, we focus on investigating the\nmagnetization dynamics for a wide range of phenomeno-\nlogical damping parameter.\nWe will initially assume the single-ion anisotropy K=\n0 and we will see later the consequence of considering\nit. Figure 3 shows the dynamics of magnetization when\ndamping parameter is on the higher side, e.g., 0.1. Wesee that magnetization has switched successfully in the\nend (see Fig. 3d), while the AFM vector did not switch\n(see Fig. 3c). Note that the spins S1andS2are deflected\nfrom the x-yplane, in the z-direction due to rotational\nmotion of magnetization. Also, note that magnetization’s\nx- andz-component and AFM vector’s y-component have\nnot changed at all due to the complimentary dynamics of\nthe spins S1andS2. This corresponds to the case (1) in\nFig. 1b.\nFigure 4 plots the dynamics when magnetization damp-\ningα= 0.01. We see that magnetization has failed to\nswitch(seeFig.4d), while theAFMvectorisswitchedsuc-\ncessfully (see Fig. 4c). Magnetization was on the way to\nchange its direction, but eventually, magnetization came\nback to its initial state. This corresponds to the case (2)\nin Fig. 1b. Due to low damping, we notice ringing in all\nthe plots in the Fig. 4.\nFor the lower damping of α= 0.01, from the simulation\nresults as shown in the Fig. 4, the positions of the two\nspins have just got interchanged, which is depicted as the\ncase (2) in Fig. 1b. Since the canting angle ofthe spins are\nsmall (θc≃5◦), one can say that the spins have rotated\nmuch more than that of the case for the higher damping\nofα= 0.1 [case (1) in Fig. 1b and the simulation results\nas shown in the Fig. 3]. While both the cases as shown in\nthe Fig. 1b respect the DM invariant at steady-state , the\ndynamics of magnetization dictates the final state that is\nreached. With a lower damping, the spins get deflected\nout-of-plane more (see the z-components of the spins S1\nandS2intheFigs.3and4)andthisout-of-planeexcursion\neventuallycanleadthe spins tointerchangetheir positions\nas can be noticed from the Fig. 4. The interchange of the\nspinsS1andS2indeed respects the DM invariant [case\n(2) of Fig. 1b], but the magnetization Mfailsto switch in\nthis case, while the AFM vector Lgets switched.\np-3Kuntal Roy\nFig. 3: Dynamics of magnetization for damping parameter α=\n0.1. Magnetization does switch successfully. (a) Dynamics of\nS1, (b) Dynamics of S2, (c) Dynamics of AFM vector L, and\n(d) Dynamics of magnetization M.\nTo understand the magnetizationdynamics further that\nhow switching may be successful even at magnetization\ndamping α= 0.01, we first investigate its dependence on\nthe polarization dynamics. Simulation results show that\nmagnetization switches successfully if we make the polar-\nization damping 200 times faster [see Fig. 5a]. Basically,\ndue to the coupling between the polarization and mag-\nnetization, if polarization is switched faster, magnetiza-\ntion is also switched faster, which makes the switching\nsuccessful. We further investigate the effect of single-ion\nanisotropy parameter Kon magnetization dynamics. The\nion-anisotropy basically adds an extra field that tries to\nkeep the magnetization in-plane (i.e., x-yplane) and the\nsimulation results show that magnetization switches suc-\ncessfully if we take the single-ion-anisotropy into account\nand the ringing in the magnetization dynamics does not\nFig. 4: Dynamics of magnetization for damping parameter α=\n0.01. Magnetization fails to switch successfully. (a) Dynami cs\nofS1, (b) Dynamics of S2, (c) Dynamics of AFM vector L,\nand (d) Dynamics of magnetization M.\nshow up in this case [see Fig. 5b]. However, if the single-\nion-anisotropy is reduced to a value of K=−0.003µeV,\nit is noticed that the magnetization fails to switch success-\nfully.\nWe further study the effect of varying the DM interac-\ntion strength Dson the switching dynamics. It is found\nquiteobviouslythat as Dsdecreasesforafixedcantingan-\ngleθc, i.e., polarization-magnetization coupling weakens,\nthe AFM vector Ldeflects more, which can be interpreted\nas that the magnetization Mis more prone to switching\nfailures. For α= 0.1, ifDsis reduced 10 times, the mag-\nnetization Mstill switches successfully.\nAn interesting investigation would be to see whether\nmagnetization, being a rotational body, oscillates for a\ncertain range of damping parameter. For example, a spin-\npolarized current can spawn oscillatory states in a nano-\np-4Dynamical systems study in single-phase multiferroic materials\nFig. 5: Magnetization switches successfully at damping par ameterα= 0.01. (a) Polarization is switched 200 times faster by\nchanging the polarization damping. The magnetization swit ches successfully. (b) The single-ion anisotropy constant Kis taken\ninto account. In this case, magnetization switches success fully too.\nmagnet [50]. Figure 6a shows the magnetization dynam-\nics when damping parameter α= 0.001. The single-ion-\nanisotropy is included here and magnetization was able\nto get past towards the + y-direction, however, could not\nsettle there and oscillates with a time period of 3.3 ps.\nWith further lowering of the damping parameter, magne-\ntizationstilloscillates, howeverwith alowerfrequency[see\nFig. 6b].\nThe oscillatory mode of magnetization too can be ex-\nplained from the out-of-plane excursion ofthe spins due to\nlow damping. The spins get deflected out-of-plane (i.e., z-\ndirection) and when they go completely out-of-plane, they\ncontinue rotating and reach the out-of-plane in the oppo-\nsite directions than the previous ones. Therefore the spins\nsustaina self-oscillation . The DM interactionensuresthat\nthex- andz-component of the spins are canceled out and\nthey-components are added due to symmetry, as can be\nnoticed in the Fig. 6. Note that such self-oscillation oc-\ncursandsustainsevenintheabsenceofanexternalelectric\nfield, making the system unstable. Such spontaneous self-\noscillation is not uncommon in electronic systems having\nnegative damping due to positive feedback leading to in-\nstabilities. The oscillationtime-period increasesat a lower\ndamping (see Fig. 6b) since it takes more time for magne-\ntization to traverse for a lower damping parameter.\nThe research on single-phase multiferroic materials is\nstill emerging, and the search for a room-temperature sys-\ntem that requires a low enough electric field for switch-\ning the polarization is still underway. It will be interest-\ning to incorporate the thermal fluctuations in the model\nto understand the consequence on magnetization dynam-\nics [14,51]. Also for a shape-anisotropic single-domain\nnanomagnet, the corresponding anisotropy needs to be in-\ncluded for detailed simulation [39].\nConclusions. – We have investigated the electric\nfield induced magnetization switching dynamics in single-\nphase multiferroic materials. The dynamical system anal-\nysis, contrary to steady-state analysis, revealed important\nintriguing phenomena of switching failures and oscillatorymode of magnetization. The key parameters that can\nshape the dynamics of magnetization are identified. The\nphenomenological magnetization damping turns out to be\na key parameter that can prevent successful switching of\nmagnetization. Hence, the present analysis puts forward\nanimportantstep towardanalyzingmagnetizationswitch-\ning dynamics between 180◦symmetry equivalent states in\nthe emerging multiferroic materials. Moreover, the anal-\nysis identifies the oscillatory mode of magnetization that\ncan act as a source of microwave signals. The present\nanalysis can facilitate designing multiferroic materials for\nrelevant technological purposes.\nREFERENCES\n[1]Schmid H. ,Ferroelectrics ,162(1994) 317.\n[2]Eerenstein W., Mathur N. D. andScott J. F. ,Na-\nture,442(2006) 759.\n[3]Scott J. F. ,Nature Mater. ,6(2007) 256.\n[4]Daraktchiev M., Catalan G. andScott J. F. ,Phys.\nRev. B,81(2010) 224118.\n[5]Spaldin N. A. andFiebig M. ,Science,309(2005) 391.\n[6]Fiebig M. ,J. Phys. D: Appl. Phys. ,38(2005) R123.\n[7]Ramesh R. andSpaldin N. A. ,Nature Mater. ,6(2007)\n21.\n[8]Fiebig M., Lottermoser T., Fr ¨ohlich D., Goltsev\nA. V.andPisarev R. V. ,Nature,419(2002) 818.\n[9]Pyatakov A. P. andZvezdin A. K. ,Eur. Phys. J. B ,\n71(2009) 419.\n[10]Pyatakov A. P. andZvezdin A. K. ,Physics-Uspekhi ,\n55(2012) 557.\n[11]Glinchuk M. D., Eliseev E. A., Morozovska A. N.\nandBlinc R. ,Phys. Rev. B ,77(2008) 024106.\n[12]Glinchuk M. D., Morozovska A. N., Eliseev E. A.\nandBlinc R. ,J. Appl. Phys. ,105(2009) 084108.\n[13]Hill N. A. ,Annu. Rev. Mater. Res. ,32(2002) 1.\n[14]Roy K.,SPIN,3(2013) 1330003.\n[15]Roy K.,Appl. Phys. Lett. ,103(2013) 173110.\n[16]Roy K.,Appl. Phys. Lett. ,104(2014) 013103.\n[17]Roy K.,J. Phys. D: Appl. Phys. ,47(2014) 252002.\n[18]Roy K.,J. Phys.: Condens. Matter ,26(2014) 492203.\np-5Kuntal Roy\nFig. 6: Magnetization oscillation for low damping paramete rs. The electric field is withdrawn at 0.5 ns. (a) Damping para meter\nα= 0.001. Magnetization has got past from −y-axis to + y-axis but could not settle there rather oscillates around th at position\nwith a time period of 3.3 ps. (b) Damping parameter α= 10−4. Magnetization got past from −y-direction to + y-direction but\ncould not settle there rather oscillates around that positi on with a time period of 5.3 ps. With the removal of electric fie ld, the\nmagnetization traverses toward −y-direction and oscillates around there. The three-dimensi onal dynamics of the spins S1and\nS2are shown at the lower part of the figure for both the cases (a) a nd (b).\n[19]Roy K., Bandyopadhyay S. andAtulasimha J. ,Appl.\nPhys. Lett. ,99(2011) 063108.\n[20]Roy K., Bandyopadhyay S. andAtulasimha J. ,Sci.\nRep.,3(2013) 3038.\n[21]Khomskii D. I. ,J. Magn. Magn. Mater. ,306(2006) 1.\n[22]Tokura Y. ,J. Magn. Magn. Mater. ,310(2007) 1145.\n[23]Fennie C. J. ,Phys. Rev. Lett. ,100(2008) 167203.\n[24]Birol T., Benedek N. A., Das H., Wysocki A. L.,\nMulder A. T., Abbett B. M., Smith E. H., Ghosh S.\nandFennie C. J. ,Curr. Opin. Sol. St. Mater. Sci. ,165\n(2012) 227.\n[25]Rondinelli J. M., May S. J. andFreeland J. W. ,\nMRS Bulletin ,37(2012) 261.\n[26]Chappert C., Fert A. andDau F. N. V. ,Nature\nMater.,6(2007) 813.\n[27]Lottermoser T., Lonkai T., Amann U., Hohlwein\nD., Ihringer J. andFiebig M. ,Nature,430(2004) 541.\n[28]Roy K.,J. Phys. D: Appl. Phys. ,47(2014) 422001.\n[29]Dzyaloshinsky I. ,J. Phys. Chem. Sol. ,4(1958) 241.\n[30]Moriya T. ,Phys. Rev. ,120(1960) 91.\n[31]Anderson P. W. ,Phys. Rev. ,115(1959) 2.\n[32]Fox D. L. andScott J. F. ,J. Phys. C: Solid St. Phys. ,\n10(1977) L329.\n[33]Ederer C. andSpaldin N. A. ,Phys. Rev. B ,74(2006)\n020401(R).\n[34]Ederer C. andSpaldin N. A. ,Phys. Rev. B ,71(2005)\n060401(R).\n[35]Ederer C. andFennieC. J. ,J. Phys.: Condens. Matter ,\n20(2008) 434219.\n[36]Bandyopadhyay A. K., Ray P. C. andGopalan V. ,\nJ. Appl. Phys. ,100(2006) 114106.\n[37]Landau L. andLifshitz E. ,Phys. Z. Sowjet. ,8(1935)\n153.[38]Gilbert T. L. ,IEEE Trans. Magn. ,40(2004) 3443.\n[39]Chikazumi S. ,Physics of Ferromagnetism (Oxford Uni-\nversity Press, USA) 1997.\n[40]Bhattacharjee S., Rahmedov D., Wang D., niguez\nJ. I.andBellaiche L. ,Phys. Rev. Lett. ,112(2014)\n147601.\n[41]Rahmedov D., Wang D., niguez J. I. andBellaiche\nL.,Phys. Rev. Lett. ,109(2012) 037207.\n[42]Mochizuki M. andNagaosa N. ,Phys. Rev. Lett. ,105\n(2010) 147202.\n[43]Kalinin S. V., Rodriguez B. J., Jesse S., Chu Y. H.,\nZhao T., Ramesh R., Choudhury S., Chen L. Q.,\nEliseev E. A. andMorozovska A. N. ,Proc. Natl.\nAcad. Sci. U.S.A. ,104(2007) 20204.\n[44]Varga T., Droubay T. C., Bowden M. E.,\nNachimuthu P., Shutthanandan V., Bolin T. B.,\nShelton W. A. andChambers S. A. ,Thin Sol. Films ,\n520(2012) 5534.\n[45]Stokes H. T., Kisi E. H., Hatch D. M. andHoward\nC. J.,Acta Crystallogr. Sect. B ,58(2002) 934.\n[46]Li J., Nagaraj B., Liang H., Cao W., Lee C. and\nRamesh R. ,Appl. Phys. Lett. ,84(2004) 1174.\n[47]Gilmore K., Idzerda Y. U. andStiles M. D. ,Phys.\nRev. Lett. ,99(2007) 027204.\n[48]Bailey W., Kabos P., Mancoff F. andRussek S. ,\nIEEE Trans. Magn. ,37(2001) 1749.\n[49]Kajiwara Y., Harii K., Takahashi S., Ohe J.,\nUchida K., Mizuguchi M., Umezawa H., Kawai H.,\nAndo K. andTakanashi K. ,Nature,464(2010) 262.\n[50]Zeng Z., Finocchio G. andJiang H. ,Nanoscale ,5\n(2013) 2219.\n[51]Brown W. F. ,Phys. Rev. ,130(1963) 1677.\np-6"    },    {        "title": "0905.4544v2.Hydrodynamic_theory_of_coupled_current_and_magnetization_dynamics_in_spin_textured_ferromagnets.pdf",        "content": "Hydrodynamic theory of coupled current and magnetization dynamics in\nspin-textured ferromagnets\nClement H. Wong and Yaroslav Tserkovnyak\nDepartment of Physics and Astronomy, University of California, Los Angeles, California 90095, USA\nWe develop the hydrodynamic theory of collinear spin currents coupled to magnetization dynamics\nin metallic ferromagnets. The collective spin density couples to the spin current through a U(1)\nBerry-phase gauge \feld determined by the local texture and dynamics of the magnetization. We\ndetermine phenomenologically the dissipative corrections to the equation of motion for the electronic\ncurrent, which consist of a dissipative spin-motive force generated by magnetization dynamics and\na magnetic texture-dependent resistivity tensor. The reciprocal dissipative, adiabatic spin torque\non the magnetic texture follows from the Onsager principle. We investigate the e\u000bects of thermal\n\ructuations and \fnd that electronic dynamics contribute to a nonlocal Gilbert damping tensor in\nthe Landau-Lifshitz-Gilbert equation for the magnetization. Several simple examples, including\nmagnetic vortices, helices, and spirals, are analyzed in detail to demonstrate general principles.\nPACS numbers: 72.15.Gd,72.25.-b,75.75.+a\nI. INTRODUCTION\nThe interaction of electrical currents with magnetic\nspin texture in conducting ferromagnets is presently a\nsubject of active research. Topics of interest include\ncurrent-driven magnetic dynamics of solitons such as do-\nmain walls and magnetic vortices,1,2,3,4as well as the\nreciprocal process of voltage generation by magnetic\ndynamics.5,6,7,8,9,10,11,12This line of research has been\nfueled in part by its potential for practical applications\nto magnetic memory and data storage devices.13Funda-\nmental theoretical interest in the subject dates back at\nleast two decades.5,6,14It was recognized early on6that\nin the adiabatic limit for spin dynamics, the conduction\nelectrons interact with the magnetic spin texture via an\ne\u000bective spin-dependent U(1) gauge \feld that is a local\nfunction of the magnetic con\fguration. This gauge \feld,\non the one hand, gives rise to a Lorentz force due to\n\\\fctitious\" electric and magnetic \felds and, on the other\nhand, mediates the so-called spin-transfer torque exerted\nby the conduction electrons on the collective magnetiza-\ntion. An alternative and equivalent view is to consider\nthis force as the result of the Berry phase15accumulated\nby an electron as it propagates through the ferromagnet\nwith its spin aligned with the ferromagnetic exchange\n\feld.8,10,16In the standard phenomenological formalism\nbased on the Landau-Lifshitz-Gilbert (LLG) equation,\nthe low-energy, long-wavelength magnetization dynamics\nare described by collective spin precession in the e\u000bective\nmagnetic \feld, which is coupled to electrical currents via\nthe spin-transfer torques. In the following, we develop\na closed set of nonlinear classical equations governing\ncurrent-magnetization dynamics, much like classical elec-\ntrodynamics, with the LLG equation for the spin-texture\n\\\feld\" in lieu of the Maxwell equations for the electro-\nmagnetic \feld.\nThis electrodynamic analogy readily explains various\ninteresting magnetoelectric phenomena observed recently\nin ferromagnetic metals. Adiabatic charge pumping bymagnetic dynamics17can be understood as the gener-\nation of electrical currents due to the \fctitious electric\n\feld.18In addition, magnetic textures with nontrivial\ntopology exhibit the so-called topological Hall e\u000bect,19,20\nin which the \fctitious magnetic \feld causes a classical\nHall e\u000bect. In contrast to the classical magnetoresis-\ntance, the \rux of the \fctitious magnetic \feld is a topo-\nlogical invariant of the magnetic texture.6\nDissipative processes in current-magnetization dynam-\nics are relatively poorly understood and are of central\ninterest in our theory. Electrical resistivity due to quasi-\none-dimensional (1D) domain walls and spin spirals have\nbeen calculated microscopically.21,22,23More recently, a\nviscous coupling between current and magnetic dynam-\nics which determines the strength of a dissipative spin\ntorque in the LLG equation as well the reciprocal dis-\nsipative spin electromotive force generated by magnetic\ndynamics, called the \\ \fcoe\u000ecient,\"2was also calcu-\nlated in microscopic approaches.3,24,25Generally, such\n\frst-principles calculations are technically di\u000ecult and\nrestricted to simple models. On the other hand, the num-\nber of di\u000berent forms of the dissipative interactions in the\nhydrodynamic limit are in general constrained by sym-\nmetries and the fundamental principles of thermodynam-\nics, and may readily be determined phenomenologically\nin a gradient expansion. Furthermore, classical thermal\n\ructuations may be easily incorporated in the theoretical\nframework of quasistationary nonequilibrium thermody-\nnamics.\nThe principal goal of this paper is to develop a (semi-\nphenomenological) hydrodynamic description of the dis-\nsipative processes in electric \rows coupled to magnetic\nspin texture and dynamics. In Ref. 11, we drew the anal-\nogy between the interaction of electric \rows with quasis-\ntationary magnetization dynamics with the classical the-\nory of magnetohydrodynamics. In our \\spin magnetohy-\ndrodynamics,\" the spin of the itinerant electrons, whose\n\rows are described hydrodynamically, couples to the lo-\ncal magnetization direction, which constitutes the col-\nlective spin-coherent degree of freedom of the electronicarXiv:0905.4544v2  [cond-mat.mes-hall]  16 Nov 20092\n\ruid. In particular, the dissipative \fcoupling between\nthe collective spin dynamics and the itinerant electrons\nis loosely akin to the Landau damping, capturing cer-\ntain kinematic equilibration of the relative motion be-\ntween spin-texture dynamics and electronic \rows. In our\nprevious paper,11we considered a special case of incom-\npressible \rows in a 1D ring to demonstrate the essential\nphysics. In this paper, we establish a general coarse-\ngrained hydrodynamic description of the interaction be-\ntween the electric \rows and textured magnetization in\nthree dimensions, treating the itinerant electron's degrees\nof freedom in a two-component \ruid model (correspond-\ning to the two spin projections of spin-1 =2 electrons along\nthe local collective magnetic order). Our phenomenology\nencompasses all the aforementioned magnetoelectric phe-\nnomena.\nThe paper is organized as follows. In Sec. II, we use a\nLagrangian approach to derive the semiclassical equation\nof motion for itinerant electrons in the adiabatic approx-\nimation for spin dynamics. In Sec. III, we derive the\nbasic conservation laws, including the Landau-Lifshitz\nequation for the magnetization, by coarse-graining the\nsingle-particle equation of motion and the Hamiltonian.\nIn Sec. IV, we phenomenologically construct dissipative\ncouplings, making use of the Onsager reciprocity princi-\nple, and calculate the net dissipation power. In particu-\nlar, we develop an analog of the Navier-Stokes equation\nfor the electronic \ruid, focusing on texture-dependent\ne\u000bects, by making a systematic expansion in nonequi-\nlibrium current and magnetization consistent with sym-\nmetry requirements. In Sec. V, we include the e\u000bects of\nclassical thermal \ructuations by adding Langevin sources\nto the hydrodynamic equations, and arrive at the central\nresult of this paper: A set of coupled stochastic di\u000ber-\nential equations for the electronic density, current, and\nmagnetization, and the associated white-noise correlators\nof thermal noise. In Sec. VI, we apply our results to\nspecial examples of rotating and spinning magnetic tex-\ntures, calculating magnetic texture resistivity and mag-\nnetic dynamics-generated currents for a magnetic spiral\nand a vortex. The paper is summarized in Sec. VII and\nsome additional technical details, including a microscopic\nfoundation for our semiclassical theory, are presented in\nthe appendices.\nII. QUASIPARTICLE ACTION\nIn a ferromagnet, the magnetization is a symmetry-\nbreaking collective dynamical variable that couples to the\nitinerant electrons through the exchange interaction. Be-\nfore developing a general phenomenological framework,\nwe start with a simple microscopic model with Stoner in-\nstability, which will guide us to explicitly construct some\nof the key magnetohydrodynamic ingredients. Within a\nlow-temperature mean-\feld description of short-ranged\nelectron-electron interactions, the electronic action isgiven by (see appendix A for details):\nS=Z\ndtd3r^ y\u0014\ni~@t+~2\n2mer2\u0000\u001e\n2+\u0001\n2m\u0001^\u001b\u0015\n^ :(1)\nHere, \u0001( r;t) is the ferromagnetic exchange splitting,\nm(r;t) is the direction of the dynamical order param-\neter de\fned by ~h^ y^\u001b^ i=2 =\u001asm,\u001asis the local spin\ndensity, and ^ (r;t) is the spinor electron \feld operator.\nFor the short-range repulsion U > 0 discussed in ap-\npendix A, \u0001( r;t) = 2U\u001as(r;t)=~and\u001e(r;t) =U\u001a(r;t),\nwhere\u001a=h^ y^ iis the local particle number density.\nFor electrons, the magnetization Mis in the opposite di-\nrection of the spin density: M=\r\u001asm, where\r <0 is\nthe gyromagnetic ratio. Close to a local equilibrium, the\nmagnetic order parameter describes a ground state con-\nsisting of two spin bands \flled up to the spin-dependent\nFermi surfaces, with the spin orientation de\fned by m.\nWe will focus on soft magnetic modes well below the\nCurie temperature, where only the direction of the mag-\nnetization and spin density are varied, while the \ructu-\nations of the magnitudes are not signi\fcant. The spin\ndensity is given by \u001as=~(\u001a+\u0000\u001a\u0000)=2 and particle den-\nsity by\u001a=\u001a++\u001a\u0000, where\u001a\u0006are the local spin-up/down\nparticle densities along m.\u001ascan be essentially constant\nin the limit of low spin susceptibility.\nStarting with a nonrelativistic many-body Hamilto-\nnian, the action (1) is obtained in a spin-rotationally\ninvariant form. However, this symmetry is broken by\nspin-orbit interactions, whose role we will take into ac-\ncount phenomenologically in the following. When the\nlength scale on which m(r;t) varies is much greater than\nthe ferromagnetic coherence length lc\u0018~vF=\u0001, where\nvFis the Fermi velocity, the relevant physics is captured\nby the adiabatic approximation. In this limit, we start\nby neglecting transitions between the spin bands, treat-\ning the electron's spin projection on the magnetization\nas a good quantum number. (This approximation will\nbe relaxed later, in the presence of microscopic spin-\norbit or magnetic disorder.) We then have two e\u000bec-\ntively distinct species of particles described by a spinor\nwave function ^ 0, which is de\fned by ^ =^U(R)^ 0. Here,\n^U(R) is an SU(2) matrix corresponding to the local spa-\ntial rotationR(r;t) that brings the z-axis to point along\nthe magnetization direction: R(r;t)z=m(r;t), so that\n^Uy(^\u001b\u0001m)^U= ^\u001bz. The projected action then becomes:\nS=Z\ndtZ\nd3r^ 0y\"\n(i~@t+ ^a)\u0000(\u0000i~r\u0000^a)2\n2me\n\u0000\u001e\n2+\u0001\n2^\u001bz\u0015\n^ 0\u0000Z\ndtF[m];(2)\nwhere\nF[m] =A\n2Z\nd3r(@im)2(3)\nis the spin-texture exchange energy (implicitly summing\nover the repeated spatial index i), which comes from the3\nterms quadratic in the gauge \felds that survive the pro-\njection. In the mean-\feld Stoner model, the ferromag-\nnetic exchange sti\u000bness is A=~2\u001a=4me. To broaden our\nscope, we will treat it as a phenomenological constant,\nwhich, for simplicity, is determined by the mean electron\ndensity.26The spin-projected \\\fctitious\" gauge \felds are\ngiven by\na\u001b(r;t) =i~h\u001bj^Uy@t^Uj\u001bi;\na\u001b(r;t) =i~h\u001bj^Uyr^Uj\u001bi: (4)\nChoosing the rotation matrices ^U(m) to depend only on\nthe local magnetic con\fguration, it follows from their\nde\fnition that spin- \u001bgauge potentials have the form:\na\u001b=\u0000@tm\u0001amon\n\u001b(m); a\u001bi=\u0000@im\u0001amon\n\u001b(m);(5)\nwhere amon\n\u001b(m)\u0011 \u0000i~h\u001bj^Uy@m^Uj\u001bi. We show in Ap-\npendix B the well known result (see, e.g., Ref. 27) that\namon\n\u001bis the vector potential (in an arbitrary gauge) of\na magnetic monopole in the parameter space de\fned by\nm:\n@m\u0002amon\n\u001b(m) =q\u001bm; (6)\nwhereq\u001b=\u001b~=2 is the monopole charge (which is ap-\npropriately quantized).\nBy noting that the action (2) is formally identical to\ncharged particles in electromagnetic \feld, we can imme-\ndiately write down the following classical single-particle\nLagrangian for the interaction between the spin- \u001belec-\ntrons and the collective spin texture:\nL\u001b(r;_r;t) =me_r2\n2+_r\u0001a\u001b(r;t) +a\u001b(r;t); (7)\nwhere _ris the spin-\u001belectron (wave-packet) velocity. To\nsimplify our discussion, we are omitting here the spin-\ndependent forces due to the self-consistent \felds \u001e(r;t)\nand \u0001( r;t), which will be easily reinserted at a later\nstage. See Eq. (29).\nThe Euler-Lagrange equation of motion for v=\n_rderived from the single-particle Lagrangian (7),\n(d=dt)(@L\u001b=@_r) =@L\u001b=@r, gives\nme_v=q\u001b(e+v\u0002b): (8)\nThe \fctitious electromagnetic \felds that determine the\nLorentz force are\nq\u001bei=@ia\u001b\u0000@ta\u001bi=q\u001bm\u0001(@tm\u0002@im);\nq\u001bbi=\u000fijk@ja\u001bk=q\u001b\u000fijk\n2m\u0001(@km\u0002@jm):(9)\nThey are conveniently expressed in terms of the tensor\n\feld strength\nq\u001bf\u0016\u0017\u0011@\u0016a\u001b\u0017\u0000@\u0017a\u001b\u0016=q\u001bm\u0001(@\u0017m\u0002@\u0016m) (10)\nbyei=fi0andbi=\u000fijkfjk=2.\u000fijkis the antisymmet-\nric Levi-Civita tensor and we used four-vector notation,de\fning@\u0016= (@t;r) anda\u001b\u0016= (a\u001b;a\u001b). Here and\nhenceforth the convention is to use Latin indices to de-\nnote spatial coordinates and Greek for space-time coor-\ndinates. Repeated Latin indices i;j;k are, furthermore,\nalways implicitly summed over.\nIII. SYMMETRIES AND CONSERVATION\nLAWS\nA. Gauge invariance\nThe Lagrangian describing coupled electron transport\nand collective spin-texture dynamics (disregarding for\nsimplicity the ordinary electromagnetic \felds) is\nL(rp;vp;m;@\u0016m)\n=X\np \nmev2\np\n2+vp\u0001a\u001b+a\u001b!\n\u0000A\n2Z\nd3r(@im)2\n=X\np \nmev2\np\n2+v\u0016\npa\u001b\u0016!\n\u0000A\n2Z\nd3r(@im)2:(11)\nv\u0016\np\u0011(1;vp),vp=_r, and\u001bhere is the spin of indi-\nvidual particles labelled by p. The resulting equations\nof motion satisfy certain basic conservation laws, due to\nspin-dependent gauge freedom, space-time homogeneity,\nand spin isotropicity.\nFirst, let us establish gauge invariance due to an ambi-\nguity in the choice of the spinor rotations ^U(r;t)!^U^U0.\nOur formulation should be invariant under arbitrary di-\nagonal transformations ^U0=e\u0000ifand ^U0=e\u0000ig^\u001bz=2on\nthe rotated fermionic \feld ^ 0, corresponding to gauge\ntransformations of the spin-projected theory:\n\u000ea\u001b\u0016=~@\u0016fand\u000ea\u001b\u0016=\u001b~@\u0016g=2; (12)\nrespectively. The change in the Lagrangian density is\ngiven by\n\u000eL=j\u0016@\u0016fand\u000eL=j\u0016\ns@\u0016g; (13)\nrespectively, where j=j++j\u0000andjs=~(j+\u0000j\u0000)=2\nare the corresponding charge and spin gauge currents.\nThe action S=R\ndtd3rLis gauge invariant, up to sur-\nface terms that do not a\u000bect the equations of motion,\nprovided that the four-divergence of the currents vanish,\nwhich is the conservation of particle number and spin\ndensity:\n_\u001a+r\u0001j= 0;_\u001as+r\u0001js= 0: (14)\n(The second of these conservation laws will be relaxed\nlater.) Here, the number and spin densities along with\nthe associated \rux densities are\n\u001a=X\npnp\u0011\u001a++\u001a\u0000;\nj=X\npnpvp\u0011\u001av; (15)4\nand\n\u001as=X\npq\u001bnp\u0011~\n2(\u001a+\u0000\u001a\u0000);\njs=X\npq\u001bnpvp\u0011\u001asvs; (16)\nwherenp=\u000e(r\u0000rp) and\u001bp=\u0006for spins up and down.\nIn the hydrodynamic limit, the above equations deter-\nmine the average particle velocity vand spin velocity\nvs, which allows us to de\fne four-vectors j\u0016= (\u001a;\u001av)\nandj\u0016\ns= (\u001as;\u001asvs). Microscopically, the local spin-\ndependent currents are de\fned, in the presence of electro-\nmagnetic vector potential aand \fctitious vector potential\na\u001b, by\nme\u001a\u001bv\u001b= Reh y\n\u001b(\u0000i~r\u0000a\u001b\u0000ea) \u001bi; (17)\nwheree<0 is the electron charge.\nB. Angular and linear momenta\nOur Lagrangian (11) contains the dynamics of m(r)\nthat is coupled to the current. In this regard, we note\nthat the time component of the \fctitious gauge poten-\ntial (B4),a\u001b=\u0000~@t'(1\u0000\u001bcos\u0012)=2, is a Wess-Zumino\naction that governs the spin-texture dynamics.4,6,28The\nvariational equation m\u0002\u000emL= 0 gives:\n\u001as(@t+vs\u0001r)m+m\u0002\u000emF= 0: (18)\nTo derive this equation, we used the spin-density con-\ntinuity equation (14) and a gauge-independent identity\nsatis\fed by the \fctitious potentials: their variations with\nrespect to mare given by\n\u000ema\u001b\u0016(m;@\u0016m) =q\u001bm\u0002@\u0016m; (19)\nwhere\n\u000em\u0011@\n@m\u0000X\n\u0016@\u0016@\n@(@\u0016m): (20)\nOne recognizes that Eq. (18) is the Landau-Lifshitz (LL)\nequation, in which the spin density precesses about the\ne\u000bective \feld given explicitly by\nh\u0011\u000emF=\u0000A@2\nim: (21)\nEquation (18) also includes the well-known reactive spin\ntorque:\u001c= (js\u0001r)m,3which is evidently the change\nin the local spin-density vector due to the spin angular\nmomentum carried by the itinerant electrons. One can\nformally absorb this spin torque by de\fning an advective\ntime derivative Dt\u0011@t+vs\u0001r, with respect to the\naverage spin drift velocity vs.\nEquation (18) may be written in a form that explicitly\nexpresses the conservation of angular momentum:27,29\n@t(\u001asmi) +@j\u0005ij= 0; (22)where the angular-momentum stress tensor is de\fned by\n\u0005ij=\u001asvsjmi\u0000A(m\u0002@jm)i: (23)\nNotice that this includes both quasiparticle and collective\ncontributions, which stem respectively from the trans-\nport and equilibrium spin currents.\nThe Lorentz force equation for the electrons, Eq. (8),\nin turn, leads to a continuity equation for the kinetic\nmomentum density.6To see this, let us start with the\nmicroscopic perspective:\n@t(\u001avi) =@tX\npnpvp=X\np( _npvp+np_vp): (24)\nUsing the Lorentz force equation for the second term, we\nhave:\nmeX\npnp_vp=X\npq\u001bnp(ei+\u000fijkbkvpj) =X\npq\u001bnpfi\u0016v\u0016\np\n=\u001asm\u0001(@tm\u0002@im) +\u001asvsjm\u0001(@jm\u0002@im)\n= (@im)\u0001(\u000emF) =\u0000A(@im)\u0001(@2\njm); (25)\nutilizing Eq. (18) to obtain the last line. Coarse-graining\nthe \frst term of Eq. (24), in turn, we \fnd:\nX\np_npvp=\u0000@jX\np\u000e(r\u0000rp)vpivpj!\u0000@jX\n\u001b\u001a\u001bv\u001biv\u001bj:\n(26)\nPutting Eqs. (25) and (26) together, we can \fnally write\nEq. (24) in the form:\nme@t(\u001avi) +@j \nTij+meX\n\u001b\u001a\u001bv\u001biv\u001bj!\n= 0;(27)\nwhere\nTij=A\u0014\n(@im)\u0001(@jm)\u0000\u000eij\n2(@km)2\u0015\n(28)\nis the magnetization stress tensor.6\nA spin-dependent chemical potential ^ \u0016=^K\u00001^\u001agov-\nerned by local density and short-ranged interactions can\nbe trivially incorporated by rede\fning the stress tensor\nas\nTij!Tij+\u000eij\n2^\u001aT^K\u00001^\u001a: (29)\nIn our notation, ^ \u0016= (\u0016+;\u0016\u0000)T, ^\u001a= (\u001a+;\u001a\u0000)Tand ^Kis\na symmetric 2\u00022 compressibility matrix in spin space,\nwhich includes the degeneracy pressure as well as self-\nconsistent exchange and Hartree interactions. In general,\nEq. (29) is valid only for su\u000eciently small deviations from\nthe equilibrium density.\nUsing the continuity equations (14), we can combine\nthe last term of Eq. (27) with the momentum density\nrate of change:\n@t(\u001a\u001bv\u001bi) +@j(\u001a\u001bv\u001biv\u001bj) =\u001a\u001b(@t+v\u001b\u0001r)v\u001bi;(30)5\nwhich casts the momentum density continuity equation\nin the Euler equation form:\nmeX\n\u001b\u001a\u001b(@t+v\u001b\u0001r)v\u001bi+@jTij= 0: (31)\nWe do not expect such advective corrections to @tto\nplay an important role in electronic systems, however.\nThis is in contrast to the advective-like time derivative\nin Eq. (18), which is \frst order in velocity \feld and is\ncrucial for capturing spin-torque physics.\nC. Hydrodynamic free energy\nWe will now turn to the Hamiltonian formulation and\nconstruct the free energy for our magnetohydrodynamic\nvariables. This will subsequently allow us to develop a\nnonequilibrium thermodynamic description. The canon-\nical momenta following from the Lagrangian (11) are\npp\u0011@L\n@vp=mevp+ap;\n\u0019\u0011@L\n@_m=X\npnp@a\u001b\n@_m=X\npnpamon\n\u001b(m): (32)\nNotice that for our translationally-invariant system, the\ntotal linear momentum\nP\u0011X\nppp+Z\nd3r(\u0019\u0001r)m=meX\npvp; (33)\nwhere we have used Eq. (5) to obtain the second equality,\ncoincides with the kinetic momentum (mass current) of\nthe electrons. The latter, in turn, is equivalent to the lin-\near momentum of the original problem of interacting non-\nrelativistic electrons, in the absence of any real or \fcti-\ntious gauge \felds. See appendix A. While Pis conserved\n(as discussed in the previous section and also follows now\nfrom the general principles), the canonical momenta of\nthe electrons and the spin-texture \feld, Eqs. (32), are\nnot conserved separately. As was pointed out by Volovik\nin Ref. 6, this explains anomalous properties of the lin-\near momentum associated with the Wess-Zumino action\nof the spin-texture \feld: This momentum has neither\nspin-rotational nor gauge invariance. The reason is that\nthe spin-texture dynamics de\fne only one piece of the\ntotal momentum, which is associated with the coherent\ndegrees of freedom. Including also the contribution as-\nsociated with the incoherent (quasiparticle) background\nrestores the proper gauge-invariant momentum, P, which\ncorresponds to the generator of the global translation in\nthe microscopic many-body description.\nPerforming a Legendre transformation to Hamiltonianas a function of momenta, we \fnd\nH[rp;pp;m;\u0019] =X\npvp\u0001pp+Z\nd3r_m\u0001\u0019\u0000L\n=X\np(pp\u0000a\u001b)2\n2me+A\n2Z\nd3r(@im)2\n\u0011E+F; (34)\nwhereEis the kinetic energy of electrons and Fis the\nexchange energy of the magnetic order. As could be\nexpected,Eis the familiar single-particle Hamiltonian\ncoupled to an external vector potential. According to\na Hamilton's equation, the velocity is conjugate to the\ncanonical momentum: vp=@H=@ pp. We note that ex-\nplicit dependence on the spin-texture dynamics dropped\nout because of the special property of the gauge \felds:\n_m\u0001@_ma\u001b=a\u001b. Furthermore, according to Eq. (19), we\nhavem\u0002\u000emE= (js\u0001r)m, so the LL Eq. (18) can be\nwritten in terms of the Hamiltonian (34) as11\n\u001as_m+m\u0002\u000emH= 0: (35)\nSo far, we have included in the spin-texture equa-\ntion only the piece coupled to the itinerant electron de-\ngrees of freedom. The purely magnetic part is tedious\nto derive directly and we will include it in the usual LL\nphenomenology.29To this end, we rede\fne\nF[m(r)]!F+F0; (36)\nby adding an additional magnetic free energy F0[m(r)],\nwhich accounts for magnetostatic interactions, crystalline\nanisotropies, coupling to external \felds, as well as energy\nassociated with localized dorforbitals.30Then the to-\ntal free energy (Hamiltonian) is H=E+F, and we in\ngeneral de\fne the e\u000bective magnetic \feld as the thermo-\ndynamic conjugate of m:h\u0011\u000emH. The LL equation\nthen becomes\n%s_m+m\u0002h= 0; (37)\nwhere%sis the total e\u000bective spin density. To enlarge\nthe scope of our phenomenology, we allow the possibility\nthat%s6=\u001as. For example, in the s\u0000dmodel, an extra\nspin density comes from the localized d-orbital electrons.\nMicroscopically, %s@tmterm in the equation of motion\nstems from the Wess-Zumino action generically associ-\nated with the total spin density.\nIn the following, it may sometimes be useful to separate\nout the current-dependent part of the e\u000bective \feld, and\nwrite the purely magnetic part as hm\u0011\u000emF, so that\nh=hm\u0000m\u0002(js\u0001r)m (38)\nand Eq. (37) becomes:\n%s_m+ (js\u0001r)m+m\u0002hm= 0: (39)6\nFor completeness, let is also write the equation of motion\nfor the spin- \u001bacceleration:\nme(@t+v\u001b\u0001r)v\u001bi=q\u001b[m\u0001(@tm\u0002@im)\n+v\u001bjm\u0001(@jm\u0002@im)]\u0000r\u0016\u001b;(40)\nretaining for the moment the advective correction to\nthe time derivative on the left-hand side and reinserting\nthe force due to the spin-dependent chemical potential,\n^\u0016=^K\u00001^\u001a. These equations constitute the coupled re-\nactive equations for our magneto-electric system. The\nHamiltonian (free energy) in terms of the collective vari-\nables is (including the elastic compression piece)\nH[\u001a\u001b;p\u001b;m] =X\n\u001bZ\nd3r\u001a\u001b(p\u001b\u0000a\u001b)2\n2me\n+1\n2Z\nd3r^\u001aT^K\u00001^\u001a+F[m]; (41)\nwhere p\u001b=mev\u001b+a\u001bis the spin-dependent momentum\nthat is locally averaged over individual particles.\nD. Conservation of energy\nSo far, our hydrodynamic equations are reactive, so\nthat the energy (41) must be conserved: P\u0011_H=_E+\n_F= 0. The time derivative of the electronic energy Eis\n_E=Z\nd3rX\n\u001b\u0014\nme\u001a\u001bv\u001b_v\u001b+ _\u001a\u001b\u0012mev2\n\u001b\n2+\u0016\u001b\u0013\u0015\n=Z\nd3rX\n\u001b\u0014\nme\u001a\u001bv\u001bj_v\u001bj\u0000@j(\u001a\u001bv\u001bj)\u0012mev2\n\u001b\n2+\u0016\u001b\u0013\u0015\n=Z\nd3rX\n\u001b\u001a\u001bv\u001bj[me(@t+v\u001b\u0001r)v\u001bj+@j\u0016\u001b]\n=Z\nd3rX\n\u001bq\u001b\u001a\u001bv\u001b\u0001(e+v\u001b\u0002b)\n=Z\nd3rX\n\u001bq\u001b\u001a\u001bv\u001b\u0001e=Z\nd3rjs\u0001e: (42)\nThe change in the spin-texture energy is given, according\nto Eq. (39), by\n_F=Z\nd3r_m\u0001\u000emF=Z\nd3r_m\u0001hm\n=Z\nd3r_m\u0001[%sm\u0002_m+m\u0002(js\u0001r)m)]\n=\u0000Z\nd3rjs\u0001e: (43)\nThe total energy is thus evidently conserved, P= 0.\nWhen we calculate dissipation in the rest of the paper,\nwe will omit these terms which cancel each other. The\ntotal energy \rux density is evidently given by\nQ=X\n\u001b\u001a\u001b\u0012mev2\n\u001b\n2+\u0016\u001b\u0013\nv\u001b: (44)IV. DISSIPATION\nHaving derived from \frst principles the reactive cou-\nplings in our magneto-electric system, summed up in\nEqs. (39)-(41), we will proceed to include the dissipa-\ntive e\u000bects phenomenologically. Let us focus on the lin-\nearized limit of small deviations from equilibrium (which\nmay be spin textured), so that the advective correction\nto the time derivative in the Euler Eq. (40), which is\nquadratic in the velocity \feld, can be omitted. To elimi-\nnate the quasiparticle spin degree of freedom, let us, fur-\nthermore, treat halfmetallic ferromagnets, so that \u001a=\u001a+\nand\u001as=q\u001a, whereq=~=2 is the electron's spin.31From\nEq. (40), the equation of motion for the local (averaged)\ncanonical momentum is:32\n_p=q\n\u001aj\u0002b\u0000r\u0016; (45)\nin a gauge where a\u001b= 0, so that _p=me_v\u0000qe.33\n\u0016=\u001a=K. The Lorentz force due to the applied (real)\nelectromagnetic \felds can be added in the obvious way\nto the right-hand side of Eq. (45). Note that since we\nare now interested in linearized equations close to equi-\nlibrium,\u001ain Eq. (45) can be approximated by its (ho-\nmogeneous) equilibrium value.\nIntroducing relaxation through a phenomenological\ndamping constant (Drude resistivity)\n\r=me\n\u001a\u001c; (46)\nwhere\u001cis the collision time, expressing the \fctitious\nmagnetic \feld in terms of the spin texture, Eq. (45) be-\ncomes:\n_pi=\u0000q\n\u001a(m\u0002@im)\u0001(j\u0001r)m\u0000@i\u0016\u0000\rji: (47)\nAdding the phenomenological Gilbert damping34to\nthe magnetic Eq. (37) gives the Landau-Lifshitz-Gilbert\nequation:\n%s(_m+\u000bm\u0002_m) =h\u0002m; (48)\nwhere\u000bis the damping constant. Eqs. (47) and\n(48), along with the continuity equation, _ \u001a=\u0000r\u0001j,\nare the near-equilibrium thermodynamic equations for\n(\u001a;p;m) and their respective thermodynamic conjugates\n(\u0016;j;h) = (\u000e\u001aH;\u000epH;\u000emH). This system of equations of\nmotion may be written formally as\n@t0\n@\u001a\np\nm1\nA=b\u0000[m(r)]0\n@\u0016\nj\nh1\nA: (49)\nThe matrix ^\u0000 depends on the equilibrium spin texture\nm(r). By the Onsager reciprocity principle, \u0000 ij[m] =\nsisj\u0000ji[\u0000m], wheresi=\u0006if theith variable is even\n(odd) under time reversal.7\nIn the quasistationary description of a nonequilibrium\nthermodynamic system, the entropy S[\u001a;p;m] is for-\nmally regarded as a functional of the instantaneous ther-\nmodynamic variables, and the probability of a given con-\n\fguration is proportional to eS=kB. If the heat conduc-\ntance is high and the temperature Tis uniform and con-\nstant, the instantaneous rate of dissipation P=T_Sis\ngiven by the rate of change in the free energy, P=_H=R\nd3rP:\nP=\u0000\u0016_\u001a\u0000h\u0001_m\u0000j\u0001_p=\u000b%s_m2+\rj2; (50)\nwhere we used Eq. (47) and expressed the e\u000bective \feld\nhas a function of _mby taking m\u0002of Eq. (48):\nh=%sm\u0002_m\u0000\u000b%s_m: (51)\nNotice that the \fctitious magnetic \feld bdoes not con-\ntribute to dissipation because it does not do work.\nSo far, there is no dissipative coupling between the\ncurrent and the spin-texture dynamics, and the macro-\nscopic equations obey the global time-reversal symme-\ntry. However, we know that dissipative couplings ex-\nists due to the misalignment of the electron's spin with\nthe collective spin texture and spin-texture resistivity.3,22\nFollowing Ref. 11, we add these well-known e\u000bects phe-\nnomenologically by making an expansion in the equations\nof motion to linear order in the nonequilibrium quanti-\nties _mandj. To limit the number of terms one can write\ndown, we will only add terms that are spin-rotationally\ninvariant and isotropic in real space (which disregards,\nin particular, such e\u000bects as the angular magnetoresis-\ntance and the anomalous Hall e\u000bect). To second order in\nthe spatial gradients of m, there are only three dissipa-\ntive phenomenological terms with couplings \u0011,\u00110, and\f\nconsistent with the above requirements, which could be\nadded to the right-hand side of Eq. (47).35The momen-\ntum equation becomes:\n_pi=\u0000q\n\u001a(m\u0002@im)\u0001(j\u0001r)m\u0000@i\u0016\u0000\rji\n\u0000\u0011(@km)2ji\u0000\u00110@im\u0001(j\u0001r)m\u0000q\f_m\u0001@im:(52)\nIt is known that the \\ \fterm\" comes from a misalignment\nof the electron spin with the collective spin texture, and\nthe associated dephasing. It is natural to expect thus\nthat the dimensionless parameter \f\u0018~=\u001cs\u0001, where\u001cs\nis a characteristic spin-dephasing time.3The \\\u0011terms\"\nevidently describe texture-dependent resistivity, which\nis anisotropic with respect to the gradients in the spin\ntexture along the local current density. Such term are\nalso naturally expected, in view of the well-known giant-\nmagnetoresistance e\u000bect,36in which noncollinear magne-\ntization results in electrical resistance. The microscopic\norigin of this term is due to spin-texture misalignment,\nwhich modi\fes electron scattering.\nThe total spin-texture-dependent resistivity can be putinto a tensor form:\n\rij[m] =\u000eij\u0002\n\r+\u0011(@km)2\u0003\n+\u00110@im\u0001@jm\n+q\n\u001am\u0001(@im\u0002@jm): (53)\nThe last term due to \fctitious magnetic \feld gives a Hall\nresistivity. Note that ^ \r[m] = ^\rT[\u0000m], consistent with\nthe Onsager theorem. We can \fnally write Eq. (47) as:\n_pi=\u0000\rij[m]jj\u0000@i\u0016\u0000q\f_m\u0001@im: (54)\nAs was shown in Ref. 11, since the Onsager relations\nrequire thatb\u0000[m] =b\u0000[\u0000m]Twithin the current/spin-\ntexture \felds sector, there must be a counterpart to the\n\fterm above in the magnetic equation, which is the well-\nknown dissipative \\ \fspin torque:\"\n%s(_m+\u000bm\u0002_m) =h\u0002m\u0000q\fm\u0002(j\u0001r)m:(55)\nThe total dissipation Pis now given by\nP=\u000b%s_m2+ 2q\f_m\u0001(j\u0001r)m+\u0002\n\r+\u0011(@km)2\u0003\nj2\n+\u00110[(j\u0001r)m]2\n=\u000b%s\u0014\n_m+q\f\n\u000b%s(j\u0001r)m\u00152\n+\u0002\n\r+\u0011(@km)2\u0003\nj2\n+\u0012\n\u00110\u0000(q\f)2\n\u000b%s\u0013\n[(j\u0001r)m]2: (56)\nThe second law of thermodynamics requires the total dis-\nsipation to be positive, which puts some constraints on\nthe allowed values of the phenomenological parameters.\nWe can easily notice, however, that the dissipation (56)\nis guaranteed to be positive-de\fnite if\n\u0011+\u00110\u0015(q\f)2\n\u000b%s; (57)\nwhich may serve as an estimate for the spin-texture re-\nsistivity due to spin dephasing. This is consistent with\nthe microscopic \fndings of Ref. 23.\nV. THERMAL NOISE\nAt \fnite temperature, thermal agitation causes \ruc-\ntuations of the current and spin texture, which are cor-\nrelated due to their coupling. A complete description\nrequires that we supplement the stochastic equations of\nmotion with the correlators for these \ructuations. It\nis convenient to regard these \ructuations as being due\nto the stochastic Langevin \\forces\" ( \u000e\u0016;\u000ej;\u000eh) on the\nright-hand side of Eq. (49). The complete set of \fnite-\ntemperature hydrodynamic equations thus becomes:\n_\u001a=\u0000r\u0001~j;\n_p+q\f_mirmi=\u0000^\r[m]~j\u0000r~\u0016;\n%s(1 +\u000bm\u0002)_m=~h\u0002m\u0000q\fm\u0002(~j\u0001r)m:(58)8\nwhere (~\u0016;~j;~h) = (\u0016+\u000e\u0016;j+\u000ej;h+\u000eh). The simplest\n(while possibly not most realistic) case corresponds to\na highly compressible \ruid, such that K!1 . In this\nlimit,\u0016=\u001a=K!0 and the last two equations com-\npletely decouple from the \frst, continuity equation. In\nthe remainder of this section, we will focus on this special\ncase. The correlations of the stochastic \felds are given\nby the symmetric part of the inverse matrix b\u0007 =\u0000b\u0000\u00001,37\nwhich is found by inverting Eq. (58) (reduced now to a\nsystem of two equations):\n~j=\u0000^\r\u00001(_p+q\f_mirmi);\n~h=%sm\u0002_m\u0000\u000b%s_m\u0000q\f(~j\u0001r)m: (59)\nWriting formally these equations as (after substituting ~j\nfrom the \frst into the second equation)\n\u0012~j\n~h\u0013\n=\u0000b\u0007[m(r)]\u0012\n_p\n_m\u0013\n; (60)\nwe immediately read out for the matrix elements\nb\u0007(r;r0) =b\u0007(r)\u000e(r\u0000r0):\n\u0007ji;ji0(r) =(^\r\u00001)ii0;\n\u0007ji;hi0(r) =q\f(^\r\u00001)ik@kmi0;\n\u0007hi0;ji(r) =\u0000q\f(^\r\u00001)ki@kmi0;\n\u0007hi;hi0(r) =\u000b%s\u000eii0+%s\u000fii0kmk\n\u0000(q\f)2(@kmi)(^\r\u00001)kk0(@k0mi0):(61)\nAccording to the \ructuation-dissipation theorem, we\nsymmetrize b\u0007 to obtain the classical Langevin\ncorrelators:37\nh\u000eji(r;t)\u000eji0(r0;t0)i=T=gii0;\nh\u000eji(r;t)\u000ehi0(r0;t0)i=T=q\fg0\nik@kmi0;\nh\u000ehi(r;t)\u000ehi0(r0;t0)i=T=\u000b%s\u000eii0\n\u0000(q\f)2gkk0(@kmi)(@k0mi0); (62)\nwhereT= 2kBT\u000e(r\u0000r0)\u000e(t\u0000t0) and\n^g= [^\r\u00001+ (^\r\u00001)T]=2;^g0= [^\r\u00001\u0000(^\r\u00001)T]=2 (63)\nare, respectively, the symmetric and antisymmetric parts\nof the conductivity matrix ^ \r\u00001. The short-ranged, \u000e-\nfunction character of the noise correlations in space stems\nfrom the assumption of high electronic compressibility.\nContrast this to the results of Ref. 11 for incompressible\nhydrodynamics. A presence of long-ranged Coulombic\ninteractions and plasma modes would also give rise to\nnonlocal correlations. These are absent in our treatment,\nwhich disregards ordinary electromagnetic phenomena.\nFocusing on the microwave frequencies !characteris-\ntic of ferromagnetic dynamics, it is most interesting to\nconsider the regime where !\u001c\u001c\u00001. This means that\nwe can employ the drift approximation for the \frst of\nEqs. (59):\n_pi=me_vi\u0000qei\u0019\u0000qei=q_m\u0001(m\u0002@im): (64)Substituting this _pin Eq. (59), we can easily \fnd a closed\nstochastic equation for the spin-texture \feld:\n%s(1 +\u000bm\u0002)_m+m\u0002\u001c$_m= (hm+\u000eh)\u0002m;(65)\nwhere we have de\fned the \\spin-torque tensor\"\n\u001c$=q2(^\r\u00001)kk0(m\u0002@km\u0000\f@km)\n\n(m\u0002@k0m+\f@k0m): (66)\nThe antisymmetric piece of this tensor modi\fes the e\u000bec-\ntive gyromagnetic ratio, while the more interesting sym-\nmetric piece determines the additional nonlocal Gilbert\ndamping:\n\u000b$=\u001c$+\u001c$T\n2%s=q2\n%sG$; (67)\nwhere\nG$=gkk0\u0002\n(m\u0002@km)\n(m\u0002@k0m)\u0000\f2@km\n@k0m\u0003\n+\fg0\nkk0[(m\u0002@km)\n@k0m\u0000@km\n(m\u0002@k0m)]:\n(68)\nIn obtaining Eq. (65) from Eqs. (59), we have separated\nthe reactive spin torque out of the e\u000bective \feld: h=\nhm\u0000qm\u0002(j\u0001r)m. (The remaining piece hmthus re\rects\nthe purely magnetic contribution to the e\u000bective \feld.)\nThe total stochastic magnetic \feld entering Eq. (65),\n\u000eh=\u000eh+qm\u0002(\u000ej\u0001r)m; (69)\ncaptures both the usual magnetic Brown noise38\u000eh\nand the Johnson noise spin-torque contribution39\u000ehJ=\nqm\u0002(\u000ej\u0001r)mthat arises due to the substitution j=~j\u0000\u000ej\nin the reactive spin torque q(j\u0001r)m. Using correla-\ntors (62), it is easy to show that the total e\u000bective \feld\n\ructuations \u000ehare consistent with the nonlocal e\u000bec-\ntive Gilbert damping tensor (68), in accordance with the\n\ructuation-dissipation theorem applied directly to the\npurely magnetic Eq. (65).\nTo the leading, quadratic order in spin texture, we can\nreplacegkk0!\u000ekk0=\randg0\nkk0!0 in Eq. (68). This ad-\nditional texture-dependent nonlocal damping (along with\nthe associated magnetic noise) is a second-order e\u000bect,\nphysically corresponding to the backaction of the magne-\ntization dynamics-driven current on the spin texture.11\nIt should be noted that in writing the modi\fed LLG\nequation (55), we did not systematically expand it to\ninclude the most general phenomenological terms up to\nthe second order in spin texture. We have only included\nextra spin-torque terms, which are required by the On-\nsager symmetry with Eq. (52). The second-order Gilbert\ndamping (68) was then obtained by solving Eqs. (52) and\n(55) simultaneously. (Cf. Refs. 11,40.) This means in\nparticular, that this procedure does not capture second-\norder Gilbert damping e\u000bects whose physical origin is\nunrelated to the longitudinal spin-transfer torque physics\nstudied here. One example of that is the transverse spin-\npumping induced damping discussed in Refs. 41.9\nVI. EXAMPLES\nA. Rigidly spinning texture\nTo illustrate the \u0011resistivity terms in the electron's\nequation of motion (52), we \frst consider 1D textures.\nTake, for example, the case of a 1D spin helix m(z)\nalong thezaxis, whose spatial gradient pro\fle is given by\n@zm=\u0014^ z\u0002m, where\u0014is the wave vector of the spatial\nrotation and m?^ z. See Fig. 1. It gives anisotropic re-\nsistivity in the xyplane,r(\u0011)\n?, and along the zdirection,\nr(\u0011)\nk:\nr(\u0011)\n?=\u0011(@zm)2=\u0011\u00142; r(\u0011)\nk= (\u0011+\u00110)\u00142: (70)\nFIG. 1: (Color online) The transverse magnetic helix, @zm=\n\u0014^ z\u0002m, with texture-dependent anisotropic resistivity (70).\nWe assume here translational invariance in the transverse ( xy)\ndirections. Spinning this helix about the vertical zaxis gen-\nerates the dissipative electromotive forces f(\f)\nz, which is spa-\ntially uniform and points everywhere along the zaxis. A\nmagnetic spiral, @zm=\u0014^'\u0002m=\u0014^\u0012, spinning around the z\naxis, on the other hand, produces a purely reactive electromo-\ntive forceez, as discussed in the text, which is oscillatatory\nin space along the zaxis.\nThe \fctitious electric \feld and dissipative \fforce re-\nquire magnetic dynamics. A general texture globally ro-\ntating clockwise in spin space in the xyplane according\nto_m=\u0000!^ z\u0002m(which may be induced by applying a\nmagnetic \feld along the zdirection) generates an electric\n\feld\nei= (m\u0002_m)\u0001@im=\u0000!(m\u0002^ z\u0002m)\u0001@im\n=\u0000!@imz=\u0000!@icos\u0012 (71)and a\fforce\nf(\f)\ni=\u0000\f_m\u0001@im=\f!^ z\u0001(m\u0002@im)\n=\f!sin2\u0012@i'; (72)\nwhere (\u0012;') denote the position-dependent spherical an-\ngles parametrizing the spin texture. The reactive force\n(71) has a simple interpretation of the gradient of the\nBerry-phase15accumulation rate [which is locally deter-\nmined by the solid angle subtended by m(t)]. In the\ncase of the transverse helix discussed above, \u0012=\u0019=2,\n'=\u0014z\u0000!t, so thatez= 0 whilef(\f)\nz=\u0000\f!\u0014 is \fnite.\nAs an example of a dynamical texture that does not\ngenerate f(\f)while producing a \fnite e, consider a spin\nspiral along the zaxis, described by @zm=\u0014^'\u0002m=\u0014^\u0012,\nand rotating in time in the manner described above. It is\nclear geometrically that the change in the spin texture in\ntime is in a direction orthogonal to its gradients in space.\nSpeci\fcally, \u0012=\u0014z,'=\u0000!t, so thatf(\f)\nz= 0 while the\nelectric \feld is oscillatory, ez=!\u0014sin\u0012.\nB. Rotating spin textures\nWe show here that a vortex rotating about its core in\norbital space generates a current circulating around its\ncore, as well as a current going radially with respect to\nthe core. Consider a spin texture with a time depen-\ndence corresponding to the real-space rotation clockwise\nin thexyplane around the origin, such that m(r;t) =\nm(r(t);0) with _r=!^ z\u0002r=!r^\u001e, where we use polar co-\nordinates (r;\u001e) on the plane normal to the zaxis in real\nspace [to be distinguished from the spherical coordinates\n(\u0012;') that parametrize min spin space], we have\n_m= (_r\u0001r)m=!@\u001em: (73)\nForm(r;\u001e) in polar coordinates, the components of the\nelectric \feld are,\ner=!m\u0001(@\u001em\u0002@rm); e\u001e= 0; (74)\nwhile the components of the \fforce are\nf(\f)\nr=\u0000\f!(@rm)\u0001(@\u001em); f(\f)\n\u001e=\u0000\f!(@\u001em)2\nr:(75)\nIn order to \fnd the \fctitious electromagnetic \felds, we\nneed to calculate the following tensors (which depend on\nthe instantaneous spin texture):\nbij\u0011m\u0001(@im\u0002@jm) = sin\u0012(@i\u0012@j'\u0000@j\u0012@i');\ndij\u0011@im\u0001@jm=@i\u0012@j\u0012+ sin2\u0012@i'@j': (76)\nAs an example, consider a vortex centered at the ori-\ngin in thexyplane with winding number 1 and positive\npolarity, as shown in Fig. 2. Its angular coordinates are\ngiven by\n'= (\u001e+!t) +\u0019\n2; \u0012=\u0012(r); (77)10\nwhere\u001e= arg( r) and\u0012is a rotationally invariant func-\ntion such that \u0012!0 asr!0 and\u0012!\u0019=2 asr!1 .\nEvaluating the tensors in equation (76) for this vortex in\npolar coordinates gives drr= (@r\u0012)2,d\u001e\u001e= (sin\u0012=r)2,\ndr\u001e= 0, andbr\u001e=\u0000(@rcos\u0012)=r. The radial electric\n\feld is then given by\ner=\u0000!rbr\u001e=!@rcos\u0012: (78)\nThe\fforce is in the azimuthal direction:\nf(\f)\nr= 0; f(\f)\n\u001e=\u0000\f!rd\u001e\u001e=\u0000\f!sin2\u0012\nr: (79)\nWe can interpret this force as the spin texture \\dragging\"\nthe current along its direction of motion. Notice that the\nforces in Eqs. (78) and (79) are the negative of those in\nEqs. (71) and (72), as they should be for the present case,\nsince the combination of orbital and spin rotations of our\nvortex around its core leaves it invariant, producing no\nforces.\nFIG. 2: Positive-polarity magnetic vortex con\fguration pro-\njected on the xyplane. mhas a positive (out-of-plane) z\ncomponent near the vortex core. Rotating this vortex about\nthe origin in real space generates the current in the xyplane\nshown in Fig. 3.\nThe total resistivity tensor (53) is (in the cylindrical\ncoordinates)\n^\r=\r+\u0011(drr+d\u001e\u001e) +\u00110^d+q\n\u001a^b=\u0012\n\rr\r?\n\u0000\r?\r\u001e\u0013\n;(80)\nwhere\n\rr=\r+ (\u0011+\u00110)(@r\u0012)2+\u0011\u0012sin\u0012\nr\u00132\n;\n\r\u001e=\r+\u0011(@r\u0012)2+ (\u0011+\u00110)\u0012sin\u0012\nr\u00132\n;\n\r?=\u0000q\n\u001a@rcos\u0012\nr: (81)Here, the two diagonal components, \rrand\r\u001e, describe\nthe (dissipative) anisotropic resistivity, while the o\u000b-\ndiagonal component, \r?, captures what is called the\ntopological Hall e\u000bect.19\nIn the drift approximation, Eq. (64), the current-\ndensity \feld j=jr^ r+j\u0012^\u0012is given by\nj= ^\r\u00001q(e+f(\f));\u0012\njr\nj\u001e\u0013\n=q!^\r\u00001\u0012@rcos\u0012\n\u0000\fsin2\u0012=r\u0013\n=\u0000q!sin\u0012\n\rr\r\u001e+\r2\n?\u0012\n\r\u001e\u0000\r?\n\r?\rr\u0013\u0012\n@r\u0012\n\fsin\u0012=r\u0013\n:(82)\nMore explicitly, we may consider a pro\fle \u0012=\u0019(1\u0000\ne\u0000r=a)=2, whereais the radius of the vortex core. The\ncorresponding current (82) is sketched in Fig. 3.\nFIG. 3: We plot here the current in Eq. (82) (all parameters\nset to 1). Near the core, the current spirals inward and charges\nbuild up at the center (which is allowed for our compressible\n\ruid).\nWe note that the \fctitious magnetic \feld \u000fijkbjk=2\npoints everywhere in the zdirection, its total \rux\nthrough the xyplane being given by\nF=Z\nd\u001edr (rbr\u001e) =\u0000Z\nd\u001edr (@\u001e'@rcos\u0012) = 2\u0019:\n(83)\nNote that the integrand is just the Jacobian of the map\nfrom the plane to the sphere de\fned by the spin-texture\n\feld:\n(\u0012(r);'(r)) :R2!S2: (84)\nThis re\rects the fact that the \fctitious magnetic \rux is\ngenerally a topological invariant, corresponding to the \u00192\nhomotopy group of the mapping (84).6,42\nC. Anisotropic resistivity of a 3D spiral\nConsider the texture described by @im=\u0014i^ z\u0002m,\nwhere the spatial rotation stays in the xyplane, but the11\nwave vector\u0014can be in any direction. The spin texture\nforms a transverse helix in the zdirection and a planar\nspiral in the xandydirections. Fig. 4 shows such a\ncon\fguration for \u0014pointing along ( x+y+z)=p\n3. The\n\fctitious magnetic \feld bvanishes, but the anisotropic\nresistivity still depends nontrivially on the spin texture:\n\rij=\u0002\n\r+\u0011(@km)2\u0003\n\u000eij+\u00110@im\u0001@jm\n= (\r+\u0011\u00142)\u000eij+\u00110\u0014i\u0014j; (85)\nwhich, according to j= ^\r\u00001E, would give a transverse\ncurrent signal for an electric \feld applied along the Carte-\nsian axesx,y, orz.\nFIG. 4: (Color online) A set of spin spirals which is topo-\nlogically trivial because r\u0012= 0 (and equivalent to the spin\nhelix, Fig. 1, up to a global real-space rotation), hence the\n\fctitious magnetic \feld b, Eq. (76), is zero. There is, how-\never, an anisotropic texture-dependent resistivity with \fnite\no\u000b-diagonal components, Eq. (85).\nVII. SUMMARY\nWe have developed semi-phenomenologically the hy-\ndrodynamics of spin and charge currents interacting with\ncollective magnetization in metallic ferromagnets, gener-\nalizing the results of Ref. 11 to three dimensions and\ncompressible \rows. Our theory reproduces known re-\nsults such as the spin-motive force generated by mag-\nnetization dynamics and the dissipative spin torque, al-\nbeit from a di\u000berent viewpoint than previous microscopic\napproaches. Among the several new e\u000bects predicted,\nwe \fnd both an isotropic and an anisotropic texture-\ndependent resistivity, Eq. (53), whose contribution to theclassical (topological) Hall e\u000bect should be described on\npar with that of the \fctitious magnetic \feld. By calculat-\ning the dissipation power, we give a lower bound on the\nspin-texture resistivity as required by the second law of\nthermodynamics. We \fnd a more general form, includ-\ning a term of order \f, of the texture-dependent correction\nto nonlocal Gilbert damping, predicted in Ref. 11. See\nEq. (68).\nOur general theory is contained in the stochastic hy-\ndrodynamic equations, Eqs. (58), which we treated in\nthe highly compressible limit. The most general situ-\nation is no doubt at least as rich and complicated as\nthe classical magnetohydrodynamics. A natural exten-\nsion of this work is the inclusion of heat \rows and re-\nlated thermoelectric e\u000bects, which we plan to investigate\nin a future work. Although we mainly focused on the\nhalfmetallic limit in this paper, our theory is in principle\na two-component \ruid model and allows for the inclu-\nsion of a fully dynamical treatment of spin densities and\nassociated \rows.31Finally, our hydrodynamic equations\nbecome amenable to analytic treatments when applied to\nthe important problem of spin-current driven dynamics\nof magnetic solitons, topologically stable objects that can\nbe described by a small number of collective coordinates,\nwhich we will also investigate in future work.\nAcknowledgments\nWe are grateful to Gerrit E. W. Bauer, Arne Brataas,\nAlexey A. Kovalev, and Mathieu Taillefumier for stimu-\nlating discussions. This work was supported in part by\nthe Alfred P. Sloan Foundation and the NSF under Grant\nNo. DMR-0840965.\nAPPENDIX A: MANY-BODY ACTION\nWe can formally start with a many-body action, with\nStoner instability built in due to short-range repulsion\nbetween electrons:25\nS[\u0016 \u001b(r;t); \u001b(r;t)] =Z\nCdtZ\nd3r\n\u0014\n^ +\u0012\ni~@t+~2\n2mer2\u0013\n^ \u0000U\u0016 \"\u0016 # # \"\u0015\n;(A1)\nwhere time truns along the Keldysh contour from \u00001\nto1and back. \u0016 \u001band \u001bare mutually independent\nGrassmann variables parametrizing fermionic coherent\nstates and ^ += (\u0016 \";\u0016 #) and ^ = ( \"; #)T. The four-\nfermion interaction contribution to the action can be de-\ncoupled via Hubbard-Stratonovich transformation, after12\nintroducing auxiliary bosonic \felds \u001eand\u0001:\neiSU=~= exp\u0012\n\u0000i\n~Z\nCdtZ\nd3rU\u0016 \"\u0016 # # \"\u0013\n=Z\nD[\u001e(r;t);\u0001(r;t)] exp\u0012i\n~Z\nCdtZ\nd3r\n\u0014\u001e2\n4U\u0000\u00012\n4U\u0000\u001e\n2^ +^ +\u0001\n2^ +^\u001b^ \u0015\u0013\n:(A2)\nIn obtaining this result, we decomposed the interaction\ninto charge- and spin-density pieces:\n\u0016 \"\u0016 # # \"=1\n4(^ +^ )2\u00001\n4(^ +m\u0001^\u001b^ )2; (A3)\nwhere mis an arbitrary unit vector. It is easy to\nshow thath\u001e(r;t)i=Uh^ +(r;t)^ (r;t)iandh\u0001(r;t)i=\nUh^ +(r;t)^\u001b^ (r;t)i, when properly averaging over the\ncoupled quasiparticle and bosonic \felds.\nThe next step in developing mean-\feld theory is to\ntreat the Hartree potential \u001e(r;t) and Stoner exchange\n\u0001(r;t)\u0011\u0001(r;t)m(r;t) \felds in the saddle-point approx-\nimation. Namely, the e\u000bective bosonic action\nSe\u000b[\u001e(r;t);\u0001(r;t)] =\u0000i~lnZ\nD[^ +;^ ]ei\n~S(^ +;^ ;\u001e;\u0001)\n(A4)\nis minimized, \u000eSe\u000b= 0, in order to \fnd the equations\nof motion for the \felds \u001eand\u0001. In the limit of suf-\n\fciently low electron compressibility and spin suscepti-\nbility, the charge- and spin-density \ructuations are sup-\npressed, de\fning mean-\feld parameters \u0016\u001eand \u0016\u0001. Since\na constant \u0016\u001eonly shifts the overall electrochemical po-\ntential, it is physically inconsequential. Our theory is de-\nsigned to focus on the remaining soft (Goldstone) modes\nassociated with the spin-density director m(r;t), while\n\u001e(r;t) and \u0001( r;t) are in general allowed to \ructuate\nclose to their mean-\feld values \u0016\u001eand \u0016\u0001, respectively.\nThe saddle-point equation of motion for the collective\nspin direction m(r;t) follows from \u000emSe\u000b[m] = 0, after\nintegrating out electronic degrees of freedom. Because\nof the noncommutative matrix structure of the action\n(A2), it is still a nontrivial problem. The problem sim-\npli\fes considerably in the limit of large exchange split-\nting \u0001, where we can project spins on the local magnetic\ndirection m. This lays the ground to the formulation dis-\ncussed in Sec. II, where the collective spin-density \feld\nparametrized by the director m(r;t) interacts with the\nspin-up/down free-electron \feld. The resulting equations\nof motion constitute the self-consistent dynamic Stoner\ntheory of itinerant ferromagnetism.\nIn the remainder of this appendix, we explicitly show\nthat the semiclassical formalism developed in Secs. II-\nIII B is equivalent to a proper \feld-theoretical treatment.\nThe equation of motion for the spin texture follows from\nextremizing the e\u000bective action with respect to variations\ninm. Because of the constraint on the magnitude of m,\nits variation can be expressed as \u000em=\u000e\u0012\u0002m, with\u000e\u0012being an arbitrary in\fnitesimal vector, so that the\nequation of motion is given by m\u0002\u000emSe\u000b= 0:\n0 =m\u0002\u000emSe\u000b\n=1\nZZ\nD[^ +;^ ] (m\u0002\u000emS)ei\n~S[^ +;^ ;\u001e;\u0001]\n=X\n\u001b\u0016(m\u0002\u000ema\u001b\u0016)\u001c@S\n@a\u001b\u0016\u001d\n\u0000m\u0002\u000emF; (A5)\nwhereZ=R\nD[^ +;^ ]ei\n~S[^ +;^ ;\u001e;\u0001]and we have used\nthe path-integral representation of the vacuum expecta-\ntion value. a\u001b\u0016are the spin-dependent gauge potentials\n(4) andFthe spin exchange energy, appearing after we\nproject spin dynamics on the collective \feld \u0001. Equa-\ntion (A5) may be expressed in terms of the hydrody-\nnamic variables of the electrons. De\fning spin-dependent\ncharge and current densities, j\u0016\n\u001b= (\u001a\u001b;j\u001b), by\n\u001a\u001b=\u001c@S\n@a\u001b\u001d\n=h\u0016 \u001b \u001bi;\nj\u001b=\u001c@S\n@a\u001b\u001d\n=1\nmeRe\n\u0016 \u001b(\u0000i~r\u0000a\u001b) \u001b\u000b\n=\u001a\u001bv\u001b;\n(A6)\nEq. (A5) reduces to the Landau-Lifshitz Eq. (18). Min-\nimizing action (A4) with respect to the \u001eand \u0001 \felds\ngives the anticipated self-consistency relations:\n\u001e(r;t) =Uh^ +(r;t)^ (r;t)i=U(\u001a++\u001a\u0000);\n\u0001(r;t) =Uh^ +(r;t)^\u001bz^ (r;t)i=U(\u001a+\u0000\u001a\u0000):(A7)\nAPPENDIX B: THE MONOPOLE GAUGE FIELD\nLet (\u0012;') be the spherical angles of m, the direction\nof the local spin density, and ^ \u001f\u001bbe the spin up/down\n(\u001b=\u0006) spinors given by, up to a phase,\n^\u001f+(\u0012;') =\u0012\ncos\u0012\n2\nei'sin\u0012\n2\u0013\n;\n^\u001f\u0000(\u0012;') = ^\u001f+(\u0019\u0000\u0012;'+\u0019) =\u0012\nsin\u0012\n2\n\u0000ei'cos\u0012\n2\u0013\n:(B1)\nThe spinors are related to the spin-rotation matrix ^U(m)\nby ^\u001f\u001b=^Uj\u001bi. The gauge \feld in mspace, which enters\nEq. (5), is thus given by\namon\n\u001b(\u0012;') =\u0000i~^\u001fy\n\u001b@m^\u001f\u001b=~\n2\u00121\u0000\u001bcos\u0012\nsin\u0012\u0013\n^';(B2)\nwhere we used the gradient on a unit sphere: @m=\n^\u0012@\u0012+^'@'=sin\u0012. The magnetic \feld corresponding to\nthis vector potential [extended to three dimensions by\na(m)!a(\u0012;')=m] is given on the unit sphere by\n@m\u0002amon\n\u001b=@m\u0002(a'^') =m\nsin\u0012@\u0012(sin\u0012a') =\u001b~\n2m:\n(B3)13\nIt follows from Eqs. (5) and (B2) that the spin-dependent\nreal-space gauge \felds are given by\na\u001b\u0016=\u0000~\n2@\u0016'(1\u0000\u001bcos\u0012): (B4)\nNotice that the \u001b=\u0006monopole \feld (B2), as well as the\nabove gauge \felds, are singular on the south/north pole(corresponding to the Dirac string). This is what allows a\nmagnetic \feld with \fnite divergence. Any other choice of\nthe monopole gauge \feld (B2) would correspond to a dif-\nferent choice of the spinors (B1), translating into a gauge\ntransformation of the \felds (B4). This is immediately\nseeing by noticing that amon\n\u001b(m)!amon\n\u001b(m) +@mf\u001b(m)\ncorresponds to a\u001b\u0016(r;t)!a\u001b\u0016(r;t) +@\u0016f\u001b(m(r;t)).\n1L. Berger, J. Appl. Phys. 55, 1954 (1984); M. Kl aui,\nC. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer, G. Faini,\nE. Cambril, and L. J. Heyderman, Appl. Phys. Lett.\n83, 105 (2003); E. Saitoh, H. Miyajima, T. Yamaoka,\nand G. Tatara, Nature 432, 203 (2004); Z. Li and\nS. Zhang, Phys. Rev. Lett. 92, 207203 (2004); S. E. Barnes\nand S. Maekawa, ibid. 95, 107204 (2005); L. Thomas,\nM. Hayashi, X. Jiang, R. Moriya, C. Rettner, and\nS. S. P. Parkin, Nature 443, 197 (2006); M. Yamanouchi,\nD. Chiba, F. Matsukura, T. Dietl, and H. Ohno, Phys.\nRev. Lett. 96, 096601 (2006); M. Yamanouchi, J. Ieda,\nF. Matsukura, S. E. Barnes, S. Maekawa, and H. Ohno,\nScience 317, 1726 (2007); L. Heyne, M. Kl aui, D. Backes,\nT. A. Moore, S. Krzyk, U. R udiger, L. J. Heyderman, A. F.\nRodr\u0013 \u0010guez, F. Nolting, T. O. Mentes, et al., Phys. Rev.\nLett. 100, 066603 (2008).\n2S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004);\nA. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Euro-\nphys. Lett. 69, 990 (2005);\n3Y. Tserkovnyak, H. J. Skadsem, A. Brataas, and G. E. W.\nBauer, Phys. Rev. B 74, 144405 (2006); Y. Tserkovnyak,\nA. Brataas, and G. E. Bauer, J. Magn. Magn. Mater. 320,\n1282 (2008).\n4G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468, 213\n(2008).\n5L. Berger, Phys. Rev. B 33, 1572 (1986).\n6G. E. Volovik, J. Phys. C: Sol. State Phys. 20, L83 (1987).\n7A. Stern, Phys. Rev. Lett. 68, 1022 (1992); Y. Aharonov\nand A. Stern, ibid.69, 3593 (1992).\n8S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 98, 246601\n(2007).\n9W. M. Saslow, Phys. Rev. B 76, 184434 (2007); R. A.\nDuine, ibid.77, 014409 (2008).\n10Y. Tserkovnyak and M. Mecklenburg, Phys. Rev. B 77,\n134407 (2008).\n11Y. Tserkovnyak and C. H. Wong, Phys. Rev. B 79, 014402\n(2009).\n12S. A. Yang, G. S. D. Beach, C. Knutson, D. Xiao, Q. Niu,\nM. Tsoi, and J. L. Erskine, Phys. Rev. Lett. 102, 067201\n(2009).\n13S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M.\nDaughton, S. von Molna, M. L. Roukes, A. Y. Chtchelka-\nnova, and D. M. Treger, Science 294, 1488 (2001); S. S. P.\nParkin, U.S. Patent 7031178 (2006); H. Dery, P. Dalal,\nL. Cywi\u0013 nski, and L. J. Sham, Nature 447, 573 (2007); and\nM. E. Flatt\u0013 e, Nature Phys. 3, 153 (2007).\n14F. D. M. Haldane, Phys. Rev. Lett. 57, 1488 (1986).\n15M. V. Berry, Proc. R. Soc. London A 392, 45 (1984).\n16This subject has recently been revived in a broader context\nof semiclassical wave-packet dynamics in the presence of\nBerry curvatures including momentum space texture dueto spin-orbit coupling and band-structure e\u000bects. See, e.g.,\nG. Sundaram and Q. Niu, Phys. Rev. B 59, 14915 (1999);\nS. A. Yang, D. Xiao, and Q. Niu, cond-mat/0709.1117;\nR. Shindou and L. Balents, Phys. Rev. B 77, 035110\n(2008).\n17T. Moriyama, R. Cao, X. Fan, G. Xuan, B. K. Nikoli\u0013 c,\nY. Tserkovnyak, J. Kolodzey, and J. Q. Xiao, Phys. Rev.\nLett. 100, 067602 (2008).\n18Y. Tserkovnyak, T. Moriyama, and J. Q. Xiao, Phys. Rev.\nB78, 020401(R) (2008).\n19P. Bruno, V. K. Dugaev, and M. Taillefumier, Phys. Rev.\nLett. 93, 096806 (2004).\n20B. Binz and A. Vishwanath, Phys. Rev. B 74, 214408\n(2006); A. Neubauer, C. P\reiderer, B. Binz, A. Rosch,\nR. Ritz, P. G. Niklowitz, and P. B oni, Phys. Rev.\nLett. 102, 186602 (2009);M. Lee, W. Kang, Y. Onose,\nY. Tokura, and N. P. Ong, ibid.102, 186601 (2009).\n21M. Viret, D. Vignoles, D. Cole, J. M. D. Coey, W. Allen,\nD. S. Daniel, and J. F. Gregg, Phys. Rev. B 53, 8464\n(1996); G. Tatara and H. Fukuyama, Phys. Rev. Lett. 78,\n3773 (1997); A. Brataas, G. Tatara, and G. E. W. Bauer,\nPhys. Rev. B 60, 3406 (1999).\n22C. H. Marrows, Adv. Phys. 54, 585 (2005).\n23C. Wickles and W. Belzig, Phys. Rev. B 80, 104435 (2009).\n24H. Kohno, G. Tatara, and J. Shibata, J. Phys. Soc. Jpn.\n75, 113706 (2006).\n25R. A. Duine, A. S. N\u0013 u~ nez, J. Sinova, and A. H. MacDonald,\nPhys. Rev. B 75, 214420 (2007).\n26Coupling of this exchange energy to the electronic density\n\ructuations can also be treated systematically. However,\nsince it does not lead to any signi\fcant e\u000bects, while at\nthe same time unnecessarily complicating our discussion,\nwe will disregard it in the following.\n27Y. B. Bazaliy, B. A. Jones, and S.-C. Zhang, Phys. Rev. B\n57, R3213 (1998).\n28H.-B. Braun and D. Loss, Phys. Rev. B 53, 3237 (1996).\n29E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part\n2, vol. 9 of Course of Theoretical Physics (Pergamon, Ox-\nford, 1980), 3rd ed.\n30Generally, the phenomenological free energy of a ferromag-\nnet is given by F[m] =A(@im)2=2 +Uani(m)\u0000Mhext\u0001m,\nwhereAis the e\u000bective sti\u000bness, Uani(m) =Uani(\u0000m)\nis an anisotropic potential due to magnetostatic and crys-\ntalline \felds, Mis the equilibrium magnetization, and hext\nis an external magnetic \feld.\n31In the more general two-band model of spin-up and spin-\ndown electrons, we would have to develop a two-\ruid ef-\nfective theory, with spin-\rip scattering between the two\n\ruid components. The phenomenology simpli\fes, however,\nreducing formally to the halfmetallic case, in the long-\nwavelength low-frequency limit: !\u001csf\u001c1 andk\u0015sd\u001c1,14\nwhere!,kare the characteristic frequency, wave number\nof the magnetohydrodynamics and \u001csf,\u0015sd/p\u001csfare the\nspin-\rip time, spin-di\u000busion length. In this limit, it may be\npossible to describe the hydrodynamic state of the system\nby the spin-texture \feld, the charge-density distribution,\nand the charge-current \feld. If any out-of-equilibrium spin\nimbalance decays su\u000eciently fast, therefore, we only need\nto retain a one-\ruid description for the charge \rows. The\nkey phenomenological modi\fcation is then to introduce a\nmaterial-dependent dimensionless \\spin-polarization\" pa-\nrameterp, such that q!pqin the following equations\nof motion. Namely, the e\u000bective charge that couples the\nelectronic particle-number \rux densities jwith the spin-\ntexture gauge \feld is renormalized by p. While in the\nhalfmetallic limit p= 1 and in normal metals p= 0, we\nmay expect some intermediate value in realistic multiple-\nband ferromagnets with fast spin relaxation.\n32To make contact with Ref. 11, de\fne the canonical current\nasJ=H\nCp\u0001dl, for an arbitrary closed curve C. Its equa-\ntion of motion is given by @tJ=qH\nCdl\u0001(v\u0002b). If the\ncurveCcoincides with a quasi-1D wire, then vkdland we\nrecover the reactive equation of Ref. 11: @tJ= 0.\n33Similarly to qthat should generally be viewed as\na material-dependent phenomenological parameters, the\nelectron mass mefrom now on is also an e\u000bective parame-ter, which is not necessarily identical with the free-electron\nmass.\n34T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n35We are not including the ordinary hydrodynamic viscosity\nin our treatment.\n36M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van\nDau, F. Petro\u000b, P. Etienne, G. Creuzet, A. Friederich, and\nJ. Chazelas, Phys. Rev. Lett. 61, 2472 (1988); G. Binasch,\nP. Gr unberg, F. Saurenbach, and W. Zinn, Phys. Rev. B\n39, 4828 (1989).\n37L. D. Landau and E. M. Lifshitz, Statistical Physics, Part\n1, vol. 5 of Course of Theoretical Physics (Pergamon, Ox-\nford, 1980), 3rd ed.\n38W. F. Brown, Phys. Rev. 130, 1677 (1963).\n39J. Foros, A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. B 78, 140402(R) (2008).\n40S. Zhang, and S. S.-L. Zhang, Phys. Rev. Lett. 102, 086601\n(2009).\n41E. M. Hankiewicz, G. Vignale, and Y. Tserkovnyak, Phys.\nRev. B 78, 020404(R) (2008); Y. Tserkovnyak, E. M. Han-\nkiewicz, and G. Vignale, ibid.79, 094415 (2009).\n42A. A. Belavin and A. M. Polyakov, JETP Lett. 22, 503\n(1975)."    },    {        "title": "2009.11708v1.Magnetic_winding__what_is_it_and_what_is_it_good_for_.pdf",        "content": "rspa.royalsocietypublishing.org\nResearch\nArticle submitted to journal\nSubject Areas:\nmagnetohydrodynamics, topological\nfluid dynamics\nKeywords:\nmagnetohydrodynamics, magnetic\ntopology, helicity, winding\nAuthor for correspondence:\nD. MacTaggart\ne-mail:\ndavid.mactaggart@glasgow.ac.ukMagnetic winding: what is it\nand what is it good for?\nC. Prior1and D. MacTaggart2\n1Department of Mathematical Sciences, Durham\nUniversity, Durham, DH1 3LE, UK\n2School of Mathematics and Statistics, University of\nGlasgow, Glasgow G12 8QQ, UK\nMagnetic winding is a fundamental topological\nquantity that underpins magnetic helicity and measures\nthe entanglement of magnetic field lines. Like\nmagnetic helicity, magnetic winding is also an\ninvariant of ideal magnetohydrodynamics. In this\narticle we give a detailed description of what magnetic\nwinding describes, how to calculate it and how\nto interpret it in relation to helicity. We show\nhow magnetic winding provides a clear topological\ndescription of magnetic fields (open or closed) and we\ngive examples to show how magnetic winding and\nhelicity can behave differently, thus revealing different\nand imporant information about the underlying\nmagnetic field.\n1. Introduction\nThe title of this paper pays homage to the now\nclassic article by Finn and Antonsen (FA) [1] which,\ntogether with the seminal work of Berger and Field (BF)\n[2], introduced relative magnetic helicity - an important\ntopological invariant of ideal magnetohydrodynamics\n(MHD). The “what is it” of FA’s title describes how\nhelicity can be defined for a magnetic field with non-\ntangential components on domain boundaries. BF show\nthat by extending the magnetic field so that it becomes\nclosed outside the domain (adding a closure), a relative\nmeasure of (gauge-invariant) helicity can be found which\ncompares two different magnetic fields with the same\nbounday conditions and closure. The general formula\nfor relative helicity, that is most widely used today, is\npresented in FA. For a (simply connected) domain \n, the\nrelative magnetic helicity HRis given by\nHR=Z\n\n(A+A0)\u0001(B\u0000B0) d3x; (1.1)\nwhereBandB0are divergence-free fields (magnetic\nfields) with the same boundary conditions on @\nwith\nc\rThe Authors. Published by the Royal Society under the terms of the\nCreative Commons Attribution License http://creativecommons.org/licenses/\nby/4.0/, which permits unrestricted use, provided the original author and\nsource are credited.arXiv:2009.11708v1  [physics.plasm-ph]  24 Sep 20202rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .B=r\u0002AandB0=r\u0002A0.\nThe “what is it good for” is described in FA for fusion applications. However, equation (1.1)\nhas been used heavily in solar physics to understand the topological properties of magnetic fields\nin the solar atmosphere. A recent review of calculating relative helicity in solar applications can\nbe found in [3].\nSolar observations cannot determine HRdirectly as it is presented in equation (1.1). This is\nbecause three-dimensional (3D) information about the magnetic field is not available throughout\nthe solar atmosphere. This information can only be found at the solar boundary, the photosphere,\nand so a full 3D magnetic field needs to be constructed based on a particular model. The chosen\nmodel is normally a force-free field [4–6], which is only strictly justified in the corona and not in\nthe photosphere. That being said, other non-force-free models have also been considered [7,8].\nAnother way that the relative helicity can be determined from solar observations is to\nintegrate the rate of change of relative helicity through the photosphere (where the magnetic field\ncomponents can be observed). As shown in BF, the rate of change of relative helicity through a\nhorizontal plane P(representing the photosphere), in ideal MHD, can be written as\ndHR\ndt= 2Z\nP[(A0\u0001B)uz\u0000(A0\u0001u)Bz]d2x; (1.2)\nwhereezis normal to P,uis the velocity of the flow and A0satisfies the following\nchosen properties: r\u0002A0\u0001ez=Bzandr?\u0001(ez\u0002A0\u0002ez) = 0 onP, wherer?refers to the\nhorizontal gradient on P.\nAlthough equation (1.2) is not the most general expression of the rate of relative helicity\nthrough a plane, it does have particular advantages that will lead us to uncover the underlying\ntopological structure of relative helicity. Before stating what these advantages are, let us quickly\nrecap some useful topological insights concerning classical helicity. Moffat [9] showed that\nclassical helicity,\nH=Z\n\nA\u0001BdV; (1.3)\nwhereB\u0001n= 0on@\n, has a topological interpretation in terms of the Gauss linking number.\nConsider two linked loops (a pair of closed and linked magnetic field lines) given by dx=ds=\nB(x)anddy=ds=B(y)(the following can be extended to nloops and also ergodic field lines,\nbut two loops will suffice for this demonstration). Then the Gauss linking number [10] is given by\nLk(x;y) =1\n4\u0019Z\nx(s)Z\ny(\u001b)dx\nds\u0001dy\nd\u001b\u0002x\u0000y\njx\u0000yj3dsd\u001b: (1.4)\nIf around each loop we identify a solid magnetic torus, as in [11], then since the magnetic field\nis tangential to the toroidal boundary, we can consider the domain to be R3, where the magnetic\nfields in the tori are ‘extended by zero’ outside of the tori volumes [12]. Thus equation (1.4) can\nlead to classical helicity formula\nH=1\n4\u0019Z\nR3Z\nR3B(x)\u0001B(y)\u0002x\u0000y\njx\u0000yj3d3xd3y: (1.5)\nEquation (1.5) is equivalent to equation (1.3) but is written in terms of a particular gauge, namely\nr\u0001A= 0. However, the purpose of “deriving” equation (1.5) in the way above is to show that\nhelicity has an underlying topological structure (here measured by the Gauss linking number)\nthat is not dependent on the magnetic field strength. The magnetic field can be constructed (at\nleast in a formal way) from the underlying geometric structure of the field lines, whose topology\nis described by Gauss linkage. Once the loops are turned into genuine magnetic field volumes,\nthe classical helicity Hbecomes a measure that combines magnetic field strength andtopological\ninformation (linking).\nReturning to equation (1.2), although it is not the most general expression of the relative\nhelicity rate (in the sense that it is based on a particular gauge), the choices made for A0reveal an\nunderlying topological structure similar to that for classical helicity described above. Berger [13]3rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .was the first to notice this underlying structure and his ideas were developed further by Prior and\nYeates [14]. We refer to this underlying topological structure as magnetic winding .\nJust as classical helicity can be thought to be based on the more fundamental property of Gauss\nlinkage, in this work, we will show that relative helicity (through equation 1.2) can be based\non the more fundamental property of winding. We show that magnetic winding can provide\nthe underlying topological structure for both open and closed magnetic fields and we will give\nexamples of how this property can be used in applications. In particular, we will show that\nthe calculation of magnetic winding in solar flux emergence can be used to identify regions of\ncomplex field line topology (see also [19,20]). This result can then be used to aid the prediction of\nsolar eruptions, which depend on non-trivial field line topology.\nThe outline of the paper is as follows: we first will define magnetic winding, which provides\nthe underlying topological descriptions for quantities to be introduced later. We then introduce\nwinding helicity and show how this provides a link bewteen winding, relative helicity and\nrelative helicity flux. We then renormalize the helicity expressions to provide analogous winding\nexperssions that are not dependent on the magnetic field strength. We then discuss properties of\nmagnetic winding and give demonstrations of how it can be used, in conjunction with magnetic\nhelicity, to understand the topological evolution of magnetic fields.\n2. Magnetic winding - what is it?\n(a) Basic definitions\nIn order to show how winding can describe the underlying topology of magnetic fields, we\nbegin with some basic definitions. Consider a Euclidean space with the standard Cartesian\nbasisfe1;e2;ezg(this notation highlights the importance of the ez-direction that will become\napparent later). Consider, further a horizontal plane Pwith normalez. Let\r: [a;b]!R2nf0gbe\na suitably smooth (Lipschitz continuity will be assumed here) plane parametric curve on Pgiven\nin Cartesian coordinates by \r(t) = (x(t);y(t)). In polar coordinates, the path can be written as\n\r(t) =r(t)(cos\u0012(t);sin\u0012(t)): (2.1)\nThe distance from the (chosen) origin is r(t) =p\nx(t)2+y(t)2and the angle \u0012(t)can be\nmultivalued for any t(by multiples of 2\u0019). In order to define \u0012(t)to be a unique function, we\nfirst choose a reference angle \u0012a(fort=a). It can be checked that\nd\u0012\ndt=_\u0012(t) =\u0000y(t) _x(t) +x(t) _y(t)\nx(t)2+y(t)2: (2.2)\nTherefore, the angle function with initial condition \u0012aand the above derivative is\n\u0012(t) =\u0012a+Zt\na\u0000y(u) _x(u) +x(u) _y(u)\nx(u)2+y(u)2du: (2.3)\nWith the angle function \u0012(t), the total (signed) change in angle of the path \ris\u0012(b)-\u0012(a). The\nwinding number can now be defined as the net number of times \rgoes around the origin, where\nanti-clockwise motion is positive and clockwise motion negative. The winding number of \r\naround the origin is thus given by\nL(\r;0) =1\n2\u0019Z\n\r\u0000ydx+xdy\nx2+y2=1\n2\u0019Zb\nad\u0012(\r)\ndtdt: (2.4)\nWhat we have described here is a standard definition of winding about a point (taken as the\norigin here although translating to another point is trivial). Now let us consider the net winding\nbetween twopaths, rather than a path relative to a specific point.\nConsider two parametric curves \r;\r0: [a;b]!R2onPthat are distinct for all t2[a;b]. For\neacht, setr(t) =\r(t)\u0000\r0(t). Therefore, the winding number between two curves is just the net4rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n(a)\n (b)\nFigure 1: (a) A domain \nconsisting of stacked horizontal slices that are orthogonal to ez. Two\ncurves (field lines) xandyare shown. Some slices are shown together with horizontal vectors r,\nat each height, connecting xandy. Magnetic field in \nis tangent to the side of the domain but\ncan be arbitrary on the top and bottom horizontal boundaries. For the calculation of helicity and\nwinding fluxes, each slice can represent the boundary at a different instant of time rather than a\nlocation in space. (b) The rvectors projected onto one horizontal plane. The resulting parametric\ncurve allows for the calculation of the pairwise winding of two field lines.\nrotation ofrabout the origin, i.e.\nL(\r;\r0) =L(r;0) =1\n2\u0019Zb\nad\u0012(r)\ndtdt: (2.5)\nIt is now straightforward to generalize the winding in equation (2.5) to describe the net rotation\nof two three-dimensional curves (which we will shortly identify with magnetic field lines)\nabout each other. Consider two smooth and distinct parametric curves x;y: [0;h]!R3that\nare monotonically increasing the ez-direction, as shown in Figure 1(a). The vertical range of\nthe curves is [0;h]. For eachz2[0;h], we define the horizontal vector r(z) =x(z)\u0000y(z). We\nthen project each r(z)for allz2[0;h]onto one horizontal plane. Due to the smoothness and\ncontinuity of the curves xandy, the resulting projection of r(z)forz2[0;h]is a suitably smooth\nplanar parametric curve, as indicated in Figure 1(b). Hence, we can make use of the expression\nfor winding given in equation (2.5) and write\nL(x;y) =1\n2\u0019Zh\n0d\ndz\u0012(x(z);y(z)) dz: (2.6)\nNote that performing the integration in equation (2.6) leads to\nL(x;y) =1\n2\u0019[\u0012(x(h);y(h))\u0000\u0012(x(0);y(0))] +n; (2.7)\nwheren2Zis the number of full rotations (in the sense described above) of the projected r\naround the origin. If the positions of the curves at z= 0andz=hremain fixed, then any smooth\ndeformation of the curves xandythat does not result in cuts or reconnection, preserves the value\nofnand, hence,L[15].\nIf we now consider the curves xandyto follow magnetic field lines, Lrepresents a topological\nconstraint on the magnetic field. Just as the Gauss linking number describes the pair-wise linkage\nof closed curves, the winding number Ldescribes the pair-wise winding of two open curves.5rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .From the above discussion, the winding represents a two-dimensional description of the\ntopology of curves (as opposed to the inherently three-dimensional description of Gauss linkage).\nAlthough this is fine for curves monotonically increasing in the ez-direction, more work is\nrequired to define an adequate measure of winding for curves that bend backwards and are\nnot monotonically increasing in the ez-direction. Berger and Prior [15] found such a measure\nby splitting the curves into regions separated by turning points. Suppose xandyhavenandm\ndistinct turning points respectively, that is points where dxz=dz= 0 (x\u0001ez=xz)ordyz=dz= 0\n(y\u0001ez=yz). Now splitxinton+ 1 regions andyintom+ 1 regions. In each region, curve\nsectionsxiandyjshare a mutual z-range [zmin\nij;zmax\nij]. Hence, in each section, equation (2.6) can\nbe applied and the total winding can be written as\nL(x;y) =n+1X\ni=1m+1X\nj=1\u001b(xi)\u001b(xj)\n2\u0019Zzmax\nij\nzmin\nijd\ndz\u0012(xi(z);yj(z)) dz; (2.8)\nwhere\u001b(xi)is an indicator function marking where the curve section ximoves up or down in z,\ni.e.\n\u001b(xi) =8\n><\n>:1 if dxz=dz>0;\n\u00001 if dxz=dz<0\n0 if dxz=dz= 0:(2.9)\nAgain, if the curves are fixed on the horizontal boundaries at z= 0andz=h, and are deformed\nsmoothly without cuts or reconnection, the generalized winding in equation (2.8) is conserved.\nIt was noted in [14,15] that if the domain containing closed curves is divided into horizonal\nslices, as we have described in this section, the Gauss linkage of these curves is equal to the\nwinding given in equation (2.8). Thus, winding can be considered to be a more extensive\ntopological description of field line entanglement than the Gauss linking number. We will now\ndescribe how winding forms the fundamental topological description of the helicity of open\nmagnetic fields.\n(b) Winding gauge\nAs mentioned earlier, relative helicity is a common measure of helicity for open magnetic fields.\nFor a restricted (but still very general) domain, the magnetic helicity of open magnetic fields can\nbe expressed without the need of a reference field [14]. Consider a simply connected domain\n\nconstructed of “stacked” horizontal planes, as portrayed in Figure 1(a). Field lines can be\nconnected to the top and bottom horizontal boundaries (or neither for closed field) and are\ntangent to the side boundaries. This domain can also be turned into the half-space by pushing the\ntop and side boundaries to infinity and assuming that the magnetic field decays suitably quickly\nwith distance from its source on the bottom plane.\nConsider a particular gauge, known as the winding gauge ,\nAW=1\n2\u0019Z\nSzB(y1;y2;z)\u0002r\njrj2d2y; (2.10)\nwherer= (x1\u0000y1;x2\u0000y2;0)andSzis a horizontal surface at height z.AWis a suitable vector\npotential for Bin\n. Also, the winding gauge satisfies r?\u0001AW= 0 on any surface Sz. This\ngauge can be thought of as a two-dimensional equivalent of the usual Coulomb gauge, which\nMoffatt used to show that the topological structure of closed-field helicity is described by Gauss\nlinkage. As its name suggests, the winding gauge leads to a description of open-field helicity\nwhose underlying topological structure is encoded in the winding of its field lines.\nPrior and Yeates [14] defined a winding helicity which can be written as\nHW(B) =Z\n\nAW\u0001BdV=1\n2\u0019Zh\n0Z\nSzZ\nSzd\ndz\u0012(x;y)Bz(x)Bz(y)d2xd2ydz: (2.11)6rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Notice from the right-hand side of equation (2.11) that the winding helicity is the average pair-\nwise winding of field lines (see equation (2.6)) weighted by magnetic flux. Again, this topological\ndescription of open-field helicity is analogous to the closed-field case (see equations (1.4) and\n(1.5)). For applications to be discussed later, it will be useful to consider magnetic helicity as\nhaving a geometric structure (field line curves) with a topological description (winding) “clothed”\nby magnetic field.\nThere are immediate consequences for the value of helicity due to the combined topology-\nfield strength description given above. First, the helicity can be zero even if field strength is large\neverywhere but the field line topology has an equal number of positive and negative windings.\nSecond, even if a field has a highly complex (non-cancelling) topology, the value of helicity can be\nsmall if the field strength is weak. From these simple observations, it would be useful to calculate\nthe field line topology as well as the helicity in order to get a better picture of the total magnetic\nfield topology. We will return to this important point later.\nA related invariant of ideal MHD is the field line helcity [16] which, for a field line curve xand\ngeneral vector potential A, is written as\nH(B) =Z\nxA\u0001dl; (2.12)\nBy applying the winding gauge, we can show that the field line winding represents the average\nwinding of all other field lines with the field line in question, weighted by magnetic flux. For\na magnetic field in the domain \n, the field line helicity of a curve x, which can be split into n\nmonotonic sections with ranges z2[zmin\ni;zmax\ni]byn\u00001vertical turning points, can be written\nas\nH=Z\nxAW\u0001dl=nX\ni=0Zzmax\ni\nzmin\niAW\u0001Bi\njBzijdz=1\n2\u0019nX\ni=0Zzmax\ni\nzmin\niZ\nSzd\ndz\u0012(xi;y)\u001b(xi)Bz(y) d2ydz:\n(2.13)\n(c) Relative helicity\nThere is always a trade-off in helicity calculations between generality and topological\ninformation. Although relative helicity (equation (1.1)) can be written in any suitable gauge, it\nis difficult to attach any clear topological interpretation to a general formulation. Although the\nwinding helicity is not general, in the sense that it is based on a specific gauge, it allows for a much\ndeeper topological interpretation compared to a more general formulation. One clear connection\nbetween winding helicity and relative helicity is that in the domain \n,\nHR(B;B0) =HW(B)\u0000HW(B0): (2.14)\nA reference field B0can always be found so that HW(B0) = 0 , thus equating the relative and\nwinding helicities [14].\nThe connection between relative helicity and winding helicity does not end there, however.\nThe practical injection of relative helicity, as mentioned in the Introduction, is performed by\nintegrating the input of helicity through a boundary in time. In ideal MHD, the time-integrated\nrate of change of the relative helicity of any magnetic field passing through a horizontal planar\nboundaryP(orS0in\n) in the time range [0;T]can be made equivalent to the winding helicity\ngiven in equation (2.11).\nIt was shown in [17,18] that the time-integrated input of relative helicity through P(from\nequation 1.2) can be written as\nHR=\u00001\n2\u0019ZT\n0Z\nPZ\nPd\ndt\u0012(x;y)Bz(x)Bz(y) d2xd2ydt: (2.15)\nThe ‘advantages’ of equation (1.2) that were mentioned earlier are now clear in equation (2.15),\nwhich reveals that the time-integrated helicity flux measures the winding of field lines in time\nweighted by magnetic flux. Equation (2.15) was the first expression found that directly connects7rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .helicity and winding [13]. With the introduction of the winding helicity above, comparing this\nwith the relative helicity flux reveals that they have a common underlying topological structure.\nUpon inspection, equation (2.15) can be transformed to equation (2.11) by identifying z$t\nand[0;T]$[h;0]. To understand why the order of integration needs to be swapped, consider\nmagnetic field (e.g. a flux tube), initially below P, which then rises rigidly through Puntil a time\nT. At timeT, the helicity of the magnetic field would be integrated, using equation (2.11), from\nz= 0up to the maximum height of the emerged magnetic field, at z=hsay. Using equation (2.15),\nhowever, the integration is performed in reverse, since the slice corresponding to the top of the\nmagnetic region is counted at Pfirst (it is the first to pass through P). This process continues until\nthe last slice, which corresponds to z= 0at timet=T.\nThe identification made above can also be considered as a mapping between \nand\nt, where\nthe latter domain is the same as the former but with the z-coordinate replaced by time. That is, \nt\ncomprises of stacks of Pat different times in the range [0;T]. If a magnetic field passes through P\nin a complex manner (e.g. a mixture of emergence and submergence in different parts of P), the\nfield integrated in \ntwill no longer be equivalent to the magnetic field in \n, although the winding\nhelicity in\nwill still be equal to the time-integrated relative helicity in \nt. As a simple example,\nconsider the emergence and then complete submergence of a magnetic loop (e.g. magnetic field in\na semi-torus). At the end of the submergence, there is no magnetic field in \nand so the winding\nhelicity is trivially zero. In \ntthe emergence and submergence of every field line creates closed\nloops which are unlinked with all other loops. Hence, the helicity is also zero in this case.\n(d) Separating winding and helicity\nSo far, we have demonstrated that, through the use of the winding gauge, the helicity of open\nmagnetic fields can be interpreted as the field line winding weighted by the magnetic flux. In\nideal MHD, however, the winding of the field lines themselves is also an invariant to deformations\nwhich vanish on the boundaries. It, therefore, makes sense to seek a purely topological measure of\nthe field lines that is independent of the field strength. In studying flux emergence, [19,20] defined\na time-integrated magnetic winding flux LR1\nLR=\u00001\n2\u0019ZT\n0Z\nPZ\nPd\ndt\u0012(x;y)\u001b(x)\u001b(y) d2xd2ydt: (2.16)\nHere, theBzterms from equation (2.15) have been replaced with corresponding indicator\nfunctions\u001b. Equation (2.16) provides topological information about the magnetic field that is\nunbiased by the magnetic field strength\nWe can also define the magnetic winding in \nby constructing a purely geometric version of\nthe winding gauge. Consider the tangent vector T=B=jBjto the magnetic field and let G=\nT=jTzj=\u001bdx=dz. We defineCto be a geometric analogue of the winding gauge as\nC=1\n2\u0019Z\nSzG(y)\u0002r\njrj2d2y: (2.17)\nAnalogous to the calculations in [14], it can be shown that\nC\u0001G=1\n2\u0019Z\nSzd\ndz\u0012(x;y)\u001b(x)\u001b(y) d2y: (2.18)\nIntegrating this quantity over the volume of \ngives\nL=1\n2\u0019Zh\n0Z\nSzZ\nSzd\ndz\u0012(x;y)\u001b(x)\u001b(y) d2xd2ydz: (2.19)\n1The symbolLis used for winding (and LRfor winding flux) due to its connection to Lk(see [15]). Capital Roman letters\nrefer to total measures of helicity and winding and calligraphic letters refer to the field line versions of these quantities. All\nfurther references to magnetic winding correspond to the definitions in this subsection.8rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Similar to the field line helicity, we can also define a field line winding\nL(B) =Z\nxC\u0001dl: (2.20)\nThe field line winding of curve x, which can be split into nmonotonic sections with ranges z2\n[zmin\ni;zmax\ni]byn\u00001turning points, can be written as\nL=1\n2\u0019nX\ni=0Zzmax\ni\nzmin\niZ\nSzd\ndz\u0012(xi;y)\u001b(xi)\u001b(y) d2ydz; (2.21)\nwhich is just the average pair-wise winding of all field lines in \nwithx. Prior and Yeates [21]\nshowed thatLcan be used to topologically categorize braided magnetic fields, where the field\nline helicity cannot. This was used to quantify the changing connectivity of magnetic flux rope\nexperiments carried out at the UCLA basic plasma facility.\n3. Winding - what is it good for?\nIn this section we will demonstrate that magnetic winding can provide different and more\ndetailed information about field line topology than the helicity.\n(a) Resistive magnetostatics\n(i) Linear force-free fields\nThe magnetic induction equation can be written as\n@B\n@t=r\u0002 (u\u0002B)\u0000r\u0002 (\u0011r\u0002B); (3.1)\nwhere\u0011is the magnetic diffusion. Only the second term on the right-hand side of equation (3.1)\ncan change the values of HWandL(and their associated field line partners HandL) so we will\nfocus on this term by considering the quasi-static evolution with u=0. We will demonstrate\nthat different components affect helicity and winding in distinct ways, thus emphasising the\nimportance of separating winding from helicity. For the sake of simplicity, we assume that \u0011is\nconstant in what follows.\nWe begin with one of the simplest possible current carrying magnetic fields, a linear force-free\nfield, that satisfies r\u0002B=\u000bBfor constant \u000b, in the domain \n(we set\u00160= 1in this work for\nconvenience). The top and bottom boundaries are fixed or periodic. The magnetic field remains\ntangent to the side boundary. The following theorem tells us that only the helicity decays for such\nfields.\nTheorem 3.1. The winding helicity and field line helicity of a linear force-free field in \nsubject to constant\nmagnetic diffusion \u0011and no flow obey\nHW[B(t)] =HW[B(0)] exp(\u00002\u000b2\u0011t) andH[B(t)] =H[B(0)] exp(\u0000\u000b2\u0011t); (3.2)\nwhere\u000bis the force-free constant. Under the same assumptions, the winding quantities L[B(t)]and\nL[B(t)]are constant.\nProof. We start by proving that L[B(t)]andL[B(t)]are constant under such an evolution. The\nquasistatic induction equation is\n@B\n@t=\u0000r\u0002 (\u0011r\u0002B): (3.3)\nJette [22] (see also [23]) proved that in resistive magnetohydrostatics, the only force-free fields\nBthat remain force-free in time are those with constant \u000b, i.e. linear force-free fields. Using the9rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .linear force-free equation,\nr\u0002 (\u0011r\u0002B) =r\u0002 (\u0011\u000bB) =\u000b2\u0011B: (3.4)\nFrom this result, the magnetic field behaves as\nB(t) =B(0) exp(\u0000\u000b2\u0011t); (3.5)\nwhereB(0)is the magnetic field at the start of the quasistatic decay . Therefore, the field line\nstructure ofBremains constant for all t>0, that isT(t) =T(0), and soL[B(t)]andL[B(t)]are\nfixed in time. Then from equations (2.11) and (2.13) and the fact that\nBz=Bz(0) exp(\u0000\u000b2\u0011t); (3.6)\nwe have\nHW[B(t)] =HW[B(0)] exp(\u00002\u000b2\u0011t) andH[B(t)] =H[B(0)] exp(\u0000\u000b2\u0011t): (3.7)\nThis particularly simple situation shows that even when there is no flow, the helicity and the\nwinding can behave differently, despite being so intimately related. Therefore, each quantity can\nprovide different information on the overall behaviour of the evolving magnetic field.\n(ii) General differences in the decay of HWandL\nWe now explore more general differences between the winding and the helicity and relate them\nto specific physical properties of the magnetic field. To do so we write the curl of the magnetic\nfield as the sum of force-free and Lorentz force ( Fl) generating components. Using Ampère’s law\nJ=r\u0002BandFl=J\u0002B, it can be checked that\nr\u0002B=B\u0002Fl\njBj2+\u000bB; \u000b =(r\u0002B)\u0001B\njBj2: (3.8)\nIn the second term, \u000brepresents the component of the axial current weighted by the field strength.\nThe parameter \u000balso represents a topological quantity: the mean twisting of the field around the\nfield line passing locally through the point at which the field Bis anchored [15]. In the first term,\nthe vectorFl=jBj2points along the direction of the Lorentz force but with a magnitude that is\nthe ratio of the Lorentz force strength to the square of the field strength. Thus we can write (3.8)\nas\nr\u0002B=\u0015Bf?+\u000bB; (3.9)\nwhere\nBf?=B\u0002^Fl;^Fl=Fl\njFlj; (3.10)\nand\n\u0015=r\u0002B\u0001Bf?\njBj2: (3.11)\nFrom a geometric perspective, \u0015measures the rotation of the magnetic field around the direction\nof the vectorB\u0002^Fl, a vector normal to both the Lorentz force and magnetic field. If we specify\nthe function \u0015then the Lorentz force can be written as\nFl=\u0015Bf?\u0002B=\u0015jBj2^Fl; (3.12)\nand\u0015can be seen to represent the relative magnitude of the Lorentz force to the magnetic field\nstrength. Thus the representation of the magnetic field’s varying local geometry through (3.9)\nhas two scalar parameters, \u0015and\u000b, which represent, respectively, the relative magnitude of the\nLorentz force and axial current and hence measure their relative effect on the local geometry of\nthe magnetic field through its curl.10rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .As shown in the Appendix, the (quasistatic) induction equation can be written in terms of\ncomponents parallel and perpendicular to the magnetic field,\n@B\n@t=\u0000\u0011\u0010\nC?+Ck\u0011\n; (3.13)\nCk=\u0010\nr\u0015\u0001^Fl+\u000b2+\u0015!b\u0011\nB; (3.14)\nC?=\u0000(r\u0015\u0001B)^Fl+\u000b\u0015Bf?+r\u000b\u0002B+\u0015!f^Fl+\u0015!flBf?; (3.15)\nwhere!brepresents the rotation of the Lorentz force vector around a field line, !frepresents\nthe rotation of the magnetic field Baround the direction of the Lorentz force and !flrepresents\nthe rotation of the pair (B;^Fl)around the direction of the field Bf?. The expressions for these\nscalars can be found in the Appendix.\nBefore discussing the interpretation of individual terms of C?andCk, we highlight the\nimportance of decomposing the vector @B=@tinto components parallel and perpendicular to the\nmagnetic field. First, the topology of the field, which can be found from the unit tangent vector\nT=B=jBjor from the vector G(thez-derivative if the curve x), is only affected by C?, i.e.\n@T\n@t=\u0000\u0011C?\njBj;@G\n@t=\u0000\u0011\u001bC?Bz\u0000(C?\u0001ez)B\nB2z; (3.16)\nwhere the second expression uses the fact we can write G=\u001bB=Bz. By contrast, the magnetic\nfield’s magnitude is only changed by Ck,\n@jBj\n@t=\u0000\u0011Ck\u0001B\njBj: (3.17)\nThese properties indicate that the change in magnetic winding is only affected byC?, whilst the\nchange in helicity depends on both C?andCk. To give compact representations of their changes\nwe define the operator\nW(N) =1\n2\u0019Z\nSzN(y)\u0002r\njrjd2y; (3.18)\nfor any vector field N, e.g.C=W(G)andAW=W(B)are examples. Then the change in\nhelicity is\n@HW\n@t=Z\n\n@B\n@t\u0001W(B) dV+Z\n\nB\u0001W\u0012@B\n@t\u0013\ndV;\n=\u0000\u0011\u0014Z\n\n(C?+Ck)\u0001W(B) dV+Z\n\nB\u0001W\u0010\nC?+Ck\u0011\ndV\u0015\n; (3.19)\nwhere we have used (2.11) in conjunction with (3.13). Similarly, using (2.18) in conjunction with\n(3.16) we find\n@L\n@t=\u0000\u0011\u0014Z\n\n\u001b(x)\u0012C?Bz\u0000(C?\u0001ez)B\nB2z\u0013\n\u0001W(G) dV\n+Z\n\nG\u0001W\u0012\n\u001b(y)C?Bz\u0000(C?\u0001ez)B\nB2z\u0013\ndV\u0015\n: (3.20)\nUsing (2.13) and (2.20) in conjunction with (3.13) and (3.16), the changes in field line helicity and\nfield line winding are\n@H\n@t=\u0000\u0011\"nX\ni=1Zzmax\ni\nzmin\ni\u001b(xi)\u0012C?iBzi\u0000(C?i\u0001ez)Bi\nB2\nzi\u0013\n\u0001W(B) dz\n+nX\ni=1Zzmax\ni\nzmin\niGi\u0001W(C?+Ck) dz#\n; (3.21)11rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .@L\n@t=\u0000\u0011\"nX\ni=1Zzmax\ni\nzmin\ni\u001b(xi)\u0012C?iBzi\u0000(C?i\u0001ez)Bi\nB2\nzi\u0013\n\u0001W(G) dz\n+nX\ni=1Zzmax\ni\nzmin\niGi\u0001W\u0012\n\u001b(y)C?Bz\u0000(C?\u0001ez)B\nB2z\u0013#\ndz: (3.22)\nNote that there is no contribution from derivatives of the integral boundaries (which would result\nfrom the Leibniz rule) as contributions cancel from connected integrals.\nWith these expressions in place, we can now review what physical effects will alter these\nquantities. By considering equations (3.13) to (3.15), the terms which change only the field strength\nand, hence, only affect the helicity are\n(i) The square of the strength of the axial current relative to the magnetic field strength: \u000b2.\n(ii) Gradients of the relative strength of the Lorentz force along the direction of the Lorentz\nforce:r\u0015\u0001^Fl.\n(iii) The rate of rotation of the Lorentz force around the curve:\n\u0015!b=\u0015r\u0002Bfl\u0001B\njBj2: (3.23)\n(sinceBf?andFform an orthogonal pair spanning the plane normal to B)\nTerms (i-ii) represent variations of the relative strength of the Lorentz force and the axial current.\nTerm (iii) accounts for the rotating geometry of the Lorentz force vector along a field line. Terms\nwhich change only the field topology , and which affect the winding, are\n(i) Gradients of the axial current normal to the magnetic field: r\u000b\u0002B.\n(ii) The product of the relative strength of the axial current and Lorentz force: \u000b\u0015. That is\nto say if both axial current and Lorentz forces are present there must be some change in\ntopology.\n(iii) Gradients of the relative strength of the Lorentz force along the direction of the magnetic\nfield:r\u0015\u0001B.\n(iv) The rotation of field Baround the direction of the Lorentz force: \u0015!f.\n(v) The rotation of the pair (B;^Fl)aroundBf?:\u0015!fl.\nTerms (i-iii) represent variations of the relative strength of the Lorentz force and the axial current.\nTerms (iv-v) account for the varying (relative) geometry of the fields BandF. These last two\ncomponents are difficult to visualize but, as we shall see in the following toy examples, are\npotentially significant in magnetic flux ropes.\n(iii) Toy examples\nWe consider simple toy examples of a radially symmetric magnetic field in a cylindrical geometry\n(i.e. a simple flux rope model). The field is assumed to take the form\nB=B\u0012(r)e\u0012+Bz(r)ez: (3.24)\nwhere (r;\u0012;z )is a cylindrical coordinate system with unit vectors fer;e\u0012;ezg,z2[0;1]andr2\n[0;rm]. Due to the cylindrical symmetry and the fact the magnetic field has no er-component, the\nfield lines are helical curves lying on concentric cylinders of radius r. The ratioBz=B\u001eis their\nhelical pitch. Thus we can study all field line topological quantities as functions of r.\nThe curl of (3.24) is\nr\u0002B=\u0000dBz\ndre\u0012+\u0012dB\u0012\ndr+B\u0012\nr\u0013\nez: (3.25)\nSince this quantity has no er-component, the Lorentz force must be directed along er-direction,\ni.e.^Fl=er. ThusBf?=B\u0002er=Bze\u0012\u0000B\u0012ez. Then, if we specify the functions \u0015(r)and\u000b(r),12rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n(a)\n (b)\nFigure 2: Distributions of quantities associated with the fields BlandBnl. (a) The axial twisting\ndistribution \u000b(r)forBnl. (b) TheBzandB\u0012components of the fields Bl(dashed) and Bnl\n(solid).\nwe can equate equations (3.25) and (3.9) to obtain the following ordinary differential equations\ndBz\ndr+\u000bB\u0012+\u0015Bz= 0;dB\u0012\ndr+B\u0012\nr\u0000\u000bBz+\u0015B\u0012= 0; (3.26)\nwhich, if solved, give the field components BzandB\u0012with the required \u0015and\u000bbehaviour. In\nthis study we consider the initial conditions B\u0012(0) = 0 andBz(0) = 1 . The condition on B\u0012means\nthat\u0015must be zero at r= 0for a valid solution.\nUsing the fact that\nr\u0002Bf?=\u0012\n\u000b\u00002B\u0012Bz\nrB2\u0013\nBf?+ \n\u0000\u0015+B2z\u0000B2\n\u0012\nrB2!\nB; (3.27)\nwe have, using equation (3.13),\n@B\n@t=\u0000\u0011\"\u0012\n2\u000b\u0015\u0000d\u000b\ndr\u0000\u00152B\u0012Bz\nrB2\u0013\nBf?+ \nd\u0015\ndr\u0000\u00152+\u000b2+\u0015B2z\u0000B2\n\u0012\nrB2!\nB#\n: (3.28)\nWe will consider cases where the field strength jBjand its axial twisting \u000bareO(1). We further\nassume that \u0015\u001cO(1)but that its gradient d\u0015=drisO(1). With these assumptions, we can\napproximate the behaviour away from the core as\n@B\n@t\u0019\u0000\u0011\u0014\n\u0000d\u000b\ndrBf?+\u0012d\u0015\ndr+\u000b2\u0013\nB\u0015\n: (3.29)\nFrom this we can see that an axial current through \u000band a Lorentz force gradient though the\nderivative d\u0015=drwill tend to alter the magnetic field strength. The field line topology, however,\ncan only change if there are gradients in the axial current. The terms\n\u00152B\u0012Bz\nrB2and\u0015B2z\u0000B2\n\u0012\nrB2; (3.30)\ncan be prominent near the core centre, even if \u0015is small (the condition that \u0015(0) = 0 means the\nsecond term is bounded). This discussion highlights the fact that small and local variations in the\ngeometry of the magnetic field and the Lorentz force can lead to significant changes in helicity and\nwinding, even if the magnitude of the Lorentz force is small. Since the the helicity and winding\nquantities involve integration over the plane, they can have a non-local effect on field lines away\nform the core.13rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n(a)\n (b)\n(c)\n (d)\nFigure 3: Field line helicity and winding distributions and their temporal changes for force-free\nfields. (a)H(r)for the fields BlandBnl. (b)L(r)for the fields BlandBnl. (c)@H=@tfor the\nfieldsBlandBnl, as a function of r. (d)@L=@tfor the fieldsBlandBnl, as a function of r. Note\nthat@L(Bl)=@t= 0for this case, as expected from Theorem 3.1.\n(iv) Force-free example\nWe present some numerical results to highlight the effect of varying the \u000band\u0015functions. First we\nconsider force-free fields ( \u0015= 0). We consider two fields on a domain r2[0;2], a linear force-free\nfield (constant \u000b)Bland a nonlinear force-free field Bnlwith a decaying axial current, specifically\n\u000b(r) =tan\u00001(20\u000017:5r) +\u0019\n2\n\u0019\n2+ tan\u00001(20): (3.31)\nThis distribution is shown in Figure 2(a). It has a region at the magnetic field’s core which\nhas almost constant axial current which then drops of sharply with a significant gradient. The\nmagnetic field components for the two cases are shown in Figure 2(b). The major feature is the\nexpected decay of the B\u0012component for the field Bnl.\nThe field line helicities H(r)for both fields are shown in Figure 3(a). In both cases there is some\nvariation but only as a relatively small percentage of the total value. There is less field line helicity\nin the nonlinear field as its twist decays where the linear case does not. The field line winding\ndistributions, shown in Figure 3(b), are qualitatively very similar.\nIn the following we set \u0011= 1. The changes @H=@tare shown in Figure 3(c) and the distributions\ndiffer significantly for both fields. In the inner core, r2[0;0:8], where the fields are very similar,\nthe distributions for the linear and nonlinear cases show a (radially) constant rate of loss. After\nthis, the gradient d\u000b=drdrives a rapid sipke in the decay of the nonlinear field’s Hdistribution,\nresulting from the changing field topology related to the helical relaxation of the field. This decay\nthen drops to zero as the twisting decays. In the linear case there is no such spike but a steady\nincrease towards the edge of the cylinder. This behaviour is due to the fact that there is a continual\n\u000b2decay and the fact that the length of the field lines increases towards the edge of the field - since14rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\n(a)\n (b)\nFigure 4: Temporal changes in the field line helicity and winding distributions for fields with\na weak and highly oscillatory Lorentz force. (a) Changes @H=@tfor the fields BlandBnl, as a\nfunction of r. (b) Changes @L=@tfor the fieldsBlandBnl, as a function of r.\nBzdecreases and B\u0012increases (Figure 2(b)) the field line curves become increasingly tightly-\ncoiled helices. Intriguingly, despite the fact that the two fields’ Hdistributions are qualitatively\nsimilar, their temporal evolution is not.\nIn Figure 3(d) we see the change in winding @L=@t, qualitatively, nearly identical to @H=@tfor\nthe nonlinear field, owing to the dominance of the topology-changing d\u000b=dtgradient. There is no\nchange in the linear force-free field as expected from Theorem 3.1.\n(v) Adding a Lorentz force\nWe now consider the effects of including a small Lorentz force, through \u0015, of the form\n\u0015= 0:05 sin(20r): (3.32)\nThis choice implies that the Lorentz force is between one and two orders of magnitude smaller\nthan the magnetic field strength but that its gradient is O(1).\nFigure 4 displays @H=@tand@L=@tafter re-solving for the fields BlandBnlwith the above\nchoice of\u0015. For brevity, we do not plot the distributions of HorLas they are very similar to\nthose in Figures 3(a) and (b). The distributions for Bnlare also almost identical to those of the\nforce-free case (these are dominated by the gradient in \u000b) and the major changes appear for Bl.\nThe@H(Bl)=@tdistrubution in Figure 4(a) has the same general trend as that in Figure 3(c) but\nwith oscillation added to it. The @L(Bl)=@tdistribution in Figure 4(b) now exhibits a constant\nnegative rate of field line winding in the inner region r2[0;1]. By reperforming the calculations\nwithout the various terms it was established this change arises due to the changing geometry and\nmagnitude of the Lorentz force along the field through the term 2\u000b\u0015in equation (3.28), which\nleads to a significant contribution from the tightly coiled curves towards the flux rope’s edge.\nA pronounced oscillatory component of @L(Bl)=@tdevelops in the outer region r>1. This is\ndue to the oscillation in the Lorentz force magnitude \u0015and the effect is magnified as field lines\nbecome increasingly tightly coiled, to the point where there is a region where the winding is being\nincreased rather than decaying.\n(vi) General conclusions\nWe have demonstrated that there can be significant differences between the evolution of the field\nline winding and helicity. Further, we have linked these differences to various physical properties\nof the field, specifically components and magnitudes of axial currents and the Lorentz force. We\nhave also given some examples of how gradients in both the axial current and weak Lorentz\nforces might affect magnetic fields with flux rope-type geometries. Of course, our analysis is only\nstrictly valid when the Lorentz force is suitably small and decoupled from the fluid pressure\ngradient and inertia terms. Such a situation would be valid, for example, in the solar corona (at15rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .least before the onset of significant plasma motion). For a discussion of the effects of non-magnetic\nterms in quasistatic decay, the reader is directed to [24].\n(b) Flux emergence and submergence\nThe emergence of magnetic field into the solar atmosphere is one of the main drivers of\nsolar activity. It has long been appreciated that helicity is important for solar eruptions, so\nunderstanding how topologically complex field emerges into the solar atmosphere provides\nus with information on what kinds of eruptions can form. Recently, MacTaggart and Prior\n[19,20] performed a detailed analysis on how helicity and winding is transported into the solar\natmosphere in flux emergence simulations. In order to understand what topological information\nis being transported by emergence, we now present a simple but informative model of flux\nemergence that illustrates one of the main results of [19,20].\n(i) Magnetic field construction\nEmerging magnetic fields in the solar atmosphere have two footpoints (sunspots) which move\napart from each other until a certain distance. This behaviour suggests that the basic geometry of\nthe emerging magnetic field is toroidal [25]. To construct a toroidal flux tube, we need to define a\nmagnetic field in a toroidal domain. That is, we wish to create a divergence-free field with a field\nline structure of our choosing in a toroidal domain.\nTo construct such a field, we will consider transforming a magnetic field defined in a cylinder\nto one in a semi-torus. The reason for this is that it is much easier to define magnetic flux tubes of\narbitrary complexity in cylinders.\nFor a toroidal flux tube, the axis of the tube can be written as the parametric curve\nr(s) =\u0000Rcos(s=R)e1+ (Rsin(s=r) +z0)ez; (3.33)\nwheresis the arclength along the tube axis, Ris the major radius of the torus and z0is the\nheight at which the footpoints of the tube meet the photosphere. The unit tangent vector of r(s)is\nT= dr(s)=dsand by taking the (unit) normal and binormal vectors, d1andd2say, we can define\nan orthonormal triad fT;d1;d2g. This basis can be used to define a tubular coordinate system\nthrough the mapping\nf(s;x1;x2) =r(s) +x1d1+x2d2; (3.34)\nwith\nd1= cos(s=R)e1\u0000sin(s=R)ez;d2=e2:\nThe metric tensor can be written as\ngij=@f\n@qi\u0001@f\n@qj; (3.35)\nwherei;j= 1;2;3withq1=s,q2=x1andq3=x2. The Jacobian of the mapping is given by\npg=q\ndet(gij) =R\u0000x1\nR: (3.36)\nIfBrepresents a magnetic field, it must obey the solenoidal constraint, i.e.\n@\n@qi(pgBi) = 0: (3.37)\nIt follows, therefore, that any magnetic field defined in a cylinder can be converted to one in terms\nof the above tubular coordinates by simply dividing the components bypg.\nFor our application here, we will consider a twisted flux tube that has a small concentration of\neven higher twist at the centre of the tube, localized at the apex. To construct such a magnetic flux16rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .tube, we first write the magnetic field as\nB=Bs@f\n@s+B1@f\n@x1+B2@f\n@x2: (3.38)\nThen, following [19], the magnetic field with twist at its centre (x1;x2) = (0;0)can be written as\nB=b0pg\u00142\nwexp\u0012\n\u0000x2\n1+x2\n2\nw2\u0000(z\u0000z0)2\nl2\u0013\n+tw\u0015\u0012\n\u0000x2@f\n@x1+x1@f\n@x2\u0013\n+b0pg@f\n@s; (3.39)\nwhereb0is the axial field strength, twis the background twist and wandlcontrol the size of the\nlocalized twist region. The flux tube is then defined as the expression given in equation (3.39) for\nx2\n1+x2\n2\u0014a2and zero for x2\n1+x2\n2>a2. For the choice b0= 5,w= 0:9,l= 0:1andtw= 0:1, the\nmagnetic flux tube is visualized in Figure 5. Blue field lines illustrate twisted field lines near the\nboundary of the tube. Red field lines illustrate the localized region of strong twist at the centre of\nthe tube at the apex.\nFigure 5: A toroidal shaped magnetic flux tube. Blue field lines are traced at the edge of the tube.\nRed field lines are traced a the centre of the tube and show a localized region of strong twist near\nthe apex of the tube. The plane is at ‘height’ z0, on which white indicates positive magnetic field\nand black, negative.\n(ii) Helicity and winding inputs\nTo mimic emergence and submergence, we push the magnetic field defined by equation (3.39)\nthrough a horizontal boundary. For the emergence phase, we push the (rigid) magnetic flux tube\nthrough the photospheric plane until this plane is coincident with z0(as illustrated in Figure 5).\nFor the submergence phase, the magnetic field is pulled back down through the photosphere\nat the same rate as it emerged. The submergence phase stops before the strong twist region,\nillustrated in Figure 5 by the red field lines, submerges beneath the photosphere.\nWe calculate HRandLRvarying in time and normalize these quantities by their maxima, for\nease of comparison. In determining these quadratures we impose a cut-off of jBzj= 0:01- any\nvalue beneath this cut-off is ignored to avoid numerical errors in the winding calculation [19,20].\nThe results are displayed in Figure 6.\nFocussing on the winding input in the emergence phase first, there is sharp increase followed\nby a levelling out after t\u00190:5. The sharp rise is dominated by the detection of the localized region\nof strong twist. After this region has passed through the photosphere, the remaining twist in the\nemerging magnetic field is much weaker and its effect on the winding is thus much weaker. The17rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .\nFigure 6: The helicity and winding time-integrated fluxes. Each quantity is weighted by its\nmaximum value for presentation purposes. The vertical line separates the emergence and\nsubmergence phases.\nhelicity input in the emergence phase also shows an increase in time, with the gradient reaching\na maximum just before t\u00190:5and then taking a smaller and almost constant value. The part of\nthe curve with the highest gradient corresponds to the localized region of strong twist passing\nthough the photosphere. The rest of the curve, with lower gradients, corresponds to the more\nweakly twisting field passing through the photosphere. Since helicity is weighted by magnetic\nflux, the identification of the input of topologically complex magnetic field above the photosphere\nis not as simple to detect as it is for the winding input.\nIn the submergence phase, where part of the flux tube is pushed back down through the\nphotosphere, the behaviour of the helicity and the winding is very different. The part of the\nflux tube that is pushed back down through the photosphere corresponds to the more weakly\ntwisted field (the blue field lines in Figure 5). Since the helicity is weighted by magnetic flux,\neven if the topological complexity of the field lines is weak in this region, the helicity can still be\nlarge. Therefore, a large reduction is seen in the helicity due to submergence. For the winding,\nhowever, we see very little change due to submergence. This is because the winding is not biased\nby magnetic flux and so if only weak field line topology submerges, only a marginal decrease in\nthe winding is recorded.\nThis example has been set up to show that magnetic winding can identify particular regions of\ntopological complexity which the helicity cannot. Thus a combination of winding and helicity can\nprovide a more complete picture of the structure of emerging magnetic fields and, importantly,\nboth quantities can be calculated in observations. More realistic emergence studies in [19,20]\nfurther demonstrate the potential of the winding input rate time series as a metric for immediate\nevent detection, the event being the emergence/submergence of sub-regions of highly twisted\nfield emerging into the corona.\n(c) Influence of a moving boundary\nSo far, we have considered winding and helicity fluxes through a stationary and flat boundary P\nthat represents the solar photosphere. It is likely, however, that in solar observations, the magnetic\nfield components recorded in planar magnetograms are actually at different heights. That is,\nthe photosphere is likely to be a moving and non-uniform surface. In that case, what are the\nconsequences for the formulae that we have presented? For the magnetic winding flux, we will\nshow that it is independent of the geometry of a moving boundary and so the calculations using a\nflat plane can still be used in this situation.\nFor the application of flux emergence through the photosphere, we will consider a\nphotospheric boundary whose projection onto a horizontal plane is one to one, as in Figure 7.18rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .We can consider more complicated boundary surfaces that fold back on themselves by defining\ncoordinate systems on them and mapping them to flat planes viaconformal (angle-preserving)\nmappings. This process, however, is not necessary for the applications we have in mind and the\nfollowing (simpler) description will be suitable for our purposes.\nP0\nPezezN\nFigure 7: A representation of a non-uniform boundary P0(which can also change in time) and the\nstandard flat boundary P. Horizontal surfaces are orthogonal to ezandNrepresents the normal\ntoP0, which is different at different locations. Two points on P0are highlighted together with\ntheir projections onto P.\nLetP0represent a non-uniform simple surface that evolves in time. As shown in Figure 7, P0is\nextended so that its boundary is horizontal. The purpose of it is to define an axis, the vector ezin\nthis case, about which the winding of field lines is measured. Since winding is a two-dimensional\n(2D) measure, it requires a specific fixed normal vector. The vector Nwould not be suitable as it\nchanges from point to point and N\u0011ezonly near the boundary of P0.\nConsider two points on P0, as shown in Figure 7. These two points (and any others on P0) are\nlocations where field lines can intersect the surface. This could be an instantaneous moment of\nflux emergence, the movement of the boundary itself or a combination of both. The non-uniform\nboundary determines what part of the field line is used to calculate winding but its geometry\ndoes not enter explicitly into the winding rate calculation. This fact is simply because the relative\nangle of these points about a given origin is a purely 2D calculation, as is evident from\nd\u0012\ndt=d\ndtarctan\u0014(x\u0000y)\u0001e2\n(x\u0000y)\u0001e1\u0015\n=ez\u0001(x\u0000y)\njx\u0000yj2\u0002\u0012dx\ndt\u0000dy\ndt\u0013\n; (3.40)\nwherexandyare horizontal vectors. Thus, to calculate the winding rate, every point on P0can\nbe projected orthogonally onto Pand the standard formula can be used, i.e.\ndLR\ndt=\u00001\n2\u0019Z\nP0Z\nP0d\ndt\u0012(x;y)\u001b(x)\u001b(y) d2xd2y=\u00001\n2\u0019Z\nPZ\nPd\ndt\u0012(x;y)\u001b(x)\u001b(y) d2xd2y:\n(3.41)\nIn this sense, the winding rate through P0does not depend explicitly on the geometry of P0. The\ngeometry of P0selects what part of the field line is recorded but does not add any weighting to\nthe winding rate calculation.\nThe input of magnetic helicity through the photosphere is dependent on the shape of P0. This\nis simply because helicity depends of the magnetic flux through P0which depends on the shape\nofP0. Despite this dependency, however, calculations of helicity flux through different moving\nboundaries in flux emergence simulations [20] have demonstrated that the qualitative behaviour\nof helicity input in time is not strongly affected by the choice of boundary (assuming it is not\nplaced unrealistically far from the photosphere region).19rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4. Summary\nIn this work we have shown that magnetic winding is the topological underpinning of magnetic\nhelicity for open magnetic fields. In essence, magnetic winding can be thought of as helicity\nwithout the magnetic flux weighting, thus representing a more fundamental topological quantity.\nThrough examples of quasistatic resistive decay and flux emergence, we have demonstrated\nthat, despite their close connection, helicity and winding can behave differently. Therefore,\nmagnetic winding can provide different, and more detailed, information on field line topology\nthan magnetic helicity. Due to the similarity, in terms of its mathematical expression, of winding\nto helicity, there are exciting opportunites to use winding practically, in both simulations and\nobservations, to provide a deeper understanding of magnetic field topology.\nAppendix\nThe quasistatic induction equation is defined by the following identity:\nr\u0002 (\u0015Bf?+\u000bB) =r\u0015\u0002Bf?+\u0015r\u0002Bf?+r\u000b\u0002B+\u000br\u0002B: (4.1)\nThe aim is to decompose this into components parallel to the field and those perpendicular, which\nmust either be along the direction of Bf?or along the direction of the Lorentz force ^Fl. To do so\nwe use the following identities:\n\u000br\u0002B=\u000b\u0015Bf?+\u000b2B; (4.2)\n(r\u0015\u0002Bf?)\u0001Bf?= 0; (4.3)\n(r\u0015\u0002Bf?)\u0001B=r\u0015\u0001(Bf?\u0002B) =jBj2r\u0015\u0001^Fl; (4.4)\n(r\u0015\u0002Bf?)\u0001^Fl=\u0000r\u0015\u0001B: (4.5)\nFinally we consider the vector r\u0002Bf?. This is the rotation of the vector normal to both the\nLorentz force and the magnetic field. We can express this vector in the local orthogonal basis\n(B;^F;Bf?)as follows:\nr\u0002Bf?=!bB+!f^Fl+!flBf?; (4.6)\nwhere\n!b=r\u0002Bf?\u0001B\njBj2(4.7)\nrepresents the rotation of the Lorentz force vector around the field line,\n!f=r\u0002Bf?\u0001^Fl; (4.8)\nrepresents the rotation of field Baround the direction of the Lorentz force, and\n!fl=r\u0002Bf?\u0001Bf?\njBj2; (4.9)\nrepresents the rotation of the pair (B;^Fl)around the direction of the field Bf?.\nEthics. Not applicable.\nData Accessibility. This article has no additional data.\nAuthors’ Contributions. CP contributed to mathematical modelling, performed numerical calculations,\nplotted the figures and reviewed the manuscript; DM contributed to mathematical modelling, performed\nnumerical calculations and drafted the manuscript. Both authors approved the final version and agree to be\naccountable for all aspects the work.\nCompeting Interests. No competing interests.\nFunding. Not applicable.\nAcknowledgements. We thank Anthony Yeates for helpful discussions.20rspa.royalsocietypublishing.org Proc R Soc A 0000000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .References\n1. Finn J, Antonsen Jr T. 1985 Magnetic helicity: What is it and what is it good for?. Comments on\nPlasma Physics and Controlled Fusion 9, 111–126.\n2. Berger MA, Field GB. 1984 The topological properties of magnetic helicity. Journal of Fluid\nMechanics 147, 133–148.\n3. Pariat E. 2020 Using Magnetic Helicity, Topology and Geometry to Investigate Complex\nMagnetic Fields. In MacTaggart D, Hiller A, editors, Topics in Magnetohydrodynamic Topology,\nReconnection and Stability Theory (CISM Series) vol. 591 pp. 145–176. Springer.\n4. Wheatland M, Sturrock P , Roumeliotis G. 2000 An optimization approach to reconstructing\nforce-free fields. The Astrophysical Journal 540, 1150.\n5. Guo Y, Ding M, Liu Y, Sun X, DeRosa M, Wiegelmann T. 2012 Modeling magnetic field\nstructure of a solar active region corona using nonlinear force-free fields in spherical geometry.\nThe Astrophysical Journal 760, 47.\n6. Wiegelmann T, Sakurai T. 2012 Solar force-free magnetic fields. Living Reviews in Solar Physics\n9, 5.\n7. Wiegelmann T, Neukirch T, Nickeler DH, Solanki SK, Barthol P , Gandorfer A, Gizon L,\nHirzberger J, Riethmüller TL, van Noort M, Blanco Rodríguez J, Del Toro Iniesta JC, Orozco\nSuárez D, Schmidt W, Martínez Pillet V , Knölker M. 2017 Magneto-static Modeling from\nSunrise/IMaX: Application to an Active Region Observed with Sunrise II. The Astrophysical\nJournal Supplement Series 229, 18.\n8. Neukirch T, Wiegelmann T. 2019 Analytical Three-dimensional Magnetohydrostatic\nEquilibrium Solutions for Magnetic Field Extrapolation Allowing a Transition from Non-\nforce-free to Force-free Magnetic Fields. Solar Physics 294.\n9. Moffatt HK. 1969 The degree of knottedness of tangled vortex lines. Journal of Fluid Mechanics\n35, 117–129.\n10. Ricca RL, Nipoti B. 2011 Gauss’ linking number revisited. Journal of Knot Theory and Its\nRamifications 20, 1325–1343.\n11. Ricca RL. 2008 Topology bounds energy of knots and links. Proceedings of the Royal Society of\nLondon Series A 464, 293–300.\n12. MacTaggart D, Valli A. 2019 Magnetic helicity in multiply connected domains. Journal of\nPlasma Physics 85, 775850501.\n13. Berger MA. 1986 Topological invariants of field lines rooted to planes.. Geophysical and\nAstrophysical Fluid Dynamics 34, 265–281.\n14. Prior C, Yeates A. 2014 On the helicity of open magnetic fields. The Astrophysical Journal 787,\n100.\n15. Berger MA, Prior C. 2006 The writhe of open and closed curves. Journal of Physics A:\nMathematical and General 39, 8321.\n16. Berger M. 1988 An energy formula for nonlinear force-free magnetic fields. Astronomy and\nAstrophysics 201, 355–361.\n17. Berger MA. 1984 Rigorous new limits on magnetic helicity dissipation in the solar corona.\nGeophysical and Astrophysical Fluid Dynamics 30, 79–104.\n18. Pariat E, Démoulin P , Berger MA. 2005 Photospheric flux density of magnetic helicity.\nAstronomy & Astrophysics 439, 1191–1203.\n19. Prior C, MacTaggart D. 2019 Interpreting magnetic helicity flux in solar flux emergence.\nJournal of Plasma Physics 85.\n20. MacTaggart D, Prior C. 2020 Helicity and winding fluxes as indicators of twisted flux\nemergence. Geophysical & Astrophysical Fluid Dynamics pp. 1–40.\n21. Prior C, Yeates AR. 2018 Quantifying reconnective activity in braided vector fields. Physical\nReview E 98, 013204.\n22. Jette AD. 1970 Force-free magnetic fields in resistive magnetohydrostatics. Journal of\nMathematical Analysis and Applications 29, 109–122.\n23. Lo Surdo C. 1979 Pseudostatic Resistive Hydromagnetic Equilibria in a Toroid. Journal of\nApplied Mathematics and Physics (ZAMP) 31, 647–653.\n24. Low B. 1974 Resistive diffusion of force-free magnetic fields in a passive medium. IV-The\ndynamical theory. The Astrophysical Journal 193, 243–252.\n25. Hood AW, Archontis V , MacTaggart D. 2012 3D MHD Flux Emergence Experiments: Idealised\nModels and Coronal Interactions. Solar Physics 278, 3–31."    },    {        "title": "1803.01750v1.Generation_and_stability_of_dynamical_skyrmions_and_droplet_solitons.pdf",        "content": "Generation and stability of dynamical skyrmions and droplet\nsolitons\nNahuel Statuto,1, 2Joan Manel Hern\u0012 andez,2Andrew D. Kent,3and Ferran Maci\u0012 a1,\u0003\n1Institut de Ci\u0012 encia de Materials de Barcelona (ICMAB-CSIC),\nCampus UAB, 08193 Bellaterra, Spain\n2Dept. of Condensed Matter Physics,\nUniversity of Barcelona, 08028 Barcelona, Spain\n3Center for Quantum Phenomena, Department of Physics,\nNew York University, New York, New York 10003 USA\n(Dated: March 6, 2018)\nAbstract\nA spin-polarized current in a nanocontact to a magnetic \flm can create collective magnetic\noscillations by compensating the magnetic damping. In particular, in materials with uniaxial\nmagnetic anisotropy, droplet solitons have been observed a self-localized excitation consisting of\npartially reversed magnetization that precesses coherently in the nanocontact region. It is also\npossible to generate topological droplet solitons, known as dynamical skyrmions . Here we study\nthe conditions that promote either droplet or dynamical skyrmion formation and describe their\nstability in magnetic \flms without Dzyaloshinskii-Moriya interactions. We show that Oersted\n\felds from the applied current as well as the initial magnetization state can determine whether\na droplet or dynamical skyrmion forms. Dynamical skyrmions are found to be more stable than\ndroplets. We also discuss electrical characteristics that can be used distinguish these magnetic\nobjects.\n1arXiv:1803.01750v1  [cond-mat.mes-hall]  5 Mar 2018The control of magnetic states in nanostructures without using magnetic \felds is now pos-\nsible with the discovery of the spin-transfer torque (STT) e\u000bect1{3. A spin-polarized current\ncan transfer angular momentum to a magnetic material4and modify its magnetization. The\nSTT e\u000bect is used in the control of both static and dynamic magnetic states; one can switch\nthe magnetization direction of a magnetic layer within a nanopillar or create coherent spin\nwaves in an extended thin \flm5. In particular the STT e\u000bect can be used to nucleate and\ncontrol solitonic modes|magnetization states that behave as particles. These self-localized\nmagnetic objects include magnetic domains, vortices, bubbles, or skyrmions6{13and they\nare receiving a growing interest since they can be topological and, thus, more stable against\nperturbations such as thermal \ructuations or fabrication defects14{16. Besides the possibil-\nity of nucleation and control of static solitonic modes, the STT e\u000bect is also used to excite\ntheir dynamical counterparts consisting in oscillating modes that are unstable in dissipative\nmaterials|damping is present in all magnetic materials and suppresses these excitations.\nHowever, damping can be now compensated locally by the STT e\u000bect, for example, with an\nelectrical point contact providing a spin-polarized current17{20.\nDissipative magnetic droplet solitons (droplets hereafter) are nonlinear localized wave\nexcitations consisting of partially reversed precessing spins that can be created in \flms with\nperpendicular magnetic anisotropy (PMA)21. Droplets have been experimentally created\nusing the STT e\u000bect in electric nanocontacts to PMA \flms22{28. Droplets are magnetic nano-\noscillators and have a growing interest as key elements in neuromorphic computation29,30\nand in communication devices31. Droplets are topologically trivial objects|they can be\ncreated continuously from a uniform ferromagnetic state where all spins are aligned in the\nsame direction. A similar magnetic object having topologically non-trivial spin texture could\nbe created in a similar experimental geometry: a dynamical skyrmion (DS). Zhou et al.32\nhave shown with micromagnetic simulations that DS can be nucleated and sustained with\na spin-polarized current in a nanocontact and are, indeed, fundamental solutions for the\nmagnetization excitations in a \flm with PMA33. Liu et al.34presented an experimental\nobservation of a solitonic mode modulation that could indicate the existence of a DS. So\nfar, the topology modi\fcation of droplets has been associated to the Dzyaloshinskii-Moriya\ninteraction (DMI) present in some magnetic \flms35.\nA schematic plot of both solitonic modes, droplet and DS, is shown in Fig. 1 where the\nblue region represents magnetization pointing out-of-plane and the brown, in the opposite\n2direction. The magnetization of a droplet or a DS is precessing, with a small amplitude\nnear its center and with a larger amplitude at the boundaries. The lower panels of Figs.\n1a and 1b show the magnetization orientation in a transversal cut of both droplet and\nDS. The main di\u000berence is in the region separating the center of soliton from the rest\nof \flm's magnetization; droplets have no topology (the magnetization shown in Fig. 1a\ncan be transformed continuously into a ferromagnetic state with all spins aligned in any\narbitrary direction) whereas the DS has topology (such a transformation is not possible).\nThe topology in two dimensions can be described by the skyrmion number ( S), which is\ncalculated mathematically as S=\u00001\n4\u0019Z\nm\u0010\n@xm\u0002@ym\u0011\n. A droplet has S= 0, and a DS\nhasS= 1.\nHere, we investigate the conditions that lead to either droplet or DS formation and we\nstudy their stability in nanocontacts to ferromagnetic thin \flms with PMA and without\ninterfacial DMI. Our micromagnetic simulations show that the Oersted \felds associated\nwith the localized electrical current, the initial magnetization state, and the rise time of\nthe injected current, play a key role on determining whether droplet or DS form. DS are\nmore stable to perturbations and can be sustained with much lower currents than droplets.\nWe also provide characteristic features of droplets and DS that could distinguish the two\nmagnetic objects experimentally.\nMagnetic film\nDS\nDW DW DW DWa b\nDropletMagnetic film\nFIG. 1. Schematic representation of droplet (a) and dynamical skyrmion (DS) (b)\nmagnetic con\fguration . Magnetization within droplet or DS is reversed with respect to the\n\flm's magnetization and is precessing with a small amplitude at the center and with a larger\namplitude at the boundaries. Lower panels show a transversal cut of the spin con\fguration for the\ndroplet (S= 0) in a) and DS ( S= 1) in b).\n3RESULTS\nSimulations details\nWe consider a circular nanocontact to a ferromagnetic thin \flm with PMA. The param-\neters for the material are taken from experiments using Co and Ni multilayers25,36. Magne-\ntization saturation, Ms= 5\u0002105A/m, damping constant, \u000b= 0:03, uniaxial anisotropy\nconstant,Ku= 2\u0002105J/m3, exchange sti\u000bness constant, A= 10\u000012J/m, and a nanocon-\ntact diameter of 150 nm for most of the presented results. We modeled the magnetization\ndynamics in the nanocontact by solving the Landau-Lifshitz equation adding the STT term1\nwith a constant spin polarization. We performed micromagnetic simulations using the open-\nsource MuMax code37using a graphics card with 2048 processing cores. We considered the\ne\u000bects of Oersted \felds but we did not include interfacial DMI or temperature e\u000bects (full\ncodes are available in Supplementary Materials).\nCreation Process\nTo excite a droplet in a ferromagnetic layer with PMA using a spin polarized current in\na nanocontact, the spin-transfer torque must compensate the damping. There is a thresh-\nold current that depends on the NC size, the magnetization, the spin polarization of the\ncurrent, and the external \feld21,26,28,38. For currents above the threshold, the magnetization\nin the NC forms a droplet state in a process that can take less than a nanosecond39. Once\nthe droplet is created, the current in the nanocontact is still required to sustain the mag-\nnetic excitation|although smaller current values than the threshold current are needed22,25.\nDroplet states can be inferred by measuring the dc resistance of the nanocontact|a rever-\nsal of the magnetization produces a change in the nanocontact resistance22{28. Further, the\nmagnetization dynamics of droplets can be detected experimentally through the ac electrical\nresistance oscillations in the nanocontact22{25,27,28caused by the precessing magnetization\nin the droplet.\nDS may also form in a NC to a ferromagnetic layer with PMA32,34when a su\u000eciently large\ncurrent is applied. The di\u000berence between droplet and DS is in the topology of the spins on\nthe boundary that might provide additional stability. For this reason, we are interested in\ndetermining the di\u000berences in stability between droplet and DS as well as the experimental\n410151719\nCurrent (mA)f (GHz) Droplet DS\n01S\n-101\nI  = 30 mA\nθI = 2.5º\nθI = 0.8º\n1 2 3\nTime (ns)θI  = 1.5º\nI  = 26 mA \nI  = 36 mA \n-101 mz θI = 1.5º\n30 40 20\n0.10.30.5\nAmplitude\n0 deg±180 dega b\ncmzFIG. 2. Droplet and dynamical skyrmion creation process . a) Resonance frequency (in\nblue dots) and amplitude (red dots) as a function of the applied current for the nanocontact overall\nmagnetization. Both frequency and amplitude correspond to the average over the nanocontact of\none of the in-plane components of the magnetization, mx;y. The current values are always applied\nfrom a same initial magnetization angle in an applied \feld of 0.5 T and with a polarization of\np= 0:45. At current values below the threshold (below 10 mA) the nanocontact magnetization\nprecesses close to the ferromagnetic resonance frequency with a small amplitude. A \frst current\nthreshold at 10 mA corresponds to a droplet formation and shows a much larger amplitude (red\ncurve) and a frequency jump down to a lower value|that remains almost constant with increasing\nthe applied current. A second current threshold at 28 mA corresponds to the DS formation and\nhas a similar precession frequency and a smaller amplitude. The bottom panel show the skyrmion\nnumber,S, at each current step. b) and c) Time evolution of the normalized magnetization inside\nthe NC for droplet (yellow line) and DS (red line) for the same external applied \feld of 0 :5 T. In\nb) both solitons are excited at an initial magnetization angle, \u0012I= 1:5\u000ebut using di\u000berent applied\ncurrents. In c) both solitons are exited at 30 mA but changing the initial magnetization angle.\nconditions under with DS form.\nIn simulations we choose an initial magnetization that is close to equilibrium|all spins\naligned with the applied \feld|and then we apply a spin-polarized current and record the\nevolution of the magnetization in an area that is 5 times larger than the contact diameter.\nFigure 2a shows the magnetization precession frequency within the NC as a function of the\napplied spin-polarized current under an applied \feld of 0.5 T.\nFor values of current below 10 mA the magnetization in the NC (the average value) has a\nsmall oscillation with a frequency close to the ferromagnetic resonance frequency. Above 10\n5mA there is an abrupt decrease of the frequency together with an increase of the precession\namplitude, which corresponds to the creation of a droplet. If we continue applying current\nof larger amplitude (always starting from a same initial state), we reach a second threshold\nat 28 mA where DS form, having an almost identical magnetization precession frequency\n(blue dots) but a much smaller precession amplitude (red dots). The precession amplitude\nof spins is much larger at the boundary of the soliton than in the central part. Thus,\nthe average nanocontact precession amplitude is mostly driven by the edge precession. In\nDS the spins at the boundary precess at a similar amplitude than in droplets but the fact\nthat they are not in phase causes a cancellation of the e\u000bect when measuring the average\ncontact electrical characteristics|which is a feature to identify DS experimentally. The\nsame arguments applies to describe the smooth decrease in the precession amplitude of the\nNC magnetization in the droplet state as the current increases from 10 mA to 28 mA; the\nphase of droplet becomes less and less uniform along the overall droplet edge when increasing\nthe applied polarized current39.\nThe second threshold was also identi\fed by Zhou et al.32through mciromagnetic simula-\ntions where a DS was excited from an initial ferromagnetic state obtained after a relaxation\nprocess. A relaxation process leads to a state with magnetization almost perpendicular to\nthe \flm with an angle \u0012I\u00190\u000e. In our simulations we indeed study the e\u000bect of initial\nmagnetization states on the formation of droplet and DS. The initial magnetization angle,\n\u0012I, is \fxed and treated as a parameter in simulations. Figure 2a is done using an initial\nstate with\u0012I= 1:5\u000e.\nWe next study the time evolution of magnetization during the process of droplet and DS\nformation. Figure 2b shows the magnetization evolution in the NC region, mz, as a response\nof an applied current for a droplet (yellow line) at 26 mA and for the DS (red line) at 36\nmA; both time traces correspond to points in Fig. 2a having an initial magnetization state\nwith\u0012I= 1:5\u000e. We see that the higher applied current has a faster magnetization reversal,\nwhich is something that occurs no matter whether the \fnal state is a droplet or a DS and is\ncaused by a larger STT e\u000bect{which is proportional to the applied current39. We can also\nobserve that the DS (red line) presents a larger oscillation of the magnetization indicating\nthere is a breathing of the localized object at the precession frequency32,34. We note here\nthat the magnetization mzaverage over the NC (plotted in Fig.2b) is a relevant quantity\nfor experiments as it can be directly associated to the NC resistance.\n6The initial state determines whether the response to an applied current is a droplet or\na DS. In Fig. 2c we plot time traces for the magnetization, mz, in the NC for a same\napplied current, 30 mA, but di\u000berent initial magnetization states, \u0012I= 2:5\u000e(yellow line)\nand\u0012I= 0:8\u000e(red line). We note that the initial state with \u0012I= 2:5\u000eevolves to a droplet\nstate whereas the initial state with \u0012I= 0:8\u000eevolves to a DS state. The current threshold\nfor DS formation thus has a dependence on the magnetization initial state. Additionally,\nwe measured the precession frequency of droplet and DS for the case presented in Fig. 2c.\nFor the same current both the droplet and DC have nearly the same precession frequency:\nf= 14:60 GHz for droplet and f= 14:58 GHz for DS, which is not seen in the transition\nat 27 mA of Fig. 2a due to the small di\u000berence. We attribute such a small variation in\nfrequency to the small changes in the size of the magnetic object and therefore in the value\nof internal magnetic \felds|mainly dipolar \felds.\nIn order to understand how the threshold current for DS formation depends on the initial\nmagnetization state, we repeat the process used in Fig. 2a with di\u000berent initial magnetization\nstates (di\u000berent of \u0012I) and we identify the current that result in a droplet or a DS. Figure 3a\nshows the phase diagram of droplet and DS formation as a function of the applied current and\nthe initial magnetization angle. We see that the threshold for droplet formation is always the\nsame independent of the initial magnetization state; di\u000berent initial states cause the process\nof droplet formation to become faster or slower (see traces for time evolution in the insets\nof Fig. 3a)39. On the other hand, the threshold for DS formation has a strong dependence\non the initial magnetization angle, \u0012I, increasing with larger angles. An additional map\nis provided in the Supplementary materials showing the phase diagram of droplet and DS\nformation as a function the polarization of the applied current and the initial magnetization\nangle (\u0012I) for a \fxed current of 30 mA. In that case the Oersted-\feld e\u000bects are \fxed and\nonly STT e\u000bects vary with spin-polarization. At a small polarization, there is a small STT\ne\u000bect and no excitations are present independently of the initial values of magnetization.\nAs the current polarization increases we found \frst the onset of droplet states and with a\nfurther increase the onset of DS. Again the droplet threshold does not depend on the initial\nstate whereas the DS has a strong dependence requiring larger values of polarization at\nlarger angles of the initial magnetization angle, \u0012I.\nWe have used a polarization of p= 0:45 for the phase diagram of Fig. 3 but a di\u000berent\nvalue would shift both droplet and DS thresholds. An increase of polarization from p= 0:45\n7top= 0:6 produces a shift of 2 mA in the droplet threshold and a shift of 5 mA in the\nDS threshold. The contact size determines the net current required to excited solitonic\nmodes. We computed droplet and DS thresholds for contact diameters of 50 and 100 nm\nand obtained values of 4 and 6 mA for the droplet threshold|which represents a decrease\nof 3 and 5 mA with respect to the diameter of 150 nm presented in Fig. 3. Here we note\nthat the thresholds does not scale exactly with the current density beacasue there are always\nOersted \felds associated with the currents that depend on the contact size. We observed a\nlarger reduction of 5 and 9 mA for the DS formation. Both diagrams are presented in the\nSupplementary materials.\n−101\nTime (ns)1\n−101\nTime (ns)1 33θI(deg.)\n10 20 30 40 50543210\nDS\nDropletFM\n−101\nTime (ns)1 3\n−101\nTime (ns)1 33\n3θI=0.1º\nI =15 mA\nθI=3º\nI =15 mAθI=0.1º\nI =40 mA\nθI=3º\nI =40 mA\nCurrent (mA)mz mz\nmz mz\nFIG. 3. Phase diagram of the droplet and DS formation Creation of both solitonic modes\nas a function of the applied current, I(with polarization p= 0:45), and the initial magnetization\nangle,\u0012I. For currents below 10 mA neither droplet nor DS can be excited, orange region. When\nthe current is higher than 10 mA a droplet is excited and droplet's threshold current does not\ndepend on \u0012I, yellow region. If the current is further increased, DS are created, pink region.\nThe current threshold for DS (red line) is higher than the droplet and depends on the initial\nmagnetization state, \u0012I. Insets correspond to time evolution curves of nanocontact magnetization\nat di\u000berent conditions.\n8Stability\nBoth droplet and DS exhibit magnetic bistability over considerable ranges of applied\ncurrent and magnetic \feld22{28,32,34. We investigate here the conditions that produce the\nannihilation of the solitonic modes when a lower degree of spin transfer torque|a lower\ncurrent|is applied. In Fig. 4a we show two curves corresponding to the average magneti-\nzation within the NC, mz, as the applied current decreases from an initial value of I= 30\nmA. A droplet and a DS are created at 30 mA (using \u0012I= 3\u000eand\u0012I= 0:1\u000erespectively).\nThe droplet collapses at about 9 mA whereas the DS requires a much lower current value of\n4 mA to vanish revealing that the DS remains stable over a larger range of applied currents\nor in other words, the DS requires smaller current values to be sustained.\nNext, we investigate the e\u000bect of a magnetic \feld gradient in the NC. A small constant in-\nplane \feld, a small change in anisotropy, or a variation in the \flm's thickness combined with\nthe Oersted \felds from the charge current could result in a gradient of e\u000bective magnetic\n\feld in the NC that could dephase the precession of magnetization in di\u000berent locations of\nthe NC and eventually annihilate the magnetic excitation. Experiments revealed that the\nlow frequency noise in droplets26,28,38is associated with a periodic process of shifting, anni-\nhilation, and creation. Simulations showed that an asymmetry of the e\u000bective \feld causes\na drift instability resulting in an oscillatory signal of hundreds of MHz|drift resonances.\nWe excite droplet and DS states at 30 mA using di\u000berent initial states (same as in Fig. 4a)\nand after a stabilization period we reduce the applied current until 10 mA, black squares in\nFig. 4a. We then apply a small in-plane \feld of 50 mT in order to destabilize the solitonic\nmodes. The combination of a \fxed in-plane \feld with the Oersted \felds creates an in-plane\n\feld gradient in the nanocontact. Figure 4b shows the time evolution of the magnetization\nfor a droplet (red line) and a DS (blue line). The small in-plane applied \feld causes a shift\nof the droplet away from the NC followed by a re-nucleation of a droplet state. The process\nof creation and annihilation is repeated at a frequency in the MHz range ( \u001880 MHz)26.\nThe e\u000bect of an in-plane \feld to the DS is di\u000berent; DS has an initial change as a result of\nthe abrupt change of the magnetic \feld but later on the DS stabilizes again. Full videos of\nthe evolution of droplet and DS in Fig. 4b are available in the Supplementary Materials.\n90 25 50 75 100 125−0.500.51\nTime (ns)Droplet\nDS\n-1\n0 10 20 30\nDroplet DS5 mAI (mA)−0.500.51\n-1a bmzFIG. 4. Stability of Droplet and DS a) curves of annihilation of droplet (red line) and DS\n(blue line) with decreasing the applied current. The curves correspond to the averaged normalized\nmagnetization within the nanocontact, mz, as the applied current decreases from an initial value of\nI= 30 mA. Both droplet and a DS are created at 30 mA (using \u0012I= 3\u000eand\u0012I= 0:1\u000erespectively).\nThe droplet collapses at about 9 mA whereas the DS does it at a lower current of about 4 mA. b)\nTime evolution of mzfor a droplet (red line) and a DS (blue line) in the presence of an in-plane\nmagnetic \feld. Both droplet and DS states are created at 30 mA using di\u000berent initial states (same\nas in a)) and after stabilization the applied current is reduced to 10 mA, black squares in a), and a\nsmall in-plane \feld of 50 mT is applied. The magnetization of droplet and DS behaves completely\ndi\u000berently; the droplet's magnetization oscillates caused by a drift resonance ( \u001840 MHz) while\nthe DS's magnetization, although it initially oscillates, it stabilizes after \u001880 ns and remains with\nthe initial topology having S= 1.\nDISCUSSION\nTwo main e\u000bects are involved in the magnetization dynamics when a spin-polarized cur-\nrent \rows through a nanocontact to a magnetic \flm. On the one hand, a spin-polarized\ncurrent of the appropriate polarity interacts with the magnetization via the STT e\u000bect try-\ning to align the magnetization in the opposite direction of the applied \feld. The STT e\u000bect\nis proportional to the non-collinear component of the magnetization with respect to the\npolarization of the current ( i.e., if the magnetization is precisely aligned in the direction of\nthe polarized current, say zfor the studied case, there is no e\u000bect). On the other hand,\nthe electrical current \rowing through the nanocontact causes Oersted \felds that curl the\nmagnetization. Here we note that in absence of other e\u000bects the magnetization of a PMA\nlayer adopts a con\fguration with S= 1 in presence of Oersted \felds. The skyrmion con\fg-\nuration provides a topological protection in two dimensions, which is valid for variations of\nthe in-plane components of the magnetization.\n10In the creation process of solitonic modes there is a competition between the two men-\ntioned e\u000bects. The STT e\u000bect increases rapidly as the magnetization tilts from the state\nperpendicular to the \flm plane, \u0012I= 0\u000e, and thus if the initial magnetization state is su\u000e-\nciently far from such a state, the solitonic mode forms without topology resulting in a droplet\nstate, states with \u0012I6= 0 can be prepared by increasing the temperature or by applying a\nshort in-plane \feld pulse. On the other hand if the initial state is closer to \u0012I= 0\u000ethe e\u000bect\nof STT produces a much slower variation of the magnetization and there is a time lapse\nwhere the e\u000bect of the Oersted \felds provides topology to the magnetization in the NC,\nwhich eventually results in the formation of a DS. In summary, the farther from equilibrium\nthe initial magnetization state is, the larger it is the required current density to create a DS.\nThere is another ingredient that plays a role in de\fning whether a droplet or a DS forms;\nthe speed of ramping the polarized current from zero, or from a small value, to a high value\nthat nucleates solitonic modes. Simulations in the diagram shown in Fig. 2b and c are done\nwith a sharp step of current. However, we have seen that using ramping currents that are\nlarger than 700 ps suppresses the formation of DS in favor of droplets.\nWe, thus, speculate about the possibility of observing DS experimentally. Typically, an\nexperimental setup used for the study of droplets contains a free layer with PMA where the\nsolitonic modes may form, which corresponds to our simulated CoNi layer, and a \fxed layer\nthat is used as a spin polarizer for the current,22{28. To create a DS instead of a droplet we\nneed to either depart from an initial magnetization state close to all perpendicular or produce\ntorques associated with the Oersted \felds larger than those associated with the polarized\ncurrents. In the \frst case we can try to apply large out-of-plane \felds in order to set\nand appropriate initial magnetization state or lower the temperature to reduce the thermal\nnoise that might produce \ructuations of the magnetization. It could be that experiments\nperformed at low temperatures25,38have already created DS. The second case consists in\nproviding a large current that is not polarized, producing large Oersted \felds but no STT\ne\u000bect. With the same con\fguration, the current polarization has to increase so that the STT\nbecomes predominant and promotes the creation of a solitonic mode. If the magnetization\nwas already curled due to the Oersted \felds it could result in the creation of a solitonic mode\nwith topological protection: a DS. This realization is feasible by using in-plane polarizers\nthat provide a spin polarization that depends on the out-of-plane applied \feld. We added in\nthe Supplementary materials simulations where the polarization is varied at a \fxed current\n11valued and found that again to create a DS one needs to vary the polarization of the current\nfast enough|as shown for the applied current, we need pulses of less than 1 ns.\nIt is necessary however to distinguish experimentally the two solitonic modes once they\nare created. The di\u000berences in precession frequency are two small to serve as a signature\nof droplet or DS. Instead, studying the stability of the solitonic modes is the best option.\nOne could study the hysteretic response or the response to small in-plane \felds and the\nappearance of low frequency noise as seen in Fig. 4.\nIn conclusion we have shown that both droplet and DS can be created with a same\ncon\fguration of applied \feld and spin-polarized current by controlling the initial magne-\ntization state, the degree of spin polarized current, or the speed at which the current{or\nthe polarization|is changed. We also studied the di\u000berence in stability between droplet\nstates and DS and found that DS is not only more stable against e\u000bective \feld variations\nbut DS also requires much lower currents to be sustained. Our results provide a pathway\nfor experimental studies of DS and their stability.\nACKNOWLEDGMENTS\nF.M. acknowledges \fnancial support from the Ram\u0013 on y Cajal program through RYC-\n2014-16515 and from MINECO through the Severo Ochoa Program for Centers of Excel-\nlence in R&D (SEV-2015-0496). JMH and NS acknowledge support from MINECO through\nMAT2015-69144. Research at NYU was supported by Grant No. NSF-DMR-1610416.\n12SUPPLEMENTARY MATERIAL FOR: GENERATION AND STABILITY OF\nDYNAMICAL SKYRMIONS AND DROPLET SOLITONS\nI. VIDEO DESCRIPTION\nThe video shows the time evolution of droplet (left-hand-side panel), and a DS (left-\nhand-side panel) in the presence of an in-plane magnetic \feld. Both droplet and DS states\nare created at 30 mA using di\u000berent initial states and after stabilization the applied current\nis reduced to 10 mA, and a small in-plane \feld of 50 mT is applied. Droplet and DS behave\ncompletely di\u000berently; the droplet has a drift resonance, being annihilated and created again\nat a frequency of \u001840 MHz while the DS, although it initially oscillates, it stabilizes after\n\u001880 ns and remains with the initial topology having S= 1.\n13II. ADDITIONAL SIMULATIONS\nWe calculated a phase diagram of droplet and DS formation as a function the polarization\nof the applied current and the initial magnetization angle ( \u0012I) for a \fxed current of 30 mA.\nThe Oersted-\feld e\u000bects are \fxed|given by the current of 30 mA|and only STT e\u000bects\nvary with spin-polarization. At small polarizations, there is a small STT e\u000bect and no\nexcitations are present independent of the initial magnetization. As the current polarization\nincreases we found \frst the onset of droplet states and with a further increase the onset of\nDS.\n0 0.2 0.4 0.6 0.8 1\nCurrent PolarizationI =30 mA\n543210\nDropletDSFMθI(deg.)\nFIG. 5. Phase diagram of the droplet and DS formation. Creation of both solitonic modes\nas a function of the polarization of the applied current and the initial magnetization angle ( \u0012I)\nfor a \fxed current of 30 mA. As the current is \fxed the Oersted \feld e\u000bects are \fxed and only\nspin-transfer torque (STT) e\u000bects vary with spin-polarization.\nNext we calculated phase diagrams of droplet and DS formation similar to those presented\nin Fig. 3 in the main manuscript for di\u000berent contact sizes and di\u000berent current polarizations.\nIn Fig. 6a we plot diagrams for di\u000berent contact size. The contact size determines the total\namount of current required to excited solitonic modes. We observe that the total amount\nof current required for droplet nucleation in contact diameters of 50, 100 and 150 nm is\n4, 6, and 9 mA. The threshold for DS creation is also reduced with decreasing the size of\nthe nanocontact with approximately 5 and 9 mA for 50 and 100 nm in comparison with\n150nm. Figure 6b compares the same diagram for the 150 nm nanocontact at di\u000berent\n14current polarization values. An increase of polarization from p= 0:45 top= 0:6 reduced by\n2 mA the droplet threshold and about of 5 mA in the DS threshold.\nθI(deg)\nCurrent (mA)0 10 20 30 40543210\n0 10 20 30 40543210\nCurrent (mA)a b\n50 nm\n100 nm\n150 nmp=0.6\np=0.45\np=0.45 150 nm\nFIG. 6. Phase diagrams of the droplet and DS formation as a function of NC diameter\nand polarization. a) Creation of droplet and DS as a function of applied current for nanocontact\ndiameters of 50, 100, and 150 nm for a \fxed current polarization of 0.45. b) Creation of droplet and\nDS as a function of applied current for current polarizations of 0.45 and 0.6 at a \fxed nanocontact\ndiameter of 150 nm.\nFinally we show in Fig. 7 a simulation where we depart from an equilibrium con\fguration\nconsisting of having an applied current of 30 mA with no polarization and then changing\nthe polarization to a given value. At low polarizations (below 0.16) no solitonic excitation\noccurs. The \frst threshold is at 0.16 and corresponds to a droplet formation whereas the DS\nformation requires 0.26. The results are similar to what we obtained with increasing current\nwith a \fxed polarization but here we always depart from an initial equilibrium con\fguration.\nThis realization is feasible by using in-plane polarizers that provide a spin polarization that\ndepends on the out-of-plane applied \feld and varying the out of the plane \feld to vary the\ncurrent poalrization.\n15−1−0.500.51mz\n0 0.1 0.2 0.3 0.4 0.500.51\nCurrent PolarizationSI=30 mA\nFM Droplet DSFIG. 7. Droplet and dynamical skyrmion creation process . The upper panel shows the\naveraged nanocontact magnetization as a function of the current polarization and the lower panel\nshows the corresponding skyrmion number, S >, of the magnetization con\fguration. A constant\ncurrent of 30 mA is applied from the beginning for a few ns to allow magnetization relax before a\nchange in polarization is applied. As the polarization increases we cross a \frst threshold at 0.16\nthat corresponds to the creation of a droplet. A further increase of polarization creates a DS (above\n0.26).\nIII. SUPPLEMENTARY NOTE 2: MICROMAGNETIC CODE\n// mumax3 is a GPU-accelerated micromagnetic simulation open-source software\n// developed at the DyNaMat group of Prof. Van Waeyenberge at Ghent University.\n// The mumax3 code is written and maintained by Arne Vansteenkiste.\n//GRID\nNumCells := 256\nCellSize:=4.e-9\nSetGridSize(NumCells, NumCells, 1)\nSetCellSize(CellSize, CellSize, CellSize)\nSETPBC(4, 4, 0)\n//REGIONS\nsetGeom(layer(0))\ndiam circ:= 150e-9\nrcirc:= diam circ/ 2\nAcirc:= pi * pow(r circ, 2)\nDefRegion(1, layer(0).intersect(circle(diam circ)))\n//MATERIAL PARAMETERS FOR STANDARD CoNi\nlambda = 1\nepsilonprime = 0\n16Msat = 500e3\nKu1 = 200e3\nAex = 10e-12\nalpha = 0.03\nanisU = vector(0, 0, 1)\n\fxedlayer = vector(0., 0., 1.)\n//OERSTED FIELDS\ncurrent := vector(0., 0., 1.)\nposX := 0.\nposY := 0.\nmask := newSlice(3, NumCells, NumCells, 1)\nfor i := 0; i <NumCells; i++f\nfor j := 0; j <NumCells; j++f\nr := index2coord(i, j, 0)\nr = r.sub(vector(posX, posY, 0))\nb := vector(0, 0, 0)\nif r.len()>= r circf\nb = r.cross(current).mul(mu0 / (2 * pi * r.len() * r.len()))\ngelsef\nb = r.cross(current).mul(mu0 / (2 * pi * r circ* rcirc))\ng\nfor k := 0; k < 1; k++ f\nmask.set(0, i, j, k, b.X())\nmask.set(1, i, j, k, b.Y())\nmask.set(2, i, j, k, b.Z())\ng\ng\ng\n//RUNNING\nBext.RemoveExtraTerms()\nCurr := -30e-3\nBext= vector(0, 0, 0.5)\nPol = 0.45\nAngle := 89.9\nmy := cos(angle * pi / 180)\nmz := sin(angle * pi / 180)\nm = Uniform(0, my, mz)\nj.SetRegion(1, vector(0, 0, Curr/A circ))\nBext.RemoveExtraTerms()\nBext.add(mask, Curr)\nRun(20e-9)\n17\u0003fmacia@icmab.es\n1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n2L. Berger, Phys. Rev. B 54, 9353 (1996).\n3J. A. Katine, F. J. Albert, R. A. Buhrman, E. B. Myers, and D. C. Ralph, Phys. Rev. Lett.\n84, 3149 (2000).\n4D. Ralph and M. Stiles, J. Magn. Magn. Mater. 320, 1190 (2008).\n5A. Brataas, A. D. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012).\n6E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A. Buhrman, Science 285, 867\n(1999).\n7J. Shibata, Y. Nakatani, G. Tatara, H. Kohno, and Y. Otani, Phys. Rev. B 73, 020403 (2006).\n8M. Klaui, M. Laufenberg, L. Heyne, D. Backes, U. Rudiger, C. A. F. Vaz, J. A. C. Bland, L. J.\nHeyderman, S. Cheri\f, A. Locatelli, T. O. Mentes, and L. Aballe, Appl. Phys. Lett. 88, 232507\n(2006).\n9A. V. Khvalkovskiy, V. Cros, D. Apalkov, V. Nikitin, M. Krounbi, K. A. Zvezdin, A. Anane,\nJ. Grollier, and A. Fert, Phys. Rev. B 87, 020402 (2013).\n10Y. Yamane and J. Sinova, J. Appl. Phys. 120, 233901 (2016).\n11S. Emori, U. Bauer, S. M. Ahn, E. Martinez, and G. S. Beach, Nat. Mater. 12, 611 (2013).\n12K. S. Ryu, L. Thomas, S. H. Yang, and S. Parkin, Nat. Nanotechnol. 8, 527 (2013).\n13N. Perez, E. Martinez, L. Torres, S.-H. Woo, S. Emori, and G. S. D. Beach, Appl. Phys. Lett.\n104, 092403 (2014).\n14U. K. Roszler, A. N. Bogdanov, and C. P\reiderer, Nature 442, 797 (2006).\n15N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8, 899 (2013).\n16J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nat. Nanotechnol. 8, 839 (2013).\n17A. Slavin and V. Tiberkevich, Phys. Rev. Lett. 95, 237201 (2005).\n18V. E. Demidov, S. Urazhdin, and S. O. Demokritov, Nat. Mater. 11, 1028 (2012).\n19D. Backes, F. Maci\u0012 a, S. Bonetti, R. Kukreja, H. Ohldag, and A. D. Kent, Phys. Rev. Lett.\n115, 127205 (2015).\n20S. Bonetti, R. Kukreja, Z. Chen, F. Maci\u0012 a, J. M. Hern\u0012 andez, A. Eklund, D. Backes, J. Frisch,\nJ. Katine, G. Malm, S. Urazhdin, A. D. Kent, J. St ohr, H. Ohldag, and H. A. D urr, Nat.\n18Commun. 6, 8889 (2015).\n21M. A. Hoefer, T. J. Silva, and M. W. Keller, Phys. Rev. B 82, 054432 (2010).\n22S. M. Mohseni, S. R. Sani, J. Persson, T. N. A. Nguyen, S. Chung, Y. Pogoryelov, P. K. Muduli,\nE. Iacocca, A. Eklund, R. K. Dumas, S. Bonetti, A. Deac, M. A. Hoefer, and J. \u0017Akerman,\nScience 339, 1295 (2013).\n23S. Chung, S. M. Mohseni, S. R. Sani, E. Iacocca, R. K. Dumas, T. N. Anh Nguyen, Y. Pogo-\nryelov, P. K. Muduli, A. Eklund, M. Hoefer, and J. \u0017Akerman, J. Appl. Phys. 115, 172612\n(2014).\n24S. M. Mohseni, S. R. Sani, R. K. Dumas, J. Persson, T. N. Anh Nguyen, S. Chung, Y. Pogo-\nryelov, P. K. Muduli, E. Iacocca, A. Eklund, and J. \u0017Akerman, PHYSICA. B 435, 84 (2014).\n25F. Maci\u0012 a, D. Backes, and A. D. Kent, Nat. Nanotechnol. 9, 992 (2014).\n26S. Lend\u0013 \u0010nez, N. Statuto, D. Backes, A. D. Kent, and F. Maci\u0012 a, Phys. Rev. B 92, 174426 (2015).\n27S. Chung, S. M. Mohseni, A. Eklund, P. D urrenfeld, M. Ranjbar, S. R. Sani, T. N. Anh Nguyen,\nR. K. Dumas, and J. \u0017Akerman, Low Temp. Phys. 41, 833 (2015).\n28S. Chung, A. Eklund, E. Iacocca, S. M. Mohseni, S. R. Sani, L. Bookman, M. A. Hoefer, R. K.\nDumas, and J. Akerman, Nat. Commun. 7, 11209 (2016).\n29J. Grollier, D. Querlioz, and M. D. Stiles, Proc. IEEE 104, 2024 (2016).\n30R. M. Jacob Torrejon, J. and, F. A. Araujo, S. Tsunegi, G. Khalsa, D. Querlioz, P. Bortolotti,\nV. Cros, K. Yakushiji, A. Fukushima, H. Kubota, S. Yuasa, M. D. Stiles, and J. Grollier,\nNature 547, 428 (2017).\n31M. D. Maiden, L. D. Bookman, and M. A. Hoefer, Phys. Rev. B 89, 180409 (2014).\n32Y. Zhou, E. Iacocca, A. A. Awad, R. K. Dumas, F. C. Zhang, H. B. Braun, and J. \u0017Akerman,\nNat. Commun. 6, 8193 (2015).\n33B. A. Ivanov and A. M. Kosevich, Zh. Eksp. Teor. Fiz. 24, 495 (1976).\n34R. H. Liu, W. L. Lim, and S. Urazhdin, Phys. Rev. Lett. 114, 137201 (2015).\n35M. Carpentieri, R. Tomasello, R. Zivieri, and G. Finocchio, Sci. Rep. , 16184 (2015).\n36F. Maci\u0012 a, P. Warnicke, D. Bedau, M.-Y. Im, P. Fischer, D. Arena, and A. Kent, J. Magn.\nMagn. Mater. 324, 3629 (2012).\n37A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez, and B. V. Waeyen-\nberge, AIP Advances 4, 107133 (2014).\n38S. Lend\u0013 \u0010nez, J. Hang, S. V\u0013 elez, J. M. Hern\u0013 andez, D. Backes, A. D. Kent, and F. Maci\u0012 a, Phys.\n19Rev. Applied 7, 054027 (2017).\n39J. Hang, C. Hahn, N. Statuto, F. Maci\u0012 a, and A. D. Kent, (submitted) (2017).\n20"    },    {        "title": "2003.04653v1.Spin_dynamics_of_a_hydrogen_atom_in_a_periodic_magnetic_structure.pdf",        "content": "arXiv:2003.04653v1  [physics.atom-ph]  10 Mar 2020Spin dynamics of a hydrogen atom in a periodic magnetic\nstructure\nA.I. Milstein,1,∗Yu.V. Shestakov,1,†and D.K. Toporkov1,‡\n1Budker Institute of Nuclear Physics of SB RAS, 630090 Novosi birsk, Russia\n(Dated: March 11, 2020)\nAbstract\nThe spin dynamics of a hydrogen atom during the passage of a pe riodic magnetic structure is\ndiscussed. The occupation numbers of the components of the h yperfine structure are considered\nas a function of time. The characteristic low-frequency osc illations are visible, which have a direct\nanalogue in the effect of nuclear magnetic resonance. An envel ope forms of these oscillations are\nfound using the Krylov-Bogolyubov-Mitropol’skii method. The dependence of spin dynamics on\nthe parameters of the magnetic structure is investigated. I t is shown that this dependence is very\nsensitive to the structure of the magnetic field.\n∗A.I.Milstein@inp.nsk.su\n†Yu.V.Shestakov@inp.nsk.su\n‡D.K.Toporkov@inp.nsk.su\n1I. INTRODUCTION\nAt present, there are quite a few works in which the spin dynamics of atoms and\nmolecules are studied during the passage of various magnetic struc tures, see, e.g. [ 1–6]\nand the references therein. The study of this dynamics is importan t, first of all, from a\npractical point of view. There are various interpretations of the o btained experimental\nresults. Using the spin dynamics of a hydrogen atom passing throug h a periodic magnetic\nstructure as an example, we show that in this problem there is a direc t analogy between\nthe time dependence of the level population and the phenomenon of nuclear magnetic reso-\nnance. We restrict ourselves to the simplest case of an electron in t he 1sstate, taking into\naccount the hyperfine interaction of the electron and the nucleus .\nConsider the magnetic moment µof a particle with spin S= 1 in a magnetic field,\nwhich is a superposition of a constant magnetic field H0directed along the zaxis and a\nfieldH1rotating with a frequency ωin thexyplane. Choosing the zaxis as the spin\nquantization axis, it is easy to calculate from the Pauli equation [ 7] the probabilities w1,\nw0, andw−1to find a particle in the state with the corresponding projection Sz, ifw1= 1\nandw0=w−1= 0 in the initial time. One has\nw1= (1−κ)2, w0= 2κ(1−κ), w−1=κ2,\nκ=(µH1)2\n(/planckover2pi1Ω)2sin2(Ωt/2),Ω =1\n/planckover2pi1/radicalbig\n(µH0−/planckover2pi1ω)2+(µH1)2. (1)\nIn resonance, /planckover2pi1ω=µH0. IfH1≪H0, then near the resonance Ω ≪ω. IfH1≫H0, then\nnear the resonance Ω ≫ω.\nLet us now consider a hydrogen atom in the 1 sstate, moving with velocity vin a time-\nindependent but non-uniform periodic magnetic field. Taking into acc ount the hyperfine\ninteraction of an electron and a proton, we have four states: thr ee states with a total spin\nof electron and proton to be S= 1 and projections Sz=±1,0, and one state with a total\nspinS= 0. Let us consider the magnetic field much smaller than the magnetic field of the\nproton magnetic moment at a distance aB(Bohr radius), i.e. H≪100G. In this case,\none can neglect the transitions between the states with S= 1 andS= 0. The simplest\nazimuthally symmetric configuration of a magnetic field periodic along zaxis, satisfying\nthe equation div H= 0 has the form\nHz=H0sin(kz), Hρ=−1\n2kρH0cos(kz), (2)\n2whereH0andkare some constants. Passing to the rest frame of the hydrogen a tom, we\nhave a time-dependent magnetic field\nHz(t) =H0sin(kvt), Hρ=−1\n2kρH0cos(kvt). (3)\nThe atomic magnetic moment operator is µH= 2µese+2µpsp, whereseandspare the\nelectron and proton spin operators, µeandµpare their magnetic moments. The matrix\nelement of µHover the states with a fixed S=se+spcoincides with the matrix element\nof the operator µH= (µe+µp)S≈ −µBS, whereµBis the Bohr magneton. Using the\nvariableτ=kvt, we write the Pauli equation [ 7], which describes the spin dynamics, in\nthe form\ni∂\n∂τψ=B[Szsinτ−√\n2λSxcosτ]ψ, B=µBH0\n/planckover2pi1kv, λ=kρ\n2√\n2, ψ=\na1\na2\na3\n,(4)\nwhereSzandSxarematrices corresponding to the zandxcomponents of the spin operator\nforS= 1. We rewrite the equation ( 4) as follows\ni˙a1=−B[−sinτ·a1+λcosτ·a2],\ni˙a2=−Bλcosτ·(a1+a3),\ni˙a3=−B[sinτ·a3+λcosτ·a2], (5)\nwhere ˙ai=∂ai/∂τ. Forv= 1500m/s, H0= 1G,k= 2πcm−1we haveB= 9. Therefore,\nfor the fields H0= 0.01÷0.1G we obtain B <1. In our work we assume λ<1. Our task\nis to analyze the solutions of the equations ( 5) for various values of the parameters Band\nλ<1.\nII. SPIN DYNAMICS AT B/lessorsimilar1ANDB/greaterorsimilar1ATλ <1.\nDenote byW1(B,λ) =|a1|2,W2(B,λ) =|a2|2, andW3(B,λ) =|a3|2the probabilities to\nhave, respectively, spin projections 1, 0 and −1 at timeτ. A typical dependence of Wionτ\nforB/lessorsimilar1 andB/greaterorsimilar1 atλ<1 is shown in Fig. 1. We chose the boundary conditions a1= 1,\na2= 0, anda3= 0 atτ= 0. However, the characteristic behavior of the probabilities Wi\ndoes not depend much on the initial value of τ. Note that for λ= 0 the probabilities Wi\nare independent of τ. It is seen from Fig. 1that the probabilities Wiare periodic functions\n30 5 10 15 200.00.20.40.60.81.0\nΤ\n2ΠWi/LParen10.7, 0.4/RParen1\n0 5 10 15 200.00.20.40.60.81.0\nΤ\n2ΠWi/LParen16, 0.1/RParen1\nFigure 1. Dependence of Wi(B,λ) onτfor various values of the parameters Bandλ. The solid\nline corresponds to the function W1, the dashed line corresponds to the function W0, and the\ndotted line corresponds to the function W−1.\nofτ, and for the oscillation period Twe haveT/(2π)≫1. Shallow ripples with a period\nof a magnetic structure are superimposed on the smooth envelope s of probabilities. The\nobserved picture fully corresponds to the time dependence of pola rization described above\nin the case of nuclear magnetic resonance. We checked that a spec ific form of the periodic\ndependence of HzandHρdoes not change the qualitative picture of spin dynamics.\nSinceT/(2π)≫1, it is possible to find the envelope form Wiof the probabilities Wi\nusing the Krylov-Bogolyubov-Mitropol’skii method [ 8] of averaging over high-frequency\noscillations. To do this, we write the amplitudes aiin Eq. (5) asai=|ai|exp(−iφi),\n|ai|=Xi+xi, andφi= Φi+ϕi, whereXiand Φ icorrespond to low-frequency oscillations,\nandxiandϕicorrespondtohigh-frequencyoscillations. Then, intheleadingapp roximation\ninλ<1 we obtain:\n˙X1=−λBcosτsinϕ12X2,\n˙X2=λB[cosτsinϕ12X1+cosτsinϕ32X3],\n˙X3=−λBcosτsinϕ32X2,\n˙ϕ12=Bsinτ,˙ϕ32=−Bsinτ,\nϕ12=ϕ1−ϕ2, ϕ32=ϕ3−ϕ2, (6)\nwhereAmeans the averaging over high-frequency oscillations. Thus,\nϕ12=−Bcosτ, ϕ 32=Bcosτ.\n4Using the integral\n1\n2π/integraldisplayπ\n−πcosτsin(Bcosτ)dτ=J1(B),\nwhereJn(x) is the Bessel function, we perform averaging over high-frequen cy oscillations\nin Eq. (6) and obtain\n˙X1=G0X2,\n˙X2=G0[−X1+X3],\n˙X3=−G0X2, (7)\nwhereG0=λBJ1(B). Finally, we arrive at the probabilities Wi=X2\ni:\nW1= cos4(Ω0τ/2),W0= 2sin2(Ω0τ/2)cos2(Ω0τ/2),W−1= sin4(Ω0τ/2),(8)\nwhere the oscillation frequency Ω 0reads\nΩ0=√\n2λBJ1(B). (9)\nThis formula is valid at Ω 0≪1. ForB≪1 we have Ω 0=λB2/√\n2. A comparison of Wi\nandWiis made in Fig. 2for a few values of B <1 andλ<1. One can see from Fig. 2an\n0 5 10 15 200.00.20.40.60.81.0\nΤ\n2ΠWi/LParen10.7, 0.4/RParen1\n0 5 10 15 200.00.20.40.60.81.0\nΤ\n2ΠWi/LParen10.8, 0.5/RParen1\nFigure 2. Comparison of the functions Wi(B,λ) andWi(Ω0) for a few values of B <1 and\nλ <1. The solid lines correspond to the functions W1andW1, the dashed lines correspond to\nthe functions W0andW0and the dotted lines correspond to the functions W−1andW−1. Lines\nwith ripples correspond to the functions Wi, and smooth lines correspond to the functions Wi.\nexcellent agreement between WiandWi.\nLet us consider the case B/greaterorsimilar1 atλ<1. Fig.3shows the dependence of WiandWion\nτfor a few values of B >1 atλ= 0.07. It is seen that this dependence is very sensitive\n5to the value of B. In the vicinity of zeros of the Bessel function J1(B) the frequency Ω 0\nvanishes (these zeros are B∗= 3.83,7.02,10.17,13.32...).\n0 5 10 15 200.00.20.40.60.81.0\nΤ\n2ΠWi/LParen13.5, 0.07/RParen1\n0 5 10 15 200.00.20.40.60.81.0\nΤ\n2ΠWi/LParen13.82, 0.07/RParen1\n0 5 10 15 200.00.20.40.60.81.0\nΤ\n2ΠWi/LParen16, 0.07/RParen1\n0 5 10 15 200.00.20.40.60.81.0\nΤ\n2ΠWi/LParen16.99, 0.07/RParen1\nFigure 3. Comparison of the functions Wi(B,λ) andWi(Ω0) for a few values of B >1 and\nλ= 0.07. The solid lines correspond to the functions W1andW1, the dashed lines correspond to\nthe functions W0andW0, and the dotted lines correspond to the functions W−1andW−1. Lines\nwith ripples correspond to the functions Wi, and smooth lines correspond to the functions Wi.\nThe expressions ( 8) forWicoincide with ( 1) forwiin the case of nuclear magnetic\nresonance at /planckover2pi1ω=µH0after the substitutions Ω 0→µH1andτ→t.\nAbove, we investigated the dependence of Wionτfor various values of Bandλ <1.\nA fixed value of λmeans a fixed value of the impact parameter ρ. Since the frequency\nΩ0of the envelopes Widepends on λ, averaging over impact parameters can distort the\noscillation pattern. To elucidate this statement, let us consider a be am with a transverse\nsizeρ0and uniform density. Then the average value of /angbracketleftWi/angbracketrightof probabilities Wiis\n/angbracketleftWi/angbracketright=2\nρ2\n0/integraldisplayρ0\n0Wi(B,λ)ρdρ=2\nλ2\n0/integraldisplayλ0\n0Wi(B,λ)λdλ, λ 0=kρ0\n2√\n2.(10)\nThe dependence of /angbracketleftWi/angbracketrightonτforB= 0.5 andλ0= 0.5 is shown in Fig. 4. For comparison,\nthe same figure shows the envelopes obtained from Eqs. ( 8) and (10).\n60 5 10 15 20 25 300.00.20.40.60.81.0\nΤ\n2Π/LessWi/GreaΤer\nFigure 4. Dependence of the functions /angbracketleftWi/angbracketright, Eq. (10), and/angbracketleftWi/angbracketrightonτforB= 0.5 andλ0= 0.5;\nthe solid lines correspond to the functions /angbracketleftW1/angbracketrightand/angbracketleftW1/angbracketright, the dashed lines correspond to the\nfunctions /angbracketleftW0/angbracketrightand/angbracketleftW0/angbracketright, and the dotted lines correspond to the functions /angbracketleftW−1/angbracketrightand/angbracketleftW−1/angbracketright.\nLines with ripples correspond to the functions /angbracketleftWi/angbracketrightand smooth lines correspond to the functions\n/angbracketleftWi/angbracketright.\nAs it follows from Eqs. ( 8) and (10), atτ→ ∞the oscillations in /angbracketleftWi/angbracketrightfade out and\n/angbracketleftW1/angbracketright →3/8,/angbracketleftW2/angbracketright →1/4, and/angbracketleftW3/angbracketright →3/8 regardless of the values of Bandλ0.\nIII. CONCLUSION\nIt is shown that in a hydrogen atom moving along the zaxis in a periodic magnetic field,\nthe probabilities Wito have certain projections of the total spin (electron and proto n) on\nthezaxis are also periodic functions of z. The periodof these functions is much larger than\nthe period of the magnetic field. Using the Krylov-Bogolyubov-Mitro pol’skii method, we\nfound the envelopes of the functions Wiand showed that the oscillation period is a function\nof the amplitude and period of the magnetic field, the velocity of the a tom, and the impact\nparameter of the atom relative to the axis of the magnetic system. We also showed that\naveraging over the impact parameter of atoms in the beam leads to o scillation damping.\nTherefore, to observe the effect of oscillations and to control th e polarization using this\neffect, it is necessary to prepare a beam in which all atoms fly at the s ame distance from\ntheaxisofthemagneticsystem. Wehavedemonstratedthatthes pindynamicsinaperiodic\nmagnetic system have a direct analogy with the effect of nuclear mag netic resonance. High\nsensitivity of the period of low-frequency oscillations to values of th e parameter Bmay, in\n7principle, be used in applications.\nAcknowledgements.\nWe are grateful to R. Engels for useful discussions.\n[1] N.F. Ramsey, Molecular beam (Oxford, the Clarendon Press, 1956).\n[2] A.D. Cronin, J. Schmiedmayer, and D.E. Pritchard, Rev. M od. Phys. 81.3, 1051 (2009).\n[3] Yu.L. Sokolov, Uspekhi Fizicheskikh Nauk 169, 559 (1999) [Physics Uspekhi, 42, 481 (1999)].\n[4] R. Engels, M. Gai?er, R. Gorski et al., Phys. Rev. Lett., 115, 113007 (2015).\n[5] P.G. Sona, Energia Nucleare 14, 295 (1967).\n[6] D.K. Toporkov, PoS (PSTP 2013) 064.\n[7] L. D. Landau and E. M. Lifshitz, Quantum Mechanics: NonRelativistic Theory, Course of\nTheoretical Physics (Elsevier Science, New York, 1981).\n[8] N.N. Bogolyubov, Yu.A. Mitropol’skii, Asymptotic methods in the theory of nonlinear oscil-\nlations(Gordon and Breach , Delhi, 1961).\n8"    },    {        "title": "2305.09151v1.Non_periodic_input_driven_magnetization_dynamics_in_voltage_controlled_parametric_oscillator.pdf",        "content": "Non-periodic input-driven magnetization dynamics in\nvoltage-controlled parametric oscillator\nTomohiro Taniguchi\nNational Institute of Advanced Industrial Science and Technology (AIST), Research\nCenter for Emerging Computing Technologies, Tsukuba, Ibaraki 305-8568, Japan,\nAbstract\nInput-driven dynamical systems have attracted attention because their dy-\nnamics can be used as resources for brain-inspired computing. The recent\nachievement of human-voice recognition by spintronic oscillator also utilizes\nan input-driven magnetization dynamics. Here, we investigate an excita-\ntion of input-driven chaos in magnetization dynamics by voltage controlled\nmagnetic anisotropy e\u000bect. The study focuses on the parametric magnetiza-\ntion oscillation induced by a microwave voltage and investigates the e\u000bect of\nrandom-pulse input on the oscillation behavior. Solving the Landau-Lifshitz-\nGilbert equation, temporal dynamics of the magnetization and its statistical\ncharacter are evaluated. In a weak perturbation limit, the temporal dynam-\nics of the magnetization are mainly determined by the input signal, which\nis classi\fed as input-driven synchronization. In a large perturbation limit,\non the other hand, chaotic dynamics are observed, where the dynamical re-\nsponse is sensitive to the initial state. The existence of chaos is also identi\fed\nby the evaluation of the Lyapunov exponent.\nKeywords:\nspintronics, chaos, input-driven dynamical system, voltage controlled\nmagnetic anisotropy e\u000bect\n1. Introduction\nAfter the successful reports on human-voice recognition by spin-torque os-\ncillator [1], associative memory operation by three-terminal magnetic mem-\nory [2], and pattern recognition by an array of spin-Hall oscillators [3] in\n2017, the application of spintronics technology to emerging computing has\nPreprint submitted to Journal of Magnetism and Magnetic Materials May 17, 2023arXiv:2305.09151v1  [cond-mat.mes-hall]  16 May 2023become an exciting topic in magnetism [4, 5]. The works bridge the research\n\feld to the others such as computer science, statistical physics, and nonlin-\near science. Among them, the input-driven dynamical theory [6] has gained\ngreat attention because most models related to emerging computing, such as\nmachine learning and robotics, are input-driven. For example, the human-\nvoice recognition task can be solved using spin-torque oscillator [1] if there is\none-to-one correspondence between the input electric voltage, converted from\nhuman voice, and the output power originated from nonlinear magnetization\ndynamics. The correspondence as such is classi\fed as input-driven synchro-\nnization [7, 8, 9, 10, 11, 12], where the dynamical output from the oscillator\nis solely determined by the input data and is independent of the initial state\nof the magnetization; therefore, by learning the correspondence, the system\ncan recognize the input data. Another example of the input-driven dynamics\nis chaos, which has a sensitivity to the initial state and has been found in\nbrain activities and arti\fcial neural networks [13, 14]. Contrary to the input-\ndriven synchronization in magnetization dynamics [1, 15, 16, 17, 18, 19, 20],\nhowever, the input-driven chaotic dynamics in spintronics devices have not\nbeen fully investigated yet [20].\nThe input-driven magnetization synchronization has been mainly stud-\nied in spin-torque oscillator [1, 15, 16, 17, 18, 19, 20], where electric current\ndrives the dynamics. From viewpoint of energy-saving computing, it would\nbe preferable to drive magnetization dynamics by voltage controlled magnetic\nanisotropy (VCMA) e\u000bect [21, 22, 23, 24, 25, 26, 27, 28, 29]. The VCMA\ne\u000bect arises from the modi\fcation of electron states [24, 25] and/or the in-\nduction of magnetic moment [29] near the ferromagnetic/insulator interface\nby an application of electric voltage, and is expected to provide low-power\nwriting scheme in magnetoresistive random access memory. A recognition\ntask of the random input signal by using the relaxation dynamics of the\nmagnetization caused by VCMA e\u000bect was reported recently [30]. Remind\nthat recognition tasks are solved in terms of input-driven synchronization.\nIn such circumstances, it is of interest to investigate a possibility to induce\nthe input-driven chaos in magnetization dynamics manipulated by VCMA\ne\u000bect.\nIn this work, we propose a method to excite the input-driven chaotic mag-\nnetization dynamics in a parametric oscillator maintained by a microwave\nVCMA e\u000bect. Note that the relaxation dynamics of the magnetization caused\nby a direct VCMA e\u000bect may not be suitable for inducing chaos because the\ndynamics saturates to a \fxed point, while chaos, on the other hand, must be\n2Figure 1: (a) Schematic illustration of a magnetic multilayer. The unit vector pointing\nin the magnetization direction in the free layer is denoted as m. An external magnetic\n\feldHapplis applied in the xdirection. In parametric oscillation state, the magnetization\nrotates around the xaxis, as schematically shown by the yellow arrow. (b) Time evolution\nofmxin the presence of a microwave voltage. The horizontal axis represents the ratio of\nthe frequency fof the voltage with respect to the Larmor frequency fL. (c) Examples of\nmx(red) andmz(black) in steady states. The solid and dotted lines correspond to the\nmicrowave frequency of f= 2:0fLandf= 2:5fL, respectively.\nsustained. To overcome the issue, we focus on the parametric magnetization\noscillation caused by a microwave VCMA e\u000bect, which was recently demon-\nstrated experimentally [31, 32]. Speci\fcally, we study the modulation of the\nparametric oscillation caused by the injection of input signal and solving the\nLandau-Lifshitz-Gilbert (LLG) equation. It is shown that the magnetization\ndynamics in the presence of random input signal become sensitive to the ini-\ntial state, indicating the appearance of input-driven chaos. The appearance\nof chaos is also investigated by evaluating the Lyapunov exponent.\n2. Temporal dynamics\nHere, we show the temporal dynamics of the magnetization in the pres-\nence of time-dependent inputs.\n2.1. Parametric oscillation\nFigure 1(a) shows a schematic view of a ferromagnetic multilayer con-\nsisting of free and reference layers separated by a thin nonmagnetic spacer.\nThe unit vector pointing in the magnetization direction in the free layer is\ndenoted as m. Thezaxis is normal to the \flm plane. It was experimentally\ncon\frmed [31] that the magnetization dynamics driven by VCMA e\u000bect is\n3well described by the macrospin LLG equation,\ndm\ndt=\u0000\rm\u0002H+\u000bm\u0002dm\ndt; (1)\nwhere\rand\u000bare the gyromagnetic ratio and the Gilbert damping constant,\nrespectively. The magnetic \feld Hconsists of the in-plane external magnetic\n\feldHappland the perpendicular magnetic anisotropy \feld HKas [31]\nH=Happlex+HKmzez; (2)\nwhere ei(i=x;y;z ) is the unit vector in the i-direction and we assume\nthat the external magnetic \feld points to the xdirection. The values of the\nparameters are similar to those used in Refs. [31, 32], where \r= 1:764\u0002107\nrad/(Oe s), \u000b= 0:005, andHappl= 720 Oe. Note that, when HapplandHK\nare constants, the magnetization dynamics described by Eq. (1) are relax-\nation dynamics towards the minima of the energy density E=\u0000MR\ndm\u0001H,\ni.e., the magnetization saturates to a \fxed point. Therefore, to excite sus-\ntainable dynamics such as an oscillation or chaos, Happland/orHKshould\nbe time-dependent.\nLet us \frst show the parametric oscillation of the magnetization [31, 32].\nBefore applying voltage, the magnetic anisotropy \feld HKhas a value de-\ntermined by the competition between the shape and interfacial magnetic\nanisotropy \felds [33, 34, 35]. Next, both direct and microwave voltages\nare applied, which make the magnetic anisotropy \feld as HK=HKd+\nHKasin(2\u0019ft) by VCMA e\u000bect, where HKaandfare the amplitude and\nfrequency of the microwave component in VCMA \felds. For simplicity, we\nassume that the direct component HKdinHKin the presence of VCMA e\u000bect\nis zero [32], while HKa= 100 Oe. Note that the value of HKa=Happlshould be\nlarger than 2 \u000bto excite a sustainable oscillation [32]. Figure 1(b) shows the\ntime evolution of mxfor various f. The magnetization basically saturates to\na \fxed point mx= +1 due to the relaxation to the direction of the external\nmagnetic \feld. An exception occurs when the input frequency fis close to\n2fL, wherefL=\rHappl=(2\u0019) is the Larmor precession frequency. Initially,\nmxoscillates around mx= 0 and \fnally tends to mx'0. Figure 1(c) sum-\nmarizes the time evolution of mx(red) andmz(black) for f= 2:0fL(solid)\nand 2:5fL(dotted). A steady precession is excited for f= 2fL, where the\nmagnetization oscillates almost in the yzplane (mx'0); see also Appendix\nA showing the spatial trajectory of the oscillation. Since the input frequency\nis two times larger than the Larmor precession frequency, the oscillation is\nclassi\fed to the parametric oscillation.\n4Figure 2: (a) Time evolution of the di\u000berence of two solutions of Eq. (1) with di\u000berent\ninitial conditions. (b) Examples of the uniformly distributed random input signal rk. (c)\nTime evolution of mx. The random input signal is injected from t= 5:0\u0016s. The strength\nand the pulse width of the random input signal are \u0017= 0:8 andtp= 2:0 ns, respectively.\n2.2. Input-driven dynamics\nNext, let us consider the input-driven dynamics. The microwave volt-\nage inducing the parametric oscillation is input signal of one kind. In fact,\nit causes a synchronized motion of the magnetization with respect to the\nmicrowave voltage, where the relative phase between them saturates to one\nof two stable values [32]; see also Appendix A. Multistability and chaotic\nbehavior were also found very recently [36]. Such a periodic input-driven dy-\nnamics has been studied for a long time [37]. Note, however, that the input\nsignal used in emerging computing is often non-periodic, as in the case of\nhuman voice. A main focus in recent input-driven dynamical theory [6] is to\nstudy whether the dynamical response caused by non-periodic input is solely\ndetermined by the input signal or depends on the initial state of the physical\nsystem. The former is the input-driven synchronization. In the latter case,\nthe dynamics might be the case of the input-driven chaos.\nTherefore, there are two requirements for studying the input-driven dy-\nnamics. First, it is necessary to compare the solutions of Eq. (1) with\ndi\u000berent initial conditions. Second, non-periodic input signal should be\nadded to VCMA e\u000bect. For the \frst requirement, we prepare natural ini-\ntial conditions in the absence of VCMA e\u000bect from thermal equilibrium\ndistribution [12]; see Appendix B. We solve the LLG equations for these\ninitial conditions with HK=HKasin(2\u0019fLt) fromt= 0 tot= 5:0\u0016s,\nwhere non-periodic input is not injected yet. For convenience, let us de-\nnote two solutions of Eq. (1) with slightly di\u000berent initial conditions as m1\nandm2. Figure 2(a) shows the time evolution of their di\u000berence, jm1\u0000\nm2j=p\n(m1x\u0000m2x)2+ (m1y\u0000m2y)2+ (m1z\u0000m2z)2, in the presence of a\nmicrowave voltage. The di\u000berence decreases with time increasing because\n5the microwave voltage tends to \fx the phase of the magnetization oscillation\n[32]. Simultaneously, we should note that a tiny di\u000berence still remains be-\ncause the phase \fxing by the microwave voltage is achieved only in the limit\noft!1 . Next, for the second requirement, we add uniformly distributed\nrandom-pulse number rk(\u00001\u0014rk\u00141) as input signal, which is used in\na recognition task of physical reservoir computing [5, 20]. The su\u000ex krep-\nresents the order of the input signal. Thus, from t= 5:0\u0016s, the magnetic\nanisotropy \feld becomes\nHK=HKa(1 +\u0017rk) sin(2\u0019fLt); (3)\nwhere the frequency fis \fxed to 2 fL. The dimensionless parameter \u0017de-\ntermines the modulation of VCMA e\u000bect by the input signal. Figure 2(b)\nshows an example of the random input signal rk, where the pulse width is 2 :0\nns. The input signal modulates the magnetic anisotropy \feld and induces\ncomplex dynamics of the magnetization, as shown in Fig. 2(c), where \u0017is\n0:8.\nNow let us investigate the sensitivity of the magnetization dynamics with\nrespect to the initial state. Figure 3(a) shows the temporal di\u000berence between\ntwo solutions of Eq. (1) with di\u000berent initial conditions, where \u0017= 0:2. As\nmentioned, for t\u00145:0\u0016s, only the microwave voltage is applied, and the\ndi\u000berence tends to be zero. There is, however, still a tiny di\u000berence, as shown\nin Fig. 2(a). This di\u000berence can be regarded as the di\u000berence given to the\ninitial state for the dynamics in the presence of the random input signal.\nNote that, even after the injection of the random input signal from t= 5:0\n\u0016s, the di\u000berence remains negligible for this weak ( \u0017= 0:2) perturbation\nlimit. The result indicates that the synchronization caused by the microwave\nVCMA e\u000bect is maintained. The conclusion can be veri\fed from a di\u000berent\nviewpoint shown in the inset of Fig. 3(a), where two solutions of the LLG\nequation are almost overlapped. However, when the strength of the random\ninput signal becomes large as \u0017= 0:8, a tiny di\u000berence at t= 5\u0016s is enlarged\ndue to the excitation by the random input, as shown in Fig. 3(b). Remind\nthat the LLG equation conserves the norm of the magnetization as jmj= 1;\ntherefore, the maximum value of the di\u000berence between two solutions is 2,\nat which two magnetizations point to the opposite direction. Therefore, the\ndi\u000berence shown in Fig. 3(b), which is larger than 1, is regarded as non-\nnegligible. The temporal dynamics of two solutions shown in the inset of\nthe \fgure also indicate that the synchronization caused by the microwave\nVCMA e\u000bect is broken. These results indicate that, although the di\u000berence\n6Figure 3: Di\u000berence jm1\u0000m2jof the solutions of the LLG equation with slightly di\u000berent\ninitial conditions for (a) \u0017= 0:2 and (b)\u0017= 0:8. The insets show temporal dynamics of\nm1xandm2x.7of two solutions at t= 5\u0016s is negligibly small, as shown in Fig. 2(a), it is\nexpanded by the injection of the random-pulse input signal. In other words,\nthe dynamics are sensitive to the di\u000berence at t= 5\u0016s. Such a sensitivity\nimplies that the dynamics in Fig. 3(b) is chaos.\n2.3. Validity of parameters\nWe note that the value of the parameters used in this work is in a reason-\nable range realized in experiments. The perpendicular magnetic anisotropy\nenergy density, K, consists of the bulk magnetic anisotropy energy density\nKV, the interfacial magnetic anisotropy energy Ki, the contribution from the\nVCMA e\u000bect as Kd=KVd+Ki\u0000\u0011E, wheredis the thickness of the free\nlayer. The electric \feld Erelates to the voltage VviaE=V=d I, wheredI\nis the thickness of the insulating barrier. In typical magnetic multilayers,\nwhere the free layer and insulating barrier are CoFeB and MgO, respec-\ntively,Kiis the dominant contribution to Kand its value increases with the\ncomposition of Fe increasing [33]. It can reach on the order of 1 :0 mJ/m2,\nwhich corresponds to, typically, on the order of 1 T in terms of magnetic\n\feld, 2Ki=(Md), whereMis the saturation magnetization and is about 1000\nemu/cm3. On the other hand, the VCMA e\u000eciency \u0011reaches 300 fJ/(Vm)\n[38, 39]. Regarding typical values of the thickness of the insulating barrier\n(about 2:5 nm) and applied voltage (0 :5 V at maximum) [40], the tunable\nrange of the magnetic anisotropy \feld by voltage is on the order of 1 :0 kOe\nat maximum. Summing these values, it is possible to generate an oscillating\ncomponent of the magnetic anisotropy \feld on the order of 100 Oe. It should\nalso be noted that a series of random-pulse input signal with the pulse width\nof nanoseconds was applied to magnetic multilayers in experiments of phys-\nical reservoir computing [15, 16]. Therefore, the proposal made here will be\nexperimentally examined.\n2.4. Comment on LLB equation\nThe results shown in this work are derived by solving the LLG equation.\nThere is another equation of motion, Landau-Lifshitz-Bloch (LLB) equation,\ndescribing the magnetization dynamics. Here, let me mention their di\u000ber-\nences brie\ry.\nThe LLG equation assumes the conservation of the magnetization mag-\nnitude, i.e.,jmj= 1, which is valid at temperature su\u000eciently lower than\nCurie temperature. The relaxation of the magnetization is characterized by\n8the dimensionless damping parameter \u000b. Note that the number of inde-\npendent variables in the LLG equation is two, although the vector mhas\nthree components in the Cartesian coordinate. This is because the condition\njmj= 1 acts as a constraint and reduces the number of independent variables.\nOn the other hand, the LLB equation does not conserve the magnetization\nmagnitude, and is valid at high temperature. There are two parameters,\nthe longitudinal and transverse relaxation times, characterizing the magne-\ntization relation. The number of independent variables is three in the LLB\nequation.\nWe should note that chaos appears in a high-dimensional system. In\nfact, chaos is prohibited in a dynamical system whose dimension is less or\nequal to two, according to the Poincar\u0013 e-Bendixson theorem. Therefore, chaos\nmight be easily excited in a system described by the LLB equation than that\ndescribed by the LLG equation. However, since the number of the parameters\ndescribing the relaxation are di\u000berent between two equations, it is di\u000ecult\nto compare chaos in these two equations on an equal footing. Therefore, we\nwould like to leave chaos in the LLB equation for further study in future.\n3. Statistical analysis of Lyapunov exponent\nIn Sec. 2.1, we study the existence of chaos from temporal dynamics. To\nidentify chaos from di\u000berent perspectives, here, we evaluate the Lyapunov\nexponent.\nThe Lyapunov exponent is an expansion rate of the di\u000berence between two\nsolutions of an equation of motion with slightly di\u000berent initial conditions.\nThe Lyapunov exponent is negative when the solution saturates to a \fxed\npoint. The input-driven synchronization is an example of the dynamics with\na negative Lyapunov exponent because the temporal dynamics are solely\ndetermined by the input signal and independent of the initial condition.\nWhen the Lyapunov exponent is zero, the di\u000berence remains constant. An\nexample of the dynamics corresponding to a zero Lyapunov exponent is a\nlimit-cycle oscillation. The corresponding dynamics thus depends on the\ninitial state but is not chaos. When the Lyapunov exponent is positive, the\ndi\u000berence is expanded and thus, the dynamics are sensitive to the initial\nstate. A positive Lyapunov exponent indicates an existence of chaos. Note\nthat the sensitivity to the initial state in dynamics is a necessary condition\nof chaos but is not a su\u000ecient condition because the dynamics with a zero\nLyapunov exponent also depends on the initial state. The evaluation of the\n9Figure 4: Lyapunov exponent as a function of the dimensionless input strength \u0017.\n10Lyapunov becomes a measure of chaos because its sign provides an evidence\nof chaos. Here, we evaluate the Lyapunov exponent by Shimada-Nagashima\nmethod [41], where the exponent is de\fned as\n\u0003= lim\nN!11\nN\u0001tNX\ni=1lnD\n\u000f; (4)\nwhere \u0001tis the time increment of the LLG equation and is 1 ps in this work.\nIn the Shimada-Nagashima method, the solution of an equation of motion at\na certain time t0is shifted with a tiny distance \u000fin phase space. Then, the\noriginal and shifted solutions are evolved from t=t0tot=t0+ \u0001tby the\nequation of motion. The distance between these solutions at t=t0+\u0001tisD.\nIfD=\u000f < (>)1, the di\u000berence given at the time t0shrinks (expanded), and\nthus, the temporal Lyapunov exponent is negative (positive). The Lyapunov\nexponent is a long-time average of such a temporal Lyapunov exponent, as\nimplied by Eq. (4); see also Appendix C for details. Figure 4 summarizes\nthe Lyapunov exponent as a function of the strength of the input signal,\n\u0017. For small \u0017, the Lyapunov exponent is negative, indicating that the\ndynamical state of the magnetization is determined by the input signal and\nis insensitive to the initial sate. The Lyapunov exponent changes its sign\naround\u0017= 0:5 and becomes positive for large \u0017, indicating that the dynamics\nbecomes sensitive to the initial state. The positive Lyapunov exponents are\nanother evidence of the appearance of input-driven chaos in the parametric\noscillator.\n4. Conclusion\nIn conclusion, the input-driven magnetization dynamics in the parametric\noscillator were studied by solving the LLG equation. The microwave volt-\nage induces a sustainable oscillation of the magnetization around an exter-\nnal magnetic \feld through VCMA e\u000bect. Adding non-periodic input signal\nchanges the dynamical behavior, depending on its magnitude. In a weak\nperturbation limit, the temporal dynamics of the magnetization were deter-\nmined by the input signal and are insensitive to the initial state. On the\nother hand, in a large perturbation limit, the dynamics become sensitive to\nthe initial state. Such a chaotic behavior was revealed by comparing the dif-\nference of two solutions of the LLG equation with di\u000berent initial conditions.\nThe evaluation of the Lyapunov exponent also identi\fed the appearance of\nchaos in the magnetization dynamics.\n11The existence of chaos in the input-driven spintronics systems will be\nof interest for emerging computing technologies. For example, it has been\nempirically shown that the computing performance of physical reservoir com-\nputing is maximized at the edge of chaos [42, 43], although it does not seem\na general conclusion [5]. Therefore, a tunability of the dynamical state in\nphysical systems is required for an enhancement of the computing capability.\nThe result shown in Fig. 4 shows, for example, that the dynamical state of\nspintronics devices can be tuned between input-driven synchronization and\nchaos by tuning the input strength. As emphasized in Sec. 2.3, the values\nof the parameters used in this work are in a reasonable range available in\nexperiments, and therefore, the results in this work will provide a direction\nto design the emerging computing devices based on spintronics technologies.\nThe input-driven chaotic magnetization dynamics might also have some ap-\nplications because chaos was found in brain activities [14] and theoretical\nmodels emulating the neural dynamics of squid [13]. Developing the present\nresults to brain-inspired computing will be, therefore, an interesting future\nwork.\nAcknowledgments\nThe work is supported by JSPS KAKENHI Grant Number 20H05655.\nAppendix A. Parametric oscillation by microwave voltage\nIn the main text, two time-dependent inputs are added to the magnetic\nanisotropy \feld. One is a microwave voltage and the other is uniformly dis-\ntributed random numbers. The former induces a parametric oscillation [31].\nFigure A.5(a) shows the spatial trajectory of the magnetization oscillation in\na steady state. As mentioned in the main text, the magnetization oscillates\naround the xaxis. The solid lines in Fig. A.5(b) show examples of the mag-\nnetization oscillation with di\u000berent initial conditions, whereas the dotted line\nrepresents the oscillation of the microwave voltage, sin(2 \u0019fLt). It indicates\nthat the oscillation frequency is a half of the microwave frequency.\nThe microwave voltage \fxes the phase of the magnetization with respect\nto the voltage oscillation. There are more than one solution of the magne-\ntization phase [32]. The phase depends on the initial conditions, as implied\nin Fig. A.5(b); see also Sec. Appendix B below. When we study chaos in\nthe main text, we choose the solutions of the LLG equation with the same\nphases because chaos is characterized by the sensitivity to the initial state.\n12Figure A.5: (a) Spatial trajectory of the parametric oscillation induced by a microwave\nvoltage. (b) Temporal evolution of mzwith di\u000berent initial conditions. Dotted line repre-\nsents the oscillation of the microwave voltage for comparison.\nAppendix B. Preparation of initial state\nChaotic dynamics are sensitive to the initial state. Therefore, to identify\nthe existence of chaos, it is necessary to study the dependence of the temporal\ndynamics on the initial state. We prepare natural initial states by solving\nthe LLG equation in the absence of the input signal. The value of HKis\nthat in the absence of external voltage and is 6 :28 kOe [31]. Also note\nthat, at zero temperature, the solution of the LLG equation saturates to\nthe minimum energy state, sin \u0012=Happl=HK, where\u0012relates tomzvia\nmz= cos\u0012. To obtain natural distribution of the initial state [12], we add a\ntorque,\u0000\rm\u0002h, due to thermal \ructuation to the right-hand side of Eq.\n(1). Here, the components of hsatisfy the \ructuation-dissipation theorem\n[44],\nhhk(t)h`(t0)i=2\u000bkBT\n\rMV\u000ek`\u000e(t\u0000t0); (B.1)\nwhere the saturation magnetization Mis assumed to be 955 emu/cm2[31].\nThe temperature Tis 300 K, while the volume is V=\u0019\u000250\u000250\u00021:1 nm3,\nwhich is typical for VCMA experiments. The thermal \ructuation excites\na small-amplitude oscillation of the magnetization around the energetically\nminimum state with the ferromagnetic resonance frequency. We pick up the\ntemporal directions of the oscillating magnetization and use them as the\nnatural initial states.\nFigure B.6(a) shows the spatial distribution of the initial states, where\n13Figure B.6: (a) Spatial distribution of the initial states prepared by solving the LLG\nequation with thermal \ructuation. (b) The samples of mx,my, andmzcorresponding to\nthe small-amplitude oscillation of the magnetization around the energetically minimum\nstate excited by thermal \ructuation.\nwe prepared 60 samples. Figure B.6(b) summarize the values of mfor these\nsamples. For example, the dynamics shown in Fig. 3 in the main text are\nderived by using the sample numbers 1 and 2 as the initial states, where the\nsolutions of the magnetization in both samples have the same phase when\nthe dynamics are driven by a microwave voltage. On the other hand, in Fig.\nA.5(b), the red and blue lines correspond to the sample number 1 and 15.\nThey are unsuitable to study chaos because the dynamical states at which\nthe random input signal is injected are greatly di\u000berent.\nAppendix C. Evaluation method of Lyapunov exponent\nThe Lyapunov exponent is evaluated by Shimada-Nagashima method [41].\nAs written in the main text, we add the random input signal from t= 5\u0016s.\nLet us denote the solution of the LLG equation at this time as m(t). We in-\ntroduce the zenith and azimuth angles, \u0012and', asm= (mx;my;mz) =\n(sin\u0012cos';sin\u0012sin';cos\u0012), i.e.,'= tan\u00001(my=mx) and\u0012= cos\u00001mz.\nThen, we also introduce m(1)(t) = (sin\u0012(1)cos'(1);sin\u0012(1)sin'(1);cos\u0012(1)).\nHere,\u0012(1)and'(1)satisfy\u000f=p\n[\u0012\u0000\u0012(1)]2+ ['\u0000'(1)]2, where\u000f= 1:0\u000210\u00005\nis a \fxed value. For convenience, let us introduce a notation,\nD[m(t);m(1)(t)] =q\n[\u0012(t)\u0000\u0012(1)(t)]2+ ['(t)\u0000'(1)(t)]2(C.1)\n14Solving the LLG equations of m(t) and m(1)(t), we obtain m(t+ \u0001t) and\nm(1)(t+ \u0001t). From them, we evaluate\nD[m(t+\u0001t);m(1)(t+\u0001t)] =q\n[\u0012(t+ \u0001t)\u0000\u0012(1)(t+ \u0001t)]2+ ['(t+ \u0001t)\u0000'(1)(t+ \u0001t)]2\n(C.2)\nThen, a temporal Lyapunov exponent at t+ \u0001tis given by\n\u0003(1)=1\n\u0001tlnD(1)\n\u000f; (C.3)\nwhere D(1)=D[m(t+ \u0001t);m(1)(t+ \u0001t)].\nNext, we introduce m(2)(t+\u0001t) = (sin\u0012(2)cos'(2);sin\u0012(2)sin'(2);cos\u0012(2)),\nwhere\u0012(2)and'(2)are de\fned as\n\u0012(2)(t+ \u0001t) =\u0012(t+ \u0001t) +\u000f\u0012(1)(t+ \u0001t)\u0000\u0012(t+ \u0001t)\nD[m(t+ \u0001t);m(1)(t+ \u0001t)]; (C.4)\n'(2)(t+ \u0001t) ='(t+ \u0001t) +\u000f'(1)(t+ \u0001t)\u0000'(t+ \u0001t)\nD[m(t+ \u0001t);m(1)(t+ \u0001t)]: (C.5)\nAccording to these de\fnitions, we notice that\nD[m(t+ \u0001t);m(2)(t+ \u0001t)] =\u000f: (C.6)\nIn other words, m(2)(t+ \u0001t) is de\fned by moving m(t+ \u0001t) to the direction\nofm(1)(t+ \u0001t) with a distance \u000fin the (\u0012;') phase space. Then, we solve\nthe LLG equations for m(t+ \u0001t) and m(2)(t+ \u0001t) and obtain m(t+ 2\u0001t)\nandm(2)(t+ 2\u0001t). The temporal Lyapunov exponent at t+ 2\u0001tis\n\u0003(2)=1\n\u0001tlnD(2)\n\u000f; (C.7)\nwhere D(2)=D[m(t+ 2\u0001t);m(1)(t+ 2\u0001t)].\nThese procedures are generalized. At t+n\u0001t, we have m(t+n\u0001t) =\n(sin\u0012(t+n\u0001t) cos'(t+n\u0001);sin\u0012(t+n\u0001t) sin'(t+n\u0001t);cos\u0012(t+n\u0001t)) and\nm(n)(t+n\u0001t) = (sin\u0012(n)(t+n\u0001t) cos'(n)(t+n\u0001);sin\u0012(n)(t+n\u0001t) sin'(n)(t+\nn\u0001t);cos\u0012(n)(t+n\u0001t)). Then, we de\fne m(n+1)(t+n\u0001t) = (sin\u0012(n+1)(t+\nn\u0001t) cos'(n+1)(t+n\u0001);sin\u0012(n+1)(t+n\u0001t) sin'(n+1)(t+n\u0001t);cos\u0012(n+1)(t+\nn\u0001t)) by moving m(t+n\u0001t) to the direction of m(n)(t+n\u0001t) with a distance\n15\u000fin the phase space as\n\u0012(n+1)(t+n\u0001t) =\u0012(t+n\u0001t) +\u000f\u0012(n)(t+n\u0001t)\u0000\u0012(t+n\u0001t)\nD[m(t+n\u0001t);m(n)(t+n\u0001t)];(C.8)\n'(n+1)(t+n\u0001t) ='(t+n\u0001t) +\u000f'(n)(t+n\u0001t)\u0000'(t+n\u0001t)\nD[m(t+n\u0001t);m(n)(t+n\u0001t)]:(C.9)\nNote thatD[m(t+n\u0001t);m(n+1)(t+n\u0001t)] =\u000f. Then, solving the LLG\nequations of m(t+n\u0001t) and m(n+1)(t+n\u0001t), we obtain m(t+ (n+ 1)\u0001t)\nandm(n+1)(t+ (n+ 1)\u0001t). A temporal Lyapunov exponent at t+ (n+ 1)\u0001t\nis\n\u0003(n+1)=1\n\u0001tlnD(n+1)\n\u000f; (C.10)\nwhere D(n+1)=D[m(t+(n+1)\u0001t);m(n+1)(t+(n+1)\u0001t)]. The Lyapunov ex-\nponent is de\fned as a long-time average of the temporal Lyapunov exponent\nas\n\u0003= lim\nN!11\nNNX\ni=1\u0003(i): (C.11)\nNote that m(1)(t) given at the initial time can point in arbitrary direc-\ntions, although it should satisfy the condition D[m(t);m(1)(t)] =\u000f. Shimada-\nNagashima method assumes that, even if the initial perturbation points to in\narbitrary directions, the di\u000berence of two solutions will arrive at the mostly\nexpanded direction by repeating the procedure. Since the random input sig-\nnals are injected from t= 5:0\u0016s tot= 15:0\u0016s and the time increment is 1\nps, we evaluate (15 :0\u00005:0)\u0016s=1ps= 107temporal Lyapunov exponents and\nevaluate the average.\nReferences\n[1] J. Torrejon, M. Riou, F. A. Araujo, S. Tsunegi, G. Khalsa, D. Quer-\nlioz, P. Bortolotti, V. Cros, K. Yakushiji, A. Fukushima, H. Kub-\nota, S. Yuasa, M. D. Stiles, J. Grollier, Neuromorphic computing with\nnanoscale spintronic oscillators, Nature 547 (2017) 428.\n[2] W. A. Borders, H. Akima, S. Fukami, S. Moriya, S. Kurihara, Y. Horio,\nS. Sato, H. Ohno, Analogue spin-orbit torque device for arti\fcial-neural-\nnetwork-based associative memory operation, Appl. Phys. Express 10\n(2017) 013007.\n16[3] K. Kudo, T. Morie, Self-feedback electrically coupled spin-Hall oscillator\narray for pattern-matching operation, Appl. Phys. Express 10 (2017)\n043001.\n[4] J. Grollier, D. Querlioz, K. Y. Camsari, K. Everschor-Sitte, S. Fukami,\nM. D. Stiles, Neuromorphic spintronics, Nat. Electron. 3 (2020) 360.\n[5] K. Nakajima, I. Fischer (Eds.), Reservoir Computing: Theory, Physical\nImplementations, and Applications, Springer, Singapore, 2021.\n[6] G. Manjunath, P. Ti\u0014 no, H. Jaeger, Theory of input driven dynamical\nsystems, in: ESANN 2012 proceedings, 2012.\n[7] Z. F. Mainen, T. J. Sejnowski, Reliability of Spike Timing in Neocortical\nNeurons, Science 268 (1995) 1503.\n[8] R. Toral, C. R. Mirasso, E. Hern\u0013 andez-Garc\u0013 \u0010a, O. Piro, Analytical and\nnumerical studies of noise-induced synchronization of chaotic systems,\nChaos 11 (2001) 655.\n[9] J. N. Teramae, D. Tanaka, Robustness of the Noise-Induced Phase Syn-\nchronization in a General Class of Limit Cycle Oscillators, Phys. Rev.\nLett. 93 (2004) 204103.\n[10] D. S. Goldobin, A. Pikovsky, Synchronization and desynchronization\nof self-sustained oscillators by common noise, Phys. Rev. E 71 (2005)\n045201(R).\n[11] H. Nakao, K. Arai, Y. Kawamura, Noise-Induced Synchronization and\nClustering in Ensembles of Uncoupled Limit-Cycle Oscillators, Phys.\nRev. Lett. 98 (2007) 184101.\n[12] Y. Imai, S. Tsunegi, K. Nakajima, T. Taniguchi, Noise-induced synchro-\nnization of spin-torque oscillators, Phys. Rev. B 105 (2022) 224407.\n[13] K. Aihara, T. Takabe, M. Toyoda, Chaotic neural networks, Phys. Lett.\nA 144 (1990) 333.\n[14] S. H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to\nPhysics, Biology, Chemistry, and Engineering, 1st Edition, Westview\nPress, 2001.\n17[15] S. Tsunegi, T. Taniguchi, S. Miwa, K. Nakajima, K. Yakusjiji,\nA. Fukushima, S. Yuasa, H. Kubota, Evaluation of memory capacity\nof spin torque oscillator for recurrent neural networks, Jpn. J. Appl.\nPhys. 57 (2018) 120307.\n[16] S. Tsunegi, T. Taniguchi, K. Nakajima, S. Miwa, K. Yakushiji,\nA. Fukushima, S. Yuasa, H. Kubota, Physical reservoir computing based\non spin torque oscillator with forced synchronization, Appl. Phys. Lett.\n114 (2019) 164101.\n[17] M. Riou, J. Torrejon, B. Garitaine, F. A. Araujo, P. Bortolotti,\nV. Cros, S. Tsunegi, K. Yakushiji, A. Fukushima, H. Kubota, S. Yuasa,\nD. Querlioz, M. D. Stiles, J. Grollier, Temporal Patter Recognition with\nDelayed-Feedback Spin-Torque Nano-Oscillators, Phys. Rev. Applied 12\n(2019) 024049.\n[18] T. Yamaguchi, N. Akashi, S. Tsunegi, H. Kubota, K. Nakajima,\nT. Taniguchi, Periodic structure of memory function in spintronics reser-\nvoir with feedback current, Phys. Rev. Research 2 (2020) 023389.\n[19] T. Yamaguchi, N. Akashi, K. Nakajima, H. Kubota, S. Tsunegi,\nT. Taniguchi, Step-like dependence of memory function on pulse width\nin spintronics reservoir computing, Sci. Rep. 10 (2020) 19536.\n[20] N. Akashi, T. Yamaguchi, S. Tsunegi, T. Taniguchi, M. Nishida,\nR. Sakurai, Y. Wakao, K. Nakajima, Input-driven bifurcations and in-\nformation processing capacity in spintronics reservoirs, Phys. Rev. Re-\nsearch 2 (2020) 043303.\n[21] M. Weisheit, S. F ahler, A. Marty, Y. Souche, C. Poinsignon, D. Givord,\nElectric Field-Induced Modi\fcation of Magnetism in Thin-Film Ferro-\nmagnets, Science 315 (2007) 349.\n[22] C.-G. Duan, J. P. Velev, R. F. Sabirianov, Z. Zhu, J. Chu, S. S. Jaswal,\nE. Y. Tsymbal, Surface Magnetoelectric E\u000bect in Ferromagnetic Metal\nFilms, Phys. Rev. Lett. 101 (2008) 137201.\n[23] T. Maruyama, Y. Shiota, T. Nozaki, K. Ohta, N. Toda, M. Mizuguchi,\nA. A. Tulapurkar, T. Shinjo, M. Shiraishi, S. Mizukami, Y. Ando,\nY. Suzuki, Large voltage-induced magnetic anisotropy change in a few\natomic layers of iron, Nat. Nanotechnol. 4 (2009) 158.\n18[24] K. Nakamura, R. Shimabukuro, Y. Fujiwara, T. Akiyama, T. Ito, A. J.\nFreeman, Giant Modi\fcation of the Magnetocrystalline Anisotropy in\nTransition-Metal Monolayers by an External Electric Field, Phys. Rev.\nLett. 102 (2009) 187201.\n[25] M. Tsujikawa, T. Oda, Finite Electric Field E\u000bects in the Large Perpen-\ndicular Magnetic Anisotropy Surface Pt/Fe/Pt(001): A First-Principles\nStudy, Phys. Rev. Lett. 102 (2009) 247203.\n[26] Y. Shiota, T. Maruyama, T. Nozaki, T. Shinjo, M. Shiraishi, Y. Suzuki,\nVoltage-Assisted Magnetization Switching in Ultrahin Fe 80Co20Alloy\nLayers, Appl. Phys. Express 2 (2009) 063001.\n[27] T. Nozaki, Y. Shiota, M. Shiraishi, T. Shinjo, Y. Suzuki, Voltage-\ninduced perpendicular magnetic anisotropy change in magnetic tunnel\njunctions, Appl. Phys. Lett. 96 (2010) 022506.\n[28] M. Endo, S. Kanai, S. Ikeda, F. Matsukura, H. Ohno, Electric-\n\feld e\u000bects on thickness dependent magnetic anisotropy of sputtered\nMgO/Co 40Fe40B20/Ta structures, Appl. Phys. Lett. 96 (2010) 212503.\n[29] S. Miwa, M. Suzuki, M. Tsujikawa, K. Matsuda, T. Nozaki, K. Tanaka,\nT. Tsukahara, K. Nawaoka, M. Goto, Y. Kotani, T. Ohkubo, F. Bonell,\nF. Tamura, K. Hono, T. Nakamura, M. Shirai, S. Yuasa, Y. Suzuki,\nVoltage controlled interfacial magnetism through platinum orbits, Nat.\nCommun. 8 (2017) 15848.\n[30] T. Taniguchi, A. Ogihara, Y. Utsumi, S. Tsunegi, Spintronic reservoir\ncomputing without driving current or magnetic \feld, Sci. Rep. 12 (2022)\n10627.\n[31] T. Yamamoto, T. Nozaki, H. Imamura, S. Tamaru, K. Yakushiji,\nH. Kubota, A. Fukushima, S. Yuasa, Voltage-Driven Magnetization\nSwitching Controlled by Microwave Electric Field Pumping, Nano Lett.\n20 (2020) 6012.\n[32] H. Imamura, R. Matsumoto, Large-Angle Precession of Magnetization\nMaintained by a Microwave Voltage, Phys. Rev. Applied 14 (2020)\n064062.\n19[33] S. Yakata, H. Kubota, Y. Suzuki, K. Yakushiji, A. Fukushima, S. Yuasa,\nK. Ando, In\runence of perpendicular magnetic anisotropy on spin-\ntransfer switching current in CoFeB/MgO/CoFeB magnetic tunnel\njunctions, J. Appl. Phys. 105 (2009) 07D131.\n[34] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan, M. Endo,\nS. Kanai, J. Hayakawa, F. Matsukura, H. Ohno, A perpendicular-\nanisotropy CoFeB-MgO magnetic tunnel junction, Nat. Mater. 9 (2010)\n721.\n[35] H. Kubota, S. Ishibashi, T. Saruya, T. Nozaki, A. Fukushima,\nK. Yakushiji, K. Ando, Y. Suzuki, S. Yuasa, Enhancement of perpendic-\nular magnetic anisotropy in FeB free layers using a thin MgO cap layer,\nJ. Appl. Phys. 111 (2012) 07C723.\n[36] S. Contreras-Celada, M. G. Clerc, S. Coulibaly, R. G. Rojas, A. O. Leon,\nVoltage-driven multistability and chaos in magnetic \flms, J. Magn.\nMagn. Mater. 562 (2022) 169793.\n[37] A. Pikovsky, M. Rosenblum, J. Kurths, Synchronization: A universal\nconcept in nonlinear sciences, 1st Edition, Cambridge University Press,\n2003.\n[38] T. Nozaki, A. Kozio l-Rachwa l, M. Tsujikawa, Y. Shiota, X. Xu,\nT. Ohkubo, T. Tsukahara, S. Miwa, M. Suzuki, S. Tamaru, H. Kubota,\nA. Fukushima, K. Hono, M. Shirai, Y. Suzuki, S. Yuasa, Highly e\u000bcient\nvoltage control of spin and enhanced interfacial perpendicular magnetic\nanisotropy in iridium-doped Fe/MgO magnetic tunnel junctions, NPG\nAsia Mater. 9 (2017) e451.\n[39] T. Nozaki, M. Endo, M. Tsujikawa, T. Yamamoto, T. Nozaki,\nM. Konoto, H. Ohmori, Y. Higo, H. Kubota, A. Fukushima, M. Hosomi,\nM. Shirai, Y. Suzuki, S. Yuasa, Voltage-cotrolled magnetic anisotropy\nin an ultrathin Ir-doped Fe layer with a CoFe termination layer, APL\nMater. 8 (2020) 011108.\n[40] A. Sugihara, T. Nozaki, H. Kubota, H. Imamura, A. Fukushima,\nK. Yakushiji, S. Yuasa, Evaluation of higher order magnetic anisotropy\nin a perpendicularly magnetized epitaxial ultrathin Fe layer and its ap-\nplied voltage dependence, Jpn. J. Appl. Phys. 58 (2019) 090905.\n20[41] I. Shimada, T. Nagashima, A Numerical Approach to Ergodic Problem\nof Dissipative Dynamical Systems, Prog. Theor. Phys. 61 (1979) 1605.\n[42] N. Bertschinger, T. Natschl ager, Real-Time Computation at the Edge of\nChaos in Recurrent Neural Networks, Neural. Comput. 16 (2004) 1413.\n[43] J. Nakayama, K. Kanno, A. Uchida, Laser dynamical reservoir comput-\ning with consistency: an approach of a chaos mask signal, Opt. Express\n24 (2016) 8679{8692.\n[44] W. F. B. Jr, Thermal Fluctuations of a Single-Domain Particle, Phys.\nRev. 130 (1963) 1677.\n21"    },    {        "title": "1311.1668v1.Dynamics_of_two_dimensional_complex_plasmas_in_a_magnetic_field.pdf",        "content": "Dynamics of two-dimensional complex plasmas in a magnetic \feld\nT. Ott1;2, H. L owen1, and M. Bonitz2\n1Institut f ur Theoretische Physik II: Weiche Materie,\nHeinrich-Heine-Universit at D usseldorf, Universit atsstra\u0019e 1, D-40225 D usseldorf, Germany\n2Christian-Albrechts-Universit at zu Kiel, Institut f ur Theoretische\nPhysik und Astrophysik, Leibnizstra\u0019e 15, 24098 Kiel, Germany\n(Dated: September 3, 2021)\nWe consider a two-dimensional complex plasma layer containing charged dust particles in a per-\npendicular magnetic \feld. Computer simulations of both one-component and binary systems are\nused to explore the equilibrium particle dynamics in the \ruid state. The mobility is found to\nscale with the inverse of the magnetic \feld strength (Bohm di\u000busion) for strong \felds. For bidis-\nperse mixtures, the magnetic \feld dependence of the long-time mobility depends on the particle\nspecies providing an external control of their mobility ratio. For large magnetic \felds, even a two-\ndimensional model porous matrix can be realized composed by the almost immobilized high-charge\nparticles which act as obstacles for the mobile low-charge particles.\nPACS numbers: 52.27.Gr, 52.27.Lw, 52.25.Fi\nI. INTRODUCTION\nTransport properties in liquids are relevant for various\napplications ranging from solvation of tablets [1] the pen-\netration of salt ions into fresh river water [2] to imbibition\nproblems [3]. Hence, there is a need for a basic under-\nstanding of particle di\u000busion on the most fundamental\nlevel of the individual particles. The particle trajecto-\nries, as governed by Newton's equation of motion with\nthe interparticle forces, are the natural starting point to\nunderstand and predict the transport properties [4]. Al-\nready in equilibrium this is still a nontrivial problem of\nclassical statistical mechanics.\nComplex plasmas [5{7] which contain mesoscopic dust\ngrains are ideal model systems to follow the dynamics\non the time and length scale of the individual particles.\nTypically the dust particles are highly charged such that\nthere are strong repulsive e\u000bective interactions between\nneighbors. At high densities, the system can therefore\nexhibit both \ruid and solid phases [8]. In this paper, we\nstudy the particle dynamics in a two-dimensional com-\nplex plasma which is exposed to an external magnetic\n\feld of strength B. Though the presence of a mag-\nnetic \feld does not alter the equilibrium static proper-\nties, such as phase transitions, the dynamics are strongly\na\u000bected [9]. Due to the Lorentz force, the charged parti-\ncles exhibit a circular motion [10{12] which is expected\nto slow the dynamics down. Therefore the magnetic \feld\nopens the fascinating possibility to change the dynamical\nproperties of the system externally without changing the\nsystem itself, e.g., Ref. [13].\nSome aspects of the one-component dust particle dy-\nnamics in a magnetic \feld have been considered in pre-\nvious studies [5, 9, 11, 14{17]. In particular, the motion\nof few-particle clusters in magnetic \felds has been stud-\nied both by experiments and simulation [10, 12]. The\ndynamics of three-dimensional ionic binary mixture have\nalso been under investigation in early as well as recent\nresearch [18{21], with a particular focus on astrophysical\naa\ng(r)a\ng(r)\n0.01.02.0\nr/a 0 1 2 3 4 5 6g11g12g22\n0.01.02.0\nr/a 0 1 2 3 4 5 6FIG. 1. (Color online) Pair distribution functions and par-\nticle trajectories during !pt= 30 at \u0000 = 30, \f= 0:5.Left:\nOne-component system. Right: Binary mixture with Qr= 4\nandnr= 1=3. The highly charged particles are shown in\nred/light.\nplasmas and those encountered in inertial con\fnement\nfusion.\nHere, we focus on two-dimensional systems and ex-\nplore the long-time dynamics by computer simulations.\nWe con\frm the 1 =B-scaling of the long-time di\u000busion\ncoe\u000ecient for strong magnetic \felds [11] for the two-\ndimensional system. We moreover consider binary sys-\ntems composed of high-charge and low-charge parti-\ncles [22{24]. Our motivation to study a binary system\ncomes from the fact that the magnetic \feld a\u000bects the\ndynamics of the particle species di\u000berently. Thereby, the\nindividual particle dynamics can be steered externally\nvia the magnetic \feld. One important parameter is the\nmobility ratio of the two species which governs the mu-\ntual di\u000busion and is one key parameter for the nature\nof the kinetic glass transition in mixtures [25{27]. ThisarXiv:1311.1668v1  [physics.plasm-ph]  7 Nov 20132\nu(t)\nωpt∼t2∼t\nβ= 0\nβ= 1\nβ= 3\nβ= 10\n10−510−410−310−210−1100101102103\n0.1 1 10 100 1000\nFIG. 2. (Color online) MSD of a one-component system with\n\u0000 = 100 at di\u000berent magnetic \feld strengths.The straight\nlines indicate linear and quadratic growth.\nratio is typically \fxed by the mass ratio [28] and the\ninteractions [29] and can therefore not easily be tuned.\nHere we show that this ratio can be controlled by an\nexternal magnetic \feld insofar as the high-charge par-\nticles are more immobilized than the low-charge parti-\ncles. For large magnetic \felds, it is even conceivable that\nthe high-charge particles are almost immobilized while\nthe low-charge particles are still mobile. This opens the\nway to realize a porous model matrix in two dimensions.\nRecently a similar matrix has been created by adsorb-\ning colloidal particles to a substrate [30]. Our approach,\nhowever, is more \rexible as everything can be controlled\nexternally.\nThis paper is organized as follows: in section II we de-\nscribe our model of complex plasmas in a magnetic \feld.\nIn III we describe results for the one-component case.\nThe binary mixtures is then considered in section IV.\nFinally, we conclude in section V.\nII. MODEL\nWe consider one-component systems and charge-\nasymmetric binary mixtures of uniform mass m, charge\nratioQr=q2=q1, and density ratio nr=n2=n1, where\nthe numeric indices label the particles species. The par-\nticles are situated in a two-dimensional quadratic simula-\ntion box of side length L, giving rise to partial densities\nn1;2=N1;2=L2, and interact via a screened Coulomb\ninteraction with screening length \u0015,\nVij(ri;rj) =qiqj\njri\u0000rjjexp\u0010\n\u0000jri\u0000rjj=\u0015\u0011\n: (1)\nIn addition, we consider the in\ruence of an external\nmagnetic \feld Bperpendicular to the particle plane, giv-\ning rise to the cyclotron frequency !c;1;2=jq1;2jB=(mc)\n(cis the speed of light).\nD∗[a2ωp]\nβ=ωc/ωpΓ = 10\nΓ = 30\nΓ = 50∼β−1\n∼β−1\n10−310−210−1100\n0 0.01 0 .1 1 10Γ = 80\nΓ = 100\nΓ = 130\nΓ = 160\nΓ = 180\nD∗/D∗\n0\nβ=ωc/ωp∼β−1\n∼β−1\n1+1\n3β\n1+7\n4β+β210−1100\n0 0.01 0 .1 1 10FIG. 3. (Color online) Top:D\u0003as a function of \ffor values\nof \u0000 as indicated in the \fgure. The dotted lines show a decay\n\f\u00001as a guide for the eye. Bottom:D\u0003normalized by the\n\feld free value D\u0003\n0=D\u0003(0). The normalized values fall on a\nuniversal curve for all values of \u0000.\nThe system is fully described by a set of \fve pa-\nrameters: Qr,nr,\u0014, \u0000, and\f. Here, the screening\nstrength is de\fned as \u0014=a=\u0015 with the Wigner-Seitz\nradiusa= (\u0019(n1+n2))\u00001=2, \u0000 = \u0000 1=Q2\n1=(akBT) (T\nis the temperature), and \f=\f1=\f2=!c;1;2=!p;1;2,\nwhere!p;1;2=\u0000\n2q2\n1;2=(a3m)\u00011=2is the nominal Coulomb\nplasma frequency. In the following, we normalize lengths\nbyaand times by the inverse of !p_ =!p;1.3\nOur investigations are carried out by molecular dynam-\nics simulation for N= 16 320 particles and encompass a\nmeasurement time of !pt= 100 000 which is preceded\nby an equilibration period. The simulation is carried out\nin the microcanonical ensemble at \u0014= 1; typical trajec-\ntory snapshots are shown in Fig. 1. Notice the familiar\ncircular paths induced by the magnetic \feld and the dif-\nferent mobility of the particle species in the binary sys-\ntem. An external magnetic \feld does not in\ruence the\nequilibrium structure of one-component systems or bi-\nnary mixtures (Bohr-van Leeuwen theorem). The charge\nratio, on the other hand, has a strong in\ruence on the\nstructure as quanti\fed by the pair distribution function\ng\u000b\f(r), see upper graphs in Fig. 1. In the binary mixture,\nthe lightly charged particles exhibit a smaller correlation\ngap at small distances and a lower peak height, indicat-\ning a smaller degree of correlation in this subsystem. The\nhighly charged subsystem is considerably more correlated\n(seeg22(r) in Fig. 1), and the cross-correlation between\nthe particles species ( g12(r)) falls in-between.\nThe study of the dynamics of the system is undertaken\nby calculating the mean-squared displacement (MSD)\nu(t) de\fned as\nu(t) =hjri(t)\u0000ri(t0)j2ii;t0; (2)\nwhere the averaging is over all particles and all starting\ntimest0. According to classical transport theory, the\ndi\u000busion coe\u000ecient follows as\nD=1\n4lim\nt!1u(t)\nt: (3)\nSince the existence of Fickian di\u000busion is doubtful for\nstrongly coupled two-dimensional Yukawa systems [31,\n32], we evaluate Eq. (3) at a \fxed time instant t!p= 4850\nand denote it D\u0003, keeping in mind that this measure of\nthe mobility should not be identi\fed with the long-time\ndi\u000busion coe\u000ecient.\nIII. ONE-COMPONENT SYSTEM\nBefore investigating the binary system, we \frst estab-\nlish the general di\u000busion trends in a magnetized, one-\ncomponent 2D Yukawa system, which are laid out here\nfor the \frst time [33]. The behavior of the MSD in such\na system at \u0000 = 100 is shown in Fig. 2 for di\u000berent mag-\nnetic \feld strengths. For \f= 0, the ballistic regime with\na quadratic increase at small times is followed by a quasi-\ndi\u000busive regime in which the MSD grows almost linearly\nwith time. With increasing magnetic \feld, the signa-\nture of the circular paths is visible in the MSD curves\nas an oscillatory growth. The localization of individual\nparticles at high magnetic \feld values gives rise to an\nadditional regime at very small time delays during which\nthe MSD is subdi\u000busive, i.e, during which u(t) grows less\nthan linearly with time.\nThe scaling of the di\u000busivity D\u0003as a function of the\nmagnetic \feld strength is of central interest with regard\nu1(t),u2(t)\nωpt∼t2∼t\nβ= 0\nβ= 1\nβ= 3\nβ= 10\n10−510−410−310−210−1100101102103\n0.1 1 10 100 1000FIG. 4. (Color online) MSD of a binary system with nr= 1,\nQr= 0:5, and \u0000 = 100 at di\u000berent magnetic \feld strengths.\nThe lower of each pair of curves corresponds to the more\nhighly charged particles. The straight lines indicate linear\nand quadratic growth.\nto the dynamics of the system. This scaling is shown in\nFig. 3. For small values of \f, the rapidity of the di\u000busive\nmotion is una\u000bected, regardless of the coupling constant\n\u0000. Only when magnetic \feld e\u000bects become important at\n\f&0:1 doesD\u0003begin to decay. At \f\u00191, the scaling be-\ncomes the familiar Bohm type di\u000busion, D\u0003/1=\f[36].\nThis is the same behavior that was found in the di\u000bu-\nsion perpendicular to the \feld in strongly coupled three-\ndimensional OCPs [11].\nThe functional form of the D\u0003(\f) dependence is quite\ninsensitive to \u0000, as demonstrated in the lower graph of\nFig. 3. This is in contrast with the corresponding be-\nhavior of a three-dimensional OCP [11] which shows a\nclear \u0000-dependence both in \feld-parallel and perpendic-\nular di\u000busion. The reason for the more complex behavior\nin 3D systems is the mutual interference between the two\ndi\u000busion directions (mediated by the strong coupling be-\ntween the particles), which is absent in 2D systems.\nIV. BINARY SYSTEM\nIn this section, we expand on the previous investigation\nand consider charge-asymmetric binary Yukawa systems\nwith a repulsive interaction. The density ratio is \fxed to\nnr= 1, i.e.,N1=N2, while the charge ratio Qrand the\nmagnetic \feld strength \fare varied.\nFigure 4 shows the MSD of such a binary system at\nQr= 0:5 and di\u000berent magnetization; for each value of\n\f, there are two curves, re\recting the two particle species.\nEvidently, the particles carrying a lower charge are more\nmobile, regardless of the magnetic \feld strength. For in-\ncreasing\f, however, the disparity in mobility between\nthe two species grows steadily, as evidenced by the in-\ncreasing gap between the two MSD curves when going\nfrom zero magnetic \feld to \f= 10.4\nD∗[a2ωp]\nβ=ωc/ωp∼β−1\nΓ = 30Qr= 0.5\nΓ = 100Qr= 0.5\nΓ = 100Qr= 0.2\n10−310−210−1100\n0 0.01 0 .1 1 10\nFIG. 5. (Color online) D\u0003as a function of \ffor binary systems\nwith charge ratio Qr= 0:5 andQr= 0:2. The lower one of\neach pair of curves corresponds to the more highly charged\nspecies. The dotted lines show a decay \f\u00001as a guide for\nthe eye.\nMore data are presented in Fig. 5 which shows D\u0003as\na function of \ffor two values of Qr. The functional form\nof the data is comparable to the one-component case con-\nsidered in the previous section and is well described by\nBohmian di\u000busion for both species for \f&1. A closer\nlook, however, reveals that the response of the less highly\ncharged (more mobile) species is shifted to higher values\n\fwhich results in an increase in the mobility ratio be-\ntween the two species.\nIn the upper graph of Fig. 6, this is demonstrated\nfor strong coupling, \u0000 = 100; 160, by plotting the ratio\nD\u0003\n2=D\u0003\n1. While a modest charge ratio of Qr= 0:8 results\nonly in a small variation of this ratio, the in\ruence of\nthe magnetic \feld grows with decreasing Qr, so that at\nQr= 0:2, the magnetic \feld alone can be used to ma-\nnipulate the mobility ratio by a factor of two for \f= 10\n(see right-hand scale in Fig. 6). Since the mobility ratio\nplays a crucial role during the glass transition, this e\u000bect\ncan be leveraged to investigate the conditions for glass\nformation in one and the same system by controlling the\nmobility ratio by the external magnetic \feld.\nThe relatively simple, monotonic dependence of\nD\u0003\n2=D\u0003\n1on\ffor strongly coupled plasmas shown in the\nupper part of Fig. 6 has to be contrasted with the more\nintricate behavior of the same ratio for \u0000 = 30 (lower\ngraph in Fig. 6). Here, a highly non-monotonic depen-\ndence of the ratio D\u0003\n2=D\u0003\n1is observed, which becomes\nmore strongly pronounced for more disparate charge ra-\ntios. The strong growth of D\u0003\n2=D\u0003\n1, which unfolds undis-\nturbed at large values of \u0000, is suppressed at magnetic \feld\nstrengths surpassing \fc\u00191, leading to the formation of\na pronounced peak at \fc.\nThe microscopic reason for the suppression at large\nD∗\n2/D∗\n1\nβ=ωc/ωpD∗\n2/D∗\n1\nβ=ωc/ωp\nQr= 0.2\nQr= 0.5\nQr= 0.8\nQr= 1.0\n0.51.01.52.02.53.0\n0 0.01 0.1 1 10Γ = 100\nΓ = 160\n0.51.01.52.02.53.0\n0 0.01 0.1 1 101.01.11.21.31.41.51.61.71.81.92.02.1\nD∗\n2/D∗\n1\nβ=ωc/ωpQr=\n0.05\n0.10\n0.15\n0.20\n0.5\n0.8D∗\n2/D∗\n1\nβ=ωc/ωpQr=\n0.05\n0.10\n0.15\n0.20\n0.5\n0.8\n1.02.03.04.05.06.07.0\n0 0.01 0.1 1 101.02.03.04.05.06.07.0\n0 0.01 0.1 1 10FIG. 6. (Color online) Ratio D\u0003\n2=D\u0003\n1for \u0000 = 100; 160 (top)\nand \u0000 = 30 (bottom) at di\u000berent charge ratios Qras a func-\ntion of\f. The right axis in the top graph shows the relative\nchange forQr= 0:2;\u0000 = 100, normalized to \f= 0.\nmagnetic \felds lies in the reduced mobility of the lightly\ncharged species. In a qualitative way, it can be traced\nback to a geometric origin by considering the ratio of\nlength scales \u0001 = rL;2=pn1between the Larmor ra-\ndiusrL;2of the lightly charged species and the aver-\nage nearest-neighbor distancepn1of the highly charged\nspecies. A lightly charged particle situated between two\nhighly charged ones will be forced on a curved trajectory\nby the magnetic \feld. At \u0001 = 1 =4, this trajectory leads,5\nD∗\n2/D∗\n1\n∆ =rL,2/√n1\n√n1rL,2Γ, Qr∆ = 0 .25\n80,0.1\n50,0.1\n40,0.1\n30,0.1\n30,0.15\n30,0.2\n1.52.02.53.03.54.04.55.0\n0.1 1 10 100\nFIG. 7. (Color online) Ratio D\u0003\n2=D\u0003\n1as a function of \u0001 =\nrL;2=pn1. Note that small values of \u0001 correspond to large\nmagnetic \felds, and vice versa.\nat thermal velocity of the particle, to a collision with one\nof the highly charged particle, e\u000bectively preventing the\ndi\u000busion of the lightly charged particle (see schematic in\nFig. 7). This results in a reduction of D\u0003\n2=D\u0003\n1at \u0001 = 1=4.\nWe have tested this simple geometric reason by plotting\ndata for the mobility ratio as a function of the geometric\nparameter \u0001, see Fig. 7 (notice that rL;2is a function\nof both \u0000 and Qr). In fact, a resonant dip in the mo-\nbility ratio is formed around \u0001 = 1 =4 supporting the\nunderlying picture. This geometric resonance e\u000bect per-\nsists across di\u000berent parameter regimes, but becomes less\npronounced as Qror \u0000 are increased, since an increased\nparticle coupling leads to stronger caging e\u000bects.\nV. CONCLUSION\nIn conclusion, we have explored the dynamics of\ncharged particles in a complex plasma layer exposed to\na perpendicular magnetic \feld which allows for an addi-tional external control parameter for the particle trans-\nport. Our simulation results can be veri\fed in experi-\nments of dusty plasmas either in magnetic \felds [37] or\nin rotating electrodes which formally lead to the same\nequations of motion [10, 12, 13]. Binary systems can also\nbe realized in dusty plasmas, e.g., Ref. [38].\nWe have demonstrated that the mobility in two-\ndimensional systems adheres to the same 1 =B-Bohm scal-\ning as in three-dimensional systems. In contrast to three-\ndimensional systems, however, the functional form of the\nscaling is largely independent of the coupling \u0000, indicat-\ning a decoupling of magnetic and interaction e\u000bects.\nOur main focus has been on the response of a charge-\nasymmetric binary mixture to an external magnetic \feld.\nSince the two subsystems are a\u000bected di\u000berently by the\nmagnetic \feld, the mobility ratio between them can be\ncontrolled by the strength of the magnetic \feld. For less\nstrongly coupled systems and high charge-asymmetry, we\nhave found that the circular trajectories of the lightly\ncharged particles can be in resonance with the positional\ncon\fguration of the highly charged particles, which leads\nto a distinct reduction of the mobility of the former. This\nis an interesting realization of a porous model matrix in\na \ruid system.\nFor future studies, as regards binary systems, a sys-\ntematic understanding of the two-dimensional glass tran-\nsition in binary mixtures is lying ahead where the mag-\nnetic \feld is exploited as a steering wheel to change the\nmobility ratio between the particle species. Moreover, it\nis known that the crystallization process out of an un-\ndercooled melt depends sensitively on the mobility ratio\nin binary systems [39] such that the magnetic \feld can\nbe used to tune crystal nucleation in mixtures [40{43].\nACKNOWLEDGMENTS\nWe thank Zoltan Donk\u0013 o and Peter Hartmann (Bu-\ndapest) for numerous stimulating discussions in the early\nstages of this work. This work is supported by the\nDeutsche Forschungsgemeinschaft via SFB TR 6 and\nSFB TR 24 and grant shp0006 at the North-German Su-\npercomputing Alliance HLRN.\n[1] P. Colombo, R. Bettini, and N. A. Peppas, J. Control.\nRelease 61, 83 (1999).\n[2] L. D. Wright and J. M. Coleman, J. Geophys. Res. 76,\n8649 (1971).\n[3] S. Gruener, Z. Sadjadi, H. E. Hermes, A. V. Kityk,\nK. Knorr, S. U. Egelhaaf, H. Rieger, and P. Huber, Proc.\nNatl. Acad. Sci. USA 109, 10245 (2012).\n[4] T. Voigtmann and J. Horbach, Phys. Rev. Lett. 103,\n205901 (2009); D. N. Perera and P. Harrowell, J.\nChem. Phys. 111, 5441 (1999); A. Winkler, J. Horbach,\nW. Kob, and K. Binder, ibid.120, 384 (2004).\n[5] M. Bonitz, Z. Donk\u0013 o, T. Ott, H. K ahlert, and P. Hart-mann, Phys. Rev. Lett. 105, 055002 (2010).\n[6] G. E. Mor\fll and A. V. Ivlev, Rev. Mod. Phys. 81, 1353\n(2009).\n[7] A. Ivlev, G. Mor\fll, H. L owen, and C. P. Royall,\nComplex Plasmas and Colloidal Dispersions: Particle-\nresolved Studiies of Classical Liquids and Solids , Series in\nsoft condensed matter (World Scienti\fc Publishing Com-\npany, Incorporated, 2012).\n[8] H. Thomas, G. Mor\fll, V. Demmel, J. Goree, B. Feuer-\nbacher, and D. M ohlmann, Phys. Rev. Lett. 73, 652\n(1994).\n[9] T. Ott, D. A. Baiko, H. K ahlert, and M. Bonitz, Phys.6\nRev. E 87, 043102 (2013).\n[10] H. K ahlert, J. Carstensen, M. Bonitz, H. L owen,\nF. Greiner, and A. Piel, Phys. Rev. Lett. 109, 155003\n(2012).\n[11] T. Ott and M. Bonitz, Phys. Rev. Lett. 107, 135003\n(2011).\n[12] M. Bonitz, H. K ahlert, T. Ott, and H. L owen, Plasma\nSources Sci. Technol. 22, 015007 (2013).\n[13] P. Hartmann, Z. Donk\u0013 o, T. Ott, H. K ahlert, and\nM. Bonitz, Phys. Rev. Lett. 111, 155002 (2013).\n[14] T. Ott, H. K ahlert, A. Reynolds, and M. Bonitz, Phys.\nRev. Lett. 108, 255002 (2012).\n[15] L.-J. Hou, P. K. Shukla, A. Piel, and Z. L. Mi\u0014 skovi\u0013 c,\nPhys. Plasmas 16, 073704 (2009).\n[16] T. Ott, H. L owen, and M. Bonitz, Phys. Rev. Lett. 111,\n065001 (2013).\n[17] T. Ott, M. Bonitz, P. Hartmann, and Z. Donk\u0013 o, Phys.\nRev. E 83, 046403 (2011); T. Ott, M. Bonitz, Z. Donko,\nH. K ahlert, and P. Hartmann, IEEE T. Plasma. Sci. 39,\n2768 (2011).\n[18] J. Hansen, F. Joly, and I. McDonald, Phys. A 132, 472\n(1985).\n[19] S. Bastea, Phys. Rev. E 71, 056405 (2005).\n[20] J. Daligault, Phys. Rev. Lett. 108, 225004 (2012).\n[21] S. W. S. Apolinario and F. M. Peeters, Phys. Rev. E 83,\n041136 (2011).\n[22] L. Assoud, R. Messina, and H. L owen, J. Chem. Phys\n129, 164511 (2008).\n[23] J. Dzubiella, G. P. Ho\u000bmann, and H. L owen, Phys. Rev.\nE65, 021402 (2002).\n[24] J. J. Gray and R. T. Bonnecaze, Langmuir 17, 7935\n(2001).\n[25] A. Imhof and J. K. G. Dhont, Phys. Rev. E 52, 6344\n(1995).\n[26] E. Zaccarelli, H. L owen, P. Wessels, F. Sciortino,\nP. Tartaglia, and C. N. Likos, Phys. Rev. Lett. 92,225703 (2004).\n[27] C. Mayer, E. Zaccarelli, E. Stiakakis, C. Likos,\nF. Sciortino, A. Munam, M. Gauthier, N. Hadjichris-\ntidis, H. Iatrou, P. Tartaglia, et al. , Nature Mater. 7,\n780 (2008).\n[28] J.-L. Barrat and A. Latz, J. Phys.: Condens. Matter 2,\n4289 (1990).\n[29] L. Assoud, F. Ebert, P. Keim, R. Messina, G. Maret,\nand H. L owen, Phys. Rev. Lett. 102, 238301 (2009).\n[30] S. Deutschl ander, T. Horn, H. L owen, G. Maret, and\nP. Keim, Phys. Rev. Lett. 111, 098301 (2013).\n[31] Z. Donk\u0013 o, J. Goree, P. Hartmann, and B. Liu, Phys.\nRev. E 79, 026401 (2009).\n[32] T. Ott and M. Bonitz, Phys. Rev. Lett. 103, 195001\n(2009); Contrib. Plasma Phys. 49, 760 (2009).\n[33] The magnetized 2D Coulomb ( \u0014= 0) system was \frst in-\nvestigated in Ref. [34]. Similar results have been obtained\nby Z. Donk\u0013 o [35].\n[34] S. Ranganathan, R. Johnson, and C. Woodward, Phys.\nChem. Liq. 40, 673 (2002).\n[35] Z. Donk\u0013 o, private communication (2009).\n[36] J. Spitzer, Lyman, Phys. Fluids. 3, 659 (1960).\n[37] E. Thomas, R. L. Merlino, and M. Rosenberg, Plasma\nPhys. Contr. Fusion 54, 124034 (2012).\n[38] P. Hartmann, Z. Donk\u0013 o, G. J. Kalman, S. Kyrkos, K. I.\nGolden, and M. Rosenberg, Phys. Rev. Lett. 103, 245002\n(2009).\n[39] H. L owen, E. Allahyarov, A. Ivlev, and G. E. Mor\fll, J.\nPhys.: Condens. Matter 24, 284125 (2012).\n[40] L. Assoud, R. Messina, and H. L owen, Mol. Phys. 109,\n1385 (2011).\n[41] H. L owen and E. Allahyarov, J. Chem. Phys 135, 134115\n(2011).\n[42] S. L. Taylor, R. Evans, and C. P. Royall, J. Phys.: Con-\ndens. Matter 24, 464128 (2012).\n[43] S. R. Ganagalla and S. N. Punnathanam, J. Chem. Phys\n138, 174503 (2013)."    },    {        "title": "1604.08780v2.Ferroelectrics_Manipulate_Magnetic_Bloch_Skyrmions_in_a_Composite_Bilayer.pdf",        "content": "Ferroelectrics Manipulate Magnetic Bloch Skyrmions in a Composite Bilayer\nZidong Wang\u0003and Malcolm J. Grimsony\nDepartment of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand\n(Dated: March 7, 2022)\nTheoretical investigation demonstrates that the composite bilayer (i.e., chiral-\nmagnetic/ferroelectric bilayer) o\u000bers the possibility of electric-induced magnetic Skyrmions\n[Phys. Rev. B 94, 014311 (2016)]. In this Article, we propose a micromagnetic model to\nphysically manipulate magnetic Bloch Skyrmions propagating in a chiral-magnetic thin \flm\nwith a polarized ferroelectric essential to drive the system through the converse magnetoelectric\ne\u000bect. E\u000bects caused by di\u000berent velocities of the propagation, sizes of the thin \flm, and strength\nof the magnetoelectric couplings strongly impact on quality and quantity of the magnetic Skyrmions.\nE-print: http://arxiv.org/abs/1604.08780\nI. INTRODUCTION\nMagnetic Bloch Skyrmions behave as stable particle-\nlike spin textures in the chiral-magnetic crystals [1],\nsuch as B20 compound metallic MnSi [2], FeGe [3],\nFe1\u0000xCoxSi [4], MnGe and Mn 1\u0000xFexGe [5], and mul-\ntiferroic Cu 2OSeO 3[6]. These materials have no inver-\nsion symmetry that can allow the emergence of mag-\nnetic Bloch Skyrmions, due to their non-centrosymmetric\nlattice structures [7]. In micromagnetics, this phe-\nnomenon is caused by two components: the nearest-\nneighbor (symmetric exchange) interaction and the in-\nherent Dzyaloshinskii-Moriya (asymmetric exchange) in-\nteraction [8]. The competition between them stabilizes\nthe helicity of magnetic Skyrmions [9]. Mathematically,\nthe Dzyaloshinskii-Moriya interaction Hdmiis the contri-\nbution of a non-linear exchange interaction between two\nneighboring magnetic spins, S1andS2[10, 11], written\nas\nHdmi=D\u0001[S1\u0002S2]; (1)\nwhereDis an oriented vector, which indicates the con-\nstrained helicity to the symmetric state. The nearest-\nneighbor interaction Hintcommonly exists in ferromag-\nnets as a linear exchange interaction,\nHint=J[S1\u0001S2]; (2)\nwhereJis the termed exchange coupling. Magnetic\nSkyrmion holds great potential for applications in spin-\ntronic memory devices, due to their self-protection be-\nhavior.\nSo far, magnetic Bloch Skyrmions have been observed\nin insulating multiferroics, i.e., Cu 2OSeO 3[6]. This mul-\ntiferroism o\u000bers an opportunity to generate magnetic\nSkyrmions by electric polarization [12]. It is due to the\nconverse magnetoelectric e\u000bect, which is the phenomenon\n\u0003Zidong.Wang@auckland.ac.nz\nym.grimson@auckland.ac.nzof inducing magnetization by applying an external elec-\ntric \feld [13]. Unfortunately, the multiferroic insulators\nrequire a low transition temperature, and have a limited\nmagnetic response, which is adverse for applications [14].\nBut composite multiferroics, which are an arti\fcially syn-\nthesized heterostructure of ferromagnetic and ferroelec-\ntric order, have a remarkable magnetoelectric coupling\ndue to the microscopic mechanism of the strain-stress ef-\nfect [15]. This coupling describes the in\ruence of electric\npolarization on the magnetization at interface [16].\nA previous investigation has discussed the magnetic\nBloch Skyrmions induced by an electric driving \feld in\na composite chiral-magnetic (CM) and ferroelectric (FE)\nbilayer [17]. In this Article, we demonstrate a micromag-\nnetic model in Sec. II, for generating and manipulating\nthe magnetic Bloch Skyrmions in a CM thin \flm, which\nis driven by a piece of mobile and polarized FE thin \flm.\nBoth of \flms are glued by a strong magnetoelectric cou-\npling. The dynamical behaviors in the CM layer and the\ndynamics of the FE layer are described in Sec. III. Results\nin Sec. IV show the creation and propagation of magnetic\nBloch Skyrmions, including e\u000bects of the propagation ve-\nlocity in Sec. IV A, the size of thin \flm in Sec IV B, and\nthe strength of magnetoelectric coupling in Sec. IV C.\nThe paper concludes with a discussion in Sec. V.\nII. MODEL\nFigure 1 illustrates the model of a composite bilayer,\nwhich consists a CM thin \flm at top, and a mobile FE\nthin \flm attached at bottom. The CM \flm can hold the\nmagnetic Bloch Skyrmions. The FE \flm has a smaller\nsize, and can be physically driven by the technology of\nmicroelectromechanical systems under the CM \flm. The\ncombination between them induces the converse magne-\ntoelectric e\u000bect. The animation of this dynamics is shown\ninMovie 1 in the Supplemental Material [18].\nThe magnetic component of the system is described\nby the classical Heisenberg model in a two-dimensional\nrectangular lattice. The magnetic spin is represented by\nSi;j= (Sx\ni;j;Sy\ni;j;Sz\ni;j), which is a normalized vector, i.e.,\nkSi;jk= 1, andi;j2[1;2;3;:::;N ] characterizes the lo-arXiv:1604.08780v2  [cond-mat.mes-hall]  26 Aug 20162\nFIG. 1. Schematic of the CM/FE heterostructure\nbilayer. The top layer represents the CM thin \flm, which\ncan construct magnetic Bloch Skyrmions, as shown in the red\ncircles; The smaller FE thin \flm is movable and coupled with\nthe CM thin \flm. See Movie 1 in the Supplemental Material\n[18].\ncation of each spin in the \flm. The Hamiltonian His\nde\fned by\nH=\u0000JX\ni;j[Si;j\u0001(Si+1;j+Si;j+1)]\n\u0000DX\ni;j[Si;j\u0002Si+1;j\u0001^x+Si;j\u0002Si;j+1\u0001^y]\n\u0000KzX\ni;j(Sz\ni;j)2\n\u0000gX\n~i;~j(Sz\n~i;~jP): (3)\nThe \frst term represents the nearest-neighbor exchange\ninteraction, and J\u0003=J=kBTis the dimensionless ex-\nchange coupling coe\u000ecient. The second term repre-\nsents the two-dimensional Dzyaloshinskii-Moriya inter-\naction [8] , and D\u0003=D=k BTis the dimensionless\nDzyaloshinskii-Moriya coe\u000ecient, and ^ xand ^yare the\nunit vectors of the x- and y-axes, respectively. The\nthird term represents the magnetic anisotropy, and K\u0003=\nK=k BTrepresents the dimensionless uniaxial anisotropic\ncoe\u000ecient in the z-direction. The last term repre-\nsents the magnetoelectric coupling, which is generally\ndescribed as a linear spin-dipole interaction [19], where\ng\u0003=g=kBTis the dimensionless strength of the magne-\ntoelectric coupling. The analytic expression of the mag-\nnetoelectric coupling can be linear or non-linear, particu-\nlarly with respect to the thermal e\u000bect [20]. A non-linear\nexpression has not been studied here, for simplicity and\ndue to their minor e\u000bects in the micromagnetic modeling.\nNote that, the magnetoelectric coupling was discussed by\nSpaldin et al. [21]. The strength of coupling is, however,\nunknown. Hence, we only use the low-energy excitations\nbetween the CM and FE layers. So we restrict ourselves\nto the linear expression of the magnetoelectric interac-\ntion. The coupling sites of magnetic spins, ~i;~j, to the FE\nlayer are variable. This is a result of a polarization pulse\npropagating through the CM layer. Beach et al. have\nphysically built a similar model in metallic ferromagnets\nwith electric current-driven dynamics [22].III. METHOD\nThe dynamics of magnetic spins in the CM layer has\nbeen studied by the Landau-Lifshitz equation [23], which\nnumerically solves the rotation of a magnetic spin in re-\nsponse to its torques [24],\n@Si;j\n@t=\u0000\r[Si;j\u0002He\u000b\ni;j]\u0000\u0015[Si;j\u0002(Si;j\u0002He\u000b\ni;j)];(4)\nwhere\ris the gyromagnetic ratio which relates the spin\nto its angular momentum, and \u0015is the phenomenological\ndamping term. He\u000b\ni;jis the e\u000bective \feld acting on each\nmagnetic spin, which is the functional derivative of the\nsystem Hamiltonian [Eq. (3)] with respect to the mag-\nnitudes of the magnetic spin in each direction [25], as\nHe\u000b\ni;j=\u0000\u000eH\n\u000eSi;j. This Landau-Lifshitz equation is solved\nby a fourth-order Range-Kutta method with a dimension-\nless time increment \u0001 t\u0003= 0:0001 of in all simulations.\nThe CM layer is large and stationary, the polarized\nFE layer is under the CM layer has a much smaller size.\nThus FE layer is transversely traveling along the CM\nlayer with a certain velocity. This technology refers to\nthe microelectromechanical systems. In simulations, the\nelectric dipoles in the FE layer are coupled locally with\nmagnetic spins in the CM layer. So the FE layer moves\nat a certain rate which characterizes the propagation ve-\nlocity of Skyrmions in the CM layer. Movie 1 shows the\nanimation of this dynamics in the Supplemental Material\n[18].\nIV. RESULTS\nTo investigate the dynamical manipulation of magnetic\nBloch Skyrmions, we implement a dimensionless param-\neter set:J\u0003= 1,D\u0003= 1,K\u0003= 0:1,g\u0003= 0:5,\r\u0003= 1,\nand\u0015\u0003= 0:1. Note that \\\u0003\" characterizes dimensionless\nquantity. The CM layer with NCM= 20\u0002100 magnetic\nspins, andNFE= 20\u000220 electric dipoles in the FE layer\nare used. Free boundary conditions and a random initial\nstate are applied. The propagation step-time is measured\nas the non-dimensional time period stopped on each po-\nsition, like an intermittent pulse, with a magnitude of\nT\u0003= 20=step.\nFigure 2 summarizes a time evolution that generates\na magnetic Bloch Skyrmion and manipulates it propagat-\ning along the CM layer. The CM layer is contacted with\na polarized FE thin \flm, which starts from the left, then\nmoves to right as shown in Fig. 1 . Initially, the mag-\nnetic domain walls are randomly located without any\nexternal energy addition on the system in Fig. 2(a) .\nFigures 2(b)!(c)!(d)!(e)show the generation of\na Skyrmion from natural alignment. Subsequently, this\nSkyrmion propagates though the CM layer, as shown in\nFigs. 2(e)!(f)!(g)!(h). Eventually, this Skyrmion\nstops at the right-hand side of thin \flm [ Fig. 2(h) ]. The3\nFIG. 2. Generation and propagation of a mag-\nnetic Bloch Skyrmion in the CM layer. (a) An ran-\ndomly helimagnetic state at start. (b)!(c)!(d)!(e)im-\nages show details of a Skyrmion generation in the CM \flm.\n(e)!(f)!(g)!(h) images show details of the Skyrmion\npropagating through the CM \flm. The color scale represents\nthe magnitude of the local z-componential magnetization. See\nMovie 2 in the Supplemental Material [18].\nfully dynamical process is shown in Movie 2 in the Sup-\nplemental Material [18].\nAs seen in Figs. 2(d)!(e)!(f)!(g), another par-\ntial Skyrmion at the edge been devoured, due to the free\nboundary condition. This can be avoided by using peri-\nodic boundary conditions. This will be studied later in\nSec. IV B: Sizes of Thin Film . Additionally, passage of\nthe FE \flm leaves a spin spiral alignment in the CM layer.\nThis is due to the existence of a \fnite Dzyaloshinskii-\nMoriya interaction in equilibrium.\nA. Velocity of Propagation\nWe next discuss the e\u000bects due to the propagation ve-\nlocity to the Skyrmions. Since the transverse travel of\nFE thin \flm can be manually controlled, Skyrmions are\ntracking this FE layer with the velocity with a short re-\nlaxation time. In this case, a large propagation velocity\nrepresents short time period of the FE layer to stay in\none position (or one step), i.e., small traveling step-time.\nFigure 3(a) shows two schematics demonstrating the\ngeneral issues of traveling (left) and \fnishing (right) in\nthe following results. In Fig. 3(b) , a slow propaga-\ntion velocity with the traveling step-time T\u0003= 30=step.\nThree Skyrmions have been generated and propagated.\nAs we drop the step-time to T\u0003= 20=step in Fig. 3(c) ,\nthe number of Skyrmions reduces to two, but they are\nhave similar size as in Fig. 3(b) . Subsequently, only one\nFIG. 3. Propagating Skyrmions with di\u000berent trav-\neling step-time, T\u0003. (a) Schematic of two issues: traveling\nand \fnishing. (b)T\u0003= 30=step, (c)T\u0003= 20=step, (d)\nT\u0003= 10=step, and (e)T\u0003= 5=step. The color scale repre-\nsents the magnitude of the local z-componential magnetiza-\ntion. See Movie 3 in the Supplemental Material [18].\nSkyrmion survived in Fig. 3(d) withT\u0003= 10=step. If a\nstep-time shorter than this value is used, it is too fast to\ncreate and propagate Skyrmions in the CM layer. This\nhas been shown in Fig. 3(e) withT\u0003= 5=step. The\ndynamical processes and the velocities comparison are\nshown in Movie 3 in the Supplemental Material [18].\nB. Size of Thin Film\nLarger size of the thin \flm o\u000bers more space to allow\nmore Skyrmions. Figure 4 exempli\fes four cases for\ndi\u000berent sizes of the composite bilayer. Figure 4(a)\nshows the CM layer contains NCM= 10\u0002100 magnetic\nspins, which generates one Skyrmion. Subsequently, a\nlarger layer with NCM= 20\u0002100 magnetic spins shows\nthree Skyrmions in Fig. 4(b) .Figure 4(c) shows \fve\nSkyrmions with a layer size of NCM= 30\u0002100 magnetic\nspins, and Fig. 4(d) shows seven Skyrmions with a layer\nsize ofNCM= 40\u0002100 magnetic spins. Consequently, the\nquantity of Skyrmions increased as the \flm size increases,\nbut the quality of Skyrmions is the same. Movie 4 in\nthe Supplemental Material animates these cases [18].\nInterestingly, Skyrmions are found to collect together\nnear the top of thin \flm in Figs. 4(b) ,(c)and(d).\nThis occurs due to the helimagnetically ordered struc-\nture which points to the upper-right corner, and gener-\nates Skyrmions in this direction. Remember that the left\nhand side in each image is the structure after FE \flm has\npassed. This behavior is also observed in Figs. 3(c) and\n(d).4\nFIG. 4. E\u000bects by di\u000berent sizes of the CM thin \flm,\nNCM. (a) NCM= 10\u0002100,(b)NCM= 20\u0002100,(c)NCM=\n30\u0002100, and (d)NCM= 40\u0002100. Each subplot shows two\nissues as shown in Fig. 3(a) . The color scale represents the\nmagnitude of the local z-componential magnetization. See\nMovie 4 in the Supplemental Material [18].\nIn another simulation, we replace the free boundaries\nby a periodic boundary condition, which linked the spins\nat the top and the bottom. Now the dynamics shows\nSkyrmions on a race track moving longitudinally. Dif-\nferent to the behavior shown from the system with free\nboundary condition. See Movie 6 in the Supplemental\nMaterial [18].\nC. Strength of Magnetoelectric Coupling\nThe electric-induced magnetic Bloch Skyrmions result\nfrom the magnetoelectric coupling between the electric\ndipoles to the magnetic spins. It is noteworthy that the\nstrength of magnetoelectric coupling plays an important\nrole in mediating the energy transfer to sustain the mag-\nnetic Skyrmions [17]. Therefore, we examined di\u000berent\nmagnetoelectric coupling strength in Fig. 5 . Such as,\ng\u0003= 0:25 in Fig. 5(a) ,g\u0003= 0:5 inFig. 5(b) ,g\u0003= 0:75\ninFig. 5(c) andg\u0003= 1 in Fig. 5(d) . Firstly, Fig. 5(a)\nshows an insu\u000ecient strength of the magnetoelectric cou-\npling to generate Skyrmions. The magnetic domain\nshows a spin spiral alignment. With increased strength\nof the magnetoelectric coupling in Figs. 5(b) and(c),\nboth of them generate and propagate two Skyrmions in\nthe CM layer. Figure. 5(b) has larger size of Skyrmions\nthan in Fig. 5(c) , since the magnetoelectric interaction\nin the former is weak compared to its Dzyaloshinskii-\nMoriya interaction. In Fig. 5(d) , the ample strength\nof magnetoelectric coupling dominates the response in a\nsaturated FM state. Movie 5 in the Supplemental Ma-\nterial animates these cases [18].\nTo determine the size of a Skyrmion, we use the\nspin-plot, and count the number of magnetic spins in\nFIG. 5. E\u000bects by di\u000berent strength of the magne-\ntoelectric couplings, g\u0003. (a) g\u0003= 0:25,(b)g\u0003= 0:5,(c)\ng\u0003= 0:75 and (d)g\u0003= 1. Each subplot shows two issues as\nshown in Fig. 3(a) . See Movie 5 in the Supplemental Mate-\nrial [18]. (e)A spin-plot image shows the Skyrmion part from\nright image in (c). The color scale represents the magnitude\nof the local z-componential magnetization. (f)The total size\nof Skyrmions versus the strength of magnetoelectric coupling,\ndetermined from the \fnishing image of each dynamics. The\ncurve through the points is a guide to the eye. See Fig. 1 in\nthe Supplemental Material [18].\na Skyrmion. An example of spin-plot is shown in\nFig. 5(e) , which involves two Skyrmions. The num-\nber of magnetic spins contributing to these Skyrmions\ncan be counted, which is the size of a Skyrmion. There-\nfore, a phase diagram parameterized by the total size of\nSkyrmions versus the strength of magnetoelectric cou-\nplingg\u0003is presented in Fig. 5(f) . In this \fgure, four\nkinds of regime are apparent. (1) For smaller g\u0003, the size\nof Skyrmions is zero due to the magnetoelectric energy\nbeing insu\u000ecient to generate Skyrmions, but the lattice\nforms spin spiral structure. (2) Slightly larger g\u0003gives a\nmixed regime of spin spirals and Skyrmions co-existing\nin the lattice. (3) Certain larger magnitudes of g\u0003gen-\nerate stable Skyrmions. This region shows a non-linear\ndecrease of the size of a Skyrmion with the increased\nmagnetoelectric coupling. (4) For larger g\u0003, the uniform\nmagnetization appears in this stage, due to the magne-\ntoelectric energy dominating energy contribution in the\nFM system. More details are shown in the Supplemental\nMaterial Fig. 1 [18].5\nV. CONCLUSION\nThis Article has shown that by using the converse\nmagnetoelectric e\u000bect in a composite CM/FE bilayer,\nmagnetic Bloch Skyrmions can be induced and manipu-\nlated by attaching a mobile polarized FE thin \flm. The\nquantity and quality of these Skyrmions correspond to\nthe conditions: (1) Propagation velocity. High speed of\nthe polarized FE layer restricts the number of Skyrmion\nformations. (2) Thin \flm size. Larger space requires\nmore Skyrmions to minimize the local energy contribu-\ntion; (3) Magnetoelectric coupling strength. The compe-\ntition between the magnetoelectric interaction and the\nDzyaloshinskii-Moriya interaction may occur di\u000berent\nstates in chiral-magnets. Excessive strength favors in the\ncentrosymmetric structure, like in ferromagnetism; insuf-\n\fcient strength shows a spin spiral state. Only a delicatebalance of the magnetoelectric coupling can o\u000ber the ex-\nistence of magnetic Skyrmions.\nSingle-phase multiferroics have persistent coupling be-\ntween the magnetic moments and the electric moments,\ndue to their solid crystallographic structures. But,\nthe coupling in the composite multiferroics can be eas-\nily varied by di\u000berent material combinations, external\nstress or heat. Therefore, it opens a novel approach on\nmagnetoelectric-induced Skyrmions in the composite bi-\nlayer. From an application viewpoint, our proposal has\nthe potential lead to a unique technology for the future\nSkyrmion based memory devices.\nACKNOWLEDGMENTS\nThe authors thank X.C. Zhang and F. Xu for discus-\nsions. Z.W. gratefully acknowledges Wang Yuhua, Zhao\nBingjin, Zhao Wenxia and Wang Feng for support.\n[1] U. R o\u0019ler, A. Bogdanov, and C. P\reiderer, Nature 442,\n797 (2006).\n[2] S. M uhlbauer, B. Binz, F. Jonietz, C. P\reiderer,\nA. Rosch, A. Neubauer, R. Georgii, and P. B oni, Sci-\nence323, 915 (2009).\n[3] X. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Zhang,\nS. Ishiwata, Y. Matsui, and Y. Tokura, Nat. Mater. 10,\n106 (2011).\n[4] X. Yu, Y. Onose, N. Kanazawa, J. Park, J. Han, Y. Mat-\nsui, N. Nagaosa, and Y. Tokura, Nature 465, 901 (2010).\n[5] K. Shibata, X. Yu, T. Hara, D. Morikawa, N. Kanazawa,\nK. Kimoto, S. Ishiwata, Y. Matsui, and Y. Tokura, Nat.\nNanotech. 8, 723 (2013).\n[6] S. Seki, X. Yu, S. Ishiwata, and Y. Tokura, Science 336,\n198 (2012).\n[7] X. Zhang, M. Ezawa, D. Xiao, G. P. Zhao, Y. Liu, and\nY. Zhou, Nanotech. 26, 225701 (2015).\n[8] N. Nagaosa and Y. Tokura, Nat. Nanotech. 8, 899 (2013).\n[9] S. Buhrandt and L. Fritz, Phys. Rev. B 88, 195137\n(2013).\n[10] I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958).\n[11] T. Moriya, Phys. Rev. 120, 91 (1960).\n[12] J. S. White, K. Pr\u0014 sa, P. Huang, A. A. Omrani,\nI.\u0014Zivkovi\u0013 c, M. Bartkowiak, H. Berger, A. Magrez, J. L.\nGavilano, G. Nagy, J. Zang, and H. M. R\u001cnnow, Phys.\nRev. Lett. 113, 107203 (2014).\n[13] Y.-H. Liu, Y.-Q. Li, and J. H. Han, Phys. Rev. B 87,\n100402 (2013).[14] J.-W. G. Bos, C. V. Colin, and T. T. M. Palstra, Phys.\nRev. B 78, 094416 (2008).\n[15] C.-W. Nan, Phys. Rev. B 50, 6082 (1994).\n[16] F. Xue, J. Hu, S. X. Wang, and J. He, Appl. Phys. Lett.\n106, 262902 (2015).\n[17] Z. D. Wang and M. J. Grimson, Phys. Rev. B 94, 014311\n(2016).\n[18] See Supplemental Material at\n10.13140/RG.2.1.2015.8326/1, for animations about\nthe dynamical results.\n[19] C.-L. Jia, T.-L. Wei, C.-J. Jiang, D.-S. Xue, A. Sukhov,\nand J. Berakdar, Phys. Rev. B 90, 054423 (2014).\n[20] L. Chotorlishvili, S. R. Etesami, J. Berakdar, R. Khome-\nriki, and J. Ren, Phys. Rev. B 92, 134424 (2015).\n[21] F. Th ole, M. Fechner, and N. A. Spaldin, Phys. Rev. B\n93, 195167 (2016).\n[22] S. Woo, K. Litzius, B. Kruger, M.-Y. Im, L. Caretta,\nK. Richter, M. Mann, A. Krone, R. M. Reeve,\nM. Weigand, P. Agrawal, I. Lemesh, M.-A. Mawass,\nP. Fischer, M. Klaui, and G. S. D. Beach, Nat. Mater.\n15, 501 (2016).\n[23] Y. Li, K. Xu, S. Hu, J. Suter, D. K. Schreiber, P. Ra-\nmuhalli, B. R. Johnson, and J. McCloy, J. Phys. D Appl.\nPhys. 48, 305001 (2015).\n[24] Z. D. Wang and M. J. Grimson, J. Appl. Phys. 118,\n124109 (2015).\n[25] Z. D. Wang and M. J. Grimson, J. Appl. Phys. 119,\n124105 (2016)."    },    {        "title": "1112.1857v3.Dynamics_of_artificial_spin_ice__continuous_honeycomb_network.pdf",        "content": "arXiv:1112.1857v3  [cond-mat.mes-hall]  20 Feb 2012Dynamics of artificial spin ice: continuous\nhoneycomb network\nYichen Shen1\nOlga Petrova2\nPaula Mellado3\nStephen Daunheimer4\nJohn Cumings4\nOleg Tchernyshyov2\n1Department of Physics, Massachusetts Institute of Technol ogy, Cambridge,\nMassachusetts 02139, USA\n2Department of Physics and Astronomy, Johns Hopkins Univers ity, Baltimore,\nMaryland 21218, USA\n3School of Engineering and Applied Sciences, Harvard Univer sity, Cambridge,\nMassachusetts 02138, USA\n4Department of Materials Science and Engineering, Universi ty of Maryland,\nCollege Park, 20742 Maryland, USA\nE-mail:cumings@umd.edu, olegt@jhu.edu\nAbstract. We model the dynamics of magnetization in an artificial analo g of\nspin ice specializing to the case of a honeycomb network of co nnected magnetic\nnanowires. The inherently dissipative dynamics is mediate d by the emission,\npropagation and absorption of domain walls in the links of th e lattice. These\ndomain walls carry two natural units of magnetic charge, whe reas sites of the\nlattice contain a unit magnetic charge. Magnetostatic Coul omb forces between\nthese charges play a major role in the physics of the system, a s does quenched\ndisorder caused by imperfections of the lattice. We identif y and describe different\nregimes of magnetization reversal in an applied magnetic fie ld determined by the\norientation of the applied field with respect to the initial m agnetization. One of\nthe regimes is characterized by magnetic avalanches with a 1 /ndistribution of\nlengths.\n1. Introduction\nSpin ice[1,2]isafrustratedferromagnetwithIsingspinsthatposs essesratherpeculiar\nproperties. First, as a consequence of strong frustration, it ha s a massively degenerateDynamics of artificial spin ice 2\nground state and retains a finite entropy density even at very low t emperatures [3].\nSecond, its low-energy excitations are neither individual flipped spin s, nor domain\nwalls, but are point defects acting as sources and sinks of magnetic fieldH[4, 5]. The\nconcept of magnetic charges, while not exactly new [6, 7, 8], has pro ven very useful in\nelucidating the static and dynamic properties of spin ice [9, 10, 11, 12 , 13]. It is worth\nnoting that these objects are magnetic analogs of excitations with fractional electric\ncharge found in the familiar water ice [14].\nArtificialspiniceisanarrayofnanomagnetswithsimilarlyfrustrated interactions.\nThe original system made by Schiffer’s group had disconnected elong ated islands (80\nnm by 220 nm laterally and 25-nm thick) made of permalloy and arrange d as links\nof a square lattice [15]. Later versions included a connected honeyc omb network of\nflat magnetic wires [16, 17, 18, 19], in which the centers of the wires f orm a kagome\nlattice, hence the sometimes used name “kagome spin ice” [17]. Wher eas it had been\noriginally intended as a large-scale replica of natural spin ice, it becam e clear very\nsoon that artificial spin ice has a number of its own peculiar features . For example,\nbecause the magnetic moments in artificialspin ice are extremely larg e, on the orderof\n108Bohr magnetons, the energy scale of shape anisotropy due to dipo lar interactions,\n105K in temperature units [20], effectively freezes out thermal fluctu ations of the\nmacrospins meaning that the system is not in thermal equilibrium. Dyn amics of\nmagnetization has to be induced by the application of an external ma gnetic field [15].\nElaborate experimental protocols involving a magnetic field of varyin g magnitude and\ndirection [21] havebeen proposedto simulate thermal agitationinvo kingparallelswith\nfluidized granular matter. It remains to be seen whether the induce d dynamics yields\na thermal ensemble with an effective temperature. The analogy with granular matter\nis further reinforced by recent observations of magnetic avalanc hes in the process of\nmagnetization reversal [18, 22].\nIn this paper we present a model of magnetization dynamics in artific ial spin ice\nsubject to an external magnetic field. Two sets of physical variab les are used: an Ising\nvariableσ=±1 encodes the magnetic state of a spin, whereas an integer qquantifies\nthemagneticchargeofanodeatthejunctionofseveralspins. Ma gnetizationdynamics\nare mediated by the emission of domain walls carrying two units of magn etic charge\nfrom a lattice node, their subsequent propagation through a magn etic element, and\nabsorption at the next node. We specialize to the case of kagome sp in ice, in which\nmagnetic elements form a connected honeycomb lattice [18, 16, 17, 19]. The model\ncan be readily extended to other geometries and lattices with discon nected magnetic\nelements [15, 22, 23, 24]. Some of the results presented here have been outlined\npreviously [25].\n2. Basic features of the model\nOur model is specialized toward an experimental realization describe d previously [17].\nThat artificial spin ice is a connected honeycomb network of permallo y nanowireswith\nsaturation magnetization M= 8.6×105A/m and the following typical dimensions:\nlengthl= 500 nm, width w= 110 nm, and thickness t= 23 nm. Three nanowires\ncome together at a vertex in the bulk. At the edge of the lattice, a v ertex may have\none or two links coming in.Dynamics of artificial spin ice 3\ni j\nlk−1\n+1 +1+1−1−1−1\n+1−3+1+1\n+1\n−1\nFigure 1. Magnetization variables σij=±1 (arrows) live on links ijof the\nhoneycomb lattice. Charges qi=±1,±3 live on nodes i.\n2.1. Basic variables: magnetization and magnetic charge\nWe label nodes of the lattice by a single index iand nanowires connecting adjacent\nnodes by the indices of its two nodes, ij. In equilibrium, the vector of magnetization\nMpoints parallel to the long axis of the wire, so we can encode the two s tates of\na nanowire by using an Ising variable σij=±1. In our convention, σij= +1 when\nthe vector of magnetization points from node iinto node j. This definition implies\nantisymmetry under index exchange, σij=−σji.\nWe define the dimensionless magnetic charge at node ias\nqi=/summationdisplay\njσji, (1)\nwhere the sum is taken over the three neighboring sites j. This definition is quite\nnatural: since magnetic induction B=µ0(H+M) is divergence-free, the magnetic\nchargeQiof nodeiequals the flux of magnetic field Hout of the node, which in turn\nequals the flux of magnetization Minto it:\nQi=/contintegraldisplay\nH·dA=−/contintegraldisplay\nM·dA=−Mtw/summationdisplay\njσij=Mtwq i.(2)\nThusqiis indeed magnetic charge measured in units of Mtw.\nThe Bernal-Fowler ice rule [2] enforcing minimization of the absolute va lue of\ncharge|Qi|is usually justified from the energy perspective: the magnetostat ic energy\nof spin ice can be written as the energy of Coulomb interaction of mag netic charges,\nE≈µ0\n8π/summationdisplay\ni/negationslash=jQiQj\n|ri−rj|+/summationdisplay\niQ2\ni\n2C. (3)\nThe dominant second term—the charging energy of a node—forces minimization of\nmagnetic charges in natural spin ice. The “capacitance” Cis determined by the\ndipolar and exchange couplings energies of adjacent spins [5].Dynamics of artificial spin ice 4\nAlthough we will see below that these energy considerations are not relevant to\nartificial spin ice within our model, for the moment we will simply adopt th e result\nto it. In honeycomb ice, where the coordination number is 3, dimensio nless charge qi\ncan take on values ±1 and±3. Minimization of node self-energy would select states\nwith\nqi=±1. (4)\nIndeed, triplemagnetic chargeshaveneverbeen observedin ours amplesofhoneycomb\nice. Ladak et al.[18, 19] have found nodes with triple charges. The difference is likely\ndue to a higher amount of quenched disorder arising from random imp erfections of\nthe lattice [26] in the samples of Ladak et al.\nWe will find it convenient to use the following notation. A site with a unit c harge\nqi=±1 has two majority links with σji=qiand one minority link with σji=−qi.\nFor siteiin Figure 1, the minority link is ij.\n2.2. Basic dynamics: emission of a domain wall\nTo reverse the magnetization in a nanowire, one must apply a sufficien tly strong\nexternal magnetic field. The reversal begins when one of the node s, sayi, emits\na domain wall (w) into link ij, Figure 2(a). If the link initially has magnetization\nσij=±1, a domain wall can traverse it from itojonly if it has charge of the right\nsign, i.e., qw= 2σij=±2. Once the domain wall passes through the link, σijchanges\nits sign. Now a domain wall with the same charge qwcan only traverse the link in the\nopposite direction.\nThe critical field Hc, at which a domain wall is emitted from a node, can be\nestimated as follows [27]. Suppose a node with magnetic charge qi=±1 emits a\ndomain wall with magnetic charge qw=±2 [8, 25]. Conservation of magnetic charge\nmeans that the charge of the site turns to qi=∓1. The emission process can thus be\nviewed as pulling a charge qw=±2 away from a charge of the opposite sign qi=∓1.\nThe maximum force between the two charges is achieved when the se paration between\nthem is of the order of their sizes a, which is roughly equal to the width of the wire w:\nFmax=µ0|QiQw|/(4πa2). This force must be overcome by the Zeeman force applied\nto the domain wall by the external magnetic field, Fext=µ0|Qw|Hext. Hence the\nestimate of the critical field,\nHc=|Qi|\n4πa2=Mtw\n4πa2≈Mt\n4πw. (5)\nFor the system parameters used in our previous work [17] and listed above, this\nestimate yields µ0Hc= 18 mT. The critical value observed experimentally [28] is\n35 mT.\nOne can envision another possible process, wherein the reversal is triggered when\na site with charge qi=±1 emits a domain wall of charge qw=∓2 and change its\ncharge to qi=±3. Considerations along the same lines as above show that the critica l\nfield required to pull apart charges qi=±3 andqw=∓2 is 3Hc. As we will see below,\nmagnetization reversal in samples with low quenched disorder occur s well before the\nexternal field has a chance to reach this value. This explains why trip le charges are\nnever generated as a result of the emission of a domain wall.\nThe estimate for the critical field was obtained under the assumptio n that the\nexternal magnetic field Hextis applied along the link into which the domain wall is\nemitted. When the field makes angle θwith the link, it is reasonable to suppose thatDynamics of artificial spin ice 5\n(a)i j\n−1\n−1 −1+2\n−1 −1 +2\n−1 +1+2 −1 −1+1\n(b)+2−1+1 +1\n−1 −1\n−1 −1\n−1 +1+2\n+2\n+2+1\nFigure 2. Magnetization reversal in a single link. At the end of the rev ersal, the\ndomain wall encounters a node with magnetic charge of the opp osite sign (a) or\nof the same sign (b). In panel (b), the emission of the domain w all from the left\nnode and its propagation along the horizontal link are omitt ed for brevity.\nonly the longitudinal component of the field Hextcosθpulls the domain wall away\nfrom the node. We thus expect the following angular dependence of the critical field:\nHc(θ) =Hc/cosθ. (6)\nAs we will see later in Sec. 3, our educated guess is almost right and th at Eq. (6)\nrequires only a minor correction: the angle θshould be measured not from the axis of\nthe link but from a slightly offset direction. This effect is caused by an a symmetric\ndistribution of magnetization around a node, which was missed by the simplified,\nmesoscopic model of this section.\n2.3. Basic physics: absorption of a domain wall\nOnce a domain wall is emitted into link ij, it quickly propagates to the other end of\nthe link, toward node j. Theoretical and experimental studies of domain wall motion\nin permalloy nanowires [29, 30] show that walls move at speeds of the o rder of 100\nm/s in an applied field of just 1 mT. This corresponds to a propagation time of the\norder of 10 ns, which is too short to be observed in most current ex perimental setups.Dynamics of artificial spin ice 6\nWhen the domain wall reaches the opposite end of the link ij, its further fate\ndepends on whether the magnetic charge at node jhas the same or opposite sign of\nmagnetic charge. We consider the two cases in turn.\nIf the domain wall and node jat which it arrives have opposite charges,\nqw=±2 =−2qj, as in Figure 2(a), the domain wall is attracted to the node. It\nis easily absorbed by the node, whose charge changes to qj=±1. A new domain\nwall with the same charge qw=±2 may be subsequently emitted into one of the\nadjacent links jkif two conditions are met: (i) the link has the right direction of its\nmagnetization, qw= 2σjkand (ii) the external field is sufficiently strong to trigger the\nemission.\nNote that condition (ii) is sensitive to the orientation of the field relat ive to link\njk. It also rests on an implicit assumption that the critical field for a new domain wall\nis not affected by the just completed absorption of the previous on e. This assumption\nis reasonable if the dynamics of domain walls are strongly dissipative an d the energy\ngenerated during the absorption process is quickly dissipated as he at. Experiments\nwith domain walls in nanowires indicate that they possess non-negligible inertia [8],\nand therefore our assumption of strongly overdamped dynamics m ay not be fully\njustified. Nonetheless, for the sake of simplicity, we shall assume t hat the dynamics\nare strongly dissipative and that the extra energy brought by the arrival of a domain\nwall does not by itself cause the emission of another domain wall from the same node.\nConsider now the other case, where the domain wall and the arrival node have\ncharges of the same sign, qw=±2 = 2qj, as in Figure 2(b). The two charges now\nrepel and the repulsion grows stronger as the domain wall approac hes the node.\nUnder the assumption of overdamped dynamics, the wall stops whe n the Coulomb\nrepulsion between the charges reaches the level of the Zeeman fo rce from the external\nfield. One might think that this may be an equilibrium situation, but we sh ow\nas follows that this is not the case. The arriving domain wall generate s a strong\nfield at the node, whose magnitude is easy to estimate. Since the dom ain wall is in\nequilibrium, the force applied to it by the external field, F=µ0|Qw|Hc, is balanced\nby the Coulomb repulsion of the node. By Newton’s third law, the doma in wall\napplies an equal force to the node. The field created by the wall at t he node is\nH=F/|µ0Qj|=|Qw/Qj|Hc= 2Hc. This field is added to the externally applied\nfieldHc. The resulting field is sufficiently strong to trigger the emission of ano ther\ndomain wallfrom the node. (This worksforanyrelevant directionof the applied field.)\nThe charge of node jchanges sign, qj=∓1 =−qw/2, and subsequently absorbs the\nstopped domain wall.\n2.4. Basic physics: quenched disorder\nImperfections ofmagneticlinks andjunctions createlocalvariatio nsofthe criticalfield\nHc. If the variations of Hcresult from a large number of small errors, one expects a\nGaussian distribution of critical fields ρ(Hc) with a mean ¯Hcand a width δHcgiven\nby\nρ(Hc) =1√\n2πδHcexp/parenleftbigg\n−(Hc−¯Hc)2\n2δH2c/parenrightbigg\n. (7)\nIn the limit of strong disorder, when the distribution width δHcis comparable to the\naverage ¯Hc, nodes with the highest critical fields may fail to follow the scenario s hown\nin Figure 2(b) and remain in a state with a triple charge until the field be comes strongDynamics of artificial spin ice 7\nenough. Nodes with triple charges have been observed by Ladak et al.[18, 19]. In\ncontrast, other samples have never shown triply charged defect s [17], indicating that\nthese samples are in the low-disorder limit, δHc≈0.04¯Hc[28].\nThe distribution width δHccan be compared to another characteristic field\nstrength, the magnetic field generated by an adjacent node, H0=Mtw/(4πl2). With\nthe aid of Equation (5), we estimate\nH0/Hc= (a/l)2≈(w/l)2. (8)\nIfH0≪δHc, the Coulomb fields produced by adjacent and more distant nodes c an\nbe ignored to a first approximation. The Coulomb contribution to the net field on a\ngiven site is small, but occasionally the redistribution of magnetic char ges on nearby\nsites may trigger the emission of a domain wall if the net field is close to t he critical\nvalue. See Sec. 4.1 for further details. In the opposite limit, H0≫δHc, these internal\nfields must be taken into account. The reversal of magnetization o n one link alters the\nmagnetic charges on its ends. The resulting increments of the tota l magnetic field at\nnearby nodes, of order H0, may be sufficient to trigger the emission of domain walls\nfrom them. Samples we studied previously [17, 28] appear to be in the regime where\nH0andδHcare comparable.\n3. Microscopic basis for the model\nTo test the basic model of magnetization dynamics presented in Sec . 2, we performed\nnumerical simulations of magnetization dynamics in a small portion of t he honeycomb\nnetwork by using the micromagnetic simulator OOMMF [31].\nThe typicalnumericalexperiment involvedajunction ofthreeperm alloymagnetic\nwires of length l= 500 nm, width w= 110 nm, and thickness t= 23 nm [17]. We used\nthe two-dimensional version of the oommfcode with cells 2 nm ×2 nm×23 nm. (The\nlateral size of the unit cell should not exceed the minimal length in the micromagnetic\nproblem, the magnetic exchangelength obtainedfrom exchangean d dipolarcouplings.\nIn permalloy, it is about 5 nm [29].) The magnetization field M(r) was allowed to\nrelax to an equilibrium state with magnetic charge q=±1 at the junction, Figure 3.\nAn external magnetic field was then applied in a fixed direction and its m agnitude was\nslowly increased keeping the system in a state of local equilibrium. Eve ntually, the\nmagnet reached a point of instability when a domain wall was emitted fr om either the\ncentral junction or one of the peripheral ends of the wires, depe nding on the direction\nof the applied field. The wall then propagated to the opposite end of the link reversing\nthe link’s magnetization. Using those orientations of the field for whic h a domain wall\nis emitted from the junction, we determined the dependence of the critical field Hon\nthe angle θbetween the field and the link in which the reversal occurs, Figure 4.\nTwo features of the angular dependence in Figure 4 stand out. Firs t,H(θ) is not\nan even function of the angle θ, and contrary to our expectations, the critical field is\nnot at its lowest when the field is parallel to the link. Second, the critic al fields for\nthree different links in the experiment have the same shape but differ in the overall\nscaleHc.\nWe have traced the physical origin for the asymmetric dependence of the critical\nfieldH(θ) to an asymmetric distribution of magnetization at the junction, Fig ure 3.\nThe energetics of the emission process shown in the figure can be de scribed in the\nlanguage of collective coordinates [32]. The soft mode associated wit h the emission ofDynamics of artificial spin ice 8\n(a) (b) (c)\n(d) (e) (f)\nFigure 3. Reversal of magnetization in a magnet consisting of three jo int links in\nan applied magnetic field (vertical arrow). In panels (a) thr ough (c), the strength\nof the field slowly increases from 0 to a critical value as the m agnetization adjusts\nadiabatically. In panels (c) through (f), a domain wall deta ches from the node and\nquickly propagates through the vertical link; the field valu e remains essentially\nunchanged. Micromagnetic simulation (oommf).\n 50 60 70 80 90 100 110µ0H, mT\n−20  0  20  40  60  80θLink 1\nLink 2\nLink 3\nFigure 4. The dependence of the critical field Hon the angle θbetween the\nfield and the link. The lines are best fits to Equation (12). Lin ks 1, 2, and 3\nhadHc= 53.6 mT, 54.7 mT, and 55.3 mT and α= 19.3◦, 19.4◦, and 19 .4◦,\nrespectively. The same numerical experiments were repeate d three times, with\nthe field initially lined up with Link 1, 2, or 3, and then rotat ed through 180◦+θ\nfrom that direction.Dynamics of artificial spin ice 9\na domain wall into the vertical link is the domain wall displacement Xalong the link.\nTo the first order in the applied field Hand to the second order in X, the energy is\nU(H,X) =U(0,0)−µ0X(QxxHx+QxyHy)+kX2/2, (9)\nwhereQxx,Qxyandkare phenomenological constants. Generally speaking, the off-\ndiagonal component Qxydoes not vanish unless the magnetization distribution is\nsymmetric under the reflection y/ma√sto→ −y. The equilibrium position of the wall depends\non the direction of the applied field H= (Hcosθ,Hsinθ,0) as follows:\nXeq= (µ0/k)(QxxHx+QxyHy) = (µ0/k)˜QHcos(θ−α),(10)\nwhere the offset angle αand effective charge ˜Qare defined through\nQxx=˜Qcosα, Q xy=˜Qsinα. (11)\nAccording to Equation (10), the relevant component of the magne tic fieldHis found\nby projecting the field onto the easy axis of a (majority) link, which is rotated\nthrough angle αtoward the minority link. These considerations suggest the following\nmodification for the postulated field dependence of the critical field (6):\nHc(θ) =Hc/cos(θ−α). (12)\nAsFigure4shows,thisequationprovidesagooddescriptionofthea ngulardependence\nof the critical field with the offset angle α≈19◦. The overall scale of the critical\nfieldHcshowed variations reflecting small imperfections of links in the simulat ion.\nFor instance, the square lattice of magnetic moments used in oommf simulations is\nincommensuratewith linkspointingat60◦toalattice axisandcreatesedgeroughness.\nThis observation confirms the proposed model of disorder introdu ced in Sec. 2.4.\n4. Numerical simulations\nThe heuristic considerations of Section 2 and the micromagnetic simu lations of\nSection 3 suggested a coarse-grained model of magnetization dyn amics in which the\nbasic degreesof freedom are Ising variablesof magnetization σijon links and magnetic\nchargesqion sites of the honeycomb lattice. Each link has its own fixed critical fi eld\nHc. The critical fields form a Gaussian distribution (7) of width δHcaround the\nmean¯Hc. The average, ¯Hc= 50 mT, was chosen on the basis of our micromagnetic\nsimulations, whereas the relative width was set to δHc/¯Hc= 0.05, a value inspired\nby our experimental observations [28]. Simulations were performe d in a rectangular\nsample with 937 links. The edge consisted of “dangling” links with no oth er links\nattached to their external ends. We choose the initial state with a maximum total\nmagnetization that can be obtained by placing the system in a strong magnetic field\nalong one set of links, Figure 5(a). Simulation details are described in A ppendix A.\nFollowing initialization, the external field is switched off and reapplied alo ng a\ndifferent direction, at an angle θto its initial orientation, Figure 5(b-d). To stimulate\nmagnetization dynamics, the rotation angle must be large enough so thatHwould\nhave a negative projection onto at least some of the majority links. When|θ|is\nbetween 30◦and 90◦, only one of the three sublattices of links will reverse. Two\nsublattices reverse when |θ|is between 90◦and 150◦. The entire lattice undergoes a\nreversal when |θ|>150◦.\nAside from the number of active sublattices, there are marked diffe rences in the\ndynamics of the reversal. For small angles of rotation, |θ|<131◦, the reversals\noccur in a gradual and uncorrelated manner, with individual links swit ching when theDynamics of artificial spin ice 10\n(a) −1−1+1−1+1\n+1H\n+1−1+1−1\n−1\n−1\n+1 (b) −1−1+1−1+1\n+1+1H\n−1+1−1\n−1\n−1\n+1\n(c)+1\n+1\n−1 +1+1−1+1−1\n−1H\n−1+1−1 +1\n(d)−1−1\n−1\n−1\n+1 −1−1+1−1+1\n+1H\n+1+1\nFigure 5. Magnetization reversal in an applied magnetic field. (a) The system is\ninitially magnetized in a strong horizontal field. (b-d) The field is then switched\noff and applied at 120◦to the original direction with a gradually increasing\nmagnitude. Open arrows denote links with reversed magnetiz ation.\napplied field reaches the link’s critical field. For larger angles, |θ|>131◦, we observed\navalanches in which long chains of links reverse magnetization simultaneously.\nThis kind of switching happens when the sublattice whose magnetizat ion is most\nantiparallel to the applied field cannot switch first because it consist s entirely of\nminority links and must wait for one of the other sublattices to begin it s reversal. If\nthat happens in a higher field, the former sublattice acts like a loaded spring, making\nthe reversal nearly instantaneous. A diagram depicting different r egimes as a function\nof the field rotation angle θis shown in Figure 6.\nIn the simplest case, the reversal of magnetization in a link occurs w hen the\nmagnetic field reaches the critical value for that link. The links thus r everse on an\nindividual basis, largely independently from the others (but see belo w). To be more\nprecise, the two ends of a link have different critical fields and the re versal begins\nfrom the end with the lower critical field and stops at the other end. The effectiveDynamics of artificial spin ice 11\nA+B,C\nA\nCB\nHB,A B\nCC,AθA+B\nA+CA+C,B\n 0 0.1 0.2 0.3 0.4 0.5\n−4−3−2−1 0 1 2 3 4Gaussian\nmodified\nbest fit\nFigure 6. Left panel: Regimes of magnetization reversal. 0 < θ <30◦: no\nreversal. 30◦< θ <90◦: sublattice B only. 90◦< θ <131◦: B, then A. 131◦<\nθ <150◦: A and B reverse together. 150◦< θ <180◦: A and B, then C. Similar\nregimes obtain for negative θ, with sublattices B and C exchanged. Right panel:\nIllustration to Equation (13). The Gaussian distribution e xp(−x2/2)/√\n2π(blue\nsquares), the modified distribution erfc( x/√\n2) exp(−x2/2)/√\n2π(red circles),\nand the best Gaussian approximation, exp( −(x−δ)2/2σ2)/√\n2πσwith the mean\nδ=−0.54 and width σ= 0.82 (solid line).\nprobability density of the critical fields thus changes from a Gaussia n distribution to\nf′(Hc) = 2ρ(Hc)/integraldisplay∞\nHcdHρ(H)\n=1√\n2πδHcexp/parenleftbigg\n−(Hc−¯Hc)2\n2δH2c/parenrightbigg\nerf/parenleftbiggHc−¯Hc\nδHc√\n2/parenrightbigg\n.(13)\nIt can be seen in Figure 6 (right panel) that the resulting distribution is very close to\na Gaussian with renormalized mean and width,\n¯H′\nc=¯Hc−0.54δHc, δH′\nc= 0.82δHc. (14)\nIn our simulation, the renormalized values are ¯H′\nc= 48.7 mT and δH′\nc/¯H′\nc= 0.042.\n4.1.30◦< θ <131◦: gradual reversal\nWith the field rotated through θ= 120◦, two sets of links have a negative projection\nof magnetization onto the field. In Figure 5(b), they are the horizo ntal minority\nlinks and the majority links parallel to the field. Because emission of a d omain wall\ninto a minority link requires a very high field, it is the majority links that u ndergo\nmagnetization reversal first. The field makes an angle α≈19◦with their easy axes,\nso the reversal is expected to occur around the field H1=¯H′\nc/cos(−19◦) = 51.5 mT.\nMagnetizationreversalin the links parallelto the field altersthe mag netic charges\non all sites, Figure 5(c). As a result of this change, horizontal links join the majority\nand become capable of reversing their magnetization. The externa l field makes\nan angle 60◦−α≈41◦with their easy axes, so their magnetization reversal is\nexpected to occur when the field reaches a higher value, H2=¯H′\nc/cos41◦= 64.5\nmT. In the presence of disorder, the reversal regions are expec ted to have finite\nwidths,δH1/H1=δH2/H2=δH′\nc/¯H′\nc. For a Gaussian distribution of critical fields,Dynamics of artificial spin ice 12\n−0.5 0 0.5 1M\n 45 50 55 60 65 70H, mTsimulation\ntheory\n 10102103104\n 1 2 3 4 5 −0.5 0 0.5 1M\n 30  35  40  45  50H, mTexperiment\n best fit\nFigure 7. Magnetization reversal curve M(H) in an applied field rotated\nthrough 120◦. Left: Simulated magnetization curve M(H) (red circles) is well\napproximated by the theoretical curve (15) (solid black lin e). Inset: semi-log plot\nof the number of avalanches as a function of their length. Rig ht: Experimental\nmagnetization curve M(H) (red circles) [28] and the best fit to Eq. (15) (solid\nblack line).\nmagnetization measured along the applied field is expected to be a sup erposition of\nerror functions:\nM(H)\nMmax=1\n2erf/parenleftbiggH−H1\nδH1√\n2/parenrightbigg\n+1\n4erf/parenleftbiggH−H2\nδH2√\n2/parenrightbigg\n+1\n4. (15)\nThe three terms reflect the contributions of the three groups of links with different\norientations.\nThe simulated dependence M(H) is shown in Figure 7 along with the theoretical\ncurve (15) that takes into account the renormalization of the Gau ssian distribution\nparameters (14).\nA close inspection of the simulated curve M(H) shows that on occasion several\nadjacent links reverse simultaneously due to a positive feedback du ring the reversal.\nWhen magnetization of a link is reversed, magnetic charges at its end s are switched.\nThe net magnetic field on an adjacent site, projected onto its easy axis, increases by\n∆H= 2H0cos41◦−(2H0/3)cos11◦= 0.86H0= 0.74 mT. (16)\nThe extra field is not negligible on the scale of the critical-field distribut ion width\nδH′\nc= 2.0 mT. It can help to stimulate the emission of a domain wall at an adjace nt\nsite if that site’s critical field is not too high. This kind of positive feedb ack causes\navalanches , in which magnetization reversals occur nearly simultaneously in links\nresiding along a one-dimensional path determined by the orientation s of easy axes.\nFor example, an avalanche occurring in the background of a fully mag netized state of\nFigure 5(b) would travel along the vertical direction. In the limit of s mall feedback,\n∆H≪δH′\nc, the distribution of avalanche lengths is exponential. Indeed, if the link\nstarting an avalanche of length nhas a critical field H,n−1 of its neighbors must\nhave critical fields in the range between HandH+∆H. The probability to find such\na collection of links is\nPn∼n/integraldisplay\n[ρ(H)∆H]n−1ρ(H)dH=n1/2/parenleftbigg∆H√\n2πδH′c/parenrightbiggn−1\n(17)Dynamics of artificial spin ice 13\nfor a Gaussian distribution of critical fields (7). The distribution of a valanches seen in\nthe simulation is shown in the inset of Figure 7 along with the theoretica l distribution\n(17).\nThese results can be directly compared to the experimental rever sal curve\nmeasured in the same geometry [28], Figure 7 (right panel). Although the overall\nscale of the magnetic field is substantially lower, the data are well fit b y Eq. (15) with\nH1= 35.9 mT and H2= 45.9. The ratio of the reversal fields, H2/H1= 1.28, agrees\nwell with the theoretical value H2/H1= cos(−19◦)/cos41◦= 1.25. The relative\nwidths are δH1/H1= 0.037 and δH2/H2= 0.046.\nThe magnetization curve M(H) was also measured experimentally [28] and\nsimulated for θ= 100◦, with similar results. The experimentally measured reversal\nfields were H1= 34.7 mT and H2= 91.5 mT and relative widths δH1/H1= 0.033\nandδH2/H2= 0.047. The reversal field ratio was H2/H1= 2.64 in the experiment,\nsomewhat off the theoretical value H2/H1= cos1◦/cos61◦= 2.06.\nOverall, it appears that our model provides a reasonably good desc ription of\nmagnetization reversalwhen the field is reapplied at θ= 120◦to the direction of initial\nmagnetization. In this regime, the reversal proceeds in two well-de fined stages, each\ninvolving one subset of links. During each stage, links reverse largely independently,\nalthough sometimes the reversal in one link changes the field on a nea rby site and\ntriggers magnetization reversal there. The reversal fields are g iven approximately by\nthe equations\nH1=¯H′\nc/cos(120◦−θ−α), H2=¯H′\nc/cos(180◦−θ−α).(18)\nThereversalfollowsthetwo-stagescenarioaslongas H1< H2, orθ <150◦−α= 131◦.\nFor larger field rotation angle θ, the reversal proceeds in a very different manner.\n4.2.131◦< θ <180◦: reversal with avalanches\nWhen the field is rotated through θ= 170◦relative to the direction of magnetization,\nthe theory described in Section 4.1 no longer applies. Because H2, the reversal field\nof horizontal links, is lower than H1, these links should reverse first. However, that\nis impossible because in the initial, fully magnetized state, Figure 8(a), these are\nminority links whose critical field is roughly 3 Hc(Section 2.2), i.e., much higher than\nH2=Hc/cos11◦≈Hc. For this reason, a horizontal link does not reverse until\none of its neighbors, a majority link, reverses and in the process alt ers the charge at\none of the horizontal link’s ends. This converts the horizontal link in to a majority\nlink enabling it to reverse magnetization. It turns out that this mode of reversal is\naccompanied by long magnetic avalanches.\nIn the simplest scenario, the dynamics begins with the reversal of t he weakest\nlink with the critical field near H1, Figure 8(b). The reversal turns the horizontal link\nnext to it into a majority link, which is now ready to reverse since the a pplied field\nexceeds its critical field: H≈H1> H2. Aq=−2 domain wall emitted from its left\nend travels to the right end where it encounters a site with charge −1, Figure 8(b).\nAs discussed in Section 2.3, the arriving domain wall induces the emissio n of another\ndomain wall into an adjacent link, Figure 2(b). The magnetization of t hat link gets\nreversed, bringing us to the state shown in Figure 8(d). The cycle r epeats creating an\navalanche. In effect, we have a q= +2 charge moving along a zigzag path parallel to\nthe applied field and reversing magnetization of the links along the way . The process\ncontinues until the moving charge reaches the edge of the system so that an avalanche\nextends from edge to edge.Dynamics of artificial spin ice 14\n(a) −1−1+1−1+1\n+1+1H\n−1+1−1\n−1\n−1\n+1 (b)+1−1H\n−1+1−1\n−1\n−1\n+1 −1−1+1\n+1−1\n(c)H\n+1−3\n−1\n−1\n+1 −1−1+1−1+1\n+1+1 −1\n(d)−1−1\n−1\n−1\n+1 −1−1+1−1+1\n+1+1H\n−1\nFigure 8. Magnetization reversal after the applied field is rotated th rough 170◦.\n−1.0−0.5 0 0.5 1M\n 40  50  60  70  80H, mT 100 1000\n 1  10\nFigure 9. Simulated magnetization reversal curve M(H) in an applied field\nrotated through 170◦. Inset: a log-log plot of the number of avalanches versus\ntheir length (red circles) and a fit to the power-law distribu tion (19) (solid line).Dynamics of artificial spin ice 15\nA different scenario may take place if the system has “weak” links tha t trigger\nthe reversal when the applied field is at or below H2. These can be links at the edge\nof the system or some sort of defects. Their reversal converts one of the horizontal\nlinks (critical field Hc1) to the majority status, as shown in Figure 8(b). When the\napplied field reaches a value sufficient to induce the reversal of that link, an adjacent\nlink is also reversed as described above, Figure 8(c-d). The next ho rizontal link down\nthe line (critical field Hc2) will switch immediately if Hc1> Hc2. The switching will\ncontinue until the avalanche comes to a stubborn link whose critical field exceeds Hc1.\nIts reversalwill happen in a higher applied field, possibly triggeringan otheravalanche.\nIf the first reversal occurs in a link whose critical field Hc1is at the lower end\nof the critical field distribution, the first avalanche will be short bec ause it is unlikely\nthat a large number of subsequent links will have even lower critical fi elds. As further\navalanches get terminated at links with higher critical fields, their len gths will tend\nto increase. Toward the end of the reversal, avalanches will begin w ith links whose\ncritical fields are near the higher end of the distribution. These ava lanches will be\nparticularly long. The last avalanche in a given string of links will termina te at the\nedge or will meet an avalanche traveling in the opposite direction. The se qualitative\nconsiderations anticipate a wide distribution of avalanche lengths. I ndeed, we show in\nAppendix B that the avalanches have a power-law distribution of leng ths,\nPn=C/n. (19)\nRemarkably, this resultapplies toanydistributionofcriticalfields, n otjust aGaussian\none, and numerical simulations confirm this picture.\nAs can be seen in Figure 9, magnetization reversal begins in an applied field\nH≈H2−δH2= 47 mT, where H2is given by Eq. (18). At that point, the\nreversals include single pairs of links from two sublattices. Long avala nches, involving\nas many as n= 10 and more links, are observed by the time the applied field reaches\nH≈H2+δH2= 51 mT. The length distribution is well fit by the power law (19) as\ncan be seen in the inset of Figure 9.\nThe third sublattice reverses in much higher fields, H≈H3= 77 mT, where\nH3=H′\nc/cos(240◦−θ−α), (20)\nThis stage of the reversal proceeds in the gradual manner descr ibed previously.\n5. Discussion\nThe dynamics of magnetization in artificial spin ice is a complex problem. In this\npaper, we have presented a simple model for this system in terms of coarse-grained\nphysical variables (Figure 1), Ising spins σijliving on the links of the spin-ice lattice\nandmagneticcharges qiresidingonitssites. Inspiredbyourearlierstudiesofmagnetic\nnanowires [33, 32], where magnetization reversal is mediated by the propagation of\ndomain walls, we have expressed the magnetization dynamics in spin ice in similar\nterms. Magnetization reversal in individual links of the lattice proce eds through\nthe emission, propagation, and absorption of domain walls with magne tic charge\nqw=±2. Coulomb-like interactions between the magnetic charges of the w alls\nand lattice sites play a major role in the dynamics. For example, the ma gnitude\nof the critical field, required for the emission of a domain wall, is set by the strength\nof magnetostatic attraction between a domain wall and the magnet ic charge of the\nlattice site. These heuristic considerations have been confirmed an d refined throughDynamics of artificial spin ice 16\nmicromagnetic simulations of a small portion of the spin-ice lattice con taining a few\nlinks.\nQuenched disorder is another major element affecting the magnetiz ation\ndynamics. Small imperfections of the artificial lattice are expected to produce a\nGaussian distribution of critical fields. The experimentally measured curve [28] is\nconsistent with a Gaussian shape and width δHc/¯Hc≈0.05.\nThe dynamics of magnetization reversal strongly depends on the d irection of\nthe external magnetic field. If the field is applied at a small angle relat ive to the\nmagnetization of a (fully magnetized) sample, θ <131◦for the parameters we used,\nthe reversalproceeds in a gradual way, with links reversingmore o r less independently\nof each other, when the strength of the applied field exceeds the t hreshold of a given\nlink. For larger angles of rotation, the reversal proceeds in one-d imensional avalanches\nthat can easily span the entire length of the system. The reversal in one link with a\ncritical field Htriggers the reversal in several others along the chain. The avala nche\nstopswhenitencountersalinkwhosecriticalfieldexceeds H. Inthisregime,avalanche\nlengths are distributed as a power law, Pn=C/n.\nIt should be pointed out that we model the magnetization dynamics in artificial\nspin ice as a purely dissipative process, in which the system moves str ictly downhill\nin the energy landscape. Such a picture is very different from an ear lier approach\nextending the notion of an effective temperature to these far-fr om-equilibriumsystems\n[20, 34]. Whereas energy of a microstate plays a major role in the effe ctive thermal\napproach, our method puts the focus on energy gradients , or forces between magnetic\ncharges.\nThis study hasa limited scope. We focus on acontinuously-connecte dhoneycomb\nnetwork realized in several experimental studies [18, 16, 17] and c over only the basic\nregimes of its magnetization dynamics, Figure 6. Interesting pheno mena arise at\nthe boundaries between different regimes, particularly when the fie ld is completely\nreversed, θ= 180◦. In this case, avalanches lose their unidirectional character and\nbecome random walks. As the magnetization reversal proceeds, a valanches can begin\nto intersect and block one another.\nOur method can be easily extended to connected networks with oth er geometries\nsuch as square spin ice [15]. Budrikis, Politi and Stamps used a similar he uristic\napproach to study the dynamics of disconnected magnetic islands [3 5].\nAcknowledgments\nOT and PM thank the Max Planck Institute for the Physics of Complex Systems in\nDresden, where part of this work was carried out. The authors ac knowledge support\nof the Johns Hopkins University under the Provost Undergraduat e Research Award\n(YS) and of the US National Science Foundation under Grants No. D MR-0520491\n(OP and PM), DMR-1056974 (SD and JC), and DMR-1104753 (OT).\nAppendix A. Simulation procedure\nFor a given applied external field, the total magnetic field Hfor each site is computed\nas a sum of the applied field and the Coulomb fields generated by the ch arges at the\nneighboring sites and domain walls (see Section 2.4). For simplicity, we o nly include\nthe fields from first and second-neighbor sites. Fields of further n eighbors decrease\nrapidly and tend to oscillate in sign. For each link attached to a given sit e, theDynamics of artificial spin ice 17\nprogram checks whether the net field has a negative projection He=Hcos(θ−α)\nonto the link’s easy axis, Eq. (12). If He<0, the program calculates the weakness\nof the site and link, W=|He| −Hc. The site and link with the largest Win the\nsample are considered to be the weakest. As the applied field increas es, the largest\nWbecomes positive, triggering the emission of a domain wall from the we akest site\ninto the weakest link. The domain wall propagates to the other end o f the link where\nit is absorbed, either immediately or after the emission of another do main wall as\ndescribed in Section 2. Once the reversal process that started w ith the weakest site is\ncomplete, the program looks for the next weakest site. The proce ss is repeated until\nthere are no positive Win the system. Spin ice rules are satisfied at each site at all\ntimes. No thermal effects are considered.\nAppendix B. Statistics of avalanches in the presence of a wea k link\nHere we derive the statistics of avalanches discussed in Section 4.2. In this case,\nthe reversal begins in Link 1 (critical field H) and spreads to consecutive Links\n2,3,...,n(of the same sublattice) as long as their critical fields are lower than H.\nThe avalanche stops when it encounters link n+ 1 whose critical field exceeds H.\nThe probability density of the critical-field distribution is ρ(H) and the cumulative\nprobability distribution is\nP(H) =/integraldisplayH\n−∞ρ(H′)dH′. (B.1)\nConsider an avalanche beginning on link kwith a critical field between Hand\nH+ dH. Thek−1 preceding links must have critical fields less than H. If the\navalanche has length nthen links k+1,k+2,...,k+n−1 must have critical fields\nless than H, whereas link k+nmust have a higher critical field. The probability of\nsuch a distribution is\nfk\nn(H)dH= [P(H)]k−1ρ(H)dH[P(H)]n−1[1−P(H)]. (B.2)\nHowever, if the avalanche terminates on link L, the last link of the chain, the factor\n1−P(H) drops out because there is no link L+1:\nfL−n+1\nn(H)dH= [P(H)]L−nρ(H)dH[P(H)]n−1. (B.3)\nThe probability to find an avalanche of length nis found by summing this distribution\nover the initial position of the avalanche kand integrating over the critical field H.\nPerforming the sum first, we find\nfn=L−n+1/summationdisplay\nk=1fk\nn=ρPL−1+L−n/summationdisplay\nk=1ρ(Pk+n−2−Pk+n−1) =ρPn−1.(B.4)\nThe integration of the resulting expression yields the expected num ber of avalanches\nof length n,\nFn=/integraldisplay∞\n−∞fn(H)dH\n=/integraldisplay∞\n−∞[P(H)]n−1ρ(H)dH=/integraldisplay1\n0Pn−1dP= 1/n. (B.5)\nNote that Fnis an expectation number of avalanches, not a probability distributio n\nnormalized to 1. Observing an avalanche of length ndoes not exclude the possibilityDynamics of artificial spin ice 18\nof observing an avalanche of a different length n′in the same chain during the same\nreversal process. The probability that an avalanche will have lengt hnis\nPn=Fn/L/summationdisplay\nn=1Fn∼1/(nlnL) (B.6)\nfor large L.\n[1] Harris M J, Bramwell S T, McMorrow D F, Zeiske T and Godfrey K W 1997 Phys. Rev. Lett.\n792554–2557\n[2] Gingras M J P 2011 Spin ice Introduction to frustrated magnetism ed Lacroix C, Mendels P and\nMila F (Springer Verlag) arXiv:0903.2772\n[3] Ramirez A P, Hayashi A, Cava R J, Siddharthan R and Shastry B S 1999 Nature399333–335\n[4] Ryzhkin I A 2005 JETP101481–486\n[5] Castelnovo C, Moessner R and Sondhi S L 2008 Nature45142–45\n[6] Landau L D and Lifshitz E M 1984 Electrodynamics of continuous media 2nd ed (Oxford:\nButterworth-Heinemann) Ch. 44\n[7] Jackson J D 1998 Classical electrodynamics 3rd ed (New York: Wiley) Ch. 5.9\n[8] Saitoh E, Miyajima H, Yamaoka T and Tatara G 2004 Nature432203–206\n[9] Jaubert L D C and Holdsworth P C W 2009 Nat. Phys. 5258–261\n[10] Morris D, Tennant D, Grigera S, Klemke B, Castelnovo C, M oessner R, Czternasty C, Meissner\nM, Rule K, Hoffmann J, Kiefer K, Gerischer S, Slobinsky D and Pe rry R 2009 Science 326\n411–414\n[11] Bramwell S T, Giblin S R, Calder S, Aldus R, Prabhakaran D and Fennell T 2009 Nature461\n956–959\n[12] Fennell T, Deen P, Wildes A, Schmalzl K, Prabhakaran D, B oothroyd A, Aldus R, Mcmorrow\nD and Bramwell S 2009 Science 326415–417\n[13] Kadowaki H, Doi N, Aoki Y, Tabata Y, Sato T J, Lynn J W, Mats uhira K and Hiroi Z 2009 J.\nPhys. Soc. Jpn. 78103706\n[14] Petrenko V F and Whitworth R W 1999 Physics of ice (Oxford University Press)\n[15] Wang R F, Nisoli C, Freitas R S, Li J, McConville W, Cooley B J, Lund M S, Samarth N,\nLeighton C, Crespi V H and Schiffer P 2006 Nature439303–306\n[16] Tanaka M, Saitoh E, Miyajima H, Yamaoka T and Iye Y 2006 Phys. Rev. B 73052411\n[17] Qi Y, Brintlinger T and Cumings J 2008 Phys. Rev. B 77094418\n[18] Ladak S, Read D E, Perkins G K, Cohen L F and Branford W R 201 0Nat. Phys. 6359–363\n[19] Ladak S, Read D, Tyliszczak T, Branford W and Cohen L 2011 N. J. Phys. 13023023\n[20] Nisoli C, Wang R, Li J, McConville W F, Lammert P E, Schiffe r P and Crespi V H 2007 Phys.\nRev. Lett. 98217203\n[21] Wang R F, Li J, McConville W, Nisoli C, Ke X, Freeland J W, R ose V, Grimsditch M, Lammert\nP, Crespi V H and Schiffer P 2007 J. Appl. Phys. 10109J104\n[22] Mengotti E, Heyderman L J, Rodr´ ıguez A F, Nolting F, H¨ u gli R V and Braun H B 2010 Nat.\nPhys.768–74\n[23] Morgan J P, Stein A, Langridge S and Marrows C H 2011 Nat. Phys. 775–79\n[24] Rougemaille N, Montaigne F, Canals B, Duluard A, Lacour D, Hehn M, Belkhou R, Fruchart\nO, El Moussaoui S, Bendounan A and Maccherozzi F 2011 Phys. Rev. Lett. 106057209\n[25] Mellado P, Petrova O, Shen Y and Tchernyshyov O 2010 Phys. Rev. Lett. 105187206\n[26] Budrikis Z, Politi P and Stamps R L 2011 Phys. Rev. Lett. 107217204\n[27] Tchernyshyov O 2010 Nat. Phys. 6323–324\n[28] Daunheimer S A, Petrova O, Tchernyshyov O and Cumings J 2 011Phys. Rev. Lett. 107167201\n[29] Thiaville A and Nakatani Y 2006 Domain-wall dynamics in nanowires and nanostrips Spin\nDynamics in Confined Magnetic Structures III (Topics in applied physics vol 101) (Berlin:\nSpringer) pp 161–205\n[30] Atkinson D, Faulkner C C, Allwood D A and Cowburn R P 2006 D omain-wall dynamics in\nmagnetic logic devices Spin dynamics in confined magnetic structures III (Topics in Applied\nPhysics vol 101) ed Hillebrands B and Thiaville A (Berlin: Springer) pp 207–223\n[31] Donahue M J and Porter D G 1999 Oommf user’s guide, versio n 1.0 Tech. Rep.\nNISTIR 6376 National Institute of Standards and Technology Gaithersburg, MD\nhttp://math.nist.gov/oommf\n[32] Clarke D J, Tretiakov O A, Chern G W, Bazaliy Y B and Tchern yshyov O 2008 Phys. Rev. B\n78134412\n[33] Tretiakov O A, Clarke D, Chern G W, Bazaliy Y B and Tcherny shyov O 2008 Phys. Rev. Lett.\n100127204Dynamics of artificial spin ice 19\n[34] Nisoli C, Li J, Ke X, Garand D, Schiffer P and Crespi V H 2010 Phys. Rev. Lett. 105047205\n[35] Budrikis Z, Politi P and Stamps R L 2010 Phys. Rev. Lett. 105017201"    },    {        "title": "2402.18885v1.Quantum_droplets_with_magnetic_vortices_in_spinor_dipolar_Bose_Einstein_condensates.pdf",        "content": "arXiv:2402.18885v1  [cond-mat.quant-gas]  29 Feb 2024Quantum droplets with magnetic vortices in spinor dipolar B ose-Einstein condensates\nShaoxiong Li1and Hiroki Saito1\n1Department of Engineering Science, University of Electro- Communications, Tokyo 182-8585, Japan\n(Dated: March 1, 2024)\nMotivated by the recent experimental realization of a Bose- Einstein condensate (BEC) of europium\natoms, we investigate the self-bound droplet state of a euro pium BEC with spin degrees of freedom.\nUnder a sufficiently weak magnetic field, the droplet has a toru s shape with circulating spin vectors,\nwhich is referred to as a magnetic vortex. The ground state tr ansforms from the torus to cigar\nshape through bistability with an increase in the magnetic fi eld. Dynamical change of the magnetic\nfield causes the torus to rotate due to the Einstein-de Haas eff ect. The magnetic vortices form a\nsupersolid in a confined system.\nA magnetic flux-closure structure is a stable configura-\ntion of a ferromagnetic material, in which magnetization\nvectors form closed loops to reduce the magnetostatic\nenergy. This structure can be observed in a ferromag-\nnetic material with multiple magnetic domains below the\nCurie temperature [1]. The magnetic flux-closure struc-\nture has also been realized within nanoscale particles [2–\n10], in which the magnetization vectors circulate along\na toroidal loop. This state is referred to as a magnetic\nvortex. The nanoparticles with magnetic vortices can be\nused for, e.g., data storage [11], cancer therapy [12], and\nneuromorphic computing [13]. Such an isolated object\nwith a magnetic vortex has so far been restricted to solid\nmaterials. Is it possible to produce a liquid or gas ana-\nlogue of this state of matter, i.e., a self-bound droplet\nwith a magnetic vortex? Although permanent-magnetic\nliquid droplets have been produced recently [14], the\nmagnetic-vortex structure has not been observed.\nHere we propose a self-bound superfluid droplet that\ncontains a magnetic vortex. Self-bound states of Bose-\nEinstein condensates (BECs) have attracted much inter-\nest recently and are referred to as quantum droplets [15].\nIn a quantum droplet, the attractive mean-field inter-\naction balances with the repulsive beyond-mean-field ef-\nfect [16], which stabilizes the system against collapse and\nexpansion in free space. This novel state of matter was\nrealized in a BEC with magnetic dipole-dipole interac-\ntion (DDI) [17–21] and a Bose-Bose mixture [22–24], and\nvarious theoretical studies have been performed on these\nsystems [25–43]. However, in the experiments of quan-\ntum droplets with DDI to date, the magnetization of the\nsystem has been frozen into the direction of the strong\nexternal magnetic field. If the external magnetic field is\nsuppressed sufficiently, spin degrees of freedom in a dipo-\nlar BEC are liberated, which allows for a spinor dipolar\nBEC [44–69]. However, self-bound quantum droplets of\nspinor dipolar BECs have not yet been studied.\nIn this paper, we will show that there exists a stable\nself-bound droplet of a spinor dipolar BEC that contains\na magnetic vortex with a torus-shaped density distribu-\ntion. This state is stable under a weak external mag-\nnetic field. As the the magnetic field is increased, theground state changes from the magnetic vortex state to\nthe well-known cigar-shaped droplet, and these two states\nexhibit bistability. If the external magnetic field is sud-\ndenly changed, the torus-shaped droplet begins to rotate\nto conserve the total angular momentum, which resem-\nbles the Einstein-de Haas effect. The ground state of a\nconfined system exhibits periodic alignment of the torus-\nshaped droplets, which can be regarded as a supersolid.\nIn the present study, we restrict ourselves to the BEC\nof151Eu atoms, which was recently realized experimen-\ntally [70]. A peculiar feature of151Eu is its wide range\nof the hyperfine spin ( F= 1,···,6) with small spin-\ndependent contact interactions, which could be smaller\nthan the DDI. The spin state in this system is there-\nfore determined mainly by the DDI under a weak exter-\nnal magnetic field, and the DDI-dominant spinor dipo-\nlar phenomena can be investigated. To observe such\nphenomena, the magnetic Feshbach resonance cannot be\nused to tune the contact interaction. Although the s-\nwave scattering length of the F= 6hyperfine state mea-\nsured in Ref. [70] does not satisfy the condition for the\ndroplet formation, the scattering lengths for the other\nhyperfine spins Fand those for153Eu are unknown, for\nwhich the spinor dipolar droplet may be possible. Fur-\nthermore, the contact interaction may be controllable us-\ning microwave-induced Feshbach resonance [71], which\nenables the formation of the spinor dipolar droplet.\nWe consider a BEC of151Eu atoms with hyperfine spin\nFat zero temperature using the beyond-mean-field ap-\nproximation [25, 26]. The total energy consists of five\nterms,E=Ekin+Es+Eddi+ELHY+EB.The kinetic\nenergy is given by Ekin= ¯h2/(2M)/summationtext\nm/integraltext\ndr|∇ψm(r)|2,\nwhereψm(r)is the macroscopic wave function for the\nmagnetic sublevels m=−F,−F+ 1,···,F, andMis\nthe mass of an atom. The wave function is normal-\nized as/summationtext\nm/integraltext\n|ψm(r)|2dr=N, whereNis the total\nnumber of atoms. The spin-independent contact inter-\naction has the form Es= 2π¯h2asM−1/integraltext\nρ2(r)dr, where\nasis the spin-independent s-wave scattering length and\nρ(r) =/summationtext\nm|ψm(r)|2is the total density. The DDI en-\nergy is given by Eddi=µ0(gµB)2/(8π)/integraltext\ndrdr′{f(r)·\nf(r′)−3[f(r)·e][f(r′)·e]}/|r−r′|3, whereµ0is2\nthe magnetic permeability of the vacuum, gis the hy-\nperfinegfactor,µBis the Bohr magneton, f(r) =/summationtext\nmm′ψ∗\nm(r)(S)mm′ψm′(r)withSbeing the spin ma-\ntrix, and e= (r−r′)/|r−r′|. The relative strength\nof the DDI is characterized by εdd=add/as, where\nadd=µ0µ2M/(12π¯h2)is the dipolar length. The mag-\nnetic moment µ=gµBFand the dipolar length addfor\neach spinFof151Eu are given in the Supplemental Ma-\nterial. The spin distribution is mainly determined by the\nDDI, ifaddis much larger than the spin-dependent scat-\ntering lengths that consist of the differences ∆aamong\nthe scattering lengths a0,2,···,2Fin collisional spin chan-\nnels. Since the values of ∆aare predicted to be relatively\nsmall for the europium atoms [70, 72, 73], we ignore the\nspin-dependent contact interaction.\nAs will be confirmed numerically, the spin state is al-\nmost fully polarized in the droplet, and we can use the\nLee-Huang-Yang (LHY) correction for the fully polarized\ndipolar BEC. Under the local density approximation, the\nLHY correction is written as [25, 26, 74, 75]\nELHY=2\n532\n3√π4π¯h2\nMa5/2χ(εdd)/integraldisplay\nρ5/2(r)dr,(1)\nwhereχ(εdd)is the real part of/integraltextπ\n0dθsinθ[1 +\nεdd(3cos2θ−1)]5/2/2. In the presence of an external\nmagnetic field B(r), the linear Zeeman energy has the\nformEB=−gµB/integraltext\nf(r)·B(r)dr. The ratio of the\nquadratic Zeeman energy to the linear Zeeman energy is\nestimated to be µB/∆hf∼10−4at most for the present\nmagnetic field ∼0.1mG, and the quadratic Zeeman en-\nergy can be neglected even for the relatively small hyper-\nfine splitting ∆hf/¯h∼100MHz of a europium atom [76].\nThe Gross-Pitaevskii (GP) equation is given by the\nfunctional derivative of the total energy as i¯h∂ψm/∂t=\nδE/δψ∗\nm. To obtain the ground state or metastable state,\nthe GP equation is propagated in imaginary time, in\nwhichion the left-hand side of the GP equation is re-\nplaced with −1. The GP equation is numerically solved\nusing the pseudospectral method with typical spatial and\ntime stepsdx∼0.01µmanddt∼0.1µs.\nFirst, we consider the case in which the external mag-\nnetic fieldBis zero. Figure 1(a) shows a typical ground\nstate, which has a torus shape, in contrast to the usual\ncigar-shaped droplet in a strong magnetic field [17–21].\nThe magnetization vectors fcirculate along the torus,\nas shown by the arrows in Fig. 1(a). Although such\na magnetic vortex state has already been proposed for\na trapped spinor dipolar BEC [49, 51], this is the first\nexample of a self-bound droplet of a fluid containing a\nmagnetic vortex.\nIn the LHY energy in Eq. (1), we assumed that the spin\nstate is fully polarized, and here we examine the validity\nof this assumption. Figure 1(c) shows the distributions\nof the atomic density ρand magnetization density |f|,\nwhich indicates that the spin is almost fully polarized,\nxy(d)\nz0\n03\n3\n\u0001\n0\n- \u0001\u0001\n0\n-\u0001\u0001\n0\n- \u0001\n(a) (b)\n(c)\nxxy\n\u0001\u0002\u0003\u0004m\n\u0005\u0004mm\u00061\nm\u00060\nm\u0006\u00071\nx / μm\b\nf\b, f\nFIG. 1. Self-bound ground state with a magnetic vortex\nforB= 0. The system has translational and rotational sym-\nmetry, and the origin and zaxis are taken as the center and\nsymmetry axis of the torus, respectively. (a) Density dis-\ntribution on the z= 0plane for F= 1,N= 15000 , and\nεdd= 1.2. The arrows represent the magnetization f. The\ninset shows the isodensity surface. (b) Phase distribution s of\ncomponents m= 1,0,−1on thez= 0plane. (c) Distribu-\ntions ofρand|f|along the xaxis. (d) Density distributions\non thez= 0plane (upper panels) and y= 0plane (lower\npanels) for (F,N,ε dd) = (1,15000,1.2),(1,80000,1.2), and\n(6,15000,1.3). The unit of density in (a), (c), and (d) is\nNµm−3.\ni.e.,|f|/ρ≃F= 1, except near the center. This result\njustifies the use of Eq. (1), since the LHY correction is im-\nportant only in the high-density region to counteract the\ncollapse. The central hole of the torus is mainly occupied\nby them= 0component, since the m/ne}ationslash= 0components\nhave the topological defects at the center, as shown in\nFig. 1(b).\nFigure 1(d) shows the parameter dependence of the\ndensity profile. The size of the droplet increases with\nthe number of atoms N, while the aspect ratio between\nthe major and minor radii appears almost unchanged.\nFor larger spin F= 6, on the other hand, the hole of\nthe torus is enlarged and the aspect ratio is significantly\nchanged. This is due to the kinetic energy that arises\nfrom the spin winding, which is proportional to F. Such3\n(a)\n(b)r / \u0001m\u0002\u0003/\u0003\u0004N\u0001m\u0005\u0006\u0007\n00.20.40.60.81\n1.1 1.2 1.3 1.4 1.5F = 1F = 600.020.040.060.080.10.12\n1.2 1.3 1.4 1.5N / 105\n\bdd0123\n0 0.2 0.4 0.6 0.8 1 1.2 1.4GP\nvariationalF = 1\nN = 15000\nF = 1\nN = 80000 F = 6\nN = 15000\nFIG. 2. (a) Density distribution ρ(r,z= 0) obtained by\nthe imaginary-time evolution of the GP equation (solid line s)\nand the variational method (dashed lines) for (F,N,ε dd) =\n(1,15000,1.2),(1,80000,1.2), and(6,15000,1.3). (b) Lines:\nthe critical number of atoms above which the droplet is stabl e,\nobtained by the variational method. The six lines represent\nF= 1,···, 6 from left to right. The circles (crosses) indicate\nthat the droplet is stable (unstable) for F= 1(blue or dark\ngray) and F= 6(red or light gray), obtained by the GP\nequation. The inset shows a magnification of the main panel.\nbehavior is analyzed in the Supplemental Material using\nthe variational method.\nWe employ the variational wave function as,\nΨv(r) =/radicalbig\nρv(r,z)e−iSzφζ(y), (2)\nwhere(r,φ,z)is the cylindrical coordinate, and ζ(y)rep-\nresents the spin state fully polarized in the ydirection\nwith/summationtext\nm|ζ(y)\nm|2= 1. Thezaxis is taken as the sym-\nmetry axis of the torus. The matrix e−iSzφrotates the\nspin vector to make a magnetic vortex. We propose a\ntorus-shaped variational density as\nρv(r,z) =N\nπ3/2σ2λ+2rσzΓ(λ+1)rλe−r2\nσ2r−z2\nσ2z,(3)\nwhereσr>0,σz>0, andλ >0are variational pa-\nrameters and Γis the gamma function. The variational\nenergy for Eqs. (2) and (3) is derived in the Supplemen-\ntal Material, which is numerically minimized with respect\nto the variational parameters using the Newton-Raphson\nmethod.\nFigure 2(a) compares the density distributions ρ(r,z=\n0)obtained by the GP equation and by the variational\nmethod. The two distributions agree well with each\n(a) (b)\n(c)xz\nxz\n0.10.20.30.40.50.6\n0 0.1 0.2 0.3 0.4unstabletoruscigar\nBz / mG\nBz / mGN / 105\n-15-10-50\ncigar\ntorus\n00.20.40.60.81\n0 0.1 0.2cigar\ntorusFz5\u0001m\n\u0002\u0001mE / (Nh2M\u0003\u0002\u0001m\u0003\u0004)\nFIG. 3. Effects of magnetic field Bzobtained by the GP\nequation for F= 1andεdd= 1.2. (a) Cross-sectional ρand\nfdistributions and isodensity surface of the ground state fo r\nN= 50000 andBz= 0.2mG. (b) Stability diagram. Param-\neter sets marked by circles and squares respectively indica te\nthat the torus-shaped and cigar-shaped droplets are stable or\nmetastable. (c) Bzdependence of the energy Eand aver-\naged magnetization Fzof the torus-shaped and cigar-shaped\ndroplets with N= 50000 . The inset shows ρandfdistribu-\ntions of the torus-shaped droplet for Bz= 0.16mG, where the\ncross section is taken for the symmetry plane ( y= 0plane).\nother. ForN= 80000 , the distribution of the GP result\nbecomes broader than that of the variational method,\nsince the flat-top tendency of a large droplet is not taken\ninto account in Eq. (3). The density at r= 0must van-\nish for the fully-polarized assumption in Eq. (2), whereas\nthe center is slightly occupied for the GP results. Fig-\nure 2(b) shows the critical number of atoms above which\nthe droplet is stable, where the lines are obtained by the\nvariational method and the plots by the GP equation.\nThe variational method can predict the critical number\nof atoms very well, which facilitates the study of this\nsystem, because the numerical cost for the GP simula-\ntion is much higher than that for the variational method,\nespecially near the critical line of the stability where con -\nvergence of the imaginary-time evolution is slow.\nWe next examine the effect of the external magnetic4\n-0.06-0.04-0.0200.020.040.06\n00.511020 3 \u00004050\n(a)\n(b)t = 0 ms\nt  / mst = 20 ms t = 40 ms\n01.2\nFz\nFz+Lz\nLzxxz\ny\n\u0001\u0002m\n0.1mG0.05mG}\n}\nFIG. 4. Einstein-de Haas effect of the torus-shaped droplet\nforF= 1,εdd= 1.2, andN= 15000 . The initial state is\nthe ground state for zero magnetic field (the same state as\nFig. 1(a), except that the symmetry axis of the torus is taken\nto be the ydirection). (a) Time evolution of the density dis-\ntribution on the z= 0plane (main panels) and isodensity\nsurface observed from the −ydirection (insets), where the\nmagnetic field Bz= 0.1mG is switched on at t= 0. The unit\nof density is Nµm−3. See the Supplemental Material for the\nmovie of the dynamics. (b) Time evolution of the orbital an-\ngular momentum Lz(solid lines), spin angular momentum Fz\n(dotted lines), and total angular momentum Fz+Lz(dashed\nlines) for Bz= 0.05(blue or dark gray) and 0.1 mG (red or\nlight gray). The first 1 ms is magnified.\nfield applied in the zdirection. Figure 3(a) shows a typ-\nical ground state for a large magnetic field Bz, where\nthe droplet has a cigar-shape, as observed experimen-\ntally. The spin is almost polarized in the zdirection,\nwhereas it is slightly tilted around both edges of the cigar-\nshape and exhibits the flower-like structure [49]. Fig-\nure 3(b) shows the stability diagram with respect to N\nandBz. There is a critical magnetic field, above which\nthe torus-shaped droplet becomes unstable, whereas the\ncigar-shaped droplet becomes unstable below some crit-\nical magnetic field. In Fig. 3(b), there is a bistabil-\nity region in which both torus-shaped and cigar-shaped\ndroplets are stationary (both circles and squares are\nmarked). Figure 3(c) reveals the bistability with plots\nof the energy Eand the averaged magnetization in the z\ndirectionFz=/integraltextfzdr/Nfor both droplets. The bistabil-\nity ranges from Bz≃0.03to≃0.17mG, and the energies\nof the two droplets cross at Bz≃0.14mG. The direction\nof the torus-shaped droplet is fixed by the magnetic field\nin such a way that the toroidal plane is parallel to the z\ndirection, as shown in the inset of Fig. 3(c). In this inset,\nthe right-hand side of the torus becomes slightly thicker\nthan the left-hand side, which results in the increase of\nFz.\nThe increase of the magnetization Fzwith the mag-\nnetic fieldBzimplies the emergence of the Einstein-de\nHaas effect [77, 78]; if the applied magnetic field Bzis in-\n0 0.8xz\ny( \u0002 \u0003\n( \u0004 \u0003\nxy2\u0001m\nFIG. 5. Ground state for F= 1,N= 4×105,εdd=\n1.4, andBz= 0in a harmonic potential M(ω2\nxx2+ω2\nyy2+\nω2\nzz2)/2with(ωx,ωy,ωz) = 2π×(100,1500,6000) Hz. (a)\nIsodensity surface and (b) distributions of the density ρand\nmagnetization fon thez= 0plane. The unit of density is\nNµm−3.\ncreased dynamically, the spin angular momentum Fzwill\nincrease, which must be accompanied by a decrease in the\norbital angular momentum Lz=−i/integraltext/summationtext\nmdrψ∗\nm(x∂y−\ny∂x)ψ∗\nmto conserve the total angular momentum. Fig-\nure 4 demonstrates the dynamics of the Einstein-de Haas\neffect, where the initial droplet state is prepared for zero\nmagnetic field with the symmetry axis in the ydirection,\nand the magnetic field Bzis turned on at t= 0. As ex-\npected, the droplet begins to rotate around the zaxis,\nwhere the total angular momentum Fz+Lzis maintained\nto be zero. We note that such mechanical rotation of the\ntorus is a clearer manifestation of the Einstein-de Haas\neffect than that in the trapped system [45].\nNext we consider the ground state of the system con-\nfined in a “surfboard-shaped” trap [79], where the trap is\nweak, moderate, and tight in the x,y, andzdirections,\nrespectively. Figure 5 shows the density and spin distri-\nbutions of the ground state, in which multiple droplets\nwith magnetic vortices are aligned along the xaxis with\nalternate circulations of the magnetic vortices. This stat e\ncan be called a one-dimensional supersolid, because the\nground state has the (quasi)periodicity that breaks the\n(quasi)translation symmetry (in the xdirection), while\neach droplet is connected with adjacent droplets, which\nenables superflow between them. We note that the dipo-\nlar supersolid of cigar-shaped droplets [80–94] also re-\nquires confinement in one or two directions, i.e., re-\nstricted geometry is required for the dipolar BEC to split\ninto multiple droplets. If the trap in the yorzdirection\nin Fig. 5 is removed, the ground state becomes a large\nsingle droplet with a magnetic vortex.\nThe torus-shaped droplet may be generated experi-\nmentally by the following procedure. First, the conden-\nsate atoms are prepared in the |F,m= 0/an}b∇acket∇i}hthyperfine state\nin an optical trap. Since this nonmagnetized state is dy-\nnamically and energetically unstable, spontaneous mag-\nnetization occurs [55]. If the BEC is confined to the ex-\npected droplet size, the spontaneous magnetization will5\nform the magnetic vortex with the lowest energy [49].\nAfter some relaxation time, the optical trap is switched\noff, which results in the self-bound torus-shaped droplet.\nWhen the hyperfine spins with higher energies ( F/ne}ationslash= 6)\nare used, the above experimental procedure must be ac-\ncomplished within the lifetime due to hyperfine exchang-\ning collisions, which has not been measured for a eu-\nropium BEC.\nTo summarize, we have investigated a self-bound\ndroplet in a spinor dipolar BEC. For a large number\nof atoms, large εdd, and a small magnetic field, there\nexists a stable self-bound torus-shaped droplet that con-\ntains a magnetic vortex (Fig. 1). For some range of the\nmagnetic field, the system exhibits bistability between\nthe torus and cigar-shaped droplets (Fig. 3). The torus-\nshaped droplet exhibits the Einstein-de Haas effect by\nthe change of the applied magnetic field (Fig. 4). Multi-\nple droplets form a supersolid in a surfboard-shaped trap\n(Fig. 5).\nThis work was supported by JSPS KAKENHI Grant\nNo. JP23K03276.\n[1] For example, C. Kittel, Introduction to Solid State\nPhysics , 8th ed., Chap. 12 (John Wiley & Sons, New\nJersey, 2005).\n[2] A. Tonomura, T. Matsuda, J. Endo, T. Arii, and K. Mi-\nhama, Direct observation of fine structure of magnetic\ndomain walls by electron holography, Phys. Rev. Lett.\n44, 1430 (1980).\n[3] R. P. Cowburn, D. K. Koltsolv, A. O. Adeyeye, M. E.\nWelland, and D. M. Tricker, Single-domain circular nano-\nmagnets, Phys. Rev. Lett. 83, 1042 (1999).\n[4] T. Shinjo, T. Okuno, R. Hassdorf, K. Shigeto, and T.\nOno, Magnetic vortex core observation in circular dots\nof permalloy, Science 289, 930 (2000).\n[5] J. Rothman, M. Kläui, L. Lopez-Diaz, C. A. F. Vaz, A.\nBleloch, J. A. C. Bland, Z. Cui, and R. Speaks, Obser-\nvation of a bi-domain state and nucleation free switching\nin mesoscopic ring magnets, Phys. Rev. Lett. 86, 1098\n(2001).\n[6] S. P. Li, D. Peyrade, M. Natali, A. Lebib, Y. Chen, U.\nEbels, L. D. Buda, and K. Ounadjela, Flux closure struc-\ntures in cobalt rings, Phys. Rev. Lett. 86, 1102 (2001).\n[7] A. Wachowiak, J. Wiebe, M. Bode, O. Pietzsch, M. Mor-\ngenstern, and R. Wiesendanger, Direct observation of in-\nternal spin structure of magnetic vortex cores, Science\n298, 577 (2002).\n[8] M. J. Hÿtch, R. E. Dunin-Brokowski, M. R. Scheinfein, J.\nMoulin, C. Duhamel, F. Mazaleyrat, and Y. Champion,\nVortex flux channeling in magnetic nanoparticle chains,\nPhys. Rev. Lett. 91, 257207 (2003).\n[9] C.-J. Jia, L.-D. Sun, F. Luo, X.-D. Han, L. J. Heyderman,\nZ.-G. Yan, C.-H. Yan, K. Zheng, Z. Zhang, M. Takano,\nN. Hayashi, M. Eltschka, M. Kläui, U. Rüdiger, T.\nKasama, L. Cervera-Gontard, R. E. Dunin-Borkowski, G.\nTzvetkov, and J. Raabe, Large-scale synthesis of single-\ncrystalline iron oxide magnetic nanorings, J. Am. Chem.Soc.130, 16968 (2008).\n[10] G. R. Lewis, J. C. Loudon, R. Tovey, Y.-H. Chen, A. P.\nRoberts, R. J. Harrison, P. A. Midgley, and E. Ringe,\nMagnetic vortex states in toroidal iron oxide nanoparti-\ncles: combining micromagnetics with tomography, Nano\nLett.20, 7405 (2020).\n[11] R. P. Cowburn, Magnetic nanodots for device applica-\ntions, J. Magn. Magn. Mater. 242-245 , 505 (2002).\n[12] D.-H. Kim, E. A. Rozhkova, I. V. Ulasov, S. D. Bader,\nT. Rajh, M. S. Lesniak, and V. Novosad, Biofunctional-\nized magnetic-vortex microdiscs for targeted cancer-cell\ndestruction, Nat. Mater. 9, 165 (2009).\n[13] J. Torrejon, M. Riou, F. A. Araujo, S. Tsunegi, G.\nKhalsa, D. Querlioz, P. Bortolotti, V. Cros, K. Yakushiji,\nA. Fukushima, H. Kubota, S. Yuasa, M. D. Stiles, and J.\nGrollier, Neuromorphic computing with nanoscale spin-\ntronic oscillators, Nature (London) 547, 428 (2017).\n[14] X. Liu, N. Kent, A. Ceballos, R. Streubel, Y. Jiang,\nY. Chai, P. Y. Kim, J. Forth, F. Hellman, S. Shi, D.\nWang, B. A. Helms, P. D. Ashby, P. Fischer, and T.\nP. Russell, Reconfigurable ferromagnetic liquid droplets,\nScience 365, 264 (2019).\n[15] D. S. Petrov, Quantum mechanical stabilization of a col -\nlapsing Bose-Bose mixture, Phys. Rev. Lett. 115, 155302\n(2015).\n[16] T. D. Lee, K. Huang, and C. N. Yang, Eigenvalues and\neigenfunctions of a Bose system of hard spheres and\nits low-temperature properties, Phys. Rev. 106, 1135\n(1957).\n[17] H. Kadau, M. Schmitt, M. Wenzel, C. Wink, T. Maier, I.\nFerrier-Barbut, and T. Pfau, Observing the Rosensweig\ninstability of a quantum ferrofluid, Nature (London) 530,\n194 (2016).\n[18] M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut,\nand T. Pfau, Self-bound droplets of a dilute magnetic\nquantum liquid, Nature (London) 539, 259 (2016).\n[19] I. Ferrier-Barbut, H. Kadau, M. Schmitt, M. Wenzel, and\nT. Pfau, Observation of quantum droplets in a strongly\ndipolar Bose gas, Phys. Rev. Lett. 116, 215301 (2016).\n[20] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler,\nL. Santos, and F. Ferlaino, Quantum-fluctuation-driven\ncrossover from a dilute Bose-Einstein condensate to a\nmacrodroplet in a dipolar quantum fluid, Phys. Rev. X\n6, 041039 (2016).\n[21] I. Ferrier-Barbut, M. Wenzel, F. Böttcher, T. Langen, M .\nIsoard, S. Stringari, and T. Pfau, Scissors mode of dipolar\nquantum droplets of dysprosium atoms, Phys. Rev. Lett.\n120, 160402 (2018).\n[22] C. R. Cabrera, L. Tanzi, J. Sanz, B. Naylor, P. Thomas,\nP. Cheiney, and L. Tarruell, Quantum liquid droplets in\na mixture of Bose-Einstein condensates, Science 359, 301\n(2018).\n[23] G. Semeghini, G. Ferioli, L. Masi, C. Mazzinghi, L. Wol-\nswijk, F. Minardi, M. Modugno, G. Modugno, M. In-\nguscio, and M. Fattori, Self-bound quantum droplets of\natomic mixtures in free space, Phys. Rev. Lett. 120,\n235301 (2018).\n[24] P. Cheiney, C. R. Cabrera, J. Sanz, B. Naylor, L. Tanzi,\nand L. Tarruell, Bright soliton to quantum droplet tran-\nsition in a mixture of Bose-Einstein condensates, Phys.\nRev. Lett. 120, 135301 (2018).\n[25] F. Wächtler and L. Santos, Quantum filaments in dipolar\nBose-Einstein condensates, Phys. Rev. A 93, 061603(R)\n(2016).6\n[26] F. Wächtler and L. Santos, Ground-state properties and\nelementary excitations of quantum droplets in dipolar\nBose-Einstein condensates, Phys. Rev. A 94, 043618\n(2016).\n[27] H. Saito, Path-integral Monte Carlo study on a droplet\nof a dipolar Bose-Einstein condensate stabilized by quan-\ntum fluctuation, J. Phys. Soc. Jpn. 85, 053001 (2016).\n[28] R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie,\nGround-state phase diagram of a dipolar condensate with\nquantum fluctuations, Phys. Rev. A 94, 033619 (2016).\n[29] A. Macia, J. Sánchez-Baena, J. Boronat, and F. Maz-\nzanti, Droplets of trapped quantum dipolar bosons, Phys.\nRev. Lett. 117, 205301 (2016).\n[30] M. Wenzel, F. Böttcher, T. Langen, I. Ferrier-Barbut,\nand T. Pfau, Striped states in a many-body system of\ntilted dipoles, Phys. Rev. A 96, 053630 (2017).\n[31] D. Baillie, R. M. Wilson, and P. B. Blakie, Collective\nexcitations of self-bound droplets of a dipolar quantum\nfluid, Phys. Rev. Lett. 119, 255302 (2017).\n[32] A. Cidrim, F. E. A. dos Santos, E. A. L. Henn, and T.\nMacrí, Vortices in self-bound dipolar droplets, Phys. Rev.\nA98, 023618 (2018).\n[33] Y. Li, Z. Chen, Z. Luo, C. Huang, H. Tan, W. Pang,\nand B. A. Malomed, Two-dimensional vortex quantum\ndroplets, Phys. Rev. A 98, 063602 (2018).\n[34] Y. V. Kartashov, B. A. Malomed, and L. Torner,\nMetastability of quantum droplet clusters, Phys. Rev.\nLett.122, 193902 (2019).\n[35] F. Böttcher, M. Wenzel, J.-N. Schmidt, M. Guo, T. Lan-\ngen, I. Ferrier-Barbut, T. Pfau, R. Bombín, J. Sánchez-\nBaena, J. Boronat, and F. Mazzanti, Dilute dipolar quan-\ntum droplets beyond the extended Gross-Pitaevskii equa-\ntion, Phys. Rev. Res. 1, 033088 (2019).\n[36] G. Ferioli, G. Semeghini, L. Masi, G. Giusti, G. Mod-\nugno, M. Inguscio, A. Gallemí, A. Recati, and M. Fattori,\nCollisions of self-bound quantum droplets, Phys. Rev.\nLett.122, 090401 (2019).\n[37] X. Zhang, X. Xu, Y. Zheng, Z. Chen, B. Liu, C.\nHuang, B. A. Malomed, and Y. Li, Semidiscrete quan-\ntum droplets and vortices, Phys. Rev. Lett. 123, 133901\n(2019).\n[38] R. Ołdziejewski, W. Górecki, K. Pawłowski, and K.\nRzążewski, Strongly correlated quantum droplets in\nquasi-1D dipolar Bose gas, Phys. Rev. Lett. 124, 090401\n(2020).\n[39] R. N. Bisset, L. A. Peña Ardila, and L. Santos, Quan-\ntum droplets of dipolar mixtures, Phys. Rev. Lett. 126,\n025301 (2021).\n[40] J. C. Smith, D. Baillie, and P. B. Blakie, Quantum\ndroplet states of a binary magnetic gas, Phys. Rev. Lett.\n126, 025302 (2021).\n[41] A.-C. Lee, D. Baillie, and P. B. Blakie, Numerical calcu la-\ntion of dipolar-quantum-droplet stationary states, Phys.\nRev. Res. 3, 013283 (2021).\n[42] T. A. Yoğurt, A. Keleş, and M. Ö. Oktel, Spinor bo-\nson droplets stabilized by spin fluctuations, Phys. Rev.\nA105, 043309 (2022).\n[43] J. Kopyciński, M. Łebek, W. Górecki, and K. Pawłowski,\nUltrawide dark solitons and droplet-soliton coexistence i n\na dipolar Bose gas with strong contact interactions, Phys.\nRev. Lett. 130, 043401 (2023).\n[44] S. Yi, L. You, and H. Pu, Quantum phases of dipolar\nspinor condensates, Phys. Rev. Lett. 93, 040403 (2004).\n[45] Y. Kawaguchi, H. Saito, and M. Ueda, Einstein-de Haaseffect in dipolar Bose-Einstein condensates, Phys. Rev.\nLett.96, 080405 (2006).\n[46] L. Santos and T. Pfau, Spin-3 chromium Bose-Einstein\ncondensates, Phys. Rev. Lett. 96, 190404 (2006).\n[47] R. B. Diener and T.-L. Ho,52Crspinor condensate: a\nbiaxial or uniaxial spin nematic, Phys. Rev. Lett. 96,\n190405 (2006).\n[48] S. Yi and H. Pu, Spontaneous spin textures in dipolar\nspinor condensates, Phys. Rev. Lett. 97, 020401 (2006).\n[49] Y. Kawaguchi, H. Saito, and M. Ueda, Spontaneous\ncirculation in ground-state spinor dipolar Bose-Einstein\ncondensates, Phys. Rev. Lett. 97, 130404 (2006).\n[50] Y. Kawaguchi, H. Saito, and M. Ueda, Can spinor dipolar\neffects be observed in Bose-Einstein condensates?, Phys.\nRev. Lett. 98, 110406 (2007).\n[51] M. Takahashi, S. Ghosh, T. Mizushima, and K. Machida,\nSpinor dipolar Bose-Einstein condensates: Classical spin\napproach, Phys. Rev. Lett. 98, 260403 (2007).\n[52] K. Gawryluk, M. Brewczyk, K. Bongs, and M. Gajda,\nResonant Einstein-de Haas effect in a rubidium conden-\nsate, Phys. Rev. Lett. 99, 130401 (2007).\n[53] B. Sun and L. You, Observing the Einstein-de Haas effect\nwith atoms in an optical lattice, Phys. Rev. Lett. 99,\n150402 (2007).\n[54] M. Yasunaga and M. Tsubota, Spin echo in spinor dipolar\nBose-Einstein condensates, Phys. Rev. Lett. 101, 220401\n(2008).\n[55] S. Hoshi and H. Saito, Symmetry-breaking magnetiza-\ntion dynamics of spinor dipolar Bose-Einstein conden-\nsates, Phys. Rev. A 81, 013627 (2010).\n[56] T. Świsłocki, M. Brewczyk, M. Gajda, and K. Rzążewski,\nSpinor condensate of87Rbas a dipolar gas, Phys. Rev.\nA81, 033604 (2010).\n[57] J. A. M. Huhtamäki, M. Takahashi, T. P. Simula, T.\nMizushima, and K. Machida, Spin textures in conden-\nsates with large dipole moments, Phys. Rev. A 81,\n063623 (2010).\n[58] F. Deuretzbacher, G. Gebreyesus, O. Topic, M. Scherer,\nB. Lücke, W. Ertmer, J. Arlt, C. Klempt, and L. San-\ntos, Parametric amplification of matter waves in dipo-\nlar spinor Bose-Einstein condensates, Phys. Rev. A 82,\n053608 (2010).\n[59] K. Kudo and Y. Kawaguchi, Hydrodynamic equation of\na spinor dipolar Bose-Einstein condensate, Phys. Rev. A\n82, 053614 (2010).\n[60] T. P. Simula, J. A. M. Huhtamäki, M. Takahashi, T.\nMizushima, and K. Machida, Rotating dipolar spin-1\nBose-Einstein condensates, J. Phys. Soc. Jpn. 80, 013001\n(2011).\n[61] K. Gawryluk, K. Bongs, and M. Brewczyk, How to ob-\nserve dipolar effects in spinor Bose-Einstein condensates,\nPhys. Rev. Lett. 106, 140403 (2011).\n[62] B. Pasquiou, E. Maréchal, G. Bismut, P. Pedri, L.\nVernac, O. Gorceix, and B. Laburthe-Tolra, Sponta-\nneous demagnetization of a dipolar spinor Bose gas in\nan ultralow magnetic field, Phys. Rev. Lett. 106, 255303\n(2011).\n[63] Y. Eto, H. Saito, and T. Hirano, Observation of dipole-\ninduced spin texture in an87RbBose-Einstein conden-\nsate, Phys. Rev. Lett. 112, 185301 (2014).\n[64] G. E. Marti, A. MacRae, R. Olf, S. Lourette, F. Fang,\nand D. M. Stamper-Kurn, Coherent magnon optics in\na ferromagnetic spinor Bose-Einstein condensate, Phys.\nRev. Lett. 113, 155302 (2014).7\n[65] H. Saito and M. Kunimi, Energy shift of magnons in\na ferromagnetic spinor-dipolar Bose-Einstein condensate ,\nPhys. Rev. A 91, 041603(R) (2015).\n[66] T. Oshima and Y. Kawaguchi, Spin Hall effect in a\nspinor dipolar Bose-Einstein condensate, Phys. Rev. A\n93, 053605 (2016).\n[67] M. O. Borgh, J. Lovegrove, and J. Ruostekoski, Inter-\nnal structure and stability of vortices in a dipolar spinor\nBose-Einstein condensate, Phys. Rev. A 95, 053601\n(2017).\n[68] B. Liao, S. Li, C. Huang, Z. Luo, W. Pang, H. Tan, B. A.\nMalomed, and Y. Li, Anisotropic semivortices in dipolar\nspinor condensates controlled by Zeeman splitting, Phys.\nRev. A 96, 043613 (2017).\n[69] S. Lepoutre, K. Kechadi, B. Naylor, B. Zhu, L. Gabar-\ndos, L. Isaev, P. Pedri, A. M. Rey, L. Vernac, and B.\nLaburthe-Tolra, Spin mixing and protection of ferromag-\nnetism in a spinor dipolar condensate, Phys. Rev. A 97,\n023610 (2018).\n[70] Y. Miyazawa, R. Inoue, H. Matsui, G. Nomura, and M.\nKozuma, Bose-Einstein condensation of europium, Phys.\nRev. Lett. 129, 223401 (2022).\n[71] D. J. Papoular, G. V. Shlyapnikov, and J. Dalibard,\nMicrowave-induced Fano-Feshbach resonances, Phys.\nRev. A 81, 041603(R) (2010).\n[72] A. A. Buchachenko, G. Chałasiński, and M. M.\nSzcz¸ eśniak, Europium dimer: van der Waals molecule\nwith extremely weak antiferromagnetic spin coupling, J.\nChem. Phys. 131, 241102 (2009).\n[73] Y. V. Suleimanov, Zeeman relaxation of magnetically\ntrapped Eu atoms, Phys. Rev. A 81, 022701 (2010).\n[74] A. R. P. Lima and A. Pelster, Quantum fluctuations in\ndipolar Bose gases, Phys. Rev. A 84, 041604(R) (2011).\n[75] A. R. P. Lima and A. Pelster, Beyond mean-field low-\nlying excitations of dipolar Bose gases, Phys. Rev. A 86,\n063609 (2012).\n[76] W.-G. Jin, T. Endo, T. Wakui, H. Uematsu, T. Mi-\nnowa, and H. Katsuragawa, Measurements of the Hy-\nperfine Structure and ∆F= +2 Transitions in Eu I by\nHigh-Resolution Diode-Laser Spectroscopy, J. Phys. Soc.\nJpn.71, 1905 (2002).\n[77] O. W. Richardson, A mechanical effect accompanying\nmagnetization, Phys. Rev. Ser. I 26, 248 (1908).\n[78] A. Einstein and W. J. de Haas, Experimental proof of\nAmpére’s molecular currents, Verh. Dtsch. Phys. Ges.\n17, 152 (1915).\n[79] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalat-\ntore, and D. M. Stamper-Kurn, Spontaneous symme-\ntry breaking in a quenched ferromagnetic spinor Bose-\nEinstein condensate, Nature (London) 443, 312 (2006).\n[80] L. Tanzi, S. M. Roccuzzo, E. Lucioni, F. Famá, A.\nFioretti, C. Gabbanini, G. Modugno, A. Recati, and S.\nStringari, Supersolid symmetry breaking from compres-\nsional oscillations in a dipolar quantum gas, Nature (Lon-\ndon)574, 382 (2019).\n[81] F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M.Guo, T. Langen, and T. Pfau, Transient supersolid prop-\nerties in an array of dipolar quantum droplets, Phys. Rev.\nX9, 011051 (2019).\n[82] L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Traut-\nmann, C. Politi, G. Durastante, R. M. W. van Bijnen, A.\nPatscheider, M. Sohmen, M. J. Mark, and F. Ferlaino,\nLong-lived and transient supersolid behaviors in dipolar\nquantum gases, Phys. Rev. X 9, 021012 (2019).\n[83] L. Tanzi, S. M. Roccuzzo, E. Lucioni, F. Famá, A.\nFioretti, C. Gabbanini, G. Modugno, A. Recati, and S.\nStringari, Supersolid symmetry breaking from compres-\nsional oscillations in a dipolar quantum gas, Nature (Lon-\ndon)574, 382 (2019).\n[84] M. Guo, F. Böttcher, J. Hertkorn, J.-N. Schmidt, M.\nWenzel, H. P. Büchler, T. Langen, and T. Pfau, The low-\nenergy Goldstone mode in a trapped dipolar supersolid,\nNature (London) 574, 386 (2019).\n[85] G. Natale, R. M. W. van Bijnen, A. Patscheider, D. Pet-\nter, M. J. Mark, L. Chomaz, and F. Ferlaino, Excitation\nspectrum of a trapped dipolar supersolid and its experi-\nmental evidence, Phys. Rev. Lett. 123, 050402 (2019).\n[86] L. Tanzi, J. G. Maloberti, G. Biagioni, A. Fioretti, C.\nGabbanini, and G. Modugno, Evidence of superfluidity in\na dipolar supersolid from nonclassical rotational inertia ,\nScience 371, 1162 (2021).\n[87] P. Ilzhöfer, M. Sohmen, G. Durastante, C. Politi, A.\nTrautmann, G. Natale, G. Morpurgo, T. Giamarchi, L.\nChomaz, M. J. Mark, and F. Ferlaino, Phase coherence in\nout-of-equilibrium supersolid states of ultracold dipola r\natoms, Nat. Phys. 17, 356 (2021).\n[88] J. Hertkorn, J.-N. Schmidt, F. Böttcher, M. Guo, M.\nSchmidt, K. S. H. Ng, S. D. Graham, H. P. Büchler,\nT. Langen, M. Zwierlein, and T. Pfau, Density fluctua-\ntions across the superfluid-supersolid phase transition in\na dipolar quantum gas, Phys. Rev. X 11, 011037 (2021).\n[89] M. Sohmen, C. Politi, L. Klaus, L. Chomaz, M. J. Mark,\nM. A. Norcia, and F. Ferlaino, Birth, life, and death of a\ndipolar supersolid, Phys. Rev. Lett. 126, 233401 (2021).\n[90] M. A. Norcia, C. Politi, L. Klaus, E. Poli, M. Sohmen,\nM. J. Mark, R. N. Bisset, L. Santos, and F. Ferlaino,\nTwo-dimensional supersolidity in a dipolar quantum gas,\nNature (London) 596, 357 (2021).\n[91] T. Bland, E. Poli, C. Politi, L. Klaus, M. A. Norcia, F.\nFerlaino, L. Santos, and R. N. Bisset, Two-dimensional\nsupersolid formation in dipolar condensates, Phys. Rev.\nLett.128, 195302 (2022).\n[92] G. Biagioni, N. Antolini, A. Alaña, M. Modugno, A.\nFioretti, C. Gabbanini, L. Tanzi, and G. Modugno, Di-\nmensional crossover in the superfluid-supersolid quantum\nphase transition, Phys. Rev. X 12, 021019 (2022).\n[93] M. A. Norcia, E. Poli, C. Politi, L. Klaus, T. Bland,\nM. J. Mark, L. Santos, R. N. Bisset, and F. Ferlaino,\nCan angular oscillations probe superfluidity in dipolar\nsupersolids?, Phys. Rev. Lett. 129, 040403 (2022).\n[94] J. Sánchez-Baena, C. Politi, F. Maucher, F. Ferlaino, a nd\nT. Pohl, Heating a dipolar quantum fluid into a solid,\nNat. Comm. 14, 1868 (2023)."    },    {        "title": "2001.02602v2.Non_equilibrium_spin_dynamics_in_the_temperature_and_magnetic_field_dependence_of_magnetization_curves_of_ferrimagnetic_Co___1_75__Fe___1_25__O__4__and_its_composite_with_BaTiO__3_.pdf",        "content": "1\n \n \nN\non\n-\nequilibrium spin dynamics in \nthe \ntemperature and magnetic field dependen\nce of\n \nmagnetization curves of \nferrimagnetic Co\n1.75\nFe\n1.25\nO\n4\n \nand \nits composite with \nBaTiO\n3\n \n \nR.N. Bhowmik\n*\n1\n, and R.Ranganathan\n2\n      \n \n1\nDepartment of Physics, Pondicherry University, R\n. V Nagar, Kalapet, Pondicherry\n-\n605014, India.\n \nCondensed \nM\natter \nP\nhysics \nD\nivi\ns\nion, Saha Institute of Nuclear Physics, 1/AF \nBidhannagar, Kolkata\n-\n700064\n \n*\nCorresponding author: Tel.: +91\n-\n9944064547; E\n-\nmail: rnbhowmik.phy@pondiuni.edu.in\n \nAbstract\n \nA\n \ncomparative \nstudy of the non\n-\nequilibrium magnetic phenomena (magnetic blocking, memory, \nexchange bias and aging effect) has been presented for \nferrimagnetic \nCo\n1.75\nFe\n1.25\nO\n4\n \n(CFO) and \nits composite with \nnon\n-\nmagnetic \nBaTiO\n3\n \n(BTO). \nS\nynchrotron X\n-\nRay diffraction \npatterns h\nave \nconfirmed \ncoexistence \nof \nCFO and BTO structures \nin composite\n, but \nmagnetic spin dynamics \nhave \nbeen \nremarkabl\ny\n \nmodifi\ned\n. The blocking \nphenomenon \nof ferrimagnetic \ndomains below \nthe \nroom temperature \nhas been \nstudied \nby \ndifferent \nmodes of \n(\nzero field coole\nd and field cooled\n)\n \nmagnetic \nmeasurements \nin \ncollaboration with \nmagnetic fields\n \nON and OFF modes and time \ndependent magnetization\n. \nThe \napplications of \nunconventional pr\notocols \nduring \ntime dependent \nmagneti\nzation\n \nmeasurement \nat \ndifferent stages of \nthe \ntempe\nrature and field dependence of \nthe \nmagnetization curves\n \nhave been useful to \nreveal\n \nt\nhe non\n-\nequilibrium dynamics of magnetic spin \norder\n. \nThe \napplying\n \nof\n \noff\n-\nfield relaxation experiments\n \nhas made possible to tune \nthe \nmagnetic \nstate and coercivity of the \nsyst\nems\n.\n \nThe role of interfacial coupling between magnetic and non\n-\nmagnetic particles has been understood on different\n \nmagnetic \nphenomena\n \n(\nmeta\n-\nstable magnetic \nstate, exchange bias\n \nand \nmemory effect\n)\n \nby comparing the experimental results of \nCo\n1.75\nFe\n1.25\nO\n4\n \nspin\nel oxide \nand \nit’s\n \ncomposite with \nBaTiO\n3\n \nparticles\n.\n \nKeywords\n:\n \nSpinel\n \nferrite, \nBaTiO\n3\n, \nComposite magnet, Exchange bias\n, \nMemory \nand aging \neffect\n.\n 2\n \n \n1. \nIntroduction\n \nThe \nnon\n-\nequilibrium spin dynamics \nin magnetic materials strongly \ndepend\nent\n \non \nspin \ndisorder \nand \nm\nanifested by \nmany unusual \nmagnetic \nphenomena, e.g., spin glass, super\n-\nspin glass\n \n/\ncluster spin glass, superparamagnetic blocking, exchange bias, domain wall pinning, memory \nand training \neffect [\n1\n-\n7\n].\n \nEach of these phenomena has their own characteristics. \nT\nhe s\npin glass\nes\n \nare \ndefined by \na typical \ncompet\nit\nion \nbetween \nferromagnetic (FM) and antiferromagnetic (AFM) \nexchange \ninteractions \nand \nfrustrat\nion\n \nof \nthe \nspins\n \nin lattice structure\n. \nThe spin dynamics below \na \ncharacteristic \nfreezing\n \ntemperature\n \nbecomes slow \ndue to \nincreasing \ninter\n-\nspin interaction\ns\n. \nThe \nsuperparamagnetic blocking\n \nof non\n-\ninteracting \nmagnetic \nparticles (group of spins) \noccurs below \na typical temperature\n \ndue to relaxation of the particles along \ntheir \nlocal anisotropy\n \naxes. \nTaking \ninto account th\ne existence of strong inter\n-\nparticle interactions, the freezing of \nnanoparticles \nassembly\n \nis defined as super\n-\nspin glass or cluster\n-\nspin glass\n \n[\n3,7\n-\n8\n]\n. \nIt is practically difficult to \ndistinguish the features of super\n-\nspin glass from superparamagnetic block\ning in magnetic \nnanoparticles, having finite inter\n-\nparticle interactions, and distribution in size and anisotropy. In \nsuch systems, t\nhe aging effect \n(relaxation phenomenon) \nplays an important role in determining \nspin dynamics below the\nir\n \nfreezing/blocking \ntemperature. \nThe \nmagnetic exchange bias effect \nwas primarily modeled for \nFM and AFM \nbi\n-\nlayers\n \n[\n9\n], \nbut \nit has been found in many particulate \nsystems where interfacial exchange coupling between FM (core) and weak FM/AFM (shell) \nstructure control \nthe \nshape o\nf magnetic hysteresis loop \n[\n2\n, \n1\n0\n-\n1\n1\n]. \nThe \nmemory \neffect is another \nform\n \nof non\n-\nequilibrium\n \nspin dynamics, where \nnew \nspin configuration/\nmeta\n-\nstable state \nachieved \nduring\n \nintermediate stops \nof \nzero field or field cool\ned \nmagnetization curves \ncan be retrieved\n \nduring re\n-\nheating\n \nprocess\n \n[\n1\n-\n3\n]\n.\n \nThe memory effect \nhas been \nobserved \nin a wide range of \nmaterials, irrespective of \nstrong\nly\n \ninteracti\nng \n[\n12\n-\n13\n]\n \nand non\n-\ninteracting \nspin sy\ns\ntems\n \n[\n14\n-\n15\n]. 3\n \n \nThe artificially designed \nferrimagnetic\n-\nferroelectric composite \nand h\netero\n-\nstructured \nspin\n \nsystems \nalso \nshow\ned\n \nexchange bias and memory effect \n[\n10\n-\n11, 16\n-\n18\n].\n \nThe\n \nexchange bias effect \ndominates \nat lower temperatures \nand \nmemory \neffect \ndominates at higher temperatures\n \n[1\n0\n, 1\n3\n, \n1\n9\n], and both are \nnot free from spin glass\n \nfreezi\nng\n, superparamagnetic blocking, anisotropy and \ndomain wall pinning effect. \nThe training effect, on the other hand, \nis related to \nan irreversible \nchange in spin structure pinned at domain walls or at the interfaces of FM\n-\nAFM structure or at \nthe interfaces o\nf ferromagnetic and ferroelectric systems [\n20\n-\n21\n]. \nThe disorder induced by \ncoexisting crystalline phases\n \nalso played \nrole\n \non \nspin \ndependent electronic conductivity \n[\n22\n]\n. \nApart from basic understanding, t\nhe \nstudy \nof non\n-\nequilibrium \nspin dynamics is\n \nuseful f\nor \napplications of \nstrongly interacting electronic spin systems\n,\n \nsuch as random alloy [\n3, 7\n], \nperovskite [\n2\n,\n \n6\n,\n \n15\n], and spinel ferrite [\n3\n-\n5\n], \nin spin valves, spins filter, read\n-\nwriting devices, \nmagneto\n-\nresistive random access memories, \nsensors and magneti\nc switches [23\n-\n24\n]. This \nrequires an effective strategy \nfor \ntun\ning\n \nthe \nferro/ferri\nmagnetic parameters\n \nby controlling \nthe \neffects of \nspin disorder\n \ninside the domains or at interfaces \nof \nthe composite \nmaterials\n.\n \n \nThe present work focuses on s\npinel ferrites\n, \nwhich \nare defined by \na \ngeneral \nformula \nunit \nAB\n2\nO\n4\n,\n \nwhere \ncat\nions occupy \nthe \ntetrahedral (A) and octahedral (B) coordinated \nlattice sites \nwith\n \nanion\ns\n \n(\nO\n2\n-\n) \nat \nfcc \npositions\n \nof\n \nthe \nlattice \nstructure\n. \nIn long range ferrimagnetic \n(FiM) \nspinel \nferrite\n, antiferr\nomagnetic (AFM) superexchange interactions between A and B site moments \n(J(A\n-\nO\n-\nB) are expected to be strong in comparison to intra\n-\nsublattice interactions (J(B\n-\nO\n-\nB) and \n(J(A\n-\nO\n-\nA))\n \n[2\n5]\n. \nIn this work, we will study \nthe effects of intrinsic disorder in ferri\nmagnetic \nCo\n1.75\nFe\n1.25\nO\n4 \nparticles\n \n[2\n6\n] and extrinsic \nspin \ndisorder \n(interfacial effect) \nin its composite with \nnon\n-\nmagnetic BaTiO\n3 \n[2\n7\n]\n \nto control the non\n-\nequilibrium magnetic phenomena, e.g., exchange \nbias, memory and aging effect\n.\n \n 4\n \n \n2. \nExperimental\n \n2.1. Ma\nterial Preparation\n \nT\nhe \nmaterial preparation and characterization of the \nCo\n1.75\nFe\n1.25\nO\n4\n \n(CFO) ferrite and its \ncomposite with BaTiO\n3\n \n(BTO) were \ndescribed\n \nin \nearlier works [\n2\n6\n-\n2\n7\n]. \nThe \nferrite \npowder \nwas \nprepared by \nchemical \nco\n-\nprecipitation route and \nthermal\n \nanneal\ning \nat \n8\n00\n \n0\nC\n \n(CF80) and 9\n00 \n0\nC\n \n(CF90) \nfor 2 hrs\n. \nT\nhe CF80 sample formed bi\n-\nphased cubic spinel structure\n, unlike single phase \nstructure in \nCF90 \nsample. \nThe composite \nsample \nCF80_BTO was prepared by mixing \nof \nCF\n80\n \nferrite and \nBTO\n \npowders \nwith mass r\natio 50:50\n, and final \nheat\n \ntreatment was performed\n \nat 1000 \n0\nC for 4 hrs.\n \nS\nynchrotron X\n-\nray diffraction \npattern\n \nconfirmed the \ncoexist\nence of \ncubic spinel \nstructure\n \nof\n \nCFO and \ntetragonal phase\n \nof\n \nBTO\n \nin \nthe composite CF80_BTO sample\n \nwithout any \nintermediate \nphase formation\n.\n \nInterestingly, \nbi\n-\nphased nature of CF80 sample (as seen from \nsplit\n \nof \nX\n-\nray diffraction peaks of \ncubic spinel phase) disappeared in \nCF80_BTO composite\n. \nThe \nspin \nstructure in \nCF90 ferrite and CF80_BTO composite samples \nare schematically mod\neled in Fig. \n1(a\n-\nb) \nand origin of the \nspin disorder for non\n-\nequilibrium spin dynamics \nare summarized below.\n \nThe single phase ferrite sample CF90 is \nmodeled as \nconsisting of average \nparticle \nsize \n\n \n40 nm \nand each magnetic particle is assumed to be consistin\ng of core\n-\nshell spin structure [1\n0\n, 1\n9\n]. The \ncore (interior) part is consisting of more than one domain (multi\n-\ndomain structure)\n. The\n \nspins \ninside each domain are ferrimagnetically (\n\n\n\n\n)\n \nordered \nand disordered or pinned at the \ndomain\n-\nwalls. \nEffectively, t\nhe shell (outer) part of a particle spreads over few lattice parameters \nwhose length is more than domain\n-\nwall thickness and spins \ntherein \nare more disordered than the \ncore\n \nspins\n. \nThe magnetic exchange interactions inside the core are stronger than \nthe \nshel\nl\n \nand \nparticles are strongly interacted in \nC\nF90\n. In case of \nCF\nO\n_BTO\n \ncomposite, \nthe ferrite particle \nof \nsize\n \n\n \n90 nm\n \nare dispersed in matrix of BTO of \nmicron size\nd\n \nparticle\ns\n. \nThe presence non\n-5\n \n \nmagnetic (NM) \nBTO \nparticle dilutes the \nmagnetic exchange interact\nions \nbetween two CFO \n(FiM) \nparticles and it \nincrease\nd\n \nferrimagnetic softness in composite sample\n \n[2\n7\n]\n. \nThe interfacial \nexchange interactions are affected by \npossibl\ne\n \nmagneto\n-\nelectric coupling [21, 28\n] \nand\n \nhidden \nexchange coupling [\n29\n]\n \nbetween FiM\n \nand \nferro\nelectric (FE) \nBTO particles\n. \nAlthough\n, both CF90 \nand CF\nO\n_BTO are \nhetero\n-\nstructured spin systems\n, but the nature of interfacial \nspin disorder \nis \ndifferent. \nIn case of hetero\n-\nstructured spin systems, the time evolution of \nspin \nvector \ninside an \nordered magnet\nic domain \ncan be re\n-\nwritten as \n, where \n \n= \n \n[\n30\n]. \nA competition between the free spin torque under external field (first \nterm) and \nintrinsic \ndamping torque (second term) under internal field \nand meta\n-\nstable states \ndete\nrmines the relaxation/orientation of spin vectors towards its nearest \nnew \nmagnetic state. \nThe \ninternal field \nis controlled by spin disorder, frustration and inter\n-\nparticle interactions. In case of \nCF90 sample, the spin disorder is contributed by intrinsic \ndisorder at core (ordered FiM) and \nextrinsic disorder at shell (disordered FiM). \nI\nn the spinel \nferrite\n \nCo\n1.75\nFe\n1.25\nO\n4\n,\n \nintrinsic spin \ndisorder is expected due to \ndistribution of magnetic moment and magneto\n-\ncrystalline anisotropy \nof the cations among A and \nB sites of the spinel structure (\nFe\n3+\n \nions at A and B sites in \nhigh spin \nstate\n \nand low anisotropic, \nCo\n2+\n \nions at A and B sites in high spin state and highly anisotropic,\n \nand Co\n3+\n \nions \nat B sites\n \nare \nnon\n-\nmagnetic \nand isotropic)\n \n[\n25\n]. \nIn case of CF\nO\n_BTO comp\nosite, \nadditional \nextrinsic \nspin disorder \nis introduced at the interfaces of \nshell (disordered FiM\n \nof CFO\n) \nand shell (non\n-\nmagnetic and ferroelectric\n-\nBTO).\n \nThis\n \nis produced \ndue \nto \nstructural and magnetic \nmismatch at the interfaces of two \nphases [\n31\n].\n \nHence,\n \nthe change of both external magnetic field \n(ON\n/\nOFF) and internal field control the \ntime response of magnetization in the temperature and \nfield dependence of magnetization curves.\n \nThe basic difference is that \nthe \nmagnetization will be 6\n \n \nwell below of the sat\nuration level in case of the temperature dependence of the magnetization \ncurves, where as \nthe \nmagnetization will be close to the saturation \nlevel \n(high magnetic state) \nin \ncase of the field dependence of magnetization curves at the starting of relaxation\n \npr\nocess\n.\n \n2.2. Measurement protocols\n \nPhysical\n \nproperty measurement system (PPMS\n-\nEC2\n, Quantum\n \nDesign\n, USA\n)\n \nwas used \nfor magnetic measurements\n.\n \nThe \ntemperature dependence of \nmagneti\nzation \nwas \nrecorded using \nzero field cool\ned \n(ZFC) and field cool\ned \n(FC) mode\ns\n \nwi\nth \nconventional and \nunconventional\n \nprotocols\n \n(PCs)\n. \nThe \nPC1\n \nis a c\nonventional \nZFC mode\n \n(Fig. 1(\nc\n))\n, where t\nhe \nsample \nwa\ns \ncooled \nfrom \n33\n0\n \nK \nto \n10\n \nK in the absence of external \nmagnetic \nfield or \napplying \na \nsmall field to \nmaintain \nthe \nresidual magnetization \ncl\nose\n \nto zero\n \nduring cooling\n. \nThis \nwa\ns followed by \nmagnetic \nmeasurement at set \n(constant) \nmagnetic field while \ntemperature of \nthe sample \ni\ns warming up to \n300 K/3\n3\n0 K.\n \nThe \nPC2\n \nis the \nconventional \nFC mode\n \n(Fig. 1(\nd\n))\n, where t\nhe sample \nwa\ns \ncooled \nfrom 300\n \nK\n/3\n3\n0\n \nK\n \nto 10 K\n \nin the presence of constant magnetic field\n.\n \nThe \nmagnetization \nwa\ns \nrecorded during \nfield \ncooling \n(MFCC(T)) \nof the sample \nfrom higher temperature \nor \nwarming \n(MFCW(T)) of \nthe \nsample \nfrom 10 K to 300 K\n/3\n3\n0 K\n \nwithout changing the field that was \nappli\ned during pre\n-\ncooling down to 10 K\n. \nThe conventional (ZFC and FC) measurement \nprotocols \nprovide\n \ngeneral features (magnetic blocking and anisotropy effect) of the \nmagnetic \nparticles. \nThe \nnon\n-\nequilibrium spin dynamics \n(memory and aging effect) \nwe\nre examined \nby \nadopting \nfew \nunconventional \nprotocols \nto \nrecord\n \ntime\n \ndependent magnetization during \nintermediate stop on \ntemperature and field dependence of\n \nmagneti\nzation\n \n[\nM(T, H\n, t\nw\n)\n]\n \ncurves \n[\n2, \n5\n, \n14\n-\n15\n]\n.\n \nWe followed \nFC protocol (PC3) (Fig. 1(\ne\n)) for studying \nthe \nmem\nory effects. In FC\n-\nPC3\n, \nMFCC(T) \ncurve was recorded \nwith \nintermediate \nstop\ns\n \nat \n250 K, 150 K and 50 K\n \nby \nswitching off the cooling field for time t\nw\n. T\nhe M(t\nw\n) data \nat the stopping temperature \nwere\n 7\n \n \nrecorded before \nswitching \nthe cooling field again ON\n \nand \nres\numing the \nMFCC(T) \nmeasurement\n \non lowering the temperature\n \ndown to 10 K\n.\n \nAfter reaching the temperature 10 K, the MFCW(T) \ncurve was recorded from 10 K to 300 K without changing the cooling field and without \nintermediate stops. \nThe \nPC\n4\n \nis the\n \nconventional fi\neld dependence of magnetiz\nation (M(H)) \nmeasurement (Fig. 1(\nf\n))\n, \npre\n-\ncooled under ZFC and FC modes\n \nfrom 300 K\n \nto \nthe \nset \ntemperature\n. The shift of FC\n-\nM(H) loop with respect to ZFC\n-\nM(H) loop \ncan be used \nto \nstudy \nexchange bias effect. \nThe \nprotocol PC5 \nin \nFig\n.\n \n1(\ng\n)\n \nis the super\nposition of PC4 \nwith \nM(t\nw\n) \nmeasurement\n, where \nt\nhe\n \nM (T = constant, H, t\nw\n) curve\n \nwa\ns \nrecorded by varying the \nmagnetic \nfield \nwith \nan \nintermediate \nstop\n \nfor waiting time (t\nw\n) \nat \nmagnetic \nf\nield to\n \nzero or \nbefore \ncoercive \nfield point in\n \nnegativ\ne \nfield \naxis \nand M(t\nw\n) data \nwe\nre \nrecorded\n. \nAfter M(t\nw\n) measurement, the \nM(H) measurement is continued\n \nin negative field side\n. The steps of \nM(H) measurement\ns\n \nare\n \nrepeated \nwith different \nt\nw\n \nvalues.\n \nThe protocol PC6 in Fig. 1(h) is \nsimilar to \nthe protocol \nPC5\n. \nThe \nonly exception is that M(t\nw\n) \nmeasurements were\n \ncarried out at \nmultiple \npoints \n(at zero field \nor points close to \ncoercive fields \nboth \non negative \nand \npositive field axes\n) \nof \nthe \nM(H) curve\ns\n. \n \n3. Result\ns\n \nand discussion\n \n3.1. \nTemperature and field depend\nen\nt\n \nmagnetization [M(T,H\n, t\nw\n)]\n \nfor CF90 sample\n \n \nFirst, we\n \nshow basic properties \nof \nthe temperature dependence of magnetization \nin \nCF90 \nsample. \nT\nhe \nMZFC(T) and MFC\nW\n(T) curves\n \nat +500 Oe\n \n(\nFig. 2(a)\n) were\n \nmeasured\n \nusing \nPC1 \nand \nPC\n2\n. The MZFC\n(T)\n \ncurve exhibit\ns\n \nmagnetic blocking temperature (T\nm\n) at \n\n \n300 K. A wide \nbifurcation between MZFC\n(T)\n \nand \nMFCW(T) \ncurves \nbelow T\nm\n, where \nMZFC curve \ndecreased \nrapidly \nbelow \nT\nm\n \nand \nbecomes \nnearly temperature independen\nce\n \nbelow \n150 K\n, and \nMFC curve \nslowly increased\n. \nThe behavio\nr \ns\nhow\ns high anisotropic \neffect at low temperatures\n.\n \nTo overcome \nthe anisotropic effect, we measured\n \nMZFC(T) curve\ns by increasing the \nmagnetic fields \nup to \n\n \n2 8\n \n \nkOe\n \n(\nFig. 2(b)\n).\n \nThe \nMZFC(T) curves \nshow\ned\n \nmore or less symmetric \nrespons\ne \nof the spin\n-\nclusters \nunder field reversal\n. \nM\nagnitude \nof \nthe \nMZFC(T) curve\ns\n \nincrease\nd\n \nwith a shift \nof  peak \nposition \nto low temperature \non \nincreasing \nthe \nmagnetic \nfield\n. \nIn highly anisotropic sample, the \nenergy density in ferrimagnetic state \nin \nthe presence of magnetic field is\n \nE = \nK\nH\n \n+\n \nK\nA\n, where \nK\nH\n \n(\n= \n\n \nM\nsat \nH cos(θ\n–\n \nφ)) is the \nZeeman energy \nand \nK\nA\n \n(\n= K\neff \n(T) sin\n2\nθ) is the crystalline \nanisotropy \nenergy\n \n[\n20\n]. \nThe\n \nMZFC(T) curves below 150 K are \nalso \nnot significantly affected \nwithin \n\n \n2 \nkOe\n, except some\n \nminor difference\ns\n. It indicates that \nZeeman energy \nis not \neno\nugh \nin this field \nrange \nto overcome the anisotropy\n \nenergy\n \nand\n \ndomain\n-\nwall pinning effect \ncontrols the shape of \nmagnetization curves \n[\n32\n]\n. \nOn the other hand, \nbroad\n \npeak \nin MZFC(T) \ncurve\ns\n \ndescribes \na \ndi\nstribution of anisotropy in the system and it \ncan\n \nbe \nquantified from \nfirst order derivative of \nthe \nMZFC(T) curves (Fig. 2(c))\n. \nThe \npeak profile in \nthe \ndM/dT vs. T \ncurves\n \n(\nFig. 2(d\n-\ne)\n)\n \nw\nas\n \nfit\nted\n \nwith Lorenzian \nshape\n \nto determine the peak parameters\n. \nThe intercept of the dM/dT curve on \ntemperature axis (> T\np\n)\n \ndefines the blocking temperature (T\nm\n).\n \nThe peak \ntemperature \n(T\np\n) of \ndM/dT vs. T curve corresponds to the inflection point of the MZFC(T) curve below T\nm\n. \nFig. 2(f) \nshows \na symmetrically decrease of \nt\nhe \npeak \nparameters (T\np\n, width, \nT\nm\n) \nabout the zero point\n \no\nf \nmagnetic field axis\n \nwith the increase of field magnitude\n. \nThe increase of peak height\n, along with \ndecrease of peak width,\n \narises due to field induced clustering of \nsmall\n \nparticles [1\n3\n].\n \nThe T\nm\n(H) \ncurve is fitted with \na power law:\n \nT\nm\n(H) = \na\n-\nb\nH\nn\n \n(\na\n \nand \nb\n \nc\nonstants) \nwith \nexponent\n \nn\n \n\n \n0.25 and \n\n \n0.2\n1\n \nfor \npositive and \nnegative fields\n, respectively\n. \nThe exponent values for T\np\n(H) curve are \n\n \n0.29 and \n\n \n0.27 for positive and negative fields, respectively. \nThe\n \nvalues of \nn\n \nin CF90 sample are \nsuggestive of magnetic \nspin\n-\nclusters coexisting in ferri\nmagnetic \nstate\n \n[\n33\n]. \nThe \nM(T) curves \nshow \nbulk \nresponse of a ferrimagnet\n \nwithout \nmuch \ninformation of \nlocal \nspin dynamics\n. \n 9\n \n \n \nIn order to get information of local spin dynamics, t\nhe memory effect \nwa\ns tested using\n \nprotocol \nPC\n3\n.\n \nFig. 3(a)\n \nshows the \ncorresponding \nMFCC(T) curve \nat \ncooling field +200 Oe \nwith \nintermediate stops and subsequent\n \nMFCW(T) curve.\n \nThe appearance of kinks in the MFCW(T) \ncurve\n \nat \nthe previously \nintermittent stop\ns \n(\nfield off condition \nat 250 K, 150 K and 50 K\n \nduring \nMFCC(T) process\n) suggests a \nrecover\ny/memory of \nthe magnetic \nspin states\n \nthat w\nere\n \nimprinted \nthrough redistribution of energy barriers during the cooling process.\n \nThe\n \nmemory \nis \nreduce\nd\n \non \nlower\ning\n \nthe stopping \ntemperature\ns\n \nand \nneg\nl\nigible \nat 10 K\n. \nTh\ne magnetization that is recovered \non re\n-\napplying the cooling \nfield depends on \nthe response of spins in relaxed/quasi\n-\nrelaxed state\n. \nIn a \nstrongly interacted \nspin\n-\nsystem\n, \nan increasing slow down of the spin \ndynamics \non \ndecreasing the sample temperature belo\nw its spin \nfreezing\n/blocking \ntemperature\n \nhinder\n \nthe \nrecovery \nof initial magnetic state. It \nlead\ns\n \nto a large step in \nMFCC(T) curve immediately after \nswitching OFF and re\n-\napplying (ON) of the cooling field at the temperatures (e.g., 250 K in our \ncase). The s\ntep in MFCC(T) decreases as sample temperature decreases far away from its spin \nfreezing temperature. It is due to increasing inter\n-\nspins interactions in a strong spin\n-\npinning state \n(e.g., 50 K)\n. \nInterestingly, \nthe \nMFCW(T) curve overshoots the MFCC(T) curv\ne \nat \ntemperatures \nabove 300 K. It \nshow\ns \nin\n-\nfield growth of magnetization due to non\n-\nequilibrium spin state of the \nmagnetic particles below the\nir\n \ntrue blocking temperature\n \n(\nabove 300 K\n)\n. \nThe \nmeasurement\n \nof \nMFCC(T) and MFCW(T) \nat 200 Oe \nwithout \nfield\n-\noff\n \nat \nintermediate temperatures \nformed a \nthermal hysteresis loop\n \n(Fig. 3(b))\n. \nI\nt \nis a characteristic feature of first order magnetic \nphase \ntransition (short range spin order coexists in long range spin order) \nin the sample\n \n[\n34\n]\n.\n \nIn our \nsample, t\nhe \nin\n-\nfield \nMFCW(\nT) \ncurve \nstarts with \nthermal activated de\n-\npinning of the spins \nthat \nwere\n \nin \npinn\ning \nin intrinsically disordered ferrimagnetic state \nat 10 K after completing the \nMFCC(T) \nmeasurement\n. \nHowever, \na\n \ndifference between MFCW(T) and MFCC(T) curves (right 10\n \n \nY axis of \nFig. 3(b)) showed \na \nmaximum at about 210 K\n \nand it marked different spin dynamics at \nlower and higher temperatures\n. \nM\nagnitude of the difference \ndecreases \nat higher temperatures \ndue \nto approaching of spin system\n \ntowards blocking temperature (less \ninteracting\n/\npinning effect) and \nat low\n \ntemperatures\n \ndue to \napproaching towards \na \nstrongly \nspin\n-\npinning\n/interacting\n \nregime. \nOn \nincreasing the magnitude of cooling field to 500 Oe \n(Fig. 3(c))\n, the memory effect is observed \nonly at 250 K and \nsuppressed \nat low temperatur\nes (50 K and 150 K)\n. \nT\nh\nis \nis \ndue to \nclustering of \nsmaller \nmagnetic \nparticles\n \nand de\n-\npinning of\n \nthe spins \n(domain wall motion)\n \nat higher magnetic \nfield\n. In this process\n, the distribution of\n \nexchange interactions and anisotropy barriers related to \ncluster si\nze\n \ndistribution\n \nis \nnarrow\ned down\n. I\nt results in strongly spin\n-\ninteracting clusters\n \nduring \nfield cooling process and reduces the memory effect at 500 Oe\n.\n \nOn the other hand, \nspin system\n \nswitch\nes\n \nits magnetic state \nfrom high to \nlow \ni\nmmediately after switching\n \noff the \ncooling \nfield\n. \nThe \nspins in \nlow \nmagnetic state \n(non\n-\nzero remanent magnetization) \nrelax \nfor sample temperature \nin blocking state (T < T\nB\n)\n \n[\n5\n-\n6, 15\n]. \nThe \ntime \ndependence of \nFC\n-\nremanent magnetization (\nM\n \n(t, \nH= 0)\n)\n \ncurves\n \nhave been \nanalyzed \nby \nvarious\n \nequations, e.g., \nstretched exponential\n \nform [\n7\n], \na \ncomplicated form of equation that consists of essentially two power law terms [\n3\n]\n.\n \nIn our sample, \n \nM(t)\n \n(\nnormalized \nby initial value \nM(t\n0\n))\n \ncurves \nduring \nfield\n-\noff \ncondition\n \n(t = \nt\nw\n \n=\n \n1500 s\n)\n \n(\nFig. \n3(d\n-\ne\n)\n)\n \nare best fitted \nwith a function\n, consisting \nof \na constant and \ntwo exponential decay terms.\n \n                \nM(t) = \n\n0\n \n\n \n\n1\nexp(\n-\nt/\n\n1\n) \n\n \n\n2\nexp(\n-\nt/\n\n2\n)                   \n        \n       \n(1)\n \nS\nign of \nthe \npre\n-\nfactors \n\n1 \nand \n\n2\n \nis \ntaken as\n \npositive and negative \nto\n \nrepresent \nthe magnetization \ndecay and growth, respectively. \nOut of the two exponential terms, one represents fast relaxation \n(initial \nprocess\n) and other one represents a slow relaxation \n(secondary process \nat higher time\ns)\n. \nSimilar \nspin relaxation \nproces\ns\ne\ns \nwe\nre found in \nmagnetic systems with \nheterogene\nous spin \nstructure \n[\n35\n]\n.\n \nThe \nfit of \nM(t) data at 50 K with \na \nlogarithmic decay M(t) = \n\n0\n \n–\nm\n*lnt\n \n(with \nm\n \n= \n 11\n \n \n0.0002 and 0.0160 at cooling fields \n200 Oe and 500 Oe, respectively) \nrepresents \na\nn extremely\n \nslow \nspi\nn systems \nand generally \nrepresent\ns\n \na distribution \nof \nactivation energy\n \nin spin glass state\n \n[\n1\n,\n3\n, \n36\n]\n. \nA comparative fit of the M(t) data \nduring \nOFF condition of \n500 Oe \nat 250 K \n(Fig. 3(f)) \nsuggest\ns\n \nthat logarithmic decay \nis s\natisfied \nfor \nlimited portion of\n \nthe \nM(t) curves\n, \nbut \n \nequation \n(1) \nwidely \nmatched \nto the\n \nM(t) curves. Hence, \nequation (1) \nis more acceptable in fitting the \nM(t) \ncurves \nduring \nfield\n-\noff condition of \nM(T) and M(H) measurements\n. \n \nThe \nnon\n-\nequilibrium spin dynamics \nduring \nZFC\n-\nM(H) loop \nmeasu\nrement \nwithin field \n\n \n70 kOe at 10 K (Fig. 4(a))\n \ncan be studied using PC\n4\n \nand \nPC\n5\n. \nThe \nM(H) loop\n \nunder zero field \ncooled mode \nwas recorded \nat 10 K \nusing PC\n4\n. Next, \nM(H) \nmeasurement \nbetween \n+70 kOe \nto \n-\n10 \nkOe \nwas repeated 6 times \nwith intermediate \nwait\n \nat 0\n \nOe\n \nto record \nthe \nM(t\nw\n) curve\ns\n \nfor \ndifferent \nt\nw\n. \nIn principle, spins in ferrimagnetic state is expected to relax \nduring waiting, irrespective of \nsweeping field ON or OFF conditions\n, if finite amount of disorder coexists in spin order\n, and it \ncould produce \nnew meta\n-\nstable state in the M(H) path\n. \nAs shown in\n \nFig. 4(b)\n, the \nM(H) curve\ns\n \nbetween 0 Oe and \n-\n10 kOe\n \nare\n \nextremely sensitive to spin relaxation\n \nduring \nt\nw\n \nat 0 Oe\n. T\nhe M(H) \ncurve\ns\n \nafter waiting at 0 Oe \nmove upward with the increase of \nt\nw\n \nwith reference t\no \nthe \nfirst \ncurve\n \n(default \nt\nw\n \n= 10 s)\n.\n \nThe \nM(\nt\nw\n) \ncurve\ns\n \nat 0 Oe (\nFig. 4(c)\n)\n \nslow\ned\n \ndown\n \nfor \nhigher \nt\nw\n \nand \nfollowed \n \nequation (1)\n. Fig. 4(d\n-\nf) shows \nthe \nwaiting time dependence \nof \nthe fit \nparameters (\nH\nC\n, \nM\n0\n,\n \n\n1\n, \n\n1\n,\n \n\n2\n, \n\n2\n)\n \nfrom M(H) curves (0 Oe to \n–\n \n10 kO\ne) and M(\nt\nw\n) curves at 0 Oe\n. \nC\noercivity \n(H\nC\n) \nof the \nCF90 \nsample\n \nsignificantly \nincreased \n(\n6628\n \nOe to \n6954\n \nOe) \nwith the increase of \nt\nw\n \nfrom \n100 s to \n7200 s \nat 0 Oe\n, unlike \na \ndecrease of the \nfit parameter M\n0\n \n(\n35.3473\n \nemu/g to \n35.304\n \nemu/g)\n. This\n \nis associated\n \nwith faster relaxation of initial process (increasing \n\n1\n \nand smaller \n\n1\n) and slower \nrelaxation of secondary process (\ndecreasing \n\n2\n \nand larger \n\n2\n)\n \nwith the \nincreas\ne of \nt\nw\n. \nA wide \ndifference\n \nbetween \n\n1\n \nand \n\n2 \nconfirms\n \nthe existence of \ntwo relaxation mechani\nsms\n \nin the sample\n. \n 12\n \n \nIn \norder to \nstudy the non\n-\nequilibrium spin dynamics in FC\n-\nM(H) \nloops\n, we have \nrecorded \n \nM(H) loop\ns\n \nat 10 K\n \nusing \nFC\n-\nPC\n4\n \nat cooling field\ns\n \n+70 kOe and \n-\n70 kOe\n. \nT\nhe \nM(H) curve\n \nstart\ned \nfrom \nfield \nsweeping\n \n+70 kOe to \n-\n70 kOe and back to +70\n \nkOe\n \nfor the \nFC loop (\ncooling \n@ \n+70 kOe)\n \nand in reverse way for the \nFC loop (\ncooling \n@ \n-\n70 kOe)\n. \nAs shown in \nFig. 5(a)\n, t\nhe \nFC \nloops\n \nexhibit widening and shifting along field and magnetization directions \nwith \nimproved \nsquare\nness\n \nin comparison to ZFC loop\n \na\nt 10 K\n.\n \nIt occurs due to exchange coupling of hetero\n-\nstructured spins at the interfaces \nor frozen in the system \nthat favor ordering along cooling field \ndirection and irreversible under reversal of the field \ndirection\n \n[Khur].\n \nT\nhe centers (H\nC0\n, M\nR0\n) and \ncoer\ncivity (H\nC\n) of the FC and ZFC loops \nhave been used \nto calculate the shift of coercivity (ΔH\nC \n= |H\nC\nFC\n \n-\n \nH\nC\nZFC\n|), exchange bias field (H\nEB \n= H\nC0\nFC\n \n–\n \nH\nC0\nZFC\n) and magnetization (ΔM\nR \n= M\nR\nFC\n \n-\n \nM\nR\nZFC\n)\n.\n \nT\nhe FC loop \n(@\n \n+70 kOe\n)\n \nis \nnearly symmetric with \nminor \nexchan\nge bias \nshift (\nH\nEB\n \n\n \n+ \n8 Oe)\n \nand\n \nits\n \nH\nC \n\n \n8295 Oe\n. \nHowever, \na \nlarge \npositive shift of \nmagnetization (ΔM\nR\n \n\n+ 1.55 \nemu/g) and coercivity (ΔH\nC\n \n\n \n+ 1\n5\n95\n \nOe)\n \nare noted \nwith respect to ZFC loop with H\nC \n\n \n6\n760\n \nOe\n. \nAs compared in the inset of Fig. 5(a), \nFC loop (@\n \n+\n70 kOe)\n \nand \nFC loop (@ \n-\n70 kOe)\n \nshowed similar features\n, \nbut \nFC loop (@ \n-\n70 kOe)\n \nshows \nhigher widening \n(H\nC \n\n \n8495 Oe, ΔH\nC\n \n\n \n+ \n1\n735\n \nOe, ΔM\nR\n \n\n \n+ 1.99 emu/g) \nand squareness\n.\n \nThis means spin dynamics is \nanisotropic to the\n \nreversal of high field cooling\n \nand \ni\nt could be \nrelated to spin pinning in ferrimagnetic domains \n[\n1\n1\n]\n. \nIn \norder to \nstudy\n \nthe \naging effect \nat intermediate point of the FC\n-\nM(H) \ncurves\n, \nwe repeated \nM(H) \nmeasurement \nwithin field range \n+ 70 kOe to \n-\n10 kOe \nfor\n \n6 times \nwith wait\ning\n \nat \n-\n2.5 kOe\n \nby ad\nopting PC5\n. Fig. 5(b) demonstrates that \nthe\n \nshap\ne (\nupward \nincrease\n)\n \nof \nthe \nM(H) curve \nin \nthe field range \n-\n2.5 kOe to \n-\n10 kOe\n \nis controlled by \nspin\n \nrelaxation at \n-\n2.5 kOe during \nt\nw\n \n(\n140 s \nto 7200 s\n)\n. \nAs shown in \nFig. 5(c)\n, the \nM(t\nw\n) curve\ns\n \nat \n-\n2.5 kOe \nalso \nfollowed \nequation (1)\n. \nThe \nincrease of \nt\nw\n \nat \n-\n2.5 kOe\n \nof the FC\n-\nM(H) loop (@+70 kOe) has \nincrease\nd the\n \nH\nC\n \n(Fig. 5(d)\n-\nleft 13\n \n \nY axis), the \noverall \nmagnetization after \n-\n2.5 kOe \nand \nfit parameter M\n0\n \n(Fig. 5(d)\n-\nright Y axis). \nThe\n \nfit parameters (Fig. 5(e\n-\nf))\n \nfor \nfast relaxation and slow relaxation\n \nprocesses showed similar \nfeatures as observed \nwith t\nw\n \nin case of ZFC\n-\nM(H) loop experiment \n(Fig. \n4\n(e\n-\nf\n)). \n \nNext, w\ne tested \nthe \nspin relaxation \non the M(H) curves \nat 150 K, \na temperature \njust above \nthe magnetization blocki\nng temperature \n\n \n125 K in MZFC(T) curve \nfor field \n50 kOe (Fig. 6(a)). \nAt this \ntemperature\n, \ndomain wall pinning is less effective\n \nbut magnetic clusters are not free from \nmutual interactions\n. T\nhe\n \nZFC\n-\nM(H) curve\n \nwas measured \nby sweeping\n \nfield \nfrom +70 kOe to \n-\n6\n \nkOe \nand \nintermediate waiting at \n-\n1 kOe\n. \nAfter measurement of \nthe \nfirst M(H) curve, the field \nwas made to zero and back to +70 kOe before starting the next curve and repeated \nit \n7 times. \nFig. \n6(b) \nshows\n \nall \nthe \nrelaxation regime of M(H) curves at \n-\n1 kOe \nduring waiting time \nand \nsubsequent \nfield dependent regime \n(\n-\n \n1 \nkOe to \n-\n5 kOe\n)\n. \nIt is noted (\nFig. 6(c)\n)\n \nth\nat\n \nmagnitude of \nthe \nM(H) curves \nfor H < \n-\n1 kOe \nis\n \nsystematically \nsuppressed\n \non increasing the \nwaiting at \n-\n1 kOe\n. \nThis trend is in contrast to the i\nncre\nasing \nincrement \nfor similar experiment \nat 10 K (Fig. 4(b)). \nThe M(t\nw\n) \ncurves\n \nin the relaxation regime \n(inset of Fig. 6(c))\n \nat 150 K also \nfollow equation (1)\n \nand \nfit parameters are shown in Fig. 6(d\n-\nf). \nThe \nH\nC\n \nhas shown \na small increment, whereas \nM\n0\n \ndecreas\ne\ns\n \nwith the increase of \nt\nw\n. \nThe\n \nvalues of \nthe \npre\n-\nfactors (\n\n1\n, \n\n2\n) and time constants (\n\n1\n, \n\n2\n) \nat 150 K are relatively larger than the values at 10 K. It indicates \na fast\ner\n \ndecay of magnetization \nat 150 K\n, \nwhere \nspin dynamics\n \nis still slow \ndue \nto strong in\ntra\n-\ncluster spin interactions\n. \n \n \n3.2 Temperature and field dependen\nt\n \nmagnetization [M\n \n(T,\n \nH\n, t\nw\n)] for CF80_BTO sample\n \n \nFig. 7(a)\n \nshows the \nfeatures of \nthe \nMZFC(T) \nand \nM\nFC\n(T)\n \ncurves at \n500 Oe\n \nin composite \nsample\n. \nIt is seen that \nbasic magnetic\n \nfeatures of \nt\nhe \nferrite particles,\n \ne.g., \nblocking temperature \n(T\nm\n) at about 300 K\n, \nwide \nmagnetic \nbifurcation at low temperatures\n, and \na weak temperature \ndependent MZFC\n(T) curve\n \nbelow 150 K, are retained \nin \nthe \nBTO matrix\n \n[2\n7\n]\n. \nMZFC\n(T)\n \ncurves 14\n \n \n(Fig. 7(b))\n \nalso \nshow\ned fie\nld induced magnetic changes, including \nincreasing \nmagnetization\n \nand \nshift of the \nbroad maximum \nto lower\n \ntemperature\ns\n.\n \nF\nirst order derivative of the MZFC\n(T)\n \ncurves \n(\n\nMZFC/\n\nT) at different magnetic fields\n \n(Fig. 7(c)) \nshowed an asymmetric shape \nabout \nthe peak\n \ntemperature (T\np\n)\n, which is the inflection point below the broad maximum of MZFC(T) curves\n. \nT\nhe \npeak profile\n \nof \n\nMZFC/\n\nT curves\n \nwere \nfitt\ned \nwith \nLorentzian\n \ncurve\n \nand the peak parameters \nare shown in Fig. 7(d)\n. \nThe peak temperature (T\np\n) decreases at higher \nfield by following a power \nlaw: T\np\n(H) = \na\n-\nb\nH\nn\n \nwith exponent (\nn\n) \n\n \n0.31\n, which is close to that obtained for CF90 sample\n. \nIt \n \nsuggests the retaining of the \nglass\ny\n \nbehavior \nof spin\n-\nclusters \nin composite system\n \n[\n33\n]\n. \nIt \nis\n \nnoted \nthat \npeak height of the \n\nMZFC/\n\nT curves initially increased for field up to 2.5 kOe, followed a \ngradual decrease at higher fields. This corresponds to \na \nminimum peak width at 2.5 kOe\n, along \nwith an increase of peak width both at low\ner\n \nand higher magnetic fields. \nThe features are \nconsis\ntent to \nfield induced \nnucleation\n \nof \nsmall particles \nby \nde\n-\npinning the spins \nat domain wall\n \nor \nat the interfaces of \nferrimagnetic and ferroelectric particles (via grain boundary)\n \nat low field \nregime\n. The \nincreas\ne of b\nroadness \nin the \nfirst order derivative c\nurves\n \nfor fields higher than 2\n.5 \nkOe \nis \nattributed \nto \nan \nincrease of \nintrinsic \ndisorder, arising from\n \na \ncompetition \nof \nanisotropy\n \nconstants\n \nand exchange interactions \ninside \nthe clusters\n, where as the\n \nreduced peak height\n \nis \nattributed to \nquasi\n-\nsaturation st\nates\n \nof magnetization curves at higher fields\n. \n \nThe \nretaining of \nmemory effect \nof the ferrite particles \nin composite sample \nis confirmed \nfrom \nMFCC(T) \nand \nMFCW(T) \ncurves\n \n(Fig. 8(a\n-\nd\n)), measured\n \nat \ncooling field\n \nrange \n200 Oe\n \nto \n10 kOe\n.\n \nThe MFCW(T) curves sho\nw\ned\n \nkinks\n \nat the temperatures \n(250 K, 150 K, 50 K) \nwhere \nfield was switched off during FCC mode\n. T\nhe \nkinks\n \nare more pronounced than the CF90 sample\n. \nThe \nmemory effect \nin \nCF80_BTO sample \nalso \nreduc\ned\n \nat low \ntemperature\ns\n \nand \nat higher \nmagnetic fields\n, simila\nr to the features in CF90 sample\n.\n \nIn Fig. 8 (e\n-\nf), we have compared the \n% 15\n \n \ndrop and relax\nation of remanent \nmagnetization \nfor \nboth \nthe samples \nduring field off condition \nof \nthe \nMF\nC\nC(T)\n \ncurves\n.\n \nT\nhe \n% \nof\n \ndrop represents fraction of \nthe \nreversible spins in the \nsystem\n \nimmediately after switching off the cooling field\n. It is 100 % for non\n-\ninteracting paramagnetic \nspins and less than 100 % for \nexistence of \nfinite interactions among \nthe \nspins or cluster of spins. \nIn case of interacting spins, the relaxation componen\nt is non\n-\nzero\n \nand i\nt represents the fraction of \nirreversible spins in the system that shows aging \neffect\n. \nT\nhe \n% \ndrop \nis \nseen to be \nhigh\ner\n \nthan the \nrelaxation \npart\n \nduring waiting time\n, as schematized in the inset of Fig. 8(e)\n. \nIt is seen that \n% of \ndrop and \nrelaxation both are drastically reduce on lowering the measurement temperatures from \n250 K to 50 K. \nHowever, \ndistinct differences \ncan be noted \nin the \noff\n-\nfield properties between \nCF90 and CF80_BTO \ncomposite sample\ns\n.\n \nFor example, t\nhe \n% of \ndrop\n \ndecreased on \nincreasing \nthe \ncooling field \nfrom 200 Oe to \n500 Oe in case of CF90 sample, whereas a monotonic increase \nnoted \nwith the increase of \ncooling field\ns\n \nfrom \n200 Oe \nto 10 kOe\n \nin \ncomposite sample\n. The higher \n% drop of \nmagneti\nzation\n \nindicates\n \nweak\nening of\n \nmagnetic \nspin \ninteraction\ns\n \nat the interfaces of \nCFO (ferrimagnetic) and BTO (non\n-\nmagnetic) particles\n. At the same time, \nincreas\ning \ndrop \nat \nhigher cooling field\ns\n \nis \nattributed to fast de\n-\nnucleation of \nlarge\nr\n \nclusters\n \ninto small\ner\n \nclusters \n(\ncomposed \nof magnetic ferri\nte and non\n-\nmagnetic BTO particles)\n \non switching off the cooling \nfield.\n \nThe \nde\n-\nnucleation of the large clusters is slow \nin CF90 sample \ndue to strong \ninter\n-\nparticle \ninteractions\n.\n \nIt gives rise to \nrelatively \nlow values of % drop and relaxation at all the meas\nurement \ntemperatures for CF90 sample. \nB\nased on the data for cooling field 500 Oe\n,\n \nit can be mentioned \nthat composite sample is consisting of approximately 2\n9\n% \nparamagnetic/non\n–\ninteracting spins \nand 3% of \nthe \ninteracting spins relaxed during waiting time 16\n00 s \nat \n250 K\n. I\nt \nis \nreduced to 1 % \n(paramagnetic spins) and 0.1\n4\n \n% (relaxation of interacting spins) for temperature 50 K. In case \nof CF90 sample, the paramagnetic spins (22%) \nand relaxation of \nthe \ninteracting spins\n \n(0.44 %) at 16\n \n \n250 K are reduced to 0.52 %\n \n(paramagnetic spins) and 0.03 % (relaxation of interacting spins) at \n50 K.\n \nThe\n \nM(t\nw\n) \ncurves\n \nduring \ncooling field off \ncondition of \nMFCC(T) measurement\ns\n \nwere fitted\n \nusing \nequation 1\n \n(Fig. \n9\n \n(a\n-\nd)) \nand \nthe \nfit parameters \nare shown in \nFig. \n9\n(e\n-\ni)\n.\n \nV\nariation o\nf \nthe \nparameter \nM(0) \nis \nconsistent to\n \nthe \ntemperature and field \ndependence of magnetization \ncurves\n. \nThe fit parameters \nassociated with relaxation processes \n(\n\n1\n, \n\n2\n, \n\n1 \nand \n\n2\n) \nare \nless sensitive to \nhigher magnetic fields (5 kOe and 10 kOe)\n \nand suggest quas\ni\n-\nequilibrium state due to nucleation \nof clusters\n.\n \nHowever, t\nhe \ntime constants (\n\n1\n, \n\n2\n) \nare relatively high at 50 K and further increased \nfor higher cooling fields. \nIt indicates slowing down of the spin dynamics at low temperature due \nto increasing interac\ntions among the spins/cluster of spins and intra\n-\ncluster interactions \n(domain \nwall motion) \ndominate at higher cooling fields. Most importantly, magnitude of the \ntime \nconstants in composite sample is nearly one order less than the values in CF90 sample. Thi\ns is an \nevidence of faster relaxation in composite sample due to less inter\n-\ncluster exchange interactions.\n \nThe \nvariation of \ncoercivity \nin \nthe \ncomposite s\nample\n \nas the function of\n \nin\n-\nfield \nwaiting \ntime \non \nFC\n-\nM(H) \ncurve\n \nhas been examined \nby \nusing PC5 \nat \n10 K \n(\nFig. 1\n0\n \n(\na\n)\n) and 150 K (\nFig. \n1\n0\n(b)\n)\n.\n \nT\nhe sample was first \nzero field cooled \nfrom 300 K to 10 K/150 K\n \nand \nM(H) curve \n(N = \n1) \nwas measured \nduring sweeping of \nthe \n \nfield from \n+\n70 kOe to \n-\n20 kOe\n \n(10 K) or complete loop \nwas measured at 150 K\n. \nAfter first round\n \nmeasurement,\n \nthe \napplied field was made \nto \nzero\n \nbefore \nincreasing the temperature to 300 K. \nNext\n, \nthe \nsample was \ncooled \nunder \n+\n70 kOe down to 10 K/\n \n150 K. After temperature\n \nstabilization\n, \nthe \nM(H) curve (N = 2) \nwas recorded \nfrom \n+\n70 kOe to \n-\n20 kOe with \nan\n \nintermediate \nwaiting\n \nat \n-\n \n7 kOe for 10 K and at \n-\n2150 Oe for 150 K\n.\n \nT\nhe M(t\nw\n) \ncurve \nwas recorded \nfor different in\n-\nfield \nwaiting time\ns by repeating the \nFC\n-\nM(H) curve \nin the \nfield range +70 kOe to \n-\n20 kOe\n. The (\nnormalized\n) in\n-\nfield\n \nM(t\nw\n) curves are shown fo\nr 10 K\n \n(Fig. \n1\n0\n(c))\n \nand for\n \n150 K\n \n(\nFig. 1\n0\n(d)\n)\n. \nT\nhe inset of Fig. 1\n0\n(a)\n \nshows that \nH\nC\n \nat 10 K is \nenhanced \nin 17\n \n \nFC\n-\nM(H)\n \ncurve\n \nand the \nH\nC\n \ni\ns further enhanced by repeating the\n \nFC loop \nwith higher \nwaiting \ntime at \n-\n \n7 kOe. \nThe t\nw\n \nat 10 K was made high enough to o\nbserve an appreciable relaxation. The \nin\n-\nfield M(t\nw\n) curves at 10 K \nwere \nfitted with logarithmic decay law M(t\nw\n) = \n\n0\n \n–\nm\n*lnt\nw\n. The \ninset of Fig. 1\n0\n(c) shows the decrease of both \n\n0\n \nand slope (\nm\n) with the increase of waiting time \nat \n-\n \n7 kOe of the FC\n-\nM(H) c\nurves. In contrast, in\n-\nfield M(t\nw\n) curves at 150 K \nwere \nfitted with \nexponential law (1). The fitted parameters are not shown in the \ngraph\n.\n \nOn th\ne\n \nother hand, \nin\n-\nfield \n(\n-\n21\n50 \nOe) magnetic relaxation \nof the FC curve \nat 150 K is fast\ner \n(over coming spin pinni\nng) \nthan the \nslow relaxation\n \n(\nstrong domain wall pinning\n) \nat 10 K\n. \nT\nhe \nin\n-\nfield \nmagnetization \nat 150 \nK \nswitche\nd\n \nfrom positive to negative \nfor\n \nt\nw\n \n> \n180 s\n \nand\n \nwait\n-\nin field \nH\nC\n \nvalue (\n-\n2150 Oe\n)\n \nbecomes \nsmaller than the \nH\nC\n \nof ZFC curve (\n-\n2533 Oe)\n. \nThis propert\ny can be used in magnetic \nswitching/sensor applications. \nThe \ndecrease of \nH\nC\n \nat 150 K with the increase of waiting time at \n-\n21\n50 \nOe\n \nis \ncharacteristically \nopposite with respect to the increment \nof \nH\nC\n \nat 10 K\n. \n \nWe \nused PC\n6\n \nto study aging effect on the ZFC\n-\nM(H\n) loop of the composite sample at 10 \nK with the field sequence +70 kOe to 0 Oe and M(t\nw\n) measurement \nfor \nt\nw\n \n= 3600 s was carried \nout \nat 0 Oe (P1). This is followed by resumption of \nthe \nM(H) measurement down to \n-\n7 kOe (P2), \nwhere the sample was waited to re\ncord the second M(t\nw\n) curve. Then, recording of M(H) curve \nwas continued down to field at \n-\n70 kOe\n \nand \nreversed back to +7 kOe (P3) where third M(t\nw\n) data \nwere recorded. Finally, M(H) curve \ncontinued for \nfield\ns\n \nup to +20 kOe. A similar M(H) \nmeasurement prot\nocols were used to record the M(t\nw\n) curves at 10 K after cooling the sample in \nthe presence of +70 kOe from 300 K. The same experiments were carried out at \nrelatively higher \ntemperature \n100 K\n \nat field points 0 Oe and \n\n \n4 kOe\n. Fig. 1\n1\n \n(a\n-\nb) shows the \nrecord\ned \nZFC and \nFC\n-\nM(H) curves\n \nat 10 K and 100 K\n. \nThe\n \nM(t\nw\n) \ncurves\n \nat poin\nts\n \nP2 and P3 started to relax \ntowards the magnetization state \nat zero field\n. The behavior is consistent to the \nreverse torque 18\n \n \nacting on the \nspins when the field is reduced to zero and rot\nate over a time toward the positive or \nnegative direction depending on the local easy\n-\naxis orientation\n \n[\n20, 37\n]\n. The FC loop at 10 K \nshows nearly symmetric widening along the field and magnetization axes. \nThis showed field \ncooled induced enhancement of coe\nrcivity and magnetization in the composite sample\n, which \nshowed small exchange bias shift \n\n \n+ 64 Oe at 10 K [2\n7\n]\n. \nThe \nfield cooled induced \nwidening \nand \nexchange bias shift are\n \nnegligible \nat 100 K. \nOn the other hand, \nM(t\nw\n) curves\n \nat 10 K and 100 K \n(\nFig. 1\n1\n(\nc\n-\nd)\n) followed equation (1) with \nM(0) values positive and negative for points P2 and P3, \nrespectively. The M(t\nw\n) curves\n,\n \nmeasured on \nZFC and FC\n-\nM(H) curves, do \nno\nt show much\n \ndifferences \nat \n10 K and 100 K for \npoint P1 (\nwait\ning\n \nat 0 Oe\n)\n. The normalized M(t\nw\n)\n \ncurves \nmeasured on \nFC\n-\nM(H) curves\n \nsignificantly enhanced for field in\n-\nwait \n\n \n7 kOe at 10 K and \n\n \n4 \nkOe at 100 K in comparison to the \nmeasurement on \nZFC\n-\nM(H) curves\n. The differences between \nM(t\nw\n) curves measured \non \nFC\n-\nM(H) curves \nsubstantially decreased at\n \n100 K\n, unlike the case on \nZFC\n-\nM(H) curves at 10 K\n. The M(t\nw\n) curves in positive field side (+7 kOe at 10 K and + 4 kOe \nat 100 K) are found to be higher than their counter parts in the negative field side. This indicates \nthe effect of cooling field induced\n \nunidirectional anisotropy \non interfacial spin ordering [\n37\n] and \nit is reflected in the variation of time constants. \nFig. 1\n1\n(e\n-\nf) shows the time constants (\n\n1\n, \n\n2\n) at \ndifferent fields in\n-\nwait in FC process are larger than that in ZFC process both at 10 K a\nnd 100 K. \nThis \nshows a \nslow\ning\n \ndown of \nspin \nrelaxation \ndue to cooling field induced nucleation of clusters. \n \n \nFinally, \nwe repeated the \nmeasurement of M(H) curves \nfrom +70 kOe to \n-\n15 kOe at 10 K \n(Fig. 1\n2\n(a)) \nand \n+\n70 kOe to \n-\n1 kOe at 300 K\n \n(Fig. 1\n2\n(b))\n \nto ex\namine \nthe \ntraining effect in \nthe \ncomposite sample\n.\n \nT\nhe sample was zero field cooled from \n300 K (\n330 K\n)\n \nbefore repeating the \nM(H) \nmeasurement\ns\n \nat 10 K (300 K)\n \nfor \ndifferent \nfield \nsweeping rate without further heating\n \nthe \nsample \nto higher temperature\n. \nThe M(\nH) curves at 10 K are practically independent of the field 19\n \n \nsweeping rate, wh\nereas \nM(\nH) curves at 300 K\n \nshowed \nmagnetic \nrelaxation\n \nat higher fields\n. \nM\nost \nof the systems with training effect\n \nhave shown decrease of\n \ncoercivity [1\n1\n, 3\n8\n]\n,\n \nbut coercivity\n \nof \nthe c\nomposite sample is independent of sweeping rate\n \nin the temperature range 10 K to 300 K\n. \n \n4. \nC\nonclusions\n \nThe \nferrimagnetic\n \nCo\n1.75\nFe\n1.25\nO\n4\n \nferrite \nand its composite with non\n-\nmagnetic BaTiO\n3\n \n(BTO) \nparticles \nare modeled in core\n-\nshell spin structure. \nThe \nexiste\nnce of \nintrinsic \nspin \ndisorder \ninside the \nferri\nmagnetic \nferrite particles\n \nis confirmed by \nexchange bias, \nmemory and relaxation \neffects. The magnetic memory effect \nin ferrite particles dominated \nat higher temperatures\n, unlike \nobservation of \nexchange bias ef\nfect at low temperature. \nThe magnetic exchange interactions \nbetween ferrimagnetic particles are diluted and modified due to \npresence \nof intermediating \nnon\n-\nmagnetic BTO particles. \nHowever, \nbasic magnetic \nproperties \n(\nblocking of ferrimagnetic \nclusters\n, \nnon\n-\ne\nquilibrium\n \nferrimagnetic \nstate, memory, exchange bias and aging) \nof the ferrite particles \nare retained in \nthe BTO matrix of composite sample\n.\n \nThe \nslow \nspin dynamics \nlow temperature \ndue to strong spin pinning/inter\n-\ncluster interactions becomes faster on inc\nreasing the \ntemperature \nclose to the \nspin freezing\n/blocking \ntemperature\n \nof the samples at \n\n \n300 K (due to increasing \nfraction of non\n–\ninteracting/paramagnetic spins). The fraction of paramagnetic spins in composite \nsample is found \nto be \nmore\n \n(\nshowing\n \npronou\nnced \nmemory effect\n \nand faster spin relaxation\n) \nthan \nthat in \nthe \nferrite sample that exhibited relatively \nsmall \nmemory dip and slow relaxation. \nThe \nrelaxation of magnetization during field off condition\ns\n \nof the temperature and field dependence \nof magnetizat\nion \ncurves \nconfirm\ned \ntwo relaxation mechanisms\n \nin both the samples\n.\n \nThe fast \nrelaxation process (initial stage of the relaxation) \nis attributed to \nloosely bound shell/interfacial \nspins, whereas the slow relaxation process (later stage of the relaxation) \nis\n \nattributed to \nstrongly \ninteracting core/interior spins in the systems. \nThe \nM(H) curves are not much affected by the 20\n \n \nvariation of field sweeping rate\n, but magnetic state and coercivity \nof the samples \nare strongly \ndependent on in\n-\nfield or off\n-\nfield waiting \ntime during M(H) measurement\ns\n. \nWe showed \nvarious \noptions for tuning the \nferri\nmagnetic state and parameters, irrespective of the magnetic \nferrite\n \nand \nit’s \ncomposite in non\n-\nmagnetic matrix. The \ntuning capability of ferrimagnetic parameters is\n \npromising for t\nechnological applications\n. It may\n \nbe generalized by applying similar protocols of \nmeasurements on different systems. \n \n \nAcknowledgment\n \nRNB acknowledges the Research support from \nUGC (\nF.\n \nNo. F.42\n-\n804/2013 (SR)\n), Gov\nt. \nof \nIndia \nand UGC\n-\nDAE\n-\n \nCSR (No M\n-\n252/2017\n/1022), Gov. of India \nfor \ncarrying out \nthis \nresearch \nwork\n.\n \nRNB also acknowledges the PPMS facility at CIF, Pondicherry University.\n \n \nReferences\n \n[\n1\n] K. Jonason, P. Nordblad, E. Vincent, J. Hammann, and J.\n-\nP. Bouchaud, Eur. Phys. J. B \n13\n, 99 \n(2000).\n \n[\n2\n] A.\n \nRos\ntamnejadi, M.\n \nVenkatesan, H.\n \nSalamati, K.\n \nAckland, H.\n \nGholizadeh, P.\n \nKameli, and J. \nM. D.\n \nCoey, J. Appl. Phys.\n121\n, 173902 (2017).\n \n[\n3\n] D. Parker, V. Dupuis, F. Ladieu, J.\n-\nP. Bouchaud, E. Dubois, R. Perzynski, and E. Vincent, \nPhys. Rev. B \n77\n, 104428 (2008).\n \n[\n4\n] L. M.\n-\nGarcía, G. Chen, Y. Montaña, A. Mascaraque, B.M. Pabón, A.K. Schmid, J.de la \nFiguera, Sci. Rep. \n8\n, 5991 (2018).\n \n[\n5\n] H\n.\n \nKhurshid, P\n. \nL\n.\n-\nKelley, Òscar Iglesias, J\n.\n \nAlonso, M\n.\n-\nH\n.\n \nPhan, C\n.\n-\nJ\n.\n \nSun, M\n.\n-\nL\n.\n \nSaboungi\n,\n \nand\n \nH\n.\n \nSrikanth, S\nci. Rep.\n  \n5\n, \n15054 \n(2015).\n \n[\n6\n] A. Kumar\n,\n \nand D. Pandey, J\n.\nMagn\n.\n \nMagn\n.\n \nMater\n.\n \n511\n,\n \n166964\n \n(2020)\n.\n 21\n \n \n[\n7\n] P\n.\n \nBag, P. R. Baral, and R. Nath, P\nhys. \nR\nev. \nB \n98\n, 144436 (2018)\n.\n \n[\n8\n] O. Cador, F. Grasset, H. Haneda, and J. Etourneau, J. Magn. Magn. Mater. \n268\n, 232 (2004).\n \n[\n9\n] J. Nogues,\n \nJ. Sort, V. Langlais, V. Skumryev, S. Surinach, J.S. Munoz, and M. D. Baro, \n \n     \nPhys. Rep. \n422\n, 65 (2005).\n \n[1\n0\n] K. Nadeem, H. Krenn, D.V. Szabó, J. Magn. Magn. Mater. \n393\n, 239 (2015).\n \n[1\n1\n] P.K. Manna, and S.M. Yusuf, Phys. Reports \n535\n, 61 (2014).\n \n[\n12\n] Y\n. Sun, M.B. Salamon, K. Garnier, R.S. Averback, Phys. Rev. Lett. \n91\n, 167206 (2003).\n \n[\n13\n] \nZ. Tian\n, \nL. Xu\n, \nY. Gao\n, \nS. Yuan\n, \nZ. Xia\n, \nAppl. Phys. Lett.\n \n111\n, 182406 (2017).\n \n[\n14\n] \nM\n.\nSasaki\n,\nP\n.\nE\n.\nJ ̈o\nnsson, H. Takayama, H. Mamiya, Phys.Rev. B71, 104405 (2005).\n \n[\n15\n] T. Zhang, X.G.\n \nLi, X.P.\n \nWang, Q.F.Fang, and M. Dressel, Eur. Phys. J. B\n \n74\n, 309 (2010).\n \n[\n16\n] S. Polisetty, S. Sahoo, A. Berger, and Ch. Binek, Phys. Rev. B \n78\n, 184426 (2008).\n \n[\n17\n] J. Barman,\n \nT. Bora, S. Ravi, J. Magn. Magn. Mater. \n385\n, 93 (2015).\n \n[\n18\n] R. Skomski, J. Phys.: Condens. Matter \n15\n, R841 (2003).\n \n[1\n9\n] G. J. Kumar, and C. Rath, J. Magn. Magn. Mater. \n466\n, 69(2018)\n \n[\n20\n]\n \nJ. Carrey, B. Mehdaoui, and M. Respaud J. Appl. Phys. \n109\n, 083921 (\n2011).\n \n[\n2\n1] C.\n \nSen, W.A.S\n \nAldulaimi, O. Mohammadmoradi, and I.B.\n \nMisirlioglu, J. Phys. D: Appl. \nPhys. \n52\n, 015305 (2019).\n \n[\n22\n] J\n.\n \nReindl, H\n. \nVolker, N\n.\nP. Breznay\n, \nand M\n.\n \nWuttig, npj Quantum Mater\n.\n \n4\n, 57 \n(2019)\n.\n \n[\n23\n] K. Inomata, N. Ikeda, N. Tezuka, R. Goto,\nS. Sugimoto, M. Wojcik, and E. Jedryka, Sci. \nTechnol. Adv. Mater. \n9\n, 014101 (2008).\n \n[2\n4\n] S. Hoffman, Y. Tserkovnyak, P.K. Amiri, and K.L. Wang, Appl. Phys. Lett.\n100\n, 212406 \n(2012)\n 22\n \n \n[2\n5\n] A. Walsh, Su\n-\nH. Wei, Y. Yan, M.M. Al\n-\nJassim, and J.A. Turner, Phys.  Re\nv.  B \n76\n, 165119 \n(2007).\n \n[2\n6\n] R.N. Bhowmik, S. Kazhugasalamoorthy, R. Ranganathan, and A.K. Sinha, J. Alloys \nCompd. \n680\n, 315 (2016).\n \n[2\n7\n] R.N. Bhowmik, S. Kazhugasalamoorthy, and A.K. Sinha, J. Magn. Magn. Mater. \n444\n, 451 \n(2017).\n \n[\n28\n] \nT. Vimal\n, \nS. Pandey\n, \nS.K. Gupta\n, \nD.P. Singh\n, \nK. Agrahari\n, \nG. Pathak\n, \nS. Kumar\n, \nP.K. \nTripathi\n, \nR. Manohar\n, \nLiquid Crystals\n \n45, \n \n687 (2018).\n \n[\n29\n] \nV. Skumryev, V. Laukhin, I. Fina, X\n. Martı´, F. Sa´nchez, M. Gospodinov, and J.       \nFontcuberta, Phys. Rev. Lett. \n106\n, 057206 (2011).\n \n[3\n0\n] T. R. Gao, Z. Shi, S. M. Zhou, R. Chantrell, P. Asselin, X. J. Bai, J. Du, and Z. Z. Zhang, J. \nAppl. Phys. \n105\n, 053913 (2009).\n \n[\n31\n] \nJ\n.\nA\n. \nGonzález\n, J.\nP\n. \nAndrés\n, \nR\n. \nL\n. \nAntó\nn\n, \nJ\n. \nDe Toro\n, \nP\n.\nS\n. \nNormile\n, \nP\n. \nMuñiz\n, \nJ\n. \nM\n.  \nRiveiro\n, \nand J\n. \nNogué\ns, Chem. Mater. \n29\n, 5200 (2017).\n \n[\n3\n2] G. Catalan, J. Seidel, R. Ramesh, and J.F. Scott, Rev. Modern. Phys. \n84\n, 119 (2012).\n \n[\n33\n] R.N. Bhowmik, R. Ranganathan, and R.\n \nNagarajan, J\n. \nMagn\n.\nMagn\n.\n \nMater\n.\n \n299\n, 327 (2006)\n.\n \n[\n34\n] S. Pramanick, S. Chattopadhyay, S. Giri, S. Majumdar, and S. Chatterjee, J. Appl. Phys.\n116\n, \n083910 (2014).\n \n[\n35\n] Y. Takagaki, J. Herfort, L. Däweritz, and K. H. Ploog, Phys. Rev. B \n74\n, 224417 (2006).\n \n[\n3\n6\n] R.V. Chamberlin, K.D. Humfeld, D. Farrell, S. Yamamuro, Y. Ijiri, and S.A. Majetich, J. \nAppl. Phys. \n91\n, 6961 (2002).\n \n[3\n7\n] M. D. Stiles, and R. D. McMichael, Phys. Rev. B \n63\n, 064405 (2001).\n \n[3\n8\n] V. Kumari, K. Dey, S. Giri, and A. Bhaumik, \nRSC Adv.\n \n6\n, 457\n01 (2016).\n \n \u0000 \u0001 \u0002 \u0003 \u0004 \u0005 \u0006 \u0007\b \t \n \u000b \u0001 \u0006 \f \r \u000b \u0007 \b \u0006 \f \u000e \b \u000f \u0010 \u0007\b \u0011 \u0011 \u0010 \u0012 \u0001 \u0013 \u0010 \u000b \u000e \u0014\u0006 \u000b \u0014 \u000e \b \r \f \u000e \u0015\u0000 \u0016 \u0017 \u0018 \n \u0019 \n \u0013\u001a \u0015\u0000 \u001b \u000f \u001c \u001d \u001b \u0006 \f \t \u0012\f \u0010 \u0001 \u000b \b \u0018 \u001e\u0019 \u001f \u0001 \u000b \u0007 \u0010 \u0012 \n \u0006 \b ! \" # ! $ # % & % ' ( ) * + ! & , ( # & $ ( \" ! * $ # % & - ./ 0\n1 2 3 4 5 6 7 8 4 9 4 :; 6 < 8 = ; = > = ? 6 @ = 8 ; 4 9 < 4 8 5 ; 7 8 4 A > B 4 1 5 : C @ D 4 ? C A @ B E1F G H G I FG IJ G K L MN O I G P Q R N P Q K I S T U G V G I G T H V K P K J K W X T G V G Q I J K V H K V N P G F L K V V G J K V F Q IO P UG P Q Y G V G X H K I X GZ [ \\ ] ^ _ ` a b ^ \\ c d ` \\ c Z b ef g c b a [ c ^ h e Z [ [ i Z b ec \\ c Z b Z [ \\ ] ^ _ ` a b ^ \\ c d ` \\ c Z b i f g j ^ k l m ^ \\ ` c h k e^ k i g c n ^ e c b \\ ] ^ \\ ^ o\\ l\n\u0000 \u0001 \u0000 \u0002 \u0000 \u0000 \u0002 \u0001 \u0000 \u0003 \u0000 \u0000 \u0003 \u0001 \u0000 \u0004 \u0000 \u0000 \u0004 \u0001 \u0000\n\u0005 \u0006 \u0007 \b\t \n \u000b \f\n\r \u000e \r\u000f \u0010 \u0011\u0012 \u0013 \u0014\n\u0015\u0016 \u0017\u0018\u0019 \u001a\u001b\u001c \u001d\n\u001e\n\u001f  ! \"# $% &\n' ( ' ) ' ' ) ( ' * ' ' * ( ' + ' ' + ( '\n, - . /0 1 2 3\n4 5 4\n6 7 68 6 9 : 9 8 7\n;\n< =\n> ? @A A BC D DE F G\nH IHJ KLM NO\nP Q P R P P R Q P S P P S Q P T P P T Q P\nU\nV WX YZ [\n\\] ^\n_`a_bcdefagahijk l mn\nopqorstuvqwqxyz {| }\n~                                ¡ ¢ £¤ ¥ ¦ ¦ §¨© ª « « ¬ ® ¯ ° ± ² ³´ µ ¶ · ¸¹ º » ¼ ½ ¾ ¿À Á  à ÄÅ Æ Æ ÇÈ\nÉ Ê Ë ÌÍ Î Ï Ï Ð ÑÒ Ó Ô Õ Ö × Ø Ù Ú Û Ò Ü Ý × Þ ß à á Ú Ò Üâ Ý× Þ á ß ã ß ß ã ä å å æ Ù Ýß Þç Ú Û Ò Ü Ý × Þ á ß ã ß ß ã è Ù é Ù ê ã Ù á ë Ó Ù é á èìí îï ð ñ ò ó ô õ òö ÷ ò ö ÷ òñ ø ù ô ñ ø ÷ õ ð ô ú ÷ û ü ý þìÿ î ö ù ô ù ñ \u0000 ì í î ï ñ \u0001 ÷ \u0001 ï ö û \u0002 ö ÿ \u0003 \u0004 ò ø ÷ ó ì \u0003 î ï ð ñ ô õ ð ö û \u0002 ö ÿ\u0005 \u0006 \u0007 \b \t \n \u000b \f \r \u000e \u000f \u0010 \t \u0011\u0012 \u0013 \u000b \u0014 \u0012 \u000b \f \u0015 \u000e \u000f \u0010 \t \u0011 \t \u0013 \u0016 \u0017 \u0018 \t \u0019 \t \u000b \u000f \u0019 \u000b \u0007 \u000b \u001a \t \f \t \u0007 \n \u000b \f \u0012 \u001b \u001c \u001c \t \u0007 \t \u0014 \f \u001c \u001b \t \u001d \u0012 \n \u0011\u001c \u0013\u0016\n\u001e\u001f !\"#$%&'(\n)* +\n,- ./ 01 2\n3 4 5 46 7 89 : ;<= > ? @ A ? ABC D\nEFGHIJKLMLJNGNOPLOMQRSTU V W X Y Z\n[ \\ ] ^ _`abcd efg\nhijklimnlopiqrsnstu\nvw x\ny z { | } ~  |                \n \n     ¡¢£ ¤ ¥ ¦ ¦ § ¨\n©ª\n«¬  ® ¯ ° ± ² ³ ´\u0000 \u0001\u0002 \u0003\u0004 \u0005\n\u0006 \u0007 \b\t \n \u000b\n\f \f\r \u000e\n\u000f \u0010 \u0011 \u0012\u0013 \u0014 \u0015 \u0015\n\u0016 \u0017 \u0016 \u0016 \u0018 \u0016 \u0016 \u0019 \u0016 \u0016\n\u001a \u001b\u001c \u001c\u001d \u001e\u001f  ! \"\n# $ # # % # # # % $ # #\n& ' ( ( )* + , , ,\n- . - . . - . . .\n/ 0 1 1 23 4 5 5 67 8 9 9 9\n:;<=>?@A\nBCDEFGHI\nJK LMNO P Q R R S TUVWXYZ[\\\n] ^ __\n`abcde``fghijkff\nl mm n o o p qr st u v v wxyz { | } ~   z z \n        \n  \n  \n\n\n ¡ ¢ £¤ ¥ ¦ §¨ © ª« ¬   ® ¯ ¯ ° ± ² ³ ´ µ ´ ± ² ¶· ¸ ¹ º » ¼· ½ ½ ¾¿ À Á ¸  à ¸ Ä Â Å ÆÇÅ È ¸ É Â Å Ê ¸ Å Ë È Ì · · É Ä È ¼· ½ Í ¾ ¿ À É Â Î Ï Ï Ì Å ¾ É À º ¼ · ½½ ¾ ¿ À É Ä ÈÐ Ñ ÒÓ Ô Õ Ö × Ø Ù Ú Û Ü Ý Þ ß à à á Û Ô â Ö Ý ã ä ä å æ æ Û Ù Û ã × Û â Û Þ Û ç Û Û ã ÐÑ Ò Ó Ý ã ä ÐÑ ÒÒ × Ø ÙÚ Û Ü Ý Þè é ê ë ì í î ï ð ñ ò ó ôõ ö÷ ø ôì ö ù ï ì ï ôú ò è û ï ð é ü ñ ù ö ù ý è é ú ê ó é ñ ð ù ò ó ó ï ì þ ÿ ÿ \u0000 ñ ô ù ö ï ú ù \u0001 ÿ ÿ \u0000 ñ ô ñ ö ÷\n\u0002 \u0003 \u0004 \u0005 \u0003 \u0004 \u0006 \u0007 \b \t \n \u000b \f\r \u000e \u000f \u0010 \u0011 \u0012 \u0013 \u000f\u0014 \u0015 \u0016\u0017 \u0018 \u0019 \u001a\n\u001b \u001c\u001c \u001d \u001e \u001e \u001f  ! \"# $ % % & '() * + , - . / 0 . ) ) 1\n2 3 456789567\n:\n;< = >? @ A B\nC D E FG H I JK L MN O P P Q R Q S S T U V W X Y X U V Z[\\]^_[\\]\n`\nab c d e f g h c i j k l g m n j e\no p q r s t u v w t x y z x r {\n| } ~     ~          \n    \n\n   \u0000 \u0001 \u0002 \u0003 \u0004\n\u0005 \u0006 \u0005 \u0005\u0007 \b \u0007 \t\n \u000b \f \n\r \u000e \u000f \u0010\n\u0011 \u0012 \u0013 \u0014 \u0015\n\u0016\u0017\u0018\n\u0019 \u001a \u001b \u0019\u001c \u001b \u0019 \u001d \u001b \u001b \u001d \u001b \u001c \u001b \u001a \u001b\n\u001e\u001f  !\" #\n$\n% &' (\n) * + , -\n. / .0 1 23 4 56 7 89 : ;\n<= <> <? <@ A\nB C D EF G H IJ K L MN O P O\nQ R S T U\nV W X VY Z [ \\] ^ _ ^\n`a\nbc defghijklmfnogpqrst\nuvwxyz\n{ | }~\n\n                      ¡ ¢ £ ¤ ¥  ¦ § ¤  ¤        ¨     © ª § « ¤ ¬ ¨ §     ¡ ¢ £ ¤    ® ¯ ¢ £ ¤    ° ± ² ³ ° ´ µ ¶ · ³ ¸ ¹ º · ¶ ± » µ ¼ º » µ ³ ¶ ¸ ³ ° ´ µ ¸ ¸ ± ° ± ¶ » ½ º µ » µ ¶ · » µ ¹ ± º » ¾ ¿ ± À º Á  à º ° µ º » µ ³ ¶ ³ ¸ Ä À Å Á ² Æ ° Ç ± È º ¸ » ± °É Ê Ë Ì Ë Í Î Ì Ë Ï Ð Ñ Ò Ó Ô Õ Ñ Ì Ó Ö Ê Ì Ê Ö × Ø Ë Í Î É Ê Ë Ì Ë Í Î Ì Ë Ï Ð Ù Ê Ú Û Í Î É Ë Ì Ü Ý Ë Ì Ö Ê Ì Ê Ñ Þ Ó Ô ß Ê Ë Ì Ë Í Î Ì Ë Ï Ð Ö Ð à Ð Í Ö Ð Í Þ Ðá â ã á ä å ã æ ç æ è é ê ë ì æ ë æ è æ ê í î ê ï ë ä è æ ð ê è æ á ë ñ ì ò ê ë ì â æ è ó ê å ê î ä è ä å ô á â õ ö ÷ ø ô ê î ó í ä ê è ù ø ú ñ ä û â ò ü\ný þÿ \u0000 \u0001 \u0002 \u0002 \u0003 \u0004\u0005 \u0006\u0007\b\t \n \u000b\f \r \u000e\u000f\u0010 \u0011 \u0010 \u0012 \u0013 \u0010 \u0014\u0015 \u0016 \u0017 \u0018 \u0015 \u0019 \u0016 \u0017 \u0016 \u001a \u001b\u001c \u001d \u001e \u001f  \u001c ! \" \u001f\n#$%&'()*+ , - . / 0 1 2\n3 4 5\n67 89 : ; < ; = > < ; ? @ A B C\nDE F\nG H I J I K L J I M N O P Q\nRS T\nU\nV\nWXYZ[\\ ] ^ _ ^ ` a _ ^ b c d e f\ngh ij k l\nm n opqrstuvwx y z { | }\n~\n  \n                         ¡ ¢ £ ¢ ¤ ¥ £ ¢ ¦§ ¨ © ª\n«¬ \n® ¯ ° ± ² ³ ° ³ ³ ° ´ µ ¶ ³ ° · ´ ¸\n¹ º º » ¼ ½ ¾ ¿À Á  ÃÄ Å Å Æ Ç È É ÊË Ì Í ÎÏ Ð Ð Ð Ñ Ò Ó Ô Õ Ö × Ø ÙÚ Û Û Û Ü Ý Þ ß à á â ã äå æ æ æ ç è é ê ë ì í î ïð ñ ò ò ó ô õ ö ÷ ø ù ð úû ü ý þ ÿ \u0000 \u0001 þ \u0002 ü \u0003 \u0004 \u0000 \u0003 \u0000\n\u0005 \u0006 \u0007 \u0005 \b\t \n \u000b \t \f\n\r\n\u000e\n\u000f\u0010\u0011\u0012\u0013\u0014\u0015\u0000 \u0001 \u0002 \u0003 \u0000 \u0001 \u0002\u0004 \u0005 \u0006 \u0005 \u0000 \u0005 \u0006 \u0000 \u0000 \u0000 \u0005\n\u0007 \b\t \t\n \u000b\n\f \r \f \u000e \f \u000f \u0010\n\u0011 \u0012\u0013 \u0014\u0015 \u0016\n\u0017 \u0018 \u0019 \u001a \u001b\n\u001c \u001d \u001e\u001f  !\" # $\n% & ' ( )\n* + *, - ./ 0 12 3 4\n5 6 7 58 7 7 8 7 6 7 9 7\n: ; <= > ?\n@\nA BC D\nE F G E H H F G\nI J K\nL\nM NO P Q R S\nT UV V\nW X YZ[\\][^_`ab]\nc d e fg h i jk l m l no p q p rs t u t vw x y y z\n{ | }~  \n \n                \n ¡ ¢\n£ ¤ ¥¦§¨©ª«¬®¯°©±²ª³´µ¶·¸¹º » ¼½¾¿ÀÁÂÃÄÅÆÇÈÅÇÆÉÊË\nÌ\nÍÎÏÐÑ Ò Ó Ô Õ Ö × Ø Ù Ú Û ÜÖ ÝØ Þ ß ß à Ò á â Ñ ã × á ä Ñ ã å æ ç è éÛ ê ß ä Û ë × ì í ç î × á ä Ò á ë Û ì ë Ú ß ï ë ì Ú Ûð ñ ò ò ó ô ó õ ö ó ñ õ ÷ ø ù ú ú û ü õ ð ó ô ý þ ÿ \u0000 \u0001 ó \u0002 õ ð \u0003 þ ÿ \u0000 \u0001 ó \u0004 \u0005 \u0006 ó \u0007\b\t\n ö ü ô \u000b ó ñ õ \f ÷ ø \r ú ð ó \u000e \u0002 \u000f\u0010 \u0011 \u0012 \u0011 \u0013 \u0014 \u0011 \u0015 \u0016 \u0017 \u0014 \u0018 \u0015 \u0017 \u0019 \u0019 \u0011 \u0010 \u0011 \u001a \u0014 \u0016 \u0013 \u0017 \u0014 \u0017 \u001a \u001b \u0014 \u0017 \u001c \u0011 \u0013 \u0014 \u001d \u001e \u001f  ! \"\u0011 \u0013 \u001a \u0015 \u0014 \u0018 \u0011 \u0011 \u0019 \u0019 \u0011 # \u0014 \u0017 $ $ \u0018 % \u0016 \u001a \u0017 \u001a & ' (\u001f ) \u0018 \u0011 \u0019 \u0017 \u0014 % \u0019* + , - . / , / . 0 1 2 3 4 5 / 2 , 2 3 4 , 2 6 7 +8 - / 3 . , 9 7 : / 12 / , 2 ; 3 ; < < 2 , = / 1/ 6 7 , 7 1 > / 1 7 > 9 ; 5 3 2 3 +. ? < - @\nA B C D E F\nGHIJKLMNO P Q R S T\nUVWXYZ[\\]^_\n`\na b c def g\nh\ni\nj\nk\nl\nm\nn\nop q r st u\nv\nw\nx\ny\nz\n{\n|\n}~  \n          \n \n\n          \n¡ ¢\n£¤\n¥ ¦ § ¨ © ª « ¬  ®¯ °\n±² ³\n´µ\n¶ · ¸¹ º » ¼ ½¾ ¿À Á\nÂà Ä\nÅÆÇ\nÈ É ÊË Ì Í\nÎ\nÏ\nÐÑÒÓÔÕ\nÖ\n×ØÙÚÛÜÝÞß\u0000 \u0001 \u0000 \u0002 \u0000 \u0001 \u0002 \u0002 \u0002 \u0000\n\u0003 \u0004\u0005 \u0006\u0007 \b\n\t \n \u000b \f \r\n\u000e\u000f\u0010\u0011\n\u0012 \u0013 \u0014 \u0015 \u0016\n\u0017\u0018\u0019\u001a\n\u001b \u001c \u001d \u001e \u001f\n !\" #\n$% & ' $ % & ( $ ) & ' $) & ( $ ( & ' ( & (\n*\n+ ,- ./ 0\n1 2 1 1 3 1 1 4 1 1\n5 67 89 :\n; < = >\n? @ AB C D\nE\nF GH I\nJ K L M N O P Q R S J TU O V W X YZ Q [ Q \\ ] X Y Q ^ \\ _ ` a b cQ U\\ V M O P Q _ K [ Q YQ d \\ e \\ _ K f g \\ g ^ h K Q d ^i j k j l i j l m j n j o p q j r s q t o l j u p v t u p r l p l u w j x y z{ m | n } j ~   w p m w w t } j  j j l n j k j t n j i                                                                   ¡ ¢ £ ¤ ¥ ¦ § § £  ¨ £ © ª « ¨ ¬ £ ¨\n® ¯ °± ² ³ ´ µ ¶ · ¸ ¹ º ¯ » ¼ ¯ ½ ´ ¾ µ ¯ · ¿ ® ¯ ® À Á º  à ´ Ä Å »  ´  ¯ º ® Æ Æ ® ÇÈÉ Ê Ë Ì Í Î Ï Ð Ñ Ò Ê Ó Ï É Í Ô Ò Õ Ò Ö Ì Í Ì Õ Ë ÈÓ × Ï ÑØ\nÙÚ\nÛ Ü Ý\nÞß à\náâãäâåæçèéä\nê ë ì\níîïðñòóôõö÷ðøùñúûüýþÿ\u0000\u0001 \u0002 \u0003\u0004\u0005\u0006\u0007\n\b \t \n\u000b\f \r\n\u000e\u000f\u0010\u0011\u0012\u0013\u0014\u0015\u0016\u0017\u0014\u0016\u0015\u0018\u0019\u001a\n\u001b\n\u001c\u001d\u001e\u001f\n !\n\"# $\n%& '\n()\n* + ,\n-. /\n01 2\n3\n4\n56789\n:\n;<=>?@AB C D EF G H I\nJKLMNOPQRSTUVWXYZ [ \\]^_`abcde f g hi j\nk lm n no p pq r r rs t t tu v v vw x y y\nz { | } ~   ~   \n              \n         ¡  ¢ £   ¤     ¡ ¥ ¦ § ¨ © ª \n« ¬  ® ¯ ° ± « ² ³ ´ µ¶ · ¸ ¹ º\n»¼\n½ ¾ ¿ À ÁÂ Ã Ä Å Æ Ç È\nÉÊ Ë\nÌ Í Î Ï Ð Ð Î Ñ Î Ò Ð Î Ò Ó Ô Î Õ Í Ö Î × Ø Ó Î Ô ÓÙ\nÚ Û Ü Ý Þ ßàáâ ã\u0000 \u0001 \u0002 \u0003 \u0004 \u0005 \u0000\n\u0006 \u0007\n\b \t \t\n \u000b \n \n \f \n \n \r \n \n\u000e\u000f\n\u0010 \u0011\u0012 \u0013\u0014 \u0015\u0016 \u0017\n\u0018 \u0019 \u0018 \u001a \u0018 \u0018 \u001a \u0019 \u0018 \u001b \u0018 \u0018 \u001b \u0019 \u0018 \u001c \u0018 \u0018 \u001c \u0019 \u0018\n\u001d\u001e\u001f !\n\" #$ %& '\n( ) *+,-./01234356/07.879:;<= > ? @ A B\nC D DE F F\nGHIJKLMNONLPLQRSTRUVWXYHZ[\\\n] ^ _ ` ab c d e fg h i j kl m n op q r s t uv w xyz{|}~             \n    \n    \n¡¢£¤¥£¤¦¡§¢¨©ª«¬  ®¯  ° ± ² ³ °  ´ µ ¶\n· ¸ ¸ ¹ º » ¼ ½ º » ¹ ¼ ¾ ¿ ¿ À »\nÁ  ÃÄÅ\nÆ Ç È È ÉÊ Ë Ì Í Î Ï Ð Ê Ñ Ò Ó Ô Õ Ö × Ï Ê Ñ Ò Ó Ô Ø Ù ÚÛ Ü Ý Õ Þ ß à à á Ü ÒÕ Ô Í Ó â Ü Ï Ð Ê Ñ Ò Ó Ô Ø Ù Ú Û Ü Ý Õ Þ × Ë ã ã Ü ÚÜ Ö Þä å å æ ç è é ê ç è æ é ë ì í îï ð ñ è ò è óå è ôä ò õ ôè é è ô ç ö ä ò ç ö è ÷ ê ø ù ú û ü õ ôö è ë ì ç ý í î ä æ ÷ ý þ ÿ ç ò ñ ê ç ò ÷ ê\u0000 \u0001 \u0002 \u0003 \u0002 \u0004 \u0005 \u0006 \u0007 \b \u0000 \u0001 \t \n \u000b\u0002 \f \u0000 \r \b \u0004 \f \u0006 \u0001 \u0004 \u0003 \u0002 \u000e \u000f \u0010 \u0004 \f \u0011 \u0012 \u0004 \u000b \b \u0004 \u0000 \b \n \f \n \u0013 \u0000 \u0001 \u0002 \u0003 \u0002 \u0004 \u0005 \u0003 \u0004 \u000b\u0004 \u0014 \u0002 \u0000 \u0002 \u000b \u0006 \u0007 \b \u0000 \u0001 \u0013 \b \u0002 \u0015 \u0011 \u000e\u0011 \u0010 \u0016\n\u0017 \u0018 \u0017 \u0019 \u0017 \u0017 \u0019 \u0018 \u0017 \u001a \u0017 \u0017 \u001a \u0018 \u0017 \u001b \u0017 \u0017\n\u001c \u001d \u001c\u001e \u001f  ! \" #$ % &\n' ( ) * +, - . / 0\n1 2 3 4 5\n6 7 89\n: ; < = > ?@ A B CD E E F G\nH I JKLMNOPQRKSMTUVRWRXQY Z[ \\\u0000 \u0001 \u0002 \u0003\n\u0004 \u0004\u0005 \u0006 \u0007\b \t \n\u000b \f \r\u000e \u000f \u0010\u0011 \u0012 \u0011\n\u0013 \u0014 \u0013 \u0015 \u0013 \u0013 \u0015 \u0014 \u0013 \u0016 \u0013 \u0013 \u0016 \u0014 \u0013 \u0017 \u0013 \u0013\n\u0018 \u0019\u001a \u001b\n\u001c \u001d \u001c \u001e \u001c \u001c \u001e \u001d \u001c \u001f \u001c \u001c \u001f \u001d \u001c  \u001c \u001c\n! \"# $% %\n&'()*+,-'+,.)/*012(3\n456789:;59:<7=8>?@6A\nBC D E FG H H IJKL M N OP Q Q RS\nT U VW U X Y Z [ X U \\ ] ^\n_ ` a b c d e f g\nhi j k l m n o pqr s tu s v w x y v s z { |} ~     }        }                   }             ~                ¡ ¢ £ ¤ ¥ £ ¡ ¦ § ¨ © £ ¢ ª ¥ £ ¡ ¦ § «¬ ¬  ¢  ® ¯ ° ± ² ® ¯ °   § ® ¯ ° § ³ ´ £   ¬ µµ  ¶ § ¡   § ¥ £ ¡ ¦ §· ¸ ¹ º » ¹ ¼ » º ½ » ¹ ¾ ¿À Á ¸  à ÄÅ Æ ÇÈ É Ê Ã Â Ë ¸ Ì Å ¹ Â Ë Ã Í Å Î Å º » ¸ ¹ ¸ Ï Á Å Ð ¹ à º » Ñ Å º » ¸ ¹ » ¹ Ï » Ã Í Â Ò ¾ ¾Ó Ô Õ Ö × Ø × Ô Õ Ù Ú Û Û Ü Ý Ø Ó Þ Ý Ö × Õ Ø Þ Ý × Õ Û Ý Ø Ô ß àÝ á Ù Ú Ø â ã ä å Ù æ ã ä å Ú Õ Ö ã ä å Ú çÝ Ó Ô è é Ú ç Ý Ö × Õê ë ì í î ë ï ì ë ð ñ ò ì óô õ ö ÷ ë ð ø ù ú ûú ü ë ý í óô þ ö ÿ \u0000 \u0001 \u0002 \u0003 ò ð ø ò î \u0004 ú ÷ ë ðø ù ú û ñ ü \u0005\n\u0006\n\u0007 \b\t \n\u000b \f\r \u000e\u000f \u0010\u0011 \u0012\n\u0013 \u0014 \u0015\u0016\u0017\u0018\u0019\u001a\u001b\n\u001c \u001d \u001e \u001f  ! \" \" # $% & ' ( ) * + , , - &. / 0 1 2 3 4 4 5 67 8 9 : ; < = > > ? 8@ A B C D E F G HI J K L M N O P Q JR S T U V W X Y Z[ \\ ] ^ _ ` a b c \\d e f g h i j k l mn o p q r s t u v w ox y z { x y z | x y z\n}~\n      \n\n                ¡ ¡ ¢ £ ¤ ¥ ¦ § ¨ © © ª «¬  ® ¯ ° ± ² ³ ³ ´ µ ¶ · ¸¹º » ¼ ½¾ ¿ À\nÁ  à Ä\nÅ Æ Ç È\nÉÊ Ë Ì Í ÎÏ\nÐÑ\nÒÓÔÔ\nÕ Ö ×\u0000 \u0001\u0002 \u0003\n\u0004 \u0005 \u0006 \u0007\b \t \n \n\u000b \f \r \u000e\u000f \u0010 \u0011 \u0012\u0013 \u0014 \u0015 \u0015\n\u0016 \u0017 \u0018 \u0019\u001a \u001b \u001c \u001d\u001e \u001f   \n! \" # #\n$ % & '( ) * *\n+,\n-.\n/ 0 / /1 2 1 3\n4 5 6 4\n7 8 7 79 : 9 ;\n< = > ? @ A\nB C D E\nF G H IJKLMKNOPQRMS T UV W X Y Z V W [ \\ ] [ ^ _ ` \\ a b c d W e _ f g h [i\nj k l m l n o m p q r n s t n u v m q u l st w q x l yk j z {q r n { k q k l m o v q w k|} } ~               }                                                         ¡  ¢ £\n¤ ¥ ¦§¨ © ª « ¬  ¬ ® ®\n¯\n°\n±² ³ ´ µ µ ¶· ¶ ¸ ¸¹º » ¼ ½ ¾ ¿ ¾ À ÀÁ\nÂ\nÃ Ä Å Æ Ç È É Ê É Ë ËÌÍ\nÎ Ï Ð\nÑÒ Ó Ô Õ Ö × Ø Ò Ù Ú Ö Û\nÜÝ Þß\nà\náâã\nä\nåæ\nç è é êë ì í îï ð ñò ó ôõ\nö\n÷øù\nú\nûüý þ ÿ\u0000\u0001 \u0002\n\u0003\u0004\u0005\u0006\u0007\b\t\n\u000b\f\r\u000e\u000f\u0010\f\u0007\u000f\u0007\u0000\u0001\u0002\u0003\u0004\u0005\u0006\u0007\b\t\u0007\n\u000b\f\r\u000e\u000f\u0010\u0010\n\u0011\u0012\u0013\u0011\u0014\u0013\u0013\u0015\u0016\u0017\u0018\u0019\u001a \u001b\u001c\u001d\u001e\u001f\n ! \"\"#$%&'(\n)*+,-.)/01/2344\n56575859:;<= >?@AB\nCDEEDEFGHIJK LMNOP\nQRST\nUV W X X X X YZ\n[\n\\] ^ _ ` a b c d e f g\nh\ni\nj\nklm\nn\nop\nq\nrs t t u u v\nwxy\nz { | } ~   \n\n\n\n\n\n\n\n\n\n¡¡¢¢£\n¤¥¦\n§\n¨©ª««¬\n®¯\n°\n±²³´´µ\n¶·¸\n¹\nº»¼¼¼¼½¾¿ÀÁÂÃÄÅÆÇÈÉÊËÌÍÎÏÐÑÒÓÇÈÔÍÕÖÐÎ×ÃØÒÕÖÃØÙÚÖÐÑÛÒÜÆÝÕÃØÙÚÖÐÞÓßÐÉÛÓ\nà\náâãäåæçèæäæäæâéäêæêëìíîïðæñòóôõôö\n÷\nøùùúûüúýþÿ\u0000\u0001ûþú\u0002û\u0003\u0004\u0003ü\u0005\u0006\u0003ÿ\u0007úüþùø\b\t\n\u0003\u000b\u0003\fø\u0002\u000búø\u0007\r\u0002ú\u000búÿü\u0007þù\u000e\u0006\u000f\bû\r\u0002\u0004ú\u0007\rÿýú\u0002\u0010\u0000\u0001øÿý\u0000\u0001\u000bþýú\u0007\u0011\u0012\u0013\u0012\u0014\u0015\u0013\u0013\u0016\u0012\u0017\u0018\u0019\u001a\u0015\u001a\u001b\u001c\u001d\u001e\u001f\u0018\u0013\u0011\u0015\u0016\u001a \u0016!\u001f\u0016\u0012\"\u0017 #\u001b\u001d\u001d$\u0012%&'()*\u0012\u0016! \u001f\u0016\u0012\"\u0017+%\u001a\n,\n-./0123456784029:552;<:5=>?@40:5=A:.BCDEFGHIDJKLIMDNOKJPGKJQERESCJCRTMKITCJNUVPCCGGCDJLGJ\nW\nXYZ[\\]X^_]`a`bcdYefghi]j_a^kd_^klXmY`Ygein\nopq\nrst\nu\nvw x y z {\n|\n}\n~\n\n\n\n       \n\n\n\n¡¢££¤¥¦\n§\n¨© ª « ¬  ® ¯\n°±²³ ´ µ ¶ · ¸ ¹ º ´ ¸ ¹ » ¶ ¼ · ½ ¾ ¿ µ À\nÁÂÃÄÅÆ\nÇ\nÈ\nÉÊËÌ\nÍ\nÎÏÐÑ\nÒÓÔÕÖÖ\n×ØÙÚÛØÜ\nÝÞßàáâã\näåæçååèè\néêëìíéîê\nïðñòóïïô\nõöö÷÷÷\nø ù ú û ü ý þ ÿ ù ý þ \u0000 û \u0001 ü \u0002 \u0003 \u0004 ú \u0005\n\u0006\u0007\b\t\n\u000b\n\f\r\u000e\n\u000f\u0010\u0011\u0012\u0013\u0014\u0015\u0016\u0017\u0018\u0019\u001a\u001b\u001c\u0019\u001d\u001b\u001e\u001f\u001f !\"#$%&\u0000\u0001 \u0002 \u0000 \u0003 \u0002 \u0003 \u0001 \u0002\n\u0004 \u0005 \u0006\u0007 \b \t\n \u000b \f\n\r \u000e \u000f \r \u0010 \u000f \u0010 \u000e \u000f\n\u0011\u0012\u0013\u0014\n\u0015\u0016 \u0017 \u0015 \u0018 \u0017 \u0017 \u0018 \u0017 \u0016 \u0017\n\u0019\u001a \u001b\u001c\u001d \u001e\n\u001f\n !\" #\n$ % & $' & & ' & % &\n( ) *+ , -\n.\n/ 01 2\n3 4 5 6 7\n8 9 :; < => ? @A B CD E F\nG H I J K\nL M NO P QR S T\nUV WX\nY\nZ[\\\n]\n^_` a b cd e\nf g h i j k l mn o p q r s tu v v w x y z { |} ~ ~     \n  \n\n\n\n    \n \n   ¡¢£¡¢¤¥¦§¨©ª«¬ ® ¯\n° ± ² ³ ´ µ\n¶ ·\n¸ ¹º»¼½¾¿ÀÁ»Á¿Ã»\nÄ Å\nÆ Ç\nÈ ÉÊËÌÍÎÏÐÑÒËÑÏÓË\nÔ Õ Ö× ØÙÚÛÜÝÞßàÚÞßáÜâÝãäåÛæç è é ê ë ì\níî ï ð ñ ñ ò\nóô õ ö ÷ ÷ ø\nù ú û ü ýþ ÿ \u0000 \u0001 \u0002\n\u0003 \u0004 \u0005 \u0006 \u0007 \b \t \n \b \t \u000b\f \r \u000e \u000f \u0010\u0011 \u0012 \u0013 \u0014\u0015\n\u0016\u0017\u0018\u0019\u001a\u0016\u0017\u001b\u0019\u001c \u001d \u001e \u001f  ! \"\n#$ %& ' ( ) * * + & , - . / & , 0 1 1 2 - 3 * 4 5 6 - 7 - . / * 4 4 5 6 8 7 9 ' 3 : ' . 3 ; <=; / ' - 3 ; 9 - 0 3 ' . ( > 3 ' = ; * : <? @ A A B C @ AD E F G H IJ K D L J L L J M @ E F J N O P Q O R L F D O S T U E V B ? E B V D T L N C W C V E F G X Y U B H I J K D L J LZ [ \\ ] ^ _ ` a b c d e _ f g h i Zj e _ [ d g h h i Zd e k c ]c l a f f c d m n o \\ k c ] ` _ k k a f p f k\\ f a ^ c j \\ [ q f _ [ f qrst\nu v w xy\nz { | } ~        ~    \n      \n                   ¡ ¢   ¡  £   ¢ ¤  £ ¡       ¢ ¥\n¦ § ¨\n© ª\n« ¬ ® ¯\n° ±² ³ ´ µ ¶ · µ ¶ ¸¹ º » ¼ ½¾ ¿ À Á ÂÃ Ä Å ÆÇÈ É Ê Ë Ì\nÍÎÏÐÑÍÎÒÐÓ Ô ÕÖ × Ø Ù\nÚÛ Ü\nÝÞ ß à á â\u0000 \u0001 \u0002 \u0000 \u0002 \u0001 \u0003 \u0000 \u0003 \u0001 \u0004 \u0000\n\u0005 \u0006\u0007 \b\t \n\u000b \f\r \u000e\n\u000f\u0010 \u0011 \u0011 \u0010 \u0011 \u0012 \u0011 \u0013 \u0011 \u0014 \u0011 \u0015 \u0011 \u0016 \u0011 \u0017 \u0011\n\u0018\u0019\n\u001a \u001b\u001c \u001d\u001e \u001f !\n\" # $ % & ' () *+ , - . / 0 10 2 3 - .0 10 4 5 2 5 & 6 7 ) 2 + 2 4 8 9 6 6 7 ): + ; 2 3 . 0 < 0 2 5 0 8 ; # 5 = 8 # > > 0 . 0 4 5? @ A B C D E A A F @ G H I J K A ? L I M L NF L D @ K A D J NF B A O\nP Q R S T U VW X Y Z[ \\\n] ^ _ ` ab c de f gh i jk l mn o pq r st u vw x yz { { | } ~       \n     \n    \n    \n¡¢£¤¥¦§¨ © ª « ¬ \n® ¯ ° ± ² ³´ µ ¶ · ¸ ¹º » ¼ ½ ¾ ¿À Á Â Ã Ä ÅÆ Ç Ç È É Ê Ë"    },    {        "title": "1603.00072v1.Gyrotropic_skyrmion_modes_in_ultrathin_magnetic_circular_dots.pdf",        "content": "arXiv:1603.00072v1  [nlin.PS]  29 Feb 2016Gyrotropic skyrmion modes in ultrathin magnetic circular d ots\nKonstantin Y. Guslienko,1,2Zukhra V. Gareeva,3\n1Depto. Fisica de Materiales, Facultad de Quimica,\nUniversidad del Pais Vasco, UPV/EHU, 20018 San Sebastian, S pain\n2IKERBASQUE, the Basque Foundation for Science, 48013 Bilba o, Spain and\n3Institute of Molecule and Crystal Physics, Russian Academy of Sciences, 450075 Ufa, Russia\n(Dated: September 25, 2018)\nWe calculate low-frequency gyrotropic spin excitation mod es of the skyrmion ground state ul-\ntrathin cylindrical magnetic dots. The skyrmion is assumed to be stabilized at room temperature\nand zero external magnetic field due to an interplay of the iso tropic and Dzyaloshinskii-Moriya ex-\nchange interactions, perpendicular magnetic anisotropy a nd magnetostatic interaction. We consider\nBloch- and Neel-type magnetic skyrmions and assume that the dot magnetization does not depend\non the thickness coordinate. The skyrmion gyrotropic frequ encies are calculated in GHz range as a\nfunction of the skyrmion equilibrium radius, dot radius and the dot magnetic parameters. Recent\nexperiments on magnetic skyrmion gyrotropic dynamics in na nodots are discussed.\nPACS numbers: 75.75.+a,75.60.Jk,75.30.Gw\nI. INTRODUCTION\nMagnetic skyrmions received recently considerable at-\ntention due to their unusual physical properties and\npromising applications in nanoelectronics, information\ndata storage, spintronics etc.1,2. Nanoscale dimensions,\nconsiderable stability, high mobility, several controllable\nparameters (polarity, chirality, topological charge) make\nthem attractive for the implementations in information\nstorage and processing devices on nanoscale.\nBeing a kind of magnetic topological solitons3in 2D\nspin systems, skyrmionsexhibit a wide varietyof unusual\nproperties that are related to their non-trivial topology.\nThe first theoretical estimation of the stability of lo-\ncalized topological solitons in infinite ferromagnets has\nbeen presented by Dzyaloshinskii et al.4Then, it was\nnoticed5that the Dzyaloshinskii- Moriyaexchangeinter-\naction (DMI) stabilizes 2D magnetic vortices (skyrmions\nin modern terminology) in magnetic systems whose sym-\nmetry group lacks of the space inversion symmetry oper-\nation. Twentyyearslater2Dhexagonalskyrmionlattices\nwereexperimentally detected in magnetic films with bulk\n(cubic B20 compounds, like MnSi, FeGe)6–8, and inter-\nfacial types9,10of DMI. It is accepted that the bulk DMI\nstabilizes Bloch skyrmions in B20 magnets6–8, whereas\nthe interfacialDMIleadstoformationofNeel skyrmions9\nin ultrathin multilayer films. The B20 skyrmions are sta-\nble predominantly at low temperatures and finite mag-\nnetic fields6–9. However, metastable skyrmion lattices\ncan also exist at zero magnetic fields in thin epitaxial\nFeGe films8and monolayer Fe films on Ir(111) surface10.\nJust few years ago simulations2and\nexperiments11–14showed that skyrmions can be stabi-\nlized at room temperatures (RT) in ultrathin multilayer\nstructures deposited by sputtering (Co/Pt11–14, and\nIr/Co/Pt12,13), including magnetic dots. Moreau-\nLuchaire et al.12,13reported the observation of RT\nskyrmions in multilayer films Pt/Co/Ir composed\nof heavy metal and ferromagnetic layers where thesingle skyrmions are stabilized by chiral DMI. Very\nrecently, RT skyrmions were observed in bulk CoZnMn\nalloys15, in Fe/Ni/Cu/Ni/Cu multilayer films16, and\nin tri-layer stripes Ta/CoFeB/TaO17. In the most\ncases, the existence of magnetic skyrmions is related to\nDMI. However, the alternative approaches referring to\nartificial RT skyrmion crystals18–22, skyrmions stabilized\nby perpendicular magnetic anisotropy23without DMI,\ndynamically stabilized skyrmions24, etc., have been also\ndeveloped.\nThe complex structure of magnetic skyrmions repre-\nsenting particle-like nanosize objects containing thou-\nsands of spins results in their reach dynamics. A num-\nber of associated extraordinary findings including emer-\ngent electromagnetic fields, topological Hall effect, ul-\ntralow densities of spin currents driving skyrmion mo-\ntion, skyrmion breathing and rotation excitation modes\netc. were reported in recent years25–32.\nLow and high frequency spin dynamics over a\nskyrmion background being of particular interest has\nbeen actively explored in 2D skyrmion lattices26–31.\nMochizuki26showed the existence of two in-plane ro-\ntational eigenmodes (clockwise and anticlockwise) of\nskyrmion spin texture unit cell and one breathing mode\n(oscillations of the skyrmion radius) in the GHz fre-\nquency range. Later on, the experiments detecting in-\nternal skyrmion modes have been carried out27–31. How-\never, the microwavemeasurements27,28confirmed the ex-\nistence of only one of the skyrmion rotating modes and a\nbreathing mode. Contrary, two rotating spin eigenmodes\nand one breathing mode were detected by time-resolved\nmagneto-optics in Cu 2OSeO331. Very recent broadband\nferromagnetic resonance measurements of MnSi, FeCoSi\nand Cu 2OSeO330showed existence ofonly two excitation\nmodes in the spectra of these skyrmion crystals. There-\nfore, there is no clear understanding of the B20 skyrmion\nspin excitation modes to the moment.\nTo understand the complicated skyrmion spin exci-\ntation spectra researches appeal to simplified systems:\nisolated skyrmions and magnetic bubbles in ferromag-2\nnetic films and dots. Additional reason for that are\nperspective applications of the RT isolated skyrmions in\nnanoscale devices33. Isolated skyrmions, conditions re-\nquired for their existence in patterned films11–14,23,34,35,\nand the dynamics of isolated skyrmions in magnetic\nfilms and nanodots have been simulated29,33,36,37and\nexperimentally explored38. Lin et al.29simulated\ndynamics of isolated skyrmion in an infinite film and\nfound one gyrotropic mode, several breathing and\nnon-symmetric skyrmion shape modes in the area of\nthe skyrmion stability. The skyrmion gyrotropic mode\nexcited by the spin polarized current in a nanopillar was\nsimulated in Ref.33. Simulations of the skyrmion breath-\ning modes in cylindrical dots were presented in Ref.36.\nSpin excitation spectrum in presence of the magnetic\nvortex (half-skyrmion) ground state was investigated\nin detail in Ref.39. It includes a low-frequency gy-\nrotropic mode (sub-GHz range) and high-frequency\nradially/azimuthally symmetric spin waves classified\nby integer indices - number of nodes of the dynamical\nmagnetization in the radial/azimuthal directions39.\nSimilar classification of the spin eigenmodes can be\napplied to isolated skyrmions in magnetic dots and\ninfinite films. The spin waves on the Bloch skyrmion\nbackground were recently calculated in Ref.37within a\nsimple model. In general, the skyrmion magnetization\noscillations can be represented as a superposition of\nspin eigenmodes - spin waves and gyrotropic modes.\nGyrotropic modes correspond to translation motion of\nthe skyrmion center around its equilibrium position.\nThere are also standing and travelling spin waves\nwith quantized eigenfrequencies in a confined system.\nThe frequencies of excited spin eigenmodes depend\non a variety of factors (system geometry (e.g., nan-\nodot diameter), magnetic anisotropy, exchange etc.).\nThe experimental investigations of spin dynamics of\nthe RT isolated skyrmions are in a beginning stage.\nSometimes it appears to be difficult to recognize exper-\nimentally the type and origin of a particular skyrmion\nexcitation mode and a theoretical input is crucial. Two\nspin excitation modes, presumably gyrotropic ones,\nof magnetic bubble skyrmions in CoB/Pt multilayer\ndots with CoB layer thickness of 0 .4 nm were recently\nmeasured by Buettner et al.38. They detected clockwise\n(CW) and counterclockwise(CCW) rotating spin modes,\nand an unusual pentagon like trajectory of the bubble\nskyrmion core.\nIn this article we focus on skyrmion gyrotropic\nexcitation modes considering the skyrmion as a ground\nstate of thin cylindrical magnetic dot. We calcu-\nlate the mode eigenfrequencies within rigid skyrmion\nmodel accounting the exchange, DMI, magnetostatic\ninteraction and uniaxial magnetic anisotropy. We\nshow that for the given dot parameters there exists\njust one gyrotropic mode, which can be CCW or CW\ndepending on the skyrmion core magnetization direction.II. THEORY\nFor consideration of magnetic skyrmion dynamics in\nthe dots we use theoretical approach based on the\nLandau-Lifshitz (LL) equation of magnetization ( M)\nmotion, ˙M=−γM×Heff, whereHeff=−δw/δM,\nγis the gyromagnetic ratio and wis the magnetic energy\ndensity\nw=A(∂µmα)2+wD−κm2\nz−1\n2Msm·Hm(1)\nwhereAis the isotropic exchange stiffness constant pro-\nportional to the Heisenberg exchange integral, K >0\nis the constant of uniaxial magnetic anisotropy, α,µ=\nx,y,z,m=M/Msis the unit magnetization vector, Ms\nis the saturation magnetization, and Hmis the magne-\ntostatic field. The DMI term in Eq. (1) can be repre-\nsented in two forms: 1. wD=D(m·rotm)1,9,25–31and\n2.w∗\nD=D[mz(∇·m)−(m·∇)mz], see Refs.2,11–14,16,38\nand references therein. Here Dis the constant of the\nDzyaloshinskii Moriya exchange interaction. The first\nform is used for bulk DMI in the B20 cubic compounds\n(MnSi, FeGe, etc.), and the second is used for interfa-\ncial induced DMI in the case of an interface of ultra-\nthin metallic ferromagnetwith a nonmagneticmetal hav-\ning a strong spin orbit coupling2, i.e., Co/Pt, Co/Pd.\nThe DMI term w∗\nDfavors to stabilization of so-called\nφ−skyrmions, where the skyrmion core and peripheries\nare separated by a Bloch domain wall (Fig. 1a). The in-\nterface DMI term leads to stabilization of ρ−skyrmions,\nwheretheskyrmioncoreandperipheriesareseparatedby\na Neel domain wall (Fig. 1b). The DMI is not of prin-\ncipal importance for φ−skyrmions in dots, which can be\nstable even at D= 023,38. However, the interface DMI\nessentially contributes to the ρ−skyrmion stabilization\nand dynamics, the skyrmions are not stable if the DMI\nstrength is lower than some critical value2,34.\nIt is convenient to rewrite the LL equation of motion\nof the dot magnetization M(ρ,t) =M(ρ,X(t)) in the\nform of Thiele equation of motion for the skyrmion core\nposition X= (X,Y) in the dot via complex variables\ns=sx+isy,s=X/R:\niGz˙s=2\nR2∂W\n∂¯s, (2)\nwhere∂/∂¯s= (∂/∂sx+i∂/∂sy)/2,Gz=−p|G|are the\nz-projection and absolute value of the gyrovector, Lis\nthe dot thickness, Ris the dot radius, W=L/integraltext\nd2ρwis\nthe total dot magnetic energy, p=±1 is the skyrmion\ncore polarization. The skyrmion magnetic energy can be\ndecomposed in series on small parameter |s|<<1 as\nW(s) =W(0) +κ|s|2/2, where κis the stiffness coef-\nficient. Let us use the cylindrical coordinates ( ρ,φ) to\ndescribe the in-plane radius vector ρ(x,y). We assume\nthat the dot is thin enough, so there is no dependence\nof the skyrmion magnetization on the thickness z- coor-\ndinate and the φ−andρ−skyrmion ground states are\nclearly defined. The problem is reduced to calculation3\na\nb\nFIG. 1. Magnetic skyrmion textures: a) Bloch ( φ) skyrmion\n(magnetization rotates in the plane perpendicular to the ra -\ndial direction), b) Neel ( ρ) skyrmion (magnetization rotates\nalong the in-plane radial direction).\nof the skyrmion magnetic energy W(s) as a function of\nthe skyrmion displacement s. We use complex variables\nz= (x+iy)/Rand an analytic function f(z) to de-\nscribe the skyrmion dynamics in 2D ferromagnet. The\nlinearized system of Eqs. (2) has the same form for the\nφ−skyrmion and ρ−skyrmion backgrounds if the mag-\nnetostatic energy related to the volume and side surface\ncharges is neglected. The magnetization components can\nbe expressed as\nmx+imy=2f(z)\n1+|f(z)|2\nmz=1−|f(z)|2\n1+|f(z)|2(3)\nThe isotropic exchange and DMI energy densities are\nwritten in the form\nwex=1\nR28A\n(1+|f(z)|2)2|∂f\n∂z|2\n¯wD=1\nR4D\n(1+|f(z)|2)2∂f\n∂z(4)\nThen, the interface (bulk) DMI densities are equal to\nw∗\nD=Re(¯wD) orwD=Im(¯wD), respectively. The typi-\ncal DMI parameter Dis about of 1 mJ/m2. Accounting\nultrathin dot thickness about 1 nm(L << R) we apply\nthe rigid skyrmion approximation f(z) =eiΦ0(z−s)/c,\nwhich is asymptotically exact within the limit of infi-\nnite film with dominating exchange interaction, L/R→\n0, and corresponds to the skyrmion equilibrium profilecosΘ0= (R2\nc−ρ2)/(R2\nc+ρ2). Here,c=Rc/Ris the re-\nduced skyrmion radius, and the skyrmion phase is Φ 0=\nCπ/2 (C=±1 is the skyrmion chirality) for φ-skyrmion\nor Φ0= 0,πforρ-skyrmion. To unify description of\nφ- andρ-skyrmions we define the generalized skyrmion\nchirality Cas follows. C=sign(mφ) = sin(Φ 0) forφ-\nskyrmions and C=sign(mρ) = cos(Φ 0) forρ-skyrmions.\nThe static DMI energy density is proportional to the\nproductDC, and is minimal at DC=−|D|<0 defining\nthe chiral skyrmion ground state. Nonzero DMI strength\nlifts the energy degeneracy with respect to the skyrmion\nchirality C. It is evidentthat in the exchangeapproxima-\ntion (4) the DMI energy just renormalizes the isotropic\nexchange energy, A→A′=A+DCRc/2.\nThe exchange contribution to the stiffness coefficient\nis\nκex(c) =−32πALc2\n(1+c2)3(5)\nThe anisotropy energy wa=−Km2\nzin Eq. (1) contribu-\ntion to the stiffness is calculated to be\nκa(c) =−8πKR2Lc2(1−c2)\n(1+c2)3(6)\nTo calculate the skyrmion magnetostatic energy wm\nwe distinguish the energy of the bulk (non–zero for\nρ−skyrmions), side surface and face dot surface mag-\nnetic charges. The energy cannot be simply expressed\nvia the analytical function f(z), therefore, we used a di-\nrect calculation of wmvia the magnetization bulk divm\nand surface divergence ( m·n), where the vector nis\nnormal to the dot surface ( n=ˆz,ˆρfor the face and side\nsurfacecharges,correspondingly). Themagnetostaticen-\nergy of the face and side surface charges of the displaced\nskyrmion can be calculated by the equation\nWm(s) =1\n2M2\ns/integraldisplay\ndS/integraldisplay\ndS′mn(r,s)mn(r′,s)\n|r−r′|(7)\nwheremn= (m·n) is the surface divergence. The side\nsurface charges energy is proportional to the function\nF(β) =/integraltext∞\n0dkf(βk)J2\n1(k)/k, whereJ1(x) is the first or-\nder Bessel function, f(x) = 1−(1−exp(−x))/x, and\nβ=L/R. Detailed calculations of the skyrmion magne-\ntostatic energywill be published elsewhere. Here we note\nthat the bulk and side surface charges energies are pro-\nportional to the small dot aspect ratio L/Rand can be\nneglected for ultrathin dots. Whereas, the face charges\nenergy main term renormalizes the uniaxial anisotropy\nconstant, K→K−2πM2\ns. We can write the total stiff-\nness coefficient for ultrathin dot as\nκ(c) = 16πM2\nsR2Lc2\n(1+c2)3[π(1−Q)(1−c2)−\n(1\nr)2(1+1\n2rdc)] (8)\nwhereQ=K/2πM2\nsis the dot magnetic material quality\nfactor,Le=√\n2A/Msis the exchangelength, r=R/Leis4\nthe reduced dot radius and dimensionless DMI parame-\nterd=DCLe/Aforφ- andρ-skyrmions. The gyrovector\nGz=−p|G|of the centered skyrmion ( s= 0) is propor-\ntional to the skyrmion topological charge23and can be\ncalculated using the definition\nGz=MsL\nγ/integraldisplay\nd2ρm[∂xm×∂ym], (9)\nor via the complex variables as\n|G|=MsL\nγ/integraldisplay\nd2ρ4\n(1+|f|2)2|∂f\n∂z|2\n|G(c)|=MsL\nγ4π\n1+c2. (10)\nThe skyrmion gyrotropic frequency calculated from the\nThiele equation of motion (2) using eqs. (9)-(10) is\nω(c) =κ(c)/|G(c)|R2, or explicitly is given by\nωG=ωMc2\n1+c2[(1−Q)(1−c2)−1\nπ1\nr2(1+1\n2rdc)],(11)\nwhereωM= 4γπMs, and the reduced equilibrium\nskyrmion radius c=Rc/Ris a function of the dot sizes\nand magnetic parameters A,D,K,M s.\nIII. RESULTS AND DISCUSSION\nWe calculated a low-frequency skyrmion dynamics of\nρ−andφ−skyrmions differing by the type of a domain\nwall(Bloch and Neel) in the skyrmionspin configuration.\nAn ultrathin circular ferromagnetic dot with radius R\nabout of 100 nmand thickness Labout of 1 nanometer\nwas considered. For such small dot thickness the magne-\ntostatic energy is reduced to the magnetic energy of the\nface charges, which can be accounted in the simplified\nform of an effective easy-plane anisotropy.\nIn the main approximation the skyrmion eigenmodes\ncan be conventionally divided into internal (low fre-\nquency) and external (high frequency) modes. The in-\nternal modes related to weak skyrmion deformations are\nlocalized close to the skyrmion center and include trans-\nlation (gyrotropic) and breathing modes. The highfre-\nquency spin wave modes are delocalized and occupy the\nwhole dot volume. We calculate the skyrmion spin ex-\ncitation modes that are closely related to the skyrmion\ntopological charge, i.e., translation or gyrotropic modes.\nThese collective spin modes correspond to the skyrmion\nrotation around an equilibrium position corresponding\nto minimum of the total magnetic energy. Based on\nthe Thiele collective coordinate approachwe describe the\nmotion of skyrmion center position in a magnetic dot\nwithin rigid skyrmion model accounting the exchange,\nDMI, magnetostatic interactions and uniaxial magnetic\nanisotropy, calculate the frequency of circular skyrmion\nrotation(gyrotropicfrequency)andcompareresultswith\nrecentexperimentaldataandmicromagneticsimulations.The skyrmion gyrotropic frequency of ultrathin cylin-\ndrical dots given by Eq. (11) is represented via the\nskyrmionradius c. Theequilibriumvalueoftheskyrmion\nradiusccan be found using minimization of the total\nmagnetic energy given in Ref.23within the ultrathin dot\nlimitL/R→0 adding the DMI term by the substitu-\ntion of the exchange stiffness A→A′=A+DCRc/2.\nThe gyrotropic frequency (11) is proportional to the in-\nverse in-plane magnetic susceptibility similarly to the\nvortex gyrotropic frequency39and is positive in the sta-\nble skyrmion state. Otherwise, the skyrmion will escape\nfrom the dot lowering its energy. The line ωG(c) = 0\nmarks the border of the skyrmion stability within the\nmodel. Accounting the reduced skyrmion radius c <1\nwe note that if the DMI parameter dof any sign is\nsmall (|d|< dc= 2Le/Rc) the skyrmion in an ultra-\nthin dot is stabilized by the face magnetostatic inter-\naction at moderate perpendicular anisotropy Q <1.\nThis is typical for φ−skyrmions23,38, but is valid also for\nρ−skyrmionsin ultrathin dots. If the parameter dis neg-\native (d=−|d|<0 corresponds to the skyrmion ground\nstate) and its magnitude is large (above the critical value\n|d|> dc), then the ρ−skyrmions might be stabilized by\nthe DMI in ultrathin dots having a large perpendicular\nanisotropy Q >12,11–14. The gyrotropic frequency ωG\nfor such ρ−skyrmions decreases approximately as 1 /R\nat the dot radius Rincreasing. We note that the sim-\nilar dependence ωG(R)≈1/Rwas calculated by Gus-\nlienko et al.39for the vortex gyrotropic frequency in thin\ncylindrical dots due to the dominating magnetostatic in-\nteraction. The φ−andρ−skyrmion states are stable in\ncircular magnetic dots within some range of the param-\neters according to Refs.2,30,32. Estimation of the critical\nvalue of the DMI parameter Dc= 2A/Rcusing typi-\ncal parameters A=10pJ/mandRc=10nmyieldsDc\n= 2mJ/m2in good agreement with experiments12–14.\nX-ray imaging of magnetic skyrmions in the ultrathin\nmultilayer circular dots Co/Pt11–14, Ir/Co/Pt12,13and\nCoB/Pt38showed that the skyrmion radius Rcis sev-\neral tens of nm, much smaller than the dot radius and\nthe typical ratio c=Rc/R= 0.1−0.2 is small. This is\nin some disagreement with micromagnetic simulations of\ntheρ−skyrmion stability in the ultrathin circular dots,\nwherethelargervaluesof c= 0.3−0.5stronglydependent\non the DMI strength, dot radius2, and dot thickness14\nhave been found.\nThepositiveskyrmioncorepolarization p=mz(0)cor-\nresponds to the counter clock-wise(CCW) skyrmioncore\ngyrotropic rotation and negative p=−1 corresponds\nto the gyrotropic mode rotating clock-wise (CW). I.e.,\nthere is always only one gyrotropic mode for the given\nskyrmionpolarization pandcorrespondingsignofthegy-\nrovector (see Theory). The second low-frequency mode\n(CW for p= +1) simulated by Mochizuki26,32cannot\nbe interpreted as a second gyrotropic eigenmode. In the\nsimulations26,32the sign of pwas determined by the sign\nofthe bias magnetic field perpendicular to the film plane.\nIt was noticed by Mochizuki et al.32that the CCW low-5\nfrequency mode has intensity much larger than intensity\nof the higher-frequency CW mode. I.e., the CCW gy-\nrotropic mode is a sole resonance mode for p= +1, or\neigenmode of the system.\nMore precisely, moving skyrmion cannot be consid-\nered as an absolutely rigid object, its dynamical pro-\nfile is deformed that can be represented as a hybridiza-\ntion with azimuthal spin wavesexcited overthe skyrmion\nbackground37and resulting in a finite skyrmion inertia\nterm in the Thiele equation of motion. The low fre-\nquency gyrotropic eigenmodes are closely related to non-\nzero skyrmion mass38,40. Recently two gyrotropic modes\nin the CoB/Pt circular dots modes were simulated and\nmeasured by X-ray imaging technique38. According to\nthe interpretation of Ref.38existence of two gyrotropic\nmodes (rotating in opposite directions) corresponds to a\nfinite skyrmion mass. However, the second higher fre-\nquency gyrotropic mode can be interpreted as azimuthal\nspin wave37,39,40, and there is only one gyrotropic mode\nin the dot spin excitation spectrum. Nevertheless, the\nskyrmions can have a considerable mass accounting for\ntheir magnetic energy change increasing skyrmion veloc-\nity. The skyrmionmassmight alsodepend on mechanism\ndriving skyrmion motions (external field gradient, tem-\nperature, current, etc.).\nWe apply Eq. (11) to estimate the gyrotropic fre-\nquency of the bubble φ−skyrmion measured in Ref.38.\nThe dot radius is R=275nm,Ms= 1190G,Q=0.866,\nandD= 0. Using the reduced value of A=15pJ/m\ntypical for ultrathin Co-films2,12,13,34we get with these\nparameters ωM/2π=43.5GHz(assuming typical value\nofγ/2π= 2.9MHz/Oe ) andLe=14.5nm. The total\nmagnetic material thickness is 12 nm (30 repeats of ul-\ntrathin CoB layer), and the magnetostatic energy in Eq.\n(11) should be accounted more precisely. This energy for\ndisplaced φ−skyrmion appears due to side surface mag-\nnetic charges and can be accounted by adding the term\nF(β)(1+c2) (see definition of the function F(β) in Sec.\n2) to the square bracket in Eq. (11). The gyrotropic fre-\nquency calculated by corrected Eq. (11) is equal to 1.11\nGHz, whereas the value of 1.00 GHzwas detected for\nthe low frequency CCW mode in Ref.39. The agreement\nis reasonablegood accounting that the relaxation time in\nRef.39is comparable with the magnetization oscillation\nperiod. That did not allow determining the skyrmion\neigenfrequencies with good accuracy and therefore, the\ngyrotropic frequencies and skyrmion mass obtained in\nRef.39are just semi-quantitative estimations.\nThe gyrotropic frequency of the skyrmion state ul-\ntrathin dots given by Eq. (11) is plotted in Figure\n2. It increases/decreases with dot radius increasing for\nd >0/d <0 (Fig. 2a). However, the frequency at d >0\ndecreasesat Rincreasingifthesidesurfacemagnetostatic\ninteraction for a moderate dot thickness is accounted.\nThe eigenfrequency of the gyrotropic mode (11) is split-\nted due to DMI in the chiral φ−andρ−skyrmion state\nmagnetic dots. There are two distinct gyrotropic fre-\nquencies for the skyrmions due to possible different signs5 10 15 20 2520030040050060070080090010001100\nGyrotropic frequency/c119G, MHz\nReduced dot radius, r12a\n0 2 4 6 8 10400500600700800900100011001200\nGyrotropic frequency/c119G, MHz\nReduced strength of DMI, IdI12b\nFIG. 2. The gyrotropic frequencies of ultrathin skyrmion\nground state circular dot calculated by Eq. (11): (a) the fre -\nquencies vs. the reduced dot radius r=R/Le,|d|= 2.9 (or\n|D|= 3mJ/m2using the magnetic parameters from Ref.38);\n(b) the frequencies vs. the reduced DMI strength |d|=\n|D|Le/Aat the reduced dot radius r= 19 (or R= 275nm\nas in Ref.38). The solid green (1) and red (2) lines correspond\ntod >0 andd <0 (d=−|d|<0 for the skyrmion dot ground\nstate). The parameters Ms= 1190G,Q=0.866 were taken\nfrom Ref.38. The exchange length is Le=14.5nm.\nof the skyrmion chirality Cof the dots in an array. The\nmρmagnetization component plays a role of the chiral-\nity forρ−skyrmions. The frequency splitting between\nthe corresponding gyromodes is determined by the DMI\nstrength ∆ ωG(c) =ωM(L2\ne|d|/R)c3/(1+c2)2and might\nbe of several hundred MHz(see Fig. 2b). However, the\nDMI energy for the skyrmion state dots is lower for such\nsign of chirality that the parameter d=−|d|<0 and\ncorresponding gyrotropic frequency is higher (the curve\n2 in Fig. 2a,b). The gyrofrequency (11) is determined\nby the exchange stiffness A, intrinsic DMI parameter D,\nthe perpendicular anisotropy constant K, as well as the\ndot saturation magnetization and radius R. The recent6\nsimulations by Zhang et al.33of the gyrotropic dynam-\nics of the skyrmion in a free layer of circular nanopillar\nare in qualitative agreement with Eq. (11). There is\none gyrotropicfrequency ωG(R), which rapidly decreases\nfrom 1.4 GHzto zero with the dot radius Rincreas-\ning from 20 to 70 nmfor Co dot with the thickness 0.6\nnm(the magnetic parameters are the same as in Ref.2).\nEq. (11) for the gyrotropic frequency is applicable as\nwellto2DskyrmiontriangularlatticesinB20compounds\nin the case of in-phase motions of the skyrmions in the\nlattice unit cells (the dot diameter 2 Rshould be substi-\ntuted to the skyrmion lattice period in this case). The\neigenfrequency is about 1 GHzfor typical B20 skyrmion\nlattice parameters.\nWe note that the skyrmion gyrotropic frequency in\nmagnetic dots has not been observed yet. The reason\nfor that is the stable skyrmions in ultrathin ferromag-\nnetic dots were obtained very recently and there was no\nchance to detect the skyrmion excitations in such mag-\nnetic nanostructures. Another difficulty we foresee is\nlarge value of the magnetization damping (the resonance\nlinewidth) in ultrathin Co films that does not allow ap-\nplyingaprecisebroadbandferromagneticresonancetech-\nnique for the gyrotropic frequencies detection in φ−and\nρ−skyrmion state magnetic dots. The ultrathin Fe/Ni\nfilms16with expected low resonance linewidth are more\npromising to detect the skyrmion dynamics. This is achallenge for the future experiments.\nIV. CONCLUSION\nWe calculatedlow-frequencygyrotropicspin excitation\nmodes of the skyrmion ground state cylindrical magnetic\ndots. The skyrmion was assumed to be stabilized at\nroom temperature and zero external magnetic field\ndue to interplay of the isotropic and Dzyaloshinskii-\nMoriya exchange interactions, perpendicular magnetic\nanisotropy and magnetostatic interaction. We consid-\nered the Bloch- and Neel-type magnetic skyrmions and\nassumed that the magnetic dot is thin enough (magne-\ntization does not depend on the thickness coordinate).\nThe skyrmion gyrotropic eigenfrequencies are calculated\nwithin the rigid skyrmion model as a function of the\nskyrmion equilibrium radius, dot radius and the dot\nmagnetic parameters. Now the challenge is to detect\nexperimentally the predicted gyrotropic eigenfrequncies.\nACKNOWLEDGMENTS\nK.G. acknowledges support by IKERBASQUE (the\nBasque Foundation for Science) and the Spanish\nMINECO grant MAT2013-47078-C2-1-P. Z.G. acknowl-\nedges support by the Russian Foundation for Basic Re-\nsearch (Grant No. 16-02-00336 A).\n1A. Fert, V. Cros, and J. Sampaio, Nature nanotechnology\n8, 152 (2013).\n2J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert,\nNature nanotechnology 8, 839 (2013).\n3A. M. Kosevich, B. Ivanov, and A. Kovalev, Physics Re-\nports194, 117 (1990).\n4I. Dzyloshinskii and B. Ivanov, JETP LETTERS 29, 540\n(1979).\n5A. Bogdanov and D. Yablonskii, Zh. Eksp. Teor. Fiz 95,\n182 (1989).\n6S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch,\nA. Neubauer, R. Georgii, and P. B¨ oni, Science 323, 915\n(2009).\n7X.Yu, Y.Onose, N.Kanazawa, J. Park, J. Han, Y.Matsui,\nN. Nagaosa, and Y. Tokura, Nature 465, 901 (2010).\n8S.HuangandC.Chien,Physicalreviewletters 108,267201\n(2012).\n9A. Sonntag, J. Hermenau, S. Krause, and R. Wiesendan-\nger, Physical review letters 113, 077202 (2014).\n10S. Heinze, K. Von Bergmann, M. Menzel, J. Brede, A. Ku-\nbetzka, R. Wiesendanger, G. Bihlmayer, and S. Bl¨ ugel,\nNature Physics 7, 713 (2011).\n11S. Woo, K. Litzius, B. Kr¨ uger, M.-Y. Im, L. Caretta,\nK. Richter, M. Mann, A. Krone, R. Reeve, M. Weigand,\net al., arXiv preprint arXiv:1502.07376 (2015).12C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sampaio,\nN. VanHorne, C. Vaz, K. Bouzehouane, K.Garcia, C. Der-\nanlot, P. Warnicke, et al., arXiv preprint arXiv:1502.07853\n(2015).\n13C. Moreau-Luchaire, C. Moutafis, N. Reyren, J. Sampaio,\nC. Vaz, N. Van Horne, K. Bouzehouane, K. Garcia, C. De-\nranlot, P. Warnicke, et al., Nature Nanotechnology (Jan.\n18, 2016).\n14O. Boulle, J. Vogel, H. Yang, S. Pizzini, D. d. S. Chaves,\nA. Locatelli, T. O. M. A. Sala, L. D. Buda-Prejbeanu,\nO. Klein, M. Belmeguenai, et al., Nature Nanotechnology\n(Jan. 25, 2016).\n15Y. Tokunaga, X. Yu, J. White, H. Rønnow, D. Morikawa,\nY. Taguchi, and Y. Tokura, Nature communications 6,\n7638 (2015).\n16G. Chen, A. Mascaraque, A. K. Schmid, et al., Applied\nPhysics Letters 106, 242404 (2015).\n17W. Jiang, P. Upadhyaya, W. Zhang, G. Yu, M. B.\nJungfleisch, F. Y. Fradin, J. E. Pearson, Y. Tserkovnyak,\nK. L. Wang, O. Heinonen, et al., Science 349, 283 (2015).\n18L. Sun, R. Cao, B. Miao, Z. Feng, B. You, D. Wu,\nW. Zhang, A. Hu, and H. Ding, Physical review letters\n110, 167201 (2013).\n19B.Miao, L.Sun, Y.Wu, X.Tao, X.Xiong, Y.Wen, R.Cao,\nP. Wang, D. Wu, Q. Zhan, et al., Physical Review B 90,\n174411 (2014).7\n20J. Li, A. Tan, K. Moon, A. Doran, M. Marcus, A. Young,\nE. Arenholz, S. Ma, R. Yang, C. Hwang, et al., Nature\ncommunications 5, 4704.\n21D. A. Gilbert, B. B. Maranville, A. L. Balk, B. J. Kirby,\nP. Fischer, D. T. Pierce, J. Unguris, J. A. Borchers, and\nK. Liu, Nature communications 6, 8462.\n22M. Sapozhnikov and O. Ermolaeva, Physical Review B 91,\n024418 (2015).\n23K. Y. Guslienko, IEEE Magnetics Letters 6, 4000104\n(2015).\n24Y. Zhou, E. Iacocca, A. Awad, R. Dumas, F. Zhang,\nH. Braun, and J. ˚Akerman, Nature communications 6,\n8193.\n25X. Yu, N. Kanazawa, W. Zhang, T. Nagai, T. Hara, K. Ki-\nmoto, Y. Matsui, Y. Onose, and Y. Tokura, Nature com-\nmunications 3, 988 (2012).\n26M. Mochizuki, Physical review letters 108, 017601 (2012).\n27Y. Onose, Y. Okamura, S. Seki, S. Ishiwata, and\nY. Tokura, Physical review letters 109, 037603 (2012).\n28Y. Okamura, F. Kagawa, M. Mochizuki, M. Kubota,\nS. Seki, S. Ishiwata, M. Kawasaki, Y. Onose, and\nY. Tokura, Nature communications 4, 2391.\n29S.-Z. Lin, C. D. Batista, and A. Saxena, Physical Review\nB89, 024415 (2014).30T. Schwarze, J. Waizner, M. Garst, A. Bauer,\nI. Stasinopoulos, H. Berger, C. Pfleiderer, and\nD. Grundler, Nature materials 14, 478 (2015).\n31N. Ogawa, S. Seki, and Y. Tokura, Scientific reports 5,\n9552.\n32M. Mochizuki and S. Seki, Journal of Physics: Condensed\nMatter27, 503001 (2015).\n33S. Zhang, J. Wang, Q. Zheng, Q. Zhu, X. Liu, S. Chen,\nC. Jin, Q. Liu, C. Jia, and D. Xue, NewJournal of Physics\n17, 023061 (2015).\n34S. Rohart and A. Thiaville, Physical Review B 88, 184422\n(2013).\n35M. Beg, R. Carey, W. Wang, D. Cort´ es-Ortu˜ no, M. Vous-\nden, M.-A. Bisotti, M. Albert, D. Chernyshenko, O. Hov-\norka, R. L. Stamps, et al., Scientific reports 5, 17137.\n36J.-V. Kim, F. Garcia-Sanchez, J. Sampaio, C. Moreau-\nLuchaire, V. Cros, and A. Fert, Physical Review B 90,\n064410 (2014).\n37Z. V. Gareeva and K. Y. Guslienko, physica status solidi\n(RRL)-Rapid Research Letters 10(Jan.5, 2016).\n38F. B¨ uttner, C. Moutafis, M. Schneider, B. Kr¨ uger,\nC. G¨ unther, J. Geilhufe, C. v. K. Schmising, J. Mohanty,\nB. Pfau, S. Schaffert, et al., Nature Physics 11, 225 (2015).\n39K. Y. Guslienko, Journal of nanoscience and nanotechnol-\nogy8, 2745 (2008).\n40K. Guslienko, G. Kakazei, J. Ding, X. Liu, and A. Adey-\neye, Scientific reports 5, 13881."    },    {        "title": "1901.07444v1.Ultrashort_magnetic_impulses_driven_by_coherent_control_with_vector_beams.pdf",        "content": "Ultrashort Magnetic Impulses Driven by Coherent Control with Vector Beams\nS. Sederberg1, F. Kong1, and P. B. Corkum1\n1Joint Attosecond Science Laboratory, University of Ottawa and National Research Council of Canada,\n100 Sussex Drive, Ottawa K1N 5A2, Ontario, Canada\n(Dated: November 20, 2021)\nWe introduce a new technique for the generation of magnetic impulses. This technique is based\non coherent control of electrical currents using cylindrical laser beams with azimuthal polarization.\nWhen used to ionize a medium, in this case atomic hydrogen is considered, an azimuthal current\nimpulse is driven. The spatial distribution of this current bears close resemblance to that of a\nsolenoid, and produces a magnetic field impulse. The excitation and relaxation dynamics of this\ncurrenttemporallyconfinetheresultingmagneticfieldtoaTesla-scale, terahertzbandwidthimpulse.\nImportantly, the magnetic fields are spatially isolated from electric fields. This all-optical approach\nwill enable ultrafast time-domain spectroscopy of magnetic phenomena.\nMagnetic fields play a central role in many areas of\nphysics, including magnetic materials and devices, su-\nperconductivity, electron spin manipulation, magnetic-\nfield-induced phase transitions, quantum and topological\nsystems, quantum critical points, plasma physics, and\nplasma confinement for nuclear fusion. The Biot-Savart\nlaw, formalized in 1820, provides a comprehensive clas-\nsical description of static magnetic fields generated by\ncharged particles in motion. Confining current to a care-\nfully chosen path, typically a variation of a solenoid, en-\nables strong, uniform magnetic fields to be generated in\nthe region enclosed by the circulating current. The mag-\nnetic field present at the center of an ideal solenoid is\ndetermined by the simple expression: B=\u0016nI, where\n\u0016is the permeability of the core medium, n=N=Lis\nthe number of wire turns per unit length, and Iis the\ncurrent flowing through the wire.\nElectrical conductors have been quintessential to\nsolenoids, and technical challenges associated with their\nresistance have hindered advancement of the amplitude\nand bandwidth of magnetic field sources. For example,\nthe latency associated with the electrical circuitry com-\nprising conventional electromagnets prevents temporal\nresolution of sub-picosecond magnetic-field-induced dy-\nnamics and as a result, the microscopic origins responsi-\nbleforthemagneticresponseofasamplemustbeinferred\nfrom static measurements.\nWires imposed comparable challenges on the study of\nelectric field phenomena in solid state systems. Phase-\nlocked terahertz electromagnetic pulses emerged as the\nfirst metrological tool capable of resolving electronic phe-\nnomena on timescales rendered prohibitive by electrical\ntransmissionlines[1–4]. Usingelectricfieldsderivedfrom\nshort laser pulses not only enabled picosecond and fem-\ntosecond temporal resolution, but also dramatically re-\nduced avalanche breakdown and thermal damage, en-\nabling new regimes of electric field strength to be ex-\nplored. Over the last two decades, our ability to inter-\nrogate matter using short electromagnetic pulses has ex-\npanded to the near-infrared and optical spectral regions,\nenabling electron dynamics to be probed on their nat-\nural, attosecond timescale [5–8]. Complementary mag-\nnetic field sources would permit insight into magneticfield phenomena on unprecedented timescales.\nRelaxingtherequirementforasolenoidfabricatedfrom\nwires, one can envisage a solenoid composed of a con-\ntinuous conductive sheet. In this case, the magnetic\nfield at the center of the solenoid would be expressed\nas:B=\u0016IL, whereILis the current per unit length cir-\nculating through the sheet. When a sufficiently intense\nlaser pulse is incident on a medium, such as a gas, it will\nionize the medium and drive the electron population in\nits optical fields.\nIn this Letter, we show that the excitation of a gas by\nintense laser pulses can produce a current distribution\nthat closely resembles that from a conventional solenoid.\nTwo cylindrical vector beams with azimuthal polariza-\ntion are used to ionize atomic hydrogen and drive an\nelectrical current circulating the longitudinal axis of the\nbeam, closely resembling current flowing in a solenoid.\nThe direction and amplitude of this current, and of the\ninduced magnetic field, can be precisely controlled by ad-\njusting the relative phase between the two light waves.\nThese calculations demonstrate a conceptually simple\ntechnique for exciting and probing samples with isolated\nmagnetic impulses of picosecond duration, providing a\nnew approach for studying magnetic phenomena on pi-\ncosecond and femtosecond timescales. We demonstrate\na simple approach to scale the magnetic field amplitude\nand investigate the influence of back electromotive force\nas magnetic fields on the Tesla scale are excited on a\nfemtosecond timescale. These proof-of-principle calcula-\ntions demonstrate the feasability of scaling the magnetic\namplitude to at least 0:42T, and approaches for further\nscaling are proposed.\nThe aforementioned currents have been routinely gen-\nerated in solids and gases through coherent control. In\ngeneral, coherent control is an optical technique whereby\na laser pulse with angular frequency, !;is applied to\na system simultaneously and collinearly with its second\nharmonic, 2![9–12]. The system can be driven into an\nexcited state by each of the two pulses independently.\nApplying the two pulses coherently and adjusting their\nrelativephaseenablescontrolofquantuminterferencebe-\ntween the two processes, which can be detected from any\nobservable related to the excited state. In atomic sys-arXiv:1901.07444v1  [physics.optics]  22 Jan 20192\nFIG. 1. (a) The azimuthal electric field component of the\nlaser beam. (b) A snapshot of the electric field from a cylin-\ndrical beam with azimuthal polarization propagating through\nthe beam waist. The laser fields produce an azimuthal cur-\nrent density bearing close resemblance to current flowing in\na solenoid, which is shown for a tranverse cross-section in (c)\nand a longitudinal cross-section in (d). (e) This current den-\nsity produces a uniform magnetic field that is enclosed by the\nsolenoidal current. (f) Magnetic vector field including both\nBzandBrcomponents.\ntems, the two laser pulses are commonly used to ionize\nan atom, and their relative phase controls the direction\nand energy of the emitted photoelectrons. In solid state\nsystems, the relative phase can be used to affect asym-\nmetry on the momentum distribution of a photoexcited\nconduction band population, driving a current through\nthe material [13, 14]. The ability to drive currents using\noptical fields, without the need for electrical conductors,\nholds potential technological and metrological applica-\ntions, and plays a central role in the work presented here.\nThe increasing availability of structured laser modes\nprovides the possibility to transfer intricate features from\nthe laser beam mode profile, polarization, and orbital\nangular momentum to the collective electron motion\nthrough the coherent control process. As a specific case,\nwe consider a cylindrical vector beam with azimuthal po-\nlarization, which is depicted in Fig. 1(a) [15]. We numer-\nically investigate strong field ionization in atomic hydro-\ngen,wherethetunnellingionizationratecanbeexpressed\nas:\nw(t) = 4!0 \nEa\nE(t)!\nexp\"\n\u00002\n3 \nEa\nE(t)!#\n;(1)\nwhere!0=me4=\u0016h3is the atomic unit of frequency,\nEa=m2e5=\u0016h4is the atomic unit of the electric field,\nandE(t)is the time-varying electric field incident on\nthe medium. Ionization and classical trajectories are cal-\nculated for a laser pulse at \u0015= 1800nm and its sec-ond harmonic at \u0015= 900nm with peak electric field\nstrengths of Ep;!= 3:0V/Å(Ip;!= 1:2\u00021014W/cm2)\nandEp;2!= 1:5V/Å(Ip;2!= 0:6\u00021014W/cm2), respec-\ntively, and identical pulse durations of \u001cp= 18fs.\nWe simulate this physical scenario using an approach\nthat is closely related to particle in cell calculations. We\npropagatethetwopulsesthroughtheRayleighlengthofa\nGaussian focusing geometry (beam waist, w0= 3\u0016m) us-\ningthree-dimensionalfinite-differencetime-domainsimu-\nlationsincylindricalcoordinates, asdepictedinFig. 1(b)\n[16]. We spatially discretize the simulation space with\n\u0001r= \u0001z= 40nm and \u0001\u0012=\u0019=150, and use a timestep\nof\u0001t= 76as. At each time-step, the ionization rate is\nevaluated at every mesh point in the simulation space,\nelectric fields are interpolated to the position of each ex-\nisting electron, and all existing electron trajectories are\nupdated. The existing trajectories are distributed into\ntheir nearest mesh points to produce an effective current\ndensity mapping, which is included in the next time-step\nof Maxwell’s equations for self-consistency. In particular,\nthis ensures that the back-action of the time-dependent\nmagnetic fields (i.e. back electromotive force) on the ex-\ncited currents is accounted for.\nIn the vicinity of the beam waist, the extremely non-\nlinear photoionization process spatially gates the ioniza-\ntion process affected by the azimuthal beams, confining\nthe excitation of an azimuthal current to a thin circu-\nlar loop in the transverse plane, as shown in Fig. 1(c).\nWe emphasize that this is an impulsively excited cur-\nrent, whereby electrons at each azimuthal position, \u000e\u0012,\nare briefly imparted with momentum in the azimuthal\ndirection, producing a pulsed collective current excita-\ntion that resembles a solenoidal current. This current\ninduces and encompasses a uniform magnetic field, and\na tranverse slice of the Bzfield is shown in Fig. 1(e).\nThe full-vectorial magnetic field distribution is depicted\nin Fig. 1(f), and demonstrates that the field is predomi-\nnantly longitudinal.\nThe temporal waveform of the electric field excitation\nat the beam waist is shown in Fig. 2(a). With the\nhighest ionization rate occurring at the peak of the laser\npulse, shown in Fig. 2(b), the current density dynamic\nis confined mainly to the latter half of the laser pulse,\nas shown in Fig. 2(c). A magnetic field is induced by\nthis current, and its dynamic build-up is presented in\nFig. 2(d).This magnetic field will exist for as long as the\nazimuthal current maintains coherence. After the laser\npulses have left the ionized atoms, a collective response\narising from the screening potential of each charge car-\nrier will develop over a timescale on the order of the in-\nverse of the plasma frequency [17]. For an ionization den-\nsity ofn= 1017cm\u00003, we calculate a plasma frequency,\n!p=q\nne2\nme\"0= 2:8THz, and a corresponding time con-\nstant on the order of \u001cr= 350fs. The excitation ( \u001cp)\nand relaxation ( \u001cr) temporally confine the current to a\nunipolartransient, whichradiatesasingle-cycleterahertz\npulsewithaspace-timestructurecloselyresemblingafly-3\nFIG. 2. Temporal dynamics. (a) An exemplary !=2!electric\nfield waveform at the beam waist. (b) Growth of the ionized\npopulation density when this field is incident on a hydrogen\ngas target with a density, N0= 1017cm\u00003. (c) The current\ndensity dynamics that occur as the exicted population inter-\nacts with the remaineder of the laser pulse. (d) The magnetic\nfield produced by this current density.\ning electromagnetic doughnut [18]. This teraherz pulse\ncan easily be spectrally isolated from the near-infrared\nexcitation pulses using a long-pass filter.\nAdjusting the relative phase between the !and 2!\nlaser pulses imposes coherent control on the excited cur-\nrent density and the resulting magnetic fields. Figure\n3(a) shows the peak magnetic field at the beam waist as\nthe relative phase between the two pulses is adjusted.\nFigures 3(b)-(e) show the magnetic field distribution\nin the longitudinal plane of the focusing geometry for\n\u0001\u001e=\u0019=8;5\u0019=8;9\u0019=8;and13\u0019=8;respectively. When\nthe relative phase is adjusted to produce a magnetic field\nmaximum in the beam waist, the magnetic field distribu-\nFIG. 3. Coherent control of magnetic fields. (a) Adjusting\nthe phase offset between the !and2!beams enables pre-\ncise control of the direction and amplitude of the generated\nmagnetic field. Longitudinal slices of the magnetic fields gen-\nerated in the focusing geometry are shown for (b) \u0001\u001e=\u0019=8,\n(c)\u0001\u001e= 5\u0019=8, (d) \u0001\u001e= 9\u0019=8, and (e) \u0001\u001e= 13\u0019=8.\ntion closely resembles that from a solenoid. Conversely,\nwhen the relative phase is adjusted for a minimum at the\nbeam waist, the magnetic field changes sign due to the\nGouy phase shift of the focused fields.\nNotably, the spatial distribution of the current density\nshown in Figs. 1(c)-(d) bears strong resemblance to that\nin a solenoid. The Rayleigh length of a focused laser\nbeam increases quadratically with the beam waist, i.e.\nzR=\u0019w2\n0=\u0015, and hence, for loose focusing conditions\nthe current density distribution converges to that of an\nideal solenoid. Increasing w0while keeping the peak elec-\ntric field constant (i.e. by increasing the laser pulse en-\nergy) produces a larger solenoidal current excitation per\nunit length, as depicted in Fig. 4(a). Due to the inde-\npendence of the solenoidal magnetic field on the solenoid\nradius, this provides a simple and convenient means to\nscale the magnetic field amplitude. With the availabil-\nity of energetic femtosecond laser pulses, a crucial ques-\ntion that must be addressed is, to what magnetic field\namplitude may this scheme be scaled before the electric\nfield induced by a rapidly excited magnetic field is suffi-\ncienttosuppresstheexcitedcurrentdensityandresulting\nmagnetic field? In other words, when does the opposing\nelectricfieldamplitudeexcitedbyatime-dependentmag-\nnetic field as determined by Faraday’s Law,\n\f# \u0014E\u0001d# \u0014l=\u0000\u0002@# \u0014B\n@t\u0001d# \u0014A; (2)\nbecome significant compared to the electric fields of the\nlaser pulse?\nTo investigate the influence of this opposing electric4\nFIG. 4. (a) Increasing the beam waist, w0of the azimuthal\nlaser beam increases the radial extent of the current density\nthat is produced at the focus, thereby increasing the current\nper unit length flowing through the solenoid. (b) Magnetic\nfield for a constant beam waist, w0= 200\u0016m, and varying\ngas density. Two simulations are performed for each gas den-\nsity: one neglecting back electromotive force and the other\nincluding it.\nfield on the excited electron trajectories, we increase the\nbeam waist to w0= 200\u0016m which, according a sim-\nple application of the solenoidal relationship ( B=\u0016IL)\ncould generate Tesla-scale magnetic fields. We consider\na fundamental laser pulse at \u0015= 4000nm with intensity\nI= 1:2\u00021014W/cm2, and a corresponding second har-\nmonic at\u0015= 2000nm with intensity I= 3\u00021013W/cm2.\nEach pulse has a duration of \u001c= 50fs. Keeping the ex-\ncitation parameters fixed, we increment the gas density\nfromn= 1\u00021014cm\u00003ton= 7:5\u00021016cm\u00003and per-\nform two simulations for each gas density: one with back\nelectromotiveforceincludedandtheotherneglectingthis\neffect. We note that in the absence of back electromo-\ntive force, one would expect linear scaling of the current\ndensity and magnetic field with respect to the gas den-\nsity. The result of these simulations is shown in Fig.\n4(b). In close agreement with the solenoidal relation-\nship (B=\u0016IL), magnetic field amplitudes up to 5:2Tare observed in the absence of back electromotive force.\nHowever, the simulations including back electromotive\nforcedemonstrateconsiderablesuppressionoftheexcited\nmagnetic field amplitude as the gas density is increased\nbeyond 1015cm\u00003, and the highest gas density produces\na magnetic field amplitude of 0:43T. Although back elec-\ntromotive force appears to be the dominant saturation\neffect in this magnetic field scaling scheme, these calcula-\ntions demonstrate the prospect for all-optical generation\nof spatially isolated magnetic fields with amplitudes that\nhave technological and metrological significance.\nWe note two routes that could potentially enable fur-\nther scaling of the magnetic field amplitude. The higher\nionization potential of helium ( Ip= 24:6eV) compared to\nhydrogen (Ip= 13:6eV) would permit higher laser inten-\nsities incident on the gas, driving the photoelectrons to\na greater final velocity. Alternatively, we envision that\na variation of this experiment incorporating a circularly\npolarised beam, which has been shown to produce high-\nenergy above threshold ionization electrons [19], and an\norbital angular momentum beam could enable an order-\nof-magnitude increase in the magnetic field amplitude.\nIn conclusion, we have proposed an all-optical scheme\nto generate magnetic field impulses. Solenoidal electrical\ncurrents are generated through coherent control of strong\nfield ionization, where an azimuthal electric field drives\nan azimuthal current. The amplitude and direction of\nthis current can be precisely controlled by changing the\nrelative phase between the !and2!light waves. Mag-\nnetic fields approaching the Tesla scale can be produced\nby using loose focusing conditions and mid-infrared laser\npulses. While the modulation bandwidth of magnetic\nfields in integrated magneto-optical circuits has been lim-\nitedtotheGHzrangeimposedbyelectricalcircuitry, this\nprovides a route to magneto-optical modulation beyond\nTHz frequencies.\nPerhaps the most intriguing property of the calculated\nmagnetic fields is their spatial localization. The longitu-\ndinal polarization of the generated magnetic field allows\nthe magnetic field direction introduced to a system, such\nas an electron spin, to be modulated simply by changing\nits angle of incidence on the region of interest, providing\nremarkable flexibility for spin logic devices. The purely\nlongitudinal magnetic field is naturally spatially isolated\nfrom coupled electric fields, granting access to metrology\nand technology based purely on magnetic fields.The in-\ntroduction of a short, intense magnetic impulse in the\nform of an electromagnetic wave would also enable opti-\ncalsynchronizationwithsophisticateddetectionschemes,\nsuch as magneto-optical sampling [20], high-harmonic\nmagnetic circular dichroism spectroscopy [21], or spin-\npolarized attosecond electron diffraction [22, 23], en-\nabling temporal resolution of magnetic-field-induced mi-\ncroscopic dynamics. Magnetic field metrology based on\nultrafast optical techniques will enable new frontiers in\nmagnetism, particularly scaling to higher magnetic fields\nand resolution of femtosecond and attosecond magneti-\nzation dynamics.5\nThis research was supported by the Natural Sciences\nand Engineering Research Council of Canada (NSERC)\nDiscovery Grant Program, the Canada Research Chairsprogram, and the United States Defense Advanced Re-\nsearchProjectsAgency(“TopologicalExcitationsinElec-\ntronics (TEE)”, agreement #D18AC00011).\n[1] R. Ulbricht, E. Hendry, J. Shan, T. F. Heinz, and M.\nBonn, “Carrier dynamics in semiconductors studied with\ntime-resolved terahertz spectroscopy,” Rev. Mod. Phys.\n83, 543-586 (2011).\n[2] D. H. Auston, K. P. Cheung, and P. R. Smith, “Picosec-\nondphotoconductingHertziandipoles,” Appl. Phys. Lett.\n45, 284-286 (1984).\n[3] Q. Wu and X. -C. Zhang, “Free-space electro-optic sam-\npling of terahertz beams”, Appl. Phys. Lett. 67, 3523-\n3525 (1995).\n[4] A. Leitenstorfer, S. Hunsche, J. Shah, M. C. Buss, and\nW. H. Knox, “Detectors and sources for ultrabroadband\nelectro-optic sampling: experiment and theory,” Appl.\nPhys. Lett. 74, 1516-1518 (1999).\n[5] E. Goulielmakis, M. Uiberacker, R. Kienberger, A. Bal-\ntuska, V. Yakovlev, A. Scrinzi, Th. Westerwalbesloh, U.\nKleineberg, U. Heinzmann, M. Drescher, and F. Krausz,\n“Direct measurement of light waves,” Science 305, 1267-\n1269 (2004).\n[6] K. T. Kim, C. Zhang, A. D. Shiner, B. E. Schmidt, F.\nLegare, D. M. Villeneuve, and P. B. Corkum, “Petahertz\noptical oscilloscope,” Nature Photon. 7, 958-962 (2013).\n[7] S. Keiber, S. Sederberg, A. Schwarz, M. Trubetskov,\nV. Pervak, F. Krausz, and N. Karpowicz, “Electro-optic\nsampling of near-infrared waveforms,” Nature Photon.\n10, 159-162 (2016).\n[8] A. Sommer, E. M. Bothschafter, S. A. Sato, C. Jakubeit,\nT. Latka, O. Razskazovskaya, H. Fattahi, M. Jobst, W.\nSchweinberger, V. Shirvanyan, V. S. Yakovlev, R. Kien-\nberger, K. Yabana, N. Karpowicz, M. Schultze, and F.\nKrausz, “Attosecond nonlinear polarization and light-\nmatter energy transfer in solids,” Nature 534, 86-90\n(2016).\n[9] J. C. Miller, R. N. Compton, M. G. Payne, and W.\nW. Garrett, “Resonantly enhanced multiphoton ioniza-\ntion and third-harmonic generation in Xenon gas,” Phys.\nRev. Lett. 45, 114-116 (1980).\n[10] D. J. Jackson, J. J. Wayne, and P. H. Kes, “Resonance-\nenhancedmultiphotonionization:Interferenceeffectsdue\nto harmonic generation,” Phys. Rev. A 28, 781-794\n(1983).\n[11] C. Chen, Y. -Y. Yin, and D. S. Elliott, “Interference be-\ntween optical transitions,” Phys. Rev. Lett. 64, 507-510(1990).\n[12] Y. -Y. Yin, C. Chen, and D. S. Elliott, “Asymmetric\nphotoelectron angular distributions from interfering pho-\ntoionization processes,” Phys. Rev. Lett. 69, 2353-2356\n(1992).\n[13] E. Dupont, P. B. Corkum, H. C. Liu, M. Buchanan, and\nZ. R. Wasilewski, “Phase-controlled currents in semicon-\nductors,” Phys. Rev. Lett. 74, 3596-3599 (1995).\n[14] A. Hache, J. E. Sipe, and H. M. van Driel, “Quantum\ninterference control of electrical currents in GaAs,” IEEE\nJ. Quantum Electron. 34, 1144-1154 (1998).\n[15] Q. Zhan, “Cylindrical vector beams: from mathematical\nconcepts to applications,” Adv. Opt. Photonics 1, 1-57\n(2009).\n[16] A. Taflove, Computational Electromagnetics: The Finite-\nDifference Time-Domain Method (Artech House, Nor-\nwood, 1995).\n[17] R. Huber, F. Tauser, A. Brodschelm, M. Bichler, G. Ab-\nstreiter, and A. Leitenstorfer, “How many-particle inter-\nactions develop after ultrafast excitation of an electron-\nhole plasma,” Nature 414, 286-289 (2001).\n[18] R. W. Hellwarth and P. Nouchi, “Focused one-cycle elec-\ntromagnetic pulses,” Phys. Rev. E 54, 889-895 (1996).\n[19] P. B. Corkum, N. H. Burnett, and F. Brunel, “Above-\nthreshold ionization in the long-wavelength limit,” Phys.\nRev. Lett. 62, 1259 - 1262 (1989).\n[20] J. A. Riordan, F. G. Sun, Z. G. Lu, and X. -C. Zhang,\n“Free-space transient magneto-optical sampling,” Appl.\nPhys. Lett. 71, 1452 (1997).\n[21] F. Willems, C. T. L. Smeenk, N. Zhavoronkov, O. Ko-\nrnilov, I. Radu, M. Schmidbauer, M. Hanke, C. von Korff\nSchmising, M. J. J. Vrakking, and S. Eisebitt, “Prob-\ning utrafast spin dynamics with high-harmonic mag-\nnetic circular dichroism spectroscopy,” Phys. Rev. B 92,\n220405(R) (2015).\n[22] C. Kealhofer, W. Schneider, D. Ehberger, A. Ryabov, F.\nKrausz, and P. Baum, “All-optical control and motrology\nof electron pulses,” Science 352, 429-433 (2016).\n[23] Y. Morimoto and P. Baum, “Diffraction and microscopy\nwith attosecond electron pulse trains,” Nature Phys. 14,\n252-256 (2018)."    },    {        "title": "2205.02308v2.Exceptional_points_as_signatures_of_dynamical_magnetic_phase_transitions.pdf",        "content": "Exceptional points as signatures of dynamical magnetic phase transitions\nKuangyin Deng,1,\u0003Xin Li,1and Benedetta Flebus1,y\n1Department of Physics, Boston College, 140 Commonwealth Avenue, Chestnut Hill, Massachusetts 02467, USA\nOne of the most fascinating and puzzling aspects of non-Hermitian systems is their spectral\ndegeneracies, i.e., exceptional points (EPs), at which both eigenvalues and eigenvectors coalesce\nto form a defective state space. While coupled magnetic systems are natural hosts of EPs, the\nrelation between the linear and nonlinear spin dynamics in the proximity of EPs remains relatively\nunexplored. Here we theoretically investigate the spin dynamics of easy-plane magnetic bilayers in\nthe proximity of exceptional points. We show that the interplay between the intrinsically dissipative\nspin dynamics and external drives can yield a rich dynamical phase diagram. In particular, we \fnd\nthat, in antiferromagnetically coupled bilayers, a periodic oscillating dynamical phase emerges in\nthe region enclosed by EPs. Our results not only o\u000ber a pathway for probing magnetic EPs and\nengineering magnetic nano-oscillators with large-amplitude oscillations, but also uncover the relation\nbetween exceptional points and dynamical phase transitions in systems displaying non-linearities.\nIntroduction . The degeneracies of Hermitian Hamilto-\nnians are diabolic points, i.e., points at which two (or\nmore) real eigenenergies coalesce, while the eigenstates\nstill span the full Hilbert space. Non-Hermitian degen-\neracies, i.e., exceptional points (EPs), display properties\nthat are radically di\u000berent from their Hermitian counter-\npart. At an EP, two (or more) complex eigenvalues and\nthe corresponding eigenvectors simultaneously coalesce,\nresulting into a defective Hamiltonian that cannot span\nthe entire Hilbert space [1{3]. The incompleteness of the\neigenbases at second-order EPs leads to a square root de-\npendence on external perturbations, resulting in a giant\nsensitivity-factor enhancement [4{7].\nAs non-Hermitian systems are recently under compre-\nhensive research [8{12], intense e\u000borts have been put for-\nward to explore the properties of EPs. Particular empha-\nsis has been placed on PT-symmetric systems [6, 13{15],\nwhere EPs signal a PT-symmetry-breaking transition at\nwhich a system's eigenvalues turn from real to complex\nconjugate pairs. The emergence of EPs does not, how-\never, require a \fne-tuned balance of gain and loss [16].\nEPs have been reported in a plethora of open systems,\nranging from optics and photonics [4, 6, 13, 17, 18] to su-\nperconducting quantum circuits [19], semimetals [20{23],\nand magnetic systems [24{35].\nMagnetic systems are intrinsically open due to the\nubiquitous dissipation of magnetization dynamics [35{\n37]. The gain can be tuned via experimentally es-\ntablished techniques such as, e.g., spin current injec-\ntion [35, 38{42]. Exceptional points naturally emerge in\nthe description of coupled magnetization dynamics and\nhave been recently observed in magnonic PT-symmetric\ndevices [35]. Second-order and higher-order EPs display-\ning higher-order roots singularities [43{49], which can\nyield further ultra-sensitivity, have been reported in mag-\nnetic multilayers [29]. While the potential of EPs in mag-\nnetic sensing has been under intense scrutiny, the role\nthat EPs play in dynamical magnetic phase transitions\nis yet relatively unexplored.\nCoupled magnetization dynamics can be described,in the long-wavelength limit, via the coupled Landau-\nLifshitz-Gilbert (LLG) equations [50]. By linearizing the\nLLG equations of motion, one can derive an e\u000bective non-\nHermitian Hamiltonian quadratic in second-quantized\nmagnon operators. The EPs appear as singularities of\nthe quadratic Hamiltonian, signaling a dynamical phase\ntransition of the linearized dynamics due to a width bi-\nfurcation [51{54]. If signatures of such transition survive\nin the nonlinear LLG-like classical dynamics, the analy-\nsis of the corresponding quadratic magnon Hamiltonian\ncan unveil unforeseen dynamical regimes as function of\nexperimentally tunable parameters.\nIn this work, we explore the connection between lin-\near and nonlinear spin dynamics in proximity of EPs\nby taking an easy-plane magnetic bilayer as an exam-\nple. The ratio between gain and loss is modulated by\nspin injection in the bottom layer and the loss of mag-\nnetization dynamics is taken to be larger than the over-\nall gain. As a function of the interlayer coupling, we\n\fnd that the linearized spectrum displays two regions\nencircled by exceptional points, emerging around, respec-\ntively, vanishing and strong antiferromagnetic (AFM) in-\nterlayer coupling. The non-linear dynamics in proximity\nof the region with vanishing interlayer coupling displays\na ferromagnetic (FM)-to-AFM dynamical phase transi-\ntion. Such transition has been reported in a magnonic\nPT-symmetric system [28]: our results show that \fne-\ntuned balance of gain and loss is not necessary for the\ntransition to take place.\nFurthermore, we unveil a distinct dynamical phase\ntransition occuring in the AF-coupled region encircled\nby the EPs. Simulations of the nonlinear dynamics\nshow, that upon crossing the EP in parameter space,\nthe damped magnetization dynamics enters a regime\nof steady self-oscillations with large amplitude that can\nbe described by a supercritical Hopf-Bifurcation [55{57].\nAccording to our estimates, this dynamical phase tran-\nsition might be observed in van der Waals and synthetic\nAFM bilayers [58, 59], which could open up a route to\nengineer magnetic nano-oscillators [42, 60{66] with large-arXiv:2205.02308v2  [cond-mat.mes-hall]  6 Mar 20232\nFIG. 1. (a) Magnetic bilayer with interlayer coupling Jin\nan external magnetic \feld B0. In the long-wavelength limit,\nthe uniform magnetization of the top (bottom) layer can be\ntreated as a macrospin SA(B). (b,c): Dependence on Jof the\nreal and imaginary energy, respectively, for K= 0. Region I\nis enclosed by EPs. The red dashed line separates a collinear\nfrom a non-collinear ground state. (d) The time evolution of\nSABfor di\u000berent values of the interlayer coupling J. The FM-\nto-AFM dynamical phase transition emerges in region I for\nsmall interlayer coupling, e.g., J= 0:1\u0016eV. Instead, for val-\nues ofJfurther away from region I, the relative alignment of\nthe macrospins remains the one of the corresponding ground\nstate. In each \fgure, the parameters are set to B0= 0:1 T,\nK= 0,\u000bA= 0:06 and\u000bB=\u00000:04.\namplitude oscillations. Our \fndings have also the poten-\ntial to shed light on the interplay between EPs and dy-\nnamical phase transition in other dissipative-driven sys-\ntems displaying non-linearities.\nModel . We consider the magnetic bilayer shown in\nFig. 1(a), whose spin Hamiltonian can be written, in the\nlong-wavelength limit, as\nH=X\ni=A;B\u0000\nKSz2\ni+\rB0\u0001Si\u0001\n+JSA\u0001SB; (1)\nwhere SA(B), withjSA;Bj=S, is the (dimensionless)\nmacrospin operator of the top (bottom) layer, B0the\napplied magnetic \feld, \r > 0 the gyromagnetic, Jthe\ninterlayer coupling, and K\u00150 parametrizes the easy-\nplane anisotropy. Here we set ~= 1 by adopting its\nunit to other parameters. To introduce loss and gain, we\nrecast the magnetization dynamics in the form of coupledLandau{Lifshitz{Gilbert (LLG) equations [50], i.e.,\ndSA\ndt=\u0000\rSA\u0002Be\u000b\nA\u0000\u000bA\nSSA\u0002dSA\ndt; (2)\ndSB\ndt=\u0000\rSB\u0002Be\u000b\nB\u0000\u000bB\nSSB\u0002dSB\ndt; (3)\nwhere we have introduced the e\u000bective \feld \rBe\u000b\ni=\n@H=@Si, withi=A;B. Here\u000bA>0 (\u000bB<0) repre-\nsents the e\u000bective damping (gain) parameter of the top\n(bottom) layer.\nTo investigate the non-Hermitian spin-wave spectrum\nas function of the exchange coupling Jand magnetic \feld\nB0, we orient the spin-space Cartesian coordinate system\nsuch that the ^zaxis locally lies along the classical orien-\ntation of the macrospin ~Si. The latter can be related to\nthe spin operator Siin the global frame of reference via\nthe transformation [67]\nSi=Rz(\u001ei)Ry(\u0012i)~Si; (4)\nwhere the matrix Rz(y)(\u0011) describes a right-handed ro-\ntation by an angle \u0011about the ^ z(^y) axis, and \u0012i(\u001ei)\nis the polar (azimuthal) angle of the classical orienta-\ntion of the spin Si. We then solve self-consistently\nEqs. (2) and (3) in the linear approximation, i.e., we\nconsider ~Si=\u0010\n~Sx\ni;~Sy\ni;S\u0011\n. Next, we introduce the com-\nplex variable ~S+\ni=~Sx\ni+i~Sy\niand invoke the Holstein-\nPrimako\u000b transformation ~S+\nA(B)\u0019p\n2Sa(b), where the\nsecond-quantized operator a(b) annihilates a magnon in\nthe top (bottom) layer and obeys bosonic commutation\nrelations [68]. By invoking the Heisenberg equation for\na(b), we obtain the non-Hermitian Hamiltonian Hnh.\nThe resulting Hamiltonian is not block-diagonal and a\nBogoliubov transformation is required to obtained the\nspin-wave spectrum [69]\nAntiferromagnetic to ferromagnetic transition. As a\n\frst instructive example, we turn o\u000b the easy-plane\nanisotropy, i.e., K= 0, and we take a damping coe\u000ecient\nof the same order of magnitude of the ones reported for\nchromium trihalide crystals [70], i.e., \u000bA= 0:06, while\nwe set\u000bB=\u00000:04 [71]. We set B0= 0:1 T and take\nB0k^x. It is worth noting that our results do not de-\npend on the \feld direction since the Hamiltonian (1) is\nSO(3)-symmetric for K= 0. The real and imaginary\nenergy spectra of Hnhas a function of Jare shown, re-\nspectively, in Fig. 1(b) and 1(c). Near J= 0, region I\nis enclosed by EPs. On the left side of the red dashed\nline, the ground state of the Hermitian Hamiltonian (i.e.,\nEq. (1) for\u000bA(B)= 0) is collinear and oriented along the\nmagnetic \feld. On the right side of the dashed line, the\ninterplay between the magnetic \feld and the antiferro-\nmagnetic coupling Jleads to a noncollinear ground state,\nwhile increasing Jfurther yields an AFM ground state.\nTo investigate how the degeneracies of the non-\nHermitian linear spectrum a\u000bect the non-linear magne-3\nFIG. 2. Real (a) and imaginary (b) energy for B0= 0:14 T,\nK= 45:9\u0016eV,\u000bA= 0:06 and\u000bB=\u00000:04. Here, region\nI is in direct correspondence with region I of Fig. 1. The\nred dashed line marks the transition from a collinear to a\nnoncollinear con\fguration. Region II is enclosed by another\npair of EPs in the noncollinear con\fguration. (c) - (e) The\ntime evolution of SAB(t) for di\u000berent values of the interlayer\ncouplingJ. A periodic dynamical phase emerges only within\nregion II.\ntization dynamics, we simulate Eqs. (2) and (3) by set-\nting the initial direction of the spins slightly away (2\u000e)\nfrom their ground-state equilibrium position. We solve\nEqs. (2) and (3) for di\u000berent values of Jand track the\ntime evolution of the product of the macrospins, i.e.,\nSAB(t) =SA(t)\u0001SB(t)=S2. As shown in Fig. 1(d), the\nrelative alignment SABbetween the macrospins remains\nFM or AFM for values of Jfurther away from the excep-\ntional point. Instead when we chose Jwithin region I, we\nobserve a switch from a FM to an AFM con\fguration.\nOur result agrees with the observations of Ref. [28], in\nwhich the authors analyze the PT-symmetric case (i.e.,\n\u000bA=\u0000\u000bB) of Eqs. (1-3) for K= 0. Here, we propose\na simple explanation for this dynamical phase transition,\nwhich occurs when the coupling Jis close to 0. In this\nregime, the spins are barely coupled and, thus, eventu-\nally, each macrospin obeys its individual dynamics. The\nmacrospin experiencing gain \rips, while the lossy one re-\ncovers its equilibrium orientation, leading to an AFM\norientation. As we have shown, PTsymmetry is not\nrequired for the FM-to-AFM switching to occur.\nA magnetic nano-oscillator . To explore the dynamical\nphase diagram of our model, we now turn on the easy-\nplane anisotropy, i.e., K > 0. With CrCl 3in mind, we set\nK= 45:9\u0016eV [58]. We consider a U(1)-symmetry break-\ning magnetic \feld B0k^xand setB0= 0:14 T,\u000bA= 0:06\nand\u000bB=\u00000:04. The real and imaginary parts of the\nmagnon energy are shown in Fig. 2(a) and 2(b), respec-\ntively. We \fnd two regions enclosed by EPs: region I nearJ= 0 and region II near J= 12:2\u0016eV, i.e., the exchange\ninteraction of CrCl 3[58]. Region I corresponds to region\nI shown in Figs. 1(b) and 1(c). Region II emerges instead\nin correspondence with a noncollinear ground state and,\nas we will show in details, its nonlinear magnetization\ndynamics (2,3) display very di\u000berent features from the\nones observed in region I.\nFigures 2(c) - 2(e) show the time evolution of the rela-\ntive alignment of the macrospins SAB(t) for, respectively,\nJ= 9, 12:2, and 16\u0016eV. Similarly to region I, pass-\ning through the EPs yields a dynamical phase transition.\nHowever, around region II, the exchange interaction is\ntoo strong for a FM-to-AFM switching to take place. In-\nstead, while for J= 9:0\u0016eV andJ= 16:0\u0016eV we ob-\nserve damped dynamical phases, see Figs. 2(c) and 2(e),\ninside region II (i.e., J= 12:2\u0016eV) a periodic dynamical\nphase emerges, as shown in Fig. 2(d). Within the peri-\nodic dynamical phase, the value of SABranges from 0 :7\nto\u00000:7, signaling unusual large-amplitude oscillations.\nOur results show that, although the overall loss is larger\nthan the e\u000bective gain, i.e., \u000bA>j\u000bBj, the system can\nstill survive in a steady periodic state in a EP-enclosed re-\ngion. The dynamical phase transition can be understood\nas a supercritical Hopf-Bifurcation [55{57]. When cross-\ning the EPs and entering in region II, the \fxed point of\nthe dynamical system, which corresponds to the damped\nmagnetization dynamics, bifurcates into a stable orbital.\nWe have veri\fed numerically that the large-amplitude\noscillations persist at long times.\nTunability. We proceed to investigate the dependence\nof the periodic stable magnetization dynamics on the sys-\ntem's parameters. Not surprisingly, the stability of the\nperiodic solution strongly depends on the ratio between\nthe e\u000bective gain and loss. Setting J= 12:2\u0016eV and\n\u000bA= 0:06, in Fig. 3(a-d) we show the time evolution\nofSA(upper panel) and SB(lower panel) on the Bloch\nsphere decreasing the e\u000bective gain j\u000bBjfrom 0:055 to\n0:01. The colors in Fig. 3(a-d) are in direct correspon-\ndence with the time intervals of the time-evolution of\nSABshown in Fig. 2(c-e). For larger values of gain,\ne.g.,\u000bB=\u00000:055, the dynamics of both macrospins SA\nandSB\row to a \fxed point, as shown by Fig. 3(a).\nWe have veri\fed that the same scenario is realized at\nthePT-symmetric point. For lower values of the gain,\nthe spin dynamics evolve into a steady-state oscillations,\nsee Figs. 3(b-d). Since the macrospin SBis directly\nsubjected to gain while SAexperiences it indirectly via\nthe coupling to SA, the amplitude of oscillations of the\nmacrospin SAis smaller than the one of SB. For decreas-\ning\u000bB, the amplitude of both limit cycles shrink.\nIn an experimental setup, the e\u000bective gain \u000bBcan\nbe controlled via the injection of spin current Jsinto the\nbottom layer. As shown in a very recent work [72], swap-\nping the dynamical gain in Eq. (3) with a spin-transfer\ntorque term, i.e., \u0000\u000bB\nSSB\u0002dSB\ndt!JsSB\u0002(SB\u0002^z)\ndoes not a\u000bect the emergence of an oscillatory phase in4\nFIG. 3. (a) - (d) The spin evolution on the Bloch spheres for di\u000berent values of the e\u000bective gain \u000bBin the region II of Fig. 2\nforB0= 0:14 T,K= 45:9\u0016eV,J= 12:2\u0016eV, and\u000bA= 0:06. The above (below) panels shows the time evolution of SA(SB).\nThe color on curves are in direct correspondence with the time intervals of the time-evolution of SABin Figs.2(c)-(d), i.e., they\nlabel the earliest to the latest time by ordering purple, blue, gray, green, yellow, orange, and red. (a) For \u000bB=\u00000:055, the\ndynamics of SAandSB\row into \fxed points (FP). (b-d): When j\u000bBj\u00140:05, the system drops on periodic orbitals (PO)\nthrough the supercritical Hopf-Bifurcation. SAwith larger loss than the gain in SBwould form smaller orbitals to maintain the\nsteady periodic oscillation. (e) Frequency fof the coupled oscillations SABas a function of the e\u000bective gain \u000bBfor di\u000berent\nvalues ofJ. ForJ= 4:2\u0016eV (J= 8:2\u0016eV), steady periodic dynamical phases exist only for j\u000bBj\u00140:035 (j\u000bBj\u00140:045).\n(f) The dependence of the square of the overlap of the two right eigenvectors, i.e., PEP\u0011\f\f\n R\n1\f\f R\n2\u000b\f\f2, on the magnetic \feld\nstrengthB0and polar angle \u0002.\ncorrespondence of EP crossing.\nThe ratio\u000bA=\u000bBis determined by the spin current\ntransport e\u000eciency through the magnetic layers which,\nto our knowledge, has not been yet thoroughly inves-\ntigated in van der Waals magnets. It is worth noting\nthat here we take CrCl 3as an example; in practice,\nthe high degree of tunability o\u000bered by synthetic AFMs\nmight make them a more desirable platform for engi-\nneering non-Hermitian phenomena [73]. To avoid spin\ncurrent injection in the top layer of a synthetic AFM\nbilayer, one could sandwich a good spin sink, e.g., Pt\nthin \flm [35, 74], between the two magnetic layers. In\nthis case, the strength of the (RKKY) interlayer cou-\npling can be controlled by tuning the Pt layer thick-\nness [35]. Synthetic AFM based on permalloy magnetic\nelements display an easy-plane anisotropy consistent with\nour model (1) [73].\nWe \fnd that the periodic oscillatory phase does not\nrequire \fne-tuning but it can instead be accessed within\na relative broad range of \u000bA=\u000bBvalues. As shown in\nFig. 3(e), the strength of the interlayer coupling con-\ntrols the frequency fof the periodic oscillations (found\nby changing B0) of the coupled dynamics SAB. For\nCrCl 3[58], the interlayer coupling strength J= 12:2\u0016eV\nyields large-amplitude oscillations with frequencies in the\n1\u000010 GHz range.Finally, we explore the dependence of the onset of re-\ngion II on the strength and direction of the applied mag-\nnetic \feld. In Fig. 3(f), we plot PEP\u0011\f\f\n R\n1\f\f R\n2\u000b\f\f2,\nwhere R\n1;2are the two right eigenvectors of the non-\nHermitian Hamiltonian Hnh. While approaching an ex-\nceptional point, the two eigenstates coalesce, i.e., PEP!\n1. The two red regions in Fig. 3(f) appear in proximity of\nthe EPs: the region comprised between them, which cen-\nters on white and blue, corresponds to region II, i.e., it\ndisplays periodic oscillatory coupled spin dynamics. As\nshown by Fig. 3(f), accessing the region II does not re-\nquire \fne-tuning: there is a broad range of values of\nthe magnetic \feld's strength and polar angle \u0002, with\nB0\u0001^z=B0sin \u0002, for which the steady-state oscillations\nappear.\nDiscussion and outlook . In this work, we investigate\nthe interplay between the linear and nonlinear spin dy-\nnamics in proximity of exceptional points. We show that\nthe emergence of EPs in the linearized magnon Hamilto-\nnian underlies a dynamical phase transition of the non-\nlinear spin dynamics. As an example, we consider on an\neasy-plane bilayer in which, while one layer experiences\ne\u000bective gain, the other layer keeps larger loss rate. An\nanalysis of the linearized long-wavelength magnetization\ndynamics of the bilayer shows that two regions encircled\nby EPs can appear as function of the interlayer coupling.5\nOne region, characterized by small values of the interlayer\ncoupling, displays an interlayer FM-to-AFM dynamical\nphase transition. The second region, appearing for larger\nvalues of the AFM interlayer coupling, displays large-\namplitude steady-state oscillations without \fne-tuning or\nPTsymmetry. We argue that this oscillatory dynamical\nregime might be accessed via spin injection in CrCl 3or\nsynthetic AFM bilayers, opening a concrete route for ex-\nperimentally probing magnetic EPs and for engineering\nlarge-amplitude magnetic nano-oscillators.\nOur theory has the potential to shed light onto the rela-\ntion between non-Hermitian singularities and dynamical\nphase transitions in a plethora dissipative-driven systems\nwhose dynamics display non-linearities, e.g., molecular\nspin dimers [75, 76], quantum dots [77{79] and microwave\nresonators [34, 80].\nAcknowledgments . K. Deng thanks B. Li for help-\nful discussions. This work was supported by the Na-\ntional Science Foundation under Grant No. NSF DMR-\n2144086.\n\u0003dengku@bc.edu\ny\rebus@bc.edu\n[1] W. Heiss, Physical Review E 61, 929 (2000).\n[2] W. Heiss, Journal of Physics A: Mathematical and The-\noretical 45, 444016 (2012).\n[3] C. Dembowski, H.-D. Gr af, H. Harney, A. Heine,\nW. Heiss, H. Rehfeld, and A. Richter, Physical Review\nLetters 86, 787 (2001).\n[4] W. Chen, S \u0018. Kaya Ozdemir, G. Zhao, J. Wiersig, and\nL. Yang, Nature 548, 192 (2017).\n[5] J. Wiersig, Physical Review Letters 112, 203901 (2014).\n[6] M.-A. Miri and A. Alu, Science 363, eaar7709 (2019).\n[7] J.-H. Park, A. Ndao, W. Cai, L. Hsu, A. Kodigala,\nT. Lepetit, Y.-H. Lo, and B. Kant\u0013 e, Nature Physics 16,\n462 (2020).\n[8] Z. Gong, Y. Ashida, K. Kawabata, K. Takasan, S. Hi-\ngashikawa, and M. Ueda, Physical Review X 8, 031079\n(2018).\n[9] H. Shen, B. Zhen, and L. Fu, Physical Review Letters\n120, 146402 (2018).\n[10] S. Yao and Z. Wang, Physical Review Letters 121,\n086803 (2018).\n[11] K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Phys-\nical Review X 9, 041015 (2019).\n[12] E. J. Bergholtz, J. C. Budich, and F. K. Kunst, Reviews\nof Modern Physics 93, 015005 (2021).\n[13] S \u0018. K. Ozdemir, S. Rotter, F. Nori, and L. Yang, Nature\nmaterials 18, 783 (2019).\n[14] L. Xiao, T. Deng, K. Wang, Z. Wang, W. Yi, and P. Xue,\nPhysical Review Letters 126, 230402 (2021).\n[15] V. Achilleos, G. Theocharis, O. Richoux, and V. Pag-\nneux, Physical Review B 95, 144303 (2017).\n[16] Y. Zhiyenbayev, Y. Kominis, C. Valagiannopoulos,\nV. Kovanis, and A. Bountis, Physical Review A 100,\n043834 (2019).\n[17] S. Longhi and G. Della Valle, Physical Review A 89,052132 (2014).\n[18] P. Renault, H. Yamaguchi, and I. Mahboob, Physical\nReview Applied 11, 024007 (2019).\n[19] M. Partanen, J. Goetz, K. Y. Tan, K. Kohvakka,\nV. Sevriuk, R. E. Lake, R. Kokkoniemi, J. Ikonen,\nD. Hazra, A. M akinen, et al. , Physical Review B 100,\n134505 (2019).\n[20] R. A. Molina and J. Gonz\u0013 alez, Physical Review Letters\n120, 146601 (2018).\n[21] K. Kawabata, T. Bessho, and M. Sato, Physical Review\nLetters 123, 066405 (2019).\n[22] J. Gonz\u0013 alez and R. A. Molina, Physical Review B 96,\n045437 (2017).\n[23] D. Chowdhury, A. Banerjee, and A. Narayan, Physical\nReview B 105, 075133 (2022).\n[24] S. Komineas, arXiv preprint arXiv:2209.01572 (2022).\n[25] X. Li, K. Deng, and B. Flebus, Physical Review B 106,\n214432 (2022).\n[26] J. Lee, T. Kottos, and B. Shapiro, Physical Review B\n91, 094416 (2015).\n[27] A. Galda and V. M. Vinokur, Physical Review B 94,\n020408 (2016).\n[28] H. Yang, C. Wang, T. Yu, Y. Cao, and P. Yan, Physical\nReview Letters 121, 197201 (2018).\n[29] T. Yu, H. Yang, L. Song, P. Yan, and Y. Cao, Physical\nReview B 101, 144414 (2020).\n[30] H. M. Hurst and B. Flebus, Journal of Applied Physics\n132, 220902 (2022).\n[31] X.-g. Wang, G.-h. Guo, and J. Berakdar, Physical Re-\nview Applied 15, 034050 (2021).\n[32] T. Je\u000brey, W. Zhang, and J. Sklenar, Applied Physics\nLetters 118, 202401 (2021).\n[33] B. Flebus, R. A. Duine, and H. M. Hurst, Physical Re-\nview B 102, 180408 (2020).\n[34] P. M. Gunnink, B. Flebus, H. M. Hurst, and R. A. Duine,\nPhys. Rev. B 105, 104433 (2022).\n[35] H. Liu, D. Sun, C. Zhang, M. Groesbeck, R. Mclaughlin,\nand Z. V. Vardeny, Science advances 5, eaax9144 (2019).\n[36] Y. Tserkovnyak, Physical Review Research 2, 013031\n(2020).\n[37] K. Deng and B. Flebus, Physical Review B 105, L180406\n(2022).\n[38] B. Flebus, Journal of Applied Physics 129, 161101\n(2021).\n[39] D. C. Ralph and M. D. Stiles, Journal of Magnetism and\nMagnetic Materials 320, 1190 (2008).\n[40] J. Sun and D. Ralph, Journal of Magnetism and Magnetic\nMaterials 320, 1227 (2008).\n[41] J. Katine and E. E. Fullerton, Journal of Magnetism and\nMagnetic Materials 320, 1217 (2008).\n[42] T. Chen, R. K. Dumas, A. Eklund, P. K. Muduli,\nA. Houshang, A. A. Awad, P. D urrenfeld, B. G. Malm,\nA. Rusu, and J. \u0017Akerman, Proceedings of the IEEE 104,\n1919 (2016).\n[43] G.-Q. Zhang and J. You, Physical Review B 99, 054404\n(2019).\n[44] I. Mandal and E. J. Bergholtz, Physical Review Letters\n127, 186601 (2021).\n[45] S. Wang, B. Hou, W. Lu, Y. Chen, Z. Zhang, and C. T.\nChan, Nature communications 10, 1 (2019).\n[46] H. Hodaei, A. U. Hassan, S. Wittek, H. Garcia-Gracia,\nR. El-Ganainy, D. N. Christodoulides, and M. Kha-\njavikhan, Nature 548, 187 (2017).\n[47] Y. Wu, P. Zhou, T. Li, W. Wan, and Y. Zou, Optics6\nExpress 29, 6080 (2021).\n[48] Z. Xiao, H. Li, T. Kottos, and A. Al\u0012 u, Physical Review\nLetters 123, 213901 (2019).\n[49] M. Zhang, W. Sweeney, C. W. Hsu, L. Yang, A. Stone,\nand L. Jiang, Physical Review Letters 123, 180501\n(2019).\n[50] M. Lakshmanan, Philosophical Transactions of the Royal\nSociety A: Mathematical, Physical and Engineering Sci-\nences 369, 1280 (2011).\n[51] I. Rotter, Journal of Physics A: Mathematical and The-\noretical 42, 153001 (2009).\n[52] H. Eleuch and I. Rotter, Physical Review A 93, 042116\n(2016).\n[53] H. Eleuch and I. Rotter, Physical Review E 87, 052136\n(2013).\n[54] I. Rotter, Acta Polytechnica 50(2010).\n[55] J.-V. Kim, in Solid State Physics , Vol. 63 (Elsevier, 2012)\npp. 217{294.\n[56] S. H. Strogatz, Nonlinear dynamics and chaos: with ap-\nplications to physics, biology, chemistry, and engineering\n(CRC press, 2018).\n[57] I. D. Mayergoyz, G. Bertotti, and C. Serpico, Nonlinear\nmagnetization dynamics in nanosystems (Elsevier, 2009).\n[58] D. MacNeill, J. T. Hou, D. R. Klein, P. Zhang, P. Jarillo-\nHerrero, and L. Liu, Physical Review Letters 123,\n047204 (2019).\n[59] J. Sklenar and W. Zhang, Physical Review Applied 15,\n044008 (2021).\n[60] J.-R. Chen, A. Smith, E. A. Montoya, J. G. Lu, and\nI. N. Krivorotov, Communications Physics 3, 1 (2020).\n[61] D. Markovi\u0013 c, N. Leroux, M. Riou, F. Abreu Araujo,\nJ. Torrejon, D. Querlioz, A. Fukushima, S. Yuasa,\nJ. Trastoy, P. Bortolotti, et al. , Applied Physics Letters\n114, 012409 (2019).\n[62] D. Houssameddine, U. Ebels, B. Dela et, B. Rodmacq,\nI. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-P.\nMichel, L. Prejbeanu-Buda, et al. , Nature materials 6,\n447 (2007).\n[63] S. Kaka, M. R. Pufall, W. H. Rippard, T. J. Silva, S. E.\nRussek, and J. A. Katine, Nature 437, 389 (2005).\n[64] V. Tiberkevich, A. Slavin, and J.-V. Kim, AppliedPhysics Letters 91, 192506 (2007).\n[65] I. Firastrau, L. Buda-Prejbeanu, B. Dieny, and U. Ebels,\nJournal of Applied Physics 113, 113908 (2013).\n[66] R. Cheng, D. Xiao, and A. Brataas, Physical Review\nLetters 116, 207603 (2016).\n[67] See Supplementary Material for the coordinate transfor-\nmation.\n[68] T. Holstein and H. Primako\u000b, Phys. Rev. 58, 1098\n(1940).\n[69] See Supplementary Material for a detailed derivation.\n[70] X. Shen, H. Chen, Y. Li, H. Xia, F. Zeng, J. Xu, H. Y.\nKwon, Y. Ji, C. Won, W. Zhang, et al. , Journal of Mag-\nnetism and Magnetic Materials 528, 167772 (2021).\n[71] These values can be chosen with a certain \rexibility as\nlong as the loss is larger than the gain, as we discuss in\ndetail later.\n[72] R. Duine, V. Errani, and J. Harms, arXiv preprint\narXiv:2302.07607 (2023).\n[73] M. M. Subedi, K. Deng, Y. Xiong, J. Mongeon,\nM. T. Hossain, P. Meisenheimer, E. Zhou, J. Heron,\nM. B. Jung\reisch, W. Zhang, et al. , arXiv preprint\narXiv:2301.07311 (2023).\n[74] B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas,\nR. Urban, and G. E. Bauer, Physical Review Letters 90,\n187601 (2003).\n[75] A. Ardavan, A. M. Bowen, A. Fernandez, A. J. Field-\ning, D. Kaminski, F. Moro, C. A. Muryn, M. D. Wise,\nA. Ruggi, E. J. McInnes, et al. , npj Quantum Informa-\ntion1, 1 (2015).\n[76] C. M. Ramsey, E. Del Barco, S. Hill, S. J. Shah, C. C.\nBeedle, and D. N. Hendrickson, Nature Physics 4, 277\n(2008).\n[77] A. F. Amin, G. Li, A. H. Phillips, and U. Kleinekath ofer,\nThe European Physical Journal B 68, 103 (2009).\n[78] P. Zhang, Q.-K. Xue, and X. Xie, Physical Review Let-\nters91, 196602 (2003).\n[79] X. Yang and Y. Liu, Journal of Applied Physics 113,\n164310 (2013).\n[80] M. Parto, S. Wittek, H. Hodaei, G. Harari, M. A.\nBandres, J. Ren, M. C. Rechtsman, M. Segev, D. N.\nChristodoulides, and M. Khajavikhan, Phys. Rev. Lett.\n120, 113901 (2018)."    },    {        "title": "1407.3188v1.Dynamic_phase_transition_properties_for_the_mixed_spin__1_2__1__Ising_model_in_an_oscillating_magnetic_field.pdf",        "content": "1 \n Dynamic phase transition properties for the mixed spin -(1/2, 1) Ising model \nin an oscillating magnetic field  \n \nMehmet Ertaş* and Mustafa Keskin  \n Department of Physics, Erciyes University, 38039 Kayseri, Turkey  \n \nAbstract  \nWe study  the dynamic phase transition  properties for the mixed spin -(1/2, 1) Ising  model  on a square \nlattice  under a time -dependent  magnetic field by means  the effective -field theory (EFT) based on the \nGlauber dynam ics. We present t he dynamic phase diagrams in the reduced magnetic field ampli tude \nand reduced temperature plane  and find that the phase diagrams exhibit the dynamic tricitical \nbehavior , the multicritical and zero -temperature critical points  as well as reentrant behavior . We also \ninvestigate  the influence of the frequency (w) and observe that for small values of w the mixed phase \ndisappears, but high values it appears and the system displays reentrant behavior as well as critical end \npoint.  \nKeywords:  mixed spin-(1/2, 1) Ising model , dynamic phase transition , effective -field theory , Glauber \ndynam ics \nPACC:  05.50.+q, 05.70.Fh, 64.60.Ht, 75.10.Hk  \n \n1. Introduction  \n \nThe mixed spin -(1/2, 1) Ising system has been studied extensively in past three \ndecades due  to the reasons that it provides a good model to investigate molecular -based \nmagnetic ma terials and ferrimagnetism as well as exhibits new critical phenomena that cannot \nbe seen in the single -spin Ising system s. The system has been used to study the equilibrium \nproperties of different physical systems within the well -known methods in the equi librium \nstatistical physics  [1-11] and references therein) . The equilibrium critical behavior of the \nmixed spin-(1/2, 1) Ising  (see [12 -22] and references therein)  and Heisenberg model [ 23-26] \nhas been also investigated, extensively . The exact solutio n of the system was studied on \ndifferent lattice s, such as  honeycomb lattice, a bathroom -tile or diced lattices , a Bethe lattice , \ntwo-fold Cayley tree , etc. (see  [27- 36] and references therein) .  \nOn the other hand, the nonequlibrium properties of the mixed spi n-1/2 and spin -1 Ising \nsystem  have  not been as thoroughly investigated . Godoy and Figueiredo studied the \nnonequilibrium behavior of the system , the Hamiltonian with only the bilinear interaction  [37-\n39] and also including the crystal field interaction [40-41], by using the dynamical pair \napproximation  and MC simulations . Buendía and Machado [ 42], and Keskin et al. [ 43] \nstudied the dynamic phase transition properties  and presented the dynamic phase diagrams of \nthe mixed spin -(1/2, 1) Ising model in the prese nce of a time -dependent oscillating magnetic \nfield by means the mean -field approximation  (MFA) based on the Glauber -type stochastic  \ndynamıcs.  Since the spin -spin correlations  are not considered in the MFA some of the first -\norder lines and also tricritical points in the phase diagram might be artifact of the MFA. \nTherefore, the dynamic phase transition properties of the system should be  studied more \naccurate methods. Thus, the aim of this paper is to study the dynamic phase transition \nproperties for the mixed spin -(1/2, 1) Ising model under a time -dependent magnetic field by \n                                                 \n*Corresponding author  \nTel: +90 352 207 66 66#33134  \nE mail: mehmetertas@erciyes.edu.tr  (M. Ertaş)  *Manuscript\nClick here to view linked References2 \n means the effective -field theory (EFT), which considers partial ly spin -spin correlation,  based \non the Glauber -type stochastic  dynamics. In particular, we stud y and obtain the dynamic \nphase transition (DPT) temperatures and present the dynamic phase diagrams.   \nWe should also emphasize that i n recent years, the EFT based the Glauber -type \nstochastic dynamics  have been applied to study the  DPT and present the dynamic phase \ndiagrams in the spin-1/2 [44 -47], spin -1[48-50], spin-3/2 [53] , spin-2 [52] and mixed spin -(2, \n5/2) [53] Ising systems in detail.  Moreover, in the past two decades, both experimental (see \n[54-58] and references therein) and theoretical  (see [4 4-53, 59 -71] and references therein) \ninvestigations  of the nonequilibrium critical phenomena, especially the DPT and dynamic \nphase diagrams,  have received a  great  deal of attention due to the reason that besides the \nscientific interests the study of DPT can also inspired new methods in materials  and \nmanufacturing process and processing  as well as in nanotechnology [ 72].  \nThe remainder of this article is organized as f ollows. In Section 2, we describe the \nmodel and its formulation.  The detailed numerical results and discussions are  given  in Section \n3. Finally Section 4 is devoted  to a summary and a brief conclusion.  \n2. Model and formulation  \nThe mixed spin -1/2 and spin -1 Ising model on a square lattice is described as a two -\nsublattice system, with spin variables σ i = ±1/2 and S j = ±1, 0 on the sites of sublattices A and \nB, respectively. The Hamiltonian of the system is given by  \nH\n \n2\ni j j i j\nij j i j= -J σ S -D S -h t σ + S\n\n   ,      (1) \nwhere <ij> indicates a summation over all pairs of nearest -neighboring sites. J is the \nexchange interaction  parameter between the two nearest -neighboring sites . D is Hamiltonian \nparameter and stand for the single -ion anisotropy (i.e. crystal  field). h(t) is a time -dependent \nexternal oscillating magnetic field and is given by  \nh(t) = h 0 sin(ωt),         (2) \nwhere h 0 and ω = 2πν are the amplitude and the angular frequency of the oscillating field, \nrespectively. The system is in contact with an isothermal heat bath at absolute temperature.  \nWithin the framework of the EFT with correlations, one can easily find the magnetizations \nmA, m B and the quadruple moment q B as coupled equations  for the mixed spin -(1/2, 1) Ising \nsystem as follows  \n                \n 24\nij jx=0Am 1 sinh J S cosh S F , J σ -1 x             (3a) \n                  \n     \n j i 10B=4\nxm cosh(J / 2)+2 sinh(J / 2) S σ G x ,       (3b) \n             \n     \n 4 2\nBj i2x=0σ q S cosh(J / 2)+2 sinh(J / 2) G x ,            (3c) \n \nhere \nx   is the differential operator. Hence, we obtained the set of coupled self -\nconsistent equations. On the other hand, the self -consistent equations for m and q that are \nobtained by using the MFT are not coupled in the spin -(1/2, 1) model  [43]. We should also \nmention that  we have not investigated the thermal behavior of q, since we do not include the 3 \n biquadratic exchange interaction parameter in Eq. (1).  The functions F(x),  G1(x), and G 2(x) \nare defined as  \n \n 11F x   tanh x  h22  \n,        (4a) \n \n12sinh[ (x h)]Gxexp(- D) 2cosh[ (x h)]   \n,                 (4b) \n \n            \n22cosh[ (x h)]Gxexp(- D) 2cosh[ (x h)]    .                  (4c) \n \nwhere \nBB β= 1/k T, k is the Boltzman factor.  Expanding the rig ht-hand sides of Eq s. (3a)-(c), \none can obtain the following equation:  \n \n \n2 3 4 5 6 7 8\nA 0 1 B 2 B 3 B 4 B 5 B 6 B 7 B 8 Bm a a m a m a m a m a m a m a m a m ,           (5) \n   \n2 3 4\nB 0 1 A 2 A 3 A 4 Am b b m b m b m b m .             (6) \nIn order to obtain the dynamic equations of motion for the average magnetizations , we \napply the Glauber -type stochastic dynamics [ 73] based on the master equation as follows:  \n               \n2 3 4 5 6 7 8\nA A 0 1 B 2 B 3 B 4 B 5 B 6 B 7 B 8 Bdm m a a m a m a m a m a m a m a m a m ,dt               (7) \nand  \n                                    \n2 3 4\nB B 0 1 A 2 A 3 A 4 Adm m b b m b m b m b m .dt                        (8) \nThe coefficients a i (i = 0, 1, …, 8) and b j (j = 0, 1, …, 4) can be easily calculated employing a \nmathematical relation \n exp ( ) ( ).  f x f x  \n3. Numerical results and discussions  \n \n In this section , first , we study the time dependence of average magnetizations in  the \nmixed spin -(1/2, 1) Ising with the  crystal field . The stationary solution of the dynamic \nequations is a periodic function of  \n, where \nwt,  with period 2π. T he time dependence \nmagnetization s\nAm ( )  and \nBm ( )  can be one of two types according to whether they comply  \nwith the following property or not: \nAAm ( 2 ) m ( )    and \nBBm ( 2 ) m ( )    . At the same \ntime, they can be one of two types according to whether they have or do not have the property\n  \n                                               \nAAm ( 2 ) m ( )     and \nBBm ( 2 ) m ( )    .      (9) \n \nIn order to obtain the dynami c phases , we solved Eqs. (7) and (8) by using the Adams -\nMoulton predictor -corrector method for a given set of parameters and initial values, as \npresented Fig. 1. From Fig. 1(a), one can see the paramagnetic phase (p) or solutions and this \nsolution satisfie s Eq. 9. The submagnetizations m A and m B are equal to each other and 4 \n oscillate around zero and are delayed with respect to the oscillating magnetic field.  Fig. 1(b) \nshows ferrimagnetic phase (i) or solution and this solution does not satisfy Eq. 9. In thi s \nsolution, the submagnetizations m A and m B are not equal to each other and m A and m B \noscillate around ±1/2 and ±1, respectively. In Fig. 1(c),  as m A oscillates around ±1/2, m B \noscillates around ±1, which corresponds to the i phase, with the initial values  of m A = 1/2 and \nmB = 1; also, m A and m B are equal to each other and they oscillate around zero, which \ncorresponds to the p phase with the initial values of m A= 0.0 and m B = 0.0. Thus, we obtain \ncoexistence solution (i + p) or the  i + p mixed phase.    \nIn order to obtain the dynamic phase boundaries among these phases and characterize \nthe nature of the dynamic phase transitions (continuous and discontinuous) , we have to \ninvestigate the thermal behavior of dynamic magnetizations (M A,B). They are defined as  \n \n2\nA,B A,B\n01M m ( )d .2\n  \n                                (10) \nwhere \n  represents wt. The thermal behavior of M and Q for several values of D/zJ and h 0/zJ \nare examined by combining the numerical methods of Adams -Moulton predictor corrector \nwith the Romberg integration  and the ir behaviors give the DPT point and the type of the  \ndynamic  phase transition.  A few interesting results are given in Figs. 2(a)-(c). In these figures, \nTc is second order phase transition temperature and T t is a first-order phase transition \ntemperature . In Fig. 2(a), MA and M B decrease  to zero continuously as the reduced \ntemperature (T/zJ)  increases,  a second -order  phase transition  temperature ( Tc = 0.3175 ) \noccurs . Figs. 2(b) and 2(c) have been obtained for D/zJ = -0.375, h0/zJ = 0. 5375  parameters \nand for two different initial values, namely M A = 1/2, M B = 1.0 for Fig. 2(a) and M A = 0.0, \nMB = 0.0 for Fig. 2(b) . In Fig. 2(b), both M A and MB undergo a first -order phase transition, \nbecause M A and MB decrease to zero discontinuously as the reduced temperature (T/zJ) \nincreases and the phase transition is from the i phase to the p phase. Fig. 2(c) demonstrates \nthat system does not undergo any phase transitions. Moreover, the p phase always occurs in \nFig. 2(c). Therefore,  the i + p mixed phase occurs below T t = 0.06, in which this fact that can \nbe seen  Fig. 3(b) for h 0/zJ = 0.5375 , explicitly.  \nWe can now construct  the dynamic phase diagrams of the model. The calculated \ndynamic phase diagrams are presented in the (T/zJ,  h0/zJ) planes  for various values of the \nreduced cry stal-field interaction ( D/zJ), illustrated  in Fig. 3. We also investigated the \ninfluence of long itudinal field frequency for ω = 0.25π and ω = 15π, respectively and plotted \none figure, namely Fig. 4.  In the figures, the solid and dashed lines represent the second - and \nfirst-order phase transition lines, respectively; a filed circle denotes the dynamic tricritical \npoint. Z, A, E, and TP are the dynamic zero -critical, multicritical , critical end, and triple  \npoints, respectively. The behavior of the phase diagrams are strongly depending on interaction \nparameters.  As seen from Fig. 3, the following  six main topological different types of phase \ndiagrams are found and we observed four interesting phenomena. (1) The phase diagrams \nexhibit the p, i, and i + p phase s in addition to A  and Z  special dynamic critical point s. (2) The \nsystem displays one dynamic tricritical behavior, seen in Figs. 3(a) -(c) as well as the re -\nentrant behavior, seen in Fig. 3(b) and 3(e).  We sh ould also mention that s everal weakly \nfrustrated ferromagnets, such as in manganite LaSr 2Mn 2O7 by electron and x -ray diffraction, \nin the bulk bicrystals of the oxide superconductor BaPb 1-xBixO3 and Eu xSr1-xS and \namorphous -Fe1-xMn x, demonstrate  the reentran t ph enomena  [74-76]. (3) The dynamic \ntricritical behavior is not exist for large negative values of D/zJ , seen in Figs. 3(d) -(f). (4) The \ndynamic phase boundaries among the p and i phase s are always second -order phase lines \nexcept the small values of the r educed temperature in  Fig. 3( c) and the  dynamic phase 5 \n boundaries among the p and i + p phase are always first -order phase lines except the high er \nvalues of the reduced temperatures in  Fig. 3(b).  \nWe also studied  the effect of longitudinal field frequency a nd presented in Figs. 4 (a) \nand (b) for D/zJ = -0.375, ω = 0.25π and D/zJ = -0.375, ω = 15π, respectively. If one \ncompares Fig. 4 (a) with Fig. 3(b), one can be see that i phase region become s smaller and the \ndynamic tricritical point occurs for low values of h 0/zJ and T/zJ . Moreover,  the i + p mixed \nphase and a special points disappear , as seen clearly in Fig. 4( a). For large values of w, the A \npoint disappears and i + p mixed phase region also occurs for high values of T/zJ . Moreover, \nthe system illustrate s one E and TP special  points as well as reentrant behavior , as seen clearly \nin Fig. 4(b).  \n4. Conclusions  \nThe dynamic phase transition and dynamic phase diagrams of the kinetic mixed spin -\n(1/2, 1)  model under a time oscillating longitudinal field are investi gated using the EFT with \ncorrelations. The EFT equation s of motion s for the average magnetization s are  obtained for \nthe square lattice by utilizing the Glauber -type stochastic process. The dynamic phase \ndiagrams contain the p and i fundamental phases and the i  + p mixed phase as well as Z, A, E, \nand TP special  points that strongly depend on the values of D/zJ. The system also shows \ndynamic tricritical and reentrant behaviors. We also find that the longitudinal field frequency \n(ω) greatly affects the dynamic behaviors  of the system . For example, if the value of ω is \nhigh,  the system shows TP and E special points instead of  A special points . \nFinally , in order to see the influence of the correlations,  by comparing the system with \nthe DMFT [ 43], the following features can be singled out: (i)  While one or two dynamic \ntricritical point s occur  within the DMFT  as seen in  Figs. 3(a), 3(b) and Figs. 3(d) -(f) of Ref. \n[43], only one dynamic tricritical point  exhibi ts within DEFT seen in Figs.  3(a)-(c). (ii) The \nsystem illustrates only Z and E special points within the DMFT but the system demonstrates \nZ, A, E, and TP special  points . (iii) The reentrant behavior is observed by using the DEFT \ncalculation, but not in the DMFT . (iv) The system does  not undergoes a dynamic phase \ntransition within DMFT for high negative values of D/zJ . (v) S ome of the first -order phase \nlines either disappear or shorten within the DEFT. These facts indicate partial spin -spin \ncorrelations, which mean that thermal fluctu ations , play an important role in the dynamic \ncritical behaviors of the systems. Lastly, we hope this study will contribute to the theoretical \nand experimental research on the dynamic magnetic properties of kinetic mixed Ising systems \nas well as to researc h on magnetism.  \nAcknowledgments  \nThis work was supported by Erciyes University Research Funds, Grant No.  FBA -2013-4411.  \nReferences  \n[1] Kaneyoshi  T 1989 Solid State Commun.  69 91 \n[2] Kaneyoshi  T, Jăšcur M 1993  Physica A  195 474 \n[3] Buendía  G.M., Novotny  M. A. 1997  J. Phys.: Condens. Matter  9 5951  \n[4] Benayad N, Dakhama  A 1997  Phys. Rev. B  55 12276  \n[5] Dakhama A, Benayad N 2000  J. Magn. Magn. Mater  213 117 \n[6] Boechat B , Filgue iras R A, Cordeiro C , and  Branco  N S 2002  Physica A  304 429 \n[7] Duh A, Liu J and Wang B 2004 Physica  A 342 583  \n[8] Kasama T , Muraoka  Y and Idogaki  T 2006 Phys. Rev. B  73 214411  \n[9] Strečka  J, Dely J and Canova  L 2009 Physica  A 388 2394    6 \n [10] Canko O , Erdinç  A, Taşkın  F and Atış M 2011  Phys. Lett. A  375 508 \n[11] Şarlı  N 2013 Physica B 411 12 \n[12] Schofield  S L and Bowers  R G, J. Phys. A: Math. Gen.  14 2163   \n[13] Kaneyoshi T 1987 J. Phys. Soc. Jpn. 56 2886   \n[14] Plascak  J A 1993 Physica A  198 655 \n[15] Yan S L, Yang  C Z 1998 Phys. Stat. Sol. (b)  208 151 \n[16] Vieira A P, de Carvalho  J X and Salinas  S R 2001  Phys. Rev. B  63 184415  \n[17] Htoutou  K, Ainane  A and  Saber  M 2004  J. Magn. Magn. M ater. 269 245  \n[18] Kasama T, Muraoka  Y and  Idogaki  T 2006  Phys. Rev. B  73 214411  \n[19] Zhang  Y F, Yan S L 2008  Phys. Lett. A  372 2696  \n[20] Wu H, Wei  G, Zhang  P, Yi G and  Gong  W 2010  J. Magn. Magn. Mater . 322 3502  \n[21] Benayad N and  Ghliyem M 2012 Physica B  407 6 \n[22] Xu  C Q and  Yan S L 2013  J. Magn. Magn. Mater . 345 261 \n[23] Hida K, Takano  K and  Suzuki  H 1993  J. Phys. Soc. Jpn. 79 1554  \n[24] Chandra  V R, Ivanov  N B and Richter  J 2010  Phys. Rev. B  89 024409  \n[25] Gong  S S, Zhao  Y and Su G 2010  Phys. Rev. B  81 214431  \n[26] Galisova L and Strečka  J 2011  Condensed Matter Physics  14 13002  \n[27] Gonçalves  L L 1986  Phys. Scr. 33 192 \n[28] da Silva  N R and Salinas S R 1991 Phys. Rev. B  44 852 \n[29] Dakhama A 1998  Physica A  252 225 \n[30] Albayrak E and Keskin  M 2003  J. Magn. Magn. Mater . 261 196 \n[31] Jaščur M and Strečka J 2005  Cond. Mat. Phys.  8 869 \n[32] Ekiz C and Erdem R 2006  Phys. Lett. A  352 (2006)  291. \n[33] Albayrak E, Bulut  T and  Yilmaz  S 2007  J. Stat. Phys.  127 967 \n[34] Albayrak E and Yilmaz  S 2008  Physica A  387 1173  \n[35] Ekiz C, Strečka  J and Jaščur M 2011  J. Magn. Magn. Mater  323 493 \n[36] Strečka  J and Ekiz C 2012 Physica A  391 4763  \n[37] Godoy M and Figueiredo W 2000  Phys. Rev. E  61 218 \n[38] Godoy M  and Figueiredo W  2002  Phys. Rev. E  65 026111  \n[39] Godoy M  and Figueiredo W  2002  Phys. Rev. E  66 036131 . \n[40] Godoy M  and Figueiredo W  2004 Braz. J. Phys.  34 422 \n[41] Godoy M and Figueiredo  W 2004 Physica A  339 392 \n[42] Buendía  G M and Machado  E 1998  Phys. Rev. E  58 1260  \n[43] Keskin M, Canko O and Polat  Y 2008  J. Korean Phys. Soc.  53 497 \n[44] Shi X and Wie G 2010  Phys. Lett. A 374 1885  \n[45] Deviren B , Canko O and Keskin M 2010  Chin. Phys. B  19 050518  \n[46] Yüksel Y, Vatansever  E, Akıncı  Ü and  Polat  H 2012  Phys. Rev. E  85 051123  \n[47] Akıncı  Ü, Yüksel Y , Vatansever  E and Polat  H 2012  Physica A 391 5810  \n[48] Shi X and  Wie G 2012  Physica A  391 29 \n[49] Deviren B and Keskin M  2012  J. Magn. Magn. Mater. 324 1051  \n[50] Gulpinar G, Vatanseve r E and  Agartioglu  M 2012  Physica A  391 3574  \n[51] Ertaş M , Kocakaplan Y and  Keskin  M 2013  J. Magn. Magn. Mater.  348 113 \n[52] Ertaş M, Deviren B and Keskin M  2012  J. Magn. Magn. Mater.  324 704  \n[53] Ertaş M, Deviren B and Keskin M  2012  Phys. Rev. E  86 05111 0 \n[54] Gedik N, Yang  D S, Logvenov  G, Bozovic I and Zewail  A H 2007  Science 316 425 \n[55] Robb  D T, Xu Y H, Hellwing  O, McCord  J, Berger  A, Novotny  M A and Rikvold  P A  \nPhys. Rev. B  78 134422  \n[56] Arregi  J J, Idigoras O, Vavassori  P and Berger A 2012 Appl. Phys. Lett.  100 262403  \n[57] Berger A, Idigoras O and  Vavassori  P 2013  Phys. Rev. Lett.  111 190602  7 \n [58] Chan L, Li H, Feng  B, Ding  Z, Qui  J, Cheng  P, Wu  K and Meng  S 2013  Phys. Rev. Lett.  \n110 085504  \n[59] B.K. Chakrabarti, M. Acharyya, Rev. Mod. Phys. 71 (19 99) 847.  \n[60] Korniss G , White  C J, Rikvolt  P A and  Novotny  M N 2000 Phys. Rev.  63 01612   \n[61] Keskin M , Canko  O and Temizer U 2005  Phys. Rev. E  72 036125   \n[62] Acharyya M 2005  Int. J. Mod. Phys. C  16 1631  \n[63] Robb D T, Rikvold PA, Berger  A and  Novotny  M A 2007 Phys. Rev. E  76 021124   \n[64] Deviren B, Keskin M and  Canko  O 2009  J. Magn. Magn. Mater. 321 458  \n[65] M. Keskin, E. Kantar, O. Canko, Phys. Rev. E 77 (2008) 051130.  \n[66] Keskin M and  Ertaş  M 2009  Phys. Rev. E  80 061140  \n[67] Deviren S A and Albayrak  A 2010  Phys. Rev. E  82 022104  \n[68] Ertaş M and Keskin M  2010  J. Stat. Phys.  139 333 \n[69] Acharrya M 2011 J. Magn. Magn. Mater.  323 2872 \n[70] Ertaş M and Keskin M  2012 Phys. Lett. A 376 2455  \n[71] Park H and  Pleimling  M 2013  Phys. Rev. E  87 032145  \n[72] Cross  M C and Hohenberg  P C 1993  Rev. Mod. Phys.  65 851 \n[73] Glauber R J 1963  J. Math. Phys.  4 294 \n[74] Li J Q, Matsui Y, Kimura T and  Tokura  Y 1998  Phys. Rev. B  57 R3205  \n[75] Sata T, Yamaguchi  T, Matsusaki  K and Membr  J 1995 Science 100 229 \n[76] Binder K and Young  A P 1986  Rev. Mo dern Phys.  58 801 \n \nList of the figure captions  \n \nFig. 1.  (Color online)  Time variations of the average magnetization s (mA and m B):  \na) Exhibiting a paramagnetic phase ( p), D/zJ = 0.25, h 0/zJ = 0.625, and T/zJ = 0. 25. \nb) Exhibiting a ferrimagnetic  (i) phase, D/zJ = -0.375, h0//zJ = 0. 375, and T/zJ = 0.125 .  \nc) Exhibiting a mixed  (i + p) phase, D/zJ = -2.2, h0//zJ = 0.175 , and T/zJ = 0.0 3. \n \nFig. 2.  (Color online) The reduced temperature dependence of the dynamic magnetization s \nMA and M B and Tt and T C are the first -order and second -order phase tra nsition \ntemperatures from the i phase to the p phase . \na) Exhibiting a second -order phase transition from the i phase to the p phase for D/zJ = -\n1.0, h/zJ = 0.1, ω = 2.0π; Tc is found as 0.3175 .  \nb) and c) Exhibiting a first-order phase transition from the i + p  phase to the p phase for \nD/zJ = -0.375 , h0/zJ = 0. 5375 ; 0.06 is found T t.  \nFig. 3.  Dynamic phase diagrams of the mixed spin -(1/2, 1)  model  in the (T/zJ,  h/zJ) plane. \nThe paramagnetic ( p), ferrimagne tic (i), fundamental phases and the i + p  mixed phase \nare obtained. Dashed and solid lines represent the first - and second -order phase \ntransitions, respectively and the dynamic tricritical points are indica ted with solid \ncircles. Z, A, E, and TP special po ints are the dynamic zero temperature, multicritical, \ncritical end point, and triple points, respectively. For ω = 2.0π and a) D/zJ = 0.25, b) \nD/zJ =-0.375, c) D/zJ = -1.0, d) D/zJ = -2.0, e) D/zJ = -2.2, and f) D/zJ = -2.5.  8 \n Fig. 4.  Same as Fig. 3(b), but  a) for ω = 0.25π and b) for ω = 25π.  \n\n\n\n\n\n\n\n\n\n\nFigure 1 0 1 02 03 04 05 06 0mA(), mB()\n-1.0-0.50.00.51.0Ferrimagnetic Phase0 1 02 03 04 0mA(), mB()\n-1.0-0.50.00.51.0\nmA = mB\nmA\nmAmB\nmBParamagnetic Phase\n0 1 02 03 04 05 06 0mA(), mB()\n-1.0-0.50.00.51.0Mixed Phase\nmA\nmAmB\nmBmA = mB( a )\n( b )\n( c )Fig.1T/zJ0.00 0.10 0.20 0.30MA, MB\n0.000.250.500.751.00\nT/zJ0.00 0.08MA, MB\n0.00.51.0\nT/zJ0.00 0.08MA, MB\n0.00.5\nMAMB\nMA = MBMAMB(a)\n(b) (c)TC\nTt\n\nFigure 2 Fig.20.000 0.125 0.250 0.375h0/zJ\n0.0000.1250.2500.3750.5000.625(a)\n0.000 0.125 0.250 0.3750.0000.1250.2500.3750.5000.625\nii + p\nip p\ni + pA\ni + p\nA\n0.000 0.125 0.250 0.375h0/zJ\n0.0000.1250.2500.3750.500\nip\nT/zJ0.0000 0.0625 0.1250h0/zJ\n0.0000.1250.250\npi + pp(b)\n(c)\n0.000 0.053 0.105 0.158 0.2100.0000.1250.2500.375\nipZ(d)\n(e)\nT/zJ0.000 0.025 0.0500.0000.1250.250\ni + p p(f)\n\n   Figure 3 Fig.3\nT/zJ0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40h0/zJ\n0.0000.1250.2500.3750.5000.625\n0.0 0.1 0.20.500.55T/zJ0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40h0/zJ\n0.0000.1250.2500.3750.500\nip\npw = 0.25\nw = 15\nETP\ni i + pi + p\ni + p\n \nFigure 4 Fig.4"    },    {        "title": "2211.08048v2.Nonlinear_sub_switching_regime_of_magnetization_dynamics_in_photo_magnetic_garnets.pdf",        "content": "1 \n Nonlinear s ub-switching regime of magnetization  dynamics in photo -magnetic  garnets  \nA. Frej, I. Razdolski, A. Maziewski, and A. Stupakiewicz  \nFaculty of Physics, University of Bialystok, 1L Ciolkowskiego, 1 5-245 Bialystok, Poland  \nAbstract.  We analyze, both experimentally and numerically, the nonlinear regime of the \nphoto -induced coherent magnetization dynamics in cobalt -doped yttrium iron garnet films. \nPhoto -magnetic excitation with femtosecond laser pulses reveals a strongly nonlinear \nrespo nse of the spin subsystem with a significant increase of the effective Gilbert damping. By \nvarying both laser fluence and the external magnetic field, we show that this nonlinearity \noriginates in the anharmonicity of the magnetic energy landscape. We numer ically map the \nparameter workspace for the nonlinear photo -induced spin dynamics below the photo -\nmagnetic switching threshold. Corroborated by numerical simulations of the Landau -Lifshitz -\nGilbert equation, our results highlight the key role of the cubic sy mmetry of the magnetic \nsubsystem in reaching the nonlinear spin precession regime. These findings expand the \nfundamental understanding of laser -induced nonlinear spin dynamics as well as facilitate the \ndevelopment of applied photo -magnetism.  \n1. INTRODUCTION  \nRecently,  a plethora of fundamental mechanisms for magnetization dynamics induced by \nexternal stimul i at ultrashort time scale s has been actively d iscussed  [1-5]. The main interest \nis not only in the excit ation of  spin precession but in the switching of ma gnetization between \nmultiple stable states, as it open s up rich possibilities for non-volatile magnetic  data  storage \ntechnology . One of t he most intriguing example s is the phenomenon  of ultrafast switching of \nmagnetization with laser pulses. Energy -efficie nt, non -thermal mechanisms of laser -induced \nmagnetization switching require a theoretical understanding of coherent magnetization \ndynamics in a strongly non -equilibrium environment  [6]. This quasiperiodic motion  of \nmagnetization  is often mode led as  an oscillator where the key parameters , such as frequency \nand damping , are considered within the framework of  the Landau -Lifshit z-Gilbert  (LLG)  \nequation  [1, 7] . Although it is inhe rently  designed to describe small -angle spin precession \nwith in the  linear approximation, there are attempts to extend this formalism into the \nnonlinear regime where the precession parameters become angle -dependent  [8]. This is \nparticularly important in light of the discovery of the so -called p recessional switching , where  \nmagnetization , having been  impulsively driven  out of equilibrium, ends its precessional \nmotion in a different minimum of the potential energy  [6, 9 -11]. Obviously,  such \nmagnetization trajectories are characterized by very large precession angles  (usually on the \norder of tens of  degrees ). It is, however, generally believed that the magnetization excursion \nfrom the equilibrium of about 10 -20 degrees is already sufficient for the violat ion of the linear \nLLG approach  [12, 13] . Thus, an intermediate regime under the switching stimulus threshold \nexists, taking a large area in the phase space and presenting an intriguing c hallenge in \nunderstanding  fundamental spin dynamics.  \nAn impulsive optical stimulus often results in a thermal excitation mechanism, inducing \nconcomitant temperature variations , which can impact the parameters of spin precession  [14-\n16]. This highlights the  special role of the non -thermal optical mechanisms  of switching  [17-2 \n 19]. Among those , we outline photo -magnetic excitation , which has been recently \ndemonstrated in dielectric Co -doped YIG  (YIG:Co)  films  [6, 11] . There, laser photons at a \nwavelength of 1300 nm resonantly excite the 5E → 5T2 electron transition s in Co -ions, resulting \nin an emerging photo -induced magnetic anisotropy  and thus in a highly efficient excitation of \nthe magnetic subsystem  [6]. This photo -induced  effective  anisotropy field features  a near ly \ninstant aneous  rise time (within the femtosecond pump laser pulse  duration), shifting the \nequilibrium direction for the magnetization and thus triggering its l arge -amplitude precession.  \nIn the sub -switching regime  (at excitation strengths just below the switching threshold ), the \nfrequency of the photo -induced magnetization precession has been shown to depend on the \nexcit ation wavelength  [20]. However, nonlinearities in magnetization dynamics in the sub-\nswitching regime have  not yet been described in detail, and the underlying mechanism  for the \nfrequency variations  is not understood.  \nIn this work , we systematically examine the intermediate sub -switching regime characterized \nby large angles of magnetization precession and the nonlinear response of the spin system to \nphoto -magnetic excitations. We show a strong increase of the effective Gilbert damping at \nelevated lase r-induced excitation levels and quantify its nonlinearity within the existing \nphenomenological formalism  [8]. We further map the nonlinear regime in the phase space \nformed by the  effective  photo -induced anisotropy field  and the external magnetic field.   \n \nFig. 1. Sketch of m agnetization dynamics at various  stimulus  levels . Owing to the highly nonlinear \nmagnetization dynamics in the switching regime, the nonlinearity onset manifests in the sub -switching \nregime too . \nThis paper is organized in the following order: in the first part, we describe the details of the \nexperiment for laser -induced large -amplitude magnetization precession. Next, we present the \nexperimental results, followed  by the fitting analysis . Then, we complement our findings with \nthe results of  numerical simulation of the photo -magnetic spin dynamics. Afterward , we \ndiscuss the workspace of parameters for the sub-switching regime of laser -induced \nmagnetization precession. The paper ends with c onclusions.  \n \n3 \n 2. EXPERIMENTAL DETAILS  \nThe experiments were done on  a 7.5 μm -thick YIG:Co  film with a composition of \nY2CaFe 3.9Co0.1GeO 12. The Fe ions at the tetrahedral and octahedral sites are replaced by Co -\nions [21]. The sample was grown by liquid -phase epitaxy on a 400  μm-thick  gadolinium gallium \ngarnet (GGG) substrate. It exhibits eight possible magnetization states along the garnet’s cubic \ncell diagonals due to its cubic magnetocrystalline anisotropy ( 𝐾1=−8.4×103 𝑒𝑟𝑔/𝑐𝑚3) \ndominating the energy landscape over the uniaxial anisotropy ( 𝐾𝑢=−2.5×103 𝑒𝑟𝑔/𝑐𝑚3). \nOwing to the 4 ° miscut, additional in -plane anisotropy is introduced, tilting the magnetization \naxes and  resulting in slightly lower energy of half of the magnetiz ation states in comparison \nto the others. In the absence of the external magnetic field, the equilibrium magnetic state \ncorresponds to the magnetization in the domains close to the <111> -type directions in YIG:Co \nfilm.  Measurements of the Gilbert damping  𝛼 using the fe rromagnetic resonance technique \nresulted in  𝛼≈0.2. This relatively high damping is inextricably linked to the C o dopants  [22-\n24]. \nThe n onlinearity  of an oscillator is usually addressed by varying the intensity of the stimulus \nand comparing the response of the system under study. Here , we investigated the nonlinear \nmagnetization dynamics by varying the optical pump fluence and , thus , the strength of the \nphoto -magnetic effective field driving the magnetization out of the equilibrium. We \nperfor med systematic studies in various magnetic states of YIG:Co governed by the magnitude \nof the  external magnetic field s. The magnetic field  𝐻⊥ was applied perpendicular to the sample \nplane and in -plane magnetic field  𝐻 was applied along the [110] direction of the YIG:Co crystal \nby means of an electromagnet. Owing to the introduced miscut, the studied YIG:Co exhibits \nfour magnetic domains at 𝐻=0 [25]. The large jump at an in-plane magnetic f ield close to \nzero shows the magnetization switching in the domain structures between four magnetic \nphases. The optical spot size in this experiment was around 100 μm while the size of smaller \ndomains was around 5 μm, resulting in the spatial averaging of the domains in the \nmeasurements. This behavior of magnetic domains was dis cussed and visualized in detail  by \nmagneto -optical Faraday effect in our previous papers  [6, 25] . With an increase of the \nmagnetic field up to a round 𝐻=0.4 kOe, larger and smaller domains are formed due to the \ndomain wall motion, eventually resulting in a formation of a single domain in a noncollinear \nstate.  Upon further increase, the magnetization rotates towards the direction of the applied \nfield until a collinear state  with in -plane magnetization orientation  is reached  at about 2 kOe \n(see Fig. 2) . 4 \n  \nFig. 2. Magnetization reversal using static magneto -optical Faraday effect under perpendicular H (a) \nand in -plane H (b) magnetic fields. The grey area indicates the magnetization switching in magnetic \ndomain structure  [25]. The green area shows the saturation range with a collinear state of \nmagnetization.  \nDynamic nonlinearities in the magne tic response were studied employing the pump -probe \ntechnique relying on the optical excitation of the spin precession in YIG :Co film. The pumping \nlaser pulse  at 1300 nm , with  a duration of 50 fs and a repetition rate of 500 Hz , induce d spin \ndynamics through  the photo -magnetic mechanism  [6]. The transient Faraday rotation of the  \nweak probe beam  at 625 nm was used to monitor  the dynamics of the  out-of-plane  \nmagnetization  component Mz. The diameter  of the pump spot was around 1 40 μm , while  the \nprobe beam was focused within the pump spot  with a size of around 50 μm . The fluence of \nthe pump beam was varied in the range  of 0.2  6.5 mJ/cm2, below the switching threshold of \nabout 39 mJ/cm2 [20]. At 1300 nm pump wavelength, the optical absorption in our garnet is \nabout 12%. An estimation of the temperature increase ΔT due to the heat load for the laser \nfluence of 6.5 mJ/cm2 results in ΔT <1 K (see Methods of Ref. 6). The polarization of both \nbeams was linear and set along  the [100] crystallographic direction in YIG:Co for the pump and \nthe [010]  direction  for the probe  pulse . The experiments  were done  at room temperature . At \neach magnetic field, we performed a series of laser fluence -dependent pump -probe \nexperiments measuring the transients of an oscillating magnetization component  normal to \nthe sample plane . We then used a phenomenological damped oscillator response function to \n5 \n fit the experimental data and retrieve the fit  parameters such as a mplitude, frequency, \nlifetime and effective damping. In what follows, we analyze the obtained nonlinearities in the \nresponse of the magnetic system and employ numerical simulations to reproduce the \nexperimental findings.  \n \n3. RESULTS  \nA. Time -resolved photo -magnetic dynamics  \nIn order to determine the characteristics of the photo -magnetic precession , we carried out \ntime -resolved measurements  of a transient Faraday rotation  ∆𝜃𝐹 in YIG:Co  film. Fig. 3(a-d) \nexemplifies a few typical datasets obtained for four v arious pump fluences (between 1.7 and \n6.5 mJ/cm2) in magnetic fields of various strength s. A general trend demonstrating a decrease \nof the precession amplitude and an increase of  its frequency is seen  upon the magnetic field \nincrease . To get further insights into the magnetization dynamics, these datasets were fitted \nwith a damped sine function on top of a non -oscillatory, exponentially decaying background : \n∆𝜃𝐹(∆𝑡)=𝐴𝐹sin(2𝜋𝑓∆𝑡+𝜙)exp (−∆𝑡\n𝜏1)+𝐵exp (−∆𝑡\n𝜏2),   (1) \nwhere 𝛥𝑡 is pump and probe time difference, 𝐴𝐹 is the amplitude, 𝑓 is the frequency , 𝜙 is the \nphase, 𝜏1 is the decay time of precession, and 𝜏2 is the decay time of the background with an \namplitude 𝐵. \n \nFig. 3. Time -resolved Faraday rotation  at different  magnetic fields  H (a-d) and laser fluence s (I1-I4 \ncorrespond  to 1.7, 3.2, 5.0,  and 6.5 mJ/cm2, respectively) . The normalized MZ on the vertical axis is \ndefined as  ΔF/max, where max is obtained for saturation magnetization rotation at H (see. Fig. 2a). \nThe curves are offset  vertically without rescaling.  The s olid lines are fittings with the  damped sine \nfunction  (Eq. 1) .  \n6 \n  \nFig. 4. Photo -magnetic  precession parameters as a  function of p ump fluence  in different  external \nmagnetic field  H: a) amplitude  of the Faraday rotation AF, b) frequency  of the precession , and c) \neffective damping. Different colors correspond to different external magnetic fields. The s olid lines are \nthe linear fits  where applicable , while the dashed lines are the visual guides.  Some of the error bars \nare smaller than the data point symbols.  \nAt low applied fields  𝐻<1 kOe, where the photo -magnetic  anisotropy field ( 𝐻𝐿) contribution \nto the total effective magnetic field is the strongest, the largest magnetization precession \namplitude is observed. Figure 4 show s the most important parameters of the magnetiza tion \nprecession, that is, amplitude, frequency and effective damping  (Fig. 4a -c). The latter is \nobtained from the frequency and the lifetime as  (2𝜋𝑓𝜏1)−1. Although the amplitude \ndependence on the pump fluence is mostly linear, the other two parameters exhibit a more \ncomplicated dependence, which is indicative of the noticeable nonlinearity in the magnetic \nsystem. In particular,  at 𝐻=0.4 and 0.5 kOe, we observe d an increase in the effective \ndamping with laser fluence, resulting in a faster decay of the magnetic precession. This is \nfurther corroborated by the frequency decrease seen in Fig. 4b. It is seen that the behavior of \nthe magnetic subsystem is noticeably dissimilar at low ( below 1  kOe) and high  (above 2 kOe)  \nmagnetic fields. At higher magnetic fie lds 𝐻>1 kOe we were unable to observe nonlinear \nmagnetization response at pump fluences up to 10 mJ/cm2. This is indicative of a significant \ndifference in the dynamic response in the collinear and noncollinear states of the magnetic \nsubsystem.  \n \n4. Nonlinear precession of magnetization in anisotropic cubic crystal s \nThe data shown in Fig. 4c clearly indicates the nonlinearity in the magnetic response \nmanifesting in the increase of the effective damping with the excitation (laser) fluence. \nPreviously, similar behavior was found in a number of metallic systems  [26-29] and quickly \nattributed to laser heating. Interestingly, Chen et al . [30] found a decrease of the effective \ndamping with laser fluence in FePt, while invoking the temper ature dependence of magnetic \ninhomogeneities to explain the results. There, the impact of magnetic inhomogeneity -driven \ndamping contribution exhibits a similar response to laser heating and an increase in the static \nmagnetic field. A more complicated mecha nism relying on the temperature -dependent \n7 \n competition between the surface and bulk anisotropy contributions and resulting in the \nmodification of the effective anisotropy field has been demonstrated in ultrathin Co/Pt \nbilayers  [31, 32] . \nNonlinear spin dynamics is a rapidly developing subfield enjoying rich prospects for ultrafast \nspintronics  [33]. Importantly, all those works featured thermal excitation of magnetization \ndynamics in metallic, strongly absorptive systems. In  stark contrast, we argue that the \nmechanism in the Co-doped YIG studied here is essentially non -thermal. This negligible \ntemperature change  ΔT is unable to induce significant variations of the parameters in the \nmagnetic syst em of YIG:Co (T N=450 K), thus ruling out the nonlinearity mechanism discussed \nabove. Rather, we note the work by M üller et al.  [34], where the non -thermal nonlinear \nregime of magnetization dynamics in CrO 2 at high laser fluences was ascribed to the spin -wave \ninstabilities at large precession amplitudes  [35]. We also note the recently debated and \nphysically rich mechanisms of magnetic nonlinearities, such as spin inertia  [36-39] and \nrelativistic effects  [40, 41] . Yet, we argue that in our case of a cubic magnetic anisotropy -\ndominated energy landscape, a much simpler explanation for the nonlinear spin dynamics can \nbe suggested. In particular, we  attribute the amplitude -dependent effective dampin g to the \nanharmonicity of the p otential well for magnetization . \n \nFig. 5. Energy landscape  as a function of the polar angle  𝜃𝜑=45°in the linear (𝐻=2.5 kOe, green) \nand nonlinear (𝐻=0.4 kOe, red) precession regime s. The d ashed lines are the parabolic fits in the \nvicinity of the minima . 𝜃 is the polar angle of magnetization  orientation  measured from the normal to \nthe sample plane  along  the [001] axis in YIG:Co .  \nWe performed  numerical calculations of the energy  density  landscape 𝑊(𝜃,𝜑):  \n𝑊(𝜃,𝜑)=𝑊𝑐+𝑊𝑢+𝑊𝑑+𝑊𝑧    (2) \ntaking into account the following terms in the free energy of the system:  the Zeeman energy  \n𝑊𝑧=−𝑴∙𝑯, demagnetizing field  term 𝑊𝑑=−2𝜋𝑀𝑠2sin2𝜃, cubic 𝑊𝑐=𝐾1∙\n(sin4𝜃sin2𝜑cos2𝜃+sin2𝜃cos2𝜃cos2𝜑+sin2𝜃cos2𝜃sin2𝜑) and uniaxial anisotropy \n𝑊𝑢=𝐾𝑢sin2𝜃 (𝜃 and 𝜑 are the polar and azimuthal angles, respectively ). In the calculations, \nwe assume 𝐾1=−9∙ 103 erg/cm3, 𝐾𝑢=−3∙103 erg/cm3, and 𝑀𝑠 is the saturation \n8 \n magnetization of 7.2 Oe [25]. Then, following  [8] and [42], we calculate the precession \nfrequency 𝑓 and the effective damping 𝛼𝑒𝑓𝑓: \n𝑓=𝛾\n2𝜋𝑀𝑠sin𝜃√𝛿2𝑊\n𝛿𝜃2𝛿2𝑊\n𝛿𝜑2−(𝛿2𝑊\n𝛿𝜃𝛿𝜑)2\n,    (3) \n𝛼𝑒𝑓𝑓=𝛼0𝛾(𝛿2𝑊\n𝛿𝜃2+𝛿2𝑊\n𝛿𝜑2sin−2𝜃)\n8𝜋2𝑓𝑀𝑠,     (4) \nwhere  the 𝛾 is gyromagnetic ratio , and 𝛼0 is the Gilbert damping in YIG:Co  [23, 24] . In Fig. 5, \nwe only show the total energy as a function  of the polar angle 𝜃, to illustrate the \nanharmonicity of the potential at small external in -plane magnetic fields. Experimental data \nand calculations of the energy  𝑊(𝜃,𝜑) have been published in Refs.  [25, 43] . There, it is seen \nthat at relative ly small external magnetic fields  canting  the magnetic state , the proximity of a \nneighboring energy minimum (to the right) effectively modifies the potential well for the \ncorresponding o scillator (on the left) , introducing an anharmonicity . On the other hand, at \nsufficiently large magnetic fields, whic h, owing to the Zeeman energy term, modify the \npotential such that a single minimum emerges (shown in Fig. 5 in green), no nonlinearity is \nexpected. This is also in line with the decreas ing impact of the cubic symmetry in the magnetic \nsystem, which is res ponsible for the anharmonicity of the energy potential.  \nTo get yet another calculated quantity that can be compare d to the experiment, we \nintroduced the photo -magnetically in duced effective anisotropy term  𝐾𝐿. This contribution \ndepends on the  laser fluence  I through the effective light -induced field 𝐻𝐿∝𝐼 as: \n𝐾𝐿=−2𝐻𝐿𝑀𝑠cos2𝜃     (5) \nThe presence of this term displaces the equilibrium for net magnetization. The equilibrium \ndirection s can be obtained by minimizing the total energy with and without the photo -\nmagn etic anisotropy term. Then, k nowing the angle between the perturbed and unperturbed \nequilibrium directions for the magnetization, we calculate d the precession amplitude 𝐴. We \nnote the difference between the amplitudes 𝐴𝐹, which refers to the Faraday rotation of the \nprobe beam, and 𝐴 standing for the opening angle of magnetization precession. Alth ough both \nare measured in degrees, their meaning is different.  \nHaving repeated this for a few levels of optical excitation, we obtain ed a linear slope of the \namplitude vs  excitation  strength  dependence. Figure 6 (a-c) illustrates the amplitude, \nfrequency , and (linear ) effective  damping as a function of the external magnetic field. The \nagreement between the calculated parameters and those obtained from fitting the \nexperimental data is an impressive indication of the validity of our total energy approach. \nFurther, the linear effective damping value of 𝛼≈0.2 obtained in the limit of  strong field s, is \nin good agreement with the values known for our Co -doped YIG from previous works  [6, 24] . \nIn principle, the effective damping  in garnets can increase  towards lower magnetic fields.  \nConventionally attributed to the extrinsic damping contributions, this behavior has been \nobserved in rare -earth iron garnets before as well and ascribed to the generation of the \nbackward volume spin w ave mode by ultrashort laser pulses  [44]. It is worth noting that there \nis no nonlinearity phenomenologically embedded in the approach given above.   9 \n  \nFig. 6. Photo -magnetic precession parameters  at various magnetic fields: amplitude (a), frequency (b) , \nand (linear) effective damping (c). The points are from the experimental data, the solid lines are \ncalculated as described in the text. The dark rectangular points are obtained in the FMR experimen ts. \nThe g rey shaded  area indicates the presence of a  domain st ate (DS). The g reen  shaded  area show s the \nmagnetization saturation state . \nYet, the data presented in Fig. 4c indicates the persistent nonlinear behavior of the effective \ndamping. To clarify the role of the potential anharmonicity, we fitted the potentials 𝑊(𝜃,𝜑) \nusing a parabolic function with an anharmonic term : \n𝑊(𝑥)=𝑊0+𝑘[(𝑥−𝑥0 )2+𝛽𝑥(𝑥−𝑥0)4]    (6) \nHere 𝑥=𝜃 or 𝜑, and 𝛽𝑥 is the anharmonicity parameter. We calculated it independently for \n𝜃 and 𝜑 for each dataset of 𝑊(𝜃,𝜑) obtained at different values of the external magnetic \nfield 𝐻 by fitting the total energy with Eq. (6) in the vicinity of the energy minimum  (Fig. 5) . \nThis anharmonicity should be examined on equal footing with the no nlinear damping \ncontribution. To quantify the latter, we follow the approach by Tiberkevich & Slavin  [8] and \nanalyze the effective damping dependencies on the precession amplitude by means of fitting \na second -order polynomial to them : \n𝛼=𝛼0+𝛼2𝐴2.      (7) \nThe examples of th e fit curves are shown in Fig. 7 a, demonstrating a good quality of the fit \nwithin a  certain range of the amplitudes  𝐴 (below 45 ). It should, however, be noted that the \nmodel in Ref.  [8] has been developed for the in -plane magnetic anisotropy, and thus its \napplicability for our case is limited. This is the re ason why we do not go beyond the amplitude \ndependence of the effective damping and do not analy ze the frequency dependence on 𝐴 in \n10 \n the limit of strong effective fields.  We note that the amplitude  𝐴, the opening angle of the \nprecession, should be understood as a mathematical parameter only, and not as a true \nexcursion angle of magnetization obtained in the real experimental conditions. There, large \neffective Gilbert damping values and a short decay t ime of the photo -magnetic anisotropy \npreclude the excursion of magnetization from its equilibrium to reach these 𝐴 values.  \n \nFig. 7. a) Effective damping in the linear and nonlinear precession regimes  of the precession amplitude \n𝐴. The lines are the second -order polynomial fits with Eq. (7). b ) Magnetic field dependence of the \nnonlinearity parameters: n onlinear damping coefficient 𝛼2 (points, obtained from experiments)  and \nthe 𝑊(𝜃) potential  anharmonicity  normalized  𝛽𝜃 (red line , calculated ).  \nWe note that the anharmonicity parameter 𝛽𝑥 calculated for the  W(θ) profiles was found to \nbe a few orders of magnitude larger than that obtained for W(𝜑). This difference in the \nanharmonicity justifies our earlier decision to focus on the shape of W(θ) potential only (cf. \nFig. 5). This means that the potential for magnetization in the azimuthal plane is muc h closer \nto the parabolic shape  and much larger amplit udes of the magnetization precession are \nrequired for it to start manifesting nonlinearities in dynamics. As such, we only consider the \nanharmonicity 𝛽𝑥 originating in the W(θ) potential energy. I n Fig. 7b, we compare the 𝛽𝜃 (red \nline) and 𝛼2 (points) dependencies on the external in -plane magnetic field. It is seen that its \ngeneral shape is very similar, corroborating our assumption that the potential anharmonicity \nis the main driving force behind the obse rved nonlinearity. We argue that thanks to the c ubic \nmagnetic anisotropy in YIG:Co  film, the potential anharmonicity -related mechanism of \nnonlinearity allows for reaching the nonlinear regime at moderate excitation levels.  \n \n5. Simulation s of laser -induced magnetization dynamics  \n11 \n To further prove that the ob served nonlinearities in magnetization dynamics do not require \nintroducing additional inertial or relativistic terms  [33], we complemented our experimental \nfindings with numerical simulations  of the LLG equation:  \n𝑑𝐌\n𝑑𝑡=−𝛾[𝐌×𝐇eff(𝑡)]+𝛼\n𝑀𝑠(𝐌×𝑑𝐌\n𝑑𝑡),    (8) \nwhere 𝐻𝑒𝑓𝑓 is the effective field derived from Eq. (2) as : \n𝐇eff(𝑡)=−∂𝑊𝐴\n∂𝑴+𝐇L(𝑡),     (9) \nWe employ ed the simulation model  from Ref.  [11] and added a term corresponding to the \nexternal magnetic field  𝐻. Calculations performed for a broad range of laser fluence s and \nexternal field values allowed us to obtain a set of  traces  of the magnetization dynamics . Figure \n8 show s a great deal of similarity  between simulations and experimen tal data (cf.  Fig. 3). It is \nseen that t he frequency increases with  increasing external  field  𝐻 while  the amplitude \ndecreases  (see Fig.  8a). The simulations for various stimulus strengths show the expected \ngrowth of the precession amplitude  (see Fig.  8b). \n \nFig. 8. Photo -magnetic  precession obtained in numerical simulations of the LLG equation  for: a) field \ndependence  at moderate excitation level  and b) power dependence  (I=4, 10, 16, and 22 arb. units ) at \n𝐻=0.4 kOe. \nWe further repeated our fit procedure with Eq.(1) to obtain the precession parameters from \nthese data. Figure 9 show s the values of the amplitude  and frequency of the precession in the \npower regime.  At a low field  𝐻=0.4 kOe (red) , the nonlinearity is clearly visible and \ncomparable with experimental data, as seen in Fig. 4. Similarly, at high field s (green) , the \nbehavior is mostly linear.  Figure 9a shows a great deal of similarity between simulations  \n(amplitude parameter)  and experi mental data  (normalize d value AF/max) (cf. Fig. 4a). The \nanalysis of the damping parameter (Fig. 9c) also confirms the exp erimental findings (as in Fig. \n7a), revealing the existence of two regimes, linear and nonlinear . The results of the s imulations \nconfirm that the observation of  the nonlinear response of the magnetic system  can be \nattributed  to the anharmonicity of the energy landscape.  \n12 \n Notably, in the simulations , as well as in the experimental data, we not only observe a second -\norder co rrection to the effective damping 𝛼2, but also a deviation from Eq.(7)  at even larger \namplitudes  (cf. Fig. 7 a and Fig. 9c). The latter manifests as a reduction of the effective damping \ncompared to the expected 𝛼0+𝛼2𝐴2 dependence shown with dashed lines. This higher -order \neffect is unlikely to originate in the multi -magnon scattering contribution since the latter \nwould only further increase the effective damping  [8]. We rather believe that th is is likely an \nartifact of the used damped oscillator model  where  in the range of  𝛼𝑒𝑓𝑓≈1 the quasiperiodic \ndescription of magnetization precession ceases to be physically justified.    \n \nFig. 9. Power dependence of  the a) amplitude  and b) frequency as obtained in the simulations  for low \n(red dataset) and high (green dataset) external magnetic field s. c) Effective damping  in the linear and \nnonlinear  precession regimes . \n \n6. Photo -induced phase diagram of sub -switching regime  \nIt is seen from both experimental and numerical results above that the cubic symmetry of the \nmagnetic system is key for the observed nonlinear magnetization dynamics. To quantify the \nparameter space for the nonlinearity, we first estimate the realistic values of the effective \nlight -induced magnetic field 𝐻𝐿. Throughout a number of works on photo -magnetism in Co -\ndoped garnets, a single -ion approach to magnetic anisotropy is consistent ly utilized. We note \n13 \n that in YIG:Co, it is the Co ions at tetrahedral sites that are predominantly responsible for the \ncubic anisotropy of  the magnetic energy landscape  [22]. In the near -IR range, these ions are \nresonantly excited at the 1300 nm wavelength, resulting in improved efficiency of the photo -\nmagnetic stimulus , as compared to previous works  [45]. Further, we note that at the \nmagnetization switching threshold, about 90% of the Co3+ ions with a concentration on the \norder of 1020 cm-3 are excited with incident photons  [11, 46] . Taking into account the single -\nion contribution to the anisotropy 𝛥𝐾1~105 erg/cm3 [47], and assuming a linear relation \nbetween the absorbed laser power (or fluence) and the effective photo -magnetic field 𝐻𝐿, for \nthe latter we find that 𝐻𝐿~1 kOe is sufficient for the magnetization switching. This means that \nthe sub -switching regime of magnetiz ation dynamics (cf. Fig. 1) refers to the laser fluences (as \nwell as wavelengths) , resulting in smaller effective fields.  \nWe reiterate that in previous works, the impact of the external magnetic field on the photo -\nmagnetically driven magnetization precess ion has not been given detailed attention.  To \naddress this gap , we plotted  the amplitude of the precession  𝐴 calculated in the same way as \nabove in the  sub-switching regime (Fig. 10) . As expected, the amplitude generally increases  \nwith 𝐻𝐿. However, we n ote a critical external field of about 0.5 kOe at which the desired \namplitudes can be reached at smaller light -induced effective fields 𝐻𝐿. At this field, where the \nsystem enters a single domain state, the potential curvature around the energy minimum \ndecreases, thus facilitating the large -angle precession. In other words, external magnetic field s \ncan a ct as leverage for the effective field of the photo -induced anisotropy, thus reducing the \nmagnetization switching threshold.  An exhaustive study of magneti zation switching across the \nparameter space shown in Fig. 10 remains an attractive perspective for future studies.  \n \nFig. 10. Calculated amplitude m ap of the photo -induced magnetization  precession in YIG:Co film. \nIn our analysis, we only considered a truly photo -magnetic excitation and neglected the laser -\ninduced effects of thermal origi n. It is, however, known that laser -driven heating can introduce \nan additional, long -lasting modification of magnetic anisotropy in iron garnets  [48, 49] . The \n14 \n relatively long relaxation times associated with cooling are responsible for the concomitant \nmodulation of the precession parameters and thus facilitate nonlinearities in the response of \nthe magnetic system. Yet, 1300 nm laser excitation of magnetization dynamics in YIG :Co film  \nwas shown to be highly polarization -dependent  [6], thus indicating the dominant role of the \nnon-thermal excitation mechanism. On the other hand, the unavoid able laser -induced heating \nwith experi mental values of laser fluence in YIG:Co film  has been estimated to not exceed 1 K  \n[6]. As such, we do not expect modification of the Gilbert damping associated with the \nproximity of the ma gnetization compensation or N éel temperature in the ferromagnetic \ngarnet  [50]. However, a detailed investigation of the temperature -dependent nonlinear \nmagnetization dynamics in the vicinity of the compensation point or a magnetic phase \ntransition  [51, 52]  represents another promising research direction.  Further , exploring the \nnonlinear  regime  in the response of the magnetic system to intense THz stimul i along the lines \ndiscussed in  [33] enjoys a rich potentia l for spintronic applications.  \n  \n7. CONCLUSIONS  \nIn summary, we studied, both experimentally and numerically, the nonlinear regime of \nmagnetization dynamics in photo -magn etic Co -doped YIG film. After  excitation with \nfemtosecond laser pulses at fluences below the magnetization switching threshold, there is a \nrange of external magnetic field where the magnetic system demonstrates strongly non linear \nprecession characterized by a significant increase of t he effective Gilbert damping. We \nattribute this nonlinearity to the anharmonicity of the potential for the magnetic oscillator \nenhanced by the dominant role of the cubic magnetocrystalline anisotropy. The effective \ndamping and its nonlinear contribution, a s obtained from numerical simulations, both \ndemonstrate a very good agreement with the experimental findings. Simulations of the \nmagnetization dynamics by means of  the LLG equation further confirm the nonlinearity in the \nmagnetic response below the switchi ng limit. Finally, we provide estimations for the realistic , \neffective photo -magnetic fields 𝐻𝐿 and map the workspace of the parameters in the sub -\nswitching, nonlinear regime of photo -induced magnetization dynamics.  \n \nACKNOWLEDGMENTS  \nThis work has been fu nded by the  Foundation for Polish Science ( Grant No. POIR.04.04.00 -00-\n413C/17)  and the National Science Centre Poland (Grant No. DEC -2017/25/B/ST3/01305) . \n \nREFERENCES  \n \n[1]  A. Kirilyuk, A. V. Kimel and T. Rasing, \"Ultrafast optical manipulation of magnetic order,\" \nRev. Mod. Phys., 82 (3), 2731 (2010).  15 \n [2]  J. Walowski and M. Münzenberg, \"Perspective: Ultrafast magnetism and THz \nspintronics,\" J. Appl. Phys., 120, 140901 (201 6).  \n[3]  K. Carva, P. Baláž and I. Radu, \"Laser -induced Ultrafast Magnetic Phenomena,\" in \nHandbook of Magnetic Materials 26 , Elsevier, (2017), pp. 29 -463.  \n[4]  E. Y. Vedmedenko, R. K. Kawakami, D. D. Sheka, P. Gambardella, A. Kirilyuk, A. Hirohata, \nC. Binek, O. Chubykalo -Fesenko, S. Sanvito and B. J. Kirby, \"The 2020 magnetism \nroadmap,\" J. Phys. D: Appl. Phys., 53, 453001 (2020).  \n[5]  J. Lloyd -Hughes, P. M. Opp eneer, T. Pereira dos Santos, A. Schleife, S. Meng, M. A. Sentef, \nM. Ruggenthaler, A. Rubio, I. Radu and M. Murnane, \"The 2021 ultrafast spectroscopic \nprobes of condensed matter roadmap,\" J. Phys.: Condens. Matter, 33, 353001 (2021).  \n[6]  A. Stupakiewicz , K. Szerenos, D. Afanasiev, A. Kirilyuk and A. V. Kimel, \"Ultrafast \nnonthermal photo -magnetic recording in a transparent medium,\" Nature, 542, 71 -74 \n(2017).  \n[7]  J. Stöhr and H. C. Siegmann, Mangetism From Fundamentals to Nanoscale Dynamics, \nSpringer, (2006).  \n[8]  V. Tiberkevich and A. Slavin, \"Nonlinear phenomenological model of magnetic \ndissipation for large precession angles: Generalization of the Gilbert mode l,\" Phys. Rev. \nB, 75, 014440 (2007).  \n[9]  G. V. Astakhov, A. V. Kimel, G. M. Schott, A. A. Tsvetkov, A. Kirilyuk, D. R. Yakovlev, G. \nKarczewski, W. Ossau, G. Schmidt, L. W. Molenkamp and T. Rasing, \"Magnetization \nmanipulation in (Ga,Mn)As by subpicosecon d optical excitation,\" Appl. Phys. Lett., 86, \n152506 (2005).  \n[10]  F. Hansteen, A. Kimel, A. Kirilyuk and T. Rasing, \"Femtosecond photomagnetic switching \nof spins in ferrimagnetic garnet films,\" Phys. Rev. Lett., 95, 047402 (2005).  \n[11]  A. Stupakiewicz, K. Szerenos, M. D. Davydova, K. A. Zvezdin, A. K. Zvezdin, A. Kirilyuk and \nA. V. Kimel, \"Selection rules for all -optical magnetic recording in iron garnet,\" Nat. \nComm., 10, 612 (2019).  \n[12]  A. Baral, S. Vollmar and H. C. Schneider, \"Mag netization dynamics and damping due to \nelectron -phonon scattering in a ferrimagnetic exchange model,\" Phys. Rev. B, 90, 014427 \n(2014).  \n[13]  D. Afanasiev, I. Razdolski, K. M. Skibinsky, D. Bolotin, S. V. Yagupov, M. B. Strugatsky, A. \nKirilyuk, T. Rasing and A. V. Kimel, \"Laser excitation of lattice -driven anharmonic \nmagnetization dynamics in dielectric FeBO3,\" Phys. Rev. Lett., 112, 147403 (2014).  16 \n [14]  E. Carpene, E. Mancini, D. Dazzi, C. Dallera, E. Puppin and S. De Silvestri, \"Ultrafast three -\ndimensi onal magnetization precession and magnetic anisotropy of a photoexcited thin \nfilm of iron,\" Phys. Rev. B, 81, 060415(R) (2010).  \n[15]  J. A. de Jong, A. V. Kimel, R. V. Pisarev, A. Kirilyuk and T. Rasing, \"Laser -induced ultrafast \nspin dynamics in ErFeO3,\"  Phys. Rev. B, 84, 104421 (2011).  \n[16]  W. He, B. Hu, Q. -F. Zhan, X. -Q. Zhang and Z. -H. Cheng, \"Probing nonlinear magnetization \ndynamics in Fe/MgO(001) film by all optical pump -probe technique,\" Appl. Phys. Lett., \n104, 142405 (2014).  \n[17]  N. P. Duong,  T. Satoh and M. Fiebig, \"Ultrafast manipulation of antiferromagnetism of \nNiO,\" Phys. Rev. Lett., 93, 117402 (2004).  \n[18]  O. Kovalenko, T. Pezeril and V. V. Temnov, \"New concept for magnetization switching by \nultrafast acoustic pulses,\" Phys. Rev. Lett., 110, 266602 (2013).  \n[19]  V. S. Vlasov, A. M. Lomonosov, A. V. Golov, L. N. Kotov, V. Besse, A. Alekhin, D. A. Kuzmin, \nI. V. Bychkov and V. V. Temnov, \"Magnetization switching in bistable nanomagnets by \npicosecond pulses of surface acou stic waves,\" Phys. Rev. B, 101, 024425 (2020).  \n[20]  A. Frej, A. Maziewski and A. Stupakiewicz, \"All -optical magnetic recording in garnets \nusing a single laser pulse at L -band telecom wavelengths,\" Appl. Phys. Lett., 118, 262401 \n(2021).  \n[21]  S. Geller, \"Crystal chemistry of the garnets,\" Z. Kristallogr. Cryst. Mater., 125, 1 -47 \n(1967).  \n[22]  Landolt -Börnstein, Numerical Data and Functional Relationships in Science and \nTechnology, New Series, Group III, vol 12, Berlin: Springer -Verlag, (1978) .  \n[23]  M. Maryško and J. Šimšova, \"Ferromagnetic resonance study of Y 3 -z Ca z Fe 5 -x-y Co x \nGe y O12 films,\" s, Czech. J. Phys. B, 34, 1125 (1984).  \n[24]  A. Stupakiewcz, M. Pashkevich, A. Maziewski, A. Stognij and N. Novitskii, \"Spin \nprecession modul ation in a magnetic bilayer,\" Appl. Phys. Lett., 101, 262406 (2012).  \n[25]  A. Maziewski, „Unexpected magnetization processes in YIG+Co films,” J. Magn. Magn. \nMater., 88, 325 -342 (1990).  \n[26]  S. Mizukami, E. P. Sajitha, D. Watanabe, F. Wu, T. Miyazaki, H. Naganuma, M. Oogane \nand Y. Ando, \"Gilbert damping in perpendicularly magnetized Pt/Co/Pt films investigated \nby all -optical pump -probe technique,\" Appl. Phys. Lett., 96, 152502 (2010).  \n[27]  S. Qiao, W. Yan, S. Nie, J. Zhao and X. Zhang, \"The in -plane anisotropic magnetic damping \nof ultrathin epitaxial Co2FeAl film,\" AIP Adv., 5, 087170 (2015).  17 \n [28]  B. Liu, X. Ruan, Z. Wu, H. Tu, J. Du, J. Wu, X. Lu, L. He, R. Zhang and Y. Xu, \"Transient \nenhancement of magnetization damping in CoFeB film via pulsed laser excitation,\" Appl. \nPhys. Lett., 109, 042401 (2016).  \n[29]  S. Mondal and A. Barman, \"Laser controlled spin dynamics of ferromagnetic thin film \nfrom femtosecond to nanosecond timescale,\" Phys. Rev. Appl., 10, 054037 (2018).  \n[30]  Z. Chen, M. Yi, M. Chen, S. Li, S. Zhou and T. Lai, \"Spin waves and small intrinsic damping \nin an in -plane magnetized FePt film,\" Appl. Phys. Lett., 101, 222402 (2012).  \n[31]  J. Kisielewski, A. Kirilyuk, A. Stupakiewicz, A. Maziewski, A. Kimel, T. Rasing, L. Baczewski \nand A. Wawro, \"Laser -induced manipulation of magnetic anisotropy and magnetization \nprecession in an ultrathin cobalt wedge,\" Phys. Rev. B, 85, 184429 (2012).  \n[32]  J. Kisielewski, W. Dobrogowski, Z. Kurant, A. Stupakiewicz, M. Tekielak, A. Kirilyuk, A. V. \nKimel, T. Rasing, L. T. Baczewski, A. Wawro, K. Balin, J. Szade and A. Maziewski, \n\"Irreversible modification of magnetic properties of Pt/Co/Pt ultrathin film s by \nfemtosecond laser pulses,\" J. Appl. Phys., 115, 053906 (2014).  \n[33]  J. Li, C. -J. Yang, R. Mondal, C. Tzschaschel and S. Pal, \"A perspective on nonlinearities in \ncoherent magnetization dynamics,\" Appl. Phys. Lett., 120, 050501 (2022).  \n[34]  G. M. Müller, M. Münzenberg, G. -X. Miao and A. Gupta, \"Activation of additional energy \ndissipation processes in the magnetization dynamics of epitaxial chromium dioxide \nfilms,\" Phys. Rev. B, 77, 020412(R) (2008).  \n[35]  H. Suhl, \"The theory of ferromagnetic res onance at high signal powers,\" J. Phys. Chem. \nSolids, 1, 209 -227 (1957).  \n[36]  A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev, A. Kirilyuk and T. Rasing, \"Inertia -\ndriven spin switching in antiferromagnets,\" Nature, 5, 727 -731 (2009).  \n[37]  M. Cherkasskii, M. Farle and A. Semisalova, \"Nutation resonance in ferromagnets,\" Phys. \nRev. B, 102, 184432 (2020).  \n[38]  K. Neeraj, N. Awari, S. Kovalev, D. Polley, N. Z. Hagström, S. S. P. K. Arekapudi, A. \nSemisalova, K. Lenz, B. Green, J. -C. Deinert, I. Ilyakov, M. Chen, M. Bawatna, V. Scalera, \nM. d’Aquino, C. Serpico, O. Hellwig, J. -E. Wegrowe, M. Gensch and S. Bonetti, \"Inertial \nspin dynamics in ferromagnets,\" Nat. Phys., 17, 245 –250 (2021).  \n[39]  M. Cherkasskii, I. Barsukov, R. Mondal, M. Farle and A. Semisalova, \"Theory of inertial \nspin dynamics in anisotropic ferromagnets,\" Phys. Rev. B, 106, 054428 (2022).  \n[40]  R. Mondal, M. Berritta and P. M. Oppeneer, \"Relativistic theory of spin relaxation  \nmechanisms in the Landau -Lifshitz -Gilbert equation of spin dynamics,\" Phys. Rev. B, 94, \n144419 (2016).  18 \n [41]  T. G. Blank, K. A. Grishunin, E. A. Mashkovich, M. V. Logunov, A. K. Zvezdin and A. V. \nKimel, \"THz -scale field -induced spin dynamics in ferrimag netic iron garnets,\" Phys. Rev. \nLett., 127, 037203 (2021).  \n[42]  S. Mizukami, F. Wu, A. Sakuma, J. Walowski, D. Watanabe, T. Kubota, X. Zhang, H. \nNaganuma, M. Oogane, Y. Ando and T. Miyazaki, \"Long -lived ultrafast spin precession in \nmanganese alloys film s with a large perpendicular magnetic anisotropy,\" Phys. Rev. Lett., \n106, 117201 (2011).  \n[43]  R. Jabłoński, A. Maziewski, M. Tekielak and J. Desvignes, \"FMR study of Co -substituted \nyttrium iron garnet films,\" J. Magn. Magn. Mater., 160, 367 -369 (1996).  \n[44]  S. Parchenko, A. Stupakiewicz, I. Yoshimine, T. Satoh and A.Maziewski, \"Wide \nfrequencies range of spin excitations in a rare -earth Bi -doped iron garnet with a giant \nFaraday rotation,\" Appl. Phys. Lett., 103, 172402 (2013).  \n[45]  F. Atoneche, A. M . Kalashnikova, A. V. Kimel, A. Stupakiewicz, A. Maziewski, A. Kirilyuk \nand T. Rasing, \"Large ultrafast photoinduced magnetic anisotropy in a cobalt -substituted \nyttrium iron garnet,\" Phys. Rev. B, 81, 214440 (2010).  \n[46]  Z. Šimša, \"Optical and magnetooptical properties of Co -doped YIG films,\" Czech. J. Phys. \nB, 34, 78 -87 (1984).  \n[47]  J. C. Slonczewski, \"Origin of magnetic anisotropy in cobalt -substituted magnetite,\" Phys. \nRev. 110, 1341, 110, 1341 (1958).  \n[48]  L. A. Shelukhin, V. V. Pavlov, P. A. Usachev, P. Y. Shamray, R. V. Pisarev and A. M. \nKalashnikova, \"Ultrafast laser -induced changes of the magnetic anisotropy in a low -\nsymmetry iron garnet film,\" Phys. Rev. B, 97, 014422 (2018).  \n[49]  C. S. Davies, K. H. Prabh akara, M. D. Davydova, K. A. Zvezdin, T. B. Shapaeva, S. Wang, A. \nK. Zvezdin, A. Kirilyuk, T. Rasing and A. V. Kimel, \"Anomalously damped heat -assisted \nroute for precessional magnetization reversal in an iron garnet,\" Phys. Rev. Lett., 122, \n027202 (2019).  \n[50]  F. Schlickeiser, U. Atxitia, S. Wienholdt, D. Hinzke, O. Chubykalo -Fesenko and U. Nowak, \n\"Temperature dependence of the frequencies and effective damping parameters of \nferrimagnetic resonance,\" Phys. Rev. B, 86, 214416 (2012).  \n[51]  M. Tekielak, A. Stupakiewicz, A. Maziewski and J. M. Desvignes, \"Temperature induced \nphase transitions in Co -doped YIG films,\" J. Magn. Magn. Mater., 254-255, 562 (2003).  \n[52]  T. Satoh, Y. Terui, R. Moriya, B. A. Ivanov, K. Ando, E. Saitoh, T. Shimura and K. Kuroda,  \n\"Directional control of spin -wave emission by spatially shaped light,\" Nat. Photon., 6, \n662-666 (2012).  \n "    },    {        "title": "2107.00869v1.Off_axis_gyration_induces_large_area_circular_motion_of_anisotropic_microparticles_in_a_dynamic_magnetic_trap.pdf",        "content": "Off-axis gyration induces l arge -area circular motion  of \nanisotropic microparticles  in a dynamic magnetic trap  \n \nYuan Liu,1† Gungun Lin,1† and Dayong Jin1,2 \n \n1Institute for Biomedical Materials and Devices, Faculty of Science, The University of Technology Sydney, Ultimo, New \nSouth Wales 2007, Australia  \n2UTS-SUStech Joint Research Centre for Biomedical Materials & Devices, Department of Biomedical Engineering, \nSouthern University of Science and Technology, Shenzhen, 518055, China  \n†: these authors contributed equally to this work.  \n*Dr. Gungun Lin.  \nEmail: gungun. lin@uts.edu.au  \nAbstract:  Magnetic tweezers  are crucial for single -molecule and atom ic characterization , and biomedical isolation  of \nmicroparticle carriers . The trapping component of magnetic  tweezing  can be relying on a magnetic potential well  that can \nconfine the relevant species  to a localized region . Here, we report  that magnetic microparticles with tailored anisotropy can  \ntransition from  localized  off-axis gyration  to large -area locomotion  in a rotating magnetic trap. The microparticles , consisting \nof assemblies of magnetic cores , are observed to either rotate about its structural geometr ic center or  gyrate about one of the \nmagnetic cores, the switch ing of which can be  modulated by the external field. Raising  the magnetic field strength above a \nthreshold, the particle s can go  beyond the traditional synchronous -rotation and asynchronous -oscillation modes , and into a \nscenario of large- area circular  motion . This results in p eculiar  retrograde locomotion  related to  the magnetization maxima of \nthe microparticle . Our finding suggests  the important role of the microparticle’s magnetic morphology in  the controlled \ntransport of microparticles and developing  smart micro -actuators and micro -robot devices.\nMagnetic t weezing is a powerful technique to manipulate single or multiple nano/microparticles  on \ndemand1,2, to concentrate particles from surrounding media3,4, study the mechanical properties of \nbiological samples5,6 and characterize the physical properties of single molecules and atoms7. Microscale \nmagnetic t weezing  can be reli ant on magnetic potential well s represented by local maxima of magnetic \nfields. The gradient  magnetic  fields can be produced by the pole shoes of  electromagnets 8,9, permanent \nmagnets10 and any other ferromagnetic micro - and nanostructures11,12.   \nThe d ynamic  form of magnetic t weezing  can leverage the spatial- temporal organization of  the magnetic \nfields from more sophisticated  electromagnet s, to enable  applications in magnetic resonance imaging13 \nand magnetic particle imaging14, and microrobotic control15,16. On the other hand, a s imple  form  of \ndynamic magnetic tweezing can be constructed by a rotating permanent magnet . It has been  known that \nsuch a rotating trap  can drive magnetic microspheres17,18 or dimer- type structures19,20, into synchronous \nrotation at low frequencies  and asynchronous oscillation at high frequencies  when the particle s cannot \ncatch up with  the pace of the rotating field . Despite the rotation transitio n, the range of the particles’ \nmotion  is typically limited to a small area. Moreover, in conventional scenarios , the influence of the  \nmorphology of a  particle on the tweezing effect  is typically overlooked.  \nHere we  report that the off-axis gyration of a n anisotropic microparticle can lead to  large -area circular \nlocomotion  using a pair of rotating permanent magnet s (Figure 1 a). The microparticles are fabricated with \nmicro -sized magnetic core (s)-polymeric shell structures with sizes comparable to that of a magnetic \npotential well, using a two- step microfluidic emulsification approach described previously19,21. We find  \nthat the morphology of the m agnetic particles  can play a crucial role in modulat ing the magnetic trapping \npoint on the microparticles . The rotation center  of a microparticle is observed to switch between its \nstructural geometric  center and the magnetic core center  at varied trapping  potential. The mechanism \nenables the motion stance of a microparticle to  switch among synchronous off-axis gyration/symmetric \nrotation , asynchronous rotation , and large -area retrograde motion  (Figure 1b) . The motion modes can be  \nmodulated by multiple parameters , such as magnetic field gradient, strength , frequency and the magnetic \narchitecture of the micro particles .  2 \n  \nFigure 1 . Tweezing anisotropic magnetic microparticles using rotating permanent magnets. (a) S chematic  illustration  of the \nexperimental scenario of rotating a dual -core microparticle under a gradient magnetic field. The  distance is the height of the \nmicroparticle to the magnets’ plane. The speed of rotated magnets can be modulated from 0 to 1500 rpm. (b) Schematic \nillustration showing that a microparticle could undergo  motion mode  transition, including transition between symmetric \nrotation and off -axis gyration, and switching between off -axis gyration and large -area circular motion. The rotation centre  of \nsymmetric rotation locates at the geometry centre. The off -axis gyration has a n off-axis gyrating radius. The large -area motion \nis hierarchical, consisting of gyration and circular revolution . \nA dynamic magnetic trap can be constructed from a pair of rotating magnets embedded in a stirrer device. \nUsing a 3D Hall sensor,  the distribution of the magnetic fields (denoted by three major components) of \nthe magnets  were measured  (data not shown) . Computational  simulation  (COMSOL Multiphysics)  of the \nmagnet s shows that the gradient field component, ∇ Bx, is represented by a dipol ar-shaped pattern with \nlocal maximum and minimum (Figure 2 a). This resulted in a magnetic potential well located at the center \nbetween the magnets. In a static state, th e potential well, acting as a magnetic trap, attracts a magnetic \nparticle to wards  its center. A comparable gradient field component along the z axis , i.e. ∇Bz, can be found \nto superimpose on the x-component  (Figure 2b) . Different from ∇ Bx, the ∇Bz has a peak -like pattern and \nis symmetric with respect to the center of the magnets. The depth and width of the potential well can be \nadjusted by  the distance of the microparticle ’s working surface from the magnets. While reducing distance, \nthe potential well becomes deep er and narrower , indicating a stronger and more localized  trapping  force . \nThis dependency suggests that when the size of a microparticle is comparable with that of the potential \nwell, the particle can  experience uneven magnetic forces . In this case, the  particle’s  morphology may \ncome into play and two force -equilibrium positions could be established . Specifically , a sizable particle \ncould be trapped to the center of the potential well with its geometric center overlapping with the well \ncenter  (here referring to Mode 1 , as shown in Figure 2c ). Alternatively, the particle could have its end \ntrapped by the potential well, with t he other end slightly off the surface, with a larger mismatch of its \ngeometric center with the well center  (here referring to Mode 2, as shown in Figure 2d ). \n3 \n  \n \n \nFigure.2 Mechanism of transition between off- axis gyration  and symmetric  trapping under magnetic field-structure coupling. (a) X-axis \ncomponent of magnetic field gradient. (Upper) Top view of  the experimental configuration. ( Bottom ) X-axis component of the magnetic \nfield gradient , ∇Bx. Inserted curve shows the profile of the field gradient with different distance between the microparticle and magnets’ \nplane . (B) Z-axis component of magnetic field gradient. (Upper) Side view of the experimental configuration.  (Bottom) Z -axis component \nof the field gradient  for different distance between the microparticle and the magnets’ plane. Illustration of the equilibrium modes of \nsymmetric trapping (C) and off- axis trapping (D). The former is featured with symmetric trapp ing where the particle is located with its \ngeometric center overlapping with the center of the potential well. The latter is shown with the particle lying in an off -axis position with \nrespect to the well center. The two modes correspond to two different po tential well depth s and width s.  \n \nTo elaborate further the theoretical basis , the major magnetic tweezing force exerted on a purposely-\ndesigned dual -magnetic core microparticle  can be decomposed into two components : gradient -field \nmagnetic force and magnetic anisotropy force as approximated by a dipole -dipole coupling force .20 For \na larger separ ation distance from the magnets and a low magnetic field strength , the former component \ncould become inappreciable in comparison with the latter. In this case, the dipolar -coupling induced \ntorque can be balanced with the hydrodynamic torque of the fluid, 𝜏𝜏h=−𝜀𝜀𝜀𝜀𝜀𝜀𝜔𝜔��⃗𝑚𝑚𝑚𝑚, at a critical rotation \nfrequency , as typically observed in previous studies . As the particle is mainly driven by th e anisotropy \n4 \n induced torque , the particle will rotate around its geometric center  at low field frequencies , as described \nin Figure 2c , Mode  1. Both permanent and induced magnetic moments may contribute to the torque22,23. \nWe have demonstrated previously that the curre nt system under investigation is majorly governed by the \ninduced magnetic moments19. \nWhen the field strength is raised, the gradient -field force component, namely, the trapping force, could \nplay a dominating role  over the dipolar coupling force . A likely scenario is off- axis trapping of a \nmicroparticle  as described by Figure 2d, Mode  2. This mode  can be justified when a particle, in practice, \nmay deviate from an ideally symmetric structure  and isotropic composition. Positive ∇Bz tends to lift the  \nparticle  at one end adjacent to the center , causing asymmetric trapping forces exerted on the microparticles. \nThe difference in the trapping force is balanced by the friction force of the particle to establish an \nequilibrium state of magnetic trapping to one end of the particle.  \nThe off-axis gyration, featured with a significant mismatch between the geometry center and rotation \ncenter, indicates that an increased centrifugal force can be associated with the gyration at high rotating \nspeed s. In this case, at a certain threshold rotating speed, centrifugal force may overcome the gradient \nforce and the friction force, causing the particle to slide  off the center of the magnetic potential well. \nConsequently, an equilibrium state of circular motion c an be achieved  when  the centrifugal force is \nbalanced by the sum of magnetic trapping force and friction force exerted on the microparticle. Worth \nnoting, t he friction force  is expressed as Fs=μsFn, where  μs is the coefficient of kinetic friction and F n \nis the normal force directly perpendicular to their surface. During the sliding- off process, Fn is a dynamic \nforce decided by the magnetic lift force:  \n𝐹𝐹𝑛𝑛=G−𝐹𝐹𝑧𝑧−𝐹𝐹𝑏𝑏                                                                [ 1] \nwhere G is the gravity, F z is the lift force induced by the B z, and F b is the buoyancy force.  \nThe anisotropy force may drive the particle to gyrate while circling  with a large radius . By b alancing  \nthe period of the circulat ing motion with that of the rotating field, one may obtain the  radius  of large -\narea circular motion , which can be expressed as24: \nRcm≈klωmg�1−pslip�\n2πωcm                                                         [ 2] \nwhere ω mg, ωcm is the angular speed of the external magnetic field  and the circular motion, respectively;  \nk the geometric factor, l is the length of the particle, pslip is the slipping probability defined by the error \nfunction of the ratio between the centrifugal force and magnetic force 24. pslip is 0 at relatively slow \nrotation and approaches 1 at high rotational speed of magnetic field, for which the centrifugal force may \nfar exceed the magnetic force. k is equal to ½ for symmetric rotation and may take on other values for off-\naxis gyration, depending on the position of the gyration centre. The above relationship suggests the \nfeasibility to regulate the motion area by changing the rotation frequency.   \nTo validate the above hypothesis , we designed  two types of particle s: one with two similar -sized cores \nwith 300 µm in diameter  and the other with two uneven cores with 200 and 400 µm in diameter . Under \na low rotation frequency of 3 Hz, both particles are observed to exhibit synchronous rotation  about their \nstructural center at a low field strength of 10 mT  rotating at 3Hz  (Figure 3a) . This is evidenced by the \nrotation radius equal to the sum of the radii of the cores. When the phase lag is smaller than 𝜋𝜋/2, rotation \nin this mode is synchronous with the external magnetic field below a  critical frequency  that scales  with \nthe magnetic field strength (Figure 3b). Above the critical frequency (phase lag  > 𝜋𝜋/2), the particle may \nrotate backwards in a rotating cycle  when it cannot catch up with the pace of the field (Figure 3c). This \nrotation mode switches to synchronous off-axis gyration, namely, rotating about one of its cores’ center, \nat a larger field strength of  30 mT  when the separation distance from the magnets is reduc ed (Figure 3 d). 5 \n Remarkably , the particle with an uneven core  size is shown to gyrate about the center of the larger core . \nThis suggests that the magnetic potential well tends to grasp  the core with stronger magnetization.   \n \nFigure. 3  Experimental evidence of switching  between  symmetric gyration and off -axis gyration. (a)  Tracking the rotation of \ntwo types of particles : one with two similar -sized cores with 300 -µm diameter and the other wit h two uneven cores with 200 \nand 400 µm in diameter.  Here r 1 and r 2 is the radius of each core, respectively. The red lines indicate the trajectories of \nsymmetric rotation . The field strength of B x at the field centre is 10 mT  and the field frequency is 3 Hz. ( b) Dependence of the \nrotation frequency of the particle  on the rotating field frequency  at two magnetic field strengths: 10 mT and 15 mT.  Error bar \ndenotes the measurement accuracy.  (c) Polar plot representation of the time -lapsed motion trajectory of a particle under \ndifferent cycling frequencies of 5 Hz (green colored) and 7.5 Hz (blue colored).  (d) Tracking of the off -axis gyration of the \nparticles at a higher field strength of 30 mT.  Scale bar  in (a) and (d) :100 µm.  \nFigure  4a shows that the microparticle can  be driven to large -area circular motion at increased \nfrequencies. At low frequencies such as below 7 Hz, it is shown that the radius of motion is constant, \nwhich indicates  that the centrifugal force is not sufficient to overcome the magnetic trap ping force. Large-\narea circular motion occurs at the rotating field frequenc y larger than 7 Hz, upon which the centrifugal \nforce exceeds 0.3 nN . The motion is featured with  a unique  retrograd ing pace. The gyration  center is \nobserved to switch between two cores of the microparticles , resembling brachiation- like locomotion  \n(Figure 4c) . During the first half cycle (0o~180o), one core contacts the substrate with the other slightly \nlifted. The stance results in gyration around the center of the substrate -contacting core, and switches to \nthe other during the second half gyration cycle. The clockwise gyration and gyration -centre switching \nresult in the anticlockwise retrograde locomotion of the microparticle.  A dynamic range of  frequency \nregulation  exist s to alter the motion radius between 0 mm to 2.75 mm. Correspondingly, t he centripetal \nforces of magnetic particles are derived to be 4.65 nN and 7.50 nN for 2.25 -mm (4.30 s period) and 2.75-\nmm (3.70 s period) orbital radii, respectively.  At even higher frequencies, an increased chance of slipping \nmay occur, leading to a reduc ed locomotion speed for the dual -core particle (Dual -P). The  three-core \nparticle  (Tri- P), however, is observed to locomote circularly at larger radii at all rotating field speeds . \nThis could be ascribed to  that the longer  length of the particle can  be featured with  an increased length \nof pace during each gyration cycle  at zero slipping probability, as described by Equation ( 3) (Figure 4b) . \n6 \n This is further confirmed by the fact that the  locomotion  speed  of the Tri -P scales with that of the  rotating \nfield in the corresponding range  up to 1400 rpm , as is in contrast with that of the Dual -P. Figure 4d \ndisplays the map of the  locomotion stance based on the field frequency and field strength of H y at the \ncentre of the magnets . Under low rotating speed and strength of the magnetic field , the microparticles \nexhibit  the synchronous rotation (SR). We found that due to the  difference  in the magnetic trapping force , \nthe synchronous rotation region may consist of two states: symmetric rotation  and off -axis gyration. Once  \nincreasing the frequency, the off -axis gyration transits to the large -area circular motion (LAM) due to \nthe large the centrifugal force, while the symmetric rotation transits to asynchronous rotation (SR) due \nto the large phase lag between the field ’s orientation and the particle’s magnetization direction  (Figure \n4d).  \n \nFigure. 4  Characteristics of large -area circular motion. (a) Locomotion trajectories of  local motion (LM) and  large -area \nretrograde motion  (LAM)  of microparticles at different rotating frequencies . (b) Motion radius and speed changing with field \nspeed.  The results are obtained from the average values of 5 particles with identical core ratios.  (c) Retrograde locomotion . \nThe gyration center , marked by the triangular shape, switches in between two cores of the microparticles  during the gyration \ncycle. (d) Mapping of motion modes of a particle with regard to the variation of  field strength and field frequency.  Three \nlocomotion modes are included: synchronous  (SR) , asynchronous (AR) and large -area retrograde motion  (LAM) . The field \nstrength of B x at the center is change d from 5 to 40 mT , and the field frequency ranges from 50 rpm to 1500 rpm.  \nIn conclusion, we have presented  the systematic study of the motion modes of anisotropic microparticles \nunder a  pair of  rotating magnet s. It has been found that t he cooperative effect of the in -plane trapping \nforce and out -of-plane lift force can  unlock the off-axis gyration of anisotropic magnetic microparticle s. \nThis is manifested by  the rotation center of a particle to  switch from its  geometric center and the center of \none of its  magnetic core s at a high trapping force . The o ff-axis gyration of the particles at high speeds can \nlead to sustainable and maneuverable large -area circular motion , which provides insight into the controlled \ntransport of microparticles  using magnetic traps.  \nACKNOWLEGMENTS  \n7 \n G.L. acknowledges financial support from the National Health and Medical Research Council \n(GNT1160635). G.L. and D.J. thank the financial support of the ARC Industry Transformational Research \nHub Scheme (grant IH150100028). Y. L. thanks the financial support of China Scholarship Council \n(201608140100). \n DATA AVAILABILITY  Data available on request from the authors.  \n Reference  \n1 J. Lipfert, J.W.J. Kerssemakers, T. Jager, and N.H. Dekker, Nat. Methods 7, 977 (2010). \n2 Q. Xin, P. Li, Y. He, C. Shi, Y. Qiao, X. Bian, J. Su, R. Qiao, X. Zhou, and J. Zhong, Anal. Methods 9, \n5720 (2017). \n3 A. Ali -Cherif, S. Begolo, S. Descroix, J.L. Viovy, and L. Malaquin, Angew. Chemie -  Int. Ed. 51 , \n10765 (2012). \n4 H. Lee, B. Ahn, K. Lee, and K.W. Oh, 15th Int. Conf. Miniaturized Syst. Chem. Life Sci. 2011, \nMicroTAS 2011 2, 948 (2011). \n5 J. Lipfert, M. Wi ggin, J.W.J. Kerssemakers, F. Pedaci, and N.H. Dekker, Nat. Commun. 2, (2011). \n6 I. De Vlaminck and C. Dekker, Annu. Rev. Biophys. 41, 453 (2012).  \n7 K.C. Neuman and A. Nagy, Nat. Methods 5, 491 (2008).  \n8 W. Hu, G.Z. Lum, M. Mastrangeli, and M. Sitti, Natur e 554 , 81 (2018).  \n9 X. Fan, X. Dong, A.C. Karacakol, H. Xie, and M. Sitti, Proc. Natl. Acad. Sci. U. S. A. 117, 27916 \n(2020).  \n10 X. Dong and M. Sitti, Int. J. Rob. Res. 39, 617 (2020).  \n11 M. Poudineh, P.M. Aldridge, S. Ahmed, B.J. Green, L. Kermanshah, V. Nguyen, C. Tu, R.M. \nMohamadi, R.K. Nam, A. Hansen, S.S. Sridhar, A. Finelli, N.E. Fleshner, A.M. Joshua, E.H. Sargent, and S.O. Kelley, Nat. Nanotechnol. 12 , 274 (2016). \n12 L. Kermanshah, M. Poudineh, S. Ahmed, L.N.M. Nguyen, S. Srikant, R. Makonnen, F. Pe na Cantu, \nM. Corrigan, and S.O. Kelley, Lab Chip 18, 2055 (2018).  \n13 X. Yan, Q. Zhou, M. Vincent, Y. Deng, J. Yu, J. Xu, T. Xu, T. Tang, L. Bian, Y.X.J. Wang, K. \nKostarelos, and L. Zhang, Sci. Robot. 2, (2017).  \n14 E.Y. Yu, M. Bishop, B. Zheng, R.M. Ferguso n, A.P. Khandhar, S.J. Kemp, K.M. Krishnan, P.W. \nGoodwill, and S.M. Conolly, Nano Lett. 17, 1648 (2017).  \n15 J. Rahmer, C. Stehning, and B. Gleich, Sci. Robot. 2, 1 (2017). \n16 F. Berndt, G. Shah, R.M. Power, J. Brugués, and J. Huisken, Nat. Commun. 9, 5025 (2018).  \n17 Y. Li, D.T. Burke, R. Kopelman, and M.A. Burns, Biosensors 4, 76 (2014). \n18 I. Sinn, P. Kinnunen, T. Albertson, B.H. McNaughton, D.W. Newton, M.A. Burns, and R. Kopelman, \nLab Chip 11, 2604 (2011). \n19 G. Lin, Y. Liu, G. Huang, Y. Chen, D. Makarov, J. Lin, Z. Quan, and D. Jin, Adv. Intell. Syst. 3, \n2000205 (2021). \n20 H. Le Ferrand, S. Bolisetty, A.F. Demirörs, R. Libanori, A.R. Studart, and R. Mezzenga, Nat. \nCommun. (2016). \n21 Y. Liu, G. Lin, Y. Chen, I. Mönch, D. Makarov, B.J. Walsh, and D. Jin, Lab Chip 20, 4561 (2020). \n22 M.M. Van Oene, L.E. Dickinson, F. Pedaci, M. Köber, D. Dul in, J. Lipfert, and N.H. Dekker, Phys. \nRev. Lett. 114 , 218301 (2015). \n23 X.J.A. Janssen, A.J. Schellekens, K. van Ommering, L.J. van IJzendoorn, and M.W.J. Prins, Biosens. \nBioelectron. 24, 1937 (2009). \n24 S. Won, S. Kim, J.E. Park, J. Jeon, and J.J. Wie, N at. Commun. 10, (2019). 8 \n  "    },    {        "title": "2101.06121v2.Spin_dynamics_from_a_constrained_magnetic_Tight_Binding_model.pdf",        "content": "Spin dynamics from a constrained magnetic Tight-Binding model\nRamon Cardias\u0003and Cyrille Barreteauy\nSPEC, CEA, CNRS, Universit\u0013 e Paris-Saclay,\nCEA Saclay F-91191 Gif-sur-Yvette, FRANCE\nPascal Thibaudeauz\nCEA, DAM, Le Ripault, BP 16, F-37260, Monts, FRANCE\nChu Chun Fux\nUniversit\u0013 e Paris-Saclay, CEA, Service de Recherches de\nM\u0013 etallurgie Physique, F-91191, Gif-sur-Yvette, FRANCE\n(Dated: April 29, 2021)\n1arXiv:2101.06121v2  [cond-mat.mtrl-sci]  28 Apr 2021Abstract\nA dynamics of the precession of coupled atomic moments in the tight-binding (TB) approxi-\nmation is presented. By implementing an angular penalty functional in the energy that captures\nthe magnetic e\u000bective \felds self-consistently, the motion of the orientation of the local magnetic\nmoments is observed faster than the variation of their magnitudes. This allows the computation\nof the e\u000bective atomic magnetic \felds that are found consistent with the Heisenberg's exchange\ninteraction, by comparison with classical atomistic spin dynamics on Fe, Co and Ni magnetic\nclusters.\nI. INTRODUCTION\nNowadays, the coupling between structural and magnetic properties in 3d based magnetic\nmaterials plays a key role in the manufacture of high performance spintronics devices [1].\nMoreover, it is also central in numerous anomalous evolutions of structural parameters [2]\nwith pressure. For instance, one of its salient consequence is that the bcc phase of \u000b-Fe\nis stabilized by its magnetic properties [3]. Thus, to accurately describe the dynamics of\n3d metals and their alloys, a fully coupled spin-lattice dynamics with an ab initio level of\nprecision is highly desirable. Unfortunately and despite notable progress [4, 5], no such tool\nis available so far.\nHowever the theory of magnetism is fundamentally a theory of electronic structure.\nAntropov et al. \frst presented a description of the motion of local magnetic moments\nin magnetic materials [6], in the framework of \frst-principles methods. Their idea was mo-\ntivated by the fact that the interatomic exchange energy among atomic magnetic moments\nis small compared to intra-atomic exchange and bandwidth energy. Thus, this adiabatic\nspin density approximation allows them to treat the angles de\fning the directions of these\nmagnetic moments as su\u000eciently slow varying degrees of freedom, to separate them from\nthe individual motion of their underlying electrons, exactly like the nuclear coordinates in\nthe Born-Oppenheimer adiabatic approach to molecular dynamics [7]. Moreover, by assum-\ning that the magnetization density in the immediate vicinity of each atom has a uniform\n\u0003ramon.cardias@cea.fr\nycyrille.barreteau@cea.fr\nzpascal.thibaudeau@cea.fr\nxchuchun.fu@cea.fr\n2orientation, each direction of every magnetic moment can be followed in time according\nto a precession equation, as it is the case of classical atomistic spin dynamics [8]. Conse-\nquently, the initial many-electron system is mimicked by this system of classical moments,\nwhen the directions and amplitudes are determined self-consistently from the requirement\nof minimizing a given free energy. Thus for each moment, the e\u000bective \feld that enters in\nthe precession equation depends only on the variation of the spin-dependant free electronic\nenergy as a functional of the magnetization direction only. Moreover, by assuming that the\nrelevant electronic correlation hole is essentially in the inner part of each atomic volume, for\nthis type of adiabatic transformation, the longitudinal moment dynamics is nonadiabatic in\nthis approach. It is governed by individual electronic spin \rips like Stoner excitations, which\nare also fast [9]. Thus, even if the amplitude of each moment cannot be globally constant in\ntime, for a small temporal excursion fast enough to keep the adiabatic approximation, the\nlongitudinal dynamics can be often neglected.\nThe paper is organized as followed. In Sec. II, we review the framework used to de-\nrive non-collinear magnetism within the tight-binding (TB) approximation. Angular mag-\nnetic constraints are imposed by penalty functionals that are solved equally during the\nself-consistently computation of the electronic structure. In Sec. II D, the derivation of an\nequation of precession of the local magnetic moments that involves constrained magnetic\n\felds is presented that allows considerations both transverse and longitudinal dampened\ntorques. The dynamics of various magnetic dimers and trimers of Fe, Co and Ni is studied\nin details in Secs. III A and III B to access the validity of the isotropic Heisenberg exchange\napproximation, that is commonly assumed. Lastly, in Sec. III C we analyse in depth the\nexample of an Fe dimer exposing the strength of our method as opposed to the limitations\nintroduced by describing this system in the global Heisenberg picture.\nII. METHODOLOGY\nWhen an Hamiltonian His a functional of the magnetization M, the e\u000bective \feld is\nnothing else than the functional derivative of Hwith the respect of the magnetization [10].\nTo calculate such an e\u000bective \feld acting on the atomic magnetic moments, the atomistic spin\ndynamics (ASD) uses a parameterized spin-Hamiltonian, where ab initio methods calculate it\nat every self-consistent iteration with various methods. One of the ab initio approach consists\n3in the use of constrained density functional theory (cDFT) [11], where a full accountability of\naccomplishments of calculations can be found now in many references [12]. The accuracy of\nthe cDFT methods requires an extremely high computational price that scales quickly with\nthe dimension and size of the studied system. In contrast, spin-Hamiltonian methods rely on\nspatial distributions of classical magnetic moments and o\u000ber an option with a computational\ncost tuned by the accuracy and how interatomic exchange parameters are treated. We o\u000ber\na method that relies in between, with a lower computational cost compared with the full ab\ninitio aspects of the cDFT method without having to rely on a correct description of the\nparameters inside a spin-Hamiltonian for a given system.\nA. Magnetic tight-binding model\nIn this work we have used a magnetic TB model that has been described in a review\narticle [13] and has been extensively benchmarked and validated in many di\u000berent mag-\nnetic systems of various dimensionalities (bulk, surfaces, interfaces, wires, clusters) [14{16],\nincluding complex magnetic structures such as spin density wave [17] and non collinear\ncon\fgurations [18].\nIt is based on a parametrized spddescription of the electronic structure where in practice\nthe parameters of the Hamiltonian are determined by a careful \ft of the total energy and\nband structures obtained from ab-initio data over a large range of lattice constants of dif-\nferent crystallographic structures. The magnetism is described via a Stoner-like interaction\nterm. The Stoner parameter Iof each element being also determined from a comparison to\nab-initio calculations at several lattice constants. This TB model describes the electronic,\nmagnetic and energetic properties with a precision close to Density Functional Theory but\nat a much smaller computational e\u000bort.\nTo avoid a too lengthy derivation, we will present a simpli\fed version of the TB formalism\nthat focuses on the most salient features of the model. Let us consider a non-magnetic TB\nHamiltonian H0written in a local basis set jii. The site index iis a composite object\nthat also includes an orbital index reference which can be dropped for simplicity. H0is\ndecomposed into onsite energy terms \"0\ni=hijH0jiiand hopping integrals \fij=hijH0jji. The\neigenfunctions of the system are written as a combination of atomic orbitals j\u000bi=P\niC\u000b\nijii\nand the density matrix between sites reads \u001aij=Pocc\n\u000bC\u000b\niC\u000b?\njwhere the summation runs\n4over the occupied energy levels \"\u000b< E0\nFwhereE0\nFis the Fermi level such thatP\ni\u001aiiis\nequal to the total number of electrons Neof the system. The total energy of a non-magnetic\nsystem is here reduced to the band energy only [19]\nE0\ntot=occX\n\u000b\"0\n\u000b= Tr(\u001aH0) =X\nij\u001aijH0\nji\n=X\nijoccX\n\u000bC\u000b\niC\u000b?\njH0\nji: (1)\nTo this non-magnetic framework, both the magnetic interaction and the local charge\nneutrality can be added by appropriate constraints, such as the total energy can be written\nin a formalism where each electronic spins are treated collinear, i.e.\nEtot=E0\ntot+X\niUi(ni\u0000n0\ni)2\u00001\n4X\niIim2\ni; (2)\nwhereni=\u001aii=ni\"+ni#andmi=ni\"\u0000ni#are respectively the charge and magnetization\nof sitei, whereasIiis the Stoner parameter and Uia large positive quantity. By minimizing\nEq.(2) with respect to the normalized coe\u000ecient C\u000b\ni, with the conditionP\ni(C\u000b\ni)2= 1,\nthis leads to a Schr odinger equation for a renormalized Hamiltonian H\u001bfor\"or#spins\nseparately. This Hamiltonian simply reads as\nH\u001b=H0+X\nijii\u0012\nUi(ni\u0000n0\ni)\u00001\n2Iimi\u001b\u0013\nhij; (3)\nwhere\u001b=\u00061 is the spin\"or#. In this Stoner picture only the onsite terms \"0\ni!\n\"0\ni+ (Ui(ni\u0000n0\ni)\u00001\n2Iimi\u001b) are a\u000bected by both the local charge neutrality and magnetism.\nThe generalization to non-collinear magnetism is straightforward. First the previous\nexpressions is extended to spin-orbitals with spin-dependent coe\u000ecients ( Ci\";Ci#) on each\nsite. Then an onsite density matrix ~ \u001aiis manipulated as a 2 \u00022 matrix with components\n\u001a\u001b\u001b0\ni=Pocc\n\u000bC\u000b\ni\u001bC\u000b?\ni\u001b0, in order to write it more conveniently as ~ \u001ai=1\n2ni\u001b0+1\n2mi\u0001\u001b, where\u001b0\nis the identity matrix \u0011Iand\u001b= (\u001bx;\u001by;\u001bz) is a vector of Pauli matrices, mi= Tr(~\u001ai\u001b).\nAs a consequence, the Hamiltonian Hthen reads as\nH=Hn\u001b0+Hm:\u001b; (4)\n5where the components of the vector Hamiltonian H= (Hn;Hm) are\nHn=X\ni\u0000\n\u000f0\ni+Ui(ni\u0000n0\ni)\u0001\njiihij+X\nij\fijjiihjj; (5)\nHm=\u00001\n2X\ni\u0001ijiihij: (6)\nwith\u0001i=Iimi. When the total energy of the system is written as the sum of the occu-\npied eigenvalues (band energy term) of the renormalized Hamiltonian, one has to take into\naccount the so-called double counting terms\nEtot=occX\n\u000b\"\u000b\u00001\n2X\niUi((ni)2\u0000(n0\ni)2) +1\n4X\niIikmik2; (7)\nwhere\"\u000bare the eigenvalues of the renormalized Hamiltonian.\nB. Magnetic constraints in TB\nWhen dealing with magnetic systems it is often interesting to be able to explore the\nenergetics of various magnetic con\fgurations. This can achieved by trying several starting\nmagnetic con\fgurations but remains a relatively limited strategy since this produces few self-\nconsistent solutions to compare with. It can be very interesting to consider the situation\nwhere magnetic constraints are imposed on any given atom iof the system. Appendix A\nsummarizes the \fxed spin method that is limited to collinear magnetism. However, among\nall the practical methods of optimization under constraints [20], the penalty method is a\nvery handy way to proceed.\nThis consists to supplement the total energy with a penalty term in a similar way that\nhas been done for the local charge neutrality constraint. There exists many possible ways\nto impose constraints on a magnetic system [11, 21, 22], which have been carefully reported\nin the reference [23].\nThere also exists various types of penalty functional depending on the quantity to impose.\nOne can impose a given moment mpen\nion a given atomic site ias presented in appendix B\nbut it is also possible to constrain only the polar angle \u0012ibetween the atomic moments of\natomiand thez-axis, a penalty functional of the form \u0015(\u0012i\u0000\u0012pen\ni)2can be considered. An\nequivalent expression can apply to the azimuthal angle \u001eitoo. To constraint simultaneously\nboth angles, we could simply add these two functionals. However as reported by Ma and\n6Dudarev [22], a combined angular penalty functional can be constructed, based on the\ndot product of miandepen\ni, here considered as a unit vector of given spherical angles\n(\u0012pen\ni;\u001epen\ni). This penalty function reads Epen\ni=\u0015(kmik\u0000epen\ni\u0001mi), and leaves the norm\nof the magnetization kmikfree to vary while the direction of the magnetic moment is\nconstraint to be the direction of epen\ni. Consequently, this introduces a renormalization of\nthe on-site terms of the TB Hamiltonian of the form \u0000Bpen\ni\u0001\u001bwithBpen\ni=\u0000\u0015(ei\u0000epen\ni),\nwheremi=kmikei. Therefore the on-site term \u0001iof the magnetic Hamiltonian Hm(see\nEq. (6)) reads:\n\u0001i=Iimi+ 2Bpen\ni (8)\nThis is exactly Eq. (1.9) of Ref. 24. The spin splitting \feld \u0001iis the sum of the Stoner-\nlike exchange \feld Iimiand the penalization \feld. This penalty scheme has many speci\fc\nproperties. For example by noting that \u0000Bpen\ni\u0001mi=Epen\ni, it can be shown that there\nare no double counting terms associated to the the renormalization. Consequently the total\nenergy can we written as in Eq. (B1) but without the last term. Moreover when \u0015!1 ,\nei\u0019epen\niandBpen\ni\u0001mi= 0 and the penalization \feld becomes perpendicular to the local\nmagnetization.\nTo be more speci\fc, let us now consider the variation of the total energy with respect\nto the polar and azimuthal angles. By considering a variation of angle d\u0012on siteiand\nby using the Force Theorem, it is straightforward to show that dE=\u0000dBpen\ni\nd\u0012\u0001mid\u0012=\n\u0000kmikdBpen\ni\nd\u0012i\u0001eid\u0012i. Now by taking the derivative of Bpen\ni\u0001ei= 0, and by noting that\nde\nd\u0012=e\u0012, we \fnd a relationship between the polar angle variation of the energy, which is the\ne\u000bective \feld up to a sign, and the penalty \feld\n1\nkmik@E\n@\u0012i=Bpen\ni\u0001ei;\u0012=Bpen\ni;\u0012; (9)\nand similarly with the azimuthal angle variation of the energy\n1\nkmik1\nsin\u0012i@E\n@\u001ei=Bpen\ni\u0001ei;\u001e=Bpen\ni;\u001e: (10)\nOr in a more compact formulation\nBpen\ni=@E\n@mi=1\nkmik@E\n@ei: (11)\nThanks to these penalty functionals, it becomes possible to target any local arbitrary mag-\nnetic con\fguration to \fnd the corresponding local e\u000bective \feld, which is an extremely useful\n7technique to explore the magnetic energy landscape. It is also possible to assign \u0015as a site-\ndependent parameter, by setting it to zero to constraint some atoms and let the others to\nadapt, during the self-consistency cycles.\nIn the following section we will use the penalty formalism to map the TB model onto an\nHeisenberg Hamiltonian and to derive a spin dynamics equation of motion that directly use\nthe penalty \feld hence derived.\nC. Exchange parameters in TB\nIn this section the general features to map the total energy of an electronic structure\nmethod onto a classical Heisenberg model is presented, that describes a system of atomic\nspin, characterized by local magnetic moments miat siteiinteracting via bare isotropic\ninteractions J0\nij:\nEHeis=\u00001\n2X\ni6=jJ0\nijmi\u0001mj;\n=\u00001\n2X\ni6=jJ0\nijkmikkmjkei\u0001ej;\n=\u00001\n2X\ni6=jJijei\u0001ej;(12)\nWithin this approach the amplitude of the magnetization kmikof siteican be incorporated\ne\u000bectively into the bare exchange interaction to produce a dressed exchange interaction, once\nassumed that thekmikbecome independent of the magnetic con\fguration. This assumption\nseems rather drastic but in many magnetic systems, where the magnetic moments are not so\ndependent on the magnetic con\fguration or for small rotations around a given angle, which\nis the case treated here. By keeping this assumption in mind, we can safely dropped the\ndressed reference.\nHowever in systems that break globally the symmetry of space rotation (particularly of\nnanometer size), this fails and the classical Heisenberg model is only valid for a limited\nrange around a given magnetic stable (or metastable) con\fguration C, that preserves the\ninvariance by point rotation only locally. In such systems the Heisenberg model can only be\nused to explore the dynamic around con\fguration C, that does not alter substantially the\ninvariance by point rotation, that are often found for low temperatures. Consequently for\nhigher temperatures or space transitions that reduce the point symmetry, the Jij's become\n8usually very sensitive to the structural parameters such as the interatomic distances and\nlocal environments, preventing their transferability to various atomic structures. This point\nis well illustrated in Appendix D.\nSince numerical implementations of the Heisenberg model are by far simpler than elec-\ntronic structure approaches, it is tempting to extract the desired exchange parameters Jij\nfrom electronic structure calculations. To do so, several methods have been reported in the\nliterature. i) The simplest method is based on a \ft of the total energy obtained by multiple\nmagnetic collinear con\fgurations, which do not necessitate any non-collinear numerical im-\nplementations neither penalty constraints [25]. ii) Another approach consists in performing\n\fnite di\u000berence calculations of the total energy between various magnetic non-collinear con-\n\fgurations [26], which can enlarged signi\fcantly the space of the magnetic con\fgurations to\nspan. In addition by varying the relative angle between the magnetic sites, it is possible to\ntest the range of validity of the Heisenberg picture [27, 28]. iii) Based on this \fnite di\u000berence\npicture, in a seminal work Liechtenstein et al derived an explicit expression of the exchange\nparameters, based on second order variation of the band energy term relying on the magnetic\nForce Theorem and Green's function formalism [29]. The latter one has shown big success in\npredicting various magnetic properties such as magnon excitation, critical temperature and\nalso used to perform dynamical calculation of magnetic moments [30]. In this work, we have\nused the approach ii), where we rotated one magnetic moment of an angle \u0012and developed\nan equation for E(\u0012) for each case, e.g. dimers (Sec. III A) and trimers (Sec. III B). We\nhave found that the energy curve between the TB model and the Heisenberg model agree\nquite well, which leads to a good agreement between the spin dynamics of the two di\u000berent\nmethods, shown later in Secs. III A and III B.\nDetails of the derived expression for both cases and the \ftting of the energies to \fnd the\nrespective exchange coupling parameter Jijfor each case is explored in more details at the\nAppendix D.\nD. Spin-dynamics in TB\nThe change in direction of each of the local magnetic moments mi= Tr(~\u001ai\u001b) with time\nis given by the transverse torque of this moment only with the e\u000bective pulsation, which is\n9in return precisely Be\u000b\ni\u0011\u0000Bpen\ni=\u0000@E\n@mi,\ndmi\ndt=mi\u0002Be\u000b\ni\n~=Bpen\ni\n~\u0002mi (13)\nBecauseBe\u000b\niis constructed orthogonal to mi,Be\u000b\niis itself a cross product of a functional of\nmi, bymi. Eq. (13) is nothing else than the Larmor's precession equation, which is itself a\nnon-relativistic limit of a more complex motion of spinning particles in a co-moving frame\n[31].\nIn practice, TB SCF calculations are \frst performed without any constraint to identify\nthe stable magnetic (or metastable) states meq\ni. Such a magnetic state is not necessarily\nunique and the process has to be repeated in frustrated systems that produce degenerate\nstates. However this process can be systematized by considering methods for \fnding mini-\nmum energy paths of transitions in magnetic systems [32]. Moreover if a precession around\nthe equilibrium magnetization is considered, the longitudinal term vanishes because Be\u000b\niis\nconstructed orthogonal to mi. Then a given spin direction mi(0) is chosen in the neighbor-\nhood of this equilibrium state and a constrained SCF calculation is performed according to\nthe chosen penalty method described above, to get the local e\u000bective \feld. Thus, a spin dy-\nnamics is produced by solving Eq.(13) in time by using an explicit solver. In this case, each\nlocal moment may have di\u000berent starting amplitude, that remains constant over time and\ntheir motion evolve on local spheres, according to the Rodrigues' rotation formula, that is\npresented in Appendix C. The procedure is repeated for each time step of the spin dynamics.\nIII. SPIN DYNAMICS OF MAGNETIC CLUSTERS\nIn this section, we study the dynamics of the magnetic moments under two di\u000berent\nscenarios: using an \"in house\" atomic spin dynamics (ASD) as implemented in Ref. [33]\nbased on an Heisenberg Hamiltonian and the tight-binding spin dynamics (TBSD) method\ndescribed in the previous Sec. II. This is applied for the most simple cases, i.e. dimers\nand equilateral triangle trimers of equivalent atoms for which the corresponding e\u000bective\nexchange interaction Jis obtained from our TB model and then used in the ASD for com-\nparison with TBSD. Note that since in the ASD code the dynamics is expressed in terms of\nunit vectors and the e\u000bective \feld is written as \u0000@E\n@ei(with nokmikfactor) we have used in\nthe TBSD an e\u000bective \feld given by \u0000kmikBpen\ni.\n10We would like to highlight that Ref. [34] have explored aspects of the results presented in\nthis paper, in parallel. Most of their e\u000borts was to verify if the e\u000bective \feld is exactly the\nnegative of the constraining \feld, which acts as a Lagrange multiplier to stabilize an out-\nof-equilibrium, noncollinear magnetic con\fguration, a point raised in Ref. 21. However, the\nquality of the derived e\u000bective \feld by constrained method is very sensitive to the numerical\nlimit of the Lagrange multiplier, a point we have carefully monitored. It is noteworthy to\nsay that our results are complementary and do not overlap in any way, specially in the\nspin-dynamics aspect of this work.\nA. Magnetic dimers\nMany studies have already addressed the spin dynamics of both quantum and classical\nHeisenberg dimers [35], not always systematically by looking the temporal dynamics of each\nof their individual moments. Using the method described in Sec. II D, we studied the time\nevolution of the net magnetic moments, here treated as a classical tridimensional vectors,\nfor magnetic dimers of Fe, Co and Ni. First, Eq. (13) is solved and the precession of these\nmagnetic moments is analyzed without damping, by starting from a tilted angle of 10\u000efrom\nthez-axis for each atomic site, as the initial con\fguration. Then by using the method\npresented in the Appendix D, our \fndings are compared with an atomistic spin dynamics\napproach using the exchange coupling Jextracted from the angular dependence of the total\nenergy. Our results, depicted in Fig. 1, show that all the three dimers behave well as under\nthe Heisenberg interaction in the studied limit, i.e. the e\u000bective \feld Beff\nican be described\nby a constant isotropic exchange, Eq. (12), that does not depend on the instantaneous\nmagnetic con\fguration. As shown in Appendix D, between \u0012= 0\u000eand\u0012= 10\u000ethe \ft\nbetween the energy calculated from the TB onto a Heisenberg Hamiltonian works perfectly,\nbut that does not hold true for higher angles. It means that a simple bi-linear Heisenberg\nHamiltonian is not enough to describe the system globally, but only locally with respect to\nthe magnetic con\fguration. Because the z-component of the magnetization is constant in\ntime, thez-component of the ASD torque is exactly zero, which is not the case in the TB\ndynamics. However, this can be consistently monitored by decreasing the timestep used to\nintegrate the precession equation, Eq. (13).\nWe can monitor that the precession frequency, as calculated in the appendix C, is well\n11-0.4-0.200.20.4mx(1)\n-0.4-0.200.20.4mx(2)\n-0.4-0.200.20.4my(1)\n-0.4-0.200.20.4my(2)\n01234mz(1)\n01234mz(2)\n-0.4-0.200.20.4Tx(1)\n-0.4-0.200.20.4Tx(2)\n-0.4-0.200.20.4Ty(1)\n-0.4-0.200.20.4Ty(2)\n0 2 4 6 8 10\ntime (fs)-0.00100.001Tz(1)\n0 2 4 6 8 10\ntime (fs)-0.00100.001Tz(2)Figure 1. (color online) Magnetization and torque dynamics of individual moments for for\ndimers of Fe (black), Co (red) and Ni (green). TBSD (resp. ASD) results are in solid lines\n(resp. circles). Unit of torques is PHz. Initial conditions are m1=g(\u0000sin(10\u000e);0;cos(10\u000e)),\nm2=g(sin(10\u000e);0;cos(10\u000e)), wheregare the SCF Land\u0013 e factor for each atom (see Appendix D).\nreproduced by the TB calculations.\nB. Magnetic trimers\nIt is known in the literature that in some speci\fc situations, the exchange coupling and\nDzyaloshinskii-Moriya interactions calculated from the ferromagnetic (FM) state are not a\ngood \ft for predictions of magnetic properties, e.g. close to the paramagnetic state [36] or\nthe transition from the FM to the skyrmion phase [37]. This is mainly because that in these\nscenarios, interactions of higher order play an important role and even sometimes a central\nrole, such as the value of considering the 4-spin interaction in case of stabilizing the skyrmion\nphase in hexagonal Fe \flm of one-atomic-layer thickness on the Ir(111) surface [38]. These\n12higher order interactions can be seen as if the exchange constants become kinetic functions of\nthe magnetization state, a possibility theorized long time ago [39]. One could argue that it is\nonly needed a high-order more speci\fc spin-Hamiltonian to describe the problem, but in some\nother cases the so called beyond-Heisenberg interactions can also be present, i.e. interactions\nthat cannot be mapped into a spin-Hamiltonian [40] or cases where the Heisenberg picture\nis simply broken [41]. Our goal here is to explore the limits and di\u000berences between the\nspin dynamics features using a spin-Hamiltonian and our presented here TB spin dynamics\nmethod.\nIn order to do that, the magnetization dynamics of magnetic equilateral triangle trimers\nof Fe, Co and Ni is explored, as can be seen in Fig. 2 The magnetization dynamics of Fe,\nCo and Ni triangle trimers, are depicted in Fig. 3 as well as the torques in Fig. 4.\nIn order to evaluate the exchange coupling between the magnetic moments in this case, an\nanalogous procedure to what was done to the dimer is performed, more precisely described in\nthe Appendix D. Fitting with the energy obtained from the TB calculation, the parameters\nare reported in Appendix D. Note that in this particular case J12=J23=J31due to the\nsymmetry. Initially, self-consistent calculations under the angular penalty function were\nperformed in order to determine the magnetic moments of each atom in the system. With\nthat information, one performs simulations of the magnetization dynamics using the spin\nHamiltonian, Eq. (12). Parallel to it, the process described in Sec. II D is followed, the\nmagnetization dynamics is calculated and the comparison between the di\u000berent methods is\nshown in Fig. 3. Similarly to the dimers case, the systems here presented show themselves as\nHeisenberg systems within the studied limit, e.g. \u0012= 10\u000e, when calculating the precession\nof the magnetic moments around the z-axis.\nSo far, these limits have served to prove the reliability of our method, and not to justify\nthe extra computational cost introduced to reproduce the behavior of an ASD approach. In\nthe next section we exhibit the simplest situation that demonstrate its relevance.\nC. Con\fguration dependence of the exchange coupling parameters Jij\nThe task of \fnding a reliable Hamiltonian to describe variations of magnetic con\fgura-\ntions is not straightforward. Continuous e\u000borts have been made throughout the years in\nthe attempt to understand the microscopic origin of these exchange parameters and their\n13Figure 2. (color online) a) Schematic representation of the the equilateral triangle trimer.\nconsequences [42]. Recently, a method to calculate the exchange coupling parameter Jijfor\nany given magnetic con\fguration, via \frst-principles simulations, was developed and ap-\nplied to study these interactions on Fe-bcc [43]. In fact, these con\fguration dependent Jij's\n14-0.6-0.4-0.200.20.40.6 mxAtom 1 Atom 2 Atom 3\n-0.6-0.4-0.200.20.40.6 my\n0 5 10 15 20\ntime (fs)11.522.53mz\n0 5 10 15 20\ntime (fs)0 5 10 15 20\ntime (fs)Figure 3. (color online) Magnetization dynamics of Fe (black), Co (red) and Ni (green) tri-\nangle trimers. TBSD (resp. ASD) results are in solid (resp. circles) lines. Initial condi-\ntions arem1(0) =g(\u00000:17365;0:0;0:98481),m2(0) =g(0:08682;\u00000:15038;0:98481),m3(0) =\ng(0:08682;0:15038;0:98481), where gare the SCF Land\u0013 e factor for each atom (see text).\nsigni\fcantly improved the spin-wave dispersion comparison between the theory and the ex-\nperiment. Within the TB approximation, Ref. [44] reports a con\fguration dependence of the\nexchange parameters by comparing various e\u000bective \feld Beffbetween the Heisenberg model\nand direct TB calculations. Moreover, it is crucial to understand the relevance of higher or-\nder parameters in the expansion of the magnetic Hamiltonian, e.g. and bi-quadratic terms,\n3-spins, 4-spins, etc., as can be seen in works like Refs. [28, 38] and [45]. Lastly Ref. [46]\nas implemented in Ref. [47], o\u000bers an attractive solution to the problem of a statistically\nunder-represented magnetic reference state, but at a cost of a span of the entire magnetic\ncon\fguration space. In principle, this allows the derivation of e\u000bective exchange coupling\nconstants that average the e\u000bect of more than 2 independent con\fgurations of spins. Un-\n15-0.4-0.200.20.4 TxAtom 1 Atom 2 Atom 3\n-0.4-0.200.20.4 Ty\n0 5 10 15 20\ntime (fs)-0.001-0.000500.00050.001\nTz\n0 5 10 15 20\ntime (fs)0 5 10 15 20\ntime (fs)Figure 4. (color online) Torque dynamics of Fe (black), Co (red) and Ni (green) triangle trimers.\nTBSD (resp. ASD) results are in solid (resp. circles) lines. Units of the torques are in PHz. Initial\nconditions are identical than those in Fig. 3.\nfortunately this statistical method is more suitable in the dilute magnetic limit and appears\nnot adequate to capture the magnetic behavior of a single speci\fc dimer or trimer. Moreover\nits implementation for alloys is complex.\nSo far, we have calculated the exchange coupling parameters by \ftting the energy from\nthe TB calculations around the ground state, i.e. FM for Fe, Co and Ni. These past studies\nhave revealed the non-Heisenberg behavior of Fe in particular and in order to illustrate our\nargument, we picked up the Fe dimer as an example. For a dimer, one can express the total\nTB energy as an expansion on a basis of Legendre polynomials up to a given order N, such\nas\nE(\u0012)\u0000E(0) =NX\nn=1J(n)\n12Pn(cos(\u0012)): (14)\n16When this series ends to N= 1,J(1)\n12is just the usual intensity of the Heisenberg coupling\nconstant. If this series ends to N\u00152, we can interpret J(2)\n12as a biquadratic component\nof the intensity of the magnetic coupling, characterized by a beyond-Heisenberg behavior.\nIn the Fig. 5 we show on the left, the total energy of Fe dimer as a function of the angle\n\u0012between the magnetic moments of each Fe atom, along with the exchange coupling J(1)\nij\ncalculated by \ftting the Heisenberg model around the local \u0012(at every step of \u0012= 10\u000e), on\nthe right.\n0 20 40 60 80 100 120 140 160 180\nangle (°)8.28.48.68.899.29.4Energy (eV)N = 1\nN = 2\nN = 6\nTB\n-2-1012\nJ12 (eV)\nFigure 5. (color online) TB total energy as a function of the angle between the two magnetic\nmoments of an Fe dimer on the left y-axis and the N= 1 exchange coupling parameter derived\nlocally for each angle, on the right y-axis. In addition, the TB total energy is globally \ftted\nby expansion in Legendre polynomials in terms of cos( \u0012). Here,N= 1 would be the bi-linear\nHeisenberg Hamiltonian, N= 2 includes the bi-quadratic term and so on so forth.\nIt is clear from the total energy calculations that, for that case, it cannot be \ftted by\na simple bi-linear Heisenberg model. We tried then to add a bi-quadratic correction to the\n17model as cos2(\u0012), as done in Ref. [48], by analyzing the P2(cos(\u0012)) =1\n2(3 cos2(\u0012)\u00001) part\nof the Legendre expansion, and then reported in the Fig. 5 along the N= 2 curve. One can\nnote that this N= 2 term improves globally the model curve, but quite not match the TB\ncalculations rigorously, in particular in the range of angles when the FM order is not the\npreferred magnetic ground state. It is needed to go up to the 6-th order to get a reasonable\n\ft that captures all the energetic features, including the reversal in the sign of the energy\nbehavior at intermediate angles. It is noteworthy to mention that the magnetic moment\nof each of the Fe atoms changes throughout the rotation of about 40% (data not shown),\nfrom 3\u0016B(FM con\fguration) to \u00191:8\u0016B(AF con\fguration); a feature that is also not\ncovered by the Heisenberg model. The parametric derivation of such a simple con\fguration\nspace indicates the magnitude of the task at hand in much more complex systems, such as\nalloys and materials with non-collinear magnetic con\fgurations as ground state. However we\nargue that properties strongly dependent on small variations around the ground state, such\nas spin-wave spectra, are well described with a local Heisenberg Hamiltonian, as already\nanticipated by Holstein and Primako\u000b [49], but we need a more precise electronic structure\nbehavior, in order to compute the correct e\u000bective \feld far from the ground state and not\nnecessarily represented by the magnon state of lowest energy. In that scenario the e\u000bective\n\feld directly derived from the electronic structure, produces the correct dynamics in time\nfor any directions of any local magnetic moments, without prior knowledge of any exchange\nvalues and represents, by construction, a direct solution to avoid such issue.\nIV. CONCLUSION\nIn this paper, we have presented a method that o\u000bers an alternative between full ab\ninitio and spin-Hamiltonian based spin-dynamics. Our approach uses a penalty functional\non the magnetic moments of each site in order to calculate self-consistently, at every time\nstep, the respective e\u000bective magnetic \feld. We have solved the precession equation on\neach site, without damping, for dimers and trimers of Fe, Co and Ni, and compared our\n\fndings with an ASD approach, where the magnetic e\u000bective \feld is not calculated directly\nfrom the electronic structure, but from a parameterized spin-Hamiltonian. The exchange\ncoupling interaction J, as a parameter, was calculated by \ftting the TB total energy with a\nparameterized spin-Hamiltonian for a range of directions of the atomic magnetic moments.\n18Our results showed that within this limit, they can be seen as good Heisenberg systems\nlocally and the comparison between the TB and ASD are fairly good. That is not the\ncase where the same set of magnetic moments connect di\u000berent magnetic extrema, meaning\nthat di\u000berent parametric local representations have to be calculated, which breaks the whole\nHeisenberg picture. For those systems, one cannot map globally the electronic structure onto\na single Heisenberg model, although these parameters still can predict with good accuracy\nproperties of their local ground states. We have illustrated this situation by studying the\ndependance of the total energy of an Fe dimer, as a function of the angle between the\natomic magnetic moments, and proved that this cannot be mapped globally into a bi-\nlinear Heisenberg Hamiltonian only. In fact, a high-order expansion in power of the angular\ndirections between the atomic magnetic moments is mandatory to match the landscape\nof the TB energy adequately. Finally, the TBSD here presented is a satisfying solution,\nwith a reasonable computational cost, to study the spin-dynamics of systems that are not\ndominated by the pair Heisenberg's interaction only, because the construction of the ab initio\ne\u000bective \feld is free from such hypothesis. This technique may serve also to investigate the\ndynamics of more complex magnetic systems that include spin-orbit mediated interactions\nin low dimensional symmetries, and appears to be both versatile and general.\nACKNOWLEDGMENTS\nWe gratefully thank to the Programme Transversal de Competences for \fnancial support\nwith the project DYNAMOL.\nDATA AVAILABILITY STATEMENT\nThe data that support the \fndings of this study are available from the corresponding\nauthor, upon reasonable request.\nAppendix A: Fixed spin moment\nThe \fxed spin moment calculation is probably the most straightforward method, but is\nlimited to the case of collinear magnetism and is independent of the site index. This is to\n19impose exactly a total magnetization of the system and therefore the total number of \"and\n#electrons. One therefore needs to de\fne two separate Fermi levels E\u001b\nF. For a homogeneous\nsystem where each atom carries the same charge and the same magnetization, the total\nenergy is\nEtot=j\"\"\n\u000bj<E\"\nFX\n\u000b\"\"\n\u000b+j\"#\n\u000bj<E#\nFX\n\u000b\"#\n\u000b+1\n4Im2; (A1)\nwhere\"\u001b\n\u000b=\"0\n\u000b\u00001\n2Im\u001b. Then the total energy can be rewritten as\nEtot=j\"0\n\u000bj<E\"\nF+1\n2ImX\n\u000b\"0\n\u000b+j\"0\n\u000bj<E#\nF\u00001\n2ImX\n\u000b\"0\n\u000b\u00001\n4Im2: (A2)\nConsequently, the derivative of the total energy with the magnetization becomes simply\nproportional to the di\u000berence of Fermi's energies\ndEtot\ndm=(E\"\nF\u0000E#\nF)\n2: (A3)\nAn e\u000bective \feld Be\u000b=\u0000(E\"\nF\u0000E#\nF)=2, aligned to these moments, can be de\fned. It comes\nout that at the extrema of Etot, the two Fermi levels are equal and the e\u000bective \feld becomes\nzero. By looking at the sign of the second derivative of the energy around m= 0, this is\nsimple to recover the Stoner criterion as described in the reference [50]. Although useful,\nthe \fxed spin moment method is limited to rather homogeneous systems.\nAppendix B: Penalty method for atomic spin moment\nLet us consider the case where a given magnetization mpen\niis imposed on each atom.\nA quadratic penalty term as Epen\ni=\u0015\n2kmi\u0000mpen\nik2can be added to each site, where\n\u0015is a large positive number. In principle \u0015should go to in\fnity, but in practice a good\ncompromise is found by increasing its value and to check the convergence of the desired\nquantity computed with. However, this problem can be circumvented by implementing\nthe Augmented Lagrangian Method, that introduces a quadratic constraint term in the\nrenormalized Hamiltonian, such as the \u0015parameter remains \fnite [51]. This is at the cost\nof an additional computational complexity and the penalization approach with a su\u000ecient\nlarge\u0015term is preferred.\nThis consists to supplement Eq.(6) with the term \u0015(mi\u0000m0\ni)jiihij. Consequently, the\non-site diagonal renormalization term can formally be written Ui\u0001ni\u001b0\u0000(BStoner\ni +Bpen\ni)\u0001\u001b\n20withBStoner\ni =1\n2Iimi,Bpen\ni=\u0000\u0015(mi\u0000mpen\ni) and \u0001ni= (ni\u0000n0\ni). The total energy should\nbe corrected accordingly by the double counting terms and reads\nEtot[fm0\nig] =occX\n\u000b\"\u000b\u00001\n2X\niUi((ni)2\u0000(n0\ni)2)\n+1\n4X\niIikmik2\u0000\u0015\n2X\ni(kmik2\u0000kmpen\nik2): (B1)\nIn the limit \u0015! 1 ,\u0000\u0015(mi\u0000mpen\ni)\u0019Bpen1\ni andmi\u0019mpen\ni. Consequently, the\ncorresponding double counting term \u0000\u0015\n2(kmik2\u0000kmpen\nik2) can be rewritten as Bpen1\ni\u0001mpen\ni.\nThe \fxed spin moment can be seen as a special case of the penalty method applied for\ncollinear magnetism with only one type of atom. The term \u0000Bpen\u001bin the renormalized\nHamiltonian just shifts rigidly the eigenvalues by \u0000Bpenfor\"spin andBpenfor#spin,\nsuch as\"\u000b=\"0\n\u000b\u00001\n2Im\u001b\u0000Bpen\u001b. The total energy of Eq.(A2) is recovered once provided\nE\u001b\nF=EF+\u001bBpen. Then one gets Bpen=1\n2(E\"\nF\u0000E#\nF) =\u0000Be\u000b.\nAppendix C: Solution of the spin dynamics of ferromagnetic dimers\nThe motion of each individual moments of ferromagnetic dimers within the Heisenberg\ninteraction is a two-body problem admitting an exact solution. Let's \n0\ns\u0011J0=~the mag-\nnitude of the exchange pulsation and E=\u0000J0m1\u0001m2its interaction energy, with J0>0.\nThe motion of each undamped moment is the solution of a set of 2 coupled equations of\nprecession, which are\ndm1\ndt= \n0\nsm2\u0002m1;\ndm2\ndt= \n0\nsm1\u0002m2;(C1)\nwith the given initial conditions m1(0) andm2(0).\nEquivalently when using an Heisenberg Hamiltonian with normalized vectors E=\u0000Je1\u0001\ne2, withJ=J0m2(wheremis the amplitude of the magnetization) we get the coupled\nevolution equations:\nde1\ndt= \nse2\u0002e1;\nde2\ndt= \nse1\u0002e2;(C2)\nwith \ns\u0011J=~. This motion is decoupled in the frame of the magnetization e\u0011(e1+e2).\nIn this frame, by combining Eqs.(C2) together, one \fndsde\ndt=0and consequently eis a\n21constant vector given by the initial conditions e= (e1(0) +e2(0)). By noting that \n se2\u0002\ne1= \ns(e1+e2)\u0002e1= \nse\u0002e1, Eqs. (C2) become fully decoupled:\nde1\ndt= \nse\u0002e1;\nde2\ndt= \nse\u0002e2:(C3)\nThen the motion of each of these unit vectors eiis simply the motion of a vector in a constant\n\feld. Its solution is given by the Rodrigues' formula [52]\nei(t) = cos(\n st)ei(0) + sin(\n st)e+ (1\u0000cos(\nst))\u001fiei(0)\u0002e; (C4)\nwhere\u001fi\u0011ei(0)\u0001e.\nThe same reasoning can be derived for trimers of identical atoms with the same exchange\nparameters applied up to the \frst neighboring shell, in between. In that very speci\fc case,\neach atomic spin follows the same equation of precession, namely\ndei\ndt= \nse\u0002ei; (C5)\nwithe\u0011P3\ni=1ei(0), whereeis found to be constant of motion. Consequently for trimers\nwith identical atoms and interactions, the precession frequency, and thus the value of the\nexchange parameter, can be measured from a single motion of any spins, as depicted in\nFigs. 3 and 4.\nAppendix D: Calculation of the exchange coupling parameters\nThe macroscopic nature of the exchange coupling parameters and how they are in\ruenced\nby the various circumstances have been widely discussed in the literature. The Bethe-\nSlater [53] (BS) curve explains in an insightful way, by means of direct exchange and the\ndistance between nearest-neighbor (NN) atoms, the trends followed by ferromagnetism (FM)\nand antiferromagnetism (AFM) ground state of the 3d transition metals from bcc Cr to\nhcp Co. Recent studies [54] have shown that, even for the bulk case of such elements,\nthe BS curve reveals a complicated background behind the macroscopic picture. Such NN\ninteractions depend not only on the distance but also the symmetry and their bonds, i.e.\nin\ruenced by the crystal \feld. That kind of dependence has also been seen in supported\nnanoclusters [55], where for the same distance, di\u000berent values for the exchange coupling\n22parameter can be found. In case of small clusters, like the dimers and trimers studied\nhere, the local density of states of each atom is very localized, which set apart the majority\nband from the minority band. It implies in a large band splitting that directly a\u000bects the\nvalue of the of the exchange coupling parameter [56, 57]. As coordination number increases,\nthe hybridization results in the broadening of such bands, shifting the center of it closer\nto the Fermi energy, thus decreasing the value of the exchange coupling parameter as the\ncoordination number increases [58, 59]. Moreover, the results here presented follow this\nlogic, as well as the BS curve trend.\nFor each of the magnetic con\fgurations, the total energy is computed with the TB param-\neters found in reference [13]. When only one rotating single magnetic moment is considered,\nthe total energy in the Heisenberg model can be written as a function of the angle with the\nz-axis, labelled \u0012. For the dimer it reads\nEdimer(\u0012)\u0000Edimer(0) =Jdimer(1\u0000cos(\u0012)); (D1)\nand for the trimer\nEtrimer(\u0012)\u0000Etrimer(0) = 2Jtrimer(1\u0000cos(\u0012)): (D2)\nAs seen in Fig. 6, Eqs. (D1) and (D2) can be \ftted with the total energy computed in the\nTB approximation, in order to \fnd the respective exchange coupling parameters J. For the\ndimer, it is obvious that J12=J21\u0011Jdimer and for the trimer, because of the C3symmetry,\nJ12=J23=J31\u0011Jtrimer also. The fact that the \ftting and the energy curve fall on top of\neach other, means that both Jdimer andJtrimer are constants within the limit considered of \u0012,\ni.e. the electronic interaction in these systems is dominated mainly by the Heisenberg's pair\ninteraction (12) in that range. The computed values taken for an equal distance d= 2\u0017A\nbetween atoms are reported in the tables I and II.\nFinally another strategy has been tested to evaluate the exchange parameters. Instead\nof considering the total energy variations E(\u0012) as the reference quantity, we have \ftted\nthe variation of the e\u000bective \feld Bpenas a function of the deviation angle \u0012. Indeed it is\nstraightforward to show that kBpenkkmkis equal to Jsin\u0012for the dimer and 2 Jsin\u0012for\nthe trimer, respectively. The results are reported in parenthesis in the tables I and II. The\nagreement between the two approaches is good and could be systematically improved by\nincreasing the penalization constant \u0015.\n23g (\u0016B)Jdimer (eV)\nFe 30.616 (0.605)\nCo 20.574 (0.561)\nNi 10.341 (0.312)\nTable I. Values of the computed SCF magnetization and exchange parameter for dimers (inter-\natomic distance of 2 \u0017A) calculated in the TB approximation. In parenthesis is shown the result\nobtained from the \ft of the e\u000bective \feld.\ng (\u0016B)Jtrimer (eV)\nFe2.6666 0.442 (0.463)\nCo1.6666 0.279 (0.273)\nNi0.6666 0.089 (0.103)\nTable II. Values of the computed SCF magnetization and exchange parameter for equilateral trian-\ngle trimers calculated in the TB approximation. In parenthesis is shown the result obtained from\nthe \ft of the e\u000bective \feld.\n[1] J. H. Lee, L. Fang, E. Vlahos, X. Ke, Y. W. Jung, L. F. Kourkoutis, J.-W. Kim, P. J. Ryan,\nT. Heeg, M. Roeckerath, V. Goian, M. Bernhagen, R. Uecker, P. C. Hammel, K. M. Rabe,\nS. Kamba, J. Schubert, J. W. Freeland, D. A. Muller, C. J. Fennie, P. Schi\u000ber, V. Gopalan,\nE. Johnston-Halperin, and D. G. Schlom, Nature 466, 954 (2010).\n[2] G. Grimvall, B. Magyari-K ope, V. Ozoli\u0018 n\u0014 s, and K. A. Persson, Rev. Mod. Phys. 84, 945 (2012);\nD. Antonangeli, L. R. Benedetti, D. L. Farber, G. Steinle{Neumann, A.-l. Auzende, J. Badro,\nM. Han\rand, and M. Krisch, Appl. Phys. Lett. 92, 111911 (2008); M. \u0014Cern\u0013 y, Materials Science\nand Engineering: A International Symposium on Physics of Materials, 2005, 462, 432 (2007);\nP. S oderlind, R. Ahuja, O. Eriksson, J. M. Wills, and B. Johansson, Phys. Rev. B 50, 5918\n(1994).\n[3] H. C. Hsueh, J. Crain, G. Y. Guo, H. Y. Chen, C. C. Lee, K. P. Chang, and H. L. Shih,\nPhys. Rev. B 66, 052420 (2002); O. Mathon, F. Baudelet, J. P. Iti\u0013 e, A. Polian, M. d'Astuto,\nJ. C. Chervin, and S. Pascarelli, Phys. Rev. Lett. 93, 255503 (2004); A. Monza, A. Me\u000bre,\n240 2 4 6 8 10\nθ [degrees]0246810E(θ)-E(0) [meV]Fe2\nCo2\nNi2\nz\nθ\n0 2 4 6 8 10\nθ[degrees]00.050.10.15m|Bpen| [eV]Fe2\nCo2\nNi2\n0 2 4 6 8 10\nθ [degrees]0246810E(θ)-E(0) [meV]Fe3\nCo3\nNi3\nθ\n0 2 4 6 8 10\nθ[degrees]00.050.10.15m|Bpen| [eV]Fe3\nCo3\nNi3Figure 6. (color online) Calculated total energy (left) and e\u000bective \feld (right) from the tight-\nbinding model for dimers and trimers of Fe, Co and Ni (interatomic distance of 2 \u0017A), in black, red\nand green, respectively. Dimers on top and trimers on the bottom. The corresponding \fts from\nan Heisenberg Hamiltonian fall exactly on top of the tight-binding curves (and are not shown),\nsuggesting that the Heisenberg's pair interaction dominates these systems within that range of \u0012.\nF. Baudelet, J.-P. Rue\u000b, M. d'Astuto, P. Munsch, S. Huotari, S. Lachaize, B. Chaudret, and\nA. Shukla, Phys. Rev. Lett. 106, 247201 (2011).\n[4] S. V. Halilov, H. Eschrig, A. Y. Perlov, and P. M. Oppeneer, Phys. Rev. B 58, 293 (1998).\n[5] P.-W. Ma, S. L. Dudarev, and J. S. Wr\u0013 obel, Phys. Rev. B 96, 094418 (2017); J. Tranchida,\nS. J. Plimpton, P. Thibaudeau, and A. P. Thompson, J. Comp. Phys. 372, 406 (2018).\n[6] V. P. Antropov, M. I. Katsnelson, M. van Schilfgaarde, and B. N. Harmon, Phys. Rev. Lett.\n75, 729 (1995); V. P. Antropov, M. I. Katsnelson, B. N. Harmon, M. van Schilfgaarde, and\nD. Kusnezov, Phys. Rev. B 54, 1019 (1996).\n[7] J. M. Combes, P. Duclos, and R. Seiler, in Rigorous Atomic and Molecular Physics, edited by\n25G. Velo and A. S. Wightman (Springer US, Boston, MA, 1981) pp. 185{213.\n[8] O. Eriksson, A. Bergman, L. Bergqvist, and J. Hellsvik,\nAtomistic Spin Dynamics: Foundations and Applications, 1st ed. (Oxford University Press,\nOxford, 2017).\n[9] T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism (Springer Berlin, Berlin, 2014);\nN. B. Melnikov and B. I. Reser, Dynamic Spin-Fluctuation Theory of Metallic Magnetism\n(Springer Nature Springer, Cham, 2018).\n[10] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena, 4th ed., International Series\nof Monographs on Physics No. 113 (Clarendon Press ; Oxford University Press, Oxford : New\nYork, 2002).\n[11] G. M. Stocks, B. Ujfalussy, X. Wang, D. M. C. Nicholson, W. A. Shelton, Y. Wang, A. Can-\nning, and B. L. Gy or\u000by, Philosophical Magazine B 78, 665 (1998); B. \u0013Ujfalussy, X.-D. Wang,\nD. M. C. Nicholson, W. A. Shelton, G. M. Stocks, Y. Wang, and B. L. Gyor\u000by, Journal of\nApplied Physics 85, 4824 (1999); R. Gebauer and S. Baroni, Phys. Rev. B 61, R6459 (2000).\n[12] M. F ahnle, R. Drautz, R. Singer, D. Steiauf, and D. V. Berkov, Computational Materials\nScience 32, 118 (2005); B. \u0013Ujfalussy, B. Lazarovits, L. Szunyogh, G. M. Stocks, and P. Wein-\nberger, Phys. Rev. B 70, 100404 (2004).\n[13] C. Barreteau, D. Spanjaard, and M.-C. Desjonqu\u0012 eres, Comptes Rendus Physique 17, 406\n(2016).\n[14] G. Aut\u0012 es, C. Barreteau, D. Spanjaard, and M.-C. Desjonqu\u0012 eres, J. Phys.: Condens. Matter\n18, 6785 (2006).\n[15] C. Barreteau and D. Spanjaard, J. Phys.: Condens. Matter 24, 406004 (2012).\n[16] L. Le Laurent, C. Barreteau, and T. Markussen, Phys. Rev. B 100, 174426 (2019).\n[17] R. Soulairol, C.-C. Fu, and C. Barreteau, J. Phys.: Condens. Matter 22, 295502 (2010).\n[18] R. Soulairol, C. Barreteau, and C.-C. Fu, Phys. Rev. B 94, 024427 (2016).\n[19] M. Finnis, Interatomic Forces in Condensed Matter, Oxford Series on Materials Modelling\nNo. 1 (Oxford Univ. Press, Oxford, 2003).\n[20] R. Fletcher, Practical Methods of Optimization, 2nd ed., A Wiley-Interscience Publication\n(Wiley, Chichester, 2008).\n[21] P. H. Dederichs, S. Bl ugel, R. Zeller, and H. Akai, Phys. Rev. Lett. 53, 2512 (1984).\n[22] P.-W. Ma and S. L. Dudarev, Phys. Rev. B 91, 054420 (2015).\n26[23] B. Kaduk, T. Kowalczyk, and T. Van Voorhis, Chem. Rev. 112, 321 (2012).\n[24] L. M. Small and V. Heine, J. Phys. F: Met. Phys. 14, 3041 (1984).\n[25] J. J. Phillips and J. E. Peralta, J. Chem. Phys. 138, 174115 (2013); D. Vaclavkova, A. Del-\nhomme, C. Faugeras, M. Potemski, A. Bogucki, J. Su\u000bczy\u0013 nski, P. Kossacki, A. R. Wildes,\nB. Gr\u0013 emaud, and A. Sa\u0013 ul, 2D Mater. 7, 035030 (2020).\n[26] L. M. Sandratskii, Physica Status Solidi (b) 136, 167 (1986); Advances in Physics 47, 91\n(1998); O. Grotheer, C. Ederer, and M. F ahnle, Phys. Rev. B 62, 5601 (2000); Phys. Rev. B\n63, 100401 (2001).\n[27] N. M. Rosengaard and B. Johansson, Phys. Rev. B 55, 14975 (1997).\n[28] S. Brinker, M. d. S. Dias, and S. Lounis, New J. Phys. 21, 083015 (2019).\n[29] A. I. Liechtenstein, M. I. Katsnelson, and V. A. Gubanov, J. Phys. F: Met. Phys. 14, L125\n(1984); A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, and V. A. Gubanov, Journal of\nMagnetism and Magnetic Materials 67, 65 (1987).\n[30] M. Pajda, J. Kudrnovsk\u0013 y, I. Turek, V. Drchal, and P. Bruno, Phys. Rev. B 64, 174402 (2001).\n[31] V. Bargmann, L. Michel, and V. L. Telegdi, Phys. Rev. Lett. 2, 435 (1959).\n[32] P. F. Bessarab, V. M. Uzdin, and H. J\u0013 onsson, Computer Physics Communications 196, 335\n(2015); A. V. Ivanov, D. Dagbartsson, J. Tranchida, V. M. Uzdin, and H. J\u0013 onsson, J. Phys. :\nCondens. Matter 32, 345901 (2020).\n[33] D. Beaujouan, P. Thibaudeau, and C. Barreteau, Phys. Rev. B 86, 174409 (2012).\n[34] S. Streib, V. Borisov, M. Pereiro, A. Bergman, E. Sj oqvist, A. Delin, O. Eriksson, and\nD. Thonig, Phys. Rev. B 102, 214407 (2020).\n[35] D. Mentrup, J. Schnack, and M. Luban, Physica A: Statistical Mechanics and its Applica-\ntions 272, 153 (1999); D. V. Efremov and R. A. Klemm, Phys. Rev. B 66, 174427 (2002);\nA. K. Kolezhuk, V. N. Glazkov, H. Tanaka, and A. Oosawa, Phys. Rev. B 70, 020403 (2004);\nA. Cabot, G. L. Giorgi, F. Galve, and R. Zambrini, Phys. Rev. Lett. 123, 023604 (2019).\n[36] S. Shallcross, A. E. Kissavos, V. Meded, and A. V. Ruban, Phys. Rev. B 72, 104437 (2005);\nA. V. Ruban, S. Khmelevskyi, P. Mohn, and B. Johansson, Phys. Rev. B 75, 054402 (2007).\n[37] S. M uhlbauer, B. Binz, F. Jonietz, C. P\reiderer, A. Rosch, A. Neubauer, R. Georgii, and\nP. B oni, Science 323, 915 (2009).\n[38] S. Heinze, K. von Bergmann, M. Menzel, J. Brede, A. Kubetzka, R. Wiesendanger,\nG. Bihlmayer, and S. Bl ugel, Nature Physics 7, 713 (2011).\n27[39] K. A. Chao, J. Spalek, and A. M. Oles, J. Phys. C: Solid State Phys. 10, L271 (1977); K. A.\nChao, J. Spa lek, and A. M. Ole\u0013 s, Phys. Rev. B 18, 3453 (1978).\n[40] Y. O. Kvashnin, R. Cardias, A. Szilva, I. Di Marco, M. I. Katsnelson, A. I. Lichtenstein,\nL. Nordstr om, A. B. Klautau, and O. Eriksson, Phys. Rev. Lett. 116, 217202 (2016).\n[41] S. Mankovsky, S. Polesya, and H. Ebert, Phys. Rev. B 101, 174401 (2020).\n[42] J. Adler and J. Oitmaa, J. Phys. C: Solid State Phys. 12, 575 (1979).\n[43] A. Szilva, M. Costa, A. Bergman, L. Szunyogh, L. Nordstr om, and O. Eriksson, Phys. Rev.\nLett. 111, 127204 (2013).\n[44] S. Streib, A. Szilva, V. Borisov, M. Pereiro, A. Bergman, E. Sj oqvist, A. Delin, M. I. Kat-\nsnelson, O. Eriksson, and D. Thonig, arXiv:2103.04726 [cond-mat] (2021), arXiv:2103.04726\n[cond-mat].\n[45] M. d. S. Dias, S. Brinker, A. L\u0013 aszl\u0013 o\u000by, B. Ny\u0013 ari, S. Bl ugel, L. Szunyogh, and S. Lounis,\narXiv:2101.00463 [cond-mat] (2021), arXiv:2101.00463 [cond-mat].\n[46] R. Drautz and M. F ahnle, Phys. Rev. B 69, 104404 (2004).\n[47] L. Szunyogh, L. Udvardi, J. Jackson, U. Nowak, and R. Chantrell, Phys. Rev. B 83, 024401\n(2011).\n[48] A. T. Costa, R. B. Muniz, and D. L. Mills, Phys. Rev. Lett. 94, 137203 (2005).\n[49] T. Holstein and H. Primako\u000b, Phys. Rev. 58, 1098 (1940).\n[50] S. Blundell, Magnetism in Condensed Matter, reprint ed., Oxford Master Series in Condensed\nMatter Physics No. 4 (Oxford Univ. Press, Oxford, 2014).\n[51] C. Li, W. Yin, H. Jiang, and Y. Zhang, Comput Optim Appl 56, 507 (2013).\n[52] O. Rodrigues, J. Math. Pures Appl. 5, 380 (1840); P. Thibaudeau and D. Beaujouan, Physica\nA391, 1963 (2011).\n[53] J. C. Slater, Phys. Rev. 36, 57 (1930); Phys. Rev. 35, 509 (1930); S. Chikazumi,\nPhysics of Ferromagnetism, International Series of Monographs on Physics (Oxford Science\nPublications, 1997).\n[54] R. Cardias, A. Szilva, A. Bergman, I. D. Marco, M. I. Katsnelson, A. I. Lichtenstein, L. Nord-\nstr om, A. B. Klautau, O. Eriksson, and Y. O. Kvashnin, Scienti\fc Reports 7, 4058 (2017).\n[55] P. Mavropoulos, S. Lounis, and S. Bl ugel, physica status solidi (b) 247, 1187 (2010); D. C.\nd. M. Rodrigues, M. Pereiro, A. Bergman, O. Eriksson, and A. B. Klautau, J. Phys.: Con-\ndens. Matter 29, 025807 (2016); A. Belabbes, G. Bihlmayer, F. Bechstedt, S. Bl ugel, and\n28A. Manchon, Phys. Rev. Lett. 117, 247202 (2016).\n[56] A. Bergman, L. Nordstr om, A. Burlamaqui Klautau, S. Frota-Pess^ oa, and O. Eriksson, Phys.\nRev. B 73, 174434 (2006).\n[57] S. Frota-Pess^ oa, R. B. Muniz, and J. Kudrnovsk\u0013 y, Phys. Rev. B 62, 5293 (2000).\n[58] M. M. Bezerra-Neto, M. S. Ribeiro, B. Sanyal, A. Bergman, R. B. Muniz, O. Eriksson, and\nA. B. Klautau, Scienti\fc Reports 3, 3054 (2013).\n[59] R. N. Igarashi, M. M. B. Neto, L. T. F. Eleno, A. Bergman, A. B. Klautau, O. Eriksson, and\nH. M. Petrilli, J. Phys.: Condens. Matter 26, 206003 (2014).\n29"    },    {        "title": "1809.04921v1.On_the_role_of_magnetic_fields_in_star_formation.pdf",        "content": "arXiv:1809.04921v1  [astro-ph.SR]  13 Sep 2018On the role of magnetic fields in star formation\nC. J. Nixona,∗, J. E. Pringlea,b\naDepartment of Physics and Astronomy, University of Leicest er, Leicester, LE1 7RH,\nUK\nbInstitute of Astronomy, Madingley Road, Cambridge, CB3 0HA , UK\nAbstract\nMagnetic fields are observed in star forming regions. However simula tions of\nthe late stages of star formation that do not include magnetic fields provide\na good fit to the properties of young stars including the initial mass f unc-\ntion (IMF) and the multiplicity. We argue here that the simulations tha t do\ninclude magnetic fields are unable to capture the correct physics, in partic-\nular the high value of the magnetic Prandtl number, and the low value of\nthe magnetic diffusivity. The artificially high (numerical and uncontro lled)\nmagnetic diffusivity leads to a large magnetic flux pervading the star f orm-\ning region. We argue further that in reality the dynamics of high magn etic\nPrandtl number turbulence may lead to local regions of magnetic en ergy dis-\nsipation through reconnection, meaning that the regions of molecu lar clouds\nwhich are forming stars might be essentially free of magnetic fields. T hus the\nsimulations that ignore magnetic fields on the scales on which the prop erties\nof stellar masses, stellar multiplicities and planet-forming discs are de ter-\nmined, may be closer to reality than those which include magnetic fields , but\ncan only do so in an unrealistic parameter regime.\nKeywords: accretion, accretion discs, dynamo, magnetic fields,\nmagnetohydrodynamics (MHD), planets and satellites: formation, stars:\nformation\n∗Corresponding author\nEmail address: cjn@leicester.ac.uk (C. J. Nixon)\nPreprint submitted to New Astronomy September 14, 20181. Introduction\nIn two papers, Mestel (1965a,b) argued for the importance of th e role\nof magnetic fields in star formation. He pointed out that an average region\nof the interstellar medium (ISM) containing a stellar amount of mass c an-\nnot simply collapse to stellar densities, because it contains too much a ngular\nmomentum. He argued that magnetic fields are likely to play a vital role\nin removing that angular momentum. At the same time, he pointed out\nthat the average region of interstellar medium containing a stellar ma ss also\ncontains too much magnetic flux for it to be able to collapse to stellar d ensi-\nties. Therefore the magnetic field has to find a balance between ena bling the\nremoval of angular momentum, and itself escaping from the collapsin g mate-\nrial. He proposed that ambipolar diffusion might provide such a mechan ism\n(see also Mestel and Spitzer 1956).\nIn contrast, Bate (2012) started with a self-gravitating, turbu lent cloud\ncore of mass M= 500M⊙, density n≈105cm−3and temperature T= 10K,\nandfollowedthesubsequent evolution. Hewasabletoreproduceth eobserved\ninitial mass function, and also the observed properties of binary an d multiple\nstars, forstarslessthanaroundasolarmass. Similarresultswere obtainedby\nKrumholz et al. (2012) using a grid based code. Moreover, Bate (20 18) has\nshown that his simulations also produce plentiful and massive discs ar ound\nhis protostars, of the kind required for planet formation (Nixon et al., 2018)\nand beginning to be seen around the youngest (Class 0 and I) proto stars\n(e.g. Tobin et al., 2015; P´ erez et al., 2016). None of the simulations b y Bate\n(2012); Krumholz et al. (2011, 2012) included magnetic fields.\nIn the light of all this McKee (Reipurth, 2017) commented: “ How is that\npossible when it is known that magnetic fields...have a major effect in extract-\ning angular momentum from the accreting gas? In fact, in our c urrent un-\nderstanding, magnetic fields are so effective at extracting a ngular momentum\nthat many simulations of the formation of protostellar disk s fail to produce\ndisks nearly as large as observed. ”\nInfact, McKee’s comments illustratevery well theproblemwithmagn etic\nfields. If we do not put them into the simulations, then we can get res ults\nquite close to the observations. But if we include magnetic fields, the n we\ndo not. Application of Occam’s Razor suggests a simple conclusion. Bu t\nthe question then is: how do we reconcile this with the observed pres ence of\nmagnetic fields in and around regions of star formation (see the rev iew by\nCrutcher, 2012)? It is this apparent contradiction that we addre ss in this\n2paper.\n2. Do we need magnetic fields?\nThe presence and influence of magnetic fields has been thought to p lay a\nmajor role in two aspects of the star formation process. First mag netic fields\nare able to transfer angular momentum efficiently and so are a poten tial\nsolution of Mestel’s angular momentum problem. Second, magnetic fie lds\nare able to provide additional support to cloud material against gra vitational\ncollapse, and so can mediate, and in particular reduce, the rate at w hich star\nformationcanproceed indense interstellar material. Wediscuss eac hofthese\nin turn.\n2.1. Is there an angular momentum problem?\nThe picture of star formation envisaged by Mestel was that of the for-\nmation of a single star, such as the Sun, from the monolithic gravitat ional\ncollapse of an amount of interstellar material. This concept was later devel-\noped in more detailed form, with single core, monolithic collapse calculat ions\nleading to the view of star formation summarized in the review by Shu e t al.\n(1987) (and also promulgated in reviews by Stahler and Palla 2005, an d by\nMcKee and Ostriker 2007). It is clear that if one views the star form ation\nprocess in terms of forming one star at a time from the interstellar m edium\nwhich is of necessity rotating, then the need for the removal of an gular mo-\nmentum from the forming protostar becomes paramount.\nThe problem with this approach from the point of view of star format ion\nis that it always leads to the formation of single stars. This is not a goo d\nresult for typical solar mass stars of which only 50 ±10 per cent are single\n(Raghavan et al., 2010).\nIn view of this it is possible to make the case (Pringle, 1989, 1991;\nClarke and Pringle, 1991; Reipurth and Clarke, 2001) that, contra ry to the\nsingle core collapse picture, the formation of binary (and multiple) st ars is\nin fact the way to understand the formation of all stars. The point is that in\norder to account for the occurrence of numbers of binary and mu ltiple sys-\ntems it is necessary that essentially all stars have to form in the pre sence of\ncompanions. If all stars form in groups, then many of these will be e jected as\nsingle stars (see the reviews by Zinnecker 2001; Reipurth et al. 201 4). And\ngiven that single stars are in a minority, it follows that most stars mus t form\n3in groups. The observational case for the veracity of this conclus ion is re-\nviewed by Lada and Lada (2003). This leads to the current model of chaotic\nstar formation crystallized by Bate (2012).\nIn this picture, it is to be expected that the angular momentum prob -\nlem is to a large extent overcome by gravitational interactions alone (e.g.\nLarson, 2010), and this expectation is confirmed by the simulations . Thus it\nis clear that while magnetic fields may be present, they are not requir ed to\nsolve Mestel’s fundamental angular momentum problem of removing a ngular\nmomentum from the interstellar medium.\nNote, however, that the presence of magnetic fields is likely require d at\nsome level in the very late stages in order to help drive the final stag es of\ndisc evolution and the formation of jets, although Hartmann and Ba e (2018)\nmake the case that the importance of disc magnetic winds may have b een\noverestimated. The early stages of disc evolution occur while the dis c is\nself-gravitating (e.g. Nixon et al., 2018) and around 90 percent of t he stellar\nmass is accumulated in this way. However, the late stages, involving a ngular\nmomentum from the last few per cent of the stellar mass, and the inn er\ndisc regions, from where the proto-stellar jets are driven, both in volve discs\nthat are ionized enough to support dynamo activity (MRI). Howeve r, the\nmagnetic fields in these instances are unlikely to have been dragged in by\naccreting material (Lubow et al., 1994). Local dynamo activity, ac ting on\nseed fields, is capable of generating the necessary viscosity throu gh MRI,\nas well as generating larger scale, sufficiently ordered fields, that c an drive\ndynamic outflows (Tout and Pringle, 1996; Fendt and Gaßmann, 201 8).1\nWe conclude that the problem of removing angular momentum from\ninterstellar material in order to allow the formation of stars does no t re-\nquire a significant presence of large-scale magnetic fields. Indeed, it has\nbeen widely demonstrated (Li and McKee, 1996; Myers et al., 2013, 2014;\nLi et al., 2014; Tomida et al., 2015; Hennebelle et al., 2016; Masson et a l.,\n2016;K¨ uffmeier et al.,2017,2018;K¨ uffmeier and Nauman,2018;G ray et al.,\n2018) that introducing additional angular momentum transport (b y intro-\nducing magnetic fields to the calculations) leads to the two major pro blems\nmentioned by McKee:\n1An important distinction here is that in contrast to hydrodynamic tu rbulence, MHD\nturbulence can give rise to an inverse cascade whereby it is able to ge nerate magnetic fields\non lengthscales much larger than the driving lengthscale of the turb ulence.\n4(i) it is difficult to reproduce the observed number of stars that are in\nbinary and multiple systems, let alone the properties of the systems , and\n(ii) it is difficult to produce the fraction of stars with massive enough\ndiscs to give rise to planet formation. Winn and Fabrycky (2015) find that\nat least one half of solar-type single stars have planetary systems ; and to\nform planets the disc masses need to be well above the minimum mass s olar\nnebula of around ∼0.01M⊙(Nixon et al., 2018).\n2.2. Is there a star formation rate problem?\nTheoriginalperceptionofmolecularcloudswasthattheyareself-g ravitating,\nisolated long-lived entities (e.g. Solomon et al., 1987; Blitz, 1991, 1993 ). In\nthat picture the observed supersonic turbulent support of the c loud was nec-\nessary in order to prevent the high star formation rate that would result\nfrom the gravitational contraction of the cloud on its free-fall or dynam-\nical timescale. Moreover, it was thought that the turbulence need ed to\nbe strongly magnetic in order to cushion the shocks and so prevent rapid\ndissipation of the turbulence (Arons and Max, 1975; Lizano and Shu , 1989;\nBertoldi and McKee, 1992; Allen and Shu, 2000). However, it turne d out\nthat inclusion of magnetic fields has a minimal effect on the dissipation r ate\noftheturbulence(Ostriker et al.,1999;Mac Low et al.,1998). Thisid eathat\nmagneticintervention isrequired inmolecular clouds inorder toslow th erate\nof star formation is indeed still prevalent (Ballesteros-Paredes et al., 2005;\nV´ azquez-Semadeni et al.,2005;Padoan and Nordlund,2011;Fed errath and Klessen,\n2013; Myers et al., 2014; Padoan et al., 2014; Federrath, 2016).\nIn recent times, this picture of molecular clouds has given way to a re al-\nisation that molecular clouds are much more transient entities.\nFirst, Elmegreen (2000), and others (for example Beichman et al., 1 986;\nLee et al.,1999;Jessop and Ward-Thompson,2000;Ballesteros-P aredes et al.,\n1999) have given cogent observational arguments that the star formation\nwithin a giant molecular cloud (GMC) occurs within one or two crossing\ntimes of its formation, that is within a few Myr. Similarly comparisons of\nthe ages of young clusters and their association with molecular gas b oth\nin our Galaxy (Leisawitz et al., 1989) and in the Large Magellanic Cloud\n(Fukui et al., 1999) indicate that the dispersal of a cloud in which sta r for-\n5mation has occurred takes a time-scale of only 5 −10Myr.2Thus, molecular\nclouds are far more ephemeral than was previously postulated, an d there-\nfore the rate of star formation within them cannot be as high as pre viously\nenvisaged.\nSecond, it has become apparent that GMCs as a whole are not self-\ngravitating (Heyer et al., 2009; Dobbs et al., 2011b).3Numerical simulations\nof the evolution of the interstellar medium within disc galaxies show tha t\nthe denser regions (the giant molecular clouds) are dynamic and tra nsient\nstructures (Dobbs and Pringle, 2013; Dobbs, 2015; Baba et al., 20 17). They\nare not isolated objects and their evolution is highly complex. The larg er\nclouds (where most of the star formation takes place) form by cum ulation of\nsmaller clouds as well as directly from the denser regions of the ISM ( conver-\ngent flows), and tend to disrupt because of galactic shear and fee dback from\nstar formation (cf. Meidt et al., 2015; Dobbs et al., 2018). They are predom-\ninantly not self-gravitating, except for small regions within the clou ds which\ngive rise to star formation events and, hence, disruptive feedbac k. None of\nthese simulations contains magnetic fields, but nevertheless, the o verall star\nformation rates in such models are in line with those observed (Dobbs et al.,\n2011a).\nThus, we conclude that the idea that magnetic fields are required to play\na dominant, or even significant, role within molecular clouds in order to\nmoderate the star formation rate is no longer tenable.\n3. Numerical simulations of magnetic fields in turbulent mol ecular\nclouds\nWe have argued above that the presence of significant magnetic fie lds\nwithin the dense, star-forming interstellar gas is not required to ex plain the\nobserved general properties of star formation.\n2Incidentally, it follows from these observationsthat, contraryto what is often assumed\n(Walch and Naab, 2015; Padoan et al., 2016; K¨ ortgen et al., 2016) s ince the vast majority\nof massive main-sequence lifetimes of stars that give rise to supern ovae, ieM≥8M⊙, are\n≥5−10Myr (Crowther, 2012), supernova explosions cannot provide a n internal source\nof turbulent energy in GMCs. It has also been shown that supernov a explosions cannot\nprovidean externalsourceofturbulent energyeither(see fore xampleSeifried et al., 2018).\n3This implies that the discussion of the properties of such clouds in ter ms of “free-fall\ntimes” (e.g. Padoan et al., 2014) not only has no meaning, but stems from the previous\noutdated physical picture (see also Kennicutt and Evans, 2012).\n6It is, however, clear (see the review by Crutcher, 2012) that mag netic\nfields are to be found in almost all regions of dense molecular gas in whic h\nstar formation is occurring. The field strengths appear to be signifi cant,\nin that the magnetic energy density is a substantial fraction of the energy\nassociated with internal turbulent (or random) cloud motions, but are not\ndominant in that they are not strong enough to prevent global gra vitational\ncollapse of the molecular complex. Since the internal cloud motions ar e\ntypically highly supersonic (with Mach numbers around 10 −20) this implies\nthat the mean magnetic energy density strongly exceeds the ther mal energy\ndensity. For typical cloud parameters, in order for the Jeans mas s (given by\na balance between thermal pressure and self-gravity) to be arou nd a solar\nmass, it is therefore necessary for the cloud material that is actu ally forming\nstars to have shed much of its original magnetic flux (cf. Lubow and Pringle,\n1996).4The main question then is how this is achieved.\nThere are many simulations in the literature of the effects of driven, su-\npersonic, but trans-Alf´ enic, turbulence on the gas density and m agnetic field\nstructures within model molecular clouds (e.g. Padoan and Nordlund , 1999,\n2011;Lemaster and Stone,2009, seethereviewsbyBallesteros- Paredes et al.\n2005 and Padoan et al. 2014). In these simulations it is found that th e tur-\nbulent motions create a range of densities, with the most dense reg ions in the\ntail of the distribution being subject to gravitational collapse (and presumed\nstar formation). These dense regions still contain appreciable mag netic flux\n(βmag=Pgas/Pmag∼0.4, Padoan and Nordlund 2011). In none of these\nsimulations was it possible to consider the formation of individual star s, let\nalone multiple stars or planet-forming discs.\nTo remedy this, a step in the direction of extending the simulations of\nBate (2012) and Krumholz et al. (2012) to include the presence of m agnetic\nfields has been reported in a series of papers (K¨ uffmeier et al., 2017 , 2018;\nK¨ uffmeier and Nauman, 2018). In these simulations (cf. Padoan et al., 2016,\nsee also Myers et al. 2013; Gray et al. 2018) the initial conditions con sist of\nuniform density and uniformly magnetized molecular cloud material wh ich\nis stirred by driven turbulence for some 10 Myr. At that time self-gr avity is\nintroduced and the cloud as a whole becomes self-gravitating. Ther eafter the\n4It is worth noting that other authors, for example Federrath and Klessen (2012), also\nargue that the magnetic field plays at most a weak role in determining t he final stellar\nmasses.\n7denser regions are subject to gravitational collapse, delineated b y the occur-\nrence of sink particles. The eventual stellar masses are around a s olar mass.\nAt this stage the minimum grid cell size is 126au, which is too large to re-\nsolve binary or multiple star formation (median binary separation 20 −40au;\nFig. 7 of Duquennoy and Mayor 1991 and Fig. 13 of Raghavan et al. 20 10),\nlet alone the presence of protostellar discs. Regions of about (40 ,000)3au3\naround a few (six or nine, depending on the paper) of the sink partic les are\nfocussed on in the calculation, and the evolution of these regions ar e then\nfollowed for a further ∼104yr at higher resolution, down to a minimum\ncell size of ∼2au, although a region of (14)3au3is excised around the sink\nparticle itself. In K¨ uffmeier et al. (2017) disc formation is reported , and it\nis found that in these flows the outward transport of angular mome ntum\nis predominantly magnetic, rather then gravitational. Of the six sink par-\nticles studied in more detail in K¨ uffmeier et al. (2018), only two are fo und\nto have steady massive discs ( Mdisc∼0.01M⊙andRdisc∼50−100au),\nand of these one forms a companion with a separation of ∼1500au. Two\nhave no disc at all. The evolution of these discs is followed in more detail\nin K¨ uffmeier and Nauman (2018), where it is shown that all of the disc s are\nstrongly magnetic, and do not fragment.5\nThus, taken at face value, these simulations imply that the ubiquitou s\npresence of magnetic fields in the molecular gas which is collapsing to fo rm\nstars, seems to prevent the desired outcome in terms of both the nature and\nproperties of the resultant stars and of the properties of proto stellar discs\nrequired for planet formation.\n3.1. Additional physics\nIt may be, of course, that other physical effects can ameliorate t he prob-\nlem. We discuss two possibilities here. But at the same time it is worth\ndiscussing the extent to which sets of numerical simulations, using c urrent\ncomputer resources, are capable of representing physical realit y. We do this\nin Section 4.\n5For example the simple binary star formation mechanism discussed by Bonnell (1994)\nwhereby a companion is formed by the interaction between gravitat ional instability in the\nprotostellar disc and continuing accretion, cannot work if the disc is strongly magnetic.\n83.1.1. Turbulent diffusivity\nOne way of enhancing the diffusivity is to appeal to turbulent motions\nwithin the magnetised gas which might lead to an enhanced value of the\neffective diffusivity. This concept (turbulent diffusivity) is appealed t o in\nvarious other branches of astrophysics, for example accretion d isc theory\n(Shakura and Sunyaev,1973)andgalaxydiscdynamotheory(Ruz maikin et al.,\n1988; Shukurov, 2004). In both these examples, the source and properties\nof the turbulence are readily identified (the magneto-rotational in stability\nin accretion discs (Balbus and Hawley, 1991), and the observed tur bulent\nmotions in the ISM for galaxy dynamos). Various authors (for exam ple\nFatuzzo and Adams, 2002; Kim and Diamond, 2002; Zweibel, 2002) d iscuss\nthe possible enhancement of the effective diffusivity by adding turbu lence\nin the context of large-scale star formation. In addition the same c oncept,\nunder the nomenclature of “reconnection-diffusion”, has been int roduced by\n(Lazarian, 2005) and applied to the final stages of collapse to form a star\nby Le˜ ao et al. (2013). However, in the case of star formation the source and\nproperties of the small-scale turbulence required to provide the en hancement\nin the effective diffusivity are neither readily identified nor discussed. Indeed\nit is questionable as to whether such a source exists. It is further q uestion-\nable as to whether what is observed is actually “turbulence” in the us ual\nfluid sense (see Section 4.3).\n3.1.2. Ambipolar diffusion\nAmbipolardiffusion(orion-neutraldrift)wasdiscussedbyMestel( 1965a,b)\nas a mechanism whereby gas in the final phase of collapse to form a st ar\nmight be able to shed itself of magnetic field. This is because at this sta ge\nthe gas can be dense enough and cold enough to be predominantly ne utral;\nsee however the additional points raised by Norman and Heyvaerts (1985).\nOn the large scale in molecular clouds it is recognised that the effect is\nsmall. For example, Balsara et al. (2001a,b) find that in this context a m-\nbipolar drift does not play a significant role, and note further the fin dings\nof Mouschovias (1991) that ambipolar diffusion is mainly important in th e\nlast stages of collapse. Many recent authors (for example Li and M cKee,\n1996; Chen and Ostriker, 2014; Masson et al., 2016; Wurster et al., 2016;\nAuddy et al., 2017; Gray et al., 2018; Vaytet et al., 2018) have come t o simi-\nlar conclusions. In addition Heitsch and Hartmann (2014) also conclu de that\nin molecular clouds as a whole, neither ambipolar diffusion nor turbulent dif-\nfusion is likely to control the formation of cores or stars.\n94. How realistic are the turbulent MHD simulations?\nWe consider the answer to this question in two parts. First we consid er\nthe extent to which the numerical simulations are able to simulate the rel-\nevant physical properties of the cloud material. We show, as recog nised by\nthose undertaking the simulations, that they are not. We then disc uss the\nconsequences of this disparity. Second, we consider the initial con ditions as-\nsumed forthe simulations compared to thecurrent picture of molec ular cloud\nformation.\n4.1. Physical properties of the cloud material\nThe numerical simulations of MHD turbulence in molecular cloud ma-\nterial are, of necessity, restricted by what is numerically possible. The two\nparameters of immediate relevance are the Reynolds number ( Re) and the\nmagnetic Prandtl number ( PM).\nAsnotedbyKritsuk et al.(2011, seealsoKritsuk et al.2009)there levant\nReynolds number is given by Re≈uL/ν, whereuisthe r.m.s. velocity inthe\nturbulence, Lis the relevant length scale (of order the energy injection scale)\nandνis the fluid kinematic viscosity. These authors note that the largest\nvalues of Rethat can be reached are typically ∼104whereas realistic values\nfor molecular clouds can be as high as ∼108. Physically what this implies\nis that the viscosity inherent in the numerical codes is too high by sev eral\norders of magnitude. The main effect of this is that the smallest scale s likely\nto be present in the turbulence are severely overestimated in the o utput of\nthe simulations.\nThe magnetic Prandtl number is given by PM=ν/ηwhereηis the mag-\nnetic diffusivity. For typical molecular cloud material Kritsuk et al. (2 011)\nfind that we may expect PM≈2×105(xi/10−7)(n/1000cm−3)−1≫1. Here\nxiis the ionization fraction and nthe particle number density. In contrast,\nnumerical simulations without explicit viscosity and explicit magnetic diff u-\nsivity, and which therefore rely on the grid scale to control both vis cosity and\ndiffusivity, generally and naturally have PM∼1. Since the numerical codes\noverestimate the viscosity by factors of order ∼104, and underestimate the\nmagnetic Prandtl number by of order ∼2×105, it follows that they over-\nestimate the magnetic diffusivity by factors of order ∼2×109. Thus, to\na first approximation, the simulations overestimate the rate at whic h cloud\nmaterial can both divest itself of, and acquire, magnetic flux by almost ten\n10orders of magnitude . It would be surprising if such a disparity did not have\nserious consequences.\n4.2. Nature of the driven turbulence\nIn the numerical simulations the freeing of material from magnetic fi eld\noccurs through driven turbulence coupled with a large magnetic diffu sivity.\nThus it is no surprise that those regions, in which gravity is just able t o\novercome magnetic fields and so enable collapse, still have near maxim al\nfield strength, viz. βmag∼1. However, it is well known that the properties\nof MHD turbulence differ substantially between the PM∼1 and the PM≫1\nregimes (Schekochihin et al. 2002a,b)6.\nIn hydrodynamic turbulence, turbulent energy is put into the flow a t\nlarge scales. The energy is then transferred through a cascade o f eddy sizes\ndown to the smallest eddies whose size is controlled by the magnitude o f the\nviscosity, ν(e.g. Tennekes and Lumley, 1972). The smaller the viscosity, i.e.\nthe higher the Reynolds number, the larger the range of eddy sizes , i.e. the\nsmaller the scales at which kinetic energy is turned into heat. In (driv en)\nMHD turbulence, with PM∼1, so that ν∼ηmagnetic energy loss and\nkinetic energy loss are able to take place at the same small scales. Th is is\nwhat is occurring in the simulations. However, when η≪ν, so that PM≫1,\nthis is no longer the case. As shown in Schekochihin et al. (2002a,b) in t heir\nmodel of a kinematic dynamo there is much more power in the magnetic field\nstructure at small scales. In effect the magnetic field is stretched and folded\ninto long thin structures, and it is the thinness of the structures t hat enables\nthe magnetic energy dissipation to take place.\nThus in the high magnetic Prandtl number regime, this gives rise to th e\nconcept that the magnetic field structure is better imagined as a se ries of flux\nropes. These ideas have been applied by Baggaley et al. (2009) to inc om-\npressible MHD . In their model of a fluctuating dynamo the magnetic fi eld is\nconfined to thin flux ropes, advected by turbulence. Dissipation of magnetic\nfield occurs predominantly through reconnection of flux ropes; bu t note that\nonce reconnection occurs, magnetic energy is reconverted to kin etic as the\nfield configuration rearranges itself. A similar, but cruder, model f or similar\nprocesses occurring in supersonic magnetic turbulence in (theref ore highly\n6This distinction has been shown to have important consequences in a ccretion disc\ninstability theory (Potter and Balbus, 2017)\n11compressible) molecular clouds was developed by Lubow and Pringle (1 996).\nThey argued that reconnection processes in a 3D geometry lead ine vitably\nto the formation of closed loops of field. This creates O-type neutr al points\nwhich then enable the field to diffuse and dissipate. This leads through out\nthe cloud to a steady generation of dense material which has been f reed from\nthe direct influence of any permeating magnetic field. They conclude d that\nsuch material would preferentially be the material within the cloud th at par-\ntakes in star formation. Similar ideas were formulated by Shu (1987) , and\nhave been revisited by Lazarian (2005), Krasnopolsky and Gammie ( 2005)\nand by Heitsch and Hartmann (2014).\nWe note that it might seem reasonable to assume that overestimatin g\nthe diffusivity would lead to underestimating the effect of the magnet ic field.\nHowever, while this is the case in regions of high field strength, this is n ot\nthe case in regions of low field strength. Regions of low field strength are,\nin the simulations, overwhelmed with large flux from the high strength re-\ngions due to artificially high (numerical and uncontrolled) diffusivity. I n\nsimulations that have too high a diffusivity, star formation will only pro ceed\nwhen the magnetic field is just low enough, meaning that all star form a-\ntion takes place with near-maximal field strengths (Padoan and Nor dlund,\n2011). We argue that this cannot happen in reality as the real diffus ivity is\nmuch lower than applied in the simulations. Thus cloud material which is\nhighly magnetic, cannot free itself from fields and so cannot form st ars (cf.\nK¨ ortgen and Banerjee, 2015). Conversely, the material which is able to form\nstarsisthatmaterialwhichisnon-magnetic(eitherbecauseitwasa lreadynot\nthreaded by field when the cloud formed (see below), or because it m anaged\nto shed field by reconnection in high PMturbulence). Such non-magnetic\nmaterial cannot occur in the simulations, because if it were present , the arti-\nficially high (numerical) diffusivity would feed large magnetic flux back int o\nit from neighbouring regions.\n4.3. Initial conditions and nature of the turbulence\nWe have noted that simulations of star formation within magnetic clou ds\ngenerally assume that all of the cloud material is initially uniformly thre aded\nwith magnetic field, and that it is then then subjected to driven turb ulence.\nIt seems unlikely that either of these assumptions is correct.\n124.3.1. Initial magnetic field distribution\nAs we have noted above, molecular clouds appear to be ephemeral o b-\njects7which readily form and disperse on timescales comparable to their\nkinematic crossing times (Dobbs and Pringle, 2013). Most of the mat erial\nwithin them is not dominated by self-gravity. It is only small portions o f the\ncloudthatarecompact enoughtobesubject toself-gravity, and so areableto\ncollapse and formstars. The material from which the clouds formis e xpected\nto be denser than average ISM material (Pringle et al., 2001; Dobbs et al.,\n2012) but since it is less dense, and more highly ionized, than cloud mat erial\nit is to be expected that the magnetic diffusivity of the pre-cloud mat erial\nis even smaller. Thus there is no reason to assume that the material from\nwhich clouds form is uniformly threaded with magnetic fields. Indeed it is\nmore likely that such material would contain a large range of flux to ma ss ra-\ntios. For this reason it seems quite plausible that when gravitational collapse\nsets in, although some of the collapsing material is threaded by magn etic\nfields, some of it may not be. If that were the case, it would be expec ted\nthat star formation would be more likely to occur from the material le ast\nthreaded by magnetic flux.\n4.3.2. Cloud turbulence\nAlthough the velocity dispersions observed within molecular clouds ar e\nusually referred to as “turbulence” is it not at all clear that the mot ions\nrepresent well-developed turbulence in the standard fluid dynamica l sense\n(e.g. Batchelor, 1953; Tennekes and Lumley, 1972). We have alrea dy com-\nmented that such motions cannot be driven by supernovae from eit her inside\nor outside the clouds.\nIndeed in view of the current picture of molecular clouds in terms of\nephemeral entities formed in regionsof converging ISM material, of tendriven\ntogetherinthecontext ofspiralarms, itseemsmorelikelythatthe supersonic\nvelocity dispersions generally assumed to be “turbulence” are the r esult of\nenergy released in the formation process. Bonnell et al. (2006) de monstrate\nthat if two clouds of ISM material, each of which has a non-uniform density\nstructure, and each of which has zero velocity dispersion , are made to collide\ninashock then theeffect of theoriginalclumpiness isto giverise toa v elocity\n7In this respect molecular clouds are much more like atmospheric cloud s than the\noriginal proposers of the nomenclature envisaged.\n13dispersion within the post-collision gas. In the astronomical literatu re such\na velocity dispersion is invariably referred to as “turbulence”. Within these\nclouds both the time-scale for the decay of these motions, and the time-\nscale for forming stars, are comparable to the clouds’ dynamical lif etimes. In\nthis model there is no need for any internal or external continuou s driving\nmechanism for the “turbulence”.\nIt is important to stress that it is the clumpiness of the pre-collision g as\nwhich gives rise to the post-collision velocity dispersion. That clumpine ss,\nwell observed within the ISM, is of course generated by instabilities a nd en-\nergysourceswithintheISM,presumablyincludingsupernovae. The ideathat\nclumpiness needs to be generated post-collision from a pre-collision s mooth\nflow(e.g.Heitsch et al.,2008;Banerjee et al.,2009;Micic et al.,2013; Fogerty et al.,\n2016, 2017) is unnecessarily restrictive.\nIt evident that in all of these pictures the initial conditions in a cloud\nat the onset of gravitational collapse (and subsequent star form ation) are\nunlikely to be close to those generated by driven homogeneous turb ulence as\nfound in the simulations.\n5. Discussion and Conclusions\nWe have considered the apparent contradiction between the relat ive suc-\ncess of those models of the late, dynamical stages of star format ion that\ndo not include magnetic fields, and the observed presence of magne tic fields\nwithin molecular clouds and cloud cores where stars form.\nWe have argued that the earlier concept of star formation in terms of sin-\ngle star collapse, which led to the notion of an angular momentum prob lem\n(and therefore to the need for, and importance of, magnetic field s), has been\nreplaced by the more recent concept of chaotic star-formation ( e.g. Bate,\n2012), where stars form predominantly in groups. In this picture g ravita-\ntional interactions provide a solution to the angular momentum prob lem,\nexcept for the late-stage evolution of the inner regions of the pro tostellar\ndiscs where MHD turbulence is likely involved. Hartmann and Bae (2018 )\nand Simon et al. (2017) make the case that while there is little evidence for\nmagnetic activity in the outer regions of protostellar discs, there is evidence\nof magnetic activity (magnetic winds, jets) in the inner regions of th ose discs,\nwhere the temperatures are high enough for an MRI-driven dynam o to be\npresent. Because of velocity considerations (outflow velocities ar e compa-\nrable to escape velocities from the central object), it has long bee n argued\n14that the major components of outflows are driven from close to th e inner disc\nradii(Konigl,1986; Pringle, 1993;Livio,1997;Price et al.,2003). In addition\nthe strongest protostellar outflows are found to occur among th e youngest\n(strongly accreting, and often heavily embedded) objects (e.g. B ally, 2016).\nMagnetic winds and jets do require the presence of a global field, bu t there\nis no need for this to have been advected by the disc – indeed that in it self\nis unlikely (Lubow et al., 1994). In such strongly ionized disc regions (s uch\nas the inner regions, and in hot strongly accreting discs) it is possible for the\nMRI-dynamo itself to create a sufficiently large global field for jet-la unching\n(Tout and Pringle, 1996).\nWe have also noted that the idea that magnetic fields play an importan t\nroleonalargerscale, preventingthegravitationalcollapseofmolec ularclouds\nand slowing down the rate of star formation within them comes from t he\nearlier concept that molecular clouds are self-gravitating isolated e ntities.\nThe more modern view is that this is not the case. Thus magnetic fields are\nno longer needed to play a significant role in slowing down the star-for mation\nprocess.\nIn this context, we have considered the ability of current numerica l simu-\nlations to emulate the early stages of star formation from molecular material.\nWe have noted problems with two aspects of this work. First, in many simu-\nlations the material is assumed to be uniformly threaded with magnet ic field\nand then subjected to prolonged driven turbulence. We have argu ed that\nthis may not be a good representation for the initial stages of grav itational\ncollapse in a star-forming cloud. Second, and more seriously, we hav e noted\nthat the physical conditions of the MHD being simulated (especially wit h\nregard to the magnetic Prandtl number and the magnetic diffusivity ) differ\nbetween the simulations and physical reality by many orders of magn itude.\nIn particular, the magnetic diffusivity, which provides the timescale o n which\ncloud material is able to lose, and to acquire, magnetic flux, is overes timated\nby almost ten orders of magnitude. Since the region of parameter s pace (in,\nfor example, the magnetic Prandtl number – diffusivity plane) that r epre-\nsents physical reality is so far removed from what is amenable to num erical\nsimulation, it is reasonable to question the usefulness of proceeding along\nthese lines. In any case it is clear that those papers which present s uch nu-\nmerical simulations do need to include some justification and discussio n of\nthe extent to which such simulations can be expected to represent physical\nreality.\nIn view of all this, we have advanced the hypothesis that there is a m uch\n15larger scale of flux to mass ratios present in the relevant molecular m aterial\nthan can, at present, be simulated numerically. If so, we suggest t hat it\nwould be predominantly the material in the cloud that is relatively free of\nmagneticfieldthat partakesintheformationofstars(cf. Lubow a nd Pringle,\n1996). Thus simulations that ignore magnetic fields on the scales on w hich\ntheproperties ofstellar masses, stellar multiplicities andplanet-for ming discs\nare determined, may be closer to reality.\nAcknowledgments\nWethankClareDobbs,CharlesGammie,LeeHartmann, MichaelK¨ uff meier\nandChrisMcKee foruseful correspondence. WethankBobCarsw ell andAn-\ndrew King for support and encouragement. CJN is supported by th e Science\nand Technology Facilities Council (grant number ST/M005917/1).\nReferences\nAllen, A., Shu, F.H., 2000. A Toy Model of Giant Molecular Clouds. ApJ\n536, 368–379. doi: 10.1086/308912 .\nArons, J., Max, C.E., 1975. Hydromagnetic Waves in Molecular Clouds.\nApJL 196, L77. doi: 10.1086/181748 .\nAuddy, S., Basu, S., Kudoh, T., 2017. Magnetic Ribbons: A Minimum\nHypothesis Model for Filaments. ArXiv e-prints arXiv:1703.05632 .\nBaba, J., Morokuma-Matsui, K., Saitoh, T.R., 2017. Eventful evolut ion of\ngiant molecular clouds in dynamically evolving spiral arms. MNRAS 464,\n246–263. doi: 10.1093/mnras/stw2378 ,arXiv:1609.00097 .\nBaggaley, A.W., Barenghi, C.F., Shukurov, A., Subramanian, K.,\n2009. Reconnecting flux-rope dynamo. Phys. Rev. E 80, 055301.\ndoi:10.1103/PhysRevE.80.055301 ,arXiv:0906.1095 .\nBalbus, S.A., Hawley, J.F., 1991. A powerful local shear instability in we akly\nmagnetized disks. I - Linear analysis. II - Nonlinear evolution. ApJ 37 6,\n214–233. doi: 10.1086/170270 .\n16Ballesteros-Paredes, J., Hartmann, L., V´ azquez-Semadeni, E., 1 999. Tur-\nbulent Flow-driven Molecular Cloud Formation: A Solution to the\nPost-T Tauri Problem? ApJ 527, 285–297. doi: 10.1086/308076 ,\narXiv:astro-ph/9907053 .\nBallesteros-Paredes, J., V´ azquez-Semadeni, E., Kim, J., 2005. St ar Forma-\ntion Efficiency in Driven, Supercritical, Turbulent Clouds, in: Protost ars\nand Planets V Posters, p. 8630.\nBally, J., 2016. Protostellar Outflows. ARA&A 54, 491–528.\ndoi:10.1146/annurev-astro-081915-023341 .\nBalsara, D., Ward-Thompson, D., Crutcher, R.M., 2001a. A\nturbulent MHD model for molecular clouds and a new method\nof accretion on to star-forming cores. MNRAS 327, 715–720.\ndoi:10.1046/j.1365-8711.2001.04787.x ,arXiv:astro-ph/0105327 .\nBalsara, D.S., Crutcher, R.M., Pouquet, A., 2001b. Turbulent Flows\nwithin Self-gravitating Magnetized Molecular Clouds. ApJ 557, 451–4 63.\ndoi:10.1086/323679 .\nBanerjee, R., V´ azquez-Semadeni, E., Hennebelle, P., Klessen,\nR.S., 2009. Clump morphology and evolution in MHD simu-\nlations of molecular cloud formation. MNRAS 398, 1082–1092.\ndoi:10.1111/j.1365-2966.2009.15115.x ,arXiv:0808.0986 .\nBatchelor, G.K., 1953. The Theory of Homogeneous Turbulence.\nBate, M.R., 2012. Stellar, brown dwarf and multiple star proper-\nties from a radiation hydrodynamical simulation of star cluster form a-\ntion. MNRAS 419, 3115–3146. doi: 10.1111/j.1365-2966.2011.19955.x ,\narXiv:1110.1092 .\nBate, M.R., 2018. On the diversity and statistical properties of pro -\ntostellar discs. MNRAS 475, 5618–5658. doi: 10.1093/mnras/sty169 ,\narXiv:1801.07721 .\nBeichman, C.A., Myers, P.C., Emerson, J.P., Harris, S., Mathieu, R., Ben -\nson, P.J., Jennings, R.E., 1986. Candidate solar-type protostars in nearby\nmolecular cloud cores. ApJ 307, 337–349. doi: 10.1086/164421 .\n17Bertoldi, F., McKee, C.F., 1992. Pressure-confined clumps in magnet ized\nmolecular clouds. ApJ 395, 140–157. doi: 10.1086/171638 .\nBlitz, L., 1991. StarFormingGiantMolecularClouds, in: Lada, C.J., Kyla fis,\nN.D. (Eds.), NATO Advanced Science Institutes (ASI) Series C, p. 3 .\nBlitz, L., 1993. Giant molecular clouds, in: Levy, E.H., Lunine, J.I. (Eds .),\nProtostars and Planets III, pp. 125–161.\nBonnell, I.A., 1994. A New Binary Formation Mechanism. MNRAS 269.\ndoi:10.1093/mnras/269.3.837 .\nBonnell, I.A., Dobbs, C.L., Robitaille, T.P., Pringle, J.E., 2006.\nSpiral shocks, triggering of star formation and the velocity\ndispersion in giant molecular clouds. MNRAS 365, 37–45.\ndoi:10.1111/j.1365-2966.2005.09657.x ,arXiv:astro-ph/0509809 .\nChen, C.Y., Ostriker, E.C., 2014. Formation of Magnetized Prestel-\nlar Cores with Ambipolar Diffusion and Turbulence. ApJ 785, 69.\ndoi:10.1088/0004-637X/785/1/69 ,arXiv:1403.0582 .\nClarke, C.J., Pringle, J.E., 1991. The role of discs in the forma-\ntion of binary and multiple star systems. MNRAS 249, 588–595.\ndoi:10.1093/mnras/249.4.588 .\nCrowther, P., 2012. Birth, life and death of massive stars. Astron omy and\nGeophysics 53, 4.30–4.36. doi: 10.1111/j.1468-4004.2012.53430.x .\nCrutcher, R.M., 2012. Magnetic Fields in Molecular Clouds. ARA&A 50,\n29–63. doi: 10.1146/annurev-astro-081811-125514 .\nDobbs, C.L., 2015. The interstellar medium and star formation on kpc\nsize scales. MNRAS 447, 3390–3401. doi: 10.1093/mnras/stu2585 ,\narXiv:1412.2911 .\nDobbs,C.L., Burkert, A., Pringle, J.E., 2011a. Thepropertiesofthe interstel-\nlar medium indisc galaxies with stellar feedback. MNRAS 417, 1318–133 4.\ndoi:10.1111/j.1365-2966.2011.19346.x ,arXiv:1107.0154 .\nDobbs, C.L., Burkert, A., Pringle, J.E., 2011b. Why are most molec-\nular clouds not gravitationally bound? MNRAS 413, 2935–2942.\ndoi:10.1111/j.1365-2966.2011.18371.x ,arXiv:1101.3414 .\n18Dobbs, C.L., Pettitt, A.R., Corbelli, E., Pringle, J.E., 2018. Simulations of\nthe flocculent spiral M33: what drives the spiral structure? MNRA S 478,\n3793–3808. doi: 10.1093/mnras/sty1231 ,arXiv:1805.04443 .\nDobbs, C.L., Pringle, J.E., 2013. The exciting lives of giant molecular clou ds.\nMNRAS 432, 653–667. doi: 10.1093/mnras/stt508 ,arXiv:1303.4995 .\nDobbs, C.L., Pringle, J.E., Burkert, A., 2012. Giant molecular clouds: w hat\nare they made from, and how do they get there? MNRAS 425, 2157– 2168.\ndoi:10.1111/j.1365-2966.2012.21558.x ,arXiv:1206.4904 .\nDuquennoy, A., Mayor, M., 1991. Multiplicity among solar-type stars in\nthe solar neighbourhood. II - Distribution of the orbital elements in an\nunbiased sample. A&A 248, 485–524.\nElmegreen, B.G., 2000. Star Formation in a Crossing Time. ApJ 530, 27 7–\n281. doi:10.1086/308361 ,arXiv:astro-ph/9911172 .\nFatuzzo, M., Adams, F.C., 2002. Enhancement of Ambipolar Diffusion R ates\nthrough Field Fluctuations. ApJ 570, 210–221. doi: 10.1086/339502 ,\narXiv:astro-ph/0201131 .\nFederrath, C., 2016. The role of turbulence, magnetic fields and fe edback\nfor star formation, in: Journal of Physics Conference Series, p. 012002.\ndoi:10.1088/1742-6596/719/1/012002 ,arXiv:1606.03121 .\nFederrath, C., Klessen, R.S., 2012. The Star Formation Rate of Tur bulent\nMagnetized Clouds: Comparing Theory, Simulations, and Observatio ns.\nApJ 761, 156. doi: 10.1088/0004-637X/761/2/156 ,arXiv:1209.2856 .\nFederrath, C., Klessen, R.S., 2013. On the Star Formation\nEfficiency of Turbulent Magnetized Clouds. ApJ 763, 51.\ndoi:10.1088/0004-637X/763/1/51 ,arXiv:1211.6433 .\nFendt, C., Gaßmann, D., 2018. Bipolar Jets Launched by a Mean-field Ac-\ncretion Disk Dynamo. ApJ 855, 130. doi: 10.3847/1538-4357/aab14c .\nFogerty, E., Carroll-Nellenback, J., Frank, A., Heitsch, F., Pon, A., 2 017.\nReorienting MHD colliding flows: a shock physics mechanism for gen-\nerating filaments normal to magnetic fields. MNRAS 470, 2938–2948 .\ndoi:10.1093/mnras/stx1381 ,arXiv:1609.02918 .\n19Fogerty, E., Frank, A., Heitsch, F., Carroll-Nellenback, J., Haig, C., A dams,\nM., 2016. Molecular cloud formation in high-shear, magnetized col-\nliding flows. MNRAS 460, 2110–2128. doi: 10.1093/mnras/stw1141 ,\narXiv:1602.01417 .\nFukui, Y., Mizuno, N., Yamaguchi, R., Mizuno, A., Onishi, T., Ogawa,\nH., Yonekura, Y., Kawamura, A., Tachihara, K., Xiao, K., Yamaguchi,\nN., Hara, A., Hayakawa, T., Kato, S., Abe, R., Saito, H., Mano,\nS., Matsunaga, K., Mine, Y., Moriguchi, Y., Aoyama, H., Asayama,\nS.i., Yoshikawa, N., Rubio, M., 1999. First Results of a CO Sur-\nvey of the Large Magellanic Cloud with NANTEN; Giant Molecular\nClouds as Formation Sites of Populous Clusters. PASJ 51, 745–749.\ndoi:10.1093/pasj/51.6.745 .\nGray, W.J., McKee, C.F., Klein, R.I., 2018. Effect of angular momen-\ntum alignment and strong magnetic fields on the formation of proto-\nstellar discs. MNRAS 473, 2124–2143. doi: 10.1093/mnras/stx2406 ,\narXiv:1709.05350 .\nHartmann, L., Bae, J., 2018. How do T Tauri stars accrete? MNRAS 474,\n88–94. doi: 10.1093/mnras/stx2775 ,arXiv:1710.08718 .\nHeitsch, F., Hartmann, L., 2014. Accretion and diffusion time-scales in\nsheets andfilaments. MNRAS443, 230–240. doi: 10.1093/mnras/stu1147 ,\narXiv:1406.2191 .\nHeitsch, F., Hartmann, L.W., Burkert, A., 2008. Fragmentation of\nShocked Flows: Gravity, Turbulence, and Cooling. ApJ 683, 786–79 5.\ndoi:10.1086/589919 ,arXiv:0805.0801 .\nHennebelle, P., Commer¸ con, B., Chabrier, G., Marchand, P., 2016. M agnet-\nically Self-regulated Formation of Early Protoplanetary Disks. ApJL 830,\nL8. doi:10.3847/2041-8205/830/1/L8 ,arXiv:1608.02525 .\nHeyer, M., Krawczyk, C., Duval, J., Jackson, J.M., 2009. Re-Examinin g\nLarson’s Scaling Relationships in Galactic Molecular Clouds. ApJ 699,\n1092–1103. doi: 10.1088/0004-637X/699/2/1092 ,arXiv:0809.1397 .\nJessop, N.E., Ward-Thompson, D., 2000. A far-infrared\nsurvey of molecular cloud cores. MNRAS 311, 63–74.\ndoi:10.1046/j.1365-8711.2000.03011.x ,arXiv:astro-ph/9908230 .\n20Kennicutt, R.C., Evans, N.J., 2012. Star Formation in the\nMilky Way and Nearby Galaxies. ARA&A 50, 531–608.\ndoi:10.1146/annurev-astro-081811-125610 ,arXiv:1204.3552 .\nKim, E.j., Diamond, P.H., 2002. Turbulent Diffusion of Magnetic Fields in\nWeakly Ionized Gas. ApJL 578, L113–L116. doi: 10.1086/344634 .\nKonigl, A., 1986. Stellar and galactic jets - Theoretical issues. Cana dian\nJournal of Physics 64, 362–368. doi: 10.1139/p86-063 .\nK¨ ortgen, B., Banerjee, R., 2015. Impact of magnetic fields on\nmolecular cloud formation and evolution. MNRAS 451, 3340–3353.\ndoi:10.1093/mnras/stv1200 ,arXiv:1502.03306 .\nK¨ ortgen, B., Seifried, D., Banerjee, R., V´ azquez-Semadeni, E., Z amora-\nAvil´ es, M., 2016. Supernova feedback in molecular clouds: global e volu-\ntion and dynamics. MNRAS 459, 3460–3474. doi: 10.1093/mnras/stw824 ,\narXiv:1603.09593 .\nKrasnopolsky, R., Gammie, C.F., 2005. Nonlinear Criterion for the Sta -\nbility of Molecular Clouds. ApJ 635, 1126–1135. doi: 10.1086/497561 ,\narXiv:astro-ph/0512629 .\nKritsuk, A.G., Nordlund, ˚A., Collins, D., Padoan, P., Norman, M.L., Abel,\nT., Banerjee, R., Federrath, C., Flock, M., Lee, D., Li, P.S., M¨ uller, W.C .,\nTeyssier, R., Ustyugov, S.D., Vogel, C., Xu, H., 2011. Comparing Nume r-\nical Methods for Isothermal Magnetized Supersonic Turbulence. ApJ 737,\n13. doi:10.1088/0004-637X/737/1/13 ,arXiv:1103.5525 .\nKritsuk, A.G., Ustyugov, S.D., Norman, M.L., Padoan, P.,\n2009. Simulating supersonic turbulence in magnetized molecu-\nlar clouds, in: Journal of Physics Conference Series, p. 012020.\ndoi:10.1088/1742-6596/180/1/012020 ,arXiv:0908.0378 .\nKrumholz, M.R., Klein, R.I., McKee, C.F., 2011. Radiation-hydrodynamic\nSimulations of the Formation of Orion-like Star Clusters. I. Implica-\ntions for the Origin of the Initial Mass Function. ApJ 740, 74.\ndoi:10.1088/0004-637X/740/2/74 ,arXiv:1104.2038 .\nKrumholz, M.R., Klein, R.I., McKee, C.F., 2012. Radiation-hydrodynamic\nSimulations of the Formation of Orion-like Star Clusters. II. The Init ial\n21Mass Function from Winds, Turbulence, and Radiation. ApJ 754, 71.\ndoi:10.1088/0004-637X/754/1/71 ,arXiv:1203.2620 .\nK¨ uffmeier, M., Frimann, S., Jensen, S.S., Haugbølle, T., 2018. Episodic\naccretion: the interplay of infall and disc instabilities. MNRAS 475, 26 42–\n2658. doi: 10.1093/mnras/sty024 ,arXiv:1710.00931 .\nK¨ uffmeier, M., Haugbølle, T., Nordlund, ˚A., 2017. Zoom-in Simula-\ntions of Protoplanetary Disks Starting from GMC Scales. ApJ 846, 7 .\ndoi:10.3847/1538-4357/aa7c64 ,arXiv:1611.10360 .\nK¨ uffmeier, M., Nauman, F., 2018. The effects of large scale mag-\nnetic fields around young protostars and their disks. ArXiv e-print s\narXiv:1710.11195 .\nLada, C.J., Lada, E.A., 2003. Embedded Clusters in Molecular Clouds.\nARA&A 41, 57–115. doi: 10.1146/annurev.astro.41.011802.094844 ,\narXiv:astro-ph/0301540 .\nLarson, R.B., 2010. Angular momentum and the formation of stars\nand black holes. Reports on Progress in Physics 73, 014901.\ndoi:10.1088/0034-4885/73/1/014901 ,arXiv:0901.4325 .\nLazarian, A., 2005. Astrophysical Implications of Turbulent Recon nec-\ntion: from cosmic rays to star formation, in: de Gouveia dal Pino,\nE.M., Lugones, G., Lazarian, A. (Eds.), Magnetic Fields in the Uni-\nverse: From Laboratory and Stars to Primordial Structures., pp . 42–53.\ndoi:10.1063/1.2077170 ,arXiv:astro-ph/0505574 .\nLe˜ ao, M.R.M., de Gouveia Dal Pino, E.M., Santos-Lima, R., Lazarian, A.,\n2013. The Collapse of Turbulent Cores and Reconnection Diffusion. A pJ\n777, 46. doi: 10.1088/0004-637X/777/1/46 ,arXiv:1209.1846 .\nLee, C.W., Myers, P.C., Tafalla, M., 1999. A Survey of Infall Motions to ward\nStarless Cores. I. CS (2-1) and N 2H+(1-0) Observations. ApJ 526, 788–\n805. doi:10.1086/308027 ,arXiv:astro-ph/9906468 .\nLeisawitz, D., Bash, F.N., Thaddeus, P., 1989. A CO survey of regions\naround 34 open clusters. ApJS 70, 731–812. doi: 10.1086/191357 .\n22Lemaster, M.N., Stone, J.M., 2009. Dissipation and Heating in Su-\npersonic Hydrodynamic and MHD Turbulence. ApJ 691, 1092–1108.\ndoi:10.1088/0004-637X/691/2/1092 ,arXiv:0809.4005 .\nLi, Z.Y., Banerjee, R., Pudritz, R.E., Jørgensen, J.K., Shang,\nH., Krasnopolsky, R., Maury, A., 2014. The Earliest Stages\nof Star and Planet Formation: Core Collapse, and the For-\nmation of Disks and Outflows. Protostars and Planets VI\n, 173–194doi: 10.2458/azu_uapress_9780816531240-ch008 ,\narXiv:1401.2219 .\nLi, Z.Y., McKee, C.F., 1996. Hydromagnetic Accretion Shocks around Low-\nMass Protostars. ApJ 464, 373. doi: 10.1086/177329 .\nLivio, M., 1997. The Formation Of Astrophysical Jets, in: Wickramas inghe,\nD.T., Bicknell, G.V., Ferrario, L. (Eds.), IAU Colloq. 163: Accretion Phe -\nnomena and Related Outflows, p. 845.\nLizano, S., Shu, F.H., 1989. Molecular cloud cores and bimodal star fo rma-\ntion. ApJ 342, 834–854. doi: 10.1086/167640 .\nLubow, S.H., Papaloizou, J.C.B., Pringle, J.E., 1994. Magneticfielddragg ing\nin accretion discs. MNRAS 267, 235–240. doi: 10.1093/mnras/267.2.235 .\nLubow, S.H., Pringle, J.E., 1996. Magnetic reconnection and star for mation\nin molecular clouds. MNRAS 279, 1251. doi: 10.1093/mnras/279.4.1251 .\nMac Low, M.M., Klessen, R.S., Burkert, A., Smith, M.D., 1998. Ki-\nnetic Energy Decay Rates of Supersonic and Super-Alfv´ enic Turb u-\nlence in Star-Forming Clouds. Physical Review Letters 80, 2754–27 57.\ndoi:10.1103/PhysRevLett.80.2754 ,arXiv:astro-ph/9712013 .\nMasson, J., Chabrier, G., Hennebelle, P., Vaytet, N., Commer¸ con, B .,\n2016. Ambipolar diffusion in low-mass star formation. I. General com -\nparison with the ideal magnetohydrodynamic case. A&A 587, A32.\ndoi:10.1051/0004-6361/201526371 ,arXiv:1509.05630 .\nMcKee, C.F., Ostriker, E.C., 2007. Theory of Star Formation.\nARA&A 45, 565–687. doi: 10.1146/annurev.astro.45.051806.110602 ,\narXiv:0707.3514 .\n23Meidt, S.E., Hughes, A., Dobbs, C.L., Pety, J., Thompson, T.A., Garc´ ıa -\nBurillo, S., Leroy, A.K., Schinnerer, E., Colombo, D., Querejeta, M.,\nKramer, C., Schuster, K.F., Dumas, G., 2015. Short GMC Life-\ntimes: An Observational Estimate with the PdBI Arcsecond Whirlpoo l\nSurvey (PAWS). ApJ 806, 72. doi: 10.1088/0004-637X/806/1/72 ,\narXiv:1504.04528 .\nMestel, L., 1965a. Problems of Star Formation - I. QJRAS 6, 161.\nMestel, L., 1965b. Problems of Star Formation - II. QJRAS 6, 265.\nMestel, L., Spitzer, Jr., L., 1956. Star formation in magnetic dust clo uds.\nMNRAS 116, 503. doi: 10.1093/mnras/116.5.503 .\nMicic, M., Glover, S.C.O., Banerjee, R., Klessen, R.S., 2013. Cloudforma tion\nin colliding flows: influence of the choice of cooling function. MNRAS 432 ,\n626–636. doi: 10.1093/mnras/stt489 ,arXiv:1303.4751 .\nMouschovias, T.C., 1991. Cosmic Magnetism and the Basic Physics of t he\nEarly Stages of Star Formation, in: Lada, C.J., Kylafis, N.D. (Eds.),\nNATO Advanced Science Institutes (ASI) Series C, p. 61.\nMyers, A.T., Klein, R.I., Krumholz, M.R., McKee, C.F., 2014. Star clus-\nter formation in turbulent, magnetized dense clumps with radiative a nd\noutflow feedback. MNRAS 439, 3420–3438. doi: 10.1093/mnras/stu190 ,\narXiv:1401.6096 .\nMyers, A.T., McKee, C.F., Cunningham, A.J., Klein, R.I., Krumholz, M.R.,\n2013. The Fragmentation of Magnetized, Massive Star-forming Co res with\nRadiative Feedback. ApJ 766, 97. doi: 10.1088/0004-637X/766/2/97 ,\narXiv:1211.3467 .\nNixon, C.J., King, A.R., Pringle, J.E., 2018. The Maximum Mass Solar\nNebula and the early formation of planets. MNRAS 477, 3273–3278.\ndoi:10.1093/mnras/sty593 ,arXiv:1803.04417 .\nNorman, C., Heyvaerts, J., 1985. Anomalous magnetic field diffusion d uring\nstar formation. A&A 147, 247–256.\n24Ostriker, E.C., Gammie, C.F., Stone, J.M., 1999. Kinetic and Structura l\nEvolution of Self-gravitating, Magnetized Clouds: 2.5-dimensional S imu-\nlations of Decaying Turbulence. ApJ 513, 259–274. doi: 10.1086/306842 ,\narXiv:astro-ph/9810321 .\nPadoan, P., Federrath, C., Chabrier, G., Evans, II, N.J., John-\nstone, D., Jørgensen, J.K., McKee, C.F., Nordlund, ˚A., 2014.\nThe Star Formation Rate of Molecular Clouds. Protostars and\nPlanets VI , 77–100doi: 10.2458/azu_uapress_9780816531240-ch004 ,\narXiv:1312.5365 .\nPadoan, P., Nordlund, ˚A., 1999. A Super-Alfv´ enic Model of Dark Clouds.\nApJ 526, 279–294. doi: 10.1086/307956 ,arXiv:astro-ph/9901288 .\nPadoan, P., Nordlund, ˚A., 2011. The Star Formation Rate\nof Supersonic Magnetohydrodynamic Turbulence. ApJ 730, 40.\ndoi:10.1088/0004-637X/730/1/40 ,arXiv:0907.0248 .\nPadoan, P., Pan, L., Haugbølle, T., Nordlund, ˚A., 2016. Supernova\nDriving. I. The Origin of Molecular Cloud Turbulence. ApJ 822, 11.\ndoi:10.3847/0004-637X/822/1/11 ,arXiv:1509.04663 .\nP´ erez, L.M., Carpenter, J.M., Andrews, S.M., Ricci, L., Isella, A., Linz, H.,\nSargent, A.I., Wilner, D.J., Henning, T., Deller, A.T., Chandler, C.J.,\nDullemond, C.P., Lazio, J., Menten, K.M., Corder, S.A., Storm, S., Testi,\nL., Tazzari, M., Kwon, W., Calvet, N., Greaves, J.S., Harris, R.J., Mundy ,\nL.G., 2016. Spiral density waves in a young protoplanetary disk. Scie nce\n353, 1519–1521. doi: 10.1126/science.aaf8296 ,arXiv:1610.05139 .\nPotter, W.J., Balbus, S.A., 2017. Demonstration of a magnetic Prand tl\nnumber disc instability from first principles. MNRAS 472, 3021–3028.\ndoi:10.1093/mnras/stx2055 ,arXiv:1704.02485 .\nPrice, D.J., Pringle, J.E., King, A.R., 2003. A comparison of the accel-\neration mechanisms in young stellar objects and active galactic nucle i\njets. MNRAS 339, 1223–1236. doi: 10.1046/j.1365-8711.2003.06278.x ,\narXiv:astro-ph/0211330 .\nPringle, J.E., 1989. On the formation of binary stars. MNRAS 239, 36 1–370.\ndoi:10.1093/mnras/239.2.361 .\n25Pringle, J.E., 1991. Binary Star Formation, in: Lada, C.J., Kylafis, N.D.\n(Eds.), NATO Advanced Science Institutes (ASI) Series C, p. 437.\nPringle, J.E., 1993. in: Burgarella, D., Livio, M., O’Dea, C. (Eds.), Astro -\nphysical Jets, Cambridge University Press. p. 1.\nPringle, J.E., Allen, R.J., Lubow, S.H., 2001. The formation of molecular\nclouds. MNRAS 327, 663–668. doi: 10.1046/j.1365-8711.2001.04777.x ,\narXiv:astro-ph/0106420 .\nRaghavan, D., McAlister, H.A., Henry, T.J., Latham, D.W., Marcy, G.W.,\nMason, B.D., Gies, D.R., White, R.J., ten Brummelaar, T.A., 2010. A\nSurvey of Stellar Families: Multiplicity of Solar-type Stars. ApJS 190,\n1–42. doi: 10.1088/0067-0049/190/1/1 ,arXiv:1007.0414 .\nReipurth, B., 2017. The Star Formation Newsletter 293, 5.\nReipurth, B., Clarke, C., 2001. The Formation of Brown Dwarfs as\nEjected Stellar Embryos. AJ 122, 432–439. doi: 10.1086/321121 ,\narXiv:astro-ph/0103019 .\nReipurth, B., Clarke, C.J., Boss, A.P., Goodwin, S.P., Rodr´ ıguez,\nL.F., Stassun, K.G., Tokovinin, A., Zinnecker, H., 2014. Mul-\ntiplicity in Early Stellar Evolution. Protostars and Planets\nVI , 267–290doi: 10.2458/azu_uapress_9780816531240-ch012 ,\narXiv:1403.1907 .\nRuzmaikin, A.A., Sokolov, D.D., Shukurov, A.M. (Eds.), 1988. Magnetic\nfields of galaxies. volume 133 of Astrophysics and Space Science Library .\ndoi:10.1007/978-94-009-2835-0 .\nSchekochihin, A.A., Boldyrev, S.A., Kulsrud, R.M., 2002a. Spectra and\nGrowth Rates of Fluctuating Magnetic Fields in the Kinematic Dy-\nnamo Theory with Large Magnetic Prandtl Numbers. ApJ 567, 828– 852.\ndoi:10.1086/338697 ,arXiv:astro-ph/0103333 .\nSchekochihin, A.A.,Maron, J.L.,Cowley, S.C., McWilliams, J.C.,2002b. The\nSmall-Scale Structure of Magnetohydrodynamic Turbulence with La rge\nMagnetic Prandtl Numbers. ApJ 576, 806–813. doi: 10.1086/341814 ,\narXiv:astro-ph/0203219 .\n26Seifried, D., Walch, S., Haid, S., Girichidis, P., Naab, T., 2018. Is Molecula r\nCloud Turbulence Driven by External Supernova Explosions? ApJ 85 5,\n81. doi:10.3847/1538-4357/aaacff ,arXiv:1802.00973 .\nShakura, N.I., Sunyaev, R.A., 1973. Black holes in binary systems. Ob serva-\ntional appearance. A&A 24, 337–355.\nShu, F.H., 1987. Summary of symposium: Low luminosity sources, in: L ons-\ndale Persson, C.J. (Ed.), NASA Conference Publication.\nShu, F.H., Adams, F.C., Lizano, S., 1987. Star formation in\nmolecular clouds - Observation and theory. ARA&A 25, 23–81.\ndoi:10.1146/annurev.aa.25.090187.000323 .\nShukurov, A., 2004. Introduction to galactic dynamos. ArXiv e-pr ints\narXiv:astro-ph/0411739 .\nSimon, J.B., Bai, X.N., Flaherty, K.M., Hughes, A.M., 2017. A New\nModel for Weak Turbulence in Protoplanetary Disks. ArXiv e-prints\narXiv:1711.04770 .\nSolomon, P.M., Rivolo, A.R., Barrett, J., Yahil, A., 1987. Mass, luminosity ,\nand line width relations of Galactic molecular clouds. ApJ 319, 730–741 .\ndoi:10.1086/165493 .\nStahler, S.W., Palla, F., 2005. The Formation of Stars. John Wiley & Son s.\nTennekes, H., Lumley, J.L., 1972. First Course in Turbulence. MIT Pr ess.\nTobin, J.J., Looney, L.W., Wilner, D.J., Kwon, W., Chandler, C.J.,\nBourke, T.L., Loinard, L., Chiang, H.F., Schnee, S., Chen, X.,\n2015. A Sub-arcsecond Survey Toward Class 0 Protostars in Pers eus:\nSearching for Signatures of Protostellar Disks. ApJ 805, 125.\ndoi:10.1088/0004-637X/805/2/125 ,arXiv:1503.05189 .\nTomida, K., Okuzumi, S., Machida, M.N., 2015. Radiation Magnetohydro -\ndynamic Simulations of Protostellar Collapse: Nonideal Magnetohydr ody-\nnamic Effects and Early Formation of Circumstellar Disks. ApJ 801, 11 7.\ndoi:10.1088/0004-637X/801/2/117 ,arXiv:1501.04102 .\nTout, C.A., Pringle, J.E., 1996. Can a disc dynamo generate large-sca le\nmagnetic fields? MNRAS 281, 219–225.\n27Vaytet, N., Commer¸ con, B., Masson, J., Gonz´ alez, M., Chabrier, G ., 2018.\nProtostellar birth with ambipolar and ohmic diffusion. ArXiv e-prints\narXiv:1801.08193 .\nV´ azquez-Semadeni, E., Kim, J., Ballesteros-Paredes, J., 2005. St ar For-\nmation Efficiency in Driven, Supercritical, Turbulent Clouds. ApJL 630 ,\nL49–L52. doi: 10.1086/491650 ,arXiv:astro-ph/0507637 .\nWalch, S., Naab, T., 2015. The energy and momentum input of supern ova\nexplosions in structured and ionized molecular clouds. MNRAS 451, 27 57–\n2771. doi: 10.1093/mnras/stv1155 ,arXiv:1410.0011 .\nWinn, J.N., Fabrycky, D.C., 2015. The Occurrence and Ar-\nchitecture of Exoplanetary Systems. ARA&A 53, 409–447.\ndoi:10.1146/annurev-astro-082214-122246 ,arXiv:1410.4199 .\nWurster, J., Price, D.J., Bate, M.R., 2016. Can non-ideal magnetohy drody-\nnamics solve the magnetic braking catastrophe? MNRAS 457, 1037– 1061.\ndoi:10.1093/mnras/stw013 ,arXiv:1512.01597 .\nZinnecker, H., 2001. Binary Stars: Historical Milestones, in: Zinnec ker, H.,\nMathieu, R. (Eds.), The Formation of Binary Stars, pp. 1–12.\nZweibel, E.G., 2002. Ambipolar Drift in a Turbulent Medium. ApJ 567,\n962–970. doi: 10.1086/338682 ,arXiv:astro-ph/0107462 .\n28"    },    {        "title": "1507.06081v1.Spin_current_emission_governed_by_nonlinear_spin_dynamics.pdf",        "content": "arXiv:1507.06081v1  [cond-mat.mes-hall]  22 Jul 2015Spin-current emission governed by nonlinear spin dynamics\nTakaharu Tashiro,1Saki Matsuura,1Akiyo Nomura,1Shun Watanabe,2\nKeehoon Kang,2Henning Sirringhaus,2and Kazuya Ando∗1,3\n1Department of Applied Physics and Physico-Informatics, Ke io University, Yokohama 223-8522, Japan\n2Cavendish Laboratory, University of Cambridge,\nJ. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom\n3PRESTO, Japan Science and Technology Agency, Kawaguchi, Sa itama 332-0012, Japan\nCoupling between conduction electrons and localized magne tization is responsible for a variety\nof phenomena in spintronic devices. This coupling enables t o generate spin currents from dynam-\nical magnetization. Due to the nonlinearity of magnetizati on dynamics, the spin-current emission\nthrough the dynamical spin-exchange coupling offers a route for nonlinear generation of spin cur-\nrents. Here, we demonstrate spin-current emission governe d by nonlinear magnetization dynamics\nin a metal/magnetic insulator bilayer. The spin-current em ission from the magnetic insulator is\nprobed by the inverse spin Hall effect, which demonstrates no ntrivial temperature and excitation\npower dependences of the voltage generation. The experimen tal results reveal that nonlinear mag-\nnetization dynamics and enhanced spin-current emission du e to magnon scatterings are triggered by\ndecreasing temperature. This result illustrates the cruci al role of the nonlinear magnon interactions\nin the spin-current emission driven by dynamical magnetiza tion, or nonequilibrium magnons, from\nmagnetic insulators.\nDynamical magnetization in a ferromagnet emits a\nspin current,1,2enabling to explore the physics of spin\ntransport in metals and semiconductors.3–22The dy-\nnamical spin-current emission has been achieved utiliz-\ning ferromagnetic metals, semiconductors, and insula-\ntors.23–26In particular, the discovery of the spin-current\nemission from a magnetic insulator yttrium iron gar-\nnet, Y 3Fe5O12, has drawn intense experimental and the-\noretical interests, opening new possibilities to spintronics\nbasedonmetal/insulatorhybrids, whereangularmomen-\ntum can be carried by both electrons and magnons.\nA ferrimagnetic insulator yttrium iron garnet,\nY3Fe5O12, is characterized by the exceptionally small\nmagnetic damping, making it a key material for the de-\nvelopment of the physics of nonlinear magnetization dy-\nnamics.27–29The nonlinear magnetization dynamics in\nY3Fe5O12has been extensively studied both experimen-\ntally and theoretically in the past half a century, bene-\nfited by the exceptional purity, high Curie temperature,\nand simplicity of the low-energy magnon spectrum.28–31\nRecently, thenonlinearmagnetizationdynamicshasbeen\nfound to affect the spin-current emission from the mag-\nnetic insulator; the spin-current emission is enhanced by\nmagnon scattering processes [see Fig. 1(a)], triggered by\nchangingthe excitationfrequencyorpowerofthe magne-\ntization dynamics.14,15,32These findings shed new light\nonthelong-standingresearchonnonlinearmagnetization\ndynamics, promising further development of spintronics\nand magnetics based on the magnetic insulator.\nIn this work, we demonstrate that the spin-current\nemission from Y 3Fe5O12is strongly affected by nonlinear\nmagnetization dynamics at low temperatures. The spin-\n∗Correspondence and requests for materials should be addres sed to\nando@appi.keio.ac.jpcurrent emission is probed by the inverse spin Hall effect\n(ISHE) in aPt film attachedto the Y 3Fe5O12film,11,33,34\nwhich enables to measure temperature dependence of the\nspin-current emission from the magnetic insulator under\nvarious conditions. In spite of the simple structure of the\nmetal/insulator bilayer, we found nontrivial variation of\nthe spin-current emission; the temperature dependence\nof the spin-current emission strongly depends on the mi-\ncrowave frequency and excitation power. This result re-\nveals that nonlinear spin-current emission due to three\nand four magnon scatterings emerges by decreasing tem-\nperature, even at constant magnon excitation frequency\nand power. This finding provides a crucial piece of infor-\nmation for understanding the spin-current emission from\nferromagneticmaterialsandinvestigatingthe magnonin-\nteractions in the metal/insulator hybrid.\nA single-crystal Y 3Fe5O12(111) film (3 ×5 mm2) with\na thickness of 5 µm was grown on a Gd 3Ga5O12(111)\nsubstrate by liquid phase epitaxy (purchased from In-\nnovent e.V., Jena). After the substrates were cleaned by\nsonication in deionized water, acetone and isopropanol, a\npiranha etching process, a mixture of H 2SO4and H 2O2\n(with the ratio of 7 : 3), was applied, then to be able\nto remove any residuals an oxygen plasma cleaning was\nperformed outside a sputtering chamber. On the top of\nthe film, a 10-nm-thick Pt layer was sputtered in an Ar\natmosphere. Prior to sputtering 10-nm-thick Pt layer,\nan argon plasma cleaning was also performed in-situ.\nThe Pt/Y 3Fe5O12bilayer film was placed on a copla-\nnar waveguide, where a microwave was applied to the\ninput of the signal line as show in Fig. 1(b). Two elec-\ntrodes were attached to the edges of the Pt layer. The\nsignal line is 500 µm wide and the gaps between the sig-\nnal line and the ground lines are designed to match to\nthe characteristic impedance of 50 Ω. An in-plane exter-\nnal magnetic field Hwas applied parallel to the signal\nline, or perpendicular to the direction across the elec-2\n(a) \nf (GHz) (b) \n0 10 -5 05\n  \n-10 \n(H - H R) (mT) µ0dV/dH  (  V/mT) µ(e) Pt/YIG \ncoplanar waveguide \n10 210 410 5\nk (cm -1 )10 310 210 410 510 3f = f 0/2 \nf = f min f = f 0(d) \nVISHE +H\n-H02\n-2V (  V) µ\n(c) \nPabs \n02P (mW) \n-10 0\n(H - H R) (mT) 10 \nµ0uniform magnon \nFIG. 1: Detection of spin-current emission. (a) The magnon dispersion in Y 3Fe5O12, wherefandkare the frequency\nand wavenumber of magnons, respectively. The dispersion of the first 40 thickness modes propagating along and opposite t o\nthe magnetic field is shown. The blue and red arrows represent the four and three magnon scatterings. The magnon dispersio n\nshows that both the three and four magnon scatterings create secondary magnons with small group velocity. The lowest\nfrequency is f=fmin. (b) The experimental setup. The Pt/Y 3Fe5O12film placed on the coplanar waveguide was cooled using\na Gifford-McMahon cooler. (c) Magnetic field ( H) dependence of the microwave absorption Pfor the Pt/Y 3Fe5O12film at\nf0= 7.6 GHz and Pin= 10 mW. µ0HR= 183 mT is the resonance field. Pabsis the definition of the magnitude of the microwave\nabsorption intensity. The absorption peak structure compr ises multiple signals due to spin-wave modes. (d) Hdependence of\nthe electric voltage V.VISHEis the magnitude of the electric voltage. The blue and red dat a were measured with the in-plane\nmagnetic field Hand−H, respectively. (e) Hdependence of dV(H)/dHfor the Pt/Y 3Fe5O12film. The damping constant of\nthe Pt/Y 3Fe5O12film was roughly estimated to be 5 ×10−4fromf0dependence of the linewidth at 5 mW.\ntrodes.11Figure 1(c) shows the in-plane magnetic field\nHdependence of the microwave absorption Pmeasured\nby applying a 10 mW microwave with the frequency of\nf0= 7.6 GHz at T= 300 K. Under the ferromagnetic\nresonance condition H=HR, dynamical magnetization\nin the Y 3Fe5O12layer emit a spin current jsinto the\nPt layer, resulting in the voltage generation through the\nISHE as shown in Fig. 1(d).1,2The sign of the voltage is\nchanged by reversing H, consistent with the prediction\nof the spin-current emission from the magnetic insula-\ntor.35Here, the absorption spectrum comprises multiple\nresonancesignalsduetospin-wavemodes, includingmag-\nnetostatic surface waves and backward-volume magneto-\nstatic waves in addition to the ferromagnetic resonance.\nTo extract the damping constant for the Pt/Y 3Fe5O12\nfilm, we have plotted dV/dHin Fig. 1(e), which allows\nrough estimation of the damping constant, α∼5×10−4.\nFigure 2(a) shows temperature dependence of\nVISHE/Pabs, whereVISHEandPabsare the magnitude of\nthe microwave absorption and electric voltage, respec-\ntively;VISHE/Pabscharacterizes the angular-momentum\nconversion efficiency from the microwaves into spin cur-\nrents. Notably, VISHE/Pabsincreases drastically below\nT= 150 K by decreasing Tatf0= 4.0 GHz. This\ndrastic change is irrelevant to the temperature depen-\ndence of the spin pumping and spin-charge conversion\nefficiency in the Pt/Y 3Fe5O12bilayer, such as the spin\nHall angle θSHE, the spin pumping conductance geff, the\nspin diffusion length λ, and the electrical conductivity\nσ. Figure 2(b) shows the temperature dependence of\nthe electrical conductivity σand the spin Hall conduc-\ntivityσs. The spin Hall conductivity was obtained from\nthe temperature dependence of VISHE/Pabsat 10 mW for\nf0= 7.6 GHz shown in Fig. 2(a); the value of VISHE/Pabs\nis insensitive to the excitation power from 5 to 15 mW,indicating that the spin-current emission is reproduced\nwith a liner spin-pumping model:36\nVISHE\nPabs=2ewFσsf0λgefftanh(d/2λ)\nµ0σ2dvFMs∆H/radicalBig\n(γµ0Ms)2+(4πf0)2,(1)\nwherewF= 3.0 mm and vF= 7.5×10−11m3are the\nwidth and volume of the Y 3Fe5O12film.d= 10 nm is\nthe thickness ofthe Pt layer. µ0∆His the half-maximum\nfull-width of the ferromagnetic resonance linewidth. For\nthe calculation of σs, we used the measured parameters\nof the electrical conductivity σand saturation magneti-\nzationMs. The spin-diffusion length37λ= 7.7 nm and\nspin pumping conductance38geff= 4.0×1018m−2were\nassumed to be independent of temperature, as demon-\nstratedpreviously.39The spin Hall conductivity ofthe Pt\nlayer shown in Fig. 2(b) increases with decreasing tem-\nperature above 100 K. Below 100 K, the spin Hall con-\nductivity decreases with decreasing temperature. This\nfeature is qualitatively consistent with the previous re-\nport.39Although the spin Hall conductivity varies with\ntemperature, the variation of the spin Hall conductiv-\nity alone is not sufficient to explain the drastic increase\nofVISHE/Pabsforf0= 4 GHz shown in Fig. 2(a). Thus,\nthe drasticchangein VISHE/Pabsacross150K at f0= 4.0\nGHzcanbeattributedtothechangeinthemagnetization\ndynamicsintheY 3Fe5O12layer. Infact, bydecreasing T,\nthemicrowaveabsorptionintensity Pabsdecreasedclearly\nacrossT= 150 K as shown in Fig. 2(c), suggesting the\nchange of the magnetization dynamics in the Y 3Fe5O12\nlayer across T= 150 K.\nThe origin of the temperature-induced drastic change\nof the spin-conversion efficiency VISHE/Pabsshown in\nFig. 2(a) is enhanced spin-current emission triggered by\nthe three magnon splitting. The three-magnon splitting3\n 7.6 GHz \n 4.0 GHz \nPin  = 10 mW\n0VISHE /Pabs  ( µV/mW) \n369\n100 200\nT (K)300 100 200\nT (K) 300 (10 6 Ω-1 m-1 )\n1.0 1.5 2.0 (b) (a)\n(c)σPabs /Pin \n50 100 150 200\nT (K) 250 3000.20\n0.15\n0.10 f0 = 4.0 GHz6\n4\n2\n0 (10 5 Ω-1 m-1 )\nσS\nFIG. 2: Temperature evolution of spin-current emis-\nsion.(a) Temperature ( T) dependence of VISHE/Pabsfor the\nPt/Y3Fe5O12film atf0= 7.6 (the black circles) and 4.0 GHz\n(the red circles). The data were measured with Pin= 10\nmW microwave excitation. (b) Tdependence of the electri-\ncal conductivity σand the spin Hall conductivity σsfor the\nPt/Y3Fe5O12film. (c) Tdependence of Pabs/Pin, wherePabs\nis the microwave absorption intensity, for Pin= 10 mW and\nf0= 4.0 GHz.\ncreates a pair of magnons with the opposite wavevec-\ntors and the frequency f0/2 from the uniform magnon\nwithf0[see also Fig. 1(a)]. The splitting process redis-\ntributes the magnons and changes the relaxation rate of\nthe spin system, increasing the steady-state angular mo-\nmentum stored in the spin system, or resulting in the\nstabilized enhancement of the spin-current emission.14,32\nThe splitting is allowed only when f0/2> fmin, where\nfminis the minimum frequency of the magnon disper-\nsion, because of the energy and momentum conservation\nlaws. This condition can readily be found by finding fmin\nfor the thin Y 3Fe5O12film from the lowest branch of the\ndipole-exchangemagnondispersion forthe unpinned sur-\nface spin condition:40\nf=/radicalbig\nΩ(Ω+ωM−ωMQ), (2)\nwhere Ω = ωH+ωM(D/µ0Ms)k2,ωH=γµ0H,ωM=\nγµ0Ms, andQ= 1−[1−exp(−kL)]/(kL).D=\n5.2×10−13Tcm2is the exchange interaction constant,\nL= 5µm is the thickness of the Y 3Fe5O12layer, and kis\nthe wavenumber of the magnons. γ= 1.84×1011Ts−1is\nthe gyromagnetic ratio. In Figs. 3(a) and 3(b), we show\nthe lowest branch of the magnon dispersion at different\ntemperatures for the Pt/Y 3Fe5O12film, calculated us-\ning Eq. (2). For the calculation, we used the saturation\nmagnetization Msat each temperature [see Fig. 3(c)],\nestimated from the resonance field data with Kittel’s for-\nmula. We assumed that Dis independent of tempera-(b)(a)\n(c)\nMs (mT) \nµ0300\n240\n180\n120\n300 200 100\nT (K) 300 K\n270 K\n240 K\n210 K\n180 K150 K\n120 K\n105 K90 K\n75 K\n50 K\n2.0 2.5 3.0 3.5 4.0 4.5 f (GHz) \n1.5 \n10 210 4\nk (cm -1 )10 310 5\nf = f0/2 \nk (cm -1 )1×1041×1052.0 2.2 2.4 f (GHz) \nFIG. 3:Magnon dispersion. (a) The lowest-energy branch\nof the magnon spectra for the Pt/Y 3Fe5O12film calculated\nfor the resonance condition at f0= 4.0 GHz. The dispersions\nwere calculated using γ= 1.84×1011Ts−1. The dotted\nred line denotes f=f0/2 = 2.0 GHz. (b) The magnified\nview of the lowest-energy branch of the magnon spectra. (c)\nTemperature dependence of the saturation magnetization Ms\nestimated from the resonance field data.\nture, as demonstrated in literature.32,41,42Although D\ncan slightly depend on temperature,43the shape of the\nmagnon dispersion is not sensitive to the small varia-\ntion ofD. Figures 3(a) and 3(b) demonstrate that the\nminimum frequency fmindecreases with decreasing tem-\nperature and the splitting condition f0/2> fminis sat-\nisfied below T= 150 K; the magnon redistribution is\nresponsible for the enhancement of VISHE/Pabs. Thus,\nthis result demonstrates that the enhanced spin-current\nemission can be induced not only by changing the excita-\ntion frequency or power of the magnetization dynamics,\nbut also by changing temperature.\nFigures 4(a) and 4(b) show temperature dependence\nof the spin-conversion efficiency VISHE/Pabsat different\nmicrowave excitation powers Pinforf0= 7.6 and 4.0\nGHz, respectively. At f0= 4.0 GHz, the enhancement\nofVISHE/Pabsdue to the three-magnon splitting below\n150 K is observed for all the excitation powers as shown\nin Fig. 4(b). The drop in VISHE/PabsatT= 50 K for\nf0= 4.0 GHz is induced by the decrease of the spin\nHall conductivity shown in Fig. 2(b); below 100K, the\nspin Hall conductivity, or the spin Hall angle, decreases\nwith decreasing temperature, whereas the spin-current\nenhancement through the magnon splitting increases by\ndecreasing temperature. The competition gives rise to\nthe peak structure in VISHE/Pabsaround 70 K for 4.04\n50 100 150 200\nT (K)250 300f0 = 7.6 GHzVISHE /Pabs  ( µV/mW) \n0.51.01.52.0\n0VISHE /Pabs  ( µV/mW) -0.5\n-1.0\n-1.5\n-2.0\nVISHE /Pabs  ( µV/mW) -3 \n-6 \n-9 \n-120 100 mW\n 80 mW\n 60 mW\n 40 mW\n 20 mW 15 mW\n 12.5 mW\n 10 mW\n 7.5 mW\n 5 mWf0 = 4.0 GHzVISHE /Pabs  ( µV/mW) \n36912 \n50 100 150 200\nT (K) 250 300(a) (b)\n2.5\n2.0\n1.5\n1.0\n150 300\nT (K) [VISHE /Pabs ] 100 mW  / [VISHE /Pabs ] 5 mW (c)\n+H +H\n-H -H\nFIG. 4: Temperature evolution of spin-current emission for differe nt microwave powers. (a) Temperature T\ndependence of VISHE/Pabsatf0= 7.6 GHz for the in-plane magnetic field H(the upper panel) and reversed in-plane magnetic\nfield−H(the lower panel). (b) Tdependence of VISHE/Pabsatf0= 4.0 GHz for the in-plane magnetic field H(the upper panel)\nand−H(the lower panel). (c) Tdependence of [ VISHE/Pabs]100 mW/[VISHE/Pabs]5 mWatf0= 7.6 GHz. [ VISHE/Pabs]100 mW\nand [VISHE/Pabs]5 mWareVISHE/Pabsmeasured at Pin= 100 mW and 5 mW, respectively.\nGHz. This result also shows that the enhancement factor\nis almost independent of the excitation power. In con-\ntrast, notably, the variation of VISHE/Pabsdepends on\nthe excitation power, especially below 150 K, at f0= 7.6\nGHz as shown in Fig. 4(a). These features for f0= 7.6\nand 4.0 GHz were confirmed in VISHE/Pabsmeasured\nwith the reversed external magnetic field [see the exper-\nimental data for −Hin Figs. 4(a) and 4(b)], indicating\nthat the change of the spin-current emission from the\nmagnetic insulator is responsible for the nontrivial be-\nhavior of VISHE/Pabsat low temperatures.\nTo understand the temperature and power depen-\ndences of VISHE/Pabsatf0= 7.6 GHz in details, we\nplot [VISHE/Pabs]100 mW/[VISHE/Pabs]5 mWin Fig. 4(c).\nFor the spin-current emission in the linear magnetiza-\ntion dynamics regime, VISHE/Pabsis constant with Pin,\nor[VISHE/Pabs]100 mW/[VISHE/Pabs]5 mW= 1becausethe\nemitted spin current is proportional to Pin.35Since the\nthree-magnon splitting is prohibited at f0= 7.6 GHz,\n[VISHE/Pabs]100 mW/[VISHE/Pabs]5 mW≈1.2, atT= 300\nK, demonstrates enhanced spin-current emission without\nthe splitting of a pumped magnon.\nThe observed enhancement of the spin-current emis-\nsion atT= 300 K is induced by the four magnon scat-\ntering, where two magnons are created with the annihi-\nlation of two other magnons [see also Fig. 1(a)].44,45The\nfour-magnon scattering emerges at high microwave exci-\ntation powers Pin> Pth, known as the second order Suhl\ninstability,46wherePthisthe thresholdpowerofthescat-\ntering. Although this process conserves the number of\nmagnons, the magnon redistribution can decrease the re-\nlaxation rate of the spin system through the annihilation\nof the uniform magnons with large damping η0and cre-\nationofdipole-exchangemagnonswithsmalldamping ηq.Thisresultsin the steady-stateenhancement ofthe angu-\nlarmomentum storedinthe spinsystem, ortheenhanced\nspin-current emission.32In the Pt/Y 3Fe5O12film, the\ndamping η0oftheuniformmagnonatlowexcitationpow-\ners is mainly dominated by the two-magnon scattering;\nthe temperature dependence of the ferromagnetic reso-\nnance linewidth is almost independent of temperature as\nshown in the inset to Fig. 5, indicating that the damping\nη0isnotdominatedbythetemperaturepeakprocessesor\ntheKasuya-LeCrawmechanism.47Incontrast,thedamp-\ningηqof the secondary magnons created by the four-\nmagnon scattering is dominated by the Kasuya-LeCraw\nmechanism, since the two-magnon scattering events are\nsuppressed due to the small group velocity; the group\nvelocity of the secondary dipole-exchange magnons cre-\natedatthesamefrequencyastheuniformmagnoncanbe\nclosetozerobecauseofthe exchange-dominatedstanding\nspin-wave branches [see Fig. 1(a)].44,48–50The exchange-\ndominated branches, i.e. the thickness modes, show the\nenergyminimum notonly atthe bottom ofthe dispersion\nbut also at the excitation frequency. Therefore, in the\npresent system, the damping η0of the uniform magnonis\ndominated by the temperature-independent two-magnon\nscattering, whereas the damping ηqof the secondary\nmagnon is dominated by temperature-dependent three-\nparticle confluences, such as the Kasuya-LeCraw pro-\ncess.47In the presence of the four magnon scattering,\nthe total number of the nonequilibrium magnons Ntis\nexpressed as32\nNt\nPabs=1\n2πηq/planckover2pi1f0/bracketleftbigg\n1−χ′′\n2γMs(η0−ηq)/bracketrightbigg\n,(3)\nwhereηqis defined as the average decay rate to the\nthermodynamic equilibrium of the degenerate secondary5\n 120 K\n 105 K\n 90 K\n 75 K\n 50 K 300 K\n 270 K\n 240 K\n 210 K\n 180 K\n 150 K[VISHE /Pabs ] / [VISHE /Pabs ] 5 mW 2.4\n2.0\n1.6\n1.2\n-2.4-2.0-1.6-1.2\n56789\n10 2 3 456789\n100[VISHE /Pabs ] / [VISHE /Pabs ] 5 mW \nPin  (mW)+H\n-H\n00.20.4\n  \n100 200 \nT (K)300H (mT) \nµ0∆\nFIG. 5: Microwave power dependence of spin-current\nemission at different temperatures. Microwave excita-\ntion power Pindependence of [ VISHE/Pabs]/[VISHE/Pabs]5 mW\natf0= 7.6 GHz for different temperatures. The in-plane\nmagnetic field is Hfor the upper panel and −Hfor the lower\npanel, respectively. The inset shows Tdependence of the\nhalf-maximum full-width µ0∆Hof ferromagnetic resonance\nfor the Pt/Y 3Fe5O12film.\nmagnons for simplicity. The imaginary part of the sus-\nceptibility is expressed as\nχ′′=2γMs\nη0+ηspf(Pin), (4)\nwhere\nf(Pin) =1/radicalbig\n1−[χ′′(η0+ηsp)/(2γMs)]4(Pin/Pth)2.(5)\nHere,ηspis the decay constant of the uniform precession\nto degenerate magnons at f0due to scattering on sample\ninhomogeneities. Under the assumption that the spin-\npumping efficiency is insensitive to the wavenumber kof\nthe nonequilibrium magnons, that is VISHE∝js∝Nt,\nEq. (3) is directly related to the spin-conversion effi-\nciency:VISHE/Pabs∝Nt/Pabs.\nThe above model reveals that the spin-current en-\nhancement due to the four-magnon scattering is re-\nsponsible for the nontrivial behavior of the volt-\nage generation shown in Fig. 4(a). As shown in\nFig. 4(c), the nonlinearity of the spin-current emis-\nsion is enhanced by decreasing temperature, from\n[VISHE/Pabs]100 mW/[VISHE/Pabs]5 mW≈1.2 atT= 300\nK to [VISHE/Pabs]100 mW/[VISHE/Pabs]5 mW≈2.4 at 50\nK.Figure5showsmicrowaveexcitationpower Pindepen-\ndenceofVISHE/Pabsforf0= 7.6GHzatdifferenttemper-\natures. Thisresultclearlyshowsthatthethresholdpower 75 K[VISHE /Pabs ] / [VISHE /Pabs ] 5 mW \n 300 K2.0\n1.5\n1.01.2\n1.1\n1.0\nPin  (mW)0 20 40 60 80 100[VISHE /Pabs ] / [VISHE /Pabs ] 5 mW \nFIG. 6: Threshold power of spin-current enhance-\nment. Microwave excitation power Pindependence of\n[VISHE/Pabs]/[VISHE/Pabs]5 mWatf0= 7.6 GHz for T= 300\nK andT= 75 K.\nPthof the spin-current enhancement decreases with de-\ncreasingtemperature, whichisthe originofthenontrivial\nbehavior of the temperature dependence of VISHE/Pabs\nshown in Figs. 4(a) and 4(c). The threshold power of the\nspin-current enhancement through the four-magnon pro-\ncess is very low at low temperatures, making it difficult\nto observe the threshold behavior. In fact, VISHE/Pabs\ndeviates from the prediction of the linear model even\nat the lowest microwave excitation power that is nec-\nessary to detect the ISHE voltage in the Pt/Y 3Fe5O12\nfilm atT= 75 K [see the orange circles in Fig. 6]. At\nT= 300K, a clear threshold is observedaround Pin= 40\nmW. The threshold power of the four-magnon scattering\nis given by47Pth∝h2\nth= (η0/γ)2(2ηq/σq), where hth\nis the threshold microwave field and σqis the coupling\nstrength between the uniform and secondary magnons.\nForsimplicity, weneglectthesurfacedipolarinteractions,\norL→ ∞. Under this approximation, the ferromag-\nnetic resonance condition is given by f0=γµ0Hand the\ncoupling strength can be approximated as σq=γµ0Ms.\nThus, thethresholdpowerforthefour-magnonscattering\nis proportional to\nh2\nth=/parenleftbiggη0\nγ/parenrightbigg2/parenleftbigg2ηq\nγµ0Ms/parenrightbigg\n. (6)\nEquation (6) predicts that the threshold power of the\nspin-currentenhancementdecreaseswithdecreasingtem-\nperature, since Msincreases by decreasing temperature\nas shown in Fig. 3(c). Although the damping η0of the\nuniform magnon is almost independent of temperature\nas shown in the inset to Fig. 5, the damping ηqof the\ndipole-exchange magnon tends to decrease the thresh-\nold power, since ηq, dominated by the Kasuya-LeCraw\nprocess is approximately proportional to temperature.47\nAt high power excitations, the competition between the\nincreaseofthe spin-current enhancement due to the four-\nmagnonscatteringandthedecreaseofthespinHalleffect\nby decreasing temperature gives rise to the peak struc-\nture inVISHE/Pabsaround 100 K for f0= 7.6 GHz [see\nFig. 4(a)].\nIn summary, we have demonstrated that the spin-6\ncurrent emission from a Y 3Fe5O12film is strongly af-\nfected by nonlinear magnetization dynamics at low tem-\nperatures. The spin-current emission has been demon-\nstrated to be enhanced even in the absence of the three-\nmagnon splitting.15The experimental results presented\nin this paper are consistent with this result and further\nextend the physics of the nonlinear spin-current emis-\nsion from the magnetic insulator. Our study reveals that\nthe spin-current enhancement arises from both the three\nand four magnon scatterings depending on the excitation\nfrequency and temperature. We show that the enhanced\nspin-currentemissioncanbetriggeredbydecreasingtem-\nperature, which is evidenced by our systematic measure-\nments for the Pt/Y 3Fe5O12film; the spin-current emis-sion can be enhanced not only by changing the magnon\nexcitation frequency or power, but also by changing tem-\nperature. This result demonstrates the generality of the\ncrucial role of magnon interactions in the spin-current\nemission, combining the long-standing research on non-\nlinear spin physics with spintronics.\nThis work was supported by JSPS KAKENHI Grant\nNumbers 26220604, 26103004, 26600078, PRESTO-JST\n“Innovative nano-electronics through interdisciplinary\ncollaboration among material, device and system lay-\ners,”theMitsubishiFoundation,theAsahiGlassFounda-\ntion, the Noguchi Institute, the Casio Science Promotion\nFoundation, and the Murata Science Foundation.\n1Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n2A. Brataas, Y. Tserkovnyak, G. E. W. Bauer, and B. I.\nHalperin, Phys. Rev. B 66, 060404(R) (2002).\n3Y. Sunet al., Phys. Rev. Lett. 111, 106601 (2013).\n4B. Heinrich et al., Phys. Rev. Lett. 107, 066604 (2011).\n5C. Hahn et al., Phys. Rev. B 87, 174417 (2013).\n6M. Weiler et al., Phys. Rev. Lett. 111, 176601 (2013).\n7S. M. Rezende, R. L. Rodriguez-Suarez, and A. Azevedo,\nPhys. Rev. B 88, 014404 (2013).\n8O. d’Allivy Kelly et al., Appl. Phys. Lett. 103, 082408\n(2013).\n9C. W.Sandweg et al., Phys.Rev.Lett. 106, 216601 (2011).\n10H. Jiao and G. E. W. Bauer, Phys. Rev. Lett. 110, 217602\n(2013).\n11E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl.\nPhys. Lett. 88, 182509 (2006).\n12E. Shikoh et al., Phys. Rev. Lett. 110, 127201 (2013).\n13K. Ando et al., Nature Mater. 10, 655 (2011).\n14H. Kurebayashi et al., Nature Mater. 10, 660 (2011).\n15V. Castel, N. Vlietstra, B. J. van Wees, and J. B. Youssef,\nPhys. Rev. B 86, 134419 (2012).\n16A.V. Chumak et al., Appl.Phys.Lett. 100, 082405 (2012).\n17M. B. Jungfleisch et al., Appl. Phys. Lett. 99, 182512\n(2011).\n18K. Ando et al., Nature Mater. 12, 622 (2013).\n19M. B. Jungfleisch et al., Appl. Phys. Lett. 103, 022411\n(2013).\n20C. Du, H. Wang, F. Yang, and P. C. Hammel, Phys. Rev.\nApplied 1, 044004 (2014).\n21K. Ando and E. Saitoh, Phys. Rev. Lett. 109, 026602\n(2012).\n22C. H. Du et al., Phys. Rev. Lett. 111, 247202 (2013).\n23S. Mizukami, Y. Ando, and T. Miyazaki, Phys. Rev. B 66,\n104413 (2002).\n24F. D. Czeschka et al., Phys. Rev. Lett. 107, 046601 (2011).\n25L. Chen, F. Matsukura, and H. Ohno, Nature Commun.\n4, 2055 (2013).\n26Y. Kajiwara et al., Nature 464, 262 (2010).\n27S. Demokritov, B. Hillebrands, and A. Slavin, Phys. Rep.348, 441 (2001).\n28A. A. Serga, A. V. Chumak, and B. Hillebrands, J. Phys.\nD43, 264002 (2010).\n29B. Lenk, H. Ulrichs, F. Garbs, and M. Munzenberg, Phys.\nRep.507, 107 (2011).\n30C. E. Patton, Phys. Rep. 103, 251 (1984).\n31V. S. L’vov, Wave Turbulence Under Parametric Excita-\ntion(Springer-Verlag, Berlin, 1994).\n32H. Sakimura, T. Tashiro, and K. Ando, Nature Commun.\n5, 5730 (2014).\n33T. Kimura et al., Phys. Rev. Lett. 98, 156601 (2007).\n34S.O.ValenzuelaandM.Tinkham, Nature 442, 176(2006).\n35K. Ando et al., J. Appl. Phys. 109, 103913 (2011).\n36R. Iguchi et al., Jpn. J. Appl. Phys. 51, 103004 (2012).\n37H. Nakayama et al., Phys. Rev. B 85, 144408 (2012).\n38X. Jia, K. Liu, K. Xia, and G. E. W. Bauer, Europhys.\nLett.96, 17005 (2011).\n39S. Meyer et al., Appl. Phys. Lett. 104, 242411 (2014).\n40B. Kalinikos, Microwaves, Optics and Antennas, IEE Pro-\nceedings H 127, 4 (1980).\n41R. C. LeCraw and L. R. Walker, J. Appl. Phys. 32, S167\n(1961).\n42C. M. Srivastava and R. Aiyar, J. Phys. C: Solid State\nPhys.20, 1119 (1987).\n43F. Heider and W. Williams, Geophys. Res. Lett. 15, 184\n(1988).\n44G. de Loubens, V. V. Naletov, and O. Klein, Phys. Rev.\nB71, 180411 (2005).\n45H. Schultheiss, K. Vogt, and B. Hillebrands, Phys. Rev. B\n86, 054414 (2012).\n46H. Suhl, J. Phys. Chem. Solids 1, 209 (1957).\n47M. Sparks, Ferromagnetic-Relaxation Theory (McGraw-\nHill, New York, 1964).\n48G. A. Melkov, V. I. Vasyuchka, Y. V. Kobljanskyj, and\nA. N. Slavin, Phys. Rev. B 70, 224407 (2004).\n49M. B. Jungfleisch et al., Appl. Phys. Lett. 99, 182512\n(2011).\n50A. V. Chumak, A. A. Serga, and B. Hillebrands, Nature\nCommun. 5, 4700 (2014)."    },    {        "title": "2109.13511v1.Dynamical_Switching_of_Magnetic_Topology_in_Microwave_Driven_Itinerant_Magnet.pdf",        "content": "arXiv:2109.13511v1  [cond-mat.str-el]  28 Sep 2021Dynamical Switching of Magnetic Topology in Microwave-Dri ven Itinerant Magnet\nRintaro Eto1and Masahito Mochizuki1\n1Department of Applied Physics, Waseda University, Okubo, S hinjuku-ku, Tokyo 169-8555, Japan\nWe theoretically demonstrate microwave-induced dynamica l switching of magnetic topology in\ncentrosymmetric itinerant magnets by taking the Kondo-lat tice model on a triangular lattice, which\nis known to exhibit two types of skyrmion lattices with differ ent magnetic topological charges of\n|Nsk|=1 and|Nsk|=2. Our numerical simulations reveal that intense excitati on of a resonance mode\nwith circularly polarized microwave field can switch the mag netic topology, i.e., from the skyrmion\nlattice with |Nsk|=1 to another skyrmion lattice with |Nsk|=2 or to a nontopological magnetic order\nwith|Nsk|=0 depending on the microwave frequency. This magnetic-top ology switching shows vari-\nous distinct behaviors, that is, deterministic irreversib le switching, probabilistic irreversible switch-\ning, and temporally random fluctuation depending on the micr owave frequency and the strength\nof external magnetic field, variety of which is attributable to different energy landscapes in the\ndynamical regime. The obtained results are also discussed i n the light of time-evolution equations\nbased on an effective model derived using perturbation expan sions.\nPACS numbers:\nINTRODUCTION\nFIG. 1: Schematics of a skyrmion lattice in the itinerant mag -\nnet irradiated with circularly polarized microwave field. T he\nthin solid arrows represent local magnetizations, whereas the\nlight blue spheres with thick solid arrows represent itiner ant\nelectrons with spins.\nTopologicalmagnetismsexemplifiedbyseveraltypesof\nskyrmions [1–6], merons [7], hedgehogs [8–12], and hop-\nfions [13] are currently attracting enormous research in-\nterest from the viewpoints of both fundamental sciences\nandpotentialapplications[14–21]. Thesenontrivialmag-\nnetic textures with spatially modulated magnetization\nare often caused by the Dzyaloshinskii-Moriya interac-\ntions [22–24], which have a relativistic origin and are ac-\ntive when the system has no spatial inversion symmetry.\nTherefore, the above topological magnetic textures are\nusually hosted in magnets having non-centrosymmetric\ncrystal structures or magnetic heterostructures having\ninterfaces. In such magnets, structural symmetries and\nanisotropies determine the way of magnetization align-\nment in magnetic textures through governing the spa-\ntial configuration of Dzyaloshinskii-Moriya vectors [25–27]. Consequently, the helicity, chirality, and vorticity of\nthe magnetic textures are inherently fixed and thus are\nnot variable [15].\nExchange coupling between itinerant electrons and lo-\ncal magnetizations is another source of topological mag-\nnetisms [28–43]. Recent theoretical studies have pre-\ndictedpossibleemergenceoftopologicalmagnetismssuch\nas skyrmion lattices [30–37], meron lattices [28, 37],\nand hedgehog lattices [28, 42, 43] in the Kondo-lattice\nmodel and its effective model which describe localized\nmagnetizations on a lattice coupled to itinerant electron\nspins via (anti)ferromagnetic exchange interactions. In\nsuch magnets, spatial modulation of magnetization is\ncaused by effective long-range interactions among local\nmagnetizations mediated by conduction electrons [e.g.,\nthe Ruderman-Kittel-Kasuya-Yosida (RKKY) interac-\ntions [44–46]]. Therefore, in the case of the Kondo-\nlattice magnets, even the centrosymmetric systems with\nspatial inversion symmetry can host topological mag-\nnetisms, despite the Dzyaloshinskii-Moriya interactions\nare absent [28, 30–39]. These magnetic textures are su-\nperpositions of magnetic helices with multiple propaga-\ntion vectors ( Qvectors), which are determined by the\neffective electron-mediated interactions among magneti-\nzations governedby multiple nesting vectorsof the Fermi\nsurfaces. Thisfactenablesustohaveavarietyoftopolog-\nical magnetic textures and their controllability via mate-\nrial variations or by tuning material parameters[47]. Re-\ncentexperimentshaveindeed observedthree-dimensional\nhedgehog lattices in SrFeO 3[12], triangular skyrmion\nlattices in Gd 2PdSi3[48–51] and Gd 3Ru4Al12[52, 53],\nand square skyrmion lattices in GdRu 2Si2[54, 55], all of\nwhich are itinerant magnets with centrosymmetric crys-\ntal structure.\nIn addition to their rich variety, topological magnetic\ntextures in centrosymmetric metallic magnets have an\ninteresting feature, that is, several degrees of freedom2\nsuch as helicity and vorticity remain to be unfrozen be-\ncause the Dzyaloshinskii-Moriya interaction is absent.\nThereby, their low-energy excitations and continuous\nvariations are possible, which provide us a unique op-\nportunity to control and switch the magnetic topology\nwith an external stimulus such as static magnetic field,\nelectric currents [56], and microwave magnetic field. It\nhas been revealed that topological magnetisms such as\nskyrmionsand skyrmiontubes in chiralmagnetshavepe-\nculiar collective modes or spin-wave modes at microwave\nfrequencies[57–62], andthey haveturned outto causein-\nteresting physical phenomena and potential device func-\ntions[63–79]. Weexpectthatthetopologicalmagnetisms\nin centrosymmetric magnets can also host interesting dy-\nnamical phenomena and functionalities associated with\ntheir microwave-active collective modes.\nIn this paper, we theoretically demonstrate that mag-\nnetic topology in a centrosymmetric itinerant magnet\ncan be switched dynamically by application of circu-\nlarly polarized microwave field by taking the Kondo-\nlattice model on a triangular lattice as an example.\nThis model is known to exhibit two distinct skyrmion-\nlattice phases [Fig.1] with different skyrmion numbers or\nmagnetic topological charges |Nsk|=1 and|Nsk|=2 [31].\nWe numerically simulate microwave-driven magnetiza-\ntion dynamics in this model using a combined technique\nof the micromagnetic simulation and the kernel poly-\nnomial method. We first find that a single spin-wave\nmode can be activated by an in-plane microwave field in\nall the three phases. We then demonstrate that by in-\ntenselyactivatingthisspin-wavemodewithcircularlypo-\nlarizedmicrowavefield, theskyrmionlatticewith |Nsk|=1\ncan be switched to that with |Nsk|=2 or a nontopo-\nlogical magnetic state of Nsk=0 depending on the mi-\ncrowave frequency. During the switching processes, we\nobserve emergent topological magnetic patterns charac-\nterized by half-integer skyrmion numbers of |Nsk|=1/2\nand|Nsk|=3/2 (i.e., meron lattices) as transient states.\nWe examine such dynamical transitions for various ini-\ntial magnetic configurations in equilibrium and find sev-\neral different behaviors, that is, deterministic irreversible\nswitching, probabilistic irreversible switching, and tem-\nporally random fluctuations under continuous microwave\nirradiation. This variety of behaviors is attributable\nto difference of the energy landscape in the dynamical\nregime. We also discuss the obtained results on the ba-\nsis of an effective model derived using perturbation ex-\npansions [32]. Note that magnetic frustration is another\nimportant mechanism to realize topological magnetism\nin centrosymmetric magnets. The phenomena revealed\nin this work can also be expected in the frustrated sys-\ntems although fine tuning of the exchange interactions\nis required to produce topological magnetic textures of\nfrustration origin [80–88].MODEL\nKondo-Lattice Model\nWe consider the Kondo-lattice model on a triangular\nlattice. The Hamiltonian is given by\nH=HKL+HZeeman, (1)\nwith\nHKL=/summationdisplay\nijσti,jˆc†\niσˆcjσ+JK/summationdisplay\ni,σ,σ′ˆc†\niσσσσ′ˆciσ′·Si,(2)\nHZeeman=−/summationdisplay\ni[Hext+H(t)]·Si. (3)\nHere ˆc†\niσ(ˆciσ) denotes the creation (annihilation) opera-\ntor of an itinerant electron with spin σ(=↑,↓) on the ith\nsite, and Sidenotes localized magnetization on the ith\nsite. The first term of HKLrepresents kinetic energies of\nitinerantelectronswhere the nearest-neighborhopping t1\nand the third-nearest-neighbor hopping t3are set to be\nt1=−1 andt3= 0.85, respectively. The second term of\nHKLrepresents the exchange coupling between itinerant\nelectron spins and local magnetizations where the cou-\npling constant is set to be JK=−0.5. The term HZeeman\nrepresents the Zeeman couplings associated with both a\nstatic external magnetic field Hext= (0,0,Hz) and a\ntime-dependent magnetic field H(t) acting on the local\nmagnetizations. For the time-dependent field H(t), we\nconsider a short-period pulse or circularly polarized mi-\ncrowave field in the present study. Note that we neglect\nthe coupling between the magnetic fields and the itiner-\nant electrons. In fact, we have examined the effects of\nthe coupling and have found that consideration of the\ncoupling does not alter the results even quantitatively.\nWe set the chemical potential µ=−3.5, which corre-\nsponds to the electron filling of 0.2 approximately. The\nabove parameter values are the same as those used in the\nprevious work [31].\nThis model is known to exhibit various magnetic or-\nders including topological ones as superpositions of three\nmagnetic helices. The propagation vectors of the three\nhelices are Q1= (π/3,0),Q2=ˆR(2π/3)Q1, andQ3=\nˆR(4π/3)Q1whereˆR(φ) is an operator to rotate the vec-\ntor by the angle φaround the z-axis. A ground-state\nphase diagram was studied in Ref. [31] as a function of\nthe strength of external magnetic field Hzwhen the mi-\ncrowave field H(t) is absent [Fig. 2(a)]. We find that\nthreemagnetic phases, i.e., askyrmion-latticephase with\n|Nsk|= 2, another skyrmion-lattice phase with |Nsk|= 1\nand a nontopological phase with Nsk= 0 successively\nemerge as Hzincreases. Note that the skyrmion-lattice\nwith|Nsk|= 2 emerges even at Hz= 0 in striking con-\ntrast to the case of the Dzyaloshinskii-Moriya magnets\nin which topological magnetisms usually appear in the3\nHz\nSkL (| Nsk |=2) SkL (| Nsk |=1) Nontopological ( Nsk =0)0.00325 0.0065 \n(a)\nNeel-type ( Nsk =ʵ1) Bloch-type ( Nsk =ʵ1) Antivortex-type ( Nsk =+1) SkL ( Nsk =ʵ2) SkL ( Nsk =+2)Siz -1                  0                   1 0\n(b) (c) (d) (e) (f)Local magnetization Si Scalar spin chirality Ci Ci-1.5                 0                 1.5 \nFIG. 2: (a) Ground-state phase diagram of the Kondo-lattice model in Eq. (1) on the triangular lattice as a function of sta tic\nmagnetic field Hzperpendicular to the lattice plane when the microwave magne tic field is absent (i.e., H(t) = 0) [31]. The\nmodel parameters are set to be t1=−1,t3= 0.85,JK=−0.5, andµ=−3.5. Successive two phase transitions among three\nphases, i.e., the skyrmion-lattice (SkL) phase with |Nsk|= 2, another SkL phase with |Nsk|= 1, and the nontopological phase\nwithNsk= 0 take place at Hz= 0.00325 and Hz= 0.0065. Several degrees of freedom of magnetic textures such a s helicity\nand vorticity are not frozen in the centrosymmetric system w ithout Dzyaloshinskii-Moriya interactions, which result s in infinite\ndegeneracy of magnetic textures (see text). (b), (c) Spatia l profiles of local magnetizations (upper panels) and local s calar spin\nchiralities (lower panels) of two degenerate SkLs with |Nsk|= 2, i.e., (b) SkL with Nsk=−2 and (c) SkL with Nsk= +2.\n(d)-(f) Those of SkLs with |Nsk|= 1, i.e., (d) N´ eel-type SkL with Nsk=−1, (e) Bloch-type SkL with Nsk=−1, and (f)\nantivortex-type SkL with Nsk= +1.\npresenceofanexternalmagneticfield. Moreimportantly,\nseveraldegreesoffreedom remain unfrozen in the present\nsystem with spatial inversion symmetry. For example,\nthe helicity and the signs of vorticity are not frozen for\nthe skyrmion-latticephase with |Nsk|= 1 in the presence\nofthe spatialinversionsymmetry. Thereby, the magnetic\nstructures in the present system have infinite degeneracy\n[see Figs. 2(b)-(f)].\nIt should be mentioned that the present triangular-\nlattice system is favorable for the emergence of the\nskyrmion lattices with a higher topological number of\n|Nsk|= 2. In the centrosymmetric Kondo-lattice sys-\ntem, the skyrmion lattices emerge as a superposition of\nthree spiral or sinusoidal states of local magnetizations,\nwhich are stabilized by the long-ranged and frustrated\nRKKY interactions. The RKKY interactions originate\nfromthe couplingbetweenconductionelectronsandlocal\nmagnetizations and thus are governed by the electronic\nstructure of conduction electrons, e.g., the Fermi-surface\ngeometry and the density of states. Consequently, the\nmodulation vectors Qνdetermined by the RKKY inter-\nactionsalsodepend onthe Fermi-surfacegeometry. Morespecifically, these modulation vectors correspond to nest-\ning vectors of the Fermi surface(s) and, thereby, reflect\nthe symmetry of the lattice structure. In addition, it has\nturned out that the skyrmion lattices with |Nsk|= 2 re-\nquires, at least, three magnetization spiral or sinusoidal\nstates, while the skyrmion lattices with |Nsk|= 1 can\nbe produced only with two magnetization spiral or si-\nnusoidal states. Hence, the triangular lattices and the\nKagomelatticeswith triangularorhexagonalsymmetries\nhavemore opportunity to host the skyrmionlattices with\n|Nsk|= 2 as compared to the simple square lattices.\nGlobal Symmetry in Skyrmion Lattices\nLet us discuss the degeneracy of topological magnetic\ntextures in the present centrosymmetric system in more\ndetail. We first consider the cases without external mag-\nneticfield. Thelocalmagnetizations Siforthe skyrmion-4\nlattice phases with |Nsk|= 1 and |Nsk|= 2 are given by,\nSNSk=±1\ni∝3/summationdisplay\nν=1\nsinQνcosφν\nλvsinQνsinφν\ncosQν\n,(4)\nSNSk=−2\ni∝\ncosQ1\ncosQ2\ncosQ3\n, (5)\nSNSk=+2\ni∝\ncosQ1\ncosQ3\ncosQ2\n, (6)\nwhere\nQν=Qν·ri+θν, φν= 2π(ν−1)/3.(7)\nHere the angles θν(ν=1,2,3) represent phase shifts, and\nΘ =/summationtext3\nν=1θνis an internal degree of freedom called pha-\nson [38, 39]. The variable λv(=±1) is called vorticity,\nandλv= +1(−1) corresponds to Nsk=−1(+1). Be-\ncause the variation of Θ is accompanied by a change in\nenergy, we set θν=0 hereafter. These formulae represent\nthat the skyrmion-lattice with |Nsk|= 1 is a superposi-\ntion of the three helices, while that with |Nsk|= 2 is a\nsuperposition of the three cosine waves.\nThese two skyrmion lattices break different symme-\ntries. Specifically, the skyrmion lattices with |Nsk|= 1\nbreak the U(1) symmetry associated with the in-plane\nrotational invariance and thus have a degree of freedom\ncalled helicity. The variation of magnetic texture upon\nthe helicity shift by φ1is given, e.g., by,\nSNSk=±1\ni∝ˆRz(φ1)3/summationdisplay\nν=1\nsinQνcosφν\nλvsinQνsinφν\ncosQν\n(8)\nwhereˆRγ(ϕ)isanoperatortorotatethevectorbythean-\ngleϕaroundthe γ-axis. Ontheotherhand, theskyrmion\nlattices with |Nsk|= 2 break the SO(3) symmetry. The\nvariation of magnetic texture upon the SO(3)-invariant\nrotational operations by angles θ2andφ2is given, e.g.,\nby,\nSNSk=−2\ni∝ˆRx(θ2)ˆRz(φ2)\ncosQ1\ncosQ2\ncosQ3\n.(9)\nNote that the energyof the skyrmion lattice with |Nsk|=\n1 does not change upon the variation of φ1, whereas that\nwith|Nsk|= 2 does not change upon the variations of θ2\nandφ2.\nThen we consider the effects of external magnetic field.\nWhen a magneticfield is applied, someofthe symmetries\nmentioned in the above discussion would be violated.\nThe U(1) symmetry around the z-axis in the skyrmion\nlattice with |Nsk|= 1 becomes absent when the external\nmagnetic field has in-plane components. Therefore, themagnetic structures of the |Nsk|= 1 skyrmion lattice do\nnot have any global symmetries when irradiatedwith cir-\ncularly polarized microwave field. In addition, the SO(3)\nsymmetry in the skyrmion lattice with |Nsk|= 2 is par-\ntiallyviolatedbythe externalmagneticfield, whereasthe\nU(1) symmetry around the magnetic field remains. More\nspecifically, the |Nsk|= 2 skyrmion lattice has the U(1)\nsymmetry around the total magnetic field at every mo-\nment, and thus its symmetry axis temporally varies. We\nalso note that a magnetic field also modulates a degree\nof freedom associated with phasons Θ.\nMETHOD\nWe simulate time evolution of the local magnetiza-\ntions in the Kondo-lattice system by numerically solving\nthe Landau-Lifshitz-Gilbert (LLG) equation. The LLG\nequation is given by,\ndSi\ndt=−Si×Heff\ni+αG\nSSi×dSi\ndt, (10)\nwhereαG(= 0.05) is the dimensionless Gilbert-damping\nconstant, and S(= 1) is the saturation magnetization.\nThe effectivemagnetic field Heff\niactingon the localmag-\nnetization at the ith site is calculated by,\nHeff\ni=−∂Ω\n∂Si+Hext+H(t). (11)\nHere Ω is the grand canonical potential of HKL, which is\ngiven by,\nΩ =/integraldisplay\nρ(ε,{Si})F(ε−µ)dε, (12)\nwith\nρ(ε,{Si}) =1\n2N2N/summationdisplay\nk=1δ(ε−εk({Si})).(13)\nHereF(ε−µ) is the free energy of the system, and\nρ(ε,{Si}) is the density of state of conduction electrons\nfor a given set of the local magnetizations {Si}. To cal-\nculate Ω and its magnetization-derivatives ∂Ω/∂Si, we\nadopt the kernel polynomial method, which is based on\nthe Chebyshev polynomial expansion of Ω and the auto-\nmatic differentiation [89–95].\nAll the simulations are performed at zero temperature\nwithnothermalfluctuationsinordertodemonstratethat\nthe microwaveapplicationcansolelyinduce the magnetic\ntopological switching and to capture the physics of this\nfield-induced phenomenon. For the simulations, a lattice\nwithN= 362sites, on which periodic boundary condi-\ntionsareimposed, isadopted. We use324correlatedran-\ndom vectors [95, 96] for simulating relaxation dynamics5\nto obtain initial magnetic configurations through mini-\nmizing the energy, whereas we adopt complete orthonor-\nmal basisstates forthe simulationsofmicrowave-induced\ndynamics. We use Chebyshev polynomials up to the\n2000th-orderfortheexpansionofΩandadoptthefourth-\norder Runge-Kutta method with a time slice of ∆ t= 4\nto solve the LLG equation in Eq. (10). The spatiotempo-\nral dynamics of local magnetizations Siand local scalar\nspin chiralities Ciare computed. The spin chirality Ciis\ncalculated by,\nCi=Si·Si+ˆa×Si+ˆa+ˆb+Si·Si+ˆa+ˆb×Si+ˆb,(14)\nwhere ˆaandˆbare the primitive lattice vectors of trian-\ngular lattice. We also compute time profiles of the net\nmagnetization Sand the skyrmion number Nsk, which\nare respectively calculated by [97],\nS=1\nNN/summationdisplay\ni=1Si, (15)\nNsk=1\n4πNmN/summationdisplay\ni=1/bracketleftigg\n2tan−1/parenleftigg\nSi·Si+ˆa×Si+ˆa+ˆb\n1+Si·Si+ˆa+Si+ˆa·Si+ˆa+ˆb+Si+ˆa+ˆb·Si/parenrightigg\n+ 2tan−1/parenleftigg\nSi·Si+ˆa+ˆb×Si+ˆb\n1+Si·Si+ˆa+ˆb+Si+ˆa+ˆb·Si+ˆb+Si+ˆb·Si/parenrightigg/bracketrightigg\n.\n(16)\nHereNm(= 27) is the number of magnetic unit cells\nwhere one unit cell contains 48 sites. Note that the sizes\nof magnetic unit cells are common for all the magnetic\npatterns which appear in the present simulations because\nthey are all constituted with three magnetic helices with\nthe same wavevectors Qν(ν= 1,2,3) determined by the\nFermi-surface nesting.\nRESULTS\nSpin-Wave Modes\nFirst, we study microwave-active resonance modes in\neach magnetic phase by numerically calculating the dy-\nnamical magnetic susceptibilities,\nχγ(ω) =∆Sγ(ω)\nHγ(ω)(γ=x,y,z), (17)\nwhereHγ(ω) and ∆ Mγ(ω) are Fourier components of\nthe time-dependent magnetic field H(t) and those of the\ntime-profile of total magnetization ∆ S(t) =S(t)−S(0).\nHere we particularly focus on the resonance modes ac-\ntive to the in-plane polarized microwave field and thus\nsetγ=x. To calculate these quantities, we adopt a\nIm /g70 (arb. units) \n02\n1\n012 \n6\n020 \n10 \n/g900 0.01 0.02/g900 0.01 0.02/g900 0.01 0.02\nHz0 0.005 0.01/g90res \n00.0050.01\nSkL (| Nsk |=2)\nSkL (| Nsk |=1)Nontopological \n(Nsk =0)/g90res\n/g90=HzHz=\n0.001 \n0.002 \n0.003 0 0.004 \n0.005 \n0.006 Hz= Hz=\n0.007 \n0.0085 \n0.01 \n(d)(a) (b) (c) |Nsk |=2 |Nsk |=1 Nsk =0 \nFIG. 3: (a)-(c) Microwave-absorption spectra for in-plane mi-\ncrowave magnetic fields inrespective magnetic phases, i.e. , (a)\nthe skyrmion-lattice phase with |Nsk|= 2, (b) the skyrmion-\nlattice phase with |Nsk|= 1, and (c) the nontopological phase\nwithNsk= 0. (d) Hz-dependence of resonance frequency\nωresfor spin-wave modes active to an in-plane microwave field\n(corresponding to peak positions of the microwave-absorpt ion\nspectra).\nspatially uniform short-time pulse of magnetic field for\nH(t), which is given by,\nH(t) =/braceleftigg\n(Hpulse,0,0) 0≤t≤1\n0 others(18)\nwheret= (t1//planckover2pi1)τis the dimensionless time with τandt1\nrespectively being the real time and the nearest-neighbor\nhopping integral. We compute time evolutions of local\nmagnetizations Si(t) and their sum S(t) after applying\nthispulsefieldtothesystem. Theusageoftheshort-time\npulse is advantageous because the Fourier components\nHγ(ω) become constant being independent of ωup to\nthe first order of ω∆tfor a sufficiently short duration ∆ t\n(i.e.,ω∆t≪1). The Fourier components are calculated\nas\nHγ(ω) =/integraldisplay∆t\n0Hpulseeiωtdt=Hpulse\niω/parenleftbig\neiω∆t−1/parenrightbig\n∼Hpulse∆t. (19)\nAs a result, we obtain the relationship χγ(ω)∝∆Mγ(ω).\nIn Figs. 3(a)-(c), we present the calculated microwave\nabsorption spectra, i.e., imaginary part of the dynamical\nmagneticsusceptibilityIm χx, for(a)theskyrmion-lattice\nphase with |Nsk|=2, (b) the skyrmion-lattice phase with\n|Nsk|=1, and (c) the nontopological phase with Nsk=0.6\nEach of the spectra has a single peak, indicating the ex-\nistence of a single resonance mode in each phase. The\nmode in the skyrmion-lattice phase with |Nsk|=1 is a\nrotational mode in which all the skyrmions constitut-\ning the skyrmion lattice rotate uniformly in the coun-\nterclockwise fashion. It is known that skyrmion lattices\nin the Dzyaloshinskii-Moriya magnets without inversion\nsymmetry exhibit two rotation modes with opposite ro-\ntationsenses(i.e., counterclockwiseandclockwise)atdif-\nferent frequencies [57], whereas skyrmion lattices stabi-\nlized by frustrated exchange interactions in centrosym-\nmetric Heisenberg magnets exhibit a counterclockwise\nmode only [82]. The situation in our centrosymmetric\nmetallic magnets with the RKKY interactions resembles\nthe latter case. Figure 3(d) presents the resonance fre-\nquencyωresas a function of external magnetic field Hz.\nWe find that a relation ω=Hz(i.e.,ω=gµBHz//planckover2pi1in di-\nmensionfull units) holds in the phases with |Nsk|=1 and\nNsk=0.\nMicrowave-Induced Dynamics\nNext we simulate the magnetization dynamics un-\nder irradiation with circularly polarized microwave field,\nwhich is given by,\nH(t) =Hωβ(t)(cosωt,sinωt,0),(20)\nwith\nβ(t) = tanh/parenleftbiggt\nτd/parenrightbigg\n. (21)\nHerethetime-dependentprefactor β(t)withτd= 2π/ωis\nintroduced to avoid unexpected artifacts due to impact-\nforce effects through gradually increasing the microwave\namplitude.\nFigures 4(a) and (b) present simulated time profiles of\nthe net magnetization S= (Sx,Sy,Sz) and the skrmion\nnumberNskin a system irradiated with circularly polar-\nized microwave field with (a) ω=0.005 and (b) ω=0.01\nwhenHz= 0.005. We start the simulation with a\nskyrmion-lattice configuration with Nsk=−1 as an ini-\ntial state for both cases, which is the ground state at\nHz= 0.005. We observe a microwave-induced switching\nof the magnetic topology from Nsk=−1 toNsk= 0 in\nFig. 4(a), whereas that from Nsk=−1 toNsk=−2 in\nFig. 4(b). The phase with Nsk=−1 and the phase with\nNsk= 0 in respective cases appear as nonequilibrium\nsteady phases where the net magnetizations show steady\noscillations.\nAlthough the skyrmion number Nskis constant, the\nspatial magnetic configurations in these nonequilibrium\nsteadyphasesunderirradiationwithmicrowavefieldvary\nperiodically in time. Figures 4(c) and (d) present a series\nof snapshots of the temporally varying local magnetiza-\ntionsSifor (c) the nontopological phase Nsk= 0 and (d)the skyrmion-lattice phase with Nsk=−2 under irradi-\nation with the microwave field at selected moments. In\nfact, as long as the skyrmion number is constant, these\nmagnetic configurations are connected to each other by\ncertainrotationaloperations. Forexample, thefourmag-\nnetic configurations with Nsk=−2 shown in Fig. 4(d)\nare all connected to each other via the SO(3)-invariant\nrotational operations by angles θ2andφ2represented by\nEq. (9). Note that the local scalar spin chiralities Ciare\ntime-independent in contrast to the local magnetizations\nSi. In Figs. 4(e) and (f), we present the spatial profiles\nofCiin the microwave-induced Nsk= 0 and Nsk=−2\nphases, respectively, which do not change temporally.\nThen we study nonequilibrium magnetic phases af-\nter sufficient a duration of the microwave irradiation\nby varying the microwave frequency ωand the applied\nstatic magnetic field Hz. Figure 5 presents the obtained\nnonequilibrium phase diagram in plane of ωandHzfor\na system under continuous irradiation with microwave\nfield given by Eq. (20). We trace time-evolutions of\nthe magnetizations for a sufficiently long duration up to\nt=48000 at most. Here we take the microwave ampli-\ntudeHω= 0.01 for the simulations. Note that when the\nmicrowave field is absent (i.e., Hω=0), the system ex-\nhibits ground-state phases shown in Fig. 1(a) where the\nthree magnetic phases with different skyrmion numbers\n|Nsk|=2,|Nsk|=1, and Nsk=0 successively emerge as Hz\nincreases. We select a ground-state magnetic configura-\ntion as an initial state for the time-evolution simulations\nat a given field strength of Hz. We present this ground-\nstatephasediagramalsoin Fig.5forareference. Wealso\nnote that in the static limit of ω=0, all these phases turn\nintothe nontopologicalphasewith Nsk=0 in the presence\nof static in-plane magnetic field of Hω=0.01. Thus, the\nphases on the Hz-axis are all assigned to Nsk=0.\nTo discuss the phase diagram in Fig. 5, we should first\nnote that a circularly polarized microwavefield generates\nan effective static magnetic field ±ωezperpendicular to\nthe polarization plane [77, 98]. The amplitude is equal to\nω(i.e.,/planckover2pi1ω/gµBinrealunits), whilethesignisdetermined\nby the sense of the circular polarization, i.e., positive\n(negative) for the clockwise (counterclockwise) polariza-\ntion. In the present case, the sign is negative because\nthe microwave field circulating in counterclockwise sense\nis applied. Thereby, a static component of the total mag-\nnetic field acting on the system is Htot\nz=Hz−ω. This\nmeans that the application of this microwave field effec-\ntively work to shift the system towards a low-field regime\nin the equilibrium phase diagram. Thus, the nonequilib-\nriumphasewith Nsk=0inthelow-frequencyregimetends\nto change into the skyrmion-lattice phase with |Nsk|=1\nand further into the skyrmion-lattice phase with |Nsk|=2\nasωincreases. Indeed, the skyrmion-lattice phase with\n|Nsk|=2 appears in the right area of the phase diagram\nwhereωis large, whereasthe skyrmion-latticephasewith\n|Nsk|=1 can emerge when ωis intermediate in the areas7\n-0.5 0.5 \n0Magnetization S/g74\n-2 2\n01\n-1 \n4000 8000 12000 0\nt(a) Hz=0.005, /g90=0.005\n(c) Hz=0.005, /g90=0.005\n(d) Hz=0.005, /g90=0.01t=7540 t=7852 t=8168 t=8484\nt=5028 t=5184 t=5340 t=5496Siz -1                  0                  1 -0.5 0.5 \n0Magnetization S/g74\n-2 2\n01\n-1 Skyrmion Number Nsk \n4000 8000 12000 0\ntNsk \nSx\nSy\nSz(b) Hz=0.005, /g90=0.01\nNsk =ʵ2Nsk =0 Skyrmion Number Nsk \n(e)\n(f)Ci-1.5                 0                 1.5 \nFIG. 4: Simulated time profiles of the net magnetization S= (Sx,Sy,Sz) and the skrmion number Nskin the Kondo-lattice\nmodel under irradiation with circularly polarized microwa ve field with (a) ω=0.005 and (b) ω=0.01 when Hz= 0.005. For\nboth cases, we start with a skyrmion-lattice configuration w ithNsk=−1 as an initial state, which is the ground state at\nHz= 0.005. (c),(d) Snapshots of the temporally varying spatial pr ofiles of local magnetizations Siin the microwave-induced\nnonequilibrium steady phases, i.e., (c) the nontopologica l phaseNsk= 0 and (d) the skyrmion-lattice phase with Nsk=−2 at\nselected moments indicated by inverted triangles in (a) and (b), respectively. (e), (f) Spatial profiles of the local sca lar spin\nchiralities Ciin each nonequilibrium steady phase, which do not change tem porally.\nreferred to as “the probabilistic irreversible switching”\nregime and “the temporally random fluctuation” regime\nin the phase diagram.\nImportantly, the ways of the emergence of the |Nsk|=1\nphase are distinct from those of the |Nsk|=2 phase and\ntheNsk=0 phase. The latter two phases emerge in a de-\nterministic and irreversible way under irradiation with\nmicrowave field. In contrast, the |Nsk|=1 phase emerges\nin a probabilistic but irreversible way when Hzis low,\nwhereas either the |Nsk|=1 phase or the |Nsk|=2 phase\nrandomly emerges in a temporally fluctuating manner\nwhenHzis high. These peculiar behaviors might be at-\ntributable to the energy landscapes characterized by the\nenergy minima and the energy barriers. We expect anenergy landscape in Fig. 5(b) for the areas where the\nsystem enters the |Nsk|=2 phase or the Nsk=0 phase in\na deterministic and irreversible manner. On the other\nhand, we expect energy landscapes in Figs. 5(c) and\n(d) forthe probabilistic-irreversible-switchingregimeand\nthe temporally-random-fluctuation regime, respectively,\nin which the energy minima are nearly degenerate. For\nthe probabilistic irreversible regime, the energy barrier\nis rather high [see Fig. 5(c)], with which the system be-\ncomes settled down once it falls into one of the minima,\nresulting in the probabilistic irreversible switching. In\ncontrast, we expect a small energy barrier in the tem-\nporally fluctuating regime where the system fluctuates\nbetween the two minima under irradiation with the mi-8\nHz\n/g90\n0.005 0.01 \n00.005 \n0.01 0.015 Hz=/g90\n|Nsk |=2\n|Nsk |=1\n N sk =0 \nProbabilistic\nTemporally\nfluctuating|Nsk |=2 |Nsk |=1 Nsk =0 Equilibrium\nH/g90=0 Nonequilibrium\nH/g90=0.01Energy \n|Nsk |=2|Nsk |=1\n|Nsk |=1Nsk =0 |Nsk |=1Nsk =0 (a)\n(d) (c) (b)Temporally fluctuatingProbabilistic \n& IrreversibleDeterministic \n& IrreversibleDeterministic\nFIG. 5: Nonequilibrium phase diagram in plane of the mi-\ncrowave frequency ωand the strength of static perpendicular\nmagnetic field Hzunder irradiation with circularly polarized\nmicrowave field of Hω=0.01. The Gilbert-damping coefficient\nis set to be αG=0.05 for the simulations. Equilibrium phase\ndiagram in the absence of microwave field (i.e., Hω=0) is\nalso presented at the left end. (b)-(d) Schematics of the en-\nergy landscapes for different switching behaviors, i.e., (b ) the\ndeterministic irreversible switching, (c) the probabilis tic ir-\nreversible switching, and (d) the temporally random fluctua -\ntion.\ncrowave field.\nInterestingly, we find dynamical formations of topolog-\nical magnetic patterns with half-integer skyrmion num-\nbers during the switching processes. We present snap-\nshots of their spatial configurations in Figs. 6(a) and (b).\nSpecifically, Fig. 6(a) shows the dynamical magnetic pat-\ntern with Nsk=−1/2 emerging during the process of\nmagnetic-topology switching from Nsk=−1 toNsk= 0\nin Fig. 4(a), whereas Fig. 6(b) shows that with Nsk=\n−3/2 emerging during the switching from Nsk=−1 to\nNsk= 0 in Fig. 4(b). Topological magnetic textures hav-\ning a half-integer topological charge are referred to as\nmerons or antimerons, and their crystallized states have\nbeen an issue of intensive research [7, 37, 88, 99–101].\nClarificationsofthe observeddynamicaltopologicalmag-\nnetic patternsin thetransientprocessesareleftforfuture\nstudies.\nDISCUSSION\nIn this section, we discuss our results in the light of an\neffective model introduced in literature [30, 32, 102, 103],Ci-1.5                 0                 1.5 Siz -1                  0                  1 (a)\n(b)\nFIG. 6: (a) Snapshots of the local magnetizations Si(left\npanel) and the scalar spin chiralities Ci(right panel) for\nthe dynamical magnetic pattern with a half-integer skyrmio n\nnumber of Nsk=−1/2 emerging in the transient process in\nFig. 4(a). (b) Those for the dynamical magnetic pattern with\nNsk=−3/2 emerging in the transient process in Fig. 4(b).\nwhich is called the effective bilinear-biquadratic (BBQ)\nmodel. This model is derived from the original Kondo-\nlattice model in Eq. (2) using the perturbation expan-\nsions when the hopping term dominates the Kondo-\ncoupling term. We argue that effective three-body in-\nteractions originating from the third-order perturbation\nprocesses might be of essential importance for the ob-\nserved microwave-induced switching of magnetic topol-\nogy in the centrosymmetric itinerant magnets.\nThe Hamiltonian of the effective BBQ model is given\nby,\nH=HBBQ+HZeeman, (22)\nwith\nHBBQ=3/summationdisplay\nν=1/bracketleftbigg\n−J|SQν|2+K\nN|SQν|4/bracketrightbigg\n.(23)\nThe first term of Eq. (23) represents the effective inter-\nactions originating from the second-order perturbation\nprocesses, which is often referred to as the RKKY in-\nteractions. On the other hand, the second term repre-\nsents parts of the contributions from the fourth-order\nperturbation processes [102, 103]. Note that contribu-\ntions from the odd-order perturbation processes usually\nvanish when magnetic fields are absent because of the\ntime-reversal symmetry, whereas they should appear in\nthe presence of magnetic fields. For the Zeeman-coupling9\n-2 2\n01\n-1 Skyrmion Number Nsk \n-2 2\n01\n-1 Skyrmion Number Nsk (b) Hz=0.2, H/g90=0.2, /g90=0.4 (a) Hz=0.2, H/g90=0.2, /g90=0.2\n-0.5 0.5 \n0Magnetization S/g74\n-0.5 0.5 \n0Magnetization S/g74\n200 400 0\nt200 400 0\ntNsk \nSx\nSy\nSz\nSiz -1                  0                   1 Ci-1.5                 0                 1.5 (c) (d)\nFIG. 7: Simulated time profiles of the net magnetization S= (Sx,Sy,Sz) and the skrmion number Nskin the effective BBQ\nmodel under irradiation with circularly polarized microwa ve field with (a) ω=0.2 and (b) ω=0.4 when Hz= 0.2. For both\ncases, we start with a skyrmion lattice with Nsk=−1 as an initial state, which is the ground state at Hz= 0.2. The skyrmion\nnumber remains constant to be Nsk=−1 even after a sufficient duration, indicating that the switch ing of magnetic topology\ndoes not occur in contrast to the case of the Kondo-lattice mo del. (c),(d) Snapshots of the local magnetizations Si(left panels)\nand the scalar spin chiralities Ci(right panels) in the microwave-induced nonequilibrium st eady phases at selected moments.\nThe magnetic configuration in (c) corresponds to the bimeron crystal with Nsk=−1, whereas that in (d) corresponds to the\nantiskyrmion crystal with Nsk=−1\ntermHZeeman, we consider both the static perpendicular\nmagnetic field Hex= (0,0,Hz) and the circularly po-\nlarized microwave field with amplitude of Hω. When\nthe microwave field is absent (i.e., Hω= 0), this effec-\ntive BBQ model is known to host two different skyrmion-\nlattice phases with |Nsk|= 1 and |Nsk|= 2 as well as the\nnontopological phase with Nsk= 0 in the ground state\ndepending on the strength of external magnetic field Hz.\nThe coupling constants JandKin Eq. (23) de-\npend on the electronic structures of conduction elec-\ntrons such as Fermi surfaces and density of states, which\nare governed by the lattice structure, the transfer inte-\ngrals, and the electron filling. The values of JandK\ncan be evaluated from the original Kondo-lattice model\nby the perturbation-expansion calculations in principle.\nThrough the second-order perturbation expansions, we\nobtain the following formula for the coupling constant J,\nJ=J2\nK/summationdisplay\nq∈BZχ0(q)eiq·r1. (24)Hereχ0(q) is the bare susceptibility of conduction elec-\ntrons, and r1denotes the Bravis vectors of the triangu-\nlar lattice. For the Kondo-lattice model in Eq. (2) and\nthe parameters used in the present work (i.e., t1=−1,\nt3= 0.85,JK=−0.5, andµ=−3.5), we evaluate\nthe value as J∼0.0035t1, which is much smaller than\nunityandthussupportsthe validityoftheperturbational\ntreatment. We canalsoevaluate the couplingconstant K\nmicroscopically from the Kondo-lattice model. However,\nto get a general insight into the effective BBQ model,\nwe treat the model rather phenomenologically by regard-\ningJ,K,HzandHωas parameters and by setting the\nconstant Jas energy units (i.e., J= 1) in the following\ndiscussion.\nNow we examine the microwave-induced magnetiza-\ntion dynamics in this effective BBQ model by deriving\na time-evolution equation for the Fourier components of\nmagnetization Sq. The equation is given by,10\ndSq\ndt=i\n/planckover2pi1[HBBQ+HZeeman,Sq]−\n=−3/summationdisplay\nν=1/parenleftbigg\n−J+K\nN|SQν|2/parenrightbigg\n(SQν×S−Qν+q−SQν+q×S−Qν)+Sq×[Hext+H′(t)]. (25)\nBy numerically solving the derived equation, we examine\ntwocaseswith ω= 0.005andω= 0.01,forwhichwehave\nrespectively observed the switching of magnetic topology\nin the original Kondo-lattice model, i.e., the switching\nfromNsk=−1 toNsk= 0 [Fig. 4(a)] and the switching\nfromNsk=−1 toNsk=−2 [Fig. 4(b)]. We set the\nparameter values as J= 1,K= 0.5,Hz= 0.2 andHω=\n0.4. Surprisingly, the switching of magnetic topology is\nnot observed for both cases [see Figs. 7(a) and (b)].\nThe failure of the effective BBQ model in reproduc-\ntion of the topology switching may be ascribed to in-\ngredients which are incorporated in the original Kondo-\nlattice model but missing in the effective BBQ model.\nOne missing ingredient is effective interactions originat-\ning from the third-order perturbation processes. The ef-\nfective BBQ model contains only terms bilinear and bi-\nquadratic with respect to SQνandS−Qν, which come\nfromthesecond-orderandfourth-orderperturbationpro-\ncesses, respectively. We infer that the other-order terms\nmightcontributetothetopologyswitching. Amongthose\nterms, the lowest-order terms, i.e., the third-order terms\nare most likely, which are three-body interactions with\nrespect to SQνandS−Qνand thus have no time-reversal\nsymmetry. Therefore, the third-order terms are forbid-\nden and should vanish in the absence of magnetic field.\nHowever, they are allowed to appear once a magnetic\nfield is applied, although they are not incorporated in\nthe effective BBQ model even under a magnetic field.\nContributions of the third-order terms to the magnetiza-\ntion dynamics are described by the following Heisenberg\nequation of motion for the Fourier component Sq,\ndSq\ndt=i\n/planckover2pi1/bracketleftig\nF(3),Sq/bracketrightig\n.(26)\nThere are five kinds of third-order terms F(3)as de-\nrived in Ref. [32], and they turn out to contribute to\nthe time evolution of Sqas shown in the appendix. An-\nother missing ingredient is contributions from momenta\napart from Qν. The effective BBQ model contains only\nthe Fourier components of magnetization SQνwithQν\nbeing the modulation wavevectors. However, the bilinear\nand biquadratic interactions originally have components\nof other wavevectors. These neglected components of the\ninteractions may play a key role in the topology switch-\ning. The issue requires further investigations and is left\nfor future studies.\nIn the meanwhile, the counterclockwise circularly po-larized microwave field considered in the present study\ngenerates an effective static magnetic field perpendicu-\nlar to the polarization plane −ωez[77, 98]. Therefore,\na perpendicular component of the total magnetic field is\nHtot\nz=Hz−ω. In the case of Fig. 7(a), the effective\nstatic component Htot\nzvanishes (i.e., Htot\nz= 0) because\nwe setHz=ω= 0.2. Under this circumstance, the\nbimeron crystal with |Nsk|= 1 [100, 101] appears as a\nnonequilibrium steady state after a sufficient duration of\nthe microwaveirradiation[Fig. 7(c)]. Note that the effec-\ntive BBQ model in Eq. (23) exhibits the skyrmion-lattice\nphase with |Nsk|= 1 in the equilibrium case when both\nthe static and microwave magnetic fields are absent (i.e.,\nHz=Hω=0). On the other hand, the effective static com-\nponentHtot\nzis negative (i.e., Htot\nz=−0.2) because we\nsetHz= 0.2 andω= 0.4. We observe the antiskyrmion\nlattice with |Nsk|= 1 [Fig. 7(d)] after a sufficient du-\nration of the microwave irradiation, in which the core\nmagnetizations point upwards.\nCONCLUSION\nIn summary, we have theoretically proposed possible\nmicrowave-induced switching of magnetic topology in\ncentrosymmetric itinerant magnets by numerically ana-\nlyzing the magnetization dynamics in the Kondo-lattice\nmodel on a triangular lattice using a combined method\nof the micromagnetic simulation and the kernel polyno-\nmial expansion technique. We have demonstrated that\nthe intense excitation of spin-wave mode with circularly\npolarized microwave field can switch the skyrmion lat-\ntice with |Nsk|=1 into that with |Nsk|=2 or the non-\ntopological magnetic state with Nsk=0 depending on the\nmicrowave frequency. During these switching processes,\ntransient topological magnetic patterns with half-integer\nskyrmion numbers of |Nsk|=1/2 and |Nsk|=3/2 were ob-\nserved. These fractionalizations of magnetic topological\ncharges in the dynamical regime are an issue of inter-\nest, which should be clarified in future studies. We have\nfound several different switching behaviors under con-\ntinuous microwave irradiation, that is, deterministic ir-\nreversible switching, probabilistic irreversible switching,\nand temporally randomfluctuation depending on the mi-\ncrowave frequency and the strength of external magnetic\nfield, which is attributable to difference of the energy\nlandscape in the dynamical regime. We have also ex-11\namined the effective BBQ model derived from the per-\nturbation expansions of the Kondo-lattice model and\nhave found that this model fails to reproduce the dy-\nnamical switching of magnetic topology. The failure\nof the effective model containing only the even-order\nterms conversely indicates that contributions from the\nodd-order perturbation processes, which break the time-\nreversal symmetry, are important to understand the ob-\nserved magnetic topology switching. Here we emphasize\nthat various unfrozen degrees of freedom inherent in cen-\ntrosymmetric magnets are sources of rich magnetic tex-\ntures and their controllability with external parameters.\nOur work will open a new research field to manipulate\nmagnetic topologies in centrosymmetric magnets.\nAPPENDIX\nWhen the electron hoppings dominate the Kondo ex-\nchange coupling in the Kondo-lattice model, we can de-riveeffectiveinteractionsamongthe localmagnetizations\nfrom this model by using the perturbation expansion\ntechnique. In the main text, we have argued that the ef-\nfective interactions originating from the third-order per-\nturbation processescan contribute to the time-evolutions\nof the Fourier components Sq. In Ref. [32], the effective\ninteractions from the third-order perturbation processes\nhave been derived, which have turned out to be three-\nbody interactions among the Fourier components SQν\nandS−Qνwith three different Qνvectors. The third-\norder contributions are given by,\nF(3)=F(3)\n1+F(3)\n2+F(3)\n3+F(3)\n4+F(3)\n5,(27)\nwith\nF(3)\n1=−2J3\nK√\nN/summationdisplay\nν(C1−C2)/bracketleftig\nSz\nQν/parenleftig\nSx\n0Sx\n−Qν+Sy\n0Sy\n−Qν/parenrightig\n+h.c./bracketrightig\n, (28)\nF(3)\n2=−2J3\nK√\nN/summationdisplay\nν(C3−C4)/bracketleftig\nSz\n0/parenleftig\nSx\nQνSx\n−Qν+Sy\nQνSy\n−Qν/parenrightig\n+h.c./bracketrightig\n, (29)\nF(3)\n3=−2J3\nK√\nN/summationdisplay\nν(C5−C6)Sz\n0Sz\nQνSz\n−Qν, (30)\nF(3)\n4=−2J3\nK√\nN(D1−D2)/bracketleftig\nSz\nQ1/parenleftig\nSx\nQ2Sx\nQ3+Sy\nQ2Sy\nQ3/parenrightig\n+h.c./bracketrightig\n+(Q1→Q2,Q2→Q3,Q3→Q1)\n+(Q1→Q3,Q2→Q1,Q3→Q2), (31)\nF(3)\n5=−2J3\nK√\nN(D3−D4)/bracketleftbig\nSz\nQ1Sz\nQ2Sz\n−Q3+h.c./bracketrightbig\n. (32)\nHere the coefficients CνandDν(ν= 1,2,3,4) are calculated by the convolution of Green’s functions. Detailed\nformulas of these coefficients are given in Ref. [32].\nEquations of the time evolution of Sqdue to the third-order contributions are given by,\ndSq\ndt=i\n/planckover2pi1/bracketleftig\nF(3)\n1,Sq/bracketrightig\n−\n= 2J3\nK\nN/summationdisplay\nν(C1−C2)/bracketleftig\n(Sq+Qν×ez)/parenleftig\nSx\n0Sx\n−Qν+Sy\n0Sy\n−Qν/parenrightig\n+\n−Sz\nQν/parenleftig\nSy\n0Sz\nq−Qν+Sz\nqSy\n−Qν/parenrightig\nSz\nQν/parenleftbig\nSx\n0Sz\nq−Qν+Sz\nqSx\n−Qν/parenrightbig\n−Sz\nQν/parenleftig\nSx\n0Sy\nq−Qν+Sy\nqSx\n−Qν/parenrightig\n+Sz\nQν/parenleftig\nSy\n0Sx\nq−Qν+Sx\nqSy\n−Qν/parenrightig\n\n\n+(Qν→ −Qν), (33)12\ndSq\ndt=i\n/planckover2pi1/bracketleftig\nF(3)\n2,Sq/bracketrightig\n−\n= 2J3\nK\nN/summationdisplay\nν(C3−C4)/bracketleftig\n(Sq×ez)/parenleftig\nSx\nQνSx\n−Qν+Sy\nQνSy\n−Qν/parenrightig\n+\n−Sz\n0/parenleftig\nSy\nQνSz\nq−Qν+Sz\nq+QνSy\n−Qν/parenrightig\nSz\n0/parenleftbig\nSx\nQνSz\nq−Qν+Sz\nq+QνSx\n−Qν/parenrightbig\n−Sz\n0/parenleftig\nSx\nQνSy\nq−Qν+Sy\nq+QνSx\n−Qν/parenrightig\n+Sz\n0/parenleftig\nSy\nQνSx\nq−Qν+Sx\nq+QνSy\n−Qν/parenrightig\n\n\n+(Qν→ −Qν), (34)\ndSq\ndt=i\n/planckover2pi1/bracketleftig\nF(3)\n3,Sq/bracketrightig\n−\n= 2J3\nK\nN/summationdisplay\nν(C5−C6)\nSx\nqSz\nQνSz\n−Qν+Sz\n0Sx\nq+QνSz\n−Qν+Sz\n0Sz\nQνSx\nq−Qν\n−Sy\nqSz\nQνSz\n−Qν−Sz\n0Sy\nq+QνSz\n−Qν−Sz\n0Sz\nQνSy\nq−Qν\n0\n, (35)\ndSq\ndt=i\n/planckover2pi1/bracketleftig\nF(3)\n4,Sq/bracketrightig\n−\n= 2J3\nK\nN(D1−D2)/bracketleftig\n(Sq+Q1×ez)/parenleftig\nSx\nQ2Sx\nQ3+Sy\nQ2Sy\nQ3/parenrightig\n+\n−Sz\nQ1/parenleftig\nSy\nQ2Sz\nq+Q3+Sz\nq+Q2Sy\nQ3/parenrightig\nSz\nQ1/parenleftbig\nSx\nQ2Sz\nq+Q3+Sz\nq+Q2Sx\nQ3/parenrightbig\n−Sz\nQ1/parenleftig\nSx\nQ2Sy\nq+Q3+Sy\nq+Q2Sx\nQ3/parenrightig\n+Sz\nQ1/parenleftig\nSy\nQ2Sx\nq+Q3+Sx\nq+Q2Sy\nQ3/parenrightig\n\n\n+(Q1→Q2,Q2→Q3,Q3→Q1)\n+(Q1→Q3,Q2→Q1,Q3→Q2)\n+(Q1→ −Q1,Q2→ −Q2,Q3→ −Q3)\n+(Q1→ −Q2,Q2→ −Q3,Q3→ −Q1)\n+(Q1→ −Q3,Q2→ −Q1,Q3→ −Q2), (36)\ndSq\ndt=i\n/planckover2pi1/bracketleftig\nF(3)\n5,Sq/bracketrightig\n−\n= 2J3\nK\nN(D3−D4)\nSx\nq+Q1Sz\nQ2Sz\n−Q3+Sz\nQ1Sx\nq+Q2Sz\n−Q3+Sz\nQ1Sz\nQ2Sx\nq−Q3\n−Sy\nq+Q1Sz\nQ2Sz\n−Q3−Sz\nQ1Sy\nq+Q2Sz\n−Q3−Sz\nQ1Sz\nQ2Sy\nq−Q3\n0\n\n+(Q1→Q2,Q2→Q3,Q3→Q1)\n+(Q1→Q3,Q2→Q1,Q3→Q2)\n+(Q1→ −Q1,Q2→ −Q2,Q3→ −Q3)\n+(Q1→ −Q2,Q2→ −Q3,Q3→ −Q1)\n+(Q1→ −Q3,Q2→ −Q1,Q3→ −Q2). (37)\nACKNOWLEDGEMENT\nThis work is supported by Japan Society for the Pro-\nmotion of Science KAKENHI (Grant No. 16H06345, No.\n19H00864, No. 19K21858, and No. 20H00337), CREST,\nthe Japan Science and Technology Agency (Grant No.JPMJCR20T1), a Research Grant in the Natural Sci-\nences from the Mitsubishi Foundation, and a Waseda\nUniversity Grant for Special Research Projects (Project\nNo. 2020C-269 and No. 2021C-566). A part of the nu-\nmericalsimulations wasperformed at the Supercomputer\nCenter of the Institute for Solid State Physics in the Uni-13\nversity of Tokyo.\n[1] S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A.\nRosch, A. Neubauer, R. Georgii, and P. B¨ oni, Science\n323, 915 (2009).\n[2] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H.\nHan, Y. Matsui, N. Nagaosa, and Y. Tokura, Nature\n465, 901 (2010).\n[3] S. Heinze, K. von Bergmann, M. Menzel, J. Brede,\nA. Kubetzka, R. Wiesendanger, G. Bihlmayer, and S.\nBlugel, Nat. Phys. 7, 713 (2011).\n[4] I. K´ ezsmarki, S. Bordacs, P. Milde, E. Neuber, L. M.\nEng, J. S. White, H. M. Ronnow, C. D. Dewhurst,\nM. Mochizuki, K. Yanai, H. Nakamura, D. Ehlers, V.\nTsurkan, and A. Loidl, Nat. Mater. 14, 1116 (2015).\n[5] X. Z. Yu, Y. Tokunaga, Y. Kaneko, W. Z. Zhang, K.\nKimoto, Y. Matsui, Y. Taguchi, and Y. Tokura, Nat.\nCommun. 5, 3198 (2014).\n[6] A. K. Nayak, V. Kumar, T. Ma, P. Werner, E. Pippel,\nR. Sahoo, F. Damay, U. K. R¨ osler, C. Felser, and S. S.\nParkin, Nature 548, 561 (2017).\n[7] X. Z. Yu, W. Koshibae, Y. Tokunaga, K. Shibata, Y.\nTaguchi, N. Nagaosa, and Y. Tokura, Nature 564, 95\n(2018).\n[8] P. Milde, D. K¨ ohler, J. Seidel, L. M. Eng, A. Bauer, A.\nChacon, J. Kindervater, S. M¨ uhlbauer, C. Pfleiderer, S.\nBuhrandt, C. Sch¨ utte, and A. Rosch, Science 340, 1076\n(2013).\n[9] C. Sch¨ utte and A. Rosch, Phys. Rev. B 90, 174432\n(2014).\n[10] N. Kanazawa, J.-H. Kim, D. S. Inosov, J. S. White, N.\nEgetenmeyer, J. L. Gavilano, S. Ishiwata, Y. Onose, T.\nArima, B. Keimer, and Y. Tokura, Phys. Rev. B 86,\n134425 (2012).\n[11] Y. Fujishiro, N. Kanazawa, T. Nakajima, X. Z. Yu, K.\nOhishi, Y. Kawamura, K. Kakurai, T. Arima, H. Mita-\nmura, A. Miyake, K. Akiba, M. Tokunaga, A. Matsuo,\nK. Kindo, T. Koretsune, R. Arita, and Y. Tokura, Nat.\nCommun. 10, 1059 (2019).\n[12] S. Ishiwata, T. Nakajima, J.-H. Kim, D. S. Inosov, N.\nKanazawa, J. S. White, J. L. Gavilano, R. Georgii, K.\nM. Seemann, G. Brandl, P. Manuel, D. D. Khalyavin,\nS. Seki, Y. Tokunaga, M. Kinoshita, Y. W. Long, Y.\nKaneko, Y. Taguchi, T. Arima, B. Keimer, and Y.\nTokura, Phys. Rev. B 101, 134406 (2020).\n[13] Y. Liu, R. K. Lake, J. Zang, Phys. Rev. B 98, 174437\n(2018).\n[14] S. Seki and M. Mochizuki, Skyrmions in Magnetic Ma-\nterials(Springer, Berlin, 2016).\n[15] N. Nagaosa and Y. Tokura, Nat. Nanotech. 8, 899\n(2013).\n[16] B. G¨ obel, I. Mertig, and O. A. Tretiakov, Phys. Rep.\n895, 1 (2021).\n[17] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotech. 8, 152\n(2013).\n[18] G. Finocchio, F. Buttner, R. Tomasello, M. Carpen-\ntieri, and M. Kl¨ aui, J. Phys. D: Appl. Phys. 49, 423001\n(2016).\n[19] A. Fert, N. Reyren, and V. Cros, Nat. Rev. Mat. 2,\n17031 (2017).[20] K. Everschor-Sitte, J. Masell, R. M. Reeve, and M.\nKl¨ aui, J. Appl. Phys. 124, 240901 (2018).\n[21] Y. Tokura and N. Kanazawa, Chem. Rev. 121, 2857\n(2021).\n[22] I. Dzyaloshinsky, J. Phys. Chem. Solids 4, 241 (1958).\n[23] T. Moriya, Phys. Rev. 120, 91 (1960).\n[24] T. Moriya, Phys. Rev. Lett. 4, 228 (1960).\n[25] A. N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP\n68, 101 (1989).\n[26] A. Bogdanov and A. Hubert, J. Magn. Magn. Mater.\n138, 255 (1994).\n[27] K. Tanaka, R. Sugawara, and M. Mochizuki, Phys. Rev.\nMat.4, 034404 (2020).\n[28] S. Hayami and Y. Motome, arXiv:2103.10647.\n[29] S. Hayami, T. Misawa, Y. Yamaji, and Y. Motome,\nPhys. Rev. B 89, 085124 (2014).\n[30] R. Ozawa, S. Hayami, K. Barros, G.-W. Chern, Y. Mo-\ntome, and C. D. Batista, J. Phys. Soc. Jpn. 85103703\n(2016).\n[31] R. Ozawa, S. Hayami, and Y. Motome, Phys. Rev. Lett.\n118, 147205 (2017).\n[32] S. Hayami, R. Ozawa, and Y. Motome, Phys. Rev. B\n95, 224424 (2017).\n[33] B. G¨ obel, A. Mook, J. Henk, and I. Mertig, Phys. Rev.\nB96, 060406(R) (2017).\n[34] Z. Wang, Y. Su, S.-Z. Lin, and C. D. Batista, Phys.\nRev. Lett. 124, 207201 (2020).\n[35] S. Hayami and R. Yambe, J. Phys. Soc. Jpn. 89, 103702\n(2020).\n[36] S. Hayami and Y. Motome, Phys. Rev. B 103, 024439\n(2021).\n[37] S. Hayami and Y. Motome, Phys. Rev. B 103, 054422\n(2021).\n[38] K. Nakazawa and H. Kohno, Phys. Rev. B 99, 174425\n(2019).\n[39] S. Hayami, T. Okubo, and Y. Motome,\narXiv:2005.03168.\n[40] S. Hayami and Y. Motome, Phys. Rev. Lett. 121,\n137202 (2018).\n[41] K. N. Okada, Y. Kato, and Y. Motome, Phys. Rev. B\n98, 224406 (2018).\n[42] S. Okumura, S. Hayami, Y. Kato, and Y. Motome,\nPhys. Rev. B 101, 144416 (2020).\n[43] K. Shimizu, S. Okumura, Y. Kato, and Y. Motome,\nPhys. Rev. B 103, 054427 (2021).\n[44] M. A. Ruderman and C. Kittel, Phys. Rev. 96, 99\n(1954).\n[45] T. Kasuya, Prog. Theor. Phys. 16, 45 (1956).\n[46] K. Yosida, Phys. Rev. 106, 893 (1957).\n[47] K. Shimizu, S. Okumura, Y. Kato, and Y. Motome,\nPhys. Rev. B 103, 184421 (2021).\n[48] T. Kurumaji, T. Nakajima, M. Hirschberger, A.\nKikkawa, Y. Yamasaki, H. Sagayama, H. Nakao, Y.\nTaguchi, T. Arima, and Y. Tokura, Science 365, 914\n(2019).\n[49] M. Hirschberger, L. Spitz, T. Nomoto, T. Kurumaji, S.\nGao, J. Masell, T. Nakajima, A. Kikkawa, Y. Yamasaki,\nH. Sagayama, H. Nakao, Y. Taguchi, R. Arita, T.-h.\nArima, and Y. Tokura, Phys. Rev. Lett. 125, 076602\n(2020).\n[50] M. Hirschberger, T. Nakajima, M. Kriener, T. Kuru-\nmaji, L. Spitz, S. Gao, A. Kikkawa, Y. Yamasaki, H.\nSagayama, H. Nakao, S. Ohira-Kawamura, Y. Taguchi,\nT.-h. Arima, and Y. Tokura, Phys. Rev. B 101,14\n220401(R) (2020).\n[51] T. Nomoto, T. Koretsune, and R. Arita, Phys. Rev.\nLett.125, 117204 (2020).\n[52] M. Hirschberger, T. Nakajima, S. Gao, L. Peng, A.\nKikkawa, T. Kurumaji, M. Kriener, Y. Yamasaki,\nH. Sagayama, H. Nakao, K. Ohishi, K. Kakurai, Y.\nTaguchi, X. Yu, T.-h. Arima, and Y. Tokura, Nat. Com-\nmun.10, 5831 (2019).\n[53] M. Hirschberger, S. Hayami, and Y. Tokura, New J.\nPhys.23023039 (2021).\n[54] N. D. Khanh, T. Nakajima, X. Yu, S. Gao, K. Shibata,\nM. Hirschberger, Y. Yamasaki, H. Sagayama, H. Nakao,\nL. Peng, K. Nakajima, R. Takagi, T. Arima, Y. Tokura,\nand S. Seki, Nat. Nanotech. 15, 444 (2020).\n[55] Y. Yasui, C. J. Butler, N. D. Khanh, S. Hayami, T.\nNomoto, T. Hanaguri, Y. Motome, R. Arita, T.-H.\nArima, Y. Tokura, and S. Seki, Nat. Commun. 11, 5925\n(2020).\n[56] S.-Z. Lin, C. Reichhardt, C. D. Batista, and A. Saxena,\nPhys. Rev. Lett. 113, 207202 (2013).\n[57] M. Mochizuki, Phys. Rev. Lett. 108, 017601 (2012).\n[58] T. Schwarze, J. Waizner, M. Garst, A. Bauer,\nI. Stasinopoulos, H. Berger, C. Pfleiderer, and D.\nGrundler, Nat. Mater. 14, 478 (2015).\n[59] S.-Z. Lin, C. D. Batista, and A. Saxena, Phys. Rev. B\n89, 024415 (2014).\n[60] B. F. McKeever, D. R. Rodrigues, D. Pinna, Ar.\nAbanov, J. Sinova, and K. Everschor-Sitte, Phys. Rev.\nB99, 054430 (2019).\n[61] S.-Z. Lin, J.-X. Zhu, and A. Saxena, Phys. Rev. B 99,\n140408(R) (2019).\n[62] S. Seki, M. Garst, J. Waizner, R. Takagi, N. D. Khanh,\nY. Okamura, K. Kondou, F. Kagawa, Y. Otani, and Y.\nTokura, Nat. Commun. 11, 256 (2020).\n[63] M. Mochizuki and S. Seki, J. Phys.: Condens. Matter\n27, 503001 (2015).\n[64] M. Garst, J. Waizner, and D. Grundler, J. Phys. D:\nAppl. Phys. 50, 293002 (2017).\n[65] M. Lonsky and A. Hoffmann, APL Materials, 8, 100903\n(2020).\n[66] M. Mochizuki and S. Seki, Phys. Rev. B 87, 134403\n(2013).\n[67] Y. Okamura, F. Kagawa, M. Mochizuki, M. Kubota,\nS. Seki, S. Ishiwata, M. Kawasaki, Y. Onose, and Y.\nTokura, Nat. Commun. 4, 2391 (2013).\n[68] F. Ma, Y. Zhou, H. B. Braun, and W. S. Lew, Nano\nLett.15, 4029 (2015).\n[69] X. Wang, Y. Nie, Q. Xia, and G. Guo, J. Appl. Phys.\n128, 063901 (2020).\n[70] W. Wang, M. Beg, B. Zhang, W. Kuch, and H. Fangohr,\nPhys. Rev. B 92, 020403(R) (2015).\n[71] M. Ikka, A. Takeuchi, and M. Mochizuki, Phys. Rev. B\n98, 184428 (2018).\n[72] A. Takeuchi, and M. Mochizuki, Appl. Phys. Lett. 113,\n072404 (2018).\n[73] C. Song, C. Jin, Y. Ma, J. Wang, H. Xia, J. Wang, and\nQ. Liu, J. Phys. D: Appl. Phys. 52, 435001 (2019).\n[74] H. Y. Yuan, X. S. Wang, M.-H. Yung, and X. R. Wang,\nPhys. Rev. B 99, 014428 (2019).[75] T. Koide, A. Takeuchi, and M. Mochizuki, Phys. Rev.\nB100, 014408 (2019).\n[76] A.Takeuchi, S.Mizushima, andM.Mochizuki, Sci.Rep.\n9, 9528 (2019).\n[77] M.MiyakeandM.Mochizuki, Phys.Rev.B 101, 094419\n(2020).\n[78] X. Xing, Y. Zhou, and H.B. Braun, Phys. Rev. Appl.\n13, 034051 (2020).\n[79] N. D. Ser, L. Heinen, and A. Rosch, arXiv:2012.11548\n(To appear in SciPost).\n[80] T. Okubo, S. Chung, and H. Kawamura, Phys. Rev.\nLett.108, 017206 (2012).\n[81] Y. Kamiya and C. D. Batista, Phys. Rev. X 4, 011023\n(2014).\n[82] A. O. Leonov and M. Mostovoy, Nat. Commun. 6, 8275\n(2015).\n[83] S.-Z. Lin and S. Hayami, Phys. Rev. B 93, 064430\n(2016).\n[84] S. Hayami, S.-Z. Lin, and C. D. Batista, Phys. Rev. B\n93, 184413 (2016).\n[85] S. Hayami, S.-Z. Lin, Y. Kamiya, and C. D. Batista,\nPhys. Rev. B 94, 174420 (2016).\n[86] S.-Z. Lin and C. D. Batista, Phys. Rev. Lett. 120,\n077202 (2018).\n[87] V. Lohani, C. Hickey, J. Masell, and A. Rosch, Phys.\nRev. X9, 041063 (2019).\n[88] Z. Wang, Y. Su, S. Z. Lin, and C. D. Batista, Phys.\nRev. B103, 104408 (2021).\n[89] Y. Motome and N. Furukawa, J. Phys. Soc. Jpn. 68,\n3853 (1999).\n[90] A. Weiße, G. Wellein, A. Alvermann, and H. Fehske,\nRev. Mod. Phys. 78, 275 (2006).\n[91] K. Barros and Y. Kato, Phys.Rev. B 88, 235101 (2013).\n[92] Z. Wang, K. Barros, G.-W. Chern, D. L. Maslov, and\nC. D. Batista, Phys. Rev. Lett. 117, 206601 (2016).\n[93] R. Ozawa, S. Hayami, K. Barros, and Y. Motome, Phys.\nRev. B96, 094417 (2017).\n[94] G.-W. Chern, K. Barros, Z. Wang, H. Suwa, and C. D.\nBatista, Phys. Rev. B 97, 035120 (2018).\n[95] Z. Wang, G.-W. Chern, C. D. Batista, and K. Barros,\nJ. Chem. Phys. 148, 094107 (2018).\n[96] J. M. Tang and Y. Saad, Numer. Linear Algebra Appl.\n19, 485 (2012).\n[97] J.-V. Kim, and J. Mulkers, IOP SciNotes 1, 025211\n(2020).\n[98] M. Mochizuki, K. Ihara, J. Ohe, and A. Takeuchi, Appl.\nPhys. Lett. 112, 122401 (2018).\n[99] S.-Z. Lin, A. Saxena, and C. D. Batista, Phys. Rev. B\n91, 224407 (2015).\n[100] Y. A. Kharkov, O. P. Sushkov, and M. Mostovoy, Phys.\nRev. Lett. 119, 207201 (2017).\n[101] B. G¨ obel, A. Mook, J. Henk, I. Mertig, and O. A. Tre-\ntiakov, Phys. Rev. B 99, 060407(R) (2019).\n[102] Y. Akagi, M. Udagawa, and Y. Motome, Phys. Rev.\nLett.108, 096401 (2012).\n[103] S. Hayami and Y. Motome, Phys. Rev. B 90, 060402(R)\n(2014)."    },    {        "title": "2204.12398v1.Slow_spin_dynamics_and_quantum_tunneling_of_magnetization_in_the_dipolar_antiferromagnet_DyScO__3_.pdf",        "content": "Slow spin dynamics and quantum tunneling of magnetization in the dipolar antiferromagnet\nDyScO3\nN. D. Andriushin,1S. E. Nikitin,2G. Ehlers,3and A. Podlesnyak4\n1Institut für Festkörper- und Materialphysik, Technische Universität Dresden, D-01069 Dresden, Germany\n2Paul Scherrer Institute (PSI), CH-5232 Villigen, Switzerland\n3Neutron Technologies Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA\n4Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA\nWe present a comprehensive study of static and dynamic magnetic properties in the Ising-like dipolar\nantiferromagnet (AFM) DyScO3by means of DC and AC magnetization measurements supported by classical\nMonte-Carlo calculations. Our AC-susceptibility data show that the magnetic dynamics exhibit a clear\ncrossover from an Arrhenius-like regime to quantum tunneling of magnetization (QTM) at T\u0003=10K. Below\nTN=3.2K DyScO3orders in an antiferromagnetic GxAy -type magnetic structure and the magnetization\ndynamics slow down to the minute timescale. The low-temperature magnetization curves exhibit complex\nhysteretic behavior, which depends strongly on the magnetic field sweep rate. We demonstrate that the low-\nfield anomalies on the magnetization curve are related to the metamagnetic transition, while the hysteresis\nat higher fields is induced by a strong magnetocaloric effect. Our theoretical calculations, which take into\naccount dipolar interaction between Dy3+moments, reproduce essential features of the magnetic behavior\nof DyScO3. We demonstrate that DyScO3represents a rare example of inorganic compound, which exhibits\nQTM at a single-ion level and magnetic order due to classical dipolar interaction.\nI. INTRODUCTION\nTimescale of spin dynamics – the time required to flip a\nsingle spin – in conventional magnetic materials is of the\norder of femto- to picosecond [1–3]. Anomalous slowing\ndown by multiple orders of magnitude down to the mil-\nlisecond range is known take place in some single-molecule\nmagnets and is induced by strong uniaxial anisotropy [4–\n6]. In that case, the strong crystalline electric field (CEF)\nsplits the ground-state multiplet Jof a magnetic ion and\ncreates a doublet ground state, which consists of two states\nwith maximal projection of angular momentum, pointing\nin the opposite directions, j 0\n\u0006i=j\u0006Ji. Thus, the direct\ntransition between j 0\n+iandj 0\n\u0000istates requires a change\nof the total momentum \u0001Jby more than one, and thus is\nforbidden by selection rules of many conventional single-\n(quasi)particle emission /absorption processes. Therefore,\nthe matrix element for this transition is low, see Fig. 1.\nIn this case there are two possible ways to change the spin\nmomentum: (i) via an activation process to the excited dou-\nblet with wavefunctions j 1\n\u0006i=j\u0006J\u00071i; or (ii) as a direct\ntransition between j 0\n+iandj 0\n\u0000ivia the quantum tunnel-\ning of magnetization (QTM) process [4]. The first mech-\nanism dominates in the temperature range T¦\u0001CEF=kB\n(\u0001CEFis the energy gap to the excited doublet and kBis the\nBoltzmann constant), but it becomes ineffective at lower\ntemperature where the QTM dominates the magnetic relax-\nation.\nQTM is a well-known process in single-molecule magnets,\nhowever to the best of our knowledge, among inorganic crys-\ntals there are only few well-documented examples including\nDy2Ti2O7[7–12]and Ca3Co2O6[13,14]. In both cases,\nthe strong CEF produces large uniaxial anisotropy, which\nfreezes magnetic moments below a crossover temperature\nT\u0003\u001913andT\u0003\u00199K, for Dy2Ti2O7and Ca3Co2O6respec-\ntively. However, in addition to the QTM both materials alsoshow complex collective magnetic behavior, classical spin-\nice physics in case of Dy2Ti2O7and a frustrated spin-chain\nbehavior in Ca3Co2O6, which obscure the QTM physics. An-\nother prominent example is a dipolar ferromagnet LiHoF4\nand its diluted modifications LiHoxY1\u0000xF4[15–18], where\nelectronic and nuclear spins of Ho3+ions are coupled be-\ncause of large hyperfine interaction, which produces complex\nslow dynamics at low temperatures.\nIn this work we focus on a classical Ising-like dipolar\nAFM DyScO3, whose magnetic properties were studied in\ndetails previously [19–22]. It exhibits non-collinear AFM\nordering below TN=3.2K with the propagation wavevector\nk= (001). Magnetization and neutron diffraction mea-\nsurements show that Dy exhibits strong uniaxial anisotropy\nat low temperature and the easy-axis lies in the ab-plane,\nwith a\u000628\u000eangle to the [010]direction. Inelastic neutron\nscattering (INS) measurements show that the ground state\ndoublet is well-isolated from the first excited doublet located\nat 290 K. Point-charge model calculations supported by mag-\nnetization measurements show that the wavefunction of the\nground state doublet consists of almost pure j\u000615=2istates,\nmaking DyScO3a prospective material to search for the QTM\neffect.\nIn this work we performed a comprehensive study of the\nlow-temperature magnetic behavior in DyScO3by means\nof AC and DC magnetization measurements. We observed\na clear peak in imaginary part of dynamical spin suscepti-\nbility\u001f00. It exhibits a crossover between an Arrhenius-like\nregime at high temperatures and a temperature-independent\nregime below ®10K, which is a fingerprint of QTM behavior.\nTheM(B)curves taken below TNdemonstrate complex hys-\nteretic behavior. By using classical Monte-Carlo simulations\nwith dipolar interactions we reproduced essential features\nof magnetic behavior of DyScO3: (i) the type of magnetic\nordering and the ordering temperature TN; (ii) temperature\ndependences of the magnetic specific heat and magnetiza-\ntion; (iii) behavior of magnetic correlation length above thearXiv:2204.12398v1  [cond-mat.str-el]  26 Apr 20222\nJ = L  + S = 15/23+\nDySOC\nCEF\nEnergy scheme of the \nlow-lying multiplet\n|+15/2> |-15/2> |-13/2> |+13/2>0 meV\n|etc> |etc>290 KEnergy (meV)\nTemperature-independent channel\nΔS = 15CEF activationS = 1 \nΔ\n|M>J\nFIG. 1. Sketch of the energy diagram of J=15=2multiplet\nof Dy3+in DyScO3, shown injMJi\u0000Ecoordinates. Spin-orbit\ncoupling (SOC) produces the J=15=2multiplet, which is split into\n8 doublets by CEF . The ground state doublet consists of j\u000615=2i\nwavefunctions, following by excited j\u000613=2i,j\u000611=2i, etc. While\nthe temperature dependence of the spin excitation associated with\nj\u000615=2i!j\u0006 13=2itransition follows Arrhenius law, the direct\ntransition between j\u000615=2istates is the temperature-independent\nQTM process.\nTN; (iv) kink and a broad magnetic hysteresis on the M(B)\ncurves. Our results demonstrate that the low-temperature\nbehavior of DyScO3is described by a combination of CEF-\ninduced QTM, dipolar intersite interaction and the strong\nmagnetocaloric effect.\nII. RESULTS AND ANALYSIS\nA. Slow dynamics and magnetic order at zero field\nWe start the presentation of our results with the magne-\ntization data collected as a function of temperature with\ndifferent sweep rates as shown in Fig. 2 (a). Note that\nthe magnetic field was applied along the easy direction,\nBk[010]in all measurements and calculations. All curves\ncollected upon warming up show a clear cusp anomaly as-\nsociated with AFM ordering. Noticeably, the position of the\ncusp shifts with the sweep rate dT=dt. One can also see\nthat the field-cooling (FC) curves differ considerably from\nthose collected upon warming up: (i) the cusp associated\nwith the AFM transition becomes less well-defined and is\nalmost gone for dT=dt\u00151K/min; (ii) the FC and warming\nup curves show considerable hysteresis below TN.\nThese results indicate the presence of a considerable mag-\nnetization relaxation at low temperatures, which takes place\non a timescale of minutes. It is unexpected for a conven-\ntional antiferromagnet, but rather reminiscent of the spin-\n234560120\n.1 K/min0\n.5 K/min1\n K/minCoolingMagnetization (µB/f.u.)T\nemperature (K)Warming 3 K/minW arming Cooling(\na)Experiment, VSMT heory, Monte-Carlo(\nb)0.1 K/min0\n.5 K/min1\n K/min3\n K/minMagnetization (µB/f.u.)2\n3456012345T\nemperature (K)FIG. 2. Temperature dependences of the magnetization of DyScO3\nmeasured using VSM at B=0.1T (a) and calculated by Monte-\nCarlo (b) with different temperature sweep rates as described in\nSec. II C. Red and blue lines represent data collected on warming\nand cooling respectively. Data are in panels (a) and (b) are shifted\nrespectively by +0.5\u0016B/f.u. and +1\u0016B/f.u. vertically for clarity.\nglass (SG) state. At sufficiently low temperature, SG mate-\nrials do not exhibit long-range magnetic order, but instead\nform magnetic clusters with short-range magnetic correla-\ntions [23,24]. A clear fingerprint of the SG behavior is a\nbroad peak in the imaginary part of the AC-susceptibility,\n\u001f00(f), which should follow an Arrhenius-like temperature\ndependence over a broad temperature range [25].\nTo discuss DyScO3in this context, Figures 3 and 4 display\nour temperature- and frequency-dependent AC-susceptibility\nmeasurements. Figure 3 (a) shows the temperature de-\npendence of the real part of the AC-susceptibility, \u001f0(T),\nmeasured at different frequencies along with the static spin\nsusceptibility, M=B, measured with VSM. The curves col-\nlected at f=1Hz and at static regime agree well above\nTN, and display a single peak at the transition temperature.\nThe qualitative behavior changes when the frequency is in-\ncreased. The low-temperature susceptibility measured at\nf\u001510Hz is reduced, but returns to M=Babove a frequency-\ndependent crossover temperature. The high-temperature\ntails of all curves follow the same Curie-Weiss law.\nThe\u001f00(T)curve measured at 1 Hz shows a strong diver-\ngence at TN, as expected for an AFM system, and a weak\nshoulder-like feature at \u00187K. With increasing frequency\nthe shape of the peak at the ordering temperature changes\nsignificantly and becomes similar to the one observed in\n\u001f0(T). The second anomaly also shifts with frequency. We\nquantified the positions of the second anomaly using the\ninflection point as shown for 1 Hz curve in Fig. 3 (b).\nTo further reveal the frequency dependence of the spin sus-\nceptibility in DyScO3we performed measurements of \u001f0(f)\nand\u001f00(f)at multiple temperatures, and the results are\nsummarized in Fig. 4. The \u001f0(f)demonstrates a plateau at\nlow frequencies and a gradual decrease above temperature-\ndependent crossover frequency. The \u001f00(f)curves exhibit\na strong broad peak at T\u00153K. The position of the peak\nshifts down with decreasing temperature, however between\nT=10and 4 K\u001f00(f)remains almost unchanged. When\ncooling below TNthe peak height decreases and shifts to-\nwards lower frequencies, which could not be followed fur-3\n0.0000.0050.010(\nb)dT/dt = 0.5 K/minB\nDC = 0.1 TB\nDC || BAC || [010] \nf = 1 Hz \nf = 10 Hz \nf = 100 Hz \nf = 1000 Hz \nVSM data, M/Bχ'(emu/g Oe)(a)5\n1 01 52 00.0000.0010.0020.003χ''(emu/g Oe)T\nemperature (K)\nFIG. 3. Temperature dependence of real (a) and imaginary (b)\nparts of the complex longitudinal AC susceptibility of DyScO3mea-\nsured at B=0.1T applied along [010]at different frequencies.\nThe static spin susceptibility M=Bmeasured using VSM is shown\nin panel (a). Crossed black lines in panel (b) illustrate how the\ncrossover temperature was determined.\nther with our AC setup. Figure 4(c) shows Cole-Cole plots\n\u001f00(\u001f0)at different temperatures. For a system with a single\nrelaxation channel (or symmetrical distribution of the re-\nlaxation channels), the curves should follow a semi-circular\ntrajectory. However, the curves measured with DyScO3are\nasymmetric, indicating a more complex distribution of the\nrelaxation times [26–28 ].\nTo characterize the timescale of the magnetization dynam-\nics below TNwe used a VSM magnetometer and measured\nmagnetization relaxation. To do this, we applied the follow-\ning protocol: (i) ZFC to the base temperature T=1.8K; (ii)\napply 0.01 T external field; (iii) wait for 300 s; (iv) decrease\nthe external field to zero with sweep rate of 0.07 T /s; (v)\ncollect time-dependent M(t)for 3 hours; (vi) increase tem-\nperature to the new target Tand repeat from the step (ii).\nThe relaxation curves collected at several selected tempera-\ntures above and below TNare shown in Fig. 5.\nConventionally, the relaxation process M(t)can be de-\nscribed with an exponential function, but we were not able\nto obtain good fit of our experimental data using a single\nexponent at T<3K. The measured curves were fitted with\na sum of two exponential functions:\nM(t) =M1e\u0000(t=\u001c1)\f1+M2e\u0000(t=\u001c2)\f2(1)\nwhere, M1and M2correspond to two relaxing moments,\n\u001c1,\u001c2are the relaxation times, \f1,\f2are stretching pa-\n11 01 001 000F\nrequency (Hz)11 01 001 000(c)( a)A\nC susceptibility χ''(arb. u.)AC susceptibility χ''(arb. u.)AC susceptibility χ'(arb. u.)F\nrequency (Hz)(b) T = 20 K \nT = 18 K \nT = 16 K \nT = 14 K \nT = 12 K \nT = 10 K \nT = 8 K \nT = 7 K \nT = 6 K \nT = 5K \nT = 4.5 K \nT = 4 K \nT = 3.5 K \nT = 3 K \nT = 2 KA\nC susceptibility χ'(arb. u.)FIG. 4. Frequency dependence of the AC-susceptibility measured\nat multiple temperatures between 2 and 20 K. Panels (a) and (b)\nshow the real [\u001f0(f)]and imaginary [\u001f00(f)]parts of the AC-\nsusceptibility, respectively. Panel (c) shows the Cole-Cole plot\n\u001f00(\u001f0). All data are shown with a constant vertical offset for\nclarity.\n100102104\nTime (sec)00.010.020.030.04Magnetization (B/f.u.)T = 1.8 K\nT = 2.2 K\nT = 2.6 K\n100102104\nTime (sec)01234Magnetization (B/f.u.)10-3\nT = 3.0 K\nT = 6.0 K(a) (b)\nFIG. 5. Time dependence of magnetization taken after switching\noff 0.01 T magnetic field. Panels (a) and (b) show low- and high-\ntemperature data. Note that y-scale is different in (a) and (b).\nrameters. To improve the fit quality we used \f1and\f2\nas global parameters for fitting of T=1.8,2.2and 2.6 K\ncurves [Fig. 5 (a) ]. The fitted curves are shown by solid lines\nin Fig. 5. Results of the fitting yield \f1=0.35and\f2=0.6\nand\u001c1=550s,\u001c2=25s atT=1.8K; the relaxation times\nexhibit only a minor increase with temperature up to 2.6 K.\nThe relaxation curve taken at T=3K can be described by a\nsingle exponent with \f=0.3 and\u001c=0.2 s [Fig. 5 (b) ].\nThe temperature dependence of observed relaxation times\nextracted from AC susceptibility and magnetization relax-\nation measurements is summarized in Fig. 6. Informed by\nour own INS measurements [19], we highlight three dif-\nferent regimes: (i) Arrhenius regime at high-temperature,\nT>10K; (ii) temperature-independent relaxation between\n10 K and TN; (iii) slow relaxation in the AFM phase. The tem-\nperature dependent AC-susceptibility in DyScO3was also\nstudied in Ref. [20], where the authors observed peak-like4\n0.10 .20 .30 .40 .510.05 .03 .32 .52 .0-\n6-4-2024681012A\nFM orderT\nwo-exponent fit of relaxationQuantum tunnelingTemperature (K) \n/s99''(T) from ref \nActivation \n/s99''(T) \n/s99''(f) \nVSM, 1//s116log(f)1\n/T (K-1)CEF Activation Δ = 290 KT\nransition0\n.10 .20 .30 .40 .5105 3 3 2 -\n4-20246810120\n.01830.13531.00007.389154.5982403.42882980.958022026.4658162754.79140.10 .20 .30 .40 .5-4-20246810120\n.050 .100 .150 .200 .250 .300 .35-5-4-3-2-10123456789101112\nFIG. 6. Arrhenius plot log (f)(1=T), reconstructed from our results\nalong with data from Ref. [20]. The grey dotted line shows the\ntransition temperature TN. The blue and green points show the\npeak positions extracted from the frequency and temperature de-\npendencies of imaginary part of AC susceptibility \u001f00, and the black\nsquares show the high-frequency data from Ref. [20]. Red solid\nline shows the calculated CEF activation curve with experimentally\ndetermined\u0001=290K. Red points show inverse relaxation con-\nstants 1=\u001cextracted from a fit of the magnetic relaxation curves\nwith Eq. (1).\nanomalies in\u001f00(T)curves, which shifted with frequency.\nThe authors associated this peak with an Arrhenius-like re-\nlaxation process, taking the population of the CEF level into\naccount, and extracted \u0001=kB=229K. Our own later INS\nmeasurements indicated that the first CEF level is located\nat higher energy, \u0001=kB=290 K [19]. The calculated curve\nfor a 290 K gap is shown in Fig. 6 by a red line, and one\ncan see good agreement with experimental points at high\ntemperature and a clear crossover between regimes (i) and\n(ii).\nThe behavior seen in DyScO3strongly resembles the slow-\ning down of the spin dynamics in a classical spin-ice com-\npound Dy2Ti2O7, which also shows three regimes at differ-\nent temperatures: (i) Arrhenius relaxation at T>15K;\n(ii) plateau at 1.5<T<15K; (iii) another Arrhenius\nregime below 1.5K due to development of the spin-ice\nregime [7,9,10]. The difference in the low-temperature\nregime (iii) is a consequence of the different ground states,\nAFM in DyScO3vs. spin-ice in Dy2Ti2O7.\nB. Field-induced anomalies\nWe next proceed with a description of the field-induced\nphysics in DyScO3. Authors of [19]reported magnetization\ncurves measured at T=2K along three crystallographic di-\nrections. Interestingly, the curves measured along [010]and\n01 2 0246810T\n = 5 KT = 3.5 KMagnetization (µB/f.u.)M\nagnetic field (T) 0 -> 2 T, 700 Oe/s \n2 -> 0 T, 700 Oe/s \n0 -> 2 T, 500 Oe/s \n2 -> 0 T, 500 Oe/s \n0 -> 2 T, 50 Oe/s \n2 -> 0 T, 50 Oe/s \n0 -> 2 T, 2 Oe/sT = 2 K0\n.00 .51 .01 .52 .00246810Magnetization (µB/f.u.)M\nagnetic field (T)T = 8 K0.00 .51 .01 .52 .00246810(\nd)( c)(b)Magnetization (µB/f.u.)M\nagnetic field (T)(a)0\n.00 .51 .01 .52 .00246810Magnetization (µB/f.u.)M\nagnetic field (T)FIG. 7. Magnetization curves M(B)measured at several tempera-\ntures as indicated at each panel. The curves were measured with\ndifferent sweep field rates as shown in legend.\n[100]-axes show two consecutive hystereses at low field and\njust below the saturation. These features were interpreted\nas two field-induced first-order phase transitions. Motivated\nby those observations we measured the magnetization of\nDyScO3at several temperatures with magnetic field applied\nalong the [010]axis. Figure 7 shows magnetization curves\ncollected at several temperatures below and above TN. One\ncan see that at T=2K magnetization curves measured with\nhigh sweep rates (50, 500 and 700 Oe /s) show consider-\nable hysteresis over the whole field range. However, we can\nclearly highlight two distinct transitions: the first kink at\nB\u00190.4T and the second anomaly at B\u00191T . Noticeably,\nwhen the sweep rate decreases, only the low-field features\nremains visible, while magnetization at higher fields shows\nsimple Brillouin-like behavior. Figures 7(b-d) demonstrate\nmagnetization collected above TNand one can see that the\nmagnetization is perfectly linear at the low-field regime in\nagreement for the expectation for a paramagnet. However,\nthe clear kink as well as the hysteresis at 0.7–2 T are clearly\nseen for high field-sweep rates, while the low-sweep curves\nshow simple Brillouin-like behavior.\nBased on these data we can associate the first transition\nwith the field-induced destruction of the AFM order, while\nthe high-field hysteresis is associated with single-ion physics,\nbecause it persists to temperatures up to \u00188K, which is\nmuch larger than the characteristic energy scale of magnetic\ninteractions in DyScO3. We associate this effect with the\nstrong magnetocaloric effect (MCE) in DyScO3[29]and\nin the next section we show that Monte-Carlo simulations,\nwhich take into account the MCE, are capable to reproduce\nthis effect.5\nC. Monte-Carlo simulations\nAs discussed above, DyScO3shows strong Ising-like single-\nion anisotropy of magnetic moments, meaning that Dy mo-\nments can be pointed up or down along the CEF-dictated\neasy magnetization direction at each site. In addition, a\nprevious report associated magnetic order with the dipole-\ndipole interactions between Dy moments [19]. The standard\napproach to describe physical properties of an Ising system\non a 3D lattice is by Monte-Carlo modeling and here we\nmake use of the metropolis algorithm to describe magnetic\nbehavior of DyScO3[30]. We considered a 10\u000210\u000210\u00024\ncluster of Ising spins, which are coupled by dipole-dipole\ninteraction and are in thermal contact with a reservoir at\ntemperature Tres. Most parameters of our model, such as in-\nteratomic distances, the field- and temperature dependence\nof the thermal conductivity (approximated from the data\nmeasured on isostructural DyAlO3), the magnetic moment\nof Dy3+and the direction of easy axis were fixed from the\nexperimental data [19,31]. The only free parameter is the\ncoefficient which converts the number of Monte-Carlo steps\nto the experimental time, which we fixed by comparison of\ncalculated and experimental \u001f00(f)curves and the absolute\nvalue of the thermal conductivity. See Sec. B for details.\nWe start the presentation of our calculations with the low-\ntemperature magnetic structure. First, our data show that\ntheGxAy is indeed the ground state of DyScO3and has the\nlowest energy among the four possible k=0magnetic con-\nfigurations in DyScO3, in agreement with previous estimates\nfor smaller clusters [19,29]. As the next step we calcu-\nlated the static magnetic structure factor, S(Q), and found\nthat the AFM order manifests itself with a strong magnetic\nBragg peak at Q= (001). We plot the calculated temper-\nature dependence of (001) peak in Fig. 8 along with the\nexperimental measurements of the ordered moment [19].\nFitting of calculated data near the critical temperature was\nperformed using the following expression:\nS=\u001a0, for (T>Tc)\nS0(1\u0000\u0000T\nTc\u0001\f)\u000b, for (T\u0014Tc),\u001b\nand yielded the critical temperature TN=2.853(1)K. The\nvery good agreement with the experimentally determined\nTN=3.11K indicates that the dipolar interaction is the\nprimary magnetic interaction in DyScO3. We have also mod-\neled the magnetic diffuse scattering above the transition\ntemperature. The representative diffuse pattern calculated\natT=3.5K is represented in Fig. 8 (d), which was ob-\ntained from averaging a number of Monte-Carlo runs. We\nextracted the temperature dependence of the correlation\nlength [Fig. 8 (c) ]and compared it with experimental re-\nsults measured at the CNCS instrument [19]. Clearly, the\nmeasured and the calculated curves show good agreement\nwith\u0018c>\u0018abover all temperature range. The reason is the\nmutual arrangement of crystal axes and directions of Dy3+\nmoments: the nearest neighbor Dy moments along the caxis\nhave strong antiferromagnetic interaction, while interaction\n-1.5-1.0-0.50.00.51.01.5-1.5-1.0-0.50.00.51.01.512 3 4 5 6 0.00.51.0 Experiment \nMC calculationIntensity (arb.u.)T\nemperature (K)12 3 4 5 6 7 8 02468101214 \nExperiment \nMC calculation \nNuclear contributionHeat capacity (J/moleK)T\nemperature (K)2\n4 6 8 10121416100101102 ξH exp. \nξL exp. \nξH calc. \nξL calc.Correlation length (Å)T\nemperature (K)(\n0 0 L) (r.l.u.)(\nH 0 0) (r.l.u.)00.51Intensity (arb.u.)(a)( b)T\n = 3.5 K(c)( d)FIG. 8. AFM ordering at zero field. (a) Temperature depen-\ndence of (001) Bragg peak. Red dots represent experimental data\nfor square of the ordered moment [19]and solid line is the re-\nsult of Monte-Carlo calculations. (b) Calculated and measured\nlow-temperature specific heat. The nuclear contribution was ap-\nproximated by C(T) =\fT3with\f=0.00023 J/mole\u0001K4and\nadded to the calculated curve. The nuclear contribution reaches\n0.047 J /mole K at T=6K and therefore is barely visible at this\ntemperature scale. (c) Temperature dependence of the correlation\nlength measured by neutron diffuse scattering (red and pink dots)\nand calculated by Monte Carlo (blue and light blue dots). The\nfilled area represents the estimated uncertainty of the calculation.\n(d) Simulated neutron diffraction pattern in the (H0L)scattering\nplane calculated at TN<T=3.5K showing anisotropic diffuse\nscattering around the primary AFM peak (001).\nbetween in-plane Dy moments is nearly canceled out for the\nGxAy magnetic configuration [29]. To further characterize\nthe magnetic transition we calculated the specific heat of\nDyScO3and show it along with the experimental curve in\nFig. 8 (b). Both curves show a sharp \u0015-like peak at TNdue to\nAFM transition with only weak shoulders above TN, which\nindicates weak magnetic correlations in the paramagnetic\nstate. The entropy change associated with anomaly in the\nspecific heat is in agreement with the value expected for a\nIsing system: Rln(2S+1)forS=1=2.\nTo reveal the origin of the bifurcation between the M(T)\ncurves shown in Fig. 2 (a) we performed Monte Carlo sim-\nulations of the temperature dependences of magnetization\n[see Fig. 2 (b) ]with different sweep rates of the reservoir\ntemperature, dTres=dt. We found that the calculated magne-\ntization curves are in qualitative agreement with experimen-\ntal data. Specifically, the magnetization curve calculated\nfordT=dt=0.1K/min shows a clear cusp at TNand a\nweak hysteresis below the transition. With increase sweep\nratedT=dtthe hysteresis becomes wider and the peak for\nwarming up curves shifts towards higher temperatures in\ngood qualitative agreement with the experiment. The rea-\nson for this behavior is the poor thermal stabilization due6\nto both, weak thermal conductivity and slow thermaliza-\ntion during the temperature sweeps, because of which the\nactual temperature of the sample can differ considerably\nfrom Tres. This effect causes bifurcation between cooling\nand warming M(T)curves and the sweep-rate dependence\nof the transition temperature.\nAs the next step we calculated the magnetic field depen-\ndence of the magnetization at three characteristic tempera-\ntures of the reservoir, Tres=1,2and 3.5 K and the results\nare summarized in Fig. 9. We performed two versions of cal-\nculations, (i) isothermal magnetization with TSample =Tres\nstrictly maintained, and (ii) by taking into account the strong\nmagnetocaloric effect [21], see details of calculations in\nSec. B. The field sweep rate for all curves in Fig. 9 was\n0.5 K/min.\nFirst of all, let us consider the magnetization curves calcu-\nlated at T=3.5K above the ordering temperature, which\nare shown in Fig. 9 (a1). The isothermal magnetization\ncalculated at T=3.5K shows featureless Brillouin-like\nbehavior in agreement with expectations for a paramagnet.\nIntroduction of the magnetocaloric effect changes qualitative\nbehavior of the magnetization curves and opens a broad hys-\nteresis at B\u00190.4–1.2 T , because of the field-induced change\nofTsamp, which is shown in Fig. 9 (a2). Thus we conclude\nthat the hysteresis at intermediate fields observed in our\nmagnetization measurements for dB=dt\u001550Oe/s (Fig. 7)\nis caused by the magnetocaloric effect, which however plays\na minor role when the field ramp rate is small compared to\nthe thermalization time, meaning that Tsamp=Tres.\nBelow the magnetic transition temperature the magneti-\nzation curve changes considerably. The curves calculated\natTres=2K demonstrate a metamagnetic transition and\na broad hysteresis at low fields, B<0.3T , which behaves\nsimilarly for both, the isothermal and non-isothermal curves\n[Fig. 9 (b1) ]. Above the metamagnetic transition the isother-\nmal curves merge and show the featureless Brillouin-like\nbehavior, while a narrow hysteresis emerges at B=0.5–1 T\nfor the curves calculated with magnetocaloric effect. We\nnote that the exact shape of the calculated curve depends\nconsiderably on the thermal conductivity of DyScO3, which\ndepends on both magnetic field and temperature, \u0015(B,T).\nThis quantity was not measured in the low-temperature\nregime, T<10K and in our calculations we make use\nof the data measured in the paramagnetic phase on the\nisostructural DyAlO3[31], which we approximate down to\nthe temperature range of interest (see Sec. B for details).\nThis procedure does not provide quantitatively precise val-\nues for the thermal conductivity, especially below TNwhere\nthermal conductivity can exhibit strong anomalies near the\ncritical field [32–34]. Therefore, the exact shape of calcu-\nlated and observed magnetization curves does not match\nquantitatively. However, our calculations allow us to asso-\nciate the high-field hysteresis with the magnetocaloric effect,\nwhile the low-field anomalies, including the metamagnetic\ntransition and the hysteresis of magnetization, are associated\nwith collective behavior.\nFigure 9 (c1) shows magnetization calculated at Tres=1K\nand one can see that isothermal and non-isothermal curvescoincide over all field scales, meaning that the magne-\ntocaloric effect has a minor influence on the magnetiza-\ntion curve at this temperature. Moreover, the curves cal-\nculated for B=0!2T demonstrate a narrow Ms=2\nplateau at B=0.4T . In a system with Ising-like anisotropy,\nthe magnetic moments can point parallel or antiparallel\nto the easy-axis direction, therefore below the ordering\ntemperature, one can expect to see the formation of SDW-\nlike incommensurate phases at the intermediate magnetic\nfields [35,36]. Figure 9 (d) shows the calculated static\nspin structure factor at T=1K and B=0.4T , which cor-\nresponds to the plateau field range. One can see clearly\nformation of the incommensurate peaks at q= (0 0 0 .55),\nwhich can be associated with the formation of \"\"\"# type of\norder along the caxis. Two other reflections correspond to\nthe primary magnetic reflection of the zero-field AFM phase,\nq= (0 0 1 )and ferromagnetic q= (0 0 0 )peak. We note\nthat a similar field-induced incommensurate magnetic order\nwithk= (0 0 1\u0006\u000e)was observed previously in the isostruc-\ntural material YbAlO3[37,38]. Low-temperature in-field\nneutron diffraction measurements would be required to ver-\nify the presence of the incommensurate magnetic phases in\nDyScO3.\nIII. DISCUSSION AND CONCLUSION\nWe demonstrate that DyScO3exhibits slow dynamics of\nmagnetization below T\u0003=10K, which is caused by strong\nCEF-induced uniaxial anisotropy of Dy3+moments. We note\nthat similar behavior was observed previously in multiple\norganic-based magnetic molecules [4–6], while such a be-\nhavior in inorganic materials remains relatively rare. Several\nrepresentative examples are Dy2Ti2O7and Ca3Co2O6. How-\never, in addition to the QTM, both aforementioned materials\ndisplay complex magnetic behavior due to frustration of\nmagnetic interactions. The presence of the competing inter-\nactions makes it difficult to disentangle the effects of QTM\nand magnetic frustration on spin freezing in these systems,\ncontrary to the case of DyScO3. We have characterized the\nlow-temperature spin dynamics of DyScO3using AC sus-\nceptibility and magnetization relaxation measurements and\nhave observed three regimes: Arrhenius-like behavior at\nT>T\u0003, a plateau between T\u0003andTN, and slow relaxation\nof magnetization below TN. Thus, DyScO3represents a so-far\nunique example of a classical dipolar AFM, which combines\na classical magnetically-ordered ground state with QTM.\nOur Monte Carlo simulations appear to capture the es-\nsential physics of DyScO3, including the magnetic ground\nstate, the temperature of the AFM transition, slow dynam-\nics of magnetization at low temperatures and bifurcation\nbetween M(T)curves collected upon cooling and warming.\nIn addition, by taking into account the magnetocaloric ef-\nfect we were able to describe the magnetization curves and\ndemonstrate that the hysteresis in the paramagnetic phase is\ncaused by considerable field-induced change of the sample\ntemperature during the measurements. Our simulations also\npredict the formation of an incommensurate spin-density7\n01 2 0246810M\nagnetocaloric     I\nsotermic  Magnetization (µB/f.u.)F\nield (T)Tres = 3.5 KT res = 2 KH\nysteresis width0\n.00 .51 .01 .52 .02\n345T (K)01 2 0246810M\nagnetocaloric     I\nsotermic  Magnetization (µB/f.u.)F\nield (T)Hysteresis width0\n.00 .51 .01 .52 .01\n23T (K)(a1)(\na2)(b1)(\nb2)(\nc1)(\nc2)(d)0\n1 2 0246810M\nagnetocaloric     I\nsotermic  Magnetization (µB/f.u.)F\nield (T)Tres = 1 KH\nysteresis width0\n.00 .51 .01 .52 .00\n.51.01.5T (K)-1.5-1.0-0.50.00 .51 .01 .5-1.5-1.0-0.50.00.51.01.5(0 0 L) (r.l.u.)(\nH 0 0) (r.l.u.)0\n0 .51 I\nntensity (arb.u.)B = 0.4 TT\n = 1 K\nFIG. 9. (a1-c1) The magnetization curves calculated taking into\naccount magnetocaloric effect at various temperatures of reservoir\nand corresponding system temperature dependence. The magneti-\nzation curves were calculated for external field ramped from zero\nto 2 T and backward, the shaded areas show width of the hysteresis\ncurves\u0001M(B) =M(B)\"\u0000M(B)#. Additional panels (a2-c2) show\ntemperature change during simulation, grey horizontal lines de-\nnote reservoir temperatures. Reservoir temperatures are indicated\nin each panel. The contour map (d) is the calculated magnetic\nstructural factor for the incommensurate state around B=0.4T\nandT=1 K (red circle point at (c1)).\nwave magnetic phase at low temperatures and intermediate\nmagnetic fields, similar to that observed in YbAlO3[37,38],\nwhose existence in DyScO3awaits experimental verification\nwith elastic neutron scattering measurements.\nWe note that although our model captures the main fea-\ntures of magnetic behavior in DyScO3there are minor quanti-\ntative disagreements such as the exact shape of the hysteresis\ncurves shown in Figs. 7 (a-c), which could probably be im-\nproved by including in the model exact results for the field-\nand temperature-dependence of thermal conductivity. In\naddition, the\u001f00(\u001f0)curve has an asymmetric shape indicat-\ning a complex distribution of the relaxation times, while our\nmodel implies a single relaxation channel for simplicity.\nTo conclude, we have applied AC susceptibility and DC\nmagnetization measurements, supported by specific heat,\nneutron diffuse scattering and Monte-Carlo calculations to\ncharacterize spin dynamics in DyScO3. Our results indicate\nthat DyScO3represents a rare combination of single-ion QTM\nbehavior with classical dipolar interactions and stimulatefurther search of rare-earth based condensed matter systems\nwith QTM.\nACKNOWLEDGMENTS\nAcknowledgments. We thank O. Stockert and A. S.\nSukhanov for stimulating discussions. S.E.N. acknowledges\nfinancial support from the European Union Horizon 2020\nresearch and innovation program under Marie Sklodowska-\nCurie Grant No. 884104. The work of N.D.A. was supported\nby the German Research Foundation (DFG) through grant\nNo. PE 3318 /3-1. This research used resources at the Spal-\nlation Neutron Source, a DOE Office of Science User Facility\noperated by the Oak Ridge National Laboratory.8\nAppendix A: Experimental details\nHigh-quality single-crystals of DyScO3were obtained com-\nmercially [39]. Magnetization and AC-susceptibility mea-\nsurements were performed in a temperature range 1.8 -\n100 K, using an MPMS SQUID VSM instrument by Quantum\nDesign. For the magnetic measurements we used a sample\nwith mass of 1.6 mg. Magnetic field was applied along the\n[010]direction, which is the easy axis of magnetization. Spe-\ncific heat was measured using PPMS from Quantum Design.\nThe single crystal neutron scattering measurements were\ndone at the Cold Neutron Chopper Spectrometer (CNCS)\n[40,41]. To quantify the correlation lengths \u0018H,\u0018Lwe per-\nform a two dimensional fitting of the (0 0 1) magnetic peak\nusing the following function [19]:\nSmag(Q) =sinha=\u0018H\n(cosh a=\u0018H\u0000cos\u0019qH)sinhc=\u0018L\n(cosh c=\u0018L\u0000cos\u0019qL),\n(A1)\nwhere aandcare the lattice parameters. The calculated\nmagnetic structure factor (see Sec. B 3) was fitted using the\nsame equation.\nAppendix B: Monte-Carlo simulations\n1. Magnetic Hamiltonian\nAt low temperature magnetic system of Dy3+ions can\nbe effectively described by dipole-dipole interaction. Since\nthe magnetic properties of DyScO3exhibit Ising-like behav-\nior, it is convenient to theoretically study these with Monte\nCarlo (MC) simulations. In this work we performed MC\nsimulations of a 3D Ising system of Dy3+moments using the\nclassical Metropolis single-flip algorithm [30]. The Hamilto-\nnian used in the calculations was taken as a combination of\ndipole-dipole interaction energy and a Zeeman term due to\nthe external field. The Hamiltonian reads:\nH=\u00001\n2\u00160\n4\u0019X\ni,j\u00143(~mi,~ri,j)(~mj,~ri,j)\nj~ri,jj5\u0000(~mi,~mj)\nj~ri,jj3\u0015\n\u0000BX\ni~mi,(B1)\nwhere the first term is the dipole-dipole interaction energy\nand the second one is the energy of Zeeman interaction\nof the magnetic moments and external field. Because the\nsummation goes over each interaction twofold there is the\n1\n2factor in front of the first term. The \u00160corresponds to\nvacuum permeability constant, ~miis the magnetic moment\non the isite,~ri,jis the radius vector between the isite and\njsite magnetic moments, the Bis the external field. In the\ncalculations the magnitude of magnetic moments was taken\nas 10\u0016B.\nThe crystal structure of the DyScO3was taken into account\nduring the simulations by implementing the 3D Ising system\nFIG. 10. Scheme of GxAy spin configuration.\nwith inter-atomic distances corresponding to the crystallo-\ngraphic data. This is necessary because inter-atomic radius\nvectors appear in the expression for the dipolar energy. The\nsize of the system in the simulations was 10\u000210\u000210unit\ncells each containing 4 spins (4000 spins in total). The\ndipole-dipole interaction was calculated for up 62 interac-\ntions per one site (the cutoff distance between neighbours\nwas 9.5 Å) as a compromise between calculation time and ac-\ncuracy. A test simulation showed that an inclusion of further\nneighbors produced minor effects on the magnetic behavior.\nThe periodic boundary conditions were used. Each Ising\nspin can be in the s=1ors=\u00001state corresponding to\nthe spin vector ~S=s(\u0006sin(\u001e), cos(\u001e),0), where the sign in\nfront of the sinterm depends on the position of site in unit\ncell (see Fig. 10). \u001eis the angle between spin vector and\nthe[010]or the [0¯10]direction determined by CEF and is\nequal to 28\u000e[19].\n2. AC-susceptibility\nAC susceptibility data were used in our simulations to\nadjust the time-scale transformation coefficient. For calcu-\nlations of real and imaginary parts of AC susceptibility the\nfollowing expressions were used:\n\u001f0(f) =1\nB0NtotalNtotalX\ni=0Misin(2\u0019fi\nA\u0000\u0019\n2) (B2)\n\u001f00(f) =1\nB0NtotalNtotalX\ni=0Micos(2\u0019fi\nA\u0000\u0019\n2), (B3)\nwhere\u001f0and\u001f00are real and imaginary parts of AC sus-\nceptibility, B0andfare the magnitude and the frequency of\nthe alternating external field, and Miis the magnetization of\nthe system at i-th MC step. The alternating external field at\nthei-th MC step is taken as B=B0sin(2\u0019f i=A\u0000\u0019=2). Used\nexternal field amplitude B0is0.1T. The Aparameter is the9\n11 01 001 000AC susceptibility (arb. u.)F\nrequency (Hz) χ' exp \nχ'' exp \nχ' calc \nχ'' calc\nFIG. 11. Calculated and experimental AC susceptibility at T=4K.\nconversion factor between real time and simulation time\ntsim=Atreal. In the above expressions, an averaging was\nperformed over numerous periods, with Ntotalup to 106MC\nsteps per spin. In Fig. 11 experimental and calculated AC\nsusceptibility at temperature 4 K were drawn on the same\nlayer for comparison. The calculated imaginary part of sus-\nceptibility\u001f00consists of one symmetrical peak associated\nwith a particular relaxation time. In contrast, the experi-\nmental dependence of \u001f00is more broad and asymmetric.\nAs it was discussed in Sec. II A, the shape of experimental\n\u001f00curves can be interpreted as a presence of more then\none relaxation channel. The value of the Aparameter was\nadjusted for fitting the calculated position of \u001f00maximum\nto the experimental value and later the Aparameter was\nfixed when used in further calculations.\n3. Magnetic diffraction\nThe magnetic structure factor was calculated as a spatial\nFourier transform of the spin-spin correlation function taking\ninto account the polarization factor of neutron scattering\nusing the following expression:\nSmag(~q) =X\n\u000b,\f\u0002\nf\u000e\u000b,\f\u0000q\u000bq\f\nj~qj2gX\nj,j0Sj,\u000bSj0,\fexp(i~q(~Rj\u0000~Rj0))\u0003\n,\n(B4)\nwhere~qis the wave vector, \u000band\fare Cartesian compo-\nnents - x,yandz, the jand j0are the variables which go\nover all the sites in system, the Sj,\u000bis the\u000bcomponent of\nspin vector on j-th site. The term in front of the second\nsummation is the polarisation factor. The isotropic magnetic\nform factor of Dy ions, which gives a monotonic suppres-\nsion of the intensity at large Q, was not included to the\ncalculations.\n0.00 .51 .012345T (K)F\nield (T)04 8 1 2Magnetic heat capacity (J / mole K)0\n.00 .51 .012345T (K)F\nield (T)00.20.40.60.81Heat conductivity (W K / cm)(\na)( b)FIG. 12. (a) Calculated magnetic part of the heat capacity as func-\ntion of external field and temperature. (b) The heat conductivity\nextrapolated in nonzero external field region.\n4. Thermodynamic properties\nIn order to achieve a better agreement with experimen-\ntal measurements the magnetocaloric effect was taken into\naccount for calculations of the field dependence of mag-\nnetization. The heat capacity used in the magnetocaloric\ncalculations was the combination of the lattice contribution,\nwhich was obtained from approximation of experimental\nmeasurements, and the magnetic contribution which was\ncalculated from energy fluctuations:\nCmag(T,B) =hH2i\u0000hHi2\nkBT2, (B5)\nwhere kBis the Boltzmann constant and the angle brackets\nmeans averaging over numerous MC steps. The total heat ca-\npacity can be written as Cp=Cmag+Clat, the lattice part was\napproximated from experimental data above ordering tem-\nperature TN. The magnetocaloric effect was implemented\nin calculations of the field dependence of magnetization as\na variable system temperature. The temperature change\ncomes form the magnetocaloric effect itself and also from\nthermal contact with the temperature reservoir. If the exter-\nnal field changes from B1toB2during time\u0001tthe system\ntemperature change \u0001Tcan be written as follows:\n\u0001T=T\n1\n2(Cp(B1,T) +Cp(B2,T))1\n2@M(B1,T)\n@T+\n+@M(B2,T)\n@T\ndB\u0000T\u0000Tres\n1\n2(Cp(B1,T) +Cp(B2,T))\u0002\n\u00021\n2(\u0015(B1,T) +\u0015(B2,T))\u0001t, (B6)\nwhere the first term is the magnetocalorics and the second\nexpresses thermal contact with reservoir. dB=B2\u0000B1is the10\nchange of external field, Tis the system temperature before\nthe correction, Tresis the reservoir temperature, \u0015(B,T)is\nthe thermal conductivity coefficient. The field change dBis\nassumed to be small enough in order to justify the replace-\nment of the integral from original thermodynamic formula\nby this simple expression. The heat capacity Cp(B,T)val-\nues were obtained as it was mentioned above. The time\n\u0001twas recalculated into Monte Carlo steps with help of\npreviously discussed Aparameter. The partial derivative\n@M(B,T)\n@Twas calculated beforehand from equilibrium values\nof magnetization:\n@M(B,T)\n@T=hM(B,T+\u0001T)i\u0000hM(B,T\u0000\u0001T)i\n2\u0001T. (B7)\nIn order to numerically estimate the derivative the aver-\naging was done over more 106MC steps per spin.\nThe exact shape of the field dependence of thermody-namic properties can significantly affect the magnetization\nbehavior in magnetocaloric calculations. Since the experi-\nmental data for the heat conductivity \u0015(B,T)is unknown\nin presence of external field and in low temperature for\nDyScO3, we had to use values obtained from extrapolation\nprocess and turn into account assumptions. The heat con-\nductivity\u0015(B,T)values in zero field were taken from data\nfor the isostructural DyAlO3material. We assumed the fol-\nlowing when extrapolating the values to nonzero field: (i)\nthe heat conductivity \u0015(B,T)slowly decreases in the ex-\nternal field [31], and (ii) near the region of metamagnetic\ntransition the heat conductivity \u0015(B,T)has a pronounced\ndrop, which often occur in AFM systems [32–34]. The points\nof critical field at which the heat conductivity \u0015(B,T)drops\nwere determined from anomaly on the calculated heat ca-\npacity. The heat capacity Cp(B,T)and the heat conductivity\n\u0015(B,T)as functions of external field and temperature, as\nthese were used in our calculations, can be seen in Fig. 12.\n[1]J. De Jong, A. Kimel, R. Pisarev, A. Kirilyuk, and T . Rasing,\nLaser-induced ultrafast spin dynamics in ErFeO3, Phys. Rev.\nB84, 104421 (2011).\n[2]K. Neeraj, N. Awari, S. Kovalev, D. Polley, N. Zhou Hagström,\nS. S. P . K. Arekapudi, A. Semisalova, K. Lenz, B. Green,\nJ.-C. Deinert, I. Ilyakov, M. Chen, M. Bawatna, V . Scalera,\nM. d’Aquino, C. Serpico, O. Hellwig, J.-E. Wegrowe, M. Gen-\nsch, and S. Bonetti, Inertial spin dynamics in ferromagnets,\nNature Phys. 17, 245 (2021).\n[3]A. de la Torre, D. M. Kennes, M. Claassen, S. Gerber, J. W .\nMcIver, and M. A. Sentef, Colloquium: Nonthermal pathways\nto ultrafast control in quantum materials, Rev. Mod. Phys. 93,\n041002 (2021).\n[4]D. Gatteschi and R. Sessoli, Quantum tunneling of magnetiza-\ntion and related phenomena in molecular materials, Angew.\nChem. 42, 268 (2003).\n[5]S. K. Langley, D. P . Wielechowski, V . Vieru, N. F . Chilton,\nB. Moubaraki, L. F . Chibotaru, and K. S. Murray, Modulation\nof slow magnetic relaxation by tuning magnetic exchange in\nCr2Dy2single molecule magnets, Chem. Sci. 5, 3246 (2014).\n[6]Y.-N. Guo, G.-F . Xu, Y. Guo, and J. Tang, Relaxation dynamics\nof dysprosium (iii) single molecule magnets, Dalton Transac-\ntions 40, 9953 (2011).\n[7]K. Matsuhira, Y. Hinatsu, and T . Sakakibara, Novel dynami-\ncal magnetic properties in the spin ice compound Dy2Ti2O7,\nJournal of Physics: Condensed Matter 13, L737 (2001).\n[8]J. Snyder, B. Ueland, J. Slusky, H. Karunadasa, R. Cava, and\nP . Schiffer, Low-temperature spin freezing in the Dy2Ti2O7\nspin ice, Phys. Rev. B 69, 064414 (2004).\n[9]K. Matsuhira, C. Paulsen, E. Lhotel, C. Sekine, Z. Hiroi, and\nS. Takagi, Spin dynamics at very low temperature in spin ice\nDy2Ti2O7, J. Phys. Soc. Jpn 80, 123711 (2011).\n[10]H. Takatsu, K. Goto, H. Otsuka, R. Higashinaka, K. Matsub-\nayashi, Y. Uwatoko, and H. Kadowaki, AC susceptibility of\nthe dipolar spin ice Dy2Ti2O7: Experiments and Monte Carlo\nsimulations, J. Phys. Soc. Jpn 82, 104710 (2013).\n[11]J. Gardner, G. Ehlers, P . Fouquet, B. Farago, and J. R. Stew-\nart, Slow and static spin correlations in Dy2+xTi2\u0000xO7\u0000\u000e, J.\nCondens. Matter Phys. 23, 164220 (2011).[12]G. Ehlers, A. Cornelius, M. Orendac, M. Kajnakova, T . Fennell,\nS. Bramwell, and J. Gardner, Dynamical crossover in ’hot’ spin\nice, J. Condens. Matter Phys. 15, L9 (2002).\n[13]V . Hardy, D. Flahaut, M. Lees, and O. Petrenko, Magnetic\nquantum tunneling in Ca3Co2O6studied by AC susceptibil-\nity: Temperature and magnetic-field dependence of the spin-\nrelaxation time, Phys. Rev. B 70, 214439 (2004).\n[14]N. G. Hegde, I. Levati ´c, A. Magrez, H. M. Rønnow, and\nI. Živkovi ´c, Magnetic dynamics across the in-field transition\nin Ca3Co2O6, Phys. Rev. B 102, 104418 (2020).\n[15]D. Bitko, T . Rosenbaum, and G. Aeppli, Quantum critical\nbehavior for a model magnet, Phys. Rev. Lett. 77, 940 (1996).\n[16]R. Giraud, W . Wernsdorfer, A. Tkachuk, D. Mailly, and\nB. Barbara, Nuclear spin driven quantum relaxation in\nLiY0.998Ho0.002F4, Phys. Rev. Lett. 87, 057203 (2001).\n[17]S. Bertaina, B. Barbara, R. Giraud, B. Malkin, M. Vanuynin,\nA. Pominov, A. Stolov, and A. Tkachuk, Cross-relaxation and\nphonon bottleneck effects on magnetization dynamics in\nLiYF4:Ho3+, Phys. Rev. B 74, 184421 (2006).\n[18]R. Johnson, B. Malkin, J. Lord, S. Giblin, A. Amato, C. Baines,\nA. Lascialfari, B. Barbara, and M. J. Graf, Evolution of spin re-\nlaxation processes in LiY1\u0000xHoxF4studied via ac-susceptibility\nand muon spin relaxation, Phys. Rev. B 86, 014427 (2012).\n[19]L. Wu, S. E. Nikitin, M. Frontzek, A. I. Kolesnikov, G. Ehlers,\nM. D. Lumsden, K. A. Shaykhutdinov, E.-J. Guo, A. T . Savici,\nZ. Gai, et al. , Magnetic ground state of the Ising-like antifer-\nromagnet DyScO3, Phys. Rev. B 96, 144407 (2017).\n[20]X. Ke, C. Adamo, D. G. Schlom, M. Bernhagen, R. Uecker,\nand P . Schiffer, Low temperature magnetism in the perovskite\nsubstrate DyScO3, Appl. Phys. Lett. 94, 152503 (2009).\n[21]Y.-D. Wu, Y.-L. Qin, X.-H. Ma, R.-W . Li, Y.-Y. Wei, and Z.-F . Zi,\nLarge rotating magnetocaloric effect at low magnetic fields in\nthe Ising-like antiferromagnet DyScO3single crystal, J. Alloys\nCompd. 777, 673 (2019).\n[22]M. Bluschke, A. Frano, E. Schierle, M. Minola, M. Hepting,\nG. Christiani, G. Logvenov, E. Weschke, E. Benckiser, and\nB. Keimer, Transfer of magnetic order and anisotropy through\nepitaxial integration of 3d and 4f spin systems, Phys. Rev. Lett.\n118, 207203 (2017).11\n[23]J. Kroder, K. Manna, D. Kriegner, A. Sukhanov, E. Liu, H. Bor-\nrmann, A. Hoser, J. Gooth, W . Schnelle, D. S. Inosov, et al. ,\nSpin glass behavior in the disordered half-heusler compound\nirmnga, Phys. Rev. B 99, 174410 (2019).\n[24]V . Anand, D. Adroja, and A. Hillier, Ferromagnetic cluster spin-\nglass behavior in PrRhSn3, Phys. Rev. B 85, 014418 (2012).\n[25]J. Quilliam, S. Meng, C. Mugford, and J. Kycia, Evidence of\nspin glass dynamics in dilute LiHoxY1\u0000xF4, Phys. Rev. Lett.\n101, 187204 (2008).\n[26]S. Havriliak and S. Negami, A complex plane representation\nof dielectric and mechanical relaxation processes in some\npolymers, Polymer 8, 161 (1967).\n[27]C. Topping and S. Blundell, Ac susceptibility as a probe of\nlow-frequency magnetic dynamics, J. Phys. Condens. Matter\n31, 013001 (2018).\n[28]P . Miskinis, The havriliak–negami susceptibility as a nonlinear\nand nonlocal process, Phys. Scr. 2009 , 014019 (2009).\n[29]L. Wu, S. Nikitin, M. Brando, L. Vasylechko, G. Ehlers,\nM. Frontzek, A. T . Savici, G. Sala, A. D. Christianson, M. D.\nLumsden, and A. Podlesnyak, Antiferromagnetic ordering\nand dipolar interactions of YbAlO3, Phys. Rev. B 99, 195117\n(2019).\n[30]I. Beichl and F . Sullivan, The metropolis algorithm, Comput.\nSci. Eng . 2, 65 (2000).\n[31]T . Numazawa, H. Kimura, K. Shimamura, and T . Fukuda,\nThermal conductivity of RAlO3(R=Dy, Er and Ho) in liquid\nhelium temperatures, J. Mater. Sci. 33, 827 (1998).\n[32]M. J. Metcalfe and H. M. Rosenberg, The magnetothermal\nresistivity of antiferromagnetic crystals at low temperatures.\nI. DyPO4, a nearly ideal Ising system, J. Phys. C Solid State\nPhys. 5, 450 (1972).\n[33]Z. Y. Zhao, X. G. Liu, Z. Z. He, X. M. Wang, C. Fan, W . P . Ke,Q. J. Li, L. M. Chen, X. Zhao, and X. F . Sun, Heat transport of\nthe quasi-one-dimensional ising-like antiferromagnet baco 2\nv 2 o 8 in longitudinal and transverse fields, Phys. Rev. B 85,\n134412 (2012).\n[34]G. S. Dixon, V . Benedict, and J. E. Rives, Low-temperature\nthermal conductivity of antiferromagnetic MnCl2\u00014H2O, Phys.\nRev. B 21, 2865 (1980).\n[35]M. E. Fisher and W . Selke, Infinitely many commensurate\nphases in a simple ising model, Phys. Rev. Lett. 44, 1502\n(1980).\n[36]W . Selke, The ANNNI model – theoretical analysis and exper-\nimental application, Phys. Rep. 170, 213 (1988).\n[37]L. Wu, S. Nikitin, Z. Wang, W . Zhu, C. Batista, A. Tsve-\nlik, A. Samarakoon, D. Tennant, M. Brando, L. Vasylechko,\nM. Frontzek, A. Savici, G. Sala, G. Ehlers, A. Christianson,\nM. Lumsden, and A. Podlesnyak, Tomonaga-Luttinger liquid\nbehavior and spinon confinement in YbAlO3, Nat. Commun.\n10, 698 (2019).\n[38]S. E. Nikitin, S. Nishimoto, Y. Fan, J. Wu, L. Wu, A. Sukhanov,\nM. Brando, N. Pavlovskii, J. Xu, L. Vasylechko, R. Yu, and\nA. Podlesnyak, Multiple fermion scattering in the weakly cou-\npled spin-chain compound YbAlO3, Nature Commun. 12, 1\n(2021).\n[39]https: //www.mtixtl.com /.\n[40]G. Ehlers, A. Podlesnyak, J. L. Niedziela, E. B. Iverson, and\nP . E. Sokol, The new cold neutron chopper spectrometer at\nthe Spallation Neutron Source: design and performance, Rev.\nSci. Instrum. 82, 085108 (2011).\n[41]G. Ehlers, A. Podlesnyak, and A. I. Kolesnikov, The cold neu-\ntron chopper spectrometer at the Spallation Neutron Source -\nA review of the first 8 years of operation, Rev. Sci. Instrum.\n87, 093902 (2016)."    },    {        "title": "1306.1014v2.Magnetic_chains__From_self_buckling_to_self_assembly.pdf",        "content": "Themagneto-elastica : From self-buckling\nto self-assembly\nBy Dominic Vella, Emmanuel du Pontavice, Cameron L. Hall and\nAlain Goriely\nMathematical Institute, University of Oxford, Oxford, OX2 6GG, UK\nSpherical neodymium-iron-boron magnets are perman-ent magnets that can be as-\nsembled into a variety of structures due to their high magnetic strength. A one-\ndimensional chain of these magnets responds to mechanical loadings in a manner\nreminiscent of an elastic rod. We investigate the macroscopic mechanical properties\nof assemblies of ferromagnetic spheres by considering chains, rings, and chiral cylin-\nders of magnets. Based on energy estimates and simple experiments, we introduce\nan e\u000bective magnetic bending sti\u000bness for a chain of magnets and show that, used\nin conjunction with classic results for elastic rods, it provides excellent estimates for\nthe buckling and vibration dynamics of magnetic chains. We then use this estimate\nto understand the dynamic self-assembly of a cylinder from an initially straight\nchain of magnets.\nKeywords: elasticity, discrete-to-continuum asymptotics, self-assembly\n1. Introduction\nThe self-assembly of magnetic particles is of great general scienti\fc and engineer-\ning interest appearing as it does in a range of key applications from microscopic\nswimming (Ku et al. , 2010 a), through water \fltration (Yavuz et al. , 2006) to the\nmanufacture of high-end electronic devices (Sun et al. , 2000), and the design of new\nsynthetic viruses (Perez et al. , 2003). The general scienti\fc problem is to under-\nstand the process by which magnetic particles self-assemble through a combination\nof magnetic interactions and other mechanical forces, and to explore the in\ruence\nof the initial distribution of magnets and random \ructuations on the process of as-\nsembly. However, once formed, the overall macroscopic properties of the assembly\nare also of considerable interest. Previous research has focused on modeling two\nspeci\fc cases: chains of paramagnetic beads linked by a molecular bond (Dreyfus\net al. , 2005; Roper et al. , 2006) or the preferred arrangement of large ensembles of\ndipolar nano-particles using both experiment and computation (Ku et al. , 2010 b;\nVar\u0013 on et al. , 2013).\nIn this article, we consider the collective elastic behavior of interacting dipolar\nferromagnetic spheres arranged in a chain, and we study the spontaneous self-\nassembly of such a chain into a chiral cylinder.\nWe use collections of millimetre-sized spherical neodynium-iron-boron (NdFeB)\nmagnets as a paradigm for dipolar self-assembly. These magnets can be easily ob-\ntained either as toys (under the brand names `Neocube' or `Buckyballs') or as\nwell-calibrated magnets for engineering applications. The high magnetic strength\nof these spheres can be used to create various shapes and structures that resist\nArticle submitted to Royal Society TEX PaperarXiv:1306.1014v2  [cond-mat.soft]  4 Dec 20132 D. Vella, E. du Pontavice, C. L. Hall & A. Goriely\n(a)\n(b) (c) (d)\nFigure 1. Con\fnement of the magneto-elastica . A chain consisting of N= 25 spheres (with\ndiameter 2a= 6 mm and magnetic \feld strength B= 1:195 T) is compressed by bringing\nits ends closer by a distance \u0001 Land forms a shape that is compared to the elastica with\nthe same end-end compression (solid curves). Results are shown for (a) \u0001 L=2aN= 0:19,\n(b) \u0001L=2aN= 0:56, (c) \u0001L=2aN= 0:68 and (d) \u0001 L=2aN= 0:80.\nmechanical deformation through their magnetic interaction. The simplest demon-\nstration of this resistance to deformation is seen by taking a chain of beads and\nbringing the ends closer together (see \fgure 1). The deformed chain retains a coher-\nent shape, which is remarkably reminiscent of the classic elastica that is formed by\nperforming the same experiment with an elastic rod or beam (Love, 1944). Because\nof this similarity we refer to this shape as the ` magneto-elastica ', though we stress\nthat the spheres do not have any elastic connection in the usual sense.\nInstead, the spheres will interact through their magnetic \felds, with the con-\nsequence that magnetic beads that are far apart along the chain but close to each\nother in space will potentially have strong interactions that cannot be accounted\nfor using classical elastica theory. These interactions between widely-spaced spheres\n(along with other issues such as the discreteness of the chain and imperfections in\nthe application of `clamped' boundary conditions) may explain some of the discrep-\nancy observed in Figure 1 between the con\fnement of the magneto-elastica and the\nresults of classical elastic theory. A detailed analysis of the di\u000berences between the\nmagneto-elastica and the classical elastica would involve the di\u000eculty of quantify-\ning the di\u000berences between two shapes. However, the qualitative similarity seen in\n\fgure 1 is striking, suggesting that the concept of an e\u000bective `bending sti\u000bness' for\nthe magneto-elastica may be a useful one. Further demonstrations of the elastic-\nlike nature of the magneto-elastica can readily be found. For example, chains held\nvertically buckle under their own weight, a closed ring oscillates, and a cylinder\nresists bending but recovers its shape after poking (\fgure 2). As described below,\nthese experiments give us opportunities to test quantitatively the hypothesis that\nArticle submitted to Royal SocietyThe magneto-elastica : From self-buckling to self-assembly 3\n)b( )a(\n(c)\nFigure 2. Three simple experiments that illustrate the resistance to deformation of as-\nsemblies of ferromagnetic spheres. (a) The self-buckling of a vertical chain of magnetic\nspheres as further spheres are added. (b) The prolate-oblate oscillation of a ring of mag-\nnetic spheres. Snap-shots of the motion are shown at intervals of 0 :021 s. (c) A self-assem-\nbled chiral `nano-tube'. In each case, the diameter of the spheres is 2 a= 5 mm and the\nmagnetic \feld strength is B= 1:195 T.\nthe magneto-elastica can be approximated by a chain with a magnetism-induced\nbending sti\u000bness.\nThe origin of the resistance to deformation exhibited in \fgure 2 is simple to\nunderstand in physical terms. For simplicity, we assume that the magnetic spheres\nhave a uniform internal magnetisation and recall that the external magnetic \feld\naround such a sphere is precisely dipolar (Jackson, 1999). At equilibrium, a straight\nchain of spherical magnets is therefore an oriented collection of magnetic dipoles\n(shown schematically in \fgure 3a). However, when this chain is forced to bend,\nthe dipoles cannot be aligned both with the chain and with each other (see \fgure\n3b). In this frustrated con\fguration, the magnetic \feld of each dipole exerts a\ntorque on every other dipole. The combined resistive torque that is generated acts\nto straighten the chain locally and reduce its curvature. This behaviour reminds\nus of an elastic rod, which exerts a torque proportional to the curvature \u0014to\nresist bending. However, this torque resisting bending is di\u000berent from the `tension'\nthat has previously been observed in a chain of paramagnetic spheres placed in an\nexternal magnetic \feld (Dreyfus et al. , 2005; Roper et al. , 2006).\nFor an elastic rod, the ratio of the bending moment applied and the curvature\ninduced,\u0014, is called the bending sti\u000bness, which is frequently denoted K. Provided\nArticle submitted to Royal Society4 D. Vella, E. du Pontavice, C. L. Hall & A. Goriely\n(a)\n(b) (c)\nFigure 3. The physical mechanism behind the resistance to deformation of a magnetic\nchain. (a) In a straight chain all of the magnetic dipoles are aligned. (b) When deformed\nthe dipoles can no longer be aligned. This frustration gives rise to a torque that resists\ndeformation. (c) The general problem of determining the dipole orientation for a given\ndeformation is in principle complicated (Hall et al. , 2013); here we consider a model\ncalculation based on computing the energy required to form closed rings from chains.\nthat the radius of curvature is large compared to the typical cross-sectional dimen-\nsion of the rod, Kis a constant, and, for an elastic rod of radius aand Young's\nmodulusE, it is well-known that\nK=\u0019\n4Ea4: (1.1)\nThe bending sti\u000bness has the dimensions of a force times a length squared, i.e. Nm2.\nBased on such dimensional considerations alone we might expect that a magnetic\nchain should have an e\u000bective bending sti\u000bness of the form\nKe\u000b=B2a4\n\u00160f(a\u0014;N ): (1.2)\nHereBis the magnetic \feld strength (a quantity that is speci\fed by the manufac-\nturer in Tesla where 1 T = 1 NA\u00001m\u00001),\u00160= 4\u0019\u000210\u00007NA\u00002is the permeability\nof free space, ais the radius of an individual sphere and f(a\u0014;N ) is a dimensionless\nfunction of the dimensionless chain curvature, a\u0014, and the number of spheres in the\nchain,N.\nIt is not clear a priori whether the function f(a\u0014;N ) in (1.2) is a constant,\nhow it might depend on Nor even why Ke\u000bshould not depend on derivatives of\nthe curvature. An analysis of this question for general shapes has been considered\nin detail elsewhere (Hall et al. , 2013) and shows that there is, in fact, a non-local\ncontribution to the deformation energy that cannot be explained as a result of an\nArticle submitted to Royal SocietyThe magneto-elastica : From self-buckling to self-assembly 5\n\\e\u000bective bending sti\u000bness\". However, this analysis is complicated to the extent that\neven deriving the equilibrium equations for the shape of a chain in all but some\nvery simple cases is impossible. In this paper, therefore, we adopt a more direct\napproach: we compute the energy required to deform a linear magnetic chain into\na polygonal ring and, by comparing to the corresponding elastic result determine\nthe corresponding dimensionless bending sti\u000bness. In this simpli\fed case symmetry\nindicates that the function f(a\u0014;N ) of (1.2) is a function of Nalone, ~f(N). More-\nover, we \fnd that ~f(N) rapidly approaches a constant value as Nincreases. Having\nshown this we then proceed to use the concept of a magnetic bending sti\u000bness to\nunderstand quantitatively some di\u000berent experimental scenarios, including those\nshown in \fgure 2a,b.\n2. The e\u000bective bending sti\u000bness\n(a)Theoretical background\nWe begin by considering a single magnetic sphere centred at the origin and\nrecall that, assuming the internal magnetisation is uniform, then the external \feld\nat a position rdue to this sphere is (Jackson, 1999)\nB(r) =\u00160\n4\u00193(m\u0001r)r\u0000r2m\njjrjj5: (2.1)\nHere the dipole moment of a sphere of radius aism=4\u0019a3\n3M^mwithMthe\nstrength of the internal (uniform) magnetization and ^ma unit vector in the direc-\ntion of the dipole moment. (Note that for the magnetic spheres available commer-\ncially the value of B=\u00160Mis given by the manufacturers and is typically in the\nrange 1.1 to 1.4 Tesla.) The approach we adopt here is based on the calculation\nof the magnetic energy of various con\fgurations of spheres. The energy of a single\nmagnetic dipole min an externally imposed magnetic \feld, Bextis (Jackson, 1999)\nU=\u0000m\u0001Bext: (2.2)\nSince the external \feld of a magnetic sphere is precisely the same as a dipole\nwith dipole moment m, the energy of interaction between two magnetic spheres\nmust be precisely the energy of two dipoles with the same strength, separation\nand orientation (Hall et al. , 2013). Therefore, the total energy of a collection of N\nmagnetic spheres located at riwith dipole moment miis\nUN=\u0000NX\ni=1NX\nj=1\nj6=imi\u0001Bj(ri)\n2; (2.3)\nwhere\nBj(r) =\u00160\n4\u00193\u0002\n(r\u0000rj)\u0001mj\u0003\n(r\u0000rj)\u0000jjr\u0000rjjj2mj\njjr\u0000rjjj5\nand the factor 1 =2 is introduced in (2.3) to ensure that the interaction energy is\nnot double-counted.\nArticle submitted to Royal Society6 D. Vella, E. du Pontavice, C. L. Hall & A. Goriely\nWe shall proceed by considering the energies of a chain consisting of Nspheres\nin a linear chain and in a closed ring y. Having calculated these energies we evaluate\ntheir asymptotic behaviour in the limit of large chains, N\u001d1, by making extensive\nuse of the Euler{Maclaurin formula (Knopp, 1990; Olver et al. , 2010), which are\ngiven for completeness in Appendix Appendix A.\n(b)A linear chain\nBefore calculating the energy of a \fnite linear chain, see \fgure 3a, we note\nthat it is already known, in fact, that the closed ring is energetically favourable in\ncomparison to the linear chain for all N > 3 (Prokopieva et al. , 2009; N.Vandewalle\n& S.Dorbolo, 2013). This is because of the release of energy that occurs when the\ntwo spheres at the end of such a chain are brought close to one another; as such\nthis di\u000berence in energy does not re\rect the energy required to frustrate the dipole\nalignment, which is the basis of the sti\u000bness that is of interest to us here. Rather\nthan considering the energy of a \fnite linear chain, therefore, we instead consider\nthe energy of a line of Nspheres embedded within an in\fnite chain. Orientating the\nin\fnite chain along the y-axis, the position vector of the centre of the i\u0000th bead is\nri= 2a(0;i); i2Z: (2.4)\nClearly, in equilibrium the dipole moments must be parallel to the line of the chain;\nwithout loss of generality we assume that the moments are orientated in the negative\ndirection so that the dipole moments of the beads are given by\nmi= (0;\u00001)4\u0019a3\n3M; i2Z: (2.5)\nWe consider the energy of the bead at the origin in the magnetic \feld of all of\nthe other beads, U0. The energy of a chain of length Nwill thus be U0\u0002N=2, with\nthe factor of 2 included to avoid double counting the energy, and the simple form\nto go from a single sphere to all Narising from the fact that the outer chain is\nin\fnite.\nNow, the magnetic \feld at the origin is\nB(r= 0) =\u00160\n4\u00191X\ni=\u00001\ni6=03(ri\u0001mi)ri\u0000r2\nimi\nr5\ni: (2.6)\nThis will only have a component in the ydirection (and indeed we are only interested\nin this component since m0/j); we therefore calculate\nj\u0001B(r= 0) =\u00160\n4\u00191X\ni=\u00001\ni6=0\u000016\u0019a5Mi2+ 4a2i24\u0019a3\n3M\n(2ajij)5=\u0000\u0010(3)\n6\u00160M; (2.7)\nyThese con\fgurations are chosen because their symmetry dictates that the dipole moments\nmust be aligned with the tangent vector of the deformed chain. In fact, this result holds more\ngenerally for chains whose curvature is not too large, but involves a detailed calculation that is\nnot particularly enlightening (Hall et al. , 2013).\nArticle submitted to Royal SocietyThe magneto-elastica : From self-buckling to self-assembly 7\nwhere\u0010(3) =P1\ni=1i\u00003\u00191:202. Hence\nU0=\u0000m0\u0001B(r= 0) =\u00002\u0019\u0010(3)\n9\u00160a3M2(2.8)\nand the energy of the chain of Nsuch magnets is\nU(chain)\nN =N\n2U0=\u0000\u0019\u0010(3)N\n9\u00160a3M2: (2.9)\n(c)A \fnite ring\nTo isolate the e\u000bects of bending, we consider a polygonal ring of Nspheres,\nwith the centre of the polygon being at the origin. As N!1 this approaches a\ncircle, and so will isolate any curvature-dependent energy that arises.\nLettingR=acsc(\u0019=N) be the distance of the centre of each magnetic sphere\nfrom the origin, we have that the position vector of these centres is\nri=R\u0014\ncos2\u0019\nN(i\u00001);sin2\u0019\nN(i\u00001)\u0015\n; i= 1;2;:::;N (2.10)\nand, again taking the bead i= 1 to be orientated in the negative ysense, the\nmagnetic moments are\nmi=4\u0019a3\n3M\u0014\nsin2\u0019\nN(i\u00001);\u0000cos2\u0019\nN(i\u00001)\u0015\n; i= 1;2;:::;N (2.11)\nsince symmetry dictates that mi\u0001ri= 0.\nUsing symmetry again, it is enough to calculate the energy of the bead i= 1\nand multiply this value by N=2 (again to avoid double counting). To do this, we\nnote that\njr1\u0000rij= 2Rsin\u0019\nN(i\u00001): (2.12)\nWe \fnd that\nU1=\u0000\u0019\n18a6M2\nR3NX\ni=21 + cos2\u0019\nN(i\u00001)\nsin3\u0019\nN(i\u00001)(2.13)\nand hence\nU(ring)\nN\n\u00160M2a3=\u0000\u0019\n36Nsin3\u0019\nNN\u00001X\nj=11 + cos2\u0019\nNj\nsin3\u0019\nNj: (2.14)\nWe note that an equivalent expression has been found previously in studies of the\nground state of dipolar particles (Prokopieva et al. , 2009).\nIn deriving the expression (2.14), we have not made any approximations. How-\never, to progress further we need to evaluate the sum\nS=N\u00001X\nj=11 + cos2\u0019\nNj\nsin3\u0019\nNj= 2(N\u00001)=2X\nj=11 + cos2\u0019\nNj\nsin3\u0019\nNj+O(1) = 2H+O(1) (2.15)\nin the limit N\u001d1. To do this, we would like to apply the Euler{Maclaurin formula\n(A 1). However, the summand in H(and hence also its derivatives) grows without\nArticle submitted to Royal Society8 D. Vella, E. du Pontavice, C. L. Hall & A. Goriely\nbound asjN\u00001!0. In order to overcome this di\u000eculty, we notice that, for jclose\nto 1, the summand in Hmay be approximated by a Taylor series for large N(or\nsmall\u0019=N). We therefore write H=H1+H2where\nH1=N\u000b\u00001X\nj=11 + cos2\u0019\nNj\nsin3\u0019\nNj; H 2=(N\u00001)=2X\nj=N\u000b1 + cos2\u0019\nNj\nsin3\u0019\nNj(2.16)\nand 0<\u000b< 1.\nNow, expanding the summand in H1for\u0019j=N\u001c1 (which holds for N\u001d1\nsincej\u0014N\u000b<N), we have\nH1=N\u000b\u00001X\nj=1\u00122N3\n\u00193j3+7\u0019\n60Nj+:::\u0013\n=2N3\n\u001930\n@\u0010(3)\u00001X\nj=N\u000bj\u000031\nA+7\u0019\n120N2\u000b\u00001+O(1):\n(2.17)\nThe partial sum of j\u00003in this expression may be computed using the Euler{\nMaclaurin formula, which gives\nH1=2\u0010(3)\n\u00193N3\u0000N3\u00002\u000b\n\u00193\u0000N3(1\u0000\u000b)\n\u00193+7\u0019\n120N2\u000b\u00001+O(1): (2.18)\nTo evaluate H2, we immediately make use of the Euler{Maclaurin formula,\nwhich yields\nH2=N\n\u0019Z\u0019=2\u0000\u0019=2N\n\u0019N\u000b\u000011 + cos2y\nsin3ydy+1\n\u00193N3(1\u0000\u000b)+O(1); (2.19)\nwhere the constant terms occur from the evaluation of the function at the limits\nof the integral. Calculating the integral analytically and performing Taylor series\nexpansions about y= 0 andy=\u0019=2, we \fnd that\nH2=N\n\u0019\u0014\n\u0000\u0019\n2N+O(N\u00003) +N2(1\u0000\u000b)\n\u00192\u00001\n6\u00007\n120\u00192N2(\u000b\u00001)\u0015\n+1\n\u00193N3(1\u0000\u000b)+O(1):\n(2.20)\nCombining this result with that for H1, we see that the terms involving \u000ball cancel\n(as they must, since \u000bwas arbitrary) and we are left with\nS=4\u0010(3)\n\u00193N3\u00001\n3\u0019N+O(1): (2.21)\nUsing this result in (2.14) we \fnd that\nU(ring)\nN\n\u00160M2a3=\u0000\u0019\n36N\u00193\nN3\u0012\n1\u0000\u00192\n2N2+:::\u0013\n\u00024\u0010(3)N3\n\u00193\u0012\n1\u0000\u00192\n12\u0010(3)N2\u0013\n+O(1);\n(2.22)\nor\nU(ring)\nN\n\u00160M2a3=\u0000\u0019\u0010(3)\n9N+\u00193\n18\u0014\n\u0010(3) +1\n6\u0015\nN\u00001+O(N\u00002): (2.23)\nArticle submitted to Royal SocietyThe magneto-elastica : From self-buckling to self-assembly 9\n10 50 20 30 150.101.00\n0.50\n0.200.30\n0.150.70\nN∆UN/µ0M2a30\n– 1\n– 2\n– 3\n– 4\n– 5\n0 0.2 0.4 0.6 0.8 1.0E(N)×103\n(10/N)3(a)\n0 10 20 30 40 500.200.220.240.260.280.30\nN˜f(N)(b)\nFigure 4. Numerical and asymptotic results for the e\u000bective bending sti\u000bness. (a) The\nnumerically computed change in energy \u0001 UNas a function of N(points) rapidly tends\nto the asymptotic result (2.24) (dashed line). Although the error estimate in (2.24) leads\nus to expect that E(N)\u0018N\u00002, numerical computations reveals that in fact E(N)\u0018N\u00003\n(see inset). (b) The behaviour of the dimensionless bending sti\u000bness ~f(N) =f(sin\u0019\nN;N),\nwithfas de\fned in (1.2), computed numerically (points) and in the limit N\u001d1 (dashed\nline).\n(d)Energy due to bending and the e\u000bective bending sti\u000bness\nIn determining the energy of deformation, it is the di\u000berence in energies between\nthe deformed and undeformed states that is of interest. Comparing the ring and\nthe linear chain, we \fnd that the ring has an energy that is higher by an amount\n\u0001UN\n\u00160M2a3=U(ring)\nN\u0000U(chain)\nN\n\u00160M2a3=\u00193\n18\u0014\n\u0010(3) +1\n6\u0015\nN\u00001+O(N\u00002): (2.24)\nAs expected, the ring state is energetically unfavourable compared to the straight\ncon\fguration | there is an energy penalty to be paid by bending the chain into a\nclosed loop. The energy change can be computed exactly for di\u000berent values of N\nby evaluating the sum in (2.14) numerically, which gives the energy of a closed ring,\nand comparing it to the energy of a linear chain of the same length, which is given\nby (2.9). This numerical computation of the exact energy di\u000berence between the\ndeformed and undeformed con\fgurations shows that the closed loop is energetically\nunfavourable for all N\u00154, but that for N= 3, the closed loop, i.e. triangular,\ncon\fguration is actually energetically favourable compared to the straight chain.\nFigure 4a shows the asymptotic behaviour of \u0001 UNfor largeN. We observe\nthat the leading order behaviour of the asymptotic result (2.24) is in excellent\nagreement with numerical computations of the change in energy for even quite\nmoderateN&10. Furthermore, the error in (2.24) appears to be O(N\u00003), although\nwe are only able to formally justify that this error should be O(N\u00002). This suggests\nthat a cancellation occurs in the calculation of the N\u00002term in (2.24) that we are\nnot able to access using the relatively simple calculation outlined here.\nFinally, we compare the energy of deformation in (2.24) with that of an elastic\nrod of bending sti\u000bness Kand length Lthat is bent round to form a circular ring\nof radiusR=L=2\u0019. The energy of an elastic ring is simply UB=K=2\u0002L=R2=\n2\u00192K=L. By identifying this energy with the energy (2.24) of our magnetic ring of\nArticle submitted to Royal Society10 D. Vella, E. du Pontavice, C. L. Hall & A. Goriely\nlengthL= 2aN, we obtain an e\u000bective bending sti\u000bness\nKe\u000b=\u00160M2a4\u0019\n18\u0014\n\u0010(3) +1\n6\u0015\n\u00190:239B2a4\n\u00160: (2.25)\nCompared to an elastic rod with a circular cross section, radius aand bending\nsti\u000bnessK=E\u0019a4=4 the e\u000bective bending sti\u000bness in (2.25) corresponds to an\ne\u000bective Young's Modulus Ee\u000b\u00190:956B2=\u00160\u0019\u0019400 kPa; our magnetic solid is a\nrather soft material.\nWe note that the form of (2.25) is identical to that given on purely dimensional\ngrounds in (1.2). However, we also note that the dimensional analysis that led to\n(1.2) was not able to tie down the dimensionless function for indeed to determine\nwhether it should depend on N. In fact, we have found that, to leading order in\nN\u00001, the bending sti\u000bness is a constant independent of N. This result is compared\nto numerical results in \fgure 4b where we see that, as expected, the numerically\ncomputed ~f(N)\u0018\u0019\n18[\u0010(3) + 1=6] asN!1 . While this notion of an e\u000bective\nbending sti\u000bness only holds exactly for the closed rings considered here(Hall et al. ,\n2013), the remainder of this paper is concerned with determining whether this con-\ncept is useful in gaining quantitative understanding of some more general scenarios\nbeyond that for which the above analysis is strictly valid.\n3. The heavy magneto-elastica\nA vertical column of magnetic spheres will buckle once its weight reaches a critical\nvalue, as shown in \fgure 2a. A purely numerical approach to this problem is to\ncompute the shape that minimizes the gravitational and magnetic energies of a\n\fnite chain of spheres; such an approach reveals that su\u000eciently short chains remain\nvertical, while the chain buckles above a critical length (i.e. number of spheres), as is\nalso observed experimentally. However, this minimization process does not highlight\nthe underlying physics at play in such scenarios. A complementary approach is to\nuse the classic result (Wang, 1986) that a heavy elastic rod with bending sti\u000bness\nKand linear mass density \u001a`buckles when its length\nL\u0015Lc\u00191:986(K=g\u001a`)1=3: (3.1)\nFor the chains of magnetic spheres considered here, L= 2aNand\u001a`= 2\u0019\u001asa2=3.\nUsing the e\u000bective bending sti\u000bness Ke\u000bfrom (2.25) we thus expect that buckling\nwill occur if\nN >Nc\u00190:4817G\u00001=3(3.2)\nwhere we have introduced the Magneto{Gravitational number\nG=\u00160\u001aga\nB2(3.3)\nto characterize the relative importance of the magnetic properties and the weight\nof the spheres.\nExperiments with spheres of di\u000berent radii ain the range 1 :5 mm\u0014a\u00144 mm\nand di\u000berent magnetic strengths 1 :195 T\u0014B\u00141:4 T were conducted to determine\nthe number of spheres required for self-buckling (with a vertical clamp at the base).\nArticle submitted to Royal SocietyThe magneto-elastica : From self-buckling to self-assembly 11\n10ï410ï310\n5N\nG\n3\u0002\u0001ï\u0004\u0002\u0001ï\u0003\u0002\u0001\u0002\u0001\n\u0001\u0002\u0001\u0001N\nc≈\n0.4817\nG−1/3FIG. 2:\nThere seems to be a problem with this graph —there aren’t enough ticks on the vertical axes between orders\nof magnitude!!!\nExperimental, numerical and asymptotic re-sults showing the regions of (\nG,N) parameter space for thebuckling of a chain of magnets under its own weight. Points\ncorrespond to experiments in which the chain buckled (red\nsquares) or remained straight (blue circles). The solid curve\nshows the results of direct numerical computations (using\nthe optimisation toolbox of MATLAB), while the dashed line\ngives the approximate relationship obtained using the idea of\nan e\nffective magnetic bending sti\nffness.to use the classic result [11] that a heavy elastic rod with\nbending sti\nffness\nKand linear mass density\nρ/lscriptbuckleswhen its length\nL≥Lc≈1.986(\nK/g\nρ/lscript)1/3. For thechains of magnetic spheres considered here,\nL=2\naNand\nρ/lscript=2\nπρ\nsa2/3. Using the e\nffective bending sti\nffnessK\nefffrom (8) we thus expect that buckling will occur if\nN>N\nc≈0.4817\nG−1/3\n(9)where we have introduced the Magneto-Gravitational\nnumber\nG=µ0ρgaB2\n(10)to characterize the respective energy contribution of the\nmagnetic properties versus the weight of the spheres. Ex-\nperiments with spheres of di\nfferent radii\nain the range1.5 mm\n≤a≤4 mm and di\nfferent magnetic strengths1.195 T\n≤B1.4 T were conducted to determine the num-ber of spheres require for self-buckling (with a vertical\nclamp at the base). A comparison of this prediction with\nthe results of numerical computations (based on the opti-\nmisation toolbox in MATLAB) and experiments is shown\nin Fig. 2. This comparison demonstrates that the sim-\nple idea of an e\nffective magnetic bending sti\nffness pro-vides a quantitative understanding of a complicated phe-\nnomenon.\nTo test whether the e\nffective continuum theory is auseful tool in dynamic scenarios, we perform a series of\nexperiments in which a chain of magnets is formed into\u0001\n\u0006\n\u0002\u0001\n\u0002\u0006\n\u0003\u0001\n\u0003\u0006\n\u0004\u0001\n\u0004\u0006\u0001\u0006\u0001\u0002\u0001\u0001\u0002\u0006\u0001\u0003\u0001\u0001\u0003\u0006\u0001\n1t\u0003\u0001\u0006\u0004\u0005\u0003\u0007\u0002\n\u0002\u0001ï\u0003\u0002\u0001ï\u0002\n\u0002\u0001\n\u0003\u0001\n\u0004\u0001\n\u0005\u0001\n\u0006\u00011\n2\n\u0001FIG. 3: Experimental and asymptotic results for the natural\nfrequency of oscillation of a ring of magnetic spheres. Main\nfigure: The relationship between the frequency\nωand thematerial parameters observed experimentally (points) is as\npredicted by (12) (solid line). Inset: The dimensionless oscil-\nlation frequency\nΩpredicted by (13), solid line, is also borneout by experiments (points). Here the di\nfferent colours showdifferent sized spheres (smallest red, largest magenta) and dif-ferent symbols show di\nfferent magnetic strength of spheres.a ring of radius\nR≈Na/\nπ. The ring is then perturbedand the frequency\nωof the resulting oscillations measured(using high speed video footage) for various\nN,s p h e r eradii and magnetic strengths.\nFor an elastic ring with bending sti\nffness\nK, radius\nRand linear density\nρ/lscript, the oscillation frequency of the\nn-thmode is given by [12, 13]\nω2n=EI\nρ/lscriptR4n2(n2−1)2n2+1\n.\n(11)The experiment considered here corresponds to the\nprolate-oblate oscillation mode, i.e.\nn= 2, and so, af-ter substituting\nK\nefffrom (8), we expect thatω=π2/braceleftbigg\n3\n5/bracketleftbigg\nζ(3) +1\n6/bracketrightbigg/bracerightbigg\n1/2\nB (ρµ0)1/2aN2.\n(12)or, alternatively, in dimensionless form\nΩ≡(µ0ρ)1/2aω B\n=π2/braceleftbigg\n3\n5/bracketleftbigg\nζ(3) +1\n6/bracketrightbigg/bracerightbigg\n1/2\nN−2.(13)These results are in quantitative agreement with experi-\nments (i.e. including the prefactor), as shown in Fig. 3.\nHaving considered two problems for which classicalelastic analogs exist, we now\nNext,\nweconsider the\ndy-namic\nself-assembly of\na chiral\ncylinder from\nan initiallystraight\nchain\n— a problem that, to our knowledge, doesnot have a previously studied elastic analog\n. A simpleexperiment consists in taking an initial configuration of\nFigure 5. Experimental, numerical and asymptotic results showing the regions of ( G;N)\nparameter space for the buckling of a chain of magnets under its own weight. Squares\ncorrespond to experiments in which the chain buckled while circles correspond to exper-\niments in which the chain remained straight. Crosses correspond to the results of direct\nnumerical computations (using the optimization toolbox of MATLAB) which, for a given\nnumber of spheres, gives the critical Magneto{Elastic number Gcat which buckling oc-\ncurs. The dashed line gives the corresponding approximate relationship (3.2), which is\nobtained using the idea of an e\u000bective magnetic bending sti\u000bness and the classic result for\nthe buckling of a heavy elastic column (Wang, 1986).\n(In these, and the other experiments described in this paper, we do not measure the\nvalue ofBdirectly but rather assume that the value given by the manufacturer is\ncorrect.) In these experiments, a chain that is su\u000eciently short to stand vertically\ndespite gravity is \frst formed. Additional spheres are then added to the top of\nthe chain and whether the chain remains straight or buckles is noted. The largest\n(respectively smallest) values of Nfor which the chain remained vertical (respec-\ntively buckled) are presented for a number of values of the Magneto{Gravitational\nnumberGin Fig. 5. We observe that the scaling for the critical number of spheres\nat which buckling occurs, (3.2), is in good agreement with experiments while the\npre-factor given in (3.2) overestimates Ncby around 10%.\nTo investigate whether the di\u000berence between experiments and theory observed\nin \fg. 5 is due to experimental uncertainties or rather the errors that are inevitably\nintroduced in applying a result derived for a continuous, elastic problem to this dis-\ncrete scenario, we also performed numerical computations of the discrete problem.\nIn these computations, Nidentical spheres are aligned in a linear con\fguration\nthat is slightly tilted with respect to the vertical (at an angle \u0019=100) and with\ntheir magnetic dipoles aligned in the same direction. The optimisation toolbox in\nArticle submitted to Royal Society12 D. Vella, E. du Pontavice, C. L. Hall & A. Goriely\nMATLAB was then used to \fnd the minimum energy con\fguration of these spheres\nby adjusting their position and the orientation of their magnetic moments. For a\ngiven number of spheres, this leads to a numerically computed value of the crit-\nical Magneto{Gravitational number, Gc, at which buckling occurs. These results\nare quantitatively consistent with experiments, which were performed with a \fxed\nG(Fig. 5). This suggests that the error between the approximate result (3.2) and\nexperiment is due to the approximations made in deriving this result (e.g. the dis-\ncreteness of the experimental system and long range interactions between spheres),\nrather than inaccuracies in experiments. Nevertheless, the concept of a bending\nsti\u000bness that is purely magnetic in origin gives a reasonable quantitative under-\nstanding of experiments.\n4. Two dynamic scenarios\nHaving seen the reasonable success of the theory of the classic elastica to determine\nthe behaviour of a chain of magnetic spheres in two static scenarios, it is natural to\nask whether the idea of an e\u000bective bending sti\u000bness is also useful in understanding\nsome dynamic scenarios. This is the subject of this section.\n(a)Oscillating rings\nA classic dynamic demonstration of the restoring force due to elasticity is to\npinch an elastic ring and observe the ensuing oscillations. This classic problem\nwas \frst considered by Hoppe (Love, 1944; Hoppe, 1871) who showed that, for an\nelastic ring with bending sti\u000bness K, radiusRand linear density \u001a`, the oscillation\nfrequency of the n-th mode is given by\n!2\nn=EI\n\u001a`R4n2(n2\u00001)2\nn2+ 1: (4.1)\nPerforming the same experiment with a magnetic chain is simple (see \fgure\n2b for some snap shots of this motion); a chain with Nspheres in it will form a\nring of radius R\u0019Na=\u0019 and will have \u001a`= 2\u0019\u001aa2=3. The experiment considered\nhere corresponds to the prolate-oblate oscillation mode, i.e. n= 2, and so, after\nsubstituting Ke\u000bfrom (2.25), we expect that\n!=\u00192\u001a3\n5\u0014\n\u0010(3) +1\n6\u0015\u001b1=2B\n(\u001a\u00160)1=2aN2; (4.2)\nor, alternatively, in dimensionless form\n\n\u0011(\u00160\u001a)1=2a!\nB=\u00192\u001a3\n5\u0014\n\u0010(3) +1\n6\u0015\u001b1=2\nN\u00002: (4.3)\nTo test these results, we performed a series of experiments in which magnetic\nchains of di\u000berent lengths, sphere radii and magnetic strengths are formed into rings\nand placed on a smooth horizontal table. The rings are then pinched and released.\nThe frequency !of the resulting oscillations was measured from high speed video\nfootage obtained using a Finepix HS10 camera at frame rates of up to 240 Hz. The\nArticle submitted to Royal SocietyThe magneto-elastica : From self-buckling to self-assembly 13\n0 5 10 15 20 25 30 35050100150200250\nΩω (rad/s)\n10−210−1\n10 20 30 40 501\n2\nN5\nfundamental to many problems in physics, is to relate the\nmacroscopic behaviour to the microscopic interactions.\nIndeed Cauchy and Poisson related the macroscopic elas-ticity of crystals to the microscopic interactions betweenmolecules [12]. Here, we have presented an example of\nthis by considering the simplest possible configurationsof a chain of dipoles and determined a macroscopic ef-\nfective ‘bending sti ffness’. The utility of this stiffness\nhas been demonstrated in three easily reproducible ex-\nperiments, both static and dynamic; the concept of aneffective bending sti ffness gives excellent results for the\nproblems of the buckling of a standing column and theoscillations of a ring, while providing a qualitative un-\nderstanding and the key features for the self-assembly\nproblem of a chain into a cylinder.\nA natural generalisation of the questions addressed\nhere is the elasticity of two-dimensional sheets of mag-nets for which the same ideas can be applied. Further-\nmore, other classical problems of materials science and\nsolid mechanics can be tested on assemblies of magnetic\nbeads including problems of dislocations in sheets, frac-ture, zipping of chains to name but a few. We hope thatthese systems will become both a testing ground for new\nideas in physics and mathematical methods as well as\na perfect demonstration tool for classical physical prob-\nlems.\nB//bracketleftbig\n(ρµ0)1/2aN2/bracketrightbig\nAcknowledgments: This publication is based in part\nupon work supported by Award No. KUK-C1-013-04,\nmade by King Abdullah University of Science and Tech-nology (KAUST). AG is a Wolfson/Royal Society MeritAward Holder and acknowledges support from a Reinte-gration Grant under EC Framework VII.\n[1]J. Ku, D. M. Aruguete, A. P. Alivisatos, and P. L.\nGeissler, Journal of the American Chemical Society 133,\n838 (2010).\n[2]C. F. Yavuz, J. T. Mayo, W. W. Yu, and A. Prakash,\nScience 314, 964 (2006).\n[3]S. Sun, C. Murray, D. Weller, L. Folks, and A. Moser,\nScience 287, 1989 (2000).\n[4]J. M. Perez, F. J. Simeone, Y. Saeki, L. Josephson, and\nR. Weissleder, Journal of the American Chemical Society\n125, 10192 (2003).\n[5]R. Dreyfus, J. Baudry, M. L. Roper, M. F. H. A. Stone,\nand J. Bibette, Nature 436, 862 (2005).\n[6]M. Roper, R. Dreyfus, J. Baudry, M. Fermigier, J. Bi-\nbette, and H. A. Stone, J. Fluid Mech. 554, 167 (2006).\n[7]J.-Y. Ku, D. M. Aruguete, A. P. Alivisatos, and P. L.\nGeissler, J. Amer. Chem. Soc. 133, 838 (2010).\n[8]M. Var´ on, M. Beleggia, T. Kasama, R. Harrison,\nR. Dunin-Borkowski, V. Puntes, and C. Frandsen, Sci-entific Reports 3(2013).\n[9]J. D. Jackson, Classical Electrodynamics (Wiley, 1999).\n[10]C. L. Hall, D. Vella, and A. Goriely, SIAM J. Appl. Math.\nxx, xx (2013).\n[11]C. Y. Wang, Int. J. Mech. Sci 28, 549 (1986).\n[12]A. E. H. Love, A Treatise on the Mathematical Theory of\nElasticity (Dover, 1944).\n[13]R. Hoppe, J. Reine Angewand. Math. 73, 158 (1871).\n[14]We have to do this to avoid picking up the attractive\nenergy that results from bringing the two ends of the\nchain together\n[15]See EPAPS document for details of this asymptotic cal-\nculation\n[16]See EPAPS for a movie of this process3 mm, 1.350 T2a          B\n4 mm, 1.300 T\n5 mm, 1.400 T5 mm, 1.235 T\n6 mm, 1.195 T\n8 mm, 1.195 T\nFigure 6. Experimental and theoretical results for the natural frequency of oscillation of\na ring of magnetic spheres. Main \fgure: The relationship between the frequency !and\nthe material parameters observed experimentally (points) is as predicted by (4.2) (solid\nline). Inset: The dimensionless oscillation frequency \n predicted by (4.3), solid line, is also\nborne out by experiments (points). The symbols used to encode di\u000berent sphere sizes and\nmagnetizations are as described in the legend.\nexperimental results are plotted in \fgure 6. We see excellent quantitative agreement\nbetween experiments and the simple theoretical analysis, which is based on the idea\nof an e\u000bective magnetic bending sti\u000bness developed in x2.\n(b)A self-assembling cylinder\nAs a \fnal example of the utility of the notion of an e\u000bective magnetic bending\nsti\u000bness developed in this article we consider a problem that has, to our knowledge,\nno classic analogue in the theory of elasticity. This experiment is shown schemati-\ncally in \fgure 7: a long chain of M+Pspheres is laid on a horizontal surface with\none end clamped and the other end rolled into a small cylindrical helix. The cylin-\ndrical portion, containing Pspheres, has radius R(N), and there are Nspheres in\neach circuit of the cylinder (so that any sphere kis in contact with its neighbours\nk\u00061 and the spheres k\u0006Nandk\u0006(N+ 1) along the helix). The straight chain\ninitially consists of Mmagnets lying on a table along the x-axis (see Fig. 7). For\n`seeding' cylinders of su\u000eciently large diameter, this con\fguration is unstable, and\nthe straight portion of the chain spontaneously wraps itself onto the cylinder ex-\ntending the initial cylindrical helix (see Supplementary Information for movies of\nthis process).\nArticle submitted to Royal Society14 D. Vella, E. du Pontavice, C. L. Hall & A. Goriely\nvzx\nFigure 7. Self-assembly of a cylindrical helix from a chain. The straight chain is clamped\non the right side and exerts both a torque and a force on the cylinder. For su\u000eciently large\n`seeding' cylinders, the cylinder spontaneously rolls up the chain increasing the length (but\nnot the radius) of the cylinder.\n(i)A simple scaling law\nA simple scaling argument, using the idea of a bending sti\u000bness developed in\nx2 allows us to gain some understanding of this self-assembly phenomenon. We\nbegin by noting that self-assembly occurs because it is energetically favourable\nfor the spheres within the chain to be aligned with, and brought closer to, their\nneighbours; this is achieved by wrapping up into a cylinder. However, to do this\nthey must go from being in a linear con\fguration (with dipoles aligned) to being in\na curved con\fguration (with dipoles partially frustrated). Any excess energy gain\nthat remains once this penalty has been paid can increase the kinetic energy of\nthe sphere. While in the initial stages of the motion this energy gain may also be\nused to accelerate the whole cylinder, at late times we expect the cylinder to be\ntravelling at a steady speed, v1, so that the change in energy is converted solely\ninto kinetic energy.\nConsider therefore an element of the linear chain \u0001 `as it is assimilated into\nthe helix. We assume that the energy released as the element is taken from the\nchain and brought into contact with its neighbours is some constant, \r, per unit\nlength; the total energy released for this element is then just \r\u0001`. However, there\nis a bending energy penalty,1\n2Ke\u000bR(N)\u00002\u0001`, and any remaining energy will be\nused to give this element a kinetic energy \u001a`v2\n1\u0001`(assuming the cylinder rolls, its\nkinetic energy is Mv2\n1, rather than1\n2Mv2\n1). Equating these energies we therefore\nexpect that\n\u001a`v2\n1\u0019\r\u0000\u00192\n2Ke\u000b\nN2a2; (4.4)\nwhich may be written\nv2\n1\u0018B2\n\u00160\u001a\u0012\n~\r\u00001\nN2\u0013\n=B2\n\u00160\u001a\u00121\nN2c\u00001\nN2\u0013\n: (4.5)\nIn (4.5), ~\r= 1=N2\ncrepresents a dimensionless adhesive energy gain and the signi\f-\ncance ofNcwill become apparent shortly.\nThere are two interesting, surprising and important features of the scaling law\n(4.5) that we discuss now, before going on to repeat a more detailed version of\nthis calculation. Firstly, we notice that the speed appears to depend only on the\nnumber of spheres in each winding of the seeding cylinder, N, and the strength\nof the magnetic \feld, B; in particular, v1is independent of the sphere radius, a.\nArticle submitted to Royal SocietyThe magneto-elastica : From self-buckling to self-assembly 15\nSecondly,v1becomes imaginary if N <Ncso thatNcmay be interpreted as the\nmost tightly wound cylinder that can still self-assemble. Though surprising on \frst\nreading, this latter result makes physical sense since, if the cylinder is too tightly\nwound, the energy penalty to be paid upon winding up into a helix is prohibitive\nin comparison to the energy that is released by coming into contact with one's\nneighbours.\n(ii)A more detailed calculation\nWe now estimate the forces acting on the spheres in the helical con\fguration.\nIf the straight portion of the chain is long enough ( M > 15) and the number of\nhelical repeats is large enough ( P=N > 6), the force Fxand torque \u001czacting on\nthe cylinder from the chain are well-approximated by constants. We assume that\nthe cylinder rolls without slipping and that its velocity is purely along the x-axis\n(as de\fned in \fgure 7). Since the cylinder increases in mass as it rolls, it will reach\na terminal velocity given by v2\n1= (Fx+\u001cz=R)=2\u001al. Equivalently, from an energy\nconservation principle, the energy before and after a sphere is transferred from the\nline to the cylinder is\nUstraight\nM +Ucyl\nN;P+Ecyl\nP=Ustraight\nM\u00001+Ucyl\nN;P+1+Ecyl\nP+1 (4.6)\nwhereEcyl\nP=P\u001a4\n3\u0019a3v2\n1is the kinetic energy of the rolling cylinder and Ucyl\nN;Pis\nthe total magnetic energy of a chain of Pspheres wound into a helical cylinder with\nNspheres in each winding. This energy can be estimated asymptotically for N\u001d1\nby realizing that the cylinder tends to a hexagonal lattice as Ntends to in\fnity and\nthat, to order N\u00001, the energy required to bend a straight chain is equivalent to the\nbending energy of the ring. The magnetic energy density (per sphere) in an in\fnite\nhexagonal lattice can be evaluated numerically to be \u0000\u00160M2a3\u0019\u0010(3)(2 +\u000b)=18\nwhere\u000b\u00190:295. Therefore, we have\nUcyl\nN;P\n\u00160M2a3= (4.7)\n\u0000P\u0019\n18N\u0002\n\u0010(3)(2 +\u000b)N\u0000\u00192(\u0010(3) +1\n6)N\u00001+O(N\u00002)\u0003\n:\n(Note that the the O(N\u00001) term here, which corresponds to the bending sti\u000bness\ndiscussed earlier, is not a\u000bected by the change in geometry from a ring to a helix.)\nAssuming a perfect motion along the x-axis without sliding and in the absence of\nother forms of friction, the steady velocity of the cylinder is\nv2\n1\u00160\u001a\nB2=1\n144\u0002\n6\u000b\u0010(3)\u0000\u00192N\u00002(6\u0010(3) + 1) +O(N\u00003)\u0003\n: (4.8)\nNote that the structure of (4.8) is identical to that determined from the scaling\nanalysis ofx4(b)i, particularly (4.5). Of course, this detailed analysis allows us\nto determine a numerical value for ~ \r= 1=N2\nc; we \fnd that ~ \r\u00196:17\u00002and so\nN= 7 is the minimal number of spheres in each turn of the helix before it can roll,\nindependent of all physical parameters. We again note that both (4.8) and (4.5)\npredict a linear scaling of the velocity squared with N\u00002. We now turn to some\nexperimental tests of these predictions.\nArticle submitted to Royal Society16 D. Vella, E. du Pontavice, C. L. Hall & A. Goriely\n(iii) Experimental results\nWe performed experiments for helices made of spheres with a variety of sizes\nand magnetic strengths as well as varying the size of the seeding cylinder. We\ndo indeed observe that helices only self-assemble if the radius of the cylinder is\nsu\u000eciently large, as predicted by both the scaling argument, (4.5), and the more\ndetailed calculation that led to (4.8). However, contrary to the prediction of (4.8)\nthat self assembly will occur provided that N\u00157, we \fnd experimentally that\nN\u00159\u00061 is required. In \fgure 8 we observe a linear relationship between v2and\n1=N2and very little dependence on the size of the spheres, despite a doubling of the\nsphere diameter; both of these observations are in agreement with our theoretical\narguments.\nHowever, our theoretical estimate of the speed is about 3-4 times too large since\nit does not take into account the many dissipative e\u000bects present: the rolling of a\ncylinder made of discrete spheres (raising and lowering the centre of mass at each\nstep), the rolling friction with the table and between the spheres, the audible noise\ngenerated as new spheres are added to the cylinder (see Supplementary Information\nfor a video with soundtrack demonstrating this noise), the motion of the straight\nchain on the table along the z-axis, the force component along the y-axis, and\nso on. Rather than modelling the details of these dissipative e\u000bects, we use the\nexperimentally observed number for rolling Nc\u00199 to estimate that the e\u000eciency\nof conversion of magnetic energy into kinetic energy as spheres are brought into the\nhelix isE\u0019(6:17=9)2= 47%. Modifying (4.8) by setting \u000b!\u000bEwe see that our\ntheoretical estimate captures the main trends of the data, but still overestimates\nsigni\fcantly, with vtheo\u00192vexpo. We attribute the remaining discrepancy to the\nfact that our estimate of dissipative e\u000bects accounts only for e\u000bects that prevent the\ncylinder from rolling up in the \frst place (cf. static friction) while many dynamic\ndissipative e\u000bects also exist.\n5. Conclusions\nSystems of interacting spherical magnets constitute a conceptually simple and ex-\nperimentally testable physical system for which all the interactions between the\nconstituent units are known. The question addressed here, fundamental to many\nproblems in physics, is to relate the macroscopic behaviour to the microscopic inter-\nactions. Indeed, Cauchy and Poisson related the macroscopic elasticity of crystals\nto the microscopic interactions between molecules (Love, 1944). Here, we have pre-\nsented an example of this process by considering the simplest possible con\fgurations\nof a chain of dipoles and determined a macroscopic e\u000bective bending sti\u000bness. While\nthe fact that the magnetic dipoles in a magnetic chain should resist bending of the\nchain is obvious (see \fgure 3), the analogy is only mathematically exact in certain\nvery specialised geometries (Hall et al. , 2013). We have therefore investigated the\nutility of the simple physical notion of an e\u000bective `bending sti\u000bness' by attempting\nto understand three simple experiments using it. We have found that the concept of\nan e\u000bective bending sti\u000bness gives excellent results for the problems of the buckling\nof a standing column and the oscillations of a ring, while providing a qualitative\nunderstanding and the key features for the self-assembly of a chain into a cylindri-\ncal helix. However, detailed asymptotic calculations for other con\fgurations (Hall\nArticle submitted to Royal SocietyThe magneto-elastica : From self-buckling to self-assembly 17\n6 mm, 1.195 T3 mm, 1.350 T2a          B\n4 mm, 1.300 T\n5 mm, 1.400 T\n5 mm, 1.195 T\n0510150123413rspa.royalsocietypublishing.org Proc R Soc A 0000000..........................................................\n\u0001\u0003\u00026 mm, 1.195 T3 mm, 1.350 T2a          B\n4 mm, 1.300 T\n5 mm, 1.400 T\n5 mm, 1.195 T\n05101501234\nv2µ0ρ\nB2×103\nN−2×103Figure 7. Rescaled square of the terminal velocity of a self-assembling cylinder as a function of N−2. As described in the\ntext, the squared velocity scales linearly with N−2. The dotted line is the theoretical prediction (4.8)with the pre factor\nαmodified by the efficiency E(see main text). Here the different colours show different diameter spheres (smallest red,\nlargest black) and different symbols show different magnetizations of the spheres; details are as described in the legend.\nan effective bending stiffness gives excellent results for the problems of the buckling of a standing\ncolumn and the oscillations of a ring, while providing a qualitative understanding and the key\nfeatures for the self-assembly of a chain into a cylindrical helix. However, detailed asymptotic\ncalculations for other configurations [ 9] show that, in fact, this is not the whole story: interactions\nbetween widely spaced spheres can also play a role.\nA natural generalization of the questions addressed here is the elasticity of two-dimensional\nsheets of magnets for which the same ideas can be applied. Furthermore, other classical problems\nof materials science and solid mechanics can be tested on assemblies of magnetic beads including\nproblems of dislocations in sheets [ 14], fracture, zipping of chains to name but a few. We hope\nthat these systems will become both a testing ground for new ideas in physics and mathematical\nmethods as well as a perfect demonstration tool for classical physical problems.\nAcknowledgment\nThis publication is based in part upon work supported by Award No. KUK-C1-013-04, made\nby King Abdullah University of Science and Technology (KAUST). AG is a Wolfson/Royal\nSociety Merit Award Holder and acknowledges support from a Reintegration Grant under EC\nFramework VII.13rspa.royalsocietypublishing.org Proc R Soc A 0000000..........................................................\n\u0001\u0003\u00026 mm, 1.195 T3 mm, 1.350 T2a          B\n4 mm, 1.300 T\n5 mm, 1.400 T\n5 mm, 1.195 T\n05101501234\nv2µ0ρ\nB2×103\nN−2×103Figure 7. Rescaled square of the terminal velocity of a self-assembling cylinder as a function of N−2. As described in the\ntext, the squared velocity scales linearly with N−2. The dotted line is the theoretical prediction (4.8)with the pre factor\nαmodified by the efficiency E(see main text). Here the different colours show different diameter spheres (smallest red,\nlargest black) and different symbols show different magnetizations of the spheres; details are as described in the legend.\nan effective bending stiffness gives excellent results for the problems of the buckling of a standing\ncolumn and the oscillations of a ring, while providing a qualitative understanding and the key\nfeatures for the self-assembly of a chain into a cylindrical helix. However, detailed asymptotic\ncalculations for other configurations [ 9] show that, in fact, this is not the whole story: interactions\nbetween widely spaced spheres can also play a role.\nA natural generalization of the questions addressed here is the elasticity of two-dimensional\nsheets of magnets for which the same ideas can be applied. Furthermore, other classical problems\nof materials science and solid mechanics can be tested on assemblies of magnetic beads including\nproblems of dislocations in sheets [ 14], fracture, zipping of chains to name but a few. We hope\nthat these systems will become both a testing ground for new ideas in physics and mathematical\nmethods as well as a perfect demonstration tool for classical physical problems.\nAcknowledgment\nThis publication is based in part upon work supported by Award No. KUK-C1-013-04, made\nby King Abdullah University of Science and Technology (KAUST). AG is a Wolfson/Royal\nSociety Merit Award Holder and acknowledges support from a Reintegration Grant under EC\nFramework VII.\nFigure 8. Rescaled square of the terminal velocity of a self-assembling cylinder as a function\nofN\u00002. As described in the text, the squared velocity scales linearly with N\u00002. The dotted\nline is the theoretical prediction (4.8) with the pre factor \u000bmodi\fed by the e\u000eciency E\n(see main text). The symbols used to encode di\u000berent sphere sizes and magnetizations are\nas described in the legend.\net al. , 2013) show that, in fact, this is not the whole story: interactions between\nwidely spaced spheres can also play a role.\nThe idea of a magnetic bending sti\u000bness may have consequences for modelling\nthe mechanics of chains of ferromagnetic particles along the lines of those models de-\nveloped previously for chains of paramagnetic particles (Dreyfus et al. , 2005; Roper\net al. , 2006). However, we also highlight the possibility that the heavy magneto{\nelastica and oscillating ring experiments may be useful in their own right as simple\nassays through which the strength of spherical permanent magnets may be deter-\nmined. In particular, the oscillation frequency of a circular ring grows linearly with\nthe \feld strength Band so is quite sensitive to variations in B. Similarly, the num-\nber of beads that can be formed into a straight vertical chain without buckling\nunder its weight grows like B2=3and so is also sensitive to variations in B.\nA natural generalization of the questions addressed here is the elasticity of two-\ndimensional sheets of magnets for which the same ideas can be applied. Moreover,\nthe mathematical technique of approximating sums to obtain energies is applica-\nble to a wide range of problems in other branches of materials science, from self-\nassembly of particles to understanding the mechanical properties of crystals. Not\nonly do we expect that these mathematical approaches to particle interactions will\nhave broad applications in the physical sciences, but we have con\fdence that, in\nparticular, systems of magnets will become both a testing ground for new ideas in\nphysics and a perfect demonstration tool for classical physical problems.\nArticle submitted to Royal Society18 D. Vella, E. du Pontavice, C. L. Hall & A. Goriely\nAcknowledgment\nThis publication is based in part upon work supported by Award No. KUK-C1-013-\n04, made by King Abdullah University of Science and Technology (KAUST). AG\nis a Wolfson/Royal Society Merit Award Holder and acknowledges support from a\nReintegration Grant under EC Framework VII. AG and CH acknowledges support\nsupport from the EPSRC through Grant No. EP/I017070/1.\nAppendix A. The Euler{Maclaurin formula\nGiven a function, f2C2p[m\u000f;n\u000f ], the Euler{Maclaurin formula gives the following\nasymptotic result as \u000f!0:\nnX\ni=mf(i\u000f) =\u000f\u00001Zn\u000f\nm\u000ff(x) dx\u0000B1\u0002\nf(m\u000f) +f(n\u000f)\u0003\n+pX\nk=1B2k\u000f2k\u00001\n(2k)!h\nf(2k\u00001)(n\u000f)\u0000f(2k\u00001)(m\u000f)i\n+R(A 1)\nwhereBiare the Bernoulli numbers, B1=\u00001=2,B2= 1=6;B3= 0;:::;n\u0000mis an\nincreasing function of \u000f; andRis a remainder term given by\nR=\u000f2pZn\u000f\nm\u000fP2p\u0012\u001ax\u0000m\u000f\n\u000f\u001b\u0013\nf(2p)(x) dx; (A 2)\nwherePi(x) are the Bernoulli polynomials, P1(x) =x\u00001\n2,P2(x) =x2\u0000x+1\n6,\netc.andfxg=x\u0000bxcis the fractional part of x.\nReferences\nDreyfus, R., Baudry, J., Roper, M. L., Fermigier, M., Stone, H. A. &\nBibette, J. 2005 Microscopic arti\fcial swimmers. Nature 436, 862{865.\nHall, C. L., Vella, D. & Goriely, A. 2013 The mechanics of a chain or ring\nof spherical magnets. SIAM J. Appl. Math. 73, 2029{2054.\nHoppe, R. 1871 Vibrationen eines ringes in seiner ebene. J. Reine Angewand.\nMath. 73, 158{170.\nJackson, J. D. 1999 Classical Electrodynamics . Wiley.\nKnopp, K. 1990 Theory and application of in\fnite series . Dover.\nKu, J., Aruguete, D. M., Alivisatos, A. P. & Geissler, P. L. 2010 aSelf-\nassembly of magnetic nanoparticles in evaporating solution. J. Am. Chem. Soc.\n133(4), 838{848.\nKu, J.-Y., Aruguete, D. M., Alivisatos, A. P. & Geissler, P. L. 2010 b\nSelf-assembly of magnetic nanoparticles in evaporating solution. J. Amer. Chem.\nSoc.133, 838{848.\nArticle submitted to Royal SocietyThe magneto-elastica : From self-buckling to self-assembly 19\nLove, A. E. H. 1944 A Treatise on the Mathematical Theory of Elasticity . Dover.\nN.Vandewalle & S.Dorbolo 2013 Magnetic ghosts and monopoles.\narxiv:1308.5794v1 .\nOlver, F. W. J., Lozier, D. W., Boisvert, R. F. & Clark, C. W. 2010 NIST\nHandbook of Mathematical Functions . New York: Cambridge University Press.\nPerez, J. M., Simeone, F. J., Saeki, Y., Josephson, L. & Weissleder, R.\n2003 Viral-induced self-assembly of magnetic nanoparticles allows the detection\nof viral particles in biological media. Journal of the American Chemical Society\n125(34), 10192{10193.\nProkopieva, T. A., Danilov, V. A., Kantorovich, S. S. & Holm, C. 2009\nGround state structures in ferro\ruid monolayers. Phys. Rev. E 80, 031404.\nRoper, M., Dreyfus, R., Baudry, J., Fermigier, M., Bibette, J. & Stone,\nH. A. 2006 On the dynamics of magnetically driven elastic \flaments. J. Fluid\nMech. 554, 167{190.\nSun, S., Murray, C., Weller, D., Folks, L. & Moser, A. 2000 Monodisperse\nFePt nanoparticles and ferromagnetic FePt nanocrystal superlattices. Science\n287(5460), 1989{1992.\nVar\u0013on, M., Beleggia, M., Kasama, T., Harrison, R., Dunin-Borkowski,\nR., Puntes, V. & Frandsen, C. 2013 Dipolar magnetism in ordered and\ndisordered low-dimensional nanoparticle assemblies. Scienti\fc Reports 3.\nWang, C. Y. 1986 A critical review of the heavy elastica. Int. J. Mech. Sci 28,\n549{559.\nYavuz, C. F., Mayo, J. T., Yu, W. W. & Prakash, A. 2006 Low-\feld magnetic\nseparation of monodisperse fe 3o4nanocrystals. Science 314, 964{967.\nArticle submitted to Royal Society"    },    {        "title": "0907.4417v2.Magnetic_Nanoparticle_Assemblies.pdf",        "content": "arXiv:0907.4417v2  [cond-mat.mes-hall]  14 Apr 2010Magnetic Nanoparticle Assemblies\nDimitris Kechrakos∗\nOctober 31, 2018\nDepartment of Sciences, School of Pedagogical and Technologica l Education\n(ASPETE), Athens 14121, Greece\nAbstract\nThis chapter provides an introduction to the fundamental ph ysical\nideas and models relevant to the phenomenon of magnetic hyst eresis in\nnanoparticle assemblies. The concepts of single-domain pa rticles and su-\nperparamagnetism are discussed. The mechanisms of magneti zation by\ncoherent rotation and the role of temperature in the gradual decay of\nmagnetization are analyzed in the framework of simple analy tical models.\nModern numerical techniques (Monte Carlo simulations, Mag netization\nDynamics) used to study dense nanoparticle assemblies are p resented. An\noverview of the most common experimental techniques used to measure\nthe magnetic hysteresis effect in nanoparticle assemblies a re presented and\nthe underlying principles are exposed.\nKeywords : magnetic nanoparticles; magnetic anisotropy; dipolar interaction s;\nmagnetic hysteresis; superparamagnetism; mean field theory; Mo nte Carlo;\nmagnetization dynamics\n∗Contact author: Tel : +30-210-2896705, Fax : +30-210-28967 13, E-mail :\ndkehrakos@aspete.gr\n1Contents\n1 Introduction 3\n2 Isolated magnetic nanoparticles 5\n2.1 Single-Domain Particles . . . . . . . . . . . . . . . . . . . . . . . 5\n2.2 Magnetization by Coherent Rotation . . . . . . . . . . . . . . . . 8\n2.3 Magnetic behavior at finite temperature . . . . . . . . . . . . . . 11\n2.3.1 Superparamagnetism and Blocking temperature . . . . . . 11\n2.3.2 Thermal relaxation under an applied field . . . . . . . . . 13\n3 Magnetic Measurements 16\n3.1 Field-cooled (FC) and Zero-Field-Cooled (ZFC) Magnetization . 17\n3.2 Remanent magnetization and Coercive field . . . . . . . . . . . . 17\n4 Interacting nanoparticle assemblies 19\n4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19\n4.2 Mean Field models . . . . . . . . . . . . . . . . . . . . . . . . . . 21\n4.3 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . . . 23\n4.3.1 The Monte Carlo method . . . . . . . . . . . . . . . . . . 23\n4.3.2 The Magnetization Dynamics method . . . . . . . . . . . 24\n4.3.3 Time scale of numerical methods . . . . . . . . . . . . . . 24\n4.3.4 MMC study of dipolar interacting assemblies : A case study 25\n5 Summary 27\n6 Future perspectives 27\n21 Introduction\nMagnetic nanoparticles (MNPs) are minute parts of magnetic mater ials with\ntypical size well below 10−7m. They are present in different materials found\nin nature such as rocks, living organisms, ceramics and corrosion pr oducts,\nbut they are also artificially made and used as the active component o f fer-\nrofluids, permanent magnets, soft magnetic materials, biomedical materials and\ncatalysts. Their diverse applications in geology, physics chemistry, biology and\nmedicine renders the study of their properties of great importanc e both to sci-\nence and technology.\nIn geology, the nature and origin of magnetic phenomena related to the\npresence of magnetite nanoparticles in rocks is of great interest t o the palaeo-\nmagnetist who searches for the geomagnetic record of rocks. Th e presence of\nmagnetite particles associated with the trigeminal nerve in pigeons o ffers a re-\nliable explanation to the Earth’s magnetic field detection and the cons equent\nnavigation capability. In fine arts, magnetic analysis of ancient paint ings facil-\nitates the reconstruction of the production techniques of ancien t ceramics. In\nliving organisms, the role of ferritin, a magnetic nanoparticle per se, is impor-\ntant among the iron storage proteins. MNPs are also used as contr ast agents\nin Magnetic Resonance Imaging. Recent work has involved the develo pment of\nbioconjugated MNPs, which facilitated specific targeting of these M RI probes\nto brain tumors. MNPs are also used as highly active catalysts which h as long\nbeen demonstrated by the the use of finely divided metals in several reactions.\nOwing to their high surface-to-volume ratio MNPs of iron are more effi cient at\nwaste remediation than bulk iron.\nHigh density magnetic data storage media provide a major technolog ical\ndriving force for further exploration of MNPs. It is expected that if MNPs\nwith diameter 5 nmcan be used as individually addressed magnetic bits, mag-\nnetic data storage densities of 1 Tbit/in2would be achieved, namely an order\nof magnitude higher than the present record (Moser 2002). MNPs have also\nbeen demonstrated to be functional elements in magneto-optical switches, sen-\nsors based on Giant Magneto-Resistance and magnetically controlla ble Single\nElectron Transistor devices.\nThe most common preparation methods for MNPs produce assemblie s with\ndifferent structural and compositional characteristics that dep end on the par-\nticular method adopted. Granular films, ferrofluids and cluster-as sembled films\nare characterized as assemblies with random order in MNP locations, while or-\ndered arrays are found in patterned media (known also as magnetic dots) and\nself-assembled films. The MNP preparation methods are divided to to p-down\nand bottom-up. In top-down methods , the NPs are formed from a larger sys-\ntem by appropriate physical processing, such as thermal treatm ent, etching,\netc. In bottom-up methods, the NPs are formed by an atomic nucle ation pro-\ncess that takes place either in ultrahigh vacuum or in a liquid environme nt.\nThe latter method relies on colloidal chemistry techniques and prese ntly ap-\npears to be the most promising method for production of nanopart icles with\nextremely narrow size distribution. Colloidal synthesis methods com bined with\n3self-assembly methods produce MNP samples with both size uniformit y and long\nrange structural order. It is worth noticing that structural or der in a MNP as-\nsembly is a decisive property for production of ultrahigh density sto rage media.\nOwing to their attractive features and their low cost, colloidal synt hesis meth-\nods and self-assembly attract presently intense research activit y in the field of\nMNP preparation (Petit 1998, Murray 2001, Willard 2004, Farrell 20 05, Darling\n2005).\nThe magnetic properties of MNPs and their assemblies provide a fasc inating\nfield for basic research, which is done on two different scales, the at omic and\nthe mesoscopic. In the atomic scale, the properties of individual MN Ps are\nexamined and they are revealed in samples with low particle concentra tion.\nIn the mesoscopic scale, dense samples are examined which exhibit co llective\nmagnetic behavior arising from interparticle interactions. The stud y of the\nmagnetic properties can be naturally divided in the investigation of th e ground\nstate configuration (long range order, disorder, etc) and the ex citations from it.\nExcitations can be either weak, as for example at low temperature a nd weak\nexternal magnetic field, or strong, as for example, close to a ther mal phase\ntransition or under a reversing magnetic field.\nFor individual MNPs the ground state configuration can differ remar kably\nfrom the parent bulk material in various ways. For example, owing to energy\nbalance reasons, the abundance of magnetic domains that form in a bulk magnet\ncan be replaced by a single domain in a MNP, which then becomes magnet ically\nsaturated even in the absence of an external magnetic field (N´ ee l 1949). The\napplication of an external field forces the atomic magnetic moments of a single-\ndomain MNP to rotate coherently (Stoner 1948). Also, for temper ature above a\nthreshold, the direction of particle’s magnetization fluctuates at r andom, making\nthe particle bahave as a molecule with a giant magnetic moment. The ap pli-\ncations of this effect, known as superparamagnetism (Bean 1959) , are presently\na lot, ranging from geology to medicine. Finally, we should remark that the\nabove described simplified picture of a single-domain MNP becomes inva lid if\none considers the crucial effect of the MNP surface. Reduced cry stal symmetry\nand chemical disorder close to the surface can produce variations between the\nsurface and interior magnetic structure and modify the overall re sponse of the\nMNP to an applied field (Kodama 1999).\nWhen MNPs form dense assemblies, interparticle interactions produ ce a col-\nlective behavior, by coupling the magnetic moments of individual MNPs . This\nfact renders in most cases even the determination of the ground s tate configura-\ntion an intricate physical problem. The collective behavior of dense ( interacting)\nassemblies is reflected also on the modified magnetic response of the assembly,\ncompared to isolated MNPs. The most complex behavior occurs in sam ples\nwith random morphology and long-range magnetostatic interaction s. Various\nexperimental measurements have been proposed to reveal the n ature of the in-\nterparticle interactions, and various measuring protocols probe d ifferent aspects\nof the collective behavior. On the other hand, analytical models hav e difficul-\nties in predicting or explaining the magnetic behavior of these interac ting MNP\nassemblies, and most of the curret research relies on numerical sim ulations.\n4In this chapter we provide an introduction to the fundamental idea s and\nconcepts pertaining to the magnetic properties of MNP assemblies. Emphasis\nis given to the response of MNP assemblies to an applied magnetic field a nd the\nrelated issue of magnetization reversal. The chapter is organized a s follows :\nIn Section 2 we discuss the magnetic properties of individual (isolate d) MNPs.\nFist, the condition under which a single-domain MNP is formed is derived , and\nthen the magnetic response under an applied field is examined. The pr esentation\nis based on a simple theoretical model (N´ eel 1949, Stoner 1948). In Section 3 we\ngive a brief overview of the most common magnetic characterization techniques\nand explain the information extracted from each one. In Section 4 w e discuss the\nresponse of a dense MNP assembly to a magnetic field, when the inter particle\ninteractions are important and lead to a collective behavior of the MN Ps. Mean-\nfield models are presented and an introduction to modern numerical techniques\n(Monte Carlo, Magnetization dynamics) to tackle this problem are pr esented.\nThe chapter is summarized in Section 5 and the perspectives in this fie ld are\npresented in Sections 6.\n2 Isolated magnetic nanoparticles\nIn this section we derive the criterion for formation of single-domain MNPs and\nexamine the magnetization process at zero temperature by coher ent rotation of\nmagnetization (Stoner-Wohlfarth model). The behavior of a MNP as sembly at\nfinite temperature is discussed and the related concepts of super paramagnetism\nand blocking temperature are introduced. The effects of an applied dc magnetic\nfield is examined within the simplest model assuming uniaxial anisotropy and\nbistability of particle moments (N´ eel model).\n2.1 Single-Domain Particles\nThe ground state magnetic structure of a ferromagnetic (FM) ma terial is the\noutcome of the balance between three different types of energies , namely, the\nexchange ( Uex), the magnetostatic ( Um) and the anisotropy energy ( Ua). The\nexchange interaction has its origin in the Pauli exclusion principle for e lectrons.\nLet the FM material be divided in small cubic elements each one carryin g a\nmagnetic moment− →µi. The exchange interaction between the cubic elements\nfavors parallel alignment of neighboring magnetic moments and it is wr itten in\nthe usual Heisenberg form as Uex=−(A/a2)/summationtext\nijcosθij, whereAis the stiffness\nconstant, ais the lattice constant and θijis the angle between moments at sites\niandj. The stiffness constant is related to the microscopic exchange ene rgyJ\nthrough the relation A=zJS2/a, whereSis the atomic spin and z= 1,2,4\nfor sc, fcc and bcc lattice, respectively. The magnetostatic ener gy, is the sum of\nCoulomb energies between the magnetic moments comprising the FM m aterial.\nIt can be expressed as Um=−µ0− →Hd·− →M/2, where Hdis thedemagnetizing\nfield and Mthe sample magnetization. The anisotropy energy, is the energy\nrequired to orient the magnetization at an angle ( θ) relative to certain fixed axes\n5of the system, known as the easy axes. The microscopic mechanisms leading\nto anisotropy can be quite diverse and the most common types of an isotropy\nfound in FMs are as follows:\n(i)Crystal anisotropy. It arises from the combined effects of spin-orbit cou-\npling and quenching of the orbital momentum that produce a prefer red\norientation of the magnetization along a symmetry axes of the unde rly-\ning crystal. For a uniaxial materials (e.g. hexagonal Co) it has the fo rm\nUa=K1sin2θ+K2sin4θ+..., whereK1,K2,...are the anisotropy con-\nstants, and θthe angle between the magnetization direction and the easy\naxis. Typical values for cobalt are K1= 4.5×106J/m3andK2=\n1.5×105J/m3. For cubic crystals (e.g. fcc Fe, Ni) it reads Ua=\nK1(a2\n1a2\n2+a2\n2a2\n3+a2\n3a2\n1) +K2a2\n1a2\n2a2\n3+..., wherea1,a2,a3are the direc-\ntion cosines of the magnetization direction. Typical values for Fe ar e\nK1= 4.8×104J/m3andK2=±0.5×104J/m3.\n(ii)Stress anisotropy. It is produced by the presence of stress in the sample\nand it has a uniaxial character Ua=Kσsin2θ, whereKσ=3\n2λiσ, withλi\nthe magnetically induced isotropic strain and σthe stress.\n(iii)Surface anisotropy. This is caused by the presence of sample free bound-\naries, where the reduced symmetry and the presence of defects can induce\nadditional anisotropy. It is important in MNPs because of the subst antial\nsurface-to-volume ratio.\n(iv)Shape anisotropy. This occurs because on one hand the demagnetizing\nfield depends on the shape of the magnetized body and takes the low -\nest value along the longest axis of the sample, and on the other hand ,\nUmis minimized when Mis parallel to Hd. As an example, consider a\nspecimen in the shape of prolate spheroid with major axis cand minor\naxisa, magnetized at an angle θwith respect to c-axis. Then, Um=\nµ0\n2/bracketleftbig\nNc(Mcosθ)2+Na(Msinθ)2/bracketrightbig\n=1\n2(Nc−Na)M2sin2θ, whereNcand\nNaare thedemagnetizing factors along the corresponding axes. This ex-\npression for Umhas the form of uniaxial anisotropy with Ks=1\n2(Nc−\nNa)M2. Typical cases, are a spherical specimen with Ks= 0, an infinitely\nthin planar specimen with N/bardbl= 0 (in-plane) and N⊥= 1, and a infinitely\nlong (needle-shaped) specimen with N/bardbl=1\n2(along the axis) and N⊥= 0.\nIn studies of the magnetic properties of MNPs, it is a common practic e, to\ndescribe, within the simplest approximation, the overall effect of th e various\nanisotropy types by an effective uniaxial anisotropy term Ua=Keffsin2θ.\nThe constant Keffaccounts for the total effect of crystalline, surface and shape\nanisotropy.\nA bulk FM material is composed of many uniformly magnetized regions\n(domains ). The direction of magnetization in different domains varies, and in\na bulk sample it is randomly distributed leading to a non-magnetized sam ple\neven at temperatures far below the Curie point. The formation of m agnetic\ndomains in FM materials results from the competition between the exc hange and\n6Figure 1: One-dimensional model of a FM. (a) Long-range order. ( b) An in-\nfinitely thin DW (dashed line). The increase of exchange energy at th e wall is\nhigher than the decrease of the magnetostatic energy. (c) A 1800domain wall\nspread over N= 10 sites. The gradual rotation of atomic moments produces a\nstate with lower total energy compared to (b).\nthe magnetostatic energy. The former favors perfect alignment of neighboring\nmoments and the latter is reduced by breaking a uniformly magnetize d body\ninto as many as possible regions with opposite magnetization direction s. The\noutcome of this competition is the formation of a certain number of d omains\nin a sample with a particular orientation of the magnetization direction s. A\ntypical domain size in a bulk ferromagnet is 1 µm.\nNeighboring magnetic domains are separated by a region where the lo -\ncal magnetization changes gradually direction between the two opp osite sides,\nknown as domain wall (DW). Domain walls have finite width ( δw) determined\nby the balance between the exchange and anisotropy energy. As a n example,\nconsider an one-dimensional model of a DW in a uniaxial material, wher e a 1800\nrotation of magnetization is distributed over N sites, as shown in Fig. 1. The\ntotal energy per unit area reads\nσ(N) =σex+σa=JS2(π/N)2(N/a2) +NaK1. (1)\nMinimization with respect to Nleads to\nδw=Na=π(A/K1)1/2. (2)\nFor a typical exchange stiffness value ( A≈10−11J/m), Eq.(2) predicts for iron\nδw≈0.4µmwhile for a magnetically harder material like cobalt, δw≈60nm.\nSubstituting the result of Eq.(2) into in Eq.(1) provides the areal en ergy density\nof the DW\nσw= 2π(AK1)1/2(3)\n7Consider a finite sample of a FM material, with size d. As the size of the\nsample is reduced, the number of DWs it contains decreases, becau se fewer\nregions with opposite directions of magnetization are required to re duce the\nmagnetostatic energy. Below a critical value of the system size, th e sample does\nnot contain any DW and it is in a single domain (SD) state exhibiting satu-\nration magnetization ( Ms). For a spherical particle, the critical diameter ( dc)\ncan be estimated as follows: the SD state is stable when the energy n eeded to\ncreate a DW that spans the whole particle, Uw=σwπr2, is greater than the\nmagnetostatic energy gain from the reduction to a multidomain stat e, which is\napproximately equal to the magnetostatic energy stored in a unifo rmly mag-\nnetized sphere, Um=1\n3µ0M2\nsV, withMsthe saturation magnetization and\nV=4π\n3r3. The condition Uw=Umprovides\nrc= 9(AK1)1/2\nµ0M2s(4)\nFor Fe, this approximation gives rc≈3nm, which is by far too small. The rea-\nson is that the DW is assumed to have the same one-dimensional stru cture as in\nthe bulk material. An improved calculation that considers a three-dim ensional\nconfinement of the DW provides for the critical radius:\nrc=/radicalBigg\n9A\nµ0M2s/bracketleftbigg\nln/parenleftbigg2rc\na/parenrightbigg\n−1/bracketrightbigg\n(5)\nIn the case of Fe, numerical solution of Eq.(5) gives rc≈25nm, which is\nvery close to more accurate micromagnetic calculations and the exp erimentally\nobtained value (Cullity 1972).\n2.2 Magnetization by Coherent Rotation\nThe magnetization ( M) of a bulk FM crystal that contains many magnetic do-\nmains, changes under application of an external magnetic field ( H), a process\nknown as technical magnetization . However, the value of Mis not a unique\nfunction of Hand the state of the sample prior to application of the field is\nimportant. This is the phenomenon of magnetic hysteresis , which is commonly\ndepicted by drawing the M−Hdependence under a cyclic variation of the field\nfrom a positive to a negative and back to a positive saturation value ( hysteresis\nloop). Two important characteristic values of a hysteresis loop are the rema-\nnence (Mr), namely the magnetization after removal of the saturating field, and\nthecoercivity (Hc), namely the field required for the magnetization to vanish.\nIn a bulk FM crystal, the magnetization proceeds by two basic mecha nisms,\nnamely domain wall motion (weak fields) and rotation of magnetization (strong\nfields).\nIn MNPs, the change of magnetization under an applied field proceed s only\nby rotation, because formation of DWs is energetically unfavorable . During the\nmagnetization rotation the atomic moments of the MNP remain paralle l to each\n8other and the MNP behaves as a giant molecule carrying a magnetic mo ment\nof a few thousand Bohr magnetons ( µ∼104µBfor a 5nmdiameter Fe MNP).\nThis process of magnetization is known as coherent rotation or Stoner-Wohlfarth\n(SW) model, after the authors who introduced and solved it (Stone r 1948). We\ndiscuss it briefly next. Consider a MNP with uniaxial (effective) anisot ropyK1\nalong an easy axis taken to be the z-axis (Fig. 2). For an applied field that\nmakes an angle θ0with the easy axis, we wish to determine the equilibrium\nposition of the magnetic moment µ=MsV. Let− →µmake an angle θwith the\neasy axis, then the total energy density reads\nu=−K1cos2(θ−θ0)−µ0HMscosθ (6)\nThe equilibrium condition (zero-torque) is\ndu\ndθ= 0⇒2K1sin(θ−θ0) cos(θ−θ0) +µ0HMssinθ= 0 (7)\nand introducing the dimensionless quantity h=H/Hawith the anisotropy field\nHa= 2K1/µ0Ms, Eq.(7) becomes\nsin (2(θ−θ0)) + 2hsinθ= 0. (8)\nWe define the reduced magnetization along the field m=µcosθ/MsV=cosθ\nand the solution of Eq.(8) is written as\n2m(1−m2)1/2cos 2θ0+ sin 2θ0(1−2m2) + 2h(1−m2)1/2= 0 (9)\nThe remanence ( h= 0) and coercivity ( m= 0) are readily obtained from Eq.(9)\nas\nmr= cosθ0andhc= sinθ0cosθ0. (10)\nFor non-zero field values, Eq.(9) is solved for has a function of mand the data\nare shown in Fig. 2. Consider the two extreme cases, namely for θ0= 900\n(hard-axis magnetization) and θ0= 00(easy-axis magnetization). In the former\ncase, the magnetization shows zero coercivity and a linear field depe ndence.\nIn the latter case, the magnetization remains constant until the r eversing field\nbecomes equal to the anisotropy field, and then an irreversible jump of the\nreduced magnetization from m= +1 to m=−1 is seen. These extreme cases\ndemonstrate the distinct mechanism of switching by rotation that c an occur in\nan assembly. More generally, at an arbitrary field angle, an irrevers ible jump\nof the magnetization occurs at the so called switching field (Hs) defined as the\nfield value satisfying dm/dh→ ∞ . AtH=Hsthe local minimum of the total\nenergy, corresponding to the higher energy state (magnetizatio n opposite to the\napplied field) disappears and the system jumps to the remaining minimu m that\ncorresponds to a magnetization direction along the field (see Fig. 3) . In other\nwords,Hsis an instability point of the total energy and it satisfies du/dθ = 0\nandd2u/dθ2= 0. In the SW model, the stability condition reads\nd2u\ndθ2= 0⇒cos 2(θ−θ0)±hsinθ= 0 (11)\n9/s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48/s48 \n/s51/s48/s48 \n/s54/s48/s48 /s56/s48/s48 /s32/s77/s47/s77\n/s83\n/s72/s47/s72\n/s97 /s57/s48/s48 /s72/s43/s122 \n/s77 \n/s83 \n/s111/s40/s97/s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s40/s98/s41\nFigure 2: (a) Sketch of a magnetic nanoparticle with uniaxial anisotr opy along\nthez-axis and an applied field at an angle ( θ0) with respect to the easy axis.\n(b) Magnetization curves within the Stoner-Wohlfarth model for v arious field\ndirections. The initial direction of the magnetization is taken along th e field.\nFrom Eqs.(8) and (11) we obtain for the switching field hs=Hs/Ha\nhs= (cos2/3θ0+ sin2/3θ0)−3/2(12)\nBy comparison of Eqs.(10) and (12) one finds that hc< hsfor 450< θ0<900,\nnamely switching happens after the magnetization changes sign, wh ile for field\nangles close to the easy axis, 00< θ0<450, the magnetization changes sign\nby an irreversible jump ( hc=hs). The physical distinction between hsand\nhccan be understood by the following example. Consider a SW particle un der\napplication of a reversing field h=hc, which brings the particle’s moment− →µin\na direction perpendicular to the field, so that m= 0. Then the field is switched\noff adiabatically. If hc< hs(i.e. 450< θ0<900),− →µwill return back to the\npositive remanence value ( m= +1), while if hc=hs(i.e. 00< θ0<450),− →µwill jump to the negative remanence state ( m=−1). The switching field\nof a hard (i.e. large anisotropy) magnetic material is a physical quan tity with\ngreat technological interest in magnetic recording applications. In these, the\ninformation bit is stored in the direction of magnetization and the swit ching\nfield is the field required to write or erase this information.\nStoner and Wohlfarth (Stoner 1948) also studied an assembly of iso lated\nMNPs with easy axes directions distributed uniformly on a sphere ( random\nanisotropy model , RIM). The reported values for the remanence and coercivity\nare\nmr= 0.5 and hc= 0.48. (13)\nThis result is particularly useful as random easy axis distribution is fo und\n10/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s45/s52/s45/s51/s45/s50/s45/s49/s48/s49/s50\n/s111/s61/s48\n/s48/s46/s53/s48/s46/s50/s48/s46/s48\n/s49/s46/s48\n/s32/s82/s101/s100/s117/s99/s101/s100/s32/s69/s110/s101/s114/s103/s121/s32/s40/s85/s32/s47/s32/s75\n/s49/s86/s41\n/s32/s47/s32 /s40/s114/s97/s100/s41/s49/s46/s53\nFigure 3: Dependence of total energy on the direction of the part icle’s moment\n(see Eq.(6)), for various strengths of the applied field ( h=H/Ha). The energy\nminimum at θ=πbecomes unstable at the switching field hs= 1.\nin most MNP-based materials (granular films, cluster-assembled films , self-\nassembled arrays, etc)\nAs a final remark, we remind that in the SW model thermal effects ar e\nignored ( T= 0), thus energy-minimization with respect to the magnetic moment\ndirection is a sufficient condition to determine the field-dependent ma gnetization\nat equilibrium. The magnetic behavior of SD particles at finite tempera ture is\ndiscussed in the following section.\n2.3 Magnetic behavior at finite temperature\nHow do thermal fluctuations affect the average magnetization dire ction of an\nisolated MNP ? How does the presence of an applied field modify the mag netic\nresponse at finite temperature ? Is the assembly magnetization st able in time,\nwhen the MNP moment are subject to thermal fluctuations ? These points are\nbriefly discussed next, along the lines of a model first studied by N´ e el (N´ eel\n1949).\n2.3.1 Superparamagnetism and Blocking temperature\nConsider an assembly of identical SD particles with uniaxial anisotrop y. The en-\nergy (per particle) is U=−K1Vcos2θ, whereθis the angle between the single\nparticle magnetic moment− →µand the easy axis. The energy barrier that must be\novercome for a MNP to rotate its magnetization is Eb=K1V. As first pointed\nout by N´ eel (N´ eel 1949), thermal fluctuations could provide th e required energy\nto overcome the anisotropy barrier and spontaneously (i.e. withou t externally\n11applied field) reverse the magnetization of a MNP from one easy direc tion to the\nother. This phenomenon can be thought of as a Brownian motion of a particle’s\nmagnetic moment. The assembly shows paramagnetic behavior, how ever it is\nthe giant moments of the MNPs that fluctuate rather than the ato mic moments\nof a classical bulk paramagnetic material. This magnetic behavior of t he MNPs\nis called superparamagnetism (SPM) (Bean 1959) At high enough temperature,\nkBT >> K 1V, the anisotropy energy can be neglected and the assembly magne-\ntization can be described by the well known Langevin function M=nMs/suppress L(x),\nwherenis the particle number density, and x=µ0µH/kBT. Thus, the features\nserving as signature of superparamagnetism are the scaling of mag netization\ncurves with H/T , as dictated by the Langevin function, and the lack of hys-\nteresis, i.e. vanishing remanence and coercivity. Moreover, the ma jor difference\nbetween classical paramagnetism of bulk materials and SPM is the wea k fields\n(H∼0.1T) required to achieve saturation of a MNP assembly magnetization\nM. This occurs because of the large particle moment ( µ∼104µB) compared\nto the atomic moments ( µat∼µB).\nMeasurement of magnetization curves at sufficiently high temperat ure can,\nin principle, be used to extract the particle moment µ. In practice, two compli-\ncation arise. First, the presence of different particle sizes in any sa mple produces\na convolution of the Langevin function with the volume distribution fu nction.\nSecond, interparticle interactions, modify the reversal mechanis m and the SW\nmodel needs extensions, which are discussed in the Section 4.\nAt low temperature, kBT << K 1V, the anisotropy barriers are very rarely\novercome (weak thermal fluctuations), the assembly shows hyst eresis and this\nis called the blocked state.\nOne might now ask, whether there exists a temperature value that draws the\nborder between the blocked and the SPM state. Following N´ eel’s arg uments,\nwe assume that thermal activation over the anisotropy barrier ca n be described\nwithin the relaxation time approximation (or Arrhenius law ) as\nτ=τ0exp(K1V/kBT), (14)\nwhere 1/2τis the probability per unit time for a reversal of− →µ. The intrinsic\ntimeτ0depends on the material parameters (magnetostriction constan t, Young\nmodulus, anisotropy constant and saturation magnetization). Ty pical values\nareτ0∼10−10−10−9sas obtained by N´ eel. To detect the superparamagnetic\nbehavior experimentally, the MNP must be probed for a long enough p eriod of\ntime to perform many switching events that would produce a vanishin g small\ntime-average magnetic moment. If τmis the measuring time-window, the con-\ndition for SPM behavior is τm≫τ. The strong (exponential) dependence of τ\non temperature (see Eq.(14)) permits us to define a temperature value (or more\nprecisely, a very narrow temperature range) above which the rela xation time is\nso small that SPM behavior is observed. This is called the blocking temperature\n(Tb) of the assembly, and is given by\nTb=K1V/kBln(τm/τ). (15)\n12ForT < T b, the particle moments fluctuate without switching direction (on av-\nerage) and the assembly is in the blocked state exhibiting hysteresis . ForT > T b\nthe assembly is in the SPM state, hysteresis disappears and therma l equilibrium\nis established. It is remarkable, that the value of Tbdepends on τm, which is a\ncharacteristic of the experimental technique adopted. For exam ple, in dc sus-\nceptibility measurements τm≈100s, in ac susceptibility τm≈10−8−104s,\nin M¨ ossbauer spectroscopy τm≈10−9−10−7sand in neutron spectroscopy\nτm≈10−12−10−8s. Therefore, if Tbis of interest for a particular application,\nthe measurement technique implemented must imitate the real cond itions. For\nexample, to study the reliability of magnetic storage media, dc magne tic mea-\nsurements over a wide time window ( τm∼102−104s) should be used, while\nto study magnetic recording speed, ac measurements are approp riate.\nBrown (Brown 1963) extended the treatment of thermal activat ion over the\nanisotropy barrier, allowing also for fluctuations of µtransverse to the easy axis,\nwhich N´ eel has neglected, and obtained a different expression for τ0. However,\nthe common feature of both studies is the temperature and volume dependence\nofτ, so the final result, Eq.(14), is referred to as the N´ eel-Brown model.\nIn apolydisperse assembly, the distribution of particle volumes f(V), pro-\nduces a corresponding distribution of blocking temperatures f(Tb). Then, at\na certain temperature Tthe assembly contains a mixture of blocked and SPM\nparticles. The MNPs with volumes above a critical value Vc, fulfill the require-\nment of strong thermal energy with respect to their anisotropy b arrier, and are\nSPM, while those while those with V≤Vcare blocked. From Eq.(14), the\ncritical volume reads Vc=kbTln(τm/τ0)/K1. As explained above for Tb, also\nforVcthe experimental determination depends on the technique adopte d. Most\npreparation techniques result in polydisperse samples and the prob lem of ex-\ntracting the size distribution function from magnetic measurement s, pioneered\nby Bean and Jacobs more than fifty years ago (Bean 1956) remains a difficult\ntask mainly due to the complications introduced by interparticle inter actions.\nKnobel and colleagues have recently reviewed this subject (Knobe l 2008).\n2.3.2 Thermal relaxation under an applied field\nConsider an assembly of Nidentical MNPs with uniaxial anisotropy along\nthez-axis and let their moments point initially along the + z-axis. Assume\nthat a magnetic field H, weaker than the switching field , which is equal to\nHa, is applied along the −z-axis. Then, the total energy per particle reads,\nU=−K1Vcos2(θ−θ0) +µ0HMsVcosθ. It exhibits two non-equivalent local\nminima at θ= 0,πwith values U±=−K1V±MsVH and a maximum at\nθ=π/2 withUmax=K1V(H/Ha)2, as shown in Fig. 4. The energy barriers\nand the corresponding relaxation times for the forward (+) and th e backward\n(−) rotations are\nE±\nb(H) =K1V(1∓H/Ha)2andτ±=τ0exp(E±\nb/kBT). (16)\nThe change of τ0due to the field is much weaker than the change of the expo-\nnential factor and as such it is neglected in the above equation.\n13/s75\n/s49/s86/s45\n/s32/s85\n/s109 /s97/s120\n/s85\n/s45/s85\n/s43/s69/s32/s43\n/s98 \n/s69/s32/s45\n/s98 /s43/s85\n/s48/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32\nFigure 4: Total energy of an isolated particle with uniaxial anisotrop y subject to\na negative field parallel to the easy axis with value less than the switch ing field\n(0< H < H s). Energy barriers ( Eb) and relaxation times ( τ) for the forward\n(+) and the backward ( −) process are not equal.\nThe blocking temperature, as measured within a time-window τm, is reached\nwhen the observation time equals the forward relaxation time τ+, because the\nlatter corresponds to a moment flip from the initial state along + zto the op-\nposite direction, namely a process that reduces the initial magnetiz ation. From\nEq.(16) one obtains\nTb(H) =K1V(1−H/Ha)2\nkBln(τm/τ)≡Tb(0)(1−H/Ha)2. (17)\nwhich indicates that the blocking temperature is reduced by the pre sence of\na reverse field. By completely symmetric arguments one could show t hatTb\nincreases in the presence of a field with the same direction as the initia l magne-\ntization.\nSince thermal fluctuations act in synergy to a reverse field in switch ing the\nmoment of a MNP, it is expected that the coercivity of an assembly will decay\nwith temperature. As discussed above, for a particle with its momen t along\nthe +z-axis, a reverse field (0 < H < H a) reduces the barrier for reversal to\nthe value E+\nbgiven in Eq.(16). If the field is strong enough, it will reduce\nthe barrier to the value appropriate for superparamagnetic relax ation, namely\nkBTln(τm/τ0), and the (time-average) magnetization will vanish. On the other\nhand, the reverse field that makes the magnetization vanish is by de finition the\ncoercive field. Therefore, the following relation holds\nK1V(1−Hc/Ha)2=kBTln(τm/τ0) (18)\nwhich, using Eq.(15), provides the temperature dependent coerc ivity\nHc(T) =Ha/bracketleftBig\n1−(T/Tb)1/2/bracketrightBig\n. (19)\n14The microscopic mechanism of thermal activation of the MNP moment\nover the anisotropy barrier, produces a macroscopically measure d time-decay\nof the magnetization. We derive this dependence assuming that whe n a mo-\nment switches direction it continues to remain along the easy axis (N´ eel 1949).\nThen, at time t,N+particles occupy the lower minimum at θ= 0, and the\nrestN−=N−N+particles occupy the higher minimum at θ=π. The time-\nevolution of N+is governed by the rate equation\ndN+\ndt=−N+\nτ++N−\nτ−. (20)\nThe magnetization per particle is given as M(t)≡(2N+(t)/N−1)Ms, and\nsolution of Eq.(20) provides\nM(t) =M∞+ (M0−M∞) exp(−t/τ) (21)\nwith 1/τ= 1/τ++ 1/τ−being the reduced relaxation time and\nM∞=τ+−τ\nτ++τ−MsandM0=/parenleftbigg2N+(0)\nN−(0)−1/parenrightbigg\nMs (22)\nthe time-asymptote and initial values of the particle magnetization, respectively.\nEq.(21) indicates that the magnetization decays exponentially towa rds the equi-\nlibrium value M∞, reached as t→ ∞ . In other words, equilibrium is reached\nwhen the population of the energy minima is proportional to the corr esponding\nrelaxation times ( N+/N−=τ+/τ−), as dictated by Eq.(20). When the applied\nfield is strong enough ( H > H s) to produce only one minimum, thermal equilib-\nrium is always reached. Obviously, in the absence of an external field , thermal\nequilibrium is reached when the two equivalent minima are equally populat ed\n(N+=N−), resulting in a vanishing magnetization.\nNotice that in Eq.(20) we assumed bistability of the moment direction, which\nis a valid approximation provided the anisotropy barrier is high ( K1V≫kBT).\nFor lower anisotropy barriers or elevated temperature ( K1V≈kBT), the trans-\nverse fluctuations of− →µ, or, in other words, intra-valley motion around the\nenergy minimum should be taken into account. A general treatment of thermal\nrelaxation of SD MNPs was pioneered by Brown (Brown 1963) and ext ended\nto the case of an applied external field (Aharoni 1965, Coffey 1998 , Garannin\n1999).\nIf an assembly is polydisperse, characterized by a volume distributio nf(V),\na distribution of blocking temperatures f(Tb) exists. However, it remains still\nunclear if the mean value < Tb>is the appropriate blocking temperature of\nthe assembly, which should be substituted, for example, in Eq.(19). This point\nis discussed further in the literature (Nunes et al 2004).\nIn a polydisperse assembly, a distribution of relaxation times f(τ) exists,\nwithf(τ)d(lnτ) the probability of a MNP to have ln τin the range (ln τ,lnτ+dlnτ)\nand the normalization condition/integraltext∞\n0f(τ)d(lnτ) = 1. In this case the magneti-\nzation can be obtained by a superposition of the single-particle magn etization\n15properly weighted, as follows\nM(t) =Ms/integraldisplay∞\n0[1−exp(−t/τ)]f(τ)\nτdτ, (23)\nwhere the term in brackets is the probability per unit time for a partic le not to\nflip its moment. For a broad enough distribution, the observation tim etwill\nsatisfyτ1≪t≪τ2, whereτ1andτ2are the minimum and maximum relaxation\ntimes of the assembly, respectively. Assuming a uniform distribution f(τ), it\ncan be shown that the magnetization exhibits a logarithmic relaxation\nM(H,t) =M(H,0)−S(H,T) ln(t/τ0) (24)\nwithSthemagnetic viscosity of the system. Thus, polydispersity produces a\nmuch slower decay of magnetization with time.\nThe discussion so far, refers to a field applied parallel to the easy ax is.\nHowever, random anisotropy is most commonly found in MNP assemblie s and\nthe the necessity to study the effect of a tilted field with respect to the easy axis,\narises. In this case, the calculation of the energy barriers and rela xation time\nis a much more complicated task and no analytical solution exists. Num erical\nstudies (Pfeiffer 1990) showed that the energy barrier for an app lied field at an\nangleθ0to the easy axis can be approximately written as\nEb(θ0) =K1V(1−H/Ha)0.86+1.14hs(25)\nwherehsis given by Eq.(12). In the limit of θ0= 0, Eq.(25) reduces to Eq.(16).\nThe temperature dependence of the coercivity for a monodispers e assembly\nwith random anisotropy has also been obtained numerically (Pfeiffer 1 990) as\nHc(T) = 0.48Ha/bracketleftbig\n1−(T/Tb)0.77/bracketrightbig\n, (26)\nwhich at T= 0 reduces to the SW result of Eq.(13). A detailed theoretical\nstudy of the relaxation time for a non-uniaxial applied field can be fou nd in the\nreview by Coffey and colleagues (Coffey 1993).\nAs a concluding remark, the presence of polydispersity and random anisotropy\nmakes the description of the magnetic behavior of an assembly intra ctable to\nexact analytical treatment. Instead, numerical approximations and simulation\nmethods provide the alternative theoretical tools to study these systems. Nu-\nmerical simulation approaches are introduced in Section 4.\n3 Magnetic Measurements\nThermal relaxation has a dynamic character, therefore, the rela tion between the\nvarious relaxation times of the assembly and the measurement time is a decisive\nparameter for the outcome of a measurement. Additionally, if the a ssembly\nis not at equilibrium during the measurement, or if it changes its equilibr ium\nstate (for example, by adiabatic changes of the applied field) the re sult of the\n16measurements depends on the measurement protocol followed. I n what follows\nwe discuss two very common types of static measurements, that r eveal the\ntemperature and field dependence of the magnetization and provid e evidence\nfor superparamagnetic relaxation. Dynamic measurements are no t discussed\nin this article. The interested reader can find more on the physical p rinciples\nbehind the most common magnetic measurement techniques in the re view of\nDormann and colleagues (Dormann 1997).\n3.1 Field-cooled (FC) and Zero-Field-Cooled (ZFC) Mag-\nnetization\nThis is a measurement protocol adopted for investigation of the te mperature\ndependent magnetization of an assembly and it reveals superparam agnetic be-\nhavior. It is performed in three stages. In the first, the sample is in itially at\na high enough temperature ( Tmax) to ensure a SPM state and it is cooled to\nlow temperature ( Tmin) to approach its ground state. In the second stage, a\nweak field is applied ( H≪Hsat), the sample is heated up to Tmaxand the\nmagnetization is measured as a function of temperature. This is the ZFC curve.\nIn the third stage, the system is cooled down to Tmin, without removing the\nfield, while the magnetization is recorded again, producing the FC cur ve. Dur-\ning cooling and heating the temperature changes at the same const ant rate. A\ntypical ZFC-FC curve is shown in Fig. 5. As the temperature rises, t he blocked\nmagnetic moments align easier along the applied field leading to an initial in -\ncrease of the ZFC curve. However, as soon as thermal fluctuatio ns push the\nmoments over the anisotropy barrier, thermal randomization of t he moments\nproduces a drop of the curve. Therefore, the peak of ZFC curve corresponds to\nthe blocking temperature of the assembly. Notice that above Tbthe ZFC and\nFC curves coincide, because the system is in thermal equilibrium and t he the\nheating (cooling) process is reversible. On cooling below Tbthe moments re-\nmain partially aligned along the field, and the magnetization tends to a n on-zero\nvalue. The magnetization vanishes at the ground state ( Tmin) if the measuring\nfield is very weak, a random distribution of the easy axes exists and t he as-\nsembly is non-interacting (dilute). Deviations from any of the above conditions\nproduce a non-zero value for MZFC(T= 0). For isolated MNPs, the ZFC-FC\ncurves are only weakly sensitive to the value of the applied field, prov ided that\nit is weak ( H≪Hs).\n3.2 Remanent magnetization and Coercive field\nRemanent magnetization at a certain field, Mr(H), is measured after switching\noff the previously applied field H. In an assembly of MNPs, remanence arises\nbecause the moments of some particles, which have rotated under an applied\nfield and to do so they have overcome an energy barrier, cannot ro tate back to\ntheir original direction after removal of the field. In a polydisperse assembly,\nat finite temperature T, only the blocked MNPs, namely those with Tb< T\ncontribute to the remanence. Therefore, Mr/Ms=/integraltext∞\nEb,cf(Eb)dEbwhereEb,c=\n17/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s77/s32/s47/s32/s77\n/s83/s41\n/s32/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s107 \n/s66 /s84/s47/s75 \n/s49 /s86 /s32/s41/s84\n/s98 /s40/s97/s41/s40/s98/s41\n/s32/s84/s32/s40/s48 \n/s75 /s41/s32/s77/s32/s47/s32/s77\n/s115\n/s32\n/s49/s48/s48/s32/s71\n/s32/s32 /s32/s53/s48/s48/s32/s71\nFigure 5: (a) Typical FC-ZFC magnetization curves. The curves jo in at the\npeak of the ZFC which corresponds to Tb. Arrows indicate the direction the\nmeasurements are taken. For T > T bthe system is in thermal equilibrium\nand the heating process is reversible. (b) FC-ZFC curves for a dilut e (non-\ninteracting) assembly of Fe nanoparticles ( D= 3.0nm,M s= 1720emu/cc\nandK1= 2.4×105erg/cc ). The blocking temperature (dotted line) decreases\nweakly with increasing measuring field. The non-zero values of MZFC(T= 0)\nare due to the finite value of the measuring field. Data produced by M onte\nCarlo simulations (Section 4.3)\nK1Vcis the critical barrier for SPM relaxation at temperature T. Taking into\naccount that Tb∼V(see Eq.(15)), we deduce that\ndMr(T)/dT=f(Tb) (27)\nnamely, the slope of Mr(T) provides the barrier (or blocking temperature) dis-\ntribution function of the assembly. There are three different meas urement proto-\ncols for the remanent magnetization, as first suggested by Wohlfa rth (Wohlfarth\n1958) :\n(i)Thermoremanence TRM (H,T), measured at the end of a FC process with\nfieldHfromTmaxdown to the measuring temperature T.\n(ii)Isothermal Remanence IRM (H,T), measured at the end of ZFC process\nfromTmaxdown to the measuring temperature T, at which a field His\napplied and then removed.\n(iii)DC Demagnetization remanence DcD (H,T). First, a ZFC process from\nTmaxdown to the measuring temperature Tis performed. Second, the\nsample is brought to saturation remanence IRM (∞,T). Third, a reverse\nfieldHis applied and then removed to leave the sample at the DcD (H,T)\nremanence.\n18Wohlfarth pointed out that for isolated MNPs the different remanen t magne-\ntizations are related as DcD (H) =IRM (∞)−2·IRM (H). More interestingly,\nthe deviations from this equality, defined as\n∆M(H) =DcD (H)−[IRM (∞)−2·IRM (H)] (28)\nquantify the character and strength of interparticle interaction s and are obtained\nexperimentally (O’Grady et al 1993). Positive ∆ Mvalues imply interactions\nwith magnetizing character, and negative values indicate demagnet izing inter-\nactions. We should say that this is only a phenomenological characte rization\nof the interactions, because Eq.(28) does not provide any informa tion about\ntheir microscopic origin. However, Eq.(28) has been proved a stand ard tool\nfor quantification of interparticle interactions in complex MNP assem blies such\nas those used in modern industry of magnetic recording media (gran ular films,\nparticulate media). Interparticle interactions are discussed in the Section 4.\n4 Interacting nanoparticle assemblies\n4.1 Introduction\nThe magnetic interactions that are present in bulk magnetic materia ls pertain to\nMNP assemblies and they preserve their physical origin and charact eristics. In\nparticular, (direct) exchange between atomic moments separate d by a few lat-\ntice constants can couple ferromagnetically or antiferromagnetic ally two MNPs\nvia their surface atoms. Indirect exchange or Ruderman-Kittel-K asuya-Yosida\n(RKKY) interaction exists between MNPs hosted in a metallic matrix, w hich\nprovides free electrons required to mediate the interaction betwe en the atomic\nmoments of the MNPs. Finally, magnetostatic interactions, which ar e of minor\nimportance in bulk magnets due to their weakness, become the domin ant in-\nteractions in MNP assemblies with well separated MNPs. This situation occurs\nfor two reasons. First, the exchange interactions have a very sh ort range (up to\n∼5˚A), the RKKY interactions have an oscillating FM/AFM character with a\nperiod of a few ˚A, which renders to zero their average effect on the MNP vol-\nume, so both have a weak effect in interparticle coupling. On the othe r hand,\nmagnetostatic interactions, in the lowest approximation, namely th e dipolar\ncontribution, are proportional to the magnitude of the coupled ma gnetic mo-\nments, which for SD particles has an enormously large value compare d to the\natomic moments ( µMNP∼103µB∼103µatom).\nFurther on we discuss the effects of magnetostatic (dipolar) inter actions on\nthe magnetic properties of MNP assemblies and their interplay with sin gle-\nparticle anisotropy. The complexity of this problem arises from the long-range\n(∼1/d3,dbeing the interparticle distance) and anisotropic character of the\ndipolar interactions, namely the dependence of interaction energy on the orien-\ntation of the moments relative to the bond joining the particle cente rs (Fig. 6).\n19Figure 6: Ground state configuration of magnetic nanoparticles wit h elliptic\nshape coupled by magnetostatic (dipolar) forces. The easy axis co incides with\nthe long axis of the ellipse (shape anisotropy). Dipolar coupling induce s anti-\nferromagnetic ordering when the moments are forced (by anisotr opy) to remain\nnormal to the bond, as in (a). FM ordering (nose-to-tail) is favore d when the\neasy axes are parallel to the bond, as in (b). In an assembly with ran dom\nanisotropy, as in (c), the moments are aligned along the local easy a xes. Dipo-\nlar interactions have a complex effect, leading to misalignment of the m oments\nwith respect to the local easy axis.\nUnderstanding and controlling the effects of dipole-dipole interactio ns (DDI)\nin MNP assemblies is of paramount importance to modern technology o f mag-\nnetic recording media for two opposite reasons. First, DDI couple t he MNPs\nof an assembly. The ultimate goal in magnetic recording applications is to ad-\ndress each MNP individually and treat it as a magnetic bit. In this case, DDI\nhave a parasitic role and one wishes to estimate and reduce their impa ct in the\nmagnetic properties of an assembly. On the contrary, magnetic log ic devices,\nhave been proposed and built that exploit the magnetostatic couplin g between\nordered MNP arrays (linear or planar) to transfer a magnetic bit (u sually a\nflipped moment) between two distant points in the array (Cowburn 2 006). In\nthis case, DDI are of central importance and the goal is to enhanc e and tailor\ntheir effects.\nOver the last two decades, many research groups have prepared and mea-\nsured MNP assemblies in various forms (granular films, ferrofluids, c luster as-\nsembled films, self-assembled nanoparticles, lithographic arrays of magnetic\ndots) and studied the intrinsic factors (host and particle material, particle size,\nparticle density) and the extrinsic factors (temperature, field, m easurement pro-\ntocol) that control the magnetic behavior. In many of these stud ies the presence\nof magnetostatic interactions has been confirmed. Among the abo ve mentioned\nsystems, the self-assembled MNPs prepared by a synthetic route offer the ad-\nvantage of containing well separated MNPs with a very narrow size d istribution\n(σV∼5−10%), so they are ideal systems to study DDI effects. Experimen-\ntal observations on self-assembled MNPs that have been attribut ed to DDI\ninclude, reduction of the remanence at low temperature (Held 2001 ), increase\n20of the blocking temperature (Murray 2001), increase of the barr ier distribution\nwidth (Woods 2001), deviations of the zero-field cooled magnetizat ion curves\nfrom the Curie behavior (Puntes 2001), and difference between th e in-plane\nand out-of-plane remanence (Russier 2000). Long-range ferro magnetic order\nin linear chains (Russier 2003), and hexagonal arrays (Puntes 200 4, Yamamoto\n2008) of dipolar coupled single-domain magnetic nanoparticles has be en demon-\nstrated, supporting the existence of a dipolar superferromagnetic ground state,\ncharacterized by ferromagnetic long-range order of the particle moments.\nInvestigations of the static and dynamic magnetic properties of dip olar in-\nteracting nanoparticle assemblies brought up fundamental issues related to the\nexistence of a ground state which shares common features with spin glasses ,\nsuch as slow relaxation, memory and ageing effects (Sasaki 2005). The latter\nare magnetic systems characterized by disorder and competing int eractions that\nproduce an energy landscape with many local minima, considered res ponsible for\nthe occurrence of these effects. Dipolar interparticle interaction s in dense and\nrandom nanoparticle assemblies are believed to cause a spin-glass-like behavior\n(Dormann 1997).\nTheoretical models have been developed in an effort to explain these obser-\nvations and related previous ones in assemblies with randomly located MNPs\n(granular films, cluster-assembled films). On a microscopic level, the presence of\nDDI between MNPs modifies the magnetization switching mechanism, w hich for\nan isolated MNP obeys the N´ eel-Arrhenius model. When anisotropic M NPs are\ndipolar coupled, the reversal mechanism is determined by the interp lay between\nthe single-particle anisotropy energy ( Ea∼K1V) and the dipolar interaction en-\nergy (Ed∼µiµj/r3\nij). For weak interactions ( Ed≪Ea), the moments reverse\nindependently by thermal activation over energy barriers, which are however\nmodified due to DDI. This limiting case is treated within a mean-field appr oxi-\nmation and is discussed in Section 4.2. For strong interactions ( Ed≫Ea), the\nsingle-particle reversal is no longer valid. Reversal of one particle c an excite the\nreversal of others, and the assembly behaves in a collective manne r. Many-body\nenergy barriers exist in the system, with values that depend on the configuration\nof all moments. Their evaluation becomes a formidable task and nume rical sim-\nulations offer in this case an indispensable tool. Numerical methods ar e briefly\ndiscussed in Section 4.3.\nFor a detailed review on the magnetic properties of dipolar interactin g MNP\nassemblies the reader is referred to the relevant literature (Dorm ann 1997, Far-\nrellet al 2005, Knobel et al 2008, Kechrakos and Trohidou 2008). The role of\nmagnetostatic interactions in patterned magnetic media has been r eviewed by\nMart´ ınet al (Mart´ ın 2003)\n4.2 Mean Field models\nIn an early attempt to include the effect of interparticle interaction s in the ther-\nmal relaxation of MNPs , Shtrikman and Wohlfarth (Shtrikman 1981) assumed\nthat the single-particle anisotropy barrier of a MNP is increased by t he Zeeman\nenergy due to the interaction field Hintproduced by the moments of neighbor-\n21ing particles. In this model, the N´ eel relaxation time is obtained from Eq.(16)\nwith the applied field Hreplaced by the interaction field Hint. The mean-field\napproximations consists in replacing Hintby its thermal average value, which\nin N´ eel’s model is\nHint=Hinttanh(µ0µHint/kBT)≈µ0µH2\nint/kBT (29)\nthe latter approximation being valid for weak interaction fields. Subs titution of\nEq.(29) into Eq.(16) gives\nτ=τ0exp/bracketleftBigg\nK1V+µ2\n0µ2H2\nint/kBT\nkBT/bracketrightBigg\n. (30)\nUsing the approximation 1 + x≈1/(1−x) we write Eq.(30) in the form\nτ≈τ0exp/bracketleftbiggK1V\nkB(T−T0)/bracketrightbigg\n(31)\nwithT0=µ2\n0µ2H2\nint/kBK1V. Eq.(31), also known as the Vogel-Fulcher law ,\nindicates that the relaxation time of an assembly of interacting MNPs is the\nsame as that of the isolated MNPs at a lower temperature.\nIn the Shtrikman-Wohlfarth model, the temperature T0, or equivalently the\nthermal average H2\nint, is not related to the microscopic parameters of the as-\nsembly, i.e. particle location, and is treated as a phenomenological pa rameter,\nfitted to experimental data. Dormann and colleagues (Dormann 19 88, Dor-\nmann 1997) developed a statistical model for the average barrier in a dipolar\ninteracting assembly which quantifies the interaction field and provid es for the\nsingle-particle energy barrier\nEb=K1V+n1a1M2\nsV/suppress L(a1µ2/VkBT) (32)\nwithn1the number of nearest neighbors of a particle, a1=xv/√\n2,xvthe\nvolume concentration of the particles and /suppress L( ·) the Langevin function. Eq.(32)\nindicates that the anisotropy barrier is increased due to DDI, thus the model\nof Dormann et al predicts an increase of the blocking temperature due to DDI.\nThis model behavior has been observed in almost all types of MNP ass emblies,\nwith a few exceptions (Hansen and Mørup 1998).\nMore recently, Allia et al (Allia 2001) used also a phenomenological approach\nto describe a superparamagnetic assembly with weak DDI. Namely, a n assembly\nin a regime that the remanence and coercivity vanish, but the field-d ependent\nmagnetization varies with concentration of MNPs indicating the pres ence of\nDDI. The authors (Allia 2001) suggested that the dipolar field chang es at a high\nrate and in random direction and therefore acts similar to the therm al field. The\neffect is accounted for by an apparent increase of the system tem perature. The\nmagnetization at temperature Tis given by M=Ms/suppress L [µH/kB(T+T∗)], with\nT∗related to the average dipolar energy via kBT∗=n1µ2/d3and obtained\nby a fitting procedure. This model interpreted successfully the ma gnetization\nbehavior of Co nanoparticles in Cu matrix and established the existen ce of the\ninteracting superparamagnet regime (Allia 2001).\n224.3 Numerical Techniques\nThe mean-field models have the advantage of providing analytical ex pressions\nsuitable for extracting system parameters from the experimenta l data by a fit-\nting process. However, they are not applicable to strongly dipolar s ystems and\nthey do not account for collective effects. Numerical techniques o n the other\nhand, have the major advantage that they treat rigorously the lo cal and tempo-\nral statistical fluctuations of the macroscopic quantities charac terizing the MNP\nassembly and provide an efficient interpolation scheme between the w eak and\nthe strong interaction regimes. We discuss briefly two most common numerical\napproaches, the Monte Carlo (MC) method and the Magnetization D ynamics\n(MD) method.\n4.3.1 The Monte Carlo method\nDifferent algorithms that mimic thermal fluctuations of the degrees of freedom of\na physical system by means of (pseudo)random numbers go under the umbrella\nof Monte Carlo techniques. In the case of MNPs, two widely used algo rithms\nare the Metropolis Monte Carlo (MMC) and the Kinetic Monte Carlo (KM C).\nThe former is appropriate for a description of the equilibrium behavio r of an\nassembly, while the latter also accounts, within a certain time scale, f or the tran-\nsition to equilibrium. Both algorithms provide thermal averages of ma croscopic\nquantities of interest in the canonical ensemble, i.e. at constant te mperature.\nTo do so a sampling of the phase space is performed, however the sa mpling\nprocedures differ, as outlined below.\nThe MMC algorithm samples the phase space, visiting preferentially st ates\nclose to the equilibrium states ( Importance Sampling ). This is achieved when\nsubsequently visited states form a Markov chain, meaning that the probability of\nvisiting the next state depends only on the last visited one. To do so, one chooses\nthe transition from state stos′to occur with certainty, if it reduces the total\nenergy (Es′≤Es) and with a finite probability p(s→s′) = exp(−Es′−Es\nkT), if it\nincreases the total energy ( Es′> Es). Thus the system is allowed to climb-up\nenergy barriers and slide-down toward energy minima until it reache s eventually\nthe global minimum.\nIn KMC the system jumps from a state sat a local minimum to a new state s′\nbeing also a local minimum by overcoming a barrier Eb. The jump is performed\nwithin a predefined time step ∆ twith probability p(∆t) = 1−exp(−∆t/τ)\nwhereτis the corresponding relaxation time with Arrhenius behavior, τ=\nτ0exp(Eb/kT).\nIn both algorithms, interparticle interactions are included by replac ing the\napplied field Hwith the total fieldHi=H+/summationtext\nj(/negationslash=i)Hint,ij, which includes\nthe contribution from the interaction fieldHint,ij. In contrast to mean-field\ntheories, in MC and MD (see next section) techniques the interactio n field is\ntreated exactly, meaning that its value depends on the configurat ion of all the\nmoments of the assembly and it changes at each time-step.\nAn important distinction between the MC algorithms is that KMC simu-\n23lates the relaxation of the system in physical time, while time quantific ation of\nthe MMC time step is possible only in the absence of interparticle intera ctions\n(Nowak 2000, Chubykalo 2003). However, a serious difficulty in KMC a rises\nfrom the calculation of the local energy barrier required to obtain t he transi-\ntion probability. In an interacting system the barrier depends on alldegrees of\nfreedom and its calculation is a formidable task (Chubykalo 2004, Jen sen 2006),\nusually performed in an approximate manner (Pfeiffer 1990, Chantr ell 2001).\nFurthermore, the KMC assumes that the system evolves through thermally ac-\ntivated jumps over energy barriers, an approximation that becom es invalid at\nelevated temperatures ( kT∼Eb), or when collective behavior becomes impor-\ntant, as, for example, in strongly interacting MNPs. Collective effec ts are better\ndescribed within the MMC algorithm.\nFor a detailed description and technical implementation of MC algorith ms\nthe interested reader could refer to the book by Landau and Binde r (Landau\n2000)\n4.3.2 The Magnetization Dynamics method\nIn this method, the equations of motion for the magnetic moments a re integrated\nin time and time averages of the macroscopic magnetization are reco rded. At\nzero temperature, the time-evolution of a magnetic moment µiunder a total\nfieldHiis described by the Landau-Lifshitz-Gilbert (LLG) equation\nd− →µi\ndt=−A(− →µ×− →Hi)−Bi− →µi×(− →µi×− →Hi) (33)\nwithA≡γ/(1 +α2),Bi≡αγ/(1 +α2)µi,γthe gyromagnetic ratio, and αa\ndimensionless damping parameter. The first term on the r.h.s. of Eq.( 33) is the\ntorque term leading to precession around the field axis and the seco nd one is\na phenomenological damping torque that tends to align the precess ing moment\nwith the field Hi.\nThe dynamics at finite temperature are described by introduction o f an\nadditional field ( Hf,i) term in Eq.(33) with stochastic character. Hfis assumed\nto have zero time-average ( white noise ) and its values at different sites i,jor\ndifferent instants t,t′are uncorrelated. The LLG equation augmented by the\nthermal field term is commonly referred to as the Langevin orstochastic LLG\nequation.\n4.3.3 Time scale of numerical methods\nThe MC and MD techniques are complementary since they describe th ermal\nrelaxation of magnetic properties in different time scales. In the MD m ethod,\nthe characteristic time is a fraction ( ∼10−2) of the precessional (Larmor) period\n(∼10−10s), which implies that simulation times up to ∼1nsare presently\nattainable. Thus, MD is the appropriate scheme to investigate fast -relaxation\nphenomena as, for example, the reversal path of magnetization u nder an applied\nshort field pulse (Berkov 2002, Suess 2002).\n24In KMC the characteristic time is the single-particle relaxation time (s ee\nEq.(14)), which is much larger than τ0∼10−10sin the temperature range\nof interest ( kT≪Eb), a fact that makes the method suitable to treat slow-\nrelaxation problems, as, for example the thermal decay of magnet ization in\npermanent magnets, a phenomenon that evolves within days or yea rs (Van de\nVeerdonk 2002).\nFinally, when static magnetic properties are concerned, the syste m is at a\nstable (or metastable) state and the MMC is a powerful and sufficien t scheme to\ndescribe, for example, long range order at the ground state or co llective behavior\nat finite temperature (Kechrakos 1998, Jensen 2003).\n4.3.4 MMC study of dipolar interacting assemblies : A case st udy\nIn this section we show typical results from MMC simulations of the ma gnetic\nproperties of dipolar interacting MNP assemblies (Kechrakos 1998, Kechrakos\n2002). Our system contains Nidentical SD NPs with diameter Dand uniaxial\nanisotropy in a random direction. The MNPs are located randomly in sp ace or\non the vertices of a hexagonal lattice. The former is an appropriat e model for\ngranular samples, and the latter for self-assembled MNPs. The tot al energy of\nthe system is\nE=g/summationdisplay\nij/hatwideSi·/hatwideSj−3(/hatwideSi·/hatwideRij)(/hatwideSi·/hatwideRij)\nR3\nij−k/summationdisplay\ni(/hatwideSi·/hatwideei)2−h/summationdisplay\ni(/hatwideSi·/hatwideH) (34)\nwhere/hatwideSiis the magnetic moment direction (spin) of the i-th particle, /hatwideeiis the\neasy-axis direction, and Rijis the center-to-center distance between particles\niandj. Hats indicate unit vectors. The energy parameters entering Eq.( 34)\nare: (i) the dipolar energy g≡µ2\n0µ2/4πd3, withµ=MsVthe particle moment\nanddthe minimum interparticle distance. (ii) the anisotropy energy k≡K1V,\nand (iii) the Zeeman energy h≡µ0µHdue to the applied dc field H. The\nenergy parameters ( g,k,h ) entering Eq.(34), the thermal energy kBT, and the\ntreatment history of the sample determine the micromagnetic confi guration at a\ncertain temperature and field. The freedom to choose an arbitrar y energy scale\nmakes the numerical results applicable to a class of materials with the same\nparameter ratios rather than to a specific material. The crucial pa rameter that\ndetermines the transition from single-particle to collective behavior is the ratio\nof the dipolar to the anisotropy energy ( g/k).\nWe show in Fig. 7 the concentration and temperature dependence o f the\nremanence magnetization of a random assembly. Notice in Fig. 7a, th at weak\nDDI produce an increase of the remanence with concentration, wh ile strong DDI\nhave the opposite effect. Remarkably, the presence of free samp le boundaries,\ncan reverse the increasing trend of the remanence, due to the pr esence of a de-\nmagnetizing field. When DDI are much stronger than single-particle a nisotropy\n(g/k∼10), the remanence value is sensitive to the morphology of the asse m-\nbly, as the peak around the percolation threshold indicates. This be havior is\nexplained by the anisotropic character of DDI (see Fig. 6). In Fig. 7 b, DDI\n25/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53/s48/s46/s54\n/s32/s103/s47/s107 /s61/s49/s48\n/s32/s103/s47/s107 /s61/s49\n/s32/s103/s47/s107 /s61/s48/s46/s50/s53\n/s32/s32/s82/s101/s109/s97/s110/s101/s110/s99/s101/s32/s40/s77\n/s114/s32/s47/s32/s77\n/s115 /s41\n/s86/s111/s108/s46/s32/s70/s114/s97/s99/s116/s105/s111/s110/s32/s40/s120\n/s86 /s41/s40/s97/s41/s40/s98/s41/s32/s103/s47/s107 /s61/s48\n/s32/s103/s47/s107 /s61/s49/s32/s32/s120\n/s86 /s61/s48/s46/s49/s53/s55\n/s32/s32/s82/s101/s109/s97/s110/s101/s110/s99/s101/s32/s40/s77\n/s114/s47/s77\n/s115 /s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s107\n/s66/s84/s47/s75\n/s49/s86/s41\nFigure 7: Dependence of the saturation isothermal remanence of a random\nassembly (a) on volume fraction of MNPs, at very low temperature ( t/k=0.001)\nand, (b) on temperature, for fixed volume fraction. The particles have random\nanisotropy. The data are obtained by MMC simulations.\ninteractions are shown to produce a much slower temperature dec ay, producing\nfinite remanence values above the blocking temperature of the isola ted MNPs.\nThis result supports the predictions of the mean-field theory abou t the increase\nof the measured blocking temperature in dipolar interacting system s (Dormann\n1988).\nIn chemically-prepared, self-assembled MNPs the possibility to cont rol the\ninterparticle separation by variation of the surfactant (Willard 200 4) offers the\npossibility to study the dependence of Tbon interparticle spacing while pre-\nserving the geometrical arrangement of the assembly (hexagona l). In Fig. 8\nwe show results for the ZFC magnetization ( MZFC) and the blocking tempera-\nture as obtained from the peak of the MZFC(T) curve for a hexagonal array of\ndipolar interacting MNPs with random anisotropy. Parameters corr esponding\nto Co nanoparticles are used (Kechrakos 2002). The characteris tic dependence\nofTbon the inverse cube of interparticle spacing can be used as a proof o f the\ndominant character of DDI in an assembly. Notice also that MZFC(T≈0)\nassumes a positive value that increases with dvalues. This feature arises from\nthe gradual formation of a long range ferromagnetic ground stat e, due to DDI.\nMore examples of MC or MD simulations and comparison to experiments\non MNP assemblies can be found in the relevant literature (Vedmeden ko 2007,\nKechrakos and Trohidou 2008).\n26/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s48/s40/s68 /s47/s100/s41/s51\n/s45/s45/s45/s45/s45/s45/s45/s45/s45\n/s48/s46/s51/s55/s53\n/s48/s46/s50/s50/s53\n/s48/s46/s49/s48\n/s48/s46/s48/s53\n/s48/s32/s90/s70/s67/s32/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s77\n/s90/s70/s67/s47/s77\n/s115 /s41\n/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s107\n/s66/s84/s47/s75\n/s49/s86/s41\n/s32/s32\n/s32\n/s32/s32\n/s32\n/s32\n/s32\n/s32/s32 /s32 /s32/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54\n/s40/s68/s47/s100/s41/s51\n/s32/s32/s66/s108/s111/s99/s107/s105/s110/s103/s32/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s84\n/s98 /s32/s47/s32/s84\n/s98 /s48\n/s41\nFigure 8: ZFC curves and blocking temperature for an ordered (he xagonal)\nassembly of identical MNPs with diameter Dand center-to-center distance d.\n(a) Evolution of ZFC curves with decreasing dvalues. (b) Linear scaling of Tb\nwith inverse cube of d. Data obtained by MMC simulations.\n5 Summary\nWe have discussed the main theoretical concepts that pertain to t he magnetiza-\ntion properties of isolated (non-interacting) nanoparticles and th eir assemblies.\nWe estimated the critical radius for formation of single domain partic les and\nstudied the zero-temperature magnetization reversal mechanis m of coherent ro-\ntation (Stoner-Wohlfarth model), the thermally activated revers al (N´ eel-Brown\nmodel) and the related phenomenon of superparamagnetism occur ring above the\nblocking temperature. Complications arising from size polydispersity , distribu-\ntion of easy axes directions, and applied field on the relaxation time fo r mag-\nnetization reversal were discussed. Two standard experimental techniques for\n(static) magnetic measurements, namely the field and temperatur e dependence\nof magnetization were outlined. Finally, the subject of interparticle dipolar in-\nteractions was introduced along with the most common theoretical techniques\nused to analyze interacting systems. Examples from Monte Carlo st udies of\nMNP assemblies were given.\n6 Future perspectives\nThe dynamic behavior of MNPs in the presence of interparticle intera ctions is\nexpected to remain a topic of intense scientific and technological re search in\nthe coming years. The research effort is expected to focus on bot h the atomic\nscale properties of individual magnetic nanoparticles and on the mes oscopic\n27properties of nanoparticle assemblies.\nOn theatomic scale, future goals will include : (i) Reduction of the mag-\nnetic particle size without violating thermal stability ( superparamagnetic limit )\nat room temperature . The technological benefit from progress in this direction\nwill be the development of magnetic data-storage media with higher a real den-\nsity. Given that the SPM effect is not observed below a certain size of a MNP,\ndue to disorder effects on the particle surface, the search for ne w high-anisotropy\nmaterials is required. Composite nanoparticles with a core-shell mor phology\n(Skumryev 2003) constitute an interesting perspective.\n(ii) Understanding and control of surface effects. With reduction of particle size\nthe contribution from surface moments become of increasing impor tance. The\nchemical structure of the surface (disorder, defects) contro ls the magnitude and\ntype of the surface anisotropy, which is usually much (up to ∼10 times) larger\nthan the core anisotropy. Synthetic methods can offer indispensa ble routes to\nsurface structure modification. Ab-initio electronic structure calculations are a\nvaluable tool to predict the surface anisotropy values and modeling of MNPs as\nmulti-spin system will reveal complex magnetization reversal mecha nisms be-\nyond the Stoner-Wohlfarth model (Kachkachi 2000). Experimen ts on individual\nnanoparticles (Wernsdorfer 2000) offer a unique test of the abov e theories.\nThe future task on the mesoscopic scale will be to understand and control col-\nlective magnetic behavior in ordered nanostructures (self-assem bled MNPs and\nmagnetic patterned media). Ordered nanostructures include che mically pre-\npared self-assembled MNPs and lithographically prepared magnetic p atterned\nmedia. Chemical synthesis of MNPs and self-assembly (bottom-up a pproach)\nis a very promising and cost-effective method to produce ordered M NP arrays\n(Willard 2004). However, deeper understanding and improvement o f the self-\nassembly process is required in order to achieve larger (beyond 1 mm2) sample\narea with structural coherence. There is still a remaining problem a s nanopar-\nticles self-assemble into hexagonal arrays that are incompatible wit h the square\narrangements required in industrial applications. A resolution to th is problem\ncould be the recently demonstrated templated assembly (Cheng 20 04). Litho-\ngraphic patterning (top-down approach) offers better control over the geomet-\nrical aspects of the assembly but cannot yet produce nanostruc tures with size\nbelow∼100nm(Martin 2003). Increase of lithographic resolution is demanded\nin order to achieve patterned media with smaller (below 0 .1µm) characteristic\nsize. On the measurements side, improvement of existing technique s to probe\nmesoscopic magnetic order and excitations is demanded. Recent ex amples are\nthe observation of mesoscopic sale magnetic order in self-assemble d Co nanopar-\nticles by an indirect method (small-angle neutron scattering)(Sach an 2008) and\nby direct methods such as magnetic force microscopy (Puntes 200 4) and electron\nholography (Yamamoto 2008). From the point of view of basic physic s, ordered\nnanostructures constitute model systems to study collective ma gnetic behavior\ndriven by magnetostatic interactions, because the size, the shap e and the spatial\narrangement of the magnetic nanostructures is well controlled. K nown phenom-\nena are to be demonstrated on the mesoscopic scale and new ones p ossibly to\nbe discovered. As a recent example, we refer to the observation o f magnetic\n28frustration is magnetostatically coupled magnetic microrods (Wang 2006), a\nphenomenon previously met in bulk magnetic random alloys (spin glasse s).\nFinally, progress in numerical modeling will provide methods for bridgin g the\natomic scale and the mesoscopic scale simulations. Such multi-scale sim ulations\npoint to the future of theoretical investigations in the field of relax ation in\nmagnetic nanoparticles and have only recently started to appear ( Yanes 2007,\nKazantseva 2008).\nReferences\nAharoni, A. 1965. Effect of a Magnetic Field on the Superparamagne tic Re-\nlaxation Time. Phys. Rev. 177: 793.\nAllia, P., Coisson, M., Tiberto, P., et al. 2001. Granular Cu-Co alloys as\ninteracting superparamagnets. Phys. Rev. B 64: Art. No. 144420\nBean, C. P. and Jacobs, I. S. 1956. Magnetic Granulometry and Su per-\nParamagnetism. J. Appl. Phys. 27: 1448-1452.\nBean, C. P. and Livingston, J. D. 1959. Superparamagnetism. J. Appl. Phys.\n30: 120S-129S.\nBerkov, D.V. 2002. Fast switching of magnetic nanoparticles: Simula tion of\nthermal noise effects using the Langevin dynamics IEEE Trans. Magn.\n38(5): 2489-2495\nBrown, W. F. Jr. 1963. Thermal fluctuations of single-domain part icle.Phys.\nRev. 130: 1677-1686.\nCheng J. Y., Mayes, A. M. and Ross, C. A. 2004. Nanostructure en gineering\nby templated self-assembly of block copolymers. Nat. Mater. 3: 823-828.\nChantrell, R. W., Walmsley, N., Gore J., et al. 2001 Calculations of the sus-\nceptibility of interacting superparamagnetic particles Phys. Rev. B 63\n(2): Art. No. 024410\nChubykalo, O., Nowak, U., Smirnov-Rueda, R., et al. 2003. Monte Carlo tech-\nnique with a quantified time step : Application to the motion of magnetic\nmoments. Phys. Rev. B 67: Art. No. 064422\nChubykalo-Fesenko, O. A. and Chantrell, R. W. 2004. Numerical ev aluation\nof energy barriers and magnetic relaxation in interacting nanostru ctured\nmagnetic systems. J. Magn. Magn. Mater. 343: 189-194\nCowburn, R. P. 2006. Where Have All the Transistors Gone ? Science 311 :\n183-184.\nCullity, B. D. 1972. Introduction to Magnetic Materials . Addison-Wesley Pub-\nlishing Company.\n29Coffey, W. T., Crothers, D. S. F. , Dormann, J. L., et al. 1998. Effect of an\noblique magnetic field on the superparamagnetic relaxation time : Infl u-\nence of the gyromagnetic term. Phys. Rev. B 58: 3249-3266.\nCoffey, W. T., Cregg, P. J. and Kalmykov, Yu. P. 1993. On the Theor y\nof Debye and N´ eel Relaxation of Single Domain Ferromagnetic Partic les.\nAdv. Chem. Phys. 69: 263-315.\nDarling, S. B. and Bader, S. D. 2005. A materials chemistry perspec tive on\nnanomagnetism. J. Mater. Chem. 15: 4189-4195.\nDormann, J. L., Bessais, L. and Fiorani, D. 1988. A dynamic study of small\ninteracting particles: superparamagnetic model and spin-glass law s.J.\nPhys. C: Solid State Phys. 21: 2015-2034.\nDormann, J. L., Fiorani, D., and Tronc, E. 1997. Magnetic relaxation in fine-\nparticle systems. Adv. Chem. Phys. 98: 283-494.\nFarrell, D., Cheng, Y., McCallum, R. W., et al. 2005. Magnetic interactions\nof iron nanoparticles in arrays and dilute dispersions. J. Phys. Chem. B\n109: 13409-13419.\nGaranin, A., Kennedy, E. C., Crothers, D. S. F., et al. 1999. Thermally acti-\nvated escape rates of uniaxial spin systems with transverse field: Uniaxial\ncrossovers. Phys. Rev. E 60: 6499.\nHansen, M. F. and Mørup, S. 1988. Models for the dynamics of inter acting\nmagnetic nanoparticles. J. Magn. Magn. Mater. 184: 262-274.\nHeld, G. A., Grinstein, G., Doyle, H., et al. 2001. Competing interactions in\ndispersions of superparamagnetic nanoparticles. Phys. Rev. B 64: Art.\nNo. 012408.\nJensen, P. J. and Pastor, G. M. 2003. Low-energy properties of two-dimensional\nmagnetic nanostructures: interparticle interactions and disorde r effects.\nNew J. Phys. 5: Art. No. 68\nJensen, P. J. 2006. Average energy barriers in disordered intera cting magnetic\nnanoparticles. Comp. Mater. Science 35: 288-291\nKazantseva, N., Hinzke, D., Nowak, U., et al. 2008. Towards multiscale mod-\neling of magnetic materials: Simulations of FePt. Phys. Rev. B 77: Art.\nNo. 184428.\nKachkachi H, Ezzir A, Nogues M, et al. 2000. Surface effects in nanoparticles:\napplication to maghemite γ-Fe2O3Eur. J. Phys. B 14(4): 681-689.\nKechrakos, D. and Trohidou, K. N. 1998. Magnetic properties of d ipolar in-\nteracting single-domain particles. Phys. Rev. B 58: 12169-12177.\n30Kechrakos, D. and Trohidou, K. N. 2002. Magnetic properties of s elf-assembled\ninteracting nanoparticles. Appl. Phys. Lett. 81: 4574-4576.\nKechrakos, D., and Trohidou, K.N. 2008. Dipolar interaction effects in the\nmagnetic and magnetotransport properties of ordered nanopar ticle arrays.\nJ. Nanoscience Nanotech. 8(6): 2929-2943.\nKodama, R. H. 1999. Magnetic nanoparticles. J. Magn. Magn. Mater. 200:\n359-372.\nKnobel, M., Nunes, W. C., and Sokolovsky, L. M., et al. 2008. Superparam-\nagnetism and other magnetic features in granular materials. J. Nanosci.\nNanotechnol. 8: 2836-2857\nLandau, D. P. and Binder, K. 2000. A Guide to Monte Carlo Simulations in\nStatistical Physics. Cambridge University Press.\nMart´ ın, J. I., Nogu´ es, J, Liu, K., et al. 2003. Ordered magnetic na nostructures:\nFabrication and properties. J. Magn. Magn. Mater. 256: 449-501\nMoser, A., Takano, K., Margulies, D. T., et al. 2002. Magnetic recording:\nadvancing into the future. J. Phys. D.: Appl. Phys. 35: R157-R167.\nMurray, C. B., Sun, S., Doyle, H., et al. 2001. Monodisperse 3d transition-\nmetal (Co, Ni, Fe) nanoparticles and their assembly into nanopartic le\nsuperlattices. MRS Bull. 26: 985-991.\nN´ eel, L. 1949. Influence de fluctuations thermiques sur l’ aimanta tion de grains\nferromagn´ etiques tr´ es fins. Compt. Rend. 228: 664-666\nNowak, U., Chantrell, R. W., and Kennedy, E. C. 2000. Monte Carlo sim ula-\ntion with time-step quantification in terms of Langevin dynamics. Phys.\nRev. Lett. 84: 163-166.\nNunes, W. C., Folly, W. S. D. , Sinnecker, J. P., and Novak, M. A. 2004 .\nTemperature dependence of the coercive field in single-domain part icle\nsystems. Phys. Rev. B 70: Art. No. 014419.\nO’Grady, K., El-Hilo, M., and Chantrell, R.W. 1993. The characterizatio n of\ninteraction effects in fine-particle systems. IEEE Trans. Magn. 29(6):\n2608-2613.\nPetit, V., Taleb, A., and Pileni, M.P. 1998. Self-organization of magnet ic\nnanosized cobalt particles. Adv. Mater. 10: 259-261.\nPfeiffer, H. 1990. Determination of anisotropy field distribution in pa rticle\nassemblies taking into account thermal fluctuations. Phys. Stat. Sol. (a)\n118: 295-306.\nPuntes, V. F., Krishnam, K. M., and Alivisatos, A.P. 2001. Colloidal nan ocrys-\ntal shape and size control: The case of cobalt. Science 291: 2115-2117.\n31Puntes, V. F., Gorostiza, P., Aruguete, D. M., et al. 2004. Collective behavior\nin two-dimensional cobalt nanoparticle assemblies observed by magn etic\nforce microscopy. textitNat. Mater. 3: 263-268.\nRussier, V., Petit, C., Legrand, J., et al. 2000. Collective magnetic properties\nof cobalt nanocrystals self-assembled in a hexagonal network: Th eoretical\nmodel supported by experiments. Phys. Rev. B 62: 3910-3916.\nRussier, V., Petit, V. and Pileni, M.P. 2003. Hysteresis curve of magn etic\nnanocrystals monolayers: Influence of the structure. J. Appl. Phys. 93:\n10001-10010.\nSachan M., Bonnoit, C., Majetich, S. A., et al. 2008. Field evolution of\nmagnetic correlation lengths in ǫ-Co nanoparticle assemblies. Appl. Phys.\nLett. 92(15): Art. No. 152503.\nSasaki, M., J¨ onsson, P.E., Takayama, H., et al. 2005. Aging and memory\neffects in superparamagnets and superspin glasses. Phys. Rev. B 71(10):\nArt. No. 104405.\nSkumryev, V., Stoyanov, S., Zhang, Y. , et al. 2003. Beating the superparam-\nagnetic limit with exchange bias. Nature 423: 850-853.\nStoner, E. C. and Wohlfarth, E. P. 1948. A mechanism of magnetic h ysteresis\nin heterogeneous alloys. Proc. R. Soc. London A 240: 599-642.\nShtrikman, S. and Wohlfarth, E. P. 1981. The theory of Vogel-Fulc her law of\nspin glasses. Phys. Lett. A 85(8,9): 467-470.\nSuess, D., Schrefl, T., Scholz, W., et al. 2002 Fast switching of small magnetic\nparticles J. Magn. Magn. Mater. 242: 426-429.\nVan de Veerdonk, R. J. M., Wu, X. W., Chantrell, R. W., et al. 2002. Slow\ndynamics in perpendicular media. IEEE Trans. Magn. 38(4): 1676-1681\nVedmedenko, E. Y. 2007. Competing Interactions and Patterns in Nanoworld .\nWiley-VCH Verlag.\nWang, R.F., Nisoli, C., Freitas, R.S., et al. 2006. Artificial ’spin ice’ in a\ngeometrically frustrated lattice of nanoscale ferromagnetic island s.Nature\n439(7074): 303-306.\nWernsdorfer, W., Mailly, D. and Benoit, A. 2000. Single nanoparticle m ea-\nsurement techniques. textitJ. Appl. Phys. 87(9): 5094-5096.\nWillard, M. A., Kurihara, L. K., Carpenter, E. E., et al. 2004. Chemically\nprepared magnetic nanoparticles Int. Mater. Rev. 49 : 125-170\nWohlfarth, E. P. 1958. Relations between different modes of acquis ition of the\nremanent magnetization of ferromagnetic particles. J. Appl. Phys. 29:\n595-596\n32Woods, S. I., Kirtley, J. R., Sun, S., et al. 2001. Direct investigation of\nsuperparamagnetism in Co nanoparticle films. Phys. Rev. Lett. 87:\n137205-137208.\nYamamoto, K., Majetich, S.A., McCartney, M.R., et al. 2008. Direct visu-\nalization of dipolar ferromagnetic domain structures in Co nanopart icle\nmonolayers by electron holography. Appl. Phys. Lett. 93(8): Art. No.\n082502\nYanes, R., Chubykalo-Fesenko, O., Kachkachi, H., et al . 2007. Effective\nanisotropies and energy barriers of magnetic nanoparticles with N´ eel sur-\nface anisotropy. Phys. Rev. B 76(6): Art. No. 064416\n33"    },    {        "title": "1406.4915v1.Spin_lattice_coupling_induced_weak_dynamical_magnetism_in_EuTiO_3_at_high_temperatures.pdf",        "content": "preprint(September 21, 2018)\nSpin-lattice coupling induced weak dynamical magnetism in EuTiO 3at high\ntemperatures\nZ. Guguchia,1H. Keller,1R.K. Kremer,2J. K ohler,2H. Luetkens,3T. Goko,3A. Amato,3and A. Bussmann-Holder2\n1Physik-Institut der Universit at Z urich, Winterthurerstrasse 190, CH-8057 Z urich, Switzerland\n2Max Planck Institute for Solid State Research, Heisenbergstr. 1, D-70569 Stuttgart, Germany\n3Laboratory for Muon Spin Spectroscopy, Paul Scherrer Institute, CH-5232 Villigen PSI, Switzerland\nEuTiO 3, which is a G-type antiferromagnet below TN= 5.5 K, has some fascinating properties\nat high temperatures, suggesting that macroscopically hidden dynamically \ructuating weak mag-\nnetism exists at high temperatures. This conjecture is substantiated by magnetic \feld dependent\nmagnetization measurements, which exhibit pronounced anomalies below 200 K becoming more dis-\ntinctive with increasing magnetic \feld strength. Additional results from muon spin rotation ( \u0016SR)\nexperiments provide evidence for weak \ructuating bulk magnetism induced by spin-lattice coupling\nwhich is strongly supported in increasing magnetic \feld.\nPACS numbers: 75.30.Kz,75.85.+t,63.70.+h\nPerovskite oxides are well known for their rich ground\nstates and the possibility of tuning these by doping,\npressure, and temperature. Especially, their ferroelectric\nproperties have attracted increased interest, since they\no\u000ber a broad range of technological applications [[1]\nand Refs. therein]. EuTiO 3(ETO) has \frst been\nsynthesized in the early \ffties when a major boom in\nthe search of ferroelectrics without hydrogen bonds took\nplace [2]. Since ETO did not show any ferroelectric\nproperties, but became antiferromagnetic (AFM) at\nlow temperature with TN\u00195.5 K [3, 4], it vanished\nfrom research activities and only regained substantial\ninterest lately when it was demonstrated that strong\nmagneto-electric coupling is present in this material\n[5]. This was established by dielectric permitivity\nmeasurements, where an anomaly in the dielectric\npermitivity \"sets in atTN. This anomaly vanishes upon\nthe application of a magnetic \feld. The rather large and\nstrongly temperature dependent dielectric permitivity\nhas been related to a soft optic q = 0 mode, reminiscent\nof a ferroelectric soft mode [6, 7]. However, its complete\nfreezing is inhibited by quantum \ructuations quite\nanalogous to SrTiO 3(STO) [8]. In the search for new\nmultiferroic materials ETO thus became a potential\ncandidate and enormously enhanced the research in the\nETO physical and chemical properties. In the focus\nof the research was mainly the transition to the AFM\nphase [9, 10] and its possible polar properties with novel\nspin arrangements together with electronic structure\ninvestigations [11].\nThe close analogy between ETO and STO, namely the\nsame lattice constants, the same ionic valency of the\ncation with almost equal ionic radii, and the tendency\ntowards a polar instability, has recently been shown to\nbe even closer, by establishing that ETO transforms\nfrom cubic to tetragonal caused by an oxygen octahedral\nrotation instability [12]. Amazingly, this takes place at\nTS= 282 K in ETO, in contrast to STO with TS= 105\nK. This large spread in TSbetween the two compoundscan only be caused by the di\u000berent atomic masses of\nSr and Eu and also by the Eu 4 felectrons with spin\nS= 7/2. In order to obtain a more clear picture of\nthe origin of this di\u000berence in TS, the mixed crystal\nseries SrxEu1\u0000xTiO3has been studied as a function of\nxwith focus on the development of TSandTNwith\nx[13{15]. Interestingly, both transition temperatures\nvary nonlinearly with x, re\recting the dilution e\u000bect of\nthe Eu 4fspins. Especially, the low temperature phase\ntransition line indicates the stability of the AFM order\nwhich persists up to x\u00180.25, in contrast to recent the-\noretical arguments which predict a transition from AFM\nvia ferrimagnetic to ferromagnetic order with decreasing\nx[16, 17]. The high temperature phase transition line\nhas been obtained by di\u000berent experimental techniques,\nnamely, electron paramagnetic resonance (EPR), muon\nspin rotation ( \u0016SR), electrical resistivity, and speci\fc\nheat measurements [13]. The spectacular aspect stems\nfrom EPR and \u0016SR data which test magnetic properties.\nEspecially, a \fnite \u0016SR relaxation rate signals the\npresence of some kind of exotic magnetism being present\nin the bulk sample, and indicates that dynamic magnetic\norder must be present in spatially con\fned regions of the\nceramics. This conclusion is further supported by the\nfact thatTSdepends on an applied external magnetic\n\feld which is amazing, since TSlies 275 K above TN\n[18],i:e:, deeply in the paramagnetic region. From both\nresults it must be concluded that the oxygen octahedral\nrotations in\ruence the Eu 4 fspin-spin interactions still\nat high temperatures and contribute to an e\u000bective\nsecond nearest neighbor ferromagnetic spin exchange\nwhich is otherwise small or vanishing.\nTo substantiate these conclusions, we have recently\nproposed a novel approach to test strong spin-lattice\ncouplings by magnetization measurements [19]. If a cou-\npling between the spins and the lattice is present in the\nform of a quadratic interaction ESL\u0018\u000bP\nn;i;jw2SiSj,\nwhere w is the polarizability coordinate of the TiO 6\ncluster and Si,Sjare second nearest neighbor spinsarXiv:1406.4915v1  [cond-mat.mtrl-sci]  18 Jun 20142\n05 010015020025030001234567\n T5\n T3\n T1\n T0\n.5 T0\n.002 Tm (10-3Am2) \n T\n (K)EuTiO3(a)\n1001 502 002 503 0026.426.626.827.0T\n0 \n 7\n T5\n T3\n T1\n T0\n.5 T0\n.002 TmT/(µ0H) (10-3Am2K/T)T\n (K)TSEuTiO3(b)\nFIG. 1: (Color online) (a) The magnetic moment mof\nEuTiO 3as a function of temperature and magnetic \feld. (b)\nThe quantity mT=(\u00160H) as a function of temperature and\nmagnetic \feld. The arrows indicate the structural phase tran-\nsition temperature Tsand the magnetic crossover temperature\nT0.\nalong the diagonal via the intermediate oxygen ions,\nwith\u000bbeing the coupling strength, then the magnetic\nsusceptibility attains an extra temperature dependent\ncomponent in the paramagnetic phase according to: \u001f\n=N\u00160g2\u00162\nBS(S+1)(1+\u000bhwi2\nT)/(3kBT) withN,\u0016B,g\nbeing the number of magnetic moments per volume, the\nBohr magneton and the gfactor, respectively. The po-\nlarizability coordinate refers to the relative displacement\ncoordinate between core and shell of the nonlinearly\npolarizable cluster mass at the TiO 3lattice site and its\nrole in phase transitions has been discussed in detail in\nRefs. [20{22]. Since hwi2\nTis the thermal average over\nthe polarizability coordinate into which all dynamical\ninformation enters, the temperature dependence of the\nsoft transverse acoustic zone boundary mode contributes\n15017520022525027526.426.526.6184.5185.0185.5 mT (10-3Am2K) \nT\n (K)µ0H = 1 T \nµ\n0H = 3 T \n µ\n0H = 7 TEuTiO3T\n07\n9.279.479.679.880.0 \nFIG. 2: (Color online) Magnetic moment mmultiplied by\ntemperature Tof EuTiO 3for\u00160H= 1, 3, and 7 T. The\narrows indicate the magnetic crossover temperature T0.\nessentially to its temperature dependence. Above TS\nthe squared soft mode frequency decreases linearly with\ndecreasing temperature to become zero at TS. Below\nTSthe mode recovers to increase in a Curie-Weiss type\nmanner linearly with twice the gradient as compared to\nthe para phase. This scenario has the consequence that\nthe product of susceptibility and temperature is not a\nconstant far above TNbut follows the T-dependence of\nthe soft mode which we indeed were able to demonstrate\nrecently [18]. However, in addition another consequence\nresults, since the increasing size of hwi2\nTbelowTS,\nrespectively the increasing rotation angle, leads to an\n012345670.00.51.01.52.0 \n d(mT)/dT (10-5Am2)/s61549\n0H (T)EuTiO3\nFIG. 3: (Color online) The slope of the magnetic moment\ntimes temperature data mT(Figs. 2) as a function of the\napplied magnetic \feld \u00160Hfor EuTiO 3as obtained in the\ntemperature range between 150 and 200 K. The full line in\nthe \fgure is a guide to the eye. The size of the symbols\nexceeds the experimental error bars.3\nincreasing strength in the ferromagnetic second nearest\nneighbor exchange. This supports the formation of grow-\ning dynamical magnetic clusters which can be tested by\napplying a magnetic \feld. Indirectly cluster formation\nhas already been seen in the bare magnetic susceptibility\ndata where an upturn was detected approximately 100\nK belowTS[18]. In order to substantiate this scenario,\nwe have performed magnetization measurements in\napplied magnetic \felds up to \u00160H= 7 Tesla on ceramic\nsamples of ETO which have been prepared as described\nin Ref. [12]. The magnetic moment mas a function of\ntemperature and at various magnetic \felds is shown in\nFig. 1a. While the data in Fig. 1a appear to be typical\nfor a paramagnetic system, a detailed analysis of the\ndata in terms of the product mT normalized by the\nmagnetic \feld \u00160HversusTreveals striking anomalies\nwhich - for the smallest \feld \u00160H= 0.002 T - are related\nto the structural instability as already reported before.\nWith increasing \feld strength a second temperature\nscale emerges from the data appearing around T0'\n210 K which does not depend on the \feld strength\n(Fig. 1b). In the following we denote T0as the magnetic\ncrossover temperature. The normalization of the data\nwith respect to Hhas the advantage that data for all\n\feld strengths can be shown in a single graph. However,\nthis methodology obscures important details which are\nstriking when mTis plotted as a function of temperature.\nThe details are shown in Fig. 2 for three representative\n\feld strengths. Apparently, in Fig. 2 a change in slope\n0.40 .50 .60122\n.82 .93 .03 .13 .20.00.40.81.2FT-amplitude (a.u.) Broad \nNarrow \nSum(\nb)/s61549\n0H = 3 TT\n = 100 KFT-amplitude (a.u.)/s61549\n0H = 0.5 TT\n = 100 K(a) \n Broad \n Narrow \n Sum \nB\n (T)\nFIG. 4: (Color online) Fourier transform for the \u0016SR asym-\nmetry spectra of EuTiO 3at 100 K for two magnetic \felds:\n(a)\u00160H= 0.5 T and (b) 3 T. The solid lines are the FTs of\nthe corresponding theoretical A(t) given by Eq. (1).(in the temperature range between 150 and 210 K) of\nmTversusTtakes place with increasing \feld strength\nwhich is shown in detail and for all \feld strengths in\nFig. 3. With increasing \feld strength the slope increases\nnonlinearly with the magnetic \feld, evidencing that\nsome kind of correlated magnetism is present. Since the\noxygen octahedral rotations modify the second nearest\nneighbor ferromagnetic exchange interaction only and\nhave no in\ruence on the direct nearest neighbor AFM\nexchange, we suggest that weak ferromagnetism appears\nbelow 210 K. Note, that this \fnding has nothing to do\nwith structural modulations as reported in Refs. [10]\nand [23], since the x-ray tested temperature dependent\nstructural data are only in accordance with tetragonal\nsymmetry and show no superlattice re\rections. It is\nalso important to note that the data in [10] report a TS\naround 245 K, much lower than our value, which has\nbeen corrected in a subsequent publication to arrive at\nthe same value as ours. In Ref. [23] single crystals have\nbeen investigated with TNbeing substantially smaller\nthan our and other literature values, which might stem\nfrom impurities or defects and thus can give rise to\nthe reported structural modulations. In addition, a\nrecent detailed structural study con\frmed our results\nand arrived at the conclusion that the unmodulated\ncase corresponds to the bulk structure of pure material\n[24]. The weak magnetism has been tested by bulk\nsensitive local probe experiments as provided by the\n\u0016SR technique. As has already been demonstrated in\nRef. 12, a \fnite \u0016SR relaxation rate is seen above the\nstructural phase transition temperature and persists in\nthe tetragonal phase.\nThese former data are here complemented by mea-\nsurements of the \u0016SR relaxation rate as a function of\ntemperature and magnetic \feld. Transverse-\feld (TF)\n\u0016SR experiments were performed at the \u0019E3 beamline\nof the Paul Scherrer Institute (Villigen, Switzerland), us-\ning the HAL-9500 \u0016SR spectrometer. The specimen was\nmounted in a He gas-\row cryostat with the largest face\nperpendicular to the muon beam direction, along which\nthe external \feld was applied. Magnetic \felds between\n10 mT and 5 T were applied, and the temperatures were\nvaried between 100 and 300 K. The \u0016SR time spectra\nwere analyzed using the free software package MUSR-\nFIT [25].\nFor all \felds and in the whole investigated tempera-\nture range, two-component signals were observed: a sig-\nnal with fast exponential relaxation (broad signal) and\nanother one with a slow exponential relaxation (narrow\none). As an example, the Fourier transform (FT) of the\n\u0016SR asymmetry at 100 K and for 0.5 T and 3 T is shown\nin Fig. 4. The \u0016SR time spectra were analyzed by using4\n2468102\n468105\n0100150200250300105\n01001502002503001020304050  /s61549\n0H = 0.5 T(\na)T0T0(\nb)  \nλ1 (µs-1) \n/s615490H = 1 Tλ1 (µs-1)T\n0(\nc) \n   \n/s615490H = 3 TT\n (K)T0(\nd)  \n /s615490H = 5 TT\n (K)\nFIG. 5: (Color online) \u0016SR relaxation rate \u00151of EuTiO 3\nas a function of temperature measured for various magnetic\n\felds: (a)\u00160H= 0.5 T, (b) \u00160H= 1 T, (c) \u00160H= 3 T,\nand (d)\u00160H= 5 T. The thin lines are guides to the eyes.\nThe arrow indicates the magnetic crossover temperature T0\nwhere the kink in \u00151occurs. The experimental error bars are\nsmaller than the size of the symbols.\nthe following functional form:\nA(t) =A1exp(\u0000\u00151t) cos(\r\u0016B\u00161t+')+\nA2exp(\u0000\u00152t) cos(\r\u0016B\u00162t+');(1)\nwhereA1(A2),B\u00161(B\u00162), and\u00151(\u00152) denote the assym-\nmetry, the local magnetic \feld at muon site, and the\nrelaxation rate of the fast (slow) component. \r=(2\u0019)'\n135:5 MHz/T is the muon gyromagnetic ratio, and 'is\nthe initial phase of the muon-spin ensemble. Regarding\nthe two-component signals of EuTiO 3, the signal with\nthe fast (slow) relaxation is associated with the volume\nfraction with (without or only very weak) magnetic or-\nder. Since the major fraction ( '80 %) of the \u0016SR signal\ncomes from the muons stopping in the part of the sample\nwith fast relaxation, we discuss here only the tempera-\nture and \feld dependence of the relaxation rate \u00151of the\nfast (magnetic) component.\nThe temperature dependence of \u00151for various applied\n\felds is shown in Fig. 5. It is evident that \u00151(T) exhibits\na pronounced kink at the magnetic crossover temperature\nT0'200 K for all measured \felds. Note that this value of\nT0is smaller than T0'210 K derived from magnetization\nmeasurements (see Fig. 2). This is likely due to the dif-\nferent experimental techniques (magnetization and \u0016SR)\nused to determine T0. Between 300 and 200 K \u00151(T) in-\ncreases with decreasing temperature. For T >T 0, an ad-\nditional signi\fcant increase of \u00151(T) is observed. More-\nover, a strong enhancement of \u00151with increasing \feld\nwas found at all investigated temperatures (see Fig. 6a).At low temperatures \u00151(H) increases stronger than at\nhigh temperatures. For clarity, the quantity d \u0015/dHde-\ntermined from the linear part of \u0015(H) is plotted in Fig. 6b\nas a function of temperature. A change in the slope of\nthe temperature dependence of d \u0015/dHcan be clearly\nseen atT0'200 K. The pronounced increase and the\nstronger \feld dependence of \u00151belowT0suggests the\nappearance of some kind of magnetic correlations in the\nsystem. The formation of \feld induced magnetic clusters\nbelowT0may be a possible explanation as proposed in\nRef. [19]. Unfortunately, our data do not admit to draw\nany de\fnite conclusions on the type of magnetic correla-\ntions (AFM or FM). However, we can \fgure out whether\nthe fast depolarization of the \u0016SR signal observed be-\nlowT0in EuTiO 3is either due to a broad distribution of\n01 2 3 4 5 6 0102030401\n00 K1\n25 K1\n50 K1\n75 K2\n00 K2\n50 K/s615481(µs-1)µ\n0H (T)300 KEuTiO3(a) \n \n501 00150200250300110(b)d/s61548/dH (µs-1T-1)T\n (K)EuTiO3T\n0 \nFIG. 6: (Color online) (a) \u0016SR relaxation rate \u00151of EuTiO 3\nas a function of magnetic \feld for various temperatures. The\nsolid lines are guides to the eyes. (b) The temperature de-\npendence of the quantity d \u0015/dH. The arrow indicates the\nmagnetic crossover temperature T0.5\nstatic \felds, and/or to strongly \ructuating magnetic mo-\nments. In order to discriminate between these two cases\n\u0016SR experiments in longitudinal \felds (LF) [26] are re-\nquired. Therefore, we performed LF \u0016SR experiments on\nETO in \felds up to 0.5 T below T0. Since no recovery\nof the muon polarization was observed at long times we\nconclude that in EuTiO 3dynamic magnetism is present.\nTo conclude, magnetization measurements and \u0016SR\ndata as functions of temperature and magnetic \feld pro-\nvide evidence for dynamic weak magnetic clusters form-\ning belowT0'200 K. While from magnetization an\nanomalous upturn in the product mTappears around\nT0in a magnetic \feld, \u0016SR data directly prove mag-\nnetic correlations which are \feld dependent. Our \fnding\nhas important consequences for possible spintronic ap-\nplications of EuTiO 3, since the magnetism far above TN\nshould considerably in\ruence transport properties and\nalso cause magnetic \feld induced changes or anomalies\nin dielectric constant at high temperatures which - until\nnow - has not been measured.\nThis work was supported by the Swiss National Science\nFoundation. We thank R. Scheuermann for the support\nin\u0016SR experiments and for valuable discussions.\n[1] J. F. Scott, Science 315, 954 (2007).\n[2] J. Brous, I. Frankuchen, and E. Banks, Acta Cryst. 6, 67\n(1953).\n[3] T.R. McGuire, M.W. Shafer, R.J. Joenk, H.A. Halperin,\nand S.J. Pickart, J. Appl. Phys. 37, 981 (1966).\n[4] L. Chien, S. DeBenedetti, and F.De.S. Barros, Phys. Rev.\nB10, 3913 (1974).\n[5] T. Katsufuji and H. Takagi, Phys. Rev. B 64, 054415\n(2001).\n[6] S. Kamba, D. Nuzhnyy, P. Van\u0014 ek, M. Savinov, K.\nKni\u0014zek, Z. Shen, E. \u0014Santav\u0013a, K. Maca, M. Sadowski,\nand J. Petzelt, Europhys. Lett. 80, 27002 (2007).\n[7] V. Goian, S. Kamba, J. Hlinka, P. Van\u0014 ek, A.A. Belik,\nT. Kolodiazhnyi, and J. Petzelt, Eur. Phys. J. B 71, 429\n(2009).\n[8] K.A. M uller and H. Burkard, Phys. Rev. B 19, 3593(1979).\n[9] V. Scagnoli, M. Allieta, H. Walker, M. Scavini, T. Katsu-\nfuji, L. Sagarna, O. Zaharko, and C. Mazzoli, Phys. Rev.\nB86, 094432 (2012).\n[10] M. Allieta, M. Scavini, L.J. Spalek, V. Scagnoli, H.C.\nWalker, C. Panagopoulos, S.S. Saxena, T. Katsufuji, and\nClaudio Mazzoli, Phys. Rev. B 85, 184107 (2012).\n[11] J.L. Bettis, M.-H. Whangbo, J. K ohler, A. Bussmann-\nHolder, and A.R. Bishop, Phys. Rev. B 84, 184114\n(2011).\n[12] A. Bussmann-Holder, J. K ohler, R.K. Kremer, and J.M.\nLaw, Phys. Rev. B 83, 212102 (2011).\n[13] A. Bussmann-Holder, Z. Guguchia, J. K ohler, H. Keller,\nA. Shengelaya, and A.R. Bishop, New Journal of Physics\n14, 093013 (2012).\n[14] Z. Guguchia, A. Shengelaya, H. Keller, J. K ohler, and A.\nBussmann-Holder, Phys. Rev. B 85, 134113 2012.\n[15] Z. Guguchia, H. Keller, A. Bussmann-Holder, J. K ohler,\nand R.K. Kremer, European Physical Journal B 86, 409\n(2013).\n[16] A.N. Morozovska, Y. Gu, V.V. Khist, M.D. Glinchuk,\nL.-Q. Chen, V. Gopalan, and E.A. Eliseev, Phys. Rev. B\n87, 134102 (2013).\n[17] E.A. Eliseev, M.D. Glinchuk, V.V. Khist, C.-W. Lee,\nC.S. Deo, R.K. Behera, and A.N. Morozovska, J. Appl.\nPhys. 113, 024107 (2013).\n[18] Z. Guguchia, H. Keller, J. K ohler, and A. Bussmann-\nHolder, J. Phys.: Condens. Matter 24, 492201 (2012).\n[19] K. Caslin, R.K. Kremer, Z. Guguchia, H. Keller, J.\nK ohler, and A. Bussmann-Holder, J. Phys.: Condens.\nMatter 26, 022202 (2014).\n[20] A. Bussmann-Holder, J. Phys.: Cond. Mat. 24, 273202\n(2012).\n[21] R. Migoni, H. Bilz, and D. B auerle, Phys. Rev. Lett. 37,\n1155 (1976).\n[22] H. Bilz, G. Benedek, and A. Bussmann-Holder, Phys.\nRev. B 35, 4840 (1987).\n[23] J.-W. Kim, P. Thompson, S. Brown, P.S. Normile, J.A.\nSchlueter, A. Shkabko, A. Weidenka\u000b, and P.J. Ryan,\nPhys. Rev. Lett. 110, 027201 (2013).\n[24] D.S. Ellis, H. Uchiyama, S. Tsutsui, K. Sugimoto, K.\nKato, and A.Q.R. Baron, Physica B 442, 34 (2014).\n[25] A. Suter and B.M. Wojek , Physics Procedia 3069-73,\n2012.\n[26] A. Yaouanc and P. Dalmas de R\u0013 eotier, J. Phys.: Cond.\nMat. 9, 9113 (1997)."    },    {        "title": "1301.5789v1.Optimal_conditions_for_magnetization_reversal_of_nanocluster_assemblies_with_random_properties.pdf",        "content": "arXiv:1301.5789v1  [cond-mat.mes-hall]  24 Jan 2013Optimal conditions for magnetization reversal of\nnanocluster assemblies with random properties\nP.V. Kharebov1, V.K. Henner1,2, and V.I. Yukalov3∗\n1Department of Physics, Perm State University, Perm 614990, Rus sia\n2Department of Physics, University of Louisville, Louisville, Kentucky 40292, USA\n3Bogolubov Laboratory of Theoretical Physics,\nJoint Institute for Nuclear Research, Dubna 141980, Russia\nAbstract\nMagnetization dynamics in the system of magnetic nanoclust ers with randomly\ndistributed properties are studied by means of computer sim ulations. The main at-\ntention is paid to the possibility of coherent magnetizatio n reversal from a strongly\nnonequilibrium state with a mean cluster magnetization dir ected opposite to an ex-\nternal magnetic field. Magnetic nanoclusters are known to be characterized by large\nmagnetic anisotropy and strong dipole interactions. It is a lso impossible to produce a\nnumber of nanoclusters with identical properties. As a resu lt, any realistic system of\nnanoclusters is composed of the clusters with randomly vary ing anisotropies, effective\nspins, and dipole interactions. Despite this randomness, i t is possible to find condi-\ntions when the cluster spins move coherently and display fas t magnetization reversal\ndue to the feedback action of resonator. By analyzing the infl uence of different cluster\nparameters, we find their optimal values providing fast magn etization reversal.\nPACS: 75.75.Jn; 75.78.Cd; 76.20.+q; 76.90.+d\nKeywords : magnetic nanoclusters, magnetization dynamics, magnetization r eversal\n∗Corresponding author: V.I. Yukalov (E-mail: yukalov@theor.jinr.ru )\n11 Introduction\nMagnetic nanoclusters are the objects enjoying rich and interest ing properties with a variety\nof applications, as can be inferred form the review articles [1–5]. Fo r example, they are\nused in magnetic chemistry as catalysts; in biomedical imaging for mag netic reading by\nmagnetometers that measure the magnetic field change induced by nanoclusters; in medical\ntreatment by employing alternating magnetic fields forcing the oscilla tion of the nanocluster\nmagnetic moment that produces heat destroying ill cells or bacteria ; in genetic engineering\nbyattachingananocluster toaparticularpieceofthemoleculeandt henremoving ittogether\nwith this piece; in waste cleaning by stacking nanoclusters to waste a nd then removing them\ntogether with waste; in information storage, processing, and qua ntum computing; and so\non [1–5]. This is why the methods of governing magnetization dynamics of nanoclusters are\nso much important.\nThe use of nano sizes of clusters is principal, since such clusters beh ave as large mon-\nodomain particles with a given large total spin. Typical radii of nanoc lusters are between 1\nand 100 nm, containing from 100 to 105atoms. Respectively, the effective total magnetiza-\ntion of a cluster can be of order 100 to 105µB. The maximal size of a cluster, when it is in a\nmonodomain state, such that the spins of atoms, composing the clu ster, sum up coherently,\nforming an effective total spin, is characterized by the coherence radius . The clusters, whose\nsizes are larger than the coherence radius, decompose into sever al domains with oppositely\ndirected magnetization, so that the total cluster spin is zero.\nEmploying nanoclusters for the purpose of information storage, p rocessing, and quantum\ncomputingmeetstherequirements thatcontradict eachother. F romoneside, forinformation\nstorage, it is required that the cluster magnetization could be well f rozen ina given direction,\nwhich needs the existence of a sufficiently large magnetic anisotropy . But, from another\nside, for information processing, it is necessary to be able to quickly reverse the cluster\nmagnetization, which is hindered by this anisotropy.\nNanoclusters enjoy the property to get their magnetization froz en at temperatures lower\nthan the blocking temperature that is of order of 10 - 100 K. Then t o reverse the cluster\nmagnetization, one has to overcome the magnetic anisotropy. So, for information storage,\none needs large magnetic anisotropy, while for information process ing, the anisotropy has\nto be suppressed to achieve fast magnetization reversal. To acco mplish such a reversal,\none applies external transverse fields pushing the cluster magnet ization. The discussion of\ndifferent methods of magnetization reversal by means of externa l magnetic fields can be\nfound in Refs. [2–6].\nA very efficient method for realizing fast magnetization reversal of nanoclusters is based\non the use of feedback field from a resonant electric circuit coupled to the ensemble of nan-\noclusters [6–10]. Actually, the idea that a resonant electric circuit c an drastically shorten\nthe relaxation time of a spin system was advanced by Purcell [11], who illustrated it for an\nensemble of nuclear spins. This mechanism has later been considered by Blombergen and\nPound [12]. The influence of the coupling of a magnetic sample with a res onant circuit on\nmagnetization dynamics is called the Purcell effect . One sometimes calls it the radiation\ndamping, following Bloembergen and Pound [12]. However the latter te rm is rather a mis-\nnomer, as has been stressed by a number of researchers [13–15], since the coupling of a spin\nsystem with a resonator does strongly influences spin dynamics, bu t not necessarily damping\nit. Forinstance, thiseffect enhances nuclear magneticresonance signals, which iswidely used\n2inNMR techniques [16–18]. The term”radiationdamping” isalso confus ing because ofmak-\ning impression that these are spins themselves that cooperate by in teracting with each other\nthrough a common radiation field. Such a radiation correlation betwe en radiating atoms is\nthe essence of the Dicke effect [19]. But for magnetic particles, as has already been stressed\nby Purcell [11], such a radiation collectivization of spin motion is absolut ely negligible. In\natomic physics, one clearly distinguishes these two principally differen t physical effects. And\ntheterm Purcell effect isemployed fordescribing theinfluenceofacavity resonatoronato mic\nradiation, which is the main part of the cavity quantum electrodynamics [20–22].\nThe role of the Purcell effect on the spin dynamics of nuclear spins is w ell studied. In\nresonance experiments, it enhances the NMR signals [16–18]. For st rongly nonequilibrium\nsystems of polarized nuclei, it leads to fast magnetization reversal [23–27] that has been dis-\ncovered in experiments [23] and later confirmed in other experiment al studies (see references\nin the review article [4]).\nMagnetization dynamics in an ensemble of nanoclusters is essentially d ifferent from that\nof nuclear systems. The basic differences are as follows.\n(i) Nanoclusters possess rather large magnetic anisotropy that h inders the possibility of\nsimple regulation of spin motion.\n(ii) Having large effective spins, nanoclusters also have strong spin d ipole interactions,\nwhich results in short dephasing time.\n(iii) The most important difficulty for organizing collective spin motion in a system of\nnanoclusters is the problem of cluster inhomogeneity, since nanoclu sters, being prepared by\nany of the known methods, whether by thermal decomposition, or microemulsion reactions,\nor by thermal spraying, are not identical particles, as nuclei would be. But nanoclusters\ndiffer by their shapes and sizes, which results in the difference in the v alue of their spins,\ndipole interactions, and of their anisotropies.\nIt is the aim of the present paper to investigate the magnetization d ynamics in a realistic\ninhomogeneous ensemble of nanoclusters exhibiting all their typical properties of anisotropy\nand dipole interactions. Since analytic investigation of such an inhomo geneous system is too\nmuch complicated, we resort to computer modelling. Because the pr oblem of magnetization\nreversal is of special interest for many applications, such as infor mation recording and pro-\ncessing, wepaythemainattentiontofinding theoptimal conditions, whenthemagnetization\nreversal is fast and as close to complete as possible.\n2 Realistic model of nanocluster system\nThe sample formed by a system of nanoclusters is described by the H amiltonian\nˆH=/summationdisplay\niˆHi+1\n2/summationdisplay\ni/negationslash=jˆHij, (1)\nwhere the first term characterizes single nanoclusters and the se cond term describes their\ndipole interactions.\nThe typical Hamiltonian of a nanocluster has the form\nˆHi=−µiB·Si−D(Sz\ni)2+D2(Sy\ni)2+D4/bracketleftbig\n(Sx\ni)2(Sy\ni)2+(Sy\ni)2(Sz\ni)2+(Sz\ni)2(Sx\ni)2/bracketrightbig\n,(2)\n3in which the first term is the Zeeman energy, and other terms descr ibe the energy due to\nmagnetic anisotropy, with the corresponding anisotropy paramet ers [1–6].\nThe interaction Hamiltonian is caused by dipole spin interactions\nˆHij=/summationdisplay\nαβDαβ\nijSα\niSβ\nj, (3)\nwith the dipolar tensor\nDαβ\nij=µ2\n0\nr3\nij/parenleftBig\nδαβ−3nα\nijnβ\nij/parenrightBig\n, (4)\nwhere\nrij≡ |rij|,nij≡rij\nrij,rij≡ri−rj.\nThe total magnetic field, acting on the system, is the sum\nB=B0ez+Hex (5)\nof an external constant field B0and the resonator feedback field H.\nThe considered sample is inserted into a coil of an electric circuit with a n attenuation γ\nand natural frequency ω. Moving spins of the sample create electric current in the coil that,\nin turn, produces the feedback magnetic field acting on these spins . The equation for the\nfeedback field follows from the Kirchhoff equation and can be written [4,26,27] as\ndH\ndt+2γH+ω2/integraldisplayt\n0H(t′)dt′=−4πηdmx\ndt, (6)\nwhereηis filling factor and the electromotive force is caused by the moving av erage magne-\ntization density\nmx=µ0\nV/summationdisplay\nj/angbracketleftSx\nj/angbracketright. (7)\nThe equations ofspin dynamics areobtained inthefollowing traditiona lway [4,14,28,29].\nFirst, we write the Heisenberg equations of motion for the spin comp onentsSα\nj, in which\nthe index j= 1,2,...,Nenumerates nanoclusters and α=x,y,z. Then these equations are\naveraged using semiclassical approximation, and spin attenuation is taken into account by\nmeans ofthesecond-order perturbationtheory, asisdescribed infulldetails by Abragamand\nGoldman [14,29]. This procedure results in the equations that can eff ectively be obtained\nfrom the evolution equations\ni/planckover2pi1d\ndtSx\nj= [Sx\nj,H]−iΓ2Sx\nj,\ni/planckover2pi1d\ndtSy\nj= [Sy\nj,H]−iΓ2Sy\nj,\ni/planckover2pi1d\ndtSz\nj= [Sz\nj,H]−iγ1/parenleftbig\nSz\nj−S/parenrightbig\n, (8)\nwith the straightforward use of the semiclassical approximation [4, 14,28,29]. Here Γ 2is\na transverse attenuation parameter, γ1is longitudinal attenuation parameter, and S≡\n4/summationtextN\nj=1Sj/Nis average spin value. In the case of strong initial polarization, the t ransverse\nattenuation is characterized [4,14,27,29] by the expression\nΓ2=γ2(1−s2), (9)\nwheresis the reduced average spin\ns≡1\nNN/summationdisplay\nj=1/angbracketleftSz\nj/angbracketright\nSj, (10)\nand the natural width is\nγ2≡ρµ2\n0S\n/planckover2pi1=ρ/planckover2pi1γ2\nSS . (11)\nIn the following numerical calculations, we shall use the reduced dime nsionless attenua-\ntion parameters, measured in units of the Zeeman frequency\nω0≡ −µ0B0\n/planckover2pi1=|γS|B0,(µ0=−2µB=−/planckover2pi1|γS|). (12)\nOur goal is to investigate spin dynamics of nanoclusters with realistic parameters. Thus,\nthe values of the anisotropy parameters, typical for nanocluste rs, such as Co, Fe, and Ni\nnanoclusters, are\nD\n/planckover2pi1γ2= 10−3,D2\n/planckover2pi1γ2= 10−3,D4\n/planckover2pi1γ2= 10−10. (13)\nStudying spin dynamics, we shall take into account that real nanoc lusters are not com-\npletely identical with each other, but exhibit the dispersion in their an isotropy parameters\nand spin values. The main aim of our study is to analyze how the parame ter dispersion\ninfluences spin dynamics and to find conditions allowing for effective sp in reversal.\n3 Introduction of dimensionless system parameters\nFor numerical analysis, it is convenient to pass to dimensionless quan tities. To this end, we\ndefine the dimensionless resonator feedback field\npH≡H\nB0(14)\nand the reduced transverse attenuation, caused by dipole intera ctions,\npd≡γ2\nω0. (15)\nAlso, we introduce the dimensionless anisotropy parameters\npA≡ωA\nω0, p B≡ωB\nω0, p C≡ωC\nω0, (16)\nexpressed through the anisotropy frequencies\nωA≡2SD\n/planckover2pi1, ω B≡2SD2\n/planckover2pi1, ω C≡2S3D4\n/planckover2pi1. (17)\n5In the equations of motion, we employ the reduced spin, with the com ponents\neα≡1\nNN/summationdisplay\nj=1Sα\nj\nSj, e z≡s . (18)\nSince the nanocluster spins Sjare large, of order 103, the semiclassical approximation is well\njustified. This allows us to treat eas a classical vector. The evolution will be considered\nwith respect to the dimensionless time\n/tildewidet≡ω0t . (19)\nAt the initial moment of time, there is no feedback field, which assume s the initial\nconditions\npH(0) = 0,˙pH(0) = 0, (20)\nwhere the overdot implies time derivative.\nThe initial conditions for the spin polarization ez(0) are constructed as follows [30,31].\nFor the given value of an initial polarization ez(0), a variety of the initial orientations for\nthe individual vectors Sz\nj(0) are admissible. These orientations can be prescribed by a kind\nof Monte Carlo techniques. First, a random configuration of vecto rsSz\njis taken and the\ncorresponding total polarization is evaluated. A new direction is cho sen randomly for each\nspin, and the new total polarization is calculated. If it is less than the initial one, the\narray with the changed magnetic moment direction distribution is cho sen as the second\niteration, otherwise, it is rejected. This procedure is repeated un til the system achieves the\nrequired average polarization ez(0) playing the role of the initial condition. The initial spin\npolarization is assumed to be directed opposite to the external mag netic field B0.\n4 Optimal conditions for spin reversal\nWe accomplish numerical solution of the evolution equations for N= 153= 3375 spins,\nsearching for the conditions of the optimal spin reversal. The Zeem an frequency ω0is taken\nto be in resonance with the circuit natural frequency ω.\nThe peculiarity of spin reversal depends on the system parameter s. For instance, the op-\ntimalvalueoftheresonatorattenuation γ, defined bytheresonatorcircuit quality, essentially\ndepends on the given transverse attenuation γ2, caused by spin dipole interactions.\nLet us fix γ2= 0.01ω0, which yields pd= 0.01, and let us take the anisotropy parameters\ntypical for Co, Fe, and Ni nanoclusters, as defined above. Below t he blocking temperature,\nthe longitudinal relaxation is suppressed, so that γ1is much smaller than γ2, which allows\nus to set γ1/γ2= 10−3. The coil filling factor is assumed to be close to one. The nanocluster\nspins are randomly distributed by the normal law with mean S= 1200 and variance δS=\n400. Dynamics ofspin polarization, ez, is shown in Fig. 1 for different resonator attenuations\nγ. Varying γ, we are looking for its optimal value that provides the maximal perma nent\nreversal, without superimposed oscillations. It is this regime that is o ptimal for fast and\nstable information processing [32]. As is seen, for the given ratio γ2/ω0, the optimal value\nof the resonator attenuation is γ≈0.35ω0. Below this value, the reversal is slower and the\nreversed spin value is smaller. For very low γ, there is no reversal at all. Above the optimal\nvalue ofγ, there arise oscillations, and the reversed spin value diminishes.\n6The spin reversal is caused by the resonator feedback field. This is illustrated in Fig. 2,\nwhere it is seen that the reversal occurs when the amplitude of the feedback-field oscillations\nis maximal and almost reaches the strength of the external field B0. For each given ratio\npd=γ2/ω0, the corresponding optimal value of γis chosen. The stronger external field B0,\nthat is, the larger ω0, hence, smaller pd, the more effective is the reversal, though the reversal\ntime increases. Spin magnitudes are randomly distributed, as in Fig. 1 .\nThe dependence of the optimal γ∗and the related value of the reversed spin s∗=e∗\nz, as\nfunctions of the ratio pd=γ2/ω0, are presented in Fig. 3. The larger the ratio pd, the larger\nthe optimal γ∗, but the smaller the value of the reversed spin s∗.\nInadditiontospindispersion, therecanexist thedispersionofthea nisotropyparameters.\nWe study spin dynamics, when the spins as well as all anisotropy para meters are distributed\nby the normal law. The spin mean is taken as S= 1200 and variance as δS= 960. The\nmean values of the anisotropy parameters are assumed to be typic al for the Co, Fe, and Ni\nnanoclusters, as in Eq. (13), with the relative variance\nδD\nD=δD2\nD2=δD4\nD4= 0.1.\nWe accomplish 30 realizations of spin dynamics varying the anisotropy parameters, with ran-\ndom spin distribution in each of the realizations. The results, averag ed over the realizations,\nare shown in Fig. 4. We see that a rather strong dispersion of the sy stem parameters does\nnot preclude spin reversal.\nTo understand when the spin reversal could be blocked by anisotro py, we vary the pa-\nrameters of the latter in a wide range, looking for the values at which the reversal becomes\nblocked. Varying oneof the anisotropy parameters, we keep othe rs fixed. Spins arerandomly\ndistributed with mean S= 1200 and relative variance 0.3. Fig. 5 illustrates the results, from\nwhich the blocking anisotropy parameters are evaluated as\nD\n/planckover2pi1γ2∼10−2,D2\n/planckover2pi1γ2∼0.5×10−1,D4\n/planckover2pi1γ2∼10−8.\nThese blocking parameters are essentially larger than the typical v alues in Eq. (13). Hence,\nthe typical anisotropy does not block spin reversal.\nWith increasing external field B0, that is, diminishing the ratio pd=γ2/ω0, the blocking\nanisotropy parameters decrease. This is illustrated by Fig. 6 that is to be compared with\nFig. 5. So, increasing the external field suppresses the role of the anisotropy.\nFinally, we study whether a strong spin dispersion can destroy the c oherent spin dynam-\nics and influence spin reversal. For this purpose, we consider a rand om distribution of spins,\nwith mean S= 1200 and very large relative variance δS/S= 1. The results of numerical\nsimulations are given in Fig. 7 for the ratio pd=γ2/ω0= 0.01, with the corresponding\noptimalγ/ω0= 0.3, and varying anisotropy parameters. The results demonstrate that, for\nsmall anisotropy parameters, spin dispersion plays practically no ro le. This role increases for\nlarger values of the anisotropy parameters. This fact finds straig htforward explanation from\nthe structure of the spin equations of motion. When the anisotrop y parameters are much\nsmaller than the Zeeman frequency, the spin rotation is governed b y the same ω0depending\nonly on the external field B0, but independent of the spin lengths. But when the anisotropy\nparameters are sufficiently large, approaching their blocking values , then the anisotropy\n7disturbs the effective rotation frequencies, so that different spin s rotate with different fre-\nquencies. This is equivalent to the appearance of inhomogeneous br oadening. Fortunately,\nthe typical nanocluster anisotropy parameters are much lower th an their blocking values.\nSo, for a sufficiently strong external field, all spins rotate with almo st the same frequency\nω0and their dispersion even being quite large, does not much influence t heir motion. Then\nthe coherent spin dynamics can be realized, resulting in a fast spin re versal.\n5 Conclusion\nWe have studied the magnetization dynamics in an ensemble of magnet ic nanoclusters, keep-\ning in mind a realistic situation, when the nanoclusters are character ized by a microscopic\nHamiltonian, with well defined parameters. Another peculiarity of re alistic nanocluster en-\nsembles is the dispersion in the values of their spins and anisotropy pa rameters, which we\nalso take into account. We have accomplished a series of computer s imulations varying the\nsystem properties and searching for the conditions when the magn etization reversal is opti-\nmal, in the sense of being fast, maximal, and quasi-stationary, witho ut oscillations. Such a\nprocess of magnetization reversal is necessary in a variety of app lications. For instance, for\nrealizing effective quantum information processing.\nFor numerical values, we have chosen the nanocluster parameter s typical for Co, Fe, and\nNi nanoclusters. It turnsout that iftheexternal magnetic fieldd efines theZeemanfrequency\nthat is essentially larger than the effective anisotropy frequencies , then the dispersion of spin\nmagnitudes plays practically no role. This also concerns the dispersio n in the values of the\nanisotropy parameters. The external magnetic field, sufficient fo r this effect is 1 T, which\ncorresponds to the Zeeman frequency ω0∼1011Hz.\nWe have shown that the magnetization reversal is caused by the re sonator feedback field\nthat is self-organized by moving spins. The value of the feedback fie ld, in the moment of\nspin reversal, almost reaches the value of the external field. Fast magnetization reversal\ncan be achieved solely by moving spins themselves, without involving st rong transverse\nmagnetic fields. For such nanoclusters as Co, Fe, and Ni, the rever sal time can be of order\n10−11s. This conclusion suggests a convenient mechanism for the efficient manipulation of\nnanocluster magnetization, which is of high importance for a variety of applications.\nAcknowledgment\nThe authors acknowledge financial support from the Russian Foun dation for Basic Re-\nsearch under the projects 10-02-96023, 11-02-00086, 11-07 -96007, and 12-02-00897. One of\nthe authors (V.I.Y.) is grateful to E.P. Yukalova for useful discuss ions.\n8References\n[1] B. Barbara, L. Thomas, F. Lionti, I. Chiorescu, and A. Sulpice, J . Magn. Magn. Mater.\n200, 167 (1999).\n[2] W. Wernsdorfer, Adv. Chem. Phys. 118, 99 (2001).\n[3] J. Ferre, Topics Appl. Phys. 83, 127 (2002).\n[4] V.I. Yukalov and E.P. Yukalova, Phys. Part. Nucl. 35, 348 (2004).\n[5] J. Wang and X.C. Zeng, in: Nanoscale Magnetic Materials and Applications (Springer,\nBerlin, 2009), p. 35.\n[6] V.I. Yukalov and E.P. Yukalova, J. Appl. Phys. 111, 023911 (2012).\n[7] V.I. Yukalov, V.K. Henner, P.V. Kharebov, and E.P. Yukalova, Las er Phys. Lett. 5, 887\n(2008)\n[8] V.I. Yukalov, V.K. Henner, and P.V. Kharebov, Phys. Rev. B 77, 134427 (2008).\n[9] V.I. Yukalov and E.P. Yukalova, Laser Phys. Lett. 8, 804 (2011).\n[10] V. Henner, Y. Raikher, and P. Kharebov, Phys. Rev. B 84, 144412 (2011).\n[11] E.M. Purcell, Phys. Rev. 69, 681 (1946).\n[12] N. Bloembergen and R.V. Pound, Phys. Rev. 95, 8 (1954).\n[13] A. Szoke and S. Meiboom, Phys. Rev. 113, 585 (1959).\n[14] A. Abragam, The Principles of Nuclear Magnetism (Clarendon, Oxford, 1961).\n[15] J. Jeener, A. Vlassenbroek, and P. Broekaert, J. Chem. Phy s.103, 1309 (1995).\n[16] D.J.Y. Marion and H. Desvaux, J. Magn. Res. 193, 153 (2008).\n[17] H.Y. Chen, Y. Lee, S. Bowen, and C. Hilty, J. Magn. Res. 208, 204 (2011).\n[18] V.V. Krishnan and N. Murali, Radiation damping in modern NMR exper iments:\nprogress and challenges. Prog. Nucl. Magn. Res. Spectrosc. (20 12) in press.\n[19] R.H. Dicke, Phys. Rev. 93, 99 (1954).\n[20] H. Walther, B.T.H. Varcoe, B.G. Englert, and T. Becker, Rep. Pr og. Phys. 69, 1325\n(2006).\n[21] J.T. Manassah, Phys. Rev. A 83, 025801 (2011).\n[22] J.T. Manassah, Laser Phys. 22, 738 (2012).\n[23] J.F. Kiselev, A.F. Prudkoglyad, A.S. Shumovsky, and V.I. Yukalov , Mod. Phys. Lett.\nB1, 409 (1988).\n9[24] T.S. Belozerova, V.K. Henner, and V.I. Yukalov, Phys. Rev. B 46, 682 (1992).\n[25] T.S. Belozerova, V.K. Henner, and V.I. Yukalov, Laser Phys. 2, 545 (1992).\n[26] V.I. Yukalov, Phys. Rev. B 53, 9232 (1996).\n[27] V.I. Yukalov, Phys. Rev. B 71, 184432 (2005).\n[28] D. ter Haar, Lectures on Selected Topics in Statistical Mechanics (Pergamon, Oxford,\n1977).\n[29] A. Abragam and M. Goldman, Nuclear Magnetism: Order and Disorder (Clarendon,\nOxford, 1982).\n[30] C.L. Davis, V.K. Henner, A.V. Tchernatinsky, and I.V. Kaganov, Phys. Rev. B 72,\n054406 (2005).\n[31] T.S. Belozerova, C.L. Davis, and V.K. Henner, Phys. Rev. B 58, 3111 (1998).\n[32] I. Zutic, J. Fabian, and S.D. Sarma, Rev. Mod. Phys. 76, 323 (2004).\n10Figure Captions\nFig. 1. Dynamics of the average spin polarization ez=s, for the ratio γ2/ω0= 0.01,\nas a function of dimensionless time, for random spins with the mean S= 1200 and relative\nspin variance δS/S= 0.333 for different values of the resonator attenuation: (a) γ/ω0= 0\n(dashed-dotted line), γ/ω0= 0.04 (dashed line), γ/ω0= 0.12 (solid line); (b) γ/ω0= 0.3\n(dashed-dotted line), γ/ω0= 0.4 (dashed line), γ/ω0= 0.8 (solid line).\nFig. 2. Average spin polarization ez=s(dashed line) and feedback field pH(solid line)\nas functions of dimensionless time, for different ratios pd=γ2/ω0, with the corresponding\noptimalγ: (a)pd= 0.1,γ/ω0= 0.7; (b)pd= 0.01,γ/ω0= 0.3; (c)pd= 0.001,γ/ω0= 0.1.\nSpins are randomly distributed as in Fig.1.\nFig. 3. Dependence of the optimal γ∗and the related value of the reversed spin s∗=e∗\nz,\nas functions of the ratio pd=γ2/ω0.\nFig. 4. Influence of the combined dispersion of spins and anisotropy para meters on the\ndynamics of spin reversal for pd=γ2/ω0= 0.01, with the corresponding optimal γ/ω0= 0.3:\n(a) spin polarization s=ez(dashed line) and feedback field pH(solid line) as functions\nof the dimensionless time for random spins with the relative spin varian ceδS/S= 0.8 and\nfixedanisotropyparameters; (b)Spinpolarization s=ez, averagedover 30realizationsofthe\nanisotropydistribution withtherelative variance0.1, asafunctiono fthedimensionless time;\n(c) feedback field pHas a function of the dimensionless time, averaged over 30 realization s\nof the anisotropy distribution.\nFig. 5. Spin polarization s=ez, as a function of dimensionless time, for the ratio\npd=γ2/ω0= 0.01, with the corresponding optimal γ/ω0= 0.3, for varying anisotropy\nparameters: (a) pA= 0.1 (solid line), pA= 0.2 (dashed line), pA= 0.3 (dashed-dotted line),\npA= 0.4 (dotted line); (b) pB= 0.5 (solid line), pB= 1 (dashed line), pB= 2 (dashed-\ndotted line), pB= 3 (dotted line); (c) pC= 0.1 (solid line), pC= 0.2 (dashed line), pC= 0.3\n(dashed-dotted line), pC= 0.4 (dotted line). In each case, spins are randomly distributed\nwith mean 1200 and relative variance 0.3.\nFig. 6. Spin polarization s=ez, as a function of dimensionless time, for the ratio\npd=γ2/ω0= 0.01, with the corresponding optimal γ/ω0= 0.3, for varying anisotropy\nparameters: (a) pA= 0.01 (solid line), pA= 0.05 (dashed line), pA= 0.1 (dashed-dotted\nline),pA= 0.2 (dotted line); (b) pB= 0.05 (solid line), pB= 0.1 (dashed line), pB= 0.2\n(dashed-dotted line), pB= 0.5 (dotted line); (c) pC= 0.01 (solid line), pC= 0.05 (dashed\n11line),pC= 0.1 (dashed-dotted line), pC= 0.2 (dotted line). In each case, spins are randomly\ndistributed with mean 1200 and relative variance 0.3.\nFig. 7. Spin polarization s=ez, as a function of dimensionless time, for the ratio\npd=γ2/ω0= 0.01, with the related optimal γ/ω0= 0.3, for varying anisotropy parameters.\nSolid line describes the case of strong spin dispersion, with mean spin S= 1200 and relative\nvariance δS/S= 1. Dashed line corresponds to the case without spin dispersion. Th e\nanisotropy parameters are: (a) pA= 0.01,pB= 0.01,pC= 0.001, (b)pA= 0.05,pB= 0.1,\npC= 0.05, (c)pA= 0.1,pB= 0.2,pC= 0.1, (d)pA= 0.2,pB= 0.7,pC= 0.2, (e)pA= 0.05,\npB= 0.01,pC= 0.001, (f)pA= 0.01,pB= 0.1,pC= 0.001, (g) pA= 0.01,pB= 0.01,\npC= 0.02.\n12Figure 1: Dynamics of the average spin polarization ez=s, for the ratio γ2/ω0= 0.01, as\na function of dimensionless time, for random spins with the mean S= 1200 and relative\nspin variance δS/S= 0.333 for different values of the resonator attenuation: (a) γ/ω0= 0\n(dashed-dotted line), γ/ω0= 0.04 (dashed line), γ/ω0= 0.12 (solid line); (b) γ/ω0= 0.3\n(dashed-dotted line), γ/ω0= 0.4 (dashed line), γ/ω0= 0.8 (solid line).\n13Figure 2: Average spin polarization ez=s(dashed line) and feedback field pH(solid line)\nas functions of dimensionless time, for different ratios pd=γ2/ω0, with the corresponding\noptimalγ: (a)pd= 0.1,γ/ω0= 0.7; (b)pd= 0.01,γ/ω0= 0.3; (c)pd= 0.001,γ/ω0= 0.1.\nSpins are randomly distributed as in Fig.1.\n14Figure 3: Dependence of the optimal γ∗and the related value of the reversed spin s∗=e∗\nz,\nas functions of the ratio pd=γ2/ω0.\n15Figure 4: Influence of the combined dispersion of spins and anisotro py parameters on the\ndynamics of spin reversal for pd=γ2/ω0= 0.01, with the corresponding optimal γ/ω0= 0.3:\n(a) spin polarization s=ez(dashed line) and feedback field pH(solid line) as functions\nof the dimensionless time for random spins with the relative spin varian ceδS/S= 0.8 and\nfixedanisotropyparameters; (b)Spinpolarization s=ez, averagedover 30realizationsofthe\nanisotropydistribution withtherelative variance0.1, asafunctiono fthedimensionless time;\n(c) feedback field pHas a function of the dimensionless time, averaged over 30 realization s\nof the anisotropy distribution.\n16Figure 5: Spin polarization s=ez, as a function of dimensionless time, for the ratio pd=\nγ2/ω0= 0.01, withthecorresponding optimal γ/ω0= 0.3, forvaryinganisotropyparameters:\n(a)pA= 0.1 (solid line), pA= 0.2 (dashed line), pA= 0.3 (dashed-dotted line), pA= 0.4\n(dotted line); (b) pB= 0.5 (solid line), pB= 1 (dashed line), pB= 2 (dashed-dotted line),\npB= 3 (dotted line); (c) pC= 0.1 (solid line), pC= 0.2 (dashed line), pC= 0.3 (dashed-\ndotted line), pC= 0.4 (dotted line). In each case, spins are randomly distributed with me an\n1200 and relative variance 0.3.\n17Figure 6: Spin polarization s=ez, as a function of dimensionless time, for the ratio pd=\nγ2/ω0= 0.01, withthecorresponding optimal γ/ω0= 0.3, forvaryinganisotropyparameters:\n(a)pA= 0.01 (solid line), pA= 0.05 (dashed line), pA= 0.1 (dashed-dotted line), pA= 0.2\n(dotted line); (b) pB= 0.05 (solid line), pB= 0.1 (dashed line), pB= 0.2 (dashed-dotted\nline),pB= 0.5 (dotted line); (c) pC= 0.01 (solid line), pC= 0.05 (dashed line), pC= 0.1\n(dashed-dotted line), pC= 0.2 (dotted line). In each case, spins are randomly distributed\nwith mean 1200 and relative variance 0.3.\n18Figure 7: Spin polarization s=ez, as a function of dimensionless time, for the ratio pd=\nγ2/ω0= 0.01, with the related optimal γ/ω0= 0.3, for varying anisotropy parameters. Solid\nline describes the case of strong spin dispersion, with mean spin S= 1200 and relative\nvariance δS/S= 1. Dashed line corresponds to the case without spin dispersion. Th e\nanisotropy parameters are: (a) pA= 0.01,pB= 0.01,pC= 0.001, (b)pA= 0.05,pB= 0.1,\npC= 0.05, (c)pA= 0.1,pB= 0.2,pC= 0.1, (d)pA= 0.2,pB= 0.7,pC= 0.2, (e)pA= 0.05,\npB= 0.01,pC= 0.001, (f)pA= 0.01,pB= 0.1,pC= 0.001, (g) pA= 0.01,pB= 0.01,\npC= 0.02.\n19"    },    {        "title": "2309.15568v1.Regulating_spin_dynamics_in_magnetic_nanomaterials.pdf",        "content": "arXiv:2309.15568v1  [cond-mat.mes-hall]  27 Sep 2023Regulating spin dynamics in magnetic nanomaterials\nV.I. Yukalov1,2and E.P. Yukalova3\n1Bogolubov Laboratory of Theoretical Physics,\nJoint Institute for Nuclear Research, Dubna 141980, Russia\n2Instituto de Fisica de S˜ ao Carlos, Universidade de S˜ ao Pau lo,\nCP 369, S˜ ao Carlos 13560-970, S˜ ao Paulo, Brazil\n3Laboratory of Information Technologies,\nJoint Institute for Nuclear Research, Dubna 141980, Russia\nE-mails:yukalov@theor.jinr.ru ,yukalova@theor.jinr.ru\nAbstract\nMagnetic nanomaterials can be used in the construction of de vices for information\nprocessing and memory storage. For this purpose, they have t o enjoy two contradictory\nproperties, from one side being able of keeping for long time magnetization frozen, hence\ninformation stored, and from the other side allowing for qui ck change of magnetization\nrequired for fast erasing of memory and rewriting new inform ation. Methods of resolving\nthis dilemma are suggested based on triggering resonance, d ynamic resonance tuning, and\non quadratic Zeeman effect. These methods make it possible to r ealize effective regulation\nof spin dynamics in such materials as magnetic nanomolecule s and magnetic nanoclusters.\nMagnetic nanomaterials find wide application in the creation of numero us devices in spin-\ntronics. There are several types of such materials. Among the mo st known, these are magnetic\nnanomolecules [1–9] and magnetic nanoclusters [10–14]. There exis ts the so-called magnetic\ngraphene represented by graphene flakes containing defects [1 5,16], including various magnetic\ndefects [17–20]. Trapped atoms, interacting through dipolar and s pinor forces, form clouds\npossessing effective spins [21–27]. Quantum dots, often called artifi cial nanomolecules [28],\nalso can have magnetization [29–31]. There exist as well nanomolecule s, such as propanediol\nC3H8O2and butanol C 4H9OH that, although do not have magnetization in their ground state,\nbut can be polarized and can keep magnetization for very long times, for hours and months,\ndepending on temperature [4,32]. To be concrete, in the present paper we consider magnetic\nnanomolecules and magnetic nanoclusters, although the similar cons ideration is applicable to\nother nanomaterials.\nMagnetic nanomolecules have the degenerate ground state, when the molecular spin can be\ndirected either up or down. These directions are separated by str ong magnetic anisotropy, with\nthe anisotropy barrier 10 −100 K. Below the blocking temperature 1 −10 K, the spin is frozen\nin one of the directions. The total spins of molecules can be different , between 1 /2 and 27/2.\nMagneticnanoclustershavemanypropertiessimilartomagneticnan omolecules. Amagnetic\nnanocluster behaves as a magnetic object with a large spin summariz ing the spins of particles\ncomposing the cluster. The number of particles forming a cluster ca n be up to 105. The\nmagnetization blocking temperature is 10 −100 K. A cluster radius is limited by coherence\n1radiusthat is between 1 nm to 100 nm. To form a single-domain magnet, a clus ter radius has\nto be not larger than the coherence radius, otherwise the sample b ecomes divided into several\ndomains.\nMagneticnanomolecules ornanoclusters, tobeusedformemoryde vices, need topossess two\nproperties contradicting each other. From one side, in order to ke ep memory for long time, the\nspinhastobewell frozen, which canbeachieved withstrong magnet icanisotropy. Butfromthe\nother side, in order to quickly change the magnetization, which is nec essary for memory erasing\nor for rewriting the information content, it is required to have no ma gnetic anisotropy that\nhinders spin motion. Thus magnetic anisotropy leads to the dilemma: a nisotropy is necessary\nfor being able to keep well memory, but it is an obstacle for spin regula tion. The goal of the\npaper is to suggest ways of resolving this dilemma.\nLet us consider, first, a single nanomagnet, either a nanomolecule o r a nanocluster, whose\nHamiltonians are practically the same by form, only with different value s of the corresponding\nparameters. The typical Hamiltonian of a nanomagnet reads as\nˆH=−µSB·S−DS2\nz+E(S2\nx−S2\ny), (1)\nwhereDandEare the anisotropy parameters. The total magnetic field\nB= (B0+∆B)ez+Hex+B1ey (2)\ncontains an external constant magnetic field B0, and additional magnetic field ∆ Bthat can be\nregulated, a feedback magnetic field Hcreated by a magnetic coil of an electric circuit, and a\ntransverse anisotropy field B1.\nThe existence of an electric circuit, with a magnetic coil, where the sa mple is inserted to,\nis the principal part of the setup we suggest. The action of a feedb ack field, created by the\nmoving spin itself, is the most efficient way for spin regulation [3,4,33– 35].\nLet at the initial moment of time the sample be magnetized in the direct ion of spin up.\nAt low temperature, below the blocking temperature, the spin direc tion is frozen, even if the\nexternal magnetic field is turned so that the sample is in a metastable state. The feedback field\nequation, obtained [33–35] from the Kirchhoff equation has the for m\ndH\ndt+2γH+ω2/integraldisplayt\n0H(t′)dt′=−4πηresdmx\ndt, (3)\nwhereγis the circuit attenuation, ω, resonator natural frequency, ηres≈V/Vres, filling factor,\nV,samplevolume, Vres,thevolumeoftheresonancecoil,andtheelectromotiveforceispr oduced\nby the moving average spin\nmx=µS\nV/angbracketleftSx/angbracketright. (4)\nSpin operators satisfy the Heisenberg equations of motion. Avera ging these equations, we\nare looking for the time dependence of the average spin component s\nx≡/angbracketleftSx/angbracketright\nS, y≡/angbracketleftSy/angbracketright\nS, z≡/angbracketleftSz/angbracketright\nS. (5)\nThepresence ofthemagneticanisotropyleadstotheappearance intheequations ofspinmotion\nof the terms binary in spin operators, which need to be decoupled. T he standard mean-field\n2approximation cannot be applied, being incorrect for low spins. We us e [35] the corrected\nmean-field approximation\n/angbracketleftSαSβ+SβSα/angbracketright=/parenleftbigg\n2−1\nS/parenrightbigg\n/angbracketleftSα/angbracketright/angbracketleftSβ/angbracketright (6)\nthat is exact for S= 1/2 and asymptotically exact for large spins S≫1, where we keep in\nmindα/negationslash=β.\nTo write down the equations of spin motion in a compact form, we intro duce the notations\nfor the Zeeman frequency ω0, dimensionless regulated field b, coupling attenuation γ0, and the\ndimensionless feedback field h:\nω0≡ −µS\n/planckover2pi1B0, b≡ −µS∆B\n/planckover2pi1ω0, γ 0≡πµ2\nSS\n/planckover2pi1Vres, h≡ −µSH\n/planckover2pi1γ0.(7)\nAlso, we define the anisotropy frequencies\nωD≡(2S−1)D\n/planckover2pi1, ω E≡(2S−1)E\n/planckover2pi1, ω 1≡ −µS\n/planckover2pi1B1, (8)\nand the anisotropy parameter\nA≡ωD+ωE\nω0. (9)\nThen the spin equations acquire the structure\ndx\ndt=−ω0(1+b−Az)y+ω1z ,\ndy\ndt=ω0(1+b−Az)x−γ0hz ,dz\ndt= 2ωExy−ω1x+γ0hy , (10)\ndh\ndt+2γh+ω2/integraldisplayt\n0h(t′)dt′= 4dx\ndt. (11)\nAs is seen from the equations, the effective Zeeman frequency is\nωeff=ω0(1+b−Az). (12)\nThis frequency is not constant, but depends on time, since the ter mAzvaries with time. The\nspin polarization varies in the range −1< z <1, hence the term Azvaries in the interval\n[−A,A]. This implies that the detuning is varying and large,\nωeff−ω\nω0=b−Az . (13)\nSuppose the initial setup is with spin up, z0= 1, and the external field B0turns so that the\nspin direction up corresponds to a metastable state. Nevertheles s, the spin can be kept in this\nstate for very long time being protected by the magnetic anisotrop y. Spin reversal can start\nonly when the effective Zeeman frequency is in resonance with the re sonator natural frequency\nω. However, the large detuning (13) does not allow for the resonanc e to occur.\nAssume that we need to reverse the spin at time τ. To start the spin motion, we can\norganize resonance at this time t=τby switching on the regulated field b=b(t) and setting\n30 10 20 30 40 t-1-0.500.51z(t)\n = 20\n = 10\n = 1 = 5B = 1\nFigure 1: Longitudinal spin polarization z\nas a function of dimensionless time tforω=\nω0= 10,ωE=ω1= 0.01,γ= 1, and b(τ)≡\nB=Az0, withA= 1 and z0= 1. The\ntriggering resonance is arranged at different\ntimesτ.0 5 10 15 20 25 30t-1-0.500.51z(t)\n = 10\n = 100\n = 5 = 1\n = 20 = 10\nFigure 2: Longitudinal spin polarization z\nas a function of dimensionless time tforω=\nω0= 100,ωE=ω1= 0.01,γ= 10, and\nA= 1. Dynamic resonance tuning starts at\ndifferent delay times τ.\nb(τ) =Az0, so that at this initial time detuning (13) be zero. This initial resonan ce triggers\nthe spin reversal because of which this can be called triggering resonance [36]. The reversal of\nthe longitudinal spin polarization for the triggering resonance, rea lized at different times τ, is\nshown in Fig. 1. Frequencies are measured in units of γ0and time, in units of 1 /γ0.\nAlthough the triggering resonance quickly initiates spin reversal, bu t at the last stage, when\nthere is no resonance, there appear long tails slowing down the reve rsal process. The reversal\nwould be much faster provided the resonance could be kept during t he whole process of spin\nreversal. This can be achieved by switching on the regulated field so t hat to support the\nresonance by varying b(t),\nb(t) =/braceleftbigg\n0, t < τ\nAzreg, t≥τ. (14)\nThen, till the time t=τthe spin is frozen by the anisotropy. Starting from the time τ, the\nfieldb(t) is varied by tuning zregin such a way, that zregbe close to z, thus diminishing the\ndetuning,\nb(t)−Az(t) =A[zreg(t)−z(t) ]→0. (15)\nThetimedependenceof zregcanbedefinedbythespindynamicsinasamplewithoutanisotropy.\nThis method is called dynamic resonance tuning [37]. The spin reversal under this method is\nfaster than in the method of triggering resonance, and there are no tails of spin polarization,\nas is seen from Fig. 2.\nIn the case of a sample containing many nanomagnets, it is necessar y to take into account\ntheir interactions through dipolar forces. Then the system Hamilto nian\nˆH=−µS/summationdisplay\njB·Si+ˆHA+ˆHD (16)\n4contains the Zeeman term, the term of magnetic anisotropy\nˆHA=−/summationdisplay\njD(Sz\nj)2, (17)\nand the energy of dipolar interactions\nˆHD=1\n2/summationdisplay\ni/negationslash=j/summationdisplay\nαβDαβ\nijSα\niSβ\nj, (18)\nwhereDαβ\nijis a dipolar tensor. The total magnetic field is B= (B0+∆B)ez+Hex.\nThe methods of triggering resonance and of dynamic resonance tu ning regulate spin dynam-\nics in a system of many nanomagnets in the same way as in the case of a single nanomagnet.\nThe presence of dipolar interactions produces the dephasing width γ2, hence the dephasing time\nis 1/γ2. However, the spin motion induced by the resonance coupling with a r esonant electric\ncircuit is coherent and spin reversal happens during the time much s horter than the dephasing\ntime. The reversal time can be as short as 10−11s. Thus dipolar interactions do not hinder the\npossibility of very fast spin reversal when the suggested methods are used. Moreover, under\nthe existence of dipolar interactions, there appear dipolar spin wav es triggering the initial spin\nmotion and facilitating spin reversal [4,33–35].\nOne more method allowing for the regulation of spin dynamics is based o n the use of the\nalternating-current quadratic Zeeman effect [26,27,38,39]. The n the Hamiltonian for a system\nof nanomagnets ˆH=ˆHZ+ˆHA+ˆHDcontains the same anisotropy term (17) and the dipolar\nterm (18), but the Zeeman term\nˆHZ=−µS/summationdisplay\njB·Sj+qZ/summationdisplay\nj(Sz\nj)2(19)\nincludes, in addition to the linear part, the quadratic Zeeman term. T he external magnetic\nfield can be taken in the form B=B0ez+Hex.\nWriting down the equations of spin motion shows that the effective Ze eman frequency\nbecomes ωeff=ω0(1+Az), with the effective anisotropy parameter\nA= (2S−1)qZ−D\n/planckover2pi1ω0. (20)\nThe coefficient of the alternating-current quadratic Zeeman effec t\nqZ(t) =−/planckover2pi1Ω2(t)\n4∆res(t)(21)\ncan be varied in time by varying the Rabi frequency Ω( t) and the detuning from an internal\nresonance ∆( t). By changing the quadratic Zeeman-effect coefficient qZ(t) according to the rule\nqZ(t) =/braceleftbigg\n0, t < τ\nD, t≥τ(22)\nmakes it possible to freeze the spin before the time τand, when necessary, to suppress the\neffective anisotropy term, thus realizing resonance and fast spin r eversal.\nIn conclusion, we have suggested several ways of regulating spin d ynamics in magnetic\nnanomaterials, such as magnetic nanomolecules and magnetic nanoc lusters. These methods\ncan be employed in spintronics, e.g. for creating memory devices.\n5References\n[1] B. Barbara, L. Thomas, F. Lionti, I. Chiorescu, and A. Sulpice J. Magn. Magn. Mater.\n200, 167 (1999).\n[2] A. Caneschi, D. Gatteschi, C. Sangregorio, R. Sessoli, L. Sorac e, A. Cornia, M.A. Novak,\nC.W. Paulsen, and W. Wernsdorfer, 1999 J. Magn. Magn. Mater. 200, 182 (1999).\n[3] V.I. Yukalov, Laser Phys. 12, 1089 (2002).\n[4] V.I. Yukalov and E.P. Yukalova, Phys. Part. Nucl. 35, 348 (2004).\n[5] J.R. Friedman and M.P. Sarachik, Annu. Rev. Condens. Matter Ph ys.1, 109 (2010).\n[6] J.S. Miller, Mater. Today 17, 224 (2014).\n[7] G.A. Craig and M. Murrie, Chem. Soc. Rev. 44, 2135 (2015).\n[8] S.T. Liddle and J. van Slageren, Chem. Soc. Rev. 446655 (2015).\n[9] B. Rana, A.K. Mondal, A. Bandyopadhyay, and A. Barman, Nanot echnology 33, 062007\n(2021).\n[10] R.H. Kodama, J. Magn. Magn. Mater. 200, 359 (1999).\n[11] G.C. Hadjipanayis, J. Magn. Magn. Mater. 200, 373 (1999).\n[12] W. Wernsdorfer, Adv. Chem. Phys. 118, 99 (2001).\n[13] V.I. Yukalov and E.P. Yukalova, Laser Phys. Lett. 8, 804 (2011).\n[14] J. Kudr, Y. Haddad, L. Richtera, Z. Heger, M. Cernak, V. Ada m, and O. Zitka, Nanoma-\nterials7, 243 (2017).\n[15] H. Terrones, R. Lv, M. Terrones, and M. Dresselhaus, Rep. P rog. Phys. 75, 062501 (2012).\n[16] E. Bekyarova, S. Sarkar, S. Niyogi, M.E. Itkis, and R.C. Haddon , J. Phys. D: Appl. Phys.\n45, 154009 (2012).\n[17] O.V. Yaziev, Rep. Prog. Phys. 73, 056501 (2010).\n[18] T. Enoki and T. Ando, Physics and Chemistry of Graphene (Sing apore, Pan Stanford,\n2013).\n[19] V.I. Yukalov, V.K. Henner, and T.S. Belozerova, Laser Phys. Le tt.13, 016001 (2016).\n[20] V.I. Yukalov, V.K. Henner, T.S. Belozerova, and E.P. Yukalova, J . Supercond. Nov. Magn.\n29, 721 (2016).\n[21] A. Griesmaier, J. Phys. B 40, R91 (2007).\n[22] M.A. Baranov, Phys. Rep. 464, 71 (2008).\n6[23] M.A. Baranov, M. Dalmonte, G. Pupillo, and P. Zoller, Chem. Rev. 112, 5012 (2012).\n[24] D.M. Stamper-Kurn and M. Ueda, Rev. Mod. Phys. 85, 1191 (2013).\n[25] B. Gadway and B. Yan, J. Phys. B 49, 152002 (2016).\n[26] V.I. Yukalov and E.P. Yukalova, Eur. Phys. J. D 72, 190 (2018).\n[27] V.I. Yukalov, Laser Phys. 28, 053001 (2018).\n[28] J.L. Birman, R.G. Nazmitdinov, and V.I. Yukalov, Phys. Rep. 526, 1 (2013).\n[29] D.A. Schwartz, N.S. Norberg, Q.P. Nguyen, J.M. Parker, andD.R .Gamelin, J. Am. Chem.\nSoc.125, 13205 (2003).\n[30] K.D. Mahajan, Q. Fan, J. Dorcena, G. Ruan, and J.O. Winter Biot echnol. J. 8, 1424\n(2013).\n[31] A. Tufani, A. Qureshi, and J.H. Niazi, Mater. Sci. Eng. C 118, 111545 (2021).\n[32] V.I. Yukalov and E.P. Yukalova, Phys. Part. Nucl. 31, 561 (2000).\n[33] V.I. Yukalov, Phys. Rev. Lett. 75, 3000 (1995).\n[34] V.I. Yukalov, Phys. Rev. B 53, 9232 (1996).\n[35] V.I. Yukalov, Phys. Rev. B 71, 184432 (2005).\n[36] V.I. Yukalov and E.P. Yukalova, Laser Phys. Lett. 19, 046001 (2022).\n[37] V.I. Yukalov and E.P. Yukalova, Laser Phys. Lett. 19, 116001 (2022).\n[38] V.I. Yukalov and E.P. Yukalova, J. Magn. Magn. Mater. 465, 450 (2018).\n[39] V.I. Yukalov and E.P. Yukalova, Phys. Rev. B 98, 144438 (2018).\n7"    },    {        "title": "2010.04969v1.Dynamics_of_charged_and_magnetized_particles_around_cylindrical_black_holes_immersed_in_external_magnetic_field.pdf",        "content": "Dynamics of charged and magnetized particles around\ncylindrical black holes immersed in external magnetic \feld\nRayimbaev Javlon,1, 2, 3,\u0003Demyanova Alexandra,1,yUgur Camci,4,z\nAhmadjon Abdujabbarov,1, 2, 3, 5, 6,xand Bobomurat Ahmedov1, 2, 6,{\n1Ulugh Beg Astronomical Institute, Astronomicheskaya 33, Tashkent 100052, Uzbekistan\n2National University of Uzbekistan, Tashkent 100174, Uzbekistan\n3Institute of Nuclear Physics, Ulugbek 1, Tashkent 100214, Uzbekistan\n4Department of Chemistry and Physics, Roger Williams University, One Old Ferry Road, Bristol, RI 02809, USA\n5Shanghai Astronomical Observatory, 80 Nandan Road, Shanghai 200030, P. R. China\n6Tashkent Institute of Irrigation and Agricultural Mechanization Engineers, Kori Niyoziy, 39, Tashkent 100000, Uzbekistan\n(Dated: October 13, 2020)\nThis paper investigates the motion and acceleration of a particle that is electrically and magneti-\ncally charged, which rotates around a cylindrical black hole in the presence of an external asymptot-\nically uniform magnetic \feld parallel to the zaxis. Basically, the work considers the circular orbits\nof particles rotating around the central object and studies the dependence of the most internal\nstable circular orbits (ISCO) on the so-called magnetic coupling parameters, which are responsible\nfor the interaction between the external magnetic \feld and magnetized and charged particles. It\nis also shown that the ISCO radius decreases with increasing magnetized parameter. Therefore,\ncollisions of magnetized particles around a cylindrical black hole immersed in an external magnetic\n\feld were also studied, and it was shown that the magnetic \feld can act as a particle accelerator\naround non-rotating cylindrical black holes.\nPACS numbers: 04.50.-h, 04.40.Dg, 97.60.Gb\nI. INTRODUCTION\nHigh-energy radiation processes and energy release\nnear the rotating horizon of a black hole have received\nmore attention in recent publications [1]. For example,\nthe classical Penrose e\u000bect [2] is one such process that\nhas been extensively studied in the literature, for exam-\nple. in the links [3{5]. Moreover, Banados and his co-\nauthors in [ [6]] demonstrated that for an extremely spin-\nning black hole head, collisions can generate high-energy\nparticles of the center of mass (the so-called BSW pro-\ncess). The results of [6] were commented on in [7], where\nthe authors concluded that astrophysical constraints on\nmaximum spin, back reaction e\u000bects, and sensitivity to\ninitial conditions place severe constraints on the likeli-\nhood of such acceleration. Energy release and particle\nacceleration around a rotating black hole in Ho \u0014 r Ava-\nLifshitz gravity were studied in our previous article in [8].\nParticle acceleration, circular geodesic, accretion disk,\nand high-energy collisions in Janice-Newman-Vinicourt\nspacetime were investigated [9, 10]. It was recently\nshown in [11] that Kerr primordial super-spinars, ex-\ntremely compact objects with an appearance described\nby the geometry of the bare Kerr singularity, can serve\nas e\u000ecient accelerators for extremely high-energy colli-\n\u0003Electronic address: javlon@astrin.uz\nyElectronic address: demyanova@astrin.uz\nzElectronic address: ucamci@rwu.edu\nxElectronic address: ahmadjon@astrin.uz\n{Electronic address: ahmedov@astrin.uzsions.\nIn Ref. [12] author has shown that in the existence\nof the magnetic \feld innermost stable circular orbits\n(ISCO) of charged particles can be situated near to the\nhorizon in the ground of Schwarzschild spacetime. i.e\nshifts towards to center black hole. He also shown that\nfor a collision of two particles, one of which is charged\nand rotating at ISCO and the other is neutral and falling\nfrom in\fnity, the maximal collision energy formally can\nbe arbitrary high. This result has some a\u000enity with the\nrecently discussed e\u000bect of high center-of-mass energy for\ncollision of particle in the vicinity of extremely rotating\nblack holes.\nNo-hair theorem tells that no black hole can have its\nown magnetic \feld, but one can take into account the\nassumption that a black hole immersed in an external\nmagnetic \feld generated by an electric current from the\ncharged matter of an accretion disk or a magnetized ob-\nject will accompany the black hole.After Wald's solution\nof the electromagnetic \feld equation around black holes\ntrapped in an external asymptotically uniform magnetic\n\feld [13], the di\u000berent properties of electromagnetic \felds\naround various black holes in external / intrinsic asymp-\ntotically uniform magnetic \felds and magnetized neutron\nstars with an appropriate dipolar magnetic \feld were\nstudied by several authors conditions [14{33]. The dy-\nnamics of dipolar magnetized, electrically and magnet-\nically charged particles around various black holes im-\nmersed in an external magnetic \feld, as well as electri-\ncally and magnetically charged black holes have already\nbeen widely studied in various gravitational theories [34{\n52].\nThe aim of this paper is to show that a similar e\u000bectarXiv:2010.04969v1  [gr-qc]  10 Oct 20202\nof particle collision with high center-of-mass energy is\nalso possible when a black hole is non-rotating (or slowly\nrotating) provided there exists magnetic \feld in its ex-\nterior. This study might be interesting since there exist\nboth theoretical [53] and experimental [54] indications\nthat such a magnetic \feld must be present in the vicinity\nof black holes. In what follows we assume that this \feld\nis test one and its energy-momentum does not modify\nthe background black hole geometry. For a black hole of\nmassMthis condition holds if the strength of magnetic\n\feld satis\fes to the condition [12, 53]\nB\u001cBmax=c4\nG3=2M\f\u0012M\f\nM\u0013\n\u00181019\u0012M\f\nM\u0013\nGauss:\n(1)\nBlack holes with such \feld characteristics are called\n\"weakly magnetized\". The condition (1) can be expected\nto hold for both stellar mass and supermassive black\nholes.\nThe work is structured as follows. Section II is de-\nvoted to the study of the motion of charged particles\nin the space-time of a magnetized cylindrical black hole,\nwith the main focus on the properties of the ISCO radii.\nThe mechanism of particle acceleration in the region sur-\nrounding a magnetized black hole in cylindrical coordi-\nnates is discussed in Section III. Concluding remarks and\ndiscussions are presented in section IV.\nThroughout this article, we use a space-time signature\nlike (\u0000;+;+;+) and a system of units in which G= 1 =c\n(However, for these expressions with an astrophysical ap-\nplication, we wrote the speed of light explicitly.). Greek\nindexes are accepted from 0 to 3.\nII. MAGNETIZED BLACK HOLE IN\nCYLINDRICAL COORDINATES\nIt is interesting to study gravitational objects with the\ncylindrical symmetry from astrophysical point of view,\nsince the cylindrical symmetry can be applied to study\nthe jets and cosmic strings. From theoretical point of\nview cylindrical symmetry is important to study conical\nsingularities and spacetime defects.\nThe rotating charged cylindrical black hole vacuum so-\nlution is derived by considering the Einstein-Hilbert ac-\ntion with a cosmological constant in four dimensions [55{\n57] and inserting a cylindrical symmetric generic metric\ninto this action. The explicit spacetime metric in the\ncylindrical coordinates ( t;r;\u001e;z ) is\nds2=\u0000\u0001(\rdt\u0000!\n\u000b2d\u001e)2+r2(!dt\u0000\rd\u001e)2\n+dr2\n\u0001+\u000b2r2dz2; (2)where metric coe\u000ecients take form\n\u0001 =\u000b2r2\u0000\f\n\u000br+c2\n\u000b2r2;\n\f= 4M\u0012\n1\u00003\n2a2\u000b2\u0013\n;\nc2= 4Q2\u00121\u00003\n2a2\u000b2\n1\u00001\n2a2\u000b2\u0013\n;\n\r=s\n1\u00001\n2a2\u000b2\n1\u00003\n2a2\u000b2;\n!=a\u000b2\nq\n1\u00003\n2a2\u000b2;\nwith\u000b2=\u00001\n3\u0003 results in a real \u000b,\u0015is cosmological\nconstant.\nHereafter we will consider the nonrotating black hole\n(i.e.a= 0, so that != 0) with zero electric charge Q\nimmersed in external asymptotically uniform magnetic\n\feld. Then we may rewrite the metric (2) in the following\nform\nds2=\u0000\u0001dt2+dr2\n\u0001+r2d\u001e2+\u000b2r2dz2; (3)\nwhere \u0001 = \u000b2r2\u0000\f=(\u000br).\nNow we assume that a static cylindrical black hole is\nplaced in an external asymptotically uniform magnetic\n\feld so that the magnetic \feld lines are parallel to the z\naxis. Ricci scalar of the spacetime metric (3) is propor-\ntional to the cosmological constant R= 4\u0003. According\nto [58] the nonzero component of the 4-vector poten-\ntialA mu of the electromagnetic \feld in space close to\nthe black hole, which is in an external uniform magnetic\n\feld, has the following form\nA\u0016=1\n2(0;0;B0;0): (4)\nwhereB0is the asymptotic value of the external mag-\nnetic \feld aligned along axis of symmetry, being perpen-\ndicular to the equatorial plane where z=const . The\nnon zero component of the electromagnetic \feld tensor\n(F\u0016\u0017=A\u0017;\u0016\u0000A\u0016;\u0017) is\nFr\u001e=B0r (5)\nOrthonormal components of the external magnetic\n\feld near the BH can be calculated by using the tetrad\nby proper observer (see for details Eqs.(5)-(9) in Ref.[59])\nand non-zero components of the latter have the following\nform\nB^r=B0; B^\u001e=p\n\u0001B0: (6)\nEq.(6) shows that only the angular component of the\nexternal magnetic \feld around the BH re\rects gravity\ne\u000bects, while the radial component formally has the same\nform as in Newtonian case.3\nIII. CHARGED PARTICLES MOTION\nSince, the external magnetic \feld does not break the\nsymmetry of the spacetime, using the timelike \u0018\u0016\n(t)and\nspacelike\u0018\u0016\n(\u001e)Killing vectors one may easily \fnd two\nconserved quantities associated with them that are the\nenergy (E) and the generalized angular momentum ( L):\nE=\u0000\u0018\u0016\n(t)(mu\u0016+qA\u0016) =m\u0001dt\nds; (7)\nL=\u0018\u0016\n(\u001e)(mu\u0016+qA\u0016) =mr2d\u001e\nds+qr2B0\n2;(8)\nwhereu\u0016is the 4-velocity of the test particle, sis the\na\u000ene parameter and qis the charge of the test particle.\nBy combining these equations with the timelike restric-\ntion of test particles u\u0016u\u0016=\u00001, one can easily obtain\nthe equations of motion of the charged particle at equa-\ntorial plane z=const as\n_t=E\n\u0001; (9)\n_\u001e=L\nr2\u0000!B; (10)\n_r2=E2\u0000Ve\u000b; (11)\nwhere the e\u000bective potential of radial motion\nVe\u000b= \u0001\"\n1 +\u0012L\nr\u0000!Br\u00132#\n(12)\nand!B=qB0=(2mc) is the so-called cyclotron frequency,\nwhich is responsible for the magnetic interaction between\nan electric charge and an external magnetic \feld, Eand\nLare the energy and momentum per unit mass of the\ntest particle m. Fig. 1 the radial dependence of the e\u000bec-\ntive potential of the radial motion of a charged particle\nin the equatorial plane of a cylindrical black hole im-\nmersed in a uniform magnetic \feld is shown for di\u000berent\nvalues of the magnetic interaction parameter. Now we\ncan draw conclusions about how the parameters of mag-\nnetic interaction can change the nature of the motion of\ncharged particles. The parameter of magnetic interac-\ntion is responsible for the shift of the minimum value of\nthe e\u000bective potential to the center, which means that\nthe minimum distance of charged particles to the center\ndecreases with an increase in the parameter of magnetic\ninteraction. With an increase in the magnetic interac-\ntion parameter, parabolic and hyperbolic orbits begin to\ntransform into stable circular orbits. Thus the radial\npro\fle ofVe\u000bfor the di\u000berent values of the magnetic in-\nteraction parameter, running between 0 and 5 shows that\nby increasing the magnetized parameter we also lower the\npotential barrier, as compared to the pure cylindrical case\nwhen the magnetic \feld is absent.\nNext, we will consider the rotational motion of charged\nparticles around a cylindrical black hole, which is in\ru-\nenced by external magnetic \felds. The e\u000bective potential\n0 1 2 3 4 5 601020304050\nrVeffFIG. 1: The radial dependence of e\u000bective potential for radial\nmotion of charged particles around a cylindrical black hole im-\nmersed in an external asymptotically uniform magnetic \feld\nfor the di\u000berent values of the dimensionless magnetic interac-\ntion parameter: !B= 0:1 (dot-dashed line), !B= 1 (dashed\nline),!B= 5 (solid line). The mass of central object is taken\nto beM= 1, so\f= 4 and\u000b= 1:1.\nis of minimum value at the radius of such an orbit. The\nconditions for the appearance of circular orbits are as\nfollows:\ndr\nds= 0; V0\ne\u000b= 0; V00\ne\u000b= 0; (13)\nwhere prime \"0\" denotes the radial derivative of the\nfunctions. Using the \frst two equations of the [removed]\nref condition) and the [removed] ref eom2), we can eas-\nily derive the following solution to the equation for the\nenergy and angular momentum of a charged particle mov-\ning in circular orbits around a black hole located in an\nexternal magnetic \feld:\nE2=\u0012\n\u000b2r2\u0000\f\n\u000br\u0013n\n1 +1\n9\f2h\n2!B\u0000\n\u000b3r4\u0000\fr\u0001\n(14)\n\u0006q\n4!2\nB(\u000b3r4\u0000\fr)2\u00003\f(\f+ 2\u000b3r3)i2o\nL=r\n3\fn\n!B\u0000\n2\u000b3r4+\fr\u0001\n(15)\n\u0006q\n4r2!2\nB(\f\u0000\u000b3r3)2\u00003\f(\f+ 2\u000b3r3)o\n;\nFig. 2 shows the radial dependence of both the energy\nand the angular momenta of the charged particle moving\non circular orbits at the equatorial plane. One can easily\nsee that the presence of the magnetic interaction parame-\nter forces test particle to have bigger energy and angular\nmomentum in order to be kept on the circular orbit. It is\na consequence of the existence of electromagnetic inter-\naction between uniform magnetic \feld and charged test\nparticle in the background gravitational \feld of the cen-\ntral object.\nIn the next stage we will study the ISCO of charged\nparticle around cylindrical gravitational objects im-\nmersed in external magnetic \feld. In order to \fnd the4\n1 2 3 4 5 6 70200400600800\nr/Mℒ\nM\nωB=0.1\nωB=0.5 ωB=1β=4;α=1.1\n2 3 4 5 6 7050100150\nr/Mℰ ωB=0.1\nωB=0.5\nωB=1β=4;α=1.1\nFIG. 2: The radial dependence of the angular momentum\n(upper) and energy (bottom) of the charged particle orbiting\naround a cylindrical black hole immersed external magnetic\n\feld for the di\u000berent values of the parameter !B:!B= 0:1 is\ndashed line, !B= 1 is dot-dashed line, !B= 5 is solid line.\nThe mass of central object taken to be M= 1, so\f= 4 and\n\u000b= 1:1.\nvalues of the ISCO radii rISCO one should insert the ex-\npressions (14) and (15) to the condition V00\ne\u000b= 0 and\nsolve it with respect to the radial coordinate ras\n6\f\u0000\n\f+ 5\u000b3r3\u0001\n\u00004r!Bh\n2!B\n\u0002\u0000\n5\u000b6r7\u00007\u000b3\fr4+ 2\f2r\u0001\n+\u0000\n\f+ 5\u000b3r3\u0001\n\u0002q\n4r2!2\nB(\f\u0000\u000b3r3)2\u00003\f(\f+ 2\u000b3r3)i\n\u00150(16)\nSince it is impossible to \fnd analytical solution for\nrISCO we present the results of the numerical solution\nof the equation for rISCO in the Table I. One can eas-\nily see from the results that the presence of the magnetic\n\feld decreases the ISCO radii and forces particle to come\ncloser to the central object.\nNow we will analyse the e\u000bects of the parameter of\ncylindrical black hole and magnetic interaction parame-\nters on the ISCO radius of the charged black hole in plot\nform.\nFigure 3 gives information on the in\ruence of the pa-TABLE I: The dependence of the ISCO radii, energy and\nangular momentum of the charged particle from the dimen-\nsionless magnetic interaction parameter b.\n!B 0.1 0.5 1 2 510 50\nrISCO=M6:212:952:311:931:651:551:46\nE 13:64:162:781:941:240:880:4\nL 12:74:924:434:997:8612:954:6\n0.0 0.2 0.4 0.6 0.8 1.012345678\nωBr\nM\nα=0.8α=1\nα=1.2β=4\nFIG. 3: The dependence of ISCO radius of the charged par-\nticle around a cylindrical black hole immersed in the external\nmagnetic \felds for the di\u000berent values of the parameter \u000b.\nrameters of a cylindrical black hole and magnetic inter-\naction on the ISCO radius of charged particles rotating\naround a cylindrical black hole immersed in an external\nmagnetic \feld. Looking at the \fgure, we can conclude\nthat an increase in both parameters leads to a decrease\nin the ISCO radius.\nIV. MOTION OF MAGNETIZED PARTICLES\nIn this part of the study, we focus on the dynamics of\nmagnetized particles around a cylindrical black hole im-\nmersed in an external asymptotically uniform magnetic\n\feld. The equation of motion of magnetized particles can\nbe expressed using the following Hamilton Jacobi equa-\ntion [34]\ng\u0016\u0017@S\n@x\u0016@S\n@x\u0017=\u0000 \nm\u00001\n2D\u0016\u0017F\u0016\u0017!2\n; (17)\nwhere the termD\u0016\u0017F\u0016\u0017stands for the interaction be-\ntween the magnetized particle and the external magnetic\n\feld. Here we assume that the particle has magnetic\ndipole moment- \u0016\u0017and the polarization tensor D\u000b\fsat-\nis\fes the following condition\nD\u000b\f=\u0011\u000b\f\u001b\u0017u\u001b\u0016\u0017;D\u000b\fu\f= 0; (18)5\nwhereu\u0017is four velocity of the particle. The electromag-\nnetic \feld tensorF\u000b\fcan be expressed by components of\nelectricE\u000band magnetic B\u000b\felds in the following form\nF\u000b\f= 2u[\u000bE\f]\u0000\u0011\u000b\f\u001b\ru\u001bB\r; (19)\nwhere square brackets stand for antisymmetric tensor:\nT[\u0016\u0017]=1\n2(T\u0016\u0017\u0000T\u0017\u0016) ,\u0011\u000b\f\u001b\r is the pseudo-tensorial form\nof the Levi-Civita symbol \u000f\u000b\f\u001b\r with the relations\n\u0011\u000b\f\u001b\r =p\u0000g\u000f\u000b\f\u001b\r\u0011\u000b\f\u001b\r=\u00001p\u0000g\u000f\u000b\f\u001b\r;(20)\nwhereg= detjg\u0016\u0017j=\u0000r4\u000b2for the spacetime metric\n(2).\nThe interaction term D\u0016\u0017F\u0016\u0017, can be easily de\fned\nusing the expressions (18) and (19) and we have,\nD\u0016\u0017F\u0016\u0017= 2\u0016^\u000bB^\u000b= 2\u0016B0p\n\u0001: (21)\nCircular motion of the magnetized particle around the\ncylindrical black holes is studied by assuming the mag-\nnetic moment of the particle is perpendicular to the plane\nwherez=const , with the non zero vertical components\n\u0016i= (0;0;\u0016z), respectively. The spacetime symmetry\ndoes not break the axial symmetric con\fguration in the\npresence of the magnetic \feld and, therefore, one may\nkeep the two conserved quantities: p\u001e=Landpt=\u0000E\nwhich are correspond to angular momentum and energy\nof the magnetized particle, respectively. So, the expres-\nsion for the action of magnetized particles can be written\nin the following form\nS=\u0000Et+L\u001e+Sr(r): (22)\nThe form of the action allows to separate variables in the\nHamilton-Jacobi equation (17).\nOne can easily get the expression of e\u000bective potential\nfor radial motion of the magnetized particle at the equa-\ntorial plane, with pz= 0, inserting Eq.(21) to Eq.(17)\nusing the form of the action (22) and Eq.(11)\nVe\u000b(r;\u000b;l;B) = \u0001h\u0010\n1\u0000Bp\n\u0001\u00112\n+L2\nr2i\n; (23)\nwhereB= 2\u0016B0=mis magnetic coupling parameter be-\ning responsible to the magnetic interaction term D\u0016\u0017F\u0016\u0017\nin the Hamilton-Jacobi equation (17). In the case when\nwe consider the dynamics of a neutron star with the\nmagnetic dipole moment \u0016= (1=2)BNSR3\nNS, treated as\na magnetized particle, orbiting around a supermassive\nblack hole immersed in the external magnetic \feld, the\nmagnetic coupling parameter \fcan be estimated us-\ning the neutron star's observational parameters and the\nexternal magnetic \feld where the neutron star move,\naround the SMBH\nB=BNSR3\nNSBext\nmNS'\u0019\n103\u0012BNS\n1012G\u0013\u0012Bext\n10G\u0013\n\u0002\u0012RNS\n106cm\u00133\u0012mNS\n1:4M\f\u0013\u00001\n:(24)\n2 4 6 8 10 1251015202530\nr/MVeffℬ=0.1ℬ=0.2\nℬ=0.3β=4;α=1.1FIG. 4: The radial dependence of the e\u000bective potential\nof magnetized particles orbiting around the cylindrical black\nhole immersed in the external magnetic \felds for the di\u000berent\nvalues of the parameter B. The mass of central object is taken\nto beM= 1, so\f= 4 and\u000b= 1:1.\nRadial dependence of e\u000bective potential for radial mo-\ntion of the magnetized particles for the various values\nof of the magnetic coupling parameter Bis presented in\nFig.4. One can see from the \fgure that the maximum\nand minimum values of the e\u000bective potential and the\ndistance from where the potential is maximum decrease\nwith the increase of the magnetic coupling parameter.\nOne can de\fne circular orbits of particles around a\nblack hole by the following conditions\n_r= 0;@Ve\u000b\n@r= 0: (25)\nFrom the conditions (25)\nL=r3\u00010\n2\u0001\u0000r\u00010\u0010\n1\u00003Bp\n\u0001 + 2B2\u0001\u0011\n; (26)\nE=\u00013\n2\u0010\n1\u0000Bp\n\u0001\u0011\n2\u0001\u0000r\u00010h\n2p\n\u0001\u0000B(r\u00010+ 2\u0001)i\n:(27)\nFigure 5 demonstrates the radial dependence of the\nspeci\fc energy and angular momentum for the circular\nmotion of magnetized particles around a cylindrical black\nhole placed in an external magnetic \feld, taken at dif-\nferent values of the magnetic coupling parameter. An\nincrease in the magnetic coupling parameter leads to a\ndecrease in both energy and angular momentum, which\ncan be seen from the \fgures.\nISCO equation \fnds using the standard condition\nVrr\u00150 and we have got the following equation which\nits solution with respect to the radial coordinate to be\nISCO radius\nISCO equation \fnds using the standard condition\nVrr\u00150 and we have got the following equation which\nits solution with respect to the radial coordinate to be\nISCO radius6\n1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00123456\nr/Mℰℬ=0.2\nℬ=0.25\nℬ=0.3β=4;α=1.1\n2.0 2.5 3.0 3.5 4.0 4.5 5.0012345\nr/Mℒ\nMℬ=0.2\nℬ=0.25\nℬ=0.3β=4;α=1.1\nFIG. 5: The radial dependences of the energy (top graph)\nand angular momentum (bottom graph) of magnetized par-\nticles rotating in circular orbits around a black hole in an\nexternal magnetic \feld, taken at di\u000berent values of the pa-\nrameters of the relations calB.\nBr2\u0010\n4Bp\n\u0001\u00003\u0011\n\u000102+ 2rp\n\u0001\u0010\n2\u0001\u00003Bp\n\u0001 + 1\u0011\n\u00004r\u00013=2\u0010\n2B2\u0001\u00003Bp\n\u0001 + 1\u0011\u000100\n\u00010\u00150 (28)\nFrom the equation (28) it can be seen that it is im-\npossible to solve it analytically with respect to the radial\ncoordinate and to see the e\u000bects of the interaction of the\nmagnetic dipole of magnetized particles and the external\nmagnetic \feld, as well as the parameter \u000b. One way to\nanalyze this numerically is to plot ISCO as a function of\nthe parameter \u000b.\nV. ACCELERATION OF PARTICLES NEAR\nCYLINDRICAL BLACK HOLES\nObservations show that the brightness of galactic cen-\nter where a supermassive black holes is situated, much\nmore brighter than other places of the galaxy. By the\nreason, the study of energy release processes from di\u000ber-ent black holes with di\u000berent ways is always one of in-\nteresting issues in relativistic astrophysics. For the \frst\ntime Rodger Penrose has suggested a new mechanism\nof the energetic process from rotating Kerr black hole\nwhich has ergoregion where a particle come and decays\nto two, one of the parts falls down to the central black\nhole with negative energy and other one goes to in\fnity\nhaving bigger energy than initial one [60] . The Penrose\nprocess have been developed during the past years by\nseveral authors applying rotating black holes in di\u000berent\ngravity theories and conditions [1, 61]. Another model\nfor the energetic process is proposed by Banados, Silk\nand West produced by collisions of particles near horizon\nof the black hole [6, 62]. Now, here we investigate the col-\nlisions of neutral, charged and magnetized particles and\n\fnd the energy Ecmin the center of mass of the system\nwith energy at in\fnity E1andE2in the gravitational\n\feld described by spacetime metric (3) in the presence of\nexternal asymptotically uniform magnetic \feld. We will\nuse the following expression [6]\n\u00121p\u0000g00Ecm;0;0;0\u0013\n=m1u\u0016\n(1)+m2u\u0017\n(2); (29)\nwhereu\u0016\n(1)andu\u0017\n(2)are the 4-velocities of the particles,\nproperly normalized by g\u0016\u0017u\u0016u\u0017=\u00001 andm1,m2are\nthe rest masses of the particles, to \fnd the expression for\nthe center of mass energy. We will consider two particles\nwith equal mass ( m1=m2=m) which have the energy\nat in\fnityE1=E2= 1. Thus we have\nEcm=Ecmp\n2m=q\n1\u0000g\u0016\u0017u\u0016\n(1)u\u0017\n(2): (30)\nIn below we will study e\u000bects of spacetime around\ncylindrical black holes on energetics of collisions of neu-\ntral, charged and magnetized particles in frame of proper\nobserver.\nA. Neutral particles collisions\nThe expression for center of mass energy of the\ntwo neutral particles can be calculated by substituting\nEqs.(33) for the case B= 0 in to Eq.(30) in the following\nform:\nE2\ncm= 1 +1\n\u0001\u0000l1l2\nr2\u0000s\n1\u0000\u0001\u0012\n1 +l2\n1\nr2\u0013\n\u00021\n\u0001s\n1\u0000\u0001\u0012\n1 +l2\n2\nr2\u0013\n: (31)\nNow, we will analyse e\u000bects of the parameter \u000bon the\ncenter of mass energy of collisions of two neutral particles\naround cylindrical black hole plotting Eq.(31).\nFigure 6 shows the radial dependence of the energy\nof the center of mass of neutral particles rotating around\na static black hole with cylindrical symmetry for various7\n2 4 6 8 100.00.51.01.5\nr/Mℰcmℓ1=-ℓ2=2Mα=1.1\nα=1.3\nα=1.5\nFIG. 6: Radial dependence of the center of mass energy of\ncollision of two neutral particles around cylindrical black hole\nfor the di\u000berent values of the parameter \u000b.\nvalues of the parameter \u000bfor the values of the speci\fc\nenergy of colliding particles l1= 2Mandl2= 2M. It is\nseen that the energy of the center of mass increases due\nto the growth of the parameter \u000b.\nB. Charged particles collisions\nNow consider the collisions of two charged particles in\nthe vicinity of a cylindrical black hole immersed in an\nexternal magnetic \feld. Applying equations (9)-(11), we\ncan obtain equations for the energy of the center of mass\nof colliding particles in the following form:\nE2\ncm= 1 +1\n\u0001\u0000\u0012`1\nr\u0000!(1)\nB\u0013\u0012`2\nr\u0000!(2)\nB\u0013\n\u00001\n\u0001vuut1\u0000\u0001\"\n1 +\u0012`1\nr\u0000!(1)\nBr\u00132#\n\u0002vuut1\u0000\u0001\"\n1 +\u0012`2\nr\u0000!(2)\nBr\u00132#\n: (32)\nNow we need to analyze the in\ruence of space-time\naround the black hole and the magnetic \feld on the en-\nergy of the center of mass of collisions of positive-positive,\nnegative-negative and positive-negative charged particles\nby writing the equation (32).\nIn \fg.7 shows the radial dependence of the center-of-\nmass energy of two particles taken at di\u000berent values of\nthe dimensionless parameter B. It can be seen from the\n\fgure that in the presence of a magnetic \feld, a par-\nticle can signi\fcantly increase the acceleration process\nnear the horizon to high energies. In addition, we show\nthat the radial dependence of the center of mass energy\nin the collision of positive-positive and negative-negative\ncharged particles is the same for the case when the ori-\nentation of the particles is reversed.\n1.0 1.5 2.0 2.5 3.0 3.5 4.00.00.51.01.52.02.53.0\nr/MℰcmωB(1)=ωB(2)=0.1,ℓ 1=-ℓ2=2M\nωB(1)=ωB(2)=-0.1,ℓ 2=-ℓ1=2M&α=1.1\nα=1.3\nα=1.5\n1 2 3 4 5 60.00.51.01.52.02.53.0\nr/MℰcmωB(1)=-ωB(2)=0.1,ℓ 1=-ℓ2=2Mα=1.1\nα=1.3\nα=1.5FIG. 7: The radial dependence of the center of mass on the\ncollision energy of two charged particles around a black hole\nplaced in an external asymptotically uniform magnetic \feld,\nwhich are taken for di\u000berent values of the parameter \u000b.\nC. Magnetized particles collisions\nIn this subsection we consider collisions of two mag-\nnetized particles. The four-velocity of the magnetized\nparticle at equatorial plane ( z=const; _z= 0) has the\nfollowing components:\n_t=E\n\u0001;\n_r2=E2\u0000\u0001\u0014\u0010\n1\u0000Bp\n\u0001\u00112\n+l2\nr2\u0015\n;\n_\u001e=l\nr2: (33)\nOne could \fnd the expression for center of mass energy\nof the two magnetized particles inserting Eq.(33) in to\nEq.(30) in the following form:\nE2\ncm= 1 +1\n\u0001\u0000l1l2\nr2\u0000s\n1\u0000\u0001\u0014\u0010\n1\u0000B1p\n\u0001\u00112\n+l2\n1\nr2\u0015\n\u00021\n\u0001s\n1\u0000\u0001\u0014\u0010\n1\u0000B2p\n\u0001\u00112\n+l2\n2\nr2\u0015\n: (34)\nFigure 8 shows information on the radial dependence\nof the center of mass energy of magnetized particles ro-\ntating around a cylindrical black hole in an asymptoti-\ncally uniform external magnetic \feld at di\u000berent \u000b. To8\n1 2 3 4 5 6 7 80.91.01.11.21.31.41.51.6\nr/Mℰcmℬ1=ℬ2=0.1,ℓ 1=-ℓ2=2Mα=1.1\nα=1.3\nα=1.5\n2 4 6 8 10 120.00.51.01.5\nr/Mℰcmℬ1=-ℬ2=0.1,ℓ 1=-ℓ2=2Mα=1.1\nα=1.3\nα=1.5\n1.4 1.6 1.8 2.0 2.20.00.51.01.5\nr/Mℰcmℬ1=ℬ2=-0.1,ℓ 1=-ℓ2=2M\nα=1.1\nα=1.3\nα=1.5\nFIG. 8: Radial dependence of the center of mass energy of col-\nlision of two magnetized particles with the same initial energy\nE1=E2= 1, around 4-D EGB BH for the di\u000berent values of\nthe\u000bparameter.\nplot Fig.8 values of the speci\fc angular momentum for\ncolliding magnetized particles are chosen as l1=M= 2\nandl2=M=\u00002, and also collisions of magnetized parti-\ncles with co-directional and oppositely directed magnetic\ndipoles, having values of positive and negative parame-\nters of magnetic interaction. The upper graph gives in-\nformation that the collision energy of the center of mass\nof magnetized particles with the same direction of mag-\nnetic dipoles and external magnetic \feld increases with\nincreasing value of the parameter alpha . It should be\nnoted that at a large distance the center of mass of en-\nergy disappears, this means that the collision of particles\ndoes not occur due to the predominant e\u000bect of magneticinteraction between the dipoles of the particles and the\nminimum distance at which the particles cannot collide\nwith each other and the center of mass energy disappear-\nance comes close to the central object with the increase\nof the parameter \u000bThe middle panel of the \fgure demon-\nstrates the case of collision of magnetized particles with\nthe opposite direction of magnetic dipoles, when the en-\nergy of the center of mass has a minimum and a maxi-\nmum, in other words, with distance, the energy decreases\ndue to to reduce the gravitational potential, and then it\nincreases again when the magnetic interaction between\nan external magnetic \feld and magnetic dipoles plays an\nimportant role, and \fnally it decreases and tends to zero\ndue to the dominant interaction e\u000bect between the two\ndipoles. Finally, in the case when the magnetic dipole\nmoment of both the colliding magnetized particles and\nthe directions of the external magnetic \feld are opposite,\nthe energy of the center of mass decreases faster than in\nthe cases considered above due to the repulsive proper-\nties of the magnetic interaction between the particles and\nthe external magnetic \feld.\nVI. CONCLUSION\nIn this article, we studied the motion of a charged par-\nticle around a black hole in cylindrical coordinates in\nthe presence of an external asymptotically uniform mag-\nnetic \feld. The e\u000bective potential of the radial motion\nof a test particle in the equatorial plane is studied for\nthe case of various values of the magnetized parameter\nresponsible for the interaction of the magnetic \feld and\ncharged particles in the background gravitational \feld.\nUsing the conditions for the minimum form of the e\u000bec-\ntive potential, we obtained the numerical values of the\nenergy, angular momentum, and radii of ISCO. We have\nshown that the presence of a magnetic \feld can reduce\nthe radius of the ISCO, and charged particles can move\ncloser to the center of the black hole. Decreasing the\nISCO radius is very important because the gravitational\npotential near the central object can accelerate particles\nto high energies.\nIn Fig. 1 we have shown the radial dependence of the\ne\u000bective potential of the radial motion of a charged par-\nticle at an equatorial plane of a cylindrical black hole im-\nmersed in an external asymptotically uniform magnetic\n\feld for di\u000berent values of the magnetic interaction pa-\nrameter!B. We conclude that the minimum distance\nof the charged particles to the central object decreases\nwith the increase of the magnetic interaction parameter.\nWith the increase of the magnetic interaction parameter\nparabolic and hyperbolic orbits start to become stable\ncircular orbits.\nThen we have shown in Fig. 2 the radial dependence of\nboth the energy and the angular momenta of the charged\nparticle moving on circular orbits in the equatorial plane.\nThe presence of the magnetic interaction forces charged\ntest particle to have bigger energy and angular momen-9\ntum in order to be kept on its circular orbit, which is\nthe consequence of the existence of magnetic interaction\nbetween the external magnetic \feld and the test charged\nparticle in the background gravitational \feld of the cen-\ntral object.\nWe also investigate dynamics of magnetized particles\naround the cylindrical black hole immersed in the exter-\nnal magnetic \felds and shown that the maximum values\nof the speci\fc energy and angular momentum decreases\nwith the increase of the magnetic coupling parameter.\nISCO radius decrease with increase of the parameter \u000b\nfor the magnetized particle with the magnetic coupling\nparameterB= 0:1.\nAuthors of [6] underlined that a rotating black hole\ncan, in principle, accelerate the particles falling to the\ncentral black hole to arbitrary high energies. The open\nquestion is whether there is additional e\u000bects beside the\nrotation which could play a role of particle accelerator\nnear the black hole. Here we have investigated neu-\ntral, electrically charged and magnetized particle acceler-\nation mechanism near the non-rotating cylindrical black\nhole in the presence of the external magnetic \feld. Wehave obtained the exact analytical expression for cen-\nter of mass energy Ecmof the two particles. In the\nFig. 7 the radial dependence of the Ecmhas been shown\nfor the di\u000berent values of the dimensionless magnetic\ninteraction parameter !B. One can see from this \fg-\nure that in the presence of the external magnetic \feld\nthe particles could be accelerated to much higher en-\nergies with compare to the case when the magnetic\n\feld is absent and we shown that in cases when the\ntwo positive-positive and negative-negative charged par-\nticles collisions the behaviour of the radial dependence\nof the center of mass energy has symmetry in the fol-\nlowing replacements:\u0010\n!(1)\nB;!(2)\nB\u0011\n!\u0010\n\u0000!(1)\nB;\u0000!(2)\nB\u0011\nand\nl1!\u0000l1;\u0000l2!l2.\nAcknowledgments\nThis research is supported by Grants No. VA-FA-F-2-\n008 and No. MRB-AN-2019-29 of the Uzbekistan Min-\nistry for Innovative Development.\n[1] N. Dadhich, A. Tursunov, B. Ahmedov, and\nZ. Stuchl\u0013 \u0010k, Mon. Not. R. Astron. Soc 478, L89 (2018),\narXiv:1804.09679 [astro-ph.HE] .\n[2] R. Penrose, Ann. N.Y. Acad. Sci. 224, 125 (1973).\n[3] T. Piran, J. Shaham, and J. Katz, Astrophys. J. Lett\n196, L107 (1975).\n[4] T. Piran and J. Shaham, Phys. Rev. D 16, 1615 (1977).\n[5] T. Piran and J. Shaham, Astrophys. J 214, 268 (1977).\n[6] M. Ba~ nados, J. Silk, and S. M. West, Physical Review\nLetters 103, 111102 (2009).\n[7] E. Berti, K. D. Kokkotas, and E. Papantonopoulos,\nPhys. Rev. D 68, 064020 (2003), gr-qc/0306106 .\n[8] A. Abdujabbarov, B. Ahmedov, and B. Ahmedov, Phys.\nRev. D 84, 044044 (2011), arXiv:1107.5389 [astro-ph.SR]\n.\n[9] M. Patil and P. S. Joshi, Phys.Rev.D 85, 104014 (2012),\narXiv:1112.2525 [gr-qc] .\n[10] A. N. Chowdhury, M. Patil, D. Malafarina, and P. S.\nJoshi, Phys.Rev.D 85, 104031 (2012), arXiv:1112.2522\n[gr-qc] .\n[11] Z. Stuchl\u0013 \u0010k and J. Schee, Classical and Quantum Gravity\n29, 065002 (2012).\n[12] V. P. Frolov, Phys. Rev. D. 85, 024020 (2012),\narXiv:1110.6274 [gr-qc] .\n[13] R. M. Wald, Phys. Rev. D. 10, 1680 (1974).\n[14] A. N. Aliev, D. V. Galtsov, and V. I. Petukhov, Astro-\nphys. Space Sci. 124, 137 (1986).\n[15] A. N. Aliev and D. V. Gal'tsov, Soviet Physics Uspekhi\n32, 75 (1989).\n[16] A. N. Aliev and N. Ozdemir, Mon. Not. R. Astron. Soc.\n336, 241 (2002), gr-qc/0208025 .\n[17] V. P. Frolov and P. Krtou\u0014 s, Phys. Rev. D 83, 024016\n(2011), arXiv:1010.2266 [hep-th] .\n[18] C. A. Benavides-Gallego, A. Abdujabbarov, D. Malafa-\nrina, B. Ahmedov, and C. Bambi, Phys. Rev. D 99,044012 (2019), arXiv:1812.04846 [gr-qc] .\n[19] S. Shaymatov, B. Ahmedov, Z. Stuchl\u0013 \u0010k, and A. Abdu-\njabbarov, International Journal of Modern Physics D 27,\n1850088 (2018).\n[20] Z. Stuchl\u0013 \u0010k, J. Schee, and A. Abdujabbarov, Phys. Rev.\nD89, 104048 (2014).\n[21] A. Abdujabbarov and B. Ahmedov, Phys. Rev. D 81,\n044022 (2010), arXiv:0905.2730 [gr-qc] .\n[22] A. Abdujabbarov, B. Ahmedov, and A. Hakimov,\nPhys.Rev. D 83, 044053 (2011), arXiv:1101.4741 [gr-qc]\n.\n[23] A. A. Abdujabbarov, B. J. Ahmedov, S. R. Shayma-\ntov, and A. S. Rakhmatov, Astrophys Space Sci 334,\n237 (2011), arXiv:1105.1910 [astro-ph.SR] .\n[24] A. A. Abdujabbarov, B. J. Ahmedov, and V. G. Ka-\ngramanova, General Relativity and Gravitation 40, 2515\n(2008), arXiv:0802.4349 [gr-qc] .\n[25] V. Karas, J. Kovar, O. Kopacek, Y. Kojima, P. Slany,\nand Z. Stuchlik, in American Astronomical Society Meet-\ning Abstracts #220 , American Astronomical Society\nMeeting Abstracts, Vol. 220 (2012) p. 430.07.\n[26] Z. Stuchl\u0013 \u0010k and M. Kolo\u0014 s, European Physical Journal C\n76, 32 (2016), arXiv:1511.02936 [gr-qc] .\n[27] J. Kov\u0013 a\u0014 r, O. Kop\u0013 a\u0014 cek, V. Karas, and Z. Stuchl\u0013 \u0010k,\nClassical and Quantum Gravity 27, 135006 (2010),\narXiv:1005.3270 [astro-ph.HE] .\n[28] J. Kov\u0013 a\u0014 r, P. Slan\u0013 y, C. Cremaschini, Z. Stuchl\u0013 \u0010k,\nV. Karas, and A. Trova, Phys. Rev. D 90, 044029 (2014),\narXiv:1409.0418 [gr-qc] .\n[29] M. Kolo\u0014 s, A. Tursunov, and Z. Stuchl\u0013 \u0010k, Eur. Phys. J.\nC.77, 860 (2017), arXiv:1707.02224 [astro-ph.HE] .\n[30] J. Rayimbaev and P. Tadjimuratov, Physical Review D\n102, 024019 (2020).\n[31] J. Rayimbaev, B. Turimov, and S. Palvanov, in Inter-\nnational Journal of Modern Physics Conference Series ,10\nInternational Journal of Modern Physics Conference Se-\nries, Vol. 49 (2019) pp. 1960019{209.\n[32] J. Rayimbaev, B. Turimov, F. Marcos, S. Palvanov, and\nA. Rakhmatov, Modern Physics Letters A 35, 2050056\n(2020).\n[33] J. Rayimbaev, B. Turimov, and B. Ahmedov, Inter-\nnational Journal of Modern Physics D 28, 1950128-209\n(2019).\n[34] F. de Felice and F. Sorge, Classical and Quantum Gravity\n20, 469 (2003).\n[35] F. de Felice, F. Sorge, and S. Zilio, Classical and Quan-\ntum Gravity 21, 961 (2004).\n[36] J. R. Rayimbaev, Astrophys Space Sc 361, 288 (2016).\n[37] T. Oteev, A. Abdujabbarov, Z. Stuchl\u0013 \u0010k, and B. Ahme-\ndov, Astrophys. Space Sci. 361, 269 (2016).\n[38] B. Toshmatov, A. Abdujabbarov, B. Ahmedov, and\nZ. Stuchl\u0013 \u0010k, Astrophys Space Sci 360, 19 (2015).\n[39] A. Abdujabbarov, B. Ahmedov, O. Rahimov, and U. Sa-\nlikhbaev, Physica Scripta 89, 084008 (2014).\n[40] O. G. Rahimov, A. A. Abdujabbarov, and B. J. Ahme-\ndov, Astrophysics and Space Science 335, 499 (2011),\narXiv:1105.4543 [astro-ph.SR] .\n[41] O. G. Rahimov, Modern Physics Letters A 26, 399\n(2011), arXiv:1012.1481 [gr-qc] .\n[42] K. Haydarov, A. Abdujabbarov, J. Rayimbaev,\nand B. Ahmedov, Universe 6(2020), 10.3390/uni-\nverse6030044.\n[43] K. Haydarov, J. Rayimbaev, A. Abdujabbarov, S. Pal-\nvanov, and D. Begmatova, European Physical Journal\nC80, 399 (2020), arXiv:2004.14868 [gr-qc] .\n[44] A. Abdujabbarov, J. Rayimbaev, B. Turimov, and\nF. Atamurotov, Physics of the Dark Universe 30, 100715\n(2020).\n[45] B. Narzilloev, J. Rayimbaev, S. Shaymatov, A. Abdujab-\nbarov, B. Ahmedov, and C. Bambi, Physical Review D\n102, 044013 (2020), arXiv:2007.12462 [gr-qc] .\n[46] J. Rayimbaev, M. Figueroa, Z. Stuchl\u0013 \u0010k, and B. Juraev,\nPhys. Rev. D 101, 104045 (2020).\n[47] B. Turimov, J. Rayimbaev, A. Abdujabbarov, B. Ahme-\ndov, and Z. c. v. Stuchl\u0013 \u0010k, Phys. Rev. D 102, 064052\n(2020).[48] M. De Laurentis, Z. Younsi, O. Porth, Y. Mizuno,\nand L. Rezzolla, Phys.Rev.D 97, 104024 (2018),\narXiv:1712.00265 [gr-qc] .\n[49] V. S. Morozova, L. Rezzolla, and B. J. Ahmedov,\nPhys.Rev.D 89, 104030 (2014), arXiv:1310.3575 [gr-qc]\n.\n[50] A. Nathanail, E. R. Most, and L. Rezzolla, Monthly\nNotices of the Royal Astronomical Society: Letters 469,\nL31 (2017), arXiv:1703.03223 [astro-ph.HE] .\n[51] J. Vrba, A. Abdujabbarov, M. Kolo\u0014 s, B. Ahmedov,\nZ. Stuchl\u0013 \u0010k, and J. Rayimbaev, Phys.Rev.D 101, 124039\n(2020).\n[52] J. Vrba, A. Abdujabbarov, A. Tursunov, B. Ahmedov,\nand Z. Stuchl\u0013 \u0010k, European Physical Journal C 79, 778\n(2019), arXiv:1909.12026 [gr-qc] .\n[53] M. Y. Piotrovich, N. A. Silant'ev, Y. N. Gnedin,\nand T. M. Natsvlishvili, Astrophysical Bulletin 66, 320\n(2011).\n[54] A. K. Baczko, R. Schulz, M. Kadler, E. Ros, M. Perucho,\nT. P. Krichbaum, M. B ock, M. Bremer, C. Grossberger,\nM. Lindqvist, A. P. Lobanov, K. Mannheim, I. Mart\u0013 \u0010-\nVidal, C. M uller, J. Wilms, and J. A. Zensus, Astron-\nomy & Astrophysics 593, A47 (2016), arXiv:1605.07100\n[astro-ph.GA] .\n[55] J. P. S. Lemos, Classical and Quantum Gravity 12, 1081\n(1995), arXiv:gr-qc/9407024 [gr-qc] .\n[56] J. P. S. Lemos, Physics Letters B 353, 46 (1995),\narXiv:gr-qc/9404041 [gr-qc] .\n[57] J. P. S. Lemos and V. T. Zanchin, Phys.Rev.D 54, 3840\n(1996), arXiv:hep-th/9511188 [hep-th] .\n[58] R. Wald, Annals of Physics 82, 548 (1974).\n[59] L. Rezzolla and B. J. Ahmedov, Mon. Not. R. Astron.\nSoc.352, 1161 (2004), arXiv:gr-qc/0406018 .\n[60] R. Penrose, Phys. Rev. Lett. 10, 66 (1963).\n[61] S. M. Wagh, S. V. Dhurandhar, and N. Dadhich, Astro-\nphys J. 290, 12 (1985).\n[62] A. A. Abdujabbarov, A. A. Tursunov, B. J. Ahmedov,\nand A. Kuvatov, Astrophys Space Sci 343, 173 (2013),\narXiv:1209.2680 [gr-qc] ."    },    {        "title": "1004.2429v1.Dynamics_of_strongly_magnetized_ejecta_in_Gamma_Ray_Bursts.pdf",        "content": "Dynamics of strongly magnetized ejecta in Gamma Ray Bursts\nMaxim Lyutikov\nDepartment of Physics, Purdue University,\n525 Northwestern Avenue, West Lafayette, IN 47907-2036\nABSTRACT\nWe consider dynamical scales in magnetized GRB out\rows, using the solutions to\nthe Riemann problem of expanding arbitrarily magnetized out\rows (Lyutikov 2010).\nFor high ejecta magnetization, the behavior of the forward shock closely resembles\nthe so-called thick shell regime of the hydrodynamical expansion. The exception is at\nsmall radii, where the motion of the forward shock is determined by the dynamics of\nsubsonic relativistic out\rows. The behaviors of the reverse shock is di\u000berent in \ruid\nand magnetized cases: in the latter case, even for medium magnetization, \u001b\u00181, the\nreverse shock forms at fairly large distances, and may never form in a wind-type external\ndensity pro\fle.\n1. Introduction\nMagnetic \felds may play an important dynamical role in the GRB out\rows ( e.g.Lyutikov\n2006, 2009). They may power the relativistic out\row through Blandford & Znajek (1977) process\n(e.g.Komissarov 2005), and contribute to particle acceleration in the emission regions. In this paper\nwe discuss the dynamics of the relativistic, strongly magnetized ejecta. The results are based on an\nexact solution of a one-dimensional Riemann problem of expansion of a cold, strongly magnetized\ninto vacuum and into external medium of density \u001aex(Lyutikov, submitted); they are reviewed in\nx2.\nIn application to GRBs, we assumes that the central engine produces jet with density \u001a0and\nmagnetization \u001b(\u001b=B2\n0=\u001a0; magnetic \feld is normalized byp\n4\u0019), moving with Lorentz factor\n\rw\u001d1. In fact, parameters \rwand\u001bare not always independent quantities: at small radii, when\nthe motion of the ejecta is subsonic, they should be determined together with the motion of the\nboundary, seex3.3. In a supersonic regime, relation between \rwand\u001bdepends on the details of\nthe \row acceleration ( e.g.in conical \rows we expect \rw\u0018p\u001b). For generality, we do not assume\nany relationship between \u001band\rw. The ejecta is moving into external density \u001aex.arXiv:1004.2429v1  [astro-ph.HE]  14 Apr 2010{ 2 {\n2. Riemann problem for relativistic expansion of magnetized gas\n2.1. Simple waves and forward shock dynamics\nLet us assume that the jet plasma is moving with velocity \fwtowards the external medium.\nWe found (Lyutikov, submitted) exact self-similar solution of relativistic Riemann problem for\nthe expansion of cold plasma with density \u001a0and magnetic \feld B0(magnetization parameter\n\u001b=B2\n0=\u001a0; magnetic \feld is normalized byp\n4\u0019), moving initially with velocity vwtowards the\nvacuum interface\n\u000e\f=\u000e2=3\n\u0011\u000e2=3\nA;0\u000e1=3\nw\n\u000eA=\u000e2=3\nA;0\u000e1=3\nw\n\u000e1=3\n\u0011(1)\nwhere the Doppler factors \u000ea=p\n(1 +\fa)=(1\u0000\fa) are de\fned in terms of the plasma veloc-\nity\f, local Alfv\u0013 en velocity \fA, self-similar parameter \u0011=z=t, initial wind velocity \fwand the\nAlfv\u0013 en velocity in the undisturbed plasma \fA;0=p\n\u001b=(1 +\u001b). These equations give the velocity \f,\ndensity\u001a=U2\nA\u001a0=\u001b(UA=\fA=q\n1\u0000\f2\nA) and proper magnetic \feld, B= (\u001a=\u001a0)B0as a function\nof the self-similar variable \u0011=z=t(expansion of plasma starts at t= 0;z= 0 and proceeds into\npositive direction z>0). We stress that these solutions are exact, no assumptions about the value\nof the parameter \u001band velocity vwwere made.\nParticularly simple relations are obtained for plasma initially at rest expanding into vacuum\n\fw= 0; \u000e\f= 1 (Lyutikov, submitted). The \row accelerates from rest towards the vacuum interface.\nThe bulk of the \row is moving with Lorentz factor \r0\u0018\u001b1=3. The \row becomes supersonic at \u0011= 0,\nat which point \r0= (\u001b=2)1=3. The vacuum interface moves with Lorentz factor \r0\nvac= 1 + 2\u001b. In\nthe observer frame the vacuum interface is moving with \u000e\u0011=\u000e2\nA;0\u000ew, which in the limit \u001b; \rw\u001d1\nthis gives\n\rvac= 4\rw\u001b (2)\nAs the \row expands, the local magnetization\n\u001bloc=B2\n\u001a=0\n@\u000e2=3\nA;0\n\u000e1=3\n\u0011\u0000\u000e1=3\n\u0011\n\u000e2=3\nA;01\nA (3)\ndecreases. At the sonic point \u001bloc= (\u001b=2)2=3.\nIf there is an outside medium with density \u001aex, we may identify two expansion regimes. For\nrelativistically strong forward shocks, so that the post-shock pressure is much larger than density,\nthe Lorentz factor of the CD is\n\rCD=\u00123B2\n0\r2\nw\n8\u001aex\u00131=4\n\u0019\u0012L\n\u001aexc3\u00131=4\nr\u00001=2(4){ 3 {\n(the last approximation assumes \u001b\u001d1). For weak forward shocks the velocity of the CD approaches\nthe expansion velocity into vacuum \rvac, Eq. (2). The transition between the relativistically strong\nand weak shocks occurs for\n\u001bcrit=\u00123\n2048\r2w\u001a0\n\u001aex\u00131=3\n(5)\nFor\u001b<\u001bcrit, the forward shock is weak.\n2.2. Existence of the reverse shock\nFor cold unmagnetized jets the reverse shock always exists; it is weak for \rw\u0014p\n\u001a0=\u001aexand\nstrong otherwise (Sari & Piran 1995). For magnetized jets the conditions for existence of a reverse\nshock are more complicated (see also Giannios et al. 2008; Mizuno et al. 2009). There are, in fact,\ntwo somewhat di\u000berent regimes for the existence of a RS in highly magnetized out\rows. First, if\nejecta is supersonic with respect to the CD (in term of Riemann waves, this transition corresponds\nto the case when the location of the FS coincides with the location of the rarefaction wave), a\nstrong RS must forms. Secondly, if the ejecta is subsonic with respect to the CD, but moves with\nvelocity higher than the CD, slowing of the ejecta is achieved by a compression wave, which may\nor may not turn into a reverse shock. One dimensional compression waves are always unstable to\nshock formation (Landau & Lifshitz 1959). In contrast, multidimensional subsonic out\row need\nnot form shocks. So, formally, the condition for reverse shock is \rw> \rCD, but in the range\n\rCD<\rw<2\rwp\u001bthe RS may not form, if a more complicated \row patters are allowed. In any\ncase, the RS shock, even if it exists, is weak in this regime.\nConditions for strong reverse shock (which implies a highly supersonic \row, with velocity much\nlarger than the Alfv\u0013 en velocity in the upstream plasma) were derived by Kennel & Coroniti (1984).\nIn the frame of the CD, the reverse shock is moving with (Kennel & Coroniti 1984)\n\f0;2\nRS\u00191\u00001\n\u001b;for\u001b\u001d1 (6)\nIf\rw\u0015\rCD, the reverse shock is weak (if it exists) and one should use a more detailed calculations\nof the dynamics of perpendicular shocks of arbitrary strength.\nThus, for\u001b\u00151, the existence of strong RS requires \rw>2\rCDp\u001b, which using Eq. (4) gives\n\rw>p\n6r\u001a0\n\u001aex\u001b3=2; (7)\nwhile a weak RS may exist for \rCD<\rw<2\rwp\u001b:\nr\n3\n8r\u001a0\n\u001aexp\u001b<\rw<p\n6r\u001a0\n\u001aex\u001b3=2; (8){ 4 {\nThe Lorentz factor of a strong RS with respect to the contact discontinuity isp\u001b(assuming\n\u001b\u001d1). The Lorentz factor of the reverse shock in the frame of stationary external medium is then\n\rRS=1\n2\u0012p\u001b\n\rCD+\rCDp\u001b\u0013\n=8\n><\n>:\u00003\n128\u00011=4p\rw\n\u001b1=4\u0010\n\u001a0\n\u001aex\u00111=4\nif\rCD\u001dp\u001b\n\u001b1=4\n61=4p\rw\u0010\n\u001aex\n\u001a0\u00111=4\nif\rCD\u001cp\u001b(9)\nThe two cases in Eq. (9) correspond to RS moving in the same direction as the CD, \rCD>p\u001b,\nand the RS moving in the opposite direction than the CD, \rCD<p\u001b. Condition \rCD=p\u001bgives\n\rw=r\n8\n3r\u001aex\n\u001a0p\u001b (10)\nIn this case the reverse shock is stationary in the frame of the external medium.\nLet us summarize the main results. If the the ratio of ejecta density to external density is\nf=\u001a0=\u001aex, then weak RS can form for \rw\u0015\rCD, which gives \rw\u0018pf\u001b; strong RS forms\nfor\rw>p\u001b\rCD,\rw\u0015p\nf\u001b3. RS shock propagates in the forward direction for \rCD>p\u001b,\n\rw>p\n\u001b=f. Forwards shock is relativistically weak for \rCD\u0015\u001b,\rw> \u001b3=2=pfand becomes\nnon-relativistic for \rCD\u00181,\rw<1=p\u001bf.\n3. Dynamics of magnetized \rows in GRBs\nIn this section we apply the previous relationships to consider dynamics of magnetized \rows in\nGRBs, generalizing discussion of Sari & Piran (1995) to strongly magnetized \rows. We will derive\nmain results in a thin shell approximation (not to be confused with a thin shell case, see below),\nassuming that the distances between the forward shock, the contact discontinuity and the reverse\nshock are small. The velocity of the shocks and contact discontinuity are determined from the local\nforce balance conditions. More precisely, they are determined by the local solutions to the Riemann\nproblem of the decay of the discontinuity of the \row: there is no memory in the \row. Thin shell\napproximation is likely to be applicable, since for reasonable GRB parameters the reverse shock\nnever stalls while expansion is relativistic, see discussion after Eq. (20).\nThe ejecta \row is taken to expands conically and carrying toroidal magnetic \feld. We assume\nthat the central source operates for time \u0001 ts= \u0001=c(\u0001 is the initial width of the launched shell)\nand produces a wind with magnetization \u001b\u001d1 (magnetization \u001b=B2=\u001ais twice the ratio of\nmagnetic to particle energy in plasma frame; magnetic \feld is normalized byp\n4\u0019), moving with\nthe Lorentz factor \rw. For spherical expansion (expansion along conical surfaces) of magnetized\n\rows into vacuum, the magnetization parameter \u001bremains constant outside the fast magnetosonic\nsurface (Michel 1973; Vlahakis & K onigl 2003). As we will see, the above assumption (that the\ncentral source produces a \row with a given \rwand\u001b) is not self-consistent for small radii, where\nthe reverse shock does not form. In this case of subsonic expansion, the \row dynamics cannot be\nspeci\fed ad hoc : it needs to be determined self-consistently with the motion of the boundaries.{ 5 {\nThe wind luminosity is assumed to be Liso=Eiso=\u0001tswhereEisois the isotropic equivalent\nenergy released by the central source. Luminosity is produced in a form of Poynting and particle\n\ruxes\nL= 4\u0019r2\r2\nw(B2\n0+\u001a0) = 4\u0019r2\r2\nwB2\n01 +\u001b\n\u001b(11)\nWe are interested in the case \u001b\u00151. For numerical estimates we will use the typical values for long\nGRBs:Liso= 1051erg s\u00001, \u0001ts= 100 s,Eiso= 1053erg,\rw= 300. External density is \u001aex=mpn.\n3.1. Forward shock dynamics\nIn case of magnetized ejecta, as well as in the hydrodynamical case (Sari & Piran 1995), the\nimportant scales in the problem (Sedov scale lS(13), energy scale rE(14), reverse shock formation\nscalerN(18), reverse shock crossing scale r\u0001(21) and spreading distance rS(22)) are related by a\nquantity (Sari & Piran 1995)\n\u0018=r\nlS\n\u0001\r\u00004=3\nw; rN=\u0018=rE=p\n\u0018r\u0001=\u00182rs (12)\nIn the hydrodynamical case, the parameter \u0018determines whether the reverse shock and the rar-\nefaction wave reach the whole ejecta before most of the energy is transferred to the forward shock,\n\u0018>1, or later,\u0018<1. The dynamics of magnetized ejecta generally follows the hydrodynamic thick\ncase, though the meaning of some radii change ( e.g., in case of strongly magnetized ejecta rNis\nthe scale of RS formation).\nThere is a number of typical radii where dynamics of the out\row changes. There is Sedov\nradius\nlS\u0018\u0012Eiso\n\u001aexc2\u00131=3\n= 4\u00021018cmn\u00001=3(13)\nwhere the ejecta and the swept-up ISM material become non-relativistic.\nThere is radius rE, where the ejecta deposits approximately half of the energy or momentum\nto the external medium. For supersonic \rows, which reached terminal Lorentz factor \rw, equating\nenergy in the shocked medium \r2\nw\u001aexc2r3\nEto the total energy Eiso, gives (Rees & Meszaros 1992)\nrE\u0018\u0012Eiso\n\r2w\u001aexc2\u00131=3\n=lS\n\r2=3\nw= 9\u00021016cmn\u00001=3(14)\nrEdepends exclusively on the total energy of the explosion and not on its form (magnetic or\nbaryonic). For radii smaller than rEthe ejecta's and the forward shocks' Lorentz factors remain\nconstant and equal to the initial Lorentz factor \rw. For larger radii the \row enters the self-similar\nSedov-Blandford-McKee stage, with Lorentz factor decreasing according to\n\r=\u0012ls\nr\u00133=2\n(15){ 6 {\nIn case of pure baryonic \row, and only in that case, rEis also the radius when the swept-up\nmass equals the ejecta mass divided by \rw\nrM\u0018\u0012M0\n\rw\u001aexc2\u00131=3\n=\u0012EK\n\r2w\u001aexc2\u00131=3\n(16)\nHereEK=Eiso=(1 +\u001b) is the energy associated with bulk motion of matter. Only in the case of\nzero magnetization rEequalsrM, since in that case Eiso=EK=M0\rw.1For highly magnetized\nout\rowrM\nrE=\u001b\u00001=3\u001c1.\nThe above description of the forward shock dynamics is, in fact, applicable only in the so\ncalled thin shell case, \u0018>1 (see Eq. (12). In this case the reverse shock quickly crosses the ejecta,\nwhich becomes causally connected so that all of the ejecta interacts with the external medium.\nAlternatively, in the thick shell case, \u0018 <1 (see Eq. (12), the reverse shock does not have time to\ncross the ejecta before the causally connected shocked part starts to decelerate at smaller radius\nrN, Eq. (18). The Lorentz factor starts decreasing, but since new material and new momentum is\nbeing added to the shocked part of the ejecta, the ejecta and the forward shock behave e\u000bectively\nas a self-similar shock with energy supply\n\r=\u0012L\n\u001aexc2\u00131=41pr=l3=4\ns\n\u00011=4pr(17)\nSince in the thick shell case the Lorentz factor starts to decelerate earlier than in the thin shell\ncase, the rate of energy transfer to the external medium is smaller, so that the ejecta gives most of\nits initial energy to the ISM at larger distances r\u0001>rE(Sari & Piran 1995).\n3.2. Formation and dynamics of the reverse shock\nThe weak reverse shock may form at (see Eq. (8))\nrN=1\n\r2ws\n3L\n2\u0019\u001aISMc3\u00191\n\r2wl3=2\nSp\n\u0001= 1016cmn\u00001=2(18)\nRS becomes strong at rRS;strong\u0018\u001brN(see Eq. (7)).\nIf the outside medium is stellar wind, strong RS forms immediately if\n\rw>\u00123\n2\u0019Lvwind\u001b2\nc3_M\u00131=4\n= 220\u001b1=2L1=4\n51v1=4\nwind; 8 _M\n10\u00008M\f=yr!\u00001=4\n(19)\n1The two radii rMandrEwere confused by Zhang & Kobayashi (2005), who \"de\fne the deceleration radius using\nEKalone [] where the \freball collects 1 =\rwof \freball rest mass\". According to Zhang & Kobayashi (2005) \"only the\nkinetic energy of the baryonic component ( EK) de\fnes the afterglow level\", while magnetic energy is transferred at\nunspeci\fed \"later\" time. This is incorrect (Lyutikov 2005).{ 7 {\nwherevwind; 8is the velocity of the progenitors wind in thousands kilometers per second. For smaller\n\rw, no RS forms ever (for weak shocks, one should put \u001b!1).\nThe reverse shock could stall (in the observer frame) at (assuming \u001b\u001d1)\nrRS;stall =1\n\u001bs\n3L\n8\u001aexc3\u00181\n\u001bl3=2\nSp\n\u0001= 3\u00021021cmn\u00001=2\u001b\u00001(20)\nSincerRS;stall is typically larger than lS, RS does not stall during the relativistic expansion phase;\nthus, the thin shell approximation is generally applicable.\nIn the unmagnetized case, an important quantity is the radius when the RS crosses the ejecta\nr\u0001\u0018\u0012E\u0001\n\u001aexc2\u00131=4\n=l3=4\nS\u00011=4= 1017cmn\u00001=4(21)\nIn the magnetized case, r\u0001is still a good approximation for the RS crossing radius, but with\ntwo cavities. First, a delayed onset of the RS, see (18), delays the RS crossing moment. Since\nrN=r\u0001=\u00183=2, this delay is not important for \u0018 <1 (the thick shell case, generally applicable to\nthe magnetized ejecta). Also, for subsonic out\rows (see below), r\u0001is the distance where the back\nof the out\row catches with the CD, see x3.3.\nSecond, magnetized shell is necessarily expanding, so that the tail part of the \row is moving\nwith\r\u0018\rCD=(2p\u001b). The typical shell spreading distance is\nrS\u0018\u0001\r2\nw= 3\u00021017cm (22)\nSince the spreading occurs with Alfven velocity, the tail part of the \row catches with the CD in\nthe Blandford-McKee phase at rtail\u0018p\u001br\u0001. We stress that spreading of magnetic shell, unlike of\nthe cold baryonic shell, is unavoidable consequence of the high internal pressure (spreading of cold\nbaryonic shell requires internal motion). This is the reason why magnetic out\rows are similar to\nthe thick shell case of the baryonic out\rows.\n3.3. Dynamics of subsonic expansion\nThus, for a given \rwand\u001b, at distances r <rNa rarefaction wave is launched into the \row,\nwhile the \row accelerates to Lorentz factors larger than \rw. This implies that our assumption that\na \row has a given \rwand\u001bis not justi\fed at r < rN: at these distances the \row is e\u000bectively\nsubsonic and its dynamics needs to be solved self-consistently, taking into account interaction with\nthe external medium. The subsonic out\row may be considered as a collection of outgoing fast\nmagnetosonic waves propagating from the central source, which constantly re-energize the FS. The\nout\row may be separated into two stages, which we will call \"early\" and \"late\", depending on\nwhether or not most of the fast waves emitted by the central source have caught up with the CD\nand their energy has been given to the circumburst medium. The transition between two stages{ 8 {\noccurs at the moment, which is similar to the shell crossing radius in the supersonic case, except\nthat in the case of subsonic expansion the CD is decelerating all the time, but with di\u000berent laws\nbefore and after the transition.\nAt the \"early\" stage the CD is constantly re-energized by the fast-magnetosonic waves prop-\nagating from the central source. The motion of the CD is determined by the luminosity at the\nretarded time t0:\nL\n(t0)\u0018\u001aexc3\r4\nCDr2(23)\n(this is a condition of pressure balance between the wind and the ram pressure of ISM in the frame\nof the CD). For constant luminosity Eq. (23) gives Eq. (17) for the Lorentz factor of the CD.\nThis is exactly the same estimate as for the intermediate scale in \u0018 < 1 supersonic \rows, since\nL\n\u0018Eisoc=\u0001; also this is the same scaling as in the case of relativistic \ruid reverse shock (Sari\n1997). We stress that in the limit of strong FS, Eq. (5), the FS dynamics is independent of the\ncomposition of the \row , only the total power is important, Eq. (23).\nThe early stage lasts for r < r \u0001, Eq. (21). At larger radii the \row enters the self-similar\nSedov-Blandford-McKee stage. At this stage, only a fraction of the shell is interacting with the\nexternal medium, while the newly shocked ejecta material keeps adding energy and momentum to\nthe shocked shell and the ISM, which evolve, e\u000bectively, as a \row with energy supply. Finally, for\nr>r \u0001the shock enters Blandford-McKee stage, with Lorentz factor given by Eq. (15).\n4. Discussion\nIn this paper we discuss the dynamics of strongly magnetized out\rows in GRBs. We \fnd\nthat the evolution of the forward shock driven by strongly magnetized out\rows are qualitatively\nthe same as in the case of \ruid shocks. The de\fnitions of radii rN; rEandr\u0001involve only the\ntotal energy of the ejecta, it's thickness and initial Lorentz factor, and notthe information about\nit's content, e.g., parameter \u001b. The typical radii (12) are the same for two \rows ( cf.Eq. (12) of\nthe present paper and Eq. (9-10) of Sari & Piran 1995). These similarities may be understood,\n\frst, by noting that jump conditions in perpendicular magnetized shocks may be reduced to \ruid\nshock jump conditions, with an appropriate choice of the equation of state, and, second, by the fact\nthat the thin shell approximation is applicable in our case (so that the global conservation of the\ntoroidal magnetic \rux, which modi\fes the global \row dynamics (Kennel & Coroniti 1984), is not\nimportant). Another reason for this similarity is that magnetic \feld behaves in many respects as a\n\ruid with internal pressure. The only di\u000berence in the dynamics of the forward shocks driven by\nmagnetized and \ruid \rows occurs for supersonic \rows, \rw>p\u001b\rCD, at very early stages r\u0014rN\norr\u0014rE, see Fig. (1). Qualitatively, magnetized out\rows are similar to thick shell hydrodynamic\nout\row,\u0018<1 atr>rN.\nOnly at very early times, at r < rN, the forward shock bears information about anergy\ncontent: forward shock is coasting with \rw=const in the \ruid case and decelerating \r/r\u00001=2in{ 9 {\nFig. 1.| Evolution of the Lorentz factor of the forward shock for matter-dominated (Left panel) and\nPoynting \rux-dominated models (Right panel). Matter-dominated ejecta coasts with the injection\nLorentz factor \rwuntil either rE(for\u0018>1, thin shell case) or until rN(for\u0018<1, thick shell case).\nAtrNreverse shock becomes strong. At large radii ( r > rEorr > r \u0001) the out\row enters the\nSedov-Blandford-McKee regime. For highly magnetized ejecta the Lorentz factor of the CD and\nthe FS initially decreases \u0000 ISM/r\u00001=2, changing to Sedov-Blandford-McKee regime \u0000 ISM/r\u00003=2\napproximately at r\u0001. Reverse shock is launched at rNand becomes strong at rN\u001b. Due to internal\nexpansion of the magnetized shell, the back of the shell catches with the CD at distance \u0018r\u0001p\u001b.\nThis is the reason why at distances close to r\u0001the Lorentz factor starts decreasing below the r\u00001=2\nlaw.\nthe magnetized case. Dynamics of the reverse shock is quite di\u000berent in case of high magnetization.\nFirst, the reverse shock forms at a \fnite distance from the source (Eq. 18), and may not form at\nall in a wind environment, (Eq. 19). This fact may be related to observed paucity of optical \rashes\nin the Swift era (Gomboc et al. 2009). (The standard model had a clear prediction, of a bright\noptical \rare with a de\fnite decay properties (Sari et al. 1996; Meszaros & Rees 1997). Though a\n\rare closely resembling the predictions was indeed observed (GRB990123, M\u0013 esz\u0013 aros & Rees 1999),\nthis was an exception.)\nIn addition, at distances rN<r<\u001br N, where the RS is weak, the formation of the RS shock\ndepends on the details of the \row: RS forms if the \row is strictly radial, but need not to form if the\nthe \row pattern is more complicated. We suggest that optical variability often seen in GRBs ( e.g.\nGRB021004 and most notoriously GRB080916C) is a re\rection of the non-trivial \row patterns and\nthe corresponding non-steady RS formation. Also, a recent detection of high polarization in optical\n(Steele et al. 2009) indicates a presence of an ordered magnetic \feld in the ejecta.\nI am greatly thankful to Dimitros Gianios, Sergey Komisarov and Alexandre Tchekhovskoy.{ 10 {\nREFERENCES\nBlandford, R. D., & Znajek, R. L. 1977, MNRAS, 179, 433\nGiannios, D., Mimica, P., & Aloy, M. A. 2008, A&A, 478, 747\nGomboc, A., Kobayashi, S., Mundell, C. G., Guidorzi, C., Melandri, A., Steele, I. A., Smith,\nR. J., Bersier, D., Carter, D., & Bode, M. F. 2009, in American Institute of Physics Con-\nference Series, Vol. 1133, American Institute of Physics Conference Series, ed. C. Meegan,\nC. Kouveliotou, & N. Gehrels, 145{150\nKennel, C. F., & Coroniti, F. V. 1984, ApJ, 283, 694\nKomissarov, S. S. 2005, MNRAS, 359, 801\nLandau, L. D., & Lifshitz, E. M. 1959, Fluid mechanics (Course of theoretical physics, Oxford:\nPergamon Press, 1959)\nLyutikov, M. 2005, ArXiv Astrophysics e-prints\n|. 2006, New Journal of Physics, 8, 119\n|. 2009, ArXiv e-prints\nMeszaros, P., & Rees, M. J. 1997, ApJ, 476, 232\nM\u0013 esz\u0013 aros, P., & Rees, M. J. 1999, MNRAS, 306, L39\nMichel, F. C. 1973, ApJ, 180, 207\nMizuno, Y., Zhang, B., Giacomazzo, B., Nishikawa, K., Hardee, P. E., Nagataki, S., & Hartmann,\nD. H. 2009, ApJ, 690, L47\nRees, M. J., & Meszaros, P. 1992, MNRAS, 258, 41P\nSari, R. 1997, ApJ, 489, L37+\nSari, R., Narayan, R., & Piran, T. 1996, ApJ, 473, 204\nSari, R., & Piran, T. 1995, ApJ, 455, L143+\nSteele, I. A., Mundell, C. G., Smith, R. J., Kobayashi, S., & Guidorzi, C. 2009, Nature, 462, 767\nVlahakis, N., & K onigl, A. 2003, ApJ, 596, 1080\nZhang, B., & Kobayashi, S. 2005, ApJ, 628, 315\nThis preprint was prepared with the AAS L ATEX macros v5.2."    },    {        "title": "0911.2741v1.Symmetry_breaking_magnetization_dynamics_of_spinor_dipolar_Bose_Einstein_condensates.pdf",        "content": "arXiv:0911.2741v1  [cond-mat.quant-gas]  14 Nov 2009Symmetry-breaking magnetization dynamics of spinor dipol ar Bose-Einstein\ncondensates\nShoichi Hoshi and Hiroki Saito\nDepartment of Applied Physics and Chemistry, The Universit y of Electro-Communications, Tokyo 182-8585, Japan\n(Dated: June 17, 2021)\nSymmetry-breaking magnetization dynamics of a spin-1 Bose -Einstein condensate (BEC) due to\nthe dipole-dipole interaction are investigated using the m ean-field and Bogoliubov theories. When a\nmagnetic field is applied along the symmetry axis of a pancake -shaped BEC in the m= 0 hyperfine\nsublevel, transverse magnetization develops breaking the chiral or axial symmetry. A variety of\nmagnetization patterns are formed depending on the strengt h of the applied magnetic field. The\nproposed phenomena can be observed in87Rb and23Na condensates.\nPACS numbers: 03.75.Mn,67.85.Fg,03.75.Lm,03.75.Kk\nI. INTRODUCTION\nA Bose-Einstein condensate (BEC) of atoms with spin\ndegreesoffreedom(spinorBEC)allowsthestudyofmag-\nnetism in a quantum fluid. There are two mechanisms\nof magnetization for a spinor BEC: the ferromagnetic\ncontact interaction and the magnetic dipole interaction\n(MDI) between atoms. Magnetization dynamics due to\nthe ferromagneticcontactinteraction havebeen observed\nforaspin-187RbBEC[1, 2]. However,magnetizationdy-\nnamics due to the MDI have not been studied yet, and\nthis is the subject of the present paper.\nABEC of52Cr atomswith alargemagnetic dipole mo-\nment (6µBwithµBbeing the Bohr magneton) has been\nrealized by the Stuttgart group [3] and its anisotropic\nbehaviors originating from the anisotropy of the dipole-\ndipole interaction have been observed [4, 5, 6, 7, 8].\nWhile the magnetic dipole moment of spin-1 alkali atoms\n(µB/2) is much smaller than that of52Cr, it is neverthe-\nless predicted that a small MDI can create spin textures\nin a87Rb BEC[9, 10, 11]. The crystallinemagneticorder\nobserved in a87Rb BEC is considered to be caused by\nthe MDI [12]. MDI effects have been detected in39K [13]\nand7Li [14] BECs using Feshbach resonance.\nIn this paper, we show that magnetization dynami-\ncally develops in spin-187Rb and23Na BECs due to the\nMDI. We consider a situation in which a BEC in the\nm= 0 magnetic sublevel is confined in an axisymmet-\nric pancake-shaped trap and a magnetic field is applied\nalong the symmetry axis. We numerically solve the time-\ndependent nonlocal Gross-Pitaevskii (GP) equation in-\ncluding the MDI and show that magnetization develops\nin the direction perpendicular to the magnetic field due\nto the MDI. These magnetization dynamics break the\nchiral or axial symmetry spontaneously. We find that\nvarious magnetization patterns emerge depending on the\nstrength of the magnetic field. For spin-187Rb, we can\nsuppress magnetization due to the ferromagnetic contact\ninteraction using the microwave-induced quadratic Zee-\nman effect [15], and the pure MDI effect can thus be\nobserved. We perform a Bogoliubov analysis and show\nthat the magnetization is triggered by the dynamical in-stability. We also employ the variational method with\nthe Gaussian approximation to explain the numerical re-\nsults.\nThis paper is organized as follows. Section II formu-\nlates the mean-field and Bogoliubov theories to study\nthe present system. Section III numerically studies the\nBogoliubov spectra and demonstrates the magnetization\ndynamics. Section IV analyzes the phenomena using the\nvariational method. Section V gives conclusions to the\nstudy.\nII. FORMULATION OF THE PROBLEM\nA. Mean-field theory\nWe consider a system of spin-1 bosonic atoms with\nmassMand magnetic dipole moment µB/2 confined in\nan axisymmetric harmonic potential V(r) =M(ω2\n⊥r2\n⊥+\nω2\nzz2)/2 withr⊥= (x2+y2)1/2. A uniform magnetic\nfieldBzis applied in the zdirection and the linear Zee-\nman energy is given by −µBBzm/2 for magnetic sub-\nlevelsm= 0,±1. We neglect the magnetic quadratic\nZeeman effect, since the strength of the magnetic field\nconsidered here is Bz<10 mG. Instead, we assume that\nthe microwave-induced quadratic Zeeman effect [15] lifts\nthe energy of the m=±1 states by Q.\nWe employ the mean-field approximation and the con-\ndensate is described by the macroscopic wave function\nψmwith magnetic sublevel m, which satisfies the nor-\nmalization condition/summationtext\nm/integraltext\n|ψm|2dr=N, withNbeing\nthe total number of atoms. The nonlocal GP equations\nincluding the MDI are given by\ni¯h∂ψ0\n∂t=/parenleftbigg\n−¯h2\n2M∇2+V+g0ρ/parenrightbigg\nψ0\n+g1√\n2(F+ψ1+F−ψ−1)−µB\n2Bd·/summationdisplay\nm(f)0mψm,\n(1a)2\ni¯h∂ψ±1\n∂t=/parenleftbigg\n−¯h2\n2M∇2+V∓P+Q+g0ρ/parenrightbigg\nψ±1\n+g1/parenleftbigg1√\n2F∓ψ0±Fzψ±1/parenrightbigg\n−µB\n2Bd·/summationdisplay\nm(f)±1mψm,\n(1b)\nwhereρ=/summationtext\nm|ψm|2,F=/summationtext\nmm′ψ∗\nm(f)mm′ψm′,fis\nthe vector of the spin-1 matrices, F±=Fx±iFy, and\nP=µB\n2Bz. (2)\nThe spin-independent and spin-dependent contact-\ninteraction parameters g0andg1have the forms\ng0=4π¯h2\nMa0+2a2\n3, g 1=4π¯h2\nMa2−a0\n3,(3)\nwhereaSis thes-wave scattering length for colliding\natoms with total spin S. The MDI produces an effec-\ntive magnetic field:\nBd(r) =−µ0\n4πµB\n2/integraldisplay\ndr′F(r′)−3[F(r′)·e]e\n|r−r′|3,(4)\nwhereµ0is the magnetic permeability of vacuum and\ne= (r−r′)/|r−r′|. Equation (1) is numerically solved\nin Fourier space for the kinetic term and in real space\nfor the other terms using a fast Fourier transform (FFT).\nTheFFTisalsousedtocalculatetheconvolutionintegral\nin Eq. (4).\nIn the present paper, we consider an initial state in\nwhich all the atoms are in the m= 0 sublevel. In order\nto simulate this situation, we prepare the ground state\nofψ0withψ±1= 0 by the imaginary-time evolution of\nEq. (1). Small noise (a complex random number on each\nmesh) is then applied to the initial state of ψ±1to break\nthe symmetry and trigger magnetization due to dynam-\nical instability. The small noise corresponds to quantum\nfluctuation, thermal atoms, and residual atoms in an ex-\nperiment. We note that if the initial state of m=±1isψ±1= 0, the right-hand side of Eq. (1b) vanishes and\nψ±1never develops within the mean-field theory. Mag-\nnetization thus occurs if small noise in the m=±1 state\nis exponentially amplified by dynamical instabilities.\nB. Bogoliubov analysis\nWe investigate the Bogoliubov excitation spectrum of\nmagnons for the stationary state of ψ0. Assuming that\nψ±1is small and neglecting the second and higher orders\nofψ±1in Eq. (1), we obtain\ni¯h∂ψ±1\n∂t=/parenleftbigg\n−¯h2\n2M∇2+V∓P+Q+g0|ψ0|2/parenrightbigg\nψ±1\n+g1/parenleftbig\n|ψ0|2ψ±1+ψ2\n0ψ∗\n∓1/parenrightbig\n−gd\n2√\n2/integraldisplay\ndr′1\n|r−r′|3\n×/bracketleftbig\n(1−3e2\nz)F∓(r′)+3e2\n∓F±(r′)/bracketrightbig\nψ0(r),\n(5)\nwheregd=µ0µ2\nB/(16π) ande±=ex±iey. Using the\nmode functions u±1andv±1, we write a single-mode ex-\ncitation ofψ±1as\nψ±1(r,t) =e−iµt/¯h/bracketleftBig\nu±1(r⊥,z)ei(L±1−1)φe−iωt\n+v∗\n±1(r⊥,z)e−i(L∓1−1)φeiωt/bracketrightBig\n,(6)\nwhereφ= arg(x+iy),Lis an integer, and µis the\nchemical potential,\nµ=1\nN/integraldisplay\ndrψ∗\n0/parenleftbigg\n−¯h2\n2M∇2+V+g0|ψ0|2/parenrightbigg\nψ0.(7)\nEachψ±1in Eq. (6) has 2( L−1)-fold symmetry\naround the zaxis. Substituting Eq. (6) and ψ0=\n|ψ0|exp(−iµt/¯h) into Eq. (5) yields the closed form of\nthe nonlocal Bogoliubov-de Gennes equations,\n/braceleftbigg\n−¯h2\n2M/bracketleftbigg\n∇2\n⊥z−(L±1−1)2\nr2\n⊥/bracketrightbigg\n+V∓P+Q+g0|ψ0|2−µ/bracerightbigg\nu±1+g1|ψ0|2(u±1+v∓1)\n−gd\n2/integraldisplay\ndr′|ψ0(r′)|\n|r−r′|3/braceleftbig\n(1−3e2\nz)[u±1(r′)+v∓1(r′)]+3e2\n∓[u∓1(r′)+v±1(r′)]/bracerightbig\n|ψ0(r)|= ¯hωu±1,(8a)\n/braceleftbigg\n−¯h2\n2M/bracketleftbigg\n∇2\n⊥z−(L∓1−1)2\nr2\n⊥/bracketrightbigg\n+V∓P+Q+g0|ψ0|2−µ/bracerightbigg\nv±1+g1|ψ0|2(u∓1+v±1)\n−gd\n2/integraldisplay\ndr′|ψ0(r′)|\n|r−r′|3/braceleftbig\n(1−3e2\nz)[u∓1(r′)+v±1(r′)]+3e2\n±[u±1(r′)+v∓1(r′)]/bracerightbig\n|ψ0(r)|=−¯hωv±1,(8b)\nwhere∇2\n⊥z=∂2\nr⊥+r−1\n⊥∂r⊥+∂2\nz. We numerically diag-\nonalize Eq. (8) by expanding u±1andv±1with orthog-onal bases, e.g., the harmonic-oscillator eigenfunctions.3\n00.010.020.03\n-0.1 00.10.20.30.40.5Im ω / ω⊥\n(P−Q)//104ω⊥L=1\nL=2\nL=3\nL=4\nL=5\nL=6\nFIG. 1: (color) Imaginary part of Bogoliubov frequency ω\nfor magnon excitation from the m= 0 state of87Rb atoms as\na function of P−QwithQ= 50¯hω⊥. The excitation mode\nhas 2(L−1)-fold symmetry around the zaxis, where Lis\ndefined in Eq. (6). The number of atoms is N= 105and the\nradial and axial trap frequencies are ω⊥= 2π×100 Hz and\nωz= 2π×400 Hz. The values of Pindicated by the arrows\nare used in Figs. 2 and 3.\nIf complex frequencies ωemerge, the stationary state\nψ0becomes dynamically unstable against excitations of\nmagnons.\nIII. MAGNETIZATION DYNAMICS AND\nBOGOLIUBOV SPECTRA FOR ALKALI ATOMS\nA. Spin-1 rubidium 87\nWe first consider a spin-187Rb BEC. The scattering\nlengths are a0= 101.8aBanda2= 100.4aB[16] with\naBbeing the Bohr radius. The spin-dependent contact-\ninteraction parameter g1is therefore negative and the\nground state is ferromagnetic [1]. In order to distinguish\nmagnetization by the MDI from that by the ferromag-\nnetic contact interaction, we apply microwave radiation\nto lift the energy of the m=±1 states by Q, which must\nbe much largerthan the ferromagneticinteraction energy\n|g1|ρ. The magnetization due to the ferromagnetic con-\ntact interaction is thus suppressed and the pure effect of\nthe MDI can be observed.\nTo investigate the dynamical instability against mag-\nnetization, we numerically diagonalize Eq. (8). Figure 1\nshows the imaginary part of the Bogoliubov frequencies\nωas a function of the applied magnetic field, where\nN= 105atoms are confined in a pancake-shaped trap\nwithω⊥= 2π×100 Hz and ωz= 2π×400 Hz. The\nmicrowave-induced quadratic Zeeman energy Qis cho-\nsen to be 50¯ hω⊥, which is sufficient to suppress mag-\nnetization by the ferromagnetic contact interaction. In\nfact, we have confirmed that the Bogoliubov spectrum\nis always real if the MDI is absent, gd= 0, for this00.10.2\n0 0.5 1 1.5 2Population of m=1\nt [s]p=−0.026\np=0.42\np=0.2\np=0.35\np=0.31\nFIG. 2: (color) Time evolution of the population of the m=\n1 stateR\n|ψ1|2dr/Nfor the values of p≡(P−Q)/(¯hω⊥)\nindicated by the arrows in Fig. 1. The parameters are the\nsame as those in Fig. 1. The populations of the m=−1 state\nare too small to be discerned at this scale of the ordinate.\nvalue ofQ. The linear Zeeman energy P= 50¯hω⊥cor-\nresponds to Bz≃7.15 mG. From Fig. 1, we find that\nmagnon modes with various rotational symmetries (var-\niousL) become dynamically unstable, depending on the\nlinear Zeeman energy P. There is no imaginary part for\np≡(P−Q)/(¯hω⊥)<−0.1, while many peaks in the\nimaginary part exist for p>0.5.\nFigure 2 shows time evolution of/integraltext\n|ψ1|2dr/Nfor the\nvalues ofPindicated by the arrows in Fig. 1. The tran-\nsition from the m= 0 state to the m= 1 state occurs\ndue to the dynamical instability shown in Fig. 1. From\nFig. 2, we find that the transition occurs periodically ex-\ncept forp= 0.42 (green line). The complicated behavior\nforp= 0.42 originates from the fact that the dynami-\ncally unstable modes are not only L= 2 but also L= 3\n(see Fig. 1). We note that the transition to the m=−1\nstate is negligibly small and the total spin in the zdirec-\ntion/integraltext\n(|ψ1|2−|ψ−1|2)dris not conserved, indicating that\nthe transition is not due to the spin-exchange contact\ninteraction but due to the MDI. Since the zcomponent\nof the total angular momentum must be conserved, the\nsystem acquires orbital angular momentum. The trans-\nfer of the spin angular momentum to the orbital angular\nmomentum in a spinor dipolar BEC also occurs in the\nEinstein-de Haas effect [17, 18, 19].\nFigure 3 shows transverse magnetization at the times\nof the first peaks of/integraltext\n|ψ1|2dr(the first peaks of the lines\nin Fig. 2) for the linear Zeeman energies Pindicated by\nthe arrows in Fig. 1. A variety of magnetization patterns\nwith 2(L−1)-fold symmetry emerge depending on the\nstrength of the applied magnetic field. The closure struc-\nture of the magnetization in Fig. 2 (a) is an energetically\nfavorablestructure for the MDI energy. The directions of\nthe magnetization vectors in the closure structure have\nclockwise and counterclockwise symmetry, and therefore\nthe spin-vortex generation in Fig. 3 (a) breaks the chiral4\n0.006 0\n0.004\n0.0035\n0.002(a) p=-0.026\n(b) p=0.42\n(c) p=0.2\n(d) p=0.35\n(e) p=0.31x\ny\n0.006 0\n0\n0\n0\nFIG. 3: (color) Magnitude of the integrated transverse\nmagnetization |R\nF+dz|(right panels) and its direction arg\n(R\nF+dz) (left panels) at the first peaks of the curves in Fig. 2.\nThe values of p≡(P−Q)/(¯hω⊥) used are indicated by the\narrows in Fig. 1. The unit ofR\nF+dzisNMω ⊥/¯h. The length\nof the vector is proportional to |R\nF+dz|. The field of view is\n9.7×9.7µm.00.0020.0040.0060.008\n0 0.1 0.2 0.3 0.4 0.5Im ω / ω⊥\nP/ /104ω⊥L=1\nL=2\nL=3\nL=4\nL=5\nL=6\n00.010.02\n1.5 1.6 1.7 1.8 1.9 2Im ω / ω⊥\nP/ /104ω⊥(a)\n(b)\nFIG. 4: (color) Imaginary part of the Bogoliubov frequency\nωfor magnon excitation from the m= 0 state of23Na atoms\nas a function of the linear Zeeman energy P. The range of\nPis (a) 0 ≤P/(¯hω⊥)≤0.5 and (b) 1 .5≤P/(¯hω⊥)≤2.\nThe excitation mode has 2( L−1)-fold symmetry around the\nzaxis, where Lis defined in Eq. (6). The number of atoms\nisN= 106and the trap frequencies are the same as those in\nFig. 1. The values of Pindicated by the arrows in (b) are\nused in Fig. 5.\nsymmetry. The m= 1 component of these spin vor-\ntices isψ1∝e−iφ. This situation is different from the\nspin-vortex generation by the ferromagnetic contact in-\nteraction, in which polar-core vortices of ψ±1∝e±iφand\nψ±1∝e∓iφemerge with an equal probability [20]. The\nclosure structures are also seen in Figs. 3 (b)-3 (e). The\nmagnetization in Figs. 3 (b)-3 (e) caused by the dynam-\nical instability with L≥1 exhibits a variety of patterns,\nbreaking the axisymmetry of the system.\nB. Spin-1 sodium 23\nNext we consider a spin-123Na BEC. The scattering\nlengths are given by ( a0+ 2a2)/3 = 53.4aB[21] and\na2−a0= 2.47aB[22]. The spin-dependent contact-\ninteraction parameter g1is then positive and the polar5\n0.0018 0x\ny(a) P/hω =1.63, ω t=640(b) p/hω =1.875, ω t=670⊥ ⊥ ⊥ ⊥\nFIG. 5: (color) Integrated transverse magnetization |R\nF+dz|\nat the time whenR\n|ψ1|2drbecomes the first maximum in\ntime evolution for the values of Pindicated by the arrows in\nFig. 4 (b). The unit of |R\nF+dz|isNMω ⊥/¯h. The field of\nview is 21 ×21µm.\nstate (m= 0) is energetically favorable. Spontaneous\nmagnetization due to the contact interaction is therefore\nsuppressed and the microwave-induced Zeeman effect is\nunnecessary ( Q= 0). The number of atoms is assumed\nto beN= 106and the trap frequencies are the same as\nthose in Sec. IIIA.\nFigure 4 shows the imaginary part of the Bogoliubov\nfrequency obtained by numerically diagonalizing Eq. (8).\nCompared with the case of87Rb in Fig. 1, the width and\nheight of the peaks are small for 0 ≤P/(¯hω⊥)≤0.5\n[Fig. 4 (a)]. The width and height of the main peaks\ngradually increase and saturate for P/(¯hω⊥)∼2 [Fig. 4\n(b)]. ForP <0, there is no imaginary part.\nWe numerically solve the GP equation for the values of\nPindicated by the arrows in Fig. 4 (b). The initial state\nis prepared by the same method as for87Rb. Figure 5\nshows the integrated transverse magnetization |/integraltext\nF+dz|\nat the time of the first peak of/integraltext\n|ψ1|2drin the time\nevolution. Many radial nodes in the magnetization pat-\nterns are evident, since the values of Pcorrespond to the\nhigher-order peaks in Fig. 4 (b). The population of the\nm= 1 state,/integraltext|ψ1|2dr/N, is 0.01 in Fig. 5 (a) and 0.05\nin Figure 5 (b). The population of the m=−1 state is\nvery small ∼10−4.\nIV. GAUSSIAN VARIATIONAL ANALYSIS\nTo qualitatively examine the Bogoliubov spectra ob-\ntained in Sec. III, we perform Gaussian variational anal-\nysis. The variational wave function for the m= 0 state\nhas the form\nψ0=√\nN\nπ3/4d⊥d1/2\nzexp/parenleftbigg\n−r2\n⊥\n2d2\n⊥−z2\n2d2z−iµ\n¯ht/parenrightbigg\n,(9)\nwhered⊥anddzare the variational parameters char-\nacterizing the size of the condensate in the radial and\naxial directions. Substituting Eq. (9) into Eq. (7) andthe mean-field energy\nE=/integraldisplay\ndrψ∗\n0/parenleftbigg\n−¯h2\n2M∇2+V+g0\n2|ψ0|2/parenrightbigg\nψ0,(10)\nwe obtain\nµ\n¯hω⊥=1\n2/parenleftBigg\n1\n˜d2\n⊥+˜d2\n⊥/parenrightBigg\n+1\n4/parenleftbigg1\n˜d2z+λ2˜d2\nz/parenrightbigg\n+˜g0\n˜d2\n⊥˜dz,\n(11)\nE\nN¯hω⊥=1\n2/parenleftBigg\n1\n˜d2\n⊥+˜d2\n⊥/parenrightBigg\n+1\n4/parenleftbigg1\n˜d2z+λ2˜d2\nz/parenrightbigg\n+˜g0\n2˜d2\n⊥˜dz,\n(12)\nwhereλ=ωz/ω⊥,˜d⊥=d⊥/a⊥,˜dz=dz/a⊥, and\n˜g0=g0N/[(2π)3/2¯hω⊥a3\n⊥] witha⊥= [¯h/(Mω⊥)]1/2.\nThe variational parameters ˜d⊥and˜dzare determined\nso as to minimize Eq. (12).\nForsimplicity, werestrictthe magnonexcitationto the\nform,\nψ±1(r,t) =e−iµt/¯h/bracketleftbig\nα±1e−iωtχ±(r)+β∗\n±1eiωtχ∓(r)/bracketrightbig\n,\n(13)\nwith\nχ±(r) =e±iφr⊥\nπ3/4d2\n⊥d1/2\nzexp/parenleftbigg\n−r2\n⊥\n2d2\n⊥−r2\nz\n2d2z/parenrightbigg\n,(14)\nwhichcorrespondstothelowestmodeof L= 1in Eq.(6).\nSubstitution ofEqs.(9), (11), and (13) into Eq.(8) yields\n(Λ∓˜P)α±1+(G+D1)(α±1+β∓1)+D2(α∓1+β±1)\n= ˜ωα±1, (15a)\n(Λ∓˜P)β±1+(G+D1)(α∓1+β±1)+D2(α±1+β∓1)\n=−˜ωβ±1, (15b)\nwhere˜P=P/(¯hω⊥), ˜ω=ω/ω⊥, and\nΛ =1\n2/parenleftBigg\n1\n˜d2\n⊥+˜d2\n⊥/parenrightBigg\n+Q\n¯hω⊥−˜g0\n2˜d2\n⊥˜dz, (16)\nG=Ng1\n2(2π)3/2¯hω⊥a3\n⊥˜d2\n⊥˜dz, (17)\nD1=˜gd\n2˜d2\n⊥˜dz(˜d2\n⊥−˜d2z)5/2\n×/bracketleftBigg\n(˜d2\n⊥−˜d2\nz)1/2(−4˜d4\n⊥−7˜d2\n⊥˜d2\nz+2˜d4\nz)\n+9˜d4\n⊥˜dzcot−1˜dz\n(˜d2\n⊥−˜d2z)1/2/bracketrightBigg\n, (18)\nD2=3˜gd\n2˜d2\n⊥(˜d2\n⊥−˜d2z)5/2/bracketleftBigg\n˜dz(˜d2\n⊥−˜d2\nz)1/2(−5˜d2\n⊥+2˜d2\nz)\n+3˜d4\n⊥cot−1˜dz\n(˜d2\n⊥−˜d2z)1/2/bracketrightBigg\n, (19)6\nwith ˜gd=gdN/[6(2π)1/2¯hω⊥a3\n⊥]. Diagonalizing\nEq. (15), we obtain the excitation frequency as\n˜ω2=˜P2+Λ2+2(G+D1)Λ\n±2/radicalBig\n[Λ2+2(G+D1)Λ]˜P2+Λ2D2\n2.(20)\nFor the parameters of87Rb in Fig. 1, ˜ ωin Eq. (20)\nbecomes imaginary between ˜P≃49.9 and 50.01 and the\nmaximum value of Im ωis≃0.01. For the parameters of\n23Na in Fig. 4, ˜ ωbecomes imaginary between ˜P≃0.125\nand 0.13 and the maximum value of Im ωis≃0.002.\nThese results are in qualitative agreement with the first\npeaks ofL= 1 in Figs. 1 and 4. The differences between\nthevariationalandnumericalresultscomefromtheforms\nof the variational wavefunctions assumed in Eqs. (9) and\n(13); more appropriate variational functions are needed\nfor quantitative explanation of the numerical results.\nV. CONCLUSIONS\nIn conclusion, we have studied the magnetization dy-\nnamics caused by the MDI in a spin-1 BEC in the m= 0\nhyperfine state prepared in a pancake-shaped trap anda magnetic field applied in the axial direction. We\nfound that transverse magnetization develops due to the\nMDI breaking the chiral or axial symmetry, and a vari-\nety of magnetization patterns appear depending on the\nstrength of the applied magnetic field. We showed that\nthese phenomena occur in spin-187Rb and23Na BECs.\nWe also performed Bogoliubov analysis and found that\nthe initial fluctuations in the magnetization are exponen-\ntially amplified by the dynamical instability. A Gaussian\nvariational analysis provided a qualitative explanation of\nthe results.\nOur study has shown that magnetization due to the\nMDI strongly depends on the shape of the system. Mag-\nnetization dynamics for various trapping potentials in-\ncluding cigar-shaped and lattice potentials merit further\nstudy.\nAcknowledgments\nThis work was supported by the Ministry of Educa-\ntion, Culture, Sports, Science and Technology of Japan\n(Grants-in-Aid for Scientific Research, No. 17071005and\nNo. 20540388).\n[1] M. -S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer,\nK. M. Fortier, W. Zhang, L. You, and M. S. Chapman,\nPhys. Rev. Lett. 92, 140403 (2004).\n[2] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore,\nand D. M. Stamper-Kurn, Nature (London) 443, 312\n(2006).\n[3] A. Griesmaier, J. Werner, S. Hensler, J. Stuhler, and T.\nPfau, Phys. Rev. Lett. 94, 160401 (2005).\n[4] A. Griesmaier, J. Stuhler, T. Koch, M. Fattori, T. Pfau,\nand S. Giovanazzi, Phys. Rev. Lett. 97, 250402 (2006).\n[5] T. Lahaye, T. Koch, B. Fr¨ ohlich, M. Fattori, J. Metz, A.\nGriesmaier, S.Giovanazzi, andT. Pfau, Nature(London)\n448, 672 (2007).\n[6] T. Koch, T. Lahaye, J. Metz, B. Fr¨ ohlich, A. Griesmaier,\nand T. Pfau, Nature Phys. 4, 218 (2008).\n[7] T. Lahaye, J. Metz, B. Fr¨ ohlich, T. Koch, M. Meister,\nA. Griesmaier, T. Pfau, H. Saito, Y. Kawaguchi, and M.\nUeda, Phys. Rev. Lett. 101, 080401 (2008).\n[8] J. Metz, T. Lahaye, B. Fr¨ ohlich, A. Griesmaier,T. Pfau,\nH. Saito, Y. Kawaguchi, and M. Ueda, New J. Phys. 11,\n055032 (2009).\n[9] S. Yi and H. Pu, Phys. Rev. Lett. 97, 020401 (2006).\n[10] Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev. Lett.\n97, 130404 (2006).\n[11] Y. Kawaguchi, H. Saito, and M. Ueda, Phys. Rev. Lett.\n98, 110406 (2007).\n[12] M. Vengalattore, S. R. Leslie, J. Guzman, and D. M.Stamper-Kurn, Phys. Rev. Lett. 100, 170403 (2008).\n[13] M. Fattori, G. Roati, B. Deissler, C. D’Errico, M. Zac-\ncanti, M. Jona-Lasinio, L. Santos, M. Inguscio, and G.\nModugno, Phys. Rev. Lett. 101, 190405 (2008).\n[14] S. E. Pollack, D.Dries, M. Junker, Y.P.Chen, T. A. Cor-\ncovilos, and R. G. Hulet, Phys. Rev. Lett. 102, 090402\n(2009).\n[15] S. R. Leslie, J. Guzman, M. Vengalattore, J. D. Sau, M.\nL. Cohen, and D. M. Stamper-Kurn, Phys. Rev. A 79,\n043631 (2009).\n[16] E. G. M. van Kempen, S. J. J. M. F. Kokkelmans, D. J.\nHeinzen, and B. J. Verhaar, Phys. Rev. Lett. 88, 093201\n(2002).\n[17] Y. Kawaguchi and H. Saito and M. Ueda, Phys. Rev.\nLett.96, 080405 (2006).\n[18] L. Santos and T. Pfau, Phys. Rev. Lett. 96, 190404\n(2006).\n[19] K. Gawryluk, M. Brewczyk, K. Bongs, and M. Gajda,\nPhys. Rev. Lett. 99, 130401 (2007).\n[20] H. Saito, Y. Kawaguchi, and M. Ueda, Phys. Rev. Lett.\n96, 065302 (2006).\n[21] A. Crubellier, O. Dulieu, F. Masnou-Seeuws, M. Elbs, H.\nKn¨ ockel, and E. Tiemann, Eur. Phys. J. D 6, 211 (1999).\n[22] A. T. Black, E. Gomez, L. D. Turner, S. Jung, and P. D.\nLett, Phys. Rev. Lett. 99, 070403 (2007)."    },    {        "title": "2107.12988v1.Updated_Magnetized_Transport_Coefficients__Impact_on_Laser_Plasmas_with_Self_Generated_or_Applied_Magnetic_Fields.pdf",        "content": "Updated Magnetized Transport Coe\u000ecients: Impact on Laser-Plasmas with\nSelf-Generated or Applied Magnetic Fields\nC. A. Walsh,1J. D. Sadler,2and J. R. Davies3\n1)Lawrence Livermore National Laboratory\n2)Los Alamos National Laboratory\n3)Laboratory for Laser Energetics, University of Rochestera)\n(Dated: 28 July 2021)\nErrors in the Epperlein & Haines [PoF (1986)] transport coe\u000ecients were recently found at low electron\nmagnetizations, with new magnetic transport coe\u000ecients proposed simultaneously by two teams [Sadler,\nWalsh & Li, PRL (2021) and Davies, Wen, Ji & Held, PoP (2021)]; these two separate sets of updated\ncoe\u000ecients are shown in this paper to be in agreement. The importance of these new coe\u000ecients in laser-\nplasmas with either self-generated or applied magnetic \felds is demonstrated. When an external magnetic\n\feld is applied, the cross-gradient-Nernst term twists the \feld structure; this twisting is reduced by the\nnew coe\u000ecients in the low magnetization regime. For plasmas where only self-generated magnetic \felds are\npresent, the new coe\u000ecients are found to result in the magnetic \feld moving with the Righi-Leduc heat-\n\row, enhancing the impact of MHD. Simulations of Biermann Battery magnetic \felds around ICF hot-spot\nperturbations are presented, with cross-gradient-Nernst transport increasing spike penetration.\nI. INTRODUCTION\nMagnetic \felds exist in all plasmas: in laser-driven ab-\nlation fronts41,51,61, ICF capsules54,57,72, shock fronts59,\nhohlraums50,66and Z-pinches52. Magnetic \felds can\nalso be purposefully applied to ICF implosions to im-\nprove fusion performance43,62,67,71. The transport of\nthese magnetic \felds within the plasma is critically im-\nportant to the design and interpretation of experiments.\nNon-local VFP simulations capture the transport pro-\ncesses innately by resolving the distribution of electron\nenergies54,58,66. However, these calculations are pro-\nhibitively expensive and unstable, so simpli\fed models\nwith prescribed transport coe\u000ecients are used for spatial\nand temporal scales of interest. These transport coe\u000e-\ncients are calculated by \ftting to VFP data. The sem-\ninal work in this area is that of Braginskii, who laid\nthe groundwork for the anisotropic transport of ther-\nmal energy and magnetic \felds within the magneto-\nhydrodynamic (MHD) framework40. Following on from\nBraginskii, Epperlein & Haines improved the trans-\nport coe\u000ecients, allowing for closer agreement to VFP\ncalculations49. For 35 years the transport coe\u000ecients of\nEpperlein & Haines were the agreed-upon standard to be\nimplemented into extended-MHD codes50,57,62,72,74.\nRecently, simultaneous work by two research teams\ndemonstrated errors in two of the transport coe\u000ecients\nat low electron magnetization48,65. While Epperlein &\nHaines stated low errors in \ftting the transport coef-\n\fcients to VFP data, they did not consider that their\ntransport coe\u000ecients were ill-formed. From their per-\nspective the magnetic \feld evolution was captured by:\na)Electronic mail: walsh34@llnl.gov\nFIG. 1: The cross-gradient-Nernst coe\u000ecient as a\nfunction of Hall Parameter calculated using di\u000berent\nsources. All coe\u000ecients are attempting to replicate\nthose given by VFP calculations.\n@B\n@t=r\u0002(v\u0002B)\u0000r\u0002j\u0002B\nnee\u0000r\u0002\u000b\u0001j\nn2ee2\n+r\u0002\f\u0001rTe\ne+r\u0002rPe\nnee(1)\nwhere\u000band\fare tensor transport coe\u000ecients calcu-\nlated by \ftting to 0D VFP simulations. While accurate,\nequation 1 does not make clear the impact of each term\non the movement or generation of magnetic \felds. Re-\ncent work re-arranged this equation to make the di\u000berent\nphysical processes more obvious47,63,70:arXiv:2107.12988v1  [physics.plasm-ph]  27 Jul 20212\n@B\n@t=\u0000r\u0002\u000bk\n\u00160e2n2er\u0002B+r\u0002(vB\u0002B)\n+r\u0002 \nrPe\nene\u0000\fkrTe\ne!(2)\nwhere the \frst term is di\u000busive, the second term is ad-\nvection of magnetic \feld at a velocity vBand the \fnal\nterm is a source of magnetic \rux. The advection ve-\nlocity is then a combination of the bulk plasma motion,\nthermally-driven terms and current-driven terms:\nvB=v\u0000\r?rTe\u0000\r^(^b\u0002rTe)\u0000j\nene(1+\u000ec\n?)+\u000ec\n^\nene(j\u0002^b)\n(3)\nwhere the transport coe\u000ecients have been re-written\nfrom the form given by Braginskii. The dimensionless\nforms (denoted by a super-scriptc) depend only on Hall\nParameter and e\u000bective Z70:\n\rc\n?=\fc\n^\n!e\u001ce(4)\n\rc\n^=\fc\nk\u0000\fc\n?\n!e\u001ce(5)\n\u000ec\n?=\u000bc\n^\n!e\u001ce(6)\n\u000ec\n^=\u000bc\n?\u0000\u000bc\nk\n!e\u001ce(7)\nThe dimensional forms of the thermally-driven coe\u000e-\ncients are given by:\n\r=\rc\u001ce\nme(8)\nOnce the transport coe\u000ecients have been re-written\ninto this physically-motivated form, the \fts of Epperlein\n& Haines to the VFP simulations have large errors in \rc\n^\nand\u000ec\n^at small!e\u001ce. While the errors in \ftting to \fc\n?\nare small, the \r^numerator ( \fc\nk\u0000\fc\n?) and denominator\n(!e\u001ce) both tend to zero for small !e\u001ce. This resulted in\nboth\rc\n^and\u000ec\n^being maximum for small !e\u001ce, while the\nVFP simulations calculated them as exactly zero. Figure\n1 shows how \rc\n^calculated using the values from Epper-\nlein & Haines deviate from the desired VFP data.\nNew \fts to the VFP data incorporate the knowl-\nedge of how the transport coe\u000ecients should be for-\nmulated to capture their limiting behavior at low Hall\nParameter48,65. It should be reiterated that this newer\nwork did not modify the VFP calculations, only changing\nhow the transport coe\u000ecients were \ft to the VFP data.\nFIG. 2: Ratio of thermal to magnetic transport\ncoe\u000ecients as a function of electron Hall Parameter for\nthe old and new coe\u000ecients. These are plotted for\nZ= 1.\nFigure 1 includes \rc\n^calculated by Sadler, Walsh & Li as\nwell as by Davies, Wen, Ji & Held; note that these two\nreferences give near-identical results.\nThe updated transport coe\u000ecients make clear the sim-\nilarities between thermal and magnetic transport. Look-\ning speci\fcally at the transport due to temperature gra-\ndients:\nvN =\u0000\r?r?Te\u0000\r^^b\u0002rTe (9)\nq\u0014=\u0000\u0014krkTe\u0000\u0014?r?Te\u0000\u0014^^b\u0002rTe(10)\nwhere equation 9 is magnetic transport and equation\n10 is thermal conduction. These equations are split into\ncomponents parallel ( k) to the magnetic \feld, perpen-\ndicular to the \feld ( ?) and perpendicular to both the\nmagnetic \feld and temperature gradient ( ^). In equa-\ntion 9 the?term is Nernst advection and the ^term is\ncross-gradient-Nernst. For thermal conduction the terms\nare the unrestricted thermal conduction along \feld lines,\nsuppressed thermal conduction perpendicular to the \feld\nand the Righi-Leduc heat-\row.\nThe analogy between Nernst ( \rc\n?) and perpendicular\nheat-\row (\u0014c\n?) was originally noted by Haines53. Haines\nfound that if the kinetic collision operator uses an ar-\nti\fcialv\u00002dependence, where vis the electron speed,\nthere is an exact equivalence between heat \row and mag-\nnetic \feld motion, such that \u0014c\n?=\rc\n?=\u0014c\n^=\rc\n^= 5=2.\nThe physical interpretation is that magnetic \feld is only\nfrozen into the faster electrons in the distribution, since\nit can easily di\u000buse through the slower, more collisional\nelectrons. The \feld therefore follows the faster electrons,\nwhich tend to move down temperature gradients with\nthe heat \row. For the arti\fcial collision operator, this\nintuition holds exactly for all Zand!e\u001ce.3\nHowever, this physical picture is una\u000bected by chang-\ning thevdependence of the collision operator, so the\ncharacter of magnetic \feld movement with the electron\nheat \row should still approximately hold. For example,\nif instead the collisions use the realistic Fokker-Planck\nv\u00003dependence, magnetic \feld should still move with\nthe heat \row, although they are no longer directly pro-\nportional. With the Fokker-Planck collision operator, the\nratios are no longer \fxed at 5 =2, and they have depen-\ndence onZand!e\u001ce. However, for \fxed Z, the ratios\nremain approximately constant. Figure 2 shows the ra-\ntio\u0014c\n?=\rc\n?against Hall Parameter for Z= 1; this result\nis similar using the old or new coe\u000ecients, giving little\nvariation.\nFigure 2 also plots the ratio \u0014c\n^=\rc\n^against Hall Param-\neter forZ= 1. Using the old Epperlein & Haines coe\u000e-\ncients the Righi-Leduc ( \u0014c\n^) coe\u000ecient is much smaller\nthan cross-gradient-Nernst ( \rc\n^) at low magnetization.\nWith the new transport coe\u000ecients, however, it can be\nseen that the cross-gradient-Nernst term is analogous to\nadvection of magnetic \feld with the Righi-Leduc heat-\n\row; forZ= 1, the ratio \u0014c\n^=\rc\n^varies by less than 3%\nacross magnetization.\nThe ratio of thermal to magnetic transport coe\u000ecients\ndoes vary with plasma ionization. For low Hall Param-\neter,\u0014c\n?=\rc\n?ranges between 3.63 for Z= 1 to 1.39 for\nZ= 100. This ratio is fundamental for systems where\nthermal conduction drives plasma ablation, where equa-\ntion 9 will move magnetic \felds into the colder regions,\nwhile ablation driven by equation 10 will counter-act that\ntransport39. The balance of Nernst demagnetization to\nplasma ablation has been found to be critically important\nin magnetized plasma conditions41,50,54,55,60,66,72,73. The\ndecrease of \u0014c\n?=\rc\n?with Z suggests that low-Z plasmas\nwill be less impacted by Nernst de-magnetization than\nhigh-Z plasmas.\nThe new \fts make intuitive sense. The ^terms come\nfrom electrons having curved orbits; if the magnetiza-\ntion is low, the electron trajectories should be straight\non average. The Epperlein & Haines \fts captured this\nbehavior for the heat-\row ( \u0014^) but not for the magnetic\ntransport (\r^). At large magnetization the electrons go\nthrough many orbits before colliding, meaning that there\nis no preferential direction for transport; this is captured\nin both the old and new coe\u000ecients, with \u0014^and\r^go-\ning to zero at large magnetization. For the wedge terms\nto be important, the plasma must be in a moderately\nmagnetized regime ( !e\u001ce\u00191).\nThe error in the Epperlein & Haines transport coef-\n\fcients can alternatively be explained by their focus on\nthe electric \feld. They did not consider the coe\u000ecients\nin terms of the induction equation, where the gradients\nof the coe\u000ecients are important. The \ftting function in\nthe Hall parameter that Epperlein & Haines chose has\na derivative that does not tend to zero as the magnetic\n\feld tends to zero, which is a physical requirement. The\nerrors in the \fts become more apparent when the coef-\n\fcients are reformulated in a manner that more clearlydemonstrates their physical e\u000bects in the induction equa-\ntion.\nThis paper demonstrates the importance of the new\ntransport coe\u000ecients in laboratory plasmas, focusing\non the thermally-driven cross-gradient-Nernst term. In\nsection II the new coe\u000ecient is shown to reduce mag-\nnetic \feld twisting in under-dense systems relevant to\nMagLIF preheat52. This setup was proposed in order\nto make the \frst measurements of the cross-gradient-\nNernst transport70. Section III then shows how the cross-\ngradient-Nernst term can result in \feld twisting in pre-\nmagnetized ICF capsules68; again, the new coe\u000ecients\nresult in reduced twisting at low plasma magnetizations.\nHowever, the new coe\u000ecients do not universally reduce\nthe impact of cross-gradient-Nernst transport. Section\nIV compares ICF hot-spot simulations without MHD,\nwith MHD but no cross-gradient-Nernst, full extended-\nMHD with the Epperlein & Haines coe\u000ecients and with\nthe new coe\u000ecients. Cross-gradient-Nernst transport is\nshown to be important in hot-spot cooling, increasing\nthe penetration depth of a cold spike. An example of\nhow the new \r^coe\u000ecient a\u000bects perturbation growth\nin direct-drive ablation fronts was already given in one\nof the original updated coe\u000ecients publications65\nSimulations in this paper use the Gorgon extended-\nMHD code44,45,72. The code can use either the old Ep-\nperlein & Haines coe\u000ecients49or the updated coe\u000ecients\n(see the supplementary material of reference65). The co-\ne\u000ecients calculated by Sadler, Walsh & Li are used in\nthis paper, although the coe\u000ecients from Davies, Wen,\nJi & Held have been implemented and give no notice-\nable di\u000berences48(as expected from \fgure 1). Epperlein\n& Haines provided tabulated data for the coe\u000ecients at\nspeci\fc values of Z; these values are interpolated in the\nsimulations. The new coe\u000ecients provide a polynomial\n\ft to ionization, which is preferable for reducing com-\nputations and eliminating discontinuities when gradients\nare taken in Z.\nII. UNDERDENSE PLASMAS\nThis section investigates the impact of cross-gradient-\nNernst on magnetized under-dense gases heated by a\nlaser. This setup is relevant to MagLIF67and mini-\nMagLIF preheat38, but was also investigated as a means\nof measuring speci\fc extended-MHD transport terms for\nthe \frst time70. Particularly relevant was the suggestion\nthat quanti\fcation of a twisted magnetic \feld compo-\nnent could be used to measure the cross-gradient-Nernst\nadvection velocity70. Simulations in that experiment\ndesign paper used the erroneous Epperlein & Haines\ncoe\u000ecients49, which are shown here to give excessive\nmagnetic \feld twisting at low magnetization.\nThe con\fguration used as demonstration here is as fol-\nlows. A low density (5 \u00021019atoms/cm3) deuterium gas\nis irradiated by a 5 Jbeam with a 0 :5nssquare tem-\nporal pulse. The beam has a Gaussian spatial pro\fle4\nFIG. 3: Azimuthal magnetic \feld component generated by cross-gradient-Nernst advection for an under-dense\nmagnetized plasma70using the Epperlein & Haines transport coe\u000ecients49(left) and the updated coe\u000ecients\n(right)48,65. These pro\fles are 1.0ns after the laser turns on.\nwith a standard deviation \u001b= 100\u0016m at best focus.\nThe beam is tapered along its propagation direction such\nthat atr=\u001bthe angle of the rays to the axis satis-\n\fes sin\u0012= 1=12. A 1T magnetic \feld is applied along\nthe laser propagation axis. The beam tapering is pur-\nposefully used to induce magnetic \feld twisting by cross-\ngradient-Nernst transport70.\n2-Dr,zsimulations are used, with the laser drive and\napplied magnetic \feld along z. The laser is treated as in-\ndividual rays that are traced through the plasma and de-\nposit their energy by inverse Bremsstrahlung. A 460 \u0016m\naxial extent is used, with radial and axial resolution of\n1\u0016m.\nAs the gas density is low, very little energy couples\nto the system. At 0 :1ns the total absorption along the\n460\u0016m axial extent is 2.5%, reducing to 1% by the end\nof the laser pulse when the plasma temperature is higher\n(>100eV). As the Hall Parameter is low for the chosen\nsetup (!e\u001ce<0:2 everywhere) thermal conduction ef-\nfectively transports energy radially. Simultaneously, the\nNernst term advects magnetic \feld away from the beam\ncenter, reducing the \feld strength to 0.3T by 0.5ns. The\nmagnetic \rux piles up at the thermal conduction front,\npeaking at 2T.\nThe impact of cross-gradient-Nernst can be seen by\ndecomposing the advection term in equation 2:\n\u0014@B\n@t\u0015\nN^+(vN^\u0001r)B=\u0000B(r\u0001vN^)+(B\u0001r)vN^(11)\nAs the temperature gradient is predominantly radial\nand the magnetic \feld is predominantly axial, the cross-\ngradient-Nernst velocity is purely in \u0012. Therefore, equa-\ntion 11 reduces to\u0002@B\u001e\n@t\u0003\nN^=Bz@vN^\u0012\n@z. This term rep-\nresents magnetic \feld twisting. Following a single ax-ial magnetic \feld line, if vN^= 0 at the bottom and\nvN^6= 0 at the top, a B\u0012\feld will be generated by twist-\ning in between. By tapering the laser beam, this setup\npurposefully makes this happen; near the laser edge a\nmagnetic \feld line will pass between an unheated and\na heated regime, resulting in twisting of the magnetic\n\feld70. This process can be viewed as a dynamo driven\nby electron heat-\row.\nFigure 3 compares the twisted magnetic \feld compo-\nnent att= 1:0ns using the old Epperlein & Haines trans-\nport coe\u000ecients and the updated coe\u000ecients48,65. The\nBiermann Battery generation has been neglected in these\nsimulations, which means that B\u0012can only be generated\nby cross-gradient-Nernst twisting. As the experiment\nhere is in the low magnetization regime, the di\u000berence\nin the\r^coe\u000ecient is substantial; at the laser spot edge,\nwhere much of the twisting takes place, !e\u001ce\u00190:05,\nresulting in a factor of 4 reduction in the cross-gradient-\nNernst coe\u000ecient. By 1.0ns the peak B\u0012\u00190.08T for the\nold coe\u000ecients, and \u00190.01T using the new coe\u000ecients.\nThe new coe\u000ecients change the regime where signi\f-\ncantB\u0012is generated. Designs using the old coe\u000ecients\naimed for the underdense plasma being as low magne-\ntization as possible. Now, however, it is clear that the\nplasma should be balanced in the moderately magnetized\nregime, with the cross-gradient-Nernst velocity peaking\nat around!e\u001ce\u00190:8 for deuterium65. This is still real-\nizable, with the density, laser power and applied \feld\nall acting as free variables. An experiment with the\nplasma parameters de\fned here but scanning applied\n\feld strength is expected to observe the true \r^depen-\ndence.\nHowever, with the e\u000bect of cross-gradient-Nernst re-\nduced, the Biermann Battery contribution to B\u0012becomes\nmore important. At 0.1ns for the simulations shown here5\nFIG. 4:B\u001eat 6.3ns for a warm indirect-drive\nimplosion with an applied axial \feld of 5T. Left is a\nsimulation with cross-gradient-Nernst included (using\nthe new coe\u000ecients), while right shows a case with both\ncross-gradient-Nernst and Biermann included.\ntheB\u0012\feld from Biermann Battery is of the same or-\nder as from cross-gradient-Nernst. Biermann requires a\ndensity gradient in the plasma, which only develops later\nin time, at time-scales on the order of the laser radius\ndivided by the sound speed. Therefore, probing early\nin time before the plasma hydrodynamically expands is\nimportant to distinguish Biermann from cross-gradient-\nNernst.\nIII. PRE-MAGNETIZED ICF CAPSULES\nAxial magnetic \felds can be applied to ICF cap-\nsules to reduce thermal cooling of the fuel during\nstagnation43,62,71. Throughout the implosion the mag-\nnetic \feld compresses along with the plasma; this hap-\npens dominantly at the capsule waist, where the implo-\nsion velocity is perpendicular to the magnetic \feld lines.\nAt the poles the magnetic \feld remains uncompressed,\nas the implosion velocity is along \feld lines.\nIt has been suggested that the cross-gradient-Nernst\nterm should twist the applied magnetic \feld during the\nimplosion47. Preliminary simulations showed signi\fcant\ntwisting by bang-time, with the yield increased by as\nmuch as 10% due to the extra path length required for\nheat to travel along \feld lines out of the hot-spot68.\nAgain, these calculations used the old Epperlein & Haines\nFIG. 5: Density and B\u001eat neutron bang-time\n(t= 7:3ns) using the old Epperlein & Haines\ncoe\u000ecients (left) and the updated versions (right).\ncoe\u000ecients. Here, as with the twisting seen in the under-\ndense con\fguration in section II, the twisting is found to\nbe lower when the new transport coe\u000ecients are used.\nThe simulated con\fguration is a D2gas-\flled HDC\nindirect-drive capsule that is used on the National Igni-\ntion Facility, although these results broadly apply to all\nmagnetized spherical implosions. A 5T magnetic \feld is\napplied axially. 1 \u0016m radial resolution is used throughout\nthe implosion and 180 polar cells are used in the simula-\ntion range\u0012= 0;\u0019. Before the \frst shock converges on\naxis the simulations are transferred to a cylindrical mesh.\nModerate HDC shell thickness variations are\ninitialized42, such that the capsule is mildly per-\nturbed by neutron bang-time. 1600 perturbations with\nrandom amplitude and mode are used, with each chosen\nin the range \u000f= 0;0:5nm andk= 0;180 respectively.\nWhile for the underdense con\fguration a tapered beam\nwas used to induce twisting, a spherical implosion natu-\nrally has twisting peaking at \u0012=\u0019=4;3\u0019=4. This hap-\npens because the cross-gradient-Nernst velocity \u0000^b\u0002rTe\nis maximum at the capsule waist and zero at the poles.\nTherefore, the axial magnetic \feld at the waist is dis-\nplaced in\u001ebut at the poles it is stationary; in be-\ntween the waist and the poles the magnetic \feld lines\nare twisted out of the plane.\nTwisting of magnetic \feld lines introduces a closed\n\feld line component, B\u001e. This is of great interest to the\nmagneto-inertial fusion community, as this component\nis e\u000bectively compressed during the implosion and re-\nsults in additional thermal energy containment. Schemes\nhave been considered to apply a purely B\u001ecomponent to\na capsule56, although none have been demonstrated for\nspherical implosions.\nThe cross-gradient-Nernst velocity is signi\fcant both6\nin the in-\right shock-compressed gas and in the stag-\nnating hot-spot. For the con\fguration chosen here the\nin-\right phase is not changed signi\fcantly by the new co-\ne\u000ecients, as the low density and high temperature gas is\nmagnetized even with B0= 5T. For lower applied \felds\nthe coe\u000ecients may make a signi\fcant di\u000berence, but\nthese regimes are not of interest for magneto-inertial fu-\nsion.\nFigure 4 shows the B\u001e\feld pro\fle induced by the\ncross-gradient-Nernst e\u000bect at 6.3ns, which is before the\n\frst shock converges onto the axis. While B\u001eonly\nreaches 3T at this time, the \feld twisting is signi\fcant:\nB\u001e=jBj= 1=3 in locations near the shock-front.\nAlso shown in \fgure 4 is a simulation with the Bier-\nmann Battery self-generated magnetic \felds included.\nThis term requires an asymmetric implosion to be im-\nportant, of which there are two sources in this implosion.\nFirst of all, the applied \feld results in anisotropy of the\nheat-\row; here that results in a hotter compressed gas at\nthe waist compared with the poles. Also, the HDC thick-\nness variations generate magnetic \felds. The interest\nhere is in the fuel B\u001ecomponent, which is dominated by\ncross-gradient-Nernst twisting for B0= 5T. Larger ap-\nplied \felds result in suppression of cross-gradient-Nernst\nand enhancement of the thermal conductivity anisotropy,\nenhancing the impact of Biermann.\nFigure 5 shows the density and out-of-plane magnetic\n\feld component at bang-time. Biermann Battery gener-\nation of magnetic \feld has been turned o\u000b in the code\nfor these simulations, which means that all of B\u001eis from\ncross-gradient-Nernst twisting. On the left is a case us-\ning the old cross-gradient-Nernst coe\u000ecients from Ep-\nperlein & Haines, while on the right uses the updated\ncoe\u000ecients. The new coe\u000ecients lower the peak B\u001efrom\n500T to 250T. Nonetheless, the twisting is signi\fcant,\nwithB\u001e=jBjup to 0.3 throughout the hot-spot.\nThe twisting introduces several additional e\u000bects.\nFirstly, thermal conduction and Nernst are moderately\nreduced, as these components become partially out of the\nsimulation plane; Cross-gradient-Nernst and Righi-Leduc\nthen have components within the simulation plane. If the\nmagnetic tension becomes signi\fcant then the twisted\n\feld would also be expected to induce plasma motion\nin\u001e; however, as cross-gradient-Nernst is suppressed\nfor the large applied \felds required for tension to be\nimportant62,71, this may not be a realizable regime. For\nB0= 5T the plasma \fis too large for signi\fcant motion.\nWhile the simulations here focused on indirect-drive\nimplosions, cross-gradient-Nernst also twists magnetic\n\felds in pre-magnetized direct-drive ablation fronts73,\nwhere the temperature gradients drive extreme heat-\n\rows. However, the B\u001ecomponent is e\u000bectively advected\nby the plasma and Nernst velocities, lowering its impact.\nFIG. 6: Self-generated magnetic \felds at bang-time\naround a cold spike pushing into a hot-spot. 3 cases are\nshown. Left: a simulation without cross-gradient-Nernst\nadvection included. Middle: a simulation using the\nEpperlein & Haines transport coe\u000ecients49. Right: a\nsimulation using the updated transport coe\u000ecients48,65.\nIV. SELF-GENERATED FIELDS IN ICF HOT-SPOTS\nThis section looks into the impact of cross-gradient-\nNernst on self-generated magnetic \feld pro\fles in regu-\nlar ICF hot-spots, where magnetic \felds have not been\nexternally applied.\nMagnetic \felds are generated by the Biermann Battery\nmechanism around perturbations, with estimated \feld\nstrengths up to 10kT72. These \felds are predominantly\ngenerated in the stagnation phase, when the tempera-\nture and density gradients are largest69. Recent advances\nhave been made in the theoretical understanding of mag-\nnetic \rux generation in these systems, with more \rux\nbeing generated around high mode and large amplitude\nperturbations69.\nWhile it was expected that self-generated magnetic\n\felds would reduce heat loss from ICF hot-spots, re-\nsearch found that magnetization of the electron popula-\ntion introduced Righi-Leduc heat-\row, which enhanced\ncooling72. As cross-gradient-Nernst is the magnetic\ntransport analogue of Righi-Leduc heat-\row, it is impor-\ntant in these systems.\nThe simulations here use the indirect-drive HDC de-\nsign N170601 is used, although the physics is also appli-\ncable to direct-drive hot-spots. An isolated 200nm HDC\nshell thickness variation is imposed at the capsule pole,\ncausing a cold spike to push into the hot-spot. Shell\nthickness variations have been found to be a signi\fcant\ndegradation mechanism for HDC implosions42; the vari-\nation applied here is not based on any target fabrication7\nFIG. 7: Electron temperature along the axis of a\nhot-spot spike at bang-time. Temperature pro\fles are\nshown for cases: without any self-generated \felds\nincluded; with extended-MHD but no\ncross-gradient-Nernst advection; using the Epperlein &\nHaines transport coe\u000ecients49; using the updated\ncoe\u000ecients48,65.\nspeci\fcations, and is instead used as a demonstration of\nmagnetized heat-\row in ICF implosions.\nThe simulations here are 2-D and are in spherical\ngeometry until the \frst shock converges onto the axis\n(t\u00197:6ns). The resolution up to this time is 1 \u0016m radi-\nally, with 180 cells in the polar direction from \u0012= 0;\u0019.\nThe simulations are then remapped into cylindrical ge-\nometry for the stagnation phase, when the resolution is\n1\n2\u0016m.\nFigure 6 shows the magnetic \feld distribution at neu-\ntron bang-time ( t= 8:5ns) for simulations with di\u000berent\ncross-gradient-Nernst physics included. The magnetic\n\feld pro\fles are in green (into the page) and purple (out\nof the page), plotted over the density so that the prox-\nimity to the spike can be seen. The dense fuel is at the\ntop of the \fgure and the hot-spot at the bottom.\nFirst in \fgure 6 is a case with no cross-gradient-Nernst\nadvection. The Righi-Leduc heat-\row direction is shown\nwith white arrows. Righi-Leduc acts to cool the spike\ntip, which results in regular Nernst compressing the \feld\nonto the simulation axis. This process causes numerical\nissues, advecting all of the magnetic \feld into a single\nline of cells. Once the cross-gradient-Nernst is included,\nthe magnetic \feld moves in the same direction as Righi-\nLeduc, preventing the \feld pro\fle from compressing onto\nthe axis. It can be seen in \fgure 6 that the new cross-\ngradient-Nernst coe\u000ecient does not advect as rapidly as\nthe Epperlein & Haines version.\nIt is posited that the updated coe\u000ecients will always be\nimportant in systems with self-generated magnetic \felds\nand thermal conduction; there is always a null-point of\nmagnetic \feld, resulting in no electron magnetization.Even if the peak magnetization is large enough to be\nin a regime where the Epperlein & Haines coe\u000ecients\nare valid, the magnetization will decrease to zero in the\nsurrounding regions, passing through the regime where\nthe old coe\u000ecients are invalid.\nFigure 7 shows the impact of the imposed spike on the\nhot-spot temperature for simulations with various MHD\npackages included. The electron temperature is plotted\nalong the spike axis. Without any MHD included the\nspike does not propagate as far; this suggests that cur-\nrent design calculations of ICF implosions (both directly\nand indirectly driven) are underestimating the impact of\nperturbations.\nSimulations with the new coe\u000ecients included result in\nthe greatest discrepancy with the estimates that do not\ninclude Biermann generation; at the Te= 3:5keV contour\nthe spike has penetrated 7 \u0016m further due to the mag-\nnetization of heat-\row. The simulations without cross-\ngradient-Nernst are less a\u000bected by the magnetic \felds,\nas the \felds are transported by the standard Nernst term\ninto a one-cell width (see \fgure 6).\nThe new coe\u000ecients give a greater impact of MHD\non spike penetration compared with the old coe\u000ecients.\nThis can be understood from the fact that the new co-\ne\u000ecients result in the magnetic \feld moving with the\nRighi-Leduc heat-\row. In contrast, the old coe\u000ecients\nallowed the magnetic \rux to move ahead of the Righi-\nLeduc heat-\row into regions where it would have a lower\nimpact on the plasma magnetization. This can be seen\nin \fgure 2, which shows that the ratio of Righi-Leduc\ncoe\u000ecient to cross-gradient-Nernst coe\u000ecient decreasing\nto zero for low magnetization.\nCloser analysis of the impact of MHD on capsule per-\nturbation growth for a variety of mode numbers and am-\nplitudes will be the subject of a future publication.\nV. CONCLUSIONS\nIn summary, updated magnetic transport coe\u000ecients\nthat accurately replicate kinetic simulations at low elec-\ntron magnetizations48,65have been shown to be impor-\ntant across a range of laboratory plasma conditions.\nThese coe\u000ecients are the new standard for implementa-\ntion into extended-MHD codes. While the two references\ngive di\u000berent \fts for the coe\u000ecients, they are found to\nbe practically equivalent.\nWith an external magnetic \feld applied, the cross-\ngradient-Nernst term tends to twist the magnetic\n\feld46,68; using the new coe\u000ecients reduces twisting in\nthe low magnetization ( !e\u001ce<1) regime. This result im-\npacts attempts to measure cross-gradient-Nernst for the\n\frst time70as well as the design of pre-magnetized cap-\nsule implosions43,62,68,71. Twisting is still possible in the\nmoderate magnetization regime ( !e\u001ce\u00191).\nFor systems with signi\fcant self-generated magnetic\n\felds50,64,72the new coe\u000ecients result in the magnetic\n\felds moving with the Righi-Leduc heat-\row. This is8\nfound to enhance the impact of Righi-Leduc, as shown in\ndirect-drive ablation fronts (where Righi-Leduc reduces\nperturbation growth)65and in ICF hot-spots (where\nRighi-Leduc enhances perturbation growth).\nIn addition to being more physically accurate, the new\ncoe\u000ecients have been found to increase numerical stabil-\nity, as they introduce fewer discontinuities into the sim-\nulations.\nACKNOWLEDGEMENTS\nThis work was performed under the auspices of the\nU.S. Department of Energy by Lawrence Livermore Na-\ntional Laboratory under Contract DE-AC52-07NA27344\nand by the LLNL-LDRD program under Project Number\n20-SI-002. The simulation results were obtained using\nthe Imperial College High Performance Computer Cx1..\nThis document was prepared as an account of work\nsponsored by an agency of the United States government.\nNeither the United States government nor Lawrence Liv-\nermore National Security, LLC, nor any of their employ-\nees makes any warranty, expressed or implied, or assumes\nany legal liability or responsibility for the accuracy, com-\npleteness, or usefulness of any information, apparatus,\nproduct, or process disclosed, or represents that its use\nwould not infringe privately owned rights. Reference\nherein to any speci\fc commercial product, process, or\nservice by trade name, trademark, manufacturer, or oth-\nerwise does not necessarily constitute or imply its en-\ndorsement, recommendation, or favoring by the United\nStates government or Lawrence Livermore National Se-\ncurity, LLC. The views and opinions of authors expressed\nherein do not necessarily state or re\rect those of the\nUnited States government or Lawrence Livermore Na-\ntional Security, LLC, and shall not be used for advertis-\ning or product endorsement purposes.\nResearch presented in this article was also supported\nby the Laboratory Directed Research and Develop-\nment program of Los Alamos National Laboratory, un-\nder the Center for Nonlinear Studies project number\n20190496CR. This research was supported by the Los\nAlamos National Laboratory (LANL) through its Center\nfor Space and Earth Science (CSES). CSES is funded by\nLANL's Laboratory Directed Research and Development\n(LDRD) program under project number 20180475DR.\nFinally, the information, data, or work presented\nherein was funded in part by the Advanced Research\nProjects Agency-Energy (ARPAE), U.S. Department of\nEnergy, under Award No. DE-AR0001272, by the De-\npartment of Energy O\u000ece of Science, under Award No.\nDE-FG02-04ER54746, by the Department of Energy Na-\ntional Nuclear Security Administration under Award No.\nDENA0003856, the University of Rochester, and the New\nYork State Energy Research and Development Authority.REFERENCES\n38D. H. Barnak, J. R. Davies, R. Betti, M. J. Bonino, E. M.\nCampbell, V. Yu. Glebov, D. R. Harding, J. P. Knauer, S. P.\nRegan, A. B. Sefkow, A. J. Harvey-Thompson, K. J. Peter-\nson, D. B. Sinars, S. A. Slutz, M. R. Weis, and P.-Y. Chang.\nLaser-driven magnetized liner inertial fusion on OMEGA. 24\n(5):056310. ISSN 1070-664X. doi:10.1063/1.4982692. URL\nhttps://doi.org/10.1063/1.4982692 .\n39R. Betti, M. Umansky, V. Lobatchev, V. N. Goncharov, and\nR. L. McCrory. Hot-spot dynamics and deceleration-phase\nRayleigh{Taylor instability of imploding inertial con\fnement fu-\nsion capsules. 8(12):5257{5267. doi:10.1063/1.1412006. URL\nhttp://aip.scitation.org/doi/10.1063/1.1412006 .\n40S. I. Braginskii. Transport Processes in a Plasma. In Reviews of\nPlasma Physics , volume 1, pages 205{205. URL http://adsabs.\nharvard.edu/abs/1965RvPP....1..205B .\n41Walsh C. A. Russell. B. K. Chittenden J. P. Crilly A. Fiksel\nG. Nilson P. M. Thomas A. G. R. Krushelnick K. Willingale L.\nCampbell, P. T. Magnetic signatures of radiation-driven double\nablation fronts. 145001. doi:10.1103/PhysRevLett.125.145001.\n42D. T. Casey, B. J. MacGowan, J. D. Sater, A. B. Zylstra, O. L.\nLanden, J. Milovich, O. A. Hurricane, A. L. Kritcher, M. Hohen-\nberger, K. Baker, S. Le Pape, T. D oppner, C. Weber, H. Huang,\nC. Kong, J. Biener, C. V. Young, S. Haan, R. C. Nora, S. Ross,\nH. Robey, M. Stadermann, A. Nikroo, D. A. Callahan, R. M.\nBionta, K. D. Hahn, A. S. Moore, D. Schlossberg, M. Bruhn,\nK. Sequoia, N. Rice, M. Farrell, and C. Wild. Evidence of\nthree-dimensional asymmetries seeded by high-density carbon-\nablator nonuniformity in experiments at the national ignition fa-\ncility. 126(2):025002. doi:10.1103/PhysRevLett.126.025002. URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.126.025002 .\n43P Y Chang, G Fiksel, M Hohenberger, J P Knauer, R Betti,\nF J Marshall, and D D Meyerhofer. Fusion Yield Enhance-\nment in Magnetized Laser-Driven Implosions. 035006:2{5. doi:\n10.1103/PhysRevLett.107.035006.\n44J P Chittenden, S V Lebedev, C A Jennings, S N Bland,\nand A Ciardi. X-ray generation mechanisms in three-\ndimensional simulations of wire array Z-pinches. In\nPlasma Physics and Controlled Fusion , volume 46, pages\nB457{B476. doi:10.1088/0741-3335/46/12B/039. URL\nhttp://stacks.iop.org/0741-3335/46/i=12B/a=039?key=\ncrossref.db05b373851fc67f42aa76a38351c1e0 .\n45A. Ciardi, S. V. Lebedev, A. Frank, E. G. Blackman, J. P. Chit-\ntenden, C. J. Jennings, D. J. Ampleford, S. N. Bland, S. C.\nBott, J. Rapley, G. N. Hall, F. A. Suzuki-Vidal, A. Marocchino,\nT. Lery, and C. Stehle. The evolution of magnetic tower\njets in the laboratory. In Physics of Plasmas , volume 14.\ndoi:10.1063/1.2436479. URL http://arxiv.org/abs/astro-ph/\n0611441 .\n46J. R. Davies, D. H. Barnak, R. Betti, E. M. Campbell, P.-Y.\nChang, A. B. Sefkow, K. J. Peterson, D. B. Sinars, and M. R.\nWeis. Laser-driven magnetized liner inertial fusion. 24(6):062701,\n. ISSN 1070-664X. doi:10.1063/1.4984779. URL https://aip.\nscitation.org/doi/full/10.1063/1.4984779 .\n47J. R. Davies, R. Betti, P.-Y. Chang, and G. Fiksel.\nThe importance of electrothermal terms in Ohm's law for\nmagnetized spherical implosions. 22(112703), . doi:\n10.1063/1.4935286. URL http://scitation.aip.org/content/\naip/journal/pop/22/11/10.1063/1.4935286 .\n48J. R. Davies, H. Wen, Jeong-Young Ji, and Eric D. Held. Trans-\nport coe\u000ecients for magnetic-\feld evolution in inviscid mag-\nnetohydrodynamics. 28(1):012305, . ISSN 1070-664X. doi:\n10.1063/5.0023445. URL https://doi.org/10.1063/5.0023445 .\n49E. M. Epperlein and M. G. Haines. Plasma transport co-\ne\u000ecients in a magnetic \feld by direct numerical solution of\nthe Fokker{Planck equation. 29(4). doi:10.1063/1.865901.\nURL http://scitation.aip.org/content/aip/journal/pof1/\n29/4/10.1063/1.865901 .9\n50W. A. Farmer, J. M. Koning, D. J. Strozzi, D. E. Hinkel, L. F.\nBerzak Hopkins, O. S. Jones, and M. D. Rosen. Simulation of\nself-generated magnetic \felds in an inertial fusion hohlraum en-\nvironment. 24(5):052703{052703. doi:10.1063/1.4983140. URL\nhttp://aip.scitation.org/doi/10.1063/1.4983140 .\n51L. Gao, P. M. Nilson, I. V. Igumenshchev, M. G. Haines,\nD. H. Froula, R. Betti, and D. D. Meyerhofer. Precision Map-\nping of Laser-Driven Magnetic Fields and Their Evolution in\nHigh-Energy-Density Plasmas. 114(21):215003{215003. doi:\n10.1103/PhysRevLett.114.215003. URL https://link.aps.org/\ndoi/10.1103/PhysRevLett.114.215003 .\n52M. R. Gomez, S. A. Slutz, C. A. Jennings, D. J. Ampleford,\nM. R. Weis, C. E. Myers, D. A. Yager-Elorriaga, K. D. Hahn,\nS. B. Hansen, E. C. Harding, A. J. Harvey-Thompson, D. C.\nLamppa, M. Mangan, P. F. Knapp, T. J. Awe, G. A. Chandler,\nG. W. Cooper, J. R. Fein, M. Geissel, M. E. Glinsky, W. E.\nLewis, C. L. Ruiz, D. E. Ruiz, M. E. Savage, P. F. Schmit, I. C.\nSmith, J. D. Styron, J. L. Porter, B. Jones, T. R. Mattsson,\nK. J. Peterson, G. A. Rochau, and D. B. Sinars. Performance\nScaling in Magnetized Liner Inertial Fusion Experiments. 125\n(15):155002. doi:10.1103/PhysRevLett.125.155002. URL https:\n//link.aps.org/doi/10.1103/PhysRevLett.125.155002 .\n53M G Haines. Heat \rux e\u000bects in Ohm's law. In Plasma\nPhysics and Controlled Fusion , volume 28, pages 1705{1716. doi:\n10.1088/0741-3335/28/11/007.\n54D. W. Hill and R. J. Kingham. Enhancement of Pressure\nPerturbations in Ablation due to Kinetic Magnetised Trans-\nport E\u000bects under Direct-Drive ICF relevant conditions. URL\nhttp://arxiv.org/abs/1712.02663 .\n55Ridgers C. Kingham R. J. Walsh C. A. Hill, D. W. Vlasov-\nfokker-planck simulations of pre-magnetised ablating planar tar-\ngets. Submitted.\n56M Hohenberger, P. Chang, G. Fiksel, J. P. Knauer, R. Betti,\nF. J. Marshall, F. Seguin, C. K. Li, M. J E Manuel, and R D\nPetrasso. Magnetic Flux Compression Experiments on OMEGA.\n57I. V. Igumenshchev, A. B. Zylstra, C. K. Li, P. M. Nilson,\nV. N. Goncharov, and R. D. Petrasso. Self-generated mag-\nnetic \felds in direct-drive implosion experiments. 21(6). doi:\n10.1063/1.4883226.\n58A S Joglekar, A G R Thomas, W Fox, and A Bhat-\ntacharjee. Magnetic reconnection in plasma under in-\nertial con\fnement fusion conditions driven by heat \rux\ne\u000bects in Ohm's law. 112(10):105004{105004. doi:\n10.1103/PhysRevLett.112.105004. URL http://journals.aps.\norg/prl/abstract/10.1103/PhysRevLett.112.105004 .\n59C. K. Li, V. T. Tikhonchuk, Q. Moreno, H. Sio, E. D'Humi\u0012 eres,\nX. Ribeyre, Ph. Korneev, S. Atzeni, R. Betti, A. Birkel, E. M.\nCampbell, R. K. Follett, J. A. Frenje, S. X. Hu, M. Koenig,\nY. Sakawa, T. C. Sangster, F. H. Seguin, H. Takabe, S. Zhang,\nand R. D. Petrasso. Collisionless shocks driven by supersonic\nplasma \rows with self-generated magnetic \felds. 123(5):055002.\ndoi:10.1103/PhysRevLett.123.055002. URL https://link.aps.\norg/doi/10.1103/PhysRevLett.123.055002 .\n60M. J E Manuel, M. Flaig, T. Plewa, C. K. Li, F. H. S\u0013 eguin,\nJ. A. Frenje, D. T. Casey, R. D. Petrasso, S. X. Hu, R. Betti,\nJ. Hager, D. D. Meyerhofer, and V. Smalyuk. Collisional e\u000bects\non Rayleigh-Taylor-induced magnetic \felds. In Physics of Plas-\nmas, volume 22, . doi:10.1063/1.4919392.\n61M. J E Manuel, C. K. Li, F. H. S??guin, J. Frenje, D. T. Casey,\nR. D. Petrasso, S. X. Hu, R. Betti, J. D. Hager, D. D. Mey-\nerhofer, and V. A. Smalyuk. First measurements of Rayleigh-\nTaylor-induced magnetic \felds in laser-produced plasmas. 108\n(25):1{5, . doi:10.1103/PhysRevLett.108.255006.\n62L. J. Perkins, D. D.-M Ho, B. G. Logan, G. B. Zimmerman, M. A.\nRhodes, D. J. Strozzi, D. T. Black\feld, and S. A. Hawkins. The\npotential of imposed magnetic \felds for enhancing ignition prob-\nability and fusion energy yield in indirect-drive inertial con\fne-\nment fusion. 24(6):062708{062708. doi:10.1063/1.4985150. URL\nhttp://aip.scitation.org/doi/10.1063/1.4985150 .\n63James D. Sadler, Hui Li, and Kirk A. Flippo. Magnetic \feldgeneration from composition gradients in inertial con\fnement\nfusion fuel. 378(2184):20200045, . doi:10.1098/rsta.2020.0045.\nURL https://doi.org/10.1098/rsta.2020.0045 .\n64James D. Sadler, Hui Li, and Brian M. Haines. Magnetization\naround mix jets entering inertial con\fnement fusion fuel. 27(7):\n072707, . ISSN 1070-664X. doi:10.1063/5.0012959. URL https:\n//doi.org/10.1063/5.0012959 .\n65James D. Sadler, Christopher A. Walsh, and Hui Li. Symmetric\nSet of Transport Coe\u000ecients for Collisional Magnetized Plasma.\n126(7):075001, . doi:10.1103/PhysRevLett.126.075001. URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.126.075001 .\n66M. Sherlock and J. J. Bissell. Suppression of the Bier-\nmann Battery and Stabilization of the Thermomagnetic In-\nstability in Laser Fusion Conditions. 124(5):055001. doi:\n10.1103/PhysRevLett.124.055001. URL https://link.aps.org/\ndoi/10.1103/PhysRevLett.124.055001 .\n67S. A. Slutz, M. C. Herrmann, R. A. Vesey, A. B. Sefkow, D. B.\nSinars, D. C. Rovang, K. J. Peterson, and M. E. Cuneo. Pulsed-\npower-driven cylindrical liner implosions of laser preheated fuel\nmagnetized with an axial \feld. 17(5):056303. ISSN 1070-\n664X. doi:10.1063/1.3333505. URL https://doi.org/10.1063/\n1.3333505 .\n68C. A. Walsh. Extended Magneto-hydrodynamic E\u000bects in\nIndirect-Drive Inertial Con\fnement Fusion Experiments.\n69C. A. Walsh and D. S. Clark. Biermann Battery Magnetic Fields\nin ICF Capsules: Total Magnetic Flux Generation.\n70C. A. Walsh, J. P. Chittenden, D. W. Hill, and C. Ridgers.\nExtended-magnetohydrodynamics in under-dense plasmas. 27\n(2):022103, . ISSN 1070-664X. doi:10.1063/1.5124144. URL\nhttps://aip.scitation.org/doi/10.1063/1.5124144 .\n71C A Walsh, K Mcglinchey, J K Tong, B D Appelbe, A Crilly,\nM Zhang, and J P Chittenden. Perturbation Modi\fcations by\nPre-magnetisation in Inertial Con\fnement Fusion Implosions.\n096:1{12, .\n72C.A. Walsh, J.P. Chittenden, K. McGlinchey, N.P.L. Ni-\nasse, and B.D. Appelbe. Self-Generated Magnetic Fields\nin the Stagnation Phase of Indirect-Drive Implosions on the\nNational Ignition Facility. 118(15):155001{155001, . doi:\n10.1103/PhysRevLett.118.155001. URL http://link.aps.org/\ndoi/10.1103/PhysRevLett.118.155001 .\n73C.A. Walsh, A.J. Crilly, and J.P. Chittenden. Magnetized\ndirectly-driven ICF capsules: Increased instability growth from\nnon-uniform laser drive. 60(10):106006, . ISSN 0029-5515. doi:\n10.1088/1741-4326/abab52. URL http://dx.doi.org/10.1088/\n1741-4326/abab52 .\n74H C Watkins and R J Kingham. Magnetised thermal self-\nfocusing and \flamentation of long-pulse lasers in plasmas rel-\nevant to magnetised ICF experiments. 25:92701{92701. doi:\n10.1063/1.5049229. URL https://doi.org/10.1063/1.5049229 .\n38D. H. Barnak, J. R. Davies, R. Betti, M. J. Bonino, E. M.\nCampbell, V. Yu. Glebov, D. R. Harding, J. P. Knauer, S. P.\nRegan, A. B. Sefkow, A. J. Harvey-Thompson, K. J. Peter-\nson, D. B. Sinars, S. A. Slutz, M. R. Weis, and P.-Y. Chang.\nLaser-driven magnetized liner inertial fusion on OMEGA. 24\n(5):056310. ISSN 1070-664X. doi:10.1063/1.4982692. URL\nhttps://doi.org/10.1063/1.4982692 .\n39R. Betti, M. Umansky, V. Lobatchev, V. N. Goncharov, and\nR. L. McCrory. Hot-spot dynamics and deceleration-phase\nRayleigh{Taylor instability of imploding inertial con\fnement fu-\nsion capsules. 8(12):5257{5267. doi:10.1063/1.1412006. URL\nhttp://aip.scitation.org/doi/10.1063/1.1412006 .\n40S. I. Braginskii. Transport Processes in a Plasma. In Reviews of\nPlasma Physics , volume 1, pages 205{205. URL http://adsabs.\nharvard.edu/abs/1965RvPP....1..205B .\n41Walsh C. A. Russell. B. K. Chittenden J. P. Crilly A. Fiksel\nG. Nilson P. M. Thomas A. G. R. Krushelnick K. Willingale L.\nCampbell, P. T. Magnetic signatures of radiation-driven double\nablation fronts. 145001. doi:10.1103/PhysRevLett.125.145001.\n42D. T. Casey, B. J. MacGowan, J. D. Sater, A. B. Zylstra, O. L.\nLanden, J. Milovich, O. A. Hurricane, A. L. Kritcher, M. Hohen-10\nberger, K. Baker, S. Le Pape, T. D oppner, C. Weber, H. Huang,\nC. Kong, J. Biener, C. V. Young, S. Haan, R. C. Nora, S. Ross,\nH. Robey, M. Stadermann, A. Nikroo, D. A. Callahan, R. M.\nBionta, K. D. Hahn, A. S. Moore, D. Schlossberg, M. Bruhn,\nK. Sequoia, N. Rice, M. Farrell, and C. Wild. Evidence of\nthree-dimensional asymmetries seeded by high-density carbon-\nablator nonuniformity in experiments at the national ignition fa-\ncility. 126(2):025002. doi:10.1103/PhysRevLett.126.025002. URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.126.025002 .\n43P Y Chang, G Fiksel, M Hohenberger, J P Knauer, R Betti,\nF J Marshall, and D D Meyerhofer. Fusion Yield Enhance-\nment in Magnetized Laser-Driven Implosions. 035006:2{5. doi:\n10.1103/PhysRevLett.107.035006.\n44J P Chittenden, S V Lebedev, C A Jennings, S N Bland,\nand A Ciardi. X-ray generation mechanisms in three-\ndimensional simulations of wire array Z-pinches. In\nPlasma Physics and Controlled Fusion , volume 46, pages\nB457{B476. doi:10.1088/0741-3335/46/12B/039. URL\nhttp://stacks.iop.org/0741-3335/46/i=12B/a=039?key=\ncrossref.db05b373851fc67f42aa76a38351c1e0 .\n45A. Ciardi, S. V. Lebedev, A. Frank, E. G. Blackman, J. P. Chit-\ntenden, C. J. Jennings, D. J. Ampleford, S. N. Bland, S. C.\nBott, J. Rapley, G. N. Hall, F. A. Suzuki-Vidal, A. Marocchino,\nT. Lery, and C. Stehle. The evolution of magnetic tower\njets in the laboratory. In Physics of Plasmas , volume 14.\ndoi:10.1063/1.2436479. URL http://arxiv.org/abs/astro-ph/\n0611441 .\n46J. R. Davies, D. H. Barnak, R. Betti, E. M. Campbell, P.-Y.\nChang, A. B. Sefkow, K. J. Peterson, D. B. Sinars, and M. R.\nWeis. Laser-driven magnetized liner inertial fusion. 24(6):062701,\n. ISSN 1070-664X. doi:10.1063/1.4984779. URL https://aip.\nscitation.org/doi/full/10.1063/1.4984779 .\n47J. R. Davies, R. Betti, P.-Y. Chang, and G. Fiksel.\nThe importance of electrothermal terms in Ohm's law for\nmagnetized spherical implosions. 22(112703), . doi:\n10.1063/1.4935286. URL http://scitation.aip.org/content/\naip/journal/pop/22/11/10.1063/1.4935286 .\n48J. R. Davies, H. Wen, Jeong-Young Ji, and Eric D. Held. Trans-\nport coe\u000ecients for magnetic-\feld evolution in inviscid mag-\nnetohydrodynamics. 28(1):012305, . ISSN 1070-664X. doi:\n10.1063/5.0023445. URL https://doi.org/10.1063/5.0023445 .\n49E. M. Epperlein and M. G. Haines. Plasma transport co-\ne\u000ecients in a magnetic \feld by direct numerical solution of\nthe Fokker{Planck equation. 29(4). doi:10.1063/1.865901.\nURL http://scitation.aip.org/content/aip/journal/pof1/\n29/4/10.1063/1.865901 .\n50W. A. Farmer, J. M. Koning, D. J. Strozzi, D. E. Hinkel, L. F.\nBerzak Hopkins, O. S. Jones, and M. D. Rosen. Simulation of\nself-generated magnetic \felds in an inertial fusion hohlraum en-\nvironment. 24(5):052703{052703. doi:10.1063/1.4983140. URL\nhttp://aip.scitation.org/doi/10.1063/1.4983140 .\n51L. Gao, P. M. Nilson, I. V. Igumenshchev, M. G. Haines,\nD. H. Froula, R. Betti, and D. D. Meyerhofer. Precision Map-\nping of Laser-Driven Magnetic Fields and Their Evolution in\nHigh-Energy-Density Plasmas. 114(21):215003{215003. doi:\n10.1103/PhysRevLett.114.215003. URL https://link.aps.org/\ndoi/10.1103/PhysRevLett.114.215003 .\n52M. R. Gomez, S. A. Slutz, C. A. Jennings, D. J. Ampleford,\nM. R. Weis, C. E. Myers, D. A. Yager-Elorriaga, K. D. Hahn,\nS. B. Hansen, E. C. Harding, A. J. Harvey-Thompson, D. C.\nLamppa, M. Mangan, P. F. Knapp, T. J. Awe, G. A. Chandler,\nG. W. Cooper, J. R. Fein, M. Geissel, M. E. Glinsky, W. E.\nLewis, C. L. Ruiz, D. E. Ruiz, M. E. Savage, P. F. Schmit, I. C.\nSmith, J. D. Styron, J. L. Porter, B. Jones, T. R. Mattsson,\nK. J. Peterson, G. A. Rochau, and D. B. Sinars. Performance\nScaling in Magnetized Liner Inertial Fusion Experiments. 125\n(15):155002. doi:10.1103/PhysRevLett.125.155002. URL https:\n//link.aps.org/doi/10.1103/PhysRevLett.125.155002 .\n53M G Haines. Heat \rux e\u000bects in Ohm's law. In Plasma\nPhysics and Controlled Fusion , volume 28, pages 1705{1716. doi:10.1088/0741-3335/28/11/007.\n54D. W. Hill and R. J. Kingham. Enhancement of Pressure\nPerturbations in Ablation due to Kinetic Magnetised Trans-\nport E\u000bects under Direct-Drive ICF relevant conditions. URL\nhttp://arxiv.org/abs/1712.02663 .\n55Ridgers C. Kingham R. J. Walsh C. A. Hill, D. W. Vlasov-\nfokker-planck simulations of pre-magnetised ablating planar tar-\ngets. Submitted.\n56M Hohenberger, P. Chang, G. Fiksel, J. P. Knauer, R. Betti,\nF. J. Marshall, F. Seguin, C. K. Li, M. J E Manuel, and R D\nPetrasso. Magnetic Flux Compression Experiments on OMEGA.\n57I. V. Igumenshchev, A. B. Zylstra, C. K. Li, P. M. Nilson,\nV. N. Goncharov, and R. D. Petrasso. Self-generated mag-\nnetic \felds in direct-drive implosion experiments. 21(6). doi:\n10.1063/1.4883226.\n58A S Joglekar, A G R Thomas, W Fox, and A Bhat-\ntacharjee. Magnetic reconnection in plasma under in-\nertial con\fnement fusion conditions driven by heat \rux\ne\u000bects in Ohm's law. 112(10):105004{105004. doi:\n10.1103/PhysRevLett.112.105004. URL http://journals.aps.\norg/prl/abstract/10.1103/PhysRevLett.112.105004 .\n59C. K. Li, V. T. Tikhonchuk, Q. Moreno, H. Sio, E. D'Humi\u0012 eres,\nX. Ribeyre, Ph. Korneev, S. Atzeni, R. Betti, A. Birkel, E. M.\nCampbell, R. K. Follett, J. A. Frenje, S. X. Hu, M. Koenig,\nY. Sakawa, T. C. Sangster, F. H. Seguin, H. Takabe, S. Zhang,\nand R. D. Petrasso. Collisionless shocks driven by supersonic\nplasma \rows with self-generated magnetic \felds. 123(5):055002.\ndoi:10.1103/PhysRevLett.123.055002. URL https://link.aps.\norg/doi/10.1103/PhysRevLett.123.055002 .\n60M. J E Manuel, M. Flaig, T. Plewa, C. K. Li, F. H. S\u0013 eguin,\nJ. A. Frenje, D. T. Casey, R. D. Petrasso, S. X. Hu, R. Betti,\nJ. Hager, D. D. Meyerhofer, and V. Smalyuk. Collisional e\u000bects\non Rayleigh-Taylor-induced magnetic \felds. In Physics of Plas-\nmas, volume 22, . doi:10.1063/1.4919392.\n61M. J E Manuel, C. K. Li, F. H. S??guin, J. Frenje, D. T. Casey,\nR. D. Petrasso, S. X. Hu, R. Betti, J. D. Hager, D. D. Mey-\nerhofer, and V. A. Smalyuk. First measurements of Rayleigh-\nTaylor-induced magnetic \felds in laser-produced plasmas. 108\n(25):1{5, . doi:10.1103/PhysRevLett.108.255006.\n62L. J. Perkins, D. D.-M Ho, B. G. Logan, G. B. Zimmerman, M. A.\nRhodes, D. J. Strozzi, D. T. Black\feld, and S. A. Hawkins. The\npotential of imposed magnetic \felds for enhancing ignition prob-\nability and fusion energy yield in indirect-drive inertial con\fne-\nment fusion. 24(6):062708{062708. doi:10.1063/1.4985150. URL\nhttp://aip.scitation.org/doi/10.1063/1.4985150 .\n63James D. Sadler, Hui Li, and Kirk A. Flippo. Magnetic \feld\ngeneration from composition gradients in inertial con\fnement\nfusion fuel. 378(2184):20200045, . doi:10.1098/rsta.2020.0045.\nURL https://doi.org/10.1098/rsta.2020.0045 .\n64James D. Sadler, Hui Li, and Brian M. Haines. Magnetization\naround mix jets entering inertial con\fnement fusion fuel. 27(7):\n072707, . ISSN 1070-664X. doi:10.1063/5.0012959. URL https:\n//doi.org/10.1063/5.0012959 .\n65James D. Sadler, Christopher A. Walsh, and Hui Li. Symmetric\nSet of Transport Coe\u000ecients for Collisional Magnetized Plasma.\n126(7):075001, . doi:10.1103/PhysRevLett.126.075001. URL\nhttps://link.aps.org/doi/10.1103/PhysRevLett.126.075001 .\n66M. Sherlock and J. J. Bissell. Suppression of the Bier-\nmann Battery and Stabilization of the Thermomagnetic In-\nstability in Laser Fusion Conditions. 124(5):055001. doi:\n10.1103/PhysRevLett.124.055001. URL https://link.aps.org/\ndoi/10.1103/PhysRevLett.124.055001 .\n67S. A. Slutz, M. C. Herrmann, R. A. Vesey, A. B. Sefkow, D. B.\nSinars, D. C. Rovang, K. J. Peterson, and M. E. Cuneo. Pulsed-\npower-driven cylindrical liner implosions of laser preheated fuel\nmagnetized with an axial \feld. 17(5):056303. ISSN 1070-\n664X. doi:10.1063/1.3333505. URL https://doi.org/10.1063/\n1.3333505 .\n68C. A. Walsh. Extended Magneto-hydrodynamic E\u000bects in\nIndirect-Drive Inertial Con\fnement Fusion Experiments.11\n69C. A. Walsh and D. S. Clark. Biermann Battery Magnetic Fields\nin ICF Capsules: Total Magnetic Flux Generation.\n70C. A. Walsh, J. P. Chittenden, D. W. Hill, and C. Ridgers.\nExtended-magnetohydrodynamics in under-dense plasmas. 27\n(2):022103, . ISSN 1070-664X. doi:10.1063/1.5124144. URL\nhttps://aip.scitation.org/doi/10.1063/1.5124144 .\n71C A Walsh, K Mcglinchey, J K Tong, B D Appelbe, A Crilly,\nM Zhang, and J P Chittenden. Perturbation Modi\fcations by\nPre-magnetisation in Inertial Con\fnement Fusion Implosions.\n096:1{12, .\n72C.A. Walsh, J.P. Chittenden, K. McGlinchey, N.P.L. Ni-\nasse, and B.D. Appelbe. Self-Generated Magnetic Fields\nin the Stagnation Phase of Indirect-Drive Implosions on theNational Ignition Facility. 118(15):155001{155001, . doi:\n10.1103/PhysRevLett.118.155001. URL http://link.aps.org/\ndoi/10.1103/PhysRevLett.118.155001 .\n73C.A. Walsh, A.J. Crilly, and J.P. Chittenden. Magnetized\ndirectly-driven ICF capsules: Increased instability growth from\nnon-uniform laser drive. 60(10):106006, . ISSN 0029-5515. doi:\n10.1088/1741-4326/abab52. URL http://dx.doi.org/10.1088/\n1741-4326/abab52 .\n74H C Watkins and R J Kingham. Magnetised thermal self-\nfocusing and \flamentation of long-pulse lasers in plasmas rel-\nevant to magnetised ICF experiments. 25:92701{92701. doi:\n10.1063/1.5049229. URL https://doi.org/10.1063/1.5049229 ."    },    {        "title": "1608.05587v1.Vesicles_in_magnetic_fields.pdf",        "content": "arXiv:1608.05587v1  [cond-mat.soft]  19 Aug 2016Vesicles in magnetic fields\nDavid Salac\nDepartment of Mechanical and Aerospace Engineering,\nUniversity at Buffalo SUNY,\n318 Jarvis Hall, Buffalo, NY 14260-4400, USA\n716-645-1460\ndavidsal@buffalo.edu\nLiposome vesicles tend to align with an applied magnetic fiel d. This is due to the directional\nmagnetic susceptibility difference of the lipids which form t he membrane of these vesicles. In this\nwork a model of liposome vesicles exposed to magnetic field is presented. Starting from the base\nenergy of alipidmembraneina magnetic field, theforceappli ed tothesurroundingfluidsis derived.\nThis force is then used to investigate the dynamics of vesicl e in the presence of magnetic fields.\n1 Introduction\nKnowledge about the directed motion of biological and bio-c ompatible nano- and microstructures,\nparticularly liposome vesicles, is critically important f or a number of biotechnologies. For example,\nin directed drug delivery it is critical that drug carriers b e directed towards locations where they\nare to release an encapsulated drug.[1]The directed motion of biological cells could be used to sort\ncells[2–4]and to form larger structures.[5]Numerous techniques have been proposed as possibleways\nto control the motion of these soft-matter systems. Special ly designed microfluidic devices can use\ndifferences in size,[6]shape,[7]and rigidity[8]to physically separate particles. It is also possible to\nuse light[9]and acoustic waves[10]to induce particle motion.\nOf particular interest is the use of externally controllabl e fields to direct the motion of soft\nparticles. Electric fields have been demonstrated to induce large deformation in liposome vesicles\nboth experimentally[11;12]and theoretically.[13–16]Electric fields can be used to sort cells[17–19]and\nto induce the formation of pores in vesicle membranes.[20]\nMagnetic fields also offer the opportunity to direct the motion of biologically compatible soft-\nmatter. Experiments have demonstrated that liposome vesic les tend to align with an externally\napplied magnetic field.[21]The phospholipids which compose the vesicle membrane are kn own to\nbe diamagnetically anisotropic and tend to align perpendic ularly to an external field.[22]Due to\nthe nature of the liposome membrane this results in a rotatio nal/alignment force which aligns and\nstretches a vesicle parallel to the applied field, see Fig. 1. Due to the unique nature of vesicles\nthis stretching is balanced by an increase in surface tensio n and bending energy. Using this fact\nHelfrich developed a model for the deformation of a vesicle w hen exposed to a magnetic field to\ndetermine the flexibility of the vesicle membrane.[23;24]\nUnlike electric field effects, the influence of magnetic fields o n liposome vesicles has received\nmuch less attention. In addition to the work of Helfrich ment ioned above, it has been experimen-\ntally shown that liposomes made from dipalmitoylphosphati dyl choline (DPPC) have temperature\ndependent deformation and permeability when exposed to mag netic fields.[25]In the same work a\nsimple model was developed to explore the observed behavior . The fusion of liposome vesicles for\n1B\nFigure 1: To minimize the total energy lipids will orient per pendicularly to an applied magnetic\nfield. This results in a force which aligns the vesicle to the a pplied field.\na range of applied magnetic fields has also been experimental ly demonstrated[26]while others have\nverified the alignment of vesicles to the external magnetic fi eld.[27;28]Theoretical and computa-\ntional investigation of magnetohydrodynamics of liposome vesicles are much less common. Helfrich\ninvestigated the birefringence of vesicles in magnetic fiel ds[23]and very few investigations into the\nbiomechanics of vesicles in magnetic field have been perform ed.[29]\nTo the author’s knowledge this is the first effort to model the ge neral dynamics of vesicles when\nexposed to externally driven magnetic fields. In the remaind er of this work the governing equations\nand numerical methods used to model this system will be prese nted. Sample results of a vesicle in\nthe presence of magnetic fields will also be shown.\n2 Governing Equations\nConsider a vesicle suspended in a fluid and exposed to an exter nally driven magnetic field, Fig. 2.\nThe density, electrical and magnetic properties between th e inner and outer fluid are matched while\nthe viscosity may vary. The magnetic field in static in time bu t could be spatially varying. The size\nof the vesicle is on the order of 10 µm while the thickness of the membrane is approximately 5 nm,\nwhich allows for the modeling of the membrane as a thin interf ace separating two fluids.[30]The\nvesicle membrane is impermeable to fluid molecules and the nu mber of lipids in a vesicle membrane\ndoes not change over time while the surface density of lipids at room temperature is constant.[30]\nThese conditions result in an inextensible membrane with co nstant enclosed volume and global\nsurface area, in addition to local surface area incompressi bility.\nFor any multiphase fluid system the time-scale associated wi th charges migrating towards the\ninterfaceisgivenbythechargerelaxation time, tc=ǫ/σ, whereǫandσarethethefluidpermittivity\nand conductivity, respectively.[31;32]Typical values for these in experimental vesicles investig ations\nareǫ≈10−9F/m and s≈10−3S/m.[14;33;34]This results in a charge relaxation time of tc≈10−6\ns, which is much faster than observed dynamics of vesicles wh en exposed to magnetic fields.[21]\nIt is thus valid to assume that there are no free charges in the bulk fluids and thus the leaky-\ndielectric model can be assumed.[31]This lack of free charges in the bulk fluids has implications\nwhen considering the forces on the fluid. If there are free-ch arges in the fluid then the Lorentz force\nwill drag the fluid into motion, which is common for magnetohy drodynamics using conducting\nfluids.[35–37]In the absence of free charges the Lorentz force can be ignore d in the bulk fluid and\ntherefore fluid will be driven into motion only by conditions at the membrane.\nIn general the applied magnetic field and induced electric fie ld are coupled through Maxwell’s\nequations. Under the assumption that the magnetic field is st atic in time the only possible cou-\npling between the electric field and the magnetic field is thro ugh the electric current density,\nj=σ(−∇Φ+u×B), where Φ is the induced electric potential, uis the fluid velocity and B\nis the applied magnetic field. As the induced electrical curr ent must divergence free, ∇ ·j= 0,\n2B\nm+m-\nW+W-qBqvn\nFigure 2: Schematic of a vesicle exposed to an externally dri ven magnetic field. The magnetic field\nBis at an angle of θBfrom the x-axis while the inclination angle of the vesicle is given by θv.\nThe inner, µ−, and outer, µ+, fluid viscosity may differ. The outward unit normal vector to t he\ninterface is given by n.\nany induced electric potential will obey ∇·(σ∇Φ) =∇·(σu×B). Experimental investigations of\nvesicles in a 1.5 T magnetic field demonstrate that responses take on the order of 10 s.[21]Assuming\nthat the distance traveled during this time is 20 µm this results in a velocity of 2 ×10−6m/s. In\nthe absence of an external electric field this results in an in duced electric current density of 3 ×10−9\nA/m2. At such small induced current densities the induced electr ic field will be much smaller than\nthose needed to induce vesicle deformation.[11;13;14;38]From this analysis the induced electric field\nand it’s contribution to the dynamics of the vesicle will be i gnored.\n2.1 Forces Exerted by the Membrane\nThe motion of the fluid will be driven by the conditions and for ces at the vesicle membrane. Let the\nlipid membrane be given by Γ. The total energy of the membrane is composed of four components:\nE[Γ] =Ek[Γ]+Eγ[Γ]+Em,bulk[Γ]+Em,rot[Γ], (1)\nEk[Γ] =/contintegraldisplay\nΓ/parenleftbigg1\n2kcH2+kgK/parenrightbigg\ndA, (2)\nEγ[Γ] =/contintegraldisplay\nΓγ dA, (3)\nEm,bulk[Γ] =−χ⊥d\n2µm/contintegraldisplay\nΓB2dA, (4)\nEm,rot[Γ] =−∆χd\n2µm/contintegraldisplay\nΓ(n·B)2dA. (5)\nThe first integral, Ek[Γ], provides the bending energy associated with the curren t membrane con-\nfiguration where κcis the bending rigidity, κgis the Gaussian bending rigidity, Kis the Gaussian\ncurvature, and His the total curvature, which equals the sum of the principle curvatures. In this\nwork the Gaussian curvature energy is ignored as the integra l of the Gaussian curvature around\nany closed surface is a constant.[39]The second integral, Eγ[Γ], is the energy associated with a\nnon-uniform tension, γ. These two energies are based on the Helfrich model in the abs ence of\nspontaneous curvature[24]and have been used extensively to model liposome vesicles.[14;40–42]\n3The final two integrals, Em,bulk[Γ] andEm,rot[Γ], provide the energy of a lipid membrane in\na magnetic field Bwhen the outward unit normal to the interface is given by nandB·B=\nB2.[22;43]The first energy, Em,bulk, provides the total (bulk) energy of a membrane with a magnet ic\nsusceptibility perpendicular to a lipid axis given by χ⊥, a membrane thickness of d, and where\nµmis the magnetic permeability of the membrane. As lipid molec ules are diamagnetic materials,\nχ⊥<0. The second, Em,rot, is the magnetic alignment energy, with ∆ χ=χ/bardbl−χ⊥being the\ndifference between the magnetic susceptibilities in the para llel and perpendicular direction for lipid\nmolecules. For lipid molecules, the perpendicular magneti c susceptibility is larger (less negative)\nthan the parallel one and thus ∆ χ <0, although it is possible to change this by adding biphenyl\nmoieties to the phospholipids.[44]This magnetic susceptibility difference drives the lipid mol ecules\nto become perpendicular to an applied magnetic field, which i n turn causes the lipid vesicle itself\nto align with the field.[21]\nInordertominimizetheenergy, themembranewillexert afor ceonthesurroundingfluid. These\nforces are calculated by taking the variation of the appropr iate membrane energy with respect to\na change of membrane location. For a vesicle and neglecting s pontaneous curvature the ultimate\nforms of the bending and tension forces are found to be[45]\nτk=−κc/parenleftbigg1\n2H3−2HK+∇2\nsH/parenrightbigg\nn, (6)\nτγ=γHn−∇sγ. (7)\nThe surface gradient, ∇s, and surface Laplacian, ∇2\ns, are defined using the projection operator\nP=I−n⊗n. More precisely, the surface gradient of a scalar field is giv en by∇sf=P∇fwhile\nthe surface Laplacian is ∇2\nsf=∇s·∇sf.\nThe magnetic force has not been presented in the literature a nd is derived here. In particular,\nthe method outlined by Napoli and Vergori is used to determin e the variation of the energy.[46]\nLet a generic energy functional be given by/contintegraltext\nΓw dA, wherewis an arbitrary energy functional\ndensity per unit area which only a function of the unit normal nand no other surface quantities.\nThe first variation of the energy with respect to a change of th e interface is then given by ∇s·\n(wP−n⊗(Pwn)), where wn=∂w/∂nis the derivative of the energy density with respect to the\nunit normal.\nAssume that a single lipid species is present. Therefore the material properties χ⊥, ∆χ,d, and\nµmare all constants. First consider the contribution of the bu lk energy of the lipid membrane in\na magnetic field,/contintegraltext\nΓB2dA. In this case w=B2and thus wn= 0. Using the results shown in\nAppendix A, this results in\n∇s·/parenleftbig\nB2P/parenrightbig\n=∇sB2−B2Hn. (8)\nNext consider the rotational energy contribution,/contintegraltext\nΓ(n·B)2dA. Fromw= (n·B)2=B2\nnthe\nquantity wn= 2(n·B)B= 2BnBis obtained. Thus,\n∇s·/parenleftbig\nB2\nnP−n⊗(2BnPB)/parenrightbig\n=∇s·/parenleftbig\nB2\nnP/parenrightbig\n−2∇s·(n⊗(BnPB)) (9)\nwhereBn=n·Bis the portion of the magnetic field in the normal direction. E ach of these\ncomponents are considered in turn. The first component resul ts in\n∇s·/parenleftbig\nB2\nnP/parenrightbig\n=∇sB2\nn−B2\nnHn= 2Bn∇sBn−B2\nnHn. (10)\nThe second component is\n∇s·(n⊗(BnPB)) =Bn(∇sn)PB+n∇s·(BnPB)\n=Bn(∇sn)B+n/parenleftbig\nB·∇sBn−B2\nnH+Bn∇s·B/parenrightbig\n=BnLB+n/parenleftbig\nB·∇sBn−B2\nnH+Bn∇s·B/parenrightbig\n, (11)\n4where the surface gradient of the unit normal, ∇sn=L, is called the curvature tensor, or shape\noperator, of theinterface. It is a symmetricand real matrix which characterizes thecurvatureof the\nsurface.[46–48]Oneeigenvalue of Lis zero andhas a correspondingeigenvector in the direction of the\nunit normal, n. AsLis symmetric and real, it can be decomposed as L=κtt⊗t+κbb⊗b, where\nκtandκbare the remaining eigenvalues of Lwith corresponding eigenvectors tandb, respectively.\nIn this case the eigenvalues are principle curvatures of the interface while the eigenvectors are the\nprinciple tangent directions. Thus, the first part of Eq. (11 ) can be written as\nBnLB=Bn(κtt⊗t+κbb⊗b)B=Bn(κtBtt+κbBbb), (12)\nwhereBt=t·BandBb=b·Bare the components of the magnetic field in the principle dire ctions.\nCombining the results of Eqs. (9)-(12) and simplifying resu lts in\n∇s·/parenleftbig\nB2\nnP−n⊗(2BnPB)/parenrightbig\n=\n2Bn∇sBn+B2\nnHn−2Bn(κtBtt+κbBbb)−2n(B·∇sBn)−2Bnn∇s·B.(13)\nUsing these results the force due to the magnetic field is then\nτm,bulk=−δEm,bulk\nδΓ=χ⊥d\n2µm/parenleftbig\n∇sB2−B2Hn/parenrightbig\n, (14)\nτm,rot=−δEm,rot\nδΓ=∆χd\n2µm/parenleftbig\n2Bn∇sBn+B2\nnHn\n−2Bn(κtBtt+κbBbb)−2n(B·∇sBn)−2Bnn∇s·B), (15)\nrecalling that B2=B·B,Bn=n·B,Bt=t·B, andBb=b·B.\nFor general situations, the above expressions works well. W hen the magnetic field is spatially\nconstant simplifications can be made by expanding the ∇sBn=∇s(n·B) terms:\n∇sBn=∇s(n·B) =B·∇sn+n·∇sB=B·L+n·∇sB=B·L=LB,(16)\nas∇sB= (∇B)P=0andL=LT. Beginning with Eq. (15) and using the LBform this results\nin\nτm,rot=∆χd\n2µm/parenleftbig\n2Bn∇sBn+B2\nnHn−2BnLB−2n(B·∇sBn)−2Bnn∇s·B/parenrightbig\n=∆χd\n2µm/parenleftbig\n2BnLB+B2\nnHn−2BnLB−2n(B·LB)/parenrightbig\n=∆χd\n2µm/parenleftbig\nB2\nnH−2B·LB/parenrightbig\nn\n=∆χd\n2µm/parenleftbig\nB2\nnH−2B·(κtBtt+κbBbb)/parenrightbig\nn\n=∆χd\n2µm/parenleftbig\nB2\nnH−2κtB2\nt−2κbB2\nb/parenrightbig\nn, (17)\ndue to the fact that ∇s·B=P:∇B= 0 when Bis spatially constant.\n2.2 Interface Description\nIn this work a level-set formulation is used to describe the l ocation of the vesicle membrane. Let\nthe evolving interface be given as the set of points where a le vel-set function φ(x,t) is zero: Γ( t) =\n5{x:φ(x,t) = 0}, wherexis a position in space and tis time. Instead of explicitly tracking the\nlocation of the interface Γ through time the position is impl icitly tracked by evolving φ. Following\nconvention the inner fluid, Ω−, is given by the region φ <0 while the outer fluid, Ω+, is given by\nφ >0. The entire domain is denoted as Ω = Ω−∪Ω+. Using the level-set geometric quantities are\neasily obtained. For example, the outward facing unit norma l vector and the total curvature (sum\nof principle curvatures) is given by\nn=∇φ\n/ba∇dbl∇φ/ba∇dbl, (18)\nH=∇·n. (19)\nIt is also possible to use the level set function to define vary ing material parameters using a single\nrelation. Consider the determination of the viscosity at an y location in the domain. Letting\nµ−be the inner viscosity and µ+be the outer viscosity the viscosity at a point xis given by\nµ(x) =µ−+ (µ+−µ−)H(φ(x)), where His the Heaviside function. Similar expressions hold\nfor other material quantities. In practice a smoothed versi on of the Heaviside function is used to\nensure numerical stability.[49]Finally, motion of the interface is obtained by advecting th e level set\nfunction,\n∂φ\n∂t+u·∇φ= 0. (20)\nDetails of the numerical implementation will be presented l ater.\n2.3 Fluid Flow Equations\nDefine the bulk fluid hydrodynamic stress tensor in each fluid a s\nT±\nhd=−p±I+µ±(∇u±+∇Tu±) in Ω±. (21)\nThe forces derived in Section 2.1 are balanced by a jump in the fluid stress tensor,\nn·(T+\nhd−T−\nhd) =τk+τγ+τm,bulk+τm,rot. (22)\nNote that in general there is a contribution from a jump in the Maxwell stress tensor acting on\nthe interface. In the absence of electric fields and with matc hed magnetic fluid properties this\ncontribution is zero and thus is not included.\nUsing the level set formulation it is possible to write the flu id momentum equations and the\ninterface force balance as a single equation valid over the e ntire domain:[50;51]\nρ(φ)Du\nDt=−∇p+∇·/parenleftbig\nµ(φ)/parenleftbig\n∇u+∇Tu/parenrightbig/parenrightbig\n+δ(φ)κc/parenleftbiggH3\n2−2KH+∇2\nsH/parenrightbigg\n∇φ\n+δ(φ)/ba∇dbl∇φ/ba∇dbl(∇sγ−γHn)\n−δ(φ)/ba∇dbl∇φ/ba∇dblχ⊥d\n2µm/parenleftbig\n∇sB2−B2Hn/parenrightbig\n−δ(φ)/ba∇dbl∇φ/ba∇dbl∆χd\n2µm/parenleftbig\n2Bn∇sBn+B2\nnHn\n−2Bn(κtBtt+κbBbb)−2n(B·∇sBn)−2Bnn∇s·B), (23)\n6where the full form of the force, Eqs. (14) and (15), have been used. The use of the Dirac function\nδ(φ) localizes the contributions from the membrane forces near theφ= 0 contour, which is the\nlocation of the interface.[52]Volume and surface area conservation are provided by ensuri ng that\n∇·u= 0 in Ω , (24)\n∇s·u= 0 on Γ . (25)\nNote that in the fluid formulation, Eq. (23), the tension is us ed to enforce surface area constraint\n∇s·u= 0 and is computed as part of the problem alongside the pressu re.\n2.4 Nondimensional Parameters and Equations\nAssume that the density is matched between the inner and oute r fluids while the viscosity has a\nratio ofη=µ−/µ+. Each of the forces acting on the vesicle membrane have an ass ociated time-\nscale which depends on the material properties. The time-sc ale associated with the bending of the\nmembrane is[15]\ntk=µ+(1+η)l3\n0\nκc, (26)\nwherel0is the characteristic length scale.\nThe magnetic field introduces two times scales, one associat ed with each component of the\nmagnetic field energy, Eqs. (4) and (5). In both case the magne tic forces are compared to the\nviscous forces. The first is the time scale of the bulk energy,\ntm,bulk=µ+(1+η)l0µm\n|χ⊥|dB2\n0, (27)\nwhile the magnetic rotation time scale is\ntm,rot=µ+(1+η)l0µm\n|∆χ|dB2\n0, (28)\nwhereB0is the characteristic magnetic field strength and recalling thatµmis the magnetic per-\nmeability of the membrane.\nLet the characteristic time be given by t0, which allows for the definition of the characteristic\nvelocity: u0=l0/t0. Define a capillary-like number providing the relative stre ngth of the bend-\ning, Ca = tk/t0, a magnetic Mason number indicating the strength of the bulk motion, Mn =\nsgn(χ⊥)tm,bulk/t0, and a magnetic field induced rotational force number, Rm = sg n(∆χ)tm,rot/t0,\nwhile the Reynolds number is given by Re = ρu0l0/µ+=ρ l2\n0/(µ+t0). The use of sgn( χ⊥) and\nsgn(∆χ) takes into account the fact that the values of χ⊥and ∆χcan be either positive or negative.\nTherefore, the dimensionless parameters Mn and Rm can eithe r be positive or negative, depending\non if the membrane is a paramagnetic or diamagnetic material . It is then possible to write the\n7dimensionless fluid equations as\nDu\nDt=−∇p+1\nRe∇·/parenleftbig\nµ(φ)/parenleftbig\n∇u+∇Tu/parenrightbig/parenrightbig\n+1\nCa Reδ(φ)/parenleftbiggH3\n2−2KH+∇2\nsH/parenrightbigg\n∇φ\n+δ(φ)/ba∇dbl∇φ/ba∇dbl(∇sγ−γHn)\n−1\n2 Re Mnδ(φ)/ba∇dbl∇φ/ba∇dbl/parenleftbig\n∇sB2−B2Hn/parenrightbig\n−1\n2 Re Rmδ(φ)/ba∇dbl∇φ/ba∇dbl/parenleftbig\n2Bn∇sBn+B2\nnHn−2Bn(κtBtt+κbBbb)\n−2n(B·∇sBn)−2Bnn∇s·B), (29)\nwhere all quantities are now dimensionless and the viscosit y at a point can be calculated using\nµ(φ) =η+(1−η)H(φ).\nThe energy is normalized by using the bending rigidity, κc, as the characteristic energy scale.\nWhen writing the normalized total energy, the contribution from the tension, Eq. (3), is not\nincluded as it is a co-dimension one parameter used to enforc e surface incompressibility. Therefore,\nthe normalized energy of the system is then\nE[Γ] =1\n2/contintegraldisplay\nΓH2dA−1\n2Ca\nMn/contintegraldisplay\nΓB2dA−1\n2Ca\nRm/contintegraldisplay\nΓ(n·B)2dA. (30)\nIt is useful to compare the time-scales associated with a typ ical experiment. Boroske and\nHelfrich reported the alignment of vesicles in magnetic fiel ds with a strength of 1.5 T and reported\nthat the magnetic susceptibility difference to be −3.52×10−8in SI units, assuming a membrane\nthickness of 6 nm.[21]Assuming the viscosity is matched and equal to water, µ+= 10−3Pa s, the\nmembrane magnetic permeability is that of free-space, µm= 4π×10−7H/m, and a characteristic\nlength of 10 µm (as estimated by the figures in Boroske and Helfrich) the mem brane rotation time\nistm,rot≈53 s. It was reported by Boroske and Helfrich that rotation th rough an angle of π/2\ntook approximately 100 s, which matches well with the charac teristic time calculated here. Using\nthe same values and given that the bending rigidity for the sy stem is approximately 10−19J,[53]\nthe time-scale associated with bending is 20 s, which agrees with the experimental results as no\ndeformation of the membrane was observed during rotation.\n3 Numerical Methods\nTwo numerical methods must be discussed. The first is the adve ction of the level set field. In\nthis work a new semi-implicit level set Jet scheme is used.[54]In addition to the level set function\nthe gradient of the level set are also tracked, which increas es the accuracy of the method.[55]The\nextension allows the original level set Jet scheme to be used for stiff advection problems. It is\ncomposed of three main steps. First, the level set field is adv anced using a second-order, semi-\nimplicit, semi-Lagrangian update,\n3φn+1−2φn\nd+φn−1\nd\n2∆t=β∇2φn+1−β∇2ˆφ, (31)\nwhereβ= 0.5 is a constant, φn\ndis the departure value of the level set at time tn,φn−1\ndis the\ndeparture value of the level set at time tn−1, andˆφ= 2φn−φn−1is an approximation of the\n8level set value at time tn+1. Once the smooth level set field is obtained the effect of smooth ing is\ncaptured by defining a source term,\nSφ=β∇2/parenleftBig\nφn+1−ˆφ/parenrightBig\n. (32)\nThis advection source term is used to update the level set val ues on a sub-grid which surrounds all\ngrid points,\n3φs,n+1−2φs,n\nd+φs,n−1\nd\n2∆t=Sφ. (33)\nUsing these updated sub-grid level set values, φs,n+1, finite difference approximations are used to\ncalculate the updated gradient field. It was shown that this m ethod results in an accurate and\nstable scheme for the modeling of moving interfaces under st iff advection fields.[54]\nThe fluid field is obtained using a projection-based method.[51]The first step is to calculate a\ntentative field using a semi-implicit, semi-Lagrangian met hod:\n3u∗−2un\nd+un−1\nd\n2∆t=−∇ˆp+1\nRe∇·/parenleftbig\nµ/parenleftbig\n∇u∗+∇Tˆu/parenrightbig/parenrightbig\n+fn\nk+fn\nγ+fn\nm,bulk+fn\nm,rot, (34)\nwherefn\nk,fn\nγandfn\nm,bulk+fn\nm,rotare the bending, tension, and magnetic forces while un\ndand\nun−1\ndare the departure velocities and ˆ p= 2pn−pn−1is an extrapolation of the pressure to time\ntn+1. The next step is to calculate the corrections to the pressur e and tension to enforce volume\nand surface area conservation,\n3\n2un+1−u∗\n∆t=−∇q+δ(φ)/ba∇dbl∇φ/ba∇dbl(∇sξ−ξH∇φ), (35)\nwhereqandξare the corrections needed for the pressure and tension, res pectively. The pressure\nand tension are computed simultaneously to enforce both loc al and global conservation of the\nenclosed volume and surface area. Complete details of the me thod are provided in Kolahdouz and\nSalac.[51]\n4 Results\nThe experimental results most applicable are those of Boros ke and Helfrich.[21]As such the results\npresented here will be modeled after those experiments. The characteristic time is chosen to be\nt0= 1 s. Assuming a bending rigidity of κc≈25kBT≈10−19J,[56]a vesicle radius of 10 µm, fluid\ndensity of 1000 kg/m3, and outer fluid viscosity of 10−3Pa s with matched viscosity ( η= 1) the\ncapillary-like number becomes Ca = 20 while the Reynolds is R e = 10−4.\nUsing a magnetic susceptibility difference on the order of ∆ χ=−3×10−8in SI units, with a\nmembrane magnetic permeability of µm= 4π×10−7H/m and membrane thickness of 6nm, the\nmagnetic rotation constant scales as Rm ∼ −140/B2\n0. Assuming a magnetic field strength between\nB0= 1 T and B0= 10 T, this results in a range of −150/lessorsimilarRm/lessorsimilar−1.\nThe magnetic susceptibility perpendicular to the lipid axi s,χ⊥, is not readily available in the\nliterature, and thus it is not clear what the magnitude shoul d be. In spatially constant magnetic\nfield the bulk magnetic energy contribution, Eq. (4), is cons tant for incompressible membranes\nand thus the bulk magnetic force, Eq. (14), does not need to be included. For spatially varying\nmagnetic fields it is assumed that the magnetic susceptibili ty perpendicular is of the same order as\n9the magnetic susceptibility difference and therefore the mag netic Mason number will be taken to\nbe−100/lessorsimilarMn/lessorsimilar−1\nDue to the long simulation times, and to facilitate a larger n umber of trials, the results will\nbe in the two-dimensional regime, resulting in zero Gaussia n curvature: K= 0. Unless otherwise\nstated, the computational domain is a square spanning [ −6.4,6.4]2using a 2572grid and periodic\nboundary conditions while the time step is fixed at ∆ t= 0.1h, whereh= 12.8/256 = 0.05 is the\ngrid spacing. The choice of this domain size and time step is j ustified in Sec. 4.2.\nVesicles are be characterized by several parameters. Speci fically, the viscosity ratio η=µ−/µ+,\nthe inclination angle θv, the deformation parameter D, and the reduced area ν. Inclination angles\nare determined by calculating the eigenvalues and eigenvec tors of the vesicle’s inertia tensor about\nits center of mass. The eigenvector corresponding to the lar ger of the two eigenvalues provides the\ndirection of the long axis of a vesicle. The angle between the eigenvector associated with the long\naxis of the vesicle and the x-axis is denoted as the inclination angle. The deformation p arameter\nis given by D= (a−b)/(a+b), where aandbare the long and short axes of an ellipse with the\nsame inertia tensor as the vesicle.[57;58]The vesicle reduced area indicates how deflated a vesicle\nis compared to a circle with the same interface length, and is given by ν= 4Aπ/L2, whereAand\nLare the enclosed area and interface length, respectively. A value of ν= 0.5 indicates that the\nenclosed area is one-half of a circle with the same interface length while ν= 1 denotes a circle.\nAll simulations begin with an ellipse having an interfacial length of 2 πand a reduced area of\nν= 0.71. This reduced area was estimated from Boroske and Helfric h, Fig. 1.[21]The vesicle is\nthen allowed to evolve in the absence of a magnetic field to obt ain a shape near the bending energy\nminimum.[59]This shape is then used as the initial condition for the magne tically driven results.\nThe initial orientation of all vesicles is vertical, which i s denoted as having an inclination angle of\nθv=π/2, see Fig. 2. It is also assumed throughout that the viscosit y is matched, η= 1.\n4.1 Direct comparison with Boroske and Helfrich\nTo provide the reader a better understanding of the vesicle d ynamics, a direct comparison with\nthe results of Boroske and Helfrich is performed.[21]Using Fig. 1 from that manuscript, it was\nestimated that the angle between the vesicle and the applied magnetic field is 0 .455π. In the\nsimulation, the vesicle is initially aligned with the verti cal axis and the magnetic field has an angle\nofθB= 0.045π, which matches the conditions of the experiment. As the magn etic field is spatially\nconstant the bulk magnetic energy is ignored while the dimen sionless magnetic rotation constant\nis set to Rm = −7.5. The inclination angle up to a time of 200 is shown in Fig. 3 wh ile the\ncomputationally derived vesicle shapes are compared to the experimental result in Fig. 4.\nThecomputationalresultsmatchverywellwiththeexperime ntalresults. Assumingamembrane\nthickness of d= 6 nm, the properties of water, and an applied magnetic field s trength of B0= 1.5\nT, and using the value of Rm = −7.5, the magnetic susceptibility difference is calculated to be\n∆χ=−2.48×10−7. While this value is larger than that estimated by Boroske an d Helfrich, it is\nwithin other experimentally determined values.[60]\n4.2 Verification of domain parameters\nTo verify the choice of domain size, grid size, and time step a systematic investigation is performed\nby varying each simulation parameter individually. The mag netic field is spatially constant and\nfixedat anangleof θB= 0.045π. Threemagneticrotation constants usedareRm = −1, Rm = −10,\nand Rm = −100. As the magnetic field is spatially constant, the bulk mag netic field contribution\nis neglected.\n10Time0 50 100 150 200Angle\n00.20.40.60.811.21.41.6\nFigure 3: The inclination angle as a function of time for a ves icle with reduced area of ν= 0.71 in\na magnetic field at an angle of 0 .045πwith a rotation constant of Rm = −7.5. The dots indicate\nthe angles determined from Fig. 1 of Boroske and Helfrich,[21]after an appropriate rotation is done\nto take into account the different initial angles.\nFigure 4: Comparison between experimental results of Boros ke and Helfrich and the simulation.\nDue to the different initial orientation, the simulation resu lts are first flipped about the horizontal\naxis and then rotated 0 .045πcounter-clockwise. Reprinted from Biophysical Journal, V ol 24 (3),\nBoroske and Helfrich, “Magnetic anisotropy of egg lecithin membranes”, Pages 863-868., December\n1978, with permission from Elsevier.\n11Time0 40 80 120 160 200Angle\n00.20.40.60.811.21.41.6\n1292\n1932\n2572\n3852\n5132Rm=-10\nRm=-1Rm=-100\nGrid Size\nFigure5: Theinclinationangleversustimeformagneticrot ationstrengthsofRm = −1, Rm = −10,\nandRm = −100andgridsizesrangingfrom1292to5133. Allresultsuseadomainsizeof[ −6.4,6.4]2\nwith a time step of ∆ t= 0.1h, wherehis the grid spacing. No change in the results are seen past\na grid size of 2572.\nFirst consider the influence of the grid size on the results. U sing a [−6.4,6.4]2domain, grid\nsizes ranging from 1292to 5132are used. In all cases the time step is set to ∆ t= 0.1h, whereh\nis the grid spacing. The results shown in Fig. 5 indicate that a grid size of 1292is not sufficient.\nThis shouldn’t be surprising as with this grid spacing only a pproximately 8 grid points are used\nto describe the vesicle at it’s narrowest point. The differenc e in the results using more than a grid\nsize of 2572are not noticeable for any of the three magnetic rotation con stants, which justifies that\nparticular choice.\nNext consider theinfluenceof domainsize on therotation dyn amics. Usingaconstant gridspac-\ning ofh= 0.05 and time step of ∆ t= 0.1h, various domain sizes from [ −2.4,2.4]2to [−8.0,8.0]2are\nconsidered, see Fig. 6. Clearly, boundary effects are present in the smallest domains, particularly\nwhen Rm = −10. Once the domain size reaches [ −6.4,6.4]2, only small differences are observed.\nFinally consider the influence of the time step on the rotatio n dynamics. Using a [ −6.4,6.4]2\ndomain with 2572grid points, various time steps from ∆ t= 0.02hto ∆t= 0.5hare considered.\nNote that using time steps of ∆ t=hproved unstable. There are almost no differences using time\nsteps smaller than ∆ t= 0.1h, and thus that is the time step chosen for further results.\n4.3 Influence of Rm\nThe influence of the magnetic rotation force is explored by va rying Rm within the range from 1 to\n100 up to a time of t= 200. The resulting inclination angle over time is shown in F ig. 8(a), while\nthe amount of time needed to rotate through an angle of 0 .05π, 0.25π, and 0.4πis shown in Fig.\n8(b). There are several points to be made. First, the equilib rium angle of the vesicle, given enough\ntime, will match that of the applied magnetic field. Second, t he amount of time that is required to\nrotate through a particular angle is linearly dependent on t he Rm value. It should be noted that\ndue to the definition of Rm, this is related to the quadratic of the magnetic field strength, i.e. a\n2-fold increase in the magnetic field results in a 4-fold decr ease of the Rm parameter. Therefore,\nincreasing the magnetic field strength by a factor of two redu ces the amount of time needed to\nrotate by a factor of four.\nAn investigation of the energy for three characteristic rot ation strengths, Rm = −1, Rm = −10,\n12Time0 40 80 120 160 200Angle\n00.20.40.60.811.21.41.6\nRm=-10\nRm=-1Rm=-100\n[-2.4, 2.4]2\n[-3.2, 3.2]2\n[-4.8, 4.8]2\n[-6.4, 6.4]2\n[-8.0, 8.0]2Domain Size\nFigure6: Theinclinationangleversustimeformagneticrot ationstrengthsofRm = −1, Rm = −10,\nand Rm = −100 and domain sizes ranging from [ −2.4,2.4]2to [−8.0,8.0]2. The number of grid\npoints is adjusted so that a constant grid spacing of h= 0.05 and constant time step ∆ = 0 .1his\nused for each simulation. No change in the results are seen pa st a domain size of [ −6.4,6.4]2.\nTime0 40 80 120 160 200Angle\n00.20.40.60.811.21.41.6\n0.02h\n0.05h\n0.10h\n0.25h\n0.50hRm=-10\nRm=-1Rm=-100\nT\nFigure 7: The inclination angle versus time for magnetic rot ation strengths of Rm = −1, Rm =\n−10, and Rm = −100 and time steps ranging from ∆ t= 0.02hto ∆t= 0.5h. The domain is fixed\nat [−6.4,6.4]2while the size of the domain is 2572. No change in the results are seen past a time\nstep of ∆ t= 0.1h.\n13Time0 40 80 120 160 200Angle\n00.20.40.60.811.21.41.6\n-1-2-5-10-20-40-60-80-100\n(a) Inclination angle over time for various values of Rm,\nindicated by the numbers.\n0 20 40 60 80 020406080200\npp0.40p\n(b) Time it takes to rotate through an angle of 0 .05π,\n0.25π, and 0.4π.\nFigure 8: The influence of the magnetic field-induced rotatio nal force, Rm, on the inclination angle.\nAs Rm increases it takes additional time to align with the mag netic field.\n14Time0 20 40 60 80 100Ener\n0520253035404550\nT\n(a) Rm = −1Time0 20 40 60 80 100Ener\n02468T\n(b) Rm = −10\nTime0 20 40 60 80 100Ener\n023456\nT\n(c) Rm = −100\nFigure 9: The bending, rotation, and total energy for rotati on strengths of Rm = −1, Rm = −10,\nand Rm = −100.\nand Rm = −100, is shown in Fig. 9. As expected, the total energy decreas es as the vesicle becomes\naligned with the magnetic field. The overall rotation rate is directly correlated to the initial\nrotation energy, as a higher initial magnetic rotation ener gy correlates to faster rotation time. It\nis interesting to note that when the magnetic rotation stren gth is strong, Rm = −1, the vesicle\nmembrane can not respond quickly to changes in bending energ y and thus the bending energy\ncontribution increases, Fig. 9(a). This is in contrast to th e weaker rotation forces shown in Figs.\n9(b) and 9(c), where both the rotation and bending energy are strictly decreasing.\nTo further explore the influence of Rm on the vesicle shape, th e deformation parameter for the\nthree characteristic Rm values is shown in Fig. 10. It is clea rly observed that the strong rotation\nforce given by Rm = −1 causes larger deformations than the Rm = −10 and Rm = −100 cases.\nThe shape of the vesicle using Rm = −1, as shown in Fig. 11, can be compared to that shown in\nFig. 4, and it is clear that larger deformation are observed b efore the vesicle flattens out.\nIt should be noted that the initial angle between the long-ax is of the vesicle and the applied\nmagnetic field is less than π/4. If the angle is equal to π/4, then the mechanism of alignment\nis no longer rotation, but large-scale deformation of the in terface. This can be seen in Fig. 12,\n15Time0 10 20 30 40 50Deformation Parameter0.320.340.360.380.40.420.440.460.48\n-1\n-10\n-100Rm\nFigure 10: The deformation parameter for Rm = −1, Rm = −10, and Rm = −100. Strong\nmagnetic field effects induce larger shape deformations. The c ircles on the Rm = −1 correspond\nto the interfaces shown in Fig. 11. The shapes for Rm = −10 and Rm = −100 do not look\nqualitatively different from that shown in Fig. 4.\ny\nx-1.5 -1 -0.5 0 0.5 1 1.5-1.5-1-0.500.511.5\n(a)t= 0\ny\nx-1.5 -1 -0.5 0 0.5 1 1.5-1.5-1-0.500.511.5\n(b)t= 3y\nx-1.5 -1 -0.5 0 0.5 1 1.5-1.5-1-0.500.511.5\n(c)t= 6\ny\nx-1.5 -1 -0.5 0 0.5 1 1.5-1.5-1-0.500.511.5\n(d)t= 9\nFigure 11: The shape of the vesicle at various times using Rm = −1.\n16y\nx-1.5 -1 -0.5 0 0.5 1 1.5-1.5-1-0.500.511.5\n0\n20\n4045\n50,100\n(a) Interface Location at the times indi-\ncated\nTime0 20 40 60 80 10000.050.10.150.20.250.30.350.40.450.5Deformation Parameter\n00.20.40.60.811.21.41.6Angle\n(b) Angle and Deformation Parameter\nFigure12: Sampleinterface locations, theinclination ang le, anddeformation parameter foravesicle\nin a magnetic field aligned along the x-direction. In this case Rm = −1.\nwhere the vesicle is placed in a magnetic field aligned with th ex-axis and the rotation/alignment\nstrength is Rm = −1. Up until approximately t= 30, the major axis of the vesicle is aligned\nwith the y-axis. The vesicle undergoes large deformations, as demons trated by the decrease in the\ndeformation parameter. After t= 30, the major axis is aligned with the magnetic field along th e\nx-axis and the vesicle begins to elongate to reduce both the be nding and magnetic energies. It\nshould be noted that the final deformation parameter for this case,D≈0.46, is similar to that\nshown in Fig. 10, despite the difference in the magnetic angle.\n4.4 Spatially varying magnetic field\nNext consider the influence of a spatially varying magnetic fi eld. In this case, the full magnetic\nenergy contribution must be considered, and thus both Rm and Mn will be varied. To construct\nthe variable magnetic field, the vesicle is placed inside a do main spanning [ −3.2,3.2]2using a grid\nsize of 1292so thath= 0.05. In this case wall boundary conditions are assumed. To ind uce the\nmagnetic field, two infinitely long wires are placed at the loc ations (−3.2,0) and (3 .2,0). Each of\n17Figure13: Themagneticfieldlines, initial location of thev esicle, andrepresentation ofthemagnetic\nfieldstrength (color online). Themagnetic fieldis stronges tat thecenter of thetwo current carrying\nwires located at ( −3.2,0) and (3 .2,0), indicated by the color red, and quickly decays towards th e\ncenter of the domain, indicated by blue.\nthese wires has a current of magnitude I0in the vertical z−direction and produces a magnetic field\ngiven by\nB=B0\n2π/parenleftbiggx−a\n((x−a)2+(y−b)2,−y−b\n((x−a)2+(y−b)2/parenrightbigg\n(36)\nwhere the B0=µ0I0is the strength of the induced magnetic field surrounding a wi re at (a,b).[61]\nDue to the linearity of the magnetic field, the total magnetic field is simply the summation of that\ninduced by both wires. A vesicle with a reduced area of ν= 0.71 and matched viscosity, η= 1,\nis then centered at ( −2,0). It is expected that the vesicle will migrate towards the c enter of the\ndomain, which is the location of lowest magnetic field streng th. An example of the magnetic field\nand initial vesicle location is given in Fig. 13, which shows both the magnetic field lines and the\nintensity of the magnetic field.\nThe location of the interface at times of t= 0,t= 100, and t= 200 for Mn and Rm values\nbetween 1 and 100 is shown in Fig. 14. In all cases the interfac e migrates towards the center\nof the domain. The rate of this migration and the overall defo rmation of the interface strongly\ndepends on both the Mn and Rm parameters. In general, as the st rength of the alignment and\nbulk magnetic effects increases, the rate of of migration also increases. It should also be observed\nthat for stronger rotational strengths, denoted by lower Rm values, the vesicle tends to align with\nthe local magnetic field. As the underlying local magnetic fie ld is close to circular, the interface\nadopts this configuration.\nThe location of the x-centroid and the deformation parameter of the vesicle when exposed to\nthis spatially varying magnetic field is shown in Fig. 15. It i s clear that the fastest migration is\nachieved with small values of Rm and Mn. Even in situations wh ere the bulk-magnetic field effects\nare small, such as when Mn = −100, migration can occur due to the alignment energy. This is due\nto the fact that Eq. (5) can be decreased by not only aligning t he interface with the magnetic field,\nbut also by pushing the interface towards regions of lower ma gnetic field strength.\nThe deformation parameter results mimic those seen in the sp atially constant results. As the\nalignment strength increases, the vesicle becomes more def ormed. As the value of Mn increases,\nthis deformed state persists longer. This is due to the fact t hat it takes longer for the vesicle to\n18y\nx-3 -2.5 -2 -1.5 -1 -0.5 0-1.5-1-0.500.511.5Mn=-100 Rm=-100y\nx-3 -2.5 -2 -1.5 -1 -0.5 0-1.5-1-0.500.511.5Mn=-100 Rm=-10y\nx-3 -2.5 -2 -1.5 -1 -0.5 0-1.5-1-0.500.511.5Mn=-100 Rm=-1\ny\nx-3 -2.5 -2 -1.5 -1 -0.5 0-1.5-1-0.500.511.5Mn=-10 Rm=-100y\nx-3 -2.5 -2 -1.5 -1 -0.5 0-1.5-1-0.500.511.5Mn=-10 Rm=-10y\nx-3 -2.5 -2 -1.5 -1 -0.5 0-1.5-1-0.500.511.5Mn=-10 Rm=-1\ny\nx-3 -2.5 -2 -1.5 -1 -0.5 0-1.5-1-0.500.511.5Mn=-1 Rm=-100y\nx-3 -2.5 -2 -1.5 -1 -0.5 0-1.5-1-0.500.511.5Mn=-1 Rm=-10y\nxMn=-1 Rm=-1\n-3 -2.5 -2 -1.5 -1 -0.5 0-1.5-1-0.500.511.5\nt = 0 t = 100 t = 200\nFigure 14: The location of the interface at times t= 0,t= 100, and t= 200 for a spatially\nvariable magnetic field using various values of Mn and Rm. The results show a portion of the\nentire domain, which spans [ −3.2,3.2]2. The magnetic field arises due to current carrying wires\nembedded at locations ( −3.2,0) and (3 .2,0).\n19Time0 40 80 120 160 200x-Centroid Location\n-2-1.5-1-0.5\n(a)x-centroid\nTime0 40 80 120 160 200Deformation Parameter\n0.280.40.440.48\n(b) Deformation Parameter\nFigure 15: The evolution of the x-centroid and deformation parameter of the vesicle over tim e for\nvarious values of Mn and Rm. The combinations shown here corr espond to those shown in Fig. 14.\nThe legend is common to both figures.\nmigrate towards the center of the domain when Mn is large. Thi s results in the vesicle remaining\ncloser to the stronger and more compact magnetic field center ed at (−3.2,0).\nFinally, theenergy of thesystem over timefor a bulkconstan t of Mn = −10 andthree alignment\nstrengths, Rm = −1, Rm = −10, and Rm = −100 is shown in Fig. 16. As in Sec. 4.3, the magnetic\nenergies are strictly decreasing over time. For the cases of strong magnetic field effects, particularly\nfor Mn = −10 and Rm = −1, the bending energy increases above the initial value, and remains\nelevated throughout the simulation. It should be expected t hat as the vesicle moves towards the\ncenter of the domain, where the magnetic field is weakest, the bending energy should have a larger\ninfluence.\n5 Conclusion\nIn this work a numerical model of vesicles in magnetic fields i s presented. Based on the energy\nof the membrane, the interface forces due to magnetic rotati on/alignment and the bulk magnetic\n20Time0 50 100 150 200gy\n0102040\nT\n(a) Rm = −1Time0 50 100 150 200gy\n0246810121416\nT\n(b) Rm = −10\nTime0 50 100 150 200gy\n02468101214\nT\n(c) Rm = −100\nFigure 16: The bending, bulk, rotation, and total energy for three different rotation strengths. The\nresults assume Mn = −10.\n21energy are derived. These magnetic interface forces, in add ition to the bending and tension forces\nof a vesicle, are used in conjunction with a level set descrip tion of the interface and a projection\nmethod for the fluid field to investigate the dynamics of a two- dimensional vesicle. The simulation\nis compared to the experimental results of Boroske and Helfr ich, and good agreement is achieved.\nA systematic investigation of the influence of the rotation/ alignment parameter, Rm, on the vesicle\nmembraneisperformedforspatiallyconstantmagneticfield . Ingeneral, thereisalinearrelationship\nbetween Rm and the amount of time it takes a vesicle to rotate t hrough a particular angle. It was\nalso demonstrated that if the angle between a vesicle and the magnetic field is π/4, then the\nalignment is not done through rotation, but by bulk deformat ion of the membrane.\nThe movement of a vesicle in a spatially-varying magnetic fie ld was also considered by placing a\nvesicle between two current-carrying wires. This magnetic field induced linear motion of the vesicle,\nwith the rate of migration dependent on both the alignment pa rameter Rm and the bulk magnetic\nfield parameter Mn. The particular nature of the underlying m agnetic field induced deformations\nof the membrane, with the magnitude of these deformations de pending on the particular parameter\nset.\nThe use of magnetic fields opens up new possibilities for char acterization and processing of\nnot only liposome vesicle, but also other soft-matter multi phase systems such as polymer vesicles\nor biological cells. For example, it is imagined that using t he experimental equivalent to the\nsimulations shown here it could be possible to determine mat erial properties such as the magnetic\nsusceptibilities of the membrane molecules. This knowledg e could then be used to design processing\ntechniques, possibly in conjuncture with electric fields, t o precisely control the dynamics of vesicles.\nFuture work will explore these possibilities.\nAcknowledgments\nThisworkhasbeensupportedbytheNationalScienceFoundat ionthroughtheDivisionofChemical,\nBioengineering, Environmental, and Transport Systems Gra nt #1253739.\nAppendix A Calculus on Surfaces\nOne issue with derivatives on surfaces is that operations re quire information of not only how\na function varies on the interface, but also how the interfac e itself varies. For this reason, some\nstandardvectorcalculusidentities maynothold. Inthisse ction thesurfacevector calculusidentities\nused to calculate the magnetic field force are derived.\nLet the interface be orientable with an outward unit normal n. Without loss of generality, it\nis assumed that the interface is described as the zero contou r of a function Ψ such that Ψ is the\nsolution to the Eikonal equation, |∇Ψ|= 1 within a distance of rto the interface, where rdepends\non the curvature of the interface. With this assumption the n ormal is simply n=∇Ψ. As the\nnormal is now defined in a small region surrounding the interf ace, quantities such as the gradient\nof the unit normal, ∇n, are well-defined near the interface.\nThe projection operator is given by P=I−n⊗n, or in component form Pij=δij−ninj,\nwhereδijis the Kronecker delta function. In this work, indices iandjare free indices and while\n22p,q, andrare dummy indices. The projection operator is symmetric, P=PT, and idempotent,\n[PP]ij=PipPpj\n= (δip−ninp)(δpj−npnj)\n=δipδpj−ninpδpj−npnjδip+ninpnpnj\n=δij−ninj−ninj+ninj\n=δij−ninj= [P]ij, (37)\nwhere [v]iis theithcomponent of a vector v, [A]ijis thei,jcompnent of a tensor A, and repeated\nindices indicate summation.\nThe generalized surface gradient function can be written as ∇sA= (∇A)P, whereAcan be\neither a scalar, vector, or tensor field.[47;48;62]For example, the surface gradient of a scalar field a\nin component form would be written as\n[∇sa]i= [(∇a)P]i=∂a\n∂xpPpi, (38)\nThe surface gradient of a scalar field asquared is\n/bracketleftbig\n∇sa2/bracketrightbig\ni=/bracketleftbig/parenleftbig\n∇a2/parenrightbig\nP/bracketrightbig\ni=∂a2\n∂xpPpi\n= 2a∂a\n∂xpPpi= [2a∇sa]i. (39)\nFor a vector field vthe surface gradient would be\n[∇sv]ij= [(∇v)P]ij=∂vi\n∂xpPpj. (40)\nThe surface gradient of a vector dot product is\n[∇s(v·w)]i=∂(vpwp)\n∂xqPqi\n=wp∂vp\n∂xqPqi+vp∂wp\nxqPqi\n= [w·∇sv+v·∇sw]i. (41)\nThe surface divergence of any vector vcan be written as ∇s·v= tr∇sv=P:∇v.[47]In\ncomponent form this is written as\n[∇s·v] = [P:∇v] =Ppq∂vp\n∂xq. (42)\nThe surface divergence of a tensor field Ais defined as[47]\n[∇s·A]i= [(∇A)P]i=∂Aip\n∂xqPqp. (43)\n23The surface divergence of the projection operator is given b y\n[∇s·P]i= [(∇P)P]i=∂Pip\n∂xqPqp\n=∂\n∂xq(δip−ninp)Pqp\n=−∂ni\n∂xqPqpnp−ni∂np\n∂xqPqp\n= [−(∇n)Pn−n∇s·n]i= [−Hn]i (44)\ndue to the definition of total curvature, H=∇s·n, and the fact that Pn= 0:\n[Pn]i=Pipnp= (δip−ninp)np\n=δipnp−ninpnp=ni−ni= [0]i. (45)\nLetabe a scalar field. The surface divergence of this scalar field t imes the projection operator is\n[∇s·(aP)]i= [(∇(aP))P]i=∂(aPip)\n∂xqPqp\n=∂a\n∂xqPipPqp+a∂Pip\n∂xqPqp\n=∂a\n∂xqPqpPpi+a∂Pip\n∂xqPqp\n=∂a\n∂xqPqi+a∂Pip\n∂xqPqp= [∇sa−aHn]i. (46)\nNext, consider the surface divergence of the tensor (outer) product of the unit normal nand\nany vector v:\n[∇s·(n⊗v)]i= [(∇(n⊗v))P]i=∂(nivp)\n∂xqPqp\n=∂ni\n∂xqPqpvp+ni∂vp\n∂xqPpq= [(∇sn)v+n∇s·v]i (47)\nFinally, consider the surface divergence of a scalar, the pr ojection operator, and a vector,\n[∇s·(aPv)] = [P:∇(aPv)] =Ppq∂(aPprvr)\n∂xq\n=Ppq/parenleftbigg∂a\n∂xqPprvr+a∂Ppr\n∂xqvr+aPpr∂vr\n∂xq/parenrightbigg\n=∂a\n∂xqPqrvr+a∂Ppr\n∂xqPpqvr+a∂vr\n∂xqPqr\n=∂a\n∂xqPqrvr+a∂Prp\n∂xqPqpvr+a∂vr\n∂xqPrq\n= [v·∇sa+a(∇s·P)·v+a∇s·v]\n= [v·∇sa−aHn·v+a∇s·v] (48)\n24References\n[1] D. Kagan, R. Laocharoensuk, M. Zimmerman, C. Clawson, S. Balasubramanian, D. Kong,\nD. Bishop, S. Sattayasamitsathit, L. Zhang, J. Wang, Rapid d elivery of drug carri-\ners propelled and navigated by catalytic nanoshuttles, Sma ll 6 (23) (2010) 2741–2747.\ndoi:10.1002/smll.201001257 .\n[2] M. Toner, D. Irimia, Blood-on-a-chip, Annual Review Of B iomedical Engineering 7 (2005)\n77–103.doi:10.1146/annurev.bioeng.7.011205.135108 .\n[3] N. Xia, T. P. Hunt, B. T. Mayers, E. Alsberg, G. M. Whitesid es, R. M. Westervelt, D. E.\nIngber, Combined microfluidic-micromagnetic separation o f living cells in continuous flow,\nBiomedical Microdevices 8 (4) (2006) 299–308. doi:10.1007/s10544-006-0033-0 .\n[4] P. Chen, X. Feng, W. Du, B.-F. Liu, Microfluidic chips for c ell sorting, Frontiers In Bioscience-\nlandmark 13 (2008) 2464–2483. doi:10.2741/2859 .\n[5] A. A. Solovev, S. Sanchez, M. Pumera, Y. F. Mei, O. G. Schmi dt, Magnetic control of tubu-\nlar catalytic microbots for the transport, assembly, and de livery of micro-objects, Advanced\nFunctional Materials 20 (15) (2010) 2430–2435. doi:10.1002/adfm.200902376 .\n[6] Y.-C. Tan, Y. Ho, A. Lee, Microfluidic sorting of droplets by size, Microfluidics and Nanoflu-\nidics 4 (4) (2008) 343–348. doi:10.1007/s10404-007-0184-1 .\nURLhttp://dx.doi.org/10.1007/s10404-007-0184-1\n[7] J. DuBose, X. Lu, S. Patel, S. Qian, S. Woo Joo, X. Xuan, Mic rofluidic electrical sorting of\nparticles based on shape in a spiral microchannel, Biomicro fluidics 8 (1) (2014) 014101.\n[8] G. Wang, W. Mao, R. Byler, K. Patel, C. Henegar, A. Alexeev , T. Sulchek, Stiffness De-\npendent Separation of Cells in a Microfluidic Device, PLOS ON E 8 (10) (2013) e75901.\ndoi:10.1371/journal.pone.0075901 .\n[9] M. MacDonald, G. Spalding, K. Dholakia, Microfluidic sor ting in an optical lattice, Nature\n426 (6965) (2003) 421–424, 10.1038/nature02144.\n[10] R. Zhou, C. Wang, Acoustic bubble enhanced pinched flow f ractionation for micropar-\nticle separation, Journal of Micromechanics and Microengi neering 25 (8) (2015) 084005.\ndoi:10.1088/0960-1317/25/8/084005 .\n[11] R. Dimova, N. Bezlyepkina, M. D. Jordo, R. L. Knorr, K. A. Riske, M. Staykova, P. M.\nVlahovska, T. Yamamoto, P. Yang, R. Lipowsky, Vesicles in el ectric fields: Some novel aspects\nof membrane behavior, Soft Matter 5 (2009) 3201–3212.\n[12] R. Dimova, K. A. Riske, S. Aranda, N. Bezlyepkina, R. L. K norr, R. Lipowsky, Giant vesicles\nin electric fields, Soft matter 3 (7) (2007) 817–827.\n[13] E.M.Kolahdouz, D.Salac, Dynamicsofthree-dimension alvesicles indcelectricfields, Physical\nReview E 92 (1) (2015) 012302. doi:10.1103/PhysRevE.92.012302 .\n[14] P. F. Salipante, P. M. Vlahovska, Vesicle deformation i n DC electric pulses, Soft Matter 10\n(2014) 3386–3393.\n25[15] J. T. Schwalbe, P. M. Vlahovska, M. J. Miksis, Vesicle el ectrohydrodynamics, Physical Review\nE 83 (4) (2011) 046309.\n[16] L. C. McConnell, M. J. Miksis, P. M. Vlahovska, Vesicle e lectrohydrodynamics in DC electric\nfields, IMA Journal of Applied Mathematics 78 (4) (2013) 797– 817.\n[17] L. M. Barrett, A. J. Skulan, A. K. Singh, E. B. Cummings, G . J. Fiechtner, Dielectrophoretic\nmanipulation of particles and cells using insulating ridge s in faceted prism microchannels,\nAnalytical Chemistry 77 (21) (2005) 6798–6804. doi:10.1021/ac0507791 .\n[18] E. B. Cummings, A. K. Singh, Dielectrophoresis in micro chips containing arrays of insulating\nposts: Theoretical and experimental results, Analytical C hemistry 75 (18) (2003) 4724–4731.\ndoi:10.1021/ac0340612 .\n[19] M. D. Vahey, J. Voldman, An equilibrium method for conti nuous-flow cell sorting using di-\nelectrophoresis, Analytical Chemistry 80 (9) (2008) 3135– 3143.doi:10.1021/ac7020568 .\n[20] K. A. Riske, R. Dimova, Electro-deformation and porati on of giant vesicles viewed with high\ntemporal resolution, Biophysical Journal 88 (2) (2005) 114 3–55.\n[21] E. Boroske, H. W., Magnetic-anisotropy of egg lecithin membranes, Biophysical Journal 24 (3)\n(1978) 863–868.\n[22] R. S. M. Rikken, R. J. M. Nolte, J. C. Maan, J. C. M. van Hest , D. A. Wilson, P. C. M. Chris-\ntianen, Manipulation of micro- and nanostructure motion wi th magnetic fields, Soft Matter\n10 (9) (2014) 1295–1308. doi:10.1039/c3sm52294f .\n[23] W. Helfrich, Lipid Bilayer Spheres - Deformation and Bi refringence In Magnetic-fields, Physics\nLetters A A 43 (5) (1973) 409–410. doi:10.1016/0375-9601(73)90396-4 .\n[24] W. Helfrich, Elastic Properties ofLipidBilayers - The oryandPossibleExperiments, Zeitschrift\nFur Naturforschung C-a Journal of Biosciences C 28 (11-1) (1 973) 693–703.\n[25] T. S. Tenforde, L. R. P., Magnetic Deformation of Phosph olipid-bilayers - Effects On Liposome\nShape and Solute Permeability At Prephase Transition-temp eratures, Journal of Theoretical\nBiology 133 (3) (1988) 385–396. doi:10.1016/S0022-5193(88)80329-1 .\n[26] S.Ozeki, H.Kurashima,H.Abe,High-magnetic-field effec tsonliposomesandblackmembranes\nof dipalmitoylphosphatidylcholin: Magneotresponses in m embrane potential and magnetofu-\nsion, Journal of Physical Chemistry B 104 (24) (2000) 5657–5 660.doi:10.1021/jp9934073 .\n[27] X. X. Qiu, P. A. Mirau, C. Pidgeon, Magnetically Induced Orientation of Phos-\nphatidylcholine Membranes, Biochimica Et Biophysica Acta 1147 (1) (1993) 59–72.\ndoi:10.1016/0005-2736(93)90316-R .\n[28] M. A. Kiselev, T. Gutberlet, A. Hoell, V. L. Aksenov, D. L ombardo, Orientation of the DMPC\nunilamellar vesicle system in the magnetic field: SANS study , Chemical Physics 345 (2-3)\n(2008) 181–184. doi:10.1016/j.chemphys.2007.09.002 .\n[29] H. Ye, A. Curcuru, Vesicle biomechanics in a time-varyi ng magnetic field, BMC Biophysics 8\n(2015) 2. doi:10.1186/s13628-014-0016-0 .\n[30] U. Seifert, Configurations of fluid membranes and vesicl es, Advances in Physics 46 (1) (1997)\n13–137.\n26[31] D. A. Saville, Electrohydrodynamics: The Taylor-Melc her leaky dielectric model, Annual Re-\nview of Fluid Mechanics 29 (1962) (1997) 27–64.\n[32] J. Melcher, G. Taylor, Electrohydrodynamics: A review of the role of interfacial shear stresses,\nAnnual Review of Fluid Mechanics 1 (1) (1969) 111–146.\n[33] K. A. Riske, R. Dimova, Electric pulses induce cylindri cal deformations on giant vesicles in\nsalt solutions, Biophysical Journal 91 (5) (2006) 1778–86.\n[34] D.Needham, R.Hochmuth, Electro-mechanical permeabi lization oflipidvesicles. Roleofmem-\nbrane tension and compressibility, Biophysical Journal 55 (5) (1989) 1001–1009.\n[35] T. Guo, S. Wang, R. Samulyak, Sharp Interface Algorithm for Large Density Ratio Incom-\npressible Multiphase Magnetohydrodynamic Flows, 2013 Int ernational Conference On Com-\nputational Science 18 (2013) 511–520. doi:10.1016/j.procs.2013.05.215 .\n[36] H. Ki, Level set method for two-phase incompressible flo ws under magnetic fields, Computer\nPhysics Communications 181 (6) (2010) 999–1007. doi:10.1016/j.cpc.2010.02.002 .\n[37] T. Tagawa, Numerical simulation of two-phase flows in th e presence of a mag-\nnetic field, Mathematics and Computers In Simulation 72 (2-6 ) (2006) 212–219.\ndoi:10.1016/j.matcom.2006.05.040 .\n[38] P. M. Vlahovska, R. S.Gracia, S.Aranda-Espinoza, R.Di mova, Electrohydrodynamicmodelof\nvesicle deformation in alternating electric fields, Biophy sical Journal 96 (12) (2009) 4789–4803.\n[39] M. Carmo, Differential Geometry of Curves and Surfaces, P rentice-Hall, 1976.\n[40] T. Biben, C. Misbah, Tumbling of vesicles under shear flo w within an advected-field approach,\nPhysical Review E 67 (3) (2003) 031908.\n[41] P. M. Vlahovska, R. S. Gracia, Dynamics of a viscous vesi cle in linear flows, Physical Review\nE 75 (1) (2007) 016313.\n[42] J. S. Sohn, Y.-H. Tseng, S. Li, A. Voigt, J. S. Lowengrub, Dynamics of multicompo-\nnent vesicles in a viscous fluid, Journal of Computational Ph ysics 229 (1) (2010) 119–144.\ndoi:10.1016/j.jcp.2009.09.017 .\n[43] F. Scholz, B. E., H. W., Magnetic-anisotropy of lecithi n membranes - A new anisotropy sus-\nceptometer, Biophysical Journal 45 (3) (1984) 589–592.\n[44] C. Tan, B. Fung, G. Cho, Phospholipid bicelles that alig n with their normals parallel to\nthe magnetic field, Journal of the American Chemical Society 124 (39) (2002) 11827–11832.\ndoi:10.1021/ja027079n .\n[45] U. Seifert, Fluid membranes in hydrodynamic flow fields: Formalism and an application to\nfluctuating quasispherical vesicles in shear flow, The Europ ean Physical Journal B-Condensed\nMatter and Complex Systems 8 (3) (1999) 405–415.\n[46] G. Napoli, L. Vergori, Equilibrium of nematic vesicles , Journal of Physics A: Mathematical\nand Theoretical 43 (44) (2010) 445207.\nURLhttp://stacks.iop.org/1751-8121/43/i=44/a=445207\n27[47] E.Fried, M.E.Gurtin,A continuum mechanical theory fo r turbulence: a generalized Navier–Stokes- αequation\nTheoretical and Computational Fluid Dynamics 22 (6) (2008) 433–470.\ndoi:10.1007/s00162-008-0083-4 .\nURLhttp://dx.doi.org/10.1007/s00162-008-0083-4\n[48] G. Napoli, L. Vergori, Surface free energies for nemati c shells, Physical Review E 85 (2012)\n061701.doi:10.1103/PhysRevE.85.061701 .\nURLhttp://link.aps.org/doi/10.1103/PhysRevE.85.061701\n[49] J. D. Towers, Finite difference methods for approximatin g Heaviside functions, Journal of\nComputational Physics 228 (9) (2009) 3478–3489. doi:10.1016/j.jcp.2009.01.026 .\n[50] Y. Chang, T. Hou, B. Merriman, S. Osher, A Level Set Formu lation of Eulerian Interface\nCapturing Methods for Incompressible Fluid Flows, Journal of Computational Physics 124 (2)\n(1996) 449–464.\n[51] E. M. Kolahdouz, D. Salac, Electrohydrodynamics of Thr ee-dimensional Vesicles: A Nu-\nmerical Approach, SIAM Journal on Scientific Computing 37 (3 ) (2015) B473–B494.\ndoi:10.1137/140988966 .\n[52] J. D. Towers, A convergence rate theorem for finite differe nce approximations to delta func-\ntions, Journal of Computational Physics 227 (13) (2008) 659 1–6597.\n[53] G. Beblik, S. R. M., H. W., Bilayer Bending Rigidity of So me Synthetic Lecithins, Journal De\nPhysique 46 (10) (1985) 1773–1778. doi:10.1051/jphys:0198500460100177300 .\n[54] G.Velmurugan, E.M.Kolahdouz,D.Salac, Level Set Jet S chemes for Stiff Advection Equations: The SemiJet\nComputer Methods in Applied Mechanics and Engineering 310 ( 2016) 233–251.\ndoi:10.1016/j.cma.2016.07.014 .\nURLhttp://www.sciencedirect.com/science/article/pii/S0 045782516307423\n[55] B. Seibold, R. R. Rosales, J.-C. Nave, Jet schemes for ad vection problems, Discrete and Con-\ntinuous Dynamical Systems - Series B 17 (4) (2012) 1229–1259 .\n[56] V. Vitkova, A. G. Petrov, Chapter Five - Lipid Bilayers a nd Membranes: Material Properties,\nin: A. Igliˇ c, J. Genova (Eds.), A Tribute to Marin D. Mitov, V ol. 17 of Ad-\nvances in Planar Lipid Bilayers and Liposomes, Academic Pre ss, 2013, pp. 89–138.\ndoi:10.1016/B978-0-12-411516-3.00005-X .\nURLhttp://www.sciencedirect.com/science/article/pii/B9 78012411516300005X\n[57] D. Salac, M. J. Miksis, Reynolds number effects on lipid ve sicles, Journal of Fluid Mechanics\n711 (2012) 122–146. doi:10.1017/jfm.2012.380 .\nURLhttp://journals.cambridge.org/article_S002211201200 3801\n[58] S.Ramanujan,P.C.,Deformation of liquid capsules enc losed by elastic membranes in simple shear flow: large\nJournal of Fluid Mechanics 361 (1998) 117–143. doi:10.1017/S0022112098008714 .\nURLhttp://journals.cambridge.org/article_S002211209800 8714\n[59] D. Salac, M. Miksis, A level set projection model of lipi d vesicles in general flows, Journal of\nComputational Physics 230 (22) (2011) 8192–8215.\n28[60] L. F. Braganza, B. B. H., C. T. J., M. D., The superdiamagn etic effect of magnetic-fields on\none and 2 component multilamellar liposomes, Biochimica Et Biophysica Acta 801 (1) (1984)\n66–75.doi:10.1016/0304-4165(84)90213-7 .\n[61] E. E. Tzirtzilakis, A mathematical model for blood flow i n magnetic field, Physics of Fluids\n17 (7).doi:10.1063/1.1978807 .\nURLhttp://scitation.aip.org/content/aip/journal/pof2/1 7/7/10.1063/1.1978807\n[62] M. E. Gurtin, A. Ian Murdoch, A continuum theory of elast ic material surfaces, Archive for\nRational Mechanics and Analysis 57 (4) (1975) 291–323. doi:10.1007/BF00261375 .\nURLhttp://dx.doi.org/10.1007/BF00261375\n29"    },    {        "title": "1911.08144v1.On_the_Dynamics_of_Inverse_Magnetic_Billiards.pdf",        "content": "ON THE DYNAMICS OF INVERSE MAGNETIC BILLIARDS\nSEAN GASIOREK\nAbstract. Consider a strictly convex set \n in the plane, and a homogeneous,\nstationary magnetic \feld orthogonal to the plane whose strength is Bon the\ncomplement of \n and 0 inside \n. The trajectories of a charged particle in this\nsetting are straight lines concatenated with circular arcs of Larmor radius \u0016.\nWe examine the dynamics of such a particle and call this inverse magnetic\nbilliards . Comparisons are made to standard Birkho\u000b billiards and magnetic\nbilliards, as some theorems regarding inverse magnetic billiards are consistent\nwith each of these billiard variants while others are not.\n1.Introduction\nConsider the classical motion of a particle of mass mand charge ein the plane.\nLet \n\u001aR2denote a connected, strictly convex domain, and de\fne a constant,\nhomogeneous, stationary magnetic \feld orthogonal to the plane which has strength\nBonR2n\n and 0 on \n. As such, the equations of motion for the particle of position\nqand velocity vare as follows:\n(\n_q=v\n_v=B\n(q)Jvwith J:=\u00120\u00001\n1 0\u0013\n; B \n(q) :=(\n0q2\nB q2R2n\n:\nThe solution to this initial value problem are continuous curves which are circular\narcs outside \n and straight lines inside \n. The circular arcs will have Larmor radius\n\u0016=mjvj\njeBj, and speedj_qjand energy Eare constants of motion. Without loss of\ngenerality we assume e <0 andB > 0 so that the motion along the circular arcs\nwill be traversed in the counterclockwise direction.\nFollowing the construction in [2], suppose the boundary @\n isCkwithk\u00153\nand total lengthj@\nj=L. The boundary @\n =Image (\u0000(s)) will be parametrized\nby arc length, s:\n\u0000(s) = (X(s);Y(s)); ds2=dX2+dY2; s2R=LZ:\nThe unit tangent and unit normal vectors and curvature are given by\nt(s) = (X0(s);Y0(s)) = (cos(\u001c(s));sin(\u001c(s)));\nn(s) = (\u0000Y0(s);X0(s));\n\u0014(s) =d\u001c\nds=X0(s)Y00(s)\u0000X00(s)Y0(s) =1\n\u001a(s);\nso that\u001c(s) is the polar angle between the positive x-axis andt(s), and\u001a(s) is the\nradius of curvature. Because \n is strictly convex the curvature of the boundary\nis strictly positive and \u001a(s) is bounded by positive constants, 0 < \u001amin\u0014\u001a(s)\u0014\n\u001amax<1for alls. Following the lead of [19], we will explore the dynamics of our\n1arXiv:1911.08144v1  [math.DS]  19 Nov 20192 SEAN GASIOREK\nsystem in terms of the relative sizes of the Larmor radius \u0016and the maximum and\nminimum radii of curvature of @\n. We will refer to these possibilities\n\u0016<\u001amin; \u001amin<\u0016<\u001amax; \u001amax<\u0016\nascurvature regimes . The billiard \row is hence given by the Lagrangian\nL(q;_q) =1\n2mj_qj2+eh_q;A(q)i;A(q) =1\n2(\u0000yB\n(q);xB \n(q)) =1\n2B\n(q)Jq\nwhereh\u0001;\u0001iis the standard Euclidean inner product. We call this dynamical system\ninverse magnetic billiards , following the naming by [22].\nElectron dynamics in piecewise-constant magnetic \felds are studied in [4], [9],\n[10], [18], [20], [21], and [22]. Classical, semiclassical, and quantum approaches to\nthis system are each addressed to a degree { occasionally in compact subsets and\nsometimes in unbounded regions { but none are in-depth mathematically to the\nextent of [2] with respect to magnetic billiards, for example.\nThis paper is strongly in\ruenced by the work of Berglund and Kunz in [2],\nand is organized as follows. Section 2 gives a thorough description of the billiard\n\row and describes its motion through a return map T. An exact expression is\ngiven for the Jacobian DT. The map Tis sometimes a twist map and admits a\ngenerating function G, which is given explicitly in section 3. In section 4 we address\nthe existence of periodic orbits using G. Some calculations are made in section 5\nthat are speci\fc to the ellipse. Section 6 details the existence and nonexistence of\ncaustics using approaches similar to Mather, Berglund, and Kunz.\n2.Constructing the return map\nAs the particle moves, it successively leaves and re-enters \n at the points P0;P1;\nP2;P3:::2@\n. Index these points so that points with even index P0;P2;P4;:::\nare re-entry points and points P1;P3;P5;:::of odd index are exit points. Express\nthe oriented line segment P0P1joining each entry point to its successive exit point\nas a vector `1~ v0=P1\u0000P0where~ v0is the unit vector representing the direction of\nmotion of the particle while it travels inside \n from P0toP1and where`1=jP0P1j\nis the chord distance it travels.\nThe entire dynamics is summarized by the map T:M!Mwhich takes ( P0;v0)\nto (P2;v2), sending reentry point and direction to successive re-entry point and\ndirection. The phase space Mof the map Tconsists of unit vectors ( Pi;vi) whose\nbase points Pare on@\n with inward direction v. We call this map the return map\nand will express it in terms of the Birkho\u000b coordinates used in standard billiards.\nCoordinatize P0by its arc length parameter s0and the vector v0by the negative\ncosine of the angle \u00120between the tangent to \u0000 at P0and this vector. Writing\nui=\u0000cos(\u0012i) we call (si;ui) the Birkho\u000b coordinates of the trajectory as it exits\nor re-enters \n at Pi.\nThe phase space Mcan be identi\fed with the annulus P=R=LZ\u0002[\u00001;1]\u0018=\nS1\u0002[\u00001;1], the return map Tcan then be written as a map\nT:P!P;(s2i;u2i)7!(s2i+2;u2i+2)\nso thatTis a smooth map of the closed annulus, P. Further, the restriction\nTj@P= IdP, where the boundary @PofPis the usual boundary of P, namelyON THE DYNAMICS OF INVERSE MAGNETIC BILLIARDS 3\nFigure 1. The standard picture of the return map, T.\n\u0000\nS1\u0002f\u0000 1g\u0001\n[\u0000\nS1\u0002f1g\u0001\n.\nAt times it may be easier to work with Tas a map in terms of ( si;\u0012i). In partic-\nular, we will compute Taylor expansions of Tin section 6 in terms of sand\u0012. With\nthis interpretation, we see the inverse magnetic billiard as a discrete dynamical\nsystem.\nBy construction de\fne `i=jPi\u00001Pijfor integers i. De\fneA2i;2i+2to be the area\nbetween chord P2i+1P2i+2and \u0000(s) that is also inside the Larmor circle, and let\n\r2i;2i+2be the circular arc of Larmor radius \u0016that is outside \n. Let S2i;2i+2be the\narea within the circular arc \r2i;2i+2and outside \n. De\fne \u001f2i;2i+2to be the angle\nmeasured counterclockwise from\u0000\u0000\u0000\u0000\u0000!P2iP2i+1to\u0000\u0000\u0000\u0000\u0000\u0000\u0000!P2i+1P2i+2. See Figure 1 for the case\nwheni= 0.\nRemark 1. For notational simplicity, we now omit the subscripts 2i;2i+2, assume\ni= 0, and recognize each of the described quantities below are associated to a single\niteration of the return map Tand its realized trajectory.\nConsider the magnetic arc, \r. Let the angle of such an arc be  ,\"= 2\u0019\u0000 ,\u000e\nis the angle between the chord P1P2and the radius of the arc connecting each of\nP1andP2to the center of \r. See Figure 2a. From the de\fnition of these angles\nand elementary geometry we \fnd that\n = 2\u001fand sin(\u001f) =`2\n2\u0016:\nIt is important to note that there may be two trajectories with supplementary \u001f\nfor a given chord length `2. This is a characteristic e\u000bect of magnetic billiards. See\nFigure 2b for such an example.4 SEAN GASIOREK\n(a)\n (b)\nFigure 2. (a) A magnetic arc. (b) An example of two trajectories\nwith the same `2where\u001fand\u001f0are supplementary.\nWe decompose Tinto its two distinct pieces. De\fne the map T1: (s0;u0)7!\n(s1;u1) as the analogue to the standard billiard map. The map T2: (s1;u1)7!\n(s2;u2) is the particle moving from P1along the circular arc \rof Larmor radius \u0016\nuntil intersecting @\n again at P2. ThusT=T2\u000eT1.\nProposition 1. Given the maps T1andT2, the Jacobians DT1= \n@s1\n@s0@s1\n@u0@u1\n@s0@u1\n@u0!\nandDT2= \n@s2\n@s1@s2\n@u1@u2\n@s1@u2\n@u1!\nhave components\n@s1\n@s0=\u00140`1\u0000sin(\u00120)\nsin(\u00121)@s1\n@u0=`1\nsin(\u00120) sin(\u00121)\n@u1\n@s0=\u00140\u00141`1\u0000\u00141sin(\u00120)\u0000\u00140sin(\u00121)@u1\n@u0=\u00141`1\u0000sin(\u00121)\nsin(\u00120)\n@s2\n@s1=sin(2\u001f\u0000\u00121)\u0000\u00141`2cos(\u001f)\nsin(\u00122)@s2\n@u1=`2cos(\u001f)\nsin(\u00121) sin(\u00122)\n@u2\n@s1=sin(2\u001f\u0000\u00121) sin(2\u001f\u0000\u00122)\u0000sin(\u00121) sin(\u00122)\n`2cos(\u001f)@u2\n@u1=sin(2\u001f\u0000\u00122)\u0000\u00142`2cos(\u001f)\nsin(\u00121)\u0000\u00141sin(2\u001f\u0000\u00122)\u0000\u00142sin(2\u001f\u0000\u00121) +\u00141\u00142`2cos(\u001f)\nFurthermore, det(DT1) = 1 anddet(DT2) = 1 .\nThe details of this proof are given in Appendix A. The components of DT1are\nwell-known while the components of DT2are analogous to those found in Proposi-\ntion 1 of [2].ON THE DYNAMICS OF INVERSE MAGNETIC BILLIARDS 5\n-4 -3 -2 -1 1 2 3\n-2-1123\nFigure 3. The behavior of the return map for \fxed s0and varying\nu0when\u0016<\u001amin:Also shown are the Larmor centers and points\nP1;P2for each corresponding value of u0.\nCorollary 1. LetT=T2\u000eT1. ThenDT= \n@s2\n@s0@s2\n@u0@u2\n@s0@u2\n@u0!\nwith\n@s2\n@s0=\u00140`1sin(2\u001f\u0000\u00121)\u0000sin(\u00120) sin(2\u001f\u0000\u00121)\u0000\u00140`2cos(\u001f) sin(\u00121)\nsin(\u00121) sin(\u00122)\n@s2\n@u0=`1sin(2\u001f\u0000\u00121)\u0000`2cos(\u001f) sin(\u00121)\nsin(\u00120) sin(\u00121) sin(\u00122)\n@u2\n@s0=\u00142sin(\u00120) sin(2\u001f\u0000\u00121)\nsin(\u00121)+2 sin(\u001f) sin(2\u001f\u0000\u00121\u0000\u00122)(\u00140`1\u0000sin(\u00120))\n`2sin(\u00121)\n\u0000\u00140\u0012\nsin(2\u001f\u0000\u00122) +\u00142`1sin(2\u001f\u0000\u00121)\nsin(\u00121)\u0000\u00142`2cos(\u001f)\u0013\n@u2\n@u0=\u00142`2cos(\u001f)\u0000sin(2\u001f\u0000\u00122)\nsin(\u00120)+2`1sin(\u001f) sin(2\u001f\u0000\u00121\u0000\u00122)\u0000\u00142`1`2sin(2\u001f\u0000\u00121)\n`2sin(\u00120) sin(\u00121):\nFurthermore, det(DT) = 1 .\nFrom this we conclude that Tis an area- and orientation-preserving map of the\nannulusPand that the Birkho\u000b coordinates are conjugate. Just as with Birkho\u000b\nand magnetic billiards, the map Tpreserves the symplectic area-form ds^du=\nsin(\u0012)ds^d\u0012onP.\n3.Generating Functions and Twist Maps\nTwist maps have been studied extensively ([7], [14], [15]) in the context of dy-\nnamics and symplectic geometry. Let fbe a symplectic map from the annulus\nR=Z\u0002Rto itself. To be a monotone twist map , the lift of fto its universal cover\nbfmust satisfy the following properties, where ( x0;y0) =bf(x;y):\ni)bf(x+ 1;y) =~f(x;y) + (1;0);\nii)@x0\n@y>0 (twist condition);6 SEAN GASIOREK\niii)bfadmits a periodic exact symplectic map Gcalled a generating function :\ny0dx0\u0000ydx=dG(x;x0):\nAlternately we may say y0=@G\n@x0andy=\u0000@G\n@x.\nIn Birkho\u000b billiards, the billiard map is always a monotone twist map whose\ngenerating function is the negative of the Euclidean (chord) distance between suc-\ncessive collisions with the boundary. In the magnetic billiard setting, the magnetic\nbilliard map is not always twist, but when it is the generating function also depends\nupon the area associated with an arc of a given trajectory which appears as a \rux\nterm. It is not surprising that in this problem that has elements of both standard\nand magnetic billiards, that our generating function contains a combination of these\nelements.\nTo better understand when the return map Tis a twist map, we turn to the\nfollowing theorem which we prove in Appendix B.\nTheorem 1. Let\u0000(s) =@\nbe of classCk,k\u00153, and let\u001amin be the minimum\nradius of curvature of the strictly convex boundary curve \u0000(s). Then if\u0016 < \u001amin\nthenTis a twist map whose unique generating function (up to an additive constant)\nis given by\nG(s0;s2) =\u0000`1\u0000j\rj+1\n\u0016S\nwhere`1is the length of the line segment inside \n,j\rjis the length of the circular\narc\rof Larmor radius \u0016, andSis the area inside the circular arc \rbut outside \n.\nRemark 2. This generating function need not be unique. But in general we can\nthink of the generating function as the reduced action along a solution \u0017to the Euler\nLagrange equations which connects P0toP2. See [1]and[2].\nAn interesting property of this generating function (and this problem in general)\nis as follows: In the high magnetic \feld limit (i.e. \u0016!0), bothj\rj! 0 and\n1\n\u0016S! 0. This is because j\rj=O(\u0016) andS=O(\u00162). So as\u0016!0, our generating\nfunction approaches the standard billiard generating function, and our return map\napproaches the standard billiard map for billiards inside a convex set.\nWe can decompose Ginto non-magnetic and magnetic parts,\nG(s0;s2) =\u0014\n\u0000`1\u00001\n\u0016A\u0015\n+\u0014\n\u0000j\rj+1\n\u0016Area (A[S )\u0015\n=E(s0;s2) +F\u0016(\u001f(s0;s2)):\nHereArea (A[S ) is the area ofA[S ,E(s0;s2) has quantities `1andAwhich\nare not directly dependent upon the magnetic \feld, and F\u0016is dependent upon the\nmagnetic \feld and can be written as\nF\u0016(\u001f(s0;s2)) =\u0000\u0016(\u001f+ sin(\u001f) cos(\u001f)):\nWe can also write F\u0016as a function of `2, though with caveats:\nF\u0016(`2(s0;s2)) =\u0000\u0016arccos0\n@\u0006s\n1\u0000`2\n2\n4\u001621\nA\u0000\u0006`2\n2s\n1\u0000`2\n2\n4\u00162;\nwhere (+) is used if 0 <\u001f\u0014\u0019\n2and (\u0000) is used if\u0019\n2<\u001f<\u0019 .ON THE DYNAMICS OF INVERSE MAGNETIC BILLIARDS 7\n(a)\n (b)\nFigure 4. A (2;4) (a) and (4 ;5) (b) periodic orbit in an ellipse\nwith\u0016 < \u001amin:The centers of the Larmor circles are marked in\norange and the points Piare in dark purple.\n4.Periodic Orbits\nThe study of periodic orbits and their properties is a fundamental part of any\ndynamical system. In billiards, Birkho\u000b used Poincar\u0013 e's last geometric theorem to\nshow the existence of in\fnitely many distinct orbits ([3]). One way to distinguish\ndistinct periodic orbits from one another is by the rotation number . The rotation\nnumber of a periodic orbit is the rational number\nm\nn=winding number\nminimal period2[0;1]\nwhere the winding number m > 1 is computed with respect to the orientation of\n@\n induced by the parametrization \u0000( s). A periodic orbit with rotation numberm\nn\nis sometimes referred to as having frequency (m;n).\nA continuous orientation-preserving homeomorphism of the circle S1to itself has\na well-de\fned rotation number, de\fned modulo 1, when the circle is normalized to\nhave perimeter 1. When a lift to Rof this homeomorphism is chosen, this rotation\nnumber is now a real number. By the de\fnition of a twist map, Tsends boundary\ncircles to boundary circles, so the lifted homeomorphism has a bottom and top\nrotation number, !\u0000and!+. Then the rotation numbers belong to an interval\nI(bT) = [!\u0000;!+] provided!\u0000<!+. In particular, if the map is the identity on the\nboundary circles then necessarily !\u0000;!+2Z.\nWith this idea in mind, we can extend our de\fnition of rotation number of the\norbitf(s2k;u2k)gk2Zto include irrational numbers by writing\n!=1\nLlim\nk!1sk\nk;\nprovided this limit exists. We note that in the context of Birkho\u000b billiards and\ninverse magnetic billiards, this de\fnition agrees with the geometric de\fnition in\nterms of winding number given above.\nOne particularly useful application of a generating function is in the search of\nperiodic orbits. We will denote the lift of the return map TasbT. Theorems about8 SEAN GASIOREK\nthe existence of periodic orbits for continuous area-preserving twist maps can be at-\ntributed to Poincar\u0013 e and Birkho\u000b, Aubry, Mather, and Meiss and MacKay. These\ntheorems describe the existence of ( m;n) periodic orbits that are \\maximizing\"\nand \\maximin\" when considering the quantityPl\u00001\nj=kG(s2j;s2j+2) along with char-\nacterizations of irrational !2I(bT) and their relationship to quasiperiodic orbits.\nSee, for example, [15] or the summary from section 4 of [2] for more details.\nWe take a similar approach below, and can apply the theorems Poincar\u0013 e, Birkho\u000b,\nAubry, and Mather to the map Twhile making qualitative comments about the\nbehavior of T.\nProposition 2. Consider the three curvature regimes:\n(1) If\u0016 < \u001amin, the function s2(s0;u0)is strictly monotonic in u0andTis\na twist map. For \fxed s0the curvefT(s0;u0) :\u00001< u 0<1grotates\nonce around phase space (see Figure 5a) with lim\nu0!\u00061T(s0;u0) = (s0;u0).\nThereforeI(bT) = [0;1].\n(2) If\u001amin<\u0016<\u001amax, then the map may be discontinuous due to the Larmor\ncircle becoming tangent to the boundary. The function s2(s0;u0)is not\nnecessarily monotonic in u0and is not a twist map (see Figure 5c). It is\nstill true that lim\nu0!1T(s0;u0) = (s0;u0), but not necessarily when u0!\u00001.\n(3) If\u001amax<\u0016, thens2(s0;u0)is initially decreasing in u0and then begins to\nincrease again (see Figure 5e). We still have lim\nu0!\u00061T(s0;u0) = (s0;u0),\nwhich implies that there are exactly two distinct trajectories with equal \u001f\nfor a given s0,s2.\nThis proposition is very similar to the qualitative behavior of magnetic billiards\n(c.f. section 5.1 of [2]). In particular, part (1) tells us that for any convex set\nwith smooth boundary and nonvanishing curvature, periodic orbits of every rational\nfrequency!=m\nnexist, and the earlier summary gives us information about rational\nand irrational orbits in I(bT).\n5.When@\nis an Ellipse\nWe take a quick detour and consider the case when @\n is an ellipse. Consider\nthe parametrization of @\n as\nx(\u001e) = (\u0015cos(\u001e);sin(\u001e));ds\nd\u001e=C(\u001e) =q\n\u00152sin2(\u001e) + cos2(\u001e):\nWithout loss of generality we may assume that the parameter \u0015\u00151. In such a\ncase,\u001amin=\u0015\u00001. Consider the points Pi=x(\u001ei);i= 0;1;2.\nAssuming then that \u0016 < \u001amin, an important geometric consequence is that\n\u001e2\u0000\u001e1<\u0019, which simpli\fes the calculation below. Then Tis a twist map and\nG=\u00002 sin(\u001e\u0000\n10)C(\u001e+\n10)\u00001\n\u0016\u0015\u0012\n\u001e\u0000\n21\u00001\n2sin(2\u001e\u0000\n21)\u0013\n+F\u0016(2 sin(\u001e\u0000\n21)C(\u001e+\n21))\nwhere\u001e\u0006\nab=\u001ea\u0006\u001eb\n2.\nIn the case that \n is the unit disk ( \u0015= 1), we see that C= 1 and hence\n\u001e2\u0000\u001e0= 2\u001f. Another geometric observation is that \u0012i=\u0012and\u001fare both\nconstant, and hence ui=uis constant. This is because the the diagram in FigureON THE DYNAMICS OF INVERSE MAGNETIC BILLIARDS 9\n(a)\n1 2 3 4 5 6\n-1.0-0.50.00.51.0 (b)\n(c)\n1 2 3 4 5 6\n-1.0-0.50.00.51.0 (d)\n(e)\n1 2 3 4 5 6\n-1.0-0.50.00.51.0 (f)\nFigure 5. Structure of half the phase space in the ( \u001e;u)-plane for\nan ellipse where horizontal axis is \u001e, the polar angular parameter\nused in place of arc length, s. The vertical line \u001e=\u001e\u0003is shown\nwith its image under Tin the left column, \u00001\u0014u\u00141 for: (a)\n\u0016<\u001amin; (c)\u001amin<\u0016<\u001amax; (e)\u001amax<\u0016. The right column\nis half of a typical phase portrait, 0 \u0014\u001e\u0014\u0019, of an ellipse for: (b)\n\u0016<\u001amin; (d)\u001amin<\u0016<\u001amax; (f)\u001amax<\u0016.10 SEAN GASIOREK\n1 is symmetric about the line connecting the center of \n and the center of the\ncircular arc. This in turn implies that all of our geometric quantities, `1,`2,j\rj,\nandSare constant as they only depend upon \u0012and\u001f. Using the elementary\ngeometry of a circle-circle intersection,\n\u001f=\u0012+ arcsin \n\u0016sin(\u0012)p\n1 +\u00162\u00002\u0016cos(\u0012)!\nand the return map is explicitly\nT(s;u) = (s+ 2\u001f;u):\nIt is clear that since \u0012is constant, uis a constant of motion and the system is\nintegrable (in the sense of Liouville). Further, the simplicity of the return map in\nthe circular case allows us to \fnd periodic orbits directly. Since\nT(s0;u0) = (s0+ 2\u001f;u0);\nwe see that\nbTn(s0;u0) = (s0+ 2n\u001f;u 0):\nTherefore a periodic orbit will have rotation number m=n if and only if \u001f=m\u0019=n:\n6.Existence and Nonexistence of Caustics\n6.1.Preliminary Results. In both standard and magnetic billiards, the question\nof the existence of caustics has been addressed by Lazutkin [12], Berglund and Kunz\n[2], Moser [16], [17], and more in several variants of the standard billiard problem.\nTrajectories with caustics (the \\whispering gallery modes\") correspond to invariant\ncurves (a homotopically nontrivial curve) in phase space. Lazutkin had to assume a\nhigh degree of di\u000berentiability of the boundary in order to guarantee the existence\nof caustics, though this was later reduced to degree 6 by Douady [5].\nDue to the nature of inverse magnetic billiards problem, we call a smooth closed\nconvex curve in \n with the property that each trajectory that is tangent to it stays\ntangent to it after each successive reentry an inner convex caustic . An analogous\nde\fnition holds for outer caustics which contain \n.\nFor example, in a circle of radius R, elementary geometry shows that all trajecto-\nries of the inverse magnetic billiard have both inner and outer caustics that are cir-\ncles of radii rinner =Rjcos(\u00120)j=Rju0jandrouter =\u0016+p\nR2+\u00162\u00002R\u0016cos(\u00120),\nrespectively. All of the trajectories in Figure 6 have both inner and outer circular\ncaustics.\nOur \frst result in this regard is an inverse magnetic version of Mather's theorem\n([13], [14]): If a billiard table with a smooth convex boundary curve has a point of\nvanishing curvature, then the billiard inside the curve has no caustics.\nTheorem 2. If the boundary of the billiard table @\nhas a point of vanishing\ncurvature and \u0016<\u001amin, the inverse magnetic billiard has no interior caustics.\nProof. By Birkho\u000b's Theorem ([3]), an invariant curve of an area-preserving twist\nmap is a graph of some function. If our billiard has a caustic, then we have a\none-parameter family of chords P0P1to \u0000 corresponding to points on the invariant\ncurve. The graph property of Birkho\u000b's Theorem implies that if P\u0003\n0P\u0003\n1is a nearby\nchord such that P\u0003\n0has moved along \u0000 in the positive direction from P0thenP\u0003\n1hasON THE DYNAMICS OF INVERSE MAGNETIC BILLIARDS 11\n(a)\n (b)\n(c)\n (d)\n(e)\n (f)\nFigure 6. Periodic orbits of period 9 in the unit circle with \u0016<\n\u001amin: (a) (1;9) orbit; (b) (2 ;9) orbit; (c) (4 ;9) orbit; (d) (5 ;9)\norbit; (e) (7 ;9) orbit; (f) (8 ;9) orbit. The dots along the circle are\nthe pointsPiwhile the other dots are the centers of the Larmor\narcs.12 SEAN GASIOREK\nmoved in the positive direction from P1on \u0000. These chords must intersect in the\ninterior of \u0000, and by the existence of the caustic, must be tangent to the caustic.\nFigure 7. Picture of the proof of the nonexistence of caustics if\nthe boundary has a point of vanishing curvature.\nAssume an interior caustic e\u0000 exists. The billiard portion of a trajectory forms a\nchordP0P1tangent to the caustic, moves along its magnetic arc, and reenters \n to\nform the next chord P2;P3, also tangent to the caustic. Suppose the curvature \u0000\nasP2vanishes. Consider an in\fnitesimally close chord P\u0003\n0P\u0003\n1, tangent to the same\ncaustic, as described earlier, along with its next chord P\u0003\n2P\u0003\n3. Since the curvature\natP2vanishes, the tangent line at P\u0003\n2is, in the linear approximation, the same as\nthe one at P2. Let\u00122and\u0012\u0003\n2be the angle between the linear approximation and\nthe chordsP2P3andP\u0003\n2P\u0003\n3, respectively.\nThere are three geometrically distinct cases. If \u001f<\u0019\n2, then\u00122>\u0012\u0003\n2, and so the\nchordsP2P3andP\u0003\n2P\u0003\n3will not intersect in the interior of \u0000, a contradiction. See\nFigure 7. Similarly, if \u001f >\u0019\n2,\u00122> \u0012\u0003\n2. And if\u001f=\u0019\n2, then the chords P2P3and\nP\u0003\n2P\u0003\n3are parallel and will not intersect. \u0003\nTo better understand the nature of caustics in this inverse magnetic billiard set-\nting, we seek to understand the maps T1andT2near the boundary, as they show\nqualitatively di\u000berent behavior. We also make the adjustment to the maps T,T1,\nandT2so they are de\fned on the annulus R=LZ\u0002[0;\u0019] so the second variable is\n\u0012iinstead ofui.\nLazutkin produced a well-known calculation of the Taylor expansion of the bil-\nliard mapT1up to fourth order in \u0012([12]) and Berglund and Kunz calculate the\nTaylor expansion of the inner magnetic billiard map T\u0003\n2up to \frst order in \u0012([2]).\nWhile Lazutkin proved the existence of a positive measure set of caustics su\u000eciently\nclose to the boundary, Berglund and Kunz show the existence of caustics in inner\nmagnetic billiards for three special cases by citing a version of the KAM theorems\n([17], [5]).\n6.2.Mimicking the Approach of Berglund and Kunz. We can investigate\nthe behavior of the outer magnetic billiard map T2using the same techniques in\n[2], and ultimately learn about T. For a nonzero magnetic \feld near the boundary,\nwe will be able to apply KAM theorems to show the existence of invariant curves.\nBy adapting the proof of the Taylor expansion of the inner magnetic billiard map\nT\u0003\n2in section 5.2 of [2] to the outer magnetic billiard map T2, we arrive at a similarON THE DYNAMICS OF INVERSE MAGNETIC BILLIARDS 13\nexpression. Coe\u000ecients are also calculated in appendix C directly. We state this\nresult in the following proposition.\nProposition 3. If the boundary @\nisCk, the outer magnetic billiard map T2is\nCk\u00001for small sin(\u00121)and has the form\ns2=s1+2\u0016sin(\u00121)\n1\u0000\u0016\u00141cos(\u00121)+o(sin(\u00121)) modL\n\u00122=\u00121+o(sin(\u00121)):\nTherefore, near u=\u00001, the map is of the form\ns2=s1+2\u0016\n1\u0000\u0016\u00141\u00121+o(\u00121) modL\n\u00122=\u00121+o(\u00121)\nand nearu= 1, writing\u0012i=\u0019\u0000\u0011ithe map is of the form\ns2=s1+2\u0016\n1 +\u0016\u00141\u00111+o(\u00111) modL\n\u00112=\u00111+o(\u00111):\nWe must be cautious as there are two properties we must check with regards\nto the map above. First, the map must be well-de\fned (i.e. the denominators\nmay not vanish). This is only an issue when \u0012\u001c1. The second is that the outer\nmagnetic billiard map must denote the correct intersection of the magnetic arc with\nthe boundary of our billiard table. This is only an issue if a magnetic arc intersects\nthe boundary in more than two places.\nDe\fnition 1. A closedCk,k\u00152, planar curve \u0000is said to have the \u0016-intersection\nproperty for some\u0016>0if any circle of radius \u0016intersects \u0000at most twice.\nHowever, a su\u000ecient condition for the \u0016-intersection property to be satis\fed is\nfor either\u0016 < \u001aminor\u001amax< \u0016 (Corollary to Lemma 3 in Appendix D of [2]).\nWhen satis\fed, there is a one-to-one correspondence between inner magnetic bil-\nliard trajectories and outer magnetic billiard trajectories: For every outer magnetic\narc there is a \\dual trajectory\" that is the complementary arc which completes the\nLarmor circle. This complementary arc can be interpreted as an inner magnetic\nbilliard map with no change to our magnetic \feld convention. See Figure 8.\nTherefore determining the correct intersection point from our map is only an\nissue when \u001amin< \u0016 < \u001amax, as a Larmor circle in this case may intersect @\n in\nmore than two places.\nIf we consider the three curvature regimes, we notice the following:\n(1) If\u0016<\u001amin, then\u0016\u0014(s)\u0014\u0016\u0014max<1, so 0<1\u0000\u0016\u0014(s) for alls;\n(2) If\u001amin<\u0016<\u001amax, then\u0014min\n\u0014max<\u0016\u0014min<1<\u0016\u0014max<\u0014max\n\u0014min;\n(3) If\u001amax<\u0016, then 1<\u0014min\u0016\u0014\u0014(s)\u0016, so 1\u0000\u0016\u0014(s)<0 for alls.\nThe denominators of the coe\u000ecients in the theorem above are well-de\fned in cases\n(1) and (3), but potentially not de\fned in case (2).\nProposition 4. The inverse magnetic billiard map Tadmits the following Taylor\nexpansion for \u0012inear 0:\nsi+2=si+2\n\u0014i(1\u0000\u0016\u0014i)\u0012i+O(\u00122\ni) modL14 SEAN GASIOREK\nFigure 8. The duality of the magnetic billiard map: every inner\nmagnetic billiard trajectory has a corresponding outer magnetic\nbilliard trajectory, provided the \u0016-intersection property is satis\fed.\n\u0012i+2=\u0012i+O(\u00122\ni)\nwhere we have omitted the dependence upon siand that\u0014i:=\u0014(si)\u0019\u0014(si+1).\nFor\u0012i=\u0019\u0000\u0011iwith\u0011inear 0, the map Tadmits the Taylor expansion\nsi+2=si\u00002\n\u0014i(1 +\u0016\u0014i)\u0011i+O(\u00112\ni) modL\n\u0011i+2=\u0011i+O(\u00112\ni):\nAn outline of the proof is given in appendix C. First we observe that both\nversions of this map are not well-de\fned if the curvature is allowed to vanish,\nwhich is consistent with our version of Mather's result. Further, consider the\ntwo limiting cases of \u0016: if\u0016!1 , the map Tlimits tos2=s0+O(\u00122\n0), and\n\u00122=\u00120+O(\u00122\n0) which the identity map to \frst order; And if \u0016!0+, the mapT\nlimits tos2=s0+2\n\u00140\u00120+O(\u00122\n0), which is the standard billiard map to \frst order.\nThis is consistent with the geometric observations via the generating function in\nSection 3.\nWe may now make comments about the maps above in the style of [2].\n(1) Nearu=\u00001, the map Thas the form\nsi+2=si+2\n\u0014i(1\u0000\u0016\u0014i)\u0012i+O(\u00122\ni) modL\n\u0012i+2=\u0012i+O(\u00122\ni):\nWe have already observed that the denominator will not vanish in two cases:\n\u000fIf\u0016<\u001amin, then we make the change of variables 'i=si\u0000\u0016\u001ciand\nri= 2\u001ai\u0012ito make the map of the form\n'2='0+r0+O(r2\n0) modL\u00002\u0019\u0016\nr2=r0+O(r2\n0):ON THE DYNAMICS OF INVERSE MAGNETIC BILLIARDS 15\nThis corresponds to the correct intersection of the magnetic arc with\nthe boundary, as this trajectory corresponds to a small billiard chord\nforward plus a small skip forward along the boundary from the outside.\n\u000fIf\u001amax<\u0016, then we make the change of variables 'i=\u0016\u001ci\u0000siand\nri= 2\u001ai\u0012ito make the map of the form\n'2='0\u0000r0+O(r2\n0) mod 2\u0019\u0016\u0000L\nr2=r0+O(r2\n0):\nAgain, this is the correct intersection with the boundary, because a\nlarge magnetic arc will reenter \n \\behind\" its exit point.\n(2) Nearu= 1, the map Thas the form\nsi+2=si\u00002\n\u0014i(1 +\u0016\u0014i)\u0011i+O(\u00112\ni) modL\n\u0011i+2=\u0011i+O(\u00112\ni)\nwhere we have written \u0012i=\u0019\u0000\u0011i. Observe that the denominator can\nnever vanish, so this approximation is valid for all three curvature regimes.\nMoreover, this map can be understood as a short interior billiard chord\nbackwards followed by most of a circular magnetic arc forwards, ultimately\nresulting in traveling a small distance backwards. The change of variables\n'i=si+\u0016\u001ciandri= 2\u001ai\u0011iturns the map into\n'2='0\u0000r0+O(r2\n0) modL+ 2\u0019\u0016\nr2=r0+O(r2\n0):\nEach of these three maps can be interpreted via KAM theorems ([5], pg. III-8 or\n[17] pg. 52), and to do so we want to brie\ry de\fne a relevant condition.\nDe\fnition 2. Let\u001b2R. We say\u001bsatis\fes the Diophantine condition if for\neveryp\nq2Q, there exists \r;\u00172R+such that\n\f\f\f\f\u001b\u0000p\nq\f\f\f\f\u0015\rq\u0000\u0017:\nTheorem 3. Consider the inverse magnetic billiard in a strictly convex set \nwith\nCkboundary,k\u00156. Consider the following cases:\n(1) if 0<\u0016<\u001amin, de\fne\u0010=\u0012,M=L\u00002\u0019\u0016, and\u0015= 1;\n(2) if\u001amax<\u0016<1, de\fne\u0010=\u0012,M= 2\u0019\u0016\u0000L,\u0015=\u00001;\n(3) or if 0<\u0016<1, de\fne\u0010=\u0019\u0000\u0012,M=L+ 2\u0019\u0016,\u0015=\u00001.\nThen there exists \u000f>0depending upon \u0016andkwith the following signi\fcance: if\n!2[0;\u000f)and satis\fes the Diophantine condition, then there is an invariant curve\nof the form\ns=\u0018+V(\u0018)\n\u0010=!\n2\u0016+U(\u0018);\nwhereU;V2C1,V(\u0018+M) =V(\u0018) +L\u0000M,U(\u0018+M) =U(\u0018). The induced map\non this curve has the form\n\u00187!\u0018+\u0015!:16 SEAN GASIOREK\n(a)\n-3 -2 -1 1 2 3\n-2-112 (b)\n1 2 3 4 5 6s\n-1.0-0.50.00.51.0u\n(c)\n-10 -5 5 10\n-10-5510 (d)\n1 2 3 4 5 6s\n-1.0-0.50.00.51.0u\n(e)\n-4 -2 2 4\n-3-2-1123 (f)\n1 2 3 4 5 6s\n-1.0-0.50.00.51.0u\nFigure 9. Caustics in an ellipse for the three valid regimes: (a)\nnearu=\u00001 and\u0016 < \u001amin; (c) nearu=\u00001 and\u001amax< \u0016; (e)\nnearu= 1; and their accompanying invariant curves in the ( \u001e;u)-\nplane, (b), (d), (e), respectively. The centers of the Larmor circles,\nfoci of the ellipse, and points Piare shown.ON THE DYNAMICS OF INVERSE MAGNETIC BILLIARDS 17\nSimilarly to the case of inner magnetic billiards, our theorem con\frms the existence\nof invariant curves in three cases (see Figure 9):\n(1) Nearu=\u00001,\u0012\u00190 in a strong magnetic \feld, \u0016<\u001amin. These correspond\nto short billiard chords plus short magnetic arcs, keeping the particle's\ntrajectory near the boundary.\n(2) Nearu=\u00001,\u0012\u00190 in a weak magnetic \feld, \u001amax<\u0016<1. These corre-\nspond to short billiard chords followed by long magnetic arcs encompassing\n\n and reentering behind the original starting point, still near the boundary.\n(3) Nearu= 1,\u0012\u0019\u0019for all values of the magnetic \feld. These correspond\nto backwards billiard chords followed by most of a magnetic arc, reentering\nclose to the starting point and staying near the boundary.\nThis approach does not give us any indication of the behavior of the map for the\nintermediate curvature regime, \u001amin<\u0016<\u001amaxnearu=\u00001. While numerically\nwe do not observe any invariant curves in this region in this case, we do not have\nde\fnitive proof. This is also the case in inner magnetic billiards. Moreover, our\ntheorem also indicates that provided we have a su\u000eciently smooth strictly convex\nboundary (at least C6), inverse magnetic billiards are not ergodic.\n7.Conclusions and Next Steps\nWe have found that inverse magnetic billiards shares some similarities with stan-\ndard and magnetic billiards while also showing concrete di\u000berences. The in\ruence\nof the magnetic \feld on the dynamics is signi\fcant, and we have clearly seen that\ninverse magnetic billiards is a nontrivial perturbation of the standard billiard.\nThe behavior of inverse magnetic billiards in the regimes \u001amin<\u0016<\u001amaxand\n\u001amax<\u0016are not well understood at this time. For example, numerical simulations\nseem to show the existence of a C0caustic comprised of piecewise C1curves. Of\nfurther interest is the locus of the centers of the Larmor circles in such a case, as\nsometimes these centers appear to lie on a smooth simple closed curve with two\naxes of symmetry. See Figure 10.\nAnother aspect of inverse magnetic billiards that has not been studied is the\nexistence of outer caustics. Figures 6, 9, 10 all show the existence of caustics\noutside of \n, and this phenomena is certainly worth investigating.\n8.Acknowledgements\nThe author is grateful to Richard Montgomery, Serge Tabachnikov, and Alfonso\nSorrentino for their repeated useful discussions. The author also wishes to thank\nN. Berglund and H. Kunz for their exposition [2], as they provided a thorough\nroadmap to follow when approaching this problem. This material is based upon\nwork supported by the National Science Foundation under Grant No. DMS-1440140\nWhile the author was in residence at the Mathematical Sciences Research Institute\nin Berkeley, California, during the Fall 2018 semester. A portion of this work was\nalso supported by Grant No. DP190101838 from the Australian Research Council.\nAppendix A.Proof of Proposition 1\nThe components of DT1are well-known (e.g. [8], Theorem 4.2 in Part V or [11]).\nWe provide an outline of the computation of the components of DT2.18 SEAN GASIOREK\n(a)\n (b)\n(c)\n (d)\nFigure 10. C0caustics in an ellipse for the two non-twist curva-\nture regimes and 103iterations of T: (a)\u001amin< \u0016 < \u001amax; (c)\n\u001amax< \u0016; and their accompanying invariant curves in the ( \u001e;u)-\nplane, (b), (d), respectively. The locus of centers of the Larmor\ncircles appear to lie on a continuous curve.\nConsider a single magnetic arc, as in Figure 2a. De\fne\n\u000b1= arg[(X2\u0000X1) +i(Y2\u0000Y1)];\nthe polar angle between the positive x-axis and the segment P1P2. NearP1we see\nthat\u001c1\u0000\u00121=\u000b1\u0000\u001f. A similar picture centered on P2tells us that \u001c2=\u000b1+\u001f\u0000\u00122.\nThis leads to the following equations:\n\u00121=\u001c1\u0000\u000b1+\u001f\n\u00122=\u000b1+\u001f\u0000\u001c2:ON THE DYNAMICS OF INVERSE MAGNETIC BILLIARDS 19\nBy construction, we have\n`2\n2= (X2\u0000X1)2+ (Y2\u0000Y1)2\ntan(\u000b1) =Y2\u0000Y1\nX2\u0000X1\nThese equations imply\n@\u000b1\n@s1=1\n`2\n2[(Y2\u0000Y1)X0(s1)\u0000(X2\u0000X1)Y0(s1)]\n=1\n`2\n2[`2sin(\u000b1) cos(\u001c1)\u0000`2cos(\u000b1) sin(\u001c1)] =sin(\u001f\u0000\u00121)\n`2;\n@`2\n@s1=1\n2`2[2(X2\u0000X1)(\u0000X0(s1)) + 2(Y2\u0000Y1)(\u0000Y0(s1))]\n=\u0000[cos(\u001c1) cos(\u000b1) + sin(\u001c1) sin(\u000b1)] =\u0000cos(\u00121\u0000\u001f);\n@\u001f\n@s1=1\n`2cos(\u001f)`2\n2\u0016@`2\n@s1=\u0000sin(\u001f) cos(\u00121\u0000\u001f)\n`2cos(\u001f):\nNext, we di\u000berentiate the angle formulas for \u00121,\u00122with respect to s1to get\n@\u00121\n@s1=\u00141\u0000sin(2\u001f\u0000\u00121)\n`2cos(\u001f);@\u00122\n@s1=\u0000sin(\u00121)\n`2cos(\u001f):\nRepeating this process again but with respect to s2yields\n@\u00121\n@s2=sin(\u00122)\n`2cos(\u001f);@\u00122\n@s2=sin(2\u001f\u0000\u00122)\n`2cos(\u001f)\u0000\u00142:\nThe above quantities determine d\u00121andd\u00122as a function of ds1andds2. Solve\nthis linear system for ds2andd\u00122in terms of ds1andd\u00121. Lastly, writing dui=\nsin(\u0012i)d\u0012iwe obtain the components of DT2.\nAppendix B.Proof of Theorem 1\nThe proof that Tsatis\fes the twist condition@s2\n@u0>0 whenever \u0016 < \u001aminis a\nsmall exercise in geometry and trigonometry. The full proof can be found in the\nappendices of [6].\nNext, we derive our expression for the generating function. Following the proof\nfrom [2], we take our generating function\nG(s0;s2) =\u0000`1\u0000j\rj+1\n\u0016S\nand break it into a magnetic \feld-dependent component and a magnetic \feld-\nindependent component. Recall the notation used in Figure 1, and we proceed\nwithout the \\0 ;2\" subscripts. Write S=Area (A[S )\u0000A andArea (A[S ) to\nmean the area inside the circular arc \rcut by the chord P1P2. Then\nG(s0;s2) =\u0014\n\u0000`1\u00001\n\u0016A\u0015\n+\u0014\n\u0000j\rj+1\n\u0016Area (A[S )\u0015\n:\nFrom elementary geometry, we see that\n@A\n@s2=1\n2`2sin(\u001f\u0000\u00122):20 SEAN GASIOREK\nThus\n@G\n@s2= 0\u00001\n\u0016@A\n@s2\u0000@j\rj\n@\u001f@\u001f\n@s2+1\n\u0016@Area (A[S )\n@\u001f@\u001f\n@s2\n=\u0000sin(\u001f) sin(\u001f\u0000\u00122)\u0000cos(\u001f) cos(\u001f\u0000\u00122) =u2:\nAnd the calculation of the other partial derivative is simple since all factors of G\nexcept for`1do not depend upon s0. This is just the calculation from the standard\nbilliard map, so@G\n@s0=\u0000u0:Hence\nG(s0;s2) =\u0000`1\u0000j\rj+1\n\u0016S\nis the generating function.\nAppendix C.Proof of Proposition 4\nProof. We compute the coe\u000ecients in the Taylor expansion of the map T2and\nhence forT. Omitting the dependence on s, these terms are\n@s2\n@\u00121(s;0) =2\u0016\n1\u0000\u0016\u0014@s2\n@\u00121(s;\u0019) =\u00002\u0016\n1 +\u0016\u0014\n@\u00122\n@\u00121(s;0) = 1@\u00122\n@\u00121(s;\u0019) = 1\n@s2\n@\u00120(s;0) =2\n\u0014(1\u0000\u0016\u0014)@s2\n@\u00120(s;\u0019) =2\n\u0014(1 +\u0016\u0014)\n@\u00122\n@\u00120(s;0) = 1:@\u00122\n@\u00120(s;\u0019) = 1:\nFirst we compute@s2\n@\u00121(s1;\u00121) near\u00121= 0. Using the approximations \u001f\u0019\u001f\u0003(\u00121)\nfrom the previous appendix and applying the l'Hopital rule in the second equality,\nwe get\nL:= lim\n\u00121!0+`2cos(\u001f)\nsin(\u00122)= lim\n\u00121!0+@`2\n@s2@s2\n@\u00121cos(\u001f)\u0000`2sin(\u001f)@\u001f\n@\u00121\ncos(\u00122)@\u00122\n@\u00121\n= lim\n\u00121!0+cos(\u001f\u0000\u00122)@s2\n@\u00121cos(\u001f)\u0000`2sin(\u001f)@\u001f\n@\u00121\ncos(\u00122)h\nsin(2\u001f\u0000\u00122)\nsin(\u00122)\u0000\u00142`2cos(\u001f)\nsin(\u00122)i\n=L\n2c\u00001\u0000\u00141L:\nwherec=\u001a1\n\u001a1\u0000\u0016=1\n1\u0000\u0016\u00141. This tells us that\nL=L\n2c\u00001\u0000\u00141L:\nIt follows from the convexity of \u0000( s) and [8] (Theorem 4.3 in Part V) that L<1,\nsoL= 0 orL=2c\u00002\n\u00141. We wish to show that L>0. Consider the osculating circle\nO\u0000(s2) at \u0000(s2) with radius \u001a2. Then via elementary geometry, the length of the\nchord`2that is insideO\u0000(s2) is exactly 2 \u001a2sin(\u00122). Therefore `2\u00152\u001a2sin(\u00122), and\nso\nL\u00152 cos(\u001f)\u001amin>0:\nThis means L=@s2\n@\u00121(s;0) =2c\u00002\n\u00141=2\u0016\n1\u0000\u0016\u00141.ON THE DYNAMICS OF INVERSE MAGNETIC BILLIARDS 21\nNext, we see that\n@\u00122\n@\u00121(s1;0) := lim\n\u00121!0+@\u00122\n@\u00121(s;\u00121) = lim\n\u00121!0+sin(2\u001f\u0000\u00122)\nsin(\u00122)\u0000\u00142@s2\n@\u00121\n= lim\n\u00121!0+2c\u00001 +O(\u00122\n1)\u0000\u0014\u00122c\u00002\n\u0014\u0013\n= 2c\u00001\u0000(2c\u00002) = 1:\nUsing these two computations, one can use expressions for@s2\n@\u00120and@\u00122\n@\u00120to derive\nthe expressions near 0 from the summary. Repeating analogous calculations near\n\u0019produces the expressions in the summary above.\n\u0003\nReferences\n1. N Berglund, Billiards in a potential: variational methods, periodic orbits, and KAM tori ,\nPreprint, 1996.\n2. N. Berglund and H. Kunz, Integrability and ergodicity of classical billiards in a magnetic \feld ,\nJournal of Statistical Physics 83(1996), no. 1-2, 81{126.\n3. G.D. Birkho\u000b, Dynamical Systems , American Mathematical Society / Providence, RI, Amer-\nican Mathematical Society, 1927.\n4. Giulio Casati and Toma\u0014 z Prosen, Time Irreversible Billiards with Piecewise-Straight Trajec-\ntories , Phys. Rev. Lett. 109(2012), no. 17, 174101.\n5. Raphael Douady, Applications du th\u0013 eor\u0013 eme des tores invariants , Ph.D. thesis, Universit\u0013 e\nParis VII, 1982.\n6. Sean Gasiorek, On the dynamics of inverse magnetic billiards , Ph.D. thesis, University of\nCalifornia Santa Cruz, 2019.\n7. C. Gol\u0013 e, Symplectic Twist Maps: Global Variational Techniques , Advanced series in nonlinear\ndynamics, World Scienti\fc, 2001.\n8. Anatole Katok and Jean-Marie Strelcyn, Invariant manifolds, entropy and billiards: smooth\nmaps with singularities , Lecture notes in mathematics, no. 1222, Springer, Berlin, 1986,\nOCLC: 15018884.\n9. Bence Kocsis, Gergely Palla, and J\u0013 ozsef Cserti, Quantum and semiclassical study of magnetic\nquantum dots , Physical Review B 71(2005), no. 7, 075331.\n10. A. Korm\u0013 anyos, P. Rakyta, L. Oroszl\u0013 any, and J. Cserti, Bound states in inhomogeneous mag-\nnetic \feld in graphene: Semiclassical approach , Phys. Rev. B 78(2008), no. 4, 045430.\n11. V.V. Kozlov and D.V. Treshch ev, Billiards: A Genetic Introduction to the Dynamics of\nSystems with Impacts: A Genetic Introduction to the Dynamics of Systems with Impacts ,\nTranslations of Mathematical Monographs, American Mathematical Society, 1991.\n12. V F Lazutkin, The Existence of Caustics for a Billiard Problem in a Convex Domain , Math-\nematics of the USSR-Izvestiya 7(1973), no. 1, 185{214.\n13. John N. Mather, Glancing billiards , Ergodic Theory and Dynamical Systems 2(1982), no. 3-4,\n397{403.\n14. John N. Mather and Giovanni Forni, Action minimizing orbits in hamiltomian systems , Tran-\nsition to Chaos in Classical and Quantum Mechanics: Lectures given at the 3rd Session of\nthe Centro Internazionale Matematico Estivo (C.I.M.E.) held in Montecatini Terme, Italy,\nJuly 6-13, 1991 (Sandro Gra\u000e, ed.), Springer Berlin Heidelberg, Berlin, Heidelberg, 1994,\npp. 92{186.\n15. J. D. Meiss, Symplectic maps, variational principles, and transport , Reviews of Modern\nPhysics 64(1992), no. 3, 795{848.\n16. J. Moser, On Invariant Curves of Area-preserving Mappings of an Annulus , Nachrichten der\nAkademie der Wissenschaften in Gttingen: II, Mathematisch-Physikalische Klasse, Vanden-\nhoeck & Ruprecht, 1962.\n17. ,Stable and Random Motions in Dynamical Systems: With Special Emphasis on\nCelestial Mechanics (AM-77) , Princeton Landmarks in Mathematics and Physics, Princeton\nUniversity Press, 2016.\n18. Alain Nogaret, Electron dynamics in inhomogeneous magnetic \felds , Journal of Physics: Con-\ndensed Matter 22(2010), no. 25, 253201.22 SEAN GASIOREK\n19. M Robnik and M V Berry, Classical billiards in magnetic \felds , J. Phys. A: Math. Gen. 18\n(1985), no. 9, 1361{1378.\n20. Heung-Sun Sim, G Ihm, N Kim, and S J Lee, Magnetic Edge States in a Magnetic Quantum\nDot, Physical Review Letters 80(1998), no. 7, 4.\n21. Yu Song and Yong Guo, Bound states in a hybrid magnetic-electric quantum dot , Journal of\nApplied Physics 108(2010), no. 6, 064306.\n22. Z. V or os, T. Tasn\u0013 adi, J. Cserti, and P. Pollner, Tunable Lyapunov exponent in inverse mag-\nnetic billiards , Physical Review E 67(2003), no. 6, 065202.\nSchool of Mathematics and Statistics, Carslaw Building F07, University of Sydney,\nNSW 2011 Australia\nE-mail address :sean.gasiorek@sydney.edu.au"    },    {        "title": "1612.03670v1.Symbolic_Dynamics_of_Magnetic_Bumps.pdf",        "content": "Symbolic Dynamics of Magnetic Bumps\nAndreas Knauf\u0003, Marcello Seriy\nJune 8, 2022\nAbstract\nFornconvex magnetic bumps in the plane, whose boundary has a cur-\nvature somewhat smaller than the absolute value of the constant magnetic\n\feld inside the bump, we construct a complete symbolic dynamics of a\nclassical particle moving with speed one.\n1 Introduction and notation\nThe subject of chaotic scattering is mostly about scattering by obstacles and\nby potential bumps, see [Ga, Sm]. Here we consider the motion of a classical\nparticle in the plane under the in\ruence of a magnetic potential. In the case of\nmotion in the plane, this has the form\nB:=X\n`2AB`withB`:=b`1 lC`:R2!R;\nwith the alphabet A:=f1;:::;ng, for mutually disjoint, convex and compact\ndomainsC`\u0012R2withC2boundaries@C`, and \feld strengths b`2Rnf0g.\nThe Hamiltonian system (P;!;H )with phase space P:=TR2,\nH:P!R,H(q;v) =1\n2kvk2\nand (discontinuous) symplectic form\n!:=dq1^dv1+dq2^dv2+Bdq 1^dq2\n\u0003Department Mathematik, Universit at Erlangen-N urnberg, Cauerstr. 11, D{91058 Erlan-\ngen, Germany. e-mail: knauf@math.fau.de\nyDepartment of Mathematics and Statistics University of Reading Whiteknights, PO Box\n220, Reading RG6 6AX (UK) e-mail: m.seri@reading.ac.uk\n1arXiv:1612.03670v1  [math.DS]  12 Dec 2016give rise to the Hamiltonian vector \feld X:P!TP,iX!=dH. The\ncorresponding di\u000berential equation is\n_q=v,_v=B(q)Jv; withJ:= (0\u00001\n1 0):\nThe solution of the initial value problem for magnetic \feld b2Rnf0gthus\nequals\nq(t) =q0+1\nb\u0010\nsin(bt) cos(bt)\u00001\n1\u0000cos(bt) sin(bt)\u0011\nv0,v(t) =\u0010\ncos(bt)\u0000sin(bt)\nsin(bt) cos(bt)\u0011\nv0:\nSo the particle with energy H=Ehas speedp\n2E. Outside the bumps it moves\non a straight line, whereas inside the `thbump it moves on a segment of a circle\nof Larmor radiusp\n2E=jb`j. The sense of rotation is positive (counter-clockwise),\nifb`>0. Thet{invariant center of the circle is\nc`(v;q) :=q+b\u00001\n`Jv: (1.1)\nWithout loss of generality we \fx the energy to equal E:= 1=2, so that we\nget unit speed on the energy surface\n^\u0006 :=H\u00001(E)with projection \u0019:^\u0006!R2;(q;v)7!q:\n2 Single bumps\nConsider for `2Athe set ~C`of pointsc2C`whose minimal distance to the\nboundary@C`is larger thanjb`j\u00001. This is a compact convex set, possibly void.1\nWe remove from ^\u0006the sets \u0006`:=f(v;q)2^\u0006jc`(v;q)2~C`g. These compact\nsets are invariant under the 2\u0019=jb`j{periodic Larmor \row for magnetic \feld b`.\nAt pointsx2@\u0006`corresponding to Larmor circles in the con\fguration plane\nthat touch@C`in only one point, the boundary @\u0006`isC2. In particular this is\nthe case if the curvature \u0014`2C(@C`;[0;1))of@C`is strictly smaller than jb`j.\n2.1 De\fnition\n\u000fWe set\u0014`:= minx2@C`\u0014`(x)and\u0014`:= maxx2@C`\u0014`(x)(so0\u0014\u0014`\u0014\u0014`>0).\n\u000fIfjb`j<\u0014`for all`2A, the system has weak \felds .\n\u000fIfjb`j>\u0014`for all`2A, the system has strong \felds .\n\u000fForn\u00152we setd`:= minm2Anf`gminx2C`;y2Cmkx\u0000yk(sod`>0).\n\u000fIfjb`j>1=(d`\u000bmin) + 2\u0014`for all`2A, the system has very strong \felds .2\n1If~C`has positive area, its boundary is Lipschitz. However, e.g. for a stadion C`composed\nof two half-disks of radius 1=jb`jand a rectangle, ~C`is a line segment.\n2Here\u000bminis a constant, related to the arrangement of the bumps in the plane, that will\nbe discussed in more details in Section 4.\n22.2 Example (Single disks)\nForn= 1 and a diskC\u0011C1of radiusr>0the curvature of @Cis constant and\nequals 1=r. So the \feld B=b1 lCis weak i\u000bjbj<1=rand strong i\u000bjbj>1=r.\nCompare Figures 2.2 and 2.3 for dynamics near such a bump. \u0006\nWe will mainly consider the (very) strong \felds cases.\nThe following lemma relates strong \felds to some property of a single bump.\nOur convention for the 'Jacobi equation'3along the trajectory is to use the\northonormal basis e1(t); e2(t)ofTc(t)R2, given bye1(t) :=v(t)ande2(t) :=\nJe1(t). Then writing a vector \feld along cin the formt7!I(t)e1(t)+J(t)e2(t),\nwe obtain the linearized \row with\nJ(t) =\u0000sin(bt)I(0) + cos(bt)J(0)\u0000(1\u0000cos(bt))_I(0)=b+ sin(bt)_J(0)=b\nand\n_J(t) =\u0000bcos(bt)I(0)\u0000bsin(bt)J(0)\u0000sin(bt)_I(0) + cos(bt)_J(0):\n2.3 Lemma\nIn the case of a strong \feld, along a trajectory c:I!R2entering the bump\natc(0)2@C`and leaving it at time T >0(c(T)2@C`), the parallel incoming\nJacobi \feld4\u0000\nJ(0);_J(0)\u0001\n= (1;0)has outgoing data J(T)<0,_J(T)<0.\nProof:\u000fConsider the unique disk Dof radiusr:= 1=jb`j, whose boundary is\ntangent to@C`atc(0)and which is entered by the trajectory at time t= 0. As\nthe curvaturejb`jof its boundary is strictly larger than the curvature of @C`, we\nhaveD\u0012C`, andc(0)is the only intersection between @Dand@C`. We denote\nthe center ofDbyz2R2, see Figure 2.1, left.\n\u000fWe claim that all trajectories d:I!R2enteringDat timet= 0 with\nvelocity _d(0) = _c(0)leaveDat the same point f:=z+J_c(0)=b`:\nClearlyd(0)is a solution xof the equations\nkx\u0000zk2=r2andkx\u0000d(0)\u0000J_c(0)=b`k2=r2\nof@Dand the Larmor circle (1.1). But the second intersection of these two\ncircles equals f.\n\u000fThe union of the above orbit family is a Lagrangian manifold, and the vector\ninTx\u0006,x:= (c(0);_c(0)) given Jacobi \feld data\u0000\nJ(0);_J(0)\u0001\n, is tangent to it.\nSo the Jacobi \feld along cturns vertical at f=c(tf), that is,J(tf) = 0 and\n3This is an abuse of language, since Hamilton's equation is not geodesic.\n4We assume w.l.o.g. that the component parallel to the direction _c(0)vanishes.\n3_J(tf)<0. Butf, being a boundary point of Ddi\u000berent from c(0), is an interior\npoint ofC`, so thattf2(0;T).\n\u000fConsider the lineTinR2that is tangent to @C`atc(0). If we denote byD\nthe image ofDunder the re\rection by the line through f, perpendicular to T\n(see Figure 2.1, right), then the Larmor circle through c(0)andfintersectsT\nfor the second time at the unique point in T\\@D. Again by re\rection symmetry,\nat that point the Jacobi \feld along cis parallel.\n\u000fAsC`is strictly convex, it lies entirely on one side of T. So by a comparison\nargument (see again Figure 2.1, right), J(T)<0. \u0003\nFigure 2.1: Proof of Lemma 2.3\n2.4 Remark IfC`is a disk, in the strong \feld case we conclude from Lemma\n2.3, that for parallel incoming trajectories the envelope of the solution curves is\na half-circle of radius r`\u00001=jb`j, see lower part of Figure 2.2. \u0006\n3 The degree for weak and strong \felds\nScattering by a bump is qualitatively di\u000berent for weak and for strong force \felds.\nTo see this, consider oriented lines in con\fguration space R2. The set of these\nlines is naturally isomorphic to the cylinder TS1\u0018=S1\u0002R.\nWe assume that there is just one bump (so n= 1), and consequently omit\nsubindices. There is a problem in de\fning the \row on \u0006in the case of glancing\ntrajectories, that is, trajectories tangent to @C(consider the uppermost trajectory\nin Figure 2.2). Either we continue these incoming rays by just extending them\n4Figure 2.2: Dynamics for a disk Cand a strong \feld (with b=\u00002r)\nFigure 2.3: Dynamics for a disk Cand a weak \feld (with b=\u0000r=2)\n5to a straight line, or we extend them by a segment of the Larmor circle in the\nbump. This is a complete circle for a strong \feld b. So in that case we could\nattach the outgoing ray with the incoming direction.\nNone of these prescriptions leads to a continuous \row. However, it is impor-\ntant to notice that either prescription leads to a continuous map called\nS:TS1!TS1;\nsending incoming to outgoing oriented lines.5\nAs shown in [Kn], see also [KK], such a map de\fnes a topological index\ndeg(C;b)2Z. Using the bundle projection \u0019:TS1!S1, it can be de\fned\nas follows. A family of incoming rays with the same direction is parameterized\nby the value of their angular momentum L: \u0006!R;(q;v)7!hJq;vi. This is\nobviously \row-invariant outside the bump. Smaps initial to \fnal pairs, consisting\nof directions in S1and angular momenta in R.\nGiven the initial direction '2S1, the continuous map\nT'S1\u0018=R!S1,`!\u0019\u000eS(`;')\ncan be uniquely completed (using the Alexandrov compacti\fcation R[f1g\u0018=S1\nofR) to a continuous map S1!S1. The degree of that map is independent of\n'2S1and is called the scattering degree .\n3.1 Lemma For a single bump C, the scattering degree deg(C;b)equals\n\u000fzero, for weak \felds b,\n\u000fsign(b), for strong \felds b.\nProof: The total curvature of a trajectory equals the signed angular length of\nthe segment of its Larmor circle. This is obviously zero if the trajectory does not\nintersectC. It is bounded away from 2\u0019in absolute value for weak \felds.\nFor strong \felds b>0it equals +2\u0019exactly for the glancing trajectories inter-\nsecting@CwithCon their left hand side. It then decreases to zero when the\nangular momentum value of the incoming rays it decreased, until one arrives at\na glancing trajectory with Con its right hand side.\nFor strong \felds b<0the sides are interchanged. \u0003\n5For \felds that are neither weak nor strong, it can happen that Scannot be de\fned\ncontinuously.\n64 A strictly invariant cone \feld\nAssumption:\nWe assume that no three convex sets C`\u0012R2lie on a line. For n\u00153\n\u000bmin:= min\nk6=`6=m6=k2Aminn\narccos\u0010D\ny\u0000x\nky\u0000xk;y\u0000z\nky\u0000zkE\u0011\njx2Ck;y2C`;z2Cmo\n;\nand forn= 2 we set\u000bmin:=\u0019=3, say. As we assumed that no three bumps are\nintersected by a line, the angle \u000bmin2(0;\u0019=3]is positive. For a segment of a\nsolution curve that intersects Ck,C`andCmin succession, the total curvature\ninsideC`is bounded below by \u000bmin. So the length of that sub-segment is at\nleast\u000bmin=jbmj.\nWe now consider in the very strong force regime the set \u0003\u0012\u0006of initial\nconditions belonging to bounded orbits. We denote by\nN(x)2TxR2(`2A;x2@C`)\nthe unit outward normal vectors of C`.\nH\u0006:=S\n`2AH`\u0012\u0006is the disjoint union of the Poincar\u0013 e surfaces\nH\u0006\n`:=f(v;q)2\u0006jq2@C`;\u0006hv;N(q)i>0g:\nFor arbitrary \felds, the \row induces a Poincar\u0013 e map P(i):H\u0000!H+internal\nto the bumps, which is a di\u000beomorphism. On the other hand, there are maximal\nopen subsets V\u0006\u0012H\u0006so that the \row gives rise to a di\u000beomorphism P(e):\nV+!V\u0000, external to the bumps. Setting U\u0000:= (P(i))\u00001(V+), we obtain a\ndi\u000beo\nP:=P(e)\u000eP(i):U\u0000!V\u0000:\n4.1 Lemma Assume that the \felds b`obey the very strong \felds inequalities\njb`j>1=(d`\u000bmin) + 2\u0014`(`2A): (4.1)\nThen the cone \felds\nC`(x) :=f\u0015(l)e(l)+\u0015(u)e(u)j\u0015(l);\u0015(u)2R;\u0015(l)\u0001\u0015(u)>0g(x2H`);\nde\fned by the tangent vectors e(l);e(u)2TH`,\ne(l):= (1\n0),e(u):=\u0000d`\n1\u0001\nare strictly invariant under the linearized Poincar\u0013 e map TP.\n7Proof:\n\u000fAccording to Lemma 2.3, the vector e(l)2TxU\u0000(withx2H\u0000\n`) is mapped\nbyTxPto a vector TyP(e(l))2TyV\u0000(withy:=P(x)2H\u0000\nm) of the form\n(1d\n0 1)\u0000J\n_J\u0001\nwithJ;_J < 0,dbeing the length of the trajectory segment between\nthe points\u0019\u000eP(i)(x)of exit from C`resp.\u0019\u000eP(x)of entrance in Cm.\nThis shows that e(l)is contained in the cone \feld C`(x).\n\u000fWe want to show a similar statement for the vector e(u)2TxU\u0000, but we choose\nto work backwards in time. So we consider the vector e(u)=\u0000dm\n1\u0001\n2TyU\u0000\nand show that its preimage Ty(P)\u00001(e(l))isnot contained inC`(x). First of\nall,Ty(P(e))\u00001(e(l)) =\u0000dm\u0000d\n1\u0001\nhas non-positive \frst entry, since Ty(P(e))\u00001=\n(1\u0000d\n0 1)andd\u0015dm. For comparison we consider the vector (0\n1)2Tx0with\nx0:=P(i)(x)and show that its preimage Tx0(P(i))\u00001\u0000\n(0\n1)\u0001\nis not contained in\nC`(x).\nWe choose the disk E\u0012C`of radius 1=\u0014`, whose boundary is tangent to @C`\natx0. Likewise,D\u0012E is the disk of radius 1=jb`jwhose boundary is tangent to\n@C`(and@E) atx0; see Figure 4.1.\nLike in the proof of Lemma 2.3, at the second intersection x00of@Dwith the\nLarmor circle (gray in Figure 4.1), the family of Larmor solutions crossing at\nx0has become parallel. We must show that by following the corresponding\nJacobi \feld backwards from x00tox, it is not contained in the cone \feld C(x).\nAlthough we have no direct information about the length of the Larmor circle\nsegment between x00andx, we know (by de\fnition of \u0014`) that it is longer than\nthe one between x00and the intersection x000with@E.\nSo we must bound the length of that arc from below. The length of the arc\nbetweenx0andx000equalsjb`j\u000b,\u000bbeing the angle between the normal N(x0)\nand the center Lof the Larmor circle, seen from x0.\nThe Larmor angle between x0andx000equals 2 arctan[rsin(\u000b)=(1+rcos(\u000b))],\nwithr:=\u0014`=jb`jbeing the ratio between the radii of @Dand@E. This identity\nfollows when considering the line through Land the center eofE. That line\nbisects the Larmor angle. By elementary trigonometry the angle between that\nline and the one through eandx0equals arctan[rsin(\u000b)=(1 +rcos(\u000b))].\nThus the length of the Larmor arc between x00andx000equals\nf(\u000b) :=\u000b\u00002 arctan[rsin(\u000b)=(1 +rcos(\u000b))];\nmultiplied by its radius 1=jb`j. Asf(0) = 0 ,f0(\u000b) =1\u0000r2\nr2+2rcos(\u000b)+1withf0(0) =\n1\u0000r\n1+r>0andf00(\u000b) =r(1\u0000r2)sin(\u000b)\n(r2+2rcos(\u000b)+1)2\u00150, we can bound ffrom below by\nf(\u000b)\u00151\u0000r\n1+r\u000b=jb`j\u0000\u0014`\njb`j+\u0014`\u000b (\u000b2[0;\u0019]): (4.2)\nAs the Larmor arc belongs to a solution segment that intersects Ck,C`andCm\nin consecution, the argument of fis bounded below by \u000b\u0015\u000bmin.\n8We have to check, whether then the inequality\nd\u00001\n`<tan(jb`jf(\u000b))\nholds true. By (4.2) we check the stronger inequality\nd\u00001\n`<tan\u0010\njb`jjb`j\u0000\u0014`\njb`j+\u0014`\u000b\u0011\n:\nIn fact, we argue that even\nd\u00001\n`<jb`jjb`j\u0000\u0014`\njb`j+\u0014`\u000b:\nThe corresponding equation is quadratic in x:=jb`jand has the positive solution\nx=1\n2(c+\u0014`+p\nc2+ 6c\u0014`+\u00142\n`)\u0014c+ 2\u0014`withc:= 1=(d`\u000bmin). \u0003\nFigure 4.1: Proof of Lemma 4.1\n5 Symbolic dynamics\nWe equip for the (discretely topologized) alphabet A=f1;:::;ngthe sequence\nspace\n\u0004A:=fa:Z!Aj8k2Z:ak+16=akg\nwith the product topology. Then the shift \u001b: \u0004A!\u0004A,\u001b(a)k=ak+1is known\nto be a homeomorphism.\n95.1 Theorem Let the very strong \felds assumption (4.1) be valid for the n\u00152\nbumps. Then every bounded trajectory c:R!\u0006intersectsHin\fnitely often.\nThe set \u0004 := \u0003\\H is homeomorphic to \u0004A, under the homeomorphism6\n\b : \u0004A!\u0004,a7!\\\nk2Z\b\nPk(x)jx2Ha0:Pk(x)is de\fned and inHak\t\n(5.1)\nto one-point sets. \bis a conjugacy for the Poincar\u0013 e map Pand the shift map\n\u001b, i.e.\n\b\u000e\u001b=P\u000e\b:\nProof:\u000fIt is generally true that for a continuous, strictly invariant cone \feld\nthere is at most one element in each set \b(a)of (5.1).\n\u000fThen we employ the technique of [Kn, Chapter 6] to show that the sets de\fning\n\b(a)are non-void. According to Lemma 3.1 we have degrees deg(C`;b`) =\nsign(b`)2f\u0000 1;1gin the strong \feld case. Existence and non-vanishing of the\ndegree is the condition of [Kn, Theorem 6.1].\n\u000fThe conjugacy property follows, as Pk(P(x))2H\u001b(a)kis equivalent toPk+1(x)\n2Hak+1and toPk(x)2Hak(k2Z). \u0003\n5.2 Remarks\n1. The result is quite di\u000berent from the one in [KSS]. There, for each of the\ncontinuous radially symmetric bumps, one had assumed existence of an (un-\nstable) periodic orbit. This then lead to arbitrarily large winding numbers.\nThe symbolic dynamics of scattering then gave rise to a semi conjugacy,\nsee [KSS, Theorem 3.4].\n2. Similarly, one may consider trapped orbits (with limt!\u00061kq(t)k=1but\nlim supt!\u00071kq(t)k<1and scattering orbits (with limt!+1kq(t)k=\nlimt!\u00001kq(t)k=1). There one uses half-in\fnite respectively \fnite\nsymbol sequences. Prescribing initial and/or \fnal directions of these orbits,\none again obtains symbolic conjugacies. Then, however, one has to exclude\ndirections that are given by oriented lines meeting two domains C`,Cm.\u0006\nReferences\n[Ga] Pierre Gaspard: Chaos, scattering and statistical mechanics. Cambridge\nUniversity Press, 2005\n6identifying one-element sets with their elements\n10[KK] Andreas Knauf, Markus Krapf: The non-trapping degree of scattering.\nNonlinearity 21, 2023{2041 (2008)\n[KSS] Andreas Knauf, Frank Schulz, Karl Friedrich Siburg: Positive topological\nentropy for multi-bump magnetic \felds. Nonlinearity 26, 727{743 (2013)\n[Kn] Andreas Knauf: Qualitative Aspects of Classical Potential Scattering. Reg-\nular and Chaotic Dynamics, 4, No.1, 1{20 (1999)\n[Sm] Uzy Smilansky: The Classical and Quantum Theory of Chaotic Scattering.\nLecture Notes. Les Houches, 1989\n11"    },    {        "title": "2201.04396v1.The_angular_dependence_of_magnetization_dynamics_induced_by_a_GHz_range_strain_pulse.pdf",        "content": "arXiv:2201.04396v1  [physics.app-ph]  12 Jan 2022Japanese Journal of Applied Physics REGULAR PAPER\nThe angular dependence of magnetization dynamics induced b y a GHz range strain pulse\nKakeru Tojo, Akira Nagakubo,∗and Hirotsugu Ogi\nGraduate School of Engineering, Osaka University, Suita, O saka 565-0871, Japan\nThe dynamics of magnetization is important in spintronics, where the coupling between phonon and magnon attracts much a ttention. In this\nwork, we study the angular dependence of the coupling betwee n longitudinal-wave phonon and magnon. We investigated the magnetization\ndynamics using the time-resolved magneto-optical Kerr effe ct, which allows measuring spin-wave resonances and the mag netic echo signal.\nThe frequency, mode number, and amplitude of the spin-wave r esonance change with the out-of-plane angle of the external magnetic field.\nThe amplitude of the magnetic echo signal caused by the strai n pulse also changes with the angle. We calculate these angul ar dependences\nbased on the Landau-Lifshitz-Gilbert equation and find that the angles of the external field and magnetic moment are impor tant factors for\nthe phonon-magnon coupling when phonon propagates in the th ickness direction under the out-of-plane magnetic field.\n1. Introduction\nThe interaction between phonon and magnon such as spin-\nwave,1)magnetic resonance,2)spin current,3)spin relaxation,4)\nand so on, has attracted great attention. In particular, rec ent\nlithography techniques enable us to excite GHz-range surfa ce\nacoustic waves (SAWs), whose frequencies can be matched\nto those of the ferromagnetic resonance (FMR). SAW causes\nthe FMR with the same frequency in ferromagnetic film un-\nder external magnetic field, which is enhanced at a specific\nmagnetic angle.5,6)SAW can be focused into a nano region,\nwhich enables us to investigate the magnetic properties of s in-\ngle nanomagnets without thermal excitation.7)Recent stud-\nies have shown that the smaller the size of nanomagnet, the\nlarger the amplitude of the FMR.8)SAW also can “rewrite” the\nmagnetization direction recorded in elliptical nanomagne ts.9)\nHowever,SAW frequencyis usually limited to ∼10 GHz or less\nbecause of the limitation of the interdigital transducer pe riod.\nLongitudinal ultrasound propagating in the out-of-plane d i-\nrection also can excite FMR. Bayer and coauthorssucceeded i n\nexciting 3–40 GHz FMR in a magnetic film deposited on GaAs\nsubstrate by a strain pulse launched from the back surface us -\ning femtosecond pulse lasers.10–12)A strain pulse excited in the\nmagnetic film also modulates a magnetic response,13)which\ncan be enhanced by acoustic Bragg mirrors.14)Sub-THz ul-\ntrasound can be used for investigating magnetic properties at\nultrahigh frequencies and the interaction between phonon a nd\nmagnon, which are important to understand the fast magnetic\ndynamics.\nIn this study, we generate 10–100 GHz longitudinal ultra-\nsound in a nickel (Ni) thin film using picosecond ultrasonics\nand observe the magnetic dynamics using the time-resolved\nmagneto-optical Kerr effect (TR-MOKE) measurement. Ul-\ntrashort light pulse excites the strain pulse and the spin-w ave\nresonance (SWR) simultaneously. The strain pulse modulate s\nthe magnetization dynamics, which largely depends on the an -\ngle of the external magnetic field. We measure and calculate\nthe angular dependence of the resonance frequency and am-\nplitudes of magnetization dynamics to evaluate the interac tion\nbetween ultrasound and magnetization.\n∗E-mail: nagakubo@prec.eng.osaka-u.ac.jp2. Experiment and Theory\n2.1 Measurement method\nWe excited a broadband strain pulse and SWR by a femtosec-\nond pulse laser. We observe strain and magnetization dynami cs\nby time-resolved reflectivity change15,16)and TR-MOKE17,18)\nmeasurements, respectively. A schematic of our developed o p-\ntics is shown in Fig. 1. We applied up to 0.5-T external mag-\nnetic field using a permanent magnet by changing its angle\n/u1D703/u1D43Bbetween 0 and 60 degrees. A strain pulse was excited by\nan 800 nm pump light, and a linearly-polarized 400 nm probe\nlight was incident perpendicularly on the specimen surface .\nDetails are shown in our previous studies.19–23)We detected\nthe strain pulse from the reflectivity change of the probe lig ht\nusing a balanced detector 1 (BD-1). On the other hand, the\nmagnetization dynamics is detected from the intensity diffe r-\nence between S and P polarization components of the reflected\nprobe light using a balanced detector 2 (BD-2).\nUsing the RF magnetron sputtering method, we deposited\n∼300 nm Ni film capped by 5 nm SiO 2. We used a (100) Si\nsubstrate whose surface was thermally oxidized. Base pres-\nsure, Ar pressure, sputtering power, and substrate tempera ture\nduring deposition were ∼1×10−5Pa, 0.8 Pa, 50 W, and 350\n◦C, respectively. We confirmed that Ni has a polycrystalline\nstructure by X-ray diffraction measurement.\n2.2 Frequency of spin-wave resonance\nTo calculate the frequency of SWR of thin ferromagnetic\nisotropic film, we consider the following magnetization en-\nergy density /u1D438normalized by the saturation magnetization\n/u1D440/u1D460:6)\n/u1D438=−/u1D48E·/u1D46F+/u1D440/u1D460\n2/u1D7070(/u1D48E·/u1D475)2−/u1D437Δ/u1D48E·/u1D48E,(1)\nwhere/u1D48Edenotes normalized magnetization ( /u1D48E=/u1D474//u1D440/u1D460),\n/u1D46Fis the external magnetic field, /u1D475is a unit vector paralleled\nto the out-of-plane direction, /u1D437is the exchange stiffness, and\nΔ =/u1D7152\n/u1D715/u1D4652+/u1D7152\n/u1D715/u1D4662+/u1D7152\n/u1D715/u1D4672is the Laplacian operator. The first, second,\nand third terms denote the Zeeman, shape anisotropy, and the\nexchange energy, respectively.\nWe calculate the stable angles /u1D7190\n/u1D440and/u1D7030\n/u1D440through/u1D715/u1D438\n/u1D715/u1D719/u1D440=0\nand/u1D715/u1D438\n/u1D715/u1D703/u1D440=0, leading to the following equations:\n/u1D7190\n/u1D440=/u1D719/u1D43B (2)\n1Jpn. J. Appl. Phys. REGULAR PAPER\nRemovable mirror\nDichroic MirrorAcousto-Optic\nModulatorCorner ReflectorsPolarization Beam SplitterSecond Harmonic\nGeneration\nBeam SplitterPulse Laser\nBD-2Wollaston Prism\nObjective Lenses\nSpecimen\nθΗyxz\nm1m2\nm3\nφHφMθMθH\ny\nxz\nNH\nmBD-1\nMagnet\nFig. 1. Schematic of the optical systems. Red and blue lines denote 8 00\nand 400 nm lights, respectively. BD-1 and BD-2 are used for re flectivity and\nTR-MOKE measurement, respectively. The inset shows the defi nition of\neach coordinate system and parameter.\nsin 2/u1D7030\n/u1D440=2/u1D7070/u1D43B\n/u1D440/u1D460sin/parenleftBig\n/u1D7030\n/u1D440−/u1D703/u1D43B/parenrightBig\n. (3)\nThe resonance frequency can be calculated by solving the\nfollowing Smit-Beljers resonance formula of24,25)\n/parenleftbigg/u1D714/u1D446/u1D44A/u1D445\n/u1D6FE/parenrightbigg2\n=1\nsin2/u1D7030\n/u1D440/braceleftBigg\n/u1D7152/u1D438\n/u1D715/u1D7192\n/u1D440/u1D7152/u1D438\n/u1D715/u1D7032\n/u1D440−/parenleftbigg/u1D7152/u1D438\n/u1D715/u1D719/u1D440/u1D715/u1D703/u1D440/parenrightbigg2/bracerightBigg\n.(4)\nWe obtain the resonance frequency /u1D453/u1D446/u1D44A/u1D445=/u1D714/u1D446/u1D44A/u1D445/2/u1D70Bas fol-\nlowing:26,27)\n/u1D453/u1D446/u1D44A/u1D445=/u1D6FE\n2/u1D70B/radicalbig\n/u1D43B1/u1D43B2 (5)\n/u1D43B1=/u1D43Bcos(/u1D703/u1D43B−/u1D7030\n/u1D440) −/u1D440/u1D460\n/u1D7070cos2/u1D7030\n/u1D440+/u1D437/u1D4582\n/u1D453(6)\n/u1D43B2=/u1D43Bcos(/u1D703/u1D43B−/u1D7030\n/u1D440) −/u1D440/u1D460\n/u1D7070cos(2/u1D7030\n/u1D440) +/u1D437/u1D4582\n/u1D453,(7)\nwhere/u1D6FEdenotes the Gyromagnetic ratio. /u1D458/u1D453is the wavenum-\nber vector of spin-wave in the /u1D467-axis direction, which is defined\nby a mode number /u1D45Band the film thickness /u1D43Fas/u1D458/u1D453=/u1D45B/u1D70B\n/u1D43F.\n2.3 Amplitude of spin-wave resonance\nWe evaluate the amplitudes of SWR. The dynamics of the\nmagnetic moment follows the Landau-Lifshitz-Gilbert (LLG )\nequation:28,29)\n/u1D715/u1D48E\n/u1D715/u1D461=−/u1D6FE/u1D48E×/u1D46Feff+/u1D6FC/u1D48E×/u1D715/u1D48E\n/u1D715/u1D461, (8)\nwhere/u1D6FCis damping parameter and /u1D46Feffis the effective mag-\nnetic field, which is given by /u1D46Feff=−∇/u1D438under a static ex-\nternal magnetic field. The SWR is excited by demagnetization\nΔ/u1D440/u1D460due to instantaneous temperature increase caused by the\npump light, which modulates /u1D46Feffas30)\n/u1D46F′\neff=/u1D46Feff+/u1D715/u1D46Feff\n/u1D715/u1D440/u1D460Δ/u1D440/u1D460 (9)\n=/u1D46Feff−Δ/u1D440/u1D460\n/u1D7070cos/u1D7030\n/u1D440ˆ/u1D49B, (10)where ˆ/u1D49Bis the/u1D467-direction unit vector. Assuming that this effec-\ntive field keeps constant during the resonance period, we can\nevaluate the change of the magnetic moment to be proportiona l\nto the torque of\n/u1D6FE/u1D48E×/u1D46F′\neff=/u1D6FEΔ/u1D440/u1D460\n/u1D7070cos/u1D7030\n/u1D440sin/u1D7030\n/u1D440ˆ/u1D49A (11)\n=/u1D6FEΔ/u1D440/u1D460\n2/u1D7070sin 2/u1D7030\n/u1D440ˆ/u1D49A. (12)\nWe can measure the Kerr rotation angle change Δ/u1D703/u1D43Eby the TR-\nMOKE measurement, which corresponds to the /u1D467-direction\ncomponent of the magnetic moment. Therefore, we define the\namplitude of SWR as /u1D434/u1D446/u1D44A/u1D445and evaluate it as follows:\n/u1D434/u1D446/u1D44A/u1D445=/barex/barex/u1D6FE/u1D48E×/u1D46F′\neff/barex/barexsin/u1D7030\n/u1D440(13)\n∝sin 2/u1D7030\n/u1D440sin/u1D7030\n/u1D440. (14)\n2.4 Analysis for the phonon-magnon coupling\nStrain pulse modulates magnetic moment through the magne-\ntoelastic effect. Here, we calculate the dynamics of the mag-\nnetic moment and its /u1D467-component amplitude. By considering\nthe magnetoelastic effect, its energy /u1D438MEis added to /u1D438:\n/u1D438=−/u1D48E·/u1D46F+/u1D440/u1D460\n2/u1D7070(/u1D48E·/u1D475)2−/u1D437Δ/u1D48E·/u1D48E+/u1D438ME (15)\n/u1D438ME=/u1D44F1/parenleftBig\n/u1D700/u1D465/u1D465/u1D45A2\n/u1D465+/u1D700/u1D466/u1D466/u1D45A2\n/u1D466+/u1D700/u1D467/u1D467/u1D45A2\n/u1D467/parenrightBig\n+2/u1D44F2/parenleftbig/u1D700/u1D465/u1D466/u1D45A/u1D465/u1D45A/u1D466+/u1D700/u1D466/u1D467/u1D45A/u1D466/u1D45A/u1D467+/u1D700/u1D467/u1D465/u1D45A/u1D467/u1D45A/u1D465/parenrightbig,(16)\nwhere/u1D44F1and/u1D44F2are the magnetoelastic coupling constants.\nBecause we discuss small magnetization modulation from the\nstable condition, /u1D45A1and/u1D45A2are much smaller than 1, and\nmagnetic moment in the /u1D45A1/u1D45A2/u1D45A3coordinate system can be\nwritten as\n/u1D48E=/parenlefttpA/parenleftexA\n/parenleftbtA0\n0\n1/parenrighttpA/parenrightexA\n/parenrightbtA+/parenlefttpA/parenleftexA\n/parenleftbtA/u1D45A1\n/u1D45A2\n0/parenrighttpA/parenrightexA\n/parenrightbtA+/u1D442/parenleftBig\n/u1D45A2\n1,/u1D45A2\n2/parenrightBig\n(17)\nas shown in Fig. 1. We also assume that displacement /u1D462/u1D467and\nmagnetic moment have the forms of /u1D462/u1D467=/u1D4620\n/u1D467exp{/u1D456(/u1D458/u1D467−/u1D714/u1D461)}\nand/u1D45A/u1D457=/u1D45A0\n/u1D457exp{/u1D456(/u1D458/u1D467−/u1D714/u1D461)}(/u1D457=1,2), respectively, be-\ncause we excite the plane wave strain pulse propagating in th e\n/u1D467axis. Here, /u1D43Beffwith respect to the /u1D48Edirection is given by\n/u1D46Feff=−∇/u1D48E/u1D438 (18)\n=−/parenlefttpA/parenleftexA/parenleftexA/parenleftexA\n/parenleftbtA/u1D7152/u1D438\n/u1D715/u1D45A2\n1/u1D45A1+/u1D7152/u1D438\n/u1D715/u1D45A1/u1D715/u1D45A2/u1D45A2\n/u1D7152/u1D438\n/u1D715/u1D45A1/u1D715/u1D45A2/u1D45A1+/u1D7152/u1D438\n/u1D715/u1D45A2\n2/u1D45A2\n/u1D715/u1D438\n/u1D715/u1D45A3/parenrighttpA/parenrightexA/parenrightexA/parenrightexA\n/parenrightbtA−/parenlefttpA/parenleftexA/parenleftexA/parenleftexA\n/parenleftbtA/u1D715/u1D438ME\n/u1D715/u1D45A1/u1D715/u1D438ME\n/u1D715/u1D45A2/u1D715/u1D438ME\n/u1D715/u1D45A3/parenrighttpA/parenrightexA/parenrightexA/parenrightexA\n/parenrightbtA, (19)\nwhere∇/u1D48E=/parenleftBig\n/u1D715\n/u1D715/u1D45A1,/u1D715\n/u1D715/u1D45A2,/u1D715\n/u1D715/u1D45A3/parenrightBig\nis the vector differential opera-\ntor. We define ℎ/u1D457=−/u1D715/u1D438ME\n/u1D715/u1D45A/u1D457and obtain the following relation-\nship from the LLG equation (8):\n/parenleftbigg/u1D45A1\n/u1D45A2/parenrightbigg\n=/u1D712/parenleftbiggℎ1\nℎ2/parenrightbigg\n(20)\n/u1D712=1\n/u1D436/parenlefttpA/parenleftexA\n/parenleftbtA/u1D7152/u1D438\n/u1D715/u1D45A2\n2−/u1D715/u1D438\n/u1D715/u1D45A3−/u1D456/u1D714/u1D6FC\n/u1D6FE−/u1D7152/u1D438\n/u1D715/u1D45A1/u1D715/u1D45A2−/u1D456/u1D714\n/u1D6FE\n−/u1D7152/u1D438\n/u1D715/u1D45A1/u1D715/u1D45A2+/u1D456/u1D714\n/u1D6FE/u1D7152/u1D438\n/u1D715/u1D45A2\n1−/u1D715/u1D438\n/u1D715/u1D45A3−/u1D456/u1D714/u1D6FC\n/u1D6FE/parenrighttpA/parenrightexA\n/parenrightbtA(21)\n2Jpn. J. Appl. Phys. REGULAR PAPER\n/u1D436=/parenleftBigg\n/u1D7152/u1D438\n/u1D715/u1D45A2\n1−/u1D715/u1D438\n/u1D715/u1D45A3−/u1D456/u1D714/u1D6FC\n/u1D6FE/parenrightBigg /parenleftBigg\n/u1D7152/u1D438\n/u1D715/u1D45A2\n2−/u1D715/u1D438\n/u1D715/u1D45A3−/u1D456/u1D714/u1D6FC\n/u1D6FE/parenrightBigg\n−/parenleftbigg/u1D7152/u1D438\n/u1D715/u1D45A1/u1D715/u1D45A2/parenrightbigg2\n−/parenleftbigg/u1D714\n/u1D6FE/parenrightbigg2\n. (22)\nThe measurable amplitude /u1D434/u1D45D/u1D45Ainduced by the phonon-\nmagnon coupling is the /u1D467-direction component of the mag-\nnetization dynamics. We assume that the magnetization\ndynamics is caused by longitudinal phonons of /u1D462/u1D467=\n/u1D4620\n/u1D467exp{/u1D456(/u1D458/u1D467−/u1D714/u1D461)}, resulting in only non-zero strain of /u1D700/u1D467/u1D467.\nTherefore, by substituting /u1D438in Eq. (15) into /u1D45A1in Eq. (20),\n/u1D434/u1D45D/u1D45Acan be written as follows:\n/u1D434/u1D45D/u1D45A=/u1D45A0\n1sin/u1D7030\n/u1D440 (23)\n∝/radicalbigg\n/u1D43B2\n2+/parenleftBig\n/u1D6FC/u1D714\n/u1D6FE/parenrightBig2\n/radicalbigg\n/u1D44E0+/u1D44E2/parenleftBig\n/u1D714\n/u1D6FE/parenrightBig2\n+/u1D44E4/parenleftBig\n/u1D714\n/u1D6FE/parenrightBig4sin 2/u1D7030\n/u1D440sin/u1D7030\n/u1D440(24)\n/u1D44E0=/u1D43B2\n1/u1D43B2\n2(25)\n/u1D44E2=/u1D6FC2/parenleftBig\n/u1D43B2\n1+/u1D43B2\n2/parenrightBig\n−2/u1D43B1/u1D43B2 (26)\n/u1D44E4=/u1D6FC2+1, (27)\nwhere/u1D43B1and/u1D43B2are functions of /u1D703/u1D43Band are calculated by\nEqs. (6) and (7), respectively. In Eq. (24), we omitted /u1D44F1and\n/u1D700/u1D467/u1D467because they are independent from /u1D7030\n/u1D440and/u1D714.\n3. Results and Discussion\nAt first, we observed propagation of the strain pulse from the\nreflectivity measurement. The reflectivity change represen ts\nthe electron-density change caused by strain. Figure 2(a) s hows\nmeasured reflectivity changes, which reflect fast electron d if-\nfusion, slower thermal diffusion, and strain-pulse echoes. We\nextract the echo signals as shown in Fig. 2(b), where the back -\nground changes of thermal diffusion are subtracted by fitting\npolynomial functions. Then, we evaluate each frequency am-\nplitude by applying the fast Fourier transformation (FFT) a s\nshown in Fig. 2(c). We observed echo signals every 95.4 ps,\nwhich leads to the thickness /u1D43Fof∼280 nm from the veloc-\nity of 5.8 nm/ps for polycrystalline Ni.31)We also observed\nreflectivity changes under the 0.5-T magnetic field at the an-\ngle/u1D703/u1D43Bof 20 degrees. Under the magnetic field, background\nchanges contain not only thermal diffusion but also ∼10 GHz\nsmall oscillations as shown in the corresponding FFT spectr a\nof Fig. 2(c). The appearance of strain oscillations suggest s the\nmagnetostriction effect due to SWR.\nUsing TR-MOKE measurement, we observed the magne-\ntization dynamics by changing the angle of the external field\n/u1D703/u1D43Bbetween 0 and 60 degrees. The observed SWRs are shown\nin Fig. 3(a), where background changes are subtracted by fit-\nting polynomial functions. We apply FFT to determine their\nfrequencies and also evaluate the amplitudes from the peak-\nto-peak heights of the SWR responses. The dependences of\nthem on /u1D703/u1D43Bare shown in Figs. 4(a) and (b). We observed\n8–21 GHz SWR, and its frequency increases with the increase\nin/u1D703/u1D43B. The measured SWR frequencies agree with the calcu-\nlations from Eq. (5) for the 7th-mode SWR at low /u1D703/u1D43B, and\nthey approach those of the 4th-mode SWR at higher angles as\nshown in Fig. 4(a). In the calculation, we used /u1D437=0.157 nmA,32)/u1D440/u1D460=0.61 T,32)/u1D6FE=35.2 kHz m/A with /u1D7070/u1D43B=0.5 T\nand/u1D43F=276 nm. The higher modes are excited by the local\nand large initial changes of magnetic moment angle /u1D703/u1D440: The\npenetration depth of the 800-nm pump light for Ni is ∼13.0\nnm,33,34)which causes the demagnetization near the surface\ndue to temperatureincrease and excites the higher-modeSWR .\nThe lower /u1D703/u1D43Balso contributes to the larger initial change of\n/u1D703/u1D440because/u1D703/u1D440is highly affected by the demagnetizing field\ndue to shape anisotropy at lower /u1D703/u1D43B. We calculate the changes\nof/u1D703/u1D440due to the decrease in /u1D440/u1D460from Eq. (3), which increases\nwith the decrease in /u1D703/u1D43B. The dependence of measured ampli-\ntudes of SWR on /u1D703/u1D43Bagrees with that of calculation using Eq.\n(14) as shown in Fig. 4(b). These results ensure the validity of\nour measurement and calculation. Torque caused by the initi al\nthermal strain should have the dependence on /u1D703/u1D43Bas discussed\nin the following section, however, it will be much smaller th an\nthe thermal demagnetization and will not affect the amplitud e\nof SWR.\nWe also observed a pulse signal around 95 ps in the TR-\n(c)0 50 100 150 200Time [ps]Reflectivity change [arb. unit](b)\n020 40 60 80100\nFrequency [GHz]Intensity [arb. unit](a)\n    0 T\n 0.5 T\nFig. 2. (a) Observed reflectivity changes, (b) extracted signals, a nd (c)\ncorresponding FFT spectra. Red and blue lines show the resul ts under no\nand 0.5 T magnetic fields, respectively.\n3Jpn. J. Appl. Phys. REGULAR PAPER\n80 90 100 110Intensity [arb. unit]   0\n 10\n 20\n 30\n 40\n 50\n 60\n pulse(a)\n(b)\nθH [degrees]\nTime [ps]020406080100120140\nTime [ps]Kerr-rotation [arb. unit]\nFig. 3. (a) Observed Kerr-rotation angle changes and (b) extracted echo\nsignals by changing the angle of the external magnetic field /u1D703/u1D43B.\nMOKE measurement, and extract it by subtracting other re-\nsponses using a polynomial function as shown in Fig. 3(b). We\ndetermine its amplitude /u1D434/u1D45D/u1D45Afrom the peak-to-peak height,\nwhose dependence on /u1D703/u1D43Bis shown in Fig. 5. The amplitude\nbecomes the largest at /u1D703/u1D43B=20 degrees. This pulse insists\nthat the coherent strain pulse causes the changes in a magnet ic\nmoment and the interaction has a large dependence on /u1D703/u1D43B.\nCalculated /u1D434/u1D45D/u1D45Afrom Eq. (24) agrees well with the measured\namplitudes for /u1D453=10.7 GHz and /u1D6FC=0.1, indicating that the\ncoupling effect can be increased by adjusting the angle as wel l\nas the intensity of the external magnetic field. This depende nce\nstems from the change of the torque caused by the strain: Ac-\ncording to the LLG equation (8), the torque becomes smaller a t\nlower/u1D703/u1D440because/u1D48Eand/u1D489are parallel. On the other hand, at\nhigher/u1D703/u1D440,/u1D489becomes smaller because the initial /u1D467component\nof/u1D48Ebecomes smaller, resulting in vanishment of the torque.\nTherefore, the torque caused by the /u1D467-direction magnetoelastic\nfield takes a maximum at a specific angle. Similar angle de-\npendence is reported by the in-plane SAW-FMR coupling.5,6)\nIn the out-of-plane magnetic fields, the angle and its intens ity0102030\n n = 7\n n = 6\n n = 4\n n = 0fSWR(a)\n0102030405060708090ASWR\nθH [degrees] Meas.\n Calc.(b)\nNiM\nFig. 4. The dependence of (a) SWR frequencies /u1D453/u1D446/u1D44A/u1D445and (b) SWR\namplitudes/u1D434/u1D446/u1D44A/u1D445on/u1D703/u1D43B. Black circles and solid lines denote measured\nand calculated values.\n0102030405060708090\nθH [degrees] Meas.\n Calc.Apm [arb. unit]NiM\nFig. 5. The dependence of measured pulse signal amplitudes on /u1D703/u1D43B.\nBlack circles and solid line denote measured /u1D434/u1D45D/u1D45Aand calculated /u1D434/u1D45D/u1D45A,\nrespectively.\naffect magnetic moment much more because of the demagne-\ntization field. These results suggest that the SWR at 8–21 GHz\nis excited by thermal demagnetization at 0 ps, and the mag-\nnetization oscillation is excited by phonon-magnon coupli ng\nat∼95 ps when the propagated strain pulse returns near the\nsurface. Our measurement and calculation reveal that the /u1D703/u1D43B\ndependence highly affects the phonon-magnon interaction in\nthe out-of-plane field, which contributes to the investigat ion\nof the phonon-magnon coupling in a higher frequency region.\n4Jpn. J. Appl. Phys. REGULAR PAPER\n4. Conclusions\nWe excited the strain pulse and higher-mode spin-wave res-\nonances by a femtosecond pulse laser. Measured frequencies\nand amplitudes of the spin-wave resonances agree with the\ntheoretical calculation, ensuring the validity of our meas ure-\nment and calculation. We also observed the magnetic respons e\ncaused by the strain pulse, which has a large dependence on\nthe angle of the applied external magnetic field. We experi-\nmentally and analytically clarify that the coupling effect c an\nbe maximized at certain angles. In addition, we find the contr i-\nbution of a certain frequency component of the strain pulse t o\nthe coupling. Our results will contribute to the measuremen t\nof the coupling between longitudinal phonons and magnons\nthat becomes important for higher-frequency techniques.\nAcknowledgment\nThis work was supported by KAKENHI Grant No. 18H01859\nof Grant-in-Aid for Scientific Research (B).\n1) S. Li, C. Cheng, K. Meng, and C. Chen, Jpn. J. Appl. Phys. 58, 040903\n(2019).\n2) X. Sun, X. Zhang, M. A. S. Syed, L. Qi, X. Dong, Y. Li, and F. Su , Jpn.\nJ. Appl. Phys. 58, 065005 (2019).\n3) N. L. Kang and S. D. Choi, Jpn. J. Appl. Phys. 51, 053002 (2012).\n4) Y. Yamaoka, K. Iwasaki, S. Oda, and T. Kodera, Jpn. J. Appl. Phys. 56,\n04CK07 (2017).\n5) M. Weiler, L. Dreher, C. Heeg, H. Huebl, R. Gross, M. S. Bran dt, and\nS. T. B. Goennenwein, Phys. Rev. Lett. 106, 117601 (2011).\n6) L. Dreher, M. Weiler, M. Pernpeintner, H. Huebl, R. Gross, M. S.\nBrandt, and S. T. B. Goennenwein, Phys. Rev. B 86, 134415 (2012).\n7) W. G. Yanga and H. Schmidt, Appl. Phys. Lett. 116, 212401 (2020).\n8) M. Jaris, W. Yang, C. Berk, and H. Schmidt, Phys. Rev. B 101, 214421\n(2020).\n9) V. S. Vlasov, A. M. Lomonosov, A. V. Golov, L. N. Kotov, V. Be sse, A.\nAlekhin, D. A. Kuzmin, I. V. Bychkov, and V. V. Temnov, Phys. R ev. B\n101, 024425 (2020).\n10) A. V. Scherbakov, A. S. Salasyuk, A. V. Akimov, X. Liu, M. B ombeck,\nC. Br∝ddotaccuggemann, D. R. Yakovlev, V. F. Sapega, J. K. Furdyna, and M.\nBayer, Phys. Rev. Lett. 105, 117204 (2010).\n11) M. Bombeck, A. S. Salasyuk, B. A. Glavin, A. V. Scherbakov , C.Br∝ddotaccuggemann, D. R. Yakovlev, V. F. Sapega, X. Liu, J. K. Furdyna, A.\nV. Akimov, and M. Bayer, Phys. Rev. B 85, 195324 (2012).\n12) J. V. J ∝ddotaccager, A. V. Scherbakov, T. L. Linnik, D. R. Yakovlev, M. Wang, P.\nWadley, V. Holy, S. A. Cavill, A. V. Akimov, A. W. Rushforth, a nd M.\nBayer, Appl. Phys. Lett. 103, 032409 (2013).\n13) J. W. Kim, M. Vomir, and J. Y. Bigot, Phys. Rev. Lett. 109, 166601\n(2012).\n14) J. V. J ∝ddotaccager, A. V. Scherbakov, B. A. Glavin, A. S. Salasyuk, R. P.\nCampion, A. W. Rushforth, D. R. Yakovlev, A. V. Akimov, and M.\nBayer, Phys. Rev. B 92, 020404 (2015).\n15) C. Thomsen, J. Strait, Z. Vardeny, H. J. Maris, and J. Tauc , Phys. Rev.\nLett. 53, 989 (1984).\n16) C. Thomsen, H. T. Grahn, H. J. Maris, and J. Tauc, Phys. Rev . B34,\n4129 (1986).\n17) D. Guarisco, R. Burgermeister, C. Stamm, and F. Meier, Ph ys. Rev.\nLett. 68, 1729 (1996).\n18) B. Koopmans, M. van Kampen, J. T. Kohlhepp, and W. J. M. de J onge,\nPhys. Rev. Lett. 85, 844 (2000).\n19) H. Ogi, T. Shagawa, N. Nakamura, M. Hirao, H. Odaka, and N. Kihara,\nJpn. J. Appl. Phys. 48, 07GA01 (2009).\n20) A. Nagakubo, A. Yamamoto, K. Tanigaki, H. Ogi, N. Nakamur a, and\nM. Hirao, Jpn. J. Appl. Phys. 51, 07GA09 (2012).\n21) H. K. Weng, A. Nagakubo, H. Watanabe, and H. Ogi, Jpn. J. Ap pl.\nPhys. 59, SKKA04 (2020).\n22) H. Fukuda, A. Nagakubo, and H. Ogi, Jpn. J. Appl. Phys. 60, SDDA05\n(2021).\nElectronics 42, 2Pb1-2 (2021).\n24) S. V. Vonsovskii: Ferromagnetic Resonance (Oxford, New York,\nPergamon Press, 1966).\n25) J. Smit and H. G. Beljers, Philips Res. Rep. 10, 113 (1955).\n26) H. Puszkarski, Phys. Stat. Sol. B 96, 61 (1979).\n27) J. T. Yu, R. A. Turk and P. E. Wigen, Phys. Rev. B 11, 420 (1975).\n28) L. Landau and E. Lifshitz, Phys. Z. Sowjetunion 8, 153 (1935).\n29) T. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n30) D. M. Lattery, J. Zhu, D. Zhang, J. P. Wang, P. A. Crowell, a nd X.\nWang, Appl. Phys. Lett. 113, 162405 (2018).\n31) G. A. Alers, J. R. Neighbours, and H. Sato, J. Phys. Chem. S olids 13,\n40 (1960).\n32) C. Kittel: Introduction to solid state physics , 8th Edition (J. Wiley and\nSons, New York, 2004).\n33) P. B. Johnson and R. W. Christy, Phys. Rev. B 9, 5056 (1974).\n34) A. Nagakubo, H. Ogi, and M. Hirao, Jpn. J. Appl. Phys. 54, 128001\n(2015).\n5"    },    {        "title": "1411.7146v1.The_dynamic_magnetic_behaviors_of_the_Blume_Capel_Ising_bilayer_system.pdf",        "content": " 1 \n The dynamic magnetic behaviors of the Blume -Capel Ising bilayer system  \n \nMehmet Ertaş1 \nDepartment of Physics, Erciyes University, 38039 Kayseri, Turkey  \n \nAbstract  \n \nThe dynamic magnetic behaviors of the spin -1 Blume -Capel Ising bilayer system  (BCIS)  are \nstudi ed in an oscillating external magnetic field on a two -layer square lattice  by utilizing the \nmean field theory based on Glauber -type stochastic dynamics (DMFT) . The dynamic \nequations describing the time -dependencies of the average magnetizations are obtaine d with \nthe Master equation. The d ynamic phases in this system are found by solving these dynamic \nequations. The temperature dependence of the dynamic order parameters is examined to \ncharacterize the nature (continuous  or discontinuous) of the phase transit ions and to obtain the \ndynamic phase transition points  (DPT) . The dynamic phase diagrams are shown for \nferromagnetic / ferromagnetic, antiferromagnetic / antiferromagnetic, antiferromagnetic / \nantiferromagnetic interactions in the plane of the reduced temp erature versus magnetic field \namplitude and they display dynamic tricritical and reentrant behavior as well as the dynamic \ntriple point.  \n \nPACS  number(s): 05.50.  +q; 05.70.Fh; 64.60.Ht; 75.10.Hk  \n \nKeywords:  Blume -Capel bilayer Ising system; Glauber -type stoc hastic dynamics ; Mean field \ntheory, Dynamic phase transitions, dynamic phase diagrams.  \n \n1. Introduction  \nResearch on magnetic bilayers has received much interest both from the experimental and \ntheoretical point of view in recent years [1 -10]. Experimentally, several works have been \npublished on the properties of the magnetic bilayer such as FePt/Fe [11], FM/TbFe (FM=Fe, \nPy-permalloy, FeCo) [12], SiGe, SiC and GeC [13], Ni/Au [14], Fe/Ni and Fe/Co [15]. On the \nother hand, theoretically, the equilibrium behavior s of bilayer Ising systems consisting of \nsingle ions, namely spin -1/2, spin -1 etc., have been widely studied by utilizing the well -\nknown theoretical methods used in equilibrium statistical physics , such as the mean field \ntheory (MFT) [16, 17], the effective  field theory (EFT) [18, 19], the exact recursion equation \non the Bethe lattice [6, 7], the renormalization -group (RG) method [20, 21], and Monte Carlo \n(MC) simulations [22, 23] . \nDespite these equilibrium studies, the nonequilibrium aspects of bilayer Ising  systems \nconsisting of single ions have not been as thoroughly explored. There have only been few \ninvestigations, to our knowledge, about the dynamical aspects of the bilayer Ising system.  \nThese works are related to the dynamic behavior of the spin-1/2 bil ayer system  and can be \nsummarized as follows. By using dynamic MC simulations, the short -time process and scaling \nbehavior of a two -layer Ising model was studied by Wu et al. [24]. Jang and co -workers [25, \n26] investigated the dynamic phase behaviors of ma gnetic thin ferromagnetic Heisenberg \nfilms using a dynamic MC simulation. The dynamic behavior of the spin -1/2 Ising model on a \ntwo-layer square lattices under the presence of a time -dependent oscillating external magnetic \n                                                 \n1Correspo nding author  \nE-mail address:  mehmetertas@erciyes.edu.tr   \n Manuscript\nClick here to download Manuscript: Manuscript.doc \nClick here to view linked References 2 \n field was investigated using  Glau ber-type stochastic dynamics based on the mean field theory \n(DMFT) [27] and the effective field theory (DEFT) [28]. In these studies, the dynamic phase \ntransition (DPT) temperatures and the dynamic phase diagrams of the spin -1/2 bilayer Ising \nsystem are ca lculated in detail. We should also mention that very recently, the dynamic \nmagnetic behaviors of the mixed Ising bilayer system consisting of different spin s, namely \nmixed spin s (1, 3/2), (3/2, 2), (2, 5/2)  [29-32], by using DMFT for FM/FM, AFM/FM, and \nAFM /AFM interactions  were investigated  on the bilayer square lattice . \n Unfortunately, the dynamic magnetic behaviors of the spin -1 Blume -Capel Ising \nbilayer (BCIB) system have not been studied so far. Therefore, the purpose of the present \npaper is to study th e dynamic behaviors of the spin -1 BCIB system in an oscillating external \nmagnetic field within DMFT. The DMFT equations for average magnetizations are obtained \nwith the Glauber -type stochastic dynamics. We investigate the time variation of average \nmagnetiz ations to find the phases and the thermal behavior of dynamic magnetizations to \nobtain the dynamic phase transition points , as well as to characterize the nature of the \ndynamic phase transitions. Finally, the dynamic phase diagrams are presented for FM/FM, \nAFM/FM, and AFM/AFM interactions in the plane of the reduced temperature versus \nmagnetic field amplitude.  \nIt can be noted that the physics of equilibrium phase transitions is well understood, \nand it is well established that structures arising from differe nt dynamics that obey detailed \nbalance and respect the same conservation laws exhibit universal asymptotic large -scale  \nfeatures. However, the mechanisms behind nonequilibrium phase transitions are not very well \nknown, and the dependence on the specific dynamics is still an open question [ 33]. Therefore, \nfurther efforts on these challenging time -dependent problems, especial ly calculating the DPT \npoints and constructing the phase diagram, should promise to be rewarding in the future. \nHence, the DPT in an oscillating external magnetic field has attracted considerable attention \nin recent literature, theoretically (see [3 4-39] and references therein) and analytically [ 40, 41]. \nMoreover, experimental evidence for the DPT has  been reported in polyethylene naphthalate \n(PEN) nanocomposites, in  amorphous ultrathin Co films, cuprite superconductors, [Co/Pt] 3 \nmagnetic multilayers, etc. [42-45]. Moreover, O'Malley et al. [46] studied the DPT in \necological systems in a temporally varying environment (seasonal variations ). \n The layout of this work is as follows. In Section 2, the model and the formulation are \ngiven briefly. In Section 3, the  detailed numerical results and discussion are presented. A \nsummary and conclusion are given in Section 4.  \n \n2. Model and derivation of mean field dynamic equations  \n \nThe two-layer  square lattice is an extension of its one -layer version. Two identical layers , \nnamely the  upper layer (G 1) and lower layer (G 2) of the square lattices , are placed parallel to \neach other, forming the two -layer square lattice . The schematic representation of the system is \nillustrated in Fig. 1.  Each layer has N sites and interacts with its nearest -neighbor (NN) and \nthe corresponding adjacent spins in the other layer whose sites are labeled by i, i′, j and j′ , as \nseen in Fig. 1.  Such a system may be described by the following Hamiltonian : \n \ni ' j'\ni ' j'2 2 2 2\n1 i' j' 2 i j 3 i' i j' j i j\ni' j' ij ii' jj' i' j' i j\nij\ni' j' i jJ J S S J S S D S S\nH S S ,               \n                           \n   \n     \n       \n   H\n    (1) \n  3 \n where <ij> an d <i′j′> indicate a summation over all the pairs of nearest -neighboring sites of \neach layer. J 1 and J 2 denote  exchange constants for the first and second layer  and are also \ncalled intralayer coupling constants . J3 is the interlayer coupling constant over a ll the adjacent \nneighboring sites of layers , as seen in Fig. 1 . D is the single ion anisotropy constant and H is a \ntime-dependent oscillating external magnetic field: \n0 H(t)=H cos(wt),  where H 0 and w = 2πν \nare the amplitude and the angular freque ncy of the oscillating field, respectively. The system \nis in contact with an isothermal hea t bath at absolute temperature T A.  \nOn the other hand, f or the formulation of the system, one needs to introduce the order \nparameters, namely magnetizations. From Fi g. 1, one can see that each layer of the system is \nalso a two -lattice system with spin variables σi = ±1, 0 and \ni' = ±1, 0 on  the sites of the \nsublattices A and B, respe ctively , and Sj = ±1, 0 and \nj'S = ±1, 0 on the sites of  the sublattices \nA and B, respectively. Therefore , the system can be described wi th four sublattice \nmagnetizations or four simple magnetizations. These magnetizations  are introduced as \nfollows : \nA B A B\n1 i 1 j 2 i 2 j'm , m , m S , m S ,       where \n  is the thermal expectation \nvalue . \n Now,  by utilizing Glauber -type stochastic dynam ics, the dynamic mean field equations \ncan be obtained. I n particular , we employ Glauber transition rates  to obtain the set of dynamic \nmean field equations. The system evolves according to  a Glauber -type stochastic process at a \nrate of \n1  transitions per unit time; hence the frequency of spin flipping, f, is \n1 . \n1 2 NP( , ,..., ;t)  \n and \n1 2 NP(S ,S ,..., S ;t)  are the probability functions when the system owns \n1 2 N, ,...,   \n and \n1 2 NS ,S ,..., S  spin configurations at time t. If we let \nAA\ni i iW ( )    be the \nprobability per unit time that the ith spin changes from the value \nA\ni  to \nA\ni , while the S spins \non the sublattice B are momentarily fixed , then we may write the time derivate of the \nA A A A\n1 1 2 NP ( , ,..., ;t)  \n as \n \n    \nAA\nii\nAA\niiA A A A A A A A A A A A\n1 1 2 N i i i 1 1 2 i N\ni\nA A A A A A A A\ni i i 1 1 2 i N\nidP ( , ,..., ;t) W ( ) P ( , ,..., ,..., ;t)dt\nW ( )P ( , ,..., ,..., ;t) . (2) \n            \n       \n\n \n \nThis equation is called the Master equation. \nA A A\ni i iW ( )   is the probability per unit time \nthat the ith spin cha nges from the value  \nA\ni  to \nA\ni . In this sense  the Glauber model is \nstochastic. Since the system is in contact with a heat bath at absolute temperature T A, each \nspin can change from the value \nA\ni  to \nA\ni  with the probability per unit time  \n \nA\niAA\nA A A ii\ni i iAA\niiexp( E( ))1W ( ) ,\nexp( E( ))\n         \n   (3)  4 \n where \nA\ni is the sum over the two possible values of \nA\ni = ±1, 0, and  \n                                 \nA A A A A B A\ni i i i i 1 j 3 i\njiE ( ) ( )(J J S H),\n                             (4) \n0 \ngives the change in the energy of the system when the σi-spin changes. The probabilities \nsatisfy the detailed balance condition  \n \n                                 \nA A A A A A A\n1 2 i N i i i\nA A A AA A A\n1 2 i Ni i iP( , ,..., ,..., ) W ( )\nP( , ,..., ,..., ) W ( )            ,                  (5) \n \nand by substituting the possible values of σi, we get  \n \n   \nAA\niiexp -βD 1W (1 0)=W (-1 0)=τ 2Cosh(βx)+exp(-βD) ,        (6a) \n \nAA\nii1 exp(- βx)W (1 1)=W (0 1)= ,τ 2Cosh(βx)+exp(-βD) \n                  (6b) \n \n         \nAA\nii1 exp( βx)W (0 1)=W (-1 1)= ,τ 2Cosh(βx)+exp(-βD)                                (6c) \nwhere \njBA\n1 3 i\n<j > ix = Jσ +J S +H\n\n . From the Master equation associated with the stochastic process, \nit follows that the average \nA\nk satisfies the following equation:  \n \n\nB A B A\n1 j 3 i 1 j 3 i\nj i j iAA\nkk\nBA\n1 j 3 i\nji2sinh J J S H 2exp 4 D sinh J J S H\nd.dt\n2cosh J J S H exp D\n\n\n                     \n             \n     \n   \n\n   (7) \n \nIn terms of the mean  field approach,  this dynamic equation can be written as follows:  \n \n \nB A B A 33\n1 2 1 2\n11 AA\n11\nBA 3\n12\n1JJ 2 d 12sinh zm m h cos exp sinh zm m h cosT J T T J dm m ,d J 2d2cosh zm m h cos expT J T                                           \n(8) \n  5 \n \nA A B B A A B B 1\n1 i 1 j 2 i 2 j 1 0 1 1 where m , m , m S , m S , wt, T ( J ) , h = H /J , d = D/J\nand w. We fixed z = 4 and w = 2 .\n        \n   \nThe other dynamic equations concerning  the G1 and G 2 layers can be similarly \nobtained as  follows:  \nA B A B 33\n1 2 1 2\n11 BB\n11\nAB 3\n12\n1JJ 2 d 12sinh zm m h cos exp sinh zm m h cosT J T T J dm m ,d J 2d2cosh zm m h cos expT J T                                     \n (9) \n \n \nB A B A 33 22\n2 1 2 1\n1 1 1 1 AA\n22\nBA 3 2\n21\n11JJ JJ2 d 12sinh zm m h cos exp sinh zm m h cosT J J T T J J dm m ,d J J2d2cosh zm m h cos expT J J T                                       \n(10) \n \nA B A B 33 22\n2 1 2 1\n1 1 1 1 BB\n22\nAB 3 2\n21\n11JJ JJ2 d 12sinh zm m h cos exp sinh zm m h cosT J J T T J J dm m .d J J2d2cosh zm m h cos expT J J T                                       \n  (11) \nThus a set of mean -field dynamical equations are obtained.  \n3. Numerical results and discussions  \n \n3.1. Phases in t he system  \n \nWe solved Eqs. ( 8)-(11) look for steady states then classif ied their behavior to find the \nphases as paramagnetic ( p), ferromagnetic ( f), antiferromagnetic (af), surface (sf), \ncompensated (c), mixed (m)   and  non-magnetic (nm) . The stationary sol utions of Eqs. ( 8) and \n(11) will be a periodic function of \n  with period 2π; that is  mA, B(\n+2 ) = mA, B(\n). \nMoreover, they can be one of t hree types according to whether they have or d o not have the \nproperty  \n \nA A B B\n1 1 1 1m ( + )= m ( ) and m m ,      \n     (12a) \nand \nA A B B\n2 2 2 2m ( + )= m ( ) and m m .      \n     (12b) \n \nA solution satisfying Eq s. (12a) and (12b) is called a symmetric solution which corresponds \nto a paramagnetic ( p) solution and it exists at high values of T and h.  In this solution, the \naverage magnetizations are equal to each other. They oscillate around  the zero value and are  6 \n delayed with respect to the external magnetic field. An example  is shown in Fig. 2(a). The \nsecond type of solution , which does not sat isfy Eqs. (12a) and (12b), is called a nonsymmetric \nsolution and magnetizations do not follow the external magnetic field anymore ; instead of \noscillating around the zero value, they oscillate around the nonzero value.  These fundamental \nphases are defined a s follows . (i) The ferromagnetic (f)  phase: \nA\n1m =\nB\n1m\n 0 with positive spin \nvalues and  \nA\n2m =\nB\n2m\n 0 and with posit ive spin values . (ii) The anti ferromagnetic (af) phase:  \nA\n1m\n=\nB\n1m ,\nA\n2m =\nB\n2m. (iii) The surface ferromagnetic  (sf) phase: \nA\n1m =\nB\n1m\n 0, \nA\n2m = \nB\n2m\n  \n0. (iv) The compensated (c) phase : \nA\n1m =\nB\n1m\n 0 with positive  spin values  and \nA\n2m =\nB\n2m\n 0 \nwith negative  spin values , or \nA\n1m =\nB\n1m\n 0 with negative  spin values  and \nA\n2m =\nB\n2m\n 0 with \npositive spin values . (v) The mixed (m) phase: \nA\n1m = \nB\n1m , \nA\n2m = -\nB\n2m.  These phases ar e \ndefined according to Refs. [2 7, 31, 32 ] and are illustrated in Fig. 2(b) -(f). These phases  can be \nbetter understood by examination of Table 1 . In addition to these seven  fundamental phases, \nseven coexistence solutions or mixed phases  are found, which are composed of binary \ncombinations of the fundamental phases, namely the f + p, f + c, f + m, sf + nm, nm + p, m + \np and af + p. \n \n3.2. Behaviors of dynamic magnetizations and dynamic phase transition points  \n \nThe dynamic magnetizations (\nA,B\n1M and \nA,B\n2M ), are defined as  \n22\nA A B B\n1 1 1 1\n0011M m ( )d , M m ( )d ,22\n     \n     (13) \nand \n22\nA A B B\n2 2 2 2\n0011M m ( )d , M m ( )d .22\n     \n     (14) \nIn order to determine the dynamic phase transition (DPT) temperatures among the phases we \nwill study the temperature dependences of the dynamic magnetizations, by solving Eqs. (13) -\n(14) numerically.  This investigation also leads us to characterize the nature (first - or second -\norder) of the DPT temperatures. A  few interesting results are plotted in Figs. 3(a) -(f). \nRespectively, T C and T t denotes second -and first -order phase transition temperature. Fig. 3(a) \nis calculate d for J1=1.0, \n21J / J =1.0,\n31J / J = 1.0 , d = 1.0 , h = 0.5  and illustrates a second -\norder phase transition which  is from the f pha se to the p phase at TC = 3.63.  Because, at zero \nvalue of temperature, \nA,B\n1M =\nA,B\n2M =1.0 and as the temperature increases, they reduce to zero \nvalue incessantly . Fig. 3(b) is obtained for J1=1.0, \n21J / J =1.0,\n31J / J = 1.0 , d = 1.0 , h = 3.77 \nand it shows first -order phase transition which is from the f phase to the p phase at T t = 0.75. \nSince, at zero value of temperature, \nA,B\n1M =\nA,B\n2M =1.0 and as the temperature increases, they \ndecrease to zero value transient by the time T t. Figs. 3(c) and 3( d) are plotted for J1=1.0, \n21J / J\n=1.0,\n31J / J = 1.0 , d = -1.0, h = 0.1 and  various different initial value s. The behavior \nof Fig. 3( c), is similar to that of Fig. 3( a); hence the system undergoes a second -order phase \ntransition from the f phase to the p phase at TC= 2.95.  In Fig. 3( d), the system undergoes two \nsuccessive phase transitions. The first is a first -order  transition , because discontinuity  occurs \nat T t = 0.28. The t ransition is from the p phase to the f phase. The second is a second -order \ntransition from the f phase to the p phase at T C = 2.95. This means that the coexistence region,  7 \n i.e the f + p mix ed phase , exists in the system.  Figs. 3(e) and 3( f) are obtained  for J1 = -1.0, J1 \n= -2.0, J3 = -3.0, d = -1.0, h = 0.1  and various different initial values . In Fig. 3( e), \nAB\n12MM = \n1 and  \nBA\n12MM  at the zero temperatu re and they decrease to zero continuously as the \ntemperature increases; hence the system undergoes a second -order phase transition from the  \naf phase to the p phase  at TC = 4.30. In Fig. 3( f), \nA,B\n1M =\nA,B\n2M  is alway s equal to zero; hence \nthe system does not undergo any phase transition . This figure corresponds to the p phase. \nFrom Figs. 3( e) and 3( f), one can see that the af + p mixed phase region exists until T C. It is \nworthwhile mention ing that two successive trans itions have also been experimentally \nobserved in DyVO 4 [47]. \n3.3. Dynamic phase diagrams  \nHaving  obtained the dynamic phase transition ( DPT ) points, we are now ready  to investigate  \nthe dynamic  phase diagrams  (DPD) . Figs. 4 -6 illustrate the  FM / FM, AFM / FM, AFM  / \nAFM interactions, respectively. In these figures solid and das hed lines represent the second -\norder (TC-lines) and first -order (T t-lines) phase transition lines and the filled circle (•)  \ncorresponds t o the dynamic tricritical point.  \n \n3.3.1.  The case of the FM/ FM interaction  \n \nThe dynamic phase diagram s of the BCIB system with the case of FM/ FM interactions for J1 \n= 1.0, \n2 1 3 1J / J 1.0, J / J 1.0  and various values of d are illustrated  in Fig s. 4 (a)-(e), \nnamely d= 1.0, -1.0, -2.0, -2.5, -3.0, respectively . In Fig. 4, the following five interesting \nphenomena are observed .  (i) The phase diagrams show one, two and three dynamic tricritical \npoints  that separate a second - order transition line from a first -order transition line , seen in \nFigs. 4(a)-(c), (e) and (d).  (ii) The tp dynamic special point , at which three first -order \ntransition lines meet , occur s in Fig s. 4(a), (b), (d). (iii) Figs. 4(a), (b) and (d) display  both a \ntricritical point and a triple point (TP), but Fig s. 4(c) and (e) show only a tricritical point . (iv) \nThe BCIB system always illustrates  the ferromagnetic fundamental phase. (v) The mixed \nphases usually separate with first -order phase transition lines from the fundamental phases. \nOn the other hand, t he dynamic phase boundaries among the fundamental phases are usually \nsecond -order phase transition lines.  \n  \n3.3.2.  The case of the AFM/FM interaction  \n \nThe dynamic phase diagram s of the BCIB system with the case of AFM/FM interactions for \nJ1 = -1.0, \n2 1 3 1J / J 1.0, J / J 0.1  and various values of d are illustr ated in Fig s. 5 (a)-(e), \nnamely d= 1.0, -1.0, -2.0, -2.5, -3.0, respectively . These phase diagrams in Fig. 5 are similar \nto Fig. 4 , and f rom these phase diagrams the  following four interesting phenomena were  \nobserved.  (i) Five different phase diagrams are obtained for AFM/FM interactions. (ii) The \nBCIB  system includes  the p  and nm fundamental phases  as well as  the sf + nm, nm + p  and \nnm + p mixed phases.  (iii) While the mixed phases usually separate with first -order phase \ntransition lines from the fundament al phases, the dynamic phase boundaries among the \nfundamental phases are usually second -order phase transition lines.  (iv) The phase diagrams \ndisplay  one, and two dynamic tricritical points , as seen in Figs. 5(a), 5(b ) and 5(c)-(e).  (v) \nOnly Fig. 5(c) il lustrates  the dynamic special point, namely tp.  \n \n3.3.3.  The case of the AFM/AFM interaction   8 \n  \nThe dynamic phase diagram of the BCIB system with the case  AFM/AFM  The phase \ndiagrams in Fig. 6 are not similar to those in Figs. 4 and 5. From these phase diagrams, the \nfollowing four interesting phenomena were  observed.  (i) Five  different phase diagrams are \nobtained for AFM/ AFM interactions. (ii) These dynamic phase diagrams, namely Fig. 6(a), \n(b) and (d), show a reentrant behavior. (iii) The BCIB  system contains  the p  fundamental \nphase  as well as  the f+ nm, m + p and af + p mixed phases.  (iv) One,  two or three  dynamic \ntricritical points  are seen in these phase diagrams. (v) The dynamic phase diagrams do not  \ncontain a dynamic special point.  \nIt has long been recognized t hat similar phase diagrams to those in  Figs. 4 (a), 4(b), 5(a), \n5(b), 6(d) and 6(e) have also been obtain ed in the kinetic mixed (1, 2) and mixed (2, 5/2)  \nsystems  (see [ 31, 32 , 48]  and references there in). On the other hand, the dynamic phase \ndiagrams in F igs. 4(c)–(e), 5(c) -(e), 6(a) -(c) are only observed in the BCIB system; hence \nthey were not obtained by previous studies of the kinetic Ising (see [31, 32, 48] and references \ntherein).  \n \n4. Summary and Conclusion  \n \nBy utilizing t he mean field theory based on  Glauber -type stochastic dynamics (DMFT) , the \ndynamic magnetic behaviors of the BCIS are investigated under  the presence of a time -\ndependent oscillating external magnetic field . Glauber -type stochastic dynamics were used to \ndescribe t he time evolution of t he system and the time variations of the average order \nparameters  were studied  in order to find the phases in the systems. Then, as a function of the \nreduced temperature , the behavior of the average order parameters in a period or the dynamic \norder paramet ers was investigated ; this study led us to characterize the (continuous or \ndiscontinuous) nature of the dynamic phase transitions and to obtain the dynamic phase \ntransition (DPT) points.  Finally, the dynamic phase diagrams were presented for \nferromagnetic / ferromagnetic, antiferromagnetic / antiferromagnetic, antiferromagnetic / \nantiferromagnetic interactions in the plane of the reduced temperature versus magnetic field \namplitude and they display dynamic tricritical and reentrant behavior as well as the dy namic \ntriple point.  \nWe hope that our detailed theoretical investigation may stimulate further works to study \nthe DPT and the dynamic hysteresis in the mixed Ising model using more accurate techniques , \nsuch as kinetic Monte Carlo (MC) simulations or renorma lization -group (RG) calculations, \nand that it may also be of assistance in further experimental research. Moreover, it may \nstimulate theoretical and experimental research to explore more complicated systems.  \n \nReferences  \n \n[1] T. Balcerzak and K. Szałowaki, Physica A 395, 183 -192 (2014).  \n[2] E. Kantar and M. Ertaş, Solid State Commun. 188, 71 -76 (2014).  \n[3] K. Szałowaki and T. Balcerzak, Thin Solid Films 534, 546 -552 (2013).  \n[4] I. Mazilu and B. Schmittmann, J. Stat. Phys. 113, 314 (2013).  \n[5] M.C. Figueiredo, T. M. Cotta and G. O. Pellegrino, Phys. Rev. E 81, 012104 -012108 \n(2010).  \n[6] E. Albayrak, A. Yiğit, T. Cengiz, Physica A 389, 2522 -2532 (2010).  \n[7] E. Albayrak, Phys. Status Solidi (b) 246, 226 -236 (2009).  \n[8] T. Balcerzak and I. Luźniak, Physica A 388, 357 -369 (2009).  \n[9] Y. Ma and A. Du, Phys. Status Solidi (b) 245, 740 -744 (2008).  \n[10] C. Ekiz, Physica A 387, 1185 -1199 (2008).   9 \n [11] L.S. Hung, J.F. Hu, J.S. Chen, J. Magn. Magn. Mater. 324, 1242 -1247 (2012).  \n[12] J. Li, J. Magn. Magn. Mate r. 324, 1512 -1515 (2012).  \n[13] L. Pan, H.J. Liu, Y.W. Wen, X.J. Tan, H.Y. Lv, J. Shi, X. F. Tang, Phys. Lett. A 375, \n614-619 (2011).  \n[14] G. Bayreuther, F. Bensch, and V. Kottler, J. Appl. Phys. 79, 4509 (1996).  \n[15] R. Krishnan, H.O. Gupta, H. Lassri, C. Sella, and J. Kaaboutchi, J. Appl. Phys. 70, 6421 \n(1991).  \n[16] D.L. Mills, Phys. Rev. B 3, 3887 (1971).  \n[17] T.C. Lubensky and M.H. Rubin, Phys. Rev. B 12, 3885 (1975)  \n[18] T. Kaneyoshi, Physica B 407, 4358 (2012).  \n[19] Ü. Akıncı, J. Magn. Magn. Mater. 329 , 178 (2012).  \n[20] K. Ohno, Y. Okabe, Phys. Rev. B 39, 9764 (1989).  \n[21] Q. Jiang, Z.Y. Li, J. Magn. Magn. Mater. 80, 178 (1989) . \n[22] P. Cossio, J. Mazo -Zuluaga, J. Restrepo, Physica B 384, 227 (2006).  \n[23] A. Zaim, Y. El Amraoui, M. Kerouad, L. Bihc, Fer roelectrics 372, 3 (2008).  \n[24] M Y Wu, A J Ye, Z B Li, W G Zeng, Acta Phys. Sinica 49, 1168  (2000).  \n[25] H. Jang, M. J. Grimson, Phys. Rev. E 63 (2001) 066119.  \n[26] H. Jang, M. J. Grimson, T. B. Woolf, Phys. Rev. E 70 (2004) 047101.  \n[27] O. Canko, E. Ka ntar, M. Keskin, Physica A 388 (2009) 28.  \n[28] M. Ertaş, E. Kantar and M. Keskin, J. Magn. Magn. Mater. 358 -359, 56 (2014).  \n[29] Ü. Temizer, J. Magn. Magn. Mater. 372, 47 (2014).  \n[30] Ü. Temizer, M. Tülek and S. Yarar, Physica A 415, 156 (2014).  \n[31] M. E rtaş, M. Keskin, Phys. Lett. A 376, 2455 (2012).  \n[32] M. Ertaş, M. Keskin, Chin . Phys. B 22, 120507 (2013).  \n[33] G. M. Buendía, P. A. Rikvold, Phys. Rev. E 78, 051108 (2008).  \n[34] Ü. Temizer, Chin. Phys. B 23, 070511 (2014).  \n[35] M. Ertaş and M. Keskin, P hysica A 411, 42 (2014).  \n[36] E. Vatansever and H. Polat, J. Magn. Magn. Mater. 332, 28 (2013).  \n[37] M. Ertaş, M. Keskin and B. Deviren, J. Magn. Magn. Mater. 324, 1503 (2012).  \n[38] M. Ertaş, B. Deviren and M. Keskin, Phys. Rev. E 86, 051110 (2012).  \n[39] M. Ertaş, M. Keskin and B. Deviren, Physica A 391, 1038 (2012).  \n[40] H. Fujisaka, H. Tutu and P. A. Rikvold, Phys. Rev E 63, 036109 (2001).  \n[41] E. Z. Meilikov, JETP Letters 79, 620 (2004).  \n[42] Q. Jiang, H.N. Yang, and G.C. Wang, Phys. Rev. B 52, 14911 (1 995).  \n[43] Z.A. Samoilenko, V.D. Okunev, E.I. Pushenko, V.A. Isaev, P. Gierlowski, K. Kolwas, \nand S.J. Lewandowski, Inorganic Materials 39, 836 (2003).  \n[44] N. Gedik, D.S. Yang, G. Logvenov, I. Bozovic, A.H. Zewail, Science 316, 425 (2007).  \n[45] D.T. Rob b, Y.H. Xu, O. Hellwing, J. McCord, A. Berger, M.A. Novotny, and P.A.  \nRikvold, Phys. Rev. B 78, 134422 (2008).  \n[46] L. O'Malley, G. Korniss, S.S. Praveen Mungara and T. Caraco, Evolutionary Ecology \nResearch 12, 279 (2010).  \n[47] A.H. Cooke, C.J. Ellis, K.A . Gehring, M.J.M. Leask, D.M. Martin, B.M. Wanklyn, M.R. \nWells, R.L. White, Solid State Commun. 8, 689 (1970)  \n[48] M. Keskin, M. Ertaş and O. Canko, Phys. Scr. 79, 025501 ( 2009 ). \n \nList of figure captions  \nFig. 1. Two-layer square lattice, G 1 and G 2 refer to  the upper and lower layers containing the \nspins labeled as\ni' , \nj' and \niS, \njS. \nFig. 2. Time variations of the magnetizations  (m1A, m 1B, m 2A and m 2B):  10 \n a) Exhibit ing paramagnetic phase (p), J1 = 1.0, \n2 1 3 1J / J 1.0, J / J 1.0 , h = 5.0, d = 1.0 \nand T  = 2.0; \nb) Exhibiting  ferromagnetic phase (f), J1 = 1.0, \n2 1 3 1J / J 1.0, J / J 1.0 , h = 1.0, d = 1.0 \nand T = 2. 0; \nc) Exhibiting antiferromagnetic phase (af), J 1 = -1.0, \n2 1 3 1J / J 1.0, J / J 3.0  , h = \n3.0, d = -1.0 and T = 1.52; \nd) Exhibiting surface  phase ( sf), J 1 = -1.0, \n2 1 3 1J / J 1.0, J / J 0.1 , h = 1.5, d = 1.0 and \nT = 1.0; \ne) Exhibiting  compensated phase  (c), J1 = 1.0, \n2 1 3 1J / J 1.0, J / J 1.0 , h = 2.05, d = -\n2.0 and T = 0.05; \nf) Exhibiting mixed phase (m),  J1 = -1.0, \n2 1 3 1J / J 1.0, J / J 5.0  , h = 4.0, d = 0.1 \nand T = 3.0; \ng) Exhibiting  nonmagnetic  phase  (nm), J1 = -1.0, \n2 1 3 1J / J 0.5, J / J 0.1 , h = 3, d = \n0.1 and T = 2. \nFig. 3. Reduced temperature dependences of dyna mic magnetizations \nA\n1M , \nB\n1M  and \nA\n2M , \nB\n2M\n. Tt and  TC are the critical or the first-order phase transition and the second -order phase \ntransition temperatures , respectively.  \na) Exhibiting seco nd-order phase transition from  the f phase to the p phase for J1 = 1.0, \n2 1 3 1J / J 1.0, J / J 1.0\n, d = 1.0 and h = 0. 5; TC is found at 3.63. \nb) Exhibiting first -order phase transition from the f phase to the p phase for J 1 = 1.0, \n2 1 3 1J / J 1.0, J / J 1.0\n, d = 1.0 and h = 3.77 ; Tt is found at 0. 75. \nc)  and d) Exhibiting t wo successive phase transitions, the first is a first -order phase \ntransition from the f + p phase to the f phase and the second one is a second -order \nphase transiti on from the f to the p phase for J1 = 1.0, \n2 1 3 1J / J 1.0, J / J 1.0 , d = -\n1.0 and h = 0.1; 0.28 and 2.95 are found for Tt and TC, respectively.  \ne) and f)  Exhibiting second -order phase transitions from the af + p phase to the  p phase \nfor J 1 = -1.0, J1 = -2.0,  J3 = -3.0, d = -1.0 and h = 0.1; 4.3 is  found for TC. \nFig. 4 . Phase diagrams of  spin-1 Blume -Capel  Ising  bilayer  (BCIB) system  on two-layer \nsquare lattice  in (T, h) plane  for FM/FM , i.e. J 1 > 0 and J 2 > 0. Dashed lines represent first -\norder phase trans ition, and tp represent s the dynamic triple point.  For J1 = 1.0, \n2 1 3 1J / J 1.0, J / J 1.0\n and a) d = 1.0; b) d = -1.0; c) d = -2.0, d) d = -2.5 and e) d = -3.0.  \nFig. 5 . Fig. 5 same as Fig. 4, phase diagrams of BCIB system  on two -layer square lattice in  \n(T, h) plane for AFM/FM , i.e. J 1 < 0 and J 2 > 0. For J1 = -1.0, \n2 1 3 1J / J 1.0, J / J 0.1  and a) \nd = 1.0 ; b) d = -1.0; c) d = -2.0, d) d = -2.5 and e) d = -3.0.  \nFig. 6. Fig. 6 same as Fig. 4, phase diagrams of BCIB system  on two -layer square lattice in  \n(T, h) plane for AFM/AFM , i.e. J 1 < 0 and J 2 < 0. a) J1 = -1.0, \n2 1 3 1J / J 1.0, J / J 5.0  , d =  11 \n 0.1; b) J1 = -1.0, \n2 1 3 1J / J 1.0, J / J 5.0  , d = 1.0 ; c) J1 = -1.0, \n2 1 3 1J / J 1.0, J / J 5.0  , \nd = -1.0 d) J1 = -1.0, \n2 1 3 1J / J 1.0, J / J 5.0  , d = -2.0 and e) J1 = -1.0, \n2 1 3 1J / J 1.0, J / J 3.0 \n, d = -1.0 \n \nTable caption  \nTable 1. Characteristic s of time variations of magnetizations (\nA\n1m ( ) ,\nB\n1m ( ) ,\nA\n2m ( ) ,\nB\n2m ( ) ).  \nG2i' i'j'\nFig. 1G1\nJ3\nJ3J3\nJ2J1\nSi\nSiSiSi Si\nSjSjSj\nSjj'\ni' i'i' j'\nj'Figure 1\nClick here to download Figure: Fig. 1.doc  \nFig. 2 \n0 20 40 60 80 100 120m1A (), m1B(), m2A(), m2B()\n-1.0-0.50.00.51.00 20 40 60 80m1A (), m1B(), m2A(), m2B()\n-1.0-0.50.00.51.0\nm1A ()= m1B()= m2A()= m2B()\n0 20 40 60 80 100 120m1A (), m1B(), m2A(), m2B()\n-1.0-0.50.00.51.0m1A ()= m1B()= m2A()= m2B()\n0 20 40 60 80 100m1A (), m1B(), m2A(), m2B()\n-1.0-0.50.00.51.0\nm1A ()= m2B()\n m2A()= m1B()m1A ()= m1B()= m2B()\nm2A()\n0 20 40 60 80 100 120m1A (), m1B(), m2A(), m2B()\n-1.0-0.50.00.51.0\nm1A ()= m1B()\nm2A ()= m2B()\n0 20 40 60 80m1A (), m1B(), m2A(), m2B()\n-1.0-0.50.00.51.0\nm1A ()= m2A()\nm1B ()= m2B()\n0 20 40 60 80m1A (), m1B(), m2A(), m2B()\n-1.0-0.50.00.51.0\nm1A ()= m1B() m2A()= m2B()\n m2A()= m2B()( a ) ( b )\n( c ) ( d )\n( e ) ( f )\n( g )Figure 2\nClick here to download Figure: Fig. 2.doc T0 1 2 3 4M1A, M1B, M2A, M2B\n0.00.20.40.60.81.0 ( a )\nT0.0 0.5 1.0M1A, M1B, M2A, M2B\n0.00.20.40.60.81.0\nM1A= M1B= M2A= M2B( b )\nTt TC\nT0.0 0.9 1.8 2.7M1A, M1B, M2A, M2B\n0.00.20.40.60.81.0 ( c )\nTCM1A= M1B= M2A= M2B\nM1A= M1B= M2A= M2B\nT0.0 0.9 1.8 2.7M1A, M1B, M2A, M2B\n0.00.20.40.60.81.0( d )\nTt TCM1A= M1B= M2A= M2B\nT0.0 1.0 2.0 3.0 4.0 5.0M1A, M1B, M2A, M2B\n-1.0-0.50.00.51.0\nM1A= M2B\n M1B= M2A\nTC( e )\nT0.0 1.0 2.0 3.0 4.0 5.0M1A, M1B, M2A, M2B\n-1.0-0.50.00.51.0( f )\nM1A= M1B= M2A= M2B \n \n                            Fig. 3         Fig. 3\nClick here to download Figure: Fig. 3.doc T0 1 2 3 4h\n01234\nfpf + p\ntp\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0h\n01234\nf + ptp\nff + p ( a ) ( b )\nT0.0 0.5 1.0 1.5 2.0 2.5h\n01234\nnm+ pf + pp\nff + c\nT0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4h\n012345\nnm + pf\nf + pf + ptpf + p\nf + c\nT0.0 0.2 0.4 0.6 0.8 1.0h\n0.01.53.04.56.0\nnm + pf + cff + p( c ) ( d )\n( e )\nppppp \n \n        Fig. 4  Figure 4\nClick here to download Figure: Fig. 4.doc  \n \n                                                                                                             Fig. 5  \nT0.0 0.5 1.0 1.5 2.0 2.5 3.0h\n0.00.81.52.33.03.8\np\nsf + nm\nT0.0 0.5 1.0 1.5 2.0 2.5h\n0.01.02.03.04.0\nnm+ p\nnm + psf + nmp nm + p\nT0.0 0.2 0.4 0.6 0.8 1.0 1.2h\n0.00.81.52.33.03.8\nnm\nnm + ptp\nT0.0 0.2 0.4 0.6 0.8h\n0.01.02.03.04.0\nnm\nnm + p\nnm + pp\np\nT0.0 0.2 0.4 0.6 0.8h\n0.00.81.52.33.03.84.5\nnm\nnm + pp( a ) ( b )\n( c ) ( d )\n( e )pFigure 5\nClick here to download Figure: Fig. 5.doc T0.0 1.5 3.0 4.5 6.0h\n0.01.02.03.04.05.06.07.0\nT0 1 2 3 4 5 6h\n0.01.02.03.04.05.06.07.0\nm + p\nf + mp( a ) ( b )\nm + p\n0.0 0.5 1.00.00.40.8\nf + mp\nT0 1 2 3 4 5 6h\n0.01.02.03.04.05.06.07.0\nm + pp\nf + m\nT0 1 2 3 4 5h\n0.01.02.03.04.05.06.07.0\nm + pp( c ) ( d )\nT0 1 2 3 4 5h\n0.01.02.03.04.05.06.0\naf + pp( e ) \n \nFig. 6  Figure 6\nClick here to download Figure: Fig. 6.doc Table  1  Oscillation of  Magnetizations  \nPHASES  Symbol  Configurations  m1A m1B m2A m2B \nParamagnetic \nPhase  p  0 0 0 0 \nFerromagnetic \nPhase   \nf \n  \n \n 1 1 1 1 \nAntiferromagnetic \nPhase  af  1 -1 -1 1 \nSurface Phase  sf  1 1 -1 1 \nCompensated \nPhase  c  \n1 1 -1 -1 \n-1 -1  1  1 \nMixed Phase   \nm \n  \n1 -1 1 -1 \nNon magnetic \nPhase  nm             \n            \n            \n           or 0 0 1 1 \n0 0 -1 -1 \n1 1 0 0 \n-1 -1 0 0 \n Table\nClick here to download Table: Table 1.doc  \nERCIYES UNIVERSITY  \nDEPARTMENT OF PHYSICS  \n38039 KAYSERI -TURKEY  \n \nPHONE: 90 (352) 207666 Ext:  33134 \nFAX: 90 (352) 437493 3 \nE-Mail: mehmetertas@erciyes.edu.tr       November  23, 2014 \n \nProf. Dr. Joel L. Lebowitz                       \nDirector,Center for Mathematical Sciences Research.  \nRutgers, The State University  \n110 Frelinghuysen Road  \nPiscataway, NJ  08854 -8019  \nPhone: 732 -445-3117/3923  \nFax: 732 -445-4936  \nEmail: lebowitz@math.rutgers.edu   \n \n \n            Dear Prof . Dr. Lebowitz  \n \nI hereby respectfully submit a manuscript entitled “ The dynamic magnetic behaviors of the \nBlume -Capel Ising bilayer system ” for publication in Journal of Statistical Physics .  \n \nLooking forward to hearing from you soon.  \n \n \n                                                                                                                                                         \nSincerely yours,  \n \nAssociate Professor  \nMehmet Ertaş  \n \n attachment to manuscript\nClick here to download attachment to manuscript: Cover lett.doc Research Highlights  \n \n Dynamic behaviors in the spin-1 Blume -Capel Ising  bilayer system  is investigated . \n \n The dynamic phase transitions and dynamic phase diagrams are obtained.  \n \n Dynamic phase diagrams are presented for FM/FM, AFM/FM and AFM/AFM \ninteraction.  \n Dynamic phase diagrams exhibit several ordered phases, coexistence phase regions as \nwell as a re -entrant behavior.  attachment to manuscript\nClick here to download attachment to manuscript: Research Highlights.doc "    },    {        "title": "2007.13122v3.Effect_of_the_anomalous_magnetic_moment_of_quarks_on_magnetized_QCD_matter_and_meson_spectra.pdf",        "content": "E\u000bect of the anomalous magnetic moment of quarks on magnetized QCD matter and\nmeson spectra\nKun Xu1;2,\u0003Jingyi Chao3, and Mei Huang1y\n1School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China\n2Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049, P.R. China and\n3Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, P.R. China\nWe systematically investigate the e\u000bect of the anomalous magnetic moment(AMM) of quarks on\nthe magnetized QCD matter, including the magnetic susceptibility, the inverse magnetic catalysis\naround the critical temperature and the neutral/charged pion and rho meson spectra under magnetic\n\felds. The dynamical AMM of quarks, its coupling with magnetic \feld causes Zeeman splitting of\nthe dispersion relation of quarks thus changes the magnetism properties and meson mass spectra\nunder magnetic \felds. It is found that including the AMM of quarks cannot fully understand lattice\nresults of the magnetized matter: The AMM of quarks reduces the dynamical quark mass thus causes\nthe inverse magnetic catalysis around Tc. The neutral pion mass is very sensitive to the AMM, it\ndecreases with magnetic \feld quickly, and the charged pion mass shows a nonlinear behavior, i.e.,\n\frstly linearly increases with the magnetic \feld and then saturates at strong magnetic \feld. For\nrho meson, it is observed that AMM reduces the mass of neutral rho meson mass with di\u000berent sz,\nand reduces the mass of sz= +1 ;0 component charged rho meson mass but enhances the sz=\u00001\ncomponent charged rho meson mass. The magnetic susceptibility at low temperature can be either\npositive or negative with di\u000berent AMM.\nPACS numbers: 12.38.Mh, 25.75.Nq, 25.75.-q\nI. INTRODUCTION\nUnderstanding properties of QCD matter under strong magnetic \fled is of vital importance to further explore the\ninterior of magnetar [1, 2], neutron-star merges [3, 4], non-central heavy-ion collisions [5, 6] and the evolution of the\nearly universe [7]. The study of the QCD vacuum and strongly interacting matter under external magnetic \felds has\nattracted much attention, see reviews, e.g. Refs. [8{12]. With the presence of a magnetic \feld background, the strongly\ninteracting matter shows a large number of exotic phenomena, for example, Chiral Magnetic E\u000bect(CME) [13{16],\nMagnetic Catalysis(MC) in the vacuum [17{19], Inverse Magnetic Catalysis(IMC) around the critical temperature [20{\n22].\nThe catalysis of the chiral symmetry breaking induced by the magnetic \feld, i.e., the MC e\u000bect can be easily\nunderstood from the dimension reduction. On the other hand, the IMC e\u000bect, the critical temperature of chiral phase\ntransition decreases with the magnetic \feld, which intuitively contradicts with the MC e\u000bect and still remains as\na puzzle, though there have been many literatures trying to explain the IMC by considering neutral pion \ructua-\ntions [23], chirality imbalance [24], running coupling constant [25]. Recently, lattice calculations show more interesting\nand novel properties of magnetized QCD matter: The charged pion mass shows a non-monotonic magnetic \feld de-\npendent behavior [26], and magnetized matter exhibits diamagnetism (negative susceptibility) at low temperature\nand paramagnetism (positive susceptibility) at high temperature [27, 28].\nMagnetic \felds modify the spectrum of charged particles. The point particle approximation gives the charged\npion massm2\n\u0019\u0006(B) =m2\n\u0019\u0006(B= 0) +eBincreasing linearly with the magnetic \feld, and neutral pion mass keeps\nas a constant. Calculation in the e\u000bective quark model, e.g, the Namu{Jona-Lasinio (NJL) model which takes into\naccount of quark magnetization, modi\fes the linear slope of charged pion and shows similar results for pion mass\nspectra as point-approximation results [29{39]. In the NJL model, mesons are considered as quantum \ructuations\nin Random Phase Approximation(RPA), where the mesons are introduced via the summation of an in\fnite number\nof quark loops [32, 33, 36, 37, 40{42]. However, with presence of magnetic \feld, the Schwinger phase appears in each\nquark propagator [43]. For neutral pion, the Schwinger phases cancel out for each loop, while they cannot for charged\npion. In Ref.[36, 37] the authors employed the Ritus eigenfunction method in the two-\ravor NJL model, which allows\nto properly take into account the presence of Schwinger phases in the quark propagators. They found that in the\nregioneB\u00180\u00001:5GeV2, neutral pion mass decreases slightly while charged pion mass steadily increases. And in\n\u0003xukun@mail.ihep.ac.cn\nyhuangmei@ucas.ac.cnarXiv:2007.13122v3  [hep-ph]  9 Feb 20212\nRef.[32], it is found that the charged pion becomes much heavier in the magnetic \feld and are sensitive to the \feld\nstrength, while the neutral pion still keeps as a Nambu-Goldstone particle in the region around eB\u00180\u00000:4GeV2.\nAnd Ref.[34] shows that neutral pion mass \frstly decreases and then increases with magnetic \feld, which is consistent\nwith lattice result in Ref.[44, 45]. Similar results are also found in another lattice calculation Ref.[46]. In Ref.[20],\ncharged pion mass was calculated in Lattice QCD, and they found that charged pion mass increases with magnetic\n\feld in the region of eB\u00180\u00000:3GeV2, however, for eB > 0:3GeV2, charged pion mass doesn't show a increase\ntrend with magnetic \feld. Recent lattice calculation in Ref.[26] shows that the neutral pion mass decreases with the\nmagnetic \feld while the charged pion and kaon show non-monotonic behaviors, the mass of which \frstly increases\nlinearly and then decreases as magnetic \feld increases, and all these masses show a saturation at eB&2:5GeV2,\nwhich are quite di\u000berent from point-particle approximation and previous results from e\u000bective models. It is worthy\nof mentioning that lattice results in [20, 46] show a nonlinear and saturation behavior for charged pion, no decreasing\nwith magnetic \feld is observed at strong magnetic \feld.\nThe remaining IMC puzzle and recently discovered properties of magnetized matter attract our renewed interest\nto revisit QCD vacuum and matter under external magnetic \feld and try to \fnd the underline mechanism for these\nproperties. It is known that the dynamical chiral symmetry broken is one of the most signi\fcant feature of QCD, where\nquarks obtain a dynamical mass. It has been found that anomalous magnetic moments(AMM) of quarks can also be\ngenerated dynamically along with dynamical quark mass [47, 49, 50]. In Ref.[49] the authors explained how dynamical\nchiral symmetry breaking produces a dressed light-quark with a momentum-dependent anomalous chromomagnetic\nmoment as well as the generation of an anomalous electromagnetic moment for the dressed light-quark in QED and\nQCD. In Ref.[47], the authors considered one-\ravor NJL model with a new channel \u0018(\u0016 \u00063 )2+ (\u0016 i\r5\u00063 )2in\nthe presence of magnetic \feld, and they found that the chiral condensate and new condensate h\u0016 i\r1\r2 i, which\ncorresponds to AMM for quarks, are di\u000berent from zero at vacuum with magnetic \feld simultaneously. A two-\ravor\nNJL model with tensor channel was also investigated in Ref.[54], and similar results were obtained. And in Ref.[50, 51],\nthe authors investigated the non-perturbative generation of an anomalous magnetic moment for massless fermions\nin the presence of an external magnetic \feld, and they proved that the phenomenon of magnetic catalysis of chiral\nsymmetry breaking is also responsible for the generation of the dynamical anomalous magnetic moment. Thus, once\nthe quarks acquire a dynamical mass, they should also acquire a dynamical AMM [47{51].\nThere have been some papers working on how AMM of quarks in\ruences QCD phase diagram as well as mesonic\nproperties [25, 52, 53, 55{59]. For example, in Ref.[55], the authors used two-\ravor NJL model with AMM and\nfound that the critical temperature for chiral transition decreases with the external magnetic \feld, while a sudden\njump for pion mass at and above the Mott transition temperature appears when the AMM of the quarks are taken\ninto consideration. And Ref.[52] found that Inverse Magnetic Catalysis occurs for large enough AMM. In Ref.[25]\nthe authors considered NJL model with dynamical induced AMM and found that with magnetic-dependent coupling\nconstants, the inverse magnetic catalysis can be obtained.\nIn the present paper, we focus on the dynamical quark mass as well as meson mass, e.g., pion and rho, in the\npresence of magnetic \feld with anomalous magnetic moments of quarks. This paper is organized as follows: in Sec. II\nwe introduce two-\ravor NJL model with AMM in the presence of magnetic \feld, and investigate the e\u000bect of AMM on\ndynamical quark mass and the magnetism property. Then we investigate pion and rho mass as a function of magnetic\n\feld with di\u000berent AMM In Sec.III. Finally, we discuss the results in Sec.IV.\nII. MODEL SETUP\nWe choose two-\ravor Nambu{Jona-Lasinio model including the AMM of quarks in the presence of magnetic \feld,\nthe Lagrangian of which takes form of [52, 55, 57]:\nL=\u0016 (i\r\u0016D\u0016\u0000m0+\u0014fqfF\u0016\u0017\u001b\u0016\u0017) +GSn\n(\u0016  )2+ (\u0016 i\r5~ \u001c )2o\n\u0000GVn\n(\u0016 \r\u0016~ \u001c )2+ (\u0016 \r\u0016\r5~ \u001c )2o\n:(1)\nHere are two-\ravor quark \fled  = (u;d)T,m0is current mass and we assume the current quark mass for both\n\ravors are the same: mu=md=m0. The covariant derivative D\u0016=@\u0016\u0000iqfA\u0016withqfthe electric charge of\nquarks, and A\u0016is the Abelian gauge \feld and \feld strength F\u0016\u0017=@\u0016A\u0017\u0000@\u0017A\u0016. Without loss of generality, we\nchoose external uniform magnetic \feld along z-direction, which led to A\u0016=f0;0;Bx; 0g. The term \u0016 \u0014fqfF\u0016\u0017\u001b\u0016\u0017 ,\nwith\u001b\u0016\u0017=i\n2[\r\u0016;\r\u0017], is to present the contribution of AMM, and \u0014fis de\fned as \u0014f=\u000bf\u0016Bwith\u0016B=e\n2Mthe\nBohr magneton, and Mthe constituent quark mass as de\fned below. At one-loop level we have \u000bf=\u000beq2\nf\n2\u0019, with\n\u000be=1\n137the electromagnetic \fne structure constant. However, to study how the AMM in\ruence quark mass as well\nas meson mass, we treat \u0014fas a free parameter and be \ravor-independent, e.g., \u0014u=\u0014d=\u0014. Besides,GSandGVare3\nthe coupling constants for (pseudo-)scalar and (pseudo-)vector interaction channel, respectively. Then the Lagrangian\nafter mean-\feld approximation is given by:\nL=\u0000(M\u0000m0)2+\u0019a\u0019a\n4GS+Va\n\u0016V\u0016;a+Aa\n\u0016A\u0016;a\n4GV+\u0016 (i\r\u0016D\u0016\u0000M+\u0014qfB\u001b12) ; (2)\nwhere we de\fne:\nM=m0\u00002GSh\u0016  i; ~ \u0019 =\u00002GSh\u0016 i\r5~ \u001c i;\nVa\n\u0016=\u00002GVh\u0016 \r\u0016\u001ca i; Aa\n\u0016=\u00002GVh\u0016 \r\u0016\r5\u001ca i:(3)\nIt is know that NJL model is non-renormalized, thus regularization scheme is necessary for \fnite numerical results, and\nin this paper, a soft cut-o\u000b is applied for momentum integration and Landau level summation during the numerical\ncalculation:\njqfBj\n2\u0019X\nnZdpz\n2\u0019!jqfBj\n2\u0019X\nnZdpz\n2\u0019f\u0003(pz;n) (4)\nwith\nf\u0003(pz;n) =\u000310\n\u000310+ (p2z+ 2njqfBj)5: (5)\nThere are four parameters in total: current quark mass m0, three-momentum cuto\u000b parameter \u0003 in Eq.(5), (pseudo-\n)scalar/(pseudo-)vector coupling constant GS/GV, and these parameters are determined by \ftting to experimental\ndatas at zero temperature and vanishing magnetic \feld. In this work, we have used two sets of parameters: I),\nm0= 5MeV, \u0003 = 624 :18MeV and GS\u00032= 2:014, which corresponds to pion decay constant f\u0019= 93MeV, pion\nmassm\u0019= 135:6MeV as well as the quark condensate h\u0016  i=\u0000(251:8MeV)3, and this set of parameters is used\nto investigate dynamical quark mass as well as pion mass; II), m0= 5MeV, \u0003 = 582MeV, GS\u00032= 2:388 and\nGV\u00032= 1:73, which are chosen in such a way that m\u0019= 140MeV, m\u001a= 768MeV while M= 458MeV at zero\ntemperature as used in Ref.[29, 41], which indicates the quark condensate h\u0016  i=\u0000(267MeV)3. In particular, the\ndynamical generated quark mass Mresulting from parameter set II) is deliberately chosen to be large to avoid the\ndecay process \u001a!q\u0016qat zero temperature, and this set is used for the study of rho meson.\nA. Dispersion Relation for Fermions with AMM\nIn this part, we derive the dispersion relation for positive charged fermion  with charge q, dynamical mass Mand\nthe anomalous magnetic moment \u0014in the presence of homogeneous magnetic \feld B, the Dirac equation of which is\ngiven by\n(i\r\u0016D\u0016\u0000M+1\n2\u0014q\u001b\u0016\u0017F\u0016\u0017) = 0: (6)\nSimilar with Eq.(1), we take the magnetic \feld along zdirection, and to simplify following derivation, we set T=\u0014qB,\nthen the Dirac equation becomes:\n(i\r\u0016D\u0016\u0000M+T\u001b12) = 0; (7)\nwhere\u001b12=i\r1\r2. In this case, the general solution of  has form of:\n = e\u0000iEt\u0012\n\u001e\n\u001f\u0013\n(8)\nwhere\u001eand\u001fare the two-component spinors. Inserting Eq.(8) into Eq.(7), where chiral representations of the\n\r-matrices are used, we obtain the coupled equations for \u001eand\u001f:\n(M\u0000T\u001b3)\u001e\u0000(E+i~ \u001b\u0001~D)\u001f= 0; (9)\n(E\u0000i~ \u001b\u0001~D)\u001e\u0000(M\u0000T\u001b3)\u001f= 0: (10)\nEliminating \u001ffrom Eq.(9) and Eq.(10) then the equation for \u001eis obtained:\n^A\u001e=n\n(M\u0000T\u001b3)(M2\u0000T2)\u0000(E+i~ \u001b\u0001~D)(M+T\u001b3)(E\u0000i~ \u001b\u0001~D)o\n\u001e= 0: (11)4\nIt is obvious that \u001eis a two-component spinor while ^Ais a 2\u00022 matrix, and the elements of ^Aare listed as following:\n^A12= 2T(E+iD3)(iD1+D2);\n^A21=\u00002T(E\u0000iD3)(iD1\u0000D2);\n^A11= (M+T)(M\u0000T)2\u0000(M+T)(E2+D2\n3) + (M\u0000T)(iD1+D2)(iD1\u0000D2);\n^A22= (M\u0000T)(M+T)2\u0000(M\u0000T)(E2+D2\n3) + (M+T)(iD1\u0000D2)(iD1+D2):(12)\nTo determine the form of \u001e, let's \frst consider function fk(x) de\fned as below:\nfk(x) =cke\u00001\n2(x\nl\u0000pyl)2Hk(x\nl\u0000pyl)ei(\u0018fpyy+pzz); (13)\nwhereHk(x) is the Hermite polynomials, ckis the normalized constant, with l= 1=p\njqBj,\u0018f= sgn(q), and we set\nf\u00001= 0. With a little e\u000bort the following relations for q>0 can be obtained:\n(iD1+D2)fk=\u0000ip\njqBjck;k+1fk+1; (14)\n(iD1\u0000D2)fk= 2ikp\njqBjck;k\u00001fk\u00001; (15)\nwherecn;m=cn=cm. In fact, the operators iD1+D2andiD1\u0000D2are creation and annihilation operator, respectively.\nAnd then the general form of \u001eis straightforward:\n\u001ek(x) =\u0012\nfk(x)\nfk\u00001(x)\u0013\n: (16)\nNow insert Eq.(12) and Eq.(16) into Eq.(11) and we obtain two de-coupled equations:\nn\n(M+T)(M\u0000T)2\u0000(M+T)(E2\u0000p2\nz) + 2k(M\u0000T)jqBjo\nfk\n\u00002iT(E\u0000pz)p\njqBjck\u00001;kfk= 0;(17)\nn\n(M\u0000T)(M+T)2\u0000(M\u0000T)(E2\u0000p2\nz) + 2k(M+T)jqBjo\nfk\u00001\n\u00002iT(E+pz)\u00012kp\njqBjck;k\u00001fk\u00001= 0:(18)\nCombining above two equations with relations Eq.(14) and Eq.(15), the dispersion relation for positive-charged fermion\nwith spin-sis obtained as:\nE2\nk=8\n><\n>:p2\nz+fp\nM2+ 2kjqBj\u0000sTg2;ifk\u00151\np2\nz+ (M\u0000T)2; ifk= 0(19)\nA similar relation for negative charged fermion can be obtained, however, we won't repeat here. Finally, taking\nT=\u0014qB, the dispersion relation for fermions with charge q, spin-sand anomalous magnetic moment \u0014in the\npresence of magnetic \feld is:\nE2\nk=p2\nz+fp\nM2+ (2k+ 1\u0000s\u0018)jqBj\u0000s\u0014qBg2; (20)\nwheres=\u00061 is for spin-up and spin-down,respectively, and \u0018= sgn(q). From this dispersion relation, we can see\nthat Lowest Landau Level (LLL) for both spin-up/down positive/negative charged quarks have the form of\nE2\n0=p2\nz+ (M\u0000\u0014jqfjB)2; u\";\u0016d\"\nE2\n0=p2\nz+ (M\u0000\u0014jqfjB)2;\u0016u#;d#\nE2\n0=p2\nz+ (q\nM2+ 2jqfBj+\u0014jqfjB)2;\u0016u\";d\"\nE2\n0=p2\nz+ (q\nM2+ 2jqfBj+\u0014jqfjB)2: u#;\u0016d#(21)\nTherefore, for two-\ravor quark system, for \u0014>0, the lowest energy is occupied by spin-up positive-charged fermions\nu\";\u0016d\"and spin-down negative-charged fermions \u0016 u#;d#. For higher excitations, for either positive or negative charged\nfermions, the energy spectrum exhibits a Zeeman splitting( s=\u00061), which can be seen clearly in Fig.(1).5\nFIG. 1: Energy levels of positive charged fermion for the case with(right) and without(left) AMM in the presence of\nmagnetic \feld.\nB. Inverse Magnetic Catalysis with AMM\nFrom the dispersion relation Eq.(21) in the LLL, e.g., E2\n0=p2\nz+ (M\u0000\u0014jqfjB)2, we can see that AMM reduces the\ndynamical quark mass. We numerically investigate the e\u000bect of AMM on the dynamical quark mass, and following\nRef.[40, 52, 55, 57] we obtain the one-loop level e\u000bective potential at zero baryon chemical potential and \fnite\ntemperature:\n\n =(M\u0000m0)2\n4GS\u0000NcX\nfjqfBj\n2\u0019X\nnX\ns=\u00061Zdpz\n2\u0019En;f;s\n\u00002NcTX\nfjqfBj\n2\u0019X\nnX\ns=\u00061Zdpz\n2\u0019ln(1 + e\u0000En;f;s\nT):(22)\nhere only scalar channel is considered for the study of the dynamical quark mass, which can be obtained by solving\nthe gap equation:\n@\n@M= 0; (23)\nand in explicit:\nM\u0000m0\n2GS=NcX\nfjqfBj\n2\u0019X\nnX\ns=\u00061Zdpz\n2\u0019n\n1\u00002(1 + eEn;f;s\nT)\u00001oM\nEn;f;s(1\u0000s\u0014qfB\nMn); (24)\nwhereM=m0+\u001bis quark's dynamical mass with\n\u001b= 2GSNcX\nfjqfBj\n2\u0019X\nnX\ns\u00061Zdpz\n2\u0019n\n1\u00002(1 + eEn;f;s\nT)\u00001oM\nEn;f;s(1\u0000s\u0014qfB\nMn); (25)\nwhereMn=p\nM2+ (2n+ 1\u0000s\u0018f)jqfBj\u0000s\u0014qfB. In strong magnetic \feld region, we can take LLL approximation\nthen the gap equation at zero temperature becomes:\nM=m0+ 2GSNcX\nfjqfBj\n2\u0019Zdpz\n2\u00191\nE0(1\u0000\u0014jqfBj\nM)\n=m0+ 2GSNcX\nfjqfBj\n2\u0019Zdpz\n2\u00191\nE(1\u0000\u0015\u0014+ (\u00142));(26)6\nwhereE=p\np2z+M2and\u0015=p2\nz\nM(M2+p2z). In the second line we assume a small \u0014expansion, and it's clear that a\nnone-zero\u0014reduces the quark mass Min the strong magnetic \feld region.\nThe numerical results of the dynamical quark mass as a function of the magnetic \feld with AMM is shown in\nFig.(2) and Fig.(3), where \u0014is treated as a constant and \u0014\u0018\u001bwith\u001bsolved from Eq.(25), respectively.\nκ=0.0 GeV-1κ=0.3 GeV-1κ=0.6 GeV-1\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\neB[GeV2]Mq[GeV]\nκ=0.62 GeV-1κ=0.65 GeV-1κ=0.7 GeV-1κ=0.8 GeV-1\n0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\neB[GeV2]Mq[GeV]\nFIG. 2: Dynamical quark mass as a function of magnetic \feld with di\u000berent constant \u0014at zero temperature.\n1) We \frstly treat \u0014as a free constant and calculate the dynamical quark mass as a function of the magnetic\n\feld at zero temperature with di\u000berent \u0014, and numerical results are shown in Fig.(2). It can be observed clearly\nthat the magnetic catalysis (MC) e\u000bect is competing with the mass reducing e\u000bect from the the AMM. For small\n\u0014.0:6GeV\u00001, the dynamical quark mass Mqincreases with magnetic \feld, which is known as Magnetic Catalysis,\nbut the AMM explicitly reduces dynamical quark mass as shown in the left \fgure of Fig.(2), which is consistent with\nthe dispersion relation of quark with AMM as shown in Eq.(21). For larger \u0014, as shown in the right \fgure of Fig.(2),\nthe behavior of dynamical quark mass becomes complicated. For \u0014= 0:62GeV\u00001,Mqstill increases with magnetic\n\feld but drops to zero and then jumps to none zero at a narrow region of magnetic \feld around eBc\u00180:6GeV2.\nAs\u0014increases, the region that Mq= 0 becomes larger and larger, the left edge of which decreases while the right\nedge increases. And when \u0014= 0:8GeV\u00001as the blue dashed line shown in Fig.(2), the dynamical quark mass slightly\ndecreases as magnetic \feld increases then drops to zero at eBc\u00180:3GeV2and keeps zero in the rest region we\nconsidered eB < 1GeV2.\n2) Then we take \u0014proportional to be to the quark condensate \u0014=v\u001bwithvthe ratio and \u001bsolved from Eq.(25).\nThe corresponding result for dynamical quark mass at zero temperature is shown in Fig.(3). We can see that when the\nratiovis small, as shown by the red solid line and green dashed line, Mqincreases with magnetic \feld monotonically,\nthe MC e\u000bect overcomes the mass reducing e\u000bect by the AMM. When vis large enough, the mass reducing e\u000bect\nv=0.0v=0.5v=1.0\n0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0\neB[GeV2]Mq[GeV]\nFIG. 3: Dynamical quark mass as a function of magnetic \feld with \u0014=v\u001bat zero temperature.7\nv=0.0v=0.5v=1.0\n0.05 0.10 0.15 0.20 0.25 0.300.00.10.20.30.40.50.6\nT[GeV]Mq[GeV]\n(a)\nv=0.0v=0.5v=1.0\n0.0 0.2 0.4 0.6 0.80.050.100.150.200.250.30\neB[GeV2]Tc[GeV] (b)\nFIG. 4: (a): Dynamical quark mass Mqas a function of temperature with \u0014=v\u001bat a \fxed magnetic \feld\neB= 0:5GeV2. (b): critical temperature Tcof chiral transition aas a function of magnetic \feld. With proper v,\nInverse Magnetic Catalysis can be obtained(e.g., blue dashed line).\ninduced by the AMM dominates thus one can see that the dynamical quark mass decreases smoothly with the\nincreasing magnetic \feld, no \"jump\" behavior shows up as the case of constant \u0014. Then we also consider dynamical\nquark mass at \fnite temperature, under a \fxed magnetic \feld eB= 0:5GeV2, as shown in Fig.(4a), it's clear that\nlargervcan not only reduce the dynamical quark mass, also reduce the critical temperature Tc, which is determined\nthroughTc= (\u0000@Mq=@T).Tcas a function of the magnetic \feld under di\u000berent vis plotted in Fig.(4b): for small\nv, conventional Magnetic Catalysis shows up, i.e., Tcincreases with the magnetic \feld, however, when vis large, for\nexample,v= 1 as shown by the blue dashed line, Tcdecreases with the magnetic \feld, which indicates the Inverse\nMagnetic Catalysis.\nC. Magnetic Susceptibility with AMM\nRecently, lattice calculations show more interesting and novel properties of magnetized QCD matter: magnetized\nmatter exhibits diamagnetism (negative susceptibility) at low temperature and paramagnetism (positive susceptibility)\nat high temperature [27, 28]. From the above dispersion relation, we can see that the AMM of quarks causes Zeeman\nsplitting in the dispersion relation thus changes the magnetism properties. We show the numerical results for magnetic\nsusceptibility induced by the AMM. The magnetic susceptibility is de\fned as:\n\u001f=\u0000@2\n@(eB)2jeB=0; (27)\nand we de\fne \u001f0(T) =\u001f(T)\u0000\u001f(T= 0), the numerical result of which is shown in Fig.(5), where soft cuto\u000b is applied.\nWe can see that at high temperature, there is no doubt that the magnetized matter shows paramagnetism with\n\u001f0>0. However, at low temperature, a negative \u0014gives negative susceptibility, i.e, which indicates the diamagnetism\nof magnetized QCD matter, while a positive \u0014gives positive susceptibility, i.e, the paramagnetism of magnetized\nQCD matter. Therefore, the diamagnetism property cannot be understood by considering the AMM of quarks.\nIII. MESON SPECTRA WITH AMM\nNext, we investigate meson spectra with the AMM. In the NJL model, mesons are regarded as q\u0016qbound states or\nresonances, which can be obtained from the quark-antiquark scattering amplitude [40{42]. As shown in Fig.(6), The\nfull propagator of meson can be expressed to leading order in 1 =Ncas an in\fnite sum of quark-loop chains under the\nrandom phase approximation (RPA). Following the procedure given in Ref.[9], the quark propagator in the Landau\nlevel representation is given by\nS(x;y) = ei\bf(x;y)Zd4q\n(2\u0019)4e\u0000i(x\u0000y)qeS(q); (28)8\nκ=-0.5κ=0.0κ=0.5\n0.10 0.15 0.20 0.25 0.30 0.35 0.40-0.04-0.020.000.020.040.06\nT[GeV]χ0\nFIG. 5: Magnetic susceptibility with di\u000berent constant \u0014. A soft cuto\u000b is applied during the numerical calculation.\nwhere the Schwinger phase \b f(x;y) =qf(x1+y1)(x2\u0000y2)=2 breaks the translation invariant while eS(q) is translation-\ninvariant, which takes the form of :\neSf(k) =iexp\u0012\n\u0000k2\n?\njqfBj\u00131X\nn=0(\u00001)nDn(qfB;k)Fn(qfB;k)\nAn(qfB;k); (29)\nwhere:\nFn(qfB;k) = (\u0014qfB\u0000k0\r3\r5+k3\r0\r5)2\u0000M2\u00002njqfBj; (30)\nand\nDn(qfB;k) = (k0\r0\u0000k3\r3+M+\u0014qfB\u001b12)h\n(1\u0000i\r1\r2\u0018f)Ln\u0012\n2k2\n?\njqfBj\u0013\n\u0000(1 +i\r1\r2\u0018f)Ln\u00001\u0012\n2k2\n?\njqfBj\u0013i\n+ 4(k1\r1+k2\r2)L1\nn\u00001\u0012\n2k2\n?\njqfBj\u0013\n;\n(31)\nwhere\u0018f= sign(qfB),L\u000b\nnare the generalized Laguerre polynomials and Ln=L0\nn. And the Denominator:\nAn(qfB;k) = \u0005s=\u00061n\u0010\n\u0014qfB+sp\n(k0)2\u0000(k3)2\u00112\n\u0000M2\u00002njqfBjo2\n: (32)\nWith RPA approximation, the composite \u0019propagator is written as:\nD\u0019(q2) =2GS\n1\u00002GS\u0005\u0019(q2); (33)\nwhere \u0005\u0019is the one loop polarization function for pion:\n\u0005\u0019(q2) =iZd4k\n(2\u0019)4Tr[\u001cai\r5eS(k)\u001cbi\r5eS(p)]; (34)\nwithq=k\u0000p. For neutral pion, the Pauli matrices take \u001ca=\u001c3;\u001cb=\u001c3while for charged pion they take\n\u001ca=\u001c\u0006;\u001cb=\u001c\u0007with de\fnition:\n\u001c\u0006=1p\n2(\u001c1\u0007i\u001c2): (35)\nAndeS(q) is the translation-invariant part of quark propagator in momentum space and the explicit expression is\ngiven in Eq.(29). From the pole of propagator D\u0019(q2), the corresponding meson mass can be obtained by solving:\n1\u00002GS\u0005\u0019(q2=m2\n\u0019) = 0: (36)9\nFIG. 6: Meson propagator under the Random Phase Approximation in the NJL model.\nIt's similar for the vector meson rho with corresponding one loop polarization function \u0005\u0016\u0017\nV:\n\u0005\u0016\u0017\nV(q2) =iZd4k\n(2\u0019)4Tr[\u001ca\r\u0016eS(k)\u001cb\r\u0017eS(p)]; (37)\nand it can be easily proved that the polarization function for charged rho has following structure:\n\u0005\u0016\u0017\n\u0006=0\nB@\u000500\n\u00060 0 0\n0 \u000511\n\u0006\u000512\n\u00060\n0 \u000521\n\u0006\u000522\n\u00060\n0 0 0 \u000533\n\u00061\nCA; (38)\nwhere \u000511\n\u0006= \u000522\n\u0006and \u000512\n\u0006=\u0000\u000521\n\u0006. One can also decompose it into four parts with respect to z-component of spin,\ne.g.,sz, in the rest frame [29]:\n\u0005\u0016\u0017\n\u0006(q) = \u0005sz=+1\n\u0006\u000f?;\u0016\n1\u000f\u0017\n1+ \u0005sz=\u00001\n\u0006\u000f?;\u0016\n2\u000f\u0017\n2+ \u0005sz=0\n\u0006b\u0016b\u0017+ \u0005u\n\u0006u\u0016u\u0017; (39)\nwhereu\u0016= (1;0;0;0) is the four momentum in the rest frame and spin projection operator are introduced:\n\u000f\u0016\n1=1p\n2(0;1;i;0); \u000f\u0016\n2=1p\n2(0;1;\u0000i;0); b\u0016= (0;0;0;1): (40)\nIt's worthy to point out that the last term \u0005u\n\u0006corresponds to un-physical component of charged rho polarization\nfunction. As a consequence, the charged rho propagator can be written as:\nD\u0016\u0017\n\u0006=Dsz=+1\n\u0006\u000f?;\u0016\n1\u000f\u0017\n1+Dsz=\u00001\n\u0006\u000f?;\u0016\n2\u000f\u0017\n2+Dsz=0\n\u0006b\u0016b\u0017+Du\n\u0006u\u0016u\u0017; (41)\nthen we obtain propagator for each component:\nDsz\n\u0006(q) =2GV\n1 + 2GV\u0005sz\n\u0006(q); (42)\nand the mass for each component can be obtained by solving following equations:\n1 + 2GV\u0005sz=+1\n\u0006 (q2=m\u001a\u0006;sz=+1) = 1 + 2GV(\u000511\n\u0006\u0000i\u000512\n\u0006) = 0; (43)\n1 + 2GV\u0005sz=\u00001\n\u0006 (q2=m\u001a\u0006;sz=\u00001) = 1 + 2GV(\u000511\n\u0006+i\u000512\n\u0006) = 0; (44)\n1 + 2GV\u0005sz=0\n\u0006(q2=m\u001a\u0006;sz=0) = 1 + 2GV\u000533\n\u0006= 0: (45)\nFor neutral rho, the one loop polarization function has structure:\n\u0005\u0016\u0017\n0=0\nB@\u000500\n00 0 0\n0 \u000511\n00 0\n0 0 \u000522\n00\n0 0 0 \u000533\n01\nCA; (46)10\nwhere \u000511\n0= \u000522\n0, and now the gap equations for neutral rho with sz=\u00061;0 are:\n1 + 2GV\u0005sz=\u00061\n0 (q2=m\u001a0;sz=\u00061) = 1 + 2GV\u000511\n0= 0; (47)\n1 + 2GV\u0005sz=0\n0(q2=m\u001a0;sz=0) = 1 + 2GV\u000533\n0= 0: (48)\nFor neutral pion and rho, the Schwinger phase in quark-antiquark loop cancels out while for charged pion and rho\nthey not, for the Schwinger phase led to more complicated calculation, in the present paper, we ignore the Schwinger\nphase and only consider the translation invariant part eS.\nWe present numerical results for the mass of pion and rho at zero temperature, where a soft cut-o\u000b is applied.\nThe neutral and charged pion mass as a function of magnetic \feld are shown in Fig.(7). It is found that The\npresence of AMM reduces neutral pion mass M\u00190signi\fcantly, and M\u00190is sensitive to the AMM of quarks, as\nshown in Fig.(7a). In the case without AMM, neutral pion mass decreases slightly as magnetic \feld increases then\nit increases after a in\rection point. When an appropriate AMM is applied, the previous in\rection point disappears,\nM\u00190continuously decreases and reaches zero at a critical magnetic \feld point eBc. And as\u0014increases,eBcdecreases,\nwhich indicates a quick drop of M\u00190as magnetic \feld increases: when \u0014= 0:005GeV\u00001,eBcis larger than 1 :5GeV2,\nand for\u0014= 0:008GeV\u00001,eBc\u00181:15GeV2whileeBc\u00180:95GeV2for\u0014= 0:01GeV\u00001. It has been studied in many\npapers [32, 33, 36, 37] that magnetic \feld increases charged pion mass M\u0019\u0006, and as shown in Fig.(7b), M\u0019\u0006increases\nwith magnetic \feld in both zero and non-zero AMM cases. Similar to neutral pion, AMM also reduces charged pion\nmass, however, the modi\fcation from AMM is slight enough to be ignored. Besides, comparing the results of neutral\npion with the charged pion, it's obvious that they exhibit di\u000berent sensitivities to the AMM: A very small \u0014of AMM\ncan change the behavior of neutral mass behavior in the region of eB > 0:4GeV2(dashed blue line shown in Fig.(7a))\nfrom increasing with magnetic \feld to decreasing, while the behavior of charged pion mass hardly changes even up to\n\u0014= 0:5GeV\u00001(dashed blue line shown in Fig.(7b)) at the range of magnetic \feld 0 <eB < 1:5GeV2.\nκ=0.001 GeV-1κ=0.005 GeV-1κ=0.008 GeV-1κ=0.01 GeV-1\n0.0 0.5 1.0 1.50.000.050.100.15\neB[GeV2]Mπ0[GeV]\n(a)\nκ=0.0 GeV-1κ=0.1 GeV-1κ=0.3 GeV-1κ=0.5 GeV-1\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.40.00.51.01.5\neB[GeV2]Mπ±[GeV] (b)\nFIG. 7: (a)Neutral and (b)charged pion mass as a function of the magnetic \feld with di\u000berent constant \u0014.\nNext, let's consider the e\u000bect of AMM on neutral rho mass with di\u000berent spin component sz, which is shown in\nFig.(8). In the case of \u0014= 0, all rho mass with three di\u000berent szincrease with magnetic \feld, while the presence of\nAMM reduces neutral rho mass M\u001a0regardless its spin. However AMM can reduce mass of neutral rho with sz= 0\nsigni\fcantly while the mass of neutral rho with sz=\u00061 is reduced slightly. Besides, when \u0014\u00150:7GeV\u00001,M\u001a0(sz= 0)\ndecreases with magnetic \feld, opposite to the zero AMM case.\nFor charged rho we only consider it with positive charge, and the numerical results are shown in Fig.(9). First of\nall, AMM reduces M\u001a+(sz= +1) and M\u001a+(sz= 0) while increase M\u001a+(sz=\u00001). For positive charged rho with\nsz= +1, its mass M\u001a+(sz= +1) decreases with magnetic in both zero and non-zero AMM cases, and drops to zero\nat a critical magnetic \feld eBc, similar to neutral pion, eBcdecreases as \u0014increases, which indicates that the vacuum\nis more polarized when consider the AMM of quarks. For rho with sz=\u00001 andsz= 0, the modi\fcations of mass\ninduced by AMM can be ignored compared to their mass.\nIV. DISCUSSION AND CONCLUSION\nIn the present paper, we systematically investigate the e\u000bect of the anomalous magnetic moment(AMM) of quarks\non the magnetized QCD matter in the two-\ravor Nambu{Jona-Lasinio model, including the magnetic susceptibility,11\nκ=0.0κ=0.3κ=0.5κ=0.7\n0.2 0.4 0.6 0.8 1.00.20.40.60.81.01.21.4\neB[GeV2]Mρ0,sz=0[GeV]\n(a)\nκ=0.0\nκ=0.1\nκ=0.3κ=0.5\n0.0 0.2 0.4 0.6 0.8 1.00.81.01.21.41.61.82.0\neB[GeV2]Mρ0,sz=±1[GeV] (b)\nFIG. 8: Mass of neutron rho with (a) sz= 0 and (b) sz=\u00061 as a function of magnetic \feld with di\u000berent constant \u0014.\nκ=0.0 GeV-1\nκ=0.1 GeV-1\nκ=0.3 GeV-1\nκ=0.5 GeV-1\n0.05 0.10 0.15 0.20 0.250.00.20.40.60.8\neB[GeV2]Mρ+,sz=+1[GeV]\n(a)\nκ=0.0 GeV-1\nκ=0.1 GeV-1\nκ=0.3 GeV-1\nκ=0.5 GeV-1\n0.2 0.4 0.6 0.8 1.00.51.01.52.02.53.03.54.0\neB[GeV2]Mρ+,sz=-1[GeV] (b)\nκ=0.0 GeV-1\nκ=0.1 GeV-1\nκ=0.3 GeV-1\nκ=0.5 GeV-1\n0.2 0.4 0.6 0.8 1.01.01.52.02.5\neB[GeV2]Mρ+,sz=0[GeV]\n(c)\nFIG. 9: Mass of charged rho with (a) sz= +1, (b)sz=\u00061 and (c)sz= 0 as a function of magnetic \feld with\ndi\u000berent constant \u0014.\nthe inverse magnetic catalysis around the critical temperature and the neutral/charged pion and rho meson spectra\nunder magnetic \felds. The dynamical AMM of quarks is presented by a term \u0016 \u0014fqf\u001b\u0016\u0017F\u0016\u0017 in the Lagrangian, its\ncoupling with magnetic \feld causes Zeeman splitting of the dispersion relation of quarks thus changes the magnetism\nproperties and meson mass spectra under magnetic \felds. It is found that though including the AMM of quarks\ncannot fully understand lattice results of the magnetized matter, it can partially explain lattice results. Our results12\ncan be summarized as following:\n1) The AMM of quarks reduces the dynamical quark mass which is consistent with the dispersion relation Eq.(21),\na proper choice of \u0014\u0018\u001bcan produce the inverse magnetic catalysis around the critical temperature Tc.\n2) The neutral pion mass is very sensitive to the AMM, it decreases with magnetic \feld quickly, and reaches zero\nat a critical magnetic \feld eBc, andeBcdecreases as \u0014increases. On the other hand, the charged pion mass almost\ndoes not change with \u0014, it shows a nonlinear behavior, i.e., \frstly linearly increases with the magnetic \feld and then\nsaturates at strong magnetic \feld. The lattice calculation in [26] shows that the neutral pion mass decreases with the\nmagnetic \feld and then saturate at about 0 :6M\u00190(eB= 0) at large magnetic \feld eB > 1GeV2, and the charged pion\nmass \frstly increases with magnetic \feld till eB\u00180:6GeV2and then decreases with magnetic \feld.\n3) It's more complicated for \u001abecause di\u000berent spin component szof rho behave di\u000berently. It is observed that\nAMM reduces the mass of neutral rho meson mass with di\u000berent spin component sz, and reduces the mass of sz= +1;0\ncomponent charged rho meson mass but enhances the sz=\u00001 component charged rho meson mass. Similar to pion,\nthe neutral rho meson is more sensitive to the AMM of quarks than the charged rho. AMM can reduce the mass of\nneutral neutral rho with sz= 0 andsz=\u00061, however the former one changes signi\fcantly while the latter one only\nshows a slight modi\fcation, which can be ignored compared to its mass. Besides, for \u0014\u00150:7GeV\u00001,M\u001a0(sz= 0)\ndecreases continuously with magnetic \feld, while it increases monotonously at zero AMM case. For charged rho\nwith spin component sz= +1, its mass decreases with magnetic \fled and drops to zero at a critical magnetic \feld\neBc, and the AMM reduces its mass as well as eBc, which indicates that including the AMM of quarks makes the\nmagnetized matter more easily to be polarized. Besides, AMM slightly reduces and enhances M\u001a+(sz= 0)\u001a+(sz= 0)\nandM\u001a+(sz=\u00001), respectively.\n4) The magnetic susceptibility of the magnetized QCD matter cannot be understood by including the AMM of\nquarks. It is found that at low temperature the magnetic susceptibility can be either positive (paramagnetism) or\nnegative(diamagnetism) with di\u000berent AMM.\nIn summary, taking into account of the AMM of quarks cannot fully explain lattice results on magnetized matter\nand meson spectra. More studies and other mechanism need to be investigated in the future.\nACKNOWLEDGMENTS\nThis work is supported in part by the National Natural Science Foundation of China (NSFC) Grant Nos. 11735007,\n11725523, and Chinese Academy of Sciences under Grant No. XDPB09, the start-up funding from University of\nChinese Academy of Sciences(UCAS), and the Fundamental Research Funds for the Central Universities.\n[1] T. A. Mihara et al., New observations of the cyclotron absorption feature in Hercules X-1,Nature (London)346, 250 (1990);\nR. C. Duncan and C. Thompson, Astrophys. J. 392, L9 (1992). doi:10.1086/186413; A. Lyne and F. Graham-Smith, Pulsar\nAstronomy (Cambridge University Press, Cambridge, England, 2005).\n[2] T. Tatsumi, AIP Conf. Proc. 847, no. 1, 171 (2006) doi:10.1063/1.2234399 [hep-ph/0601065].\n[3] K. Kiuchi, P. Cerd\u0013 a-Dur\u0013 an, K. Kyutoku, Y. Sekiguchi and M. Shibata, Phys. Rev. D 92, no. 12, 124034 (2015)\ndoi:10.1103/PhysRevD.92.124034 [arXiv:1509.09205 [astro-ph.HE]].\n[4] L. Baiotti and L. Rezzolla, Rept. Prog. Phys. 80, no. 9, 096901 (2017) doi:10.1088/1361-6633/aa67bb [arXiv:1607.03540\n[gr-qc]].\n[5] V. Skokov, A. Y. .Illarionov and V. Toneev, Int. J. Mod. Phys. A 24, 5925 (2009).\n[6] W. -T. Deng and X. -G. Huang, Phys. Rev. C 85, 044907 (2012).\n[7] T. Vachaspati, Phys. Lett. B 265, 258 (1991). doi:10.1016/0370-2693(91)90051-Q\n[8] J. O. Andersen, W. R. Naylor and A. Tranberg, Rev. Mod. Phys. 88, 025001 (2016) doi:10.1103/RevModPhys.88.025001\n[arXiv:1411.7176 [hep-ph]].\n[9] V. A. Miransky and I. A. Shovkovy, Phys. Rept. 576, 1 (2015) doi:10.1016/j.physrep.2015.02.003 [arXiv:1503.00732 [hep-\nph]].\n[10] X. G. Huang, Rept. Prog. Phys. 79, no. 7, 076302 (2016) doi:10.1088/0034-4885/79/7/076302 [arXiv:1509.04073 [nucl-th]].\n[11] D. E. Kharzeev, J. Liao, S. A. Voloshin and G. Wang, Prog. Part. Nucl. Phys. 88, 1 (2016) doi:10.1016/j.ppnp.2016.01.001\n[arXiv:1511.04050 [hep-ph]].\n[12] A. Bzdak, S. Esumi, V. Koch, J. Liao, M. Stephanov and N. Xu, arXiv:1906.00936 [nucl-th].\n[13] D. Kharzeev and A. Zhitnitsky, Nucl. Phys. A 797, 67 (2007).\n[14] D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl. Phys. A 803, 227 (2008).\n[15] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev. D 78, 074033 (2008).\n[16] D. E. Kharzeev and D. T. Son, Phys. Rev. Lett. 106, 062301 (2011) [arXiv:1010.0038 [hep-ph]].\n[17] S. P. Klevansky and R. H. Lemmer, Phys. Rev. D 39, 3478 (1989). doi:10.1103/PhysRevD.39.347813\n[18] K. G. Klimenko, Theor. Math. Phys. 89, 1161 (1992) [Teor. Mat. Fiz. 89, 211 (1991)]. doi:10.1007/BF01015908\n[19] V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Nucl. Phys. B 462, 249 (1996) doi:10.1016/0550-3213(96)00021-1\n[hep-ph/9509320]; V. P. Gusynin, V. A. Miransky and I. A. Shovkovy, Nucl. Phys. B 563, 361 (1999) doi:10.1016/S0550-\n3213(99)00573-8 [hep-ph/9908320].\n[20] G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz, S. Krieg, A. Schafer and K. K. Szabo, JHEP 1202 , 044\n(2012) doi:10.1007/JHEP02(2012)044 [arXiv:1111.4956 [hep-lat]].\n[21] G. S. Bali, F. Bruckmann, G. Endrodi, Z. Fodor, S. D. Katz and A. Schafer, Phys. Rev. D 86, 071502 (2012);\n[22] G. S. Bali, F. Bruckmann, G. Endrodi, F. Gruber and A. Schaefer, arXiv:1303.1328 [hep-lat].\n[23] K. Fukushima and Y. Hidaka, Phys. Rev. Lett. 110, no. 3, 031601 (2013) doi:10.1103/PhysRevLett.110.031601\n[arXiv:1209.1319 [hep-ph]].\n[24] J. Chao, P. Chu and M. Huang, Phys. Rev. D 88, 054009 (2013) doi:10.1103/PhysRevD.88.054009 [arXiv:1305.1100 [hep-\nph]].\n[25] E. J. Ferrer, V. de la Incera and X. J. Wen, Phys. Rev. D 91, no. 5, 054006 (2015) doi:10.1103/PhysRevD.91.054006\n[arXiv:1407.3503 [nucl-th]].\n[26] H. T. Ding, S. T. Li, S. Mukherjee, A. Tomiya and X. D. Wang, [arXiv:2001.05322 [hep-lat]].\n[27] G. S. Bali, F. Bruckmann, M. Constantinou, M. Costa, G. Endrodi, S. D. Katz, H. Panagopoulos and A. Schafer, Phys.\nRev. D 86, 094512 (2012) doi:10.1103/PhysRevD.86.094512 [arXiv:1209.6015 [hep-lat]].\n[28] G. S. Bali, G. Endr} odi and S. Piemonte, arXiv:2004.08778 [hep-lat].\n[29] H. Liu, L. Yu and M. Huang, Phys. Rev. D 91, no. 1, 014017 (2015) doi:10.1103/PhysRevD.91.014017 [arXiv:1408.1318\n[hep-ph]].\n[30] H. Liu, L. Yu and M. Huang, Chin. Phys. C 40, no. 2, 023102 (2016) doi:10.1088/1674-1137/40/2/023102 [arXiv:1507.05809\n[hep-ph]].\n[31] H. Liu, X. Wang, L. Yu and M. Huang, Phys. Rev. D 97, no.7, 076008 (2018) doi:10.1103/PhysRevD.97.076008\n[arXiv:1801.02174 [hep-ph]].\n[32] Z. Wang and P. Zhuang, Phys. Rev. D 97, no.3, 034026 (2018) doi:10.1103/PhysRevD.97.034026 [arXiv:1712.00554 [hep-\nph]].\n[33] S. Mao, Phys. Rev. D 99, no.5, 056005 (2019) doi:10.1103/PhysRevD.99.056005 [arXiv:1808.10242 [nucl-th]].\n[34] S. S. Avancini, W. R. Tavares and M. B. Pinto, Phys. Rev. D 93, no.1, 014010 (2016) doi:10.1103/PhysRevD.93.014010\n[arXiv:1511.06261 [hep-ph]].\n[35] S. S. Avancini, R. L. S. Farias, M. Benghi Pinto, W. R. Tavares and V. S. Tim\u0014 s ®teo, Phys. Lett. B 767, 247 (2017)\ndoi:10.1016/j.physletb.2017.02.002 [arXiv:1606.05754 [hep-ph]].\n[36] M. Coppola, D. G\u0013 omez Dumm and N. Scoccola, Phys. Lett. B 782, 155-161 (2018) doi:10.1016/j.physletb.2018.04.043\n[arXiv:1802.08041 [hep-ph]].\n[37] M. Coppola, D. Gomez Dumm, S. Noguera and N. Scoccola, Phys. Rev. D 100, no.5, 054014 (2019)\ndoi:10.1103/PhysRevD.100.054014 [arXiv:1907.05840 [hep-ph]].\n[38] S. Fayazbakhsh, S. Sadeghian and N. Sadooghi, Phys. Rev. D 86, 085042 (2012) doi:10.1103/PhysRevD.86.085042\n[arXiv:1206.6051 [hep-ph]].\n[39] S. Fayazbakhsh and N. Sadooghi, Phys. Rev. D 88, no. 6, 065030 (2013) doi:10.1103/PhysRevD.88.065030 [arXiv:1306.2098\n[hep-ph]].\n[40] S. P. Klevansky, Rev. Mod. Phys. 64(1992), 649-708 doi:10.1103/RevModPhys.64.649\n[41] Y. B. He, J. Hufner, S. P. Klevansky and P. Rehberg, Nucl. Phys. A 630, 719 (1998) [nucl-th/9712051].\n[42] P. Rehberg and S. P. Klevansky, Annals Phys. 252, 422 (1996) [hep-ph/9510221].\n[43] J. S. Schwinger, Phys. Rev. 82(1951), 664-679 doi:10.1103/PhysRev.82.664\n[44] Y. Hidaka and A. Yamamoto, Phys. Rev. D 87(2013) no.9, 094502 doi:10.1103/PhysRevD.87.094502 [arXiv:1209.0007\n[hep-ph]].\n[45] M. A. Andreichikov, B. O. Kerbikov, V. D. Orlovsky and Y. A. Simonov, Phys. Rev. D 87, no. 9, 094029 (2013)\ndoi:10.1103/PhysRevD.87.094029 [arXiv:1304.2533 [hep-ph]].\n[46] G. S. Bali, B. B. Brandt, G. Endr} odi and B. Gl a\u0019le, Phys. Rev. D 97, no.3, 034505 (2018) doi:10.1103/PhysRevD.97.034505\n[arXiv:1707.05600 [hep-lat]].\n[47] E. J. Ferrer, V. de la Incera, I. Portillo and M. Quiroz, Phys. Rev. D 89, no. 8, 085034 (2014)\ndoi:10.1103/PhysRevD.89.085034 [arXiv:1311.3400 [nucl-th]].\n[48] P. J. A. Bicudo, J. E. F. T. Ribeiro and R. Fernandes, Phys. Rev. C 59, 1107 (1999) doi:10.1103/PhysRevC.59.1107\n[hep-ph/9806243].\n[49] L. Chang, Y. Liu and C. D. Roberts, Phys. Rev. Lett. 106, 072001 (2011) doi:10.1103/PhysRevLett.106.072001\n[arXiv:1009.3458 [nucl-th]].\n[50] E. J. Ferrer and V. de la Incera, Phys. Rev. Lett. 102, 050402 (2009) doi:10.1103/PhysRevLett.102.050402 [arXiv:0807.4744\n[hep-ph]].\n[51] E. J. Ferrer and V. de la Incera, Nucl. Phys. B 824, 217-238 (2010) doi:10.1016/j.nuclphysb.2009.08.024 [arXiv:0905.1733\n[hep-ph]].\n[52] S. Fayazbakhsh and N. Sadooghi, Phys. Rev. D 90, no. 10, 105030 (2014) doi:10.1103/PhysRevD.90.105030\n[arXiv:1408.5457 [hep-ph]].\n[53] M. Strickland, V. Dexheimer and D. P. Menezes, Phys. Rev. D 86, 125032 (2012) doi:10.1103/PhysRevD.86.125032\n[arXiv:1209.3276 [nucl-th]].\n[54] S. Mao and D. H. Rischke, Phys. Lett. B 792, 149 (2019) doi:10.1016/j.physletb.2019.03.034 [arXiv:1812.06684 [hep-th]].14\n[55] N. Chaudhuri, S. Ghosh, S. Sarkar and P. Roy, Phys. Rev. D 99, no.11, 116025 (2019) doi:10.1103/PhysRevD.99.116025\n[arXiv:1907.03990 [nucl-th]].\n[56] E. J. Ferrer, V. de la Incera, D. Manreza Paret, A. P\u0013 erez Mart\u0013 \u0010nez and A. Sanchez, Phys. Rev. D 91, no. 8, 085041 (2015)\ndoi:10.1103/PhysRevD.91.085041 [arXiv:1501.06616 [hep-ph]].\n[57] N. Chaudhuri, S. Ghosh, S. Sarkar and P. Roy, [arXiv:2003.05692 [nucl-th]].\n[58] S. Ghosh, N. Chaudhuri, S. Sarkar and P. Roy, [arXiv:2004.09203 [nucl-th]].\n[59] J. Chao, Y. X. Liu and L. Chang, [arXiv:2007.14258 [hep-ph]]."    },    {        "title": "1903.00656v2.Microwave_excitations_and_hysteretic_magnetization_dynamics_of_stripe_domain_films.pdf",        "content": "1 \n Microwave excitations  and hysteretic magnetization  dynamics  of strip e domain  films  \nMeihong Liu1, Qiuyue Li1, Chengkun  Song2, Hongmei Feng2, Yawen Song1, Lei Zhong1, Lining \nPan2, Chenbo  Zhao2, Qiang Li1, Jie Xu1, Shandong Li1, Jianbo Wang2,3, Qingfang Liu2, Derang \nCao1,2* \n1College of Physics, Center for Marine Observation and Communications, Qingdao University, \n266071 Qingdao, China  \n2Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education , Lanzhou \nUniversity, Lanzhou 730000, China  \n3Key Laboratory for Special Function Materials and Structura l Design of the Ministry of the \nEducation, Lanzhou University, Lanzhou 730000, China  \n \nAbstract  \nFeNi  films with the stripe domain  (SD)  pattern are  prepared by electrodeposition and sputtering \nmethods.  The magnetic domain , static magnetic parameter s, and quality factor , as well as dynamic \nproperties of the two  films , are respectively performed . The results show t he magnetization s of the \nfilm were dependent on the direction of SD, and the rotation of the SD is lagging behind the \nmagnetization reversal . The microwave properties of the SD emerg e dynamic hysteresis  before the \nsaturation magnetic field . These microwave properties are selective ly excited  with acoustic mode , \noptical mode , and spin-wave  mode . The frequency and intensity of different resonance modes of stripe \ndomain are determined by the local magnetization . The magnetization variation s and the rotation of \nSD of different modes are further illuminate d by the micromagnetic simulation . The magnetic \nanisotropy and the resonance intensity of permeability  of different modes  were finally descri bed by  \nthe modified resonance equations.  \nKeywords : Microwave excitations , stripe domain, magnetic properties, magnetization dynamics , \nmicromagnetic simulation.  \n \n1. Introduction  \nDuring the past decades , extensive \ninvestigation s are devoted  to study ing the static \nand dynamic properties of low-anisotropy  \nferromagnetic  films both from the fundamental \npoint of view and for  numerous potential \n                                                             \n*Corresponding author: caodr@qdu.edu.cn  applications in magnetic storage media  and \nmicrowave devices.  These films can be \ndistinguished by the type of  anisotropy , i.e.,  in-\nplane and out-of-plane  anisotropy  [1]. Both the \nstatic magnetization configuration and the  \ndynamic response are con trolled by these key \nparameters. The most common stud ies of those  2 \n films are characterized  by an in-plane uniaxial  \nanisotropy [2, 3], and another class of the films \ninvestigated corresponds to the ones exhibiting  \na weak perpendicular anisotropy  [4]. It has \nbeen well ascertain ed that one of  the \nremarkable characteristic s is the existence of a \nstripe domain  (SD) pattern  for those  films with \nsuch anisotropy  [5, 6].  \nSD structure is firstly  propose d in Fe Ni film \nin 1964 b y Spain [7] and N. Saito et al [8, 9]. \nThe researchers  then provide  the formation \nmechanism  of SD  which is relate d to the \ncolumnar grain structure  induced  perpendicular  \nanisotropy  of film [10-13]. It is now widely \naccepted  [8, 11, 14, 15] that the physical origin  \nof the SD structu re is the competition between \na moderate perpendicular  magnetic anisotropy , \nexchange interaction , and the easy -plane \ndipole -dipole magneto static coupling  energy . \nThe domain structure is characterized by a \nperiodic modulation of the up and down \nmagnetization component as well as a Boloh - \nand Né el - type walls between t wo stripe s. Such \nSD pattern is now evidenced  in FeSi [16], Co \n[17], FeCo -based  [18-20], NdCo  [21], FeBSi  \n[22], FePt [23], FeGa [6, 24], and FeNi [7, 25, \n26] film. The film thickness of the SD is always \nabove  a criti cal value  tc [14, 27]. The strength  \nof SD is characterized by a quality factor , \nQ=K/2Ms2, which is defined as the ratio of \nthe perpendicular anisotropy  energy  to \nmagnetic dipole -dipole energy [1, 28]. Q<1, \nthe magnetization tends to lie in the plane of the \nfilm; Q>1, the magnetization oscillate s in the \nout of the plane .  \nDue to  the up and down magnetic  moment  \ndistribution s of the SD film,  the dynamic \nmagnetization  response  and the anisotropy of the film become more fascinating  and complex . \nThe magnetic behavior of the films is in-plane  \nisotropic  but dependent on the magnetic history . \nThis phenomenon  further  results in the multiple \ndynamic properties  under the different \nmagnetic configuration s. Most researcher s \nhave studied  the resonan t model and the \ndynamic res ponse of the SD structure film [1, \n5, 20, 23, 28-44]. Particularly  in detail , Acher \net al  [43] investigate d the microwave \npermeability  of SD film and mentioned  the \nmultiple  resonance peak s to the spin-wave  \n(SW) . Vukadinovic  et al [4, 14, 42] and Ebels et \nal [31] respectively propose d a ferromagnetic \nresonance (FMR) model and calculate d the \nsusceptibility  spectra  of SD structure films  \nusing a micromagnetic simulation . The results \nqualitative ly explain ed that the s ource of \ndifferent  resonance peaks  was related to the \ninside dom ain distribution of the SD . Chai et al \n[18, 45] and Tacchi  et al [6, 24] focused their \ninvestigation s on the rotational anisotropy of \nSD film  by mea suring the magnetic spectr a \ndependent on the  in-plane angle  between SD \nand magnetic field . Recently, Tacchi  et al [46] \nalso phenomenological ly compared  the \ndiffere nt magnetization dynamic  modes of SD \nfilm via the Brillouin light scattering  (BLS)  and \nFMR  technique s. The above -previous studies  \nconclu de the very interesting results that the \nmultiple  resonance s of magnetic  excitation  in \nthe SD film depend on the  specific pumping \nfield configuration s concerning  the stripe \ndirection  and magnetic field . The frequenc ies \ndependence of these modes  reflects the changes \nof the local and internal anisotropy fields of SD \nstructure.  In this case, however,  there are still \nsome confusions that need to b e solved.  In \nanother word , the process  and relation  between  3 \n the SD rotati on and its inside magnetization  \nreversal  as well as the quantitative  anisotropy  \nare not yet clear.  It is n ecessary  to further \npropose the evolution  processes  of SD films  \nunder different  resonan t modes . \nIn this work,  the typical parameter s of the \nFeNi SD film were first studied  and given . By \ncomparing the static magnetization  of SD film , \nthe dynamic response  was measured under \ndifferent  pumping - or applied magnetic - field \nconfiguration s concerning  both the stripe and \nmagnetization  of the SD film. The results \ndiscuss  the variation  of the dom ain, \nmagnetization  distribution , hysteresis , resonan t \nfrequency , permeability , and anisotropy  of the \nSD film in those  configuration s. The process of \nthe stripe  rotation and its inside magnetization  \nreversal  is further  demonstrate d by the  \nmicromagnetic simulation . This study  showed  \na clear evolution  of the magnetization , stripe,  \nand the relevant anisotropy  of SD  films . The \nresults  provide d a favorable  point  to further \nunderstand  the magnetization  dynamics  of the \nSD. \n2. Experiment process and method  \nThe films were deposited on Si  (100) \nsubstrates  and indium tin oxide (ITO) \nsubstrates by RF magnetron sputtering and \nelectrodeposition method respectively . The \ncomponent  is Fe65Ni35 for electrodeposit ed film \nand Fe20Ni80 for sputter ed film and the \ncorresponding  thicknesses tfilm of the two films \nare 50015 nm and 200 8 nm respectively.  We \nnote that  the composition  of electrodeposit ed \nfilm and the sputter ed film is different.  Our \nprevious composition -dependent  results [47] \nindicate FeNi films exhibit good acoustic and optic mode resonances properties when Ni at.% \nis between 25% and 38% for the \nelectrodeposited film. This does not affect the \nresearch content involved in this work.  The \ndeposition rate  of electrodeposited film and the \nsputtered film  is about 50 nm/min  and 3 \nnm/min  respectively. For the magnetron \nsputtering  technique , the target was Permalloy , \nthe base pres sure of  the v acuum chamber was \nbetter than 5× 10−5 Pa, and  the sputtering power \nand Ar  pressure were 120  W and 0.4 Pa . For the \nelectrodeposition method , the deposition \npotential was -1.2 V , and the contents of \nelectrolyte were composed of FeSO 4·7H 2O \n(0.05 mol/L), NiSO 4·7H2O (0. 05 mol/L), \nH3BO 3 (0.5 mol/L), C 6H8O6 (1 g/L), C 2H5NO 2 \n(2 g/L) and C 7H5O3NS (2 g/L) , which was also \nafforded  in our previous work  [48]. \nMagnetic force microscopy ( MFM, As ylum \nResearch  MFP3D)  was used to study the \ndoma in structures of  the films. The \ncomposition s of the samples were identif ied by \nan energy -dispersive X -ray spectrometer (EDX , \nHitachi S -4800 ). The in-plane hysteresis loops \nand magnetiz ation  curve s of samples were \nmeasured by vibrating sample magnetometer \n(VSM, Lakeshore 7304) and B -H Loop \nanalyzer  (Riken Denshi , BHV -30S) at room \ntemperature respectively . The ferromagnetic \nresonances (FMR) of the films were performed \nby electron spin resonance (ESR, JE OL, \nJESFA300) with an X -band spectrometer at 9 \nGHz. The permeability spectra of all films were \nmeasured by a vector network analyzer (VNA, \nAgilent E8363B) method from 100 MHz to 9 \nGHz  [49]. \n3. Results and discussion  4 \n A. Basic  structure  and static magnetic \nproperties  of SD film  \nFigure  1 (a-b) show s the zero-field MFM \nimages of electrodeposited  and sputtered  FeNi  \nDark and bright  regions (domains) represent an \nout-of-plane component o f the magnetization \nwith upward or downward  directions , \nrespectively , and  between the two  close  stripes \nis the domain wall. The average SD width of \nthe two films is 167 13 nm (electrodeposition) \nand 1435 nm (sputtering) respectively.  It can \nbe observed from the pictures that the \nelectrodeposit ed film presents a dispersive  SD \nstructure, while  the SD pattern  is more uniform \nand clear  in the sputter ed film.  The SD pattern \nof the film shows good regularity and \nhomogeneity domain i.e., evenly spaced and \nparallel stripe distributions with a single and \nstrong Fast Fourier transform peak . This  \nquantitative definition  has been given in our \nprevious work [50]. Our previous work of SD \npattern s prepared by these two ways always \nshow s such different domain results [48, 51], \nand the results are related to the film quality  of \nthe two methods. The film  prepared by \nsputter ed usually has better quality  than \nelectrodeposit ed film.  It has been proved that  \nthe obvious and uniform  SD pattern reveals a \nstrong and clear resonant mode [50]. As \nmentioned in the introduction, the level  of SD \nis characterized by a quality factor  Q \n(Q=K/2Ms2). The perpendicular anisotropy  \nconstant  K and saturation magnetization 4Ms \ncan be  obtained by measuring the FMR. The \ninset of figure  1 (e-f) is the o ut-of-plane angular \nof the magnetic field  (θH) dependence of the \nout-of-plane FMR field ( HR). By fitting the \nrelation between H and HR [48, 52], the 4Ms and K are 13.1 kGs and 6.3×105 erg/cm3 for \nelectrodeposited  film and 0.9 kGs and 5.5 × 105 \nerg/cm3 for sputtered  film respectively . The \ncalculate d Q is 0.09 and 0.17  for \nelectrodeposited  and sputtered films \nrespectively . The sputtered film shows a larger \nQ than that of electrodeposited  film. The results \nconfirm the  obvious SD pattern of sputtered \nfilm than electrodeposited  film.  We note that \nwhen other parameters  are fixed , the SD width \ndecreases with the increasing saturation \nmagnetization, while it increases with the \nincreasing thickness of the film.  The same \nconclusion  can be found in previous \npublication s [1, 12, 53, 54]. Such results  can be  \nalso easily seen from Equation (S1) of domain \nwidth in our Supplemental Material. For the \nquality factor, it is complicated and is related to \nthe saturat ion magnetization, thickness  \n(demagnetization) , and perpendicular \nanisotropy field. This is the reason that we \nobserved the larger perpendicular anisotropy \nbut smaller quality f actor in the \nelectrodeposited film.  In addition , according to \nthe above values of 4Ms and K, the domain \nwidth  was also calculated by the  theory \nexpression  [55, 56]. See the Supplemental  \nMaterial  for details process . The calculate d SD \nwidth  of two films using Eq. (S1) is about  154 \nnm for electrodeposit ed film  and 144 nm  for \nthe sputtered film, which  is comparable with \nthe MFM value.  It can be seen the error \nbetween the measured value and the calculated  \nvalue for the electrodeposit ed film  is large r. \nThis may be due to the dispersive  and \ninhomogeneous  SD pattern in electrodeposit ed \nfilm, which causes a large measurement error.  \nFigure  1 (c -d) display s the in -plane 5 \n hysteresis loops  recorded on the \nelectrodeposit ed and sputter ed FeNi SD films . \nBoth the two loops are characterized by a linear \ndecrease of magnetiz ation from its saturation \nfield to a moderate remanence. Thi s shape is \nthe typical  loop for thin films exhibiting a SD \nstructure  [47]. The hysteresis loops  of the SD \nfilms are in-plane isotropic wh erever the \norientation of the in-plane  applie d magnetic \nfield is (See figure  S3 of Supplemental \nMaterial ). This  reflects the so -called rotatable \nanisotropy effect  in SD  [1]. Figure  1 (e-f) are \nthe a mplifying  results of the first and fourth \nquadrant  hysteresis  loops of ( c) and ( d) \nrespectively. This step was used to describe the \nfollowing VNA measurement c onveniently . \nThe letter s a1~j1 of the upper solid line \nrepresent  the applied magnetic field H \nchanging  from 180 Oe to 0 Oe, while the letters \nj2~a2 mean that the H is from 0 Oe to 180 Oe. \nThe step size  of H is 20 Oe. It can be observed \nthat the  upper and lower lines are  dividing , and \nare starting to approach when H is larger than \n70 Oe for electrodeposited film and 90 Oe for \nthe sputtered film. We note that, d uring the \nmeasurement, both the intensity and direction \nof H for points a 1 and a 2, b1 and b 2, ··· j 1 and j 2, \nare respectively the same, but the relevant  \nmagnetizations (both the intensity and direction) \nare different. We deduce  that such result is \nrelated to the stripe or internal magnetization s \nof SD . Our following study will systematic ally \ndiscuss this process .   \nFigure  1. (a-b) Zero-field MFM images of \nelectrodeposited  and sputtered  FeNi  films . (c-d) In-\nplane hysteresis loops of the two sample  films. (e-f) \nthe a mplifying loops of the first and fourth quadrant . \nThe letter s a1~j1 of the upper solid line represent  the \napplied magnetic field H changing  from 180 Oe to 0 \nOe, while the letters j 2~a2 mean the H is from 0 Oe to \n180 Oe . The step size of H is 20 Oe.  The inset of the \nfigure ( e-f) is the corresponding out -of-plane angular \nof the magnetic field ( θH) dependence of the out-of-\nplane FMR field ( HR). \nFor the SD film, a s well known,  it is possible \nto select the easy direction  of the magnetization \nby the application of a sufficiently large in-\nplane magnetic field along this direction and \nthen remove it . To further study the static \nrotation  of the stripe and magnetization , we \nfurther studied the in-plane magnetization \ncurves of SD films  at different orientation s. The \nresults are  shown in figure 2. For each \nmeasurement,  the direction of the SD is fixed \nparallel  to the x-axis, and H is applied along the \nx-axis or y-axis. Thus, three different \n6 \n situations are carried out:  \ni) H is applied along the x-axis, i.e.,  parallel  to \nthe direction of SD and M (figure  2c). \nii) H is applied along x-axis, i.e.,  parallel  to \nthe direction of SD and anti -parallel  to the \ndirection of  M (figure  2d). \niii) H is applied along the y-axis, i.e.,  \nperpendicular  to the direction of SD and M. M \nhas two orientation s here, but the  results of the \ntwo situations  are the same (figure  2e). \nThe i n-plane m agnetization  curves  of \nelectrodeposit ed and sputter ed SD films are \nshown in figure  2 (a-b) respectively . The \nfollowing rules  are declare d from the curves :  \ni) Three different curves are observed and \nhighly related to the initial direction of H and \nSD.  \nii) For HSD, the initial  4πM value  (when H is \nzero)  of curve measured at situation  HM \nand HM is the same , but the direction is \nopposite . \niii) For HSD, the curves measured  at situation  \nHM and HM are the same. Here we \nonly present  the results of HM. Their \ninitial  4πM (when H is zero) is closed to zero . \nThis is  due to the near -zero component of \nmagnetization in the y-axis at this time . \niv) For HSD, the 4πM of the HM and \nHM show a linear increase with the \nimprovement of H (Note: for HM, this \nlinear increase happens as  H exceeds a reversal  \nfield Hrev). The two curves are similar to the \nVSM loops of figure  1 (e-f). The Hc obtained \nfrom the curves is consistent  with the VSM \nloops.   \nv) For HSD, the 4πM presents a linear \nincrease  first when H is less than a trans ition \nfield Htra. Htra is about 72 Oe for \nelectrodeposited film and 12 3 Oe for the sputtered film . The 4πM then continues  to \nincrease but with a  reduc ed slope as H further \nincreases.   \nvi) The 4πM of the three  situations start s to \noverlap  when the H is larger  than Htra, and \nfinally reach es saturat ion. \nThe results indicate that the magnetization \nprocess  of SD film is strong ly depend ent on the \ndirection of SD and its  inside  magnetization  \nconfiguration .  \nWhen HSD, for the situation HM, the \nM inside SD  is always parallel  to the direction \nof H. For the situation  HM the M is anti -\nparallel at the beginning  of the curve, and starts \nrotating  to the direction of  H when H exceeds \nHc. They final ly point  to the direction of H \nwhen H is larger than  Hrev. We note that the \ndirection of SD is still parallel  to H during the \ntwo processes ( HM and HM).  \nWhen HSD, the M inside SD is also \nperpendicular to H. Both the M and SD  start to \nrotate  with the increas e of H, but their relevant  \ncritical -rotation al field is different . The M \nbegins to move  once  the H is applied , while the \nSD rotates  when H is near Htra. These results  \nwill be demonstrate d in the following \nmicromagnetic  simulation s.  \nThe above results re veal that  the rotation of \nSD is hysteretic  compared with  its inside \nmagnetization . This is the origin  of a rotational \nanisotropy  in SD film. Some researchers call \nthis anisotropy  a pseudo -anisotropy  [16]. This  \nis due to  the anisotropy depends on  the \ndirection  of SD and is controlled by the \nexternal  magnetic field . 7 \n  \nFigure  2. (a-b) In-plane m agnetization curves  of \nelectrodepos ited and sputtered FeNi SD  films . The \ncurves were measured at  the different direction s of SD, \nmagnetization M, and applied magnetic field H. The \nred circle line  is the situation HSD and  HM; the \nblue triangle line  is the situation  HSD and  HM; \nthe black square line is  the situation  HSD and \nHM. (c-e) The orientation schematic diagram  of \nthe SD, M, and H during  measurement . The solid line \narrows  are the direction of SD, M, and H respectively.  \nB. Dynamic magnetic properties  of the film \nThe d ynamic magnetic spectr a of film s were \nperformed by VNA with an applied magnetic \nfield H. The direction of the microwave  \nmagnetic field hrf is fixed orthogonality  to H. \nThe results are shown in figure  3. Figure  3 (a) \nand (c)  show s the imaginary permeability \nspectra  of electrodeposited and sputtered FeNi \nfilms at different H. The application process of \nthe H is the same as VSM measurement. The \nletter s, a1~j1 and a 2~j2, are corresponding to  the \napplication of H in VSM . Figure  3 (b-d) show s \nthe color-coded imaginary permeability spectra of the two FeNi films  as a function of the \napplied field.  One can observe  the f ollow ing \ncomments  from figure  3: \ni) In figure  3, the application of H is the same  \nas the first and fourth quadrant o f VSM  \nmeasurement . Similar ly, we can get the same \nspectra  when the application procedure  of H \naccord s with the second  and third  quadrant o f \nVSM  measurement . \nii) Both the value and direction of H for a1~j1 \n(180~0 Oe ) and j2~a2 (0~180 Oe ) are the same \nduring the permeability spectra  measurement . \nHowever,  the corresponding  spectra  are \ndifferent  and replace d by a hysteretic behavior .  \niii) The color-coded permeability spectra  are \nasymmetric  about zero fields , and an inflection  \nappears  in the range of  15~45 Oe for \nelectrodeposited film and 45~80 Oe for the \nsputtered film . It is approximate ly symmetrical  \nwhen H exceeds about 140 Oe for \nelectrodeposited films and 120 Oe for the \nsputtered film.  \niv) In the asymmetric  region, except  zero fields \nboth the frequency and intensity of spectra  are \ndifferent when H is in a small range  \n(corresponding  to field at letters e, f, g, h, and \ni). It can be seen that the  corresponding  spectra  \nin figure  3 (a) and (c)  are non-overlapping . \nv) The law of two color -coded magnetic spectra  \n(figure  3b and 3d) is similar to the VSM loops  \n(figure  2c-d). The value of the magnetic field  in \nthe inflection region of the spectra is close to \ntheir Hc (Hc obtained from VSM are 4 0 Oe for \nelectrodeposited film and 6 4 Oe for sputtered \nfilm respectively ). The results indicate  the \nasymmetric  spectra are related to their current \nmagnetization state. \nvi) The spectra  of the sputtered film  with higher \nfrequency  (figure  3d) are attributed to  spin-\n8 \n wave (SW) [4, 40, 43], but it is not observed in \nthe electrodeposited film . This may be due to \nthe weak  SW being overlap ped with the  broad \nresonance line width  of electrodeposited film or \nthe SW can not be excited in the low-regular  SD \npattern  [50] of the electrodeposited film  (see \nMFM results) . \nThe above results indicate  that this \nasymmetric  spectrum  (both the frequency and \nintensity ) is related to the magnetization  and \nreversal magnetization  along with  SD film . The \nH-dependent  frequency and intensity  of spectra  \nwill demonstrate  in Sec. 3D. As a comparison , \nthe same measurement is performed in \nFe45Co55 and Fe 20Ni80 films  with the in-plane \nuniaxial anisotropy , and the results are  shown \nin figure  S1 of Supplemental Material . The \nresults indicate  that this dynamic hysteretic behavior  is almost disappeared . Their  \ncorresponding  spectra  are overlapping  well, \nand show  an ideal  symmetric color-coded  \nspectrum  about zero field s. Acher et al [3] \nprevious ly demonstrated that the soft magnetic \nthin film with in -plane anisotropy  showed  the \ndynamic hysteretic behavior  when the applied \nfield was smaller than  its Hc. The applied  field \nwas very small (<3 Oe) when dynamic \nhysteretic behavior  happen ed. This effect can \nbe negligible  in our in-plane films , where the \napplied  field step (10 Oe)  is relatively  large . \nHowever, the dynamic hysteretic behavior  in \nSD film is more obvious. This dynamic \nhysteretic behavior  was also observed in the \nstrong exchange biased  film [57, 58]. The film \nalways showed  the asymmetric  loops about \nzero field s. \n \nFigure  3. (a) and (c) Imaginary permeability spectra  of electrodeposited and sputtered FeNi films measured at \ndifferent  H. (b) and (d)  Color -coded imaginary permeability spectra of the two FeNi film s as a function of H; the \n9 \n short dash -dot is the maximum frequency value of permeability spectra . The letters (a1~j1 and a 2~j2) represent the \napplication of H (both intensity and direction) , and it  is the same as VSM measurement  (figure  2c-d).\nTo clarify  the dynamic hysteretic  behavior  \nand the dynamic  properties corresponding  to \nthe magnetization curve s, the magnetic  spectra  \ndependen ce of  the applied magnetic field  H and \nmicrowave  magnetic field hrf are further \ndemonstrate d in figure  4. The hrf is fixed \northogonality  to H during the measurement. \nThe application of H is the same as the \nmagnetization curves of figure  2 (a-b). The \ndirections of the M and SD  are orientated  by the \napplication of a sufficiently large in-plane H \nand then remov e it. The following  comments \ncan be obtained:  \ni) Two modes are observed in figure  4. When \nthe direction of hrf is perpendicular to  SD \n(hrfSD), the measurement is the acoustic \nmode (AM) with in -phase p recession . When \nthe direction of hrf is parallel  to SD (hrfSD), \nthe measurement is the optical mode  (OM)  with \nout-of-phase p recession  [23, 31]. The \nresonance frequency and intensity  of AM and \nOM are different. AM has a low er resonance \nfrequency but higher intensity  while O M has a \nhigher  resonance frequency but lower  intensity , \nIn AM and OM, t hey have their own \ncorresponding SW resonance .  \nii) For all AMs (figure  4a, 4b, 4d, and 4e), the \nresonan t peak  at the low fre quency is in \naccordance with the  classical  in-plane uniform \nprecession  mode  [43]. The spectra  with a \nhigher frequency  in the  sputtered film is the \nSW. The SW of AM originate s from  the spin \nresonance  inside the Bloch -type domain walls \nand the spin resonance  at the film surface  [4, 14, \n31]. iii) For the AM, the frequency fr of HM \n(figure  4a and 4d) increase s with the \nimprovement of H, while it decreases first and \nthe increas e for HM(figure  4 b and 4e). \nThis is related to the opposite  initial  direction \nbetween the H and M when H is less than their \nHc. \niv) For the OM (figure  4c and 4f), the intensity \nof OM weake ns when compar ed with AM. The \nOM floats at about 3 GHz  for electrodeposited \nfilm and about 5.5 GHz  for the sputtered film . \nThe AM emerge s when H is large r than about  \n40 Oe for electrodeposited film and 60 Oe  for \nsputtered film , and their fr is increasing with the \nincrease d H. The higher fr (>6 GHz) is SW. The \nSW of OM originate s from  the spin resonance  \ninside the  Né el -type like  domain and the spin \nresonance  in the u pper and lower part of \nvolume  [4, 14, 31]. The transitions of the OM \nand AM are related to the rotation of the  SD \nwith the increase of the H. This changes the \nrelative direction between the hrf and SD , and \nthus, different resonan t modes are excited at the \nsame time.  It can be seen multiple  resonance  \npeaks  (such mix resonance  peaks including SW, \nOM, AM, domain wall r esonance ) in the range \nof H = 90 Oe ~150 Oe , and  it is difficult to \nclearly  distinguish  these peaks . These \nresonance s contain the contributions from \nmagnetization s inside the  domain , domain wall , \nand the flux closure cap  [4, 14, 31, 46]. The \nlater m icroma gnetic simula tions also reveal a \ncomplicated  domain structure. All the mode s \nwill become the in-plane uniform precession  \nmode  when SD disappears.  10 \n v) For the AMs of HM and HM in \nfigure  4 (a-b) and (d-e), the results  are similar \nwith the law of figure  3 (b) and (d)  respectively.  \n \nFigure  4. (a-f) Color -coded imaginary permeability \nspectra  corresponding to the different direction s of the \nelectrodeposited film  (a-c) and sputtered film  (d-f) as \na function of H; the short dash -dot is the frequency \nvalue of permeability spectra . The application of the \nmagnetic field of the magnetic spectra  measurement  \nis the same as the magnetization curve s of figure  2 (a-\nb). (g) The orientation schematic diagram of the SD, \nM, H, and hrf during measurement. The solid line \narrows are the direction of SD, M, and H respectively , \nand the dash line  arrow  is the direction of hrf. The hrf \nis fixed orthogonality to H during the measurement.  \nC. Micromagnetic simulations  \nMicromagnetic simula tions have been \nperformed using  the graphic s processing unit \n(GPU) accelerated software  Mumax3  [59]. In \nmicromagnetic simulations, we assumed \neffective homogeneous material parameters in \nthe whole structure. The simula tions were \nrelax ed to compute dispersion relations for the uniformly magnetized structure and the s table \nSD pattern . The parameters in the simulations \nare as follows  [40]. The saturatio n \nmagnetization was set to 4Ms = 10 kGs . The \nexchange stiffnes s constant and the out -of-\nplane  perpendicular  anisotropy were \nAex=7.2×10-7 ergs/cm  and K = 5.0×105 \nergs/cm3, respec tively. The values of the \nquality factor  were  then calculated with Q= \nK/2Ms=0.13.  The tota l simulated size is \n1000× 1000× 200 nm3 and was discretized into \ncells with the  dimensions of 5×5×5 nm3. A \nrelaxation  method is used  iteratively  in order to \nachieve the random  alignment  of the \nmagnetization vector in the minimum energy . \n \nFigure  5. 3D micromagnetic simulation result for \nsquare structures 1000× 1000× 200 nm3. The color \nscale is  adopted for the magnetization component Mz \nalong the z-axis. Red regions have positive Mz, teal \nregions have negative Mz, and white regions have Mz \n= 0. (a) Initial state; (b) Remanence state. (c) Cross -\nsection of magnetization at the remanence state in one \nperiod. The front plane of (a -b) shows the equilibrium \nmagnetization of the film plane ( yz plane) at half of \nthe film.  The black arrows indicate  the direction of \nmagnetization.  \nFigure  5 presents the magnet ization \ndistribution simulated  at the initial state and \nremanence  state. The arrows represen t the \nprojection of the  magnetization in  the plane ( x, \ny) while the component Mz is given by a teal-\nwhite -red color -code  map. Figure  5 (a) shows a \nrandom  stripe  distribution  at the initial state \n11 \n while figure  5 (b-c) present s the periodic  \narranging stripes  at the remanence  state [figure  \n5 (c) is the amplify ing image ]. These \ndistribution diagrams  can be more  distinctly  \nseen in figure  S2 of Supplemental Material  by \n2D plotted picture s. At remanence , the \nfollowing conclusion is deserved :  \ni) It can be observed from figure  5 (b)  that the \nbasic domains in SD show an  alternatel y up and \ndown (along  the z-axis) magnetization \nconcerning  the film plane.  The result is  in \nreasonable agreement  with the  experimental \nMFM images.  At the film surface,  these up and \ndown magnetizations are oriented along  the x-\naxis (the direction of  the magnetiz ing history ) \nwith slightly  tilting concerning  the film surface  \ndue to the large demagneti zation  energy . \nii) The middle of the adjacent stripe  is separated \nby Bloch -type domain walls  magnetized along \nthe +x-axis. These walls  lead to SD parallel ing \nthe direction of the saturation  field and \npresenting the periodicity .  \niii) Due to the  moderate  value of the out -of-\nplane perpendicular anisotropy  K, a flux \nclosure cap  of Néel-type like  domains  consist s \nof a region near the film surface and  in-plane \nmagnetized along the y-axis. The flux closure cap domain is  the characteristic of the weak SD \nregime , and t he size of the  region  is in \naccordance with  the values of  the quality factor  \nQ, domain  width w, and critical thickness  tc \n[46]. Since SD has an equal size and is \nalternat ely magnetized, it can be seen  (figure  \n5b) that the sum of My at remanence  is zero . \nThis is in agreement with  the magnetization \ncurves of figure  2 (situation HSD).  \nTo validate the experim ental results and to \nunderstand the physical character istics  of the \nrotation  of SD  and magnetization, the \nmicromagnetic simulations  are further carried \nout with a different direction of SD, M, and H \nrespectively.  The film is fixed at remanence \nbefore  simulation . The H is set at the same \ndirection with the measurement of \nmagnetization curves ( figure  2), i.e., three  \nsituations: HSD and HM, HSD and \nHM, HSD and HM or HM. We \nnote that f or HSD, the results  of HM or \nHM are the same, here we only give the \nresults of HM. The simulation  results are \nshown in figure  6 (top view ) and figure  7 \n(cross -section) . The intensity of H has been \nincreased from 0  to 3000 Oe by steps of 100 Oe . 12 \n  \nFigure  6. Top view of the film plane (xy plane)  for the  micromagnetic simulations  at the various  direction s of SD, M, \nand H. The images show the equilibrium magnetization of the film plane at half of the thickness . Before applying H, \nthe initial state of films is remanence  and the direction of SD and M is parallel  to the x-axis. The simulation  and color \nscale  are the same as in figure  5. The large and bl ue arrows guide  the direction  of the in -plane (xy plane) magnetization  \nin one stripe  period . (a) HSD and HM, H is applied along the +x-axis. (b) HSD and HM, H is applied \nalong x-axis. (c) HSD and HM, H is applied along +y-axis. The blue arrow  is to clarify the change in the local \nmagnetization directions during the three different magnetization processes. The length of the blue represents the  \nquantity of the local magnetization here along this direction.  \n \nFigure  7. Cross-section of magnetization ( yz plane)  for the  micromagnetic simulations at the various direction s of \nSD, M, and H. The images show the equilibrium magnetization of the film plane at half of the film plane . The \nsimulation  and color scale  are the same as figure  5 and figure 6. The large and blue arrows guide  the direction of the \ncross -section  (yz plane)  magnetization  in one stripe  period ; “●” represents the magnetization  is “out” in the region , \n13 \n while  “” represents the magnetization  is “in” in the region . (a) HSD and  HM, H is applied along +x-axis. (b) \nHSD and  HM, H is applied along the x-axis. (c) HSD and HM, H is applied along +y-axis. The blue \narrow  is to clarify the change in the local magnetization directions during the three different magnetization processes. \nThe length of the blue represents the quantity of the local magnetization here along this direction.  \n \nThe detailed  description of figure  6 and \nfigure  7 are displayed as flown:  \nI) When H is zero , the pictures  are the same \nas figure  5 (b-c) and figure  S2 (c-d). The \ncomponent Mx comes from the \nmagnetization along the +x-axis in Bloch -\ntype domain wall  and a tilted in-plane \nmagnetization  along  x-axis near the film \nsurface . My is compose d of  the \nmagnetization  along the y-axis at a flux \nclosure cap of Né el -type like domains  \nmagnetized near the film surface . Mz is \nconsist ing of basic d omains  alternately \nmagnetized up and down along the z-axis \nconcerning  the film surface .  \nII) HSD and HM (figure  6a and 7a), H \nis applied along the +x-axis. \ni) A small H (0~400 Oe) is applied  to SD . \nWith the increasing H, the shape of the \nstripe  remain s nearly unchanged , but the \nmagnetization s at the  edge of the stripe  and \nBloch -type domain walls  are rotating to  the \n+x-axis. The magnetization s near the film \nsurface  and inside the flux closure cap  start \nrotating to the field direction . When H is \n400 Oe,  the magnetization s near film \nsurface  point to  +x-axis, and the region of \nflux closure cap  reduces  a little.  \nii) A moderate  H (400~1000 Oe) is applied  \nto SD . It can be seen the SD pattern  becomes \nweak.  Most of the marginal  magnetization s \nin basic domains and flux closure cap  are \nmoving towards  the direction  of the magnetic  field with the increasing H, and \nfurther cause an increas ing region of in-\nplane magnetization s along the + x-axis near \nthe film surface . The number of  \nmagnetization s in the critical area between \nthe stripe  and Bloch -type domain walls  is \nturning to +x axis with the increasing H. \nWhen H reaches 1000 Oe,  the \nmagnetization s at the edge of basic domains \nand flux closure  are orient ating to  the +x-\naxis.  \niii) A large H (1000~2500 Oe) is applied  to \nSD. All magnetization s are rotating to  the \n+x-axis with the increasing H. The SD  splits \ninto several narrow -weak SD by the \nincreasing number of Bloch -like domain  \nwalls  at 1500 Oe , and disappears gradually  \nwhen the film is in the saturati ng state (2500 \nOe). The magnetization of the film is finally \nalign ing to the +x axis. \nIII) HSD and HM (figure  6a and 7a), \nH is applied along x-axis. \ni) A small H (0~400 Oe) is applied . With the \nincreasing H, the basic domains  of the stripe \nremain nearly unchanged . The tilted \nmagnetization s near the film surface  and \ninside the flux closure cap  as well as the  \nboundary of Bloch -type domain walls  are \ngradually  orientat ing to  the field direction . \nWhen H is 400 Oe,  part of magnetization s \nin the critical area between t he basic \ndomains  and Bloch -type domain walls  are \nreversed  to x-axis. The region s of flux 14 \n closure cap  and Bloch -type domain walls  \ndecrease obviously  due to the rotation of the \nmagnetization at the edge of flux closure \ncap and Bloch -type domain walls  to the \ndirection of the magnetic field with the \nincreasing H. All the tilted magnetization s \nnear the film surface  orientate  to x-axis. \nii) A moderate  H (400~1000 Oe)  is applied  \nto SD.  The SD pattern  becomes weak, and  \nthe marginal  magnetization s in basic \ndomains are rotating to the direction of the \nmagnetic field with the increasing H. The \nregion s of flux closure cap  further reduce , \nand only a small flux closure core is \nobserved  when H reaches 1000 Oe . This \nfurther causes an increas ing proportion  of \nmagnetization  parallel ing to the x-axis \nnear the film surface . The magnetization s in \nthe critical area between the basic domains \nand Bloch -type domain walls  are firstly  \norientat ed to –x-axis. Part of the \nmagnetizations  along +x-axis in Bloch -type \ndomain walls  change to be along the –x-axis \nat 500 Oe , and complete ly turn towards  –x-\naxis when H exceeds 600 Oe .  \niii) A large H (1000~2500 Oe) is applied  to \nSD. All magnetization s are rotating to –x-\naxis with the increasing H. The SD  splits \ninto several narrow -weak SD by the \nincreasing number of Bloch -like domain  \nwalls  at 1500 Oe and disappears gradually  \nwhen the film is  in the saturati ng state (2500 \nOe). The magnetization of the film is finally \naligned  to x-axis. \nIV) HSD and HM, H is applied along  \nthe +y axis. \ni) A small H (0~400 Oe) is applied . The \nmagnetization s in the critical area between \nthe basic domains  and Bloch -type domain walls  show a little deviat ion along the \ndirection of the magnetic field, and become \nmore obvious with the increasing H. The \nflux closure cap s parallel  (antip arallel) to \nthe field direction  expand (shrink)  slowly  \nwith the increasing H, this results in  the \ncomponent of the My parallel to the field . My \nlinearly increases with H, which is in \nagreement with the magnetization curve  \nmeasurement  (black  square  line in figure  2). \nThe centers of Bloch -type domain walls  \nwere found to shift  alternat ely upwards and \ndownwards along  the z-axis. The above \nvariation  of the flux closure caps and Bloch -\ntype domain walls  cause s the twist  of basic \ndomains  and magnetization near  the film \nsurface . \nii) A moderate  H (400~1000 Oe)  is applied  \nto SD.  The expand ing or shrink ing of flux \nclosure caps , the shifting  of the centers of \nBloch -type domain walls , and the twist  of \nbasic domains  and magnetization near film \nsurface  are more obvious at 500 Oe . The SD \nis rotating  towards the applied field \ndirection  (+y) when the intensity of the H is \n600 Oe . The magnetization at this moment \nis similar to the simulation result of HSD \nexcept for the direction of SD . The \nmagnetization s are consist ing of a pair of \nreversed  magnetization s at the film surface  \nalong the x-axis and the tilted \nmagnetization s along the + y-axis. When the \nH is further  increased , the SD is completely \nturning to the +y-axis. \niii) A large H (1000~2500 Oe) is applied  to \nSD. The results are similar to the simulation  \nresults of HSD. All magnetization s are \nrotating to the +y-axis with the increasing H. \nThe SD  splits into several narrow -weak SD 15 \n by the increasing number of Bloch -like \ndomain  walls  at 1500 Oe, and disappears \ngradually  when the film is in the saturati ng \nstate at (2500 Oe).  The magnetization of the \nfilm is finally aligned  to the +y-axis. \nThe micromagnetic simulation results \ncan clearly explain the behavior of \nmagnetization curves ( figure  2): \ni) For HSD and HM, the component  of \nthe magnetization  parallel ing to the external \nfield increases  linear ly until its saturation \nstate, which is in agreement with the \nmagnetization curves  (red circle  line in \nfigure  2).  \nii) For HSD and HM, the \nmagnetization  is originally opposite with \nthe direction of the field , and then part of the \nmagnetization  is rotating towards the  field \ndirection . This rotati on of magnetization  \nbegins to strongly increase when H exceeds  \n500 Oe ( corresponding to the Hc in figure  2), \nand further  displays the sam e magnetization \nstate with HM when H is larger than  \n1000 Oe ( corresponding to the Hrev in figure  \n2). Finally,  the cur ves are getting to their \nsaturation state. \niii) For HSD and HM, the initial  \nmagnetization  is zero  due to the equal and \nnegative magnetization  component  My at \nremanence . When the H is increasing,  the \ncomponent  of the magnetization  parallel ing \nto the external field increases  until the \nstripes rotate towards the  field direction  at \n600 Oe ( corresponding to the Htra in figure  \n2). When H further increases, t he \nmagnetization s then change similar ly with \nHSD, and  reach  the saturation state .  \nWe n ote that the higher value of the \nreorientation field found in the  micromagnetic simulations, concerning  the \nexperiment, maybe due to edge effects \ninduced by the limited extension of the  \nsimulated cell  [24, 46]. We performed a \nlarge number of simulations  by changing  \nthe thickness, saturation magnetization, \nperpendicular anisotropy, exchange \ncoefficient , etc. (not shown) . The results \nshow that these parameters affect the \ndomain width, critical thickness, saturation \nfield, etc. of the SD, but the physical process \nand mechanism of this work are not affected. \nThe domain width w is reduced with the \nincreased applied field H, and  previous \nresearch has studied this systematically  [30, \n60], thus we do not discuss it here.  \nD. Discussion and calculation  \nThe underlying equation of motion for \nthe temporal  evolution of the magnetization \nis the Landau –Lifschitz –Gilbert (LLG)  \nequation [14, 61]: \neffdm dmm H mdt dt   \n      (1) \nwhere m is the magnetization direction,  is \nthe gyromagnetic  ratio, the dimensionless  \ncoefficient α is called the Gilbert damping \nconstant , and Heff is the effective magnetic \nfield including  the external, \ndemagnetization, and anisotropy  fields. To \nexcite the acoustic and optic mode s, certain \nexperimental conditions  have to be met \nwhich can be summariz ed by the following \ntwo general  rules  [31]: \ni) The pumping field hrf must have a \nnonzero component perpendicular to the \nstatic magnetization  M, to exert a finite \ntorque on the magnetization  and tilt it out of 16 \n its equilibrium position : \n0rf Mh\n             (2) \nii) The total dynamic moment mtot must \nhave a nonzero projection parallel to  the \ndirection of the pumping field hrf :  \n0tot rfmh\n             (3) \nThese  magnetic moment  processions  of \neach excitation mode  can finally attribute to  \nthe local  or total  effective  field in the film , \nand one can describe the  modes  by their \nresonance  frequency . First, combining the \nresults of micromagnetic simulation  (figure  \n6 and figure  7) and magnetization curves \n(figure  2), it can be sure  that the results of \nthe different  resonance  frequenc ies and \nmodes  are highly depend ent on the change \nof the magnetization  and SD . The detail ed \nconclusion is below:  \ni) For hrfSD, it is the AM. The magnetic \nmoments in the neighbor stripe  resonate in -\nphase  and are coupled by the domain \nsurface charges . This gives rise to the \ndynamic dipolar coupling field  [31]. Such  \ndynamic dipolar fields  add to an out-of-\nplane  restoring torque and induce  the \nprecession frequency.  It can be considered \nthat there is a pseudo -anisotropy  along the \ndirection of SD . The permeability  spectra  \n(figure  3b and 3 d, figure  4a-b and 4d-e) are \nin accordance with the conventional  in-\nplane uniform precession  mode . We note \nthat the permeability spectra  of figure  3 (b, \nd) and figure  4 (a, b, d, and e ) are \nsubstantially  the same.  We list the results of \nfigure  4 to discuss  here. It is worth noting \nfor the situation  hrfSD and HM that \nthe fr reduce s first due to the opposite  initial  direction between H and M. The M rotate s \nto the direction of H gradually  with the \nincreasing H (figure  6b and 7b). When  H is \nlarger than Hc, most  M turns towards  the \ndirection of H. The t otal M is further \norientat ing along  the direction of H as the H \nexceeds Hrev. We note that when the \nmagnetic field exceeds Hrev, the \nmagnetization  along the field direction in \nSD will linearly increase  i.e., the in -plane \nmagnetization curves  of HM and \nHM are overlapping  (figure 2). This \nconversion  from Hc to Hrev is very quick , \ncorrespond ing to the short inflection  in the \npermeability spectra  (figure  4b and 4e). The \nfr of HM then increases l inear ly and  \nshows the same law with that of HM \nafter H is larger than Hrev.  \nii) For hrf//SD,  this is the OM. The magnetic \nmoments in the neighbor stripe  resonate out \nof phase  and are coupled by the wall surface \ncharges,  which give rise to the  in-plane  \ndynamic dipolar coupling field  [31]. The \ndynamic dipolar fields  add to an in-plane  \nrestoring torque and enhance the resonance  \nfrequency.  Thus, the fr of the OM is higher \nthan AM. When a small H is applied \nperpendicular to  SD, although the M is \ndisturb ed towards the direction of H (figure  \n6c and 7c), the SD does not rotate . The hrf is \nstill orthogonal  to SD,  the film also \nresonate s with the OM (figure  4c and 4f). \nWith the further increasing H, SD begins \nturning to the direction of H, the OM \nweakens while  the AM appear s. The \nresonance peaks  in the field range of 90 to \n150 Oe  become  complicated  and the field at \nthis range  (Htra) is also the transition  value \nof SD  rotation . These multiple  peaks 17 \n attribute to  the domain , domain wall , and \nthe flux closure cap , which ha ve been \ncalculate d detailedly by Ebels et al  and  \nVukadinovic  et al [4, 14, 31]. When  the SD \norientat es completely  to the direction of H, \nthe OM and AM  will compose  to AM,  \nfinally continues as the uniform precession  \nmode . \nTo describe  quantitative ly the relation of \nthe resonance  frequency  depending on the \nhrf and H, we further demonstrate  the \nresonance equation  of different modes.  As \nwell know, t he resonance frequency fr of the \nfilm can be determined by Kittel equation  \n[62]: \n  42r eff s efff H M H\n      (4) \nHowever, the SD film is in-plane isotropy , \nand the presence of the response  in SD  can \nbe related to its local magnetization , where \nthe magnetization s are alternatively  up and \ndown in the stripes . Thus,  the contribution  \nof the effective magnetic field  Heff of SD is \ncomplicated , and its magnetization  is not a \nconstant  of 4Ms, but changes as a function \nof H until saturation . The Kittel formula  \nneed s to be revise d. According to previous \nresults  [33, 42, 43, 63], the free ene rgy \ndensity of periodic domains  consist s of \nmagnetostatic energy , anisotropy energy,  \nexchange energy , and magnetic field  energy , \nand th e frequencies of the excitation  modes \nare calculated by the Smit -Beljers  procedure \n[40, 42, 43, 63, 64]. The resonance \nfrequency  of AM and OM with applied \nmagnetic field H is rewritt en as follow:  \ni) hrfSD, HSD, and HM \n1/2[( 4 )( )]2AM dyn dyn\nr k kf H H M H H    \n                            (5a) \n,\n1/2[( 4 )2\n( )]AM SW dyn\nr k sw\ndyn\nk swf H H M H\nH H H   \n\n (5b)  \nii) hrfSD, HSD and HM \nWhen H<Hc, \n1/2[( 4 )( )]2AM dyn dyn\nr k kf H H M H H   \n \n       (6a) \n,\n1/2[( 4 )2\n( )]AM SW dyn\nr k sw\ndyn\nk swf H H M H\nH H H   \n\n(6b)  \nWhen H>H c, the equations of frAM and \nfrAM,SW are the same as Eqs. (5a) and (5b) \nrespectively.  \niii) hrf//SD, HSD and HM or \nHMThe equations of frAM and frAM,SW are \nthe same  as Eqs. (5a) and (5b) respectively.  \n1/2[( 4 )2\n( )]OM dyn SD\nr k ex\ndyn SD\nk exf H H H M\nH H H   \n\n   (7a) \n,\n1/2[( 4 )2\n( )]OM SW dyn SD\nr k ex sw\ndyn SD\nk ex swf H H H M H\nH H H H    \n  \n \n     (7b) \nWhere,  \n2 2= ( )swAnHMD\n           (8) \n \nHkdyn: the d ynamic anisotropy  of SD . This \nrepresent s the sum of the inner  anisotropy of \nSD film  including r otational anisotropy  of \nSD and s tatic anisotropy  of the film. In this 18 \n work,  we did not induce a static anisotropy  \nduring the deposition, and this term can be \nneglect ed. \nHsw: the spin-wave  field of SD . The spin  \nwave come s from the s tanding wave \nresonance between periodic SD . \nn: the quantization number SW.  \nD: the period  of SD.  \nHexSD: the exchange coupl ing field between \nthe adjacent  stripes .  \nM: the magnetization of film.  It can be  \nobtained from the magnetic hysteresis loops  \nor magnetization curves . The values of M \nare depending on the H, and the results are \nshown in figure  8(a-b). Ms is the saturation \nmagnetization.  \nBased on  the above Eqs. (5-8), we obtained \nthe resonance frequency  fr at each magnetic \nfield H by VNA -FMR, the corresponding  \n4M at each magnetic field obtained from \nVSM and magnetization curve . Finally, the \nHk dependent magnetic field can be \nestimate d. The calculat ed Hkdyn and Hk are \npresent ed in figure  8 (c-f). The p hysical \nprocess of the measured two modes or \nmeasured three situation s have been \ndiscussed above , and  the change of the  \ncorresponding  different Hk fits well with \nthese  discussion s. We can obtain that the \nvariation of the fr of different resonant  \nmodes with e xternal magnetic field H is not \nonly highly related  to magnetization \ndistribution  but also the SD induced \nanisotropy (i.e., the direction of SD) . The \nvalues of SD-induced anisotropy  are \ndependent on t he strength and direction of \nSD, which changes with the H. In addition, \nthe transition /rotation  field (Htra) of the SD \nis more hysteresis  than the reversal  field (Hrev) of the magnetization. It can be  seen \nthat the value of Htra is about 60 Oe larger \nthan Hc and 30 Oe  larger than Hrev for both \nsputtered  and electrodeposited  films. In \naddition, t he stronger  exchange coupling  \nanisotropy  in the sputtered SD film than the \nelectrodeposited SD film proves the  more \nregular  stripe in  sputtered film, which \nagree s with the above MFM results.  \nFigure  8. The magnetization s 4πM (a-b) and \ndifferent anisotropy  Hk (c-f) of different resonant  \nmodes depend on the applied magnetic field H. In \n(a-b), the 4πM are the experimental result obtained \nby VSM and magnetization curves. In (c -f), the \nresult is obtained by using Eqs. (5 -8). \nAs can be seen in the magnetic  spectra  \nof figure  3 and 4, the peak intensity  of \nimaginary permeability  are not equal  for the \nsame film with the different applied \nmagnetic field s. We further calculate  the \nresults and compare  them with the \nexperimental results.   \nArcher  et al  [65] has demonstrate d the \nlimitation  between fr and the initial \n19 \n permeability  in: \n2 2 2( 1) ( ) (4 )2in r s fM\n       (9) \nThe maximum  intensity of imaginary  \npermeability  peak  max can be obtained by \nsolved LLG equation  [66]:  \n''\nmax 211= ( 1) 12in\n         (10) \nmax finally turns out for SD to be  the form  \n2\n''\nmax 2(4 ) 1=12[( )(4 )]dyn dyn\nkkM\nH H M H H  \n(11) \nIt can be seen from Eq. (11), the max \ndepend s on , 4πM, and Hkdyn. For the same \nsample, the  is a constant , which is fitting \nas 0.01 for sputtering  film and 0.07 for \nelectrodeposition film in this work . The \ncalculative results of max are shown in \nfigure  9. It can be seen that changes in the \nmeasurement  and calculation data are in \ngood agreement.  This means that the \nmodified equations are suitable to explain \nthe change process of SD resonance \nintensity  and reflect  the result  of max under \nthe increased H. The results indicate that the \nresonance intensity of different modes in \nSD is determined by the local magnetization \nrather than the whole  magnetization of SD.  \nWe note that t he result of the \nelectrodeposition film and the  result of \nsputtering film in the field range of 90 to \n150 Oe do not fit very well. This is due to \nthe large error of OM for electrodeposition \nfilm (the resonance  intensity is low , the line \nwidth  is very large , and we have explained \nthe reason above ) and the low distinguish ing of the  overlapping peaks n \nthe field range of 90 to 150 Oe.  \n \nFigure  9. Experimental and calculative maximum \nintensity of imaginary  permeability  peak  max. \nThe different shape is the experimental result \nwhile the line is the calculative result . The \nexperimental result and calculative result  are in the \nsame color. Figure (a -b) c orresponds to  the \nresonance intensity  of figure 3, and F igure (c -d) \ncorresponds to  the resonance intensity  of figure 4 . \n4. Conclusion  \nIn summary, we first investigated the \nstructure  and magnetic domain  of FeNi  SD \nfilms . It is found that the electrodeposited \nfilm showed a dispersive SD pattern  and \nresulted in  a weak exchange coupling \ninteraction when compared  with sputtered \nfilm.  The static magnetic properties \nespecially the magnetization curve  reveal \nthe magnetization  distribution in SD w as \ndependent on the direction of SD . The \nmagnetization s tarts to reversal after the \napplied field exceeds Hc. The rotation of SD  \nis hysteresis  with 60 Oe larger magnetic  \nfield than Hc. Such magnetization and SD \ndistribution  determine d the selective ly \nexcited  dynamic  microwave magnetic \nproperties , which  emerged the dynamic \nhysteresis , the AM, OM, and SW resonance . \n20 \n The frequency and intensity of different \nresonance modes of stripe domain are \ndetermined by the local  magnetization . The \nSD rotation  and magnetization  reversal \nwere further certif ied by the micromagnetic \nsimulation . Based on the above results, the \nanisotropy and resonance intensity of \ndifferent modes  were calculate d by the \nmodified resonance equation s, and the \nresults fit well with the experimental data. \nThe results help to deeply understand the \nrotation  mechanism  of the SD and provide \nthe possibility of SD film  for microwave \nexcitation  applications in spintronics . \nAcknowledgments  \nThis work is supported by  the National \nNatural Science F oundation  of China \n(11704211  and 11847233 ), China \nPostdoctoral Science Foundation \n(2018M632608) , the Applied basic research \nproject  of Qingdao  (18-2-2-16-jcb), the \nbasic scientific research business expenses  \nof the central university , and O pen Project \nof Key Laboratory for Magnetism and \nMagnetic Materials of the Ministry of \nEducation, Lanzhou University . \nReferences  \n[1] J. Ben Youssef, N. Vukadinovic, D. Billet, M. \nLabrune, Thickness -dependent mag netic \nexcitations in Permalloy films with nonuniform \nmagnetization, Phys. Rev. B, 69 (2004) 174402.  \n[2] R. Lopusnik, J.P. Nibarger, T.J. Silva, Z. Celinski, \nDifferent dynamic and static magnetic anisotropy \nin thin Permalloy ™ films, Appl. Phys. Lett., 83 \n(2003) 96 -98. \n[3] O. Acher, S. Queste, M. Ledieu, K. -U. Barholz, \nR. Mattheis, Hysteretic behavior of the dynamic \npermeability on a Ni -Fe thin film, Phys. Rev. B, 68 (2003) 184414.  \n[4] N. Vukadinovic, M. Labrune, J.B. Youssef, A. \nMarty, J.C. Toussaint, H. Le Gall, Ferromagnetic \nresonance spectra in a weak stripe domain \nstructure, Phys. Rev. B, 65 (2001) 054403.  \n[5] N.A. Buznikov, K.N. Rozanov, The effect of \nstripe domain structure on dynamic permeability \nof thin ferromagnetic films with out -of-plane \nuniaxial a nisotropy, J. Magn. Magn. Mater., 285 \n(2005) 314 -326. \n[6] S. Tacchi, S. Fin, G. Carlotti, G. Gubbiotti, M. \nMadami, M. Barturen, M. Marangolo, M. \nEddrief, D. Bisero, A. Rettori, M.G. Pini, \nRotatable magnetic anisotropy in a Fe 0.8Ga0.2 thin \nfilm with stripe domains: Dynamics versus \nstatics, Phys. Rev. B, 89 (2014) 024411.  \n[7] R.J. Spain, Dense -Banded Domain Structure in \n``Rotatable Anisotropy'' Permalloy Films, Appl. \nPhys. Lett., 3 (1963) 208.  \n[8] H.F. Nobuo Saito, Yutaka Sugita, A New type of \nmagnetic domain  in thin Ni -Fe files, J. Phys. Soc. \nJapan, 19 (1964).  \n[9] N. Saito, H. Fujiwara, Y . Sugita, A new type of \nmagnetic domain structure in negative \nmagnetostriction Ni -Fe films, J. Phys. Soc. Japan, \n19 (1964) 1116 -1125.  \n[10] T. Iwata, R.J. Prosen, B.E. Gran, P erpendicular \nAnisotropy in Polycrystalline Ni ‐Fe Thin Films, \nJ. Appl. Phys., 37 (1966) 1285 -1286.  \n[11] M.M.H. D.S.Lo, Origin of Stripe Domains in \nNiFe Films, J. Appl. Phys., 38 (1967).  \n[12] Derang Cao, Zhenkun Wang, Erxi Feng, Jinwu \nWei, Jianbo Wang, Qingf ang Liu, Magnetic \nproperties and microstructure investigation of \nelectrodeposited FeNi/ITO films with different \nthickness, J. Alloys Compd., 581 (2013) 66 -70. \n[13] J.C. Li, Q.F. Zhan, M.J. Pan, L.P. Liu, H.L. Yang, \nY .L. Xie, S.H. Xie, R.W. Li, Preparation and \nmagnetic anisotropy of NiFe film with stripe \ndomains, Acta Phys. Sin., 65 (2016).  \n[14] N. Vukadinovic, O. Vacus, M. Labrune, O. \nAcher, D. Pain, Magnetic excitations in a weak -\nstripe -domain structure: A 2D dynamic \nmicromagnetic approach, Phys. Rev. Lett ., 85 21 \n (2000) 2817 -2820.  \n[15] N. Amos, R. Fernandez, R. Ikkawi, B. Lee, A. \nLavrenov, A. Krichevsky, D. Litvinov, S. \nKhizroev, Magnetic force microscopy study of \nmagnetic stripe domains in sputter deposited \nPermalloy thin films, J. Appl. Phys., 103 (2008) \n07E732.  \n[16] L.M. Alvarez -Prado, G.T. Pérez, R. Morales, \nF.H. Salas, J.M. Alameda, Perpendicular \nanisotropy detected by transversely biased initial \nsusceptibility via the magneto -optic Kerr effect \nin FeSi 1-x thin films and Fe xSi1-x/Si multilayers: \nTheory and experiment, Phys. Rev. B, 56 (1997) \n3306 -3316.  \n[17] O. Donzelli, M. Bassani, F. Spizzo, D. Palmeri, \nReorientational transition and stripe domains in \nCo films, J. Magn. Magn. Mater., 320 (2008) \ne261 -e263.  \n[18] G. Chai, N.N. Phuoc, C.K. Ong, High thermal \nstability of zero -field ferromagnetic resonance \nabove 5GHz in ferrite -doped CoFe thin films, \nAppl. Phys. Lett., 103 (2013) 042412.  \n[19] L. Feilong, Z. Fu, Y . Ge, W. Dongping, W. Fulin, \nApproximate Solution of the Magnetostatic \nProperty of FeCoZr Thin Films w ith Stripe \nDomain, Appl. Phys. Express, 6 (2013) 073003.  \n[20] N.N. Phuoc, C.K. Ong, Thermal stability of high \nfrequency properties of gradient -composition -\nsputtered FeCoHf films with and without stripe \ndomains, J. Appl. Phys., 114 (2013).  \n[21] A. Hierro -Rodriguez, R. Cid, M. Vélez, G. \nRodriguez -Rodriguez, J.I. Martín, L.M. Álvarez -\nPrado, J.M. Alameda, Topological Defects and \nMisfit Strain in Magnetic Stripe Domains of \nLateral Multilayers With Perpendicular Magnetic \nAnisotropy, Phys. Rev. Lett., 109 (2012) 1 17202.  \n[22] M. Coïsson, F. Celegato, E. Olivetti, P. Tiberto, \nF. Vinai, M. Baricco, Stripe domains and spin \nreorientation transition in FeBSi thin films \nproduced by rf sputtering, J. Appl. Phys., 104 \n(2008) 033902.  \n[23] N. Vukadinovic, H. Le Gall, J. Ben Y oussef, V . \nGehanno, A. Marty, Y . Samson, B. Gilles, \nMagnetization dynamics and relaxation in epitaxial FePd thin films with a stripe domain \nstructure, Eur. Phys. J. B, 13 (2000) 445 -450. \n[24] S. Fin, R. Tomasello, D. Bisero, M. Marangolo, \nM. Sacchi, H. Pop escu, M. Eddrief, C. Hepburn, \nG. Finocchio, M. Carpentieri, A. Rettori, M.G. \nPini, S. Tacchi, In -plane rotation of magnetic \nstripe domains in Fe 1-xGax thin films, Phys. Rev. \nB, 92 (2015) 224411.  \n[25] R.J. Prosen, J.O. Holmen, B.E. Gran, Rotatable \nAnisotrop y in Thin Permalloy Films, J. Appl. \nPhys., 32 (1961) S91 -S92. \n[26] G. Wang, C. Dong, W. Wang, Z. Wang, G. Chai, \nC. Jiang, D. Xue, Observation of rotatable stripe \ndomain in permalloy films with oblique \nsputtering, J. Appl. Phys., 112 (2012) 093907.  \n[27] E. Sallica Leva, R. Valente, F. Martínez Tabares, \nM. Vásquez Mansilla, S. Roshdestwensky, A. \nButera, Magnetic domain crossover in FePt thin \nfilms, Phys. Rev. B, 82 (2010) 144410.  \n[28] W. Tee Soh, N.N. Phuoc, C. Tan, C. Ong, \nMagnetization dynamics in permalloy  films with \nstripe domains, J. Appl. Phys., 114 (2013) \n053908.  \n[29] N. Vukadinovic, F. Boust, Three -dimensional \nmicromagnetic simulations of magnetic \nexcitations in cylindrical nanodots with \nperpendicular anisotropy, Phys. Rev. B, 75 (2007) \n014420.  \n[30] R.  Bruc̆as, H. Hafermann, M.I. Katsnelson, I.L. \nSoroka, O. Eriksson, B. Hjörvarsson, \nMagnetization and domain structure of bcc \nFe81Ni19/Co (001) superlattices, Phys. Rev. B, 69 \n(2004) 064411.  \n[31] U. Ebels, L. Buda, K. Ounadjela, P.E. Wigen, \nFerromagnetic re sonance excitation of two -\ndimensional wall structures in magnetic stripe \ndomains, Phys. Rev. B, 63 (2001) 174437.  \n[32] A.V . Svalov, I.R. Aseguinolaza, A. Garcia -\nArribas, I. Orue, J.M. Barandiaran, J. Alonso, \nM.L. Fernandez -Gubieda, G.V . Kurlyandskaya, \nStructure and Magnetic Properties of Thin \nPermalloy Films Near the \"Transcritical\" State, \nIEEE Trans. Magn., 46 (2010) 333 -336. \n[33] M. Ramesh, P.E. Wigen, 22 \n Ferromagnetodynamics of parallel stripe \ndomains - domain walls system, J. Magn. Magn. \nMater., 74 (1988) 123-133. \n[34] H.P. Winkler, H. Dötsch, B. Lührmann, S. Sure, \nDynamic conversion of optical modes in \nmagnetic garnet films induced by resonances of \nperiodic stripe domains, J. Appl. Phys., 76 (1994) \n3272 -3278.  \n[35] Y . Shimada, M. Shimoda, O. Kitakami, Multi ple \nMagnetic Resonance in Amorphous Co -Nb-Zr \nFilms with Weak Perpendicular Anisotropy, Jpn. \nJ. Appl. Phys., 34 (1995) 4786 -4789.  \n[36] U. Ebels, P.E. Wigen, K. Ounadjela, Probing \ndomain wall structures in Co(0001) thin films \nusing ferromagnetic resonance, E urophys. Lett., \n46 (1999) 94 -100. \n[37] C.A. Ramos, E. Vassallo Brigneti, J. Gómez, A. \nButera, Stripe domains in Permalloy films as \nobserved by ferromagnetic resonance and \nmagnetic force microscopy, Physica B, 404 \n(2009) 2784 -2786.  \n[38] J. Ding, N. Singh, M . Kostylev, A.O. Adeyeye, \nStatic and dynamic magnetic properties of \nNi80Fe20 anti -ring nanostructures, Phys. Rev. B, \n88 (2013) 014301.  \n[39] V . Estévez, L. Laurson, Magnetic domain -wall \ndynamics in wide permalloy strips, Phys. Rev. B, \n93 (2016) 064403.  \n[40] S. V oltan, C. Cirillo, H. Snijders, K. Lahabi, A. \nGarcía -Santiago, J. Hernández, C. Attanasio, J. \nAarts, Emergence of the stripe -domain phase in \npatterned Permalloy films, Phys. Rev. B, 94 \n(2016) 094406.  \n[41] E.F. Silva, M.A. Corrêa, R.D.D. Pace, C.C.P. \nCid, P.R. Kern, M. Carara, C. Chesman, O.A. \nSantos, R.L. Rodríguez -Suárez, A. Azevedo, S.M. \nRezende, F. Bohn, Thickness dependence of the \nmagnetic anisotropy and dynamic magnetic \nresponse of ferromagnetic NiFe films, J. Phys. D: \nAppl. Phys., 50 (2017) 18500 1. \n[42] N. Vukadinovic, J.B. Youssef, H. Le Gall, \nInfluence of magnetic parameters on microwave \nabsorption of domain mode ferromagnetic \nresonance, J. Magn. Magn. Mater., 150 (1995) 213-222. \n[43] O. Acher, C. Boscher, B. Brulé, G. Perrin, N. \nVukadinovic, G.  Suran, H. Joisten, Microwave \npermeability of ferromagnetic thin films with \nstripe domain structure, J. Appl. Phys., 81 (1997) \n4057 -4059.  \n[44] J. Wei, H. Feng, Z. Zhu, C. Song, X. Wang, Q. \nLiu, J. Wang, Understanding stripe domains in \nPermalloy films via the angular dependence of \npermeability spectra, J. Magn. Magn. Mater., 432 \n(2017) 245 -249. \n[45] G. Chai, N.N. Phuoc, C.K. Ong, Angular \ntunable zero -field ferromagnetic resonance \nfrequency in oblique sputtered CoFeBSm thin \nfilms, Appl. Phys. Express, 7 (201 4) 063001.  \n[46] I.S. Camara, S. Tacchi, L.C. Garnier, M. Eddrief, \nF. Fortuna, G. Carlotti, M. Marangolo, \nMagnetization dynamics of weak stripe domains \nin Fe –N thin films: a multi -technique \ncomplementary approach, J. Phys.: Condens. \nMatter, 29 (2017) 465803 . \n[47] D. Cao, Y . Song, L. Pan, H. Du, H. Feng, C. \nZhao, Q. Li, J. Xu, S. Li, Q. Liu, J. Wang, \nInfluence of the phases structure on the acoustic \nand optical modes ferromagnetic resonance of \nFeNi stripe domain films, J. Magn. Magn. Mater., \n475 (2019) 103 -107. \n[48] D. Cao, L. Pan, X. Cheng, Z. Wang, H. Feng, Z. \nZhu, J. Xu, Q. Li, S. Li, J. Wang, Q. Liu, \nThickness -dependent on the static magnetic \nproperties and dynamic anisotropy of FeNi films \nwith stripe domain structures, J. Phys. D: Appl. \nPhys., 51 (2018) 0 25001.  \n[49] Jinwu Wei, Jianbo Wang, Qingfang Liu, Xiaoyu \nLi, Derang Cao, Xiaojun Sun, An induction \nmethod to calculate the complex permeability of \nsoft magnetic films without a reference sample, \nRev. Sci. Instrum., 85 (2014) 054705.  \n[50] M. Liu, S. Du, F. Wang, R. Adam, Q. Li, X. Ma, \nX. Guo, X. Chen, J. Yu, Y . Song, J. Xu, S. Li, D. \nCao, Influence of surface pinning in the domain \non the magnetization dynamics in permalloy \nstriped domain films, J. Alloys. Compd., 869 \n(2021) 159327.  23 \n [51] W. Jinwu, Z. Zengtai,  F. Hongmei, D. Jinlu, L. \nQingfang, W. Jianbo, Top -down control of \ndynamic anisotropy in permalloy thin films with \nstripe domains, J. Phys. D: Appl. Phys., 48 (2015) \n465001.  \n[52] Y . Chen, A. Yang, M.R. Paudel, S. Stadler, C. \nVittoria, V .G. Harris, Microwav e \nmagnetoelectric coupling and ferromagnetic \nresonance frequency tuning of a \nCo2MnSb/GaAs/PZN -PT heterostructure, Phys. \nRev. B, 83 (2011) 104406.  \n[53] J. Wei, Z. Zhu, H. Feng, J. Du, Q. Liu, J. Wang, \nTop-down control of dynamic anisotropy in \npermalloy thin  films with stripe domains, J. Phys. \nD: Appl. Phys., 48 (2015) 465001.  \n[54] M. Huige, L. Chengyi, W. Wenfeng, C. Guozhi, \nThickness -dependent resonance frequency of \nnon-uniform procession mode in CoZr stripe -\ndomain magnetic films, J. Phys. D: Appl. Phys., \n51 (2018) 285004.  \n[55] Y . Murayama, Micromagnetics on Stripe \nDomain Films. Ⅰ. Critical Cases, J. Phys. Soc. \nJpn., 21 (1966) 2253 -2266.  \n[56] A. Hubert, R. Schäfer, Magnetic domains: the \nanalysis of magnetic microstructures, 3rd ed., \nSpringer Science & Busine ss Media, New York, \n2008.  \n[57] D. Spenato, S.P. Pogossian, Hysteretic \nresonance frequencies and magnetization \nreversal in exchange biased polycrystalline F/AF \nbilayers, J. Magn. Magn. Mater., 285 (2005) 79 -\n87. \n[58] S. Queste, S. Dubourg, O. Acher, K.U. Bar holz, \nR. Mattheis, Exchange bias anisotropy on the \ndynamic permeability of thin NiFe layers, J. Appl. \nPhys., 95 (2004) 6873 -6875.  \n[59] C. Song, C. Jin, J. Wang, H. Xia, J. Wang, Q. \nLiu, Skyrmion -based multi -channel racetrack, \nAppl. Phys. Lett., 111 (2017) 192413.  \n[60] Y . Talbi, P. Djemia, Y . Roussigné, J. BenYoussef, \nN. Vukadinovic, M. Labrune, Magnetic \nexcitation in weak stripe domains: Ferromagnetic \nresonance and Brillouin light sattering studies, J. \nPhys.: Conference Series, 200 (2010) 072107.  [61] Y . Ts erkovnyak, A. Brataas, G.E.W. Bauer, \nEnhanced Gilbert Damping in Thin \nFerromagnetic Films, Phys. Rev. Lett., 88 (2002) \n117601.  \n[62] C. Kittel, On the Theory of Ferromagnetic \nResonance Absorption, Phys. Rev., 73 (1948) \n155-161. \n[63] J.O. Artman, S.H. Charap , Ferromagnetic \nresonance in periodic domain structures, J. Appl. \nPhys., 49 (1978) 1587 -1589.  \n[64] G. Gubbiotti, G. Carlotti, S. Tacchi, M. Madami, \nT. Ono, T. Koyama, D. Chiba, F. Casoli, M.G. \nPini, Spin waves in perpendicularly magnetized \nCo/Ni(111) multi layers in the presence of \nmagnetic domains, Phys. Rev. B, 86 (2012) \n014401.  \n[65] O. Acher, A.L. Adenot, Bounds on the dynamic \nproperties of magnetic materials, Phys. Rev. B, \n62 (2000) 11324 -11327.  \n[66] Y . Ding, T.J. Klemmer, T.M. Crawford, A \ncoplanar waveg uide permeameter for studying \nhigh-frequency properties of soft magnetic \nmaterials, J. Appl. Phys., 96 (2004) 2969 -2972.  \n "    },    {        "title": "0906.5431v1.Determination_of_a_flow_generating_a_neutral_magnetic_mode.pdf",        "content": "arXiv:0906.5431v1  [nlin.CD]  30 Jun 2009Determination of a flow generating a neutral magnetic mode\nVLADISLAV ZHELIGOVSKY\nInternational Institute of Earthquake Prediction Theory\nand Mathematical Geophysics\n84/32 Profsoyuznaya St., 117997 Moscow, Russian Federation\nObservatoire de la Cˆ ote d’Azur, CNRS\nU.M.R. 6529, BP 4229, 06304 Nice Cedex 4, France\nThe problem of reconstruction of a flow of conducting incompressib le fluid generating\na given magnetic mode is considered. We use the magnetic induction eq uation to derive\nordinary differential equations along the magnetic field lines, which giv e an opportunity\nto determine the generating flow, if additional data is provided on a t wo-dimensional\nmanifold transversal to magnetic field lines, and show that an arbitr ary solenoidal vector\nfield can not be a neutral magnetic mode sustained by any flow of con ducting fluid.\n1. Introduction\nAccording to the modern scientific paradigm, magnetic fields of astr ophysical objects,\nranging from planets to galaxies, are often sustained by conductin g fluid flows, driven by\nconvection in the melted medium in their interiors [12, 11, 20, 16, 19]. T hese processes\nare governed by the Navier-Stokes and magnetic induction equatio ns (supplemented by\nother equations, such as heat equation and rheology relations, as appropriate). However,\nit is difficult to study them numerically because of the extreme parame ter values involved,\nwhich require prohibitively high resolution of simulations. Thus, applica tion of analytical\nor semi-analytical methods to the study of astrophysical dynamo s appears unavoidable.\nIn the present paper we suggest an approach, in principle enabling o ne “to separate” the\ntwo fundamental equations; hopefully, this can be useful for inve stigation of asymptotics\nof astrophysical dynamos.\nUsually the magnetic induction equation is employed for investigation o f the evolution\nof magnetic field for a given flow of incompressible conducting fluid (wh ich is predefined\nin kinematic dynamo problems, or supposed to evolve simultaneously w hen nonlinear\ndynamos are studied). We consider here an inverse problem, invest igating which conse-\nquences existence of a neutral magnetic mode bears upon the gen erating flow. We show\nhow the flow can be reconstructed uniquely up to the data which mus t be provided on\na two-dimensional manifold transversal to magnetic field lines. We de monstrate that an\narbitrary solenoidal vector field can not be a magnetic mode sustain ed be any flow of in-\ncompressible fluid, unless the field satisfies a consistency equation in the fluid volume. We\nhope that such an analysis may be useful, in particular, for examinat ion of asymptotical\nproperties of various steady magnetohydrodynamic systems and their stability.\nIn recent numerical studies of nonlinear magnetic dynamos acting in plasma [8] and\nfluid [13, 5, 6] flows with a prescribed forcing, as well as in thermal co nvection in a\nhorizontal layer of conducting fluid rotating about a vertical axis [2 5] or in the absence of\nrotation [14, 15], it was discovered that temporal evolution can res ult in emergence of a\nsteady state with a non-vanishing magnetic field. Magnetostatic eq uilibria in ideal plasma\nwere discussed in [1]. A magnetic field of a steady configuration is a neu tral magnetic\nmode, i.e., a vector field belonging to the kernel of the magnetic induc tion operator.\nNeutral magnetic modes play an important rˆ ole in large-scale dynam os [21]-[24].\n1We therefore focus on neutral magnetic modes in our analysis. How ever, a straight-\nforward modification of our approach can be applied for reconstru ction of flows for eigen-\nfunctions of the magnetic induction operator associated with any g iven eigenvalue, or for\narbitrary evolving magnetic fields.\n2. Reconstruction of flows\nConsider the magnetic induction equation\n∂tb=η∇2b+∇×(u×b). (1)\nMagnetic field is solenoidal:\n∇·b= 0. (2)\nIn a steady state magnetic field is a neutral mode of the magnetic ind uction operator.\nFor a given flow band molecular diffusivity ηthe operator is elliptic. If magnetic field\ngeneration in a bounded volume of fluid is considered and regular boun dary conditions\nfor magnetic field are imposed, it has a discrete spectrum, with the e igenvalues tending\nto−∞. For a randomly chosen pair η,uthe kernel of the operator does not contain\nmean-free magnetic fields, and generically the only mean-free solut ion isb= 0.\nThe processes bringing the system to a steady state thus can be v iewed as adjustment\nof the flow to a configuration allowing for a non-zero neutral mean- free magnetic mode.\nIt is natural therefore to treat (1) as an equation in u. “Uncurling” it, one obtains\nη∇×b=u×b−η∇a, (3)\nwhereais a scalar function (the constant factor ηis introduced for convenience).\nConsider separately the components of (3) parallel and perpendic ular tob. Scalar\nmultiplying (3) by bfind\n(b·∇)a=−b·(∇×b). (4)\nThe equation controls the magnitude of a magnetic field, whose direc tion is prescribed:\nLetiBbe a unit vector collinear with b, then (4) implies\n|b|=−(iB·∇)a\niB·(∇×iB). (5)\nIf a magnetic force line is a closed loop (including the loops emerging due to spatial\nperiodicity), then by virtue of (4)\n/contintegraldisplay\niB·(∇×b)ds=−/contintegraldisplay\n(iB·∇)ads= 0\n(the parameter son the curve is the distance along the curve from a fixed point on it),\nwhich can be also viewed as a constraint on the magnitude of the magn etic field (following\nfrom (5)).\nThe component of (3) perpendicular to bis accessed by cross-multiplication of (3) by\nb, yielding\nηb×(∇×b+∇a) =u|b|2−b(u·b).\nThe component of uparallel to bis not determined by (3), hence ( u·b) remains an\nunidentified arbitrary scalar function. We denote\n(u·b)\n|b|2= 1+ηα,\n2implying\nu= (1+ηα)b+ηe×(∇×b+∇a), (6)\nwhere\ne=b/|b|2.\n(6) and (4) together are equivalent to the equation for a neutral magnetic mode. The\nscalar field αsatisfies the equation\n(b·∇)α+∇·(e×(∇×b+∇a)) = 0, (7)\nequivalent to the solenoidality condition for the flow u.\nNow, in order to find the flow velocity (6), we need to determine ∇a, which we do\nemploying the solenoidality condition for the flow. Generically iBand∇ ×eare not\nparallel, and in this case (4) and (7) are equivalent to the equation\n∇a=AiA+BiB+CiC, (8)\nwhereiA,iB,iCis an orthonormal basis,\niC≡∇×e−(iB·(∇×e))iB\n|∇×e−(iB·(∇×e))iB|,iA≡iB×iC,\nB≡ −iB·(∇×b)≡ −|b|iB·(∇×iB), (9)\nC=(B/|b|)2−(b·∇)α−∇·(e×(∇×b))\n|∇×e−(iB·(∇×e))iB|. (10)\nWe derive from (8) individual equations in AandC.\nThe solvability condition for (8) is obtained by taking its curl:\n0 =∇A×iA+A∇×iA+∇B×iB+B∇×iB+∇C×iC+C∇×iC.(11)\nScalar multiplying it by iA,iBandiC, one finds\nA=−CiA·(∇×iC)−(iB·∇)C−BiA·(∇×iB)+(iC·∇)B\niA·(∇×iA), (12)\n0 =AiB·(∇×iA)+(iC·∇)A−B2\n|b|+CiB·(∇×iC)−(iA·∇)C, (13)\nC=−AiC·(∇×iA)+(iB·∇)A−BiC·(∇×iB)−(iA·∇)B\niC·(∇×iC), (14)\nwhereBis defined by (9). Substitution of (12) into (14) yields a second orde r differential\nequation along magnetic force lines, in principle, defining C. Initial conditions for this\nequations must be set on two-dimensional manifolds, transversal to magnetic force lines.\nThey must assure geometric consistency: the solutions along close d force lines must be\nperiodic. For force lines, intersecting with the boundary of the reg ion occupied by the\nfluid, it is naturally to set the conditions on the boundary. The data c an be provided on\ntwo manifolds, crossing a force line; in this case one obtains a bounda ry value problem\nforC. In turn, αcan be found, in principle, from (10). This completes reconstructio n\nof the flow. (The divergence of (8) yields an equation in a, which can be used to find\n3the potential itself.) Substituting A(12) and Cinto (13), one obtains an equation in b.\nThus, not every solenoidal field can be a magnetic neutral mode: Th e scalar consistency\nequation (13) constrains, together with the solenoidality condition , a neutral mode up to\na scalar field.\nImplementation of this program can become particularly difficult in the presence of\nmagneticnulls, i.e., points, wheremagneticfieldvanishes. (Thisisclear , ofcourse, already\nfrom the definition of the vector field e, which becomes singular at the nulls). Topology\nof magnetic field with null points and its bifurcations during reconnec tions are studied in\ndetailinsolarmagnetohydrodynamics [9,10,2,17,18]–theyarep resumed tobeoffunda-\nmental importanceforoccurrence of sudden explosive energyre lease events, solarflares, in\nthe Sun’s corona. In the vicinity of a null point magnetic field exhibits a n approximately\nlinear behavior controlled by the Jacobian /ba∇dbl∂bi/∂xj/ba∇dbl. Solenoidality of the magnetic field\nimplies, that the sum of the three eigenvalues of this matrix vanishes . Hence, generically\nit has two eigenvalues with real parts of the same sign, and an eigenv alue of an oppo-\nsite sign. Consequently, one can identify a two-dimensional manifold of magnetic force\nlines behaving coherently – all approaching the null point or all depar ting from it (if the\ntwo eigenvalues have negative or positive real parts, respectively ) and an one-dimensional\nmanifold (a force line), exhibiting the behavior of the opposite kind. I n the parlance of\nsolar physics, the two-dimensional manifold is the fan, and the one-dimensional manifold\nthespineof the null (see Fig. 1 in [4]). Therefore, in our problem there are infin itely many\ncharacteristics (constituting the fan), which must bring the same values of AandCto (or\ntake the same values from) the null point, implying that the problem o f consistency of the\nglobal solution for the flow arises. The situation is further complicat ed by the fact that\niBis typically discontinuous at null points (its direction is not well-defined ), and hence\niAandiCare discontinuous as well.\nThus, the presence of magnetic null points is likely to result in a discon tinuity of\nthe reconstructed flow, but they are not the only source of trou bles. More generally, our\nformalismbecomes ill-defined at thepoints, where themagnetic field bis parallel to ∇×e.\nIf a magnetic force line crosses the boundary at two points, a prob lem arises in satisfying\nthe boundary conditions for the flow at the two points.\n3. Axisymmetric magnetic neutral modes\nEquations (12)-(14) suggest that the complexity of the problem d epends considerably on\nthe geometry of magnetic force lines. For instance, reconstruct ion of the flow is difficult,\nif force lines exhibit a chaotic spatial behavior. We consider here one of the simplest\nexamples of an axisymmetric magnetic neutral mode\nb=b(ρ,z)iϕ,iB=iϕ,\n(ρ,ϕ,z) being a cylindrical coordinate system and iρ,iϕ,izthe respective unit vectors.\nBefore we formulate the system of equations (12)-(14) in the var iablesAandC, which\nwe need to solve in order to reconstruct the flow (6), we derive som e useful properties of\nthe basis iA,iB,iC. Curls of azimuthal and poloidal vector fields independent of ϕare,\nrespectively, poloidal and azimuthal; this implies the orthogonality\niA·(∇×iA) =iB·(∇×iB) =iB·(∇×e) =iC·(∇×iC) = 0. (15)\nBy a simple calculation,\niC=∇×e\n|∇×e|=h/parenleftBigg\n−∂κ\n∂ziρ+∂κ\n∂ρiz/parenrightBigg\n,\n4where\nκ(ρ,z)≡ρ\nb, h(ρ,z)≡1\n|∇κ|;\nhence\niA≡iB×iC=h∇κ.\nTherefore,\niC·(∇×iA) =iC·(∇h×∇κ) = 0,\nsince none of the factors in the triple product has an azimuthal com ponent. By vector\nalgebra identities,\niA·(∇×iC)−iC·(∇×iA) =−∇·(iA×iC) =∇·iϕ= 0,\nimplying\niA·(∇×iC) =iC·(∇×iA) = 0. (16)\nNow, scalar multiplying (11)by iA,iBandiCandemploying (15)(inparticular, B= 0)\nand (16), one obtains equations\n0 =∂C\n∂ϕ, (17)\n0 =AiB·(∇×iA)+(iC·∇)A+CiB·(∇×iC)−(iA·∇)C, (18)\n0 =∂A\n∂ϕ(19)\n(which are now significantly simpler than (12)-(14) in the general ca se). Equations (17)\nand (19) are equivalent to\nC=C(ρ,z), (20)\nA=A(ρ,z). (21)\nFor an axisymmetric magnetic field, (10) takes the form\n∂α\n∂ϕ=−|∇×e|C−∇·(e×(∇×b)).\nConsequently, (20) and geometric consistency (2 π-periodicity of αinϕ) imply that\nα=α(ρ,z) is an arbitrary function (together with the relations (20), (21) a ndB= 0,\nthis formally confirms a physically obvious fact, that a flow generatin g an axisymmetric\nmagnetic field is necessarily axisymmetric), and (20) is superceded b y\nC=−∇·(e×(∇×b))\n|∇×e|. (22)\nNowAmust be determined from (18). We introduce characteristics ( R(s),Z(s)) in\nthe (ρ,z) half-plane; they satisfy the ODE’s\ndR\nds=−h(R(s),Z(s))∂κ\n∂z(R(s),Z(s)),\ndZ\nds=h(R(s),Z(s))∂κ\n∂ρ(R(s),Z(s)).\n5Direct differentiation shows that the characteristics are isolines of the scalar field b(ρ,z).\nSince along a characteristic\niB·(∇×iA) =∂h\n∂z∂κ\n∂ρ−∂h\n∂ρ∂κ\n∂z=1\nh/parenleftBigg∂h\n∂zdZ\nds+∂h\n∂ρdR\nds/parenrightBigg\n=1\nhdh\nds,\n(18) takes the form\nd\nds(Ah) =f,\nwhere\nf≡h((iA·∇)C−CiB·(∇×iC))\nandCis given by (22). Consequently,\nA(R(s),Z(s)) =A(R(0),Z(0))h(R(0),Z(0))+/integraltexts\n0f(R(s′),Z(s′))ds′\nh(R(s),Z(s)).(23)\nIf a characteristic is a closed orbit of period S, geometric consistency implies that over\nthis orbit/integraldisplayS\n0f(R(s′),Z(s′))ds′= 0. (24)\nThus, we have determined ∇aand the flow (6) (to the extent this is permitted by the\nnatural non-uniqueness of solutions to (1) in u).\nThe well-known Cowling antidynamo theorem states that generation of smooth ax-\nisymmetricmagneticfields(includingsteadyones)offinitetotalene rgyisimpossible. Two\nproofs of the theorem (following [7] and [3]) are presented in [11]. The demonstrations\nrely on the equation of total magnetic energy balance derived for a smooth axisymmetric\nflow of incompressible fluid, provided the normal component of veloc ity vanishes on the\nboundary of the region where the fluid resides. To reconcile our res ults with the Cowling\ntheorem, we note that the flow that we obtain will not satisfy some o f these conditions. It\nmay be singular on the circles, where b= 0, orκhas extrema (and then eorhare singu-\nlar, respectively). If the volume occupied by the flow is bounded, it c annot be guaranteed\nthat the normal component of the fluid velocity vanishes everywhe re on the boundary\n(or, alternatively, enforcing this condition creates a discontinuity in the flow). Hence,\nthe standard procedure employed to establish the total magnetic energy balance equation\nwill reveal additional sources of magnetic energy, which emerge be cause the flow is not\nsmooth or the surface integral representing the contribution of the advective term does\nnot vanish; under such circumstances the Cowling theorem is unapp licable.\nWe have presented the analysis of this section mainly as an illustration of how the\nproposed formalism might be applied to reconstruct flows for less tr ivial magnetic field\nconfigurations. However, in addition, it provides useful informatio n in regard to the\nfollowing technical issue: Although we have stated at the end of the previous section that\n(13) is a constraint for a neutral magnetic mode, we have not yet p roduced any evidence,\nthatthethreeequations(12)-(14)areindependent. Eqns.(17 )-(19),whichwehavederived\nconsidering this particular example, demonstrate that (13) is not a consequence of (12),\n(14) and solenoidality of magnetic field.\n64. Concluding remarks\nWe have shown in Section 2 that reconstruction of an incompressible flow (6) from the\nstructure of a magnetic mode consists of solution of equations (12 ) and (14) in Aand\nC, followed by solution of (10) in α. These equations are ordinary differential equations\nalong magnetic force lines; thus, the problem becomes complex, if th e force lines exhibit\na chaotic behavior. For a solenoidal vector field to be a neutral mag netic mode, it must\nsatisfy the constraint (13).\nSubstitution of (6) into the momentum equation\nν∇2u+u×(∇×u)−b×(∇×b)−∇p+F= 0\nyields an equation in b:\nν∇2((1+ηα)b+ηe×(∇×b+∇a))\n+η((1+ηα)b+ηe×(∇×b+∇a))×(∇×(αb+e×(∇×b+∇a)))\n+η(αb+e×(∇×b+∇a))×(∇×b)−∇p+F= 0 (25)\ncomprising a closed system of equations together with the solenoida lity condition (2).\nRelation (13) now becomes a constraint on the acceptable fluid forc ingF.\nAnalysis of the dependence of steady or evolving magnetohydrody namic systems on\nsmall viscosity and magnetic diffusivity is a notoriously difficult problem. The structure\nof (25) may turn out to be advantageous for the study of asympt otics of MHD steady\nstates, when the force Fis of the order of small quantities ν∼η, as it is in nonlinear\ndynamos with energy equipartition [5, 6]. (The form of the scalar fac tor in front of b\nin (6) has been chosen so that all terms in (25) were in this case of th e same order of\nsmallness.)\nIn Section 3 we have considered an example of the reconstruction p roblem for axisym-\nmetric neutral magnetic modes. This particular case has proved to be highly degenerate:\nthe denominators in (12) and (14) vanish identically, and the respec tive components of\n(11) just testify that ∇ais an axisymmetric vector field. Relation (13) does not constrain\nfurther the structure of the magnetic field, but rather defines, by (23), the component A\nof∇a. Initial conditions A(R(0),Z(0)) for solutions (23) of (18) along characteristics can\nbe chosen on curves in the ( ρ,z) half-plane, which are transversal to magnetic force lines.\nThe azimuthal component of the flow velocity, (1 + ηα)b, is an arbitrary axisymmetric\nscalar field (in this case it is controlled neither by the magnetic inductio n equation, nor,\ndue to independence of ϕ, by the solenoidality condition). Thus the reconstructed flow is\nunique up to the data which must be specified on two-dimensional man ifold(s) (the scalar\nfieldαon the (ρ,z) half-plane) and on one-dimensional curve(s) on this half-plane (t he\ninitial conditions A(R(0),Z(0))).\nTheinitialdatamustbesmoothsothattheresultantfield Ahadnosingularities. Ifthe\ntopology of isolines of the magnitude of magnetic field bis non-trivial, the smoothness of\ntheinitial dataisinsufficient; forinstance, geometric consistency r equires that theintegral\n(24) over any closed magnetic force line vanishes. If the axis of sym metry intersects with\nthe volume occupied by the fluid, axisymmetry gives rise to another p roblem: regularity\nof the magnetic field implies b(0,z) = 0; consequently, the term e×(∇×b) in (6) tends\nto infinity for ρ→0. Thus, the flow is non-singular only, if initial conditions for A\ncompensate for this singularity.\n7Acknowledgments\nPart of this research was carried out during my visit to the School o f Engineering,\nComputer Science and Mathematics, University of Exeter, UK, in Ja nuary – April 2008.\nI am grateful to the Royal Society for their financial support. My research visits to\nObservatoire de la Cˆ ote d’Azur were supported by the French Minis try of Education. My\nresearch was partially financed by the grants ANR-07-BLAN-0235 OTARIE from Agence\nnationaledelarecherche, France, and07-01-92217-CNRSL afromtheRussianfoundation\nfor basic research. I am grateful to Andrew Gilbert for discussion s.\nReferences\n[1] Biskamp D. Nonlinear magnetohydrodynamics. Cambridge Univ. Pr ess (1997), 396\npp.\n[2] Biskamp D. Magnetic reconnection in plasmas. Cambridge Univ. Pre ss (2000), 380\npp.\n[3] Braginsky S.I. Self excitation of a magnetic field during the motion o f a highly con-\nducting fluid. Sov. Phys. JETP, 20 (1965), 726–735.\n[4] Brown D.S., Priest E.R. The topological behaviour of 3D null points in the Sun’s\ncorona. Astronomy & Astrophysics 367 (2001), 339–346.\n[5] Cameron R., Galloway D. Saturation properties of the Archontis d ynamo. Mon. Not.\nR. Astron. Soc. 365 (2006), 735–746.\n[6] Cameron R., Galloway D. High field strength modified ABC and rotor d ynamos.\nMon. Not. R. Astron. Soc. 367 (2006), 1163–1169.\n[7] Cowling T.G. The dynamo maintenance of steady magnetic fields. Qu art. J. Mech.\nApp. Math., vol. X (1957), 129–136.\n[8] Dorch S.B.F., Archontis V. On the saturation of astrophysical dy namos: numerical\nexperiments with the no-cosines flow. Solar Physics, 224 (2004), 1 71–178.\n[9] Lau Y.-T., Finn J.M. Three-dimensional kinematic reconnection in th e presence of\nfield nulls and closed field lines. Astrophys. J., 350 (1990) 672–691.\n[10] Lau Y.-T. Magnetic nulls and topology in a class of solar flare models . Solar Physics,\n148 (1993), 301–324.\n[11] Moffatt H.K. Magnetic field generation in electrically conducting flu ids. Cambridge\nUniv. Press (1978). 343 pp.\n[12] Parker E.N. Cosmical magnetic fields: Their origin and their activit y. Clarendon\nPress (1979). 841 pp.\n[13] Podvigina O.M. A route to magnetic field reversals: an example of a n ABC-forced\nnon-linear dynamo. Geophys. Astrophys. Fluid Dyn. 97 (2003), 14 9–174.\n8[14] Podvigina O.M. Magnetic field generation by convective flows in a pla ne layer. Eur.\nPhys. J. B, 50 (2006), 639–652.\n[15] Podvigina O.M. Magnetic field generation by convective flows in a pla ne layer: the\ndependence on the Prandtl number. Geophys. Astrophys. Fluid D yn. 102 (2008),\n409–433.\n[16] Priest E.R. Solar magneto-hydrodynamics. D.Reidel Publ. Com., D ordrecht (1984).\n469 pp.\n[17] Priest E.R., Forbes T. Magnetic reconnection. MHD theory and a pplications. Cam-\nbridge Univ. Press (2000). 600 pp.\n[18] Reconnection of magnetic fields. Magnetohydrodynamics and c ollisionless theory and\nobservations. Eds. J. Birn, E.R. Priest. Cambridge Univ. Press (20 07), 342 pp.\n[19] Ruzmaikin A.A., Shukurov A.M., Sokoloff D.D. Magnetic fields of galaxie s. Kluwer\nAcademic (1988). 313 pp.\n[20] Zeldovich Ya.B., Ruzmaikin A.A., Sokoloff, D.D. Magnetic fields in astro physics.\nGordon and Breach, New York (1990). 382 pp.\n[21] Zheligovsky V.A., Podvigina O.M., Frisch U. Dynamo effect in parity-in variant flow\nwith large and moderate separation of scales. Geophys. Astrophy s. Fluid Dyn. 95\n(2001), 227–268 [http://xxx.lanl.gov/abs/nlin.CD/0012005].\n[22] Zheligovsky V.A. Convective plan-form two-scale dynamos in a pla ne layer. Geophys.\nAstrophys. Fluid Dyn. 99 (2005), 151–175 [http://arxiv.org/abs/ physics/0405045].\n[23] Zheligovsky V.A. Mean-field equations for weakly nonlinear two-s cale perturbations\nof forced hydromagnetic convection in a rotating layer. Geophys. Astrophys. Fluid\nDyn. 102 (2008), 489–540 [http://arxiv.org/abs/0804.2326v1].\n[24] Zheligovsky V. Amplitude equations for weakly nonlinear two-sca le perturbations of\nfree hydromagnetic convective regimes in a rotating layer. Geophy s. Astrophys. Fluid\nDyn. (2009a), in print [http://arxiv.org/abs/0809.1195v2].\n[25] Zheligovsky V. Generation of a symmetric magnetic field by therm al con-\nvection in a plane rotating layer. Eur. Phys. J. B (2009b), submitte d\n[http://arxiv.org/abs/0906.5380v1].\n9"    },    {        "title": "0805.3922v2.Magnetic_field_induced_incommensurate_resonance_in_cuprate_superconductors.pdf",        "content": "arXiv:0805.3922v2  [cond-mat.supr-con]  1 Sep 2008Magnetic field induced incommensurate resonance in cuprate superconductors\nJingge Zhang and Li Cheng\nDepartment of Physics, Beijing Normal University, Beijing 100875, China\nHuaiming Guo\nDepartment of Physics, Capital Normal University, Beijing 100037, China\nShiping Feng∗\nDepartment of Physics, Beijing Normal University, Beijing 100875, China\nThe influence of a uniform external magnetic field on the dynam ical spin response of cuprate\nsuperconductors in the superconducting state is studied ba sed on the kinetic energy driven su-\nperconducting mechanism. It is shown that the magnetic scat tering around low and intermediate\nenergies is dramatically changed with a modest external mag netic field. With increasing the external\nmagnetic field, although the incommensurate magnetic scatt ering from both low and high energies\nis rather robust, the commensurate magnetic resonance scat tering peak is broadened. The part of\nthe spin excitation dispersion seems to be an hourglass-lik e dispersion, which breaks down at the\nheavily low energy regime. The theory also predicts that the commensurate resonance scattering at\nzero external magnetic field is induced into the incommensur ate resonance scattering by applying\nan external magnetic field large enough.\nPACS numbers: 74.25.Ha, 74.25.Nf, 74.20.Mn\nI. INTRODUCTION\nThe intimate relationship between the short-range an-\ntiferromagnetic (AF) correlation and superconductiv-\nity is one of the most striking features of cuprate\nsuperconductors1. This is followed an experimental\nfact that the parent compounds of cuprate supercon-\nductors are Mott insulators with the AF long-range or-\nder (AFLRO)1. However, when holes or electrons are\ndoped into these Mott insulators2, the ground state of\nthe systems is fundamentally altered from a Mott in-\nsulator with AFLRO to a superconductor with persis-\ntent short-range correlations1,3. The evidence for this\nclosed link is provided from the inelastic neutron scatter-\ning (INS) experiments4,5,6,7,8,9that show the unambigu-\nous presence of the short-rangeAF correlationin cuprate\nsuperconductors in the superconducting (SC) state.\nAt zero external magnetic field, the dynamical spin\nresponse of cuprate superconductors exhibits a number\nof universal features4,5,6,7,8,9, where the magnetic exci-\ntations form an hourglass-like dispersion centered at the\nAF ordering wave vector Q= [π,π] (in units of inverse\nlattice constant). At the saddle point, the dispersing\nincommensurate (IC) branches merge into a sharp com-\nmensurate feature, which is dramatically enhanced upon\nentering the SC state and commonly referred as the mag-\nnetic resonance scattering4,5,6,7,8,9. In particular, it has\nbeen argued that this commensurate magnetic resonance\nplays a crucial role for the SC mechanism in cuprate su-\nperconductors, since the commensurate magnetic reso-\nnance with the magnetic resonance energy scales with\nthe SC transition temperature forming a universal plot\nfor all cuprate superconductors10. To test the connec-\ntion between the commensurate magnetic resonance phe-\nnomenon and SC mechanism, it is desirable to perform\nfurther characterization. Since a uniform external mag-\nnetic field can serve as a weak perturbation helping to\nprobe the nature of the short-range AF correlation andsuperconductivity, therefore the dynamical spin response\nofcupratesuperconductorsintheSCstatehasbeenstud-\nied experimentally by application of a uniform external\nmagnetic field11,12,13,14,15. However, there is no a general\nconsensus. Some experimental results show that apply-\ning a uniform external magnetic field enhances the am-\nplitude of the IC magnetic scattering already present in\nthe system11,12. On the other hand, other experiments\nindicate that the intensity gain of the IC magnetic scat-\ntering is suppressed by application of a uniform external\nmagnetic field13. In particular, the influence of a uniform\nexternal magnetic field has been investigated on the res-\nonance scattering peak by using INS technique14,15. The\nearly INS measurement14shows that under a modest ex-\nternal magnetic field ( ∼11 Tesla), the resonance scat-\ntering peak remains almost unaffected, i.e., although a\nline broadening occurs without change of the resonance\nscattering peak amplitude, no shifting of the resonance\nscattering peak energy is observed. However, the later\nINS experiments15show that a modest external mag-\nnetic field applied to cuprate superconductors in the SC\nstate yields a very significant reduction in the commen-\nsurate magnetic resonance scattering. To the best of our\nknowledge, there are no explicit microscopic predictions\naboutthe effect ofauniformexternalmagneticfield large\nenough on the magnetic resonance scattering.\nForthecaseofzeroexternalmagneticfield, thedynam-\nical spin response of cuprate superconductors has been\ndiscussed16based on the framework of the kinetic energy\ndriven SC mechanism17, and all main features of the INS\nexperiments are reproduced, including the doping and\nenergy dependence of the IC magnetic scattering at both\nlow and high energies and commensurate magnetic reso-\nnance at intermediate energy4,5,6,7,8,9. In this paper, we\nstudy the influence of a uniform external magnetic field\non the dynamical spin response of cuprate superconduc-\ntors in the SC state along with this line. We calculate\nexplicitly the dynamical spin structure factor of cuprate2\nsuperconductorsunder auniformexternalmagneticfield,\nandshowthatthemagneticscatteringaroundlowandin-\ntermediate energies is dramatically changed with a mod-\nest external magnetic field. With increasing the external\nmagnetic field, although the IC magnetic scattering from\nboth low and high energies is rather robust, the commen-\nsurate magnetic resonance scattering peak is broadened.\nThe part of the spin excitation dispersion seems to be\nan hourglass-like dispersion, which breaks down at the\nheavily low energy regime.\nThe rest of this paper is organized as follows. The ba-\nsic formalism is presented in Sec. II, where we general-\nize the calculation of the dynamical spin structure factor\nfrom the previous zero external magnetic field case16to\nthe present case with a uniform external magnetic field.\nWithin this theoretical framework, we discuss the influ-\nence of a uniform external magnetic field on the dynam-\nical spin response of cuprate superconductors in the SC\nstateinSec. III,wherewepredictthatthecommensurate\nmagnetic resonance scattering at zero external magnetic\nfield is induced into the IC magnetic resonance scatter-\ning by an applied external magnetic field large enough.\nFinally, we give a summary and discussions in Sec. IV.\nII. THEORETICAL FRAMEWORK\nIn cuprate superconductors, the characteristic feature\nis the presence of the CuO 2plane1,3. It has been shown\nfrom ARPES experiments that the essential physics of\nthe doped CuO 2plane is properly accounted by the t-J\nmodel on a square lattice3,18. However, for discussions of\nthe influence of a uniform external magnetic field on the\ndynamical spin response of cuprate superconductors in\ntheSCstate, the t-Jmodelcanbe expressedbyincluding\nthe Zeeman term as,\nH=−t/summationdisplay\niˆησC†\niσCi+ˆησ+t′/summationdisplay\niˆτσC†\niσCi+ˆτσ+µ/summationdisplay\niσC†\niσCiσ\n+J/summationdisplay\niˆηSi·Si+ˆη−εB/summationdisplay\niσσC†\niσCiσ, (1)\nwhere ˆη=±ˆx,±ˆy, ˆτ=±ˆx±ˆy,C†\niσ(Ciσ) is the elec-\ntron creation (annihilation) operator, Si= (Sx\ni,Sy\ni,Sz\ni)\nare spin operators, µis the chemical potential, and εB=\ngµBBis the Zeeman magnetic energy, with the Lande\nfactorg, Bohr magneton µB, and a uniform external\nmagneticfield B. Thist-Jmodelwith auniformexternal\nmagnetic field is subject to an important local constraint/summationtext\nσC†\niσCiσ≤1 to avoid the double occupancy19. The\nstrong electron correlation in the t-Jmodel manifests it-\nself by this local constraint19, which can be treated prop-\nerly in analytical calculations within the charge-spin sep-\naration (CSS) fermion-spin theory20,21, where the con-\nstrainedelectron operatorsaredecoupled as Ci↑=h†\ni↑S−\ni\nandCi↓=h†\ni↓S+\ni, with the spinful fermion operator\nhiσ=e−iΦiσhirepresents the charge degree of freedom\ntogether with some effects of spin configuration rear-\nrangements due to the presence of the doped hole itself\n(chargecarrier), while the spin operator Sirepresentsthe\nspin degree of freedom (spin), then the t-Jmodel with auniform external magnetic field (1) can be expressed in\nthis CSS fermion-spin representation as,\nH=−t/summationdisplay\niˆη(hi↑S+\nih†\ni+ˆη↑S−\ni+ˆη+hi↓S−\nih†\ni+ˆη↓S+\ni+ˆη)\n+t′/summationdisplay\niˆτ(hi↑S+\nih†\ni+ˆτ↑S−\ni+ˆτ+hi↓S−\nih†\ni+ˆτ↓S+\ni+ˆτ)\n−µ/summationdisplay\niσh†\niσhiσ+Jeff/summationdisplay\niˆηSi·Si+ˆη−2εB/summationdisplay\niSz\ni,(2)\nwithJeff= (1−x)2J, andx=/an}bracketle{th†\niσhiσ/an}bracketri}ht=/an}bracketle{th†\nihi/an}bracketri}htis\nthe hole doping concentration. It has been shown that\nthe electron local constraint for the single occupancy is\nsatisfied in analytical calculations in this CSS fermion-\nspin theory20,21.\nWithin the framework of the CSS fermion-spin\ntheory20,21, the kinetic energy driven superconductivity\nhas been developed17. It has been shown that the inter-\naction from the kinetic energy term in the t-Jmodel (2)\nis quite strong, and can induce the d-wave charge car-\nrier pairing state by exchanging spin excitations in the\nhigher power of the doping concentration, then the d-\nwave electron Cooper pairs originating from the d-wave\ncharge carrier pairing state are due to the charge-spin re-\ncombination, and their condensation reveals the d-wave\nSC ground-state. Moreover, this SC-state is controlled\nby both d-wave SC gap function and quasiparticle co-\nherence, which leads to that the SC transition tempera-\nture increases with increasing doping in the underdoped\nregime, and reaches a maximum in the optimal doping,\nthen decreases in the overdoped regime16. Furthermore,\nfor the case of zero external magnetic field, the dop-\ning and energy dependent dynamical spin response of\ncuprate superconductors in the SC-state has been dis-\ncussed in terms of the collective mode in the charge car-\nrier particle-particle channel16, and the results are in\nqualitative agreement with the INS experimental data\non cuprate superconductorsin the SC state4,5,6,7,8,9. Fol-\nlowing their discussions16, the full spin Green’s function\nin the presence of a uniform external magnetic field is\nobtained as,\nD(k,ω) =1\nD(0)−1(k,ω)−Σ(s)(k,ω), (3)\nwith the mean-field (MF) spin Green’s function,\nD(0)(k,ω) =Bk\n2ωk/parenleftBigg\n1\nω−ω(1)\nk−1\nω+ω(2)\nk/parenrightBigg\n=/summationdisplay\nν=1,2(−1)ν+1Bk\n2ωk1\nω−ω(ν)\nk,(4)\nwhereBk= 2λ1(A1γk−A2)−λ2(2χz\n2γ′\nk−χ2),λ1=\n2ZJeff,λ2= 4Zφ2t′,γk= (1/Z)/summationtext\nˆηeik·ˆη,γ′\nk=\n(1/Z)/summationtext\nˆτeik·ˆτ,Zis the number of the nearest neigh-\nbor or next nearest neighbor sites of a square lattice,\nA1=ǫχz\n1+χ1/2,A2=χz\n1+ǫχ1/2,ǫ= 1+2tφ1/Jeff, the\nchargecarrier’sparticle-holeparameters φ1=/an}bracketle{th†\niσhi+ˆησ/an}bracketri}ht\nandφ2=/an}bracketle{th†\niσhi+ˆτσ/an}bracketri}ht, and the spin correlation functions\nχ1=/an}bracketle{tS+\niS−\ni+ˆη/an}bracketri}ht,χ2=/an}bracketle{tS+\niS−\ni+ˆτ/an}bracketri}ht,χz\n1=/an}bracketle{tSz\niSz\ni+ˆη/an}bracketri}ht,χz\n2=3\n/an}bracketle{tSz\niSz\ni+ˆτ/an}bracketri}ht, and the MF charge carrier excitation spec-\ntrum,ξk=Ztχ1γk−Zt′χ2γ′\nk−µ. Since a uniform exter-\nnal magnetic field is applied to the system, the MF spin\nexcitation spectrum has two branches, ω(1)\nk=ωk+2εB\nandω(2)\nk=ωk−2εB, withωkis the MF spin excitation\nspectrum at zero external magnetic field, and has been\nevaluated as16,\nω2\nk=λ2\n1[(A4−αǫχz\n1γk−1\n2Zαǫχ1)(1−ǫγk)\n+1\n2ǫ(A3−1\n2αχz\n1−αχ1γk)(ǫ−γk)]\n+λ2\n2[α(χz\n2γ′\nk−3\n2Zχ2)γ′\nk+1\n2(A5−1\n2αχz\n2)]\n+λ1λ2[αχz\n1(1−ǫγk)γ′\np−1\n2αǫ(C3−χ2γk)\n+1\n2α(χ1γ′\nk−C3)(ǫ−γp)+αγ′\nk(Cz\n3−ǫχz\n2γk)],(5)whereA3=αC1+(1−α)/(2Z),A4=αCz\n1+(1−α)/(4Z),\nA5=αC2+ (1−α)/(2Z), and the spin corre-\nlation functions C1= (1/Z2)/summationtext\nˆη,ˆη′/an}bracketle{tS+\ni+ˆηS−\ni+ˆη′/an}bracketri}ht,Cz\n1=\n(1/Z2)/summationtext\nˆη,ˆη′/an}bracketle{tSz\ni+ˆηSz\ni+ˆη′/an}bracketri}ht,C2= (1/Z2)/summationtext\nˆτ,ˆτ′/an}bracketle{tS+\ni+ˆτS−\ni+ˆτ′/an}bracketri}ht,\nandC3= (1 /Z)/summationtext\nˆτ/an}bracketle{tS+\ni+ˆηS−\ni+ˆτ/an}bracketri}ht,Cz\n3=\n(1/Z)/summationtext\nˆτ/an}bracketle{tSz\ni+ˆηSz\ni+ˆτ/an}bracketri}ht. In order to satisfy the sum\nrule of the correlation function /an}bracketle{tS+\niS−\ni/an}bracketri}ht= 1/2 in\nthe case without AFLRO, the important decoupling\nparameter αhas been introduced in the MF calculation,\nwhich can be regarded as the vertex correction22. The\nspin self-energy function Σ(s)(k,ω) in the SC-state is\nobtained from the charge carrier bubble in the charge\ncarrier particle-particle channel as16,\nΣ(s)(k,ω) =1\nN2/summationdisplay\np,q,ν=1,2(−1)ν+1Λ(q,p,k)Bq+k\nωq+kZ2\nhF\n8¯∆(d)\nhZ(p)¯∆(d)\nhZ(p+q)\nEpEp+q/parenleftBigg\nF(ν)\n1(k,p,q\nω−(Ep−Ep+q+ω(ν)\nq+k)\n+F(ν)\n2(k,p,q)\nω−(Ep+q−Ep+ω(ν)\nq+k)+F(ν)\n3(k,p,q)\nω−(Ep+Ep+q+ω(ν)\nq+k)−F(ν)\n4(k,p,q)\nω+(Ep+q+Ep−ω(ν)\nq+k)/parenrightBigg\n,(6)\nwhere Λ( q,p,k) = (Ztγk−p−Zt′γ′\nk−p)2+(Ztγq+p+k−\nZt′γ′\nq+p+k)2,¯∆hZ(k) =ZhF¯∆h(k), the charge carrier\nquasiparticle spectrum Ehk=/radicalBig\n¯ξ2\nk+|¯∆hZ(k)|2,\n¯ξk=ZhFξk,¯∆h(k) =¯∆hγ(d)\nkis the effective\ncharge carrier gap function in the d-wave symme-\ntry with γ(d)\nk= (coskx−cosky)/2,F(ν)\n1(k,p,q) =\nnB(ω(ν)\nq+k)[nF(Ep)−nF(Ep+q)]−nF(−Ep)nF(Ep+q),\nF(ν)\n2(k,p,q) = nB(ω(ν)\nq+k)[nF(Ep+q)−nF(Ep)]−nF(Ep)nF(−Ep+q),F(ν)\n3(k,p,q) =\nnB(ω(ν)\nq+k)[nF(−Ep)−nF(Ep+q)]+nF(−Ep)nF(−Ep+q),\nF(ν)\n4(k,p,q) =nB(ω(ν)\nq+k)[nF(−Ep)−nF(Ep+q)]−\nnF(Ep)nF(Ep+q), while the charge carrier quasiparticle\ncoherent weight ZhFand effective charge carrier gap\nparameter ¯∆hare determined by the following two\nself-consistent equations16,\n1 =1\nN3/summationdisplay\nk,p,q[Ztγk+q−Zt′γ′\nk+q]2γ(d)\nk−p+qγ(d)\nkZ2\nhF\n2EhkBpBq\nωpωq/parenleftbiggL1(k,p,q)\n(ωp−ωq)2−E2\nhk−L2(k,p,q)\n(ωp+ωq)2−E2\nhk/parenrightbigg\n,(7a)\n1\nZhF= 1+1\nN2/summationdisplay\np,q(Ztγp+k0−Zt′γ′\np+k0)2ZhFBpBq\n4ωpωq/parenleftbiggR1(p,q)\n(ωp−ωq−Ehp−q+k0)2+R2(p,q)\n(ωp−ωq+Ehp−q+k0)2\n+R3(p,q)\n(ωp+ωq−Ehp−q+k0)2+R4(p,q)\n(ωp+ωq+Ehp−q+k0)2/parenrightbigg\n, (7b)\nwherek0= [π,0],L1(k,p,q) = (ωp−ωq)[nB(ω(1)\nq)−\nnB(ω(1)\np) +nB(ω(2)\nq)−nB(ω(2)\np)][1−2nF(Ehk)] +\nEhk[nB(ω(1)\np)nB(−ω(1)\nq) + nB(ω(1)\nq)nB(−ω(1)\np) +\nnB(ω(2)\np)nB(−ω(2)\nq) + nB(ω(2)\nq)nB(−ω(2)\np)],\nL2(k,p,q) = ( ωp+ωq)[nB(−ω(1)\np)−nB(ω(1)\nq) +nB(−ω(2)\np)−nB(ω(2)\nq)][1−2nF(Ehk)] +\nEhk[nB(ω(1)\np)nB(ω(2)\nq) + nB(−ω(1)\np)nB(−ω(2)\nq) +\nnB(ω(2)\np)nB(ω(1)\nq) + nB(−ω(2)\np)nB(−ω(1)\nq)],\nR1(p,q) = nF(Ehp−q+k0){U2\nhp−q+k0[nB(ω(1)\nq)−\nnB(ω(1)\np)] + V2\nhp−q+k0[nB(ω(2)\nq) −4\nnB(ω(2)\np)]} − U2\nhp−q+k0nB(ω(1)\np)nB(−ω(1)\nq)−\nV2\nhp−q+k0nB(ω(2)\np)nB(−ω(2)\nq), R2(p,q) =\nnF(Ehp−q+k0){U2\nhp−q+k0[nB(ω(2)\np) −\nnB(ω(2)\nq)] + V2\nhp−q+k0[nB(ω(1)\np) −\nnB(ω(1)\nq)]} − U2\nhp−q+k0nB(ω(2)\nq)nB(−ω(2)\np)−\nV2\nhp−q+k0nB(ω(1)\nq)nB(−ω(1)\np), R3(p,q) =\nnF(Ehp−q+k0){U2\nhp−q+k0[nB(ω(2)\nq) −\nnB(−ω(1)\np)] + V2\nhp−q+k0[nB(ω(1)\nq)−\nnB(−ω(2)\np)]}+U2\nhp−q+k0nB(ω(1)\np)nB(ω(2)\nq) +\nV2\nhp−q+k0nB(ω(2)\np)nB(ω(1)\nq), R4(p,q) =\nnF(Ehp−q+k0){U2\nhp−q+k0[nB(−ω(1)\nq) −\nnB(ω(2)\np)] + V2\nhp−q+k0[nB(−ω(2)\nq)−\nnB(ω(1)\np)]}+U2\nhp−q+k0nB(−ω(2)\np)nB(−ω(1)\nq) +\nV2\nhp−q+k0nB(−ω(1)\np)nB(−ω(2)\nq), with U2\nhp−q+k0=(1 +¯ξp−q+k0/Ehp−q+k0)/2,V2\nhp−q+k0= (1 −\n¯ξp−q+k0/Ehp−q+k0)/2, and nB(ω) and nF(ω) are\nthe boson and fermion distribution functions, respec-\ntively. These two equations (7a) and (7b) must be solved\nsimultaneously with other self-consistent equations, then\nall order parameters, decoupling parameter α, and chem-\nical potential µare determined by the self-consistent\ncalculation16. In this sense, our above self-consistent\ncalculation for the dynamical spin structure factor\nunder a uniform external magnetic field is controllable\nwithout using adjustable parameters, which also has\nbeen confirmed by a similar self-consistent calculation\nfor the dynamical spin structure factor in the case\nwithout a uniform external magnetic field16.\nWith the help of the full spin Green’s function (3), we\ncan obtain the dynamical spin structure factor of cuprate\nsuperconductors under a uniform external magnetic field\nin the SC-state as,\nS(k,ω) =−2[1+nB(ω)]ImD(k,ω) =−2[1+nB(ω)]B2\nkImΣ(s)(k,ω)\n[(ω−2εB)2−ω2\nk−BkReΣ(s)(k,ω)]2+[BkImΣ(s)(k,ω)]2,(8)\nwhere ImΣ(s)(k,ω) and ReΣ(s)(k,ω) are the imaginary\nand realparts of the spin self-energyfunction (6), respec-\ntively.\nIII. MAGNETIC FIELD INDUCED\nINCOMMENSURATE MAGNETIC RESONANCE\nWe are now ready to discuss the influence of a uniform\nexternal magnetic field on the dynamical spin response\nof cuprate superconductors in the SC state. For cuprate\nsuperconductors, the commonly used parameters in this\npaper are chosen as t/J= 2.5 andt′/t= 0.3 with a\nreasonably estimative value of J∼120 meV23. At zero\nexternal magnetic field ( B= 0), we have reproduced the\nprevious results16. Furthermore, we have also performed\nthe calculation for the dynamical spin structure factor\nS(k,ω) inEq. (8) with auniformexternalmagneticfield,\nand the results of S(k,ω) in the ( kx,ky) plane for dop-\ningx= 0.15 with temperature T= 0.002Jand Zeeman\nmagnetic energy εB= 0.01J= 1.2 meV (then the corre-\nsponding external magnetic field B≈20 Tesla) at energy\n(a)ω= 0.08J= 9.6 meV, (b) ω= 0.31J= 37.2 meV,\nand (c)ω= 0.59J= 70.8 meV are plotted in Fig. 1. In\ncomparison with the previous results without a uniform\nexternal magnetic field16, our present most surprising re-\nsults involve the external magnetic field dependence of\nthe resonance scattering form, i.e., with increasing the\nexternalmagneticfield B, althoughtheICmagneticscat-\ntering from both low and high energies is rather robust,\nthe commensurate magnetic resonance scattering peak\nis broadened, and is shifted from the AF ordering wave\nvectorQto the IC magnetic scattering peaks with the\nincommensurability δr. The main difference is that theresonance response occurs at an IC in the presence of a\nuniform external magnetic field, rather than commensu-\nrate in the case of zero external magnetic field. In this\nsense, we call such magnetic resonance as the IC mag-\nnetic resonance. Experimentally, the growth of the low\nenergy IC magnetic resonance scattering due to the pres-\nenceofanexternalmagneticfield hasbeen observedfrom\nthe cuprate superconductor La 2−xSrxCuO413, which is\nqualitatively consistent with our theoretical predictions.\nFor cuprate superconductors, the upper critical magnetic\nfield at which superconductivity is completely destroyed\nis 50 Tesla or greater around the optimal doping24.\nTherefore the present result is remarkable because the\nmagnitude of the applied external magnetic field is much\nless than the upper critical magnetic field of cuprate su-\nperconductors. It has been shown that the magnetic res-\nonance scattering is very sensitive to the SC pairing, and\nthe external magnetic field induced the IC magnetic res-\nonance scattering is always accompanied with a breaking\nof the SC pairing15, this leads to a reduction of the SC\ntransition temperature in cuprate superconductors.\nHaving shown the presence of the IC magnetic reso-\nnance scatteringunder a uniform external magnetic field,\nit is importantto determineits dispersionas the outcome\nwill allow a direct comparison of the magnetic excitation\nspectra with and without a uniform external magnetic\nfield. In Fig. 2, we plot the evolution of the magnetic\nscattering peaks with energy for x= 0.15 inT= 0.002J\nwithεB= 0.01J= 1.2 meV (B≈20 Tesla) (solid line).\nFor comparison, the corresponding result for x= 0.15 in\nT= 0.002Jwith the same set of parameters except for\nεB= 0 (B= 0) is also shown in Fig. 2 (dashed line).\nAs in the previous work16, the dispersion of the magnetic\nscattering in the case of zero external magnetic field has5\nFIG. 1: The dynamical spin structure factor S(k,ω) in the\n(kx,ky) plane at x= 0.15 with T= 0.002JandεB= 0.01J\nfort/J= 2.5 andt′/t= 0.3 at energy (a) ω= 0.08J, (b)\nω= 0.31J, and (c) ω= 0.59J.\nkx0.40 0.45 0.50 0.55 0.60Energy (J)\n00.00.20.40.60.801.0\nFIG. 2: The evolution of the magnetic scattering peaks with\nenergy at x= 0.15 inT= 0.002Jfort/J= 2.5 andt′/t= 0.3\nwithεB= 0.01J(solid line) and εB= 0 (dashed line).H%00.000 0.002 0.004 0.006 0.008 0.010Gr\n00.0000.0050.0100.0150.020\nFIG. 3: The incommensurability of the incommensurate res-\nonance scattering at x= 0.15 inT= 0.002Jfort/J= 2.5\nandt′/t= 0.3 as a function of the external magnetic field.\nan hourglass shape. However, under a modest external\nmagnetic field B≈20 Tesla, although there is no strong\nexternal magnetic field induced change for the IC mag-\nnetic scattering at higher energy ω∼0.7J, the magnetic\nscattering around both intermediate and low energies is\ndramatically changed, in qualitative agreement with the\nINS experiments13,14,15. In particular, although the part\nabove 0.16J≈19 meV seems to be an hourglass-like\ndispersion, this hourglass-like dispersion breaks down at\nlower energy ω <0.16J≈19 meV. These are much dif-\nferent from the dispersion in the case of zero external\nmagnetic field.\nNow we turn to discuss that how strong external mag-\nnetic field can induce the IC resonance scattering in\ncuprate superconductors in the SC state. We have made\na series of calculations for the resonance energy at dif-\nferent external magnetic fields, and the result of the in-\ncommensurability of the IC resonance scattering δrfor\nx= 0.15 inT= 0.002Jas a function of a uniform ex-\nternal magnetic field Bis plotted in Fig. 3. Obviously,\nthe incommensurability δrincreases with increasing the\nexternal magnetic field. For a better understanding of\nthe influence of a uniform external magnetic field on the\nresonance scattering, we plot the dynamical spin struc-\nture factor S(k,ω) in the ( kx,ky) plane for x= 0.15 and\nT= 0.002Jwith (a) εB= 0.002J= 0.24 meV (then the\ncorresponding external magnetic field B≈4 Tesla) and\n(b)εB= 0.005J= 0.6 meV (then the corresponding ex-\nternal magnetic field B≈10 Tesla) at ω= 0.31J= 37.2\nmeV in Fig. 4. In comparison with Fig. 1(b), we there-\nfore find that there are two critical values of the Zeeman\nmagnetic energy ε(c)\nB1≈0.002J= 0.24meV (the corre-\nsponding critical external magnetic field Bc1≈4 Tesla)\nandε(c)\nB2≈0.005J= 0.6meV (the corresponding critical\nexternal magnetic field Bc2≈10 Tesla). When B > B c2,\nthe external magnetic field is strong enough to induce\nthe IC resonance scattering. On the other hand, when\nBc1< B < B c2, the commensurate resonance scatter-\ning peak is broadened, and remains at the same energy\nposition as the zero external magnetic field case with a\ncomparable amplitude, which is furthermore in qualita-\ntive agreement with the INS experiments14,15.6\nFIG. 4: The dynamical spin structure factor S(k,ω) in the\n(kx,ky) plane in x= 0.15 andT= 0.002Jfort/J= 2.5\nandt′/t= 0.3 with (a) εB= 0.002Jand (b)εB= 0.005Jat\nω= 0.31J.\nThe physical interpretation to the above obtained re-\nsults can be found from the property of the spin excita-\ntion spectrum. In contrast to the case of zero external\nmagnetic field, the MF spin excitation spectrum has two\nbranches, ω(1)\nk=ωk+ 2εBandω(2)\nk=ωk−2εB, in\nEq. (4) under a uniform external magnetic field as men-\ntioned in Sec. II. Since both MF spin excitation spectra\nω(1)\nkandω(2)\nkand spin self-energy function Σ(s)(k,ω) in\nEq. (6) are strong external magnetic field dependent,\nthis leads to that the renormalized spin excitation spec-\ntrum (Ω k−2εB)2=ω2\nk+ ReΣ(s)(k,Ωk) in Eqs. (3)\nand (8) also is strong external magnetic field dependent.\nAs in the case of zero external magnetic field16, the dy-\nnamical spin structure factor S(k,ω) in Eq. (8) under a\nuniform external magnetic field has a well-defined reso-\nnance character, where S(k,ω) exhibits peaks when the\nincoming neutron energy ωis equal to the renormalized\nspin excitation, i.e.,\nW(kc,ω)≡[(ω−2εB)2−ω2\nkc−BkcReΣ(s)(kc,ω)]2\n∼0, (9)\nfor certain critical wave vectors kc=k(L)\ncat low energy,\nkc=k(I)\ncat intermediate energy, and kc=k(H)\ncat high\nenergy, then the weight of these peaks is dominated by\nthe inverse of the imaginary part of the spin self-energy\n1/ImΣ(s)(k(L)\nc,ω) at low energy, 1 /ImΣ(s)(k(I)\nc,ω) at in-\ntermediate energy, and 1 /ImΣ(s)(k(H)\nc,ω) at high en-\nergy, respectively. In this sense, the essential physicsof the external magnetic field dependence of the dynam-\nical spin response is almost the same as in the case of\nzero magnetic field. However, as seen from Eqs. (6),\n(8), and (9), a modest external magnetic field mainly ef-\nfects the behavior of the dynamical spin response around\nlow and intermediate energies, and therefore leads to\nsome changes of the dynamical spin response around\nlow and intermediate energies. This is followed by a\nfact that the magnitude of the applied uniform exter-\nnal magnetic field is much less than the upper critical\nmagnetic field of cuprate superconductors, i.e., the Zee-\nman magnetic energy 2 εB/J= 0.02≪1 in Eqs. (6),\n(8), and (9), then the renormalized spin excitation spec-\ntrum at high energy can be reduced approximately as\n(ω−2εB)2=ω2\nk+ReΣ(s)(k,ω)≈ωin Eqs. (6), (8), and\n(9). This is why there is only a small influence of a mod-\nest external magnetic field on the IC magnetic scattering\nat hight energy. However, around low and intermediate\nenergies, this small Zeeman magnetic energy εBin Eqs.\n(6), (8), and (9) plays an important role that reduces the\nrange of the IC magnetic scattering at low energy and\nsplits the commensurate resonance peak at zero exter-\nnal magnetic field into the IC resonance peaks, then the\nIC magnetic resonance scattering appears. Furthermore,\nat the heavily low energy regime ω≪0.16J, the mag-\nnitude of the Zeeman magnetic energy 2 εB= 0.02Jis\ncomparable with these incoming neutron energies, where\nboth incoming lower neutron energy and Zeeman mag-\nnetic energy dominate the IC magnetic scattering, then\nthe hourglass-like dispersion breaks down.\nIV. SUMMARY AND DISCUSSIONS\nIn summary, we have discussed the influence of a uni-\nform external magnetic field on the dynamical spin re-\nsponse of cuprate superconductors in the SC state based\non the kinetic energy driven SC mechanism. Our results\nshow that the magnetic scattering around low and inter-\nmediate energies is dramatically changed with a mod-\nest external magnetic field. With increasing the ex-\nternal magnetic field, although the IC magnetic scat-\ntering from both low and high energies is rather ro-\nbust, the commensurate magnetic resonance scattering\npeak is broadened14,15. In particular, the part above\n0.16J≈19 meV seems to be an hourglass-like disper-\nsion, which breaksdown at the heavily low energyregime\nω <0.16J≈19 meV. The theory also predicts that the\ncommensurate magnetic resonance scattering at zero ex-\nternalmagneticfieldisinducedintotheICmagneticreso-\nnancescatteringbyapplyingauniformexternalmagnetic\nfield largeenough, which shouldbe verifiedby further ex-\nperiments.\nFrom the INS experimental results, it is shown that\nalthough some of the IC magnetic scattering properties\nhave been observed in the normal state, the magnetic\nresonance scattering is the main new feature that ap-\npears into the SC state4,5,6,7,8,9,10. In particular, apply-\ning a uniform external magnetic field large enough to\nsuppress superconductivity would yield a spectrum iden-\ntical to that measured at normal state25. Incorporating7\nthese experimental results, our present result seems to\nshow that the external magnetic field causes the behav-\nior of the dynamical spin response to become more like\nthat of the normal state. Moreover, in our present dis-\ncussions, the magnitude of an applied external magnetic\nfield is much less than the upper critical magnetic field\nfor cuprate superconductors as mentioned above, and\ntherefore we believe that both commensurate magnetic\nresonance scattering at zero external magnetic field and\nIC magnetic resonance scattering at an applied modest\nexternal magnetic field are universal features of cuprate\nsuperconductors.Acknowledgments\nThe authors would like to thank Professor P. Dai for\nthe helpful discussions. This work was supported by\nthe National Natural Science Foundation of China under\nGrant No. 10774015, and the funds from the Ministry\nof Science and Technology of China under Grant Nos.\n2006CB601002 and 2006CB921300.\n*To whom correspondence should be addressed (E-mail:\nspfeng@bnu.edu.cn).\n1See, e.g., the review, M.A. Kastner, R.J. Birgeneau, G.\nShiran, and Y. Endoh, Rev. Mod. Phys. 70(1998) 897.\n2J. G. Bednorz and K. A. M¨ uller, Z. Phys. B 64(1986) 189\n; Y. Tokura, H. Takagi, and S. Uchida, Nature 337(1989)\n345.\n3See, e.g., the review, A. Damascelli, Z. Hussain, and Z.-X.\nShen, Rev. Mod. Phys. 75(2003) 475.\n4See, e.g., the review, J. M. Tranquada, in Handbook of\nHigh-Temperature Superconductivity , edited by J. R. Schri-\neffer and J. S. Brooks (Springer, New York, 2007), p. 257.\n5P. Dai, H. A. Mook, R. D. Hunt, and F. Do˜ gan, Phys. Rev.\nB63(2001) 54525; M. Arai, T. Nishijima, Y. Endoh, T.\nEgami, S. Tajima, K. Tomimoto, Y. Shiohara, M. Taka-\nhashi, A. Garret, and S. M. Bennington, Phys. Rev. Lett.\n83(1999) 608.\n6K. Yamada, C.H. Lee, K. Kurahashi, J. Wada, S. Waki-\nmoto, S. Ueki, H. Kimura, Y. Endoh, S. Hosoya, G. Shi-\nrane, R. J. Birgeneau, M. Greven, M.A. Kastner, and Y.\nJ. Kim, Phys. Rev. B 57(1998) 6165; S. Wakimoto, H.\nZhang, K. Yamada, I. Swainson, H. Kim, and R.J. Birge-\nneau, Phys.Rev.Lett. 92(2004)217004; N.B.Christensen,\nD.F. McMorrow, H.M. Rønnow, B. Lake, S.M. Hayden, G.\nAeppli, T.G. Perring, M. Mangkorntong, N. Nohara, and\nH. Tagaki, Phys. Rev. Lett. 93(2004) 147002.\n7P. Bourges, B. Keimer, S. Pailh´ es, L.P. Regnault, Y. Sidis,\nand C. Ulrich, Physica C 424(2005) 45; P. Bourges, Y.\nSidis, H.F. Fong, L.P. Regnault, J. Bossy, A. Ivanov, and\nB. Keimer, Science 288(2000) 1234.\n8H. He, Y. Sidis, P. Bourges, G.D. Gu, A. Ivanov, N.\nKoshizuka, B. Liang, C.T. Lin, L.P. Regnault, E. Schoen-\nherr, and B. Keimer, Phys. Rev. Lett. 86(2001) 1610; R.\nJ. Birgeneau, C. Stock, J. M. Tranquada, and K. Yamada,\nJ. Phys. Soc. Jpn. 75(2006) 111003.\n9S.M. Hayden, H.A. Mook, P. Dai, T.G. Perring, and F.\nDo˜ gan, Science 429(2004) 531; V. Hinkov, S. Pailh´ es, P.\nBourges, Y. Sidis, A. Ivanov, A. Kulakov, C.T. Lin, D.P.\nChen, C. Bernhard, and B. Keimer, Nature 430(2004)\n650; C. Stock, W.J. Buyers, R.A. Cowley, P.S. Clegg, R.\nColdea, C.D. Frost, R. Liang, D. Peets, D. Bonn, W.N.\nHardy, and R.J. Birgeneau, Phys. Rev. B 71(2005) 24522.\n10Stephen D. Wilson, Pengcheng Dai, Shiliang Li, Songxue\nChi, H. J. Kang and J. W. Lynn, Nature 442(2006) 59.\n11B. Khaykovich, R. J. Birgeneau, F. C. Chou, R. W. Er-\nwin, M. A. Kastner, S.-H. Lee, Y. S. Lee, P. Smeibidl, P.\nVorderwisch, and S. Wakimoto, Phys. Rev. B 67(2003)\n054501; B. Khaykovich, Y. S. Lee, R. W. Erwin, S.-H.Lee, S. Wakimoto, K. J. Thomas, M. A. Kastner, and R.\nJ. Birgeneau, Phys. Rev. B 66(2002) 014528.\n12B. Lake, H. M. Ronnow, N. B. Christensen, G. Aeppli, K.\nLefmann, D. F. McMorrow, P. Vorderwisch, P. Smeibidl,\nN. Mangkorntong, T. Sasagawa, M. Nohara, H. Takagi and\nT. E. Mason, Nature 415(2002) 299.\n13J. M. Tranquada, C. H. Lee, K. Yamada, Y. S. Lee, L.\nP. Regnault, and H. M. Ronnow, Phys. Rev. B 69(2004)\n174507.\n14P. Bourges, H. Casalta, L.P. Regnault, J. Bossy, P. Burlet,\nC. Vettier, E. Beaugnon, P. Gautier-Picard, R. Tournier,\nPhysica B 234-236 (1997) 830.\n15P. Dai, H. A. Mook, G. Aeppli, S. M. Hayden and F. Doan,\nNature406(2000) 965.\n16Shiping Feng, Tianxing Ma, and Huaiming Guo, Physica\nC436(2006) 14; Shiping Feng, Tianxing Ma, and Xin-\ntian Wu, Phys. Lett. A 352(2006) 438; Shiping Feng\nand Tianxing Ma, in Superconductivity Research Horizons ,\nedited by Eugene H. Peterson (Nova Science Publishers,\nNrw York, 2007) chapter 5, p. 129.\n17Shiping Feng, Phys. Rev. B 68(2003) 184501.\n18C. Kim, P.J. White, Z.X.Shen, T. Tohyama, Y.Shibata, S.\nMaekawa, B.O. Wells, Y.J. Kim, R.J. Birgeneau, and M.A.\nKastner, Phys. Rev.Lett. 80(1998) 4245; B.O. Wells, Z.X.\nShen, A. Matsuura, D.M. King, M.A. Kastner, M. Greven,\nand R.J. Birgeneau, Phys. Rev. Lett. 74(1995) 946.\n19See, e.g., thereview, P.W.Anderson,in Frontiers and Bor-\nderlines in Many Particle Physics , edited by R. A. Broglia\nand J. R. Schrieffer (North-Holland, Amsterdam, 1987), p.\n1; P. W. Anderson, Science 235(1987) 1196.\n20Shiping Feng, Jihong Qin, and Tianxing Ma, J. Phys. Con-\ndens. Matter 16(2004) 343; Shiping Feng, Tianxing Ma,\nand Jihong Qin, Mod. Phys. Lett. B 17(2003) 361.\n21See, e.g., the review, Shiping Feng, Huaiming Guo, Yu\nLan, and Li Cheng, Int. J. Mod. Phys. B, in press,\narXiv:0710.5258.\n22J. Kondo and K. Yamaji, Prog. Theor. Phys. 47(1972)\n807.\n23S. Shamoto; M. Sato; J. M. Tranquada; B. J. Sternlib and\nG. Shirane, Phys. Rev. B 48(1993) 13817.\n24Yayu Wang, S. Ono, Y. Onose, G. Gu, Yoichi Ando, Y.\nTokura, S. Uchida, and N. P. Ong, Science 299(2003) 86.\n25G. S. Boebinger, Yoichi Ando, A. Passner, T. Kimura,\nM. Okuya, J. Shimoyama, K. Kishio, K. Tamasaku, N.\nIchikawa, and S. Uchida, Phys. Rev. Lett. 77(1996) 5417;\nX. F. Sun, Kouji Segawa, and Yoichi Ando, Phys. Rev.\nLett.93(2004) 107001."    },    {        "title": "2011.05450v1.Ultrafast_high_harmonic_nanoscopy_of_magnetization_dynamics.pdf",        "content": "Ultrafast high -harmonic nanoscopy of magnetization dynamics  \n \nSergey Zayko1, Ofer Kfir1,2, Michael Heigl3, Michael Lohmann1, \n Murat Sivis1,2, Manfr ed Albrecht3, and Claus Ropers1,2 \n1 4th Physical Institute – Solids and Nanostructures, University of G öttingen, 37077 G öttingen, Germany.  \n2 Max Planck Institute for Biophysical Chemistry, 37077 G öttingen, Germany.  \n3 Institute of Physics, University of Augsburg, Universitätsstrasse 1, 86159 Augsburg, German y \n \n \nLight-induced magnetization changes , such as all -optical switching, skyrmion nucleation , and \nintersite spin transfer, unfold on temporal and spatial scales down to femtosecond s and \nnanometers, respectively . Pump-probe  spectroscop y and diffraction  studies indicate  that \nspatio -temporal dynami cs may drastically affect  the non -equilibrium magnetic evolution . Yet , \ndirect real -space magnet ic imaging  on the relevant timescale has remained  challenging . Here, \nwe demonstrate ultrafast high -harmonic nanoscopy  employing  circularly polarized high-\nharmonic radiation  for real-space  imaging  of femtosecond magnetization dynamics . We \nobserve  the reversible and irreversible evolution of nanoscale  spin textures following \nfemtosecond laser excitation . Specificall y, we map  quenched magnetic domain s and localized  \nspin structures in Co/Pd multilayers with a sub-wavelength spatial resolution d own to 16 nm , \nand strobosocopically trace the  local magnetization dynamics  with 40 fs temporal resolution . \nOur approach enables the highes t spati o-temporal resolution  of magneto -optical imaging to \ndate. Facilitat ing ultrafast imaging with a n extreme  sensitivity to various microscopic degrees \nof freedom expressed in chiral and linear dichroism , we envisage  a wide range of applications  \nspanning magnetism, phase transitions , and carrier dynamics . \n  \nLocalized m agnetic textures are  essential elements of  hard-drives,  magnetic random -access memories  [1], \nnovel storage schemes  [2–4], and potential building blocks for next-generation logic  devices  [4–7]. \nSkyrmions and domain walls, in particular, attract considerable attention  as topological  excitations that can  \nbe nucleated, erased and controllably translated , equivalent to the writing and register -shifting of logical \nbits [3,4,7,8] . Due to the inherent ly localized nature of such magnetic features, t heir coupling with the \nenvironment  is governed by nanoscale heterogeneity , while exchange and spin -orbit interactions define the \nintrinsic femtosecond time scale of magnetic dynamics.  Whereas ultrafast demagnetization has been studied \nfor decades  [9], recent experiments based on extreme -UV and soft -X-ray scattering suggest a coupling \nbetween spatial magnetic properties and the temporal scale of the demagnetization  process  [10,11 ]. \nSimilarly, magneto -transport phenomena involve an ultrafast spatio -temporal response, such as \nsuperdiffusive spin -currents  [12,13] , hot electrons  [14], or the spin  Seebeck effect  [15,16] , ranging down \nto a few tens of femtoseconds  [17–19].  Access to combined nm -fs spatio -temporal resolution, as required for real -space observations of spin \ndynam ics at the fundamental limits, has remained a grand experimental challenge. Time-resolved \nmeasurement schemes were  developed to access the magnetic dynamics with sub -100 nm spatial and \npicosecond temporal resolution. These include implementations  of spin-polarized scanning tunneling \nmicroscopy  [20], Loren tz-contrast electron  microscopy  [21–27], scanning transmission x -ray \nmicroscopy  [28,29]  , high-resolution  ptychography  [30,31]  and tomography  [32,33] , as well as full -field \nmagneto -optical imaging approaches including Fourier transform holography (FTH)  [29,34,35]  and \ncoherent diffractive imaging (CDI)  [36,37] . Magneto -optical K err microscopy  [38] and photoemission \nelectron microscopy  [39], on the other hand , offer femtosecond snapshots of magnetic textures , albeit at  \nsomewhat lower  spatial resolution . To date, the 2014 pioneering work by von Korff -Schmising  and co -\nworkers  [40] remains the only experimental demonstration of  ultrafast  magnetization  dynamics imaged \nwith sub-100-nm spatial and 100 -fs-temporal resolution , using FT H at a free -electron laser facility.  Thus, \nexploring the frontiers of ultrafast magnetism still requires more widely accessible experimental techniques \ncapable of simultaneous nanometer spatial and femtosecond temporal resolutions.  \n \n \nFig. 1. Ultrafast hig h-harmonic nanoscopy . a A magnetic sample is excited with a femtosecond  laser pulse and probed with a circularly polarized \nhigh-harmonic beam  (wavelength of 21 nm) at a variable delay Δτ . For each time delay between the pump and probe pulses , a quantitative real -\nspace image of the  magneto -optical phase is reconstructed from the diffraction pattern of the high -harmonic beam. b Plot of the spatially -averaged \nnormalized magnetization within the field of view as a function of delay. c In-situ observa tion of laser -induced control of the magnetization state. \nImages show the resulting patterns when the sample  is quenched after excitation at high/medium fluence, and after laser annealing. Note in particular \nthe highly localized structures with diameters l ess than 100 nm.  In this work , we introduce  ultrafast dichroic  nanoscopy  based on high -harmonic radiation , reaching  \nunprecedented spatio -temporal resolution in full -field magneto -optical  imaging . Exploiting  the X-ray \nmagnetic circular dichroism (XMCD) of cobalt in Co/Pd multilayer  structures , we trace  the ultrafast \nresponse of nanoscale magnetic  textures  to femtosecond laser excitation . We captur e laser -induced spin  \ntextures, ranging from micrometer -scale domains to ~80-nm-sized  magnetic bubbles , with a sub-\nwavelength spatial resolution down to 16  nm. Ultrafast m ovies of nanoscale spin  dynamics  are recorded \nwith a temporal resolution of 40 fs .  \nThe experimental principle is depicted in Fig. 1a . Femtosecond pulses from a Ti:Sapphire laser amplifier \n(central wavelength of 800  nm, pulse energy up to 3.5 mJ , 35 fs  pulse  duration) are  split into a pump and a \nprobe arm. The p ump pulses optically excite the sample, and circularly polarized  radiation form  high-\nharmonic generation (HHG) in  a gas cell serves as the  femtosecond  probe for magnetic imaging . The 38th \nharmonic order (wavelength  of ~21 nm) , which provides XMCD  contrast near the M-edge  of cobalt , is \nselected and focused on to the sample by a toroidal grating  monochromator , whereas  the remaining \nharmonic orders are blocked.  The diffraction pattern of the radiation transmitted through the sample is \ncollected by a charge -coupled device (CCD) camera  placed a few centimeters downstream . The positioning \nof the camera determines the collected scattering angle  and the theoretical (diffraction) limit of the  spatial \nresolution . Real-space image s are retrieved algorithmically by an iterative phasing of the recorded far -field \nintensities , based on a combination of FTH  and coherent diffractive imaging ( CDI). Guiding  the CDI \nalgorithm  with FTH drastically improv es the image recovery  [41–43] and allows us to significantly increase \nthe spatial frequencies up to which the phases are consistently  reconstructed  [41]. Both  phase and \nabsorption  contrast can be achieved via the choice of  probe wavelength  near the absorption edge  [44,45] . \nImaging of reversible magnetization dynamics is conducted stroboscopically  (Fig. 1a) at 1 kHz repetition \nrate, whereas irreversible changes of the magnetic texture (Fig. 1c) are mapped after switching off the pump \nbeam.  \nThe magnetic sample is a multilayer structure of cobalt and palladium (Co/Pd)  deposited  on membranes \nmade of silicon -nitride (Si3N4) or silicon . The backside of the sample is coated with gold (180 -200 nm \nthick) which is opaque for the high-harmonic radiation . Fields of view (FOV) with diameter s of 3 to 6 µm \nare formed by removing the gold using focused ion beam ( FIB) etching . Additionally, a n array  of holes  is \nmilled through the entire sample thickness including the substrate and the magnetic film (c.f. Fig. 2 a). The \nhole-array is designed such that the strong auxiliary field emanating  from these holes  covers the entire \ndetector and interfer ometrically enhances the weak magnet o-optical  scattering  signal  from the FOV \naperture . This  signal -enhancement scheme drastically improv es the dynamic range and reduces  the required \ndose by more than one order of magnitud e compared to a conventional FTH or  CDI approach . Two of the holes  (diameters of 200 -400 nm)  are well -separated to allow for a direct recovery of low-resolution  \nholographic information  that facilitate s subsequent high-resolution iterative phase retrieval.  \n \n \nFig. 2. Dichroic imaging employing coherent signal enhancement . a Scanning electron micro scop (SEM)  image of the sample showing  a \ncircular 3.1 µm field of view (F OV) , auxiliary holes for  signal enhancement and two distant reference holes  (bottom ) for Fourier -transform \nholography (see methods) . Inset: Sample c ross section . b, c Exit-field magnitude and phase  obtained with left - (L) and right -hand ed (R) \ncircular ly polarized high -harmonic beam s. d Scatter plot of pixel -by-pixel complex exit -wave amplitudes in the FOV (top)  and of their ratio, \ni.e., the dichroic signal (bottom). e 2D-histogram of pixels with a given complex dichroic signal . f Dichroic phase contrast  image  exhibiting  \ndomains  with out-of-plane magnetiz ation.   \nExtracting  pure magnetic information  requires the j oint phase retrieval (phasing) of the far -field intensities \nrecorded with left - and right -handed circularly polarized illumination. An image  recorde d with a single \nhelicity  is dominated by non -magnetic contrast , characterized by a higher magnitude of the transmission \n(Fig. 2b) and a phase offset (Fig. 2c) in the holes compared to the film  within the circular FOV . Only weak \nvariations in the FOV are observed for single -helicity reconstructions, including fringes near the perimeter \nfrom wave -guiding and edge -diffraction effect s [46,47] . Some contrast  arises from thickness variations and \ncontaminations  of the specimen  and th e substrate . Illustrating the importance of a dichroic reconstruction, \nFig. 2d shows the complex amplitudes of all  pixels in the single -helicity exit waves  within the circular FOV , \nexhibiting considerable scatter in amplitude and phase . The dichroic signal , however , obtained by a pixel -\nby-pixel ratio of the complex amplitudes (L/R), shows a very well -defined double -lobed histogram \ndistribution , reflecting the quasi -binary contrast in the final dichroic  image ( phase contrast in Fig. 2f).  \nWe have found this dichroic microscopy approach to be very robust and have recorded hundreds of \nmagnetization maps for different samples , material systems, masks , and numerical apertures . Figure 3 \nshow s a set of dichroic phase -contrast images of magnetic textures with half-pitch spatial resolution s \nbetween 2 7 and 16 nm . Note that  these are direct reconstructions without filtering or pixel interpolation . \nWe consistently reach a  diffraction -limited  single -pixel resolution  in each of the scattering geometr ies used . \nIn real  space , the d iffraction -limited resolution is evident  in sharp contrast changes at  domain  walls  that \nspan a single pixel.  In a far-field analysis , the resolution is evaluated using the  phase retrieval transfer \nfunction (PRTF)  [48,49]  – a standard measure for the reliability of phase retrieval  procedure s. The PRTF \nvalue s, as shown in Fig 3c , stay above the 0.5 mark , indicating confidence in the retrieved phases even  at \nthe edges and the corners of the CCD  sensor , corresponding  to spatial frequencies ( half-pitch resolution s) \nof 16  nm and 12 nm, respectively . \nA high  consistency and robustness of  the demonstrated approach  allow s for in-situ imaging with only few -\nsecond  delay s between the raw data acquisition and the s ubsequent image reconstruction, providing for \nreal-time feedback on experimental parameters . We have utilized this feature in the la ser-manipulation of \ndomain structures shown in Fig. 1c. In these experiments, a high laser excitation fluence was chosen to \ndemagnetize the sample. Abrupt interruption of the excitation by blocking the laser beam leads to quench \nof the magnetic film . We have found that the resulting  spin texture s drastically depend on the excitation \nparameters  [27], ranging from large micrometer -scale domains over highly localized bubbles to rather \nfractures textures. Further elucidation of this optical domain control will be a subject of future work, as we \nwill now focus on the ultrafast capabilities of high -harmonic nanoscopy.  \n \nFig. 3  High -resolution full -field magnetic imaging . a Illustration of the diffraction data collection  at various numerical apertures  (NA).  \nb Dichroic phase contrast reconstructions for the diffraction data recorded  with 21 nm wavelength at  NA= 0.39, 0.42, 0.55 and 0.65 \ncorresponding to a half-pitch  spatial  resolution of 27 nm, 25 nm, 19 nm, and 16 nm,  respectively. Raw experimental reconstructions without  \nimage processing , averaging or interpolation . c The p hase-retrieval transfe r function (PRTF ) computed for the diffraction data recorded at  \nNA=0.65. The PRTF plot i ndicat es high consistency  of the retrieved phase s up to the edges and corners of the CCD  sensor , corresponding to \nspatial resolution s of 16 nm and 12 nm, respectively.  \nImaging with h igh-harmonic radiation inherentl y allows for  time-resolved studies on the femtosecond to \nattosecond time scale  [17,50,51] . In the following, we utilize this capability to record  ultrafast \nmagnetization movies of the multilayer films using 35-fs laser pump and sub-10 fs high -harmonic probe \npulse s (see Supplementary Materials) . Figure s 4a and b show exemplary  XMCD  phase -contrast frames of \nthe spin dynamics  movie s for two different samples  and fluences , reconstructed with a spatial resolution of \n40 nm . The pump fluence s were chosen  to induce  considerable demagnetization while minimizing  \nirreversible changes  of the domain pattern . Comparing consecutive frames  (cf. Fig s. 4a and b), the most \npronounced  observation is the overall  reduction of the magnetization immediately after the  optical pump  \n(Δ𝜏=0) with a well -resolved timescale  of 200 fs, and a subsequent partial recovery over a few \npicosecond s. Figures 4c  and d show high -fidelity reconstructions of a domain pattern before the pump and \nnear the maximum suppression of the magnetization around 1 ps.  The change of the normalized \nmagnetization is displayed in Fig. 4e , revealing a standing -wave -like pattern with features separated by \nabout half the optical wavelength, and suggesting considerable scattering of the pump beam at the mask \nedges. In a local subwavelength hotspot in the right of the FOV, a suppression down to about 20%  is \nobserved.  Segmentations in areas of different local flue nce (Fig. 4g) yield de - and remagnetization \ndynamics that strongly vary across the FOV (Fig. 4h), and which, accordingly, also differ in shape from the \nspatially -averaged dynamics evaluated for different fluences (Fig. 4f).  \nA major part of the spatially -varying magnetization dynamics observed can be attributed to the structured \nexcitation, most likely including the slower recovery at regions of higher local fluence  (Fig. 4h)  [17,52] . \nDespite the high spatiotemporal resolution of the approach, up to now, we have not found unambiguous \nevidence for a significant softening of domain walls as may have been expected from the characteristics of \nsuperdiffusive spin currents a nd previous diffraction experiments  [3,11,53 –57]. Several reasons may \naccount for this observation. First, the spatial resolution achieved may not yet be sufficient to resolve a \nbroadening if it only involves a change by a small fraction of the native domain wall width.  Secondly, \nstroboscopic imaging rel ies on reversible changes of the domain pattern. We observe that irreversible \nchanges to the real -space domain structure already occur at fluences for which the local magnetization is \nsuppressed by about  50 %. Diffraction experiments, however, suggest that  considerable domain wall -\nsoftening  may require higher fluences. Further studies , possibly  with more controlled local excitation  and \nhigher spatial gradients , may be necessary to clearly identify such spatiotemporal magnetization dynamics \nand map nanoscale  spin currents in real space.   \nFig. 4. Ultrafast high -harmonic  magnetic nano scopy. a, b Exemplary f rames from movies of magnetization  dynamics  for two samples driven \nat different fluences. Nominal incident fluences : 1.4(1) mJ/cm2 (a) 1.3(1) mJ/cm2 (b). c Interpolated averaged images over 4 frames before \nthe optical pump ing (data from panel b) . d Image after pumping  (4 frames around 1 ps) . Note the strong local variations in the suppression of \nthe magnetization . e The demagnetization map is the ratio between the magnetic -contrast before and apfter the pump highlighting the regions \nwith stronger demagnetiz ation . f Evolution of the average  magnetization (absolute value) within the field of view, for different laser pump \nfluence s. h Local magnet ization d ynamics for 4 different regions of the sample highlighted in  panel  g. \n \nIn conclusion, we have developed  ultrafast high -harmonic dichroic nanoscopy for robust full-field imaging \nof magnetic textures and their femtosecond evolution  on the nanoscale . These results represent both the \nhighest spatial and the highest spatiotemporal resolution in full -field magneto -optical imaging obtained to \ndate, irrespective of the wavelength, technique or the radiation source used. Reaching the  diffraction  limit  \nwith a high -coherence source  allows for such results  using a wavelength one to two orders of magnitude \nlonger than typically  required to access sub -50 nm resolutions.  We envisage a wide range of applications \nfor high -harmonic dichroic nanoscopy using linear or  circular dichroism, with many further examples in \nultrafast and element -specific spintronic imaging, but also extending to the tracking of hot carrier \npopulations, structural and electronic phase transitions, and chemical transformations.  \n \nMethods  \nHHG sou rce.  \nHigh -harmonic up -conversion is driven with a bi -circular two -color field tailored using a MAZEL -TOV \napparatus from a linearly polarized laser beam with the central wavelength at 800 nm. We optimize the \ngeneration conditions to improve the harmonic yield near the  maximum of the magneto -optical phase signal \nof сobalt M -edge  [44,45] . The generated harmonic spectrum is dispersed with a toroidal grating and the \n38th harmonic ( ~21 nm wavelength) is isolated with a slit before it enters the imaging chamber. We use a \ndiffraction grating to remove adjacent harmonics and ensure sufficient temporal coherence across the mask \nfor high -resolution lensless imaging  [58,59] . Moreover, the spatial coherence from the bichromatic circular \ndriver is higher compared to a conventional  HHG scheme with linearly polarized excitation  due to a reduced \nnumber of allowed electron trajectories  [60,61] . The obser vation of a full suppression of every 3rd harmonic \norder is a robust and reliable indication for generation conditions corresponding to circularly polarized \nharmonics. However, in order to maximize the magneto -optical signal, the circular polarization has to be \nprovided  at the sample  position.  This can be achieved  by adjusting the waveplate angle of the MAZEL -\nTOV to pre -compensate the polarization dependent reflection of the optical elements between the sample \nand the generation point, in this case the  diffraction grating. To optimize and verify the circular polarization \nat the sample plane , we use an extreme -UV polarization analyzer  based on nanoscale slit arrangement that \nallows for in-situ polarization measurement in a single acquisition  [62]. \nImaging scheme . \nWe employ  holographically guided lensless imaging , in which  the sample mask contains  a circular FOV \naperture , two or more  holographic reference holes , and an array of auxiliary holes . The masks are designed \naccording to the following criteria : \n1. A confinement of the exit wave  to fulfi ll the sampling requirement for phase retrieval  [58]. This \ncriterion  defines the maximum size of the FOV aperture , which  in our case was successfully tested for \ncircular apertures with diameters of  2, 3, 4, 5 and 6 µm. \n2. Sufficiently s mall holographic reference holes  (diameters betw een 200 nm and 400 nm) to  provide for \nexact information on the sample  geometry and the real-space support  via FTH directly from the \nmeasured data . Having two reference holes makes the automatic  support deconvolution process more \nstraightforward.  \n3. An array o f additional auxiliary holes of a specific shape and arrangement is introduced in the vicinity \nof the sample to ensure homogeneous coverage of the detector with a strong scattering signal that \ninterferes with weak magneto -optical signal from the FOV apertu re and coherently enhance s it above \nthe detector noise  level. This reduc es the required dose by more than an order of magnitude , depending \non required resolution.  \nFor every delay frame in the pump -probe movie, we record two sets of diffraction data using a  left- and \nright -handed circularly polarized harmonic illumination . Each data set is composed of 10 -20 diffraction \npatterns with 3 to 30 seconds exposure time (depending on the mask geometry and NA used) to increase SNR and the dynamic range of the diffrac tion data. In the case of high -resolution imaging experiments \nshown in Fig. 3, a total exposure time of up to 15 minutes was required for sub-wavelength spatial  resolution \nreconstructions. To further improve the SNR for these data sets , we acquire  short an d long exposure time \ndiffraction patterns that are subsequently merged  into a single HDR data set . Starting from a random first \nguess, deconvolved holographic support and using a modified RAAR algorithm  [63], we first reconstruct \ndata set with left -handed circular polarized probe using 1 30-150 iterations and afterwards reconstruct the \ndata set of the opposite helicity. If both data sets are accurately reconstructe d and precisely aligned, the \nratio of these two independent ly reconstructed exit wave amplitudes  of opposite helicities eliminate non -\ndichroic contributions , providing for  a pure magnetic phase  and absorption contrast image. To accelerate  \nthe iterative  process to just 10 iterations and to skip the sub -pixel alignment procedure, the reconstruction \nfor the opposite helicity (as well as for the all data sets of a given time series) can be initiated from a \npreviously obtained reconstruction. Importantly, such an approach mitigates possible phase retrieval and \nimaging artefacts for diffraction data with poor SNR, low oversampling ratio, or insufficient coherence \nproperties of the probe. Irrespective of the initial random guess, the dichroic reconstructions are virtually  \nidentical  (as evident form the PRTF plotted in Fig. 2c),  thus no averaging of several reconstructions or \nfiltering is required  for diffraction data with high SNR . Furthermore,  no additional phase retrieval \nconstraints have to be imposed , and we att ribute this to the improved spatial a nd temporal coherence of the \ndeveloped HHG source  as well as the coherent signal enhancement mechanism provided by the auxiliary \nreference holes . \nWe note that time-invariant regions of the sample, i.e., the auxiliary an d reference holes , can be enforced \nas an additional real-space constraint in the reconstruction process  to potentially relax  the SNR requirement \nfor each delay frame, following alternative approaches specifically designed for pump -probe diffraction \ndata [64]. \nResolution estimate .  \nTo estimate the spatial resolution , we first measure the accurac y of the phase retrieval process in the far -\nfield using a well -established PRTF method  [65]. Figure 3c shows the PRTF plot for 20 indivi dual \nreconstruction s (for the data with NA=0.65)  initiated from a random first guess demonstrating a consistent \nphase retrieval throughout the entire recorded far -field as expected from a diffraction limited imaging  \nsystem . Additionally, we verify the spat ial resolution in real space as the smallest resolvable feature in the \nreconstruction. We find multiple localized  domains , bubbles (and possibly skyrmions)  with sub -50 nm \ntransverse sizes that are clearly resolved. T he smallest dimensions in the investigat ed samples are magnetic \ndomain walls that are expected to be in the range between 10 and 20 nm  [66]. To estimate the sharpness of \nthe reconstructed domain walls we used a n improved  knife -edge technique that measures the average  \ntransition sharpness between all identified magnetic domain boundaries throughout the image . In this way, \nwe obtain a conserva tive estimate and the characteristic resolution, avoiding artificially sharp transitions \nfrom  a single lineout in the presence of  noise. Furthermore, to rule out any influence of the  segmentation \nalgorithm in locating domain walls , we identified the domain  wall positions in one data set, and measured \nthe final resolution using an independent data set  of the same structure . The averaged transition in contrast \noccurs over less than two pixels , consistent with the expected physical domain wall sharpness and th e 16 \nnm spatial resolution provided by the PRTF. The reconstructed image also exhibits multiple domain walls \nappearing as a single -pixel step.  \nThe temporal resolution of our approach  is conservatively estimated as 40 fs using an optical cross -\ncorrelation o f the 35 fs pump beam (verified with SPIDER) and expected sub-10 fs harmonic probe. First, \nwe measure the spatio -temporal shear introduced by the diffraction grating to the probe pulse. Second, we compute a fraction of this shear incident onto a 3µm large FOV of the object. Finally,  we consider  the non -\ncollinear excitation on the FOV and measure the convolution of the resulting pulses.  \nMagnetic film fabrication  \nThe magnetic Co/Pd ML films have the following structure: [Pd(0.75 nm)/Co(0.55 nm)] 9. The MLs were \nprepared at RT by dc magnetron sputtering from elemental targets on Silicon and Si3N4 membranes. The \nsputter process was carried out using an Ar working pressure of 3.5x10-3 mbar in an ultra -high vacuum \nchamber (base pressure <10-8 mbar). For all film s, 1.5 nm of Cr and 2.0 nm of Pd were used as seed layer. \nTo protect the films from corrosion 2 nm of Pd were used as cover layer. The thicknesses of the layers were \nestimated from the areal densities measured by a quartz balance during deposition.   \n \nRefer ences  \n1.   A. D. Kent and D. C. Worledge, \"A new spin on magnetic memories,\" Nature Nanotechnology 10, \n187–191 (2015).  \n2.   S. S. P. Parkin, M. Hayashi, and L. Thomas, \"Magnetic Domain -Wall Racetrack Memory,\" Science \n320, 190 –194 (2008).  \n3.   S. Parkin and S. -H. Yang, \"Memory on the racetrack,\" Nat Nano 10, 195 –198 (2015).  \n4.   A. Fert, V. Cros, and J. Sampaio, \"Skyrmions on t he track,\" Nat Nano 8, 152 –156 (2013).  \n5.   D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn, \"Magnetic \nDomain -Wall Logic,\" Science 309, 1688 –1692 (2005).  \n6.   A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebrands,  \"Magnon spintronics,\" Nat Phys \n11, 453 –461 (2015).  \n7.   Z. Luo, A. Hrabec, T. P. Dao, G. Sala, S. Finizio, J. Feng, S. Mayr, J. Raabe, P. Gambardella, and \nL. J. Heyderman, \"Current -driven magnetic domain -wall logic,\" Nature 579, 214 –218 (2020).  \n8.   F. Bü ttner, I. Lemesh, M. Schneider, B. Pfau, C. M. Günther, P. Hessing, J. Geilhufe, L. Caretta, D. \nEngel, B. Krüger, J. Viefhaus, S. Eisebitt, and G. S. D. Beach, \"Field -free deterministic ultrafast \ncreation of magnetic skyrmions by spin –orbit torques,\" Natur e Nanotechnology 12, nnano.2017.178 \n(2017).  \n9.   E. Beaurepaire, J. -C. Merle, A. Daunois, and J. -Y. Bigot, \"Ultrafast spin dynamics in ferromagnetic \nnickel,\" Phys. Rev. Lett. 76, 4250 –4253 (1996).  \n10.   B. Vodungbo, J. Gautier, G. Lambert, A. B. Sardinha, M. Lozano, S. Sebban, M. Ducousso, W. \nBoutu, K. Li, B. Tudu, M. Tortarolo, R. Hawaldar, R. Delaunay, V. López -Flores, J. Arabski, C. \nBoeglin, H. Merdji, P. Zeitoun, and J. Lüning, \"Laser -induced ultrafast demagnetization in the \npresence of a nanoscale magn etic domain network,\" Nature Communications 3, 999 (2012).  \n11.   B. Pfau, S. Schaffert, L. Müller, C. Gutt, A. Al -Shemmary, F. Büttner, R. Delaunay, S. Düsterer, S. \nFlewett, R. Frömter, J. Geilhufe, E. Guehrs, C. M. Günther, R. Hawaldar, M. Hille, N. Jaoue n, A. \nKobs, K. Li, J. Mohanty, H. Redlin, W. F. Schlotter, D. Stickler, R. Treusch, B. Vodungbo, M. \nKläui, H. P. Oepen, J. Lüning, G. Grübel, and S. Eisebitt, \"Ultrafast optical demagnetization \nmanipulates nanoscale spin structure in domain walls,\" Nat Com mun 3, 1100 (2012).  \n12.   M. Battiato, K. Carva, and P. M. Oppeneer, \"Superdiffusive Spin Transport as a Mechanism of \nUltrafast Demagnetization,\" Phys. Rev. Lett. 105, 027203 (2010).  \n13.   E. Turgut, C. La -o-vorakiat, J. M. Shaw, P. Grychtol, H. T. Nembach , D. Rudolf, R. Adam, M. \nAeschlimann, C. M. Schneider, T. J. Silva, M. M. Murnane, H. C. Kapteyn, and S. Mathias, \n\"Controlling the competition between optically induced ultrafast spin -flip scattering and spin \ntransport in magnetic multilayers,\" Phys. Rev. Lett. 110, 197201 (2013).  \n14.   N. Bergeard, M. Hehn, S. Mangin, G. Lengaigne, F. Montaigne, M. L. M. Lalieu, B. Koopmans, \nand G. Malinowski, \"Hot -Electron -Induced Ultrafast Demagnetization in Co/Pt Multilayers,\" Phys. \nRev. Lett. 117, 147203 (2016).  15.   K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae, K. Ando, S. Maekawa, and E. Saitoh, \n\"Observation of the spin Seebeck effect,\" Nature 455, 778 –781 (2008).  \n16.   T. Seifert, S. Jaiswal, U. Martens, J. Hannegan, L. Braun, P. Maldonado, F. Freimuth, A . \nKronenberg, J. Henrizi, I. Radu, E. Beaurepaire, Y. Mokrousov, P. M. Oppeneer, M. Jourdan, G. \nJakob, D. Turchinovich, L. M. Hayden, M. Wolf, M. Münzenberg, M. Kläui, and T. Kampfrath, \n\"Efficient metallic spintronic emitters of ultrabroadband terahertz ra diation,\" Nat Photon 10, 483 –\n488 (2016).  \n17.   S. Mathias, C. La -O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam, H. T. \nNembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann, M. M. Murnane, \nand H. C. Kapteyn, \" Probing the timescale of the exchange interaction in a ferromagnetic alloy,\" \nPNAS 109, 4792 –4797 (2012).  \n18.   S. Mathias, C. La -o-vorakiat, J. M. Shaw, E. Turgut, P. Grychtol, R. Adam, D. Rudolf, H. T. \nNembach, T. J. Silva, M. Aeschlimann, C. M. Schneider , H. C. Kapteyn, and M. M. Murnane, \n\"Ultrafast element -specific magnetization dynamics of complex magnetic materials on a table -top,\" \nJournal of Electron Spectroscopy and Related Phenomena 189, 164 –170 (2013).  \n19.   F. Willems, C. von Korff Schmising, C. S trüber, D. Schick, D. W. Engel, J. K. Dewhurst, P. Elliott, \nS. Sharma, and S. Eisebitt, \"Optical inter -site spin transfer probed by energy and spin -resolved \ntransient absorption spectroscopy,\" Nature Communications 11, 871 (2020).  \n20.   D. Wortmann, S. Hei nze, Ph. Kurz, G. Bihlmayer, and S. Blügel, \"Resolving Complex Atomic -\nScale Spin Structures by Spin -Polarized Scanning Tunneling Microscopy,\" Phys. Rev. Lett. 86, \n4132 –4135 (2001).  \n21.   M. De Graef, Introduction to Conventional Transmission Electron Micro scopy  (Cambridge \nUniversity Press, 2003).  \n22.   R. E. Dunin -Borkowski, M. R. McCartney, R. B. Frankel, D. A. Bazylinski, M. Pósfai, and P. R. \nBuseck, \"Magnetic Microstructure of Magnetotactic Bacteria by Electron Holography,\" Science \n282, 1868 –1870 (1998).  \n23.   S. D. Pollard, J. A. Garlow, J. Yu, Z. Wang, Y. Zhu, and H. Yang, \"Observation of stable Néel \nskyrmions in cobalt/palladium multilayers with Lorentz transmission electron microscopy,\" Nature \nCommunications 8, ncomms14761 (2017).  \n24.   H. S. Park, J.  S. Baskin, and A. H. Zewail, \"4D Lorentz Electron Microscopy Imaging: Magnetic \nDomain Wall Nucleation, Reversal, and Wave Velocity,\" Nano Lett. 10, 3796 –3803 (2010).  \n25.   N. Rubiano da Silva, M. Möller, A. Feist, H. Ulrichs, C. Ropers, and S. Schäfer, \"N anoscale \nMapping of Ultrafast Magnetization Dynamics with Femtosecond Lorentz Microscopy,\" Phys. Rev. \nX 8, 031052 (2018).  \n26.   G. Berruto, I. Madan, Y. Murooka, G. M. Vanacore, E. Pomarico, J. Rajeswari, R. Lamb, P. \nHuang, A. J. Kruchkov, Y. Togawa, T. La Grange, D. McGrouther, H. M. Rønnow, and F. Carbone, \n\"Laser -Induced Skyrmion Writing and Erasing in an Ultrafast Cryo -Lorentz Transmission Electron \nMicroscope,\" Phys. Rev. Lett. 120, 117201 (2018).  \n27.   M. Möller, J. H. Gaida, S. Schäfer, and C. Ropers, \" Few-nm tracking of current -driven magnetic \nvortex orbits using ultrafast Lorentz microscopy,\" Communications Physics 3, 1–7 (2020).  \n28.   A. Sakdinawat and D. Attwood, \"Nanoscale X -ray imaging,\" Nat Photon 4, 840 –848 (2010).  \n29.   M. T. Birch, D. Cortés -Ortuño, L. A. Turnbull, M. N. Wilson, F. Groß, N. Träger, A. Laurenson, N. \nBukin, S. H. Moody, M. Weigand, G. Schütz, H. Popescu, R. Fan, P. Steadman, J. a. T. Verezhak, \nG. Balakrishnan, J. C. Loudon, A. C. Twitchett -Harrison, O. Hovorka, H. Fangohr, F. Y. O grin, J. \nGräfe, and P. D. Hatton, \"Real -space imaging of confined magnetic skyrmion tubes,\" Nature \nCommunications 11, 1726 (2020).  \n30.   X. Shi, P. Fischer, V. Neu, D. Elefant, J. C. T. Lee, D. A. Shapiro, M. Farmand, T. Tyliszczak, H. -\nW. Shiu, S. Marchesini, S. Roy, and S. D. Kevan, \"Soft x -ray ptychography studies of nanoscale \nmagnetic and structural correlations in thin SmCo5 films,\" Ap pl. Phys. Lett. 108, 094103 (2016).  \n31.   W. Li, I. Bykova, S. Zhang, G. Yu, R. Tomasello, M. Carpentieri, Y. Liu, Y. Guang, J. Gräfe, M. \nWeigand, D. M. Burn, G. van der Laan, T. Hesjedal, Z. Yan, J. Feng, C. Wan, J. Wei, X. Wang, X. Zhang, H. Xu, C. Guo, H. Wei, G. Finocchio, X. Han, and G. Schütz, \"Anatomy of Skyrmionic \nTextures in Magnetic Multilayers,\" Advanced Materials 31, 1807683 (2019).  \n32.   C. Donnelly, M. Guizar -Sicairos, V. Scagnoli, S. Gliga, M. Holler, J. Raabe, and L. J. Heyderman, \n\"Three -dimensional magnetization structures revealed with X -ray vector nanotomography,\" Nature \n547, 328 –331 (2017).  \n33.   C. Donnelly, S. Finizio, S. Gliga, M. Holler, A. Hrabec, M. Odstrčil, S. Mayr, V. Scagnoli, L. J. \nHeyderman, M. Guizar -Sicairos, and J. Raabe, \" Time -resolved imaging of three -dimensional \nnanoscale magnetization dynamics,\" Nature Nanotechnology 15, 356 –360 (2020).  \n34.   S. Eisebitt, \"X -ray holography: The hole story,\" Nature Photonics 2, 529 –530 (2008).  \n35.   F. Büttner, C. Moutafis, M. Schneider, B. Krüger, C. M. Günther, J. Geilhufe, C. v K. Schmising, J. \nMohanty, B. Pfau, S. Schaffert, A. Bisig, M. Foerster, T. Schulz, C. a. F. Vaz, J. H. Franken, H. J. \nM. Swagten, M. Kläui, and S. Eisebitt, \"Dynamics and inertia of skyrmionic spin structures,\" N at \nPhys 11, 225 –228 (2015).  \n36.   A. Tripathi, J. Mohanty, S. H. Dietze, O. G. Shpyrko, E. Shipton, E. E. Fullerton, S. S. Kim, and I. \nMcNulty, \"Dichroic coherent diffractive imaging,\" PNAS 108, 13393 –13398 (2011).  \n37.   V. Ukleev, Y. Yamasaki, D. Morikawa , N. Kanazawa, Y. Okamura, H. Nakao, Y. Tokura, and T. \nArima, \"Coherent Resonant Soft X -ray Scattering Study of Magnetic Textures in FeGe,\" Quantum \nBeam Science 2, 3 (2018).  \n38.   R. Schäfer, \"Investigation of Domains and Dynamics of Domain Walls by the Ma gneto -optical \nKerr-effect,\" in Handbook of Magnetism and Advanced Magnetic Materials  (American Cancer \nSociety, 2007).  \n39.   A. Scholl, H. Ohldag, F. Nolting, J. Stöhr, and H. A. Padmore, \"X -ray photoemission electron \nmicroscopy, a tool for the investigatio n of complex magnetic structures (invited),\" Review of \nScientific Instruments 73, 1362 –1366 (2002).  \n40.   C. von Korff Schmising, B. Pfau, M. Schneider, C. M. Günther, M. Giovannella, J. Perron, B. \nVodungbo, L. Müller, F. Capotondi, E. Pedersoli, N. Mahne,  J. Lüning, and S. Eisebitt, \"Imaging \nUltrafast Demagnetization Dynamics after a Spatially Localized Optical Excitation,\" Phys. Rev. \nLett. 112, 217203 (2014).  \n41.   O. Kfir, S. Zayko, C. Nolte, M. Sivis, M. Möller, B. Hebler, S. S. P. K. Arekapudi, D. Stei l, S. \nSchäfer, M. Albrecht, O. Cohen, S. Mathias, and C. Ropers, \"Nanoscale magnetic imaging using \ncircularly polarized high -harmonic radiation,\" Science Advances 3, eaao4641 (2017).  \n42.   P. Hessing, B. Pfau, E. Guehrs, M. Schneider, L. Shemilt, J. Geilhu fe, and S. Eisebitt, \n\"Holography -guided ptychography with soft X -rays,\" Opt. Express, OE 24, 1840 –1851 (2016).  \n43.   A. S. Johnson, J. V. Conesa, L. Vidas, D. Perez -Salinas, C. M. Günther, B. Pfau, K. A. Hallman, R. \nF. Haglund Jr, S. Eisebitt, and S. Wall,  \"Mapping nanoscale charge states and phase domains with \nquantitative hyperspectral coherent diffractive imaging spectroscopy,\" arXiv:2008.11992 [physics] \n(2020).  \n44.   S. Valencia, A. Gaupp, W. Gudat, H. -C. Mertins, P. M. Oppeneer, D. Abramsohn, and C. M.  \nSchneider, \"Faraday rotation spectra at shallow core levels: 3p edges of Fe, Co, and Ni,\" New \nJournal of Physics 8, 254 –254 (2006).  \n45.   F. Willems, S. Sharma, C. v. Korff Schmising, J. K. Dewhurst, L. Salemi, D. Schick, P. Hessing, C. \nStrüber, W. D. Eng el, and S. Eisebitt, \"Magneto -Optical Functions at the 3 p Resonances of Fe, Co, \nand Ni: Ab initio  Description and Experiment,\" Phys. Rev. Lett. 122, 217202 (2019).  \n46.   S. Zayko, E. Mönnich, M. Sivis, D. -D. Mai, T. Salditt, S. Schäfer, and C. Ropers, \"Co herent \ndiffractive imaging beyond the projection approximation: waveguiding at extreme ultraviolet \nwavelengths,\" Optics Express 23, 19911 (2015).  \n47.   P. Ansuinelli, W. M. J. Coene, W. M. J. Coene, and H. P. Urbach, \"Improved ptychographic \ninspection of E UV reticles via inclusion of prior information,\" Appl. Opt., AO 59, 5937 –5947 \n(2020).  \n48.   S. Marchesini, H. N. Chapman, A. Barty, C. Cui, M. R. Howells, J. C. H. Spence, U. Weierstall, \nand A. M. Minor, \"Phase Aberrations in Diffraction Microscopy,\" arXiv :physics/0510033 (2005).  49.   H. N. Chapman, A. Barty, S. Marchesini, A. Noy, S. P. Hau -Riege, C. Cui, M. R. Howells, R. \nRosen, H. He, J. C. H. Spence, U. Weierstall, T. Beetz, C. Jacobsen, and D. Shapiro, \"High -\nresolution ab initio three -dimensional x -ray diffraction microscopy,\" J. Opt. Soc. Am. A, JOSAA \n23, 1179 –1200 (2006).  \n50.   C. La -O-Vorakiat, E. Turgut, C. A. Teale, H. C. Kapteyn, M. M. Murnane, S. Mathias, M. \nAeschlimann, C. M. Schneider, J. M. Shaw, H. T. Nembach, and T. J. Silva, \"Ultrafast \ndemagnetization measurements using extreme ultraviolet light: comparison of electronic and \nmagnetic contributions,\" Phys. Rev. X 2, 011005 (2012).  \n51.   F. Siegrist, J. A. Gessner, M. Ossiander, C. Denker, Y. -P. Chang, M. C. Schröder, A. Guggenmos, \nY. Cui, J.  Walowski, U. Martens, J. K. Dewhurst, U. Kleineberg, M. Münzenberg, S. Sharma, and \nM. Schultze, \"Light -wave dynamic control of magnetism,\" Nature 571, 240 –244 (2019).  \n52.   B. Koopmans, G. Malinowski, F. D. Longa, D. Steiauf, M. Fähnle, T. Roth, M. Cinche tti, and M. \nAeschlimann, \"Explaining the paradoxical diversity of ultrafast laser -induced demagnetization,\" \nNature Mater 9, 259 –265 (2010).  \n53.   K.-J. Kim, S. K. Kim, Y. Hirata, S. -H. Oh, T. Tono, D. -H. Kim, T. Okuno, W. S. Ham, S. Kim, G. \nGo, Y. Tserkovn yak, A. Tsukamoto, T. Moriyama, K. -J. Lee, and T. Ono, \"Fast domain wall \nmotion in the vicinity of the angular momentum compensation temperature of ferrimagnets,\" Nature \nMaterials 16, 1187 –1192 (2017).  \n54.   P. Baláž, K. Carva, U. Ritzmann, P. Maldonado, a nd P. M. Oppeneer, \"Domain wall dynamics due \nto femtosecond laser -induced superdiffusive spin transport,\" Phys. Rev. B 101, 174418 (2020).  \n55.   N. Moisan, G. Malinowski, J. Mauchain, M. Hehn, B. Vodungbo, J. Lüning, S. Mangin, E. E. \nFullerton, and A. Thia ville, \"Investigating the role of superdiffusive currents in laser induced \ndemagnetization of ferromagnets with nanoscale magnetic domains,\" Scientific Reports 4, 4658 \n(2014).  \n56.   M. Hennes, A. Merhe, X. Liu, D. Weder, C. von K. Schmising, M. Schneider, C. M. Günther, B. \nMahieu, G. Malinowski, M. Hehn, D. Lacour, F. Capotondi, E. Pedersoli, I. P. Nikolov, V. \nChardonnet, E. Jal, J. Lüning, and B. Vodungbo, \"Laser -induced ultrafast demagnetization and \nperpendicular magnetic anisotropy reduction in a Co$_{88 }$Tb$_{12}$ thin film with stripe \ndomains,\" arXiv:2006.11246 [cond -mat] (2020).  \n57.   D. Zusin, E. Iacocca, L. L. Guyader, A. H. Reid, W. F. Schlotter, T. -M. Liu, D. J. Higley, G. \nCoslovich, S. F. Wandel, P. M. Tengdin, S. K. K. Patel, A. Shabalin, N. Hua,  S. B. Hrkac, H. T. \nNembach, J. M. Shaw, S. A. Montoya, A. Blonsky, C. Gentry, M. A. Hoefer, M. M. Murnane, H. C. \nKapteyn, E. E. Fullerton, O. Shpyrko, H. A. Dürr, and T. J. Silva, \"Ultrafast domain dilation \ninduced by optical pumping in ferromagnetic CoFe /Ni multilayers,\" arXiv:2001.11719 [cond -mat] \n(2020).  \n58.   R. L. Sandberg, A. Paul, D. A. Raymondson, S. Hädrich, D. M. Gaudiosi, J. Holtsnider, R. I. \nTobey, O. Cohen, M. M. Murnane, H. C. Kapteyn, C. Song, J. Miao, Y. Liu, and F. Salmassi, \n\"Lensless diff ractive imaging using tabletop coherent high -harmonic soft -X-ray beams,\" Phys. Rev. \nLett. 99, 098103 (2007).  \n59.   J. C. H. Spence, U. Weierstall, and M. Howells, \"Coherence and sampling requirements for \ndiffractive imaging,\" Ultramicroscopy 101, 149 –152 (2004).  \n60.   S. Zayko, O. Kfir, E. Bordo, A. Fleischer, O. Cohen, and C. Ropers, \"A dynamical symmetry triad \nin high -harmonic generation revealed by attosecond recollision control,\" New J. Phys. 22, 053017 \n(2020).  \n61.   O. Kfir, E. Bordo, G. I. Haham, O. Lahav, A. Fleischer, and O. Cohen, \"In -line production of a bi -\ncircular field for generation of helically polarized high -order harmonics,\" Applied Physics Letters \n108, 211106 (2016).  \n62.   S. Zayko, M. Sivis, S. Schäfer, and C. Ropers, \"Polarization contra st of nanoscale waveguides in \nhigh harmonic imaging,\" Optica, OPTICA 3, 239 –242 (2016).  \n63.   D. R. Luke, \"Relaxed averaged alternating reflections for diffraction imaging,\" Inverse Problems \n21, 37 (2005).  64.   Y. H. Lo, L. Zhao, M. Gallagher -Jones, A. Ra na, J. J. Lodico, W. Xiao, B. C. Regan, and J. Miao, \n\"In situ coherent diffractive imaging,\" Nat Commun 9, 1–10 (2018).  \n65.   S. Marchesini, H. He, H. N. Chapman, S. P. Hau -Riege, A. Noy, M. R. Howells, U. Weierstall, and \nJ. C. H. Spence, \"X -ray image reco nstruction from a diffraction pattern alone,\" Phys. Rev. B 68, \n140101 (2003).  \n66.   H. Kronmüller and M. Fähnle, Micromagnetism and the Microstructure of Ferromagnetic Solids , \nCambridge Studies in Magnetism (Cambridge University Press, 2003).  \n \nAcknowledgments  \nWe gratefully acknowledge the support and insightful discussions with Tim Salditt, Marcell Möller, \nThomas Danz, Tobias Heinrich, Philipp Buchsteiner.  \nO.K. gratefully acknowledges funding from the European Union’s Horizon 2020 research and i nnovation \nprogramme under the Marie  Skłodowska -Curie grant agreement No. 752533.  This work was funded by the \nDeutsche Forschungsgemeinschaft (DFG) in the Collaborative Research Center “ Nanoscale Photonic \nImaging ” (DFG -SFB 755, project C08). S.Z acknowledges  funding from the Campus Laboratory for \nAdvanced Imaging, Microscopy and Spectroscopy (AIMS) at the University of Göttingen.  \nCompeting interests:  The authors declare no competing interests . \nData availability:  The data of this study is available from the co rresponding  author upon request.  \nAuthor contributions : S.Z., O.K. and C.R. conceived and designed the experiment with contributions \nfrom  M.L. and M.S. The samples were  designed by S.Z. O.K., M.S., M.H., and M.A., and fabricated by \nM.H., M.S. and M.A. Measurements and data analysis were performed by S.Z. and O.K. The manuscript \nwas written by S.Z., O.K and C.R. with contributions from all authors.   \n \n "    },    {        "title": "2104.11878v2.Theory_of_magnetic_inertial_dynamics_in_two_sublattice_ferromagnets.pdf",        "content": "Signatures of magnetic inertial dynamics in\ntwo-sublattice ferromagnets\nRitwik Mondal1;2\n1Department of Spintronics and Nanoelectronics, Institute of Physics ASCR, v.v.i.,\nCukrovarnická 10, Prague 6, 162 53, Czech Republic\n2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120\nUppsala, Sweden\nE-mail: mondal@fzu.cz\nAbstract. The magnetic inertial dynamics have been investigated for one sublattice\nferromagnets. Here, we develop the magnetization dynamics in two-sublattice\nferromagnets including the intra- and inter-sublattice inertial dynamics. First, we\nderive the magnetic susceptibility of such a ferromagnet. Next, by finding the poles of\nthe susceptibility, we calculate the precession and nutation resonance frequencies. Our\nresults suggest that while the resonance frequencies show decreasing behavior with\nthe increasing intra-sublattice relaxation time, the effect of inter-sublattice inertial\ndynamics is contrasting.\n1. Introduction\nUltrafast manipulation of electrons’ spin remains at the heart of future generation spin-\nbased memory technology [1–3]. It has been observed that a fs laser pulse is capable of\ndemagnetizing a ferromagnetic material [4–6]. On the other hand, using these ultrashort\npulses, magnetic switching has been reported in ferrimagnetic [7–9] and ferromagnetic\nmaterials [10, 11]. These observations have been explained through the spin dynamics\nwithin Landau-Lifshitz-Gilbert (LLG) equation of motion [12–15].\nThe phenomenological LLG spin dynamics consists of spin precession and a\ntransverse damping [16–18]. Such an equation of motion has been derived from a\nrelativistic Dirac theory, where the transverse damping is found to originate from spin-\norbit coupling [19–22]. However, at ultrashort timescales, the traditional LLG equation\nneeds to be supplemented by several other spin torque terms [23]. Especially, at the\nultrafast timescales, the magnetic inertia becomes particularly relevant [24]. The effect\nof magnetic inertia has been incorporated within extended LLG dynamics as a torque\ndue to the second-order time derivative of the magnetization M(r;t). The inertial LLG\n(ILLG) equation of motion reads [25–27]\n@M\n@t=M\u0002\u0014\n\u0000\rH+\u000b\nM0@M\n@t+\u0011\nM0@2M\n@t2\u0015\n; (1)arXiv:2104.11878v2  [cond-mat.other]  1 Sep 2021Signatures of magnetic inertial dynamics in two-sublattice ferromagnets 2\nwhereM0andHdefine the ground state magnetization and an effective field,\nrespectively. The first and second terms in Eq. (1) represent the traditional LLG\nequation [18]. The inertial spin dynamics in the last term of Eq. (1) gives rise to\nthe spin nutation [28, 29]. The ILLG equation signifies the fact that the dynamics of\na magnetic moment shows precession with nutation at ultrafast timescales, followed by\ntransverse damping [24]. The ILLG equation has schematically been depicted in Fig. 1.\nA simple dimension analysis shows that the transverse damping is characterized by\na dimensionless parameter \u000b, and the inertial dynamics are strengthened by inertial\nrelaxation time \u0011. The ILLG dynamics have been derived within the relativistic Dirac\nM(r, t)H\nPrecession\nNutationDamping\nFigure 1: Schematic depiction of ILLG equation of motion.\nframework as well, where it shows that the Gilbert damping \u000band inertial relaxation\ntime\u0011are tensors [30]. In particular, the relativistic theory derives that the Gilbert\ndamping dynamics is associated with the imaginary part of the susceptibility, while the\ninertial dynamics is given by the real part [31]. Such findings are found to be consistent\nwith a linear response theory of ferromagnet [32]. The inertial dynamics have also been\nderived within classical mechanics of a current loop [33]. Eq. (1) has been applied to\na single sublattice ferromagnet beyond ferromagnetic resonance (FMR), observing an\nadditional peak due to nutation resonance [34–36]. While the FMR peak appears at the\nGHz regime, the nutation resonance peak appears at the THz regime [37]. The ILLG\nequation has also been applied to antiferromagnets and ferrimagnets, and it has been\npredicted that the spin nutation should be better detected in antiferromagnets as it is\nexchange enhanced [38].\nRecently, the spin nutation resonance has been observed for ferromagnets in the\nexperiment [39]. Indeed, the nutation resonance peak has been seen at around 0.5\nTHz. Note that the experiment was performed in two-sublattice ferromagnets namely\nCoFeB and NiFe. For two-sublattice ferromagnet, the inter-sublattice exchange energies\nbecome important. Here, we describe the inertial effects in a two-sublattice ferromagnet\ncoupled by the Heisenberg exchange interaction. We follow the similar procedureSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 3\nof Ref. [38] and derive the magnetic susceptibility. We not only consider the intra-\nsublattice inertial dynamics, but alsothe inter-sublattice dynamics. Our results suggest\nthat there are two precession resonance peaks: one at GHz regime and another at THz\nregime. Similarly, two nutation peaks can also be observed, both are at the THz regime.\nBy calculating the precession and nutation resonance frequencies, we observe that the\nresonance frequencies decrease with increasing intra-sublattice relaxation time, however,\nthe scenario is different for inter-sublattice inertial dynamics.\n2. Theory of intra- and inter-sublattice inertial dynamics in two-sublattice\nferromagnets\nThe inertial dynamics for antiferromagnets have been introduced in Ref. [38]. For\ntwo-sublattice magnetic systems having magnetization MAandMB, forAandB\nrepresenting the two-sublattice, the ILLG equations of motion can be recast as\n@MA\n@t=\u0000\rA(MA\u0002HA) +\u000bAA\nMA0\u0012\nMA\u0002@MA\n@t\u0013\n+\u000bAB\nMB0\u0012\nMA\u0002@MB\n@t\u0013\n+\u0011AA\nMA0\u0012\nMA\u0002@2MA\n@t2\u0013\n+\u0011AB\nMB0\u0012\nMA\u0002@2MB\n@t2\u0013\n(2)\n@MB\n@t=\u0000\rB(MB\u0002HB) +\u000bBB\nMB0\u0012\nMB\u0002@MB\n@t\u0013\n+\u000bBA\nMA0\u0012\nMB\u0002@MA\n@t\u0013\n+\u0011BB\nMB0\u0012\nMB\u0002@2MB\n@t2\u0013\n+\u0011BA\nMA0\u0012\nMB\u0002@2MA\n@t2\u0013\n(3)\nIn each ILLG dynamics, the first term represents the spin precession around an effective\nfieldHA=B. The intra- and inter-sublattice Gilbert damping dynamics have been\ndenoted by the second and third terms, respectively. Similarly, the last two terms define\ninertial dynamics. While the intra-sublattice Gilbert and inertial dynamics have been\nweighed by \u000bAA=BBand\u0011AA=BB, the same for inter-sublattice dynamics are denoted by\n\u000bAB=BAand\u0011AB=BA. From a simple dimension analysis, it is clear to show that the\nGilbert damping parameters \u000bare dimensionless, in contrast, the inertial relaxation\ntimes\u0011have a dimension of time [26, 30]. It is worth mentioning that the Gilbert\ndamping\u000bhas been calculated for several materials within ab initio frameworks [32, 40–\n52], while there are also proposals to calculate the inertial relaxation time within\nextended breathing Fermi surface model [53–55]. These ILLG equations have been\ncontemplatedtoforecastthesignaturesofinertialdynamicsincollinearantiferromagnets\nand ferrimagnets [38].\nWe consider that the two-sublattice ferromagnet is aligned collinear at the ground\nstate such that MA=MA0^zandMB=MB0^z. The ferromagnetic system is under\nthe application of an external Zeeman field H0=H0^z. Then, the free energy of the\nconsidered two-sublattice system can be considered as the sum of Zeeman, anisotropy,Signatures of magnetic inertial dynamics in two-sublattice ferromagnets 4\nand exchange energies as\nF(MA;MB) =\u0000H0(MAz+MBz)\u0000KA\nM2\nA0M2\nAz\u0000KB\nM2\nB0M2\nBz\u0000J\nMA0MB0MA\u0001MB;\n(4)\nwhereKAandKBare anisotropy energies and Jis the isotropic Heisenberg exchange\nwithJ > 0for ferromagnetic coupling. To calculate the linear response properties of\nthe system, we consider that the small deviations of magnetization mA(t)andmB(t)\nwith respect to the ground state are induced by the transverse external field hA(t)and\nhB(t). We calculate the effective field in the ILLG equation as the derivative of free\nenergy in Eq. (4) to the corresponding magnetization\nHA=\u0000@F(MA;MB)\n@MA=\u0012\nH0+2KA\nM2\nA0MAz\u0013\n^z+J\nMA0MB0MB\n=1\nMA0(H0MA0+ 2KA+J)^z+J\nMA0MB0mB;(5)\nHB=\u0000@F(MA;MB)\n@MB=\u0012\nH0+2KB\nM2\nB0MBz\u0013\n^z+J\nMA0MB0MA\n=1\nMB0(H0MB0+ 2KB+J)^z+J\nMA0MB0mA:(6)\nWe then expand the magnetization around the ground state in small deviations,\nMA=MA0^z+mA(t)andMB=MB0^z+mB(t). Essentially, with the effective fields\nin Eqs. (5) and (6) along with the magnetization, the linear response for sublattice A\nprovides\n@mA\n@t=\u0000\rA\nMA0(H0MA0+ 2KA+J) [mAy^x\u0000mAx^y]\u0000\rAJ\nMB0[mBx^y\u0000mBy^x]\n\u0000\rAMA0[hAx^y\u0000hAy^x] +\u000bAA\u0014@mAx\n@t^y\u0000@mAy\n@t^x\u0015\n+\u000bABMA0\nMB0\u0014@mBx\n@t^y\u0000@mBy\n@t^x\u0015\n+\u0011AA\u0014@2mAx\n@t2^y\u0000@2mAy\n@t2^x\u0015\n+\u0011ABMA0\nMB0\u0014@2mBx\n@t2^y\u0000@2mBy\n@t2^x\u0015\n; (7)\nobtaining the dynamics for two components xandyas\n\rAMA0hAx=\rA\nMA0(H0MA0+ 2KA+J)mAx\u0000\rAJ\nMB0mBx+\u000bAA@mAx\n@t+\u000bABMA0\nMB0@mBx\n@t\n\u0000@mAy\n@t+\u0011AA@2mAx\n@t2+\u0011ABMA0\nMB0@2mBx\n@t2; (8)\n\rAMA0hAy=\rA\nMA0(H0MA0+ 2KA+J)mAy\u0000\rAJ\nMB0mBy+\u000bAA@mAy\n@t+\u000bABMA0\nMB0@mBy\n@t\n+@mAx\n@t+\u0011AA@2mAy\n@t2+\u0011ABMA0\nMB0@2mBy\n@t2: (9)\nIn the circular basis defined by mA\u0006=mAx\u0006imAyandhA\u0006=hAx\u0006ihAy, the equationsSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 5\ncan be put together\n\rAMA0hA\u0006=\rA\nMA0(H0MA0+ 2KA+J)mA\u0006\u0000\rAJ\nMB0mB\u0006+\u000bAA@mA\u0006\n@t+\u000bABMA0\nMB0@mB\u0006\n@t\n\u0006i@mA\u0007\n@t+\u0011AA@2mA\u0006\n@t2+\u0011ABMA0\nMB0@2mB\u0006\n@t2: (10)\nSimilarly, one can calculate the linear response of the sublattice B in the circular basis\ndefined bymB\u0006=mBx\u0006imByandhB\u0006=hBx\u0006ihByas\n\rBMB0hB\u0006=\rB\nMB0(H0MB0+ 2KB+J)mB\u0006\u0000\rBJ\nMA0mA\u0006+\u000bBB@mB\u0006\n@t+\u000bBAMB0\nMA0@mA\u0006\n@t\n\u0006i@mB\u0007\n@t+\u0011BB@2mB\u0006\n@t2+\u0011BAMB0\nMA0@2mA\u0006\n@t2: (11)\nWe define the response functions mA\u0006;mB\u0006;hA\u0006;hB\u0006/e\u0006i!tand \nA=\n\rA\nMA0(H0MA0+ 2KA+J)and \nB=\rB\nMB0(H0MB0+ 2KB+J). To simplify the\nexpressions, we introduce the following: \u0000AA=\rAMA0,\u0000BB=\rBMB0,\u0000AB=\rAMB0\nand\u0000BA=\rBMA0such that \u0000AA\u0000BB= \u0000AB\u0000BA. The linear response Eqs. (10) and\n(11) can be written in a matrix formalism\n\u0010hA\u0006hB\u0006\u0011\n=0\nBB@1\n\u0000AA(\nA\u0006i!\u000bAA\u0000!2\u0011AA\u0000!)\u00001\n\u0000AB\u0012\rAJ\nMA0\u0007i!\u000bAB+!2\u0011AB\u0013\n\u00001\n\u0000BA\u0012\rBJ\nMB0\u0007i!\u000bBA+!2\u0011BA\u00131\n\u0000BB(\nB\u0006i!\u000bBB\u0000!2\u0011BB\u0000!)1\nCCA\u0010mA\u0006mB\u0006\u0011\n:\n(12)\nFor finding the susceptibility, we recall m\u0006=\u001f\u0006\u0001h\u0006such that the susceptibility matrix\nderives as\n\u001fAB\n\u0006=1\nD\u00060\nBB@1\n\u0000BB(\nB\u0006i!\u000bBB\u0000!2\u0011BB\u0000!)1\n\u0000BA\u0012\rBJ\nMB0\u0007i!\u000bBA+!2\u0011BA\u0013\n1\n\u0000AB\u0012\rAJ\nMA0\u0007i!\u000bAB+!2\u0011AB\u00131\n\u0000AA(\nA\u0006i!\u000bAA\u0000!2\u0011AA\u0000!)1\nCCA;\n(13)\nwhere the determinant is expressed as\nD\u0006=1\n\u0000AA\u0000BB\u0000\n\nA\u0006i!\u000bAA\u0000!2\u0011AA\u0000!\u0001\u0000\n\nB\u0006i!\u000bBB\u0000!2\u0011BB\u0000!\u0001\n\u00001\n\u0000AB\u0000BA\u0012\rAJ\nMA0\u0007i!\u000bAB+!2\u0011AB\u0013\u0012\rBJ\nMB0\u0007i!\u000bBA+!2\u0011BA\u0013\n:(14)\nNote that the intra-sublattice dynamical parameters enter in the diagonal elements of\nthe susceptibility matrix, however, the inter-sublattice dynamics are reflected in the off-\ndiagonal elements. Such a susceptibility matrix has been obtained with intra- and inter-\nsublattice Gilbert damping dynamics for antiferromagnets [56]. To find the resonanceSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 6\nfrequencies, one has to solve the equation setting D\u0006= 0. Therefore, a fourth-order\nequation in frequency is obtained\nA\u0006!4+B\u0006!3+C\u0006!2+D\u0006!+E\u0006= 0; (15)\nwith the following coefficients\nA\u0006=\u0011AA\u0011BB\u0000\u0011AB\u0011BA; (16)\nB\u0006= (\u0011AA+\u0011BB)\u0007i (\u000bAA\u0011BB+\u000bBB\u0011AA)\u0006i (\u000bAB\u0011BA+\u000bBA\u0011AB); (17)\nC\u0006= 1\u0007i (\u000bAA+\u000bBB)\u0000(\nA\u0011BB+ \nB\u0011AA)\u0000\u000bAA\u000bBB\n\u0000\u0012\rA\nMA0\u0011BA+\rB\nMB0\u0011AB\u0013\nJ\u0000\u000bAB\u000bBA; (18)\nD\u0006=\u0000(\nA+ \nB)\u0006i (\nA\u000bBB+ \nB\u000bAA)\u0006i\u0012\rA\nMA0\u000bBA+\rB\nMB0\u000bAB\u0013\nJ;(19)\nE\u0006= \nA\nB\u0000\rA\rB\nMA0MB0J2: (20)\nTheanalyticalsolutionoftheabove-mentionedequationisverycumbersome. Therefore,\nwe adopt the numerical techniques for solving Eq. (15). The solution of the above\nequation results in four frequencies, two of them correspond to the precession resonance\n(!p)ofeachsublatticeandtheothertwobelongtothenutationresonance( !n). Thereal\nand imaginary parts of the resonance frequency are denoted by ReandIm, respectively.\nFor example, the precession resonance frequencies are !p=Re(!p) + i Im(!p), while\nthe nutation resonance frequencies are !n=Re(!n) + iIm(!n). Comparing Eq. (15), a\nsimilar equation has been obtained for antiferromagnets and ferrimagnets [38], however,\nwithout the inter-sublattice inertial dynamics. We mention that the inter-sublattice\nGilbert damping dynamics have extensively been discussed [56, 57]. Therefore, we will\nnot consider in the following discussions. In particular, we allow \u000bAB=\u000bBA= 0, and\ncalculate the inertial effects on precession and nutation resonances.\n3. Numerical results\nTo calculate the resonace frequencies, we numerically solve the Eq. (15) for two-\nsublattice ferromagnets having same magnetic moments in each sublattice i.e., MA0=\nMB0. We use the following parameters: \rA=\rB= 1:76\u00021011T\u00001-s\u00001,J= 10\u000021J,\nKA=KB= 10\u000023J,\u000bAA=\u000bBB=\u000b= 0:05,\u000bAB=\u000bBA= 0. Theconsideredexchange\nand anisotropy energies have similar order of magnitude as typical ferromagnets e.g., Fe\n[58]. The chosen Gilbert damping \u000b= 0:05is within the ab initio reported values [51].\nFor inertial relaxation times, even though, the ab initio calculation suggests about fs\ntimescales for transition metals [55], the recent experiment predicts it to be a higher\nvalue up to several hundreds of fs [39]. Therefore, in what follows, we have considered\nthe inertial relaxation times ranging from fs to ps.Signatures of magnetic inertial dynamics in two-sublattice ferromagnets 7\n10−1100101102103\nη(fs)10−1100101102103ωFM\n±/2π(THz)\n1/η(a)\nRe/parenleftbig\nωu\np+/parenrightbig\nRe/parenleftbig\nωl\np+/parenrightbig\nRe/parenleftbig\nωu\nn−/parenrightbig\nRe/parenleftbig\nωl\nn−/parenrightbig\n10−1100101102103\nη(fs)0.000.010.020.030.040.05Im(ω±)/Re(ω±)|FM\n(b)\nIm/parenleftbig\nωu\np+/parenrightbig\n/Re/parenleftbig\nωu\np+/parenrightbig\nIm/parenleftbig\nωl\np+/parenrightbig\n/Re/parenleftbig\nωl\np+/parenrightbig\nIm/parenleftbig\nωu\nn−/parenrightbig\n/Re/parenleftbig\nωu\nn−/parenrightbig\nIm/parenleftbig\nωl\nn−/parenrightbig\n/Re/parenleftbig\nωl\nn−/parenrightbig\nFigure 2: The calculated resonance frequencies as a function of intra-sublattice inertial\nrelaxation time for two-sublattice ferromagnets using MA0=MB0= 2\u0016B. (a) The\nprecession and nutation resonance frequencies and (b) the effective Gilbert damping\nhave been plotted.\n3.1. Intra-sublattice inertial dynamics\nTo focus on the intra-sublattice inertial dynamics, we set the inter-sublattice relaxation\ntime to zero i.e., \u0011AB=\u0011BA= 0, keeping the same inertial relaxation time in two-\nsublattice\u0011AA=\u0011BB=\u0011. Withthissetofspecifications, thecalculatedfrequencieshave\nbeen shown in Fig. 2. One can see that there exist two precession resonance frequencies\n(positive) and the corresponding two nutation resonance frequencies (negative). We\ndenote these two positive precession frequencies as !u\np+and!l\np+, while the two\nnegative nutation frequencies are !u\nn\u0000and!l\nn\u0000. The superscripts “u” and “l” denote\nthe upper and lower frequencies, respectively. These results are in contrast with the\nobservation in antiferromagnets or ferrimagnets, where one positive and one negative\nprecession (and nutation) frequencies are expected [38]. Nevertheless, the quantitative\ncomparison of the calculated frequencies agrees with those of the ferrimagnets, where\nthe upper (THz), and lower (GHz) frequency precession resonances are called an\nexchange and ferromagnetic modes, respectively [38, 59]. Similar to antiferromagnets\nand ferrimagnets [38], the resonance frequencies decrease with the intra-sublattice\ninertial relaxation time in the case of two-sublattice ferromagnets. Especially, the lower\nnutation resonance frequency scales with 1=\u0011, while the upper one shows deviation\nfrom 1=\u0011at higher relaxation times. This deviation from 1=\u0011has been noticed in\ntwo nutation modes for antiferromagnets and ferrimagnets [38]. An interesting feature\nis that the precession and nutation frequencies cross each other at certain inertial\nrelaxation times in ferromagnets. Such crossing was not observed in antiferromagnets\nand ferrimagnets [38]. The crossing happens especially with the upper precession mode\nwith lower nutation mode as seen in Fig. 2(a). However, we note that crossing of theseSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 8\n12345MA0/MB00.51.01.52.02.53.0Re°!up+¢/2º(THz)(c)¥=0.1 fs¥= 1 fs¥= 10 fs¥= 100 fs¥= 1 ps\n12345MA0/MB00.010.020.030.040.05Im°!up+¢/Re°!up+¢(d)¥=0.1 fs¥= 1 fs¥= 10 fs¥= 100 fs¥= 1 ps12345MA0/MB00.030.040.050.06Re°!lp+¢/2º(THz)(a)¥=0.1 fs¥= 1 fs¥= 10 fs¥= 100 fs¥= 1 ps\n12345MA0/MB00.030.040.05Im°!lp+¢/Re°!lp+¢(b)¥=0.1 fs¥= 1 fs¥= 10 fs¥= 100 fs¥= 1 ps\nFigure 3: The calculated precession resonance frequencies as a function of MA0=MB0\nfor two-sublattice ferromagnets, at several intra-sublattice inertial relaxation times.\n(a) The real part of the lower precession resonance frequencies, (b) the effective\ndamping of lower resonance mode, (c) the real part of the upper precession resonance\nfrequencies, (d) the effective damping of upper resonance mode, has been plotted.\ntwo modes have positive and negative frequencies, meaning that the upper precession\nmode (!u\np+) has a positive rotational sense, however, the lower nutation mode ( !l\nn\u0000) has\nthe opposite rotational sense in circular basis.\nThe inertial dynamics affect the effective Gilbert damping in a system. This has\nbeen demonstrated in Fig. 2(b) for two-sublattice ferromagnet by the ratio of imaginary\nand real parts of the calculated frequencies. We have used the same Gilbert damping\nfor both the sublattices \u000b\u00180:05and therefore, the effective damping remains the same\nat smaller inertial relaxation times. However, the effective damping decreases with\nincreased relaxation times, a fact that is consistent with the results of antiferromagnets\n[38]. It is observed that the decrease in effective damping is exactly the same for\nprecession and corresponding nutation modes. Moreover, the upper precession mode\nis influenced strongly, which has already been observed for ferrimagnets [38].Signatures of magnetic inertial dynamics in two-sublattice ferromagnets 9\nNext, we calculate the influence of different sublattice magnetic moment ( MA06=\nMB0) on inertial dynamics. In particular, we compute the precession resonance\nfrequencies as a function of the ratio of magnetic moments ( MA0=MB0), at several\ninertial relaxation times in Fig. 3. We observe that the resonance frequencies decrease\nwith increasing difference in the magnetic moments. Such reduction is less visible in\ncase of lower precession frequencies e.g., Fig. 3(a), however, more prominent in upper\nprecession frequencies in Fig. 3(c). However, the difference of frequencies calculated\nat several relaxation times are similar for MA0=MB0andMA06=MB0. The latter\nsuggests that the inertial dynamics do not get quantitatively influenced by the same or\ndifferent sublattice magnetic moments. A similar conclusion can also be made from the\ncomputation of effective damping in Figs. 3(b) and 3(d). The effective damping for the\nupper and lower precession modes remains almost constant (with a very small positive\nslope) for a higher ratio of MA0=MB0.\n3.2. Inter-sublattice inertial dynamics\n10−1100101102\nη/prime(fs)10−1100101102103ωFM\n±/2π(THz)\n(a)\nRe/parenleftbig\nωu\np+/parenrightbig\nRe/parenleftbig\nωl\np+/parenrightbig\nRe/parenleftbig\nωu\nn−/parenrightbig\nRe/parenleftbig\nωl\nn−/parenrightbig\n10−1100101102\nη/prime(fs)0.000.010.020.030.040.05Im(ω±)/Re(ω±)|FM\n(b)\nIm/parenleftbig\nωu\np+/parenrightbig\n/Re/parenleftbig\nωu\np+/parenrightbig\nIm/parenleftbig\nωl\np+/parenrightbig\n/Re/parenleftbig\nωl\np+/parenrightbig\nIm/parenleftbig\nωu\nn−/parenrightbig\n/Re/parenleftbig\nωu\nn−/parenrightbig\nIm/parenleftbig\nωl\nn−/parenrightbig\n/Re/parenleftbig\nωl\nn−/parenrightbig\nFigure 4: The calculated resonance frequencies as a function of inter-sublattice inertial\nrelaxation time for two-sublattice ferromagnets using MA0=MB0= 2\u0016B. The\nintra-sublattice inertial relaxation time was kept constant \u0011= 100fs. (a) The\nprecession and nutation resonance frequencies and (b) the effective Gilbert damping\nhave been plotted.\nTo investigate the inter-sublattice inertial dynamics, we set the intra-sublattice\nrelaxation time as \u0011AA=\u0011BB=\u0011= 100fs. Such a relaxation time is lower than the\nexperimental findings in two-sublattice ferromagnets [39]. In fact, the direct comparison\nof Eq. (2) with the Eq. (2) of Ref. [39] provides \u0011\u0018\u000b\u001c. With the experimental\nfindings for CoFeB, \u000b= 0:0044and\u001c= 72ps (see Table 1 in Ref. [39]), we calculate\n\u0011= 316fs. We compute the effect of inter-sublattice inertial dynamics as a function of\n\u0011AB=\u0011BA=\u00110in Fig. 4 considering \u00110<\u0011. As we mentioned earlier, the overlapping ofSignatures of magnetic inertial dynamics in two-sublattice ferromagnets 10\nprecession ( !u\np+) and nutation ( !l\nn\u0000) frequencies at the intra-sublattice relaxation time\n\u0011= 100fs can be seen. We observe that the upper precession resonance frequency ( !u\np+)\nincreases, while the lower one ( !l\np+) decreases very small with inter-sublattice relaxation\ntimes. A similar conclusion can be made for nutation frequencies. This is in contrast to\nthe observation of intra-sublattice inertial dynamics as discussed above. A divergence\nin the upper nutation frequency can be noticed at the limit \u00110!\u0011. Such divergence can\nbe explained through the coefficient Ain Eq. (15). At the limit \u00110!\u0011, the coefficient of\nfourth power in frequency becomes A=\u0011AA\u0011BB\u0000\u0011AB\u0011BA=\u00112\u0000\u001102!0, which brings\nthe fourth-order equation into an effective third-order equation in frequency.\nA similar observation can also be concluded from the calculation of effective\ndamping in Fig. 4(b). Similar to the intra-sublattice inertial dynamics, the effective\ndamping of the precession and corresponding nutation mode behaves exactly the same\nfor the inter-sublattice inertial dynamics. We observe that the damping of upper\nprecession and nutation modes increases with inter-sublattice inertial relaxation time,\nhowever, it is the opposite for lower precession and nutation modes. Therefore, we\nconclude that the effect of intra- and inter-sublattice inertial dynamics are contrasting.\n4. Conclusions\nTo conclude, we have incorporated the intra- and inter-sublattice inertial dynamics\nwithintheLLGequationofmotionandcalculatedtheFMRresonancefortwo-sublattice\nferromagnets. To this end, we first derive the magnetic susceptibility that is a tensor.\nTo calculate the resonance frequencies, we find the poles of the susceptibility. Without\nthe inertial dynamics, there exist two precession modes in a typical two-sublattice\nferromagnet. The introduction of inertial dynamics shows two nutation resonance\nfrequencies corresponding to the precession modes. We note that these precession\nand nutation resonances can be excited by right and left circularly polarised pulses,\nrespectively, and vice-versa within a circular basis. The precession and nutation\nfrequenciesdecreasewiththeintra-sublatticerelaxationtimeasalsohasbeenseeninthe\ncase of antiferromagnets in previous work [38]. However, at certain relaxation times, the\nprecessionandnutationfrequenciesoverlapwitheachother. Notethattheseoverlapping\nprecessionandnutationfrequencieshaveoppositerotationalsenseincircularbasis, thus,\nthey can be neatly realised in the experiments. The inter-sublattice inertial dynamics\nincrease the resonance frequencies and effective damping for upper precession mode,\nhowever, have opposite effect on lower precession mode in two-sublattice ferromagnets.\n5. Acknowledgments\nThe author acknowledges Levente Rózsa and Ulrich Nowak for valuable discussions and\nthe Swedish Research Council (VR 2019-06313) for research funding.REFERENCES 11\nReferences\n[1] Bigot J Y, Vomir M and Beaurepaire E 2009 Nat. Phys. 5515–520 URL https:\n//www.nature.com/articles/nphys1285\n[2] Stanciu C D, Hansteen F, Kimel A V, Kirilyuk A, Tsukamoto A, Itoh A and Rasing\nTh 2007 Phys. Rev. Lett. 99047601 URL http://link.aps.org/doi/10.1103/\nPhysRevLett.99.047601\n[3] Kimel A V, Kirilyuk A, Hansteen F, Pisarev R V and Rasing T 2007 J. Phys.:\nCondens. Matter 19043201 URL https://doi.org/10.1088%2F0953-8984%\n2F19%2F4%2F043201\n[4] Beaurepaire E, Merle J C, Daunois A and Bigot J Y 1996 Phys. Rev. Lett. 76\n4250–4253 URL http://link.aps.org/doi/10.1103/PhysRevLett.76.4250\n[5] Koopmans B, van Kampen M, Kohlhepp J T and de Jonge W J M 2000 Phys. Rev.\nLett.85844–847URL http://link.aps.org/doi/10.1103/PhysRevLett.85.844\n[6] Koopmans B, van Kampen M and de Jonge W J M 2003 J. Phys.: Condens. Matter\n15S723 URL http://stacks.iop.org/0953-8984/15/i=5/a=324\n[7] Mangin S, Gottwald M, Lambert C H, Steil D, Uhlíř V, Pang L, Hehn M, Alebrand\nS, Cinchetti M, Malinowski G, Fainman Y, Aeschlimann M and Fullerton E E 2014\nNat. Mater. 13286–292 URL http://dx.doi.org/10.1038/nmat3864\n[8] Hassdenteufel A, Hebler B, Schubert C, Liebig A, Teich M, Helm M, Aeschlimann\nM, Albrecht M and Bratschitsch R 2013 Advanced Materials 253122–3128 URL\nhttps://onlinelibrary.wiley.com/doi/abs/10.1002/adma.201300176\n[9] Kimel A V, Kirilyuk A, Usachev P A, Pisarev R V, Balbashov A M and Rasing\nTh 2005 Nature 435655–657 URL http://dx.doi.org/10.1038/nature03564\n[10] Lambert C H, Mangin S, Varaprasad B S D C S, Takahashi Y K, Hehn M, Cinchetti\nM, Malinowski G, Hono K, Fainman Y, Aeschlimann M and Fullerton E E 2014\nScience 3451337–1340 URL https://science.sciencemag.org/content/345/\n6202/1337\n[11] John R, Berritta M, Hinzke D, Müller C, Santos T, Ulrichs H, Nieves P,\nWalowski J, Mondal R, Chubykalo-Fesenko O, McCord J, Oppeneer P M, Nowak\nU and Münzenberg M 2017 Sci. Rep. 74114 URL https://doi.org/10.1038/\ns41598-017-04167-w\n[12] Ostler T A, Barker J, Evans R F L, Chantrell R W, Atxitia U, Chubykalo-\nFesenko O, El Moussaoui S, Le Guyader L, Mengotti E, Heyderman L J, Nolting\nF, Tsukamoto A, Itoh A, Afanasiev D, Ivanov B A, Kalashnikova A M, Vahaplar\nK, Mentink J, Kirilyuk A, Rasing T and Kimel A V 2012 Nature Communications\n3666 URL https://doi.org/10.1038/ncomms1666\n[13] Wienholdt S, Hinzke D, Carva K, Oppeneer P M and Nowak U 2013 Phys. Rev. B\n88(2) 020406 URL https://link.aps.org/doi/10.1103/PhysRevB.88.020406REFERENCES 12\n[14] GerlachS,OroszlanyL,HinzkeD,SieveringS,WienholdtS,SzunyoghLandNowak\nU 2017 Phys. Rev. B 95(22) 224435 URL https://link.aps.org/doi/10.1103/\nPhysRevB.95.224435\n[15] Frietsch B, Donges A, Carley R, Teichmann M, Bowlan J, Döbrich K, Carva K,\nLegut D, Oppeneer P M, Nowak U and Weinelt M 2020 Science Advances 6URL\nhttps://advances.sciencemag.org/content/6/39/eabb1601\n[16] Landau L D and Lifshitz E M 1935 Phys. Z. Sowjetunion 8101–114\n[17] Gilbert T L and Kelly J M October 1955 American Institute of Electrical Engineers\n(New York) pp 253–263\n[18] Gilbert T L 2004 IEEE Transactions on Magnetics 403443–3449\n[19] Hickey M C and Moodera J S 2009 Phys. Rev. Lett. 102(13) 137601 URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.102.137601\n[20] Mondal R, Berritta M, Carva K and Oppeneer P M 2015 Phys. Rev. B 91174415\nURL http://journals.aps.org/prb/pdf/10.1103/PhysRevB.91.174415\n[21] Mondal R, Berritta M and Oppeneer P M 2016 Phys. Rev. B 94144419 URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.94.144419\n[22] Mondal R, Berritta M and Oppeneer P M 2018 Phys. Rev. B 98(21) 214429 URL\nhttps://link.aps.org/doi/10.1103/PhysRevB.98.214429\n[23] Mondal R, Donges A, Ritzmann U, Oppeneer P M and Nowak U 2019 Phys.\nRev. B 100(6) 060409 URL https://link.aps.org/doi/10.1103/PhysRevB.\n100.060409\n[24] Böttcher D and Henk J 2012 Phys. Rev. B 86020404 URL http://link.aps.\norg/doi/10.1103/PhysRevB.86.020404\n[25] Ciornei M C 2010 Role of magnetic inertia in damped macrospin dynamics Ph.D.\nthesis Ecole Polytechnique, France\n[26] Ciornei M C, Rubí J M and Wegrowe J E 2011 Phys. Rev. B 83020410 URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.83.020410\n[27] Bhattacharjee S, Nordström L and Fransson J 2012 Phys. Rev. Lett. 108(5) 057204\nURL https://link.aps.org/doi/10.1103/PhysRevLett.108.057204\n[28] Wegrowe J E and Ciornei M C 2012 Am. J. Phys. 80607–611 URL http:\n//dx.doi.org/10.1119/1.4709188\n[29] Wegrowe J E and Olive E 2016 J. Phys.: Condens. Matter 28106001 URL\nhttp://stacks.iop.org/0953-8984/28/i=10/a=106001\n[30] Mondal R, Berritta M, Nandy A K and Oppeneer P M 2017 Phys. Rev. B 96(2)\n024425 URL https://link.aps.org/doi/10.1103/PhysRevB.96.024425\n[31] Mondal R, Berritta M and Oppeneer P M 2018 J. Phys.: Condens. Matter 30\n265801 URL http://stacks.iop.org/0953-8984/30/i=26/a=265801\n[32] Thonig D and Henk J 2014 New Journal of Physics 16013032 URL https:\n//doi.org/10.1088/1367-2630/16/1/013032REFERENCES 13\n[33] Giordano S and Déjardin P M 2020 Phys. Rev. B 102(21) 214406 URL https:\n//link.aps.org/doi/10.1103/PhysRevB.102.214406\n[34] Olive E, Lansac Y and Wegrowe J E 2012 Appl. Phys. Lett. 100192407 URL\nhttps://doi.org/10.1063/1.4712056\n[35] Olive E, Lansac Y, Meyer M, Hayoun M and Wegrowe J E 2015 J. Appl. Phys. 117\n213904 URL https://doi.org/10.1063/1.4921908\n[36] Makhfudz I, Olive E and Nicolis S 2020 Appl. Phys. Lett. 117132403 URL\nhttps://doi.org/10.1063/5.0013062\n[37] Cherkasskii M, Farle M and Semisalova A 2020 Phys. Rev. B 102(18) 184432 URL\nhttps://link.aps.org/doi/10.1103/PhysRevB.102.184432\n[38] MondalR,GroßenbachS,RószaLandNowakU2020 Phys. Rev. B 103(10)104404\nURL https://link.aps.org/doi/10.1103/PhysRevB.103.104404\n[39] Neeraj K, Awari N, Kovalev S, Polley D, Zhou Hagström N, Arekapudi S S P K,\nSemisalova A, Lenz K, Green B, Deinert J C, Ilyakov I, Chen M, Bawatna M,\nScalera V, d’Aquino M, Serpico C, Hellwig O, Wegrowe J E, Gensch M and Bonetti\nS 2020 Nat. Phys. URL https://doi.org/10.1038/s41567-020-01040-y\n[40] Kamberský V 1970 Can. J. Phys. 482906\n[41] Kamberský V 1976 Czech. J. Phys. B 261366–1383 URL http://dx.doi.org/\n10.1007/BF01587621\n[42] Kuneš J and Kamberský V 2002 Phys. Rev. B 65212411 URL http://link.aps.\norg/doi/10.1103/PhysRevB.65.212411\n[43] Kuneš J and Kamberský V 2003 Phys. Rev. B 68(1) 019901 URL https://link.\naps.org/doi/10.1103/PhysRevB.68.019901\n[44] Tserkovnyak Y, Brataas A and Bauer G E W 2002 Phys. Rev. Lett. 88117601 URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.88.117601\n[45] Steiauf D and Fähnle M 2005 Phys. Rev. B 72064450 URL http://link.aps.\norg/doi/10.1103/PhysRevB.72.064450\n[46] Fähnle M and Steiauf D 2006 Phys. Rev. B 73(18) 184427 URL https://link.\naps.org/doi/10.1103/PhysRevB.73.184427\n[47] Kamberský V 2007 Phys. Rev. B 76134416 URL http://link.aps.org/doi/10.\n1103/PhysRevB.76.134416\n[48] Gilmore K, Idzerda Y U and Stiles M D 2007 Phys. Rev. Lett. 99027204 URL\nhttp://link.aps.org/doi/10.1103/PhysRevLett.99.027204\n[49] Brataas A, Tserkovnyak Y and Bauer G E W 2008 Phys. Rev. Lett. 101(3) 037207\nURL https://link.aps.org/doi/10.1103/PhysRevLett.101.037207\n[50] Gilmore K, Idzerda Y U and Stiles M D 2008 J. Appl. Phys. 10307D303 URL\nhttp://dx.doi.org/10.1063/1.2832348\n[51] Ebert H, Mankovsky S, Ködderitzsch D and Kelly P J 2011 Phys. Rev. Lett. 107\n066603 URL http://link.aps.org/doi/10.1103/PhysRevLett.107.066603REFERENCES 14\n[52] Edwards D M 2016 Journal of Physics: Condensed Matter 28086004 URL\nhttps://doi.org/10.1088/0953-8984/28/8/086004\n[53] Fähnle M, Steiauf D and Illg C 2011 Phys. Rev. B 84172403 URL http:\n//link.aps.org/doi/10.1103/PhysRevB.84.172403\n[54] Fähnle M, Steiauf D and Illg C 2013 Phys. Rev. B 88219905(E) URL https:\n//link.aps.org/doi/10.1103/PhysRevB.88.219905\n[55] Thonig D, Eriksson O and Pereiro M 2017 Sci. Rep. 7931 URL http://dx.doi.\norg/10.1038/s41598-017-01081-z\n[56] Kamra A, Troncoso R E, Belzig W and Brataas A 2018 Phys. Rev. B 98(18) 184402\nURL https://link.aps.org/doi/10.1103/PhysRevB.98.184402\n[57] Yuan H Y, Liu Q, Xia K, Yuan Z and Wang X R 2019 EPL (Europhysics Letters)\n12667006 URL https://doi.org/10.1209/0295-5075/126/67006\n[58] Pajda M, Kudrnovský J, Turek I, Drchal V and Bruno P 2001 Phys. Rev. B 64(17)\n174402 URL https://link.aps.org/doi/10.1103/PhysRevB.64.174402\n[59] Schlickeiser F, Atxitia U, Wienholdt S, Hinzke D, Chubykalo-Fesenko O and Nowak\nU 2012 Phys. Rev. B 86(21) 214416 URL https://link.aps.org/doi/10.1103/\nPhysRevB.86.214416"    },    {        "title": "2402.06012v1.Balancing_a_3D_Inverted_Pendulum_using_Remote_Magnetic_Manipulation.pdf",        "content": "1\nBalancing a 3D Inverted Pendulum using Remote\nMagnetic Manipulation\nJasan Zughaibi, Bradley J. Nelson, and Michael Muehlebach\nAbstract —Remote magnetic manipulation offers wireless con-\ntrol over magnetic objects, which has important medical ap-\nplications, such as targeted drug delivery and minimally inva-\nsive surgeries. Magnetic manipulation systems are categorized\ninto systems using permanent magnets and systems based on\nelectromagnets. Electro-Magnetic Navigation Systems (eMNSs)\nare believed to have a superior actuation bandwidth, facilitating\ntrajectory tracking and disturbance rejection. This greatly ex-\npands the range of potential medical applications and includes\neven dynamic environments as encountered in cardiovascular\ninterventions. In order to highlight the dynamic capabilities\nof eMNSs, we successfully stabilize a (non-magnetic) inverted\npendulum on the tip of a magnetically driven arm. Our method\nemploys a model-based design approach, where we capture the\ndynamics that describe the interaction of the pendulum system\nand the magnetic field through Lagrangian mechanics. Using\nsystem identification we estimate the system parameters, the\nactuation bandwidth, and characterize the system’s nonlinearity.\nWe design a state-feedback controller to stabilize the inherently\nunstable dynamics, and compensate for errors arising from the\ncalibration of the magnetic field and the angle measurement\nsystem. Additionally, we integrate an iterative learning control\nscheme that allows us to accurately track non-equilibrium tra-\njectories while concurrently maintaining stability of the inverted\npendulum. To our knowledge, this is the first effort to stabilize a\n3D inverted pendulum through remote magnetic manipulation.\nIndex Terms —Inverted pendulum, remote magnetic manipula-\ntion, iterative learning control\nI. I NTRODUCTION\nRemote magnetic manipulation - often referred to as mag-\nnetic actuation - describes the use of magnetic fields to exert\nwireless control over the spatial orientation and positioning of\nmagnetic objects [1]. These objects can vary considerably in\nsize, spanning scales from nanometers to centimeters. There is\nconsiderable promise in utilizing this technology for advanced\nmedical applications [2], [3]. This includes the deployment\nof biocompatible micro- and nanorobots for targeted drug\ndelivery and magnetic continuum robots, such as catheters and\nguidewires, for minimally invasive surgeries [4], [5].\nA system that is used to control a magnetic object in space is\noften referred to as a magnetic manipulation system. Magnetic\nmanipulation systems can be broadly categorized into systems\nthat rely on the motion of permanent magnets and systems\nthat generate magnetic fields using a set of electromagnets,\nJasan Zughaibi and Bradley J. Nelson are with the Multi-Scale Robotics\nLab, ETH Zurich, 8092 Zurich, Switzerland (e-mail: zjasan@ethz.ch; bnel-\nson@ethz.ch).\nMichael Muehlebach is with the Learning and Dynamical Systems Group,\nMax Planck Institute for Intelligent Systems, 72076 T ¨ubingen, Germany (e-\nmail: michael.muehlebach@tuebingen.mpg.de).\nFig. 1. 3D inverted pendulum balanced using the OctoMag electromagnetic\nmanipulation system. The system includes an arm driven by the external\nmagnetic field, and a non-magnetic pendulum, free to rotate spherically. Both\narms are connected through a (non-magnetic) spherical joint.\nso-called electro-Magnetic Navigation Systems (eMNSs). Sys-\ntems such as the Stereotaxis Genesis® and Niobe® use robotic\narms manipulating permanent magnets with a mass of the or-\nder of several hundred kilograms [6]. The high inertia restricts\nthe physical motion and results in low actuation bandwidth\n[7]. However, there are also established systems employing\nrobotic arms with much lighter permanent magnets, with some\ndesigns featuring magnets weighing 1.5 kg or 7 kg [8], [2].\nThe potential impact of these lighter systems on actuation\nbandwidth remains an unexplored area of research, as no\nstudy has investigated or directly compared the actuation band-\nwidth of eMNSs and systems that utilize permanent magnets.\nNonetheless, eMNSs are generally understood to provide a\nhigher bandwidth, which offers several benefits, including im-\nproved trajectory tracking and superior disturbance rejection.\nThis becomes crucial in dynamic environments encountered in\ncardiovascular interventions, such as cardiac ablation therapy,\nwhere precise movement is required [9].\nIn this letter, we explore the dynamic capabilities of\neMNSs by demonstrating their capability of stabilizing a (non-\nmagnetic) inverted pendulum on the tip of a magnetically\ndriven arm (see Fig. 1). A video of the system in operation\nis available here: https://youtu.be/fNWS-9-lD84 . InarXiv:2402.06012v1  [eess.SY]  8 Feb 20242\nthe field of control theory, the inverted pendulum is a classi-\ncal and extensively studied system with numerous practical\nimplementations [10], [11], [12]. It serves as a benchmark for\nnovel control architectures and is of high educational value\n[13]. Due to its inherent instability, a successful stabilization\nnecessitates a sufficiently high actuation bandwidth.\nBeyond merely demonstrating the dynamic potentials of\neMNSs, the solution to this challenging control problem offers\nvaluable insights into magnetic control strategies, particularly\nwhen dealing with the inherent strong nonlinearities of mag-\nnetic fields. Such insights could be adapted for a variety of\ntasks, making it an intriguing testbed for the study of novel\nmagnetic control algorithms. To the best of our knowledge,\nthis study represents the first effort to stabilize a 3D inverted\npendulum using remote magnetic manipulation.\nOur objective is to control the magnetically driven arm\nusing an external magnetic field, such that the (non-magnetic)\npendulum on top of the arm maintains in its upright equilib-\nrium position. To produce the magnetic field, we utilize the\nOctoMag system [14], which comprises eight electromagnets,\nthat will be referred to as coils throughout this paper. It\nshould be noted that, theoretically, achieving this task could\nbe possible with fewer than eight coils. This is because we\nprimarily exploit the orientation of the magnetic field vector\nto exert torques, assuming a homogeneous magnetic field in\nthe proximity of the center of the workspace [15].\nA. Methodology\nWe use a model-based design approach for designing the\ncontrol algorithm. First, the dynamics describing the interac-\ntion between the magnetic field and the pendulum system are\ncaptured through Lagrangian mechanics, by incorporating the\npotential energy of the magnetic field into the Lagrangian.\nWe then perform system identification experiments in order to\nestimate the unknown parameters of our analytical model. Our\nidentification procedure enables not only a precise estimation\nof the system’s frequency response and its actuation band-\nwidth, but also provides an uncertainty estimate that captures\nthe system’s nonlinearity.\nWe then design a state-feedback controller based on our\nidentified dynamical model that stabilizes the pendulum and\nrejects disturbances. In addition, we introduce model-based\ntechniques that offer online compensation for errors arising\nfrom the calibration of the magnetic field and the calibration\nof the angle measurement system.\nOur work goes well beyond the stabilization at equilibrium\nand also designs reference tracking controllers. These are\ncapable of following dynamic non-equilibrium motions with\nthe magnetic arm, while balancing the pendulum in its upright\nposition. To that extent, we augment the control system with\nan Iterative Learning Control (ILC) scheme, that operates in\nparallel with the state-feedback controller. Our ILC scheme\nleverages information from the identified dynamic model\nand computes a feedforward correction signal based on past\ntracking error. The ILC scheme learns to compensate for all\nrepetitive tracking errors, by repeatedly executing the same\ntask.B. Related Work\nA common assumption in the field of magnetic manipu-\nlation systems is that dynamic effects can be neglected for\nmodeling and control, resulting in quasi-static models [16],\n[17]. Although this assumption may be adequate for a variety\nof medical applications, there is great potential for utilizing\ndynamic models and dynamic control algorithms.\nThe authors of [18] derive a dynamic model that captures\nthe interaction of an external, robotically steered, permanent\nmagnet and a magnetic endoscope. This model serves as\nthe foundation for synthesizing dynamic control strategies,\nenabling the magnetic tip of the tethered endoscope to be ele-\nvated while navigating along curved paths. In a complementary\nstudy, a similar experiment is demonstrated utilizing a model\npredictive controller. This predictive controller accounts for the\ndynamics and system constraints by formulating an optimal\ncontrol problem that is solved in explicit form [19].\nIn [20], the dynamics of an MRI-guided catheter are iden-\ntified and analyzed. The authors approximate the dynamics\nof the catheter using a 2D linear dynamical black-box model,\nidentified by exciting the system with a chirp signal. The work\nof [21] derives a dynamic model of a microrobot controlled\nby an eMNS. Parameter uncertainties are incorporated into\nthe model, leading to the synthesis of an H∞robust control\nalgorithm. The authors of [22] develop a dynamic model of a\nmicrorobot, actuated by a pair of Helmholtz coils. The model\nis utilized as a simulation tool to understand the influence of\ndesign parameters on the microrobot’s motion behavior.\nThe application of ILC to improve the tracking performance\nof inverted pendulum systems has been explored by several\nresearchers in the field. The work of [23] employs a frequency-\ndomain ILC scheme to enhance the tracking accuracy of a\nquadrocopter while maintaining the stability of an inverted\npendulum. Their approach leverages the periodicity of the\ntrajectories to decompose the correction and error signal using\na Fourier series, resulting in an ILC algorithm that has low\ncomputational cost. A more general approach that parametrizes\ninputs and state trajectories with basis functions has been\nproposed in [24], where similar robotic testbeds have been\nused. In [25], the potential of proportional-derivative-type ILC\nschemes is explored, implemented in both serial and parallel\nconfigurations, aiming to enhance the tracking performance of\nan inverted pendulum system mounted on a cart.\nC. Outline\nThe remainder of this document is organized as follows:\nSec. II details the experimental setup and defines all relevant\nquantities, including control inputs, parameters, and coordi-\nnates. The modeling process and parameter identification is\npresented in Sec. III. The resulting model allows synthesizing\na state-feedback controller and a compensation algorithm that\nremoves calibration errors from the actuation and sensors, as\npresented in Sec. IV. The synthesis of the ILC, which greatly\nimproves the trajectory tracking capabilities, is discussed in\nSec. V. Finally, a conclusion is drawn in Sec. VI.3\nLϕθ\nαβ\nuαuβ\n/lscriptm/lscript\nM\nmj\nm\nbmm\nexezey\ng\n|b|= const.\nFig. 2. The figure shows the parametrization of the orientations in the system,\ndescribing rotations with respect to the inertial frame of the motion capture\nsystem. The actuator is parametrized by α, β, and the inverted pendulum by\nφ, θ. The control input to the system is the orientation of the magnetic field\nvector b, parametrized by the angles uα, uβ. Rotations described by uα, α,\nandφare with respect to the inertial eyaxis, whereas uβ, β,andθcorrespond\nto the inertial exaxis. The length of the pendulum is L= 405 mm , the length\nof the actuator ℓ= 218 mm . The distance between the pivot point and the\ncenter of mass of the magnets is ℓm= 38 mm . The mass of the pendulum\nisM= 4.4 g, the mass of the actuator m= 2.4 g. The mass of the joint is\nmj= 2.0 g, the mass of the magnets mm= 12.7 g.\nII. E XPERIMENTAL PLATFORM\nThe following section describes the experimental setup used\nfor stabilizing the inverted pendulum. Furthermore, we define\nthe control inputs and their relation to the electrical currents\nof the OctoMag system.\nThe pendulum assembly consists of two commercially avail-\nable rods made from carbon-fiber reinforced thermoplastics.\nWe refer to the lower arm as the actuator and the upper\narm as the (non-magnetic) pendulum . Ten axially magnetized\npermanent magnets (Magnetkontor® R-08-03-04-N3-N, Nd-\nFeB N45, \u001f8 (3) x 4 mm) are mounted proximally to the\nactuator’s pivot point, as illustrated in Fig. 2. The actuator is\nconnected to a base plate using a non-magnetic u-joint (303\nstainless steel, McMaster-Carr® 60075K75), that allows the\nactuator to rotate spherically. We use the same type of u-\njoint to connect the actuator with the pendulum, resulting in\na total of four (angular) degrees of freedom. The actuator’s\norientation is defined by the angles, α, β, while the orientation\nof the pendulum is described by φ, θ, as illustrated in Fig. 2.\nReflective marker stripes are attached at the actuator and\nthe pendulum rod which allows us to retrieve all angles at a\nfrequency of 100 Hz using a motion capture system.\nA. Magnetic Field Allocation\nWe define the control inputs to the system as the angles,\nuα, uβ, that parametrize the orientation of the magnetic fieldvector, b∈R3, in an inertial coordinate frame (see Fig. 2).\nThe relation between uα, uβandbis given by\nΨb: (uα, uβ,|b|)7→b:\nbx\nby\nbz\n=|b|\nsin(uα) cos( uβ)\nsin(uβ)\ncos(uα) cos( uβ)\n,\nwhere |b|is a free parameter that can be specified by the user.\nWe keep the magnitude of the magnetic field vector constant\nat all times, specifically |b|= 35 mT . From a control systems\nperspective, it is advantageous to use the angles, uα, uβ, as\ncontrol inputs, since they are directly connected to the angular\ndeflections, α, β, of the actuator. This allows us to consider\nthe dynamics as decoupled, simplifying our control strategy\n(see section below).\nFor eMNSs it is commonly assumed to rely on a linear\nrelation between electrical currents, i∈R8, the field, b, and\nits gradient, g=\u0010\n∂bx\n∂x∂bx\n∂y∂bx\n∂z∂by\n∂y∂by\n∂z\u0011⊤\n, i.e.\n\u0014b\ng\u0015\n=Ai, (1)\nwhereA∈R8×8is the so-called actuation matrix at the\ncentre of the magnetic workspace [26]. Notice, that five\ngradient terms uniquely describe the gradient, as there are\nfour constraints resulting from the (static) Maxwell equations,\nnamely ∇ × b= 0 and∇ ·b= 0. In general, the matrix\nAdepends on the configuration of the coils and can be\ndetermined through a calibration procedure as described in\n[1]. Given a desired field, b, the desired currents are given by\ni=A†\u0014b\n0\u0015\n, (2)\nwhere †denotes the Moore-Penrose pseudoinverse. By setting\ng=0we achieve (approximately) a homogeneous, i.e.\ngradient-free, magnetic field in the vicinity of the centre of\nthe magnetic workspace.\nIII. M ODELING\nIn this section, the modeling process and parameter estima-\ntion for the combined actuator-pendulum system are described.\nThe modeling process consists of two distinct stages. First, we\ndevelop an analytical model that characterizes the dynamics\nof the actuator-pendulum system and its interaction with the\nexternal magnetic field. Subsequently, system identification\nexperiments are conducted in order to obtain a data-driven\nestimation of the unknown parameters in the analytical model.\nA. Analytical Model\nLinearizing the nonlinear dynamics of a 3D inverted pen-\ndulum around its upright equilibrium yields two decoupled\n2D linear models. This decoupling suggests that the full 3D\nbehavior can be effectively understood by examining these\nindividual 2D systems separately. Consequently, for clarity and\nbrevity, we focus our attention on deriving the dynamics for\none of these 2D planes, specifically the (α, φ)-plane, using\nLagrangian mechanics with (α, φ)as generalized coordinates.\nIt should be emphasized that the methods presented here can\nbe analogously applied to the dynamics in the (β, θ)-plane. We4\naccount for the system’s interaction with the external magnetic\nfield by incorporating its potential energy in the Lagrangian.\nThe potential energy of a magnetic dipole in an external (2D)\nfield is given by\nUm=−|˜m||b|cos(uα−α), (3)\nwhere ˜m(inAm2) denotes the magnetic dipole moment1.\nThe Lagrangian of the system reads as, L=T−U, where\nTandUare the kinetic and potential energy, respectively:\nT=1\n2(J+Mℓ2) ˙α2+1\n8ML2˙φ2+1\n2MℓL ˙α˙φcos(α−φ)\nU= (η+Mℓ)gcosα+MgL\n2cosφ+Um,\nwhere we introduce the abbreviations, Jandη, defined as\nJ:=mmℓ2\nm+mℓ2\n4+mjℓ2, η :=mmℓm+mℓ\n2+mjℓ.\nThe nonlinear equations of motion can be derived by applying\nthe Lagrange formalism, namely\nd\ndt\u0012∂L\n∂˙α\u0013\n−∂L\n∂α=Qnc\nα,d\ndt\u0012∂L\n∂˙φ\u0013\n−∂L\n∂φ= 0,\nwhere Qnc\nαrepresent the non-conservative generalized forces\nacting on the actuator. An essential detail to note is the\npresence of a small damping in the system. This damping\nprimarily arises from the u-joint connecting the actuator to its\nbase and is characterized as being proportional to the angular\nvelocity, i.e., Qnc\nα=−d˙α. The presence of this damping\nis essential for representing the system’s dynamic behavior.\nWe linearize the nonlinear dynamics around the (upright)\nequilibrium (α, φ, u α) = (0 ,0,0), resulting in\nM\u0014¨α\n¨φ\u0015\n+D\u0014˙α\n˙φ\u0015\n+K\u0014α\nφ\u0015\n=wuα, (4)\nwhere\nM=\u0014\nJ+Mℓ2 1\n2MℓL\n1\n2MℓL1\n4ML2\u0015\nK=\u0014−(η+Mℓ)g+|˜m||b| 0\n0 −MgL\n2\u0015D=\u0014\nd0\n0 0\u0015\nw=\u0014|˜m||b|\n0\u0015\n.\nThe term |˜m||b|in the stiffness matrix, K, can be interpreted\nas the magnetic field providing a stiffness-like effect on the\nactuator. We rewrite (4) as a first order state space model, by\nintroducing the state xα:=\u0002α φ ˙α˙φ\u0003⊤, namely\n˙xα=\u00140 I\n−M−1K−M−1D\u0015\n| {z }\n:=Acαxα+\u00140\nM−1w\u0015\n|{z}\n:=Bcαuα.(5)\nThe linearized (continuous-time) dynamics of the 3D pendu-\nlum then read as\u0014˙xα\n˙xβ\u0015\n=\u0014Ac\nα0\n0 Ac\nβ\u0015\u0014xα\nxβ\u0015\n+\u0014Bc\nα0\n0 Bc\nβ\u0015\u0014uα\nuβ\u0015\n, (6)\nwhere xβ:=\u0002\nβ θ ˙β˙θ\u0003⊤. For the sake of brevity, let x\nrepresent either xαorxβfor the remainder of this document.\n1To make the distinction clearer, we use ˜mfor the magnetic dipole moment,\ndifferentiating it from m, which represents a mass term in this paper.B. Parameter Estimation\nThe mass-related parameters in (4) can be obtained using\na weighing scale. Similarly, the geometric parameters can be\nderived from a computer-aided design model. The magnitude\nof the magnetic field vector, |b|, is (approximately) known,\nbased on the eMNS calibration. To estimate the remaining\nparameters, dand|˜m|, we perform system identification\nexperiments. An accurate estimate of these parameters is\ncrucial, as they substantially influence the system dynamics.\nDue to the unstable nature of the inverted pendulum dynamics,\nthe system identification experiments have to be performed\nwithout pendulum attached. However, it should be noted that\ndand|˜m|solely affect the actuator dynamics as shown in (4).\nThe remainder of this paragraph is closely inspired by\nthe system identification methodology presented in [27]. We\nperform a system identification in the frequency domain, using\na random-phase multisine signal to excite the system. The\nsignal is designed such that the magnitude of its Fourier\ntransform is constant in the frequency range 0-10 Hz, allowing\nto excite all frequencies that are in the range of interest. As\nthe phase shift of the sinusoids is drawn randomly, different\ntime-domain realizations can be created, while preserving an\nidentical amplitude spectrum in the frequency domain. This\ncharacteristic allows us to identify a standard deviation of the\nfrequency response, ˆσnl(jω), which can be interpreted as a\nmetric that captures the nonlinearity of the system.\nWe create r= 10 different excitation signals. To increase\nthe signal-to-noise ratio, each excitation signal is repeated\nfor ten consecutive periods, where the first four periods are\ndiscarded to minimize the influence of transients, resulting in\np= 6effective periods. Let U[i,l]\nSP(jωk)andY[i,l](jωk)denote\nthe discrete Fourier transform of u[i,l]\nSPandy[i,l], which either\nrepresents u[i,l]\nα,SPandα[i,l]oru[i,l]\nβ,SPandβ[i,l], respectively. The\nindex i= 1, . . . , p represents the index of the period within\nexcitation signal l, with l= 1, . . . , r . The averaged Fourier\ntransform for excitation signal lis then given by\nY[l](jωk) =1\nppX\ni=1Y[i,l](jωk). (7)\nNotice that U[l]\nSP(jωk) = U[i,l]\nSP(jωk),∀i= 1, . . . , p , as the\nFourier transform is applied to the (noise-free) input set-point,\nuSP, which is identical for all periods within signal l. The\nempirical transfer function estimate for excitation signal land\nits average are then given by\nˆG[l](jωk) =Y[l](jωk)\nU[l](jωk), ˆGBLA(jωk) =1\nrrX\nl=1ˆG[l](jωk),\nwhere ˆGBLA denotes the best linear approximator [28]. The\nsample variance ˆσ2\nnlreads as\nˆσ2\nnl(jωk) =1\nr(r−1)rX\nl=1\f\f\fˆG[l](jωk)−ˆGBLA(jωk)\f\f\f2\n.\nIn Fig. 3 the identified frequency response of the system,\nˆGBLA, is depicted along with its uncertainty, ˆσnl. Note that\naround the resonance frequency, the uncertainty associated\nwith the estimate is elevated. We use this information to5\nFig. 3. Bode diagram of the actuator dynamics in αdirection, which is\nobtained from system identification experiments. Similar results are obtained\nfor the βdirection. ˆσα,nlis a metric that captures the uncertainty of the\nestimate. It can be seen that the uncertainty is high in the neighborhood of\nthe resonance frequency. We map ˆσα,nlto values between (0,1], which are\nused as weights (cyan) during the curve fitting to obtain a parametric transfer\nfunction estimate (in magenta). The fitted transfer function is of second order\nand accounts for a delay e−sT, that causes additional phase shift.\nderive a frequency-dependent weight, W(ωk), for the curve-\nfitting process. Specifically, in regions of small uncertainty,\nthe values of ˆσnlare mapped to weights close to 1, while in\nhigh uncertainty zones, they approach a value close to 0, as\nshown in Fig. 3. We fit the parametric model\nG(s) =e−sT b0\ns2+a1s+a0(8)\nusing the tfest function in Matlab®. Notice that we restrict\nthe model fit to a second order model such that we can map\nthe parameters, a1, b0, to the actuator dynamics of (4) (with\nM= 0, i.e. pendulum unattached), which are directly related\nto the parameters dand|˜m|. All parameters of the model (6)\nare now fully determined.\nIV. S TATE -FEEDBACK CONTROL\nIn this section, we discuss the implementation of a state-\nfeedback controller, that stabilizes the pendulum in upright\nposition. A prefilter is designed for reducing steady-state errors\nwhen tracking non-zero setpoints. Furthermore, we analyze the\nsensitivity of the system to errors resulting from the calibration\nof the magnetic field and the motion capture system. We\nintroduce an online compensation for these errors that is\nexecuted while balancing the pendulum.\nA. Controller Design\nWe discretize the continuous-time dynamics in (6) with a\nsampling time of Ts= 10 ms using an exact-discretization for\nlinear systems, which we denote by\n\u0014xα[k+ 1]\nxβ[k+ 1]\u0015\n=\u0014Aα0\n0 A β\u0015\u0014xα[k]\nxβ[k]\u0015\n+\u0014Bα0\n0 B β\u0015\u0014uα[k]\nuβ[k]\u0015\n.The associated measurement equation is given by\n\u0014yα[k]\nyβ[k]\u0015\n=\u0014Cα0\n0 C β\u0015\u0014xα[k]\nxβ[k]\u0015\n,Cα=Cβ=\u00141 0 0 0\n0 1 0 0\u0015\n.\nFor the sake of brevity, let A,B,Crepresent either\nAα,Bα,CαorAβ,Bβ,Cβfor the remainder of this docu-\nment. For each plane (α, φ),(β, θ)(the linearized dynamics\nare decoupled) we design a Linear Quadratic Regulator (LQR)\nthat stabilizes the system around its upright equilibrium.\nFurthermore, the system can be stabilized around non-zero set-\npoints, xα,SP,xβ,SP, provided that they satisfy the equilibrium\nconditions and remain near the system’s upright equilibrium.\nThe feedback policy, at time step k, is given by\n\u0014uα[k]\nuβ[k]\u0015\n=\u0014Kα0\n0 K β\u0015\u0014xα,SP[k]−xα[k]\nxβ,SP[k]−xβ[k]\u0015\n, (9)\nwhere we calculate the angular velocities in xαandxβusing\nfinite differences. The gain matrices, Kα,Kβ, are obtained\nfrom the associated discrete-time algebraic Riccati equation.\nB. Prefilter Design\nTo ensure minimal steady-state errors while tracking non-\nzero set-points, a prefilter, F, is implemented. The main\npurpose of the prefilter is to scale the setpoints, as illustrated in\nthe blockdiagram in Fig. 5, such that the errors are minimized\nwhen the system is in steady-state.\nIt can be shown that only the actuator states, α, β, are\noutput controllable. Output controllability does not apply for\nthe pendulum states, φ, θ. Hence, we introduce the matrix,\n˜Cα=˜Cβ=\u00021 0 0 0\u0003\n, relating the state, x, to the output\ncontrollable states, αandβ. The prefilter is expressed as\nF=\u0012\n˜C¯A−1BK˜C⊤\u0013−1\n,with ¯A:=I−A+BK,(10)\nwhere Frepresents either FαorFβ.\nC. Compensation of Angle Calibration Errors\nAll angle measurements are relative to the inertial coordi-\nnate frame of the motion capture system. The frame’s posi-\ntioning is defined during the calibration procedure by placing\na precisely-manufactured tool with known reflective marker\npositions. However, during this procedure, minor deviations\ncan arise, resulting for example in the zaxis not being\nperfectly aligned with the gravitational vector. We can analyze\nthe sensitivity of such a misalignment on the steady-state\nbehavior. The misalignment angle ξcan be interpreted as a\nconstant, additive disturbance, d=\u0002\nξ ξ 0 0\u0003⊤, acting on\nthe state. The steady-state relation from dtoxssis given by\nxss=−¯A−1BKd, (11)\nEvaluating (11) for a value of ξ= 1 °, results in xss=\u0002−5.8°0.0 0.0 0.0\u0003⊤. That is, the system exhibits an am-\nplification of the disturbance, where a (constant) perturbation\nin the measurements leads to an angular deflection of the\nactuator that is almost an order of magnitude larger than the\noriginal disturbance.6\nHowever, we can estimate the misalignment, ξ, in an online\nmanner. In fact, when the pendulum is balanced in steady-\nstate, it must be perfectly aligned with the gravitational vector.\nHence, by low-pass filtering the pendulum angle, φss, we\nobtain a direct estimate of the misalignment, ξ. By subtracting\nthe learned steady-state offset, φss, from the measured angles,\nα[k]←α[k]−φss[k], φ [k]←φ[k]−φss[k] (12)\nwe greatly reduce the actuator deflection error when balancing\nthe pendulum. A similar procedure is applied in [29].\nD. Compensation of Magnetic Field Calibration Errors\nIn the process of determining the actuation matrix, A,\nminor inaccuracies can be encountered, which result in a small\ndiscrepancy between the desired and the actual magnetic field.\nWe model this discrepancy as an additive disturbance, denoted\nasud, which impacts the control inputs, uαanduβ. The\nsteady-state relation from udtoxsscan be expressed as\nxss=−¯A−1Bud. (13)\nEvaluating (13) for a value of ud= 1°, results in xss=\u0002\n−0.9°0.0 0.0 0.0\u0003⊤. This indicates that the system is\nless responsive to a disturbance in the control input than to a\ndisturbance in the control output, as detailed in the preceding\nparagraph. Using (13), we can obtain a model-based estimate\nofud, namely\nˆud[k] =−\u0000¯A−1B\u0001†xss[k] (14)\nwhere xssis the steady-state of the system, that is corrected\nusing (12). This equation essentially provides an estimation\nof the angular misalignment between the desired and the\nactual magnetic field vector, based on prior information of\nthe identified model and real-time measurements. We can\ncompensate for this misalignment by\nu[k]←u[k]−ˆud[k]. (15)\nThe experimental results of successfully stabilizing the in-\nverted pendulum are shown in Fig. 4, comparing its behavior\nwith and without the application of the presented offset com-\npensation algorithms. The offset compensation methods yield\neffects comparable to those of an integral controller, allowing\nto significantly reduce the steady-state error. However, in\ncontrast to an integral controller the methods introduced here\noffer the benefit of delivering a distinct physical understanding.\nNotice, however, that a small error remains. The compensation\nof this error, which primarily arises due to model mismatch,\nis subject of the next section.\nV. I TERATIVE LEARNING CONTROL\nIn this section, we present an ILC formulation that enables\ncompensation of any repetitive errors that arise when tracking\nperiodic reference trajectories while simultaneously balancing\nthe pendulum. While the previously introduced state-feedback\ncontroller ensures the stabilization of the system and the\nrejection of (non-repeatable) disturbances, the ILC scheme\npresented in this section enhances the performance of the sys-\ntem by counteracting all disturbances that occur repetitively.\nFig. 4. Experimental results of the stabilization of the inverted pendulum, con-\ntrasting its performance before and after the activation of offset compensation\nalgorithms. The transition to the compensation mechanism is delineated by the\ndashed line, indicating a significant decrease in the steady-state error in α. The\ndepicted experiment was replicated for twelve distinct positions of the motion\ncapture system’s calibration tool, each resulting in slightly different calibration\nerrors. Despite these discrepancies, the errors are corrected through online\nlow-pass filtering of the pendulum’s angle, φss. Consequentially, the estimated\nmagnetic field calibration error, ˆud, converges to the same values, independent\nof the motion capture calibration. Notice that the plotted variables, φ−φss\nandα−φss, represent the respective angles relative to the gravitational vector\npost-activation of the offset compensation. Results for the (α, φ)-plane are\ndepicted, with analogous outcomes observed for (β, θ).\nILC has been recognized as a powerful tool in scenarios\nwhere tasks are repetitive, such as tracking periodic reference\ntrajectories [30]. The core underlying principle is the iterative\nimprovement of the system’s performance by leveraging error\nfeedback from previous repetitions. In the following, we\nadopt a norm-optimal ILC formulation, wherein the learning\ncomputes a correction signal after the completion of each\niteration by solving a quadratic optimization problem. This\ncomputation explicitly incorporates prior knowledge derived\nfrom our identified dynamic model. The design we propose\nis implemented in a parallel architecture, with the correction\nsignal being added as a feedforward term to the state-feedback\ncontroller’s output, as visualized in Fig. 5. Drawing upon\nthe decoupled architecture of our state-feedback controller,\nit consequently follows that our ILC architecture is also\ndecoupled. This results in two independent ILC schemes,\nrunning independently for the respective planes, (α, φ)and\n(β, θ). Despite its decoupled configuration, the ILC architec-\nture maintains the proficiency to mitigate coupling effects,\nas the coupling dynamics become apparent through repetitive\nerrors. Assuming one iteration has Ntimesteps, we stack the\ncontrol inputs, measurements, and setpoints of iteration nin\nthe following vectors,\nun\nα:= [un\nα[0], . . . , un\nα[N−1]]⊤\nyn\nα:= [αn[0], φn[0], . . . , αn[N−1], φn[N−1]]⊤\nyα,SP:= [αSP[0], φSP[0], . . . , α SP[N−1], φSP[N−1]]⊤,\nand similarly for the (β, θ)-plane. Here, un\nαis the correction\nsignal and yn\nαis the measured output within iteration n.\nThe setpoints, yα,SP, are identical for each iteration and\nhence are independent of the iteration index n. Note that\nφSP[k] = 0∀k= 0, . . . , N −1, which implies that we aim at7\nILC\n/bracketleftbigg\nFα 0\n0Fβ/bracketrightbigg /bracketleftbigg\nKα0\n0 Kβ/bracketrightbigg α\nβϕ\nθ|b| g=0\nPID coil 1\ncoil 8...Ψbby\nxα\nxβA†Ψxxα,SP... ......xβ,SP−\n−− αSP\nβSPPID−actuator/\npendulum\nsystem\nf[k]−f[k−1]\nTsbx\nbzuα\nuβun\nαun\nβ\n1\nFig. 5. Block diagram of the system illustrating the cascaded control structure. The prefilters, Fα, Fβ, are designed to reduce steady-state errors by scaling the\nsetpoints appropriately. Setpoints, αSP, βSP, are mapped to the state vector using Ψx,α:αSP7→\u0002αSP 0 ˙αSP 0\u0003⊤, and similarly for βSP. The state\nfeedback controllers, Kα,Kβ, operate at 100 Hz determining the magnetic field orientation, uα, uβ, which is converted to the magnetic field vector using\nthe allocation Ψb. The magnetic field vector is then converted into electrical currents using the actuation matrix, A†. The electrical currents are controlled\nby eight independent PID controllers (drivers). Full state information is derived using finite-difference differentiation. An Iterative Learning Control (ILC)\nscheme is included calculating feedforward correction signals, un\nα, un\nβ, to counteract any repetitive error during the tracking of periodic reference trajectories.\nkeeping the pendulum in upright equilibrium, when tracking\nnon-zero setpoints, αSP[k], with the actuator. For the sake\nof brevity, let un,yn,yn\nSPrepresent either un\nα,yn\nα,yn\nα,SPor\nun\nβ,yn\nβ,yn\nβ,SPin the following. The dynamics in the lifted state\nspace can then be written as\nyn=Pun+PySP, (16)\nwhere P∈R2N×Nis defined by\nPi,j=(\nC(A−BK)i−jB,ifi≥j\n0, otherwise ,(17)\nwhere i, j= 1, . . . , N . Let en:=ySP−ynbe the error in the\nn-th iteration. Inspired by [31], we formulate the following\nobjective function\nJn+1(un+1) =ween+1⊤en+1+\n(un+1−un)⊤(un+1−un) +w˙uun+1⊤N⊤Nun+1.\nIn this formulation, we≥0andw˙u≥0are scalar, fixed\nparameters. These parameters are used to weight the relative\nimportance of the three main terms in the cost function with\nrespect to each other. The first term penalizes the predicted\nerror of the next iteration, the second term penalizes changes\nin the correction signal from iteration to iteration. The third\nterm penalizes changes within the correction signal leveraging\nthe derivative operator, N∈RN×N, that approximates the\nderivative of un+1based on finite differences, defined as\nNi,j=−δi,j+δi,j−1,∀i, j= 1, . . . , N, (18)\nwithδi,jbeing the Kronecker delta. Notice, that the optimiza-\ntion is unconstrained which allows us to calculate the optimal\nsolution in closed form,\nun+1∗= arg min\nun+1∈RN{Jn+1(un+1)} (19)\n=Qun+Len, (20)\nQ= (weP⊤P+I+w˙uN⊤N)−1(weP⊤P+I)\nL= (weP⊤P+I+w˙uN⊤N)−1P⊤we.\nHence, at the end of each iteration, the correction signal for\nthe next iteration, un+1, is calculated by evaluating (20), with\nthe precomputed matrices, Q∈RN×N, andL∈RN×2N.\nFig. 6. Experimental results tracking a circular and a figure eight trajectory,\nwithout learning (left) and with activated learning control scheme (right),\nwhere iteration 0 corresponds to no learning. The ILC scheme is able to\ncompensate for all repetitive disturbances. Due to the inherent instability of the\ninverted pendulum and the torque limitations resulting from the low magnetic\nfield of 35 mT, non-repeatable disturbances persist.\nIn Fig. 6, we present the experimental results obtained after\nfour iterations of learning with ILC. The proposed method\neffectively compensates for repetitive errors. Specifically, the\nroot-mean-square error is reduced from 1.7° during the initial\niteration (without learning) to 0.5° by the fourth iteration\nduring the circular maneuver. Notably, the dominant source of\nthese repetitive errors are unmodelled disturbances from the\nmagnetic field gradients. While nonlinearities resulting from\ndeviations from the upright linearization point also contribute,\ntheir effect is comparatively minor. In our case, for ampli-\ntudes exceeding approximately 8°, the gradient forces grow\ndisproportionately strong, challenging the corrective capability\nof the ILC. However, for movements within the 8° range,\nthe ILC demonstrates effective compensation by adjusting the\ndirection of the magnetic field vector, thereby counteracting\nthe gradient-induced errors to a notable extent.8\nVI. C ONCLUSION\nThis letter highlights the dynamic capabilities of an eMNS\nby successfully realizing and stabilizing a 3D inverted pen-\ndulum. A dynamic model is identified, which is used to\nsynthesize a feedback controller that stabilizes the inherently\nunstable pendulum system. Further, the model is used to\ncompensate for errors resulting from the calibration of the\nmagnetic field and the angle measurement system. An ILC\nscheme is implemented to improve trajectory tracking while\nsimultaneously maintaining the pendulum in upright position.\nThe dynamic modeling and control strategies presented in\nthis article generalize to tasks well beyond balancing a pendu-\nlum, offering applicability for a range of objects controlled by\nmagnetic manipulation. The techniques successfully leverage\nthe high actuation bandwidth of eMNSs. This is crucial for\nenabling new medical applications, such as cardiac ablation,\nwhere precise and responsive control is essential.\nIn this work, the direction of the magnetic field vector\nserves as the control input, whereas magnetic field gradients\nare treated as unmodelled disturbances. The proposed control\narchitecture compensates for these disturbances during non-\nequilibrium trajectory tracking, albeit within a defined radius.\nBeyond this radius, the magnetic field gradients intensify\nexcessively. Future research will explore employing magnetic\nfield gradients as additional control inputs, thereby harnessing\nthe full potential of all eight coils in the OctoMag system.\nACKNOWLEDGMENTS\nThe authors would like to thank Hao Ma for sharing his\nknowledge on system identification and the team of Mag-\nnebotiX AG for the technical support. Further thanks go to the\nMax Planck ETH Center for Learning Systems for the financial\nsupport. Michael Muehlebach thanks the German Research\nFoundation and the Branco Weiss Fellowship, administered\nby ETH Zurich, for the financial support.\nCONFLICT OF INTERESTS\nBradley Nelson is co-founder of MagnebotiX AG, which\ncommercializes the OctoMag system. The other authors de-\nclare no conflict of interest.\nREFERENCES\n[1] J. J. Abbott, E. Diller, and A. J. Petruska, “Magnetic methods in\nrobotics,” Annual Review of Control, Robotics, and Autonomous Systems ,\nvol. 3, no. 1, 2020.\n[2] Y . Kim, E. Genevriere, P. Harker, J. Choe, M. Balicki, R. W. Regenhardt,\nJ. E. Vranic, A. A. Dmytriw, A. B. Patel, and X. Zhao, “Telerobotic neu-\nrovascular interventions with magnetic manipulation,” Science Robotics ,\nvol. 7, no. 65, 2022.\n[3] J. Hwang, J.-y. Kim, and H. Choi, “A review of magnetic actuation\nsystems and magnetically actuated guidewire-and catheter-based micro-\nrobots for vascular interventions,” Intelligent Service Robotics , vol. 13,\n2020.\n[4] J. Li, B. E.-F. de ´Avila, W. Gao, L. Zhang, and J. Wang, “Mi-\ncro/nanorobots for biomedicine: Delivery, surgery, sensing, and detoxi-\nfication,” Science Robotics , vol. 2, no. 4, 2017.\n[5] Y . Kim, G. A. Parada, S. Liu, and X. Zhao, “Ferromagnetic soft\ncontinuum robots,” Science Robotics , vol. 4, no. 33, 2019.\n[6] D. V . Kladko and V . V . Vinogradov, “Magnetosurgery: Principles, design,\nand applications,” Smart Materials in Medicine , vol. 5, no. 1, 2023.[7] Z. Yang and L. Zhang, “Magnetic actuation systems for miniature robots:\nA review,” Advanced Intelligent Systems , vol. 2, no. 9, 2020.\n[8] G. Ciuti, P. Valdastri, A. Menciassi, and P. Dario, “Robotic magnetic\nsteering and locomotion of capsule endoscope for diagnostic and surgical\nendoluminal procedures,” Robotica , vol. 28, no. 2, 2010.\n[9] S. B. Kesner and R. D. Howe, “Position control of motion compensation\ncardiac catheters,” IEEE Transactions on Robotics , vol. 27, no. 6, 2011.\n[10] T. Gl ¨uck, A. Eder, and A. Kugi, “Swing-up control of a triple pendulum\non a cart with experimental validation,” Automatica , vol. 49, no. 3, 2013.\n[11] K. ˚Astr¨om and K. Furuta, “Swinging up a pendulum by energy control,”\nAutomatica , vol. 36, no. 2, 2000.\n[12] M. Muehlebach and R. D’Andrea, “Nonlinear analysis and control of\na reaction-wheel-based 3-D inverted pendulum,” IEEE Transactions on\nControl Systems Technology , vol. 25, no. 1, 2017.\n[13] O. Boubaker, “The inverted pendulum: A fundamental benchmark in\ncontrol theory and robotics,” in Proceedings of the International Con-\nference on Education and e-Learning Innovations , 2012.\n[14] M. P. Kummer, J. J. Abbott, B. E. Kratochvil, R. Borer, A. Sengul,\nand B. J. Nelson, “OctoMag: An electromagnetic system for 5-DOF\nwireless micromanipulation,” IEEE Transactions on Robotics , vol. 26,\nno. 6, 2010.\n[15] A. J. Petruska and B. J. Nelson, “Minimum bounds on the number\nof electromagnets required for remote magnetic manipulation,” IEEE\nTransactions on Robotics , vol. 31, no. 3, 2015.\n[16] J. Edelmann, A. J. Petruska, and B. J. Nelson, “Magnetic control of\ncontinuum devices,” The International Journal of Robotics Research ,\nvol. 36, no. 1, 2017.\n[17] T. Liu and M. C. C ¸ avus ¸o ˘glu, “Three dimensional modeling of an MRI\nactuated steerable catheter system,” in Proceedings of the International\nConference on Robotics and Automation , 2014.\n[18] G. Pittiglio, L. Barducci, J. W. Martin, J. C. Norton, C. A. Avizzano,\nK. L. Obstein, and P. Valdastri, “Magnetic levitation for soft-tethered\ncapsule colonoscopy actuated with a single permanent magnet: A\ndynamic control approach,” IEEE Robotics and Automation Letters ,\nvol. 4, no. 2, 2019.\n[19] B. Scaglioni, L. Previtera, J. Martin, J. Norton, K. L. Obstein, and\nP. Valdastri, “Explicit model predictive control of a magnetic flexible\nendoscope,” IEEE Robotics and Automation Letters , vol. 4, no. 2, 2019.\n[20] E. Erdem Tuna, T. Liu, R. C. Jackson, N. Lombard Poirot, M. Russell,\nand M. C. C ¸ avus ¸o ˘glu, “Analysis of dynamic response of an MRI-guided\nmagnetically-actuated steerable catheter system,” 2018.\n[21] H. Marino, C. Bergeles, and B. J. Nelson, “Robust electromagnetic\ncontrol of microrobots under force and localization uncertainties,” IEEE\nTransactions on Automation Science and Engineering , vol. 11, no. 1,\n2014.\n[22] C. Pawashe, S. Floyd, and M. Sitti, “Modeling and experimental char-\nacterization of an untethered magnetic micro-robot,” The International\nJournal of Robotics Research , vol. 28, no. 8, 2009.\n[23] M. Hehn and R. D’Andrea, “An iterative learning scheme for high\nperformance, periodic quadrocopter trajectories,” in Proceedings of the\nEuropean Control Conference , 2013.\n[24] C. Sferrazza, M. Muehlebach, and R. D’Andrea, “Learning-based\nparametrized model predictive control for trajectory tracking,” Optimal\nControl Applications and Methods , vol. 41, no. 6, 2020.\n[25] R.-E. Precup, C. Gavriluta, M.-B. Radac, S. Preitl, C.-A. Dragos, J. Tar,\nand E. Petriu, “Iterative learning control experimental results for inverted\npendulum crane mode control,” in Proceedings of the International\nSymposium on Intelligent Systems and Informatics , 2009.\n[26] Q. Boehler, S. Gervasoni, S. L. Charreyron, C. Chautems, and B. J.\nNelson, “On the workspace of electromagnetic navigation systems,”\nIEEE Transactions on Robotics , vol. 39, no. 1, 2023.\n[27] H. Ma, D. B ¨uchler, B. Sch ¨olkopf, and M. Muehlebach, “A learning-\nbased iterative control framework for controlling a robot arm with\npneumatic artificial muscles,” in Proceedings of Robotics: Science and\nSystems XVIII , 2022.\n[28] R. Pintelon and J. Schoukens, System identification: A frequency domain\napproach . John Wiley & Sons, 2012.\n[29] M. Hofer, M. Muehlebach, and R. D’Andrea, “The one-wheel Cubli: A\n3D inverted pendulum that can balance with a single reaction wheel,”\nMechatronics , vol. 91, 2023.\n[30] D. Bristow, M. Tharayil, and A. Alleyne, “A survey of iterative learning\ncontrol,” IEEE Control Systems Magazine , vol. 26, no. 3, 2006.\n[31] J. Zughaibi, M. Hofer, and R. D’Andrea, “A fast and reliable pick-and-\nplace application with a spherical soft robotic arm,” in Proceedings of\nthe International Conference on Soft Robotics , 2021."    },    {        "title": "1910.07774v1.Magnetic_textures_and_dynamics_in_magnetic_Weyl_semimetals.pdf",        "content": "Magnetic textures and dynamics in magnetic Weyl semimetals\nYasufumi Araki1\n1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai 319-1195, Japan\nRecent theoretical and experimental attemps have been successful in \fnding magnetic Weyl\nsemimetal phases, which show both nodal-point structure in the electronic bands and magnetic\norders. Beyond uniform ferromagnetic or antiferromagnetic orders, nonuniform magnetic textures,\nsuch as domain walls and skyrmions, may even more enrich the properties of the Weyl electrons in\nsuch materials. This article gives a topical review on interplay between Weyl electrons and magnetic\ntextures in those magnetic Weyl semimetals. The basics of magnetic textures in non-topological\nmagnetic metals are reviewed \frst, and then the e\u000bect of magnetic textures in Weyl semimetals\nis discussed, regarding the recent theoretical and experimental progress therein. The idea of the\n\fctitious \\axial gauge \felds\" is pointed out, which e\u000bectively describes the e\u000bect of magnetic tex-\ntures on the Weyl electrons and can well account for the properties of the electrons localized around\nmagnetic domain walls.\nI. INTRODUCTION\nMagnetism has always been a fundamental concept in\nmaterials science. Along with the development of quan-\ntum mechanics, we have understood the behavior of spins\ncontributing to magnetism, and have succeeded in de-\nsigning various magnetic materials that exhibit exotic\nfeatures useful for applications [1]. Further technolog-\nical developments have enabled us to manipulate spins\nmicroscopically, called spintronics [2, 3], which may pos-\nsibly help us design highly e\u000ecient nanoscale devices that\noperate at high speed, with low energy consumption, etc.\nThe discovery of the giant magnetoresistance (GMR) ef-\nfect and its aplication to magnetic heads in hard disks\nare perhaps the most established and successful achieve-\nments in spintronics [4, 5]. As well as writing and readout\nof magnetization in magnetic nanostructures, detection\nand manipulation of nanoscale spin textures in magnetic\nmaterials have been intensely studied, aiming to make\nuse of those tiny objects as carriers of information in fu-\nture devices [6, 7]. Various magnetic materials, including\nboth metals and insulators, have been synthesized and\ninvestigated, to \fnd peculiar features of materials that\nmay be useful for spintronics.\nRecent studies have struggled for realizing magnetism\nin topological materials, namely the materials whose\ncharacteristic electronic structures are protected by spa-\ntial and internal symmetries of the system and thus clas-\nsi\fed by topology [8, 9]. The most famous class of topo-\nlogical electronic systems is perhaps topological insulator ,\nwhich shows gapless states on the surface and is char-\nacterized by Z2topological invariants de\fned with the\nbulk electrons [10{12]. The surface electrons of topolog-\nical insulators show striking features, namely the linear\n(Dirac) dispersion robust under disorder, strong locking\nbetween electron momentum and spin degrees of free-\ndom, etc., which have been attracting interest toward\ntheir application as well [13]. By combining these sur-\nface Dirac electrons with magnetism, using magnetic het-\nerostructures or magnetic dopants in experiments [14],\nvarious new phenomena were proposed and successfullyobserved: the quantum anomalous Hall e\u000bect [15{18],\nuniversal magneto-optical response [19, 20], spin-charge\nconversion [21, 22], current-induced control of magneti-\nzation (spin-transfer torque) [23, 24], etc.\nWhile topological insulators exhibit gapless states on\nthe surface, recent studies have discovered the classes of\nmaterials showing topologically protected gapless states\nin the bulk, which are termed Dirac and Weyl semimet-\nals[25{27]. Dirac/Weyl semimetals, distinguished by de-\ngeneracy of the gapless states, are characterized by lin-\nearly dispersed bands (Dirac/Weyl cones) around cer-\ntain band-touching points in momentum space. These\nband-touching points, namely the Dirac or Weyl points,\nserve as topological objects with monopole charges in\nmomentum space [28], which give rise to the geometri-\ncal phase (Berry phase) of the electrons and thus con-\ntribute to the anomalous transport properties, such as\nthe anomalous Hall e\u000bect or the spin Hall e\u000bect [29{32].\nVarious unusual phenomena, such as the negative magne-\ntoresistance due to the chiral anomaly [33{37], quantum\noscillations related to the surface-involved Weyl orbits\ninduced by a magnetic \feld [38{42], nonlinear optical re-\nsponses [43{49], etc., have been proposed and observed in\nWeyl semimetals. Moreover, Weyl semimetals with their\nWeyl cones tilted to the Fermi surface, namely type-II\nWeyl semimetals, are also of great interest [50]. Due to\ntheir unconventional Fermi surface structure, modi\fca-\ntion of magnetic quantum oscillations [51], superconduc-\ntivity [52{60], optical activity [61{64], etc., have been\nexpected and observed in type-II Weyl semimetals.\nWeyl semimetals with broken inversion symmetry have\nbeen experimentally realized in TaAs [65{68], NbAs [69],\nTaP [70], etc. On the other hand, the Weyl semimetal\nphase without time-reversal symmetry due to magnetism\nhas been theoretically proposed from the early days\n[71, 72]. Over the last few years, several ferromagnetic\nand antiferromagnetic materials hosting Weyl electrons\nhave been numerically demonstrated and experimentally\nveri\fed [73{77]. The combination of spin-orbit coupling\nand magnetism is essential to retain the band crossing\nat each Weyl point, from which we are expecting strong\ninterplay between the magnetism and the electronic prop-arXiv:1910.07774v1  [cond-mat.mes-hall]  17 Oct 20192\nerties around the Weyl points in those \\magnetic Weyl\nsemimetals\".\nRegarding the above interests, this article reviews re-\ncently developing researches on magnetic Weyl semimet-\nals, especially focusing on the interplay between nonuni-\nform magnetic textures and behavior of Weyl electrons.\nAs seen in ordinary magnetic materials, magnetic tex-\ntures, namely spatial or temporal modulation of the\n(anti)ferromagnetic orders, enrich the electronic proper-\nties in comparison with those under uniform and static\norders. In magnetic Weyl semimetals, such e\u000bects can be\ne\u000eciently treated with the help of the idea of \fctitious ax-\nial electromagnetic \felds [78], which is the main focus of\nthis review. The idea of the axial electromagnetic \felds,\nor gauge \felds, was \frst introduced to describe physics of\nelementary particles [79], and is now frequently employed\nfor Dirac and Weyl quasiparticles in condensed matter to\ndescribe e\u000bects of various types of spatial and temporal\nmodulations in the systems [80].\nThis article is organized as follows. In Section II, I\nreview common topics about magnetic textures in con-\nventional magnetic materials, and see how they alter the\nelectron transport behavior, for later comparison with\nthe case in Weyl semimetals. In Section III, I review cur-\nrent research status about the axial electromagnetic \felds\nin Weyl semimetals by starting with a minimal model\nHamiltonian, and list up their e\u000bects on the structure\nand transport properties of Weyl electrons. In Section\nIV, I focus on the properties of magnetic domain walls in\nmagnetic Weyl semimetals, as typical magnetic textures\npossibly seen in experiments. In Section V, I summa-\nrize recent theoretical and experimental achievements to-\nward realization of magnetic Weyl semimetal phase, and\ndiscuss how the axial electromagnetic \feld picture can\nbe applied in the proposed systems. Finally, I conclude\nthis article with some future prospects about interplay\nbetween magnetism and topological electron systems in\nSection VI.\nII. MAGNETIC TEXTURES IN NORMAL\nMETALS: SPIN GAUGE FIELDS\nIn this section, I review general theories accounting\nfor the interplay between magnetic textures and elec-\ntron dynamics in normal magnetic materials, which shall\nbe compared with the treatment in Weyl semimetals for\nour better understanding. Topological magnetic textures\nare characterized by topological invariants de\fned in real\nspace, which cannot be created or unwound unless they\nare perturbed by any topological defects. Thanks to this\ntopological robustness, these magnetic textures can be\nmacroscopically described as isolated objects; in the con-\ntext of spintronics, a lot of attempts have been made\nto e\u000eciently manipulate magnetic textures, in order to\nmake use of them as carriers of information in future de-\nvices such as magnetic racetrack memories [6, 81] and\nlogic gates [82].Magnetic domain walls are perhaps the most com-\nmonly seen magnetic textures in materials, since domain\nstructure is necessary to reduce magnetostatic energy\nfrom stray \feld. Domain wall dynamics has been widely\nobserved in magnetic materials, which has given us good\nunderstanding of the spin torques that can be present\nin each system [83, 84]. Magnetic skyrmions, namely\npointlike structures characterized by swirling spin tex-\ntures inside, also play an important role in the studies on\nmagnetic materials. While skyrmions usually form lattice\nstructure, called skyrmion crystal, at ground state, detec-\ntion and manipulation of individual skyrmions have been\nsuccessfully demonstrated in recent experiments [85{87].\nFor electric manipulation and detection of such mag-\nnetic textures, we need to understand the interplay be-\ntween electron transport and magnetic textures. Elec-\ntron transport through metallic magnets is a\u000bected by\nmagnetic textures via the adiabatic phase (Berry phase)\naccumulated on the electron wave function, since the elec-\ntron spin gets eventually modulated by the localized spins\nin the magnetic textures. This Berry phase e\u000bect can be\nencoded into the \fctitious \\spin electromagnetic \felds\"\nfor the electrons, which act on the majority and minor-\nity spin states of the electrons under spin splitting by\nthe exchange interaction [88, 89]. Topological magnetic\ntextures and their dynamics can host spin electromag-\nnetic \felds, which alter the electron transport through\nthe spin textures, in a similar manner with the ordinary\nelectromagnetic \felds. Here we brie\ry review the idea of\nspin electromagnetic \felds and see some typical phenom-\nena that are twell described by this idea. (See References\n[83, 90, 91] for detailed reviews on spin electromagnetic\n\felds.)\nLet us start with the minimal model for conduction\nelectrons under magnetic textures,\nH=p2\n2m+Jn(r;t)\u0001\u001b; (1)\nwheremis the e\u000bective mass of the electron and p=\n\u0000iris the momentum operator. The second term repre-\nsents the exchange interaction between the electron spin,\ndenoted by Pauli matrices \u001b, and the local magnetic mo-\nment in the magnetic texture, with its direction given by\nthe unit vector n= (sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012), where\nJ(>0) is the exchange coupling energy. The key idea\nto obtain the spin electromagnetic \felds is to rotate the\nmagnetization nto a \fxed quantization axis, namely z-\naxis, which is achieved by the SU(2) unitary transforma-\ntion with the matrix\nU(r;t) =ei\u0012\n2\u001byei\u001e\n2\u001bz: (2)\nBy the unitary transformation\nH0=UyHU\u0000iUy@tU; (3)\nthe transformed Hamiltonian H0takes the form\nH0=(p+eAAA)2\n2m+J\u001bz\u0000eA0; (4)3\nwhere the SU(2) gauge \felds ( AAA;A0) are given by\nAAA=\u0000i\neUyrU=\u00001\n2e[(cos\u0012\u001bz\u0000sin\u0012\u001bx)r\u001e+\u001byr\u0012];\n(5)\nA0=i\neUy@tU=1\n2eh\n(cos\u0012\u001bz\u0000sin\u0012\u001bx)_\u001e+\u001by_\u0012i\n:(6)\nIn general, these gauge \felds are in 2 \u00022 matrix structure.\nIf the spin splitting Jis larger than any other energy\nscales so that the interband transition can be neglected,\ntheir diagonal components, namely the projection onto\nU(1) subspace, dominantly a\u000bect the electron transport.\nFocusing on the majority spin state, which correspond to\nthe lower component of the electron wave function, the\nprojection of (AAA;A0) onto the majority spin state reads\n~AAA=1\n2ecos\u0012r\u001e; ~A0=\u00001\n2ecos\u0012_\u001e: (7)\nThese gauge \felds yield \fctitious electromagnetic \felds\n~EEE=\u0000r~A0\u0000@t~AAA=1\n2esin\u0012\u0010\n_\u0012r\u001e\u0000_\u001er\u0012\u0011\n; (8)\n~BBB=r\u0002~AAA=\u00001\n2esin\u0012(r\u0012)\u0002(r\u001e); (9)\nwhich read in terms of the local magnetization vector n,\n~Ei=1\n2en\u0001(_n\u0002@in);~Bi=\u00001\n4e\u000fijkn\u0001(@jn\u0002@kn):\n(10)\nAs long as the electron remain on the majority spin state,\nwhich is called the adiabatic limit , these spin electromag-\nnetic \felds act on the majority-spin electrons like the or-\ndinary U(1) electromagnetic \felds. Once we incorporate\nthe e\u000bect of interband transitions, we need to consider\nthe nonadiabatic components on the o\u000b-diagonal posi-\ntions of the matrices, for which I will not go into detail\nin this article.\nFrom Equation (9), we can see that the spin magnetic\n\feld ~BBBis related to the solid angle spanned by the spatial\nmodulation of the magnetization vector n. A typical\nmagnetic texture that yields such a spin magnetic \feld\nis a skyrmion [85], since r\u0012is in the radial direction\nandr\u001eis in the azimuthal direction in a rotationally\nsymmetric skyrmion. The spin magnetic \feld attached\nto the skyrmions gives rise to the unconventional Hall\ntransport of the conduction electrons, which cannot be\ndescribed as the regular Hall e\u000bect related to the applied\nmagnetic \feld or the anomalous Hall e\u000bect related to the\nnet magnetization of the system [92, 93]. This skyrmion-\ninduced Hall e\u000bect, called the topological Hall e\u000bect , was\nsuccessfully measured in some magnetic materials (e.g.\nMnSi [94, 95], MnGe [96], etc.) as a good evidence for\nthe skyrmion crystal phase, which was originally termed\nthe A-phase.\nThe spin electric \feld ~EEEis induced by dynamics of a\nmagnetic texture. This \feld exerts a force on the elec-\ntron, which is referred to as the spin motive force , anddrives a conduction current attached to the magnetic tex-\nture [97{100]. The spin motive force was experimentally\nobserved as a current pulse induced by motion of domain\nwalls in magnetic nanowires [101, 102], and also as a dc\nvoltage induced by ferromagnetic resonance in a comb-\npatterned ferromagnetic \flm [103].\nThe form of the spin electromagnetic \felds shown\nabove applies only to the case where the electron system\nin the absence of the exchange interaction is spin-SU(2)\nsymmetric. In the presence of spin-orbit coupling for the\nelectrons, the spin electromagnetic \felds gets modi\fed\nfrom the above form; under Rashba spin-orbit coupling,\nfor example, the spin electromagnetic \felds acquire ad-\nditional terms proportional to the Rashba coupling con-\nstant [104{106]. The Weyl dispersion can be viewed as\nthe limit of extermely strong spin-orbit coupling around\nthe band crossing points, which implies that the e\u000bect\nof spin textures on the Weyl electrons should be distinct\nfrom that in normal metals, as shall be reviewed in the\nfollowing sections.\nIII. WEYL SEMIMETAL AND MAGNETIC\nTEXTURES\nNow we shall see how the interplay between electron\ntransport and magnetic textures gets altered in magnetic\nWeyl semimetals. Due to strong spin-momentum locking\naround the Weyl points, the spin gauge \feld picture in-\ntroduced in the previous section is no longer applicable\nto the Weyl electrons. Nevertheless, the e\u000bect of mag-\nnetic textures can still be mapped to \fctitious electro-\nmagnetic \felds for the Weyl electrons, which are cate-\ngorized as the axial electromagnetic \felds, coupling to\nthe pair of valleys with the opposite signs to each other\n[78, 80]. This picture is available since the location of the\nWeyl points depends on the magnetization in magnetic\nWeyl semimetal, which is the consequence of strong spin-\nmomentum locking around the Weyl points. As a result,\nspatial and temporal modulations of the magnetization\nlead to the anomalous responses in the electronic struc-\nture and transport [107], which are distinct from those\nexpected in normal metals.\nIn this section, I review the e\u000bect of magnetic tex-\ntures on the electrons in magnetic Weyl semimetals based\non the idea of the axial magnetic \felds. In the \frst\nsubsection, I review the general characteristics of Weyl\nsemimetals with broken time-reversal symmetry, start-\ning with the toy model. Although Weyl semimetals are\nintensely studied and known to show various exotic fea-\ntures, here I mainly explain the fundamental features of\nWeyl semimetals that are necessary for the discussion be-\nlow on the e\u000bect of magnetic textures. Then I introduce\nspatial and temporal inhomogeneity in the second subsec-\ntion, to see how the \fctitious axial electromagnetic \felds\nare de\fned associated with magnetic textures. With the\nidea of the axial electromagnetic \felds, I summarize in\nthe third subsection how magnetic textures modulate4\n+ − +k0 -k0\nkzkxky\nfixed kzΩ(k)bulksurface\n= gapless edge state for kz∈[-k0,+k0]Fermi arc\nFIG. 1. Schematic picture of the momentum-space struc-\nture of a Weyl semimetal. The small circles denote the Weyl\npoints, with + =\u0000inside the circles denoting the valley index\n\u0011. Gapless \\Fermi arc\" modes emerge on the surface Brillouin\nzone, which is the projection of bulk Brillouin zone onto two\ndimensions. See the main text for detail.\nthe electronic structure and transport from macroscopic\npoint of view. Under the dynamics of magnetic texture,\nin particular, we can see pumping of electric charge at-\ntached to the magnetic texture, which is reviewed in the\nfourth subsection. Finally, in the last subsection, I men-\ntion the inverse e\u000bect, namely the e\u000bect of electron distri-\nbution and transport on the magnetic texture dynamics\nvia spin torques, which can also be described by using\nthe idea of the axial electromagnetic \felds.\nA. Weyl semimetal with broken time-reversal\nsymmetry\nThe essence of Weyl semimetal is the linear band\ntouching at certain points, namely Weyl points, in mo-\nmentum space. In contrast to band touching points\n(Dirac points) in Dirac semimetals, each of which shows\nfourfold degeneracy due to the protection by time-\nreversal and inversion symmetries, each Weyl point in\nWeyl semimetals has twofold degeneracy, since either\ntime-reversal or inversion symmetry is broken [72]. Each\nWeyl point behave as a source or sink of Berry curva-\nture in momentum space, which is de\fned by \n(k) =\nihrku(k)j\u0002jrku(k)iwith the Bloch eigenstate ju(k)i,\nand thus a topological charge +1 or \u00001 is associated with\neach Weyl point [28, 108]. The net topological charge in\nmomentum space vanishes, i.e. the numbers of sources\nand sinks cancel each other over the whole Brillouin zone,\nand hence Weyl points with topological charge + and\n\u0000arise always in pair(s). This restriction is known as\nNielsen{Ninomiya's theorem [109, 110], which has origi-\nnally been employed to construct lattice models of chiral\nfermions for numerical simulations of quantum chromo-\ndynamics (QCD) [111].\nIn a Weyl semimetal with inversion symmetry pre-served but time-reversal symmetry broken [71, 72], in-\nversion symmetry demands that Weyl points at \u0006k0in\nmomentum space should be paired (see Figure 1). As-\nsuming cubic symmetry, the band dispersion in the vicin-\nity of each Weyl node \u0011k0(\u0011=\u0006) is described by the\nminimal Hamiltonian\nH\u0011(k) =\u0011vF\u001b\u0001(k\u0000\u0011k0); (11)\nwhere the Pauli matrices \u001bdenote the electron spin and\nvFis the Fermi velocity around the Weyl points [71]. (It\nshould be noted that cubic symmetry is not necessarily\npresent in general, which implies that \u001bmay consist not\nonly of spin degrees of freedom but of other degrees of\nfreedom, such as orbital, sublattice, etc., as well.) Thus\nthe Weyl-point separation 2 k0characterizes the degree of\ntime-reversal symmetry breaking in the system, arising\nfrom a magnetization, an external magnetic \feld, etc.\nIfk0= 0, the Hamiltonian reduces to that of Dirac\nsemimetal with a Dirac point degenerate at k= 0.\nThe electrons residing around the Weyl point at \u0011k0,\nwhich we call valley\u0011for short, exhibit spin-momentum\nlocking feature. The electron spin s\u0011;kand the momen-\ntum\u000ek\u0011k\u0000\u0011k0are aligned either parallel or antipar-\nallel, depending on the valley index \u0011. In other words, \u0011\nindicates the helicity (chirality) of the Weyl fermion,\ns\u0011;k\u0001\u000ek\njs\u0011;kjj\u000ekj=\u0011 (12)\nwhich distinguishes right-handed ( \u0011= +) and left-\nhanded (\u0011=\u0000) modes. The velocity operator in the\nvalley\u0011is given by\nv\u0011=@H\u0011(k)\n@k=\u0011vF\u001b; (13)\nwhose expectation value for the Bloch state at kbecomes\nproportional to the spin s\u0011;k.\nThe Weyl point \u0011possesses a topological charge \u0000\u0011,\nwith the Berry curvature\n\n\u0011(k) =\u0000\u0011\u000ek\n2j\u000ekj3(14)\nfor the electron band (with positive energy). This Berry\ncurvature distribution leads to the anomalous Hall e\u000bect\ndue to the breaking of time-reversal symmetry [29{32].\nSlicing the Brillouin zone by a plane between the two\nWeyl points, the \frst Chern number on this momentum\nplane, namely the total Berry \rux piercing this plane, be-\ncomes \fnite, which leads to the anomalous Hall e\u000bect in\nthis system. For instance, if the Weyl points are located\nat\u0011k0= (0;0;\u0006k0) onkz-axis, the Chern number on the\nplane at \fxed kz2[\u0000k0;k0] becomes unity, as shown by\nthe purple plane in Figure 1, leading to the quantized\nanomalous Hall conductivity \u001bA(2D)(kz) =e2=2\u0019on this\nplane [112]. Therefore, summing over the whole Bril-\nlouin zone, the anomalous Hall conductivity in xy-plane5\nis given by\n\u001bA=Zdkz\n2\u0019\u001bA(2D)(kz) =e2\n4\u001922k0; (15)\nwhich is directly related to the Weyl point separation\n2k0.\nThe anomalous Hall e\u000bect in the bulk is also re-\nlated with the gapless \\Fermi arc\" states on the surface\n[113, 114]. Since the plane at \fxed kz2[\u0000k0;k0] between\nthe two Weyl points host Chern number 1, as shown\nabove, there arises a gapless edge mode unidirection-\nally propagating along the one-dimensional edge of this\nplane. The emergence of such a gapless edge state is com-\nmon in two-dimensional quantum anomalous Hall insu-\nlator (Chern insulator) states, such as the gapped Dirac\nsurface state of a magnetically doped three-dimensional\ntopological insulator [15{18, 115]. Therefore, on the sur-\nface of a Weyl semimetal, the gapless states emerge for\n\u0000k0< kz< k 0, connecting the locations of the two\nWeyl points projected onto the surface. We can see\nfrom this discussion that, in a Weyl semimetal with bro-\nken time-reversal symmetry, the emergence of the gap-\nless Fermi arc state is the surface (boundary) counter-\npart of the anomalous Hall e\u000bect in the bulk, which\nshall be revisited in Section IV to see the e\u000bect of mag-\nnetic domain walls. Fermi arcs appear in time-reversal-\nsymmetric Weyl semimetals as well (although they do\nnot show the anomalous Hall e\u000bect), which have been\nclearly seen in angle-resolved photoemission spectroscopy\n(ARPES) measurements [65{68]. In Dirac semimetals,\neach Dirac point is doubly degenerate and thus serves\nas a source of two branches of Fermi arcs, which ren-\nders the surface state into closed loops in momentum\nspace, in contrast to open Fermi arcs in Weyl semimetals\n[116, 117].\nB. Magnetization and axial electromagnetic \felds\nIn Weyl semimetals with broken time-reversal symme-\ntry, the separation of the Weyl points in momentum space\ncharacterizes the e\u000bect of the breaking of time-reversal\nsymmetry, as seen in the previous subsection. If the\nbreaking of time-reversal symmetry is due to magnetiza-\ntion, modulation in the magnetization yields shift of the\nWeyl points in momentum space, which can be regarded\nas an e\u000bect of a \fctitious vector potential. Based on\nthis idea, we can consider the e\u000bect of magnetic textures\nand their dynamics in terms of \fctitious electromagnetic\n\felds for the Weyl electrons.\nLet us start from the minimal Hamiltonian Equation\n(11). If the background magnetization is coupled to the\nelectron spin \u001bby the exchange interaction J, its mod-\nulation\u000eMgives a perturbation term J\u000eM\u0001\u001b, which\ncan be incorporated in the Weyl Hamiltonian as\nH\u0011(k) =\u0011vF\u001b\u0001(k\u0000\u0011k0\u0000\u0011eA5): (16)HereA5is de\fned by\nA5=\u0000J\nvFe\u000eM; (17)\nwhich couples to the Weyl electrons in a similar man-\nner to the gauge \feld, whereas the sign of its coupling\ndepends on the valley index \u0011[78]. In the context of\nrelativistic quantum \feld theory, a gauge \feld coupled\nto the two chirality channels (right/left-handed) of Dirac\nfermions with the signs opposite to each other is termed\nthe axial gauge \feld, or the chiral gauge \feld, which\ntransformes as an axial vector under the parity opera-\ntion, in contrast to the ordinary gauge \feld as a polar\nvector [79]. Since the structure of spin-momentum lock-\ning is material-dependent, the correspondence between\nmagnetization and the axial gauge \feld should be mod-\ni\fed in some Weyl materials observed in experiments,\nwhich shall be discussed in Section V.\nJust like the ordinary gauge \feld, what a\u000bects the elec-\ntronic structure and transport is not the value of A5itself\nbut the spatial and temporal structure of A5, correspond-\ning to the electromagnetic \felds. Let us introduce spatial\nand temporal structure in \u000eM, yielding the axial gauge\npotential\nA5(r;t) =\u0000J\nvFe\u000eM(r;t): (18)\nHere the spatial and temporal variation in \u000eMshould\nbe moderate enough to rely on the axial gauge \feld pic-\nture, since a short-range or high-frequency \ructuation in\n\u000eMmay lead to hybridization of the two valleys (Weyl\nnodes). Under this condition, we can de\fne the axial\nelectric \feld\nE5(r;t) =\u0000_A5(r;t) =J\nvFe_M(r;t) (19)\nand the axial magnetic \feld\nB5(r;t) =r\u0002A5(r;t) =\u0000J\nvFer\u0002M(r;t);(20)\nwhich couple to the two valleys \u0011=\u0006with the oppo-\nsite signs\u0011[107]. The axial electric \feld comes from\ndynamics of the magnetization, while the axial magnetic\n\feld resides at a curled magnetic texture [78]. The sim-\nplest example of a magnetic texture that gives rise to the\naxial magnetic \feld is a 180-degree domain wall, as a do-\nmain wall always accompanies \rip of the magnetization\nMin real space. In the presence of these axial electro-\nmagnetic \felds, as in the case of normal electromagnetic\n\felds, the exchange term cannot be absorbed by the local\nU(1) gauge transformation, and hence they modulate the\nelectron transport at low energy.\nThe pseudo-electromagnetic \feld picture of magnetic\ntextures is also available for the two-dimensional Dirac\nelectrons on surfaces of topological insulators, as they\nshow spin-momentum locking around the Dirac point6\nin the surface Brillouin zone [118]. With this pic-\nture, electric charging of magnetic textures, arising from\nthe \fctitious magnetic \rux corresponding to the mag-\nnetic texture, was proposed on topological insulator sur-\nfaces [118, 119]. It should be noted that there are\ntwo major di\u000berences. Since the surface of topologi-\ncal insulator show only a single Dirac cone, the pseudo-\nelectromagnetic \felds from magnetic textures couple to\nthe surface Dirac electrons just like the ordinary elec-\ntromagnetic \felds. Moreover, only the in-plane two\ncomponents of magnetization contributes to the pseudo-\nelectromagnetic \felds for the surface Dirac electrons,\nsince the out-of-plane component does not shift but gaps\nout the surface Dirac point. These properties are in clear\ncontrast to those in the axial electromagnetic \feld pic-\nture, which is de\fned for the pair of Weyl points in three\ndimensions.\nAlthough the idea of the axial electromagnetic \felds in-\ntroduced here appears similar to the spin electromagnetic\n\felds mentioned in the previous section, they are di\u000ber-\nent in some aspects. Conceptually, while the spin electro-\nmagnetic \felds are obtained by projecting the exchange\ncoupling term to the magnetic textures onto the major-\nity/minority spin states, the axial electromagnetic \felds\nfor the Weyl electrons are derived by projecting them\nonto the fully spin-momentum-locked states around the\nWeyl nodes. The idea of the axial electromagnetic \felds\nis applicable to the limit of a strong spin-momentum lock-\ning and a weak spin splitting, which is opposite to the\nsituation for the spin electromagnetic \felds. From the\nphenomenological point of view, the species of magnetic\ntextures that lead to the axial electromagnetic \felds in\nWeyl semimetals is much broader than that for the spin\nelectromagnetic \felds in normal metals. For instance, the\nspin magnetic \feld ~BBBgiven by Equation (10) in normal\nmagnetic metals requires at least two-dimensional mag-\nnetic textures, such as skyrmions (see Equation (9)). On\nthe other hand, the axial magnetic \feld B5can be gener-\nated even from a one-dimensional spin texture, such as a\ndomain wall (the case for domain walls shall be discussed\nin detail in Section IV). Moreover, dynamics of even a\nuniform magnetization can lead to the axial electric \feld\nE5in a Weyl semimetal, which is in a clear contrast with\nthe spin electric \feld ~EEE, namely the spin motive force,\nrequiring dynamics of a spin texture. Such a di\u000berence\narises because the axial gauge potential A5is tied di-\nrectly to the local magnetization M, whereas the spin\ngauge potential comes from the spin connection, which\ncorresponds to the relative angle between two neighbor-\ning spins. Therefore, the e\u000bect of magnetic textures on\nthe electron transport in Weyl semimetals should be qual-\nitatively di\u000berent from that in normal magnetic metals,\nas long as the electrons are fully spin-momentum-locked\non the Fermi surface.\nWhile the idea of the axial electromagnetic \felds is in-\ntroduced to describe the e\u000bect of magnetic textures here,\nthe axial electromagnetic \felds can also be reproduced by\nlattice strain in Dirac and Weyl semimetals [80, 120{122].Since a lattice site displacement leads to the modulation\nof hopping amplitudes and shifts the Dirac/Weyl points,\na lattice strain, namely a spatially nonuniform lattice dis-\nplacement, can be regarded as the axial electromagnetic\n\felds in the vicinity of the Dirac/Weyl points. The ef-\nfect of the strain-induced axial magnetic \feld has been\nintensely studied over recent few years from the theo-\nretical point of view: Landau quantization [121], mod-\nulation of the Fermi arc structure [123], and quantum\noscillations due to the Weyl orbits connecting bulk and\nsurface [124, 125], has been predicted under the strain-\ninduced axial magnetic \feld. It is also proposed that spa-\ntial modulation in the chemical composition of antiper-\novskite Dirac materials can replicate the pseudomagnetic\n\feld for the Dirac electrons, since the locations of the\nDirac points in those materials are related to the ratio\nof the chemical composition [126]. The discussions below\nabout the e\u000bect of axial gauge \felds can be applied to\nthose systems in almost the similar manner, while this\narticle will not go into details of them.\nC. Charge and current responses to axial\nelectromagnetic \felds\nThe axial electromagnetic \felds couple to the Weyl\nelectrons in the same manner with the realistic electro-\nmagnetic \felds, as long as the Weyl nodes can well be\ntreates separately. Therefore, provided that the length\nscale of the magnetic texture and the time scale of the\nmagnetization dynamics are much longer than those cor-\nresponding to the Weyl-node separation k0, one may con-\nsider the behavior of the electrons within the individual\nvalley\u0011=\u0006, under the net electromagnetic \felds\nE\u0011=E+\u0011E5;B\u0011=B+\u0011B5: (21)\nWith these electromagnetic \felds, one can estimate the\ncharge and current responses phenomenologically for\neach Weyl node [78, 80, 107]. Here I focus on the re-\nsponses to the \felds up to their \frst order O(E\u0011;B\u0011),\nand assume that the Fermi level \u0016of the electrons is well\nde\fned in equilibrium around the Weyl nodes. The cur-\nrents induced by ( E;B) and (E5;B5) are summarized\nin Figure 2 and Table I.\n1. Equilibrium response\nLet us start from the static system, without any driv-\ning by electric \feld Eor magnetization dynamics char-\nacterized byE5. In the presence of a magnetic \feld B\u0011,\nincluding the axial \feld B5from magnetic textures, it\ninduces the Landau quantization with the cyclotron fre-\nquency!c=vFp\n2ejB\u0011j[127], just like the quantum Hall\ne\u000bect in two-dimensional Dirac electron systems, such as\ngraphene [128, 129]. In particular, the zeroth Landau\nlevel is linearly dispersed along the direction of \u0011B\u0011for\neach valley \u0011=\u0006, with the velocity vF. The density7\nof states of this zeroth Landau state (per single valley)\nis given by \u0017\u0011=ejB\u0011j=4\u00192vF, which is independent of\nthe Fermi energy \u0016due to its one-dimensional unidirec-\ntional dispersion. Therefore, if the Fermi level \u0016lies be-\nlow the \frst Landau level \u000f1(kz= 0) =!cso that it may\ncross only the zeroth Landau levels, the electric charge is\naccumulated around the magnetic \rux, with the charge\ndensity\n\u001aB=\u0000e(\u0017++\u0017\u0000)\u0016=\u0000e2\n4\u00192\u0016\nvF(B++B\u0000) (22)\nsummed over the two valleys. (Note that the background\ncharge density without the magnetic \feld is negligible\naround the charge neutrality, since the charge density of\nfree Weyl electrons is proportional to \u00163.) In particular,\neven in the absence of the realistic magnetic \feld B, the\naxial magnetic \rux B5(r) from a magnetic texture leads\nto the localized charge , with its density\n\u001aB(r) =\u0000e2\n2\u00192\u0016\nvFjB5(r)j (23)\nat the magnetic texture [121]. This localized charge is\nexplicitly derived under a one-dimensional domain wall\nin terms of the Fermi arc modes localized at the boundary\n[130, 131]; see Section IV for detail. Altough the spatially\nlocalized charge in metallic regime is inevitably screened\nonce we consider the Coulomb interaction, the screening\ne\u000bect is smaller than that in normal metals, since the\ndensity of states in topological semimetals becomes small\naround the band crossing points [132].\nSince the zeroth Landau level for each valley is dis-\npersed along the direction of \u0011B\u0011, the electrons in this\nLandau level contribute to the current\nj(C)\n\u0011=\u001aB\u0011vF\u0011^B\u0011=\u0000e2\n4\u00192\u0016\u0011B\u0011 (24)\nfor each valley \u0011[121, 133], where a bold symbol with a\n\\hat\" denotes its unit vector ( ^X\u0011X=jXj). Therefore,\nthe net current induced by the magnetic \felds reads\nj(C)=\u0000e2\n4\u00192\u0016(B+\u0000B\u0000) =\u0000e2\n2\u00192\u0016B5; (25)\nwhich depends only on the axial magnetic \feld B5(see\nFigure 2(b)). This e\u000bect is named the chiral pseudomag-\nnetic e\u000bect or the chiral axial magnetic e\u000bect in litera-\ntures [107, 134, 135].\nThe chiral pseudomagnetic e\u000bect is the axial counter-\npart of the chiral magnetic e\u000bect, namely the current\ngeneration by a magnetic \feld Bin the presence of chem-\nical potential imbalance between two valleys (left/right-\nhanded fermions), which has long been known in the con-\ntext of chiral fermions at high energy (heavy ion colli-\nsions, neutron stars, etc.) [136{140]. It is shown that\nthe chiral magnetic e\u000bect in equilibrium is absent in lat-\ntice systems, since the imbalance in the Fermi levels of\ntwo valleys within the same lattice model is not availableTABLE I. Summary of the current responses generated by the\nnormal electromagnetic \felds (EMFs) ( E;B) and the axial\nEMFs ( E5;B5). For the cell \\not applicable\", see (ii) in\nSection III C 2.\nClassi\fcation Normal EMFs Axial EMFs\nChiral magnetic e\u000bect j(C)\n5/B j(C)/B5\nDrift e\u000bect j(D)/E j(D)\n5/E5\nAnomalous Hall e\u000bect j(A)/^k0\u0002Enot applicable\nRegular Hall e\u000bect j(H)/^B\u0002E j(H)/^B5\u0002E5\nin equilibrium [141, 142]. The magnetic \feld Bin equi-\nlibrium contributes only to the axial current (see Figure\n2(a)):j(C)\n5=j(C)\n+\u0000j(C)\n\u0000=\u0000(e2=2\u00192)\u0016B[143]. The\nchiral magnetic e\u000bect is thus sought for in inequilibrium\n[144{147]; the negative magnetoresistance in Dirac/Weyl\nsemimetals is a typical inequilibrium phenomenon that\nstems from the chiral magnetic e\u000bect [33, 34].\nThe chiral pseudomagnetic e\u000bect due to the axial mag-\nnetic \feldB5, on the other hand, can locally induce\nan equilibrium current. One can qualitatively under-\nstand this local equilibrium current in connection with\nthe orbital magnetization Morbof the electron system,\nby the relation j(r) =r\u0002Morb(r) [133]. The logic\nis threefold: (i) The orbital magnetization Morbof a\nWeyl semimetal appears proportional to the spin mag-\nnetizationM. (This can be easily understood if Mis\nuniform, as the surfaces host a circulating current car-\nried by the Fermi-arc states.) (ii) If there is a spa-\ntial inhomogeneity in M(r), the orbital magnetization\nMorb(r) is also inhomogeneous, leading to the local cur-\nrentj(r) =r\u0002Morb(r) present in the bulk. (iii) Since\nr\u0002M(r) corresponds to the axial magnetic \feld B5(r),\nwe can regard this local current j(r) as the bound cur-\nrent localized at the axial magnetic \rux B5(r). (This\nbound current was calculated explicitly under a magnetic\ndomain wall [130].) It was also numerically demonstrated\non lattice models that an axial magnetic \feld correspond-\ning to lattice strain leads to an equilibrium current local-\nized at the torsion axis, although the net current over the\nwhole system is zero [121, 148].\n2. Nonequilibrium response\nWhen the electric \felds E\u0011are switched on, the elec-\ntron distribution is driven to nonequilibrium state and\nit gives rise to various current responses. E\u0011consists of\nthe normal electric \feld Eand the axial electric \feld E5\ncorresponding to the dynamics of magnetization. Up to\nthe linear response to E\u0011, the current response in Weyl\nsemimetals is classi\fed into three contributions: (i) the\ndrift current, (ii) the anomalous Hall current, and (iii) the\nregular Hall current [107]. I summarize these contribu-\ntions below, \frst using the generalized electric \feld E\u0011,\nand then limiting it to the axial electric \feld E5corre-\nsponding to the magnetization dynamics in the magnetic\nWeyl semimetal.8\nChiral magnet ic effect Drifteffect Anom alous Hall effect Regular Hall effect\nBj+(C)\nj-(C)B\n+\n−\n(a)\n+B5\n-B5(b)\n+\n−\nj+(C)\nj-(C)\n(E5 , B5)(E, B)\nE\nEj+(D)\nj-(D)\n+\n−\n(c)\n+E5\n-E5\n+\n−(d)\nj+(D)\nj-(D)(e)\n(f)(g)\nE\nj+(H)\nB\nE\nB\n+\n−j-(H)\n+E5-E5+B5\n-B5(h)\n+\n−\nj+(H)j-(H)\nEj(A)\n+\n−\n2k0\nnot applic able.\nFIG. 2. Schematic pictures of the current responses induced by the normal electromagnetic \felds ( E;B) and the axial\nmagnetic \felds ( E5;B5). Small circles located at the band crossing points denote the valley indices \u0011=\u0006. For the cell \\not\napplicable\", see (ii) in Section III C 2.\n(i) The drift e\u000bect is the current response longitudinal\nto the applied electric \feld E\u0011, given by\nj(D)\n\u0011=\u001bD\n\u0011E\u0011 (26)\nfor each valley \u0011(see Figure 2(a)). Here \u001bD\n\u0011denotes\nthe longitudinal conductivity for valley \u0011, which is well\nde\fned as long as the Fermi surfaces of the two valleys are\nwell separated in momentum space so that the intervalley\nscattering can be negligible. Under this condition, \u001bD\n\u0011\ncan be estimated semiclassically: \u001bD\n\u0011=e2v2\nFD(\u0016)\u001c=3 for\nspherically symmetric Weyl dispersion, where D(\u0016) =\n\u00162=2\u00192v3\nFis the density of states (per single Weyl cone)\nand\u001cis the transport relaxation time.\nIn the absence of the realistic electric \feld E, the cur-\nrent and the axial current induced by the axial electric\n\feldE5are given by\nj(D)=\u0000\n\u001bD\n+\u0000\u001bD\n\u0000\u0001\nE5;j(D)\n5=\u0000\n\u001bD\n++\u001bD\n\u0000\u0001\nE5(27)\nIn magnetic Weyl semimetals, the two valleys have the\nidentical structure due to its inversion symmetry. There-\nfore, the longitudinal conductivities for the two valleys\nare equal, and hence the axial electric \feld induces no\nnet currentj(D)= 0 (see Figure 2(d)); it only drives the\naxial current j(D)\n5, which corresponds to the spin accu-\nmulationh\u001biif spin and momentum are fully locked.\n(ii) The anomalous Hall e\u000bect is the current response\ntransverse to the applied electric \feld. In contrast to the\nregular Hall e\u000bect, the anomalous Hall e\u000bect does not\nrequire any magnetic \feld and arises from other time-\nreversal symmetry breaking e\u000bect, such as magnetiza-\ntion [149]. As is well known, the anomalous Hall e\u000bect\ncan be classi\fed into the extrinsic e\u000bect coming fromthe asymmetric scattering by impurities and the intrin-\nsic e\u000bect due to the anomalous velocity driven by the\nnonzero Berry curvature in momentum space. In mag-\nnetic Weyl semimetals, as shown in Section III A, the\nintrinsic anomalous Hall e\u000bect arises from the separation\nof two Weyl points: the induced current is given by\nj(A)=\u001bA^k0\u0002E; (28)\nwhere the anomalous Hall conductivity is tied to the\nWeyl-point separation 2 k0as\u001bA= (e2=2\u00192)jk0j(see Fig-\nure 2(e)). Since this intrinsic Hall e\u000bect comes from all\nthe occupied states below the Fermi level, it cannot be\nseparated to the individual valleys. Thus the \\axial elec-\ntric \feld\" picture cannot be applied to see the intrinsic\nanomalous Hall e\u000bect contribution (the cells \\not appli-\nable\" in Table I and Figure 2(f)); it may depend on the\nmodulation of band structure by the magnetic texture\ndynamics away from the Weyl points.\n(iii) The regular Hall e\u000bect is the current response in-\nduced by the electric \feld E\u0011in the presence of the mag-\nnetic \feldB\u0011: provided the regular Hall conductivity \u001bH\n\u0011\nis de\fned for the valleys \u0011separately, the current is writ-\nten as\nj(H)\n\u0011=\u001bH\n\u0011^B\u0011\u0002E\u0011: (29)\nThe Hall conductivity \u001bH\n\u0011can be estimated in either semi-\nclassical or quantum regime, depending on the magnetic\n\feld strengthjB\u0011j, the Fermi energy \u0016, and the level\nbroadening e\u000bect \u001c\u00001by impurities.\n\u000fQuantum regime : If the Fermi level \u0016and the level\nbroadening \u001c\u00001are both below the \frst Landau\nlevel\u000f1(kz= 0) =!c, only the zeroth Landau level9\ncontributs to the regular Hall e\u000bect. The Hall con-\nductivity in this regime is \u001bH(q)\n\u0011= (e2=4\u00192)(\u0016=vF),\nwhich is the three-dimensional counterpart of the\ntwo-dimensional quantum Hall conductivity e2=2\u0019\n[127].\n\u000fSemiclassical regime : If the Fermi level \u0016is far be-\nyond the Landau-level spacing \u0018!c, we can treat\nthe Hall transport unquantized. The semiclassi-\ncal Hall conductivity from the Boltzmann transport\ntheory is given by \u001bH(c)\n\u0011=\u0000(\u001c2e3\u0016=6\u00192)jB\u0011j.\nIn the absence of the realistic electromagnetic \felds,\nthe axial electromagnetic \felds ( E5;B5), corresponding\nto the dynamics and texture of the magnetization in a\nmagnetic Weyl semimetal, are the only source of the reg-\nular Hall e\u000bect. Since both E5andB5couple to the\nvalley\u0011by the sign \u0011, the regular Hall current is induced\nin the same direction for the two valleys (see Figure 2\n(h)),\nj(H)\n\u0011=\u001bH\n\u0011\u0011^B5\u0002\u0011E5=\u001bH\n\u0011^B5\u0002E5: (30)\nIn particular, if the axial magnetic \feld B5is strong\nenough to reach the quantum regime, the net regular\nHall current for the two valleys is simply given as\nj(H)=\u0000\n\u001bH\n++\u001bH\n\u0000\u0001^B5\u0002E5=e2\n2\u00192\u0016\nvF^B5\u0002E5;(31)\nwhich is independent of the \feld strength jB5j.\nIn addition to the current responses mentioned above,\nthe Weyl fermions are subject to the chiral anomaly ,\nnamely the violation of charge conservation in the pres-\nence of electromagnetic \felds. The idea of chiral anomaly\nwas originally established in the context of relativistic\n\feld theory, to account for the anomalous decay of a pion\n[150, 151], and recently it has been intensely applied to\nDirac and Weyl electrons in materials [152]. If we naively\nfocus on the electron dynamics around the Fermi surface,\nthe chiral anomaly states that charge conservation within\neach valley (chirality) \u0011=\u0006is violated by the electro-\nmagnetic \felds ( E\u0011;B\u0011) as\n@t\u001a\u0011+r\u0001j\u0011=\u0000e3\n4\u00192E\u0011\u0001B\u0011; (32)\nwhich is called the covariant anomaly [153, 154]. The\ncovariant anomaly, however, appears to violate the con-\nservation of net charge as\n@t\u001a+r\u0001j=\u0000e3\n2\u00192(E\u0001B5+E5\u0001B): (33)\nThis unphysical situation is resolved by supplementing\nthe regularization-dependent terms in the Lagrangian,\nnamely the \\Bardeen polynomials\" [155, 156]. This\ntreatment yields the charge-conserving relation\n@t\u001a+r\u0001j= 0; @t\u001a5+r\u0001j5=\u0000e3\n2\u00192\u0014\nE\u0001B+1\n3E5\u0001B5\u0015\n;\n(34)which is called the consistent anomaly. The Bardeen\npolynomials correspond to the current and charge carried\nby the occupied states away from the valleys in the con-\ntext of Weyl semimetals on lattice, including the anoma-\nlous Hall current j(A)mentioned above [123, 148]. The\nconsistent anomaly leads to the chiral charge imbalance,\nnamely the imbalance in the numbers of right-handed\nand left-handed fermions, which gives rise to several ob-\nservable phenomena; for instance, the negative magne-\ntoresistance in Dirac/Weyl semimetals is described as the\ncombined e\u000bect of the chiral charge imbalance from the\nanomaly and the current induction by the chiral magnetic\ne\u000bect [34]. While the chiral anomaly is present under the\naxial electromagnetic \felds as well, I will not go into de-\ntail in the discussions below, since it does not modulate\nthe net charge and current pro\fles at linear response to\nthose \felds.\nD. Charge pumping by magnetic texture dynamics\nBased on the forementioned list of current responses\nto the axial electromagnetic \felds, we are now ready to\ndiscuss the charge and current responses to the dynamics\nof magnetic textures in a Weyl semimetal. If there is a\nmagnetic texture M(r;t) that is spatially localized and\ntemporally modulating, we can de\fne the axial electric\n\feldE5and the axial magnetic \feld B5localized at the\nmagnetic texture [Equations(19) and (20)]. If the spatial\nand temporal modulations of the magnetic texture are\nmoderate enough compared to the mean-free path and\ntime of the electrons, we can consider the axial electro-\nmagnetic \felds locally uniform within these scales, which\nenables us to use the macroscopic response picture sum-\nmarized above in the vicinity of the magnetic texture.\nHere I focus on the local current distribution in response\nto the axial electromagnetic \felds, and derive the dynam-\nics of electric charge distribution accompanied with the\nmagnetic texture dynamics [107, 157].\nIn the absence of real electromagnetic \felds ( E;B), the\nchiral pseudomagnetic e\u000bect j(C)and the regular Hall\ne\u000bectj(H)are the only contribution to the net charge\ncurrentjind(r;t), up to linear response to the axial elec-\ntromagnetic \felds ( E5;B5). Since the chiral anomaly\nfromE5\u0001B5leads only to the chiral charge imbalance\nand does not violate the net charge conservation, we can\nuse the charge conservation relation\n@t\u000e\u001a(r;t) =\u0000r\u0001jind(r;t) =\u0000r\u0001h\nj(C)(r;t) +j(H)(r;t)i\n(35)\nto estimate the charge density pro\fle \u000e\u001a(r;t) modulated\n(pumped) by the magnetic texture dynamics. Since the\nchiral pseudomagnetic e\u000bect contribution j(C)/B5/\nr\u0002A5is divergence-free, the only contribution to the\npumped charge \u000e\u001ais from the regular Hall current j(H).\nIf the magnetic texture is well localized so that the axial10\nmagnetic \feld should be strong enough, only the zeroth\nLandau level contributes to the charge pumping; substi-\ntutingj(H)in the quantum regime [Equation (31)] to the\ncharge conservation relation Equation (35), one obtains\nthe relation\n@t\u000e\u001a(r;t) =e2\n2\u00192\u0016\nvFh\n^B5\u0001(r\u0002E5)\u0000E5\u0001(r\u0002^B5)i\n;\n(36)\nfrom which one can derive the time evolution of the\ncharge distribution \u000e\u001a(r;t) induced by the magnetic tex-\nture dynamics.\nAssuming there is no curl in the direction of the ax-\nial magnetic \feld ^B5, the second term in the right hand\nside of Equation (36) vanishes and this relation can be\nfurther simpli\fed. While it is di\u000ecult to rewrite this con-\ndition for the magnetic texture M(r) in general, we can\nconsider some extreme cases that satisfy this condition:\nifM(r) is aligned within a certain plane, which can be\nrealized under a strong easy-plane magnetic anisotropy,\nB5points perpendicular to this plane and thus ^B5be-\ncomes homogeneous. In such cases, by using the general\nrelation r\u0002E5=\u0000@tB5, we obtain a further simpli\fed\nrelation\n\u000e\u001a(r;t) =\u0000e2\n2\u00192\u0016\nvFjB5(r;t)j+ const: (37)\nThis relation is consistent with Equation (23) obtained\nin equilibrium. Therefore, we can see that the localized\ncharge\u001aBarising from the axial magnetic \rux B5moves\ntogether with the dynamics of magnetic texture.\nThe charge pumping e\u000bect in magnetic Weyl semimetal\nappears similar to the current induction by the spin mo-\ntive force in normal magnetic metals, mentioned in Sec-\ntion II. Their di\u000berence can be understood by considering\nthe work (energy transfer) on the electrons exerted by the\nmagnetic texture. The spin motive force in normal met-\nals act on an electron as the drift force \u0000e~EEEby the spin\nelectric \feld ~EEE. Since the drift force exerts a work on\nthe transported electron, the energy of the magnetic tex-\nture dynamics is eventually transferred to the ensemble\nof electrons, which is usually dissipated via the electron\nscattering by impurities. On the other hand, in magnetic\nWeyl semimetals, the driving force from the axial electro-\nmagnetic \felds ( E5;B5) is the Lorentz force \u0000e_r\u0002B5,\nwhich is perpendicular to the path of the electron. There-\nfore, it does not exert a work on the electrons, and the\nmagnetic texture dynamics does not lose its energy by\nthis pumping e\u000bect. The energy is dissipated only via the\nGilert damping of the constituent spins in the magnetic\ntexture, so that the magnetic texture dynamics does not\nsigni\fcantly heat up the Weyl electrons. In this sense,\nthis pumping e\u000bect in magnetic Weyl semimetal can be\nregarded \\adiabatic\" [107].E. Field-induced dynamics of magnetic textures\nSo far we have seen that the electron dynamics driven\nby magnetic texture dynamics can be understood with\nthe idea of the axial electromagnetic \felds. Similarly,\ndriving of magnetic texture dynamics by the electrons,\nnamely the spin transfer torque and the spin-orbit torque,\ncan also be formulated by using the idea of the axial\nelectromagnetic \felds [157].\nGenerally speaking, the spin torque is induced by the\nelectron spin accumulation h\u001bi. It gives an e\u000bective mag-\nnetic \feldJh\u001bion the magnetization vector nvia the ex-\nchange interaction J, leading to the torque T=Jh\u001bi\u0002n.\nIn a Weyl semimetal of the toy model Equation (11), in\nparticular, the spin accumulation is given equivalent to\nthe axial current,\nj5=\u0000eX\n\u0011=\u0006\u0011hv\u0011i\u0011=\u0000evFX\n\u0011=\u0006h\u001bi\u0011=\u0000evFh\u001bi(38)\nby using Equation (13), where h\u0001i\u0011denotes the expecta-\ntion value within valley \u0011. Therefore, we here need to\nfocus on the axial current response to estimate the \feld-\ninduced torques on the magnetic texture.\nWhen an electric \feld Eis applied to the magnetic tex-\nture, there arises a spin-transfer torque described by the\nregular Hall e\u000bect [157]: since the magnetic texture ac-\ncompanies the axial magnetic \feld B5, the current driven\nby the regular Hall e\u000bect is an axial current,\nj(H)\n5=j(H)\n+\u0000j(H)\n\u0000=e2\n2\u00192\u0016\nvF^B5\u0002E; (39)\nin the quantum regime. As a result, the regular Hall\ne\u000bect induces spin accumulation localized at the mag-\nnetic texture, leading to the switching of magnetization\nvia the spin torque. In contrast to the conventional spin\ntransfer torque driven by conduction current, the spin\ntorque noted here does not require a conduction current.\nTherefore, although the Weyl electrons cannot be trans-\nmitted through a sharp magnetic texture, as the valleys\nare shifted in momentum space in accordance with the\nmagnetization, this spin torque is still present and drives\na motion of the magnetic texture. This e\u000bect can also be\nregarded macroscopically as the electric driving of the lo-\ncalized charge \u001aBshown above attached to the magnetic\ntexture [130, 131].\nIn magnetic Weyl semimetals, it is also proposed that\nan external magnetic \feld Bunder a gate voltage induces\na spin torque, which is termed charge (voltage)-induced\nspin torque [158, 159]. This e\u000bect can be described in\nterms of the chiral magnetic e\u000bect: when the Fermi level\nis lifted by\u000e\u0016due to the gate voltage, the magnetic \feld\nBinduces the axial current\nj(C)\n5=j(C)\n+\u0000j(C)\n\u0000=\u0000e2\n2\u00192\u000e\u0016B; (40)\nwhich is the axial current counterpart of the chiral mag-\nnetic e\u000bect (see Figure 2(a)). Therefore, if the gate volt-\nage\u000e\u0016is applied in a limited area, the spin accumulation,11\ncorresponding to the axial current j(C)\n5, enables one to\nswitch the magnetization within this area, without driv-\ning any electric current.\nIV. EXAMPLE: MAGNETIC DOMAIN WALLS\nBased on the general theory about magnetic texture\ndynamics in Weyl semimetals, let us focus on the e\u000bect\nof magnetic domain walls in this section, as a typical ex-\nample. There have been several theoretical works on the\nelectron dynamics and transport in the presence of mag-\nnetic domain walls in magnetic Weyl semimetals. It was\nseen both analytically and numerically that a magnetic\ndomain wall in a Weyl semimetal gives rise to a large\ndomain-wall magnetoresistance, due to the mismatch of\nthe electron helicity beyond the domain wall [160, 161].\nOne of the peculiar features of magnetic domain walls\nin Weyl semimetals is the emergence of one-dimensional\nzero modes localized at the domain wall [130, 131]. These\nzero modes can be regarded as the remnant of the sur-\nface Fermi arc of Weyl semimetal. Macroscopically, this\nlocalized mode corresponds to the Landau states under\nthe axial magnetic \feld from the domain wall texture.\nLet us here see this correspondence by using a typical\none-dimensional domain wall structure.\nFor a one-dimensional magnetic domain wall, several\ntypes of internal structure are possible. If the domain\nwall is centered at x= 0 and the magnetization in each\nregion separated by the domain wall points to the direc-\ntion parallel to the wall, i.e. M(x!1 ) =\u0006M0ez, the\ninternal structure of the domain wall can be formulated\nas\nM(x) =M0\u0010\nsechx\nwcos\u000b;sechx\nwsin\u000b;tanhx\nw\u0011\n;(41)\nwhere the length scale wcorresponds to the thick-\nness of the domain wall. The internal structure is\ncharacterized by the angle \u000b: the N\u0013 eel domain wall,\nin which the magnetization is twisted in a coplanar\nmanner (within xz-plane), corresponds to \u000b= 0;\u0019,\nwhereas the Bloch domain wall, in which the mag-\nnetization is twisted transverse to x-axis (within yz-\nplane), corresponds to \u000b=\u0006\u0019=2. In realistic ma-\ngentic materials, the internal structure of the domain\nwall is governed by the magnetic anisotropy and the\nDzyaloshinskii{Moriya interaction (under the broken in-\nversion symmetry). The head-to-head domain wall\nM(x) =M0(\u0006tanh(x=w);0;sech(x=w)) can also be con-\nsidered, which we will not go into details in this article.\nUnder this domain wall, the Weyl electrons feel the\naxial magnetic \feld [Equation (20)]\nB5(x) =\u0000J\nevFr\u0002M(x) (42)\n=JM0\nevFwsech2x\nw\u0010\n0;1;sinhx\nwsin\u000b\u0011\n;\nE5\nB5\nj(H)\nM(x,t)xyz\nVDWFIG. 3. Setup of the domain wall texture M(x;t) moving\nwith velocity VDW, its corresponding axial electromagnetic\n\felds ( E5;B5), and the induced regular Hall current j(H)\nthat pumps the charge localized at the domain wall.\nwhich is localized around the domain wall at x= 0 (see\nFigure 3). As seen in the previous section, this axial mag-\nnetic \feld leads to the Landau quantization and gives rise\nto the locaized charge \u001aB(x) around the domain wall.\nIf we assume that the Fermi level \u0016is in the quantum\nregime, i.e. \u0016is between the zeroth and \frst Landau lev-\nelsthroughout the whole system , only the zeroth Landau\nlevel contributes to the localized charge and the induced\ncharge density obeys Equation (23). Thus the localized\ncharge per unit area of the domain wall can be estimated\nas\nq=e\n\u00192JM0\nv2\nF\u0016(N\u0013 eel);e\n2\u0019JM0\nv2\nF\u0016(Bloch); (43)\nfor each type of the domain wall [107]. This result\nwas veri\fed analytically for N\u0013 eel domain wall using the\nJackiw{Rebbi formalism [130], and numerically for both\ntypes of domain walls using the lattice model [131], which\nimplies that the zeroth Landau level considered in this\nmacroscopic description really has a dominant contribu-\ntion to the charging of domain walls. It was seen in these\nliteratures that the localized modes contributing to the\ncharging show the band structure similar to the Fermi\narcs on the surface, crossing the zero-energy plane by an\nopen countour that bridges two Weyl points in momen-\ntum space. The equilibrium current j(C)from the chiral\npseudomagnetic e\u000bect [Equation (25)] was also explicitly\nseen by using the wave functions of the localized modes.\nWhat will occur to this localized charge if the do-\nmain wall is moving? In order to see the dynamical\nbehavior, here I assume that the domain wall is paral-\nlelly moving with velocity VDWfor simplicity, which is\nreproduced by substituting the position xin Equation\n(41) byx\u0000VDWt(\u0011x0). Such a domain wall motion\ncan be driven by, for example, applying a magnetic \feld\nexternally. The motion of the domain wall yields both\nthe axial magnetic \feld, given by substitution x!x0in12\nEquation (42), and the axial electric \feld\nE5(x0) =J\nevF@tM(x0) =\u0000JVDW\nevF@x0M(x0) (44)\n=JM0VDW\nevFwsech2x0\nw\u0012\nsinhx0\nwcos\u000b;sinhx0\nwsin\u000b;\u00001\u0013\n:\nSinceB5andE5are perpendicular to one another, they\ngive rise to the regular Hall current j(H)localized at the\ndomain wall (see Figure 3). Among the regular Hall cur-\nrent, itsx-component\nj(H)\nx(x0) =\u0000eJM 0VDW\u0016\nv2\nFwsech2x0\nwr\n1 + sinh2x0\nwsin2\u000b\n(45)\ncontributes to the pumping of the localized charge, satis-\nfying the charge conservation @t\u001aB+@xj(H)\nx= 0. There-\nfore, we can see from the macroscopic axial electro-\nmagnetic \feld picture that a magnetic domain wall in\nthe Weyl semimetal bears a certain amount of localized\ncharge depending on the internal structure of the domain\nwall, and that the localized charge is carried along with\nthe motion of the domain wall, which can be regarded as\nthe regular Hall current driven by the axial electromag-\nnetic \felds.\nThe localized modes discussed here can also be under-\nstood in connection with the Jackiw{Rebbi mode, which\narises as the zero-energy solitonic solution of the Dirac\nequation localized at a domain wall in the Dirac mass\nterm [162]. Let us consider the case of a N\u0013 eel domain\nwall (\u000b= 0 in Equation (41)). By taking the transverse\nmomentum components kyto zero and kzto a \fxed value,\nthe Weyl Hamiltonian coupled with the magnetic texture\nreduces to a one-dimensional Dirac Hamiltonian along x,\nH\u0011;kz=h\n\u0000ivF\u0011@x+JM0sechx\nwi\n\u001bx+m\u0011;kz(x)\u001bz;\n(46)\nwith thex-dependent Dirac mass term m\u0011;kz(x) =\n\u0011vFkz+JM0tanhx\nw. By the U(1) gauge transforma-\ntion withU(x) = exph\n\u0000i\u0011JM0\nvFR\ndxsechx\nwi\n, the Hamil-\ntonian can be further reduced as H\u0011;kz=\u0000ivF\u0011@x\u001bx+\nm\u0011;kz(x)\u001bz, to which one can apply Jackiw{Rebbi's dis-\ncussion. The localized zero mode arises when m\u0011;kz(x)\nchanges its sign at x= 0, which occurs if jkzj<JM 0=vF,\ncorresponding to the condition for the surface Fermi arc.\nWhile the Jackiw{Rebbi mode interpretation is success-\nful for the N\u0013 eel domain wall, it is not applicable to the\nBloch domain wall, since the Hamiltonian cannot exactly\nbe reduced to the Jackiw{Rebbi form by the gauge trans-\nformation due to the z-component in B5(see Equation\n(42)). Thus we need correction by the pseudomagnetic\n\felds to Jackiw{Rebbi's discussion, when we investigate\nspin-momentum-locked electrons under magnetic domain\nwalls.\nIt is suggested that the localized charge on the do-\nmain wall can serve as a \\knob\" for the manipulation ofthe domain wall motion by an external electric \feld; this\nelectric manipulation was numerically simulated by solv-\ning the Landau{Lifshitz{Gilbert equation on the lattice\nmodel, using the spin torques mentioned in Section III E\n[157].\nV. MAGNETIC TEXTURES IN VARIOUS\nWEYL SEMIMETALS\nSo far we have taken a fully spin-momentum locked\nWeyl semimetal model, namely the minimal model con-\ntaining\u001b\u0001kterm, to get an intuitive understanding\non the interplay between spin-momentum locking of the\nelectrons and real-space magnetic textures. In realistic\nWeyl semimetal materials, on the other hand, the spin-\nmomentum locking structure around the Weyl points is\nmaterial-dependent, since the band crossing comes from\nspin-orbit coupling characteristic to each material. As I\nhave mentioned in the \frst Section, signi\fcant progress in\nthe material-based \frst-principle calculations and the ex-\nperimental measurement techniques has enabled us to ob-\nserve several compounds as candidates for Weyl semimet-\nals. In this section, I review the current status of the-\noretical and experimental works on realizing ferromag-\nnetic/antiferromagnetic Weyl semimetals, and see how\nmagnetic textures and the electronic properties are re-\nlated in these materials.\nA. Topological insulator multilayers\nFrom the early days of the theoretical works on Weyl\nsemimetals, it has been proposed that the ferromagnetic\nWeyl semimetal phase can be realized in magnetic multi-\nlayers composed of thin \flms of magnetically-doped topo-\nlogical insulator (TI) and normal insulator (NI) [71, 72].\nWhen a 3D TI is magnetically doped, its surface Dirac\ncone becomes gapped out due to the breaking of time-\nreversal symmetry by the ferromagnetic order, leading\nto the quantum anomalous Hall e\u000bect [16, 17, 115]. By\nstacking these TI \flms with NI \flms in between, the top\nand bottom surface states of the TI \flms are moderately\ncoupled due to intralayer and interlayer tunneling, lead-\ning to the band dispersion perpendicular to the layers.\nDepending on the ratio between the intralayer and inter-\nlayer tunneling amplitudes, the system can exhibit 3D\nTI and NI phases. Between these phases emerges the\nWeyl semimetal phase, with closing of the band gap at\nthe Weyl points [163].\nThe Hamiltonian around the Weyl points can be sym-\nbolically written as\nH(k) =vF(ez\u0002\u001b)\u0001k+m(kz)\u001bz; (47)\nwhere\u001bx;y;z are the spin Pauli matrices mixed with or-\nbital degrees of freedom. The function m(kz), governed\nby the tunneling amplitudes and the exchnge energy by13\nthe magnetic dopants, reaches zero at two kzpoints, cor-\nresponding to the Weyl points. In this Hamiltonian, the\nin-plane spin-momentum locking is inherited from that of\nthe TI surface states and hence is Rashba-like [164, 165].\nTherefore, an in-plane magnetization leads to an e\u000bective\nvector potential perpendicular to the magnetization di-\nrection. On the other hand, the out-of-plane magnetiza-\ntion gives rise to shift of the Weyl points along kz-axis via\nthem(kz)-term, yielding an out-of-plane e\u000bective vector\npotential. Since all three components of spin are cou-\npled to momentum, the magnetic textures formed by the\nmagnetic dopants in this system lead to the charge and\ncurrent pro\fle similar to those listed up in Section III,\nexcept for the in-plane spin texture here being twisted by\n90 degrees from that under the ( k\u0001\u001b)-Hamiltonian, due\nto the Rashba-like spin-momentum locking. (Since the\nspin Pauli matrices here are mixed with orbital degrees\nof freedom, the e\u000bective vector potentials mentioned here\nshould be viewed as the mixture of the normal and axial\nvector potentials.)\nSimilarly to the TI multilayers, it is proposed that a\nmagnetically doped bulk 3D TI can also exhibit the Weyl\nsemimetal phase provided that the doping ratio is prop-\nerly tuned: the Weyl phase again emerges as the inter-\nmediate phase between the TI and NI phases [166]. From\nthe mean-\feld theory, it was shown that about 10 per-\ncent doping of Cr in Bi 2(SexTe1\u0000x)3can drive the system\nfrom the TI phase into the Weyl phase. Since it is based\non the band inversion by spin-orbit coupling in TI, the\nspin-momentum locking structure in this system is sim-\nilar to that in the above multilayer system, where the\nin-plane part is given by the Rashba-like form.\nB. Layered kagome ferromagnet: Co3Sn2S2\nRecent studies suggested that a cobalt-based shandite\nmaterial Co 3Sn2S2works well as a Weyl semimetal with\nthe ferromagnetic order [75{77]. The ferromagnetic or-\nder consists of the magnetic moments in Co atoms ar-\nranged in kagome-lattice layers, with the out-of-plane\nmagnetic moment \u00180:3\u0016Bfor each Co atom [167]. The\nWeyl points due to the breaking of time-reversal symme-\ntry reside just 60meV beyond the Fermi energy, with no\nother bands crossing the Fermi level [75]. Thus Co 3Sn2S2\nmay serve as a good platform for realizing the anomalous\ntransport characteristic to Weyl semimetals: for instance,\n\frst-principle calculations and transport measurements\nshow both a large anoumalous Hall conductivity up to\n\u0018103\n\u00001cm\u00001and a large anomalous Hall angle (the\nratio of the anomalous Hall conductivity to the longitu-\ndinal conductivity) up to \u001820, which are not simulta-\nneously reached in conventional magnetic materials [75].\nThe Weyl band structure and the emergent Fermi arcs\non surface were also observed by angle-resolved photoe-\nmission spectroscopy (ARPES) measurements [77].\nIdenti\fcation of magnetic textures in kagome mag-\nnets is also an important question from the viewpointof magnetism, as the kagome lattice can host various\nkinds of unconventional spin correlations, such as the\nspin frustration and Dzyaloshinskii{Moriya interaction.\nFrom the measurement of magnetization and ac sus-\nceptibility by superconducting quantum interference de-\nvice (SQUID), an anomalous magnetic phase (A-phase)\ndistinct from the ferromagnetic phase was observed in\nCo3Sn2S2, which appears just below the Curie temper-\natureTC= 173K at low magnetic \feld [168]. It was\nsuggested that this phase may host nonuniform spin\ntextures such as stripe domains, magnetic bubbles, or\nbiskyrmions. Further observation of magnetic textures\nin the A-phase, by neutron scattering, magnetic force\nmicroscopy, etc., is now pending.\nDue to its layered structure, spin-orbit coupling in\nCo3Sn2S2is higly anisotropic, which may result in\nan anisotropy in the spin-momentum locking structure\naround the Weyl points. It was suggested by the \frst-\nprinciples calculations that a rotation of the magnetiza-\ntion from the out-of-plane direction to the in-plane direc-\ntion eventually modulates the arrangement of the Weyl\nnodes, with their complicated trajectories in the Bril-\nlouin zone [169]. It was also theoretically con\frmed that\na simpli\fed two-orbital tight-binding model on kagome\nlattice with Kane{Mele-type spin-orbit interaction well\nreproduces the Weyl-node structure observed in the \frst-\nprinciples calculations [170]; their agreement implies that\nthe out-of-plane spin component dominantly participates\nin spin-orbit coupling in Co 3Sn2S2, in the vicinity of the\nFermi level. As a result, we can roughly expect that the\nout-of-plane component of the magnetization shifts the\nWeyl nodes and contributes to the pseudo-gauge \feld, as\nwe have seen in the spin-momentum-locked model Hamil-\ntonian, while the in-plane component couples to the elec-\ntron spin conventionally as in normal metals. The e\u000bect\nof the experimentally-suggested magnetic textures men-\ntioned above on the electronic structure and transport is\nleft as an important question for future spintronics ap-\nplication of this material.\nC. Noncollinear antiferromagnet: Mn3Sn\nWhile we have so far focused on the Weyl semimetals\nwith time-reversal symmetry broken by ferromagnetism,\nwe may also consider antiferromagnets with Weyl-node\nstructure. Mn 3Sn and Mn 3Ge are known to exhibit both\nantiferromagnetism and the Weyl nodes, which was \frst\nproposed by \frst-principles calculations [73] and later\nmanifestly observed by transport and ARPES measure-\nments [74, 171{174]. Mn atoms are responsible for the\nantiferromagnetic order in these materials, arranged in\nkagome-lattice layers. The magnetic moments on these\nkagome sites form a 120-degree noncollinear antiferro-\nmagnetic order, with the N\u0013 eel temperature up to \u0018430K\nin Mn 3Sn [175{178]. It is understood that this non-\ncollinear antiferromagnetism arises from the combina-\ntion of the antiferromagnetic excchange coupling and14\nthe Dzyaloshinski{Moriya interaction. Although their\nnet magnetization is vanishingly small ( \u00180:002\u0016Bfor\neach Mn atom) [176, 177] and the scalar spin chirality\nis zero, it was numerically and experimentally observed\nthat these materials show large anomalous Hall con-\nductivity [171{173], which can be regarded as an e\u000bect\nfrom the strong Berry \rux in momentum space around\nthe Weyl points. A tight-binding model calculation on\nkagome layers con\frmed that the k-space Berry curva-\nture indeed reproduces this large anomalous Hall con-\nductivity around the Fermi level [179]. Although there\nexist trivial metallic bands in addition to the Weyl cones\nat the Fermi level so that they cannot be regarded as\n\\semimetals\", the Weyl points thus contribute strongly\nto the anomalous transport properties.\nOne can consider a spatial modulation of this 120-\ndegree pattern at macroscopic length scales much longer\nthan the lattice constant. Since there is no uni\fed or-\nder parameter such as magnetization in ferromagnets\nand N\u0013 eel vector (staggered magnetization) in collinear\nantiferromagnets, one should pick up some other vecto-\nrial quantity that characterizes the macroscopic magnetic\ntextures. One powerful quantity that we can rely on is\nthe cluster octupole moment, which was derived by the\nmultipole expansion in each cluster [180]. It has the same\nsymmetry as the magnetic dipole moment, and gives rise\nto the anomalous Hall e\u000bect in the same manner with\ndipole. Domain structure identi\fed by the cluster oc-\ntupole moment was experimentally distinguished by mea-\nsuring the magneto-optical Kerr e\u000bect (MOKE) [181].\nSince the spin-momentum locking structure is complex\ndue to the noncollinear spin texture, it seems di\u000ecult to\napply the pseudo-gauge \feld picture to treat the e\u000bect\nof macroscopic magnetic textures. One way to resolve\nthis problem theoretically is to list up the possible terms\nallowed by the symmetry of the system. In Reference\n[182], the structure of domain wall in the kagome anti-\nferromagnet was parameterized by the in-plane twisting\nangle of the noncollinear order, and the current-induced\ndomain-wall dynamics was qualitatively discussed by us-\ning the collective force on the domain wall allowed by the\nsymmetry. The orbital magnetization may be helpful to\nunderstand the qualitative e\u000bect of magnetic textures on\nthe shift of Weyl points, since they are closely related via\nthe intrinsic anomalous Hall conductivity [183, 184]. It\nwas seen by the tight-binding model calculation that the\ndirection and magnitude of the orbital magnetization sig-\nni\fcantly depend on the tilting angles of the noncollinearspins [179]. Thus one can assume that a twist in the non-\ncollinear spin texture leads to shift of Weyl points, giving\nrise to the pseudo-gauge \feld texture.\nVI. CONCLUSION\nIn this article, we have reviewed the current under-\nstanding of the interplay between magnetic textures and\nelectronic properties in magnetic Weyl semimetals. Since\nthe magnetization contributes to the shift of Weyl points\nin momentum space, the e\u000bect of magnetic textures on\nthe electrons can be e\u000bectively treated as axial gauge\n\felds, with which one can macroscopically treat the elec-\ntronic structure and transport in a similar manner with\nthose under the realistic electromagnetic \felds. In con-\ntrast to the spin gauge \felds A(r) in normal metals,\nwhich depend on spatial gradients in the magnetization\npattern rM(r) (Section II), the axial gauge \felds A5(r)\nfor the Weyl electrons is directly related to the local mag-\nnetizationM(r) due to spin-momentum locking (Section\nIII). From this di\u000berence, we have seen that the elec-\ntronic structure in Weyl semimetals gets largely altered\nby magnetic textures; even a simple one-dimensional do-\nmain wall leads to the localization and pumping of elec-\ntric charge, as we have seen in Section IV.\nWe have mainly focused on the e\u000bect of long-range and\nlow-frequency magnetic textures introduced by hand in\nthis article. On the other hand, magnetically ordered\nmaterials generally host spin-wave excitations, namely\nmagnons, which arise as collective dynamics in the pre-\ncession of spins interacting with each other. It is well\nknown that magnons carry spin current [185], and hence\ntheir properties are being intensely studied to make use\nof them as carriers in spintronics devices, generically\ntermed as magnonics [186, 187]. Since magnons can\ncouple to spin-momentum-locked Weyl electrons in mag-\nnetic Weyl semimetals, there have been several theo-\nretical proposals about magnon-induced anomalous re-\nsponses [78, 188, 189]. Interplay between electrons and\nmagnons in realistic Weyl semimetals needs to be ques-\ntioned in future studies [190].\nAcknowledgments. This work is supported by JSPS\nKAKENHI Grant Number JP17K14316. The author ac-\nknowledges K. Nomura and A. Yoshida for the related\ncollaborations. The author thanks D. Kurebayashi for\nthe fruitful discussions.\nCon\rict of interest. The author declares no con\rict\nof interest.\n[1] C. Kittel, Introduction to Solid State Physics , John Wi-\nley & Sons, New York, 1953 .\n[2] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n2004 , 76, 323.\n[3] S. D. Bader and S. S. P. Parkin, Annu. Rev. Con-\ndens. Matter Phys. 2010 , 1, 71.[4] M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van\nDau, F. Petro\u000b, P. Etienne, G. Creuzet, A. Friederich,\nand J. Chazelas, Phys. Rev. Lett. 1988 , 61, 2472.\n[5] G. Binasch, P. Gr unberg, F. Saurenbach, and W. Zinn,\nPhys. Rev. B 1989 , 39, 4828(R).\n[6] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science,15\n320, 190 (2008).\n[7] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol.\n2013 , 8, 152.\n[8] B. Yan and S.-C. Zhang, Rep. Prog. Phys. 2012 , 75,\n096501.\n[9] C.-K. Chiu, J. C. Y. Teo, A. P. Schnyder, and S. Ryu,\nRev. Mod. Phys. 2016 , 88, 035005.\n[10] M. Z. Hasan and C. L. Kane, Rev. Mod. Phys. 2010 ,\n82, 3045.\n[11] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 2011 , 83,\n1057.\n[12] B. A. Bernevig and T. L. Hughes, Topological insula-\ntors and topological superconductors , Princeton Univer-\nsity Press, Princeton, 2013 .\n[13] D. Pesin and A. H. MacDonald, Nat. Mater. 2012 , 11,\n409.\n[14] Y. Tokura, K. Yasuda, and A. Tsukazaki,\nNat. Rev. Phys. 2019 , 1, 126.\n[15] K. Nomura and N. Nagaosa, Phys. Rev. Lett. 106,\n166802 (2011).\n[16] R. Yu, W. Zhang, H.-J. Zhang, S.-C. Zhang, X. Dai,\nand Z. Fang, Science 2010 , 329, 61.\n[17] C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang,\nM. Guo, K. Li, Y. Ou, P. Wei, L.-L. Wang, Z.-Q. Ji,\nY. Feng, S. Ji, X. Chen, J. Jia, X. Dai, Z. Fang, S.-\nC. Zhang, K. He, Y. Wang, L. Lu, X.-C. Ma, and Q.-\nK. Xue, Science 2013 , 340, 167.\n[18] J. G. Checkelsky, J. Ye, Y. Onose, Y. Iwasa, and\nY. Tokura, Nat. Phys. 2012 , 8, 729.\n[19] W.-K. Tse and A. H. MacDonald, Phys. Rev. Lett.\n2010 , 105, 057401.\n[20] K. N. Okada, Y. Takahashi, M. Mogi, R. Yoshimi,\nA. Tsukazaki, K. S. Takahashi, N. Ogawa, M. Kawasaki,\nand Y. Tokura, Nat. Commun. 2016 , 7, 12245.\n[21] I. Garate and M. Franz, Phys. Rev. Lett. 2010 , 104,\n146802.\n[22] Y. Shiomi, K. Nomura, Y. Kajiwara, K. Eto, M. Novak,\nK. Segawa, Y. Ando, and E. Saitoh, Phys. Rev. Lett.\n2014 , 113, 196601.\n[23] F. Mahfouzi, N. Nagaosa, and B. K. Nikoli\u0013 c,\nPhys. Rev. Lett. 2012 , 109, 166602.\n[24] A. R. Mellnik, J. S. Lee, A. Richardella, J. L. Grab,\nP. J. Mintun, M. H. Fischer, A. Vaezi, A. Manchon, E.-\nA. Kim, N. Samarth, and D. C. Ralph, Nature 2014 ,\n511, 449.\n[25] B. Yan and C. Felser, Annu. Rev. Condens. Matter\nPhys. 2017 , 8, 337.\n[26] A. A. Burkov, Annu. Rev. Condens. Matter Phys. 2018 ,\n9, 359.\n[27] N. P. Armitage, E. J. Mele, and A. Vishwanath,\nRev. Mod. Phys. 2018 , 90, 015001.\n[28] G. E. Volovik, The Universe in a Helium Droplet ,\nClarendon, Oxford, 2003 .\n[29] A. G. Grushin, Phys. Rev. D 2012 , 86, 045001.\n[30] P. Goswami and S. Tewari, Phys. Rev. B 2013 , 88,\n245107.\n[31] A. A. Burkov, Phys. Rev. B 2014 , 89, 155104.\n[32] A. A. Burkov, Phys. Rev. Lett. 2014 , 113, 187202.\n[33] H. B. Nielsen and M. Ninomiya, Phys. Lett. B 1983 ,\n130, 389.\n[34] D. T. Son and B. Z. Spivak, Phys. Rev. B 2013 , 88,\n104412.\n[35] X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen,\nZ. Yang, H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai,and G. Chen, Phys. Rev. X 2015 , 5, 031023.\n[36] C.-L. Zhang, S.-Y. Xu, I. Belopolski, Z. Yuan, Z. Lin,\nB. Tong, G. Bian, N. Alidoust, C.-C. Lee, S.-M. Huang,\nT.-R. Chang, G. Chang, C.-H. Hsu, H.-T. Jeng, M. Ne-\nupane, D. S. Sanchez, H. Zheng, J. Wang, H. Lin,\nC. Zhang, H.-Z. Lu, S.-Q. Shen, T. Neupert, M. Zahid\nHasan, and S. Jia, Nat. Commun. 2016 , 7, 10735.\n[37] F. Arnold, C. Shekhar, S.-C. Wu, Y. Sun, R. D. dos\nReis, N. Kumar, M. Naumann, M. O. Ajeesh,\nM. Schmidt, A. G. Grushin, J. H. Bardarson,\nM. Baenitz, D. Sokolov, H. Borrmann, M. Nicklas,\nC. Felser, E. Hassinger, and B. Yan, Nat. Commun.\n2016 , 7, 11615.\n[38] A. C. Potter, I. Kimchi, and A. Vishwanath Nat. Com-\nmun. 2014 , 5, 5161.\n[39] Y. Zhang, D. Bulmash, P. Hosur, A. C. Potter, and\nA. Vishwanath, Sci. Rep. 2016 , 6, 23741.\n[40] P. J. W. Moll, N. L. Nair, T. Helm, A. C. Potter, I. Kim-\nchi, A. Vishwanath, and J. G. Analytis, Nature 2016 ,\n535, 266.\n[41] C. Zhang, A. Narayan, S. Lu, J. Zhang, H. Zhang, Z. Ni,\nX. Yuan, Y. Liu, J.-H. Park, E. Zhang, W. Wang, S. Liu,\nL. Cheng, L. Pi, Z. Sheng, S. Sanvito, F. Xiu, Nat. Com-\nmun. 2017 , 8, 1272.\n[42] C. Zhang, Y. Zhang, X. Yuan, S. Lu, J. Zhang,\nA. Narayan, Y. Liu, H. Zhang, Z. Ni, R. Liu, E. S. Choi,\nA. Suslov, S. Sanvito, L. Pi, H.-Z. Lu, A. C. Potter, and\nF. Xiu, Nature 2019 , 565, 331.\n[43] J. E. Moore and J. Orenstein, Phys. Rev. Lett. 2010 ,\n105, 026805.\n[44] I. Sodemann and L. Fu, Phys. Rev. Lett. 2015 , 115,\n216806.\n[45] T. Morimoto and N. Nagaosa, Sci. Adv. 2016 , 2,\ne1501524.\n[46] T. Morimoto, S. Zhong, J. Orenstein, and J. E. Moore,\nPhys. Rev. B 2016 , 94, 245121.\n[47] C.-K. Chan, P. A. Lee, K. S. Burch, J. H. Han, and\nY. Ran, Phys. Rev. Lett. 2016 , 116, 026805.\n[48] X.-X. Zhang, T. T. Ong, and N. Nagaosa, Phys. Rev. B\n2016 , 94, 235137.\n[49] F. de Juan, A. G. Grushin, T. Morimoto, and\nJ. E. Moore, Nat. Commun. 2017 , 8, 15995.\n[50] A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu,\nM. Troyer, X. Dai, and B. A. Bernevig, Nature 2015 ,\n527, 495.\n[51] T. E. OBrien, M. Diez, and C. W. J. Beenakker,\nPhys. Rev. Lett. 2016 , 116, 236401.\n[52] M. Alidoust, K. Halterman, and A. A. Zyuzin,\nPhys. Rev. B 2017 , 95, 155124.\n[53] M. Alidoust, Phys. Rev. B 2018 , 98, 245418.\n[54] B. Rosenstein, B. Ya. Shapiro, D. Li, and I. Shapiro,\nPhys. Rev. B 2017 , 96, 224517.\n[55] B. Rosenstein, B. Ya. Shapiro, D. Li, and I. Shapiro,\nPhys. Rev. B 2018 , 97, 144510.\n[56] B. Ya. Shapiro, I. Shapiro, D. Li, and B. Rosenstein,\nJ. Phys.: Condens. Matter 2018 , 30 335403.\n[57] Z. Hou and Q.-F. Sun, Phys. Rev. B 2017 , 96, 155305.\n[58] Q. Li, C. He, Y. Wang, E. Liu, M. Wang, Y. Wang,\nJ. Zeng, Z. Ma, T. Cao, C. Yi, N. Wang, K. Watanabe,\nT. Taniguchi, L. Shao, Y. Shi, X. Chen, S.-J. Liang,\nQ.-H. Wang, and F. Miao, Nano Lett. 2018 , 18, 7962.\n[59] R. C. Xiao, P. L. Gong, Q. S. Wu, W. J. Lu, M. J. Wei,\nJ. Y. Li, H. Y. Lv, X. Luo, P. Tong, X. B. Zhu, and\nY. P. Sun, Phys. Rev. B 2017 , 96, 075101.16\n[60] S. Das, Amit, A. Sirohi, L. Yadav, S. Gayen, Y. Singh,\nand G. Sheet, Phys. Rev. B 2018 , 97, 014523.\n[61] S. P. Mukherjee and J. P. Carbotte, Phys. Rev. B 2017 ,\n96, 085114.\n[62] S. P. Mukherjee and J. P. Carbotte, Phys. Rev. B 2018 ,\n97, 035144.\n[63] K. Halterman, M. Alidoust, and A. Zyuzin,\nPhys. Rev. B 2008 , 98, 085109.\n[64] S. Mignuzzi, S. Vezzoli, S. A. R. Horsley, W. L. Barnes,\nS. A. Maier, and R. Sapienza, Nano Lett. 2019 , 19, 3,\n1613.\n[65] S.-M. Huang, S.-Y. Xu, I. Belopolski, C.-C. Lee,\nG. Chang, B. Wang, N. Alidoust, G. Bian, M. Neupane,\nC. Zhang, S. Jia, A. Bansil, H. Lin, and M. Z. Hasan,\nNat. Commun. 2015 , 6, 7373.\n[66] S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane,\nG. Bian, C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-\nC. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez,\nB.K. Wang, A. Bansil, F. Chou, P. P. Shibayev, H. Lin,\nS. Jia, and M. Zahid Hasan, Science 2015 , 349, 613.\n[67] B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao,\nJ. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen,\nZ. Fang, X. Dai, T. Qian, and H. Ding, Phys. Rev. X\n2015 , 5, 031013.\n[68] L. X. Yang, Z. K. Liu, Y. Sun, H. Peng, H. F. Yang,\nT. Zhang, B. Zhou, Y. Zhang, Y. F. Guo, M. Rahn,\nD. Prabhakaran, Z. Hussain, S.-K. Mo, C. Felser,\nB. Yan, and Y. L. Chen, Nat. Phys. 2015 , 11, 728.\n[69] S.-Y. Xu, N. Alidoust, I. Belopolski, Z. Yuan, G. Bian,\nT.-R. Chang, H. Zheng, V. N. Strocov, D. S. Sanchez,\nG. Chang, C. Zhang, D. Mou, Y. Wu, L. Huang, C.-\nC. Lee, S.-M. Huang, BaoK. Wang, A. Bansil, H.-\nT. Jeng, T. Neupert, A. Kaminski, H. Lin, S. Jia, and\nM. Z. Hasan, Nat. Phys. 2015 , 11, 748.\n[70] S.-Y. Xu, I. Belopolski, D. S. Sanchez, C. Zhang,\nG. Chang, C. Guo, G. Bian, Z. Yuan, H. Lu, T.-\nR. Chang, P. P. Shibayev, M. L. Prokopovych, N. Ali-\ndoust, H. Zheng, C.-C. Lee, S.-M. Huang, R. Sankar,\nF. Chou, C.-H. Hsu, H.-T. Jeng, A. Bansil, T. Neupert,\nV. N. Strocov, H. Lin, S. Jia, and M. Z. Hasan, Sci. Adv.\n2015 , 1, e1501092.\n[71] A. A. Burkov and L. Balents, Phys. Rev. Lett. 2011 ,\n107, 127205.\n[72] A. A. Burkov, M. D. Hook, and L. Balents, Phys. Rev. B\n2011 , 84, 235126.\n[73] H. Chen, Q. Niu, and A. H. MacDonald,\nPhys. Rev. Lett. 2014 , 112, 017205.\n[74] K. Kuroda, T. Tomita, M.-T. Suzuki, C. Bareille,\nA. A. Nugroho, P. Goswami, M. Ochi, M. Ikhlas,\nM. Nakayama, S. Akebi, R. Noguchi, R. Ishii, N. In-\nami, K. Ono, H. Kumigashira, A. Varykhalov, T. Muro,\nT. Koretsune, R. Arita, S. Shin, T. Kondo, and\nS. Nakatsuji, Nat. Mater. 2017 , 16, 1090.\n[75] E. Liu, Y. Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao,\nS.-Y. Yang, D. Liu, A. Liang, Q. Xu, J. Kroder, V. S uss,\nH. Borrmann, C. Shekhar, Z. Wang, C. Xi, W. Wang,\nW. Schnelle, S. Wirth, Y. Chen, S. T. B. Goennenwein,\nand C. Felser, Nat. Phys. 2018 , 14, 1125.\n[76] Q. Xu, E. Liu, W. Shi, L. Muechler, J. Gayles, C. Felser,\nand Y. Sun, Phys. Rev. B 2018 , 97, 235416.\n[77] Q. Wang, Y. Xu, R. Lou, Z. Liu, M. Li, Y. Huang,\nD. Shen, H. Weng, S. Wang, and H. Lei, Nat. Commun.\n2018 , 9, 3681.\n[78] C.-X. Liu, P. Ye, and X.-L. Qi, Phys. Rev. B 2013 , 87,235306.\n[79] M. E. Peskin and D. V. Schroeder, An Introduction to\nquantum \feld theory , Westview Press, Boulder, 1995 .\n[80] R. Ilan, A. G. Grushin, and D. I. Pikulin,\narXiv:1903.11088.\n[81] R. Tomasello, E. Martinez, R. Zivieri, L. Torres, M. Car-\npentieri, and G. Finocchio, Sci. Rep. 2014 , 4, 6784.\n[82] X. Zhang, M. Ezawa, and Y. Zhou, Sci. Rep. 2015 , 5,\n9400.\n[83] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 2008 ,\n468, 213.\n[84] K.-J. Kim, Y. Yoshimura, and T. Ono,\nJpn. J. Appl. Phys. 2017 , 56, 0802A4.\n[85] N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 2013 , 8,\n899.\n[86] A. Fert, N. Reyren, and V. Cros, Nat. Rev. Mater.\n2017 , 2, 17031.\n[87] K. Everschor-Sitte, J. Masell, R. M. Reeve, and\nM. Kl aui, J. Appl. Phys. 2018 , 124, 240901.\n[88] V. Korenman, J. L. Murray, and R. E. Prange,\nPhys. Rev. B 1977 , 16, 4032.\n[89] G. E. Volovik, J. Phys. C: Solid State Phys. 1987 , 20,\nL83.\n[90] J. Ieda and S. Maekawa, in Spin Current (Second Edi-\ntion) (Eds: S. Maekawa, S. O. Valenzuela, E. Saitoh,\nand T. Kimura), Oxford University Press, Oxford,\n2017 , Ch. 7.\n[91] G. Tatara, Physica E 2019 , 106, 208.\n[92] J. Ye, Y. B. Kim, A. J. Millis, B. I. Shraiman, P. Ma-\njumdar, and Z. Te\u0014 sanovi\u0013 c, Phys. Rev. Lett. 1999 , 83,\n3737.\n[93] G. Tatara and H. Kawamura, J. Phys. Soc. Jpn. 2002 ,\n71, 2613.\n[94] M. Lee, W. Kang, Y. Onose, Y. Tokura, and N. P. Ong,\nPhys. Rev. Lett. 2009 , 102, 186601.\n[95] A. Neubauer, C. P\reiderer, B. Binz, A. Rosch, R. Ritz,\nP. G. Niklowitz, and P. B oni, Phys. Rev. Lett. 2009 ,\n102, 186602.\n[96] N. Kanazawa, Y. Onose, T. Arima, D. Okuyama,\nK. Ohoyama, S. Wakimoto, K. Kakurai, S. Ishiwata,\nand Y. Tokura, Phys. Rev. Lett. 2011 , 106, 156603.\n[97] A. Stern, Phys. Rev. Lett. 1992 , 68, 1022.\n[98] C.-M. Ryu, Phys. Rev. Lett. 1996 , 76, 968.\n[99] L. Berger, Phys. Rev. B 1999 , 59, 11465.\n[100] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 2007 ,\n98, 246601.\n[101] S. A. Yang, G. S. D. Beach, C. Knutson, D. Xiao,\nQ. Niu, M. Tsoi, and J. L. Erskine, Phys. Rev. Lett.\n2009 , 102, 067201.\n[102] P. N. Hai, S. Ohya, M. Tanaka, S. E. Barnes, and\nS. Maekawa, Nature 2009 , 458, 489.\n[103] Y. Yamane, K. Sasage, T. An, K. Harii, J. Ohe,\nJ. Ieda, S. E. Barnes, E. Saitoh, and S. Maekawa,\nPhys. Rev. Lett. 2011 , 107, 236602.\n[104] K.-W. Kim, J.-H. Moon, K.-J. Lee, and H.-W. Lee,\nPhys. Rev. Lett. 2012 , 108, 217202.\n[105] N. Nakabayashi and G. Tatara, New J. Phys., 2014 , 16,\n015016.\n[106] G. Tatara, N. Nakabayashi, and K.-J. Lee, Phys. Rev. B\n2013 , 87, 054403.\n[107] Y. Araki and K. Nomura, Phys. Rev. Applied 2018 , 10,\n014007.\n[108] G. E. Volovik, Quantum Phase Transitions from Topol-\nogy in Momentum Space , inQuantum Analogues: From17\nPhase Transitions to Black Holes and Cosmology (Eds.:\nW. G. Unruh and R. Sch utzhold) Springer, Berlin, Hei-\ndelberg, 2007 .\n[109] H. B. Nielsen and M. Ninomiya, Nucl. Phys. B 1981 ,\n185, 20.\n[110] H. B. Nielsen and M. Ninomiya, Nucl. Phys. B 1981 ,\n193, 173.\n[111] S. Chandrasekharan and U.-J. Wiese,\nProg. Part. Nucl. Phys. 2004 , 53, 373.\n[112] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and\nM. den Nijs, Phys. Rev. Lett. 1982 , 49, 405.\n[113] X. Wan, A. M. Turner, A. Vishwanath, and\nS. Y. Savrasov, Phys. Rev. B 2011 , 83, 205101.\n[114] G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang,\nPhys. Rev. Lett. 2011 , 107, 186806.\n[115] F. D. M. Haldane, Phys. Rev. Lett. 1988 , 61, 2015.\n[116] Z. K. Liu, B. Zhou, Y. Zhang, Z. J. Wang, H. M. Weng,\nD. Prabhakaran, S.-K. Mo, Z. X. Shen, Z. Fang, X. Dai,\nZ. Hussain, and Y. L. Chen, Science 2014 , 343, 864.\n[117] M. Neupane, S.-Y. Xu, R. Sankar, N. Alidoust, G. Bian,\nC. Liu, I. Belopolski, T.-R. Chang, H.-T. Jeng, H. Lin,\nA. Bansil, F. Chou, and M. Z. Hasan, Nat. Commun.\n2014 , 5, 3786.\n[118] K. Nomura and N. Nagaosa, Phys. Rev. B 2010 , 82,\n161401(R).\n[119] R. Wakatsuki, M. Ezawa, and N. Nagaosa, Sci. Rep.\n2015 , 5, 13638.\n[120] A. Cortijo, Y. Ferreir\u0013 os, K. Landsteiner, and\nM. A. H. Vozmediano, Phys. Rev. Lett. 2015 , 115,\n177202.\n[121] D. I. Pikulin, A. Chen, and M. Franz, Phys. Rev. X\n2016 , 6, 041021.\n[122] V. Arjona and M. A. H. Vozmediano, Phys. Rev. B\n2018 , 97, 201404(R).\n[123] J. Behrends, S. Roy, M. H. Kolodrubetz, J. H. Bar-\ndarson, and A. G. Grushin, Phys. Rev. B 2019 , 99,\n140201(R).\n[124] T. Liu, D. I. Pikulin, and M. Franz, Phys. Rev. B 2017 ,\n95, 041201.\n[125] D. I. Pikulin and R. Ilan, arXiv:1802.00512.\n[126] T. Kariyado, arXiv:1707.08601.\n[127] K.-Y. Yang, Y.-M. Lu, and Y. Ran, Phys. Rev. B 2011 ,\n84, 075129.\n[128] A. H. Castro Neto, F. Guinea, N. M. R. Peres,\nK. S. Novoselov, and A. K. Geim, Rev. Mod. Phys.\n2009 , 81, 109.\n[129] Eds.: H. Aoki and M. S. Dresselhaus, Physics of\nGraphene , Springer, New York, 2014 .\n[130] Y. Araki, A. Yoshida, and K. Nomura, Phys. Rev. B\n2016 , 94, 115312.\n[131] Y. Araki, A. Yoshida, and K. Nomura, Phys. Rev. B\n2018 , 98, 045302.\n[132] Y. Ominato, A. Yamakage, and K. Nomura, Condens.\nMatter 2018 , 2018, 3(4), 43.\n[133] A. G. Grushin, J. W. F. Venderbos, A. Vishwanath, and\nR. Ilan, Phys. Rev. X 2016 , 6, 041046.\n[134] J.-H. Zhou, H. Jiang, Q. Niu, and J.-R. Shi,\nChin. Phys. Lett. 2012 , 30, 027101.\n[135] Z.-M. Huang, J. Zhou, and S.-Q. Shen, Phys. Rev. B\n2017 , 96, 085201.\n[136] A. Vilenkin, Phys. Rev. D 1980 , 22, 3080.\n[137] K. Fukushima, D. E. Kharzeev, and H. J. Warringa,\nPhys. Rev. D 2008 , 78, 074033.\n[138] D. E. Kharzeev, L. D. McLerran, and H. J. Warringa,Nucl. Phys. A 2008 , 803, 227.\n[139] D. E. Kharzeev, Prog. Part. Nucl. Phys. 2014 , 75, 133.\n[140] D. T. Son and N. Yamamoto, Phys. Rev. Lett. 2012 ,\n109, 181602.\n[141] M. M. Vazifeh and M. Franz, Phys. Rev. Lett. 2013 ,\n111, 027201.\n[142] N. Yamamoto, Phys. Rev. D 2015 , 92, 085011.\n[143] K. Taguchi and Y. Tanaka, Phys. Rev. B 2015 , 91,\n054422.\n[144] Y. Chen, S. Wu, and A. A. Burkov, Phys. Rev. B 2013 ,\n88, 125105.\n[145] J. Ma and D. A. Pesin, Phys. Rev. B 2015 , 92, 235205.\n[146] A. Sekine and K. Nomura, Phys. Rev. Lett. 2016 , 116,\n096401.\n[147] S. Zhong, J. E. Moore, and I. Souza, Phys. Rev. Lett.\n2016 , 116, 077201.\n[148] E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, and\nP. O. Sukhachov, Phys. Rev. B 2017 , 96, 085130.\n[149] R. Karplus and J. M. Luttinger, Phys. Rev. 1954 , 95,\n1154.\n[150] S. L. Adler, Phys. Rev. 1969 , 177, 2426.\n[151] J. S. Bell and R. Jackiw, Nuovo Cimento A 1969 , 60,\n47.\n[152] A. A. Burkov, J. Phys.: Condens. Matter 2015 , 27,\n113201.\n[153] K. Landsteiner, Phys. Rev. B 2014 , 89, 075124.\n[154] K. Landsteiner, Acta Phys. Pol. B 2016 , 47, 2617.\n[155] W. A. Bardeen, Phys. Rev. 1969 , 184, 1848.\n[156] L. Alvarez-Gaum\u0013 e and P. Ginsparg, Nucl. Phys. B\n1984 , 243, 449.\n[157] D. Kurebayashi and K. Nomura, Sci. Rep. 2019 , 9,\n5365.\n[158] K. Nomura and D. Kurebayashi, Phys. Rev. Lett. 2015 ,\n115, 127201.\n[159] D. Kurebayashi and K. Nomura, Phys. Rev. Applied\n2016 , 6, 044013.\n[160] Y. Ominato, K. Kobayashi, and K. Nomura,\nPhys. Rev. B 2017 , 95, 085308.\n[161] K. Kobayashi, Y. Ominato, and K. Nomura,\nJ. Phys. Soc. Jpn. 2018 , 87, 073707.\n[162] R. Jackiw and C. Rebbi, Phys. Rev. D 1976 , 13, 3398.\n[163] S. Murakami, New J. Phys. 2007 , 9, 356.\n[164] H. Zhang, C.-X. Liu, X.-L. Qi, X. Dai, Z. Fang, and\nS.-C. Zhang, Nat. Phys. 2009 , 5, 438.\n[165] C.-X. Liu, X.-L. Qi, H. Zhang, X. Dai, Z. Fang, and\nS.-C. Zhang, Phys. Rev. B 2010 , 82, 045122.\n[166] D. Kurebayashi and K. Nomura, J. Phys. Soc. Jpn.\n2014 , 83, 063709.\n[167] M. A. Kassem, Y. Tabata, T. Waki, and H. Nakamura,\nJ. Phys. Soc. Jpn. 2016 , 85, 064706.\n[168] M. A. Kasssem, Y. Tabata, T. Waki, and H. Nakamura,\nPhys. Rev. B 2017 , 96, 014429.\n[169] M. P. Ghimire, J. I. Facio, J.-S. You, L. Ye, J. G. Check-\nelsky, S. Fang, E. Kaxiras, M. Richter, and J. van den\nBrink, arXiv:1903.03179.\n[170] A. Ozawa and K. Nomura, arXiv:1904.08148.\n[171] S. Nakatsuji, N. Kiyohara, and T. Higo, Nature 2015 ,\n527, 212.\n[172] N. Kiyohara, T. Tomita, and S. Nakatsuji,\nPhys. Rev. Applied 2016 , 5, 064009.\n[173] A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel,\nA. C. Komarek, C. Shekhar, N. Kumar, W. Schnelle,\nJ. K ubler, C. Felser, and S. S. P. Parkin, Sci. Adv. 2016 ,\n2, e1501870.18\n[174] M. Ikhlas, T. Tomita, T. Koretsune, M.-T. Suzuki,\nD. Nishio-Hamane, R. Arita, Y. Otani, and S. Nakat-\nsuji, Nat. Phys. 2017 , 13, 1085.\n[175] E. Kr\u0013 en, J. Paitz, G. Zimmer, and \u0013E. Zsoldos, Physica\nB1975 , 80, 226.\n[176] S. Tomiyoshi and Y. Yamaguchi, J. Phys. Soc. Jpn.\n1982 , 51, 2478.\n[177] T. Nagamiya, S. Tomiyoshi, and Y. Yamaguchi, Solid\nState Commun. 1982 , 42, 385.\n[178] P. J. Brown, V. Nunez, F. Tasset, J. B. Forsyth, and\nP. Radhakrishna, J. Phys.:Condens. Matter 1990 , 2,\n9409.\n[179] N. Ito and K. Nomura, J. Phys. Soc. Jpn. 2017 , 86,\n063703.\n[180] M.-T. Suzuki T. Koretsune, M. Ochi, and R. Arita,\nPhys. Rev. B 2017 , 95, 094406.\n[181] T. Higo, H. Man, D. B. Gopman, L. Wu, T. Koretsune,\nO. M. J. van't Erve, Y. P. Kabanov, D. Rees, Y. Li, M.-\nT. Suzuki, S. Patankar, M. Ikhlas, C. L. Chien, R. Arita,\nR. D. Shull, J. Orenstein, and S. Nakatsuji, Nat. Pho-tonics 2018 , 12, 73.\n[182] J. Liu and L. Balents, Phys. Rev. Lett. 2017 , 119,\n087202.\n[183] P. Streda, J. Phys. C: Solid State Phys. 1982 , 15, L717.\n[184] M. Koshino and I. F. Hizbullah, Phys. Rev. B 2016 , 93,\n045201.\n[185] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida,\nM. Mizuguchi, H. Umezawa, H. Kawai, K. Ando,\nK. Takanashi, S. Maekawa, and E. Saitoh, Nature 2010 ,\n464, 262.\n[186] V. V. Kruglyak, S. O. Demokritov, and D. Grundler,\nJ. Phys. D: Appl. Phys. 2010 , 43, 264001.\n[187] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. Hillebrands, Nat. Phys. 2015 , 11, 453.\n[188] J. A. Hutasoit, J. Zang, R. Roiban, and C. -X. Liu,\nPhys. Rev. B 2014 , 90, 134409.\n[189] Y. Araki and K. Nomura, Phys. Rev. B 2016 , 93,\n094438.\n[190] O. O. Shvetsov, V. D. Esin, A. V. Timonina,\nN. N. Kolesnikov, E. V. Deviatov arXiv:1904.07762."    },    {        "title": "1806.01978v1.Control_of_magnetization_dynamics_by_spin_Nernst_torque.pdf",        "content": "1 \n Control of magnetization dynamics by spin Nernst torque  \nArnab Bose, Ambika Shankar Shukla, Sutapa Dutta , Swapnil Bhuktare, Hanuman Singh, Ashwin A . \nTulapurkar  \nDepartment of Electrical Engineering, Indian Institute of Technology -Bombay, Powai, Mumbai, India 400076  \nRelativistic interaction between electron’s spin and orbital angular momentum has provided efficient \nmechanism to control magnetization of nano -magnets. Extensive research has been done to \nunderstand and improve spin -orbit interaction driven torques generated by non -magnets while \napplying electric current. In this work , we show that heat current  in non -magnet can also couple to \nits spin -orbit interaction to produce torque on adj acent ferromagnet. Hence, this work provides a \nplatform to study spin -orbito -caloritronic effect s in heavy metal/ferromagnet bi -layers.  \nSince last few years considerable attention has been drawn to control the magnetization \ndynamics by  pure spin current ge nerated by spin Hall effect  (SHE)1,2,3,4 and interfacial magnetic \nfields5,6 by Rashba effect . SHE and Rashba effects  are relativistic phenomena which couple electron ’s \nspin and orbital motion and can be used  to exert  spin-orbit torques7,8. On the other hand , thermal \ngradient in ferromagnet  can also create pure spin current9,10,11,12,13 which can further produce thermal \nspin torques14,15,16,17,18 and domain wall motion19,20. Conversion of heat current into spin current in a \nnon-magnet has been shown recently via the  spin Nernst effect (SNE)21,22,23,24. But an important \nquestion remains unanswered whether thermal gradient in non -magnet can generate spin toque owing \nto its spin -orbit coupling, which in turn could be used for manipulating magnetization. In this letter \nwe demonstrate that , interplay of heat current and spin -orbit coupling in non -magnetic P latinum (Pt) \ncan generate thermally driven spin -orbit torque (equivalent to  spin Nernst torque (SNT )). SNT has \nbeen  predicted  recently25,26 but it is lacking the experimental evidence.  Here,  we show that effective \nmagnetic damping can be controlled by SNT while creating thermal gradient in  Pt/Ni 81Fe19 bilayer . \nThis can open a new avenue to manipulate  spins in magnetic nano structures  for technological \napplications27,28,29. \n \nFig. 1. Schematic representation of the experiment.  (a) Schematic diagram of Pt/Py \nbilayer with direction of spin separation due to SHE or SNE. (b,c) Change of linewidth (or \neffective damping) on application of dc spin current  (generated by SHE or SNE) . (d) \nDirection of anti -damp ing and damping like torques. (e ) Coloured SEM image of the \nfabricated device with current and voltage terminals.  \n2 \n  \nWhile thermal gradient is established in heavy metal  (Pt in this work) , spins of opposite \npolarity separate out in a direction orthogonal to the direction of heat flow due to SNE as shown in \nFig. 1(a) . This situation is thermal analogous to SHE. Thus Pt converts heat current into pure spin \ncurrent which is then injected into neighbouring ferromagnet  (FM) . If the injected spin current \ndensity is enough,  spin torque is expected  on the FM causing enhanced or reduced damping \ndepending upon the direction of spin vectors absorbed by FM ( Fig 1(b -c)). We compare  the change \nin resonance linewidth of FM due to the spin torque generated by SNE and SHE  by performing  spin-\ntorque ferromagnetic resonance (ST -FMR) experiment2,4,6,30,31,32. Basic working principle of ST -\nFMR is the following.  Radio frequency (rf) current is applied along X axis and dc voltage is measured \nin the same terminals using a bias -T (AMR based detection of ST -FMR2,32) while external magnetic \nfield is swept  at angle θ with respect to X-axis ( Fig. 1(e) ). Pt converts rf charge current into rf spin \ncurrent which is injected to ferromagnet (Ni 81Fe19: Py here after).  Radio frequency  spin cu rrent and \ncurrent induced rf fields excite  the magnet to undergo small osc illation  around its equilibrium \nposition . Due to the AMR effect, resistance of the magnet (hence FM/HM stack) also oscillates.  \nHomodyne mixture of RF applied current and RF resistance of the sample produce s dc voltage. \nAdvantage of ST -FMR is that at resona nce large dc voltage can be obtained . This dc voltage is \ntypically combination of symmetric Lore ntzian ( VS) and anti -symmetric Lore ntzian (VA).  \n()2\n1 2 2\n0 4SVC\nHH=\n− +\n and \n()\n()0\n2 2 2\n04\n4AHHVC\nHH−=\n− + where C1 and C2 are the amplitude  of VS \nand VA respectively,  H is external ly applied  field, H0 is resonant field position  and ∆ is the resonance \nlinewidth (FWHM).  VS indicate s the contribution of spin current induced torque and VA indicates in-\nplane field induced torques. In Pt/Py bi -layers , the Oerst ed magnetic field is the dominant source of \nin-plane field.2,32. So charge to spin current conversion efficiency (or effective spin Hall angle) in \nHM can be quantified  from C1/C2 ratio as following: \n( )1/2 0 ' 1\n21/2S Pt Py\nSH Oe M t tCHHC⊥ =+\n  where \nMS is saturation magnetization, tPt, tPy are thickness of Pt and Py film respectively, \nH⊥  is \nperpendicular magnetic anisotropy  field. Now , if dc current is superimposed on the RF current then \nnon-zero dc spin current is injected which  can change the resonance linewidth of Py2,3,32. It also \nprovides direct quantification of effective spin Hall angle as following:  \n0 ' 21\ncos 4 2S Py\nSH\ncSlopeMt edHHf dJ ⊥      =+         \nwhere JC is charge current density through Pt, f \nis frequency of applied rf cur rent, γ is gyromagnetic ratio . In this work, we s have superimposed  a dc \nheat current on rf charge current. We could modulate the resonance line width of Py which  provide s \na direct evidence of control of magnetization dynamics by spin Nernst torque  (SNT) . \n We fabricated the device as shown in Fig. 1(e) . The crossbar is made of  Pt (15 nm) and a \nrectangular shaped dot of  Py (2  nm) is deposited at the centre of Pt  crossbar . Top of Py was  capped \nwith Ta (1.5 nm).  On the top and bottom lead of the Pt crossbar , two heater lines are fabricated which \nare electrically isolated by SiOx (30 nm) from Pt. Heater lines are prepared with Ta/Pt (6 0 nm). \nNumbers in bracket indicate thickness of metals. Entire fabrication is done by standard electron beam \nlithography, sputtering and lift -off technique.  Before deposition of Py , surface of Pt was cleaned by \nAr ion without breaking the vacuum. We have earlier shown that linewidth change can be sensitively \nmeasured in planar Hall structure  doing ST -FMR  [Ref.  32]. We follow the same approach in this \nwork  to compare the strength of spin Hall torque (SHT) and spin Nernst torque (SNT). Our detection 3 \n method is the following. Radio frequency current is applied along X-axis;  dc voltage is measured \nalong same direction  (hence AMR based detection) but dc heat current (or charge current) is passed \nalong Y-axis to modulate  the line width by SNT (or SHT) as shown in Fig. 1(e) . Application of dc \ncurrent  (heat current or charge current)  perpendicular to the direction of volta ge measurement reduces \nnoise and hence measurement sensitiv ity significantly increases as shown in Ref 32. \n \nFig. 2. Characterization of spin Hall torque.  (a) DC voltage generated by ST -FMR for \ndifferent frequencies of applied rf current. (b) Fit of dc vol tage by VS and VA. Inset of (b) \nshows Kittel’s fit. (c) ST -FMR spectrum when dc charge current is applied perpendicular to \napplied rf current. (d) Modulation of linewidth on application of dc charge current.   \n \n Fig. 2  shows the characterization of spin-Hall torque by measuring ST-FMR.  Fig. 2(a)  shows \nthe typical dc voltage spectrum as external magnetic field is swept for different frequencies of applied \ncurrent.  This dc voltage can be fit ted to the  sum of VS and VA (Fig. 2(b) ). Red squares in Fig. 2(b) \nshow the experimental data  and black curve shows fitting  which is sum of VS (pink curve) and VA \n(blue curve). The symmetric component confirms the spin -orbit torque generated by spin Hall effect.  \nInset of Fig. 2(b) shows the Kittel’s fit for resonant magnetic fields and frequencies. From this we \nobtain \nH⊥ =8.05 kOe  (which corresponds to Ms = 6.5×105 A/m.) Fig. 2(c)  shows the voltage spectrum \nwhen dc charge current is applied orthogonal to the d irection of rf current flow. We can clearly see \nthe dominant change in the shape of the voltage signal  as the linewidth  significantly  changes.  For the \npositive current linewidth is more in positive field value s and it is less in negative field values (vice \nversa for negative applied current).  Our detection method is so sensitive that linewidth change is \nclearly visible in Fig. 2(c)  itself.  Linewidth (∆) as a function of applied dc current is shown in top \npanel of Fig. 2(d) . It shows expected linear dependence as function of applied current.  Bottom panel \nof Fig. 2(d)  shows the difference in linewidth for θ=350 and θ=2150 (∆’=∆(350)-∆(2150)) as a function \nof applied current which also shows linear  dependence . Bottom panel of Fig. 2(d) represents  only the \ncontribution of spin current induced damping change eliminating the overall heating effect  if any . \nFrom this measurement  of linewidth modulation,  we can extract the effective spin Hall angle of Pt to \nbe 0.12±0.06 . Extracted  value o f spin Hall angle from C1/C2 ratio is somewhat lower. We have further \n4 \n noticed that in this planar Hall geometry when dc current is applied perpendicular to rf current \nmodulation in VS is higher  compared to the line -width modulation  as a function of applied dc current. \nSame behaviour  was also observed in our previous work ( Fig. 3(c) of ref 32 and Fig. 2(c) in this \nwork).  However, we have verified that the line width modulation is the same  irrespective of the \ndetection methods (AMR or  PHE based detection or  hybrid planar Hall detection  method) . So in this \nwork we quantify the strength of SHT and SNT in Pt by measuring linewidth change by injecting dc \nspin current by SHE and SN E respectively.  \n \nFig. 3. Characterization of SNT.  (a) dc voltage spectrum on application of heat current \nalong + Y axis and –Y axis. (b)  dc voltage spectrum when positive and negative field data \npoints are superimposed for dc thermal gradient along Y axis. ( c,d) Difference in linewidth  \nbetween positive an d negative field values  (∆’) for different applied heater powers and \ndifferent angles ( θ) (respectively).  \n \nNow we show the evidence of spin Nernst torque by passing  dc heat current instead of \napplying dc charge current while doing ST -FMR as discussed above.  Two different heater lines are \nfabricated on Hall bar ( Fig. 1(e) ) to create thermal gradient along ± Y-axis. When current flows in \nheaterline -1 (HL1)  it becomes hot due to Joule’s  heating and most of the heat is carried by Pt below \nthe heat line.  This creates thermal gr adient in Pt/Py bilayers along + Y axis. Similarly , when current \nis applied in heaterline -2 (HL2) the rmal gradient is created along -Y axis. Fig. 3(a)  shows the dc \nvoltage spectrum generated by ST -FMR while thermal gradient is create d along ± Y axis. We can  see \nthe difference in dc voltage spectrum for positive and negative thermal gradients.  This cannot be \nexplained by overall heating effect since overall heating would be same for both the direction of \nthermal gradients.  All these mea surements are performed when θ is 350. In Fig. 3(b)  we further show \nthat when voltage signal of positive field and negative field is superimposed , there is distinct change \nin the shape of voltage spectrum (hence the linewidth) which furthe r confirms the existence of SNT. \nDifference in the shape of voltage  signal  in Fig. 3(b) cannot be explained by overall heating effect as \nit would be same for both 350 and 2150. As shown  earlier ( Fig. 2(d) ), line-width difference for positive \nand negative f ield (∆’=∆(θ) - ∆(θ+1800)) is measured for different applied heater powers ( Fig. 3(c) ) \nand for  different angles ( Fig. 3(d) ). ∆’ is proportional to the heater power ( Fig. 3(c) ) and it closely \nfollows cosθ dependence  (Fig. 3(d) ). In our geometry ( Fig. 1 (a)) the line width modulation due to \nSNT is expected to be maximum  at θ=00, but ST -FMR signal becomes zero at θ=00. We further \nconfirm ed that polarity of heater current has negligible  effect on ∆’ as heater line is electrically \ninsulated from Pt hall bar and Py dot is quite far away (1 .5 μm) from the heater line  to get affected \nby magnetic (Oersted) field produced by the heater current.  In our control experiment we  have  \napplied heater current in both HL1 and HL2 which would  cause the same overall heating but fail to \nset up  well directed thermal gradient . We found negligible effect on the line width in this case.  Our \nobserved results shown in Fig. 3 strongly supports the evidence of spin Nernst torque in Pt.  \n5 \n   \nFig. 4.  Temperature profile . (a) Surface plot of 2D  thermal gradient. Estimated temperature \n(b) and temperature gradient (c) in Pt/Py interface.  \n \n We have followed the same approach to find thermal gradient in Pt as reported in Ref  24. \nFrom the resistance value of the heater line, its overall temperature is known (on chip temperature \ncalibration).  Once the temperature of heater line is known temperature profile of Pt/Py interface can \nbe calculated from  COMSOL simulation  (Fig. 4(b)  and 4(c)). Overall temperature rise is around 25 \nK which is fairly small and  hence we observe negligible contribution from overall heating effect.  We \nconsider the thermal gradient at the interface to be 15 K/μ m while estimating SNT.  Our estimated \ntemperature gradient in this kind of geometry is in good agreement with previous resu lts20. \nComparing the line -width modulation by SHT and SNT we can quantify that 15 K/  μm horizontal \nthermal gradient in Pt/Py interface is equivalent to the application of 4.9×109 A/m2 amount of charge \ncurrent density in Pt.  This reported value of heat current to spin current conversion efficiency by SNE \nis consistent with our previous report of SNE24 and comparable to reports by other groups21,23. \n In conclusion, we have demonstrated the control of magnetization dynamics by thermally \ndriven spin Hall torque (or spin Nernst torque). We report  approximately 0.9 % line -width change \ndue to spin Nernst torque effect.  It indicates that about 100 times more thermal gradient needs  be \ncreated to achieve switching by spin Nernst torque  which could be achieved in material having large \nspin Nernst angle and implementing efficient mechanism (such as Laser heating) to create large \nthermal  gradient  at nano scale.  Further by using ferromagnets with less out -of-plane anisotropy, the \nthermal gr adients required for switching can be reduced. These results will beneficial in energy \nharvesting sector where heat current  in non -magnet s can be utilized to write magnetic memories for \nnon-volatile memory applications.  \nReferences  \n1 J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jung wirth,  “Spin Hall effects ”. Rev. Mod. \nPhys.  87, 1213 (2015)  \n2 L. Q. Liu, T. Moriyama, D. C. Ralph, and R. A. Buhrman, “Spin-Torque Ferromagneti c Resonance \nInduced by the Spi n Hall Effect ”, Phys. Rev. Lett. 106, 036601 (2011)  \n3 K. Ando, S. Takahashi, K. Harii, K. Sasa ge, J. Ieda, S. Maekawa, and E. Saitoh, “Electric Manipulation of \nSpin Relaxation Using the Spin Hall Effect ”, Phys. Rev. Lett.  101, 036601 (2008)  \n4 L. Liu, Chi -Feng Pai, D. C. Ralph, and R. A. Buhrman,  Magnetic Oscillations Driven by the Spin Hall  \nEffect in  3-Terminal Magnetic Tunnel Junction Devices, Phys. Rev. Lett.  109, 186602 (2012).  \n5 I. M . Miron, G. Gaudin, S. Auffret, B. Rodmacq,A. Schuhl, S. Pizzini, J. Vogel, P. Gambardella. \n“Current -driven spin torque induced by the Rashba effect ina ferromagnetic metal layer ”, Nat. Mater . 9, 230 \n(2010)  \n6 A. Bose,  H. Singh, V. K. Kushwaha, S. Bhuktare, S. Dutta, and A. A. Tulapurkar. “Sign Reversal of Field -\nlike Spin -Orbit Torque in an Ultrathin Cr/ Ni Bilayer ”, Phys. Rev. Appl.  9, 014022 (2018)                                                        \n6 \n                                                                                                                                                                                \n7 Anjan Soumyanarayanan, Nicolas Reyren, Albert Fert, and Christos Panagopoulos, “Emergent phenomena \ninduced by spin -orbit coupling at surfaces, and interfaces ”, Nature (London)  539, 509 (2016)  \n8 A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine, “New perspectives for Rashba spin orbit \ncoupling ”, Nat. Mater.  14, 871 (2015)  \n9 K. Uchida, H. Adachi, T. Ota, H. Nakayama, S. Maekawa, and E. Saitoh. “Observa tion of longitudinal \nspin-Seebeck effect in magnetic insulators ”, Appl. Phys. Lett.  97, 172505 (2010)  \n10 G. E. W. Bauer, E. Saitoh and B. J. Van Wees. “ Spin Caloritronics ”, Nat. Mater.  11, 391 (2012)  \n11 M. Walter, J. Walowski, V. Zbarsky, M. M € unzenberg , M. Sch €afers, D.  Ebke, G. Reiss, A. Thomas, P. \nPeretzki, M. Seibt et al. , “Seebeck effect in  magnetic tunnel junctions ,” Nat. Mater.  10, 742 –746 (2011)  \n12 S. Jain, D. D. Lam, A. Bose, H. Sharma, V. R. Palkar, C. V. Tomy, Y.Suzuki, and A. A. Tulapurkar, \n“Magneto -Seebeck effect in spin -valve with heat current in -plane thermal gradient ”, AIP Adv.  4, 127145  \n(2014)  \n13 M. Schmid, S. Srichandan, D. Meier, T. Kuschel, J. -M. Schmalhorst, M. Vogel, G. Reiss,  C. Strunk and \nC. H. Back. “Transverse Spin Seebeck  Effect versus Anomalous and Planar Nernst Effects  \nin Permalloy Thin Films ”, Phys. Rev. Lett. 111, 187201 (2013)  \n14 H. Yu, S. Granville, D. P. Yu, and Ph. J. Ansermet, “ Evidence for thermal spin -transfer torque ”, Phys. \nRev. Lett.  104, 146601 (2010)  \n15 M. G. Choi, H. C. Moon, C. B. Min, J. K. Lee, and G. D. Cahill, “ Thermal  spin-transfer torque driven by \nthe spin -dependent Seebeck effect in metallic spin -valves ”, Nat. Phys.  11, 576 –582 (2015).  \n16 A. Pushp, T. Phung, C. Rettner, B. P. Hughes, S. H. Yang, and S. S. P.  Parkin, “ Giant thermal spin \ntorque -assisted magnetic tunnel junction  switching ,” Proc. Natl. Acad. Sci. U.S.A.  112, 6585 –6590 (2015).  \n17 A. Bose et al., “Observation of thermally driven field -like spin torque in magnetic tunnel junctions ”, Appl. \nPhys. Lett.  109, 032406 (2016)  \n18 M. B. Jungfleisch , T. An , K. Ando , Y. Kajiwara , K. Uchida , V. I. Vasyuchka , A. V. Chumak , A. A. Serga , \nE. Saitoh  and B. Hillebrands. “Heat -induced damping modification in yttrium iron garnet/platinum hetero -\nstructures ”, Appl.  Phys . Lett. 102, 062417 (2013)  \n19 W. Jiang, P. Upadhyaya, Y. Fan, J. Zhao, M. Wang, L. Chang, M. Lang, K. L. Wong, M. Lewis, Y. -T. Lin \net. al. “Direct Imaging of Thermally Driven Domain Wall Motion  in Magnetic Insulators ”, Phys. Rev. Lett.  \n110, 177 202 (2013)  \n20 P. Krzysteczko, X. Hu, N. Liebing, S. Sievers, and H. W. Schumacher. “Domain wall magneto -Seebeck \neffect ”, Phys. Rev. B  92, 140405(R) (2015)  \n21 S. Meyer, Y. -T. Chen, S. Wimmer, M. Althammer, T. Wimmer, R. Schlitz, S. Geprägs, H. Huebl, D. \nKödderitzsch, H. Ebert, G.E.W. Bauer, R. Gross, and S.T.B. Goennenwein, “Spin Nernst effect ”, Nat. Mater.  \n16, 977 (2017)  \n22 P. Sheng, Y. Sakuraba, Y. Lau, S. Taka hashi, S. Mitani, and M. Hayashi, “The spin Nernst effect in \ntungsten ”, Sci. Adv.  3, e1701503 (2017)  \n23 D.J. Kim, C.Y. Jeon, J.  G. Choi, J.W. Lee, S. Surabhi, J.R. Jeong, K.J. Lee, and B.G. Park, Nat. \n“Observation of transverse spin Nernst magnetoresistance  induced by thermal spin current in \nferromagnet/non -magnet bilayers ”, Nat. Commun.  8, 1 (2017)  \n24 A. Bose, S. Bhuktare, H. Singh, S. Dutta, V. Achanta, A. A.  Tulapurkar, “ Direct detection of spin Nernst \neffect in Pt ”, Appl. Phys. Lett.  112, 162401 (2018)  \n25 G. Geranton, F. Freimuth, S. Blugel, and Y. Mokrousov. “Spin-orbit torques in L10 -FePt/Pt thin films \ndriven by electrical and thermal currents ”, Phys. Rev. B  91, 014417 (2015)  \n26 A. A. Kovalev and V. Zyuzin. “ Spin torque and Nernst effects in Dzyaloshinskii -Moriya ferromagnets ”, \nPhys. Rev. B 93, 161106(R)  (2016).  \n27 S. Bhuktare, H. Singh, A. Bose, and A. A. Tulapurkar, “Spintronic Oscillator Based on Spin -Current \nFeedback Using the Spin Hall Effect ”, Phys. Rev. Appl.  7, 014022 (2017)  \n28 Swapnil Bhuktare, Arnab  Bose, Hanuman Singh & Ashwin A. Tulapurkar , “Gyrator Based on Magneto -\nelastic  Coupling at a Ferromagnetic/Piezoelectric Interface ”, Scientific Reports | 7: 840 | \nDOI:10.1038/s41598 -017-00960 -9 \n29 H. Singh et. al. “Integer, Fractional, and Sideband Injection Locking of a Spintronic  Feedback Nano -\nOscillator to a Mi crowave Signal” , Phys. Rev. Appl.  8, 064011 (2017)  \n30 A. A. Tulapurkar, Y. Suzuki, A. Fukushima, H. Kubota, H. Maehara, K. Tsunekawa, D. D. Djayaprawira, \nN. Watanabe, and S. Yuasa, “Spin Torq ue Diode effect in Magnetic Tunnel Junctions ”, Nature (London)  \n438, 339 (2005)  7 \n                                                                                                                                                                                \n31 A. Bose, S . Dutta, S. Bhuktare, H. Singh , and A. A.  Tulapurkar, “Sensitive measurement of spin orbit \ntorque  driven ferromagnetic resonance detected by planar Hall  geometry ”, Appl. Phys. Lett. 111, 162405 \n(2017)  \n32 A. Bose, D. D. Lam, S. Bhuktare, H. Singh , S. Miwa  and A. Tulapurkar,  “Observation of anomalous spin -\ntorque generated by a ferromagnet ”, arXiv:1706.07245 (2017)  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 8 \n                                                                                                                                                                                \nSupplementary information  \nArnab Bose, Ambika Shukla, Sutapa Dutta, Swapnil Bhuktare, Hanuman Singh, Ashwin A \nTulapurkar  \nDepartment of Electrical Engineering, Indian Institute of Technology -Bombay, Powai, Mumbai, \nIndia 400076  \n \nS1. Temperature calibration  \nThermal gradient is obtained by doing on -chip calibration.  As current is passed through heater line it becomes \nhot due to Joule’s heating and its resistance increases. Hence resistance of heater line  provides information of \naverage temperature in heater line.  Same approach w as adopted in our previous work [S1]  and by other \nresearchers  [S2,S3,S4] . Heater line was made of Ta/Pt (~50 nm).  Its resistance changes from 50 Ω to 65 Ω on \napplication of maximum  heater power (0.3 W). This corresponds to overall temperature of heater line to be \napproximately 420 K which is obtained from the temperature dependent resistance measurement of the heater \nline. We observed minor change in resistance in Pt Hall bar which contains rectangular Py dot a t the centre.  It \nindicates that the temperature of horizontal Pt line does not increase much and the heat is locally centred near \nthe heater line.  To get exact temperature profile , COMSOL simulation is done with heat transfer m odule with \nfollowing boundary conditions: (1) temperature of heater line is 420 K (experimentally obtained) and (2) \ntemperature of the bottom of Si is 293 (lab temperature).  \n \nFig. S1. (a) Temperature profile at the interface of Pt/Py.  (b) Isothermal temperature contour. (c) \nSchematics of device structure.  Arrow indicates direction of heat flow.  (d) Temperature along Y-axis. \n(e) Temperature gradient along Y-axis. \n \nFig. S1(a) shows the surface temperature. Bright yellow colour indicates  hot region and temperature of heater \nis set 420 K.  Deep red indicates the colder region.  Fig. S2(b) shows the isothermal contour which shows that \n9 \n                                                                                                                                                                                \nheat is also localized in very small region (approx. 10×10×10 μm3) as expected.  Fig. S1(d) shows the \ntemperat ure as along Y axis which shows exponential decay of thermal gradient  (Fig. S1(e)) . Temperature \nbecomes around 300 K when we go 10 μm away from the heater line.  It shows that overall temperature of Py \nis slightly more than room temperature (~325 K).  Maximu m thermal gradient along Y axis in Pt/Py interface \nis nearly 20 K/μm.  Estimated temperature profile is in good agreement with previous works  [S1-S6]. We have \nconsidered below parameters in simulation. We have also checked that slight variation of these par ameter does \nnot influence the simulated result much. Finally estimated thermal gradient will be in order tens of K/μm in \nthis kind of geometry.  \nMaterial  Pt Si SiO2  \nThermal conductivity (SI unit)  85 120 1.4 \n \nReferences  \n[S1] A. Bose, S. Bhuktare , H. Singh, S. Dutta, V. Achanta, A. A. Tulapurkar. Appl. Phys. Lett. (accepted)  \n[S2] P. Krzysteczko,  X. Hu, N. Liebing, S. Sievers, and H. W. Schumacher . Domain wall magneto -Seebeck \neffect . Phys. Rev. B 92, 140405(R) (2015)  \n[S3] S. Meyer, Y. -T. Chen, S. W immer, M. Althammer, T. Wimmer, R. Schlitz, S. Geprägs, H. Huebl, D. \nKödderitzsch, H. Ebert, G.E.W. Bauer, R. Gross, and S.T.B. Goennenwein, Nat. Mater. 16, 977 (2017)  \n[S4] A. Pushp, T. Phung, C. Rettner, B. P. Hughes, S. H. Yang, and S. S. P. Parkin, “Giant thermal spin torque -\nassisted magnetic tunnel junction switching,” Proc. Natl. Acad. Sci. U.S.A. 112, 6585 –6590 (2015).  \n[S5] A. Slachter, L. F. Bekker, J. -P. Adam, and J. B. van Wees, “Thermally driven spin injection from a \nferromagnet into a non -magnetic metal,” Nat. Phys. 6, 879 –882 (2010)  \n[S6] M. Walter, J. Walowski, V. Zbarsky, M. M€ unzenberg, M. Sch€afers, D. Ebke, G. Reiss, A. Thomas, P. \nPeretzki, M. Seibt et al., “Seebeck effect in magnetic tunnel junctions,” Nat. Mater. 10, 742 –746 (2011)  \n \n "    },    {        "title": "2001.07940v1.Imaging_three_dimensional_nanoscale_magnetization_dynamics.pdf",        "content": " \n1 \n Imaging  three -dimensional nanoscale magnetization dynamics   \nAuthors:  Claire Donnelly1,2,3 *, Simone Finizio2, Sebastian Gliga2,3,4, Mirko Holler2, \nAleš Hrabec2,3,5, Michal Odstr čil2, Sina Mayr2,3, Valerio Scagnoli2,3,  \nLaura J. Heyderman2,3, Manuel Guizar -Sicairos2*, Jörg Raabe2*. \nAffiliations:  \n1 Cavendish Laboratory, University of Cambridge, JJ Thomson Ave, Cambridge, CB3 0HE , UK.  \n2 Paul Scherrer Institute, 5232 Villigen, Switzerland.  \n3 Laboratory for Mesoscopic Systems, Department of Materials, ETH Zurich, 8093 Zurich, \nSwitzerland . \n4 SUPA, School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, UK . \n5 Laboratory for Magnetism and Interface Physics, Department of Material s, ETH Zurich, 8093 \nZurich, Switzerland  \n*Correspondence to: cd691@cam.ac.uk, manuel.guizar -sicairos@psi.ch, joerg.raabe@psi.ch.  \nAbstract : The ability to experimentally map the three -dimensional structure and dynamics in bulk  \nand patterned three -dimensional  ferromagnets is essential both for understanding fundamental \nmicromagnetic processes, as well as for investigating technologically -relevant micromagnets \nwhose functions are connected to the presence and dynamics of fundamental micromagnetic \nstructures, su ch as domain walls and vortices .  Here, we demonstrate time -resolved magnetic \nlaminography, a technique which offers access to the temporal evolution of a complex three -\ndimensional magnetic structure with nanoscale resolution . We image the dynamics of the complex \nthree -dimensional magnetization state in a two -phase bulk magnet with a lateral spatial resolution \nof 50 nm, mapping  the transition between domain wall precession and the dynamics of a uniform \nmagnetic domain  that is attributed to variations in the  magnetization state across the phase \nboundary. The capability to probe  three -dimensional magnetic structures with temporal resolution \npaves the way for the experimental investigation of novel functionalities  arising from dynamic \nphenomena in bulk and thre e-dimensional patterned nanomagnets.  \nOne Sentence Summary:  Nanoscale dynamics of a three -dimensional magnetic system are \nrevealed with pump -probe magnetic laminography.  \n \nMain Text:  \nUnderstanding and controlling the dynamic response of magnetic materials w ith a three -\ndimensional magnetization distribution is important both fundamentally and for technological \napplications . From a fundamental point of view, the internal structure and reversal phenomena in \nknown bulk materials still need to be mapped [1], along with the dynamic properties of \ntopological structures such as vortices [2], magnetic singularities [3] or skyrmi on lattices [4]. \nFrom a technological point of view, the response of inductive mate rials to magnetic fields and \nspin-polarized currents is essential for magnetic sensors and data storage devices [5].  \nIn addition to bulk systems , curved and three -dimensional magnetic structures such as nanotubes \nand nanowire -based systems have been predicted to exhibit dynamic properties markedly  \n2 \n different from those found in planar systems. These include the suppression of the Walker \nbreakdown in ma gnetic nanotubes and cylindrical nanowires  and fast domain wall velocities \nmeasured in three -dimensional magnetic microwires [6], as well as anisotropic sp in wave \npropagation [7] and the spin -Cherenkov effect [8, 9] .  \nWhile magnetization  dynamics have been observed  in 2D structures, and are well understood for \nisolated structures in thin films such as vortices   [10-13]  domain  walls  [14-17] and skyrmions  \n[18, 19] , until now insight into three -dimensional dynamics has only been possible via \ncomparison with simulation s [20]. \nStatic three -dimensional magnetic imaging has recently been achieved using hard X-ray [21, 22] , \nneutron  [23-25], electron [26] and soft X-ray tomograph ies [27, 28]  which  allow for the \ndetermination of the three -dimensional magneti c configurations . Of these methods,  X-rays are \nunique owing to their ability to provide high spatial resolution imaging of thick materials  [21, \n29], as well as element -specific information. A dditionally, the pulsed time stru cture of  \nsynchrotron X-rays provides the  possibility of performing  time-resolved measurements: pump -\nprobe measurements of material systems have proved a powerful tool for the measurement of \nMHz and GHz dynamics within a range of magnetic materials and syst ems [30]. As yet , \nhowever, it has not been possible to combine three -dimensional magnetic imaging with dynamic \nmeasurements due to the geometric constraints of tomographic imaging on sample design.  \nHere w e present an alternative geometry for 3D magnetic imaging: magnetic  laminography  [31], \nthat provides access to all three components of the magnetization vector field with a single axis \nof rotation , and is a flexible  technique  that makes possible the determination of the temporal \nevolution of three -dimensional magnetic configurations. By combining vectorial  magnetic \nimaging with a pump -probe setup, and therefore directly observing magnetization  dynam ics of \ncomplex three -dimensional magnetic configurations , we obtain  access to the temporal evolution \nof the entire three -dimensional vector field. In this way, we can study both surface excitations as \nwell as the dynamics of buried three -dimensional magnet ization textures such as magnetic \ndomains and vortex domain walls . \nWe specifically image the three -dimensional magnetic state of a GdCo microdisc  using  magnetic \nlaminography , and determine its dynamic response to a radio frequency (RF)  magnetic field . The \nsample consist s of a GdCo microdisc with a diameter of 5 μm and a thickness of 1.2 μm, \npatterned by focused ion beam milling a continuous film deposited  on the back -side of a Si 3N4 \nmembrane ( see inset of Fig. 1a for an SEM image of the sample). The  two-phase GdCo film was \ndeposited b y magnetron sputtering, first with the sample continuously rotating, and then \nstationary  for the top 1/4 of the thickness. On the front -side of the membrane a 300 nm -thick Cu \nstripline was patterned, through which an RF c urrent could be  injected to produce an in -plane \nmagnetic excitation.  \nThe experimental setup is shown schematically in Figure 1a , where t he plane of the sample is \ntilted at 29° to the propagation direction of the X-rays. During the measurement t he sample is  \nrotated about an axis perpendicular to its surface plane, and t wo-dimensional projections of the \nmagnetic structure are measured with dichroic ptychography  [29]. Two-dimensional  projections \nof the magnetic configuration of  the GdCo microdisc for the sample rotated by 0° and 45° about \nthe laminography rotation axis are shown in Figure 1b and 1c, respectively. In these projections, \na multi -domain state can be identified at the bottom of the structure, that appears to change \nthrough the thickness of the structure .   \n3 \n  \nFigure 1. Time resolved X-ray magnetic laminography. a) the experimental setup. For X-ray magnetic laminography, \nptychographic projections are measured with right - and left -handed circular polarization  with the sample rotated around an axis, \nthat is at 61° with respect to the X-ray beam propagation direction. To probe the magneti zation dynamics of the sample, the \nstructure is fabricated on a Cu stripline (see magnified view in inset , along w ith an SEM image of the structure ), and excited with \na 500 MHz RF current passing through the stripline . Magnetic laminography is then performed for a number of different delay \ntimes of the excitation with respect to the X-ray pulses, so providing temporal  resolution. XMCD projections of the magnetic \nstructure are shown in b) and c) for the sample at rotation angles of 0° and 45°, respectively. Scale bars represent 1 μm. d)  the \nconcept of pump -probe magnetic laminography  is illustrated , where the RF excitat ion is frequency and phase matched to the \nincoming X-ray pulses. By varying the time delay δt for a number of laminography measurements (two are depicted here), \ntemporal ly-resolved maps of the magnetization vector field  are obtained . \nFor magnetic laminogra phy, 2D ptychographic projections were measured at 144 angles equally \ndistributed over 360° with both right and left circularly polarized  X-rays. The three -dimensional \nmagnetic configuration was reconstructed using a GPU implementation of the gradient -based \narbitrary projection magnetic reconstruction algorithm (APMRA), detailed in [32], with a lateral \nspatial resolution of ~50 nm. The RF magnetic field used to excite the sample was frequency an d \nphase -matched to the synchrotron X -ray pulses, which are 70 ps wide X -ray flashes occurring \nevery ~2 ns (repetition frequency of 500 MHz). To probe the dynamic response of the sample, a \nmagnetic laminography dataset was acquired for a number of different  delay times of the \nexcitation field with respect to the X -ray bunches, as shown schematically in Figure 1d.  \nThe reconstructed three -dimensional magnetization distribution of the GdCo microdisc is shown \nin Figure 2. In the lower region of the structure, a multi -domain state is present, in which the \ndomains are separated by elongated vortex domain walls  (Figure 2a). As the disk thickness \nincreases, the central domain expands  and the vortex domain walls  are located increasingly \ncloser to the edges of the samp le (Figure 2b), until they are expelled  approximately mid -way \nthrough the thickness  of the sample, resulting in a single magnetic domain with a curling \nmagnetization at the edges, forming an S -state (Figure 2c). At the top of the disc, the \nmagnetization fo rms an almost uniform state (Figure 2d). The positions of the vortex cores are \nplotted using  isosurfaces in Figure 2e . Within the core of a vortex , the magnetization points out \nof the plane  of the disk, and the direction of the magnetization (i.e. ±𝑚𝑦), defines its \npolarization . Here, the isosurfaces representing the core of the vortices are colored by the 𝑚𝑦 \ncomponent, which can be seen to be positive (orange) and negative (green) , indicating that the \npair of vortices have opposite polarizations . \n \n4 \n   \nFigure 2 The reconstructed three -dimensional magnetic configuration of the GdCo microdisc . At the bottom of the disc (a), an \n“eye” state forms with a large central domain, and two vortices. With increasing height, the vortices mov e towards the edges of \nthe disc (b), until they exit the structure approximately midway through the thickness , resulting in a large magnetic domain in an \nS-state (c). At the top of the disc, the magnetic configuration is almost uniform (d). The  positions of the  cores of the vortices, \ncoloured by the y -component of the magnetization , are shown in (e) , showing that the magnetization  in the cores of the two \nvortices have opposite directions – i.e. opposite vortex polarization s. These vortices event ually exit through the edges of the \nsample approximately mid -way through its height. f) Bivariate histogram showing the number of pixels as a function of \nmagnetization  direction and thickness  through the sample. The red peak  corresponds to the central doma in, which expands in area \nwith increasing height, eventually forming a single -domain state above ~700 nm. Three main regions in the histogram are \nidentified: in the vortex state (h ≤500 nm, a,b), there is a large variation in magnetization  direction, with t wo main peaks \ncorresponding to the central and side domains. At h ~500 nm, an intermediate  S-domain state is present (c), and the \nmagnetization  directions are distributed at angles between ~50 and 90°. Above 800 nm, the single domain state ( d) results in a  \nstrong, sharper peak around 65°. g) The primary  direction of the magnetization  in the central domain  (black dots) and the  \nstandard deviation of Ѳ𝑚 (red dots) , which indicates  the magnetization  state, are given as a function of thickness  through the \nsample. The primary direction varies linearly up to a thickness of ~900 nm, after which the direction remains approximately \nconstant, consistent with the presence of in -plane magnetic anisotropy  in the top quarter of the film . Three regions of  the sample \nare identified on the plot, the vortex state (i, orange ), the S -state (ii, yellow ) and the single domain state (iii, blue). \nThe gradual transition between the multi -domain state in the lower  region of the sample to the \nsingle domain state in th e upper region can be attributed to variations in the anisotropy of the \nsystem. The rotation of the sample during sputtering prevented the formation of a well -defined \nanisotropy axis in the bottom part of the disk so that the magnetic state is predominantl y \ndetermined by the magnetostatic energy . The absence of rotation  during the latter part of the \ndeposition led to the development of a sizeable uniaxial anisotropy in the top part of the sample. \nTherefore, the resulting system can be understood as a two ph ase system with a soft (bottom) and \na hard (top) phase,  conceptually  analogous to an exchange -spring system [33].  \nWe compare the distribution of the direction of the magnetization through the thickness  of the \nsample in Figure 2f in a bivariate histogram. At the bottom of the film , there is a large spread in \nthe magnetization directions , which is  consistent with the vo rtex state, with a distinct peak at \n140° corresponding to the direction of the magnetization in the  central domain. With increasing \nheight, the spread in directions is reduced, giving rise to a single peak corresponding to the \ndirection of the magnetizatio n in the top domain. The standard deviation of the magnetization, \nwhich reflects the magnetization state, is plotted as a function of thickness  through the sample in \nFigure 2g (red dots), and three main regions can be identified: the vortex state (i), the single \n \n5 \n domain state (iii), and an intermediate region corresponding to the curled S -state (ii). The \nprimary  direction of the magnetization can be tracked through the thickness  of the sample by \nidentifying the corresponding peak in the histogram for each sl ice in Figure 2 g (black dots). The \nprimary  magnetization direction varies linearly with distance through the sample up to a height \nof 900  nm, after which the magnetization direction remains almost constant. This linear rotation  \nof the magnetization directi on along the thickness  correlates with the position of the vortex \nwalls: as the thickness increases, the vortices (plotted using isosurfaces in Fig 2e) are located \nincreasingly closer to  the edges of the sample and eventually expelled with the onset of strong \nuniaxial anisotropy in the top part of the disk.  \nWe now  probe the magnetization dynamics in response to the RF excitation field of amplitude \n~4 mT, applied in the direction shown in Figure 2, by performing magnetic laminography for \nseven delay times o ver a period of 2 ns with respect to the X -ray pulse train.  \n \nFigure 3. Dynamic response of the three -dimensional magnetization  at 500 MHz . a) The angle at which  the magnetization  \nprecesses is plotted for the heights within the sample corresponding to Fig 2 a -d (see Fig. S4 for slice locations) . In the vortex \nstate (a,b) there is significant precession in the vicinity of the vortex domain wall s (indicated with white arrows in a) , as well as at \nthe edge of the sample, indicating t he presence  of pronounced  edge modes.  In the intermediate S -state (c), an enhancement of the \nedge modes at opposite sides of the disc is observed. In the single domain state  (d), the precession is significantly reduced, \nhowever lower amplitude edge modes are present on the top left and bottom right edges of the structure ( this is visualised with a  \ndifferent colour scale  to a-c,e,f.) The three -dimensional dynamics are visualised in e, and f, by mapping the magnetisation \nprecession, where one can see that th e dynamics are limited to the vortex cores (e) and edge modes that spiral around the disc (f). \nScale bars represent 1 μm . \nWe map the spatial distribution of the magnetization dynamics by the Fourier coefficient of the \ntemporal evolution of the in -plane mag netization direction 𝜃𝑚 corresponding to 500  MHz , in \nFigure 3. The precession angle of the magnetization is plotted for the four magnetic states shown \nin Figure 2a -d in Figure 3a -d, respectively . While the direction of the applied RF field is almost \nparallel to the direction of the magnetization in the top region of the disk, magnetization \ndynamics are observed across the entire thickness, as a result of the inhomogeneous \nmagnetization structure.  At the bottom of the sample (Fig. 3 a,b – region (i)) the magnetization \noscillates in the vicinity of the vortex walls indicating their reversible displacement (red lines , \nindicated by white arrows in Fig. 3a ). The magnetization also oscillates at the bot tom left and top \nright edges of the disc, where the magnetization is almost perpendicular to the applied RF field, \n \n6 \n thereby maximizing the field -induced torque. In the intermediate S -state (region (ii)), \nmagnetization precession is observed in two regions c lose to the sample edge, which are adjacent \nto the location where the vortices are expelled from the disc (Figure 3c). In the single domain \nstate (Figure 3d) oscillations of the magnetization in the bulk of the domain are absent, and only \nlower amplitude  edge modes are present  (made visible by the more limited color scale ). Three -\ndimensional maps  of the modes are given in Figure 3e,f, showing that the magnetization \ndynamics are limited to the vortex domain walls and at the edges of the sample. Mapping the \nmagnetization dynamics gives a picture of the interaction between the different domain \nstructures  and of the resulting  modifications in the ir dynamics. Across the sample height, the \nedge oscillations are located at different positions, describing a spiral, which follows the \nevolution of the location of the vortex walls , due to the presence of the vortices giv ing rise to \nregions with a sizeable magnetization component perpendicular to the applied field direction , \nwhere the applied torque is greater . In additi on, the large anisotropy in the top region would be \nexpected to lead to the formation of an ‘onion’ state [34] characterized by dynamic edge modes \nat the extremities of th e domain, where the magnetization typically is not  parallel to the sample \nboundaries. In contrast, in our system, imprinting of the bottom vortices onto the top region \nrather leads to the formation of a ‘flower’ state  [35], thus shifting the re gions in which the \nmagnetization oscillates.  \nHaving identified the regions excited by the applied field, we now consider the details of the \ntemporal evolution of the vortex domain walls in the lower section of the disc in Figure 4. The \ncurl of the in -plane magnetization 𝛻×𝑚𝑥𝑧 is plotted for the slice at h=90 nm in Figure 4a, \nwhere the red and blue stripes correspond to the two vortex walls, which have opposite \nchiralities. In the insets, 𝛻×𝑚𝑥𝑧 is plotted in the vicinity of the domain wall with posi tive \nvorticity  (region indicated by the black dashed box) for different time delays spanning a period \nof 2 ns. Over this period, the domain wall moves about the reference position (dashed line) and \nits displacement in the direction perpendicular to the  long axis of the  wall is plotted as a function \nof time for ℎ=46 nm in Figure 4b, and for all sample heights in Supplementary Figure S6, \nwhere the vortex domain wall can be seen to consistently exhibit an oscillatory motion. The \namplitude of the domain wall os cillations is determined by calculating the Fourier coefficient of \nthe temporal domain wall displacement to be 74±4 nm over the bottom 500 nm of the sample, \nwith a maximum domain wall displacement of  200±15 nm throughout the structure, \ncorresponding to a d omain wall velocity of approximately 200 m/s.  \nThe domain wall with negative vorticity  (blue in Figure 4a) displays a similar oscillatory motion \nperpendicular to its long axis with an amplitude of (45±13) nm. Plotting the displacement of the \ndomain walls as a function of time (Figure 4b) reveals that the domain walls oscillate out of \nphase.  We quantify the phase difference between the two domain walls by calculating the phase  \nof the Fourier coefficients, and find it to be  3𝜋\n4±𝜋\n4, equivalent  to a delay of  0.75±0.25 ns. This \nout-of-phase motion corresponds to the breathing of the central dom ain, which i s due to the \ntorques exerted by the field on the two walls with opposite vorticities [12]. \n  \n7 \n  \nFigure 4. Domain wall dynamics in the lower region of the disc containing two vortex walls. a) The curl of the in -plane \nmagnetization  𝛻×𝑚𝑥𝑧 for h = 90 nm  is shown , where the two domain walls are identified by the positive ( red) and negative \n(blue) lines. The domain wall with positive curl in the region indicated by the dashed box is shown at dif ferent  times  increasing  \nclockwise around the central image , with a reference location indicated by a dashed line. b ) The dynamics of the  wall with a  \npositive curl vortex ( red, vortex 1 ) are compared with the dynamics of the wall with a negative curl vortex  (blue, vortex 2 ) for an \nexample slice of height 46 𝑛𝑚. The two oscillatory signals are out of phase by (0.75±0.25) 𝑛𝑠, consistent with the “breathing”  \nof the central domain with the applied RF field.  Scale bar represent s 1 μm.  \nBy combining  magnetic la minography with a pump -probe experimental setup , we have probe d \nthe magnetization dynamics of a three -dimensional magnetic structure in response to the \napplication of an RF magnetic field. Two  main  types of magnetization dynamics are identified: \ndomain wal l precession and magnetization  dynamics at the edges of a uniform magnetic domain \nthat are enhanced due to variations in the magnetization state across the phase boundary. At the \nboundary of two phases defined by different magnetic anisotropies, the local transfer of  \nanisotropy from the top layer to the soft layer through exchange coupling leads to the expulsion \nof the domain walls and the enhancement of edge dynamics . Since  laminography is ideal for the \nstudy of flat and thin samples, pump -probe laminograp hy is directly compatible with high spatial \nresolution soft X -ray imaging, making possible the direct application of the technique to \nnanoscale systems. The capability to experimentally measure complex magnetization dynamics \nin three -dimensional systems is  key for the investigation of promising dynamic properties \npredicted in such systems , that  have been proposed for future data storage [36], logic [37] and \nmicrowave [38, 39] applications.  \n  \n \n8 \n  \nAcknowledgments:  \nThe authors would like to thank Dario Marty for initial discussions about the sample fabrication \nand design, Elisabeth Müller for FIB patterning of the sample. All data was measured at the \ncSAXS beamline, Swiss Light Source , Paul Scherrer Institut e, Switze rland . \nFunding : C.D. acknowledge s funding from the Leverhulme Trust (ECF -2018 -016), the Isaac \nNewton Trust (18 -08) and the  L’Oréal -UNESCO UK and Ireland Fellowship  For Women \nIn Science  2019 . A.H. was funded by the European Union's Horizon 2020 research and  \ninnovation programme under Marie Sklodowska -Curie grant agreement number 794207 \n(ASIQS).  \nAuthor  contributions : C.D. and S.F. conceived the study of 3D magnetic dynamics with input \nfrom J.R.. The sample fabrication was planned by C.D. and S.F. and performed by C.D., \nA.H., S.M. and E.M.  The time -resolved setup was created by S.F. and J.R. The \nlaminography endstation was designed and built by M.H. The synchrotron measurements \nwere performed by C.D., S.F., M.G. -S., M.H., V.S., and J.R.. The non -magnetic \nlaminography reconstruction code was developed by M.O., and used by C.D. with the \nsupport of M.O. The magnetic laminog raphy reconstruction was developed and used by \nCD, with the support of M.G -S. Data analysis was performed by C.D. with the support of \nM.G-S. The time -resolved 3D magnetic data was interpreted by C.D., S.F., S.G. and \nM.G. -S. CD wrote the manuscript with con tributions from all authors.  \nCompeting interests : Authors declare no competing interests.  \nData and materials availability : All data and analysis code associated with this manuscript will \nbe made available on a repository once the manuscript is published . \nSupplementary Materials:  \nExperimental setup  \nLaminography setup  \nA gold Fresnel zone plate (FZP) with a diameter of 170  μm and an outermost zone width of \n60 nm was coherently illuminated at a photon energy of 7.247 keV, which corresponds to the \nmaximum XMCD s ignal at the G d L3 edge.  The FZP was fabricated with displaced zones, \ndesigned to introduce perturbations into the illumination wavefront and improve the image \nresolution  for ptychography  [40].  The FZP was used in combination with a 50 𝜇m-diameter \ncentral stop and 30 𝜇m-diameter order sorting aperture. Circular polarized light was generated \nby a diamond quarter wave plate phase plate positioned before the setup as presented in [21, 29] , \nwith a resulting degree of polarization  of over 99.9%, and an efficiency of approximately 60%.  \nThe sample was placed downstream of  the focal spot of the FZP where  the beam size reached \n5 𝜇m in diameter. The sample was mounted on a 2D piezo scanner with 100 x 100 µm2 scan \nrange and the scanner was installed on a rotation stage with a clear aperture allowing 360 ° \nrotation. Two linear stages with a travel range of several millimeters were used to move  the \nrotation stage in order to keep t he region of interest in the  field of view  at different rotation \nangles. The entire sample stage stack was mounted at the laminography angle of 61° (angle \nbetween axis of rotation and X-ray beam propagation ). The angle was chosen to provide a  \n9 \n balance between 3D spatial resolution and sample absorption, a nd was checked with  numerical \nsimulations of magnetic laminography that are described in the next section.  \nSimilar  to previous instrumentation  [41, 42] , closed loop sample positioning and thus positioning \naccuracy during ptychographic scans was achieved by differential laser interferometry measuring \nthe relative position between the sample and the beam -defining FZP  in both the horizontal and \nvertical directions perpendicular to the X-ray beam propagation direction  [31]. \nThe diffracted X-rays propagated in an evacuated flight tube and were detected by  an in -vacuum \nEiger detector [43] placed at a distance of 5.260 m from the sample.  \nTime resolved setup  \nThe time resolved measurements were performed with the pump -probe method, employing the \nX-ray flashes generated by the synchrotron light source as a probing signal. The time structure of \nthe Swiss Light Source is designed to provide 70 ps wide (FWHM) X-ray pulses , defining the \ntemporal resolution of our measurements,  every about 2 ns (repetition f requency of \n499 .652  MHz ). The magnetization dynamics in the sample were excited by an RF signal that \nwas synchronized to the RF electron bunch accelerators of the synchrotron light source through \nan IQ modulator, which was then employed to shift the relati ve delay between the X-ray pulses \nand the exciting RF signal.  \nThis method locks the excitation frequency and its phase to the synchrotron repetition  frequency , \nor its higher order harmonics. If a single -bunch filling pattern were  to be  employed, classical  \npump -probe measurements could be carried out, which would allow the investigation of different \nfrequency ranges, or pulsed excitations.  \nLaminography measurements  \nTo measure 2D projections of the magnetization , resonant dichroic ptychography was used. The \nenergy of the X-rays was tuned to the absorption edge of the magnetic element in question, in \nthis case the L 3 edge of gadolinium with an energy of 7.247  keV, and ptychographic projections \nwere measured with right and left circularly polarised light.  \nPtych ographic projections with a field of view of 7×12.5 μ m2 consisted of 818 diffraction \npatterns with a step size of 1 μm and an exposure time of 0.2s. The two dimensional projections \nof the magnetization  shown in Figure 1b,c consist of an average of 25 single XMCD projections, \nwhich allowed for a sufficiently high signal -to-noise ratio 2D image of the magnetic structure.  \nFor magnetic laminography, ptychographic projections were measured with both right and left \ncircularly polarised X-rays at 144 angles equally spaced over 360°. Within each dataset, six of \nthe projections were blanked by the coaxial cable carrying the RF exci tation to the sample, \nmeaning that the laminography reconstructions were performed using 138 projections of each \npolarization .  \nThe time delays of the excitation field were measured non -sequentially to be sure that any \nobserved dynamic behaviour would be i ndependent of slight sample changes occurring over time \ndue to heating or aging effects. The phases were measured in the following order:  \n𝜋,0,7𝜋\n4,3𝜋\n4,𝜋\n4,5𝜋\n4,𝜋\n2,3𝜋\n2 \nAt times of 1𝑛𝑠,0𝑛𝑠,1.75𝑛𝑠,0.75𝑛𝑠,0.25𝑛𝑠,1.25𝑛𝑠,0.5𝑛𝑠,1.5𝑛𝑠, respecti vely.   \n10 \n Unfortunately , due to experimental errors it was not possible to obtain a complete reconstruction \nfrom the 𝜋\n2 (0.5 ns) laminography dataset, which is why seven time-steps are presented in the \nmanuscript rather than eight .  \nEach laminography measurem ent consisting of 144 projections took 7 hours including overhead, \nmeaning that each magnetic laminography measurement required 14 hours in order to measure \nwith both left and right circular polarization . Therefore, i n total, seven  time steps required ~4 \ndays and 2 hours of continuous measurement  and a total of 2016 projections . \nLaminography reconstruction (non -magnetic)  \nThe measured diffraction patterns were used to reconstruct the complex -valued projections \n𝑃𝑛(𝑥,𝑦). The reconstruction was performed using 300 iterations of the difference map method \n[44] and 300 iterations of the maximum likelihood optimization [45].  Reconstructed and \nunwrapped phase  projections wer e aligned together in order to maximize their mutual \nconsistency  [46]. This is possible only because the magnetic properties  of the sample do not \naffect the phase -based alignment procedure at the selected photon energy. Aligned and \nunwrapped phase projections were reconstructed by the filtered backprojection method. The \nlaminography imaging geometry unavoidably results in missing information in the Fourier space, \nwhich is the so-called missing cone problem. Therefore, an iterative method enforcing physical \nconstraints in the real space and measured projection in the Fourier space was used to fill the \nmissing cone  [31].  \nThe non -magnetic reconstruction was used to estimate  the location of  the magnetic material, \nwhich is used as a constraint in the magnetic reconstruction detailed in the following section.  \nMagnetic Laminography  \nMagnetic laminography , as well as magnetic tomography, is a special case of arbitrary projection \n3D magnetic imaging. For completeness we repeat here the general formulation for arbitrary \nprojection magnetic tomography reconstructions [32] and later show the particular application to \nlaminography.  \nThe amplitude of the transmissivity, 𝐴𝑛, measured experimentally is related to the defined \nXMCD projection 𝑃𝑛 as follows:  \n𝐴𝑛(𝑥,𝑦)=𝑒𝑥𝑝 (−2𝜋\n𝜆𝑃𝑛(𝑥,𝑦)) \nAfter normal ization and computation of the absolute value of the transmissivity , eachprojection \nis mathematically defined in terms of rotation matrices as:  \n𝑃𝑛(𝑥,𝑦)=−𝑟𝑒\n2𝜋𝜆2(∑𝑛𝑎𝑡𝑘∫ℑ{𝑓𝑐𝑘(𝑅(𝑛)𝑟)}𝑑𝑧\n𝑘±𝑛𝑎𝑡𝑚𝑎𝑔∫ℑ{𝑓𝑚(1)}[𝑅(𝑛)†𝑚(𝑅(𝑛)𝑟)]∙𝑧^𝑑𝑧) \nWhere 𝑟𝑒 is the classi cal electron radius, 𝜆 is the X-ray wavelength, 𝑛𝑎𝑡𝑘 and 𝑛𝑎𝑡𝑚𝑎𝑔 are the atomic \ndensity of the kth element, and of the resonant magnetic element, respectively , and ℑ selects the \nimaginary part . Here, 𝑚(𝑅(𝑛)𝑟) is the magnetization  vector in the ob ject coordinate system, and \n𝑓𝑐 and 𝑓𝑚(1) are the electron density and XMCD scattering factors, respectively. The magnetic \ncontribution originates only from the resonant element, given  in the second term, whereas the  \n11 \n electronic scattering is calculated by summing over all elements present, includ ing the magnetic \nelement.  \nDuring the  magnetic laminography  reconstruction, the current reconstructed components of the \nmagnetization  are used to compute projections and compare them to the data. In this calculation, \nthe integrals are approximated by sums, and the projection is defined by:  \n𝑃̂𝑛(𝑥,𝑦)=∑{𝑐[𝑅(𝑛)†𝑚(𝑅(𝑛)𝑟)]∙𝑧̂+𝑜(𝑅(𝑛)𝑟)}\n𝑘 \nWhere 𝑚 and  𝑜 are the estimated magnetic and non -magnetic structures, respectively, and 𝑐 is a \nconstant that relates the XMCD signal to the magnetization , including  the pixel siz e of the \ndataset . At each iteration of the reconstruction, the correctness of the reconstruction is estimated \nby calculating an error metric that compares the estimated dataset 𝑃̂𝑛(𝑥,𝑦) with the measured \ndataset 𝑃𝑛(𝑥,𝑦). This error metric is defined as:  \n𝜀=∑∑[𝑃̂𝑛(𝑥,𝑦)−𝑃𝑛(𝑥,𝑦)]2\n𝑥,𝑦 𝑛 \nThe reconstruction is then updated such that the error metric is reduced, and the reconstruction \napproaches the correct solution, by calculating the gradient of the error metric with respect to the \ndifferent reconstructed va riables. In this case, the gradients for the magnetization  𝑚 and the non -\nmagnetic structure 𝑜 are analytically defined as:  \n𝜕𝜀\n𝜕𝑚(𝑅(𝑛)𝑟)=2∑[𝑃̂𝑛(𝑥,𝑦)−𝑃𝑛(𝑥,𝑦)]𝑅(𝑛)[0\n0\n1]\n𝑛 \n𝜕𝜀\n𝜕𝑜(𝑅(𝑛)𝑟)=2∑[𝑃̂𝑛(𝑥,𝑦)−𝑃𝑛(𝑥,𝑦)]\n𝑛 \nFor magnetic laminography, the rotation matrices  that define a rotation of the sample by 𝜑 \naround the rotation axis, which is in turn at an angle 𝜃𝐿 with respect to the direction of \npropagation of the X-rays (𝑧^), are given by:  \n𝑅(𝑛)†=(1 0 0\n0 𝑠𝑖𝑛𝜃𝐿 𝑐𝑜𝑠𝜃𝐿\n0−𝑐𝑜𝑠𝜃𝐿𝑠𝑖𝑛𝜃𝐿)(𝑐𝑜𝑠𝜑𝑛 0𝑠𝑖𝑛𝜑𝑛\n0 1 0\n−𝑠𝑖𝑛𝜑𝑛0𝑐𝑜𝑠𝜑𝑛) \n(𝑠𝑖𝑛𝜑𝑛 0 𝑐𝑜𝑠𝜑𝑛\n−𝑐𝑜𝑠𝜃𝐿𝑠𝑖𝑛𝜑𝑛 𝑠𝑖𝑛𝜃𝐿 𝑐𝑜𝑠𝜃𝐿𝑐𝑜𝑠𝜑𝑛\n−𝑠𝑖𝑛𝜃𝐿𝑠𝑖𝑛𝜑𝑛−𝑐𝑜𝑠𝜃𝐿𝑠𝑖𝑛𝜃𝐿𝑐𝑜𝑠𝜑𝑛) \nAs a result, the gradients for the components of the magnetization  are defined as:  \n𝜕𝜀\n𝜕𝑚(𝑅(𝑛)𝑟)=2∑[𝑃^𝑛(𝑥,𝑦)−𝑃𝑛(𝑥,𝑦)][−𝑠𝑖𝑛𝜃𝐿𝑠𝑖𝑛𝜑𝑛\n−𝑐𝑜𝑠𝜃𝐿\n𝑠𝑖𝑛𝜃𝐿𝑐𝑜𝑠𝜑𝑛]\n𝑛 \nFor 0<𝜃𝐿<90°, all three components of the gradient are non -zero, meaning that the measured \nprojections are sensitive to, and contain information about, all three components of the \nmagn etization . This in turn suggests that they can in principle be recovered with a dataset \nmeasured around a single rotation axis. To determine whether this is the case, we performed  \n12 \n numerical simulations of magnetic laminography for a complex 3D magnetic str ucture \ncontaining vortices, domain walls and Bloch points, as in [21]. For the simulations, 360 XMCD \nprojections were measured around  the laminography rotation axis, and , in order  to reconstruct \nthe magnetic signal, 50 iterations of the arbitrary pr ojection magnetic reconstruction algorithm \n(APMRA) were used. The laminography axis was varied between 0° and 90° to determine the \noptimal geometry for determining the 3D magnetization vector field with magnetic \nlaminography . \n \nFigure S1:  Numerical simulat ions of magnetic laminography for different laminography tilt angles show that there exists a range \nof angles (30° -80°) for which the 3D structure can be reconstructed with a high degree of accuracy. The accuracy of the \nreconstruction is determined by calc ulating the RMS error of the direction of the reconstructed magnetization with respect to the \noriginal structure ( ∆𝜃𝑚). The tilt angle used in th e experiment  presented in this manuscript  is 61°, which  lies comfortably within \nthis range and is indicated with an arrow.  \nThe error in the reconstructed magnetic structure with respect to the original magnetic structure \nwas defined as the error in the local magnetization 3D orientation and was  calculated for a range \nof laminography tilt angles . This error in the direction of the reconstructed  magnetization \ndirection  is shown in Figure S 1, where it can be seen that a reasonably wide range of angles \nexists for which the 3D magnetic structure is reconstructed wi th a high degree of accuracy. In \nparticular, for the angle used in this work 𝜃𝐿=61.2°, 99% of all voxels have an error less than \n6° in the direction of the magnetization  in three dimensions .   \n \nMagnetic laminography reconstruction  \nThe three -dimensional magnetic structure was reconstructed using 10 iterations of the GPU -\nimplemented version of the gradient -based arbitrary projection reconstruction code described in \n[32] and made available at  [47], using the  rotation matrices described above. During the \nmagnetic reconstruction, the magnetization  was constrained to the magnetic material using a \nmask calculated from the non -magnetic laminography reconstruction. No additional \nregularisation of the magnetization  was implemented.  \nOnce reconstructed, the three components of the magnetization  were filtered using a Hanning \nfilter with a cutoff spatial frequency of 1/6 with respect to the maximum spatial frequency in \norder to remove high frequency noise, and the magnet ization  was then normalised by its \nmagnitude to obtain a magnetic vector field of uniform amplitude.  \n \n13 \n The magnetic reconstruction for each time -step was performed independently using the same \nreconstruction parameters.  \n \nSpatial resolution and accuracy of ma gnetic reconstruction:  \nThe full -period spatial resolution of the magnetic reconstruction was estimated by taking a line \nprofile of the 𝑚𝑥 and 𝑚𝑧 components across the vortex domain wall and measuring the 25% -\n75% edge sharpness  , as shown in Figure S 2, which was found to be 50  nm. Micromagnetic \nsimulations show that at the surface of a sample, the vortex core in such a system is of the order \nof 20 -30 nm, which increases rapidly as one moves into the bulk of the disc. Considering that the \nmeasured edge sharpness is a convolution of the size of the vortex core and the spatial resolution, \nwe therefore estimate the half -period spatial resolution of the magnetic reconstruction to be \naround 50  nm.   \n \n \nFigure S2:  Line profiles of (a) the 𝑚𝑥 and ( b) the 𝑚𝑧components of the magnetization  across the vortex domain wall. The 25% -\n75% edge sharpness is measured to be on average 50  nm, indicating a half -period spatial resolution of 50  nm. c) The line along \nwhich the profile is measured is drawn in blue  \n \nA further estimate of the half -period spa tial resolution was also estimated by calculating the \nFourier Shell Correlation (FSC) of the three components of the magnetization  and the in -plane \nvector fields in the three Cartesian planes , shown in Figure S3 . The FSC indicated an average \nspatial resolu tion of 184  nm, 360  nm and 180  nm for the 𝑚𝑥, 𝑚𝑦 and 𝑚𝑧 components \nrespectively. The difference between the edge  sharpness  and FSC estimated resolution can be \nexplained by the spatial sparsity of the data, i.e. the small number of features with high spatial \nfrequencies present in the sample., and the resulting lack of high frequencies present, which \nleads to underestimation of the spatial resolution based on FSC  [48].   \n \n \nFigure S3. Fourier Shell Correlation of (a) the 𝑚𝑥, (b) the 𝑚𝑦 and (c) the 𝑚𝑧 components of the magnetization . An estimate of \nthe spatial resolution is obtained using the ½ bit threshold [36] (red dashed line) and is found to be 184  nm, 360  nm and 180  nm \nfor the 𝑚𝑥, 𝑚𝑦 and 𝑚𝑧 components of the magnetization , respectively  \n \n14 \n In the time -resolved measurement, we  are able to track displacements of the vortex domain wall \nwith an accuracy significantly higher than the spatial resolution . In this case, we track \noscillations in the domain wall location with an amplitude of 45  nm and 74  nm for the vortex \ndomain walls with negative and positive vorticity , respectively. This sub -spatial resolution \ntracking of the domain wall is possible as it is dependent on th e signal -to-noise ratio of the \nreconstruction , and not the global or local spatial resolution.  \n \nThree -dimensional magnetic structure  \nFor reference, the heights for which the magnetization  is shown in Figure 2a -d, and the 500 MHz \nmodes in Figure 3a -d, are indicated on the three -dimensional rendering of the 500 MHz modes in \nFigure S4.  \n \nFigure S4. The heights at which the magnetization  and dynamics are shown in Figs 2a -d and 3a -d are indicated by semi -\ntransparent discs, superimposed on the three -dimensional rendering of the magnetization  modes shown in Figure 3e.  \nData Analysis: Calculation of vortex domain wall displacement  \nThe displacement of the vortex domain wall was calculated using two different pr ocedures that \nboth make use of  the sub-pixel image registration algorithm described in [49]: \n \nVortex -tracking  based on 𝛻𝜃𝑚: \nThe divergence of the direction of the in -plane magnetization  was calculated for a region of the \nsample surrounding the vortex core of the domain wall, as in Figure 4a for the case of the vortex \ndomain wall with positive vorticity . As an example, the divergence of 𝜃𝑚 is shown  in Figure S 5, \nwhere one can see that a t the core of the vortex, there is a sharp peak in the gradient of 𝜃𝑚, with \na FWHM of only a few pixels. The location of the vortex core for each time -step was determined \nby aligning the image of [𝛻𝑠𝑖𝑛  (𝜃𝑚)]𝑡 with a reference two -dimensional Gaussia n with a FWHM \nof 1 pixel (46.8 nm). The same analysis was performed for all slices of the three -dimensional \nmagnetic structure in which the vortex domain wall was present (the lower ~500 nm of the disc), \nand the motion of the vortex domain wall as a functi on of height was determined .  \n \nVortex -tracking  based on sin( 𝜃𝑚): \nWhile using  𝛻𝜃𝑚 worked well for  tracking  the vortex with positive vorticity  shown in Figure 4, \nwhen applied to the second vortex domain wall with negative vorticity , the tracking  was found to \nbe more susceptible to noise due to the more complex structure of the domain wall. To reduce \nthe noise, we chose to align images with signa l that was not limited to the vortex core, but was \npresent throughout the image. For this, we calculated 𝑠𝑖𝑛(𝜃𝑚), shown in Figure S 5b, which had \na non -zero signal throughout the image, and does not exhibit phase -wrapping , i.e. the sharp \ntransition from 0→2𝜋, as it does for 𝜃𝑚. \n \n15 \n The components of the motion of the vortex domain wall parallel and perpendicular to the long \naxis of the domain wall were calculated and compared. In general, for both domain walls, the \nmotion perpendicular to the long axis of the domain wall could be identified as  oscillatory . \nHowever , the motion parallel to the domain wall was much noisier, and no clear oscillatory \nmotion could be determined. The challenges in determining the motion parallel to the long axis \nof the domain wall  are likely due to the extreme ellipticity of the vortex core due in turn to the \nanisotropy in the system. In this case, it is likely that t he much smaller displacement  of the core \nparallel to the long axis of the domain wall  is below the current accuracy of this measurement.   \n \n \nFigure S 5. Images of a) 𝛻𝑠𝑖𝑛 (𝜃𝑚) and b) 𝑠𝑖𝑛(𝜃𝑚) of the lowest slice of the magnetization . One can see that for 𝛻𝑠𝑖𝑛 (𝜃𝑚) in a), \nthe signal is limited to the domain walls, whereas for 𝑠𝑖𝑛(𝜃𝑚) the signal is non -zero throughout the structure.  \nThe motion of the second vortex domain wall perpendicular to its long axis , indicated by grey \narrows in Figure S6,  is given as a func tion of height in Figure S 6 and here  one can identify a \nsimilar periodic  temporal behaviour to the first vortex domain wall shown in Figure 4, albeit with \na smaller amplitude of (45±13)  nm, and a phase shift of (0.75±0.25) ns with respect to the first \nvortex domain wall.  \n \n \nFigure S 6. Displacement of the vortex domain wall with negative vorticity  (a, blue  in Figure 4)  and positive vorticity  (b, red in \nFigure 4)  in the direction perpendicular to the long axis of the domain wall (indicated by grey arrows) as a function of time for \ndifferent heights within the structure. The oscillatory behaviour is clearest  for both vortices  at a height of 46 nm (the bottom of \nthe structure ) and  is noisier at the top. When compar ing the two vortices , the oscillation s appear to be consistently out of phase  \nthrough the height of the sample , as shown for ℎ=46 𝑛𝑚 in Figure 4 c. Scale bar is 1 μm.  \nReferences:  \n1. Templeton, T.L., S.D. Hanham, and A.S. Arrott, Helical patterns of magnetization and magnetic charge \ndensi ty in iron whiskers.  AIP Advances, 2018. 8(5): p. 056022.  \n \n16 \n 2. Yan, M., R. Hertel, and C.M. Schneider, Calculations of three -dimensional magnetic normal modes in \nmesoscopic permalloy prisms with vortex structure.  Physical Review B, 2007. 76(9): p. 094407.  \n3. Im, M. -Y., et al., Dynamics of the Bloch point in an asymmetric permalloy disk.  Nature Communications, \n2019. 10(1): p. 593.  \n4. Milde, P., et al., Unwinding of a Skyrmion Lattice by Magnetic Monopoles.  Science, 2013. 340(6136): p. \n1076.  \n5. Hubert, A. and R . Schäfer, Magnetic Domains: The Analysis of Magnetic Microstructures . 1998: Springer.  \n6. Varga, R., et al., Supersonic domain wall in magnetic microwires.  Physical Review B, 2007. 76(13): p. \n132406.  \n7. Otálora, J.A., et al., Curvature -Induced Asymmetric S pin-Wave Dispersion.  Physical Review Letters, 2016. \n117(22): p. 227203.  \n8. Hertel, R., Ultrafast domain wall dynamics in magnetic nanotubes and nanowires.  J. Phys.: Cond. Matt., \n2016. 28(48): p. 483002.  \n9. Yan, M., et al., Fast domain wall dynamics in magn etic nanotubes: Suppression of Walker breakdown and \nCherenkov -like spin wave emission.  Applied Physics Letters, 2011. 99(12): p. 122505.  \n10. Raabe, J., et al., Quantitative Analysis of Magnetic Excitations in Landau Flux -Closure Structures Using \nSynchrotro n-Radiation Microscopy.  Physical Review Letters, 2005. 94(21): p. 217204.  \n11. Wintz, S., et al., Magnetic vortex cores as tunable spin -wave emitters.  Nature Nanotechnology, 2016. 11: p. \n948. \n12. Choe, S.B., et al., Vortex Core -Driven Magnetization Dynamics.  Science, 2004. 304(5669): p. 420.  \n13. Van Waeyenberge, B., et al., Magnetic vortex core reversal by excitation with short bursts of an alternating \nfield.  Nature, 2006. 444(7118): p. 461 -464. \n14. Baumgartner, M., et al., Spatially and time -resolved magnetization dynamics driven by spin –orbit torques.  \nNature Nanotechnology, 2017. 12: p. 980.  \n15. Finizio, S., et al., Dynamic Imaging of the Delay - and Tilt -Free Motion of Néel Domain Walls in \nPerpendicularly Magnetized Superlattices.  Nano Letters, 2019. 19(1): p. 375 -380. \n16. Stein, F. -U., et al., Time -resolved imaging of nonlinear magnetic domain -wall dynamics in ferromagnetic \nnanowires.  Scientific Reports, 2013. 3: p. 1737.  \n17. Vogel, J., et al., Direct Observation of Massless Domain W all Dynamics in Nanostripes with Perpendicular \nMagnetic Anisotropy.  Physical Review Letters, 2012. 108(24): p. 247202.  \n18. Woo, S., et al., Deterministic creation and deletion of a single magnetic skyrmion observed by direct time -\nresolved X -ray microscopy.  Nature Electronics, 2018. 1(5): p. 288 -296. \n19. Büttner, F., et al., Dynamics and inertia of skyrmionic spin  structures.  Nature Physics, 2015. 11: p. 225.  \n20. Wartelle, A., et al., Bloch -point -mediated topological transformations of magnetic domain walls in \ncylindrical nanowires.  Physical Review B, 2019. 99(2): p. 024433.  \n21. Donnelly, C., et al., Three -dimensional magnetization structures revealed with X -ray vector \nnanotomography.  Nature, 2017. 547: p. 328.  \n22. Suzuki, M., et al., Three -dimensional visual ization of magnetic domain structure with strong uniaxial \nanisotropy via scanning hard X -ray microtomography.  Applied Physics Express, 2018. 11(3): p. 036601.  \n23. Hilger, A., et al., Tensorial neutron tomography of three -dimensional magnetic vector fields in bulk \nmaterials.  Nature Communications, 2018. 9(1): p. 4023.  \n24. Kardjilov, N., et al., Three -dimensional imaging of magnetic fields with polarized neutrons.  Nature \nPhysics, 2008. 4(5): p. 399 -403. \n25. Manke, I., et al., Three -dimensional imaging of magn etic domains.  Nat Commun, 2010. 1: p. 125.  \n26. Tanigaki, T., et al., Three -Dimensional Observation of Magnetic Vortex Cores in Stacked Ferromagnetic \nDiscs.  Nano Letters, 2015. 15(2): p. 1309 -1314.  \n27. Streubel, R., et al., Retrieving spin textures on curve d magnetic thin films with full -field soft X -ray \nmicroscopies.  Nat Commun, 2015. 6: p. 7612.  \n28. Blanco -Roldan, C., et al., Nanoscale imaging of buried topological defects with quantitative X -ray \nmagnetic microscopy.  Nat Commun, 2015. 6: p. 8196.  \n29. Donne lly, C., et al., High -resolution hard x -ray magnetic imaging with dichroic ptychography.  Physical \nReview B, 2016. 94(6): p. 064421.  \n30. Magnetism and Synchrotron Radiation , ed. E. Beaurepaire, et al. 2010: Springer.  \n31. Holler, M., et al., Three -dimensiona l imaging of integrated circuits with macro - to nanoscale zoom.  Nature \nElectronics, 2019. 2(10): p. 464 -470.  \n17 \n 32. Donnelly, C., et al., Tomographic reconstruction of a three -dimensional magnetization vector field.  New \nJournal of Physics, 2018. 20(8): p. 083 009. \n33. Kneller, E.F. and R. Hawig, The exchange -spring magnet: a new material principle for permanent \nmagnets.  IEEE Transactions on Magnetics, 1991. 27(4): p. 3588 -3560.  \n34. Ha, J.K., R. Hertel, and J. Kirschner, Micromagnetic study of magnetic configura tions in submicron \npermalloy disks.  Physical Review B, 2003. 67(22): p. 224432.  \n35. Gatel, C., et al., Size-Specific Spin Configurations in Single Iron Nanomagnet: From Flower to Exotic \nVortices.  Nano Letters, 2015. 15(10): p. 6952 -6957.  \n36. Parkin, S.S.P. , M. Hayashi, and L. Thomas, Magnetic domain -wall racetrack memory.  Science, 2008. \n320(5873): p. 190 -194. \n37. Allwood, D.A., et al., Magnetic Domain -Wall Logic.  Science, 2005. 309(5741): p. 1688 -1692.  \n38. Kaka, S., et al., Mutual phase -locking of microwave spin torque nano -oscillators.  Nature, 2005. 437(7057): \np. 389 -392. \n39. Mancoff, F.B., et al., Phase -locking in double -point -contact spin -transfer devices.  Nature, 2005. \n437(7057): p. 393 -395. \n40. Odstrčil, M., et al., Towards optimized illumination for high -resolution ptychography.  Optics Express, \n2019. 27(10): p. 14981 -14997.  \n41. Holler, M. and J. Raabe, Error motion compensating tracking interferometer for the position measurement \nof objects with rotational degree of freedom . Vol. 54. 2015: SPIE. 1 -7, 7. \n42. Holler, M., et al., OMNY—A tOMography Nano crYo stage.  Review of Scientific Instruments, 2018. 89(4): \np. 043706.  \n43. Guizar -Sicairos, M., et al., High -throughput ptychography using Eiger: scanning X -ray nano -imaging of \nextended regi ons. Optics Express, 2014. 22(12): p. 14859 -14870.  \n44. Thibault, P., et al., High -resolution scanning x -ray diffraction microscopy.  Science, 2008. 321(5887): p. \n379-382. \n45. Thibault, P. and M. Guizar -Sicairos, Maximum -likelihood refinement for coherent di ffractive imaging.  New \nJournal of Physics, 2012. 14: p. 063004.  \n46. Odstrcil, M.a.H., M. and Guizar -Sicairos, M., Alignment methods for nanotomography with deep subpixel \naccuracy.  Submitted, 2019.  \n47. Donnelly, C. and M. Guizar -Sicairos, Magnetic tomograph y reconstruction algorithm 2018: \nhttp://doi.org/10.5281/zenodo.1324335.  \n48. van Heel, M. and M. Schatz, Fourier shell correlation threshold criteria.  Journal of Structural Biology, \n2005. 151(3): p. 250 -262. \n49. Guizar -Sicairos, M., S.T. Thurman, and J.R. F ienup, Efficient subpixel image registration algorithms.  \nOptics Letters, 2008. 33(2): p. 156 -158. \n "    },    {        "title": "1602.03142v1.A_problem_of_Ulam_about_magnetic_fields_generated_by_knotted_wires.pdf",        "content": "A PROBLEM OF ULAM ABOUT MAGNETIC FIELDS\nGENERATED BY KNOTTED WIRES\nALBERTO ENCISO AND DANIEL PERALTA-SALAS\nAbstract. In the context of magnetic \felds generated by wires, we study the\nconnection between the topology of the wire and the topology of the magnetic\nlines. We show that a generic knotted wire has a magnetic line of the same\nknot type, but that given any pair of knots there is a wire isotopic to the \frst\nknot having a magnetic line isotopic to the second. These questions can be\ntraced back to Ulam in 1935.\n1.Introduction\nIt is classical that the magnetic \feld generated by a closed wire is given by the\nBiot{Savart law. That is, if \r(s) is a closed curve in R3of length`and parametrized,\nsay, by the arc-length parameter, its associated magnetic \feld is given by the Biot{\nSavart integral\nB\r(x) :=Z`\n0_\r(s)\u0002(x\u0000\r(s))\n4\u0019jx\u0000\r(s)j3ds:\nIt is apparent that the vector \feld B\r, which is divergence-free, is in fact indepen-\ndent of the parametrization of the curve, although it does depend on its orientation.\nWhile the magnetic \felds created by simple curves, such as a circular wire,\nare well understood and have been utilized to describe a wide range of physical\nphenomena, our understanding of the magnetic \felds created by more complicated\nwires remains strikingly limited. This led Ulam to consider the relationship between\nthe degree of knottedness of the wire and that of the associated periodic magnetic\nlines.\nSpeci\fcally, Ulam posed the question of whether the magnetic \feld created by\na knotted wire must have a periodic magnetic line of nontrivial topology. This\nis among the \frst problems of The Scottish Book [7, Problem 18], the original\nmanuscript of which goes back to 1935. This question of Ulam was set in a broader\ncontext in his later problem collection [10, Section VII.7], where he asks whether\nthe magnetic lines topologically re\rect the knottedness of the wire.\nUnlike many of the problems collected in The Scottish Book , the progress made\non these questions has been scarce. This is particularly remarkable, on the one\nhand, because the existence of knotted \felds plays a signi\fcant role in a number\nof areas of physics (see e.g. [2, 6, 9]) and, on the other hand, because it has been\nrecently proposed to employ knotted wires to construct a new magnetic con\fnement\ndevice, called the knotatron [5]. The reason for the lack of rigorous contributions to\nthis line of research is probably that, by the very nature of the problems, the proof\nof these results must combine ideas from the theory of dynamical systems, which\n1arXiv:1602.03142v1  [math.DS]  9 Feb 20162 ALBERTO ENCISO AND DANIEL PERALTA-SALAS\nare key to study the knot types of the magnetic lines, with \fne analytic estimates\nfor the Biot{Savart integral associated with a curve of complicated geometry.\nOur objective in this paper is to provide two somehow complementary results\nconcerning the relationship between the knottedness of the wire and the topology of\nthe magnetic lines. The \frst result asserts that there are wires and magnetic lines\nwhose knot types can be prescribed at will (and independently). In particular, there\nare wires knotted in an arbitrarily complicated way with an unknotted periodic\nmagnetic line and there are wires isotopic to the unknot with a magnetic line\nisotopic to any given knot. More precisely, our \frst result can be stated as follows:\nTheorem 1.1. Let\rande\rbe any two closed curves in R3. Then there is a wire\nisotopic to \rwhose associated magnetic \feld has a periodic magnetic line isotopic\ntoe\r.\nOur second result asserts that, however, Ulam's original question can be an-\nswered in the a\u000ermative at least for C0-generic curves. That is, given any curve\nin the space, there is a smooth C0-small deformation of it which is isotopic to the\noriginal curve and has a magnetic line of the same knot type. To state this result,\nwe will say that two isotopic curves are arbitrarilyCk-close if the isotopy can be\ntaken arbitrarily close to the identity in the Cknorm.\nTheorem 1.2. Let\rbe a closed curve in R3. Given any integer k, there is a wire\nisotopic to \rand arbitrarily C0-close to it whose associated magnetic \feld has a\nperiodic magnetic line isotopic to \rand arbitrarily Ck-close to it.\nIt is worth emphasizing that the magnetic lines constructed in the above theo-\nrems are not just a mathematical curiosity, but they should be observable in actual\nexperiments. This is because they are structurally stable, so any C1small pertur-\nbation of the magnetic \feld (e.g., the magnetic \feld associated with a C1small\ndeformation of the wire) still has a magnetic line of the same topology. In this\ndirection, a feature of this magnetic line that we \fnd quite surprising is that it is in\nfact hyperbolic, not elliptic, so in particular there are no toroidal magnetic surfaces\nin a neighborhood of it.\nThe paper is organized as follows. In Section 2 we will consider magnetic \felds\ngenerated by a current density \feld de\fned on a toroidal surface and compute\nthe asymptotic behavior of the corresponding Biot{Savart integral as the width\nof the surface tends to zero. In Section 3 we will connect this kind of magnetic\n\felds with \felds created by wires. This will hinge on a measure-theoretic argument\nthat exploits the structure of divergence-free vector \felds on a torus whose integral\ncurves are all periodic. These results are put to use in Sections 4 and 5, where we\nrespectively prove Theorems 1.1 and 1.2.\nWe conclude the Introduction with a word about notation. Throughout the\npaper, all the curves are assumed to be smooth and without self-intersections unless\nstated otherwise, and the isotopies and di\u000beomorphisms are always of class C1.\nWhen it does not give rise to confusion, we will use the same notation \rfor a\nparametrized curve in space, which is a map \r(t) from the real line (or the circle)\ntoR3, and for its image \r\u0011\r(R)\u001aR3.A PROBLEM OF ULAM ABOUT MAGNETIC FIELDS GENERATED BY WIRES 3\n2.Asymptotics for magnetic fields generated by surface currents\nA key ingredient in the proof of Theorems 1.1 and 1.2 will be the use of magnetic\n\felds that are not generated by wires, but by current densities supported on toroidal\nsurfaces. Our goal in this section is to analyze the behavior of magnetic \felds of\nthis kind. We will be particularly interested in the case where the toroidal surface\nis very thin.\nTo make things precise, let us consider the toroidal surface\nS\r;\u000f:=\b\nx2R3: dist(x;\r) =\u000f\t\n;\nwhich is a smooth torus of the same knot type as the curve \rprovided that the\nwidth\u000fis small enough. We will de\fne coordinates ( s;y) on the domain bounded by\nS\r;\u000fas follows. Let sbe an arc-length parametrization of the curve \r, whose length\nwe will denote by `. This amounts to saying that the tangent \feld _ \rhas unit norm\nand\r(s) is`-periodic, so in particular we can assume that s2R=`Z. Without\nloss of generality, we will make the assumption that the curve \rdoes not have\nany in\rection points, which allows us to de\fne the normal and binormal vectors\nn(s);b(s) at any point of the curve \r(s). It is well known that this assumption is\nsatis\fed for generic curves [1, p. 184] (roughly speaking, \\generic\" here refers to\nan open and dense set, with respect to a reasonable Cktopology, in the space of\nsmooth curves in R3).\nUsing the normal and binormal vector \felds and denoting by D2the two-dimen-\nsional unit disk, we can introduce coordinates ( s;y)2(R=`Z)\u0002D2in the solid\ntorus bounded by S\r;\u000fvia the di\u000beomorphism\n(2.1) ( s;y)7!\r(s) +\u000fy1n(s) +\u000fy2b(s):\nRecall that the unit tangent vector is t(s) := _\r(s).\nIn the coordinates ( s;y), a short computation using the Frenet formulas yields\nthe formula for the Jacobian of this change of coordinates, which shows that the\nvolume measure is written in these coordinates as\n(2.2) dx:=\u000f2[1\u0000\u000f\u0014(s)y1]dsdy:\nHere and in what follows, \u0014(s) and\u001c(s) respectively denote the curvature and\ntorsion of the curve \r. We will sometimes take polar coordinates ( r;\u0012) in the\ndiskD2, which are de\fned as\ny1=rcos\u0012; y 2=rsin\u0012:\nIn terms of these coordinates, the volume reads as\ndx=\u000f2r[1\u0000\u000f\u0014(s)rcos\u0012]dsdrd\u0012:\nWe shall next consider magnetic \felds created by current densities supported\nonS\r;\u000f. The current distribution will be given by JdS, wheredSis the surface\nmeasure on the surface S\r;\u000fand the density Jis a smooth tangent vector \feld\nde\fned onS\r;\u000f. The associated magnetic \feld is then\nBJ(x) :=Z\nS\r;\u000fJ(x0)\u0002(x\u0000x0)\n4\u0019jx\u0000x0j3dS(x0):\nWe will always assume that the divergence of Jon the surface is zero (which is\nequivalent to demanding that the divergence of the vector-valued measure JdS is4 ALBERTO ENCISO AND DANIEL PERALTA-SALAS\nzero, in the sense of distributions), so that BJindeed has the physical interpretation\nof a magnetic \feld. In particular, BJsatis\fes Maxwell's equations\ncurlBJ=JdS; divBJ= 0:\nThe main result of this section is the following lemma, which provides the asymp-\ntotic behavior as \u000f!0 of the magnetic \feld BJfor a wide family of current densi-\ntiesJ. This family has two important features: \frstly, the \felds are divergence-free,\nand secondly, they have a special dependence on \u000fthat is crucial to prove the ex-\nistence of hyperbolic periodic magnetic lines isotopic to \r. Since we will only be\ninterested in the behavior of the \feld near \r, let us restrict our attention to a \fxed\nsmall neighborhood of the core curve \r(for instance, the region jyj<1\n10). In this\nregion, ifh(\u000f) is a function that tends to zero as \u000f!0 (typically \u000fmor\u000flog\u000f), let\nus agree to say that a scalar quantity q(s;y;\u000f ) is of order O(h(\u000f)) if\n(2.3a) j@j\nsDk\nyq(s;y;\u000f )j6Cj;kh(\u000f)\nfor allj;k. Likewise, given a nonnegative integer mwe will say that q(s;y;\u000f ) is of\norderO(jyjm) if\n(2.3b) j@j\nsDk\nyq(s;y;\u000f )j6Cj;kjyjmaxfm\u0000k;0g\nin the above region for all j;kand uniformly in \u000f. We will use the notations\n(2.3c) O(1)\u0011O(\u000f0) andO(h(\u000f) +jyjk) :=O(h(\u000f)) +O(jyjk):\nLemma 2.1. Let us take the divergence-free tangent vector \feld\n(2.4) J:=1\n1\u0000\u000f\u0014(s) cos\u0012\u0012\nF(\u0012)@s+G(s)\n\u000f@\u0012\u0013\n;\nde\fned onS\r;\u000fin terms of two smooth functions FandGof period 2\u0019and`,\nrespectively. We will also assume that Gdoes not vanish. Consider the current\ndistribution onS\r;\u000fgiven byJdS . Then, using the order notation (2.3) ,\nBJ=\u0002\nG(s) +O(\u000flog\u000f+jyj)\u0003\n@s+b1+b2y1\u0000a2y2+O(\u000flog\u000f+jyj2)\n2\u000f@y1\n\u0000a1+a2y1+b2y2+O(\u000flog\u000f+jyj2)\n2\u000f@y2;\nwherean;bnare the Fourier coe\u000ecients of F:\nF(\u0012) =:1X\nn=0(ancosn\u0012+bnsinn\u0012):\nProof. Let us start by recalling that Equation (2.1) and the Frenet formulas imply\nthat the vector \felds @s,@rand@\u0012corresponding to the coordinates ( s;r;\u0012 ) can be\nwritten in terms of the curvature \u0014and torsion \u001cof the curve \ras:\n@s\u0011@x\n@s= (1\u0000\u000f\u0014y1)t+\u000f\u001cy2n\u0000\u000f\u001cy1b: (2.5)\n@r\u0011@x\n@r=\u000fy1n+\u000fy2b\njyj; (2.6)\n@\u0012\u0011@x\n@\u0012=\u0000\u000fy2n+\u000fy1b: (2.7)A PROBLEM OF ULAM ABOUT MAGNETIC FIELDS GENERATED BY WIRES 5\nHere and in what follows, the curvature, torsion and elements of the Frenet basis\nare evaluated at sunless stated otherwise.\nAs the vector \feld @rhas norm\u000f, it follows from the expression (2.2) for the\nvolume in the coordinates ( s;r;\u0012 ) that the surface measure of S\r;\u000fcan be written\nas\ndS=\u000fAdsd\u0012;\nwithA(s;\u0012) := 1\u0000\u000f\u0014(s) cos\u0012. Hence the \feld Jis indeed divergence-free on S\r;\u000f,\nsince the usual formula for the divergence yields\ndivS\r;\u000fJ=1\nA(s;\u0012)@\n@s\u0012A(s;\u0012)F(\u0012)\n1\u0000\u000f\u0014(s) cos\u0012\u0013\n+1\nA(s;\u0012)@\n@\u0012\u0012A(s;\u0012)G(s)\n\u000f(1\u0000\u000f\u0014(s) cos\u0012)\u0013\n= 0:\nLet us \fx a point xin a small neighborhood of the curve \rthat we will describe\nby its coordinates ( s;y)2(R=`Z)\u0002D2, which will remain \fxed in the argument.\nLet us \fx a small number \u000eand write the equation for BJas\nBJ(x) =Z\nS\r;\u000fJ(x0)\u0002(x\u0000x0)\n4\u0019jx\u0000x0j3dx0=I+O(\u000f); (2.8)\nwhere\nI=\u000fZ2\u0019\n0Zs+\u000e\ns\u0000\u000eJ(x0)\u0002(x\u0000x0)\n4\u0019jx\u0000x0j3A(s0;\u00120)ds0d\u00120\nand (s0;\u00120) are the coordinates used to parametrize the point x02S\r;\u000f.\nTo see why this is true, it su\u000eces to observe that Jis uniformly bounded in \u000f\n(here one has to use that the norm of1\n\u000f@\u0012is 1 on the surface) and that the distance\njx\u0000x0jis at least\u000ewhenjs\u0000s0j>\u000e. This readily yields\njBJ(x)\u0000Ij=\f\f\f\f\u000fZ2\u0019\n0Z\njs0\u0000sj>\u000eJ(x0)\u0002(x\u0000x0)\n4\u0019jx\u0000x0j3A(s0;\u00120)ds0d\u00120\f\f\f\f6C\u000ejS\r;\u000fj=O(\u000f):\nAn analogous reasoning can obviously be applied to the derivatives of BJ(x)\u0000I,\nso Equation (2.8) follows.\nWe are interested in the asymptotic behavior of Ifor small\u000f. In order to compute\nit, we will need to expand the quantity x\u0000x0in the variables \u000fands0\u0000s, which\ncan be achieved using Equation (2.1) and the Frenet formulas:\nx\u0000x0=\u0000(s0\u0000s)t+\u000f(y1\u0000cos\u00120)n+\u000f(y2\u0000sin\u00120)b+O(2):\nHereafter we will use the notation O(k) to denote terms in the Taylor expansion\nthat are at least a kthpower in (\u000f;s0\u0000s). For example,\n(s0\u0000s)2j; \u000f(s0\u0000s)j; \u000f1+j\nare allO(2) ifj>1. Likewise, using the formulas (2.5){(2.7) the vector \feld J(x0)\ncan be written as\nJ(x0) =F(\u00120)t+G(s)\u0000\ncos\u00120b\u0000sin\u00120n\u0001\n+O(1);\nwhere we recall that t,nandbare evaluated at s.6 ALBERTO ENCISO AND DANIEL PERALTA-SALAS\nSince the orthonormal basis ft;n;bgis positively oriented, it then follows that\n(2.9)J(x0)\u0002(x\u0000x0) =\u000fG(s)\u0000\n1\u0000y1cos\u00120\u0000y2sin\u00120\u0001\nt\n\u0000h\n(s0\u0000s)G(s) cos\u00120+\u000f(y2\u0000sin\u00120)F(\u00120)i\nn\n\u0000h\n(s0\u0000s)G(s) sin\u00120\u0000\u000f(y1\u0000cos\u00120)F(\u00120)i\nb+O(2)\nand\n(2.10) jx\u0000x0j2= (s0\u0000s)2+\u000f2(1 +\f) +O(3);\nwith\n\f:=\u00002y1cos\u00120\u00002y2sin\u00120+jyj2:\nAs the surface measure is\ndS(x0) =\u000f(1 +O(1))ds0d\u00120;\none can now use the formulas (2.9) and (2.10) to write Ias\nI=\u000f\n4\u0019Z2\u0019\n0Zs+\u000e\ns\u0000\u000eh1t+h2n+h3b\n[(s0\u0000s)2+\u000f2(1 +\f) +O(3)]3=2ds0d\u00120: (2.11)\nHere we have set\nh1:=\u000fG(s)\u0000\n1\u0000y1cos\u00120\u0000y2sin\u00120\u0001\n+O(2);\nh2:=\u0000h\n(s0\u0000s)G(s) cos\u00120+\u000f(y2\u0000sin\u00120)F(\u00120)i\n+O(2);\nh3:=\u0000h\n(s0\u0000s)G(s) sin\u00120\u0000\u000f(y1\u0000cos\u00120)F(\u00120)i\n+O(2):\nLet us begin with the tangent component of I,\nI1:=\u000f\n4\u0019Z2\u0019\n0Zs+\u000e\ns\u0000\u000eh1\n[(s0\u0000s)2+\u000f2(1 +\f) +O(3)]3=2ds0d\u00120:\nUsing again that 1 + \fis positive for small enough y, clearly\n(s0\u0000s)2+\u000f2(1 +\f) +O(3) =\u0010\n(s0\u0000s)2+\u000f2(1 +\f)\u0011\u0000\n1 +O(1)\u0001\n;\nso one can decompose I1as\nI1=\u000f\n4\u0019Z2\u0019\n0Zs+\u000e\ns\u0000\u000e\u000fG(s)\u0000\n1\u0000y1cos\u00120\u0000y2sin\u00120\u0001\n[(s0\u0000s)2+\u000f2(1 +\f)]3=2ds0d\u00120\n+\u000fZ2\u0019\n0Zs+\u000e\ns\u0000\u000eO(2)\n[(s0\u0000s)2+\u000f2(1 +\f)]3=2ds0d\u00120\n=I11+I12: (2.12)\nSince 1 +\f >0 for small y, the second integral can be easily bounded as\njI12j6C\u000fZ2\u0019\n0Zs+\u000e\ns\u0000\u000e(s0\u0000s)2+\u000fjs\u0000s0j+\u000f2\n[(s0\u0000s)2+\u000f2(1 +\f)]3=2ds0d\u00120\n6C\u000fZs+\u000e\ns\u0000\u000e1\n[(s0\u0000s)2+\u000f2]1=2ds0\n6C\u000flog1\n\u000f:A PROBLEM OF ULAM ABOUT MAGNETIC FIELDS GENERATED BY WIRES 7\nTo study the integral I11in (2.12), let us introduce the variable\nt:=s0\u0000s\n\u000f(1 +\f)1\n2;\nin terms of which the integral reads as\nI11=1\n4\u0019Z2\u0019\n0Z\u000e=[\u000f(1+\f)1\n2]\n\u0000\u000e=[\u000f(1+\f)1\n2]G(s) (1\u0000y1cos\u00120\u0000y2sin\u00120)\n(1 +\f) (t2+ 1)3=2dtd\u00120\n=1\n4\u0019Z2\u0019\n0Z1\n\u00001G(s) (1\u0000y1cos\u00120\u0000y2sin\u00120)\n(1 +\f) (t2+ 1)3=2dtd\u00120+O(\u000f2)\n=G(s)\n2\u0019Z2\u0019\n0\u0000\n1\u0000y1cos\u00120\u0000y2sin\u00120\u0001\n(1 + 2y1cos\u00120+ 2y2sin\u00120)d\u00120\n+O(\u000f2+jyj2)\n=G(s) +O(\u000f2+jyj):\nTo pass to the third line we have expanded in yand used that\nZ1\n\u00001dt\n(t2+ 1)3=2= 2:\nNow that we are done with the computation of I1, let us consider next the normal\ncomponent of Iand decompose it as before:\nI2:=\u000f\n4\u0019Z2\u0019\n0Zs+\u000e\ns\u0000\u000eh2\n[(s0\u0000s)2+\u000f2(1 +\f)]3=2ds0d\u00120\n=\u0000\u000f\n4\u0019Z2\u0019\n0Zs+\u000e\ns\u0000\u000e(s0\u0000s)G(s) cos\u00120\n[(s0\u0000s)2+\u000f2(1 +\f)]3=2ds0d\u00120\n\u0000\u000f2\n4\u0019Z2\u0019\n0Zs+\u000e\ns\u0000\u000e(y2\u0000sin\u00120)F(\u00120)\n[(s0\u0000s)2+\u000f2(1 +\f)]3=2ds0d\u00120\n+\u000fZ2\u0019\n0Zs+\u000e\ns\u0000\u000eO(2)\n[(s0\u0000s)2+\u000f2(1 +\f)]3=2ds0d\u00120\n=:I21+I22+I23:\nArguing as in the case of I1, one immediately gets that\njI23j6C\u000flog1\n\u000f;\nand the fact that the integrand is an odd function of s0\u0000simmediately implies\nthat\nI21= 0:\nMoreover, the integral I22can be analyzed just as in the case of I11, yielding\nI22=b1+b2y1\u0000a2y2\n2+O(\u000f+jyj2):\nThe binormal component of I,\nI3:=\u000f\n4\u0019Z2\u0019\n0Zs+\u000e\ns\u0000\u000eh3\n[(s0\u0000s)2+\u000f2(1 +\f) +O(3)]3=2ds0d\u00120;8 ALBERTO ENCISO AND DANIEL PERALTA-SALAS\ncan be computed as in the case of I2, obtaining\nI3=\u0000a1+a2y1+b2y2\n2+O(\u000f+jyj2):\nAs Equations (2.5){(2.7) obviously imply that\nt= (1 +O(\u000f))@s+O(jyj)@y1+O(jyj)@y2;n=1\n\u000f@y1;b=1\n\u000f@y2;\none obtains the desired asymptotic formula for BJ.\nIt is clear that the same method yields formulas for the derivatives of the compo-\nnents ofBJ, which correspond to the derivatives of the terms that we have already\ncomputed (e.g., in the case of \frst order derivatives, @sIkand@yjIk). To illustrate\nthe reasoning, let us consider @sI1. Since the point of coordinates ( s;y) is in the\ninterior of the solid torus bounded by S\r;\u000f, one can safely di\u000berentiate under the\nintegral sign to \fnd:\n@sI1=\u000f\n4\u0019Z2\u0019\n0Zs+\u000e\ns\u0000\u000e@\n@s\u0012\u000fG(s)\u0000\n1\u0000y1cos\u00120\u0000y2sin\u00120\u0001\n[(s0\u0000s)2+\u000f2(1 +\f)]3=2\u0013\nds0d\u00120\n+\u000fZ2\u0019\n0Zs+\u000e\ns\u0000\u000e@\n@s\u0012O(2)\n[(s0\u0000s)2+\u000f2(1 +\f)]3=2\u0013\nds0d\u00120:\nHere we have used that the boundary terms cancel out by parity. Using that for\nallqof orderO(2) one can write\n@sq=@s0q1+q2\nwithqjalso of orderO(2), we can further simplify these integrals as:\n@sI1=\u000f\n4\u0019Z2\u0019\n0Zs+\u000e\ns\u0000\u000e\u000fG0(s)\u0000\n1\u0000y1cos\u00120\u0000y2sin\u00120\u0001\n+O(2)\n[(s0\u0000s)2+\u000f2(1 +\f)]3=2ds0d\u00120\n\u0000\u000fZ2\u0019\n0Zs+\u000e\ns\u0000\u000e@\n@s0\u0012\u000fG(s)\u0000\n1\u0000y1cos\u00120\u0000y2sin\u00120\u0001\n+O(2)\n[(s0\u0000s)2+\u000f2(1 +\f)]3=2\u0013\nds0d\u00120\n=\u000f\n4\u0019Z2\u0019\n0Zs+\u000e\ns\u0000\u000e\u000fG0(s)\u0000\n1\u0000y1cos\u00120\u0000y2sin\u00120\u0001\n+O(2)\n[(s0\u0000s)2+\u000f2(1 +\f)]3=2ds0d\u00120:\nHere we are using a parity argument both to get rid of the boundary terms that\nappear when one integrates by parts and to neglect the terms of O(2) that are odd\nfunctions of s0\u0000s, as they do not contribute to the integral. As the above integral\nis of the same form as I1, the previous reasoning immediately yields\n@sI1=G0(s) +O(\u000flog\u000f+jyj):\nThe derivatives with respect to y, which are in fact easier, can be handled with a\ncompletely analogous argument. \u0003\nRemark 2.2.The norm of the \feld BJis of order 1, and the reason for which the\ncomponents ofBJalong the \felds @yjare of order 1 =\u000fis simply that the norm j@yjj\nis\u000f(recall that they are simply the normal or binormal vector divided by \u000f).A PROBLEM OF ULAM ABOUT MAGNETIC FIELDS GENERATED BY WIRES 9\n3.From surface currents to closed wires\nIn this section we will derive tools that permit us to show that there are con\fg-\nurations of wires that create magnetic \felds which approximate, in a certain sense,\nmagnetic \felds generated by current densities of the form studied in Section 2. For\nsimplicity, throughout this section we will denote by Sa surface of R3di\u000beomorphic\nto a torus.\nAn important ingredient in the proof will be the idea of convergence of measures.\nWe recall that a sequence of vector-valued measures d\u0003nsupported onSconverges\nweakly tod\u0003 if, given any continuous function u:S!R3one has\nlim\nn!1Z\nSu\u0001d\u0003n=Z\nSu\u0001d\u0003:\nIn this direction, an easy but very useful result is the following:\nLemma 3.1. LetKbe a compact subset of R3. Consider a sequence of vector-\nvalued measures d\u0003nwhose supports are contained in a compact set K0and assume\nthat this sequence converges weakly to d\u0003. IfKdoes not intersect K0, then\n(3.1) lim\nn!1\r\r\r\rZ\nK0d\u0003n(x0)\u0002(x\u0000x0)\n4\u0019jx\u0000x0j3\u0000Z\nK0d\u0003(x0)\u0002(x\u0000x0)\n4\u0019jx\u0000x0j3\r\r\r\r\nCm(K)= 0\nfor any integer m.\nProof. Observe that the kernel\n(3.2) ( x;x0)7!x\u0000x0\n4\u0019jx\u0000x0j3\nis continuous in the set ( x;x0)2K\u0002K0. The convergence of the measures d\u0003nto\nd\u0003 and the fact that these measures are supported on K0imply that, uniformly for\nallx2K,\nlim\nn!1Z\nK0d\u0003n(x0)\u0002(x\u0000x0)\n4\u0019jx\u0000x0j3=Z\nK0d\u0003(x0)\u0002(x\u0000x0)\n4\u0019jx\u0000x0j3:\nSince the derivatives of the kernel (3.2) with respect to xare also continuous on K\u0002\nK0, the same argument yields the Ckconvergence (3.1) on the set K.\u0003\nThe next lemma shows how to approximate the magnetic \feld created by a\nsurface distribution JdS through the Biot{Savart integral (cf. Equation (2.8)) by\nthat of a collection of magnetic wires. Concerning the statement of the lemma, it\nis worth noting that we require the tangent vector \feld Jto be divergence-free.\nThis is not an additional condition that we impose because we want to interpret\nthis \feld as a current density, but an actual necessary technical condition for the\nstatement of the lemma to hold true. This is because the divergence of any measure\nof the form _ \rk(t)dtcan be easily seen to be zero, in the sense of distributions, so\nthe fact that there is a collection of measures d\u0003nof the form (3.3) that converge\ntoJdS automatically implies that the tangent \feld Jis divergence-free on S(or,\nequivalent, that the divergence of JdS is zero).\nLemma 3.2. LetJbe a tangent vector \feld on the toroidal surface Swhose diver-\ngence on the surface is zero. Let us assume that Jdoes not vanish and that all its\nintegral curves are periodic. Then there exist a positive constant c0and a sequence10 ALBERTO ENCISO AND DANIEL PERALTA-SALAS\nof \fnite collections of (distinct) periodic integral curves of J,\rk: (R=TkZ)!S ,\nsuch that the vector-valued measure\n(3.3) d\u0003n:=c0\nnnX\nk=1_\rk(t)dt\nconverges weakly to JdS asn!1 . HereTkis the minimal period of the integral\ncurve\rk.\nProof. It is known (cf. e.g. [3, 4.1.14]) that if Jis a non-vanishing divergence-free\n\feld on the torus whose integral curves are all periodic, then:\n(i) There is a closed transverse curve ConSwhich intersects all the integral\ncurves ofJat exactly one point.\n(ii) The period function T:S! (0;1), which maps each point on the surface\nto the minimal period of the integral curve of Jpassing through it, is\nsmooth.\nLet us now de\fne the isochronous \feld associated to J,\n(3.4) eJ:=TJ;\nwhose integral curves are all closed and of period 1. Consider a periodic coordinate\nonCof period 1,\n(3.5) \u0002 : C!T1;\nwithT1:=R=Z. Let us now construct a map \t : S!T2by setting\n(3.6) \t\u00001(\u000b;\u001b) :=\u001e\u001b[\u0002\u00001(\u000b)];\nwhere\u001e\u001bis the \row at time \u001bof the \feldeJ. SinceCintersects each integral curve\nexactly once and all integral curves of eJhave period 1, it is obvious that \t is a\ndi\u000beomorphism. Moreover, it is apparent that one can write the push-forward of eJ\nunder the di\u000beomorphism \t as\n\t\u0003eJ=@\u001b:\nAs the period function takes the same value on each integral curve of J, there is\na smooth function eT:T1!(0;1) such that\n(3.7) T[\t\u00001(\u000b;\u001b)] =eT(\u000b):\nIn particular, Tis a \frst integral of J, so it is trivial that the divergence of eJon the\nsurface is also zero. Hence, it follows that the push-forward e\u001bof the area 2-form\nonStoT2can be written as\n(3.8) e\u001b=B(\u000b)d\u000b^d\u001b;\nwhereB:T1!Ris a positive smooth function. In order to see this, it su\u000eces\nto writee\u001b=Bd\u000b^d\u001band notice that the \feld \t \u0003eJ=@\u001bis divergence-free with\nrespect toe\u001b, which implies that\n0 = (div e\u001b@\u001b)e\u001b=d(i@\u001be\u001b) =\u0000d(Bd\u000b );\nwhich shows that Bis independent of \u001b.A PROBLEM OF ULAM ABOUT MAGNETIC FIELDS GENERATED BY WIRES 11\nLet us consider a sequence of points \u000bk2T1that is uniformly distributed with\nrespect to the probability measure on T1given by\n(3.9) d\u001a:=B(\u000b)\nc0eT(\u000b)d\u000b;\nwhereB(\u000b) is the positive function de\fned in (3.8) and\nc0:=Z\nT1B(\u000b)\neT(\u000b)d\u000b:\nSpeci\fcally, this means that, for any interval IofT1,\n(3.10) lim\nn!1#fk6n:\u000bk2Ig\nn=Z\nId\u001a:\nLete\rk:T1! S be the integral curve of eJwith initial condition e\rk(0) =\n\u0002\u00001(\u000bk). We shall next check that\nd\u0003n:=c0\nnnX\nk=1_e\rk(\u001b)d\u001b\nconverges weakly to JdS. Before doing it, let us observe that this implies the\nlemma, because if \rk(t) denotes the integral curve of Jwith initial condition \rk(0) =\n\u0002\u00001(\u000bk), it is obvious from the invariance of the measure under reparametrization\n(namely, the fact that _e\rk(\u001b)d\u001b= _\rk(t)dt) thatd\u0003nis also given by the for-\nmula (3.3) provided in the statement.\nTo show that d\u0003nconverges weakly to JdS, recall that the fact that the points\nf\u000bkgare uniformly distributed with respect to the probability measure d\u001ais equiv-\nalent to saying that the measure\nd\u001an:=1\nnnX\nk=1\u000e\u000bk\nonT1converges weakly to d\u001aasn!1 . Hence,\nZ\nu\u0001d\u0003n=c0\nnnX\nk=1Z\nT1h\t\u0003u;@\u001bij(\u000bk;\u001b)d\u001b\n=c0Z\nT2h\t\u0003u;@\u001bid\u001and\u001b\nsatis\fes\nlim\nn!1Z\nu\u0001d\u0003n=c0Z\nT2h\t\u0003u;@\u001bid\u001ad\u001b\n=Z\nT2h\t\u0003u;@\u001biB(\u000b)\neT(\u000b)d\u000bd\u001b\n=Z\nSu\u0001JdS;\nwhere we have used that eJ=TJ. \u0003\nIn the proof of Theorem 1.2 we will need the following lemma, which is basically a\nre\fnement of Lemma 3.2 that provides a versatile su\u000ecient condition for a sequence12 ALBERTO ENCISO AND DANIEL PERALTA-SALAS\nof vector-valued measures supported on curves (not necessarily integral curves of\nthe \feldJ) to converge to the current distribution JdS:\nLemma 3.3. LetJbe as in Lemma 3.2 and consider the associated map \u0002 :C!T1\nde\fned in (3.5) . Suppose that \u0000n:R=TnZ!S is a sequence of periodic curves\nwithout self-intersections that satisfy the following properties, with Tnbeing the\nminimal period:\n(i)The curves \u0000nintersectCtransversally and their points of intersection are\nuniformly distributed with respect to the measure d\u001ade\fned in (3.9) , that\nis, for any open subset I\u001aC one has\nlim\nn!1#(\u0000n\\I)\n#(\u0000n\\C)=Z\n\u0002(I)d\u001a:\n(ii)The tangent vectors _\u0000n(\u001b)converge uniformly to the isochronized \feld eJ,\nde\fned in (3.4) :\nlim\nn!1sup\n\u001b2R\f\f_\u0000n(\u001b)\u0000eJ(\u0000n(\u001b))\f\f= 0:\nThen there is a positive constant c0such that the vector-valued measures\nd\u0015n:=c0\n#(\u0000n\\C)_\u0000n(\u001b)d\u001b;\nwhich are supported on \u0000n, converge weakly to JdS asn!1 .\nProof. We will use the notation introduced in the proof of Lemma 3.2 without\nfurther notice. Setting\nNn:= #(\u0000n\\C);\nlet us denote the intersection points of \u0000 nwith the transverse curve Cby\n\u0000n\\C=fpn;kgNn\nk=1;\nwhere we are labeling the points pn;kso that they correspond to consecutive in-\ntersection points. We will denote the intersection times of the curve \u0000 nby\u001bn;k2\n[0;Tn), with\u001bn;1= 0 and\npn;k= \u0000n(\u001bn;k):\nThe point in the unit circle associated to pn;kunder the map \u0002 will be denoted by\n\u000bn;k:= \u0002(pn;k).\nThe uniform convergence of _\u0000n(\u001b) toeJ(\u0000n(\u001b)) and the fact that all integral\ncurves ofeJare closed with period 1 obviously imply that the time between consec-\nutive intersections with Ctends to 1:\n(3.11) lim\nn!1max\n16k6Nnj\u001bn;k+1\u0000\u001bn;k\u00001j= 0:\nThroughout we are identifying \u001bn;Nn+1:=Tn. In particular, this implies that\nTn=Nntends to 1. Letting e\rn;k:T1!S be the integral curve of the isochronized\n\feldeJwith initial condition e\rn;k(0) =pn;k, we then infer that the integral curve e\rn;k\nis close to \u0000 nin the sense that\n(3.12) lim\nn!1max\n16k6Nn\r\r\u0000n(\u0001)\u0000e\rn;k(\u0001\u0000\u001bn;k)\r\r\nC1((\u001bn;k;\u001bn;k+1))= 0:A PROBLEM OF ULAM ABOUT MAGNETIC FIELDS GENERATED BY WIRES 13\nLet us de\fne c0as in the proof of Lemma 3.2. To prove that\nd\u0015n:=c0\nNn_\u0000n(\u001b)d\u001b\nconverges weakly to JdS, let us take an arbitrary smooth function u:S!R3,\nwhich without loss of generality can be thought of as a tangent vector \feld on S.\nNotice that\nEn:= max\n16k6Nn\f\f\f\fZ1\n\u001bn;k+1\u0000\u001bn;ku(e\rn;k(\u001b))\u0001_e\rn;k(\u001b)d\u001b\f\f\f\f\n+ max\n16k6NnZ\u001bn;k+1\n\u001bn;k\f\fu(\u0000n(\u001b))\u0001_\u0000n(\u001b)\u0000u(e\rn;k(\u001b\u0000\u001bn;k))\u0001_e\rn;k(\u001b\u0000\u001bn;k)\f\fd\u001b\ntends to zero as n!1 by (3.11) and (3.12). Setting\nd\u0003n:=c0\nNnNnX\nk=1_e\rn;k(\u001b)d\u001b;\none then has that\n\f\f\f\fZ\nSu\u0001d\u0015n\u0000Z\nSu\u0001d\u0003n\f\f\f\f6c0\nNnNnX\nk=1\f\f\f\fZ1\n\u001bn;k+1\u0000\u001bn;ku(e\rn;k(\u001b))\u0001_e\rn;k(\u001b)d\u001b\f\f\f\f\n+c0\nNnNnX\nk=1Z\u001bn;k+1\n\u001bn;k\f\fu(\u0000n(\u001b))\u0001_\u0000n(\u001b)\u0000u(e\rn;k(\u001b\u0000\u001bn;k))\u0001_e\rn;k(\u001b\u0000\u001bn;k)\f\fd\u001b\n6c0En\nalso tends to zero as n!1 . Here we have used that\nZ\u001bn;k+1\n\u001bn;ku(e\rn;k(\u001b))\u0001_e\rn;k(\u001b)d\u001b=Z\u001bn;k+1\n\u001bn;ku(e\rn;k(\u001b))\u0001_e\rn;k(\u001b)d\u001b\n+Z1\n\u001bn;k+1\u0000\u001bn;ku(e\rn;k(\u001b))\u0001_e\rn;k(\u001b)d\u001b:\nAs the points \u000bn;kare equidistributed with respect to the probability measure d\u001a,\nit follows directly from the proof of Lemma 3.2 that the measures d\u0003nconverge to\nJdS asn!1 . The lemma is then proved. \u0003\nRemark 3.4.Using the coordinates ( \u000b;\u001b) introduced in the proof of Lemma 3.2,\none can visually understand these results as follows. In these coordinates on S\n(which do not generally arise from a di\u000beomorphism of the ambient space R3, as\nthey change the knot type of the integral curves of J), the integral curves of J\nare the vertical circles f\u000b= constantg. The transverse curve Ccorresponds to the\nequatorial circle f\u001b= 0gand the collection of integral curves \rkconstructed in\nLemma 3.2 is simply a collection of vertical circles f\u000b=\u000bkgwith\u000bkdistributed\naccording to certain probability measure. The curves \u0000 nsatisfying the assumptions\nof Lemma 3.3 correspond to nearly vertical periodic curves which close after winding\nonce in the horizontal direction and Nntimes in the vertical direction.14 ALBERTO ENCISO AND DANIEL PERALTA-SALAS\n4.Magnetic lines and wires of arbitrary topology\nIn this section we will prove Theorem 1.1. For the sake of clarity, let us divide\nthe argument in several steps:\nStep 1: Construction of a surface current distribution with a hyperbolic magnetic\nline isotopic to e\r.Let us consider the toroidal surface Se\r;\u000fof core curve e\rand\nsmall width \u000f, and the divergence-free tangent vector \feld JonSe\r;\u000fgiven by\n(4.1) J:=1\n1\u0000\u000f\u0014(s) cos\u0012J0; J 0:= 2 cos 2\u0012@s+1\n\u000f@\u0012:\nNotice that the \feld Jis of the form (2.4), so Lemma 2.1 ensures that the magnetic\n\feldBJgenerated by the surface current distribution JdS is given by\nBJ=\u0002\n1 +q0(\u000f;y)\u0003\n@s\u0000y2+q1(\u000f;y)\n\u000f@y1\u0000y1+q2(\u000f;y)\n\u000f@y2:\nwhere\nq0(\u000f;y) =O(\u000flog\u000f+jyj); q 1(\u000f;y) =O(\u000flog\u000f+jyj2); q 2(\u000f;y) =O(\u000flog\u000f+jyj2):\nLet us consider the vector \feld on the domain Tbounded bySe\r;\u000fgiven by\nX:=\u0000(y2+q1(0;y))@y1\u0000(y1+q2(0;y))@y2;\nwhich vanishes identically on the curve e\r\u0011fy= 0g(of course, for \u000f= 0 we are\ntaking\u000flog\u000fas zero). In terms of the coordinates\n~y1:=y1+y2; ~y2:=y1\u0000y2;\nthe integral curves of the linearization\neX:=\u0000y2@y1\u0000y1@y2\nofXare given by\ns(t) =s0; ~y1(t) = ~y10e\u0000t; ~y2(t) = ~y20et:\nTherefore the invariant set e\r\u0011fy= 0gofXis normally hyperbolic because at\neach point of the circle there is a one-dimensional stable component (corresponding\nto the variable ~ y1with Lyapunov exponent \u00001) where the \row is exponentially\ncontracting and a one-dimensional unstable component (corresponding to ~ y2with\nLyapunov exponent 1) where the \row is exponentially expanding.\nIt is therefore well known that the invariant circle fy= 0gis preserved under\nsmall perturbations of the \feld X. More precisely, let us take any integer k>1,\na compact set K\u001aT enclosing the curve fy= 0gand de\fne the Cknorm of a\nvector \feld,\nkYkCk(K);\nas the sum of the Ck(K) norms of the components of Yin the basisf@s;@y1;@y2g\n(this is important to avoid having to deal with inessential factors of \u000f). One then\nhas [4, Theorem 4.1] that there exists some positive constant \u000e1such that any\n\feldYwith\n(4.2) kX\u0000YkCk(K)<\u000e1A PROBLEM OF ULAM ABOUT MAGNETIC FIELDS GENERATED BY WIRES 15\nhas a one-dimensional invariant set isotopic to the curve fy= 0g, and the distance\nbetween the corresponding isotopy \u0002 and the identity in the Cknorm is of order \u000e1:\nk\u0002\u0000idkCk(K)<C\u000e 1:\nSince\nkX\u0000\u000fBJkCk(K)<C\u000f log1\n\u000f;\nit follows from (4.2) that, for small enough \u000f,\u000fBJ(and thereforeBJ) has a one-\ndimensional invariant set e\r2isotopic to the curve e\r\u0011fy= 0gand contained in\na small neighborhood fjyj<C\u000f log1\n\u000fg. As the \feldBJdoes not vanish in the set\nfjyj< C\u000f log1\n\u000fgfor small enough \u000f,e\r2must be a periodic integral curve of BJ.\nThe normal hyperbolicity of the invariant set fy= 0gofXimplies thate\r2is a\nhyperbolic periodic integral curve of BJ, so in particular it is robust in the sense\nthat there is some \u000e2>0 such that any \feld Zwith\n(4.3) kBJ\u0000ZkCk(K)<\u000e2\nmust have a periodic integral curve isotopic to e\r, and the isotopy can be chosen\nclose to the identity in Ck.\nStep 2: Approximation of the magnetic \feld created by the surface current dis-\ntribution by the sum of the \felds of a \fnite collection of unknotted wires. Let us\nnext analyze the integral curves of J0, which coincide with those of Jup to a\nreparametrization of the curve. These are the solutions to the system of ODEs\n_s= 2 cos 2\u0012; _\u0012=1\n\u000f;\nwith an arbitrary initial condition ( s0;\u00120), that is,\n(4.4) s(t) =s0+\u000fsin\u0010\n2\u00120+2t\n\u000f\u0011\n; \u0012 (t) =\u00120+t\n\u000f; :\nThese curves are all periodic with period 2 \u0019\u000f. Geometrically, for small \u000fthis integral\ncurve is a small deformation of, and isotopic to, the circle fs=s0gcontained in\nthe torusfr= 1g. This curve is obviously isotopic to the unknot.\nLemmas 3.1 and 3.2 then ensure that there is a \fnite collection of periodic\nintegral curvesf\rkgN\nk=1ofJsuch that the sum of the magnetic \felds that they\ncreate,\nY:=NX\nk=1B\rk;\nis close toBJin the setKmodulo multiplication by a positive constant c:=c0=N:\n(4.5) kBJ\u0000cYkCk(K)<\u000e2\n3:\nThis collection of curves is depicted in Figure 1.16 ALBERTO ENCISO AND DANIEL PERALTA-SALAS\nFigure 1. A collection of closed wires (represented as thin blue\nlines) the sum of whose magnetic \felds has a magnetic \feld di\u000beo-\nmorphic to the core knot (in red; in this case, a trefoil).\nStep 3: Replacing the \fnite collection of unknotted wires by a single unknot \u0000\u000e.Let\nus denote by\nd\u0003 :=NX\nk=1_\rk(t)dt\nthe vector-valued measure associated with the above periodic integral curves, so\nthat the \feld Ycan be written as\nY(x) =Zd\u0003(x0)\u0002(x\u0000x0)\n4\u0019jx\u0000x0j3:\nFor concreteness, we will henceforth assume that the curves \rkare parametrized\nas in (4.4), so tranges over (0 ;2\u0019\u000f) in each curve. Notice that the measure d\u0003 is\nindependent of the way the curves are parametrized, provided that the orientation\nis preserved.\nWe will need the following observation. Let I1;:::;INbe intervals of length at\nmost\u000eand let us denote by d\u0003\u000ethe measure obtained from d\u0003 after removing the\nintervalsIkfrom the curves. That is, for any continuous vector-valued function F\nwe setZ\nF\u0001d\u0003\u000e:=NX\nk=1Z\n0<t<2\u0019\u000f;t62IkF(\rk(t))\u0001_\rk(t)dt:\nThend\u0003\u000eobviously converges weakly to d\u0003 as\u000e!0, for any choice of the inter-\nvalsIk.\nAnother useful observation is the following. Let p;qbe points in R3nK. Then,\ngiven any other pair of points ~ p;~qwith\njp\u0000~pj+jq\u0000~qj<\u000e;\nit is clear that one can choose curves \u0000 0;e\u00000: [0;1]!R3nKsuch that\n\u00000(0) =p; \u00000(1) =q;e\u00000(0) = ~p;e\u00000(1) = ~q:A PROBLEM OF ULAM ABOUT MAGNETIC FIELDS GENERATED BY WIRES 17\nFigure 2. Passing from a collection of unknots to a single unknot.\nand the distance between the curves is of order \u000eis the sense that\n(4.6) k\u00000\u0000e\u00000kCk([0;1])<C\u000e:\nAn immediate consequence of (4.6) is that, if we reverse the orientation of e\u00000by\nsettingb\u00000(t) :=e\u00000(1\u0000t), then the vector-valued measure\n_\u00000(t)dt+_b\u00000(t)dt\nconverges weakly to zero as \u000e!0.\nWith these two observations, for each \u000e >0 we will construct a closed oriented\ncurve \u0000 (piecewise smooth, although eventually we will smooth things out) such\nthat the associated measure\n_\u0000(t)dt\nconverges weakly to d\u0003 as\u000e!0. To this end, in each curve \rkwe will \fx two\ndistinct points pk;qk. For each\u000e>0, let us take two points ~ pk;~qksatisfying\njpk\u0000~pkj+jqk\u0000~qkj<\u000e:\nThe second observation above ensures that one can connect the points ( pk;~pk) with\n(qk+1;~qk+1) through curves \u0000 k;b\u0000k: [0;1]!R3that do not intersect one another\nand such that the measure\nNX\nk=1\u0000_\u0000k(t)dt+_b\u0000k(t)dt\u0001\nconverges weakly to zero as \u000e!0. Herekranges from 1 to Nand we identify\nN+ 1 with 1. Let us de\fne the piecewise smooth curve \u0000 as the union of the\ncurves \u0000k,b\u0000kand the integral curves \rk, from which we remove the 2 Narcs of\nthe curves of length of order \u000econnecting the points pkwith ~pkandqkwith ~qk. It\ncan be easily seen that one can choose the orientation of the curves \u0000 k;b\u0000kso that\nthe curve \u0000 has a well de\fned orientation, which coincides with the orientation of\neach\rk(cf. Figure 2).\nThe two observations that we made above then imply that the measure _\u0000(t)dt\nconverges weakly to d\u0003 as\u000e!0, so we infer from Lemma 3.1 that the magnetic18 ALBERTO ENCISO AND DANIEL PERALTA-SALAS\n\feld created by \u0000 is close to Yin the sense that\n(4.7) kB\u0000\u0000YkCk(K)<\u000e2\n3c\nwhenever the constant \u000eis small enough. Furthermore, since \u0000 has been constructed\nas the connected sums of the unknots \r1;:::;\rN, it is standard that it is also an\nunknot.\nStep 4: From the unknot \u0000to\rthrough a connected sum taking place far from the\nmagnetic line. Let us take a large number Rthat will be \fxed later. Translating\nthe curve\rif necessary, we can assume that the distance between \rand the set K\nis at leastR, and that\rdoes not intersect \u0000.\nLet us \fx points P2\u0000,Q2\r. For small \u000e, let us take another couple of points\n~P2\u0000,~Q2\rwith\njP\u0000~Pj+jQ\u0000~Qj<\u000e3\nfor a small enough constant \u000e3. The second observation in Step 3 ensures that there\nare oriented curves \u0000 0;b\u00000connecting the points P;~PwithQ;~Q, respectively, and\nsuch that the measure\n_\u00000(t)dt+_b\u00000(t)dt\nconverges weakly to 0 as \u000e3!0. One can obviously assume that the distance from\nthese curves to the set Kis uniformly bounded away from zero. We can now de\fne\na piecewise smooth curve \u00000as the union of the curves \u0000 0,b\u00000, \u0000 and\r, without\nthe two arcs of length of order \u000e3that connect the points PandQwith ~Pand ~Q,\nin each case. We choose the orientation of \u00000so that it coincides with that of \r\nand \u0000.\nIt follows from the construction that \u00000is isotopic to \r(because it is the con-\nnected sum of \rwith an unknot) and that\n_\u00000(t)dt!_\u0000(t)dt+ _\r(t)dt\nas\u000e3!0. By Lemma 3.1, one then has\nlim\n\u000e3!0kB\u00000\u0000B\u0000\u0000B\rkCk(K)= 0:\nSince the distance between Kand\ris at leastR, it is clear that\nkB\rkCk(K)6C\nR2;\nsoB\u00000converges to B\u0000onKasR!1 and\u000e3!0. For convenience, let us denote\nby \u000000the curve that one obtains by slightly rounding o\u000b the corners of \u00000, which\ncan then be chosen (for large Rand small\u000e3) to satisfy\nkB\u000000\u0000B\u0000kCk(K)<\u000e2\n3c:\nBy Equations (4.5) and (4.7), it then follows that\nkBJ\u0000cB\u000000kCk(K)6kBJ\u0000cYkCk(K)+ckB\u0000\u0000YkCk(K)+ckB\u0000\u0000B\u000000kCk(K)\n<\u000e2;\nso the condition (4.3) ensures that the magnetic \feld B\u000000generated by the wire \u000000\n(which is isotopic to \r) has a periodic magnetic line that is isotopic to (and actually\naCksmall deformation of) e\r. Theorem 1.1 then follows.A PROBLEM OF ULAM ABOUT MAGNETIC FIELDS GENERATED BY WIRES 19\nRemark 4.1.It is worth noticing that, although we have chosen a very concrete\ncurrentJin the proof of Theorem 1.1, the same argument works in much greater\ngenerality. In particular, the argument goes through for any current of the form\nconsidered in Lemma 2.1 provided that the function G(s) does not vanish and the\nFourier coe\u000ecients of F(\u0012) satisfy\na1=b1= 0; a2\n2+b2\n26= 0:\n5.Existence of knotted magnetic lines for generic knotted wires\nIn this section we will prove Theorem 1.2. For concreteness, let us take again\nthe tangent vector \feld Jon the toroidal surface S\r;\u000fgiven by (4.1), where \u000fis a\nsmall constant.\nOur objective is to show that there are periodic curves \u0000 n:R=nZ! S\r;\u000f\nsatisfying the hypotheses of Lemma 3.3 for the \feld Jthat are isotopic to \r. Notice\nthat, as the curve lies on S\r;\u000f, theC0norm of the di\u000berence between the isotopy\nand the identity is of order \u000f, and can therefore be made as small as one wishes.\nFurthermore, we will choose the curve so that the number of intersection points\n#(\u0000n\\C), as de\fned in Lemma 3.3, is precisely n. Lemma 3.3 then implies that\nthere is a constant c0such that the measure\nc0\nn_\u0000n(t)dt\nconverges weakly to JdS, so Lemma 2.1 ensures that for any compact set Kthat\ndoes not cutS\r;\u000fone has\n(5.1) lim\nn!1\r\r\r\rc0\nnB\u0000n\u0000BJ\r\r\r\r\nCk(K)= 0:\nSince we proved in Step 1 of Section 4 that the \feld BJas a hyperbolic periodic\nmagnetic line isotopic to \randCkclose to it, it stems from (5.1) that for large\nenoughnthe magnetic \feld B\u0000nalso has a periodic magnetic line isotopic and\nCkclose to\r. Hence Theorem 1.2 will then follow once we construct the curve \u0000 n.\nThe construction of the curve \u0000 nis simpler in the coordinates ( \u000b;\u001b) introduced\nin (3.6). We will henceforth use the notation developed in the proof of Lemma 3.2\nwithout further mention. Let us consider a sequence of points\n(5.2) f\u000bkg1\nk=1\u001aT1\nthat are uniformly distributed with respect to the probability measure (3.9). We\ncan safely assume that \u000bk6=\u000bk0for allk6=k0. For eachn, let us write\nf\u000bk: 16k6ng=f\u000bn;k: 16k6ng;\nwhere\u000bn;kis a relabelling of the n\frst points \u000bkchosen so that, identifying the\npoints in T1with numbers in [0 ;1), one has\n06\u000bn;1<\u000bn;2<\u0001\u0001\u0001<\u000bn;n:\nSince the probability measure (3.9) is absolutely continuous, the sequence (5.2) is\ndense on T1, so the di\u000berence\n[0;1)3\u0001n;k:=\u000bn;k+1\u0000\u000bn;k mod 1;20 ALBERTO ENCISO AND DANIEL PERALTA-SALAS\nunderstood as a number in [0 ;1) where we are using the convention \u000bn;n+1\u0011\u000bn;1,\nmust tend uniformly to zero in the sense that\n(5.3) lim\nn!1max\n16k6n\u0001n;k= 0:\nTake a smooth function \u001f:R!Rsuch that\n\u001f(t) =(\n0 fort60;\n1 fort>1:\nLet us de\fne smooth curves e\u0000n:R=nZ!T2in terms of the coordinates ( \u000b;\u001b) as\ne\u0000n(t) := (\u000bn(t);\u001bn(t)), where we set\n\u000bn(t) :=\u000bn;1+nX\nk=1\u0001n;k\u001f(t+ 1\u0000k) mod 1 ;\n\u001bn(t) :=t mod 1:\nIt is clear that e\u0000nhas periodnand that in each period the curve winds once along\nthe coordinate \u000bandntimes along the coordinate \u001b. Moreover,\n_e\u0000n(t) =\u0012nX\nk=1\u0001n;k\u001f0(t+ 1\u0000k)\u0013\n@\u000b+@\u001b;\nwhere\u001f0denotes the derivative of \u001f. Since at most one of the functions \u001f0(t+1\u0000k)\ncan be nonzero at any time t2R, it is apparent from (5.3) that\n(5.4) lim\nn!1sup\nt2R\f\f_e\u0000n(t)\u0000@\u001b\f\f= 0:\nFinally, let us now de\fne the curve \u0000 n:R=nZ!S\r;\u000fas\n\u0000n(t) := \t\u00001e\u0000n(t):\nIt is not hard to see that, as e\u0000nwinds once in the \u000b-direction and ntimes in the\n\u001b-direction, the curve \u0000 nis a (1 :n0) cable over the core curve \r, where\nn0:=n\u0000N0\nwithN0a \fxed number. In particular, for any large enough n, \u0000nis isotopic\nto\r. For this we will need to compute the expression of the curves in the Frenet\ncoordinates and to take into account the curve's own twist. The reason is that,\nas changes of coordinates in the torus do not necessarily come from an ambient\ndi\u000beomorphism, one cannot use arbitrary coordinates on the torus to check the\nisotopy type of a curve.\nTo check this, we start by taking the set Cas\nC:=fr=\u000f; \u0012= 0g;\nso the coordinate \u000b:C!T1can be chosen as\n\u000b:=s\n`;A PROBLEM OF ULAM ABOUT MAGNETIC FIELDS GENERATED BY WIRES 21\nwhere we recall that `denotes the length of the curve \r. The equation for the\nintegral curves of J,\n_s=2 cos 2s\n1\u0000\u000f\u0014(s) cos\u0012;\n_\u0012=1\n\u000f(1\u0000\u000f\u0014(s) cos\u0012);\nimplies that in terms of the time variable t0de\fned by the ODE\ndt0\ndt=1\n2\u0019\u000f[1\u0000\u000f\u0014(s(t)) cos\u0012(t)]; t0jt=0= 0;\nthe integral curves are given by\ns=s0+\u000fsin(2\u00120+ 4\u0019t0); \u0012 =\u00120+ 2\u0019t0:\nSince\nt0=t(1 +O(\u000f))\n2\u0019\u000f;\nit stems that the period is\nT= 2\u0019\u000f+O(\u000f2);\nso\neJ=TJ= 2\u0019@\u0012+O(\u000f):\nHence the variable \u001bcan be written in terms of ( s;\u0012) as\n\u001b=\u0012\n2\u0019+O(\u000f):\nIn view of the expression of ( \u000b;\u001b) in terms of ( s;\u0012), it is apparent that the curve\n\u0000nwinds once along the coordinate sandntimes along the coordinate \u0012. The\ncoordinates ( s;\u0012) correspond to the Frenet frame, which is well known [8] to twist\nN0times along the curve \r, where\n\u0006N0=1\n2\u0019Z`\n0\u001c(s)ds+ Writhe(\r)\nis the total torsion of the curve plus its writhe (the sign here depends of the ori-\nentation of the frame). Hence we infer that \u0000 nis a (1 :n0) cable over \r, so it is\nisotopic to \r(see Figure 3).\nBy construction, the intersection of \u0000 nwith the setCis the image under the\ndi\u000beomorphism \u0002\u00001of the pointsf\u000bk: 16k6ng, so asn! 1 they are\ndistributed with respect to the measure (3.9). Moreover, since the push-forward of\nthe \feldeJunder \t is precisely @\u001b, it follows from (5.4) that\nlim\nn!1sup\n\u001b2R\f\f_\u0000n(\u001b)\u0000eJ(\u0000n(\u001b))\f\f= 0:\nHence \u0000nis a sequence of curves that has the properties that we required above,\nso Theorem 1.2 follows.\nRemark 5.1.Although we have taken a concrete example of current \feld Jfor which\nall the computations can be made in a very explicit way, the argument carries over\nverbatim to a much more general class of \felds J. In particular, su\u000ecient conditions\nfor the argument to remain valid are the following:22 ALBERTO ENCISO AND DANIEL PERALTA-SALAS\nFigure 3. A wire (thin blue line) isotopic to the core knot (in\nthis case, a trefoil) and C0close to it that has a magnetic line (in\nred) isotopic to the core knot. The wire and the core curve can be\nchosen arbitrarily close.\n(i) The integral curves of Jare all small deformations of (and isotopic to) the\ncirclesfs= constantg, in the coordinates that we de\fned on the surface.\n(In fact, while the fact that the integral curves are all periodic is key, the\ncondition that they are small deformations of the circles fs= constantg\ncan be relaxed signi\fcantly, as it is only used to control the isotopy type\nof the curve.)\n(ii) The magnetic \feld BJhas a hyperbolic periodic magnetic line isotopic to\nandCkclose to\r. As we saw in Section 4, a su\u000ecient condition for this is\nthat the functions FandGthat appear in the \feld Jsatisfy the conditions\nof Remark 4.1.\nAcknowledgments\nThe authors are supported by the ERC Starting Grants 633152 (A.E.) and 335079\n(D.P.-S.). This work is supported in part by the ICMAT{Severo Ochoa grant SEV-\n2015-0554.\nReferences\n1. J.W. Bruce, P.J. Giblin, Curves and singularities , Cambridge University Press, Cambridge,\n1984.\n2. M.R. Dennis, R.P. King, B. Jack, K. O'Holleran, M.J. Padgett, Isolated optical vortex knots,\nNature Phys. 6 (2010) 118{121.\n3. C. Godbillon, Dynamical systems on surfaces , Springer-Verlag, Berlin, 1983.\n4. M.W. Hirsch, C.C. Pugh, M. Shub, Invariant manifolds , Springer-Verlag, New York, 1977.\n5. S.R. Hudson, E. Startsev, E. Feibush, A new class of magnetic con\fnement device in the shape\nof a knot, Phys. Plasmas 21 (2014) 010705.\n6. W.T.M. Irvine, D. Bouwmeester, Linked and knotted beams of light, Nature Phys. 4 (2008)\n716{720.\n7. R.D. Mauldin (Ed.), The Scottish book , Birkh auser, Boston, 1981.A PROBLEM OF ULAM ABOUT MAGNETIC FIELDS GENERATED BY WIRES 23\n8. W.F. Pohl, The self-linking number of a closed space curve, J. Math. Mech. 17 (1967/1968)\n975{985.\n9. U. Tkalec, M. Ravnik, S. Copar, S. Zumer, I. Musevic, Recon\fgurable knots and links in chiral\nnematic colloids, Science 333 (2011) 62{65.\n10. S.M. Ulam, Problems in modern mathematics , Wiley, New York, 1964.\nInstituto de Ciencias Matem \u0013aticas, Consejo Superior de Investigaciones Cient \u0013\u0010ficas,\n28049 Madrid, Spain\nE-mail address :aenciso@icmat.es, dperalta@icmat.es"    },    {        "title": "1002.3209v1.Isothermal_remanent_magnetization_and_the_spin_dimensionality_of_spin_glasses.pdf",        "content": "arXiv:1002.3209v1  [cond-mat.dis-nn]  17 Feb 2010November 10, 2018 17:17 Philosophical Magazine Letters IRM pap3\nPhilosophical Magazine Letters\nVol. 00, No. 00, Spring 2010, 1–6\nmanuscript\nIsothermal remanent magnetization and the spin dimensiona lity\nof spin glasses\nRoland Mathieua∗, Matthias Hudla, Per Nordblada, Yusuke Tokunagab,\nYoshio Kanekob, Yoshinori Tokurab,c, Hiroko Aruga Katorid, Atsuko Itoe\naDepartment of Engineering Sciences, Uppsala University, B ox 534, SE-751 21 Uppsala,\nSweden;\nbMultiferroics Project, ERATO-JST, Tokyo 113-8656, Japan ;\ncDepartment of Applied Physics, University of Tokyo, Tokyo 1 13-8656, Japan ;\ndMagnetic Materials Laboratory, RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan ;\neAdvanced Meson Science Laboratory, RIKEN, 2-1 Hirosawa, Wa ko, Saitama 351-0198,\nJapan.\n(2010)\nThe isothermal remanent magnetization is used to investiga te dynamical magnetic proper-\nties of spatially three dimensional spin glasses with differ ent spin dimensionality ( Ising,XY,\nHeisenberg ). The isothermal remanent magnetization is recorded vs. te mperature after inter-\nmittent application of a weak magnetic field at a constant tem perature Th. We observe that\nin the case of the Heisenberg spin glasses, the equilibrated spin structure and the direc tion\nof the excess moment are recovered at Th. The isothermal remanent magnetization thus re-\nflects the directional character of the Dzyaloshinsky-Mori ya interaction present in Heisenberg\nsystems.\nKeywords: Spin glasses; spin dimensionality; aging; memory; rejuven ation; magnetization\nmeasurements; thermal history; isothermal remanence.\n1. Introduction\nThe macroscopic response of the spin-glass phase to a magnet ic field change is lin-\near at low enough fields [1]. Experiments on the time and tempe rature dependence\nof the spin-glass magnetization can be used to learn about th e underlying spin\nconfiguration and its spontaneous re-organization towards more favorable states\n(aging) [2]. The response function, R(t,ta) reflects the spin re-organization and\nappears stationary at observation times log( t)<log (ta) and non-stationary when\nlog(t)∼log (ta), where tais the effective age of the spin glass and tthe observation\ntime of the experimental probe. The apparent age of the syste m is governed by the\ntime spent at constant temperature (wait time, tw) or in a temperature dependent\nexperiment by the cooling/heating rate.\nIn this study we have considered three model spin glasses of s patial dimensionality\nthree, but with different spindimensionality 1, 2 and 3: Fe 0.5Mn0.5TiO3(Ising) [3],\nEu0.5Sr1.5MnO4(XY) [4] and Cu(Mn) ( Heisenberg ) [5]. For all systems, we have\nstudiedtheevolution andmemoryofthespinconfigurationth athasbeenimprinted\nby a halt at constant temperature, Th, during cooling. The probe we have used in\n∗Corresponding author. Email: roland.mathieu@angstrom.u u.se\nISSN: 0950-0839 print/ISSN 1362-3036 online\nc/circlecopyrt2010 Taylor & Francis\nDOI: 10.1080/0950083YYxxxxxxxx\nhttp://www.informaworld.comNovember 10, 2018 17:17 Philosophical Magazine Letters IRM pap3\n2 Taylor & Francis and I.T. Consultant\nthis study is the isothermal remanent magnetization (IRM) t hat is attained by in-\ntermittently applying a magnetic field at the halt temperatu re. In the Heisenberg\nsystems, a local and random Dzyaloshinsky-Moriya (DM) inte raction[6] is inher-\nently at play [7]. For comparison, we have also investigated a Au(Fe) system, an-\nother three dimensional Heisenberg spin glass, however, with stronger apparent\nanisotropy [8, 9].\n2. Experimental\nThe probe materials are those listed in the introduction: an IsingFe0.5Mn0.5TiO3\nsingle crystal, Tg≈21 K), an XY(single crystal, Tg≈18 K) and two Heisenberg\n(Cu(Mn) polycrystalline, Tg≈57 K and Au(Fe) polycrystalline, Tg≈24 K ) spin\nglasses, all of spatial dimensionality three [3–5].\nThe experiments were performedin an MPMS SuperconductingQ uantum Interfer-\nence Device (SQUID) magnetometer with the low magnetic field option, allowing\na weak,<0.05 mT, residual field.\nThe temperature dependence of the zero-field-cooled (ZFC), field-cooled (FC) and\nisothermal remanent magnetization (IRM) of the samples was recorded using a\nprobe field of 1 mT. The isothermal remanent magnetization (I RM) was acquired\nbyintermittently applyinga1mTfieldduring t∆H=3000satconstanttemperature\nTh, (Th/Tg≈0.7) after two different wait times tw(3 and 3000 s), and measured\nvs. temperature in two different thermal protocols illustrat ed in Fig. 2. In the first\nprocedure (1), the sample is immediately cooled (10 K/min) t o a lowest tempera-\nture and the IRM is measured on heating (1 K/min). In the secon d procedure (2),\nthe sample is cooled immediately after the IRM has been acqui red and measured\non both cooling (1 K/min) and heating (1 K/min).\nThe magnetic field employed in these experiments (1 mT) is wea k enough to yield\nlinear response; this implies that the global evolution (ag ing) of the spin configu-\nration at constant temperature evolves “undisturbed” by th is weak perturbation\nand that the system recovers a closely similar spin state at Th, after the cooling\nprotocol (memory).\n3. Results\nFigure 1 shows the temperature dependence of the ZFC, FC and I RM magnetiza-\ntion of the four systems in reduced units. The Isingsample is measured along the\ndirection of the spins (the c-axis of the Ilmenite structure) and perpendicular to\nthis axis, in which case the susceptibility is much lower, re versible and only weakly\ntemperaturedependent(c.f. Fig. 1(a)). The XYsampleis measuredin the ab-plane\nof the layered tetragonal structure and perpendicular to th is plane, in which case,\nthe susceptibility is much lower, reversible and only weakl y temperature depen-\ndent (c.f. Fig. 1(b)). In the following we consider and perfo rm measurements on\ntheIsingandXYsystems only in the spin directions, i.e. along the c-axis for the\nIsingsystem, and perpendicular to it for the XYsystem. The two Heisenberg\nsamples have isotropic magnetic properties; the samples ha ve cylindrical shape\nand were measured along the cylinder axis, and perpendicula r to it for compari-\nson. Similar results were obtained in the two cases; only the data collected along\nthe cylinder axes areshown in Fig. 1(c-d). Theoverall behav ior of thesusceptibility\ncurves of the four samples show the characteristic spin-gla ss features: a cusp in the\nZFC curve, irreversibility between the FC and ZFC curves bel ow the cusp, and a\nweakly temperature dependent FC curve below the irreversib ility onset.November 10, 2018 17:17 Philosophical Magazine Letters IRM pap3\nPhilosophical Magazine Letters 3\nFigure 3 illustrates the temperature dependence of the isot hermal remanent mag-\nnetization for the different samples. A superposed “FC” magne tization associated\nwith the weak residual field in the magnet has been subtracted from the data in all\ncases; the magnitude of this is of the order of the IRM, see ins et of Fig. 3(d), where\nthe measured “raw” data of the magnetization of the Cu(Mn) sa mple (IRM and\nbackground) is plotted. The fact that the nominally zero fiel d in the magnetometer\nin reality is a weak finite field does not affect the IRM results of Fig. 3; since the\nprinciple of superposition is valid for the recorded magnet isation responseto a field\nchange in the linear response regime [1].\nThere are common features in the IRM for all four samples. The IRM is frozen in\non cooling and shows only a weak temperature dependence on co oling and heat-\ning at low temperatures, and fades away rapidly as the temper ature is increased\nabove the temperature whereit was acquired. A rather simila r behavior of the IRM\nwould be observed also for a system of non-interacting (or in teracting) magnetic\nnanoparticles with a distribution of particle sizes [10]. A lso in common is a wait\ntime dependence of the magnitude of the IRM; a larger magnitu de is obtained for\n3 s wait time than for 3000 s wait time at Thbefore the 1 mT field is applied for\n3000 s. This difference is explained by the wait time dependenc e of the response\nfunction; the increase of magnetization gets slower with wa it time at constant tem-\nperature in spin glasses (aging) [11]. Also in common, is tha t independently of the\nwait time, the IRM measured on heating after rapid cooling ha s larger amplitude\nthan after slower cooling.\nThere are however significant differences between the behavio r of the IRM for\nthe different samples, possibly emanating from the spin dimen sionality. The wait\ntime atTh(3 s and 3000 s) influences the magnitude of the IRM in the seque nce\nHeisenberg -XY-Ising, going from strongest to weakest influence. This is in\naccord with the observation that the influence of aging on the response function\nfollows the same sequence [2, 4]. The cooling heating curves for theXYandIsing\nsystems coalesce more rapidly below the halt temperature th an the corresponding\ncurves for the Heisenberg systems, irrespective of wait time at Th. The most strik-\ning difference is the pronounced maximum in the IRM when return ing toThin\nCu(Mn) and Au(Fe). This feature does not appear in the other t wo samples, where\nthe IRM smoothly decays with temperature through and above Th. Also, signifi-\ncantly, the heating curve in the protocol (2) where the sampl e is measured both\non cooling and heating approaches closer to the curve measur ed only on heating\n(protocol (1)) in the neighborhood of Th. This behavior of the Heisenberg system\nimplies that there is an additional memory component in the s pin structure that\nis imprinted during the halt at Th[12] (see also [13]).\n4. Discussion and Conclusions\nThe spin structure of a spin glass is continuously re-organi zed at low tempera-\nture to adjust toward a structure that is most favorable at ea ch temperature. This\nre-organization occurs always, and irrespective of whethe r a weak field is applied\nor a weak field is applied or removed. This implies that the mag netization that\nis recorded in an experiment may be composed of several indep endent superposed\ncontributions. The pure IRM (as recorded during procedure ( 1)) mirrors the excess\nmagnetization that is acquired from an intermittent field ap plication at constant\ntemperature. This magnetization is frozen in on cooling to l ower temperatures.\nDuring the time it takes to cool to low temperature and again h eat the sample\nback towards Th, the spin structure is spontaneously reorganized on short t ime-\nlength scales, however, a memory of the structure attained d uring the stop at ThNovember 10, 2018 17:17 Philosophical Magazine Letters IRM pap3\n4 Taylor & Francis and I.T. Consultant\nremains [2, 13]. During this process also parts of the IRM are lost due to relaxation\nand this is reflected in a lower value of the magnetization whe nThis recovered.\nFor theXYandIsingsystems this picture accords with the observations in Fig. 3 ,\nand one needs to make an ordinary memory experiment to realiz e that the spin\nconfiguration at Thdoes carry a memory of the equilibration that occurred durin g\nthe original cooling [2, 4].\nTheHeisenberg systems on the other hand, show an additional directional me m-\nory of the equilibrated spin structure imprinted at Th. Although a part of the\nIRM apparently (Fig. 3(a) and (b)) is lost due to the relaxati on and restructuring\nthat occurs during cooling and heating (the relaxation is la rger for Cu(Mn) than\nAu(Fe)), a large part of the original IRM magnetization is re covered when This\napproached as the sample is reheated. This pronounced incre ase of the IRM is\nsurprising since it is not observed in the other two samples, where the part of the\nmagnetization that has decayed during cooling heating rema ins lost also at Th. In\ntheHeisenberg systems on the other hand both the equilibrated spin structu re\nand the direction of the excess moment are recovered at Th. This property of the\nCu(Mn) and Au(Fe) spin glass possibly reflects a reminiscenc e of the directional\ncharacter of the Dzyaloshinsky-Moriya interaction inhere nt to the Heisenberg sys-\ntems [7].\nInterestingly, it was shown earlier that the critical expon ents associated with the\nXYspin-glass phase transition had values between those of the Heisenberg and\nIsingsystems [4], and that ac or dc memory experiments suggested d ynamical\nproperties closer to those of Isingsystems. It is interesting to note that here the\ndynamical behavior of the XYsystem probed by the IRM experiments is qualita-\ntively similar to that of the Isingsystem.\n5. Conclusion\nWe have investigated the dynamical magnetic properties of s patially three dimen-\nsional spinglasses by performingmeasurements of the isoth ermal remanent magne-\ntization (IRM). Qualitative differences in the IRM curves of t he different systems,\nwhich have different spin dimensionality (1: Ising, 2:XY, or 3:Heisenberg ), were\nobserved. Our results suggest that in Heisenberg systems, the equilibrated spin\nstructure and the direction of the excess moment of the IRM ar e kept in memory,\nreflectingthedirectional character ofthe(randomandloca l) Dzyaloshinsky-Moriya\ninteraction inherently present in Heisenberg systems.\nAcknowledgements\nWe thank the Swedish Research Council (VR) and the G¨ oran Gus tafsson Founda-\ntion for financial support.\nReferences\n[1] L. Lundgren, P. Nordblad, and L. Sandlund, Memory Behaviour of the Spin Glass Relaxation , Eu-\nrophys. Lett. 1 (1986) 529.\n[2] P. E. J¨ onsson, R. Mathieu, P. Nordblad, H. Yoshino, H. Ar uga Katori, and A. Ito, Nonequilibrium\ndynamics of spin glasses: Examination of the ghost domain sc enario, Phys. Rev. B 70 (2004) 174402.\n[3] A. Ito, E. Torikai, S. Morimoto, H. Aruga, M. Kikuchi, Y. S yono, and H. Takei, Behavior of the\nTransverse Spin Component in the Short-Range Ising Spin-Gl ass Fe 0.5Mn0.5TiO3, J. Phys. Soc.\nJpn. 59 (1990) 829.\n[4] R. Mathieu, A. Asamitsu, Y. Kaneko, J. P. He, and Y. Tokura ,Eu0.5Sr1.5MnO4: A three-dimensional\nXY spin glass , Phys. Rev. B 72 (2005) 014436.November 10, 2018 17:17 Philosophical Magazine Letters IRM pap3\nPhilosophical Magazine Letters 5\n[5] S. Nagata, P. H. Keesom, and H. R. Harrison, Low-dc-field susceptibility of CuMn spin glass , Phys.\nRev. B 19 (1979) 1633.\n[6] I. Dzyaloshinsky, A thermodynamic theory of weak ferromagnetism of antiferro magnetics , J. Phys.\nChem. Solids 4 (1958) 241; T. Moriya, Anisotropic Superexchange Interaction and Weak Ferromag-\nnetism, Phys. Rev. 120 (1960) 91.\n[7] A. Fert and P. M. Levy, Role of anisotropic exchange interactions in determining t he properties of\nspin glasses , Phys. Rev. Lett. 44 (1980) 1538.\n[8] D. Petit, L. Fruchter, and I. A. Campbell, Ordering in Heisenberg Spin Glasses , Phys. Rev. Lett. 88\n(2002) 207206.\n[9] E. Vincent, Ageing, Rejuvenation and Memory: The Example of Spin Glasse s, Lect. Notes Phys. 716\n(2007) 7.\n[10] M. Sasaki, P. E. J¨ onsson, H. Takayama, and H. Mamiya, Aging and Memory Effects in Superparam-\nagnets and Superspin Glasses Phys. Rev. B 71 (2005) 104405.\n[11] L.Lundgren, P.Svedlindh,P.Nordblad,and O.Beckman, Dynamics of the Relaxation-Time Spectrum\nin a CuMn Spin-Glass Phys. Rev. Lett. 51 (1983) 911.\n[12] R. Mathieu, M. Hudl, P. Nordblad, Memory and rejuvenation in a spin glass , cond-mat/1002.0612.\n[13] R. Mathieu, P. J¨ onsson, D. N. H. Nam, P. Nordblad, Memory and superposition in a spin glass , Phys.\nRev. B 63 (2001) 92401.November 10, 2018 17:17 Philosophical Magazine Letters IRM pap3\n6 Taylor & Francis and I.T. Consultant\n0.5 0.75 1 1.2500.20.40.60.81M/MFC(Tf )\nT/TfH//abH//c\nIsing\n(a)0.5 0.75 1 1.2500.20.40.60.81M/MFC(Tf )\nT/TfXYH//cH//ab\n(b)\n0.5 0.75 1 1.2500.20.40.60.81M/MFC(Tf )\nT/TfAu(Fe)\n(c)0.5 0.75 1 1.2500.20.40.60.81M/MFC(Tf )\nT/TfCu(Mn)\n(d)\nFigure 1. Temperature Tdependence of the zero-field cooled (ZFC) and field-cooled (F C) magnetization\nMrecorded for all systems using a small magnetic field H= 1 mT. Tis normalized by the freezing\ntemperature Tf≈Tg, whileMis normalized by the field-cooled magnetization value at Tf,MFC(Tf).\ntimeT\nH\n0∆HTminThTref\nTg\ntwt∆H1\n2\n12\n(a)\nFigure 2. Schematic representation of the thermal and magne tic field protocols used to record the\nIsothermal remanent magnetization (IRM) curves: Variatio n of the temperature Tand magnetic field H\nas a function of time according to protocol (1) (continuous l ine) and (2) (dotted line). The arrows\nindicate the time a which the magnetization starts to be reco rded in each case. The initial cooling from a\nreference temperature Trefto the lowest temperature Tminis halted at Thbelow the spin-glass\ntransition temperature Tgfor a wait time tw(3 s or 3000 s). A magnetic field of ∆ H=1 mT is then\napplied for t∆H=3000 s.November 10, 2018 17:17 Philosophical Magazine Letters IRM pap3\nPhilosophical Magazine Letters 7\n0.2 0.4 0.6 0.8 11.200.020.040.060.080.10.12M/MFC(Th)\nT/TfIsing\n(a)0.2 0.4 0.6 0.8 11.200.020.040.060.080.10.12M/MFC(Th)\nT/TfXY\n(b)\n0.2 0.4 0.6 0.8 11.200.020.040.060.080.10.12M/MFC(Th)\nT/TfAu(Fe)\n(c)0.2 0.4 0.6 0.8 11.200.020.040.060.080.10.12M/MFC(Th)\nT/Tf0.20.71.200.10.2M/MFC(Th)\nT/TfCu(Mn)\n(d)\nFigure 3. Temperature dependence of the Isothermal remanen t magnetization (IRM), recorded\nemploying the two procedures schematically illustrated in Fig. 2.Tis normalized by the freezing\ntemperature Tf, whileMis normalized by the value of the field-cooled magnetization in 1mT at the halt\ntemperature Th,MFC(Th), extracted from Fig. 1. Magnetization data collected usin g procedure (1) are\nplotted in continuous lines while the data collected using p rocedure (2) is plotted using filled ( tw=3 s)\nand open ( tw=3000 s) symbols. In all panels, the upper continuous line co rresponds to experiments\nperformed with tw=3 s, while the lower one depicts the results of experiments w ithtw=3000 s. ∆ His\n1mT and t∆His 3000 s in all experiments. In the measurements on Cu(Mn), t he reference temperature\nTrefis 70 K, the halt temperature This 40 K, and the lowest measurement temperature Tminis 10 K.\nFor Au(Fe), Tref=40 K,Th=14 K, and Tmin=5 K, while for the IsingandXYsystems, Tref=30 K,\nTh=12 K, and Tmin=5 K. The inset in panel (d) shows the “raw” magnetization dat a collected following\nprocedure (2), with wait time of 3 s (same filled symbols as in t he main frame), as well as a reference\nbackground measurement, in which the system is cooled and me asured in zero applied magnetic field\n(continuous line)."    },    {        "title": "1502.07807v1.RC_circuit_like_dynamic_characteristic_of_the_magnetic_domain_wall_in_ferromagnetic_nanowires.pdf",        "content": "arXiv:1502.07807v1  [cond-mat.mes-hall]  27 Feb 2015RC-circuit-like dynamic characteristic of the magnetic do main\nwall in ferromagnetic nanowires\nHong-Guang Piao1∗,Dong-Hyun Kim1,2†, Je-Ho Shim2, and Li Qing Pan1\n1College of Science, China Three Gorges University, Yichang 443 002, P. R. China\n2Department of Physics, Chungbuk National University, Cheong ju 361-763, South Korea\nWe have investigated dynamic behaviors of the magnetic doma in wall under\nperpendicular magnetic field pulses in ferromagnetic nanow ires using micromagnetic\nsimulations. It has been found that the perpendicular magne tic field pulse can\ntrigger the magnetic domain wall motion, where all the field t orques are kept to\nbe on the plane of nanowire strip. The magnetic domain wall sp eed faster than\nseveral hundreds meters per second is predicted without the Walker breakdown for\nthe perpendicular magnetic driving field stronger than 200 m T. Interestingly, the\ndynamic behavior of the moving magnetic domain wall driven b y perpendicular\nmagnetic field pulses is explained by charging- and discharg ing-like behaviors of an\nelectrical RC-circuit model, where the charging and the dis charging of ”magnetic\ncharges” on the nanowire planes are considered. The concept of the RC-model-like\ndynamic characteristic of the magnetic domain wall might be promising for\nspintronic functional device applications based on the mag netic domain wall motion.\n∗Corresponding author electric mail is hgpiao@ctgu.edu.cn\n†Corresponding author electric mail is donghyun@chungbuk.ac.kr2\nPrecise control of magnetic domain wall (DW) dynamic behaviors in fe rromagnetic\nnanowires has become one of the important issues for realization of spintronic devices based\non the DW motion[1–5]. In ferromagnetic nanowires, the DW motion ge nerally can be\ndriven by applying an external magnetic field[1, 2] or a spin current[3 –5]. For instance, if an\nexternal field is applied along the wire axis to drive a transverse DW, a field torque from a\ncross product between the external field and the transverse DW magnetization is pointing\nout of the wire plane, generating a demagnetizing field opposite to th e field torque. The\ndemagnetizing field rotates the transverse DW magnetization on th e wire plane, leading to\nthe translational motion of the DW along the wire axis[6]. For a strong er magnetic field,\nwhere the field torque overcomes the demagnetizing field, an inner s pin structure of a simple\ntransverse DWdeforms with a structural transformationinto an antivortex DWstructure[7–\n9]. This phenomenon has been well known as the Walker breakdown an d the threshold field\nis called as the Walker breakdown field[10].\nBy suppressing the Walker breakdown, the DW speed can be made fa ster keeping the\nsimple transverse DW structure, which implies that the DW device ope ration speed can\nbe faster with sustaining a stable transverse DW structure. Nume rous studies have been\ndevoted to exploring a suppression mechanism of the Walker breakd own[11–19]. It has\nbeen reported that the Walker breakdown can be suppressed by u sing an underlayer with a\nstrong perpendicular magnetic anisotropy[11] by applying an assis ting magnetic field along\nthetransverse orperpendicular direction[12–15], bymodulatingt henanowiregeometrywith\na periodic wavy shape[16, 17], or by using a cylindrical nanowire[18, 19 ]. It has been known\nthat the suppression of the Walker breakdown is directly related to the relative strength of\nthe demagnetizing field compared to the external field torque, whe re both of them are along\nthe perpendicular (normal) direction of flat (cylindrical) nanowires.\nIn this work, we consider the DW motion triggered by the perpendicu lar field which\nis acting not as an assisting field additional to the external magnetic field along the wire\naxis but as an only driving field , where the field torque rotating each spin is on the flat\nnanowireplaneallthetime. Wereportourmicromagneticinvestigatio nondynamicbehavior\nof the DW motion under perpendicular field pulses with systematic var iation of the flat\nnanowirewidthandthickness, whereRC-circuit-likebehaviorwithou ttheWalkerbreakdown\nphenomenon is observed.\nWe have carried out micromagnetic simulations[20] based on the Land au-Lifshitz-Gilbert3\n(LLG) equation[21].\ndM\ndt=−|γ|M×Heff+α\nMs(M×dM\ndt), (1)\nwhereγis the gyromagnetic ratio, αis the Gilbert damping coefficient, Mis the magnetiza-\ntion, and Heffis the effective field from the total energy in the micromagnetic simula tion.\nThe straight ferromagnetic nanowire is set to have a length l= 2000 nm. The nanowire\nwidthwis varied from 20 to 100 nm and the wire thickness dis varied from 2 .5 to 7.5 nm.\nThe detailed geometry and dimension of the wires are schematically illus trated in the Fig.\n1(a). The external driving field pulse strength Hextis varied from 10 to 300 mT. The\nmagnetic field pulse width was set to be 1 ns. Both the rise and fall time s were set to be\n0.1 ns. The driving pulse field is applied along the normal direction (+z) p erpendicular to\nthe nanowire plane, as shown in the Fig. 1(b). In all simulations, the m aterial parameters\nof Permalloy are used with an exchange stiffness coefficient of 13 ×10−12J/m, a saturation\nmagnetization Msof 8.6×105A/m, and zero magnetocrystalline anisotropy. The unit cell\ndimension is set to be 2 .5×2.5×2.5 nm3and the Gilbert damping constant α= 0.01. A\nhead-to-head transverse DW is initially prepared at the nanowire ce nter before applying the\nperpendicular magnetic field pulse.\nAs described in Fig. 1, all spins (dark green arrows) are kept to be o n the nanowire plane\n(xy-plane), even after the external field torque is exerted, as deno ted by the dotted black\narrows in the figure. Based on the LLG equation, it is found that the external field torque is\n/vector τH=M×Heff, so that the magnetization direction is determined bydM\ndt=−|γ|M×Hext.\nThe demagnetizing field torque is /vector τd=M×Hdemag, so thatdM\ndt=−|γ|M×Hdemag(see\nFig. 1(c)). Under a perpendicular magnetic field, inner spins of the D W are rotated by\nthe external field torque along counterclockwise direction, effect ively moving the transverse\nDW to the left ( −x) direction along the nanowire, as shown in Fig. 1(b). Due to the\ncounterclockwise rotating torque on all spins lying on the wire axis, a gradual change of\nmagnetization generates magnetic source charges acting like posit ive ’N’ and negative ’S’\n”magnetic charges”[22] at the wire edges, as denoted by magenta (N) and cyan (S) colors at\nthe wire edges in the Fig. 1(b). By the generation of magnetic charg es, demagnetizing fields\nare generated along the −y(+y) direction on the left (right) side of the DW, as denoted in\nthe Fig. 1(c). In the meantime, another torque (dotted black arr ows) is generated along the\n−zdirection bythe demagnetizing fieldfromthemagnetic charges onth ewireedges, leading4\nto the accumulation of magnetic charges on the upper and lower flat planes of the nanowire\nby the magnetization tilting toward the zaxis direction, asdenoted by ’N’ (magenta) and ’S’\n(cyan) poles in Fig. 1(c). Due to the magnetic charge accumulation, another demagnetizing\nfield is generated toward the −zdirection as depicted in the Fig. 1(d). After the magnetic\nfield pulse is applied, the role of the driving field is replaced by the demag netizing field\nfrom the accumulation of magnetic charges on the wire planes, resu lting in the DW motion\ntoward the right (+ x) direction in the nanowire, as shown in Fig. 1(d).\nTo study the DW dynamic behavior driven by the perpendicular field, w e have analyzed\nmagnetizationwithrespect tothetime, asillustrated intheFig. 2(a) , where time-dependent\nmagnetizationprofiles areplottedtogether with atemporal profile of theperpendicular mag-\nnetic field pulse. In the figure, magnetizations are plotted for a field with 50-mT strength,\n1-ns duration time, and 0.1-ns rise/fall time. The nanowire width, w = 50 nm and the thick-\nness, d = 5 nm. It is observed that the y-component magnetization Mydoesnot significantly\nchange under the pulse field driving since Mymainly comes from the magnetization of the\nstructure-stable transverse DW, which is also evidenced by the ne gligible change of exchange\nenergy curve observed in Fig. 2(b). The total energy is mainly domin ated by the energy\ngain in the Zeeman energy when the pulse field is on. While the perpendic ular pulse field is\non, the transverse DW is simply translationally displaced along the wire with keeping the\ninner spin structure stable, as confirmed in the Fig. 2(c). Convers ely, the time-dependent\nMzprofile is significantly changed under the pulse field driving, temporally consistent with\nthe applied field pulse, which means that the demagnetizing field is almos t instantaneously\ngenerated toward −zdirection and kept to be nearly constant for the field pulse duration , as\nalso observed in the demagnetization energy curve in Fig. 2(b). It s hould be noted that the\ndemagnetizing field generated toward −zdirection is originated from the magnetic charges\naccumulation on the nanowire plane, as described in Fig. 1(d).\nMxprofile exhibits a distinctive behavior compared to other magnetizat ion components.\nMoreover, very interestingly, after the pulse field is off, the DW gra dually moves back to the\noriginal position again with keeping the stable spin structure. As vivid ly demonstrated in\nthe Fig. 2(a), careful analysis reveals that the time-dependent D W position profile (open\nsquare), determined by processing the simulated DW images (see Fig . 2(c)), is exactly\nmatchingwiththetime-dependent Mxprofile(redsolidline). IncaseofFig. 2, themaximum\ndisplacement of the DW under the perpendicular field pulse is 146 nmwh ere the stable inner5\nspin structure is kept. . In the case of Fig. 2, the average speed o f the DW motion is about\n134m/sandcouldbefasterthanwhenappliedfield pulsebecomes str onger. Under thepulse\nfield with 200-mT strength and 1-ns duration time, the average DW s peed is observed to be\nabout 433 m/s, where the inner structure of the transverse DW is still stably maintained.\nReminding that the time-dependent DW position curve has the same p rofile shape with the\ntime-dependent Mxcurve, the change of the instantaneous slope from the steep one to the\nmore moderate one, looks like the curve of an exponential function . If so, the DW speed is\nexpected to be gradually decelerated during the pulsed time.\nTo examine the relation between the time-dependent DW position pro file and the driving\nfield pulse, we applied a sequential multiple field pulses perpendicular to the wire plane.\nWhere 5-times field pulses with 1-ns duration time and 100-mT streng th are successively\napplied. It is found that the profile of time-dependent Mxand DW position present a kind\nof continuous behavior, as shown in the Fig. 3(a). These disconnec ted profiles seem to be\nquite coherent and the slope change is continued even after new fie ld pulses are applied. One\ncan think that the DW motion driven by the next field pulse depends on the DW motion\ndriven by the previous field pulse, suggesting that the continuous a ccumulation of certain\nphysical quantities – ”magnetic charge”.\nBasedonaboveconfirmations andtheexponential-curve-like profi leshape, weareallowed\nto assume that the phenomenon is related to a charging/dischargin g effect of the magnetic\ncharge. The assumption is thought to be valid since there exist magn etic charges on the wire\nplanes and the magnetic charges might behave in a similar way to the ele ctrical charges.\nIn order to prove our assumption, we consider the Thiele’s equation based on the LLG\nequation[23]:\n−ˆG×˙X−ˆD·˙X+kX=u(ˆ y×Hext), (2)\nwhere,ˆGis a gyrovector and Xis a DW position. ˆDis a damping tensor (negative value)\nrelated tothe sample geometry, the DWwidth, andtheGilbert dampin g constant[24]. kand\nuare a stiffness coefficient and a constant, respectively, depending on the sample geometry\n(demagnetizing factor)[25, 26]. Hextis an external magnetic field define to be along the\nz-direction. We have found that the DW dynamic behavior is very similar to the one-\ndimensional (1D) version of dynamic magnetic vortex core behavior in the ferromagnetic\nnanodisk without the gyroforce term. Therefore, we try to use t he 1D Thiele’s equation\n−ˆD·˙X+kX=u(ˆ y×Hext) to explain the observed behavior in the present study. According6\nto the 1D Thiele’s equation, the time-dependent DW position function x[t] follows a relation\nasx[t] =C1uHext\nk(1−e−kt/|D|), where C1is a constant. It should be noted that the resultant\nfunction form is very similar to the charging function form A0[1−exp(−t/τ)] in the RC-\ncircuit model[27]. The role of |D|/kseems to be corresponding to an RC characteristic time\nτand theuHext\nkcorresponds to A0.\nIn addition, we have also found that the exponential behavior of th e DW position does\nnot significantly depend on the rise/fall time, as shown in the Fig. 3(b ), where two Mx\nprofiles with different rise time (0 and 0.2 ns) are compared. Both pro files are almost same\nand almost exactly matching if shifted by the difference of rise time. T he change of field\nstrengths does not distort the exponential behavior, either. Th us, it is concluded that the\nexponential behavior of the DW position is not originated from the fie ld parameters such\nas rise/fall time and field strength. It is also found that the expone ntial behavior of the\nDW position substantially depends on the Gilbert damping constant, a s demonstrated in\nthe Fig. 3(c), since the RC-characteristic-time-like |D|/kterm is a function depending on\nthe Gilbert damping constant[24]. Every aspect of the observed ex ponential behavior seems\nto let us firmly believe that the RC-like DW dynamic behavior is well explain ed by charging\nand discharging of magnetic charges on the nanowire.\nAccording to the RC-circuit model, the nanowire itself can be conside red as a ”magnetic\ncapacitance”, where the capacitance Cmis proportional to the area of the wire planes and\ninversely-proportional to the wire thickness. To check our scena rio, firstly, we have analyzed\nthe fully charged state corresponding to the state with the maximu m DW displacement. We\nsimply assume that the present ferromagnetic nanowire system is a nalogous to the electrical\nRC-circuit system, where an RC characteristic time constant τand the Mx/Msprofile\nare fitted with an RC-circuit-like fitting function of A0[1−exp(−t/τ)] for charging and\nA0exp(−t/τ) for discharging. A0represents the saturated state and thus, corresponding to\nthe state with the maximum DW displacement. The fitting is found to be extremely well\nmatching for both the DW position and Mx/Ms, as demonstrated in Fig. 3(d), where three\nfitting curves in the cases of different field pulse strengths of 20, 5 0, and 100 mT. Note that\nthe profiles for the case of static (not pulsed) applied field become c ontinuously exponential\nin the figure. From the fitting, A0values are determined to be 0.06, 0.16, and 0.33 for cases\nof 20, 50, and 100-mT perpendicular field respectively, which indicat es that the magnetic\nchargeatthefullychargedstate( Qm)isproportionaltotheappliedfieldstrength( H)witha7\ncoefficient of Cm=Qm\nH≈3µB˚A−1T−1[28]. Moreover, we have found that the characteristic\ntimes for different field strengths are τ≈0.79 (20 mT), 0.80 (50 mT), and 0.83 ns (100 mT)\nbecome almost same regardless of the field strength. Considering τmainly depends on the\nwire geometry, dimension, and material parameters in the magnetic RC-circuit system, the\nnegligible change of τwith different field strengths is well explained.\nIn Fig. 4(a)-(c), the time-dependent DW position profiles are plott ed with various widths\nand thicknesses of nanowires under the magnetic field pulse of 50-m T strength and 0.1-ns\nrise time. The widths are varied to be 20, 50, and 100 nm and the thick ness is changed to\nbe 2.5, 5.0, and 7.5 nm. In case of the same wire width, the DW maximum d isplacement\nbecomes smaller for thicker wires. Quantitative analysis reveals tha t the wire thickness is\ninverse-proportional to the magnetic capacitance, which is explain able by the fact that the\nthicker wire has the lager separation for a plate capacitor. In a simila r way, in case of\nthe same wire thickness, the DW maximum displacement becomes lager for wider wires,\nwhere the wire width is confirmed to be proportional to the magnetic capacitance. The\nobserved behavior is again explainable by the fact that the wider wire has the wider plane\narea for a plate capacitor. Therefore, the relation between the m agnetic capacitance and\nthe wire dimension could be simply deduced as Cm∝(w×l)/d. Ifd,l, andware fixed, the\ncapacitance will be a constant, as Cm≈3µB˚A−1T−1for the case of the Fig. 3(d).\nConsidering the relationship between the capacitance Cmand the wire dimension ( dand\nw), we have further analyzed the maximum displacement of the DW mot ion under static\nperpendicular fields with variation of the field strength from 10 to 20 0 mT. Reminding that\nthe displacement of the DWpositionis directly related to thegenerat ion ofmagnetic charges\non the wire plane,the magnetic static field is considered to act like an en ergy potential as\nin the RC-circuit model. Interestingly, we have observed that the s lope of the maximum\ndisplacement profile with respect to the static field is constant rega rdless of field strengths\nfor nanowires with various dimensions, as plotted in the Fig. 4(d). Th is also matches with\nthe RC characteristic time is mainly determined by R and C, not by the e lectrical potential.\nIn magnetic analogy, the characteristic time τis also constant for each wire regardless of the\nfield strengths (acting like the potential). To further confirm the R C-circuit characteristic of\ntheDWmotionunderperpendicularpulsefields, theresistance Riscalculatedbytherelation\nfunction τ=RCfor each wire dimension. We have found that the Rvalue also becomes\nnearly constant for each wire, not depending on the external field strength, as shown in Fig.8\n4(e), whereas, under driving field stronger than 100 mT, the Rvalue gradually increases,\nin particular, fora relatively narrower and thinner wire ( w= 20 nm and d= 2.5 nm). This\ndeviation might be explained considering that a stronger perpendicu lar field modifies the\nDW width, thereby changing the RC-characteristic-time-like |D|/kterm which depends on\nthe DW width. However, our simple RC-circuit-like model is found to be valid within a\npractical range of the field strengths not stronger than 100 mT. change.\npar\nIn conclusion, RC-circuit-like DW dynamics is discovered in ferromagn etic flat nanowires\nunder a perpendicular field pulse. With a systematic variation of the w ire width and thick-\nness as well as the field pulse strength, the RC-circuit-like dynamic D W behavior is stably\nobserved without occurence of the Walker breakdown. ”Magnetic charges” accumulated on\nthe wire plane seem to well explain the observed RC-circuit-like DW dyn amic behavior,\nwhere the displaced DW back and forth are exactly corresponding t o the charging and dis-\ncharging in the RC-circuit. The observed RC-circuit-like behavior is w ell explained as well\nbased on a simple 1D Thiele’s equation based on the LLG equation. Our s tudy opens a new\npossibility to control the fast DW motion avoiding the Walker breakdo wn in ferromagnetic\nnanowires, where the DW dynamics is exactly following the well known R C-circuit-like be-\nhavior, which is expected to be promising in designing spintronic logic de vices based on the\nDW motion along ferromagnetic nanowires.\nAcknowledgment This work was supported by the National Science Foundation of\nChina (Grant Nos. 11474183 and 51371105) and the Korea Resear ch Foundation (NRF)\n(Grant Nos. 2010-0021735). This research was also supported in part by the Leading\nForeign Research Institute Recruitment Program [Grant No 2010- 00471].\nAuthor Contributions H.G.P. designed research blueprints, did the simulation and\nwrotethemanuscript. D.H.K.contributedtoprojectdesign, theo reticalanalysis, manuscript\nwriting and whole project supervision. J.H.S. and L.Q.P. contributed t o the phenomenon\nanalysis. All authors have commented on the manuscript.\nAuthor Information The authors declare no competing financial interests. Correspon -\ndence and requests for materials should be addressed to D.H.K (don ghyun@chungbuk.ac.kr)\nor H.G.P. (hgpiao@ctgu.edu.cn).9\n[1] Allwood, D. A. et al. Magnetic domain-wall logic. Science309, 1688-1692 (2005).\n[2] Piao H.-G. et al. Ratchet effect of the domain wall by asymmetric magnetostati c potentials.\nAppl. Phys. Lett. 99, 192512 (2011).\n[3] Yamanouchi, M., Chiba, D., Matsukura, F. & Ohno, H. Curre nt-induced domain-wall switch-\ning in a ferromagnetic semiconductor structure. Nature428, 539-542 (2004).\n[4] Parkin, S. S. P., Hayashi, M. & Thomas, L. Magnetic domain -wall racetrack memory. Science\n320, 190-194 (2008).\n[5] Hayashi, M. et al. Current-controlled magnetic domain-wall nanowire shift register. Science\n320, 209-211 (2008).\n[6] De Leeuw, F. H., Van Den Doel, R. & Enz, U. Dynamic properti es of magnetic domain walls\nand magnetic bubbles. Rep. Prog. Phys. 43, 689-783 (1980).\n[7] Nakatani, Y., Thiaville, A. & Miltat, J. Faster magnetic walls in rough wires. Nat. Mater. 2,\n521-523 (2003).\n[8] Hayashi, M. et al. Direct observation of the coherent precession of magnetic domain walls\npropagating along permalloy nanowires. Nat. Phys. 3, 21-25 (2007).\n[9] Kim, S.-K. et al. Underlying mechanism of domain-wall motions in soft magne tic thin-film\nnanostripes beyond the velocity-breakdown regime. Appl. Phys. Lett. 93, 052503 (2008).\n[10] Schryer, N. L. & Walker, L. R. The motion of 180◦domain walls in uniform dc magnetic fields.\nJ. Appl. Phys. 45, 5406-5421 (1974).\n[11] Lee, J.-Y., Lee, K. -S. & Kim, S.-K. Remarkable enhancem ent of domain-wall velocity in\nmagnetic nanostripes. Appl. Phys. Lett. 91, 122513 (2007).\n[12] Bryan, M. T., Schrefl, T., Atkinson, D. & Allwood, D. A. Ma gnetic domain wall propagation\nin nanowires under transverse magnetic fields. J. Appl. Phys. 103, 073906 (2008).\n[13] Glathe, S., Berkov, I., Mikolajick, T. & Mattheis, R. Ex perimental study of domain wall\nmotion in long nanostrips under the influence of a transverse field.Appl. Phys. Lett. 93,\n162505 (2008).\n[14] Kunz, A. & Reiff, S. C. Fast domain wall motion in nanostrip es with out-of-plane fields. Appl.\nPhys. Lett. 93, 082503 (2008).\n[15] Seo, S.-M., Lee, K.-J., Jung, S.-W. & Lee, H.-W. Control lable chirality switching of a moving10\ndomain wall by oblique magnetic field. Appl. Phys. Lett. 97, 032507 (2010).\n[16] Piao, H.-G. et al. Domain wall propagation in wavy ferromagnetic nanowire. IEEE Trans.\nMagn.45, 3926-3929 (2009).\n[17] Burn, D. M. & Atkinson, D. Suppression of Walker breakdo wn in magnetic domain wall\npropagation through structural control of spin wave emissi on.Appl. Phys. Lett. 102, 242414\n(2013).\n[18] Yan, M., K´ akay, A., Gliga, S. & Hertel, R. Beating the wa lker limit with massless domain\nwalls in cylindrical nanowires. Phys. Rev. Lett. 104, 057201 (2010).\n[19] Piao, H.-G., Shim, J.-H., Djuhana, D. & Kim, D.-H. Intri nsic pinning behavior and prop-\nagation onset of three-dimensional Bloch-point domain wal l in a cylindrical ferromagnetic\nnanowire. Appl. Phys. Lett. 102, 112405 (2013).\n[20] Donahue, M. & Porter, D. G. OOMMF Users guide, Version 1.0, Interagency Report NISTIR\n6376(NIST, Gaithersburg, 1999) at http://math.nist.gov/oomm f/(Date of access: 2/7/2010\nfor Version 1.2a4).\n[21] Gilbert, T. L. A phenomenological theory of dampingin f erromagnetic materials. IEEE Trans.\nMagn.40, 3443-3449 (2004).\n[22] Hubert,A. & Sch¨ afer, R. Magnetic domains: the analysis of magnetic microstructure s,\n(Springer, Heidelberg, 2009).\n[23] Shim, J.-H. et al. Nonlinear motion of magnetic vortex under alternating-cu rrent magnetic\nfield: Dynamic correction of a gyrovector and a damping tenso r of the Thiele’s equation. Apll.\nPhys. Lett. 99, 142505 (2011).\n[24] Shim, J.-H., Piao, H.-G. & Kim, D.-H. Calculation of the gyrovector and the damping tensor\nof a magnetic vortex core based on the Thiele equation. New Physics: Sae Mulli (Korean\njournal of The Korean Physical Society) 61, 601 (2011)\n[25] Guslienko, K. Yu., Novosad, V., Otani, Y., Shima, H. & Fu kamichi, K. Field evolution of\nmagnetic vortex state in ferromagnetic disks. Apll. Phys. Lett. 78, 3848 (2001).\n[26] Guslienko, K. Yu. Low-frequency vortex dynamic suscep tibility and relaxation in mesoscopic\nferromagnetic dots. Apll. Phys. Lett. 89, 022510 (2006);\n[27] Knight, R. D. Physics for scientists and engineers , 2nd Edition (Pearson Addison-Wesley, San\nFrancisco, 2008).\n[28] Bramwell, S. T. et al. Measurement of the charge and current of magnetic monopole s in spin11\nice.Nature461, 956-959 (2009).12\nFigure 1. (Color online) (a) Schematics of a head-to-head transve rse DW spin structure\nand a ferromagnetic flat nanowire dimension under zero field. (b) Sc hematics of the mag-\nnetic charge (N and S) accumulation at the wire edges by the extern al field torque (dotted\nblack arrows) denoted as the magenta (N) and cyan (S) colors und er the perpendicular\npulse field ( Hext). (c) Schematics of the magnetic charge (N and S) accumulation at\nthe wire planes by the torque of the demagnetizing field (hollow blue ar rows) along the\ntransverse direction of the wire.(d) Schematics of the demagnetiz ing field generation along\nthe -z-direction after the field pulse is off. The rotating directions ( magnetization direction)\nof all spins are denoted as green arrows.\nFigure 2. (Color online) (a) Time-dependent magnetization and DW po sition profiles.\nHext= 50 mT field pulse profile is on the bottom. (b) Profiles of Zeeman, dem agnetiza-\ntion, exchange, and total magnetic energies. (c) DW positions at t= 1, 2, and 5 ns in\nthenanowireof w= 50nm, d= 5nm. Colorcodeontherightrepresents the Mycomponent.\nFigure 3. (Color online) (a) Time-dependent Mx/Msand DW position profiles (up)\nunder multiple sequential perpendicular field pulses with Hext= 100 mT (bottom). (b)\nMx/Msprofiles with rise times of 0 and 0.2 ns. The field pulse profiles are denot ed as\ndotted line. (c) Mx/Msprofiles with different damping parameters of 0.01, 0.05, and 0.1.\n(d) DW positions under pulse fields (solid) and static fields (open) with 20, 50, and 100-mT\nfield strengths. Fitting curves (solid line) for time-dependent Mx/Msare plotted together.\nFigure 4. (Color online) DW positions with variation of wire width and thic kness. The\nthickness is varied to be 2.5, 5.0, 7.5 nm for (a) the width of 20 nm (inse t is a zoomed-\nin figure corresponding to the dotted box region), (b) 50 nm, and ( c) 100 nm. (d) The\nmaximum DW displacement with respect to the static field strength an d (e) the resistance\nRprofile with variation of width and thickness, fixing the field pulse stre ngth in each case.\nThe fitting curves are denoted as solid lines.13\nFIG. 114\nFIG. 215\nFIG. 316\nFIG. 4"    },    {        "title": "2306.06835v1.Modeling_Magnetically_Channeled_Winds_in_3D__I__Isothermal_Simulations_of_a_Magnetic_O_Supergiant.pdf",        "content": "Modeling Magnetically Channeled Winds in 3D: I. Isothermal Simulations of a\nMagnetic O Supergiant\nSethupathy Subramanian,1Dinshaw S. Balsara,1Asif ud-Doula,2and Marc Gagn ´e3\n1University of Notre Dame, 225 Nieuwland Science Hall, Notre Dame, IN 46556, USA\n2Penn State Scranton, 120 Ridge View Drive, Dunmore, PA 18512, USA\n3West Chester University, Merion Science Center 201, West Chester, PA 19383, USA\nABSTRACT\nIn this paper we present the first set of 3D magnetohydrodynamic (MHD) simulations per-\nformed with the Riemann Geomesh code. We study the dynamics of the magnetically chan-\nneled winds of magnetic massive stars in full three dimensions using a code that is uniquely\nsuited to spherical problems. Specifically, we perform isothermal simulations of a smooth\nwind on a rotating star with a tilted, initially dipolar field. We compare the mass-loss, an-\ngular momentum loss, and magnetospheric dynamics of a template star (with the properties\nthat are reminiscent of the O4 supergiant ζPup) over a range of rotation rates, magnetic\nfield strengths, and magnetic tilt angles. The simulations are run up to a quasi-steady state\nand the results are observed to be consistent with the existing literature, showing the episodic\ncentrifugal breakout events of the mass outflow, confined by the magnetic field loops that\nform the closed magnetosphere of the star. The catalogued results provide perspective on\nhow angular-momentum loss varies for different configurations of rotation rate, magnetic field\nstrength and large magnetic tilt angles. In agreement with previous 2D MHD studies, we\nfind that high magnetic confinement reduces the overall mass-loss rate, and higher rotation\nincreases the mass-loss rate. This and future studies will be used to estimate the angular-\nmomentum evolution, spin-down time, and mass-loss evolution of magnetic massive stars as\na function of magnetic field strength, rotation rate, and dipole tilt.\nKeywords: Massive Stars – Winds, Outflows – MHD – Simulations\n1.INTRODUCTION\nAlthough fewer than 1% of stars end their lives in core-collapse supernovae, massive stars have\nhad a profound influence on the chemical, ionization and star-formation history of the universe.\nThe first very massive stars formed from the pristine H/He gas of the early universe, prior to the\nformation of the first galaxies. How massive stars evolve over cosmic time depends on understanding\nthe role that binarity, mass transfer, mergers, rotation, metallicity, mass loss, and magnetic fields\nplay on the pre-supernova evolution of individual systems (see Langer 2012, for a review).\nThe last two decades have seen the discovery of an exciting new class of stars: magnetic\nmassive stars. Strong, organized magnetic fields on O and early-B stars strongly influence angular\nmomentum evolution, and impact their late-stage evolution prior to core-collapse. The occurrence\nrate of magnetic fields in O- and early-B stars is 7 −10% (Wade et al. 2016; Grunhut et al.\n2017; Sch¨ oller et al. 2017). That said, the origin of the magnetic fields is not known. Among the\nmassive stars with detectable magnetic fields, their magnetic geometries appear to be stable on\ntimescales of many decades. Single magnetic massive stars rotate more slowly than non-magnetic\nstars, suggesting long-term magnetic braking (Wade et al. 2016).\nIn the presence of a strong, large-scale magnetic field the stellar wind is channeled toward the\nmagnetic equator, the magnetosphere is closed at relatively small radii. The wind pulls open the\nfield at high magnetic latitude, and at large radii. For weak fields, the field is unconfined and blown\nCorresponding author: Sethupathy Subramanian\nssubrama@nd.eduarXiv:2306.06835v1  [astro-ph.SR]  12 Jun 20232\nopen by the wind. ud-Doula & Owocki (2002a) showed that the effect of the field on the stellar\nwind depends largely on just a single wind magnetic confinement parameter ,η⋆, which characterizes\nthe competition between field and wind outflow energy density.\nη⋆≡B2\neqR2\n⋆\n˙MB=0v∞, (1)\nwhere the equatorial field strength Beq=1\n2Bpis half the polar field strength, R⋆is the stellar\nradius, and ˙MB=0andv∞are the mass-loss rate and terminal wind speed in the absence of a field.\nIn an extensive series of papers, 2D MHD simulations were used to understand the dynamics of\nmagnetic wind channeling and the consequent hard X-ray emission seen with Chandra (ud-Doula\n& Owocki 2002a; Owocki & ud-Doula 2004; Gagn´ e et al. 2005; ud-Doula et al. 2006, 2008, 2009,\n2013, 2014).\nSpecifically, ud-Doula et al. (2008) showed how magnetic channeling depends on rotation and\nmagnetic confinement. We hasten to point out that all this early modeling work was done within\nthe context of aligned rotators, where the rotation axis and the dipole axis coincide. Indeed, that is\nthe only case for which models can be built. Gagn´ e et al. (2005) incorporated an energy equation\nto model the X-rays from θ1Orionis C, the central star of the Orion Nebula. These early 2D models\nwere able to explain many of the observational properties of the few magnetic O stars that were\nknown at the time. More recently, Petit et al. (2013) compiled an exhaustive list of 64 confirmed\nmagnetic OB stars with Teff>16 kK, along with their physical, rotational and magnetic properties.\nMany of the observational properties of magnetic massive stars depend on the ratio of the\nAlfv´ en and the Kepler corotation radii (Petit et al. 2013). The Alfv´ en radius RA, which, at the\nmagnetic equator, is the approximate boundary between the closed magnetosphere and the open\nfield, can be expressed in terms of the magnetic confinement parameter (ud-Doula et al. 2008):\nRA\nR⋆≈0.3 + (η⋆+ 0.25)1\n4 (2)\nThe torque of the magnetic field on the wind outflow maintains rigid body rotation within the\nmagnetosphere. The rotation parameter Wis the ratio of the rotational and orbital velocities at\nthe photosphere. For near-critical rotation, the star is oblate, but for W≤0.5 we use a spherical\napproximation (ud-Doula et al. 2008):\nW≡Vrot\nVorb=ΩR⋆q\nGM⋆\nR⋆, (3)\nwhere Ω is the rotational angular frequency and W= Ω/Ωcritis the critical rotation ratio. The out-\nward centrifugal acceleration from rigid-body rotation will exactly balance the inward gravitational\nacceleration at the Kepler corotation radius.\nRK=\u0012GM\nΩ2\u00131\n3\n=W−2\n3R⋆ (4)\nMany magnetic O stars are slow rotators with approximately kG fields such that the Alfv´ en\nradius is smaller than the Kepler radius (Petit et al. 2013; Wade et al. 2016; ud-Doula & Naz´ e\n2016; Grunhut et al. 2017). In these so-called dynamical magnetospheres (DM), wind material fed\ninto the magnetosphere falls back onto the photosphere within a dynamical time scale. But for the\nmore rapidly rotating B stars, and, notably, the O7.5 III secondary in Plaskett’s star, RK< R A;\nwind mass fed into the magnetosphere builds up at the magnetic equator between the Kepler and\nAlfv´ en radii. These are the so-called centrifugal magnetospheres (CM) (Petit et al. 2013). Some\nmechanism must allow for mass and density to leak out of these centrifugal magnetospheres: slowly\nvia diffusion, or sporadically, via centrifugal breakout events, wherein magnetic tension can no3\nFigure 1. An oblique magnetic rotator. The photosphere is orange, the vertical rotation axis is shown in black, with\nthe angular velocity ⃗Ω along the z−axis. The magnetic dipole axis ⃗ mBis shown in blue with the initially dipolar\nfield lines. The magnetic tilt angle is ζ.\nlonger contain the built-up mass (Shultz et al. 2020). Owocki et al. (2020) show that the variable\nHαprofiles of centrifugal magnetosphere stars strongly favor centrifugal breakout events.\nAlthough 2D simulations have been used to successfully model aligned rotators, magnetically\nchannelled stellar winds are intrinsically 3D. The magnetic dipole axis is nearly always tilted to the\nrotation axis, with typical obliquity angles between 30◦and 90◦(Wade et al. 2016). A schematic\ndiagram of an oblique magnetic rotator is shown in Figure 1. Daley-Yates et al. (2019) use 3D\nisothermal MHD simulations performed with the PLUTO code to predict the radio emission from\nthe winds of oblique magnetic rotators. Their simulations show the formation of a two-armed spiral\nstructure.\nIn this work, our aim is to extend this modeling effort using the Riemann Geomesh MHD code\n(Balsara et al. 2019; Florinski et al. 2020). Rather than using a conventional spherical ( r, θ, ϕ ) grid,\nRiemann Geomesh employs a recursive partitioning of the spherical icosahedron based triangular\nmeshing of the sphere, and a non-linear radial grid to map the volume of the wind, typically out\nto 10−20R⋆. The resultant mesh maps the surface of the sphere as uniformly as possible, which\ntraditional Cartesian based meshing is unable to accomplish, especially near the rotational poles.\nTheRiemann Geomesh code is therefore uniquely suited to simulate oblique magnetic rotators\nin a bias free fashion.\nThe classification of centrifugal magnetosphere (CM) and dynamical magnetosphere (DM) is\nbased on an aligned magnetic rotator, as displayed in Figures 2a and 2c. This system of classifying\nmagnetospheres is especially relevant to aligned magnetic rotators but including a tilt angle compli-\ncates this simple classification scheme, as can be seen in Figures 2b and 2d. Figure 2b shows a tilted\n“dynamical” magnetosphere. Figures 2b and 2d are just conjectured sketches intended to sensitize\nthe reader that the inclusion of three dimensionality introduces a third important parameter into\nour discussion; i.e. the tilt between the rotation axis and the dipole axis. A major aim of this work\nis to examine the dynamics of these magnetospheres in 3D to understand the role of magnetic tilt.\nIn this paper, we present and analyze 3D simulations of magnetically channeled winds around\noblique magnetic rotators. The first goal of this paper is to show that the geodesic mesh MHD\ncode can perform accurate simulations with large tilt angles and rotation rates. The second goal\nof this paper is to examine the dependence of the quasi-steady state mass-loss rate with rotation,\nmagnetic confinement, and magnetic tilt angle, and to test the prediction that the overall mass-loss\nrate should decrease with the magnetic field, and increase with the rotation rate (ud-Doula et al.\n2009) . The third goal of this paper is to use the 3D simulations to look for and study the episodic\nbreakout events which are predicted to occur in centrifugal magnetospheres (ud-Doula et al. 2006;4\n(a) (b) (c) (d)\nFigure 2. Schematic magnetosphere diagrams. (a) An aligned dynamical magnetosphere (DM): the Alfv´ en radius RA\nis inside the Kepler corotation radius RK. Mass inside the magnetosphere (orange) will fall back onto the photosphere.\n(b) A sketch of a tilted DM with RA< R K. (c) An aligned centrifugal magnetosphere (CM): the Kepler corotation\nradius RKis inside the Alfv´ en radius RA(d) A conjectured sketch of a tilted magnetosphere (that would have been\na CM if it were aligned). The tilt causes RAcosζ < R Keven if we initially had RK< R A. We caution that (b) and\n(d) are just conjectures at this point in the narrative. Figures (a) and (c) are adopted from Petit et al. (2013).\nShultz et al. 2020). The corresponding angular momentum flux is also shown as a function of θ, ϕ\nat the outer most boundary of the simulation. The fourth goal is to estimate spindown as the star\nloses mass and angular momentum over its lifetime. This is achieved with the help of mass flux and\nangular momentum flux obtained at the outer boundary of the simulation. ud-Doula et al. (2008)\nanalyze angular momentum evolution and spindown for aligned magnetic rotators in 2D.\nThe outline of this paper is as follows: in section 2 we describe the geodesic mesh along with\nthe value it provides for this work. In section 3 we describe the model, the boundary conditions,\nand the numerical simulations. In section 4 we present the quasi-steady state mass-loss rate for\ndifferent magnetic tilt angles and rotation rates. In section 5 we show the episodic centrifugal\nbreakout events in 3D, as well as the mass fallback close to the star. And in section 6 we plot\nangular momentum flux as a function of θ, ϕ, and show regions of high angular momentum flux,\nand estimate characteristic stellar spindown time. Lastly, in section 7 we present the summary and\nconclusion of this work.\n2.MESH AND METHODS\nThe overarching problem consists of simulating the magnetically channeled, line driven winds\naround rotating massive stars. Due to the spherical aspect of the problem, it is best to carry out\nthe simulation on a mesh that is uniquely suited for this problem.\nOne of the innovative aspects of this paper is that we have carried out the simulations using\na geodesic mesh, and imposed a globally divergence-free formulation of the MHD equations. The\nother major advance in the simulation of line driven winds is that the magnetic field is split into two\ncomponents: one the curl free, background magnetic field B0and the other being the time evolving\ncomponent B1. This would be reflective of the fact that the primary source of the magnetic field\ni.e., the star itself, lies outside the simulation domain. The variations in the magnetic field due to\nthe material outflow is modeled with the help of the evolving field B1. The simulations are carried\nout in the rotating frame of reference of the star. Consequently, the simulations include the forces\nthat arise due to rotation, which are centrifugal and coriolis forces. The complete description of\nthe MHD equations employed in this manuscript is provided in Appendix A. In Subsection 2.1 we\ndescribe the structure of the geodesic mesh. In Subsection 2.2 we explain why it is particularly\nuseful for this problem.\n2.1. Structure of the Geodesic Mesh\nThe simulation setup is comprised of a spherical domain around the star, discretised with\nuniform triangular patches using a spherical icosahedron as the base. The uniformity of such a\ngeodesic mesh results in high accuracy and consistent time-steps, when compared to other meshing5\n(a) (b) (c) (d)\nFigure 3. Icosahedron based spherical mesh: (a) a regular icosahedron Ruen & Bulatov (2012) (b) a spherical\nicosahedron, (c) first subdivision of the spherical icosahedron, and (d) fourth subdivision of the spherical icosahedron.\nstrategies Balsara et al. (2019); Florinski et al. (2020). In the following subsections, because of the\nnovelty of the approach, the implementation of the meshing is described. Then the value of such a\nuniform mesh in a spherical domain is explained, in context with the simulation of a magnetized\nstar.\nHigh accuracy MHD computations of spherical objects require a high quality meshing of the\nspherical domain. The traditional logically Cartesian based meshing of the sphere has two severe\nshortcomings: i) smaller zones at the poles lead to a shorter time-steps and ii) the geometric\nsingularity causes a loss of accuracy at the poles. As a result, it is preferred to discretise the\nsphere by the structures emerging from the spherical icosahedron. Then recursive bisections of the\ngeodesic curves can be used to obtain the required level of angular refinement shown in Figure 3.\nThe icosahedron shown in Figure 3a is used to obtain a spherical icosahedron shown in Fig-\nure 3b. The first subdivision of the spherical icosahedron is illustrated in Figure 3c, continuing this\nsubdivision three more times, we get finer discretisation of the icosahedron-based spherical mesh,\nshown in Figure 3d. The colored patches in Figure 3d suggest that such a mesh structure leads\nnaturally to large-scale numerical parallelisation. We exploit the symmetry and the divisibility of\nthe geodesic mesh to perform these simulations efficiently on thousands of compute cores.\nThe discretisation shown in Figure 3b is referred to as level-0 sectorial division. It consists\nof 20 great triangles that subtend an angle of π/2−tan−1(1/2)rad ≈63.4◦from the center of\nthe sphere. The first subdivision shown in Figure 3c is referred to as level-1 discretisation, with\n80 great triangles, each subtending an average angle of 33 .9◦. Each colored triangle in Figure 3d\ntherefore represents a level-1 sector. In the same way, the level-4 discretisation depicted in Figure 3d\nsubtends an angle of ≈4.33◦. While it is difficult to display the level-5 subdivision, it yields an\nangular resolution of 2 .16◦. At this angular resolution, the simulations in this study cover the\nspherical surface with 20 ,480 spherical triangles.\nDue to the fast changing nature of the CAK velocity and density profiles, it is essential to\nhave a very thin radial discretisation close to the star: ∆ r= 1.0×10−3R⋆or slightly larger. This\nway, the thin radial zones near the photosphere initially expand geometrically, and then expand\nlogarithmically out to 15 stellar radii. The 3D mesh is formed by extruding the 2D mesh with\n240 radial zones. The mesh volume is therefore divided into 240 ×20,480≈5.0×106zones. The\ngeodesic coordinates 1 and 2 in Figure 4 are shown in red and blue. With the radial coordinate (in\nblack) they define access to each zone on the geodesic mesh.\n2.2. Value of Geomesh\nOur geodesic-based mapping of the sphere does not have any preferred orientation, it is highly\nregular and uniform in the angular direction. While a logically rectangular mesh in ( r, θ, ϕ ) coor-\ndinates has singularities at the poles. The drawbacks of coordinate singularities include, but are\nnot limited to, i) difficulty in doing constrained transport at the poles and ii) reduction in time-\nstep and accuracy of the solution at the poles, Balsara et al. (2019). It has also been reported in6\nFigure 4. Discretisation along the radial direction and numbering strategy for the triangulation.\nDaley-Yates et al. (2019) that there can be fictitious heating of gas at the poles. The mesh shown\nin Figure 3 is free of such singularities. Since such a mapping is uniform, a general case of magnetic\nrotation can be described by choosing a fixed axis for the rotation, and only varying the orientation\nof the magnetic field. Also, the Geomesh code employs state-of-the-art Multidimensional Riemann\nsolvers and a WENO-AO scheme, which are specifically developed for unstructured meshes (Balsara\n& Dumbser 2015; Balsara et al. 2020). The divergence free evolution of magnetic field ∇ ·B1= 0\nis enforced with the help of Yee-type collocation of facially averaged magnetic fields. This along\nwith the edge-integrated electric fields, achieve a high-order accurate numerical implementation of\nFaraday’s law; which ensures a divergence free evolution of magnetic field Balsara et al. (2019).\nPlease see a recent review by Balsara (2017) for a detailed explanation of these computational\ntechniques, as they apply to astrophysical MHD simulations.\nIn this work, the rotation axis of the star is chosen to be z-axis and the orientation of the dipole\nmagnetic field is setup with respect to the z-axis. The simulations are carried out for different tilt\nangles ( ζ), such as 0, 45, 75 degrees of angular separation between the z-axis and the magnetic\ndipole axis. The next section provides the explanation for modeling of the massive star wind and\nthe boundary conditions used.\n3.MODELING AND BOUNDARY CONDITIONS\nThe schematic of the simulation setup is shown in Figure 1. The radius of the stellar surface R⋆\nis taken as the inner boundary and the outer boundary Rois chosen as 15 R⋆. The dipolar magnetic\nfield is maintained throughout the simulation domain by means of the background magnetic field\nB0. The magnetic flux density is parametrized by η⋆(see Eq. 1).\n3.1. Initial Conditions\nOur template star is the O4 I(n)fp supergiant ζPuppis. Specifically, we use the spherical\nparameters derived by Howarth & van Leeuwen (2019) based on the Hipparcos distance d= 332 ±\n11 pc: L⋆= 4.47×105L⊙, Teff= 40 000 K , R= 13.5R⋆, M= 25M⊙, and ˙M≈1.5×10−6M⊙yr−1.\nΓe= 0.471 is the Eddington parameter for ζPup. In the case of spherically symmetric mass loss,\nthe CAK mass loss rate is given by, (Owocki 2004),\n˙MCAK=L⋆\nc2α\n(1−α)(1 + α)1/α\u0012¯QΓe\n1−Γe\u0013(1−α)/α\n(5)\nwhere, αis the CAK exponent (Castor et al. 1975), L⋆is the stellar bolometric luminosity, ¯Qis\nthe quality of the resonant absorption and cis the speed of the light. We initialize our model with7\na spherically symmetric wind with standard β-law. The radial velocity along the radial- rdirection\nis given by,\nv(r) =v∞\u0012\n1−R⋆\nr\u0013β\n(6)\nwhere, v∞is the terminal wind speed, and βis the velocity exponent. The terminal velocity speed\nis assumed to be 3 times the escape velocity, v∞= 3vescand the β= 0.8 (Friend & Abbott 1986;\nOwocki 2004). The density is initialized using the mass-loss rate and the velocity,\nρ(r) =˙MCAK\n4πr2v(r)(7)\n3.2. Boundary conditions\nOwing to the rapidly changing velocity and density profiles of the CAK wind, it was essential\nto do a resolution study with different ∆ rvalues close to star. For this, three different values of\n∆r= 1.0×10−3R⋆, 3.0×10−3R⋆and 5 .0×10−3R⋆were chosen. The resulting density and velocity\nprofiles from these simulations matched each other well, producing the expected mass loss rate ˙M.\nHence, as mentioned earlier the discretisation close to the star has ∆ r≈1.0×10−3R⋆.\nAt the inner boundary ghost zones, following ud-Doula & Owocki (2002b), we set the radial\nvelocity by constant-slope extrapolation and fix the density at a value ρcchosen to ensure subsonic\nbase outflow for the characteristic mass flux of a one-dimensional, nonmagnetic CAK model, i.e.,\nρc=˙M/4πR2\n⋆a/15. The evolving magnetic field component B1(see Appendix A) is initialized to\nzero at the inner boundary. At the outer boundary, far from the star, standard outflow conditions\nare maintained.\nIn the case of line driven winds, the acceleration just above the photosphere is high. Con-\nsequently, the density declines rapidly, while maintaining a quasi-steady mass-loss rate Owocki\n(2004). In order to numerically account for the steep density gradient, the zones near the inner\nboundary are very thin in the radial direction, increasing geometrically out to 1 .5R⋆, and increasing\nlogarithmically at larger radii. A heuristic factor kρis applied at the inner radial boundary, such\nthatρ∗=kρρsp, where ρspis the estimated sonic-point density.\nThe value of kρis taken to be 15 and it can be seen in Table 1 of section 4 that the simulated\nmass-loss rate matches within 5% of the theoretical mass-loss rate, which is 1 .43×10−6M⊙/yr. In\nthe next subsection CAK acceleration source terms are described.\n3.3. Source terms\nThe basic formalism for the acceleration of line driven winds was developed by Castor, Abbott\nand Klein in 1975 (CAK) (Castor et al. 1975). The one-dimensional CAK line acceleration along\nthe radial direction ris given by,\ngCAK=fd\n1−ακeL⋆¯Q\n4πr2c\u0012dv/dr\nρc¯Qκe\u0013α\n(8)\nwhere, fdis a finite disk correction factor, which accounts for the solid angle of the photosphere.\nThe remaining terms represent the CAK line force from a luminous point source. κeis the opacity\ndue to free electrons, and vis the velocity. There are two additional implicit acceleration terms:\ni) the radial gravitational acceleration ggravand ii) accelerations induced by the rotating reference\nframe of the simulation, namely, the Coriolis and centrifugal accelerations grot. Therefore, the total\nacceleration is:\ngtot=gCAKˆ r+ggravˆ r+grot (9)\nThe expressions for the gravitational and rotational accelerations are provided in Appendix A,\nalong with the complete set of MHD equations. The simulations are performed using these initial\nand boundary conditions and the source terms for a range of magnetic confinement parameters η⋆,\nmagnetic tilt angles ζ, and critical rotation ratios W. The next section describes the simulations,\nand the transition from initial conditions to quasi-steady state behavior.8\nTable 1. Quasi-steady-state mass-loss rates as a function of magnetic confinement η⋆, magnetic tilt angle ζ, and\ncritical rotation ratio W.RAandRKare the corresponding Alfv´ en and Kepler radii in stellar radius R⋆. CM indicates\na centrifugal magnetosphere, for which RA> R K. DM indicates a dynamical magnetosphere, for which RA≤RK.\n˙Mis the time-averaged mass-loss rate out of the outer boundary in solar masses per year.\nη⋆ζ W R KRACM/DM ˙M\n(R⋆) ( M⊙/yr)\n0 0◦0∞ - - 1.49 ×10−6\n0 0◦0.5 1.59 - - 1.70 ×10−6\n10 0◦0∞ 2.1 DM 6.92 ×10−7\n10 0◦0.5 1.59 2.1 CM 8.37 ×10−7\n10 45◦0.5 1.59 2.1 CM 7.96 ×10−7\n10 75◦0.5 1.59 2.1 CM 5.88 ×10−7\n50 0◦0∞ 2.9 DM 4.35 ×10−7\n50 0◦0.5 1.59 2.9 CM 4.58 ×10−7\n50 45◦0.5 1.59 2.9 CM 5.84 ×10−7\n50 75◦0.5 1.59 2.9 CM 4.41 ×10−7\n4.TRANSITION TO QUASI-STEADY STATE\nThe 3D Riemann Geomesh simulations described here span a range of magnetic tilt angles\nζ= 0,45◦,75◦, two rotation rates W= 0,0.5, and three magnetic confinement parameters, η⋆=\n0,10,50.The ten cases are listed in Table 1, with the corresponding Kepler and Alfv´ en radii, and\nthe corresponding aligned-magnetosphere classification: CM for centrifugal magnetospheres, and\nDM for dynamical magnetospheres. The outgoing mass-loss rates in the last column are discussed\nbelow.\nWe first note that although we do not include our simulations at W= 0.25, the behavior of the\nDMs can be seen in the W= 0 simulations. Second, we did not consider W > 0.5 because at very\nhigh rotation the photosphere becomes oblate (ud-Doula et al. 2008), and the spherical mesh is no\nlonger appropriate. Third, we note that higher values of η⋆lead to significantly longer compute\ntimes. High magnetic confinement simulations will be the subject of a future paper.\nFigure 5 shows the integrated mass loss from the outer boundary of the simulation as a function\nof time. The figures show the mass-loss rate reaching a quasi-steady value after 100 −120 ksec. The\nmass-loss spikes seen in all simulations are the initial mass blow out, which occurs as the initially\ndense inner CAK wind exits the outer boundary.\nIn the first simulation with no magnetic field and no rotation, a steady state is reached quickly,\nand, reassuringly, the density, velocity and mass-loss profiles follow those predicted by CAK theory.\nThese can be seen, respectively in Figures 6a, 6b and 6c as solid black lines (simulation) and black\nopen circles (CAK theory). In the second simulation, rapid rotation is accounted for by means\nof centrifugal and Coriolis forces, as described in Appendix A. These forces act perpendicular to\nthezrotation axis and tend to increase density and mass-loss rate, but decrease radial velocity,\napproaching the rotational equator in the xy-plane. Averaged over θ, ϕ, this can be seen as the\nyellow solid radial profiles in Figs. 6a-c. The ∼15% increase in the mass loss rate, averaged over\nθ, ϕat the outer boundary, can be seen in the last column of Table 1, and is similar to the 2D\nsimulation results obtained by ud-Doula et al. (2008). In the absence of a magnetic field, the\nincrease in ˙Mis accompanied by an average decrease inv∞. This can be understood as a decrease\nin the escape velocity vescwith increased rotation.\nFor the magnetic simulations with η⋆= 10,50,W= 0,0.5 and ζ= 0◦,45◦,75◦(see Table 1)\nthe initially dipolar field is stretched open near the magnetic poles, and remains closed near the\nmagnetic equator, out to approximately the Alfv´ en radius, channeling wind material into the closed\nmagnetosphere, thereby reducing the mass-loss rate at the outer boundary. The reduction in ˙M\nis more pronounced for higher magnetic confinement, consistent with prior 2D results (ud-Doula9\n0 100 200 300 40000.511.52\n0 100 200 300 40000.511.52\n(a) (b)\n0 100 200 300 40000.511.52\n0 100 200 300 40000.511.52\n(c) (d)\nFigure 5. Mass-loss rate at the outer boundary as a function of time showing the approach to quasi-steady state.\nShown in red (lower curve) η⋆= 10, and in blue (upper curve) η⋆50 for: (a) ζ= 0◦,W= 0, (b) ζ= 0◦,W= 0.5, (c)\nζ= 45◦,W= 0.5, and (d) ζ= 75◦,W= 0.5.Quasi-steady state is achieved typically after ≈110 ks.\net al. 2008). Table 1 also shows that, in all but one case ( η⋆= 50, ζ= 75◦), higher rotation leads\nto higher ˙M.\nFrom Table 1 for η⋆= 10, we see that as the tilt increases, the mass-loss rate is reduced. This\nbehaviour suggests a dot-product cos ζvariation involving the magnetic dipole moment mBand\nthe angular velocity Ω. That is, the moment arm of the mass outflow is reduced as tilt increases,\nthereby reducing the net mass-loss rate. The same is not true, however, at higher confinement,\nwhere the maximum ˙Mis achieved for ζ= 45◦.\nIn fact, the phenomenon depends upon the geometry of the tilt in addition to the competition\nbetween magnetic confinement and centrifugal forces. Therefore, a larger parameter study will be\nrequired to yield meaningful predictions. The complexity of the situation can be seen for η⋆= 50,\nW= 0.5 and ζ= 45◦, where the mass-loss rate is higher than both the ζ= 0◦andζ= 75◦cases.\nHere we resort to the simulations to explain the situation. At high field η⋆= 50, with no tilt\nζ= 0, and rapid rotation W= 0.5, much of the outflow close to the rotational equator is trapped\nin the magnetosphere. Because the centrifugal acceleration is greatest in the equatorial XY plane,\n˙Mis not significantly increased by rapid rotation. This is evident in rows 7 and 8 of Table 1:\nthe mass-loss rates at W= 0 and W= 0.5 are comparable. Examining Figure 9, with W= 0.5,10\n5 10-17-16-15-14-13-12-11\n5 10-5000500100015002000250030003500\n(a) (b)\n5 1000.511.522.533.5\n5 1000.511.522.5\n(c) (d)\nFigure 6. Variation of (a) log density, (b) radial velocity and (c) mass-loss rate as a function of radial distance R.\n(d) mass-loss rate for the aligned cases in comparison with empirical formula derived in ud-Doula et al. (2008) The\nquantities are averaged over θ, ϕdirections, for six sets of magnetic confinement η⋆, magnetic tilt angle ζand critical\nrotation fraction W. The non-magnetic CAK mass-loss rate is shown with open circles.\nη⋆= 50, and ζ= 45◦, most of the outflow near the rotational equator is along open field lines,\nthereby maximizing ˙Mand ˙J, the total rate of angular momentum loss. Figure 9 will be discussed\nin more detail in section 5. The total angular momentum loss is listed in Table 2 and discussed in\nsection 6 below.\nThe Figure 6 shows the variation in log density, velocity and mass-loss rate as a function of radius\nat the end of each simulation. These are averaged over the θ, ϕto illustrate their radial variation.\nOutside the Alfv´ en radii, the mass-loss rate remains constant, while the density decreases rapidly\nand the velocity increases, as expected from CAK. In general, the presence of magnetic confinement\nη⋆>1 increases the overall radial velocity, while the density profile undergoes a reduction due to\nmagnetic confinement. In the same way, in the presence of rotation, the density profile and the\nmass-loss rate increase, as the wind is flung out by rotational forces, consistent with prior 2D\nsimulations (ud-Doula et al. 2008).\nFor the aligned rotator case, there is some well-developed theory for the mass loss case even\nwhen we have rotation and magnetic fields. We compare our simulations for aligned rotators with\nthat theory in this paragraph. In figure 6d, the mass-loss rate for aligned magnetic field cases are\npresented. The mass-loss rate is normalized with respect to the non-magnetic and no-rotation case11\n(˙MB=0). The green and purple circles in the plot are obtained from the analytic formula provided\nin equations 23 and 24 of ud-Doula et al. (2008) and repeated here as:\n˙MB\n˙MB=0≈1−p\n1−R∗/Rc\n˙MB\n˙MB=0≈1−p\n1−R∗/Rc+ 1−p\n1−0.5R∗/RK\nwhere, Rcis the confinement radius of the magnetic closed loops. From our simulation results the\nconfinement radius is observed to be of the form, Rc≈R∗+ 0.6(RA−R∗). We see from figure\n6d that the agreement between our simulations and the theory is quite good. In the quasi-steady\nstate, the simulations show the magnetic confinement of the wind and the episodic breakout events\nat the magnetic equator. This is detailed in section 5 below.\n5.THE DYNAMICS OF MAGNETIC CHANNELING IN THE WIND\nThe mass outflow from the surface of the star is free to flow at the magnetic poles and it is\nconfined near the equatorial region by the closed loops of the magnetic dipole field. Due to this,\nthe wind coming from the either side of the equatorial region, guided by the magnetic field lines,\nmeet and cause a field lines breaking at the magnetic equator. This overall behaviour of closed\nmagnetic loops near the surface of the star and open field lines as we move farther can be seen in\nfigures 7.\nFigures 7a-f shows the density colorized with the magnetic field lines at different times of the\nsimulation, for the cases of magnetic field strength η⋆= 10 ,50 and no rotation. These figures\nshow the expected dynamics of a magnetically channeled line driven wind. We see that each closed\nloop has two foot-points that connect to the star at similar moment arms. Owing to the radially\noutward line driving force, both foot-points have similar amounts of matter that is driven out from\nthe surface of the star. Because Figs. 7a, b, c correspond to a smaller magnetic field than Figs\n7d, e, f, we can clearly see that the former set of figures produce a smaller magnetosphere than the\nlatter set of figures.\nThe matter driven out from the either side of the star meet at the equatorial region, forming a\ndense knot. The knot is at a much higher density than the rest of the wind, and hence experiences\na greater gravitational attraction towards the surface of the star. However, the closed field lines\njust below the knot prevents the direct fallback of matter on to the star and hence the path of the\ndensity knot fallback is dictated by the magnetic field lines. This can be seen in figures 7c, d. These\nobservations are inline with the 2D simulations of ud-Doula & Owocki (2002b). In addition, in this\n3D simulation, it can be seen that the density knots at the either end of the magnetic equator\nfallback from the opposite sides of the field lines, thereby conserving the momentum of the star\n(figures 7c-f). Also, the overall outflow is guided by the magnetic field lines from the either sides of\nthe pole, in an alternate fashion and hence the subsequent density fallbacks come from the either\nside of the magnetic equator. This is also inline with the observations made in ud-Doula & Owocki\n(2002b).\nThe next set of figures 8a-f, represent the simulations carried out with the high rotation and\nan inclined magnetic dipole. The magnetic field has the strength of η⋆= 10 and the rotation rate\nisW= 0.5, the rotation axis is the z−axis and the magnetic dipole makes a ζ= 45◦inclination\nwith the rotation axis. The dynamics are different and interesting in Figure 8, when comparing\nto that of Figure 7. Due to the tilt and high rotation, the streams of gas that come off from the\ntwo foot-points of a closed magnetic loop experiences unequal amount of centrifugal force. The\nmagnetic foot-point that is closer to the rotational equator has more centrifugal force and is flung\nout faster. The other magnetic foot-point that is closer to the pole, does not have centrifugal assist\nwhen being flung outwards. Therefore the streamer that comes off from the foot-point that is closer\nto the rotational-equator dominates and hence that one dominant streamer is doing the flailing.12\n(a) (b)\n(c) (d)\n(e) (f)\nFigure 7. Density (g cm−3) colorized with the magnetic field lines, at different times (ksec) of the simulation, for\nthe case of η⋆= 10 (a,c,e) and η⋆= 50 (b,d,f), no rotation, showing the magnetic confinement and episodic outflow.\nThexandzaxes values are in the scale of stellar radius R⋆.13\n(a) (b)\n(c) (d)\n(e) (f)\nFigure 8. Density (g cm−3) colorized with the magnetic field lines, at different times (ksec) of the simulation, for the\ncase of 45◦tilted magnetic dipole with η⋆= 10 and with the rotation of W= 0.5, showing the magnetic confinement\nand the centrifugal pull due to the rotation. The xandzaxes values are in the scale of stellar radius R⋆.14\nThis is in contrast to the case of aligned rotation, where, there are two streamers at either sides of\nthe magnetic equator that go back and forth.\nSimilar lines of observations can be made for the figures 9a-f, that corresponds to η⋆= 50,\nW= 0.5,ζ= 45◦case. In table 1, this case has RA= 2.9 and RK= 1.59 and it has been\nclassified as a centrifugal magnetosphere (CM) based on the traditional explanations that have\nbeen developed so far. Upon observing the images at different times in Figure 9a-f, it can be\nseen that the magnetosphere successively filled and with clumps falling down on the star. This\nsituation is similar to that of Figure 8a-f. Here, one would therefore be inclined to conclude that\nthe simulation in Figure 9a-f is a dynamical magnetosphere (DM) even though it was classified\nas a CM. But a little reflection allows us to understand that RAshould be modified by cosζ so\nthat RAcosζ∼RK. Here, the tilt has brought the cosine-modified Alfven radius much closer to\nthe Keplarian radius. This might explain two important features in Figure 9a-f. One, the clumps\nepisodically keep filling the magnetosphere and falling back on to the star, similar to that of a\ndynamical magnetosphere case. Second, the majority of the magnetically streamlined mass outflow\ncomes out from the magnetic foot point that is closer to the rotational-equator ( xy-plane).\nFigure 10 again shows one episode of the η⋆= 50, ζ= 75◦andW= 0.5 simulation, as\nin Figure 9. We see one clump initiated in Figure 10a. Figures 10b-e show successive times\nwhen the clump is reined in by the magnetic field and in-falling material fills the magnetosphere.\nFigure 10f shows the precise moment when the clump impacts the star. Interestingly, at outer\nradii in Figure 10f we can see the start of another episode of clump initiation. The episodes of\nclump formation and fallback can also be seen, when observing the mass-loss rate close to the\nstellar surface. Referring to Figure 11, the mass-loss rate experiences abrupt drops for the cases, i)\nη= 10, ζ= 0, W= 0 and ii) η= 10, ζ= 75, W= 0.5, where the density knot is formed by one side\nof the stream and that knot proceeds to fall onto star on the other side of the magnetic loop. The\nother cases do not show such a complete fallback and hence they do not exhibit these sharp drops\nmass-loss rate close to the stellar surface.\nIn summary, the tilted dipole simulations shown in Figs. 8, 9 and 10 suggest that the tilt\nshould be factored in when deciding whether we have a DM or CM. Larger simulated data sets will\neventually enable us to identify the boundary between DM and CM in cases where the magnetic\ndipole has a large tilt relative to the rotational axis.\nAll of the simulation results presented in this work have the rotation axis along the z−axis and\nthe magnetic tilt oriented along the xz−plane. Owing to this, much of the interesting phenomena\nare best viewed in the xz−plane. In order to illustrate the 3D nature of the simulations, three cross\nsections are presented in Figure 12, where density and magnetic field lines are presented in the XY,\nXZ and YZ planes at an intermediate simulation time, for η⋆= 50, W= 0.5 and ζ= 45◦. In the\nnext section, the angular momentum flux is described along with the calculation of mass loss rate\nand spin down time of the star.\n6.ANGULAR MOMENTUM LOSS RATE\nMass loss from rotating stars carries away angular momentum and leads to stellar spindown. In\nnon-magnetic massive stars with line-driven winds, the mass-loss rate is large, but not enough to\nsignificantly increase their rotation periods during the main sequence because the angular momen-\ntum is only carried away by gas, and the main sequence lifetime is relatively short (e.g., Maeder\n& Meynet 2000). In magnetic stars, most of the angular momentum is lost via the Poynting stress\nprovided by the magnetic field, and not by the outflowing plasma itself (Weber & Davis 1967; ud-\nDoula et al. 2009). Weber & Davis (1967) modeled the angular momentum loss of the Sun with the\nfar-field approximation of a magnetic monopole. For stronger dipole fields, ud-Doula et al. (2009)\nshowed that the angular momentum loss rate of an aligned magnetic rotator could be approximated\nas:\n˙JdWD=2\n3˙MB=0ΩR2\nA, (10)15\n(a) (b)\n(c) (d)\n(e) (f)\nFigure 9. Density (g cm−3) colorized with the magnetic field lines, at different times (ksec) of the simulation, for\nthe case of 45◦tilted magnetic dipole with η⋆= 50 and with the rotation of W= 0.5. The xandzaxes values are\nin the scale of stellar radius R⋆. While Figure 8 showed uniformly spread time intervals, this figure shows a set of\nsnapshots that are bunched in time that reveal the filling of the magnetosphere and episodic in-fall of clumps. One\nentire episode from the start of clump formation to its in-fall is shown.16\n(a) (b)\n(c) (d)\n(e) (f)\nFigure 10. Density (g cm−3) colorized with the magnetic field lines, at different times (ksec) of the simulation, for\nthe case of 75◦tilted magnetic dipole with η⋆= 50 and with the rotation of W= 0.5. The xandzaxes values are\nin the scale of stellar radius R⋆. While Figure 8 showed uniformly spread time intervals, this figure shows a set of\nsnapshots that are bunched in time that reveal the filling of the magnetosphere and episodic in-fall of clumps. One\nentire episode from the start of clump formation to its in-fall is shown.17\n15020025030035040010-710-610-5\nFigure 11. Observed mass-loss rate close to the stellar surface, at R= 1.1R⋆, as a function of time (ksec), for\ndifferent rotation rates and tilt angles. The clump formation and successfully falling back to the star happens for i)\nη⋆= 10, ζ= 0, W= 0 and ii) η⋆= 10, ζ= 75, W= 0.5 cases. The abrupt drops in the mass flow rate can be observed\nfor the same two cases, showing the clumps falling back to the star.\n(a) (b) (c)\nFigure 12. Density (g cm−3) colorized with the magnetic stream lines along the XY, XZ and YZ planes at an\nintermediate simulation time of 240 ksec, for magnetic tilt ζ= 45◦,η⋆= 50, and W= 0.5.\nwhere ˙JdWD is the total angular momentum loss rate in the dipole-modified Weber-Davis approxi-\nmation, RAis the Alfv´ en radius as defined in Eq. (2), Ω is the angular velocity, and ˙MB=0is the\nmass-loss rate in the absence of a magnetic field. To examine the angular momentum loss in our\n3D Geomesh simulations, we define the angular momentum flux:\ndjgas\ndt=ρvrvϕrsinθ, (11)\nwhere vrandvϕare the radial and azimuthal velocity, respectively, and θis the colatitude (ud-\nDoula et al. 2009). Integrating the angular momentum flux over a spherical surface yields the\nangular momentum loss rate of the outflowing gas:\n˙Jgas=I\nSdjgas\ndtdS (12)\nSimilar to Eq. (11), the angular momentum loss rate due to magnetic braking is modeled with\nthe help of the Maxwell stress tensor. The corresponding angular momentum flux is defined as,18\ndjmag\ndt=−BrBϕ\n4πrsinθ, (13)\nwhere BrandBϕare the radial and azimuthal components of the magnetic field flux density B\n(ud-Doula et al. 2009). The total magnetic flux density Bis the sum of background magnetic flux\ndensity B0and evolving magnetic flux density B1. Integrating the angular momentum flux over a\nspherical surface yields the magnetic angular momentum loss rate:\n˙Jmag=I\nSdjmag\ndtdS, (14)\nThe total angular momentum flux and total angular momentum loss rate are therefore:\ndjtot\ndt=djgas\ndt+djmag\ndt, (15)\n˙Jtot=˙Jgas+˙Jmag. (16)\nWith this preamble, the remainder of this section describes the angular momentum flux and\nangular momentum loss rate for the simulations listed in Table 1. Figure 13 displays the total\nangular momentum flux djtot/dtin the xz−plane for three tilt angles: (a) aligned rotation ζ= 0◦,\n(b)ζ= 45◦, and (c) ζ= 75◦. Figure 13a illustrates the total angular momentum flux for the case\nof aligned rotation. Far from the star, the angular momentum loss is expected to be maximal in\nthexy−plane (i.e. perpendicular to the axis of rotation) and minimal along the axis of rotation\n(i.e. the z−axis). This effect is shown in Figure 13a (i.e. in the left panel), where the large\nscale flow shows that djtot/dtis small along the rotation axis and substantially larger in the plane\nperpendicular to the rotation axis. However, in the in the zoomed right panel of Figure 13, we see\nthat the angular momentum loss perpendicular to the plane of rotation is notmaximal because\nclosed magnetic loops near the stellar surface inhibit mass loss. Focusing on the right panel of\nFigure 13a, we see that the maximal angular momentum flux comes from the footpoints of the\nopen magnetic field lines that connect to the equatorial outflow. Once the wind propagates past\nRAthe angular momentum flux is channeled towards the xy-plane.\nThus the angular momentum flux close to the surface of the star is maximal along the open field\nlines ( Bopen) in the plane of rotation; i.e., where |Bopen×Ω|is maximal. This result is especially\nevident for the case of of tilted-rotation with ζ= 45◦, shown in Figure 13b. In the zoomed panel,\nwe see that the angular momentum loss close to the star is low along the magnetic equator, i.e.\nalong the closed field lines, similar to the case of aligned rotation. Comparing the zoomed versions\nof Figs. 13a and 13b, in the ζ= 45◦case the open field lines are present on either side of the\nmagnetic equator. As a result of the magnetic tilt, the open field lines along the rotational equator,\nleading to have larger angular momentum flux close to the star. Conversely, the open field lines\nalong the rotational z−axis have smaller angular momentum flux.\nSimilar observations can also be made for the ζ= 75◦case (see Figure 13c), where the magnetic\npoles are nearly perpendicular to the axis of rotation. Hence, the total angular momentum loss\nclose to the stellar surface is maximum near the magnetic poles, where the open field lines and the\nplane of rotation coincide.\nBeyond the Alfv´ en radius, much of the angular momentum flux emerges along the magnetic\nequator because the angular momentum flux is channeled by the field lines. (see Figure 14). Figures\n14a and 14b, show the total angular momentum flux at the outer boundary of the simulation, i.e. at\na radius of R≈15R⋆. Figures 14a and 14b correspond to the case of aligned rotation. We see that\nmost of the angular momentum flux is channeled by the magnetic field lines. Figures 14c-f, show\nthe total angular momentum flux at the outer boundary of the simulation R≈15R⋆, for the case\nof tilted-rotation with ζ= 45◦,75◦. Here, similar to the aligned rotation, the angular momentum19\n(a)\n(b)\n(c)\nFigure 13. Cross section of total angular momentum flux djtot/dt(g s−2) in the xz-plane for η⋆= 10, W= 0.5:\n(a) no magnetic tilt ( ζ= 0◦), (b) tilt ζ= 45◦, (c) tilt ζ= 75◦. The panels on the left show the large scale angular\nmomentum flux in the entire xz-plane ( ±15R⋆). The panels on the right show the zoomed in angular momentum\nflux in the inner xz-plane ( ±4R⋆), with magnetic field lines overlaid.20\n(a) (b)\n(c) (d)\n(e) (f)\nFigure 14. Total angular momentum flux djtot/dt(g s−2) at the outer surface for the case of η⋆= 10 and W= 0.5\nshowing: the front (a) and back (b) view for ζ= 0◦, the front (c) and back (d) view for ζ= 45◦, and the front (e)\nand back (f) view for ζ= 75◦.21\nTable 2. Mass loss rate, characteristic mass-loss time, angular momentum loss rate and spin-down time.\nη⋆ζ W ˙M τ mass ˙Jtot\n˙JdWD˙Jtot\n˙JdWD,B=0τspin\n(M⊙yr−1) (Myr) (Myr)\n0 0◦0 1.49 ×10−616.7 . . . . . . . . .\n0 0◦0.5 1.70 ×10−614.7 1.04 1.04 2.50\n10 0◦0 6.92 ×10−736.1 . . . . . . . . .\n10 0◦0.5 8.37 ×10−729.8 1.29 5.81 0.44\n10 45◦0.5 7.96 ×10−731.4 1.35 6.07 0.42\n10 75◦0.5 5.88 ×10−742.5 1.05 4.73 0.55\n50 0◦0 4.35 ×10−757.4 . . . . . . . . .\n50 0◦0.5 4.58 ×10−754.6 0.95 7.73 0.34\n50 45◦0.5 5.84 ×10−742.8 1.24 10.12 0.26\n50 75◦0.5 4.41 ×10−756.7 0.96 7.81 0.33\nflux is influenced by the magnetic channeling of the wind and rotation. As we move farther from\nthe star, the effect of rotation overpowers the strength of the magnetic field and the same can be\nobserved from Figure 13b.\n6.1. Radial variation of angular momentum loss\nThe strength of the dipole magnetic field is maximum at the photosphere, falling steeply (1 /r3)\nas we move away from the star. Due to this steep fall off, the angular momentum loss rate is\npredominantly magnetic close to the star. As we move away from the star, the magnetic component\nof the angular momentum loss rate ˙Jmagdecreases and the gaseous component ˙Jgasincreases, while\nkeeping the total angular momentum loss rate ˙Jtotnearly constant. The total, magnetic and\ngas-borne angular momentum loss rates are plotted versus radius in Figure 15 as solid black, red\nand blue curves, respectively, for six different parameter sets. The ˙Jvalues in Figure 15, and the\ncorresponding sixth column in Table 2, are normalized to the predicted ˙JdWDin the dipole-modified\nWeber-Davis model (ud-Doula et al. 2009). The thin lines represent individual time snapshots, and\nthe bold black, red and blue lines represent the average of the thin lines. As we expect from angular\nmomentum conservation, the black ( ˙Jtot) curves are essentially flat. However, we also see ≈10%\nsnapshot-to-snapshot variations, indicating that angular momentum loss is time variable.\nThese results are in line with the dynamics of magnetically channeled winds. 2D simulations\ncarried out for the case of aligned rotation also show significant time variability in the angular\nmomentum loss (ud-Doula et al. 2008). In Figure 15 for each η⋆when the tilt angle ζ= 0, the gas\nis flung out with maximal centrifugal force. Comparing Figure 15a with 15c and also comparing\nFigure 15b with 15d, suggests that increasing the tilt angle ζreduces the moment arm, thereby\nreducing the gas-borne angular momentum loss.\nFurther examination of the left ( η⋆= 10) and right ( η⋆= 50) panels in Figure 15 suggests that\nthe angular momentum loss due to the field ˙Jmagincreases with increasing ζ, relative to the angular\nmomentum loss due to the gas ˙Jgas. This is especially evident in the higher-field panels on the left\nof Figure 15. At low ζ, the gas in the magnetic equator experiences higher centrifugal force, and\ntherefore contributes more to the total angular momentum loss. At high ζ, the open field lines\nabove the magnetic poles experience the highest centrifugal force, and therefore account for most\nof the total angular momentum loss.\n6.2. Characteristic mass-loss and spindown time\nThe characteristic mass-loss time τmassand the spin-down time τspincan be approximated from\nthe total mass loss rate ˙Mand the total angular momentum loss rate ˙Jtot.22\n0 5 1000.511.5\n0 5 1000.20.40.60.81\n(a) (b)\n0 5 1000.511.5\n0246810121400.20.40.60.811.2\n(c) (d)\n0 5 1000.20.40.60.811.2\n0 5 1000.20.40.60.81\n(e) (f)\nFigure 15. Angular momentum loss rate ˙Jas a function of stellar radius with rapid rotation ( W= 0.5) for gas in\nblue, magnetic field in red, and their total in black. ˙Jis normalized to the dipole Weber-Davis value for an aligned\nmagnetic rotator ˙JdWD. Several quasi-steady state snapshots are shown with thin lines, and their time averages are\nindicated with thicker lines for the following cases: (a) η⋆= 10 and ζ= 0◦, (b) η⋆= 50 and ζ= 0◦, (c) η⋆= 10 and\nζ= 45◦, (d) η⋆= 50 and ζ= 45◦, (e)η⋆= 10 and ζ= 75◦, and (f) η⋆= 50 and ζ= 75◦.23\nτmass=M⋆\n˙M, (17)\nτspin=J\n˙Jtot=kM⋆R2\n⋆Ω\n˙Jtot, (18)\nwhere kM⋆R2\n⋆is the star’s moment of inertia and k≈0.1 (ud-Doula et al. 2009). The characteristic\nmass-loss time and the spin-down time are presented in Table 2, along with the total angular\nmomentum loss rate and the mass loss rate for the different sets of simulations carried out in this\nwork. The total angular momentum loss rate presented in the sixth column of Table 2 is normalized\nto˙JdWD, the predicted angular momentum loss rate from the dipole-modified Weber-Davis model.\nIn column 7 we normalize ˙Jtotby˙JdWD ,B=0the prediction of the non-magnetic Weber-Davis model.\nThis last column highlights the expected increase in total angular momentum loss with increasing\nη⋆.\nComparing the non-magnetic cases, with and without rotation, the rotating case can be seen to\nhave an increased mass-loss rate and a reduced characteristic mass-loss time. With the introduction\nof a magnetic field, we see a significant increase in the mass-loss time due to the higher magnetic\nconfinement and the reduced mass-loss rate. In case of non-magnetic rotation, there is no spin-down\ndue to magnetic braking and hence it has the longest spin-down time. This is because the angular\nmomentum loss is carried by the gas. By the same token, the spin-down time is notably reduced\nas we go from η⋆= 10 to η⋆= 50, because of the increase in magnetic braking torque.\n7.SUMMARY AND CONCLUSIONS\nThe presence of a dipolar magnetic field and its tilt with respect to the rotation axis can have\nnotable influence on the mass outflow and the angular momentum outflow. The mass outflows\nfrom the O and B-type stars are very significant and thus influence the overall evolution of the\nstar. Hence, it is inevitable that the presence of magnetic field and its orientation would play a\nsignificant role in the evolution of the O and B-type stars. The detailed simulation and analysis of\nthe magnetic O and B-stars are discussed in the literature for the case of aligned rotation ud-Doula\n& Owocki (2002b); ud-Doula et al. (2008, 2009) with the help of 2D simulations.\nFor the simulation of spherical systems in full 3D, we have recently developed the Riemann\nGeomesh MHD code. The code is based on the spherical icosahedron-based meshing of the sphere.\nThe surface of the sphere is mapped as uniformly as possible, which is a significant advance com-\npared to an r, θ, ϕ mesh. This eliminates the increased discretization errors stemming from singu-\nlarities at the poles and the need for shorter time steps due to the concentration of mesh points at\nthe poles.\nIn this work, we have carried out the 3D simulations of the magnetically channeled line driven\nwinds for a template O star. For the first time, the simulations are done for a higher rotation\nrate (0 .5 Ωcrit) and larger magnetic field tilt angles (0◦,45◦,75◦); by utilizing the uniform meshing\nand state-of-the-art MHD algorithms of the Riemann Geomesh code. The simulations reach a\nquasi-steady state, as expected, and the mass-loss rate is observed to increase with rotation and\ndecrease with increased magnetic field strength, due to the magnetic confinement of the wind. For\nthe magnetized simulations, the wind exhibits the dynamics of magnetic channeling, as observed\nin previous 2D simulations. In addition, the results for the 45◦case demonstrates the interplay\nbetween the centrifugal force, which is maximum along the rotational equator, and the magnetic\nchanneling, which is prominent along the magnetic equator.\nUsing these insights, we proceed to catalogue the mass-loss rate and angular momentum loss\nrate for different magnetic-field strengths, tilt angles and rotation rates. The spatial variations in\ntotal angular momentum flux ( djtot/dt) are examined for different magnetic tilt angles, with the\nhelp of images at the cross-section plane, as well as, at the outer surface of the simulation domain.\nNear the stellar surface, the maximal total angular momentum flux ( djtot/dt) is observed where\nthe open magnetic field lines align with the plane of rotation. Farther from the star, the maximal24\nangular momentum flux ( djtot/dt) is aligned with the confinement of magnetic field lines, emanating\nfrom either sides of the magnetic equator.\nThe gaseous, magnetic and total angular momentum loss rates ˙Jare computed as a function of\nradius, and plotted for different tilt angles and magnetic field strengths. With the corresponding\nmass-loss rates, the characteristic mass-loss time and spin-down time are tabulated for the simula-\ntions carried out in this work. The results show a longer mass-loss time with increasing magnetic\nfield strength, which tracks the reduced mass-loss rate. We also find faster spin-down time with\nincreasing magnetic field strength, showing the role of magnetic braking torque on the rotation of\nthe star.\nThese simulations are the very first that have been reported with a code that can handle all\npossible tilt angles of the magnetic dipole with respect to the rotation axis. The use of a mapped\nmesh makes these simulations somewhat more expensive, but this comes with the tremendous\nadvantage of full geometric generality. While we are not restricted to dipolar fields, we have\nfocused on dipolar fields in this study. In order to develop a comprehensive understanding of the\nrelations between, η⋆, ζ, W and ˙M,˙J, a much larger set of data with more finer variations in\nη⋆, ζ, W is necessary. Therefore, this is planned for the next set of study and it will be elaborated\nin a subsequent paper.\nACKNOWLEDGMENTS\nWe acknowledge the extremely valuable inputs from Stanley Owocki. We also acknowledge\nthe computer nodes, which are provided to us on i) Compute clusters at the Notre Dame Center\nfor Research Computing, ii) Bridges-2 supercomputer at the Pittsburgh Supercomputing Center,\nand iii) Stampede-2 supercomputer at the Texas Advanced Computing Center. This work used the\nExtreme Science and Engineering Discovery Environment (XSEDE), which is supported by National\nScience Foundation grant number ACI-1548562. Support for this work was provided by the National\nAeronautics and Space Administration through Chandra Award Number TM1-22001 issued by the\nChandra X-ray Center, which is operated by the Smithsonian Astrophysical Observatory for and\non behalf of NASA under contract NAS8-03060. AuD acknowledges support from Pennsylvania\nState University Commonwealth Campuses Research Collaboration Development Program. Data\nwas generated through this support from the Institute of Computational and Data Sciences. MG\nacknowledges summer support from a Provost’s Research Grant, and a work-release grant from the\nCollege of Science and Mathematics at West Chester University.\nDATA AVAILABILITY\nThe data underlying this article are available in the article and in its online supplementary\nmaterial.\nREFERENCES\nBalsara, D. S. 2017, Living Reviews in Computational\nAstrophysics, 3, 2, doi: 10.1007/s41115-017-0002-8\nBalsara, D. S., & Dumbser, M. 2015, Journal of\nComputational Physics, 287, 269,\ndoi: 10.1016/j.jcp.2014.11.004\nBalsara, D. S., Florinski, V., Garain, S., Subramanian,\nS., & Gurski, K. F. 2019, MNRAS, 487, 1283,\ndoi: 10.1093/mnras/stz1263\nBalsara, D. S., Garain, S., Florinski, V., & Boscheri,\nW. 2020, Journal of Computational Physics, 404,\n109062, doi: 10.1016/j.jcp.2019.109062Castor, J. I., Abbott, D. C., & Klein, R. I. 1975, ApJ,\n195, 157, doi: 10.1086/153315\nDaley-Yates, S., Stevens, I. R., & ud-Doula, A. 2019,\nMNRAS, 489, 3251, doi: 10.1093/mnras/stz1982\nFlorinski, V., Balsara, D. S., Garain, S., & Gurski,\nK. F. 2020, Computational Astrophysics and\nCosmology, 7, 1, doi: 10.1186/s40668-020-00033-7\nFriend, D. B., & Abbott, D. C. 1986, ApJ, 311, 701,\ndoi: 10.1086/164809\nGagn´ e, M., Oksala, M. E., Cohen, D. H., et al. 2005,\nApJ, 628, 986, doi: 10.1086/43087325\nGrunhut, J. H., Wade, G. A., Neiner, C., et al. 2017,\nMNRAS, 465, 2432, doi: 10.1093/mnras/stw2743\nGuo, X., Florinski, V., & Wang, C. 2016, Journal of\nComputational Physics, 327, 543,\ndoi: 10.1016/j.jcp.2016.09.057\nHowarth, I. D., & van Leeuwen, F. 2019, MNRAS,\n484, 5350, doi: 10.1093/mnras/stz291\nLanger, N. 2012, ARA&A, 50, 107,\ndoi: 10.1146/annurev-astro-081811-125534\nMaeder, A., & Meynet, G. 2000, ARA&A, 38, 143,\ndoi: 10.1146/annurev.astro.38.1.143\nOwocki, S. 2004, in EAS Publications Series, Vol. 13,\nEAS Publications Series, ed. M. Heydari-Malayeri,\nP. Stee, & J. P. Zahn, 163–250,\ndoi: 10.1051/eas:2004055\nOwocki, S. P., Shultz, M. E., ud-Doula, A., et al. 2020,\nMNRAS, 499, 5366, doi: 10.1093/mnras/staa2325\nOwocki, S. P., & ud-Doula, A. 2004, ApJ, 600, 1004,\ndoi: 10.1086/380123\nPetit, V., Owocki, S. P., Wade, G. A., et al. 2013,\nMNRAS, 429, 398, doi: 10.1093/mnras/sts344\nPowell, K. 1994, An approximate Riemann solver for\nmagnetohydrodynamics (that works in more than\none dimension), CWI\nRuen, T., & Bulatov, V. 2012, Zeroth stellation of\nicosahedron.\nhttps://commons.wikimedia.org/wiki/File:\nZeroth stellation oficosahedron.pngSch¨ oller, M., Hubrig, S., Fossati, L., et al. 2017, A&A,\n599, A66, doi: 10.1051/0004-6361/201628905\nShultz, M. E., Owocki, S., Rivinius, T., et al. 2020,\nMNRAS, 499, 5379, doi: 10.1093/mnras/staa3102\nud-Doula, A., & Naz´ e, Y. 2016, Advances in Space\nResearch, 58, 680, doi: 10.1016/j.asr.2015.09.025\nud-Doula, A., Owocki, S., Townsend, R., Petit, V., &\nCohen, D. 2014, MNRAS, 441, 3600,\ndoi: 10.1093/mnras/stu769\nud-Doula, A., & Owocki, S. P. 2002a, ApJ, 576, 413\n—. 2002b, ApJ, 576, 413, doi: 10.1086/341543\nud-Doula, A., Owocki, S. P., & Townsend, R. H. D.\n2008, MNRAS, 385, 97,\ndoi: 10.1111/j.1365-2966.2008.12840.x\n—. 2009, MNRAS, 392, 1022,\ndoi: 10.1111/j.1365-2966.2008.14134.x\nud-Doula, A., Sundqvist, J. O., Owocki, S. P., Petit,\nV., & Townsend, R. H. D. 2013, MNRAS, 428, 2723,\ndoi: 10.1093/mnras/sts246\nud-Doula, A., Townsend, R. H. D., & Owocki, S. P.\n2006, ApJL, 640, L191, doi: 10.1086/503382\nWade, G. A., Neiner, C., Alecian, E., et al. 2016,\nMNRAS, 456, 2, doi: 10.1093/mnras/stv2568\nWeber, E. J., & Davis, Leverett, J. 1967, ApJ, 148,\n217, doi: 10.1086/14913826\nAPPENDIX\nA.DESCRIPTION OF THE MHD EQUATIONS\nThe governing MHD equations of this simulation study are presented below. The magnetic field\nBis split into a curl-free background field B0and a time evolving field B1. The total magnetic\nfield is defined to be, B=B0+B1. Upon imposing the vector identities, the governing equations\ntake the following forms (Powell 1994; Guo et al. 2016),\ni) conservation of mass\n∂ρ\n∂t+∇ ·(ρv) = 0 (A1)\nwhere, ρis the density and vis the velocity of the fluid\nii) conservation of momentum\n∂(ρv)\n∂t+∇ ·\u0014\nρvv+PI+1\n8πB12I+1\n4π((B0·B1)I−B1B−B0B1)\u0015\n=ρgtot (A2)\nwhere, Pis the pressure and gtotis the total net acceleration coming from the external forces. For\nthis simulation, it comprises of CAK line acceleration, gravitational acceleration and the centrifugal\nand Coriolis accelerations arising due to rotation. This can be expressed as,\ngtot=gCAKˆ r+ggravˆ r−2Ω×v−Ω×(Ω×r)\nwhere, Ωis the angular velocity vector and ˆ ris the unit vector along the radial direction.\niii) conservation of energy\n∂E\n∂t+∇\u0014\u0012\nE+P+B12\n8π+B0·B1\n4π\u0013\nv−(v·B1)B\n4π\u0015\n=ρv·gtot (A3)\nwhere, Eis the energy density. These are isothermal simulations, therefore the energy equation is\nnot used however, for the sake of completion this is included here. The pressure Pis then defined\nusing the sound speed vsasP=ρv2\ns.\niv) induction equation\n∂B1\n∂t− ∇ × (v×B) = 0 (A4)"    },    {        "title": "1707.03216v3.Magnetophononics__ultrafast_spin_control_through_the_lattice.pdf",        "content": "Magnetophononics: ultrafast spin control through the lattice\nM. Fechner,1, 2,\u0003A. Sukhov,3, 4L. Chotorlishvili,3C. Kenel,2, 5J. Berakdar,3and N. A. Spaldin2\n1Max Planck Institute for the Structure and Dynamics of Matter, 22761 Hamburg, Germany\n2Materials Theory, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093 Z ¨urich, Switzerland\n3Institut f ¨ur Physik, Martin-Luther-Universit ¨at Halle-Wittenberg, 06099 Halle/Saale, Germany\n4Forschungszentrum J ¨ulich GmbH, Helmholtz Institute Erlangen-N ¨urnberg for Renewable Energy (IEK-11), 90429 N ¨urnberg, Germany\n5Department of Materials Science and Engineering, McCormick School of Engineering,\nNorthwestern University, 2200 Campus Drive, Evanston, IL 60208, USA\n(Dated: November 13, 2018)\nUsing a combination of first-principles and magnetization-dynamics calculations, we study the effect of the\nintense optical excitation of phonons on the magnetic behavior in insulating magnetic materials. Taking the\nprototypical magnetoelectric Cr 2O3as our model system, we show that excitation of a polar mode at 17 THz\ncauses a pronounced modification of the magnetic exchange interactions through a change in the average Cr-Cr\ndistance. In particular, the quasi-static deformation induced by nonlinear phononic coupling yields a struc-\nture with a modified magnetic state, which persists for the duration of the phonon excitation. In addition, our\ntime-dependent magnetization dynamics computations show that systematic modulation of the magnetic ex-\nchange interaction by the phonon excitation modifies the magnetization dynamics. This temporal modulation\nof the magnetic exchange interaction strengths using phonons provides a new route to creating non-equilibrium\nmagnetic states and suggests new avenues for fast manipulation of spin arrangements and dynamics.\nI. INTRODUCTION\nThe field of non-linear phononics, in which high-intensity\nterahertz (THz) optical pulses are used to drive phonon ex-\ncitations, is of increasing interest1,2. The non-linear pro-\ncesses triggered by the strong phonon excitations have been\nshown repeatedly to introduce complex structural modifica-\ntions in materials, which in turn cause striking and often un-\nexpected changes in properties. Examples include the stimu-\nlation of insulator to metal transitions in correlated oxides3–5\nand the enhancement of superconducting properties in high- Tc\ncuprates6,7and other materials8. In all cases, theoretical stud-\nies combining density functional theory with phenomenolog-\nical modeling have been invaluable in interpreting the exper-\nimental results2,9–13and even in predicting new phenomena,\nsuch as the recent switching14and creation15of ferroic states,\nahead of their experimental observation16.\nIn addition to modifying electronic properties, there are\na number of examples of THz phonon excitation triggering\nmagnetic phenomena on a picosecond (ps = 10\u000012s) time-\nscale. Early results indicate that selective phonon excitations\ncan induce demagnetization processes17,18, and two-phonon\nexcitation19has been shown to excite magnons by the stimu-\nlated rotational motion of atoms13,15. We note that these be-\nhaviors are distinct from ultrafast femtosecond (fs = 10\u000015s)\nspin-flip relaxation processes induced by optical frequency\npulses, such as the pioneering experiments of Refs. [20–\n22], which heat the electronic/lattice sub-system. They are\nalso distinct from the THz excitation of electro-magnons in\nmultiferroics23, in which the electric field of the light pulse\ncouples directly to the dipole moment of the electron-magnon\nquasiparticle.\nIn this work, we address theoretically how the structural\nchanges triggered by the non-linear phononic processes af-\nfect the magnetic energy landscape. We are particularly in-\nterested in the situation in which an excited infra-red-active\nphonon mode couples quadratic-linearly to a Raman-activemode, causing a shift in the average structure that persists for\nthe duration of the phonon excitation. We show that the in-\nduced structure can have a different magnetic ordering from\nthe equilibrium structure, so that the lattice excitation can\ncause a spin-state transition. In addition, we explore the spin\ndynamics induced by the phonon coupling, and show that\nvarious complex spin-flip patterns can be selectively excited\nthrough appropriate choice of the phonon driving frequency.\nIn the next section we review the now well-established\ntheory of non-linear phononics. We then present a model\nthat combines the non-linear phononics formalism with the\nHeisenberg Hamiltonian to describe spin-phonon coupling\nthrough the changes in magnetic exchange interactions that\nare induced by changes in structure. In Section III, we ap-\nply the model to the prototypical magnetoelectric material,\nCr2O3(Fig. 1), using first-principles calculations to obtain all\nthe material-specific parameters. In Section IV we present\nand discuss the analytical solution of the non-linear phononic\nHamiltonian for the lattice dynamics and in Section V the nu-\nmerical simulations of the magnetization dynamics based on\nthe Landau-Lifshitz Gilbert equation24,25. The implications of\nour findings and suggestions for future work are discussed in\nthe Summary.\nII. THEORY\nHere we describe separately the modeling of phononic and\nmagnetic lattice systems before outlining our approach to\nmodeling their coupling. We begin with the description of\nlattice anharmonicity.\nFor large atomic displacements, such as those induced by\nintense optical pulses, the usual harmonic description of lat-\ntice phonons breaks down and higher order anharmonicities\nbecome relevant. The lattice Hamiltonian can then be writtenarXiv:1707.03216v3  [cond-mat.str-el]  5 Mar 20182\nas2,10\nHvib(\u0018IR;\u0018R) =!2\nIR\n2\u00182\nIR+!2\nR\n2\u00182\nR+g\u0018R\u00182\nIR\n+\rIR\n4\u00184\nIR+\rR\n4\u00184\nR;(1)\nwhere!IR,!Rare the frequencies of the infrared (IR) and Ra-\nman (R) modes, \u0018IRand\u0018Rare distortions, \rIR=Rare fourth\norder anharmonic constants and grepresents the coupling be-\ntween two phonon modes. (Terms in \u00183\nRare small and so\nare neglected for conciseness.) The dominant anharmonic re-\nsponse to optical pumping comes from the third-order \u0018R\u00182\nIR\nterm, which has been shown to cause a shift in the potential\nenergy to a finite value of the Raman normal mode coordinate,\ncreating in turn a quasi-static change in the structure2,9,10. For\na single optical pulse, this structural distortion decays and\nthe system relaxes back to the ground state, whereas con-\ntinous driving yields a combination of time-dependent and\ntime-independent structural distortions. We will discuss these\ndistortions later based on the analytical solution of Eqn. (1).\nTo model the magnetic structure we consider a Heisenberg\nHamiltonian with\nHmag=X\nhi;jiJi;j(Si\u0001Sj) +DNX\ni=1(Sz\ni)2; (2)\nwhereSiis a localized spin magnetic moment, Ji;jare the\nmagnetic exchange interactions between spins iandj, and\nDis the uniaxial magneto-crystalline anisotropy (MCA) en-\nergy. We introduce the coupling of the local spin moments\nFIG. 1: (a) Unit cell of Cr 2O3with the red arrows indicating the\nground state antiferromagnetic spin magnetic order. (b) Schematics\nof the phonon-driven change in magnetic ground state: The excita-\ntion of a polar phonon mode ( \u0018IR) induces an increase in the nearest-\nneighbor Cr-Cr bond length by the square-linear anharmonic phonon\ncoupling. The longer bond length results in ferromagnetic exchange\ninteraction between the Cr ions creating a transient change in the\nmagnetic state for the duration of the phonon excitation.contained in the Heisenberg Hamiltonian, to the distortion \u0018of\nthe lattice Hamiltonian by expanding the magnetic exchange\ninteractions with respect to the distortion10,26. For an expan-\nsion up to second order we obtain the following spin-phonon\ncoupling Hamiltonian:\nHsp=X\nhi;ji@Ji;j\n@\u0018(Si\u0001Sj)\u0018+X\ni;j@2Ji;j\n@\u00182(Si\u0001Sj)\u00182:(3)\nNote that the first derivatives of exchange with respect to\nmode\u0018can be zero for certain mode symmetries, and in gen-\neral only the second order derivatives are non-zero. For the\nsymmetry-conserving Raman modes, \u0018R, however, the first\norder spin-lattice coupling is non-zero; note also that these are\nthe modes that have a quadratic-linear lattice coupling with\nthe IR modes in Eqn. (1). Since the \u0018Rdistortion is symmetry\nconserving we can directly write the exchange interaction as\na function of the mode amplitude as\nJi;j(\u0018R) =Ji;j+@Ji;j\n@\u0018R\u0018R+@2Ji;j\n@\u00182\nR\u00182\nR+::: ; (4)\nwith the same labeling as in Eqn. (2). (For a phonon mode of\ngeneral symmetry, either Raman or IR active, the situation is\nmore complicated since the symmetry breaking can split de-\ngenerate exchange interactions, resulting in an increased total\nnumber of inequivalent exchange interaction parameters). In\nprinciple the MCA energy term, D, is also a function of the\nmode amplitude. However, we find that its variation is negli-\ngible for Cr 2O3.\nA. Computational details\nTo calculate the structure, phonons and magnetic ex-\nchange interactions of Cr 2O3we use density functional the-\nory with the local spin density approximation plus Hubbard\nU(LSDA +U) exchange-correlation functional. We use pa-\nrametersU=4 eV andJ=0.5 eV on the Cr- dorbitals and treat\nthe double counting correction within the fully-localized limit.\nThese parameters have been shown to give a good descrip-\ntion of Cr 2O3in earlier work27–29. We use the Vienna ab-\ninitio simulation package (V ASP)30within the projector aug-\nmented wave (PAW) method31using default V ASP PAW pseu-\ndopotentials generated with the following valence-electron\nconfigurations: Cr ( 3s23p64s13d5), O ( 2s23p4). We sam-\nple the Brillouin zone in our total energy calculations us-\ning 11\u000211\u000211 and 9\u00029\u00025k-point meshes for the primi-\ntive rhombohedral and hexagonal cells respectively, and use\na plane-wave energy cutoff of 600 eV . Finally, for computing\nthe MCA energy of Cr 2O3we use an increased k-point grid\nof 14\u000214\u000214 within the rhombohedral cell.\nPrevious theoretical studies of Cr 2O3have addressed the\nmicroscopic origin of the magnetoelectric effect28,32,33and\nthe magnetic properties27,29using a combination of first-\nprinciples density functional theory (DFT) calculations and\neffective Hamiltonian approaches. These studies demon-\nstrated that magnetoelectric properties, phonon frequencies3\nand magnetic exchange interactions, all key quantities in this\nwork, are well described by DFT calculations with technical\ndetails similar to those chosen here.\nWe calculate the atomistic spin-dynamics by solving the\nLandau-Lifshitz Gilbert equation numerically using the Heun\nmethod34with an integration time step that is one thousandth\nof the fasted period of the oscillations (\u0019\u0016\n\rD10\u00003\u00194fs).\nIII. Cr 2O3\nCr2O3crystallizes in the corundum structure which is com-\nposed of a combination of edge- and face-sharing CrO 6oc-\ntahedra. The magnitude 3 \u0016Bspin magnetic moments on\nthed4Cr3+ions order antiferromagnetically below the N ´eel\ntemperature, TN=307 K, in a collinear “ (\";#;\";#)” pattern\nwith magnetic space group R30c(161) that breaks inversion\nsymmetry (Fig. 1)35,36. The primitive unit cell, with its four\nchromium and six oxygen atoms is shown in Fig. 1(a). As a\nresult of its simultaneous breaking of time-reversal and space-\ninversion symmetry, Cr 2O3exhibits the linear magnetoelec-\ntric effect, in which a magnetic/electric field induces an elec-\ntric/magnetic polarization. Indeed, Cr 2O3is considered to be\nthe prototypical magnetoelectric, being the material in which\nthe effect was first predicted37and subsequently measured38.\nA. Calculated lattice properties of Cr 2O3\nWe begin by calculating the lowest-energy structure of\nCr2O3by relaxing its rhombohedral unit cell to obtain a force-\nfree DFT reference structure. We initialized our computations\nusing data from the experimental study of Ref. [39] and op-\ntimized the structure until the forces on each atom were less\nthan 0.01 meV/ ˚A. The resulting structure has a unit cell vol-\nume of 96.46 ˚A3, with the coordinates x= 0:152for Cr and\nx= 0:304for O at the Wyckoff positions 4cand6e, respec-\ntively, in good agreement with literature experimental39and\ntheoretical28values.\nNext, we compute the phonon frequencies and eigenvec-\ntors of our ground-state structure using density functional per-\nturbation theory43. Light radiation only excites polar phonon\nmodes close to the center of the Brillouin zone, q= (0;0;0).\nConsequently, we do not calculate the full phonon band struc-\nture but only the modes at this special point in reciprocal\nspace. Since the primitive cell of Cr 2O3contains 10 atoms,\nthere are 27 non-translational zone-center phonon modes,\nwhich span the irreducible representatives of the 30mpoint\ngroup: 2A1g\n3A2g\n2A1u\n2A2u\n10Eg\n8Eu. Of these\nmodes only the A2uandEumodes are polar, with the dipole\nmoments of the A2umodes pointing along the long rhombo-\nhedral axis ( a+b+c) and those of the Eumodes perpendicular\nto it. TheA1gmodes, which are not directly excitable by light,\nhave the symmetry of the Cr 2O3point group and consequently\nexhibit a square-linear coupling to the polar modes in the an-\nharmonic potential. We list in Tab. I the computed frequencies\nFIG. 2: Displacement pattern of the Cr 2O3phonon modes relevant\nin this work. (a) shows symmetry-conserving A1gmodes, and (b)\ndisplays IR-active A2umodes. The grey arrows show the displace-\nment direction of each atom for the specific mode, with the indicated\ndirections defining positive displacement amplitudes. The notation\nindicates the irreducible representation for the mode symmetry fol-\nlowed in brackets by the calculated mode frequency in THz, rounded\nto the nearest integer.\nofA1gand optically active modes together with available ex-\nperimental frequencies from the literature40–42, and find good\nagreement.\nIn Fig. 2 we show the displacement patterns of the A2u\nandA1gmodes with the grey arrows indicating the direction\nof displacement of the atoms for positive mode amplitude.\nWithin the Cr 2O3structure the 9.3 Thz A1g(A1g(9)) mode\nmodulates the Cr-Cr distance, whereas the higher frequency\n17.3 ThzA1g(17) mode modulates the Cr-O-Cr bond-angles\nvia a rotation of the oxygen octahedra around the rhombohe-\ndral axis. Both polar A2umodes exhibit a collective motion\nof the oxygens along the rhombohedral axis, with the 17.2 Thz\nA2u(17) mode involving the larger relative movement of the\nCr and oxygen atoms. The Cr-Cr bond lengths are unchanged\nby the movement patterns of the polar modes.\nWith our calculated phonon eigenvectors as the starting\npoint, we next compute the anharmonic phonon coupling con-\nstants by mapping the potential of Eqn. (1) onto total en-\nergy calculations of Cr 2O3structures, distorted by appropri-4\nTABLE I: Phonon frequencies of symmetry conserving Raman and\ninfrared-active modes of Cr 2O3in THz. The experimental values\n(EXP) are taken from Refs. [40–42]. The displacement patterns of\ntheA1gandA2umodes are shown in Fig. 2 (a,b).\nsym. DFT EXP\nA1g 9.3 9.0\nA1g 17.3 16.5\nA2g 8.0 –\nA2g 13.8 –\nA2g 20.7 –\nEg 9.2 8.7\nEg 10.7 10.5\nEg 12.4 12.0\nEg 16.1 15.6\nEg 19.2 18.5\nA2u 12.2 12.1\nA2u 17.2 16.0\nEu 9.3 9.1\nEu 13.5 13.2\nEu 17.0 16.1\nEu 19.0 18.2\nate superpositions of the phonon eigenvectors as in previous\nwork10,13. We are primarily interested in the quadratic-linear\ncoupling of Eqn. (1), which is only nonzero if the linear com-\nponent has the full point group symmetry, which is A1gfor\nCr2O3. For convenience, we assume that the radiation is ori-\nented along the rhombohedral axis such that only A2umodes\nare directly excited, then we compute the 2D-potential of\nEqn. (1) for all combinations of polar A2uandA1gmodes.\nIn Fig. 3 (a) we show the computed potential landscape for\nthe combination of the A1g(9) andA2u(17) phonon modes.\nDisplacement of the A2u(17) mode causes a shift of the po-\ntential minimum of the A1g(9) mode, as shown in the cuts of\nthe 2D-potential in Fig. 3 (b). The red dashed line in Fig. 3 (a)\nshows the position of the A1g(9) mode minimum within the\n2D potential landscape. For negative and positive amplitudes\nof theA2u(17) mode, the potential minimum position shifts to\npositive amplitudes of the A1g(9), corresponding to a negative\nsign of the square-linear coupling. We quantify this observa-\ntion by fitting the complete potential landscape using Eqn. (1),\nto extract all anharmonic coupling constants and repeat the\ncalculation for all combinations of A2uandA1gmodes. The\ncomputed anharmonic constants are given in Tab. II.\nWe find that the nominal value of the quadratic-linear an-\nharmonic coupling gvaries from 6 to 101 meV/(pu˚A)3and\nexhibits positive or negative sign, so that modulations of the\nCr2O3structure with positive and negative amplitudes of the\nA1gmodes can be induced by exciting the appropriate polar\nmode. (Note that the opposite choice of sign in the definition\nof the phonon eigenvectors would reverse the sign of g; the\nsigns given in Tab. II correspond to the phonons as defined in\nFIG. 3: (a) Calculated two-dimensional potential surface of the\nanharmonic phonon-phonon interaction between \u0018R=A1g(9) and\n\u0018IR=A2u(17). The red line in the three dimensional plot shows\nthe position of the potential minimum. (b) selected cuts through the\ntwo dimensional potential surface shown in (a). Note that we plot\n\u0001V(\u0018IR;\u0018R) =V(\u0018IR;\u0018R)\u0000V(\u0018IR;\u0000g\u00182\nIR=(2!2\nR)), so that the\nminimum is set to 0 meV .\nTABLE II: Upper panel: Anharmonic coupling constants g, in units\nof [meV/(pu˚A)3], between symmetry conserving A1gand IR active\nphonon modes of A2usymmetry. Lower panel: quartic anharmonic\nconstants,\r, in units of [meV/(pu˚A)4].\nmodes A2u(12) A2u(17)\nA1g(9) 6 -86\nA1g(17) -38 101\nmodes A1g(9)A1g(17)A2u(12)A2u(17)\n\rIR 1 4 4 14\nFig. 2).\nMinimization of Eqn. (1) gives the amount of induced struc-\ntural distortion to be \u0018R\u0019 \u0000g\u00182\nIR=!2\nR. Consequently, for\nthe combination of polar A2u(12) andA1g(9) modes, excita-\ntion of the polar mode induces, due to the positive coupling\nconstantg, a negative amplitude of the A1g(9) mode which\nresults in a decrease in the nearest-neighbor Cr-Cr distance.5\nIn contrast, the A2u(17) mode couples with a negative cou-\npling constant gto theA1g(9) mode and so the induced quasi-\nequilibrium structure has an increased Cr-Cr distance. The\nA1g(17) mode changes the oxygen octahedral rotation angles\naround the Cr ions. Its negative coupling to the A2u(12) mode\nresults in a decreased rotational angle, whereas the positive\ncoupling to the A2u(17) mode increases the rotational angle\nin the quasi-equilibrium structure.\nB. Calculated magnetic properties of Cr 2O3\nThe fact that the transient structure generated through the\nquadratic-linear coupling of the optically excited polar modes\nto theA1g(9) Raman mode has a modified Cr-Cr distance sug-\ngests that it might also have a different magnetic ground state.\nTo explore this possibility, we next calculate the energy dif-\nference between the AFM ground-state ordering (\";#;\";#)\nand two other magnetic orderings of the Cr spins – ferromag-\nnetic (FM) (\";\";\";\")and another antiferromagnetic (AFM 1)\n(\";\";#;#)– as a function of the A1g(9) distortion amplitude.\nFor the equilibrium structure, we find that the AFM 1state\nis 67 meV and the FM state 162 meV in energy above the\nAFM ground state. Modulating the structure with the pattern\nof atomic displacements corresponding to the A1g(9) phonon\nmode in the positive direction, so that the Cr-Cr nearest-\nneighbor distance, ( dCr\u0000Cr), is increased, significantly low-\ners both of these energy differences. For positive ampli-\ntudes larger than \u0018R\u00150.75pu˚A, corresponding to a stretch-\ning ofdCr\u0000Cr=0.06 ˚A, the energy of the AFM 1state be-\ncomes lower than the AFM ground state; at larger amplitudes\n(\u0018R\u00151.9pu˚A) the FM state becomes lower in energy than\nthe original ground state, but remains higher in energy than\nthe AFM 1state. We therefore predict that a crossover to\nthe AFM 1state should be achievable through quadratic-linear\ncoupling with appropriate choice of the polar mode excitation\nfrequency and intensity. (Note that modulating the structure\nwith a negative amplitude of A1g(9), which decreases the Cr-\nCr nearest-neighbor bond, increases the relative energies of\nthe FM and AFM 1states). In contrast, modulating the struc-\nture along the eigenvector of the second A1gmode at 17 THz,\nor along those of the polar A2umodes has only a small effect\non the magnetic energy landscape.\nTo explore the magnetic behavior further, we next calcu-\nlate the magnetic exchange interactions of the ground-state\nstructure using the Heisenberg Hamiltonian of (2), including\nmagnetic exchange interactions, Jn, up to fifth nearest neigh-\nbors, as shown in Fig. 4; this Hamiltonian has been shown to\ngive an accurate theoretical description of the magnetoelec-\ntric effect and magnetic transition temperature of Cr 2O327,28.\nSpecifically, our Heisenberg Hamiltonian for the magnetic ex-changes reads:\nHexch\nCr2O3=J1(S1\u0001S2+S3\u0001S4)\n+ 3J2(S1\u0001S4+S2\u0001S3)\n+ 3J3(S1\u0001S2+S3\u0001S4) (5)\n+ 6J4(S1\u0001S3+S2\u0001S4)\n+J5(S2\u0001S3+S1\u0001S4);\nwith theJnas shown in Fig. 4, and the labeling of spins as\nin Fig.1. We extract the magnetic exchange interactions from\nthe total energy differences between four distinct magnetic ar-\nrangements within the non-primitive hexagonal cell, using the\napproach of Ref. [44]. The resulting magnetic exchange inter-\nactions are listed in Tab. III and are in agreement with earlier\ntheoretical works27–29. We find the nearest and next-nearest\nneighbor interactions, J1andJ2, to be strongly antiferromag-\nnetic, whereas J3andJ4favor ferromagnetic arrangements.\nThe furthermost exchange interaction that we consider, J5, is\nweakly antiferromagnetic. Finally, we note that, in contrast to\nother magnetic insulators45, higher-order magnetic exchanges\nsuch as four-body interactions are not required for the descrip-\ntion of the magnetism in Cr 2O329.\nNext, we compute how the modulation of the Cr 2O3struc-\nture by the phonon mode eigenvectors changes the magnetic\nexchange interactions, using the same approach to extract the\nexchange interactions as we used above for the ground-state\nstructure. (For the A2umodes we neglect the small splittings\ninJvalues that result from the lowered symmetry.) Our cal-\nculated coefficients of the expansion of Eqn. (4), listed up\nto quadratic order in \u0018in Table III, are a measure of the\nspin-phonon coupling for each mode. In Fig. 4 (b,c), we\nplot the five nearest-neighbor magnetic exchange constants\nas a function of the A1g(9) andA2u(17) phonon mode am-\nplitudes. We find that the A1g(9) mode significantly changes\nthe nearest-neighbor exchange interaction, whereas the longer\nrange magnetic exchange interactions are less affected by the\nstructural modulation. An intriguing result is the sign change\nof the nearest-neighbor exchange interaction J1at amplitudes\n\u0018\u00150.75pu˚A, corresponding to an increase of 0.06 ˚A in the\nCr-Cr bond length, consistent with the crossover to AFM 1or-\ndering that we found above. In contrast to the A1g(9) mode\nwe see that the A2u(17) mode has minimal direct effect on\nthe magnetic exchange interactions. The other A1gandA2u\nmodes (not shown) also have minimal effect on the exchange\ninteractions. The spin-phonon coupling constants obtained by\nfitting these results to Eqn. (4) are listed in Tab. III; as ex-\npected the coefficients of J1for theA1g(9) mode are large.\nWe can understand the strong J1response by analyzing the\ndisplacement pattern of the A1g(9) mode in the context of the\norigin of the J1magnetic exchange interaction that has been\ndiscussed in the literature. Earlier analysis of the magnetic\ninteractions in the ground-state of Cr 2O327showed that the\nmain contribution to J1arises from an antiferromagnetic di-\nrect exchange interaction between the nearest Cr atoms com-\nbined with a small ferromagnetic superexchange component\nfrom the 82\u000eCr-O-Cr interaction. For positive amplitudes of\ntheA1g(9) mode, the Cr-Cr distance increases thus decreas-\ning the antiferromagnetic direct exchange interaction. At the6\nFIG. 4: (a) Illustration of the magnetic exchange interactions in Cr 2O3, from first to fifth nearest neighbor. (b,c) Changes in the magnetic\nexchange interactions due to structural modifications by the A1g(9) andA2u(17) modes. Note that for the A1g(9) mode the nearest-neighbor\nmagnetic exchange ( J1) changes sign for negative phonon mode amplitudes.\nTABLE III: Upper panel: Magnetic exchange interactions (meV) for\nthe ground-state structure of Cr 2O3. Lower panel: Spin-phonon cou-\npling constants (units meV/(pu˚A) and meV/(pu˚A)2for first/second\norder) for the A1gandA2umodes of Cr 2O3.\nJ1 J2 J3 J4 J5\n25.4 21.2 -3.9 -3.3 4.2\nn 1 2 3 4 5\nA1g(9)\n@Jn=@\u0018 -57.9 -4.4 -0.1 -0.6 -1.0\n@2Jn=@\u0018214.5 0.2 0.1 0.0 0.1\nA1g(17)\n@Jn=@\u0018 12.4 1.4 -0.0 0.1 0.3\n@2Jn=@\u001823.9 0.4 0.1 0.0 0.0\nA2u(12)\n@2Jn=@\u001820.5 0.1 -0.1 0.0 0.2\nA2u(17)\n@2Jn=@\u00182-0.7 0.1 0.0 0.0 0.0\nsame time, the Cr-O-Cr angle becomes closer to 90\u000eenhanc-\ning the ferromagnetic superexchange. The result is a change\nin sign ofJ1. We note that this observation is possibly con-\nnected to the findings of Ref. [27], in which strong modula-tions of magnetic energies induced by small changes of the\nCr2O3ground-state structure were reported. Moreover, since\nthe direct magnetic exchange interaction only affects J1, the\nmagnetic exchange interactions Jnwithn\u00152are less af-\nfected by the structural distortion.\nFinally, we calculate the MCA energy of Cr 2O3, from\nthe energy difference between alignment of the Cr spin mo-\nments along ( Ejj) and perpendicular ( E?) to the rhombohe-\ndral axis, including the spin-orbit interaction in our calcula-\ntions. We obtain an energy difference Ejj\u0000E?= -27\u0016eV; the\nexperimental35,46,47values range from -12 \u0016eV to -16\u0016eV . We\nalso calculate the change in MCA energy when the structure\nis modulated by the A1gorA2uphonon modes and find no\nsignificant change (a mode amplitude of \u0018=\u00062pu˚A lowers\nthe MCA energy by \u001910 %). In particular, the rhombohedral\neasy axis is preserved upon structural modulation. This find-\ning justifies our omission of MCA terms in our spin-phonon\nHamiltonian, Eqn. (3).\nTo summarize this section, we find a strong dependence\nof theJ1nearest-neighbor magnetic exchange interaction on\nthe structural distortion associated with the A1g(9) mode,\nwith positive mode amplitude, corresponding to increased\nCr-Cr distance, inducing a change in sign. This depen-\ndence leads to a crossover between antiferromagnetic states.\nSince theA1g(9) mode couples quadratic-linearly to the A2u\nmodes, this crossover can be induced by optical excitation of\nthe polar modes. Following the classical considerations de-\nrived in Refs. [10,13], we estimate that a pulse fluence of7\n\u001840 mJ/cm2at a frequency 17 THz should be sufficient to in-\nduce this crossover transition. A similar fluence was reported\nin Ref. [16] without damaging the sample.\nIV . NON-LINEAR LATTICE DYNAMICS\nHaving established that the structural modification induced\nvia non-linear phononic coupling can lead to a change in mag-\nnetic ordering, we next evaluate the dynamical behavior asso-\nciated with driving a phonon. We begin by calculating the\nnon-linear lattice dynamics using the vibrational crystal po-\ntential given in Eqn. (1), followed by the resulting spin dy-\nnamics. We study the case in which an IR mode is excited\nby a sinusoidal driving force F(t)with amplitude Edrive with\nfrequency \nand calculate the resulting dynamics of the cou-\npled R mode, focussing particularly on the combination of the\nA2u(17) andA1g(9) which yields a negative amplitude A1g(9)\ndisplacement and possible ferromagnetism. The time evolu-\ntion of the system described by the potential of Eqn. (1) is\nthen governed by the following set of differential equations:\n\u0018IR+!2\nIR\u0018IR+\rIR\u00183\nIR= 2g\u0018IR\u0018R+F(t); (6)\n\u0018R+!2\nR\u0018R+\rR\u00183\nR=g\u00182\nIR; (7)\nF(t) =Edrivesin(\nt): (8)\nWe derive a closed analytical solution of the dynamic equa-\ntions in the limit in which the coupling and the anharmonicity\nare small relative to the frequency, that isg\rIR\rR\n!IR!R\u001c1by\nfollowing the approach of Ref. [48]. For the case of Cr 2O3,\nourab initio values, provided in Table I and II, indicate that\nthe combination of A1gwith polarA2umodes fulfills this cri-\nterion. The dynamics of the IR mode are then given by:\n\u0018IR(t) =AIRsin[e!IRt] +A\nsin[\nt]\n+gAIRAR\n2[!2\nIR\u0000(!IR\u0000!R)2]cos[(e!IR\u0000e!R)t]\n+gAIRAR\n2[!2\nIR\u0000(!IR+!R)2]cos[(e!IR+e!R)t](9)\n+gA\nAR\n2[!2\nIR\u0000(\n\u0000!R)2]sin[(\n\u0000e!R)t]\n+gA\nAR\n2[!2\nIR\u0000(\n +!R)2]sin[(\n + e!R)t];\nand those of the R mode by:\n\u0018R(t) =\u0018R0+ARcos[e!Rt]\n+gA2\nIR\n4[!2\nR\u00004!2\nIR]cos[2e!IRt]\n+gA2\n\n4[!2\nR\u00004\n2]cos[2\nt] (10)\n+gAIRA\n2[!2\nR\u0000(\n +!IR)2]sin[(\n + e!IR)t]\n+gAIRA\n2[!2\nR\u0000(\n\u0000!IR)2]sin[(\n\u0000e!IR)t]:The time-independent displacement of the R mode oscillation\nis given by\u00180=g(A2\nIR+A2\n\n)=(4!2\nR), with the amplitude fac-\ntors,AIRandARdepending on the initial amplitudes, \u0018R(0)\nand\u0018IR(0)andA\n= 1=(!2\nIR\u0000\n2). We indicate frequencies\nwith a tilde which have been renormalized by the anharmonic\ncoupling, as given by Eqns. (A.1) and (A.2) in the Appendix.\nThe solution shows that the anharmonic potential and the\ncoupling between the phonon modes induce motions of the\noscillators which display several components given by cosine\nand sine terms. Each of these terms corresponds to a single\ncomponent of the motion with a specific amplitude and fre-\nquency – either the renormalized original frequency of each\noscillator, indicated by the tilde, or sums or differences of the\noriginal frequencies. We emphasize that these motions arise\nfrom a single mode, which exhibits multiple frequencies be-\ncause of its anharmonicity and coupling.\nNext we analyze the frequencies and amplitudes of each\nterm in Eqn. 10 for the A1g(9) R mode. In Fig. 5 (a,b) we\nshow the frequencies and relative amplitudes of the R mode\nmotions as a function of the drive frequency \n, obtained us-\ning the parameters for the A2u(17) (IR) –A1g(9) (R) coupled\nphonon modes. Note that the only effect of the external driv-\ning amplitude, Edrive, is to scale the amplitude of the mo-\ntion. We see that for drive frequencies close to the 17 THz\neigenfrequency of the IR mode, the frequencies of the R mode\ncomponents range from sub THz to 40 THz (note the logarith-\nmic scale in the lower part of Fig. 5 (a)), with the highest fre-\nquency components at around 34 THz being twice the renor-\nmalized IR mode eigenfrequency (blue line), twice the driver\nfrequency (green dashed line) and the sum of the renormal-\nized IR mode and driver frequencies (red dashed-dotted line).\nThe renormalized R mode frequency is close to 10 THz, and\nlike the renormalized IR mode frequency is independent of the\ndrive frequency. The frequency of the lowest frequency com-\nponent of the motion is given by the difference between the\nfrequency of the driver and the IR mode eigenfrequency, and\nas a result it has a strong dependence on the drive frequency,\nbecoming small as the drive frequency approaches the eigen-\nfrequency of the IR mode (note that the divergence when the\ndrive frequency equals the eigenfrequency of the IR mode is\nnot physical, and arises because of the absence of damping in\nour simulations.)\nIn Fig. 5 (b) we show the relative amplitudes of each fre-\nquency component normalized to the time-independent dis-\nplacement\u0018R0of\u0018R(t)which we set to 100 %. We see a large\nspread in amplitudes for the different components of the mo-\ntion, with the motions with the renormalized IR and R eigen-\nfrequencies having the largest amplitudes, in the order of 10 to\n20 % of\u0018R0, as well as minimal dependence on the drive fre-\nquency. The other three motion components, whose frequen-\ncies depend explicitly on the drive frequency, have strongly\ndrive-frequency-dependent amplitudes, as expected. Of these,\nthe high-frequency 2 \nmotion has the smallest amplitude fol-\nlowed by the \n +e!IRmotion, with the slow \n\u0000e!IRmo-\ntion having the largest amplitude, becoming similar in size\nto thee!IRande!Rmotions in the vicinity of the eigenfre-\nquency of IR mode. Again, the divergence when the mode\nfrequency matches the driver frequency results from the ab-8\nFIG. 5: (a) Frequencies, !, and (b) relative amplitudes, \u0018(normal-\nized to\u0018R0), of the five separate parts of the R mode motion as a\nfunction of the external driving frequency. Note the separation and\nthe linear and logarithmic scales in (a).\nsence of damping in our model, and so we do not analyze this\npoint in detail.\nTo summarize this section, we find that, in addition to the\ntime-independent offset \u0018R0of the R mode induced by its\nquadratic-linear coupling to the IR mode, the R mode has a\ncomplex oscillatory motion made up of different frequencies.\nThe largest amplitude motions have high frequencies, set by\nthee!IRande!Rfrequencies. Close to resonance between the\ndrive and IR-mode frequencies, an additional component of\nthe motion with a slower frequency \n\u0000e!IRalso develops a\nsignificant amplitude. This slow motion is particularly inter-\nesting since it is tunable in amplitude and frequency by the\nexternal driver; in the next section we will explore how it can\nbe exploited to engineer the spin dynamics.\nV . SPIN DYNAMICS\nNext we discuss how the structural modulations we de-\nscribed above drive the spin dynamics, by combining our\nfindings for the structural dynamics with those for the spin-\nphonon coupling. The time-dependent exchange modulationinduced by the structural modulation is obtained by combin-\ning Eqn. (4) for Jn(\u0018)with Eqn. (10) for \u0018(t)to yield the\nJn(\u0018(t)). We include only the modulations caused by the\nA1g(9) R mode. While this mode is driven by the excita-\ntion of theA2u(17) (IR) mode, the latter has negligible ef-\nfect on the exchange interactions, and so the time-dependent\nmagnetic exchange modulations are dominated by Jn(\u0018R(t)).\nFor the spin-dynamics we consider a single Cr 2O3unit-cell\nwith four magnetic Cr sites and periodic boundary conditions.\nRewriting the Heisenberg Hamiltonian of Eqn. (5) we obtain\nHmag;exch\nfac(\u0018R(t)) = ~J1(\u0018R(t))(S1\u0001S2+S3\u0001S4)\n+~J2(\u0018R(t))(S1\u0001S4+S2\u0001S3)(11)\n+~J3(\u0018R(t))(S1\u0001S3+S2\u0001S4);\nwhere the net magnetic exchange interactions ~Jiare given by\n~J1(\u0018R(t)) =J1(\u0018R(t)) + 3J3(\u0018R(t))\n~J2(\u0018R(t)) =3J2(\u0018R(t)) +J5(\u0018R(t)) (12)\n~J3(\u0018R(t)) =6J4(\u0018R(t))\nandS1toS4are the four classical spins in the unit cell as\nshown in Fig. 1 (a). Our full magnetic Hamiltonian is then\nHmag(t) =Hmag;exch\nfac(\u0018R(t)) +D4X\ni=1(Sz\ni)2(13)\nwhere\u0018R(t))denotes the specific time-dependent exchange\ninteraction strength (see Eqn. (12) and Table III) and Dis\nthe MCA energy which we fixed to the computed equilibrium\nvalue.\nWe next calculate the classical magnetization dynamics\nusing the Landau-Lifshitz-Gilbert equation24,25,49within an\natomistic approach50,51\ndSi\ndt=\u0000\r\n1 +\u000b2h\nSi\u0002He\u000b\ni(t)i\n\u0000\u000b\r\n1 +\u000b2h\nSi\u0002h\nSi\u0002He\u000b\ni(t)ii\n:(14)\nHereHe\u000b\ni(t) =\u00001\n\u0016B\u000eHmag;exch\nfac(t)\n\u000eSiwith\u0016Bthe Bohr magne-\nton,\ris the electron gyromagnetic ratio and \u000bis the Gilbert\ndamping. We take the value of Gilbert damping for Cr 2O3,\n\u000b\u0019\r\n2\u0019\u0017\u0001Bpp\u00190:07, estimated from spectral line widths\nmeasured at room temperature using electron paramagnetic\nresonance36.\nNext, we calculate ~Ji(\u0018R(t))when the structure is modified\nby the quadratic-linear coupling of the A1g(9) andA2u(17)\nphonon modes. We excite the latter using a continuous field\nof strength of Edrive =0.6 MV/cm oscillating at a frequency\nof 16.9 THz; the result is shown in Fig. 6 (a). Note that we\ninclude a noise corresponding to a temperature of 0.1 K in our\nspin-dynamics simulation to prevent the system from becom-\ning stuck in shallow metastable minima52.\nBefore the mode is excited (at t= 0), all ~Jiare constant,\nwith ~J1and~J2positive and ~J3negative. When the oscillat-\ning field is applied, the frequency-dependent induced struc-\ntural changes described in the previous section change ~Jicor-\nresponding to the changes in bond lengths and angles. We9\nFIG. 6: Spin dynamics in the phonon-driven state of Cr 2O3. (a) Mod-\nulation of the magnetic exchange interaction ~Ji(\u0018R(t)) with\u0018R(t)\nderived from Eqn. (10). The laser field, E(t), is switched on at time\nt= 0, withEdrive =0.6 MV/cm at \n =16.900 THz. The change in\nsign of the average exchange for ~J1is clearly visible; note that be-\ncause of the fast oscillating components ( !\u001530 THz) of the \u0018Rmo-\ntion the time-dependent exchange interaction can not be resolved on\nthis scale. (b,c) Time dependence of the z-components (normalized\nto their ground-state values) of the four spin magnetic moments in\nthe Cr 2O3unit cell with the labeling corresponding to Fig. 1 (a). The\nblue spheres (Cr atoms) and red arrows (spins) represent the Cr 2O3\nmagnetic ground state.\nsee that, while the magnetic exchange interactions oscillate,\ntheaverage magnetic exchange interactions ~J1change sign\nto negative, reflecting a net ferromagnetic interaction between\nthe nearest neighbor sites. Since the next-nearest neighbor in-\nteraction ~J2still prefers an AFM alignment the system does\nnot become fully FM but instead adopts the AFM 1state with\nits(\";\";#;#)ordering of magnetic moments on the Cr sites.\nThis is consistent with the cross-over to the AFM 1state with\nincreasingA1g(9) amplitude that we saw in the first part of\nthis paper.\nThe remaining panels of Fig. 6 show the response of the\nspin system to this modification of ~Ji, with (b) and (c) show-\ning the time evolution of the z-component of magnetization\nof the individual Cr ions and (d) that of the total spin mo-\nment of the unit cell, Stot;z(t) =1\nNPN\ni=1Si;z(t). The spins\nreact to the change in their average exchange modulation\nand reorient on the same time scale as the J-oscillations into\nthe new AFM arrangement. Note that this process is at the\nsame speed of the period of the exchange excitation oscilla-\ntions, 2\u0019=e!R\u00190.1 ps. The AFM arrangement then achieves a\nsteady state without further dynamical evolution provided that\nthe displacement of the A1g(9) continues by excitation of the\nFIG. 7: Spin dynamics in the phonon-driven state of Cr 2O3. (a)\nModulation of the magnetic exchange interaction ~Ji(\u0018R(t))with\n\u0018Rderived from Eqn. (10). The laser field E(t)is switched on at\ntimet= 0, withEdrive =0.6 MV/cm at \n = 16.995 THz. Setting\nthe excitation frequency closer to resonance induces a slow, large-\namplitude modulation component in the \u0018Rmotion, which becomes\nresolvable in the time-dependent magnetic exchange. (b,c) Time de-\npendence of the z-component of the four spin magnetic moments in\nthe Cr 2O3unit cell with the labeling corresponding to Fig. 1 (a) and\nthe spin magnitudes normalized to their static ground-state values.\nWe illustrate the Cr 2O3magnetic ground state by the blue spheres\nrepresenting the Cr atoms with the arrows showing the magnetic mo-\nments.\nA2u(17) phonon mode.\nNext, we exploit our finding from Section IV that a compo-\nnent of the R mode motion can be tuned to low frequency with\nan increased amplitude by selecting a drive frequency \nclose\nto resonance. In Fig. 7 (a) we show the time dependence of ~Ji,\ncalculated for the combination of A1g(9) andA2u(17) modes\nusing the analytical solution of Eqn. (10), this time with the\ndriving frequency, \n = 16.995 THz, close to resonance. It is\nclear that the oscillation frequency of the exchange interac-\ntions develops a significant slow component with a frequency\naround 10GHz. The resulting spin dynamics are depicted\nin the lower panels of Fig. 7. In contrast to the case shown\nin Fig. 6, a steady AFM 1state is not achieved on pumping,\nand instead the spins exhibit a flipping between up and down\nalignment. Again the spin dynamics behavior persists as long\nas the phonon mode is driven.\nIn conclusion, our calculations indicate that the quadratic-\nlinear coupling between the A1g(9) andA2u(17) modes leads\nto a reversal of the average value of the nearest-neighbor ex-\nchange between the Cr ions when the optical A2u(17) mode\nis continuously excited with sufficiently large amplitude. De-\npending on the closeness of the excitation laser frequency to10\nthe eigenfrequency of the A2u(17) mode, the additional oscil-\nlatory component of ~Ji(t)can be either fast or slow. In the\nfirst case the system responds with a steady-state change in its\nmagnetization to a ferromagnetic state; in the second an alter-\nnating switching occurs on a tens of picoseconds time scale.\nWe note that the two limits shown here represent a small frac-\ntion of spin-dynamic possibilities, with the tuning of the drive\nfrequency relative to the resonance, as well as on-off schemes\nfor the excitation, offering the potential to modulate the ex-\nchange interactions in multiple complex ways.\nVI. SUMMARY\nWe calculated the structural and magnetic responses of\nchromium oxide, Cr 2O3, to intense excitation of its optically\nactive phonon modes. Using a general spin-lattice Hamil-\ntonian, with parameters calculated from first principles, we\nshowed that the quasi-static structural distortion introduced\nthrough the non-linear phonon-phonon interaction can change\nthe magnetic state from its equilibrium antiferromagnetic to a\nnew antiferromagnetic ordering with ferromagnetically cou-\npled nearest-neighbor spins. This transition is driven by\nthe change in nearest-neighbor magnetic exchange interaction\nwhen the Cr-Cr separation is modified through non-linear cou-\npling of the optical phonons to a symmetry-conserving A1g\nRaman-active mode.The new antiferromagnetic ground state\npersists for as long as the system is continuously excited, pro-\nvided that the excitation frequency is faster than the magneticrelaxation time.\nRegarding dynamics, we find that the motion of the ex-\ncited optical modes and coupled Raman-active mode can be\ndecomposed into several different frequencies which depend\nstrongly on the difference between the excitation and reso-\nnance frequencies. This sensitivity of the response to the input\nfrequency allows selection of complex vibrational frequency\npatterns which can lead to additional components in the spin\ndynamics, for example flips of the Cr spin lattice.\nWe emphasize that we explored in this work a minimal\nmodel of phonon-driven spin dynamics, and we expect that\nextensions of the model will reveal yet more complex physics,\nsuch as dynamically frustrated or spin-spiral states. We\nhope that our work will inspire additional theoretical and ex-\nperimental studies to uncover the rich behavior of coupled\nmagneto-phononic systems.\nVII. ACKNOWLEDGMENTS\nThis work was supported financially by ETH Zurich, the\nERC Advanced Grant program, No. 291151 (MF, CK and\nNAS), the ERC under the European Union’s Seventh Frame-\nwork Programme (FP7/2007-2013) / ERC Grant Agreement\nn\u000e319286 (Q-MAC) and by the DFG through SFB762 and\nTRR227. Calculations were performed at the Swiss National\nSupercomputing Centre (CSCS) under project ID s624. MF\nand NAS thank Andrea Cavalleri and Tobia F. Nova for useful\ndiscussions.\n\u0003Electronic address: michael.fechner@mpsd.mpg.de\n1M. F ¨orst, C. Manzoni, S. Kaiser, Y . Tomioka, Y . Tokura, R. Mer-\nlin, and A. Cavalleri, Nature Phys. 7, 854 (2011).\n2A. Subedi, A. Cavalleri, and A. Georges, Phys. Rev. B 89, 220301\n(2014).\n3M. Rini, R. Tobey, N. Dean, J. Itatani, Y . Tomioka, Y . Tokura,\nR. W. Schoenlein, and A. Cavalleri, Nature 449, 72 (2007).\n4A. D. Caviglia, R. Scherwitzl, P. Popovich, W. Hu,\nH. Bromberger, R. Singla, M. Mitrano, M. C. Hoffmann,\nS. Kaiser, P. Zubko, et al., Phys. Rev. Lett. 108, 136801 (2012).\n5V . Esposito, M. Fechner, R. Mankowsky, H. Lemke, C. M, J. M.\nGlownia, M. Nakamura, M. Kawasaki, Y . Tokura, U. Staub, et al.,\nPhys. Rev. Lett. 118, 247601 (2017).\n6S. Kaiser, C. R. Hunt, D. Nicoletti, W. Hu, I. Gierz, H. Y . Liu,\nM. Le Tacon, T. Loew, D. Haug, B. Keimer, et al., Phys. Rev. B\n89, 184516 (2014).\n7W. Hu, S. Kaiser, D. Nicoletti, C. R. Hunt, I. Gierz, M. C. Hoff-\nmann, M. Le Tacon, T. Loew, B. Keimer, and A. Cavalleri, Nature\nMater. 13, 705 (2014).\n8M. Mitrano, A. Cantaluppi, D. Nicoletti, S. Kaiser, A. Perucchi,\nS. Lupi, P. Di Pietro, D. Pontiroli, M. Ricc `o, S. R. Clark, et al.,\n530, 461 (2016).\n9R. Mankowsky, A. Subedi, M. F ¨orst, S. O. Mariager, C. M, H. T.\nLemke, J. S. Robinson, J. M. Glownia, M. P. Minitti, A. Frano,\net al., Nature 516, 71 (2014).\n10M. Fechner and N. A. Spaldin, Phys. Rev. B 94, 134307 (2016).\n11R. Mankowsky, M. Fechner, M. F ¨orst, A. von Hoegen, J. Porras,T. Loew, G. L. Dakovski, M. Seaberg, S. M ¨oller, G. Coslovich,\net al., Structural Dynamics 4, 044007 (2017).\n12M. Gu and J. M. Rondinelli, Phys. Rev. B 95, 024109 (2017).\n13D. M. Juraschek, M. Fechner, and N. A. Spaldin, Phys. Rev. Lett.\n118, 054101 (2017).\n14A. Subedi, Phys. Rev. B 92, 214303 (2015).\n15D. M. Juraschek, M. Fechner, A. V . Balatsky, and N. A. Spaldin,\nPhys. Rev. Mat. 1, 014401 (2017).\n16R. Mankowsky, A. von Hoegen, M. F ¨orst, and A. Cavalleri, Phys.\nRev. Lett. 118, 197601 (2017).\n17M. F ¨orst, R. I. Tobey, S. Wall, H. Bromberger, V . Khanna, A. L.\nCavalieri, Y . D. Chuang, W. S. Lee, R. Moore, W. F. Schlotter,\net al., Phys. Rev. B 84, 241104 (2011).\n18M. F ¨orst, A. D. Caviglia, R. Scherwitzl, R. Mankowsky, P. Zubko,\nV . Khanna, H. Bromberger, S. B. Wilkins, Y . D. Chuang, W. S.\nLee, et al., Nature Mater. 14, 883 (2015).\n19T. F. Nova, A. Cartella, A. Cantaluppi, M. F ¨orst, D. Bossini, R. V .\nMikhaylovskiy, A. V . Kimel, R. Merlin, and A. Cavalleri, Nature\nPhys. 13, 132 (2016).\n20E. Beaurepaire, J. C. Merle, A. Daunois, and J. Y . Bigot, Phys.\nRev. Lett. 76, 4250 (1996).\n21A. V . Kimel, R. V . Pisarev, J. Hohlfeld, and T. Rasing, Phys. Rev.\nLett. 89, 287401 (2002).\n22C. D. Stanciu, F. Hansteen, A. V . Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett. 99, 047601\n(2007).\n23T. Kubacka, J. A. Johnson, M. C. Hoffmann, C. Vicario,11\nS. de Jong, P. Beaud, S. Gr ¨ubel, S.-W. Huang, L. Huber,\nL. Patthey, et al., Science 343, 1333 (2014).\n24L. Landau and E. Lifshitz, Phyz. Zeitsch. der Sow. 8, 153 (1935).\n25T. L. Gilbert and J. M. Kelly, Conf. Magnetism and Magnetic Ma-\nterials 14, 253 (1955).\n26C. Wang, G.-C. Guo, and L. He, Phys. Rev. B 77, 134113 (2008).\n27S. Shi, A. Wysocki, and K. D. Belashchenko, Phys. Rev. B 79,\n104404 (2009).\n28M. Mostovoy, A. Scaramucci, N. A. Spaldin, and K. T. Delaney,\nPhys. Rev. Lett. 105, 087202 (2010).\n29S. Mu, A. L. Wysocki, and K. D. Belashchenko, Phys. Rev. B 89,\n174413 (2014).\n30G. Kresse and J. Furthm ¨uller, Phys. Rev. B 54, 11169 (1996).\n31P. E. Bl ¨ochl, Phys. Rev. B 50, 17953 (1994).\n32J. Iniguez, Phys. Rev. Lett. 101, 117201 (2008).\n33A. Malashevich, S. Coh, I. Souza, and D. Vanderbilt, Phys. Rev.\nB86, 094430 (2012).\n34A. Sukhov and J. Berakdar, J. Phys. Condens. Matter 20, 125226\n(2008).\n35S. Foner, Phys. Rev. 130, 183 (1963).\n36F. S. Stone and J. C. Vickerman, Transactions of the Faraday So-\nciety 67, 316 (1971).\n37I. E. Dzyaloshinskii, JETP Lett. 10, 628 (1960).\n38D. N. Astrov, Sov. Phys. JETP 11, 708 (1960).\n39L. W. Finger and R. M. Hazen, J. Appl. Phys. 51, 5362 (1980).\n40I. R. Beattie and T. R. Gilson, J. Chem. Soc., A 5, 980 (1970).\n41G. Lucovsky, R. J. Sladek, and J. W. Allen, Phys. Rev. B 16, 4716\n(1977).\n42S. H. Shim, T. S. Duffy, R. Jeanloz, C.-S. Yoo, and V . Iota, Phys.Rev. B 69, 4253 (2004).\n43S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, Rev.\nMod. Phys. 73, 515 (2001).\n44H. J. Xiang, E. J. Kan, S.-H. Wei, M. H. Whangbo, and X. G.\nGong, Phys. Rev. B 84, 224429 (2011).\n45N. S. Fedorova, C. Ederer, N. A. Spaldin, and A. Scaramucci,\nPhys. Rev. B 91, 165122 (2015).\n46K. L. Dudko, V . V . Eremenko, and L. M. Semenenko, Phys. Status\nSolidi B 43, 471 (1971).\n47D. Tobia, E. De Biasi, M. Granada, H. E. Troiani, G. Zampieri,\nE. Winkler, and R. D. Zysler, J. Appl. Phys. 108, 104303 (2010).\n48A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic\nDynamics, 2nd Ed., Applied Mathematical Sciences (Springer,\n1992).\n49T. L. Gilbert and J. M. Kelly, IEEE Trans. Magn. 40, 3443 (2004).\n50O. Chubykalo-Fesenko, U. Nowak, R. W. Chantrell, and\nD. Garanin, Phys. Rev. B 74, 094436 (2006).\n51B. Skubic, J. Hellsvik, L. Nordstrom, and O. Eriksson, J. Phys.\nCondens. Matter 20, 315203 (2008).\n52A. Sukhov and J. Berakdar, Phys. Rev. Lett. 102, 057204 (2009).\nAppendix: Appendix\n1. Detailed expressions for the renormalized frequencies\nThe explicit expressions for the mode frequencies renor-\nmalized by the anharmonic coupling are\ne!IR=!IR\u0000g2A2\nR\n8!2\nR!IR+g2A2\n\n4!2\nR!IR\u0000g2AIR\n16!IR[!2\nR\u00004!2\nIR]\u0000g2A2\n\n4!IR[!2\nR\u0000(\n +!IR)2]\u0000g2A2\n\n4!IR[!2\nR\u0000(\n\u0000!IR)2]\n\u0000g2A2\nR\n8!IR[!2\nIR\u0000(!IR+!R)2]\u0000g2A2\nR\n8!IR[!2\nIR\u0000(!IR\u0000!R)2]+3\rIRA2\nIR\n8!IR+3\rIRA2\n\n4!IR; (A.1)\nfor the IR mode, and\ne!R=!R\u0000g2A2\nIR\n8!2\nR[!2\nIR\u0000(!IR+!R)2]\u0000g2A2\nIR\n8!R[!2\nIR\u0000(!IR\u0000!R)2]\u0000g2A2\n\n8!R[!2\nIR\u0000(\n\u0000!R)2]+3\rRA2\nR\n8!R; (A.2)\nfor the R mode."    },    {        "title": "1002.2217v1.Dynamics_of_Rotating__Magnetized_Neutron_Stars.pdf",        "content": "arXiv:1002.2217v1  [gr-qc]  10 Feb 2010August 18, 2018 6:28 WSPC - Proceedings Trim Size: 9.75in x 6. 5in main\n1\nDynamics of Rotating, Magnetized Neutron Stars\nSTEVEN L. LIEBLING\nPhysics Department–C.W. Post Campus, Long Island Universi ty,\nBrookville, NY 11548, U.S.A.\n∗E-mail: steve.liebling@liu.edu\nUsing a fully general relativistic implementation of ideal magnetohydrodynamics with no\nassumed symmetries in three spatial dimensions, the dynami cs of magnetized, rigidly ro-\ntating neutron stars are studied. Beginning with fully cons istent initial data constructed\nwith Magstar, part of the Lorene project, we study the dynami cs and stability of ro-\ntating, magnetized polytropic stars as models of neutron st ars. Evolutions suggest that\nsome of these rotating, magnetized stars may be minimally un stable occurring at the\nthreshold of black hole formation.\nKeywords : stellar evolution; MHD; neutron stars; magnetars\nIntroduction: Neutron stars are fascinating astrophysical objects melding in-\ncredibly strong gravity with large densities and pressures. Along wit h powerful\nmagnetic fields, these compact stars form magnetars which are su spected as the\nengines behind anomalous X-ray pulsars (AXPs) and soft gamma ray repeaters\n(SGRs).1There is a long history of numerical models of neutron stars, but he re\ntheir evolution is studied using a code developed for, and applied to, a number\nof astrophysical problems.2–7This code adopts a fully nonlinear scheme for the\nEinstein gravitational field equations which allows for the easy extra ction of gravi-\ntationalwavesignatures.Thisgravityiscoupledtoamagnetohydr odynamic(MHD)\ncomponent which uses high resolution shock capturing (HRSC) meth ods to evolve\na magnetized fluid in a general relativistic, finite-difference scheme. Further details\nconcerning the formulation and numerical methods are presented in Ref. 8.\nA number of tests are presented in Ref. 8. Evolutions of stable, ro tating stars\ndemonstrateconvergencewith increasingresolution.That is,con servationofquanti-\nties such as baryon mass and the z-component of the angular momentum improves\nas one moves to better resolution. Likewise, violations of the const raints remain\ncloser to zero with better resolution.\nResults: AlsodiscussedinRef.8wereevolutionsofunstablestellarsolutions. Such\nsolutions either expand, oscillating about some stable star, or they instead collapse,\neventually forming a black hole. Beginning with a particular unstable so lution,\nunavoidable numerical error will ultimately drive it to one of these end states.\nHowever, it was found in Ref. 8 that one could tune the perturbatio n via some\ngeneralized parameter p(e.g.the amplitude of some perturbation to the initial\npressure) such that for some critical value p∗, ifp > p∗then the solution collapsed\nand ifp < p∗the solution expanded. This ability to tune to threshold suggests\nthat the unstable solutions have just a single unstable mode, and th erefore that the\nsolutions represent Type I critical solutions.\nOne question remaining, among many, is whether initial data far from theseAugust 18, 2018 6:28 WSPC - Proceedings Trim Size: 9.75in x 6. 5in main\n2\nunstable solutions in some configuration space will, upon tuning, evolv e towards\nthese unstable solutions and find them at threshold. The work of Re f. 9 answers\nthis question with respect to nonrotating, spherical stars in sphe rical symmetry by\nevolving stable TOV solutions perturbed gravitationally by a pulse of s calar field.\nFor energetic enough scalar pulses, the stable star is sufficiently dis turbed that it\ncollapses to a black hole. Tuning this initial scalar pulse, they find longe r and longer\nlived unstable stellar solutions.\nBecause this code alreadyhas the capability to evolvea massless,co mplex scalar\nfield (as studied within the context of the dynamics of boson stars2,3), it is straight-\nforward to include a scalar pulse as initial data. In particular, the re al component of\nthe scalar field φ(x,y,z,t) is defined as a Gaussian pulse, φ(t= 0) =Ae(r−R0)2/δ2,\nin terms of a radial coordinate r≡/radicalbig\nǫxx2+ǫyy2+z2with real constants ǫx,ǫy,\nR0,δ, andA. The imaginary component remains zero throughout the evolution\nand we define the initial time derivative of the field such that φis approximately\nin-going. There are two significant differences with respect to the w ork of Ref. 9:\n(i) the Einstein field equations are not re-solved to account for the scalar pulse and\n(ii) the scalar pulse is located outside the star initially, but not very fa r from it.\nIn particular, two searches are carried out, both choosing R0= 30 and δ= 6, and\nusingAas the generalized parameter pwhich is tuned to threshold. The first search\nuses a spherically symmetric pulse described by ǫx= 1,ǫy= 1, and the second uses\nǫx= 1.25,ǫy= 0.75.\nIt is found that tuning a single parameter results in the evolution app roaching\nan unstable solution which once again suggests that the unstable so lution is itself a\nco-dimension one unstable solution. The time T0for which the evolution is near the\nunstable solution is expected to scale as T0=−σln|p−p∗|+Cfor some constant\nCthat depends on the particular initial-data family and some universal constant\nσ, the inverse of the growth rate of the instability. As indicated by Fig . 1, the\nresults are consistent with this expected scaling, given the relative ly large errors\nhere. These numerical errors appear to be related to effects fro m the boundary\nwhich significantly limit how close to criticality one can achieve. These tw o searches\nare both tuned to approximately one part in a million.\nAnother interesting aspect to the dynamics of magnetized stars c oncerns the\nbehavior of the magnetic field. In particular, for the case in which th e magnetic\nmoment of the star is not aligned with the rotational axis, one can loo k at the\nstructure of the dynamic magnetic field in terms of poloidal and toro idal compo-\nnents. In Fig. 2, the evolution of rotating star with a magnetic mome nt rotated\naway from the z-axis is shown. Because the initial magnetic field is confined to the\ninterior of the star, problems that generally occur when the magne tic field is strong\ncompared to the density are avoided.\nAcknowledgments: I thank J. Novak for his assistance with Magstar and\nMatthew Choptuik and Scott Noble for valuable discussions. This wor k was sup-\nported by the National Science Foundation under grant PHY-0325 224 and also\nthroughTeraGridresourcesprovidedbySDSCunderallocationawa rdPHY-040027.August 18, 2018 6:28 WSPC - Proceedings Trim Size: 9.75in x 6. 5in main\n3\nReferences\n1. S. Mereghetti, Astron. Astrophys. Rev. 15, 225 (2008).\n2. C. Palenzuela, I. Olabarrieta, L. Lehner and S. Liebling, Phys. Rev. D75, p. 064005\n(2007).\n3. C. Palenzuela, L. Lehner and S. L. Liebling, Phys. Rev. D77, p. 044036 (2008).\n4. M. Anderson et al.,Phys. Rev. D77, p. 024006 (2008).\n5. M. Anderson et al.,Phys. Rev. Lett. 100, p. 191101 (2008).\n6. C. Palenzuela, M. Anderson, L. Lehner, S. L. Liebling and D . Neilsen, Phys. Rev. Lett.\n103, p. 081101 (2009).\n7. M. Megevand et al.,Phys. Rev. D80, p. 024012 (2009).\n8. S. L. Liebling, L. Lehner, D. Neilsen and C. Palenzuela (20 10).\n9. S. Noble, PhD Thesis (2003).\nFig. 1. Results of two searches using a scalar pulse to pertur b the same, magnetic, rotating star.\nLeft:Stellar properties as functions of time in units of the perio d of rotation for near-critical\nevolutions. Right: Survival time of evolutions for the two searches demonstrat es the expected\nscaling with the distance from criticality.\nFig. 2. Evolution of a rotating solution (counter-clockwis e) with magnetic moment inclined with\nrespect to the z-axis ofπ/4 radians. Shown are the values of |/vectorB|2on thez= 0 plane for roughly\nhalf a period (from left to right). The computational domain extends in each direction a factor of\nfour times the distance shown."    },    {        "title": "2012.12763v2.Magnetic_Moment_Tensor_Potentials_for_collinear_spin_polarized_materials_reproduce_different_magnetic_states_of_bcc_Fe.pdf",        "content": "Magnetic Moment Tensor Potentials for collinear spin-polarized materials reproduce\ndi\u000berent magnetic states of bcc Fe\nIvan Novikov *,1, 2Blazej Grabowski,2Fritz K ormann,3, 4and Alexander Shapeev1\n1Skolkovo Institute of Science and Technology, Skolkovo Innovation Center, Nobel St. 3, Moscow 143026, Russia\n2Institute for Materials Science, University of Stuttgart, Pfa\u000benwaldring 55, 70569 Stuttgart, Germany\u0003\n3Computational Materials Design, Max-Planck-Institut f ur Eisenforschung GmbH, D-40237 D usseldorf, Germany\n4Materials Science and Engineering, Delft University of Technology, 2628 CD, Delft, The Netherlands\n(Dated: December 13, 2021)\nWe present the magnetic Moment Tensor Potentials (mMTPs), a class of machine-learning in-\nteratomic potentials, accurately reproducing both vibrational and magnetic degrees of freedom as\nprovided, e.g., from \frst-principles calculations. The accuracy is achieved by a two-step minimiza-\ntion scheme that coarse-grains the atomic and the spin space. The performance of the mMTPs\nis demonstrated for the prototype magnetic system bcc iron, with applications to phonon calcula-\ntions for di\u000berent magnetic states, and molecular dynamics simulations with \ructuating magnetic\nmoments.\nKeywords : magnetism, density functional theory, machine-learning interatomic potentials,\nmolecular dynamics, phonons.\n\u0003i.novikov@skoltech.ruarXiv:2012.12763v2  [physics.atom-ph]  10 Dec 20212\nINTRODUCTION\nMagnetic contributions are essential for modelling magnetic materials as they critically a\u000bect phase stability [1{3],\nvibrational properties [4{6], interstitial energies [7], local [8, 9] and extended defects [10, 11], and kinetics [12, 13].\nTaking the magnetic degrees of freedom properly into account is a prerequisite for computationally-aided design and\ndevelopment of a large number of technologically relevant materials, ranging from various steels for construction and\nsafety applications [1{6, 8{13] to hard magnets for applications in electrical transportation and renewable energy\ntechnologies [14, 15].\nOne of the most popular computational methods, capable of capturing magnetism, are \frst-principles calculations\nrealized by density functional theory (DFT). DFT calculations are, however, computationally expensive and limited\nto small system sizes and to a small number of magnetic con\fgurations. DFT calculations that sample the magnetic\ndegree of freedom explicitly, as needed for, e.g., lattice vibrations or vacancy formation energies in magnetically excited\nstates, are therefore available only for very few selected cases.\nRecent progress in machine-learning potentials has signi\fcantly accelerated accurate simulations of materials and\nmolecules [16{26]. Such potentials express the interatomic energy as a function of atomic positions alone. Ignoring\nthe electronic degrees of freedom, yet assuming a very \rexible functional form for the interatomic energies, machine-\nlearning potentials feature near-quantum mechanical accuracy at a computational e\u000eciency of the order of classical\ninteratomic potentials [27]. However, by ignoring the electronic degrees of freedom such potentials cannot distinguish\ndi\u000berent magnetic states, simply because di\u000berent magnetic states feature di\u000berent energies and the functional form\nof machine-learning potentials prohibits to capture such a magnetically-induced energy variation. In this paper we\nintroduce a strategy to overcome this fundamental shortcoming.\nRESULTS\nMagnetic Moment Tensor Potential\nThe starting point is a given set of energies, EDFT(R;S), which include the magnetic degree of freedom, e.g.,\ncomputed via DFT, and where RandSdenote a set of atomic coordinates and corresponding atomic spins. There\nare various ways to compute these energies from DFT, e.g., via fully relaxing the spin degree of freedom or, if one\nis interested in a broader sampling of EDFT(R;S), via constrained spin calculations [28{32]. We on purpose do not\ndiscuss in this work the di\u000berent approaches available and their corresponding challenges to compute EDFT(R;S) since\nthe main focus here is on an e\u000ecient parametrization for a given EDFT(R;S). We utilize standard spin-polarized\nDFT calculations where the local atomic moments are di\u000berently initialized while their longitudinal component is\nfully relaxed. The di\u000berent magnetic con\fgurations sampled are discussed further below. We note, however, that the\nproposed machine-learning potential can be straightforwardly applied with, e.g., constrained spin calculations.\nThe heart of the proposed approach is to approximate the energy EDFT(R;S) with Moment Tensor Potentials\n(MTPs) [33, 34] the idea of which is to expand the energy locally as a polynomial of its degrees of freedom, corrected\nin order to allow for a \fnite cuto\u000b of the potential. We note that there are other functional forms allowing for\napproximation of EDFTas a function of enriched degrees of freedom [35{37]. A similar functional form as utilized\nin MTPs has been recently employed within the atomic cluster expansion (ACE) [38]. Both approaches feature a\ncomplete basis of invariant polynomials that di\u000ber only in the representation of the angular terms; MTP uses tensors\nwhile ACE uses spherical harmonics.\nIn our approach the total interaction energy is partitioned into contributions of individual local atomic environments:\nEmMTP=NX\ni=1V(ni); (1)\nwhere niis the neighborhood of the i'th atom and Nis the number of atoms in the atomic con\fguration. In the\npresent paper the degrees of freedom are atomic positions R=fri; i= 1;:::;Ngand spinsS=fsi; i= 1;:::;Ng\nas opposed to the originally developed MTPs [33, 34] in which the potential energy depends only on atomic positions.\nThe atomic neighborhood of the i'th atom, ni, is hence described by the relative interatomic positions rij=rj\u0000ri,\nthe spin of the central atom, si, and the spins of the neighboring atoms sj, formally\nni=f(rij;si;sj) :j= 1;:::;Ni\nnbg;\nwhereNi\nnbis the number of neighbors of the i'th atom.3\nThe expansion of the function Vis:\nV(ni) =X\n\u000b\u0018\u000bB\u000b(ni);\nwhere\u0018=f\u0018\u000bgare the \\linear\" parameters to be optimized. The function Vis assumed to be an arbitrary polynomial\nof the corresponding degrees of freedom, modi\fed so that instead of the polynomial growth the potential Vvanishes\nbeyond some cuto\u000b distance. The potential is expanded via basis functions B\u000bde\fned through the so-called moment\ntensor descriptors\nM\u0016;\u0017(ni) =Ni\nnbX\nj=1f\u0016(jrijj;si;sj)rij\n:::\nrij|{z}\n\u0017times; (2)\nwhere \\\n\" is the outer product of vectors, and, thus, the angular part rij\n:::\nrijis a tensor of \u0017'th rank. The\nfunctionf\u0016(jrijj;si;sj) is a polynomial of jrijj,siandsj, modi\fed for a \fnite cuto\u000b radius. It has the form:\nf\u0016(jrijj;si;sj) =N'X\n\u0010=1N X\n\r=1N X\n\f=1c\f;\r;\u0010\n\u0016 \f(si) \r(sj)'\u0010(jrijj)(rcut\u0000jrijj)2; (3)\nwherec=fc\f;\r;\u0010\n\u0016gare the \\radial\" parameters to be optimized, N'is the number of polynomial basis functions\n'\u0010(jrijj) on the interval [ rmin;rcut], whererminis the minimal distance between atoms and rcutis the cuto\u000b radius\nbeyond which atoms do not interact. The term ( rcut\u0000jrijj)2ensures a smooth vanishing of the potential for jrijj>rcut.\nThe other functions,  \f(si) and \r(sj), are the polynomial basis functions of the local spins of the central and\nneighboring atoms, respectively. The number of these spin basis functions is N . They are de\fned on the interval\n[smin;smax], where the values sminandsmaxare the minimal and maximal local magnetic moments in the system\nbeing investigated.\nThe mMTP basis functions B\u000bare de\fned as all possible contractions of M\u0016;\u0017(ni) to a scalar, e.g.,\nM1;0(ni); M 0;1(ni)\u0001M1;1(ni); M 3;2(ni) :M1;2(ni); ::: ;\nwhere \\\u0001\" is the dot product of two vectors, and \\ : \" is the Frobenius product of two matrices. In principle, an\nin\fnite number of such mMTP basis functions could be constructed. In order to choose which basis functions to\ninclude in practice in the mMTP, we introduce the so-called level of each descriptor, lev M\u0016;\u0017= 2 + 4\u0016+\u0017, choose a\ncertain lev max, and include in the mMTP each basis function with lev B\u000b\u0014levmax(see Ref. [39] for details). Thus,\nthe number of the \\linear\" parameters \u0018depends on lev max, which also determines the number of radial functions,\nN\u0016. The number of the \\radial\" parameters cis equal to N\u0016N'N2\n . We denote all free parameters of an mMTP\ncollectively by \u0012=f\u0018;cg, and the total interaction energy by EmMTP=EmMTP(\u0012;R;S).\nWe note that the mMTP formalism contains the Heisenberg model as a special, limiting case. In particular, \frst-\ndegree polynomials have to be utilized for  \f(s) = \r(s) =sin Eq. (3), and '\u0010needs to \\encompass\" (i.e., be\nnonzero at) the nearest neighbors only. Such a choice of terms in the expansion Eq. (3) also leads to a model similar\nto the one proposed in Ref. [37], except that in the latter case the full vectorial spins were considered. Moreover, the\nbiquadratic terms, ( sisj)2[40, 41], adopted by Ref. [37], arise naturally when M\u0016;0is constructed with such choices\nof \fand'\u0010and gets multiplied by itself. Then the radial parameters c\f;\r;\u0010\n\u0016 correspond to the coupling constants as\nobtained from DFT data.\nThe free parameters \u0012in our approach are found by \ftting EmMTPto DFT data. We consider a training set\nincludingKmagnetic con\fgurations ( R(k);S(k)) with known DFT energies EDFT, DFT forces fDFT\ni on every atom\ni, and a 3\u00023 tensor of DFT stresses \u001bDFTand minimize the objective function:\nKX\nk=1\"\nwe\f\f\f\f\fEmMTP\u0012\n\u0012;R(k);S(k)\u0013\n\u0000EDFT\u0012\nR(k);S(k)\u0013\f\f\f\f\f2\n+wfX\ni\f\f\f\f\ffmMTP\ni\u0012\n\u0012;R(k);S(k)\u0013\n\u0000fDFT\ni\u0012\nR(k);S(k)\u0013\f\f\f\f\f2\n+ws\f\f\f\f\f\u001bmMTP\u0012\n\u0012;R(k);S(k)\u0013\n\u0000\u001bDFT\u0012\nR(k);S(k)\u0013\f\f\f\f\f2#4\n▼▼▼▼ ▼ ▼\n●\n●\n●\n●\n●\n●▼MTP\n●mMTP\n10 20 50 100 200 500 1000251020\nnumber of parameterserror(meV/atom)\nFigure 1: Convergence of the magnetic potential, mMTP, with respect to DFT energies and lack of convergence for\nthe non-magnetic MTP. The graph indicates that most variation of the energy on the training set is in the magnetic\ndegrees of freedom that can be captured only by the magnetic potential.\nwherej\u0001jis the length of a vector or the Frobenius norm of a matrix. The optimization of the parameters is carried out\nusing an iterative quasi-Newton optimization method, speci\fcally, the Broyden-Fletcher-Goldfarb-Shanno algorithm\n(BFGS) starting with a random initial guess. As opposed to mMTP, the energy of the non-magnetic MTP, proposed\nin our earlier works, does not depend on spins, i.e. EMTP=EMTP(\u0012;R), and the functions f\u0016(jrijj) do not include\nspins.\nConvergence of magnetic MTP\nWe \frst analyze the convergence behavior of the magnetic and non-magnetic MTP toward DFT energies as the\nnumber of parameters is increased. The convergence was measured on a hold-out set of about 1000 con\fgurations not\nparticipating in the \ftting of the potentials. Figure 1 shows that the mMTP exhibits a steady convergence, while the\nnon-magnetic MTP does not. This reiterates our original motivation: the space of atomic positions ( R) is not the\nright one for approximating the quantum-mechanical energy, but enriched with spins, ( R;S), this becomes a suitable\nspace for that purpose.\nBased on the convergence tests, we have chosen a well converged lev max= 24 for the subsequent tests. For both\nMTP and mMTP we took N'= 12 polynomial functions of the atomic positions with rmin= 2 \u0017A,rcut= 5:5\u0017A.\nFor the mMTP we took N = 2 polynomial functions of the local magnetic moments with smin=\u00003:5\u0016Band\nsmax= 3:5\u0016B. The total number of MTP parameters was 937 while that of mMTP was 1153. The weights in the\nobjective function were we= 1,wf= 0:01, andws= 0:001.\nFor each model we \ftted \fve potentials and selected the best (with the least training error). The validation root-\nmean-square errors are shown in Table I. We can see that adding local magnetic moments to the potential as additional\ndegrees of freedom does not signi\fcantly increase the number of parameters, but greatly improves the accuracy of\ntraining.\nPhonon spectra prediction\nWe next evaluate the performance of the best optimized MTP and mMTP potentials to predict phonon spectra\nof di\u000berent magnetic states. We consider two extreme scenarios representing the limits of magnetic con\fgurations,\nnamely the ferromagnetic state, in which all spins are aligned parallel and a paramagnetic state, treated in the\nadiabatic limit of fast \ructuating spins. Since the phonon energies were derived from small perturbations (utilizing\nthe small displacement method), this test is a very sensitive measure to detect how well even very small variations\nin interatomic forces can be captured. The results for the ferromagnetic case for both potentials are shown in\nFigure 2(a) in comparison with the data directly obtained from DFT. The agreement between the mMTP and the\nDFT data is excellent whereas the non-magnetic MTP shows signi\fcant deviations, in particular around the N-point.\nThe deviations for the non-magnetic MTP are a direct consequence of the training database which also includes\nmagnetically disordered con\fgurations responsible for pronounced phonon softening as discussed in the following.5\nmodel levelnumber of energy error force error stress error\nparameters meV/atom meV/ \u0017A (%) GPa (%)\nMTP 24 937 30.4 195 (27.3 %) 0.542 (13.3 %)\nmMTP 24 1153 1.5 64 (9.0 %) 0.087 (2.2 %)\nTable I: The best non-magnetic and magnetic MTPs. Despite the fact that the magnetic MTP has only a small\nincrease in the number of parameters, its accuracy is much higher than that of the non-magnetic MTP.\nN Γ HP Γ010203040Phonon energy (meV)mMTP\nDFT\nMTP\nN Γ HP Γ010203040\nFerromagnetic Paramagnetic (SSA)(a) (b)\nFigure 2: Computed phonon spectra for the (a) ferromagnetic and (b) paramagnetic state (modelled using the SSA\napproach) with DFT, non-magnetic MTP, and magnetic MTP.\nTo compute the phonon spectra in the paramagnetic regime we utilize the spin-space averaging (SSA) method [6]. In\nthis approach e\u000bective interatomic forces can be de\fned by averaging over various disordered magnetic con\fgurations\nweighted by a Boltzman distribution. For the actual averaging we utilized the crystal symmetries as proposed in\nRefs. [5, 6] and performed the SSA using a single random magnetic con\fguration for which each atom is displaced in\neach cartesian direction. This provides a large number of locally inequivalent magnetic con\fgurations (i.e., 54 \u00013 = 162\ncon\fgurations for the employed supercell). This procedure was shown to be robust with respect to the actually chosen\nrandom magnetic con\fguration as discussed in Ref. [6].\nThe resulting DFT based phonon spectrum shown in Figure 2(b) features a pronounced softening at the N-point\n[6]. This softening is related to the decrease of the elastic constants and constitutes an important precursor of the\nstructural transformation in iron. The non-magnetic MTP cannot distinguish the underlying atomic forces in these\ndi\u000berent magnetic states from the ferromagnetic forces. This is the reason why the MTP phonon spectrum for the\nparamagnetic state shown in Figure 2(b) is exactly the same as the one in Figure 2(a) for the ferromagnetic state.\nThe non-magnetic MTP spectra fall in-between the ferromagnetic and paramagnetic solutions and hence do not\nquantitatively reproduce the DFT data in either regime. In contrast, applying the SSA approach with the mMTP\nreveals an excellent agreement with the DFT data, reproducing quantitatively important characteristics such as, e.g.,\nthe decrease of the phonon energies near the N-point and along the H-P path.\nDisordered-local-moment molecular dynamics simulations\nTo evaluate the performance of the mMTP at \fnite temperatures and larger atomic displacements, we have per-\nformed molecular dynamics (MD) simulations. The temperature was set to 800 K and the lattice constant to 2.9\n\u0017A. To sample not only the vibrational degrees of freedom but the spin space and in particular the coupling between\nvibrations and spins, we have performed disordered-local-moment MD (DLM-MD) simulations [42]. Further, in order\nto explicitly validate the mMTP against DFT, we have utilized the concept of thermodynamic integration, similarly\nas used in the TU-TILD+MTP method previously [43]. Speci\fcally, we have introduced a linear coupling between\nDFT and mMTP forces,\nF\u0015=\u0015FDFT+ (1\u0000\u0015)FmMTP; (4)6\n(a)Density(arb. units)\n1 4 15 60 250 750\nRMSE=2.0 meV/atom\nMAE=1.5 meV/atom\n-8.05 -8.00 -7.95 -7.90 -7.85-8.05-8.00-7.95-7.90-7.85\nDFT energy(eV/atom)mMTP energy(eV/atom)(b)Density(arb. units)\n1 3 10 30 100\nRMSE=16 meV/atom\nMAE=12 meV/atom\n-8.05 -8.00 -7.95 -7.90 -7.85-8.05-8.00-7.95-7.90-7.85\nDFT energy(eV/atom)MTP energy(eV/atom)(c)Density(arb. units)\n1 20 400 8000\nRMSE=0.12μ B\nMAE=0.09μ B\n0 50 MD step-202\n-3 -2 -1 0 1 2 3-3-2-10123\nDFT spins(μ B)mMTP spins(μ B)\nFigure 3: Energy correlation (a) between magnetic MTP and DFT, (b) between non-magnetic MTP and DFT, and\n(c) spin correlation between magnetic MTP and DFT from DLM-TI calculations at 800 K.\nwith the coupling constant \u0015and DFT and mMTP forces FDFTandFmMTP. The coupled forces F\u0015were used for\nevolving the DLM-MD trajectories. The mMTP in this test was \ftted to 'pure' DFT calculations (i.e., nominally\ncorresponding to \u0015= 1) and tested independently for a new set of calculations at \u0015= 0;0:5;1. To render the\nDFT calculations feasible we employed a 16-atom supercell for the DLM-TI calculations; cross-checks for a 54-atom\nsupercell showed similar results. Further details are given in the Methods section.\nFigure 3 highlights the excellent performance of the mMTP. In the left panel we observe that the mMTP energies\nfall almost on top of the DFT energies; the root-mean-square error (RMSE) is only 2.0 meV/atom|of the same order\nas obtained previously for non-magnetic systems [43]. The middle panel clari\fes that the best possible non-magnetic\nMTP is almost an order of magnitude away in terms of energy accuracy, with an RMSE of 16 meV/atom. The right\npanel of Figure 3 shows the spin correlation between mMTP and DFT, which of course only the magnetic version\nof the MTP is capable to reproduce. We observe an RMSE of 0.12 \u0016B, which is about 5% of the magnitude of the\nabsolute spin.\nWe stress that Figure 3 includes values for all the investigated coupling constants \u0015= 0:0, 0:5, 1:0. Looking at each\n\u0015value separately, the correlations are in fact very similar. This means that there is no di\u000berence in the correlation,\nif we use pure DFT forces (cf. Eq. (4)), pure mMTP forces, or DFT-mMTP coupled forces to evolve the MD. This\nhence allows one to perform a full thermodynamic integration from the mMTP to DFT and compute the respective\nfree energy di\u000berence, which is however beyond the scope of the present work.\nDISCUSSION\nWe have developed the mMTPs, a class of magnetic machine-learning interatomic potentials capable of simulta-\nneously and accurately approximating spin and atomic degrees of freedom. This has been achieved by utilizing a\ntwo-step minimization scheme for the spin and atomic con\fgurational space. Applying the mMTP to DFT-derived\ndata for the prototypical bcc iron system reveals that the mMTPs are capable to quantitatively approximate local\nmagnetic moments, energies, and forces for various magnetic states (see the Supplementary Discussion for further\ntests). A number of applications such as the computation of phonon spectra in ferro- and paramagnetic states as well\nas molecular-dynamics simulations including spin-\rips demonstrate that mMTPs provide near DFT-accuracy without\nsigni\fcantly losing the computational e\u000eciency of classical interatomic potentials.7\nMETHODS\nDerivation of mMTP\nHere we derive the form of the MTP as a function of relative atomic positions rijand vectorial magnetic moments,\nsiandsj. Following the logic of the original paper introducing the MTP [33], an arbitrary polynomial of the positions\nand magnetic moments can be represented as all possible contractions of the following Moment Tensors,\nM\u0010;\u0017;\f;\r;\u0018;\u0011 =Ni\nnbX\nj=1Q\u0010(jrijj)jsij\fjsjj\r\u0000\nr\n\u0017\nij\u0001\n\n\u0000\ns\n\u0018\ni\u0001\n\n\u0000\ns\n\u0011\nj\u0001\n; (5)\nwhereQ\u0010is the\u0010-th radial basis function and by de\fnition\nv\nn=v\n:::\nv|{z}\nntimes\nfor an arbitrary vector v. We note that Eq. (5) directly corresponds to Ref. [36, Eq. 26] with spherical harmonics\nYm\nl(v=jvj) instead of tensors v\nn(jlj\u0014m\u0014n).\nWe next assume scalar-valued spins, i.e., that it is su\u000ecient to consider \u0018=\u0011= 0 (and adsorb the sign of the spin\ninto the radial part):\nM\u0010;\u0017;\f;\r =Ni\nnbX\nj=1Q\u0010(jrijj)s\f\nis\r\nj\u0000\nr\n\u0017\nij\u0001\n: (6)\nWe could choose to directly expand the energy over di\u000berent contractions of tensors M\u0010;\u0017;\f;\r , but instead we combine\ndi\u000berent products of radial and spin basis functions, Q\u0010(jrijj)s\f\nis\r\nj, into the functions f\u0016(rij;si;sj) with coe\u000ecients\nc\f;\r;\u0010\n\u0016 that are found from data, as explained in the main text of the manuscript. Note that in this work we use\nChebyshev polynomials  \f(s) instead of monomials s\f.\nDFT details\nAll DFT calculations were performed with vasp [44{47] utilizing the projector augmented wave (PAW) method [48]\nand the generalized gradient approximation [49]. For the training set of the 54-atom supercell we considered 70 atomic\ncon\fgurations generated from an initial ferromagnetic MD at 1000 K. For each of these atomic con\fgurations 200\ndi\u000berent arrangements of magnetic spins have been initialized of which 67% converged under the high cuto\u000b energy of\n500 eV and k-point density of 11664 k-points\u0002atoms (6\u00026\u00026 grid) chosen in combination with a convergence criterion\nof 10\u00007eV per supercell to ensure high-accurate DFT data. This resulted into in total 9351 calculations. To impose\nspin-inversion symmetry we added the same number of con\fgurations to the training with reversed spin directions.\nThe DFT calculations were performed at a lattice parameter of 2.9 \u0017A corresponding to the experimental value near\nthe Curie temperature. The phonon calculations have been performed utilizing the \fnite-displacement method with\na displacement of 0.02 \u0017A and utilizing the same set of technical parameters.\nDisordered-local moment thermodynamic integration from mMTP to DFT\nTo sample the paramagnetic state at \fnite temperatures within the framework of thermodynamic integration (TI),\nwe have employed the disordered-local-moment (DLM) MD [42]. The local magnetic moments were \ripped randomly\nevery 10 fs (= 10 MD steps) such that half of the moments was pointing up and the other half down. The timestep for\nthe MD was set to 1 fs; small enough to sample well the time development of the magnetic moments also within the\n10 fs time intervals. The temperature was controlled by the Nose thermostat [50]. Usage of the Nose thermostat was\ncritical; tests with the Langevin thermostat showed that it cannot stabilize the temperature well due to the additional\nimpact of the spin \rips on the energy of the system.\nSpin-polarized DFT calculations in general and DLM calculations for Fe in particular are very prone to convergence\nproblems, due to a \rat energy landscape with many local minima as a function of spin state. Therefore, for the8\ncalculation of the DFT energy and forces during the MD, a very tight convergence criterion of 10\u00007eV per supercell\nwas set, in order to enforce sampling of the original DFT energy landscape that served as the input to the magnetic\nMTP \ftting. To nevertheless allow for an e\u000ecient DFT MD, we have restricted the number of electronic iteration\nsteps (typically to 40). Not fully converged DFT calculations were omitted from the comparison to the magnetic\nmMTP. Likewise DFT calculations featuring local moment \rips with respect to the mMTP data were not considered\nin the comparison.\nTo increase the e\u000eciency of the DFT DLM-MD simulations we found it necessary to turn o\u000b the wave function\nextrapolation (both linear and quadratic); the reason for this lying in the randomization of the spins along the\nMD trajectory. A further e\u000eciency increase was achieved by equilibrating the MD at the temperature of interest\nby utilizing the e\u000ecient mMTP. In this way the part of the MD involving the expensive DFT calculations started\ndirectly on a well equilibrated trajectory.\nThe DLM-TI was performed at a temperature of 800 K and at a lattice constant of 2.9 \u0017A. A supercell of 2 \u00022\u00022 (in\nterms of the conventional bcc unit cell) with 16 atoms was utilized. A dense k-point sampling of 6 \u00026\u00026 corresponding\nto 3,456k-points\u0002atom, a plane wave cuto\u000b of 500 eV, and Fermi-Dirac smearing were used for the DFT calculations.\nFor the mMTP calculations, initial magnetic moments were set according to the DFT moments. Then, for every\nmMTP energy and force calculation, the magnetic moments were fully relaxed based on the mMTP energetics.\nCoupling constants of \u0015= 0:0;0:5;1:0 were utilized. At each coupling constant two di\u000berent random seeds were used\nto generate distinct trajectories. In total more than 22,000 of MD steps (22 ps) were conducted to generate statistically\nhighly reliable correlation plots as shown in Figure 3 of the main text. Test calculations for a larger 3 \u00023\u00023 supercell\nwith 54 atoms turned out to be computationally highly demanding due to the strict DFT convergence parameters.\nCorresponding results indicate however a similar performance of the mMTP also for the larger supercell.\nDATA AVAILABILITY\nThe datasets generated during and/or analyzed during the current study are available from the corresponding\nauthor on reasonable request.\nACKNOWLEDGEMENTS\nWe acknowledge support from the collaborative DFG-RFBR Grant (Grants No. DFG KO 5080/3-1, DFG GR\n3716/6-1, and RFBR 20-53-12012). B.G. acknowledges the support by the Stuttgart Center for Simulation Science\n(SimTech) and funding from the European Research Council (ERC) under the European Union's Horizon 2020 research\nand innovation programme (grant agreement No 865855).\nAUTHOR CONTRIBUTIONS\nF.K. and A.S. conceived the project. I.N. and A.S. developed the magnetic Moment Tensor Potential (mMTP).\nI.N. \ftted mMTP, investigated prediction errors and convergence of mMTP. F.K. prepared DFT datasets, calculated\nand compared the phonon spectra obtained with the magnetic MTP and DFT. B.G. conducted the disordered-local-\nmoment molecular dynamics simulations and investigated the correlation of magnetic MTP and DFT energies and\nspins. All the authors participated in analyzing the results and writing the manuscript.\nCOMPETING INTERESTS\nThe authors declare no competing interests.\n[1] H. Hasegawa and D. Pettifor, \\Microscopic theory of the temperature-pressure phase diagram of iron,\" Phys. Rev. Lett.\n50, 130 (1983).\n[2] H. Herper, E. Ho\u000bmann, and P. Entel, \\Ab initio full-potential study of the structural and magnetic phase stability of\niron,\" Phys. Rev. B 60, 3839 (1999).9\n[3] F. K ormann, T. Hickel, and J. Neugebauer, \\In\ruence of magnetic excitations on the phase stability of metals and steels,\"\nCurrent Opinion in Solid State and Materials Science 20, 77 (2016).\n[4] A. V. Ruban and V. I. Razumovskiy, \\Spin-wave method for the total energy of paramagnetic state,\" Phys. Rev. B 85,\n174407 (2012).\n[5] Y. Ikeda, A. Seko, A. Togo, and I. Tanaka, \\Phonon softening in paramagnetic bcc fe and its relationship to the pressure-\ninduced phase transition,\" Phys. Rev. B 90, 134106 (2014).\n[6] F. K ormann, A. Dick, B. Grabowski, T. Hickel, and J. Neugebauer, \\Atomic forces at \fnite magnetic temperatures:\nPhonons in paramagnetic iron,\" Phys. Rev. B 85, 125104 (2012).\n[7] S. Dudarev and P. Derlet, \\A `magnetic' interatomic potential for molecular dynamics simulations,\" Journal of Physics:\nCondensed Matter 17, 7097 (2005).\n[8] D. Boukhvalov, Y. N. Gornostyrev, M. Katsnelson, and A. Lichtenstein, \\Magnetism and local distortions near carbon\nimpurity in \r-iron,\" Phys. Rev. Lett. 99, 247205 (2007).\n[9] O. Gorbatov, P. A. Korzhavyi, A. V. Ruban, B. Johansson, and Y. N. Gornostyrev, \\Vacancy{solute interactions in\nferromagnetic and paramagnetic bcc iron: Ab initio calculations,\" Journal of nuclear materials 419, 248 (2011).\n[10] L. S. et al. , \\Stacking fault energy of c-alloyed steels: The e\u000bect of magnetism,\" Acta Mater. 122, 72 (2017).\n[11] B. Bienvenu, C. C. Fu, and E. Clouet, \\Impact of magnetism on screw dislocations in body-centered cubic chromium,\"\nActa Mater. 200, 570 (2020).\n[12] H. O. et al. , \\Atomic relaxation around defects in magnetically disordered materials computed by atomic spin constraints\nwithin an e\u000ecient lagrange formalism,\" Phys. Rev. B 102, 144101 (2020).\n[13] A. Schneider, C.-C. Fu, F. Soisson, and C. Barreteau, \\Atomic di\u000busion in \u000b-iron across the curie point: An e\u000ecient and\ntransferable ab initio{based modeling approach,\" Phys. Rev. Lett. 124, 215901 (2020).\n[14] S. H. _I.et al. , \\Ab initio phase stabilities of ce-based hard magnetic materials and comparison with experimental phase\ndiagrams,\" Phys. Rev. Mat. 3, 084407 (2019).\n[15] M. Matsumoto and H. Akai, \\Calculating curie temperatures for rare-earth permanent magnets: Ab initio inspection of\nlocalized magnetic moments in d-electron ferromagnetism,\" Phys. Rev. B 101, 144402 (2020).\n[16] J. Behler, \\First principles neural network potentials for reactive simulations of large molecular and condensed systems,\"\nAngewandte Chemie International Edition 56, 12828 (2017).\n[17] V. L. Deringer, M. A. Caro, and G. Cs\u0013 anyi, \\Machine learning interatomic potentials as emerging tools for materials\nscience,\" Advanced Materials 31, 1902765 (2019).\n[18] K. Sch utt, P.-J. Kindermans, H. E. S. Felix, S. Chmiela, A. Tkatchenko, and K.-R. M uller, \\Schnet: A continuous-\flter\nconvolutional neural network for modeling quantum interactions,\" Advances in Neural Information Processing Systems ,\n30, 991 (2017).\n[19] N. Lubbers, J. S. Smith, and K. Barros, \\Hierarchical modeling of molecular energies using a deep neural network,\" J.\nChem. Phys. 148, 241715 (2018).\n[20] V. J. et al. , \\On-the-\ry active learning of interpretable Bayesian force \felds for atomistic rare events,\" npj Comput. Mater.\n6, 20 (2020).\n[21] R. Drautz, \\Atomic cluster expansion for accurate and transferable interatomic potentials,\" Phys. Rev. B 99, 014104\n(2019).\n[22] J. S. Smith, O. Isayev, and A. E. Roitberg, \\ANI-1: an extensible neural network potential with DFT accuracy at force\n\feld computational cost,\" Chem. Sci. 8, 3192 (2017).\n[23] R. Jinnouchi, F. Karsai, and G. Kresse, \\On-the-\ry machine learning force \feld generation: Application to melting\npoints,\" Phys. Rev. B 100, 014105 (2019).\n[24] M. A. Cusentino, M. A. Wood, and A. P. Thompson, \\Explicit multi-element extension of the spectral neighbor analysis\npotential for chemically complex systems,\" J. Phys. Chem. A 124, 5456 (2020).\n[25] S. Chmiela, A. Tkatchenko, H. E. Sauceda, I. Poltavsky, K. T. Sch utt, and K.-R. M uller, \\Machine learning of accurate\nenergy-conserving molecular force \felds,\" Science advances 3, e1603015 (2017).\n[26] G. P. Pun, R. Batra, R. Ramprasad, and Y. Mishin, \\Physically informed arti\fcial neural networks for atomistic modeling\nof materials,\" Nature communications 10, 1 (2019).\n[27] Y. Zuo et al. , \\Performance and cost assessment of machine learning interatomic potentials,\" J. Phys. Chem. A 124, 731\n(2020).\n[28] P. H. Dederichs, S. Bl ugel, R. Zeller, and H. Akai, \\Ground states of constrained systems: Application to cerium impuri-\nties,\" Phys. Rev. Lett. 53, 2512 (1984).\n[29] G. Stocks et al. , \\Towards a constrained local moment model for \frst principles spin dynamics,\" Philosophical Magazine\nB78, 665 (1998).\n[30] R. Singer, M. F ahnle, and G. Bihlmayer, \\Constrained spin-density functional theory for excited magnetic con\fgurations\nin an adiabatic approximation,\" Phys. Rev. B 71, 214435 (2005).\n[31] B. Kaduk, T. Kowalczyk, and T. Van Voorhis, \\Constrained density functional theory,\" Chemical reviews 112, 321 (2012).\n[32] P.-W. Ma and S. Dudarev, \\Constrained density functional for noncollinear magnetism,\" Phys. Rev. B 91, 054420 (2015).\n[33] A. V. Shapeev, \\Moment tensor potentials: A class of systematically improvable interatomic potentials,\" Multiscale Mod-\neling & Simulation 14, 1153 (2016).\n[34] K. Gubaev, E. V. Podryabinkin, G. L. W. Hart, and A. V. Shapeev, \\Accelerating high-throughput searches for new\nalloys with active learning of interatomic potentials,\" Comp. Mater. Sci. 156, 148 (2019).10\n[35] A. Grisa\f, D. M. Wilkins, G. Cs\u0013 anyi, and M. Ceriotti, \\Symmetry-adapted machine learning for tensorial properties of\natomistic systems,\" Phys. Rev. Lett. 120, 036002 (2018).\n[36] R. Drautz, \\Atomic cluster expansion of scalar, vectorial, and tensorial properties including magnetism and charge trans-\nfer,\" Phys. Rev. B 102, 024104 (2020).\n[37] S. Nikolov et al. , \\Quantum-accurate magneto-elastic predictions with classical spin-lattice dynamics,\" arXiv preprint\narXiv:2101.07332 (2021).\n[38] G. Dusson et al. , \\Atomic cluster expansion: Completeness, e\u000eciency and stability,\" arXiv preprint arXiv:1911.03550\n(2019).\n[39] I. S. Novikov, K. Gubaev, E. V. Podryabinkin, and A. V. Shapeev, \\The MLIP package: moment tensor potentials with\nMPI and active learning,\" Machine Learning: Science and Technology 2, 025002 (2020).\n[40] N. Rosengaard and B. Johansson, \\Finite-temperature study of itinerant ferromagnetism in fe, co, and ni,\" Phys. Rev. B\n55, 14975 (1997).\n[41] A. Szilva, M. Costa, A. Bergman, L. Szunyogh, L. Nordstr om, and O. Eriksson, \\Interatomic exchange interactions for\n\fnite-temperature magnetism and nonequilibrium spin dynamics,\" Phys. Rev. Lett. 111, 127204 (2013).\n[42] B. Alling, F. K ormann, B. Grabowski, A. Glensk, I. A. Abrikosov, and J. Neugebauer, \\Strong impact of lattice vibrations\non electronic and magnetic properties of paramagnetic fe revealed by disordered local moments molecular dynamics,\" Phys.\nRev. B 93, 224411 (2016).\n[43] B. Grabowski et al. , \\Ab initio vibrational free energies including anharmonicity for multicomponent alloys,\" npj Comput.\nMater. 5, 1 (2019).\n[44] G. Kresse and J. Hafner, \\Ab initio molecular dynamics for liquid metals,\" Phys. Rev. B 47, 558 (1993).\n[45] G. Kresse and J. Hafner, \\Ab initio molecular-dynamics simulation of the liquid-metal{amorphous-semiconductor transition\nin germanium,\" Phys. Rev. B 49, 14251 (1994).\n[46] G. Kresse and J. Furthm uller, \\E\u000eciency of ab-initio total energy calculations for metals and semiconductors using a\nplane-wave basis set,\" Comput. Mater. Sci. 6, 15 (1996).\n[47] G. Kresse and J. Furthm uller, \\E\u000ecient iterative schemes for ab initio total-energy calculations using a plane-wave basis\nset,\" Phys. Rev. B 54, 11169 (1996).\n[48] P. E. Bl ochl, \\Projector augmented-wave method,\" Phys. Rev. B 50, 17953 (1994).\n[49] J. P. Perdew, K. Burke, and M. Ernzerhof, \\Generalized gradient approximation made simple,\" Phys. Rev. Lett. 77, 3865\n(1996).\n[50] S. Nos\u0013 e, \\A uni\fed formulation of the constant temperature molecular dynamics methods,\" J. Chem. Phys. 81, 511 (1984).11\nSUPPLEMENTARY DISCUSSION\nIn the article we have illustrated the use of the magnetic moment tensor potential (mMTP) as an accurate approx-\nimant to DFT in phonon and molecular dynamics simulations of bcc ferromagnetic and paramagnetic iron. Here we\nprovide the results of additional tests: predicting with mMTP the vacancy formation energy (VFE) for bcc iron in the\nferromagnetic state and the energy/volume curves for bcc, fcc, and hcp iron in the ferromagnetic and antiferromagnetic\nstates.\nTo compute the VFE in bcc iron in the ferromagnetic state we created a training set containing 71 con\fgurations of\n54 atoms (including the equilibrium one) and 47 con\fgurations of 53 atoms (i.e., the con\fgurations with the vacancy;\nthe equilibrium one was also included). The training set was calculated with VASP [44{47] using the settings and\nparameters described in the main text. We \ftted mMTP with 1153 parameters on this training set. We have obtained\na VFE of 2.28 eV with VASP and 2.19 eV with mMTP.\nIn order to \ft mMTP for calculating energy-volume curves we used the following number of con\fgurations in the\nferromagnetic state: 24 bcc con\fgurations (2-atomic unit cells), 23 fcc con\fgurations (1-atomic unit cells), and 12\nhcp con\fgurations (2-atomic unit cells). The dataset also contained antiferromagnetic con\fgurations including 24 bcc\ncon\fgurations, 23 fcc con\fgurations (cubic 4-atomic unit cells), and 12 hcp con\fgurations. A high energy cuto\u000b (500\neV) andk-point densities (always >10;000kp\u0001atom) have been chosen. We \ftted mMTP with 172 parameters and\ncalculated the energy volume/curves and magnetic moments in the equilibrium state using mMTP and DFT. They\nare shown in Supplementary Figures 1-2. Both energy-volume curves and magnetic moments computed with the\n\ftted mMTP are close to the ones computed with DFT for bcc, fcc, and hcp iron in the corresponding ferromagnetic,\nantiferromagnetic, and nonmagnetic states.12\n-8.115-8.11-8.105-8.1-8.095-8.09-8.085-8.08-8.075-8.07\n 9.5  10  10.5  11  11.5  12  12.5  13Energy per atom, eV\nVolume per atom, A3fcc fm DFT\nfcc fm mMTP\n-8.16-8.14-8.12-8.1-8.08-8.06-8.04-8.02-8-7.98-7.96\n 9.5  10  10.5  11  11.5  12  12.5  13Energy per atom, eV\nVolume per atom, A3fcc afm DFT\nfcc afm mMTP\n-8.3-8.25-8.2-8.15-8.1-8.05\n 10  10.5  11  11.5  12  12.5  13  13.5  14Energy per atom, eV\nVolume per atom, A3bcc fm DFT\nbcc fm mMTP\n-7.85-7.8-7.75-7.7-7.65-7.6\n 10  10.5  11  11.5  12  12.5  13  13.5  14Energy per atom, eV\nVolume per atom, A3bcc afm DFT\nbcc afm mMTP\n-8.18-8.16-8.14-8.12-8.1-8.08-8.06-8.04-8.02-8\n 9.5  10  10.5  11  11.5  12  12.5Energy per atom, eV\nVolume per atom, A3hcp nm DFT\nhcp fm DFT\nhcp nm mMTP\nhcp fm mMTP\n-8.18-8.16-8.14-8.12-8.1-8.08-8.06-8.04-8.02-8\n 9.5  10  10.5  11  11.5  12  12.5Energy per atom, eV\nVolume per atom, A3hcp afm DFT\nhcp afm mMTP\nSupplementary Figure 1: Energy-volume curves computed with DFT and mMTP for bcc, fcc, and hcp iron in the\nferromagnetic (fm), antiferromagnetic (afm) and nonmagnetic (nm) states. mMTP reproduces well all the\ncorresponding magnetic states, including the low-spin to high-spin transition in fcc iron and the nonmagnetic to\nferromagnetic transition in hcp iron.13\n 0 0.5 1 1.5 2 2.5 3\n 9.5  10  10.5  11  11.5  12  12.5  13magnetic moment, µB\nVolume per atom, A3fcc fm DFT\nfcc fm mMTP\n 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4\n 9.5  10  10.5  11  11.5  12  12.5  13magnetic moment, µB\nVolume per atom, A3fcc afm DFT\nfcc afm mMTP\n 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8\n 10  10.5  11  11.5  12  12.5  13  13.5  14magnetic moment, µB\nVolume per atom, A3bcc fm DFT\nbcc fm mMTP\n 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6\n 10  10.5  11  11.5  12  12.5  13  13.5  14magnetic moment, µB\nVolume per atom, A3bcc afm DFT\nbcc afm mMTP\n 0 0.5 1 1.5 2 2.5 3\n 9.5  10  10.5  11  11.5  12  12.5magnetic moment, µB\nVolume per atom, A3hcp fm DFT\nhcp fm mMTP\n 0 0.5 1 1.5 2 2.5\n 9.5  10  10.5  11  11.5  12  12.5magnetic moment, µB\nVolume per atom, A3hcp afm DFT\nhcp afm mMTP\nSupplementary Figure 2: Dependence of absolute equilibrium magnetic moments computed with DFT and mMTP\non volume per atom for bcc, fcc, and hcp iron in the ferromagnetic and antiferromagnetic states."    },    {        "title": "1907.01850v1.Dynamic_magnetic_features_of_a_mixed_ferro_ferrimagnetic_ternary_alloy_in_the_form_of_AB__p_C___1_p__.pdf",        "content": "mehmet.bati@erdogan.edu.tr  Dynamic magnetic features of a mixed ferro -ferrimagnetic ternary alloy in the \nform of AB pC1-p \nMehmet Batı1,a   Mehmet Ertaşb \naDepartment of Physics, Recep Tayyip Erdoğan University, Rize, Turkey  \nbDepartment of Physics, Erciyes University, Kayseri, Turkey  \n \nAbstract  \nDynamic magnetic features of a mixed ferro -ferrimagnetic ternary alloy in the form of ABpC1-p, \nespecially . The effect of Hamiltonian parameters on the dynamic magnetic features of the system \nare investigated. F or this aim, an ABpC1-p ternary alloy system was simulated within the mean - \nfield approximation based on a Glauber type stochastic dynamic and for simplicity, A, B and C \nions as SA = 1/2, SB = 1 and SC = 3/2, were chosen respectively.  It was found that in our dynamic \nsystem the critical temperature was always dependent on the concentration ratio of the ternary \nalloy.  \nKeywords:  ABpC1-p ternary alloys; Dynamic magnetic features ; Mean -field approximation, \nGlauber type stochastic  \n \n1- Introduction  \n \nMixed -spin Ising systems provide a good model for studying a ferro -ferrimagnetic ternary \nalloy. Ternary alloy in the form of ABpC1-p composed of Prussian blue analogs [1] have been the \ntopic of much research  because of their interesting magnetic behaviors,  such as  photo induced \nmagnetization, charge -transfer -induced spin tr ansitions, the existence of compensation \ntemperatures and h ydrogen storage capacity [2 -11]. It  has been widely studied in literature by \nvarious methods in equilibrium statistical physics,  such as exact recursion relations (ERR) on the \nBethe lattice, mean -field approximation (MFA), Monte Carlo (MC) simulations and effective -field \ntheory (EFT) with correlations [12 -29]. In the studies, ternary alloy in the form of ABpC1-p has \nbeen modelled w ith different magnitudes of spins, namely A, B and C. For instance, the magnetic \nfeatures of a mixed ferro -ferrimagnetic ternary alloy in the form of ABpC1-p consist of three \ndifferent metal ions with ternary Ising spins (1/2, 1, 3/2) [12 -16]; (3/2, 1, 1/2 ) [17]; (1/2, 1, 5/2) \n[18]; (1/2, 3/2, 5/2) [19], (1, 3/2, 5/2) [20 -23], (3/2, 1, 5/2) [24 -26] and with Ising spin s (3/2, 2, \n5/2) [27 -29]. It is worth noticing that there are many experimental studies on Prussian blue analogs \n[30–36].   \n Generally, dynamic p roperty  investigations of ABpC1-p ternary alloy s are difficult because of \ntheir structural complexity. Despite, the equilibrium magnetic properties of a mixed ferro -\nferrimagnetic ABpC1-p ternary alloy having  been studied in detail, there are only two studies  within \nour best knowledge about the dynamic properties of the ABpC1-p ternary alloy system contain ing \nspin (3/2, 1, 5/2) [37, 38]. The aim of this study was focused on studying the dynamic magnetic \nproperties of a ternary alloy system. Dynamic phas e transitions (DPT) were  obtained and the effect \nof Hamiltonian parameters on the dynamic properties of the system were  investigated. For this \naim, an ABpC1-p ternary alloy system  was simulated  by utilizing the MFA based on Glauber type \nstochastic dynamic and for simplicity, A, B and C ions as SA = 1/2 SB = 1 and SC = 3/2  were chosen, \nrespectively.  It is worth noting  that the DPT has been extensively studied theoretically [39 -45] \nand experimentally [46 -50] for different systems during the last decades.  \nThe paper is organized in the following way: Section II describe s the models and its \nformulations.  Section III presents  the numerical results. Finally, the conclusion is given in Sec. IV.   \n \n2- Model formulation  \n \nA mixed ferro -ferrimagnetic ABpC1-p ternary al loy system was considered on two \ninterpenetrating square sublattices. One of the sublattices only has spins SA = ± 1/2 as the other \nsublattices have spins  SB = ±1, 0 or SC = ± 3/2, ± 1/2. A sketch of the present model can be seen in \nFig. 1. The Hamiltonian  can be written as follows:  \n \n    1 1 , (1)\n          ΗA B C A B C\ni AB j j AC j j i j j j j\nij i jS J S J S h(t) S S S ξ ξ ξ ξ\n  \n \nwhere < ij> shows a summation over all pairs of the nearest -neighboring sites of different \nsublattices and JAB > 0 and JAC < 0 (model the ferro -ferrimagnetic interactions ) are the nearest -\nneighbor exchange constants. h(t) is the oscillating external magnetic field and is described by \nℎ(𝑡)=ℎ0cos(𝑤𝑡), where h0, w and t are the time, amplitude and angular frequency. 𝜉𝑗 is \ndistributed random variables and it takes the value of unity or zero, according to whether site j is \nfilled by an ion of B or C, respectively. So, 𝜉𝑗 is described by  \n \n                                                      \n 1 1 , (2)   j j jP p pξ ξ ξ                  \n    \n where p and (1−𝑝) are the concentration of B and C ions, respectively. A mixed ferro -\nferrimagnetic ABpC1-p ternary alloy system is in contact with an isothermal heat bath at an absolute \ntemperature Tabs and evolves according to the Glau ber-type stochastic process at a rate of 1/ τ. From \nthe master equation associated to the stochastic process, it follows that the average magnetization \nsatisfies the following equation [39 -45], \n \n              \n11 , (3a)2    A A B C\nAB j AC j 0\njjdτ S S tanh β J p S +J S p h cos wtdt\n          \n                  \n\n3\n, (3b)\n21\n  \nA\nAB i 0\ni BB\nA\nAB i 0\nisinhβ J p S +ph cos wt\ndτ S Sdtcoshβ J p S +ph cos wt                                 \n         \n \n 3 1.5 1 0.5 1, (3c)2 1.5 1 2 0.5 1                 CCsinh p βx sinh p βx dτ S Sdt cosh p βx cosh p βx                              \n \nwhere 𝑥=𝐽𝐴𝐶∑𝑆𝑖𝐴\n𝑖 +ℎ0cos(𝑤𝑡). Using  the mean -field theory; the dynamic mean -field \napproximation equations are obtained as follows  \n \n       \n  11 , (4a)2     A A AB AB A AC AC C 0dm m tanh β J z m p+J z m p h cosd                      \n  \n  3\n, (4b)\n21   AB BA A 0\nBB\nAB BA A 0sinhβ J z m p+ph cos dmmd coshβ J z m p+ph cos\n \n \n 3 1.5 1 0.5 1, (4c)2 1.5 1 2 0.5 1                 CCsinh p βy sinh p βy dmmd cosh p βy cosh p βy\n \n \nwhere 𝑦=𝐽𝐴𝐶𝑧𝐶𝐴𝑚𝐴+ℎ0cos(𝜉),  𝜉=𝑤𝑡, Ω=𝜏𝑤 and Ω=2𝜋,  zAB, zBA, zAC and zCA are taken \n4 for  a square lattice.  \n                                                                         𝑀𝑖=1\n𝜏∫𝑚𝑖(𝜉)𝑑𝜉                                                                                   (5) \n  \n where 𝑖=𝐴,𝐵 and 𝐶. In other words, 𝑀𝐴, 𝑀𝐵 and 𝑀𝐶 correspond to the dynamic order parameters \nof the magnetic components A, B and C. The total magnetization of the system is  \n \n                                                                        𝑀𝑇=(𝑀𝐴+𝑀𝐵+𝑀𝐶)\n2                                                                                   (6) \n \nThe physical parameters have been scaled in terms of 𝐽𝐴𝐵. For example, reduced temperature and \nfield amplitude are respectively   defined as 𝑇=𝑘𝐵𝑇𝑎𝑏𝑠\n𝐽𝐴𝐵, and ℎ=ℎ0\n𝐽𝐴𝐵,  throughout the  \n  \n3- Results and Discussion  \n \nThe effects of the concentration ratio p and the exchange interaction ratio 𝑅 (|𝐽𝐴𝐶|\n𝐽𝐴𝐵) on dynamic \nmagnetization and DPT  of the ternary alloy have been examined.  It should be noted that the p = 0 \ncase corresponds to a ferrimagnetic mixed spin -1/2 and spin -3/2 system while for p = 1, \ncorresponds to  a mixed spin -1/2 and spin -1 ferromagnetic system.  The phase diagram of the \nternary alloy in ( 𝑅−𝑇𝐶) and ( 𝑝−𝑇𝐶) planes  are shown  in Fig . 2 and Fig . 3, respectively. In these \nfigures, upper graphs are plotted for ℎ=0.1 and the lower ones are plotted for ℎ=0.5. The \ncritical temperature value (phase transition temperature)  is a little decrease d with increasing h. \n(𝑇𝐶=2.71 for h=0.1, 𝑇𝐶=2.67 for h=0.5)  The r eason for this situation  is that , the higher field \namplitude becomes dominant against the ferromagnetic and antiferromagnetic nearest -neighbor \nbonds.  It can be seen  that from the figures  that 𝑇𝐶 increases as 𝑅 increases and  the 𝑇𝐶 values do \nnot change with 𝑅 for 𝑝= 1.0. Because the system become  an AB alloy, there is no  AC \ninteraction. Therefore t he system becomes independent from 𝑅.  \n \nIn this section, the effects of p and R on the magnetization of a ternary alloy of the type ABpC1-p \nare discussed.   Fig. 4 shows the total magnetization  chancing with scaled temperature  for R=0.5, \n1.0 and R=2.0 values.  It is again seen  from the  figures that a ll the total magnetization curves merge \nat a unique transition temperature for p = 1.0.  For the p =0.0 case, Tc=0 at  R =0.0 (see Fig . 1 and \n2) and the dynamic critical temperature of the system increase with an increasing  of R. For p=0.25, \nthe antiferromagnetic exchange interaction between  the A and C magnetic components becomes \neffective in the system for larger R values . In other words , the 𝐽𝐴𝐶  interaction  becomes dominant \nand because  𝐽𝐴𝐶  is negative,  the 𝑀𝑇  results are negative.  It is noted  that the saturation values of \nmagnetization increase for p=0.25 and decrease for p=0.75 with the increasing of R. A second \norder phase transition occurs in the system for R=0.5 and R=1.0. But for R=2, first the  system gives  \n the first order phase transition and then the second order phase transition occurs.  𝑀𝑇 decreases as \nR increases in the range 0.5 < R <1 at p = 0.5 and after 𝑅>1 (AC interaction is more  dominant), \nas R increases, 𝑀𝑇 orientation increases by changing.  \n \nIn Fig. 5 the results have been depicted for R=1.0.  As expected, t he sign  of the 𝑀𝐴 magnetization  \nis  negative ( 𝑀𝐴=−1/2) 𝑀𝐵=0.0 and 𝑀𝐶=3/2 at T=0 for p=0.0.  For 0 .0<p<1.0 the sign  of \nthe 𝑀𝐶 magnetization is  negative  (𝑀𝐶=−3/2),  𝑀𝐴=0.5 and 𝑀𝐵=1.0 at T=0. Lastly, for \np=1.0 𝑀𝐴=1/2,  𝑀𝐵=1.0  and 𝑀𝐶=0 at T=0.  There are no dynamic magnetic properties \ninvestigation of Ternary alloy for spin (½,1,3/2). However, there is an investigation for spin \n(3/2,1,5/2) [37]. Very interesting behaviors with the variations of scaled temperature, \nconcentrations of the spins and ratio of the bilinear interactions are found in agreement with the \nliterature.  An interesting feature that has been observed in equilibrium systems is the existence of \na special interact ion ratio 𝑅𝐶=|𝐽𝐴C|/𝐽𝐴B where the critical temperature of the system becomes \nindependent of the concentration ratio 𝑝 of the ternary alloy [25]. However,  in the dynamic system \nit has been  observed  that critical temperature  is always dependent  on the  concentration ratio 𝑝 of \nthe ternary alloy.  Theoretically, t his behavior was first  reported  firstly  in literatu re by Vatansever \n[37]. Dynamic behaviors of ternary alloys have not yet been studied experimentally.  Actually, \nthere  are only a few  experimen tal studies  about dynamic phase transition in literature  [51-53], such \nas, time dependent magnetic behavior of uniaxial cobalt films in the vicinity of the dynamic phase \ntransition (DPT) as a function of an oscillating magnetic field experimentally studied  by Berger \nand coworkers  [51].   \n \n4-  Conclusions  \n \nIn this paper , the dynamic magnetic features of the  mixed ferro -ferrimagnetic  ternary alloy of the \ntype ABpC1-p on a square  lattice, composed  of spins 1/2, 1, and 3/2 has been studied . A ternary \nalloy system  was simulated within the mean - field approximation based on Glauber type stochastic \ndynamic s. The concentration ratio was obtained as well as the exchange interaction ratio \ndependence of critical temperature.  It was found that the critical temperature cou ld be controlled \nby the concentration ratio and the exchange interaction ratio. In particular, it has been  found that \nthe critical temperature is always dependent  on the  concentration ratio 𝑝 of the ternary alloy in the \ndynamic system. It is believe d that this study will shed light on future experimental and theoretical \ninvestigations of mixed ferro -ferrimagnetic compounds in the presence of a time dependent \nexternal magnetic field .  \n References  \n \n[1] J.S. Miller, M. Drillon, Magnetism: Molecules to Materials V, Wiley –VCH, Wein -heim, \n2005.  \n[2] A. Goujon, F. Varret, V. Escax, A. Bleuzen, M. Verdaguer, Polyhedron 20 (2001) 1339.  \n[3] N. Shimamoto, S. I. Ohkoshi, O. Sato, K. Hashimoto, Inorg. Chem. 41 (2002) 678.  \n[4] O. Sato, S. Hayami, Y. Einaga, Z. Gu, Bull. Chem. Soc. Jpn. 76 (2003) 443.  \n[5] E. Codjovi, H. Tokoro, S. I. Ohkoshi, K. Hashimoto, J. Phys. Chem. B 109 (2005) 8251.  \n[6] J. -H. Park, F. Frye, N. E. Anderson, D. M. Pajerowski, Y. D. Huh, D. R. Talham, M. W. \nMeisel, J. Magn. Magn. Mater. 310 (2007 ) 1458.  \n[7] O. Sato, T. Iyoda, A. Fujishima, K. Hashimoto, Science 272 (1996) 704.  \n[8] D. A. Pejakovic, J. L. Manson, J. S. Miller, A. J. Epstein, Curr. Appl. Phys. 1 (2001) 15.  \n[9] S. Ohkoshi, T. Iyoda, A. Fujishima, K. Hashimoto, Phys. Rev. B 56 (1997) 11642.  \n[10] S. Ohkoshi, Y. Abe, A. Fujishima, K. Hashimoto, Phys. Rev. Lett. 82 (1999) 1285.  \n[11] S. S. Kaye, J. R. Long, J. Am. Chem. Soc. 127 (2005) 8590.  \n[12] O. Canko, B. Deviren, M. Keskin, JETP Lett. 87 (2008) 633.  \n[13] B. Deviren, O. Canko, M. Kesk in, J. Magn. Magn. Mater. 321 (2009) 1231.  \n[14] A. Bobák, J. Dely, Physica A 341 (2004) 281.  \n[15] A. Bobák, F. O. Abubrig, D. Horváth, Physica A 312 (2002) 187.  \n[16] W. Liu, Z. Xin, X. Wang, Phys. Lett. A 372 (2008) 1114.  \n[17] H. N. Motlagh, G. Rezaei, J. Alloys Comp. 711 (2017) 677.  \n[18] S. Tsuji, T. Kasama, T. Idogaki, J. Magn. Magn. Mater. 310 (2007) e471.  \n[19] E. Albayrak, J. Magn. Magn. Mater. 324 (2012) 1809.  \n[20] G. M. Buendía, J. E.Villarroel, J. Magn. Magn. Mater. 310 (2007) e495.  \n[21] H. Hu, Z. Xi n, W. Liu, Phys. Lett. A 357 (2006) 388.  \n[22] E. Albayrak, J. Magn. Magn. Mater. 323 (2011) 992.  \n[23] J. Dely, A. Bobák, J. Magn. Magn. Mater. 305 (2006) 464.  \n[24] E. K. Çam, E. Aydiner, J. Magn. Magn. Mater. 322 (2010) 1706.  \n[25] Z. D. Vatansever, Phys. L ett. A 381 (2017) 3450.  \n[26] M. Zukovic, A. Bobák, J. Magn. Magn. Mater. 322 (2010) 2868.  \n[27] J. Dely, A. Bobák, M. Zukovic, Journal of Physics: Conference Series 200 (2010) 022005.  \n[28] J. Dely, A. Bobák, D. Horvath, Acta Phys. Pol. A 113 (2008) 461.   \n [29] J. Dely, A. Bobák, Physica B 399 (2007) 155.  \n[30] S. I. Ohkoshi, T. Iyoda, A. Fujishima, K. Hashimoto, Phys. Rev. B 56 (1997) 11642.  \n[31] A. Kumar, S. M. Yusuf, L. Keller, Phys. Rev. B 71 (2005) 054414.  \n[32] E. Manuel, M. Evangelisti, M. Affronte, M. Okubo, M. Train, M. Verdaguer, Phys. Rev. B \n73 (2006) 172406.  \n[33] W. Dong, W. Zhang, Y. -O. Yang, L. -N. Zhu, D. -Z. Liao, K. Yoshimura, Z. -H. Jiang, S. -P. \nYan, P. Cheng, J. Magn. Magn. Mater. 309 (2007) 7.  \n[34] M. Arai, M. Miyake, M. Yamada, J. Phys. Chem. C 112 (2008) 1953.  \n[35] T. Matsuda, J. E. Kim, K. Ohoyama, Y. Moritomo, Phys. Rev. B 79 (2009) 172302.  \n[36] C. P. Krap, J. Balmaseda, B. Zamora, E. Reguera, Int. J. Hydrogen Energ. 35 (2010) 10381.  \n[37] E. Vatansever, Y. Yüksel, J. Alloys Comp. 689 (2016) 446e450.  \n[38] J.D. Alzate -Cardona , E. Restrepo -Parra, C.D. Acosta -Medina, Mater. Chem. Phys.  213 \n(2018) 362e367.  \n[39] M. Ertaş, E. Kantar, M. Keskin, J. Magn. Magn. Mater. 358 (2014) 56.  \n[40] M. Ertaş, B. Deviren, M. Keskin, J. Magn. Magn. Mater. 324 (2012 ) 704.  \n[41] M. Ertaş, M. Keskin, Chin. Phys. B 22 (2013) 120507.  \n[42] M. Ertaş, Superlattices Microstruct. 85 (2015) 734.   \n[43] M. Ertaş, M. Keskin, Phys. Lett. A 379 (2015) 1576.  \n[44] M. Ertaş, O. Canko, M. Keskin, J. Magn. Magn. Mater. 320 (2008) 1765.  \n[45] M. Ertaş, O. Canko, M. Keskin, J. Exp. Theo. Phys. 105 (2007)  1190.    \n[46] Q. Jiang, H. N. Yang and G. C. Wang, Phys. Rev. B 52 (1995) 14911.  \n[47] Z. A. Samoilenko, V. D. Okunev, E. I. Pushenko, V. A. Isaev, P. Gierlowski, K. Kolwas, S. \nJ. Lewandowsk i, Inorg. Mater. 39 (2003) 836.  \n[48] N. Gedik, D. S. Yang, G. Logvenov, I. Bozovic and A. H. Zewail, Science 316 (2007) 425.  \n[49] D. T. Robb, Y. H. Xu, O. Hellwing, J. McCord, A. Berger, M. A. Novotny, and P. A. \nRikvold, Phys. Rev. B 78 (2008) 134422.  \n[50] L. O’Malley, G. Korniss, S. S. P. Mungara and T. Caraco, Evol. Ecol. Res. 12 (2010) 279.  \n[51] A. Berger, O. Idigoras, P. Vavassori, Transient Behavior of the Dynamically Ordered Phase \nin Uniaxial Cobalt Films, Phys. Rev. Lett. 111, (2013) 190602.   \n[52] P. Jurcevic, H. Shen, P. Hauke, C. Maier, T. Brydges, C. Hempel, B.P. Lanyon, M. Heyl, \nR. Blatt, and C.F. Roos , Phys. Rev. Lett.  119 (2017) 080501   \n[53] M.  Heyl , Rep. Prog. Phys. 81 (2018) 054001  \n  \n List of the Figure Captions  \n \nFig. 1: (color online) Schematic  representation of ternary alloy spin model the type of ABpC1-p \ndefined on a square lattice.  A, B and C represented in yellow, blue and red respectively . \nInteraction between A and B sites is ferromagnetic (FM) and A and C antiferromagnetic \n(AFM).   \nFig. 2: Dynamic phase diagrams of the system in a ( 𝑅−𝑇𝐶) plane for the different values of p \nwith ℎ=0.1 and ℎ=0.5. \nFig. 3: Dynamic phase diagrams of the system in a ( 𝑝−𝑇𝐶) plane for the different values of R \nwith ℎ=0.1 and ℎ=0.5. \nFig. 4: (color online) Scaled  temperature dependence of total magnetization for different values of \n𝑝 with 𝑅=0.5 (upper figure),  𝑅=1.0 (middle figure) and  𝑅=2.0 (lower figure) . \nFig. 5: (color online)  Scaled temperature dependence of magnetization ( 𝑀𝐴,  𝑀𝐵  and 𝑀𝐶) for \ndifferent values of 𝑝 with 𝑅=1.0. \n \n \n \n \n \nFig. 1: \n \nSA = 1/2  \n JAB > 0 (FM)  \nJBC < 0 (AFM)  \nSB=1 \nSC = 3/2   \n \nh=0.1\nR0.0 0.5 1.0 1.5 2.0 2.5 3.0TC\n02468\np=0.00 \np=0.25 \np=0.50 \np=0.75 \np=1.00 \nh=0.5\nR0.0 0.5 1.0 1.5 2.0 2.5 3.0TC\n02468\np=0.00 \np=0.25 \np=0.50 \np=0.75 \np=1.00  \n \nFig. 2:   \n  \nh=0.1\np0.0 0.2 0.4 0.6 0.8 1.0TC\n02468\nR=0.0 \nR=0.5\nR=1.0\nR=1.5\nR=2.0\nR=2.5\nR=3.0\nh=0.5\np0.0 0.2 0.4 0.6 0.8 1.0TC\n0246R=0.0 \nR=0.5\nR=1.0\nR=1.5\nR=2.0\nR=2.5\nR=3.0\n \n \nFig. 3:  \n  \nT0 1 2 3 4MT\n0.00.20.40.60.8\np=0.00\np=0.25\np=0.50\np=0.75\np=1.00\nR=JAC/JAB=0.5\nR=JAC/JAB=1.0\nT0 1 2 3 4MT\n-0.4-0.20.00.20.40.60.8\np=0.00\np=0.25\np=0.50\np=0.75\np=1.00\nR=JAC/JAB=2.0\nT0 1 2 3 4MT\n-0.6-0.4-0.20.00.20.40.60.81.0\np=0.00\np=0.25\np=0.50\np=0.75\np=1.00\n \nFig. 4:  \n  \np=0.00\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n-1.0-0.50.00.51.01.52.0\nMA\nMB\nMC\np=0.25\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n-2.0-1.5-1.0-0.50.00.51.01.5\nMA\nMB\nMC\np=0.50\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n-2.0-1.5-1.0-0.50.00.51.01.5\nMA\nMB\nMC\np=0.75\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n-2.0-1.5-1.0-0.50.00.51.01.5\nMA\nMB\nMC\np=1.00\nT0.0 0.5 1.0 1.5 2.0 2.5 3.0M\n0.00.20.40.60.81.01.2\nMA\nMB\nMC\n \n \nFig. 5: \n \n "    },    {        "title": "1702.04787v2.Triad_interactions_and_the_bidirectional_turbulent_cascade_of_magnetic_helicity.pdf",        "content": "Postprint version of the manuscript published in Phys. Rev. Fluids 2, 054605 (2017)\nTriad interactions and the bidirectional turbulent cascade of magnetic helicity\nMoritz Linkmann1,\u0003and Vassilios Dallas2,y\n1Department of Physics and INFN, University of Rome Tor Vergata,\nVia della Ricerca Scienti\fca 1, 00133 Rome, Italy\n2Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, United Kingdom\nUsing direct numerical simulations we demonstrate that magnetic helicity exhibits a bidirectional\nturbulent cascade at high but \fnite magnetic Reynolds numbers. Despite the injection of positive\nmagnetic helicity in the \row, we observe that magnetic helicity of opposite signs is generated between\nlarge and small scales. We explain these observations by carrying out an analysis of the magnetohy-\ndrodynamic equations reduced to triad interactions using the Fourier helical decomposition. Within\nthis framework, the direct cascade of positive magnetic helicity arises through triad interactions that\nare associated with small scale dynamo action, while the occurrence of negative magnetic helicity\nat large scales is explained through triad interactions that are related to stretch-twist-fold dynamics\nand small scale dynamo action, which compete with the inverse cascade of positive magnetic helicity.\nOur analytical and numerical results suggest that the direct cascade of magnetic helicity is a \fnite\nmagnetic Reynolds number Rme\u000bect that will vanish in the limit Rm!1 .\nI. INTRODUCTION\nThe existence of planetary and stellar magnetic \felds is currently attributed to dynamo action [1, 2]. One of the\ntheoretical arguments to explain the generation and preservation of magnetic \felds in spatial scales much larger than\nthe outer scales of planets and stars is the inverse cascade of magnetic helicity in magnetohydrodynamic (MHD)\nturbulent \rows [3]. Magnetic helicity is de\fned as the correlation between the magnetic \feld b\u0011 r\u0002 aand\nthe magnetic potential a, i.e.,Hb\u0011ha\u0001bi, with the angular brackets denoting a spatial average unless indicated\notherwise. Magnetic helicity is considered to play a critical role in the long-term evolution of stellar and planetary\nmagnetic \felds [4] and hence it is important to understand its dynamics across scales in order to shed light on the\nsaturation mechanisms of the large-scale magnetic \felds of planets and stars.\nEarly studies using turbulent closure models [5] and mean \feld theory [1, 6, 7] have shown (within the framework of\ntheir approximations) that magnetic helicity cascades from small scales to large scales in agreement with the prediction\nfrom equilibrium statistical mechanics for the ideal MHD equations [3]. In these cases, the inverse cascade of magnetic\nhelicity was associated with the \u000be\u000bect [8] of large scale dynamos, where the kinetic helicity Hu=hu\u0001!i(where u\ndenotes the velocity \feld and !=r\u0002uthe vorticity \feld) generates opposite signs of magnetic helicity between large\nand small scales. This change in sign across scales is further supported by the magnetic helicity spectra of solar wind\ndata [9], which show Hb<0 at small wavenumbers and Hb>0 at large wavenumbers. One way to understand this\nchange in sign across scales is the conceptual stretch-twist-fold (STF) mechanism [10, 11]. This mechanism proposes\nthat the advection of magnetic \feld lines by a positive helical \row leads to a positive magnetic helicity at small scales\nand to a negative magnetic helicity at large scales since large scale magnetic \feld lines are twisted in the opposite\ndirection. However, it is presently not clear if such a mechanism is directly associated with a non-linear cascade\nprocess [12].\nThe inverse cascade of magnetic helicity has been veri\fed by direct numerical simulations (DNSs) [13{15] at mod-\nerate values of magnetic Reynolds number Rm, where the small-scale magnetic helicity is dissipated quite fast. In\nthis case, in contrast with the theory, a direct cascade of magnetic helicity with smaller magnitude was also observed.\nThis bidirectional cascade of Hb, where an inverse and a direct cascade of magnetic helicity coexist, was observed\nrecently even at high Rm\rows [16, 17]. In the limit of high Rmconcerns have been raised about the e\u000bectiveness\nof the\u000be\u000bect in generating large-scale magnetic \felds with strong amplitude due to the detrimental feedback that\nfast-growing small-scale magnetic \felds have on the growth rate of the large scales [18, 19]. These concerns are sup-\nported in some sense by the bidirectional cascade of magnetic helicity, because the magnitude of the inverse cascade\n\u0003linkmann@roma2.infn.it\nyv.dallas@leeds.ac.ukarXiv:1702.04787v2  [physics.flu-dyn]  19 May 20172\nis limited by the existence of the residual direct cascade, and by the nonlocality of the inverse cascade of magnetic\nhelicity in the statistically stationary regime [15, 17].\nIn this paper, we focus on the bidirectional cascade of magnetic helicity and on the mechanism of magnetic helicity\nto generate opposite signs of helicity across the scales. By means of DNSs we inject mean magnetic helicity in our\n\rows using a helical electromagnetic forcing, while the velocity \feld is forced by a nonhelical forcing. This is in\ncontrast to dynamo studies, which typically force only the velocity \feld using a helical forcing. In dynamo studies,\nthe kinetic helicity generates opposite signs of magnetic helicity between large and small scales. This sign change\nacross scales makes the interpretation of statistics ambiguous. With our approach we attempt to have a dominant\nsign of magnetic helicity across the scales in order to avoid such ambiguities as much as possible. The interpretation of\nour numerical results is supported by analytical work on the triadic interactions of helical modes in MHD turbulence\n[20, 21]. This analysis is based on the helical decomposition of Fourier modes [22{24], which has been an important\ntool to understand the cascade dynamics of helical \rows in Navier-Stokes turbulence [25{30]. Here, the analysis of\ntriadic interactions of helical modes demonstrates that it can also be a useful tool to understand further (a) the\npresence of the direct cascade of magnetic helicity to small scales and (b) the mechanism that generates opposite\nsigns of magnetic helicity across the scales.\nThis paper is organized as follows. We begin with the description of the numerical method and the resulting\ndatabase in Sec. II. Sections III and IV present the global and spectral dynamics from our numerical simulations,\nrespectively. Our numerical results are interpreted in Sec. V in terms of the triadic interactions of helical modes,\nwhere we propose an explanation for the direct cascade of magnetic helicity and for the mechanism that generates\nopposite signs of magnetic helicity across the scales. We conclude in Sec. VI by summarizing our \fndings.\nII. NUMERICAL SIMULATIONS\nThe present work focuses on the bidirectional cascade of magnetic helicity. Thus, we consider numerical simulation\nof three-dimensional MHD turbulent \rows forced at intermediate wavenumbers in the absence of large scales conden-\nsates. Forcing at intermediate scales and aiming for a turbulent \row with high enough scale separation poses serious\ncomputational constraints, since a very large range of scales needs to be resolved. So, to circumvent the demanding\nscale separation requirement and to avoid the condensation of energy at large scales, we consider high-order dissipation\nterms acting at the small and the large scales, respectively. These dissipation terms e\u000bectively increase the extent of\nthe inertial range simulated for a given resolution by reducing the range of scales over which dissipation is e\u000bective.\nTherefore, we numerically solve the equations\n(@t\u0000\u0017\u0000(\u00001)m+1\u0001\u0000m\u0000\u0017+(\u00001)n+1\u0001n)u=u\u0002!+j\u0002b\u0000rP+fu\n(@t\u0000\u0011\u0000(\u00001)m+1\u0001\u0000m\u0000\u0011+(\u00001)n+1\u0001n)b=r\u0002(u\u0002b) + fb (1)\nwhere udenotes the velocity \feld, bthe magnetic induction expressed in Alfv\u0013 en units, !=r\u0002uthe vorticity,\nj=r\u0002bthe current density, and P=p+juj2=2, withpthe pressure and fuandfbthe external mechanical and\nelectromagnetic forces, respectively. Energy is dissipated at the small scales by the terms proportional to \u0017+and\u0011+\nand at the large scales by \u0017\u0000and\u0011\u0000. The indices nandmspecify the order of the Laplacian used. In order to obtain\na large inertial range, we choose n=m= 4. In the absence of forcing and dissipation Eqs. (1) reduce to the ideal\nMHD equations, which conserve the total energy E=Eu+Eb=1\n2hjuj2+jbj2i, the magnetic helicity Hb=ha\u0001bi\nand the cross-helicity Hc=hu\u0001bi. These conserved quantities are those that determine the turbulent cascades in our\n\rows.\nEquations (1) are solved numerically in a cubic periodic domain with sides of length 2 \u0019Lusing the standard\npseudospectral method, which ensures that r\u0001u= 0 andr\u0001b= 0. Full dealiasing is achieved by the 2 =3 rule and\nas a result the minimum and maximum wavenumbers are kbox= 1 andkmax= N/3, respectively, where Nis the\nnumber of grid points in each Cartesian coordinate. Further details of the code can be found in Refs. [31, 32].\nAs it was mentioned above, in these simulations we choose to inject mean magnetic helicity in our \rows, attempting\nto have a dominant sign of magnetic helicity across the scales in order to be able to interpret our results unambiguously\nin comparison to dynamo studies. In order to do this in a systematic way we force busing a helical forcing and u\nusing a non-helical forcing. Here, we choose to force both uandbwith the same forcing amplitude, i.e., jfuj=jfbj,\nso that both quantities are dynamically important and bhas a nonlinear feedback on the \row through the Lorenz\nforce. The forces fuandfbare constructed from a randomized superposition of eigenfunctions of the curl operator\n[14, 16, 33], resulting in Gaussian distributed and \u000ecorrelated in time forces whose helicities hfu;b\u0001r\u0002 fu;biand\ncorrelationhfu\u0001fbican be exactly controlled. The speci\fc random nature of the forces ensures that at steady state\nthe total energy input rate \"=\"u+\"b=hu\u0001fui+hb\u0001fbi/jfuj2+jfbj2is known a priori [34]. Therefore, \"can be\nused as a control parameter since the amplitudes of the external forces are input parameters. The external forces are3\nchosen such that the net cross helicity in the \row is negligible by keeping the correlation hfu\u0001fbi= 0. In summary,\nnoHcand noHuare injected into the \row, while the injection of Hbis maximal. Here, we choose to exclude the\ninjection of cross-helicity in the \row because we want to focus on the cascade dynamics of magnetic helicity, which\ncan be in\ruenced by introducing correlations between the velocity and the magnetic \feld [3, 20]. Finally, the initial\nmagnetic and velocity \felds are in equipartition with energy spectra peaked at the forcing wavenumber kfand zero\nhelicities, i.e., Hb=Hc=Hu= 0.\nFor all simulations, we set the magnetic Prandtl number for the small scales Pm+=\u0011+=\u0017+and for the large\nscalesPm\u0000=\u0011\u0000=\u0017\u0000to unity, viz. Pm+=Pm\u0000= 1. The values of \u0017\u0000and\u0011\u0000are tuned such that the inverse\ncascade is damped before the largest scales of the system are excited while the values of \u0017+and\u0011+are such that\nkmax=kd\u00151:25 is satis\fed for all the simulations, where kd\u0011(\"=(\u0017+)3)1=(6n\u00002)is the dissipation wavenumber [35].\nThe magnetic Reynolds number is de\fned based on the control parameters of the problem as Rmf\u0011Uk1\u00002n\nf=\u0011+\nwithU\u0011(\"=kf)1=3.\nThe energy input and thus the dissipation rate \"is adjusted such that Uis kept constant while the scale separation\nkfLincreases, resulting in a set of simulations with the same Reynolds number. For simulations at kfL= 10 with\ndi\u000berent Reynolds numbers we kept the energy input constant and we varied \u0017+=\u0011+. All the necessary numerical\nparameters that make these simulations reproducible are given in Table I along with their total runtime Tof the\nsimulations normalized by \u001cf\u0011(Ukbox)\u00001, a time scale de\fned based on the control parameters.\nkfL N Rm f\u0017+=\u0011+\u0017\u0000=\u0011\u0000jfujjfbjT=t f\n10 128 7\u00021036:30\u000210\u0000120.05 1 1 320\n20 256 7\u00021034:92\u000210\u0000140.05p\n2p\n2 130\n40 512 7\u00021033:68\u000210\u0000160.05 2 2 100\n10 128 7\u00021026:30\u000210\u0000110.05 1 1 90\n10 256 9\u00021044:92\u000210\u0000140.05 1 1 110\nTABLE I: Numerical parameters of the simulations. Note that kfdenotes the forcing wave number, Tis the total\nruntime in simulation units, jfujandjfbjare the mechanical and electromagnetic forcing magnitudes, respectively,\nandtf\u0011(Ukbox)\u00001a time scale de\fned based on the control parameters. All simulations are well resolved with\nkcut=kd>1:25. The hyperdissipative Reynolds number is de\fned as Rmf=Uk1\u00002n\nf=\u0011+.\nIII. TIME EVOLUTION\nBefore discussing the interaction of the helical modes in order to understand further the dynamics of magnetic\nhelicity across scales, we provide an overview of the statistics of the \rows under study. In Fig. 1a we have plotted\nthe time evolution of the normalized magnetic helicity\n\u001ab\u0011Hb=(hjaj2ihjbj2i)1=2; (2)\nwhich belongs to the range \u00001\u0014\u001ab\u00141. For\u001ab= 0 the \rows have no magnetic helicity, while for \u001ab=\u00061 the \rows\nare fully dominated by magnetic helicity, which means that the Lorentz force j\u0002bis expected to be zero on average.\nAt timet= 0, we set \u001ab= 0 but instantaneously it reaches high values because of the direct injection of positive\nHbby the helical electromagnetic force. At steady state \u001abreaches a mean value of 0.7 for all the \rows with scale\nseparations kfL= 10;20;40, indicating that the \rows are dominated to a large degree by positive mean magnetic\nhelicity.\nThe strength of the direct and the inverse cascade of magnetic helicity at steady state can be quanti\fed by the rate\nof dissipation in large scales \"\u0000\nHband small scales \"+\nHbas\n\"\u0006\nHb\u0011\u0017\u0006ha\u0001\u0001\u0006nbi: (3)\nThen, the total magnetic helicity dissipation can be computed as the sum of the dissipation at large and small scales,\nviz.,\"Hb=\"\u0000\nHb+\"+\nHb. In Fig. 1b, we plot the time-series of the ratio \"\u0000\nHb=\"Hbfor the three scale separations that we\nsimulated (see Table I). The values of \"\u0000\nHb=\"Hbcan range from 0 to 1. For \"\u0000\nHb=\"Hb= 0 there is no inverse cascade\nof magnetic helicity, while for \"\u0000\nHb=\"Hb= 1 there is no direct cascade of magnetic helicity. In the initial transient\nregime the amount of magnetic helicity that is transferred to the large scales increases monotonically until large scale4\n 0 0.2 0.4 0.6 0.8 1\n 0  20  40  60  80  100(a)ρb\nt/tfkf L=40\nkf L=20\nkf L=10\n(a)\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0  20  40  60  80  100ε-\nHb / εHb\nt/tfkf L=40\nkf L=20\nkf L=10 (b)\nFIG. 1: Time series of (a) the normalized magnetic helicity \u001aband (b) the ratio \"\u0000\nHb=\"Hbfor \rows with scale\nseparations kfL= 10;20 and 40.\ndissipation takes place and the ratio \"\u0000\nHb=\"Hbsaturates to the value of 0.9 (see Fig. 1b). This clearly shows that there\nis a bidirectional cascade of Hbwith 90% of the magnetic helicity cascading to the large scales at steady state while\n10% of the magnetic helicity cascades towards the small scales since \"+\nHb=\"Hb'0:1. The bidirectional cascade of Hb\nhad been observed qualitatively in Refs. [15, 16] and it was quanti\fed in Ref. [17]. As scale separation increases the\namount of magnetic helicity that cascades to the large and small scales remains \fxed in contrast to the cascade of the\ntotal energy, where \"\u0000=\"/(kfL)\u00001as it was shown in [17], with the total energy dissipation rate being \"=\"\u0000+\"+\nand\"\u0006\u0011\u0011\u0006hjr\u0006nbj2i+\u0017\u0006hjr\u0006nuj2i. Thus, for kfL\u001d1 we expect the ratio \"\u0000=\"!0, implying that the total\nenergy will cascade only toward the small scales while the direct cascade of magnetic helicity will not vanish because\n\"+\nHb=\"Hbis expected to remain \fnite as our numerical simulations suggest.\nHowever, for \fxed kfLit is plausible that the direct cascade of magnetic helicity vanishes in the high-magnetic-\nReynolds-number limit. This can be inferred by considering the following upper bound derived in Ref. [36] for\nthe magnitude of the total dissipation rate of magnetic helicity at small scales for the MHD equations that involve\ndissipation terms with Laplacian operators (i.e., the exponent n= 1)\nj\"Hbj\u0011\u0011jha\u0001\u0001bij=\u0011jhb\u0001jij\u0014\u0011hjbj2i1=2hjjj2i1=2\u0014\u00111=2hjbj2i1=2\"1=2; (4)\nwhere\"=\u0011hjjj2i+\u0017hj!j2iis the total energy dissipation rate in this case, \u0011is the magnetic resistivity, and \u0017is the\nkinematic viscosity. In high-Reynolds-number MHD turbulence, \"becomes \fnite and independent of \u0011and\u0017[37{39].\nThus, in the limit of \u0011!0 Eq. (4) suggests that j\"Hbj!0, unlesshjbj2i/1=\u0011, implying in\fnite magnetic energy.\nA similar inequality for the dissipation rate of magnetic helicity at small scales for Eqs. (1), which involve higher-\norder dissipation terms, can be derived as\nj\"+\nHbj\u0011\u0011+jha\u0001\u0001nbij=\u0011+jhb\u0001\u0001n\u00001jij6\u0011+hjbj2i1=2hj\u0001n\u00001jj2i1=2: (5)\nNote that for n= 1 we recover Eq. (4), however, for n>1 the termhj\u0001n\u00001jj2icannot be directly related to the small-\nscale dissipation rate of magnetic energy \"+\nb\u0011\u0011+hb\u0001\u0001nbi=\u0011+hjrn\u00001jj2i. Therefore, to make any conclusions, if\nany, for the behavior of j\"+\nHbjin the limit of \u0011+!0, we need to compute the scaling of \"+\nHbwith\u0011+as we cannot\nmake any a priori statements like in Eq. (4). For this purpose, we performed three simulations with the same scale\nseparation kfL= 10 but di\u000berent Rmf. These preliminary results are plotted in Fig. 2, which shows that \"+\nHbnormalized by the total dissipation of magnetic helicity \"Hbhas a weak power-law behavior in terms of Rmf, i.e.\n\"+\nHb=\"Hb/Rm\u00000:22\nf. This scaling suggests that the direct cascade of magnetic helicity may not persist in the limit\nofRmf!1 . However, further investigation into this issue is necessary in order to understand this scaling law and\nthe small-scale dynamics of the magnetic helicity.5\n10−210−1100\n102103104105106107Rm−0.22\nf\nε+\nHb/εHb\nRmf\nFIG. 2: Dependence of \"+\nHb=\"Hbon the magnetic Reynolds number Rmffor the \rows with scale separation\nkfL= 10.\nIV. SPECTRAL DYNAMICS\nAfter considering the global dynamics and the time evolution of our \rows, we now elaborate on the spectral dynamics\nof magnetic helicity in order to have a clear picture of the dynamics across scales. This is best demonstrated by looking\nat the spectrum of magnetic helicity Hb(k) and its normalized \rux \u0005 Hb(k)=\"Hbfor the \row with scale separation\nkfL= 40 (see Fig. 3). The magnetic helicity \rux is de\fned as\n\u0005Hb(k) =kX\nk0=1X\njkj=k0^b\u0003\nk\u0001\\(u\u0002b)k; (6)\nand the magnetic helicity spectrum as Hb(k) =P\njkj=k^a\u0003\nk\u0001^bk, where the caret denotes the Fourier modes of the\ncorresponding real vector \felds and the asterisk the complex conjugate. The gray curves denote instantaneous\n-2-1.5-1-0.5 0 0.5 1\n 0.1  1time\n(a)ΠHb(k) / εHb\nk/kfkf L = 40\n(a)\n10-710-610-510-410-310-210-1100\n 0.1  1time\n(b)(k/kf)-10/3|Hb(k)|\nk/kfkf L = 40 (b)\nFIG. 3: (a) Magnetic helicity \rux \u0005 Hb(k) normalized by \"Hband (b) absolute value of the magnetic helicity\nspectrumjHb(k)jfor a \row with scale separation kfL= 40. The light gray curves denote instantaneous quantities at\nthe transient regime of the \row for 0 :56t=tf63, while the green (dark gray) curve denotes the time-averaged\npro\fle in the steady-state regime.\npro\fles at the transient regime of the \row while the green (dark gray) curves denote the time-averaged pro\fles at the6\nsteady state regime. Negative values of the magnetic helicity \rux imply upscale transfer while positive values imply\ndownscale transfer. Thus, Fig. 3a shows clearly the bidirectional cascade behavior of magnetic helicity with most of\nHbcascading towards large scales in agreement with Fig. 1b. The gray curves indicate how the \rux builds up at the\ntransient state and of course these curves do not satisfy the balance \u0005 Hb(k) =\"Hb, which is only expected to be valid\non average over a statistically stationary regime.\nAt the transient stage of the simulation the inverse and the direct cascade of magnetic helicity is dominated by local\ninteractions, as it has already been reported [15]. Before the \row saturates to a steady-state solution a spectrum with\na negative power-law slope can be observed at low wave numbers. The scaling of this spectrum was proposed to be\nHb(k)/k\u000010=3[16, 37], which agrees with our data. However, if one integrates further in time the spectrum develops\neven more as the \row saturates to a statistically stationary regime. In this regime, most of the magnetic helicity is\nconcentrated at the largest scales [see also the spectrum of normalized helicity \u001ab(k) in Fig. 5(a)] and the inverse\ncascade of Hbbecomes nonlocal in wave-number space, i.e., Hbis transferred directly from the forced scale to the\nlargest scales of the \row, while the forward cascade remains local [14, 15, 17]. This transition to nonlocal interactions\nat steady state occurs when condensation takes place at the largest scale of the system. In this case, the large-scale\nvortex interacts directly with the small-scale vortices of the \row (i.e., nonlocally in wave-number space) [17]. This\nbehavior is also observed in con\fned two-dimensional hydrodynamic turbulent \rows [40, 41]. Finally, the scaling of\nthe time-averaged spectrum of magnetic helicity at steady state is Hb(k)/k0for wave numbers k <kf, suggesting\nthat the large scales behave as if they were in absolute equilibrium [17]. Now, the scaling for k > kfseems to be\nHb(k)/k\u00005=3. However, this is not conclusive from the plot due to the limited spectral range and the high-order\ndissipative terms that act at the largest wave numbers.\nThe behavior of the magnetic helicity \rux \u0005 Hb(k) at steady-state with increasing Rmfis presented in Fig. 4. As\ncan be seen from the \fgure, the small-scale range where \u0005 Hb(k)'\"+\nHbextends over a larger range of wave numbers\nsuccessively with increasing Rmfas expected for a cascade process. However, the height of the plateau, i.e., the value\nof\"+\nHb, decreases with increasing Rmf, as quanti\fed in Fig. 2.\n-1-0.5 0 0.5 1\n 0.1  1  10ΠHb(k) / εHb\nk/kfRmf = 700\nRmf = 7000\nRmf = 90000\nFIG. 4: Spectral \rux of magnetic helicity normalised by its dissipation rate \u0005 Hb(k)=\"Hbat di\u000berent values of Rmf\nfor the \rows with scale separation kfL= 10. The amplitude of the cascade at k=kf>1 is quanti\fed as a function of\nRmfin Fig. 2.\nThe purpose of our simulations using a positively helical electromagnetic force was to impose a mean magnetic\nhelicity in our \rows with the intention to have a dominant sign of magnetic helicity across the scales. However, at\nthe steady-state regime of our simulations there are low-wave-number modes that develop a negative sign of magnetic\nhelicity even though the mean value of magnetic helicity is positive in the \row. This is clearly depicted in Fig. 5a,\nwhere we plot the time-averaged spectra of the normalized magnetic helicity \u001ab(k) for \rows with kfL= 10;20;40.\nNote that all the \rows generate negative \u001ab(k) at wave number k= 3, which indicates that there is a consistent\nmechanism generating opposite signs of helicity across the scales even in these \rows that are dominated by positive\nnet magnetic helicity. Moreover, we observe that the number of modes at k < kfwith negative magnetic helicity\nincreases as the scale separation kfLincreases.\nDespite the nonhelical mechanical forcing, a whole spectrum of positive kinetic helicity is observed across the scales.\nThis is shown in Fig. 5b, where we plot the spectra of the relative kinetic helicity \u001au(k)\u0011Hu(k)=(hj^ukj2ihj^!kj2i)1=27\n-1-0.5 0 0.5 1\n 1  10  100ρb(k)\nkkf L = 40\nkf L = 20\nkf L = 10\n(a)\n-1-0.5 0 0.5 1\n 1  10  100ρu(k)\nkkf L = 40\nkf L = 20\nkf L = 10 (b)\nFIG. 5: (a) Time-averaged relative magnetic helicity spectra \u001ab(k) and (b) time-averaged relative kinetic helicity\nspectra\u001au(k) for \rows with scale separations kfL= 10;20, and 40.\nfor the \rows with kfL= 10;20;and 40. The values of \u001au(k) for wave numbers k < kfdecrease as the scale\nseparation kfLincreases, while they are particularly high close to the forcing scale where \u001ab(k) is also dominant.\nThis correlation seems to be related to the injection of positive magnetic helicity in the \row. In order to understand\nfurther the observations from our numerical simulation on the generation of the \u001au(k) spectrum, the direct cascade\nof magnetic helicity, and the mechanism that generates opposite signs of magnetic helicity between large and small\nscales, we study analytically the triadic interactions of helical modes in the following section.\nV. TRIAD INTERACTIONS OF HELICAL MODES\nThe three-dimensional vector \felds of the velocity and the magnetic \feld are solenoidal (viz., ik\u0001^uk= 0 and\nik\u0001^bk= 0), hence ^ukand^bkmust lie in the plane perpendicular to the wave vector k. This plane is spanned by two\neigenvectors of the curl operator in Fourier space with nonzero eigenvalues. Since these eigenvectors are by de\fnition\nfully helical, each Fourier mode ^ukand^bkcan be further decomposed into two modes with positive and negative\nhelicity\n^uk(t) =u+\nk(t)h+\nk+u\u0000\nk(t)h\u0000\nk=X\nskusk\nk(t)hsk\nk; (7)\n^bk(t) =b+\nk(t)h+\nk+b\u0000\nk(t)h\u0000\nk=X\n\u001bkb\u001bk\nk(t)h\u001bk\nk; (8)\nwhere the basis vectors h\u0006\nkare the orthonormal eigenvectors of the curl operator in Fourier space satisfying\nik\u0002hsk\nk=skjkjhsk\nkwithsk=\u0006and\u001bk=\u0006. This is the so-called helical decomposition [22{24]. To study\nthe triad interaction of helical modes, we use the formalism that was developed by Wale\u000be [26] for the Navier-Stokes\nequations and extended by Linkmann et al. [20, 21] to the MHD equations. This is essentially a linear stability\nanalysis of a dynamical system obtained from the MHD equations in the limits of \u0017!0 and\u0011!0. In the MHD\nequations that are shown here in Fourier space\n(@t+\u0017k2)^uk=\u0000F\u0014\nr\u0012\np+juj2\n2\u0013\u0015\n+X\nk+p+q=0\u0010\n(\u0000ip\u0002^up)\u0003\u0002^u\u0003\nq+ (ip\u0002^bp)\u0003\u0002^b\u0003\nq\u0011\n; (9)\n(@t+\u0011k2)^bk=ik\u0002X\nk+p+q=0^u\u0003\np\u0002^b\u0003\nq; (10)8\nwithFdenoting the Fourier transform as a linear operator, the convolutions that describe the respective Fourier\ntransforms of the inertial term u\u0002!, the Lorentz force j\u0002b, and the curl of the electromotive force r\u0002 (u\u0002b)\nare reduced to single triads of wave vectors k,p;andqsatisfying k+p+q= 0. For convenience and without loss\nof generality we impose the ordering jkj\u0014jpj\u0014jqj. In order to study the dynamics in the inertial range of scales,\nthe dissipation terms are neglected in the limits of \u0017!0 and\u0011!0. Following Lessinnes et al. [42], by substituting\nEqs. (7) and (8) into Eqs. (9) and (10) after restricting the convolutions to single triads and subsequently taking\nthe inner product with the helical basis vectors, one can then derive the following system of ordinary di\u000berential\nequations, which conserves the ideal invariants of the MHD equations [42]:\n@tusk\nk\u0003=gskspsq(spp\u0000sqq)usp\npusq\nq\u0000gsk\u001bp\u001bq(\u001bpp\u0000\u001bqq)b\u001bp\npb\u001bq\nq;\n@tusp\np\u0003=gskspsq(sqq\u0000skk)usq\nqusk\nk\u0000g\u001bksp\u001bq(\u001bqq\u0000\u001bkk)b\u001bq\nqb\u001bk\nk;\n@tusq\nq\u0003=gskspsq(skk\u0000spp)usk\nkusp\np\u0000g\u001bk\u001bpsq(\u001bkk\u0000\u001bpp)b\u001bk\nkb\u001bp\np;\n@tb\u001bk\nk\u0003=\u001bkk\u0000\ng\u001bk\u001bpsqb\u001bp\npusq\nq\u0000g\u001bksp\u001bqusp\npb\u001bq\nq\u0001\n;\n@tb\u001bp\np\u0003=\u001bpp\u0000\ngsk\u001bp\u001bqb\u001bq\nqusk\nk\u0000g\u001bk\u001bpsqusq\nqb\u001bk\nk\u0001\n;\n@tb\u001bq\nq\u0003=\u001bqq\u0000\ng\u001bksp\u001bqb\u001bk\nkusp\np\u0000gsk\u001bp\u001bqusk\nkb\u001bp\np\u0001\n; (11)\nwhere the geometric factors gdescribe the coupling between the helical basis vectors, i.e. gskspsq=hsk\nk\u0001(hsp\np\u0002hsq\nq)\n[26]. Further details on the full derivation of Eqs. (11) can be found in Refs. [20, 21, 42]. A graphical representation of\nthe system of equations (11) is shown in Fig. 6, where each convolution term corresponds to a triangle that represents\na triad of wave vectors. Note that two triads are required to describe the interactions that correspond to the term\nr\u0002(u\u0002b) due to a necessary symmetrization of the convolution in Fourier space. However, this symmetrization does\nnot allow one to disentangle the triadic interactions of the advection term u\u0001rband the stretching term b\u0001ruof\nthe magnetic \feld [20]. Di\u000berent interactions of helical modes can now be studied via Eqs. (11) by choosing di\u000berent\ncombinations of sk;sp;sqand\u001bk;\u001bp;\u001bq.\nFIG. 6: Graphical representation of the dynamical system given by Eqs. (11) consisting of the coupling among\nvelocity (left) and magnetic-velocity (right) triadic interactions as in Ref. [21]. The magnetic-velocity interactions\ncorrespond to several terms in Eqs. (11), depending on whether the time evolution of the velocity or the magnetic\n\feld modes is considered.\nA linear stability analysis of steady solutions of Eqs. (11) has been carried out by Linkmann et al. [20], where a\nlinear instability corresponds to a transfer of energy from the unstable helical mode (denoted in capital letters) of a\nsingle triad interaction to the perturbations (denoted in lower case letters), i.e., to the other two helical modes of the\ntriad. Under the assumption that the statistical behavior of the \row is controlled by the stability characteristics of\nthese isolated triads (referred to as the instability assumption) [26, 43], it is possible to draw conclusions concerning\nthe energy and helicity transfer in the MHD equations from the stability properties of Eqs. (11). Di\u000berent physical\nprocesses such as the inverse cascade of magnetic helicity or kinematic dynamo action can then be studied on the level\nof triad interactions by setting up speci\fc perturbation problems [20]. Depending on the characteristic wave numbers\nand the sign of helicities of the interacting modes, the dominant interscale energy transfers can then be identi\fed\nthrough a comparison of the growth rates of the perturbations [21].\nIf we consider a steady solution for the velocity \feld subject to magnetic perturbations, it is possible to identify\nall the triadic interactions of helical modes that produce small as well as large scale growth of the magnetic \feld (see\nRef. [20] for details). With this result Linkmann et al. [21] were able to interpret how small- and large-scale dynamos\noperate at the level of triadic interactions. In Fig. 7 we present all the triads that lead to linear instabilities and\nhence to energy transfer, where the red arrows indicate large-scale dynamo action (i.e., energy is transferred to small-\nwavenumber helical modes of the magnetic \feld), the blue arrows indicate small-scale dynamo action (i.e., energy is\ntransferred to large-wavenumber helical modes of the magnetic \feld), and the thickness of the arrows indicates the\nmagnitude of the transfer (see Refs. [20, 21] for details). More precisely, the thickness of the arrows is qualitative,9\n(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFIG. 7: Triadic interactions of helical modes leading to energy transfers when considering a steady solution for the\nvelocity \feld subject to magnetic perturbations. Blue (dark gray) arrows indicate small-scale dynamos and red (light\ngray) arrows indicate large-scale dynamos. The thickness of the arrows indicates the magnitude of the transfer.\nre\recting that for any nontrivial triad geometry the growth rates of the perturbations have a consistent ordering [see,\ne.g., Eq. (A9)].\nSimilarly, if we consider the linear stability analysis of a steady solution for the magnetic \feld subject to velocity\nand magnetic perturbations, one can obtain the triadic interactions involving unstable helical modes shown in Fig. 8,\nwhere the green arrows indicate magnetic energy transfer, the magenta arrows indicate the conversion of magnetic to\n(a)\n (b)\n (c)\n(d)\n (e)\n (f)\nFIG. 8: Triadic interactions of helical modes leading to energy transfer when considering a stable solution for the\nmagnetic \feld subject to velocity and magnetic perturbations. Green (light gray) arrows indicate an inverse transfer\nof magnetic helicity and magenta (dark gray) arrows indicate conversion of magnetic to kinetic energy due to the\nLorentz force. The thickness of the arrows indicates the magnitude of the transfer.\nkinetic energy due to the action of the Lorentz force, and the thickness of the arrows indicates the magnitude of the\ntransfer. As can be seen in Fig. 8, triadic interactions of helical modes leading to energy transfer occur only if (a)\nthe steady solution of the magnetic \feld and the magnetic perturbation have the same sign of magnetic helicity, and10\n(b) the characteristic wave number of the magnetic perturbation is smaller than the characteristic wavenumber of the\nsteady solution. In other words, the energy transfer between the magnetic modes occurs exclusively from large to\nsmall wavenumbers and only between magnetic modes with the same sign of helicity, which implies an inverse cascade\nof magnetic helicity [20]. As indicated by the thickness of the green arrows, the inverse cascade of magnetic helicity is\nstronger if the magnetic and kinetic helicities are of the same sign [21]. Following the analysis that was carried out in\nRef. [21] for the triadic interactions corresponding to kinematic dynamo action and the inverse transfer of magnetic\nhelicity, we carry out an analysis of the relative magnitude of the growth rates corresponding to triad interactions of\nthe Lorentz force in the Appendix. Our analysis shows that the conversion of magnetic to kinetic energy through the\nLorentz force occurs mainly between helical modes of the velocity and the magnetic \feld that have the same sign of\nhelicity, as indicated by the thick magenta arrows in Fig. 8.\nA. Direct cascade of magnetic helicity\nIn order to understand the presence of the direct cascade of positive magnetic helicity, observed in our simulations,\nat the level of triadic interactions, we aim to identify interactions among helical modes that transfer energy to helical\nmodes of positive magnetic helicity at large wavenumbers. The only possible way to get such interactions is via the\ntriads in Figs. 7a, 7b, and 7d. From these three triad interactions, those that dominate the small scales are shown in\nFigs. 7a and 7b, where the unstable helical modes U+\nkandU+\nptransfer energy to B+\nqvia transfers that represent a\nsmall scale dynamo. In all these cases, energy is transferred from a small-wave-number mode to a large-wave-number\nmode and this transfer originates from a positively helical unstable mode of the velocity \feld. As we have already\nobserved from our simulations, kinetic helicity has a well-de\fned spectrum [see Fig. 5b] and thus positively helical\nmodes of the velocity \feld are present and can enable such triadic interactions.\nThe generation of kinetic helicity in these \rows can be understood via the triadic interactions of Fig. 8a-8c. These\nthree triads are the dominant interactions in this linear stability analysis and in all these cases modes of positive\nmagnetic helicity generate modes of positive kinetic helicity. In other words, the analysis of triadic interactions\nsuggests that the magnetic helicity generates kinetic helicity through triad interactions associated with the Lorentz\nforce. As discussed in Sec. III, our \rows are dominated by positively helical modes of the magnetic \feld because we\nmaintain positive net magnetic helicity [see Fig. 1a] via the positively helical electromagnetic force fb.\nIn summary, since there is no direct transfer of energy between modes that have opposite sign of magnetic helicity,\nthe only way for a direct cascade of Hbto occur is via the Lorentz force, which generates modes with positive kinetic\nhelicity. These in turn excite modes with both positive and negative magnetic helicity with the dominant interactions\ncreating positively helical magnetic-\feld modes at small scales (see Figs. 7a, 7b, and 7d).\nB. Negative magnetic helicity at large scales\nHere we discuss the triad interactions that lead to the generation of modes with negative magnetic helicity at\nsmall wave numbers. As stated above, there is no direct transfer of energy between modes that have opposite sign\nof magnetic helicity. Therefore, the triad interactions that generate opposite signs of magnetic helicity between large\nand small scales have to involve a transfer of energy from positively helical modes of the velocity \feld. According\nto the results of the stability analysis of the triadic interactions shown in Figs. 7 and 8, we can conclude that there\nare two types of instabilities leading to the occurrence of large-scale magnetic helicity, namely, triadic instabilities\nthat represent large-scale dynamo, including STF dynamics [see Figs. 7b and 7c], and those that can be directly\nassociated with the inverse cascade of positive magnetic helicity [see Figs. 8b and 8c]. Here we expect a competition\nbetween triad interactions in Figs. 7b and 7c and those in Figs. 8b and 8c. Based on our numerical simulations, we\nobserve the persistent generation of negative magnetic helicity at wave number k= 3 [see Fig. 5a] despite the positive\nmean value of magnetic helicity that is maintained by the positively helical electromagnetic forcing. This observation\nindicates that the triad interactions of Figs. 7b and 7c dominate over Figs. 8b and 8c at low wavenumbers as the scale\nseparationkfLincreases. Thus, the generation of opposite signs of magnetic helicity between large and small scales\ncan only be partially explained by STF dynamics, which is essentially what the triad in Fig. 7b represents. This is\nbecause the triadic interactions of the helical modes in Fig. 7c, which represent a small scale dynamo, can play an\nimportant role. Moreover, the persistence of the triads in Figs. 7b and 7c can also explain why the scales larger that\nthe forcing scale do not become fully helical and hence the relative helicity \u001ab(k)<1 at wavenumbers 1 <k<kf.11\nVI. CONCLUSIONS\nIn this paper we investigated the bidirectional cascade of magnetic helicity Hbin homogeneous MHD turbulence\nby means of direct numerical simulations and analysis of triad interactions of helical modes. In order to be able to\nsimulate statistically stationary high- Rm\rows with large enough scale separation we considered high-order dissipation\nterms acting at small and large scales of the velocity and the magnetic \feld. To maintain positive mean magnetic\nhelicity in our \rows we used a helical electromagnetic forcing, while the velocity \feld was forced by a nonhelical\nforcing.\nOur numerical simulations demonstrated the following main results. (a) Magnetic helicity exhibits a clear bidirec-\ntional cascade even in these high- Rm\rows. This bidirectional cascade consists of a dominant inverse cascade and\na residual direct cascade of smaller magnitude, which remains invariant as the scale separation kfLincreases. (b)\nDespite the mean positive magnetic helicity measured in all our simulations, we consistently observe the generation\nof opposite signs of magnetic helicity between large and small scales.\nOur numerical results were interpreted in terms of the triad interactions of the helical Fourier modes. The residual\nforward cascade of magnetic helicity may occur due to a nonlinear small-scale dynamo, while the occurrence of large-\nscale negative helicity may result from a competition between triadic interactions associated with dynamo action and\nthose corresponding to the inverse cascade of magnetic helicity.\nThe analysis of triadic interactions of helical modes predicts that the kinetic helicity in our \rows is generated at\nall scales by the unstable modes of positive magnetic helicity. This is deduced from the stability properties of triadic\ninteractions associated with the Lorentz force, and the predictions are in agreement with our numerical simulations,\nwhere kinetic helicity becomes signi\fcant by developing a whole spectrum of predominantly positive helicity across\nthe scales. When the modes of positive kinetic helicity are unstable they preferentially generate modes of positive\nmagnetic helicity at small scales [20, 21]. This explains the existence of the residual direct cascade of magnetic helicity,\nwhich is associated with a triad interaction representing the Lorentz force and a triad that represents STF dynamics\n(i.e. the generation of opposite signs of magnetic helicity between between large and small scales due to a helical\nvelocity \feld). Moreover, since triad interactions between modes that have opposite sign of magnetic helicity are\nprohibited, the only way for the magnetic helicity to have opposite signs between large and small scales is via the\ncombined e\u000bect of triad interactions associated with STF dynamics and small-scale dynamo action, which involve\nunstable modes of positive kinetic helicity.\nA large network of triadic interactions given by the MHD equations can behave di\u000berently from a collection of\nisolated systems of triads [44]. For example, we cannot comment just from the analysis of triad interactions whether\nthe direct cascade of magnetic helicity will vanish at the limit Rmf!1 . Therefore, numerical investigations of\nthe e\u000bect of di\u000berent couplings of helical modes in DNSs like in Refs. [28, 30] would complement our analysis on the\ninterscale transfers of magnetic helicity.\nPreliminary results from numerical simulations with \fxed scale separation kfLand increasing Rmfsuggest that\nthe direct cascade of magnetic helicity may be a \fnite-Reynolds-number e\u000bect. This is because the scaling of the\nnormalized dissipation rate of Hbat small scales is \"+\nHb=\"Hb/Rm\u00000:22\nf. This weak power law implies that the direct\ncascade of magnetic helicity will persist as Rmfincreases, but it will eventually vanish in the limit of Rmf!1 .\nThis claim is supported by deriving upper bounds on the absolute value of the dissipation rate of Hbat small\nscales. However, further investigation is required in order to understand better the small scale dynamics of magnetic\nhelicity. In principle, this could be achieved through a series of high-resolution DNSs of helical MHD turbulence\ncovering a large range of magnetic Reynolds numbers from which one could extrapolate towards the high-Reynolds-\nnumber asymptotic limit. However, such a series of simulations will be computationally challenging as su\u000ecient scale\nseparation is required between the forcing scale and the box size, while the use of the Laplacian operator requires a\nlarge number of collocation points in order to achieve adequate small scale resolution.\nMagnetic helicity is considered to play a fundamental role in the dynamo action of planetary, stellar, and galactic\nmagnetic \felds. The time scale and the level of the saturation of the magnetic \feld in models of the solar activity cycle\ndepend on assumptions about the di\u000busion of magnetic helicity by small-scale velocity \ructuations [45]. Our results\nimprove our understanding of the magnetic helicity dynamics on the formation and evolution of large-scale magnetic\n\felds in astrophysical objects, which is one of the challenging problems in modern astrophysical \ruid dynamics.\nNote added. Recently, it has come to our attention that similar conclusions concerning the behavior of the direct\ncascade of magnetic helicity in the limit of in\fnite Rmhave been reached [46].\nACKNOWLEDGEMENTS\nWe thank Luca Biferale and Ganapati Sahoo for helpful conversations. This work has made use of the resources\nprovided by ARCHER, made available through the Edinburgh Compute and Data Facility (ECDF). V. D. acknowl-12\nedges support from the Royal Society and the British Academy of Sciences (Newton International Fellowship, Grant\nNo. NF140631). The research leading to these results has received funding from the European Union's Seventh\nFramework Programme under grant agreement No. 339032.\nAppendix A: Triadic analysis of the Lorentz force\nFollowing the work of Refs. [20, 21], we present here the linear stability analysis of the triadic interactions that\ncorrespond to the Lorentz force as an example. This analysis allows us to derive the additional results concerning the\nmagnitude of the magnetic-to-kinetic energy transfers depicted in Fig. 8. We consider a positively helical magnetic\nsteady solution ( B+\np06= 0;B\u0000\np0=U\u0000\np0=U+\np0= 0) of Eqs. (11) at wave vector p0=psubject to magnetic and velocity\n\feld perturbations b+\nk,b\u0000\nk,u+\nq, andu\u0000\nq,\n(\n@tb+\nk=kg+++B+\np0u+\nq;\n@tu+\nq=\u0000g+++ (p0\u0000k)B+\np0b+\nk;(A1)\n(\n@tb+\nk=kg++\u0000B+\np0u\u0000\nq;\n@tu\u0000\nq=\u0000g++\u0000(p0\u0000k)B+\np0b+\nk;(A2)\n(\n@tb\u0000\nk=\u0000kg\u0000++B+\np0u+\nq;\n@tu+\nq=g\u0000++(p0+k)B+\np0b\u0000\nk;(A3)\n(\n@tb\u0000\nk=\u0000kg\u0000+\u0000B+\np0u\u0000\nq;\n@tu\u0000\nq=g\u0000+\u0000(p0+k)B+\np0b\u0000\nk;\n(A4)\nwhere each set of equations represents a particular helical combination of velocity- and magnetic-\feld modes from\nthe system of equations (11) for a particular choice of helicities. The procedure is conceptually similar to a linear\nstability analysis of rigid body rotation [26], that is, we di\u000berentiate Eqs. (A1)-(A4) further in time and replace any\noccurrence of a \frst-order time derivative by Eqs. (11). In order to isolate the dynamical e\u000bects of the Lorentz force\non the \row we focus on the evolution of the velocity perturbations\n@2\ntu+\nq= (p0\u0000k)kjg+++j2jB+\np0j2u+\nq; (A5)\n@2\ntu\u0000\nq= (p0\u0000k)kjg++\u0000j2jB+\np0j2u\u0000\nq; (A6)\n@2\ntu+\nq=\u0000(p0+k)kjg\u0000++j2jB+\np0j2u+\nq; (A7)\n@2\ntu\u0000\nq=\u0000(p0+k)kjg\u0000+\u0000j2jB+\np0j2u\u0000\nq; (A8)\nand we observe that Eqs. (A7) and (A8) cannot have exponentially growing solutions as the term \u0000(p0+k)kis always\nnegative. Concerning Eqs. (A5) and (A6), the wavenumber ordering k<p 0<qresults in (p0\u0000k)k>0, hence these\ntwo equations admit exponentially growing solutions. In other words, linear instabilities occur only in Eqs. (A5) and\n(A6).\nFrom Eqs. (A1) and (A2) we observe that the coe\u000ecient g+++ corresponds to u+\nqandb+\nkcoupling to B+\np0. Therefore,\nthe term (p0\u0000k)kjg+++j2jB+\np0j2in Eq. (A5) describes the coupling of magnetic-\feld modes with the same signs of\nhelicity, i.e., b+\nkandB+\np0tou+\nq. Similarly, the term \u0000(p0+k)kjg\u0000++j2jB+\np0j2in Eq. (A7) describes the coupling of13\nmagnetic-\feld modes with opposite signs of helicity to u+\nq, becauseg\u0000++couplesu+\nqtoB+\np0andb\u0000\nk. Equations. (A6)\nand (A8) are analyzed analogously. Hence linear instabilities only occur in triadic interactions where the unstable\nmagnetic mode and the magnetic perturbation have the same sign of helicity [20]. In terms of the instability assumption\nthis result implies that the Lorentz force can only convert magnetic to kinetic energy if the interaction proceeds by\ntriads involving magnetic-\feld modes with the same sign of helicity [20].\nThe energy transfer due the linear instability governed by Eq. (A5) corresponds to the magenta arrow in Fig. 8b,\nwhile the linear instability described by Eq. (A7) corresponds to the magenta arrow in Fig. 8f. The remaining energy\ntransfers depicted in Fig. 8 can be derived analogously by changing the characteristic wavenumber of the steady\nmagnetic-\feld mode. The magnitude of the transfers indicated by the thickness of the arrows in Fig. 8 is determined\nby a comparison of the growth rate of the mechanical perturbations\n(p0\u0000k)kjg\u0000++j2jB+\np0j2\n(p0\u0000k)kjg+++j2jB+p0j2=jg\u0000++j2\njg+++j261; (A9)\nsince the magnitude of the geometric factors can be written as jgskspsqj=jskk+spp+sqqj[2(k2p2+p2q2+q2k2)\u0000\nk4\u0000p4\u0000q4]1=2[21, 26]. Hence the energy transfer from a positively helical magnetic \feld into a positively helical\n\row is stronger than that occurring from a positively helical magnetic \feld into a negatively helical \row. This result\nis indicated by a thicker magenta arrow in Fig. 8c than in Fig. 8f. The relative magnitudes of the remaining energy\ntransfers connected with dynamo action as shown in Figs. 7 and the inverse cascade of magnetic helicity shown by\nthe green arrows in Fig. 8 have been derived by the same method in Ref. [21].\n[1] H. K. Mo\u000batt. Magnetic Field Generation in Electrically Conducting Fluids . (Cambridge University Press, Cambridge,\n1978).\n[2] M. R. E. Proctor and A. D. Gilbert. Lectures on Solar and Planetary Dynamos . (Cambridge University Press, Cambridge,\n1994).\n[3] U. Frisch, A. Pouquet, J. L\u0013 eorat, and A. Mazure. Possibility of an inverse cascade of magnetic helicity in magnetohydro-\ndynamic turbulence. J. Fluid Mech. , 68:769{778, 1975.\n[4] A. Brandenburg. The critical role of magnetic helicity in astrophysical large-scale dynamos. Plasma Physics and Contr.\nFusion , 51(12):124043, 2009.\n[5] A. Pouquet, U. Frisch, and J. L\u0013 eorat. Strong MHD helical turbulence and the nonlinear dynamo e\u000bect. J. Fluid Mech. ,\n77:321{354, 1976.\n[6] M. Steenbeck, F. Krause, and K.-H. R adler. Berechnung der mittleren Lorentz-Feldst arke v x B f ur ein elektrisch leitendes\nMedium in turbulenter, durch Coriolis-Kr afte beein\ru\u0019ter Bewegung. Z. Naturforsch. A , 21:369, 1966.\n[7] F. Krause and K. R adler. Mean-Field Magnetohydrodynamics and Dynamo Theory . (Pergamon, Oxford, 1980).\n[8] E. N Parker. Hydromagnetic dynamo models. The Astrophysical Journal , 122:293, 1955.\n[9] A. Brandenburg, K. Subramanian, A. Balogh, and M. L. Goldstein. Scale dependence of magnetic helicity in the solar\nwind. The Astrophysical Journal , 734(1):9, 2011.\n[10] S. Childress and A. D. Gilbert. Stretch, Twist, Fold: The Fast Dynamo . (Springer, Berlin, 1995).\n[11] S. I. Vainshtein and Y. B. Zeldovich. Origin of magnetic \felds in astrophysics. Sov. Phys. Usp. , 15:159{172, 1972.\n[12] P. D. Mininni. Scale interactions in magnetohydrodynamic turbulence. Annu. Rev. Fluid Mech. , 43:377{397, 2011.\n[13] M. Meneguzzi, U. Frisch, and A. Pouquet. Helical and nonhelical turbulent dynamos. Phys. Rev. Lett. , 47:1060{1064,\n1981.\n[14] A. Brandenburg. The inverse cascade and nonlinear alpha-e\u000bect in simulations of isotropic helical magnetohydrodynamic\nturbulence. Astrophys. Journal , 550:824{840, 2001.\n[15] A. Alexakis, P. D. Mininni, and A. Pouquet. On the inverse cascade of magnetic helicity. Astrophys. J. , 640:335{343, 2006.\n[16] W. C. M uller, S. K. Malapaka, and A. Busse. Inverse cascade of magnetic helicity in magnetohydrodynamic turbulence.\nPhys. Rev. E , 85:015302, 2012.\n[17] M. Linkmann and V. Dallas. Large-scale dynamics of magnetic helicity. Phys. Rev. E , 94:053209, 2016.\n[18] S. I. Vainshtein and F. Cattaneo. Nonlinear restrictions on dynamo action. Astrophys. J. , 393:165{171, 1992.\n[19] F. Cattaneo and D. W. Hughes. Nonlinear saturation of the turbulent \u000b-e\u000bect. Phys. Rev. E , 54:R4532, 1996.\n[20] M. F. Linkmann, A. Berera, M. E. McKay and J. J ager. Helical mode interactions and spectral energy transfer in\nmagnetohydrodynamic turbulence. J. Fluid Mech. , 791:61{96, 2016.\n[21] M. Linkmann, G. Sahoo, M. McKay, A. Berera, and L. Biferale. E\u000bects of magnetic and kinetic helicities on the growth\nof magnetic \felds in laminar and turbulent \rows by helical-Fourier decomposition. Astrophys. Journal , 836:26, 2017.\n[22] A. Craya. Contribution \u0012 a l'analyse de la turbulence associ\u0013 ee \u0012 a des vitesses moyennes. Ministere de l'Air report , 1958\n(unpublished).\n[23] M Lesieur. D\u0013 ecomposition d'un champ de vitesse non divergent en ondes d'h\u0013 elicit\u0013 e. Observatoire de Nice report , 1972\n(unpublished).\n[24] J. R. Herring. Approach of axisymmetric turbulence to isotropy. Physics of Fluids , 17(5):859{872, 1974.14\n[25] P. Constantin and A. Majda. The Beltrami spectrum for incompressible \rows. Commun. Math. Phys. , 115:435{456, 1988.\n[26] F. Wale\u000be. The nature of triad interactions in homogeneous turbulence. Phys. Fluids A , 4:350{363, 1992.\n[27] Q. Chen, S. Chen, and G. L. Eyink. The joint cascade of energy and helicity in three-dimensional turbulence. Phys. Fluids ,\n15:361, 2003.\n[28] L. Biferale, S. Musacchio, and F. Toschi. Inverse energy cascade in three-dimensional isotropic turbulence. Phys. Rev.\nLett., 108:164501, 2012.\n[29] G. Sahoo, F. Bonaccorso, and L. Biferale. Role of helicity for large- and small-scale turbulent \ructuations. Phys. Rev. E ,\n92:051002, 2015.\n[30] Alexandros Alexakis. Helically decomposed turbulence. Journal of Fluid Mechanics , 812:752{770, 2017.\n[31] S. R. Yo\u000be. Investigation of the transfer and dissipation of energy in isotropic turbulence . PhD thesis, University of\nEdinburgh, 2012.\n[32] A. Berera and M. F. Linkmann. Magnetic helicity and the evolution of decaying magnetohydrodynamic turbulence. Phys.\nRev. E , 90:041003(R), 2014.\n[33] S. K. Malapaka and W.-C. M uller. Large-scale magnetic structure formation in three-dimensional magnetohydrodynamic\nturbulence. Astrophys. J. , 778:21, 2013.\n[34] E. A. Novikov. Functionals and the random-force method in turbulence theory. Soviet Physics JETP , 20:1290{1294, 1965.\n[35] V. Borue and S. A. Orszag. Self-similar decay of three-dimensional homogeneous turbulence with hyperviscosity. Phys.\nRev. E , 51:2859(R), 1995.\n[36] V. Dallas and A. Alexakis. Self-organisation and non-linear dynamics in driven magnetohydrodynamic turbulent \rows.\nPhysics of Fluids , 27(4):045105, 2015.\n[37] P. D. Mininni and A. G. Pouquet. Finite dissipation and intermittency in magnetohydrodynamics. Phys. Rev. E. , 80:025401,\n2009.\n[38] V. Dallas and A. Alexakis. The signature of initial conditions on magnetohydrodynamic turbulence. The Astrophysical\nJournal , 788:L36, 2014.\n[39] M. Linkmann, A. Berera, and E. E. Goldstraw. Reynolds-number dependence of the dimensionless dissipation rate in\nhomogeneous magnetohydrodynamic turbulence. Phys. Rev. E , 95:013102, 2017.\n[40] Vishwanath Shukla, Stephan Fauve, and Marc Brachet. Statistical theory of reversals in two-dimensional con\fned turbulent\n\rows. Phys. Rev. E , 94:061101, 2016.\n[41] H. Xia, H. Punzmann, G. Falkovich, and M. G. Shats. Turbulence-condensate interaction in two dimensions. Phys. Rev.\nLett., 101:194504, 2008.\n[42] T. Lessinnes, F. Plunian, and D. Carati. Helical shell models for MHD. Theor. Comput. Fluid Dyn. , 23:439{450, 2009.\n[43] R. H. Kraichnan. Inertial ranges in two-dimensional turbulence. Phys. Fluids , 10(10):1417, 1967.\n[44] H. K. Mo\u000batt. Note on the triad interactions of homogeneous turbulence. J. Fluid Mech. , 741:R3, 2014.\n[45] N Kleeorin, K Kuzanyan, D Moss, I Rogachevskii, D Sokolo\u000b, and H Zhang. Magnetic helicity evolution during the solar\nactivity cycle: observations and dynamo theory. Astronomy & Astrophysics , 409(3):1097{1105, 2003.\n[46] H. Aluie. Coarse-Grained incompressible magnetohydrodynamics: Analyzing the turbulent cascades. New J. Phys. ,\n19:025008, 2017."    },    {        "title": "1801.07842v1.Voltage_induced_precessional_switching_at_zero_bias_magnetic_field_in_a_conically_magnetized_free_layer.pdf",        "content": "arXiv:1801.07842v1  [cond-mat.mtrl-sci]  24 Jan 2018Voltage-induced precessional switching at zero-bias magn etic field\nin a conically magnetized free layer\nR. Matsumoto,∗T. Nozaki, S. Yuasa, and H. Imamura†\n1National Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba, Ibaraki 305-8568, J apan\n(Dated: August 12, 2018)\nAbstract\nVoltage-induced magnetization dynamics in a conically mag netized free layer with an elliptic\ncylinder shape is theoretically studied on the basis of the m acrospin model. It is found that an\napplication of voltage pulse can induce the precessional sw itching of magnetization even at zero-bias\nmagnetic field, which is of substantial importance for devic e applications such as voltage-controlled\nnonvolatile memory. Analytical expressions of the conditi ons for precessional switching are derived.\n∗rie-matsumoto@aist.go.jp\n†h-imamura@aist.go.jp\n1I. INTRODUCTION\nVoltage control of magnetic anisotropy (VCMA) in a ferromagnetic metal layer is a\npromising technology for the low-power writing in magnetoresistive r andom-access mem-\nories (MRAMs) [1–14] compared to the writing with spin-transfer to rque (STT) [15–17]. In\nthe magnetic tunnel junctions with perpendicular magnetization (p -MTJs) which have been\nthe mainstream technology for STT-MRAMs, voltage-driven writing −i.e., magnetization\nswitching −has been demonstrated by an application of a bias voltage with an app ropriate\npulse width under a bias magnetic field having an in-plane (IP) compone nt [11–14].\nFor practical applications, writing at zero-bias magnetic field is desir able to simplify\nthe device structure and reduce the fabrication cost. It is known that in magnetoresistive\ndevices the shape anisotropy field can act as a bias magnetic field in so me situations. The IP\nshape anisotropy field is obtained by microfabricating a ferromagne tic film into an elliptic\ncylinder shape. In thecase of a perpendicularly magnetized free lay er, however, the IP shape\nanisotropy field cannot move its magnetization from the perpendicu lar direction because\nthe IP shape anisotropy field is proportional to the IP component o f the magnetization.\nIt is necessary to tilt the magnetization from the perpendicular dire ction to perform the\nprecessional switching of the magnetization around the IP shape a nisotropy field.\n2m\nfree \nreference (a) (b) \nin-plane state perpendicular \nstate cone state \n0θ(0) zκ2\nκ1,eff κ(0) \nκ(0) z\nxy\nFIG. 1. (a) Magnetic tunnel junction with elliptic cylinder shape and definitions of Cartesian\ncoordinates ( x,y,z ). Thexaxis is parallel to the major axis of the ellipse. (b) Phase di agram of\nmagnetic film with uniaxial anisotropy constants κ(0)\n1,effandκ(0)\n2at equilibrium state (redrawn from\nRef. 18). The shaded area represents the cone-state phase, w here the film is conically magnetized\nwith equilibrium polar angle, θ(0). The bistable regions are hatched.\nThe titled magnetization state can be obtained by using a conically mag netized film as\na free layer [19–21]. The conically magnetized state, which is stabilized by the competition\nbetween the first- and second-order magnetic anisotropy energ ies, has been experimentally\nobserved in double-layer and multilayer systems [22, 23]. Recently, the VCMA effect [24] in\naddition to the conically magnetized state [25–27] has been observe d in a Co-Fe-B thin film\nwhich is commonly used as a free layer in an MRAM.\nIn this paper, the voltage-driven precessional switching in a conica lly magnetized free\nlayer with an elliptic cylinder shape is analyzed. Analytical expressions of the conditions for\nprecessional switching at zero-bias magnetic field are derived.\nII. MODEL\nThe system we consider is schematically shown in Fig. 1(a). The latera l size of the\nnanopillar is assumed to be so small that the magnetization dynamics c an be described by\n3the macrospin model. The direction of the magnetization in the free la yer is represented by\nthe unit vector m= (mx,my,mz) = (sinθcosφ, sinθsinφ, cosθ), where θandφare the\npolar and azimuthal angles of m. The magnetization in the reference layer is fixed to align\nin the positive zdirection.\nThe energy density of the free layer is given by [28]\nE(mx,my,mz) =1\n2µ0M2\ns(Nxm2\nx+Nym2\ny+Nzm2\nz)\n+Ku1(1−m2\nz)+Ku2(1−m2\nz)2, (1)\nwhereKu1andKu2are the first- and second-order anisotropy constants, respec tively. The\nvalues of Ku1andKu2can be varied by application of a bias voltage, V, through the\nVCMA effect. The demagnetization coefficients, Nx,NyandNzare assumed to satisfy\nNz≫Ny> Nx.µ0is the vacuum permeability, and Msis the saturation magnetization\nof the free layer. IP shape anisotropy field, Hk, is given by Hk=Ms(Ny−Nx) [29].\nNormalization by µ0M2\nsleads to the dimensionless energy density defined as [28]\nε(mx,my,mz) =1\n2(Nxm2\nx+Nym2\ny+Nzm2\nz)\n+κ1(1−m2\nz)+κ2(1−m2\nz)2, (2)\nwhereε=E/(µ0M2\ns),κ1=Ku1/(µ0M2\ns) andκ2=Ku2/(µ0M2\ns).\nBefore analyzing the switching conditions, let us show the basic prop erties of the equi-\nlibrium magnetization state at V= 0, which is the initial state of magnetization switching.\nThe direction of the magnetization of the initial state, m(0), is obtained by minimizing the\nenergy density at V= 0 as [18]\nm(0)\nz=±/radicaltp/radicalvertex/radicalvertex/radicalbt1+κ(0)\n1,eff\n2κ(0)\n2, (3)\nm(0)\nx=±/radicalBig\n1−(m(0)\nz)2, andm(0)\ny= 0. Throughout the paper, the superscript (0) indicates\nthe quantities at V= 0. Without loss of generality, the initial state is assumed to satisfy\nm(0)\nx>0 andm(0)\nz>0. The effective first-order anisotropy constant, κ1,eff, is defined as\nκ1,eff=κ1−(1/2)(Nz−Nx). Since we are interested in the voltage-induced switching of\na conically magnetized free layer, we concentrate on the cone-sta te region indicated by the\nshaded region in Fig. 1(b), where κ(0)\n1,eff<0 andκ(0)\n2>−(1/2)κ(0)\n1,eff.\n4The retention time of an MRAM is quantified by the thermal stability fa ctor, ∆(0), which\nis defined by the ratio of the energy barrier to the thermal energy ,kBT, as\n∆(0)=/bracketleftBig\nE(0)(1,0,0)−E(0)(m(0)\nx,0,m(0)\nz)/bracketrightBig\nVF\nkBT, (4)\nwhereVFrepresents the volume of the free layer, kBis the Boltzmann constant and Tis\ntemperature. The volume of the free layer is expressed as VF=πrxrytF, whererx(ry) is\nhalf the length of the major (minor) axis of an ellipse, and tFis thickness of the free layer.\nThe thermal stability factor at room temperature should be more t han 60 for the retention\ntime over 10 years.\nAlthough our analysis can be applied to a quite general situation, it is d ifficult to under-\nstand its benefit without showing specific examples. The following par ameters are assumed:\nMs= 1400 kA/m, rx= 50 nm, ry= 20 nm, tF= 1 nm, Nx= 0.0122,Ny= 0.0443,\nNz= 0.9435 [30], Hk= 566 Oe, K(0)\n1,eff=−80 kJ/m3,K(0)\nu1= 1067 kJ/m3,K(0)\nu2= 150\nkJ/m3,m(0)\nz= 0.856, ∆(0)atT= 300 K is 61.2. The equilibrium polar angle is θ(0)= 31.1◦,\nwhich corresponds to a 20% reduction of the magnetoresistance r atio from the case of the\nperpendicularly magnetized free layer. The direction of m(0)is indicated by the open circle\nin Fig. 2(a) for the contour plot of ε(0).\n5mz\nmymx\n-1 01\n-1 01-1 0 1\nenergy density, ε\nmz\nmymx\n-1 01\n-1 01-1 0 1\nenergy density, ε\nmz\nmymx\n-1 01\n-1 01-1 0 1\nenergy density, ε\nmz\nmymx\n-1 01\n-1 01-1 0 1\nenergy density, ε\n(d)(b)\n(c) (a)\nFIG. 2. (a) Energy-density contour plot of Eq. (2) for the ini tial state, m(0). The direction of\nm(0)is indicated by the open circle. (b) Typical example of the en ergy-density contour plot of\nEq. (2) in the horizontally hatched region in Fig. 3. The anis otropy constants are assumed to\nbe (κ1,eff,κ2) = (-0.080, 0.055). The open circle indicates m(0). Thick solid curves represent the\ncontour having the same energy density as ε(m(0)). (c) The same plot in the vertically hatched\nregion in Fig. 3. The anisotropy constants are assumed to be ( κ1,eff,κ2) = (-0.040, 0.025). (d) The\nsame plot in the diagonally hatched region in Fig. 3. The anis otropy constants are assumed to be\n(κ1,eff,κ2) = (-0.005, -0.005).\n6III. RESULTS\nIn voltage-driven precessional switching, only a half period of prec essional motion is used\nto switch the magnetization. Thanks to the smallness of the damping constant of the free\nlayer, the trajectory of the initial half period of precession is well r epresented by an energy-\ndensity contour including m(0)on the Bloch sphere (see Appendix A). Application of a\nbias voltage modifies the anisotropy constants, destabilizes the init ial state, and induces the\nprecessional motion. If the trajectory or energy-density cont our starting from m(0)crosses\nthe equator of the Bloch sphere, i.e., mz= 0, the magnetization can be switched by turning\noff the voltage after a half period of precession. It is important to fi nd the values of κ1,eff\nandκ2which enable precessional switching, i.e., the conditions for precess ional switching.\nThe energy-density contour having the same energy density as ε(m(0)) is expressed as\nε(m) =ε(m(0)). (5)\nThe substitution of mz= 0 into Eq. (5) yields\n1\n2(Nx−Ny)m2\nx+1\n2Ny+κ1+κ2=ε(m(0)). (6)\nRequiring 0 < m2\nx<1, one obtains the following inequality\n−ξκ1,eff−η < κ2<−ξκ1,eff, (7)\nwhere\nξ=1\n2−(m(0)\nz)2, η=ξ(Ny−Nx)\n2(m(0)\nz)2. (8)\nEquation (7) represents the condition for the energy-density co ntour to cross the equator.\nThe upper and lower boundaries of Eq. (7) are indicated by thin dott ed lines in Fig. 3.\n7κ1,effκ2\n(v) initial(iv) (i)\n(ii) (iii) \nZ+, Z - are complex Z+, Z - are in (0,1) \nZ+, Z - are not in (0,1) \nZ- is in (0,1), but Z+ is not. \nFIG. 3. Classification of the values of Z+andZ−in theκ1,effandκ2plane. The thin dotted\nlines represent the upper and lower boundaries of Eq. (7). Th e thick solid curves, which are\nsmoothly connected with each other at point (iii), represen t Eqs. (13) and (17), respectively. The\nthick solid line represents Eq. (19). In the horizontally ha tched region, both Z+andZ−are in\nthe region (0 ,1). In the vertically hatched region, both Z+andZ−are complex numbers. In the\ndiagonally hatched region, both Z+andZ−are real numbers but not in the region (0 ,1). In the\ncrosshatched region, both Z+andZ−are real numbers, but only Z−is in the region (0 ,1). The\npoint corresponding to the initial state is represented by t he open circle. The values of κ1,effat the\npoints indicated by the solid circles (i) – (v) are given in th e text.\nThere are two kinds of energy-density contour crossing the equa tor. One crosses the\n8latitude with my= 0 and surrounds the mxaxis. The other crosses the latitude with\nmx= 0 and surrounds the myaxis. The energy density ε(m) has symmetry under a sign\nchangeof mx. Even if m(0)islocatedontheenergy-density contoursurrounding the myaxis,\nit connects the initial state ( m(0)\nx,m(0)\ny,m(0)\nz) with (−m(0)\nx,m(0)\ny,m(0)\nz) and does not result in\nswitching. The region of ( κ1,eff,κ2) corresponding to such trajectories should be excluded in\nthe region defined by Eq. (7).\nThe solutions of Eq. (6) are given by\n˜mx=±/radicalBigg\n2\nNy−Nx/parenleftbigg1\n2Ny+κ1+κ2−ε(m(0))/parenrightbigg\n. (9)\nThe energy-density contour crosses the equator, at most, at f our points: ( ±˜mx,±˜my,0),\nwhere ˜my=/radicalbig\n1−˜m2x. Therefore, the energy-density contour with ε(m(0)) corresponding to\nthe precession aroundthe mxaxis does not coexist with that corresponding to the precession\naround the myaxis.\nThe value of mzat which energy-density contour crosses the latitude with mx= 0 is\nobtained as follows. Substitution of mx= 0 into Eq. (5) leads to\nκ2Z2+/bracketleftbigg1\n2(Nz−Ny)−κ1−2κ2/bracketrightbigg\nZ\n+1\n2Ny+κ1+κ2=ε(m(0)), (10)\nwhereZ=m2\nz. The solutions of Eq. (10) are obtained as\nZ±=−/bracketleftbig1\n2(Nz−Ny)−κ1−2κ2/bracketrightbig\n±√\nD\n2κ2, (11)\nwhere\nD=/bracketleftbigg1\n2(Nz−Ny)−κ1−2κ2/bracketrightbigg2\n−4κ2/bracketleftbigg1\n2Ny+κ1+κ2−ε(m(0))/bracketrightbigg\n. (12)\nThe energy-density contour crosses the latitude with mx= 0 atmz=±√Z±if 0< Z±<1.\nIn the region defined by Eq. (7), the solutions are classified into fou r groups as shown in\nFig. 3. In the horizontally hatched region both Z+andZ−are in (0, 1), which means that\nthe energy-density contour crossing the equator also crosses t he latitude with mx= 0. No\nenergy-density contour corresponding to the precessional swit ching exists in this region as\nshown in Fig. 2(b).\n9In the vertically hatched region, the solutions Z±are complex, which means that the\nenergy-density contour crossing the equator does not cross th e latitude with mx= 0 but\nsurrounds the mxaxis. Precessional switching is available, as shown in Fig. 2(c). In Fig.\n2(c), the anisotropy constants under a bias voltage are assumed to be (κ1,eff,κ2) = (-0.040,\n0.025) which is ( K1,eff,Ku2) = (-98.5, 61.6) kJ/m3in SI units.\nThe anisotropy constants, for example, can be obtained by the ap plication of V= 1 V,\ni.e., the electric field ( V/tI) when the VCMA effect is η1= 18.5 fJ/(V m) and η2= 88.4\nfJ/(V m). Here, tIrepresents the thickness of the insulator layer sandwiched betwe en the\nfree and the reference layers, and tI= 1 nm is assumed. the linear bias-voltage dependence\nofK1,effandKu2are assumed, and η1andη2represent the coefficient of the VCMA effect\nforK1,effandKu2. The anisotropy constants per unit area, K1,efftFandKu2tF, are expressed\nasK1,efftF=K(0)\n1,efftF−η1(V/tI) andKu2tF=K(0)\nu2tF−η2(V/tI).\nIt should be noted that the precessional switching is available even if Ku2is not changed\nby the bias voltage [31, 32], i.e., η2= 0. For example, ( κ1,eff,κ2) = (-0.080, 0.061) is included\nin the vertically hatched region while ( κ(0)\n1,eff,κ(0)\n2) = (-0.032, 0.061) in the initial state. In\nSI units, ( K1,eff,Ku2) = (-197, 150) kJ/m3, and it can be obtained at V= 1 V and tI= 1\nnm, i.e., the electric field ( V/tI) of 1 V/nm and η1= 117 fJ/(V m). Such linear bias-voltage\ndependence with η1∼100 fJ/(V m) has been experimentally demonstrated for the VCMA\neffect due to the modulation of charge accumulation without charge trapping [33–35].\nIn the diagonally hatched region, the solutions Z±are real but outside of (0, 1), which\nmeans that the energy-density contour crossing the equator do es not cross the latitude with\nmx= 0, and precessional switching is available, as shown in Fig. 2(d).\nTheboundaryamongthehorizontallyhatched, verticallyhatched, anddiagonallyhatched\nregions is given by\nκ2=−κ1,eff−√\nΛ\n2(1−Z(0)), (13)\nwhereZ(0)= (m(0)\nz)2and\nΛ =−κ1,eff(Ny−Nx)−1\n4(Ny−Nx)2. (14)\nThis boundary is plotted by the solid curve in Fig. 3. The curve of Eq. ( 13) crosses the\nupper boundary of Eq. (7) at point (i), where\nκ(i)\n1,eff=−(Ny−Nx)(2−Z(0))/parenleftBig\n2−Z(0)+2√\n1−Z(0)/parenrightBig\n2(Z(0))2. (15)\n10The curve of Eq. (13) is tangential to the lower boundary of Eq. (7 ) at point (ii), where\nκ(ii)\n1,eff=−(Ny−Nx)/bracketleftbig\n2−2Z(0)+(Z(0))2/bracketrightbig\n2(Z(0))2. (16)\nEquation (13) is obtained by solving D= 0, which has another solution\nκ2=−κ1,eff+√\nΛ\n2(1−Z(0)). (17)\nThe curve representing Eq. (17) smoothly connects with that of E q. (13) at point (iii),\nwhere\nκ(iii)\n1,eff=−1\n4(Ny−Nx). (18)\n11mz\nmymx\n-1 01\n-1 01-1 0 1\nenergy density, ε0.470\n0.465\n0.460εmy=0 \n0 0.5 1.0 \nmz0 0.5 1.0 \nmzεmy=0 0.474\n0.472\n0.470\nmz\nmymx\n-1 01\n-1 01-1 0 1\nenergy density, ε(c) (d) (b) (a)\nFIG. 4. (a) mzdependence of energy density with my= 0. The anisotropy constants are ( κ1,eff,\nκ2)=(0.010, -0.012). The open and solid circles indicate the i nitial state, εmy=0(m(0)\nz), and the\nmaximum point, respectively. (b) The same plot as (a) for the anisotropy constants of ( κ1,eff,\nκ2)=(0.010, -0.024). (c) Energy-density contour plot for the same parameters as in (a). The open\ncircle indicates m(0). Thick solid curves represent the contour having the same en ergy density as\nε(m(0)). (d) Energy-density contour plot for the same parameters a s in (b).\nIn the crosshatched region, only Z−is in (0, 1), and two kinds of energy-density contours\ncoexist. One crosses the latitude with mx= 0 atmz=±√Z−and corresponds to the\nprecession around the mzaxis. The other corresponds to the precession around the mxaxis.\n12The boundary between the diagonally hatched region and the cross hatched region is\ngiven by\nκ2=−κ1,eff\n1−Z(0), (19)\nwhich is represented by the thick solid line in Fig. 3. This line is tangential to the curve of\nEq. (17) at point (iv), where\nκ(iv)\n1,eff=−1\n2(Ny−Nx). (20)\nEquation (19) crosses the lower boundary of Eq. (7) at point (v), where\nκ(v)\n1,eff=(Ny−Nx)/parenleftbig\n1−Z(0)/parenrightbig\n2Z(0). (21)\nIn the crosshatched region, the precessional switching is available if the initial state is\nlocated on the energy-density contour surrounding the mxaxis. The condition for the\nprecessional switching is obtained by analyzing the mzdependence of the energy density on\nthe latitude with my= 0,εmy=0(mz). A typical example of εmy=0(mz) for parameters with\nwhich the magnetization does not switch is shown in Fig. 4(a). The anis otropy constants\nare assumed to be ( κ1,eff,κ2)=(0.010, -0.012). The initial state, εmy=0(m(0)\nz), is indicated\nby the open circle and the maximum point of εmy=0(mz) is indicated by the solid circle.\nSince the mzof the initial state is larger than that of the maximum point, the initial state\nis located on the energy-density contour surrounding the mzaxis as shown in Fig. 4(c) and\nthe magnetization does not switch.\n13κ1,effκ2initial \n(v) \n(vi) \nFIG. 5. The switching region in the crosshatched area in Fig. 3 is indicated by the shade triangle\nwith vertices at the origin and at points (v) and (vi). The thi n dotted lines and curves represents\nthe boundaries shown in Fig. 3. The thick solid line represen ts Eq. (22). The point corresponding\nto the initial state is indicated by the open circle. The valu e ofκ1,effat the point indicated by solid\ncircle (vi) are given in the text.\nA typical example of εmy=0(mz) for parameters with which precessional switching\nis available is shown in Fig. 4(b), where the anisotropy constants are assumed to be\n(κ1,eff,κ2)=(0.010, -0.024). Since the mzvalue of the initial state is smaller than that at the\nmaximum point, the initial state is located on the energy-density con tour surrounding the\nmxaxis as shown in Fig. 4(d), and precessional switching is available.\n14The boundary of the switching region is obtained by locating the initial state at the\nmaximum point as\nκ2=κ(0)\n2\nκ(0)\n1,effκ1,eff, (22)\nwhich is indicated by the thick solid line inFig. 5. The switching regionis insid e the triangle\nwith vertices at points (v) and (vi) and the origin as shown by the sha ded region in Fig. 5.\nThe value of κ1,effat point (vi) is given by\nκ(vi)\n1,eff=(Ny−Nx)(1−Z(0))\n(Z(0))2. (23)\n15κ1,eff κ2\n(vi) initial (i)\n(ii) \nFIG. 6. The total switching region and its boundary. The init ial values are indicated by the open\ncircle. The solid circles represent the vertices of the tota l switching region other than the origin.\nThe labels of the vertices are the same as those in Figs. 3 and 5 . The analytical expressions of\nthe boundaries indicated by the solid curve, the dotted line , the dashed line and the dotted-dashed\nline are given in the text.\nFigure 6 shows the total switching region, which is given by the combin ation of the\nvertically hatched region, the diagonally hatched region in Fig. 3 and t he shaded region in\nFig. 5. The boundary indicated by the solid curve is given by Eq. (13). The boundaries\nindicated by the dotted and dotted-dashed lines are the lower and u pper boundaries of Eq.\n(7), respectively. The boundary indicated by the dashed line is given by Eq. (22).\n16It should be noted that as long as Nz≫Ny> Nxand 0< θ(0)< π/2, the values κ1,effat\npoints (i), (ii), and (vi) satisfy\nκ(i)\n1,eff< κ(ii)\n1,eff<0< κ(vi)\n1,eff, (24)\nand the derived analytical expressions for the boundaries of tota l switching region are there-\nfore valid.\nLet us make some brief comments on the effects of the pulse width an d finite temperature\non switching. The pulse width which enables the precessional switchin g ranges from about\nthesecond quarter tothe thirdquarter of theprecession period . After turning off thevoltage\npulse, the magnetizationrelaxes to theequilibrium statewith preces sing around theeffective\nfield. At finite temperature, the initial state distributes around th e equilibrium direction.\nThe thermal distribution of the initial state is one of the main causes of the write error rate\n(WER). The WER of the voltage-controlled MRAM was studied in Refs. [12, 14] and is\nknown to take a minimum value at half of the precession period. Similar d ependence of the\nWER on the pulse width is expected in our system.\nIV. CONCLUSION\nIn conclusion, voltage-induced magnetization dynamics in a conically m agnetized free\nlayer with an elliptic cylinder shape is studied theoretically in this paper. It is shown that\nprecessional switching of magnetization can be performed by apply ing a voltage pulse even\nat zero-bias magnetic field. The analytical expressions of the cond itions for precessional\nswitching are derived, which is valid as long as the conically magnetized f ree layer is micro-\nfabricated into an elliptic cylinder shape. The results provide a pract ical guide for designing\na bias-field-free voltage-controlled MRAM, which simplifies the device structure and reduces\nthe fabrication cost.\nACKNOWLEDGMENTS\nThis work was partly supported by the ImPACT Program of the Coun cil for Science,\nTechnology and Innovation, and JSPS KAKENHI Grant No. JP16K17 509.\n17Appendix A: TRAJECTORY OF PRECESSIONAL SWITCHING\nOur analysis is based on the assumption that the trajectory of pre cessional switching is\nwell represented by an energy-density contour including m(0)on the Bloch sphere. In order\nto show the validity of this assumption, we perform numerical simulat ions to calculate the\nexact trajectory by solving the Landau-Lifshitz-Gilbert (LLG) eq uation [28],\ndm\ndt=−γ0m×Heff+αm×dm\ndt, (A1)\nwheretis the time, γ0is the gyromagnetic ratio, and αis the Gilbert damping constant.\nThe effective magnetic field, Heff, is defined as\nHeff=−1\nµ0Ms∇E. (A2)\nFigure7(a) shows the simulated trajectory (the blue curve) toge ther with the correspond-\ning energy-density contour (the gray curve) on the Bloch sphere . The open circle indicates\nthe initial state, m(0). The parameters are the same as in Fig. 2(c) and the damping\nconstant is assumed to be α= 0.005. One can see that within the initial half period of\nprecession, 0 ≤t≤0.56 ns, the difference between the simulated trajectory and the co rre-\nsponding energy-density contour is negligible. The scalar product b etween these two curves\nis plotted in Fig. 7(b) as a function of mz, which is not less than 99%. These results strongly\nsupport the validity of our analysis.\n18(b) \ntime evolution \nmz(a) \nmz\nmymx-1 0\n-1 01-1 \n0\n11\nFIG. 7. Time evolution of munder voltage during precessional switching. The paramete rs are the\nsame as in Fig. 2(c). (a) The simulated trajectory of m= (mx,my,mz) is plotted by the thin blue\ncurve together with the corresponding energy-density cont our (thick gray curve). The direction of\nthe precession is indicated by the arrow. The open circle ind icatesm(0). (b) The scalar product\nbetween magnetization unit vectors of the simulated trajec tory and the corresponding energy-\ndensity contour is plotted as a function of mz. The direction of the time evolution is indicated by\nthe arrow.\n[1] Martin Weisheit, Sebastian F¨ ahler, Alain Marty, Yves S ouche, Christiane Poinsignon, and\nDominique Givord, “Electric field-induced modification of m agnetism in thin-film ferromag-\nnets,” Science 315, 349–351 (2007).\n[2] T. Maruyama, Y. Shiota, T. Nozaki, K. Ohta, N. Toda, M. Miz uguchi, A. A. Tulapurkar,\nT. Shinjo, M. Shiraishi, S. Mizukami, Y. Ando, and Y. Suzuki, “Large voltage-induced\nmagnetic anisotropy change in a few atomic layers of iron,” N at. Nano. 4, 158–161 (2009).\n[3] Chun-Gang Duan, Julian P. Velev, R. F. Sabirianov, Ziqia ng Zhu, Junhao Chu, S. S.\nJaswal, and E. Y. Tsymbal, “Surface magnetoelectric effect in ferromagnetic metal films,”\nPhys. Rev. Lett. 101, 137201 (2008).\n[4] Kohji Nakamura, Riki Shimabukuro, Yuji Fujiwara, Toru A kiyama, Tomonori Ito, and A. J.\n19Freeman, “Giant modification of the magnetocrystalline ani sotropy in transition-metal mono-\nlayers by an external electric field,” Phys. Rev. Lett. 102, 187201 (2009).\n[5] Masahito Tsujikawa and Tatsuki Oda, “Finite electric fie ld effects in the large\nperpendicular magnetic anisotropy surface Pt/Fe/Pt(001) : A first-principles study,”\nPhys. Rev. Lett. 102, 247203 (2009).\n[6] T. Nozaki, Y. Shiota, M. Shiraishi, T. Shinjo, and Y. Suzu ki, “Voltage-\ninduced perpendicular magnetic anisotropy change in magne tic tunnel junctions,”\nAppl. Phys. Lett. 96, 022506 (2010).\n[7] Takayuki Nozaki, Anna Kozio/suppress l-Rachwa/suppress l, Witold Skowro ´ nski, Vadym Zayets, Yoichi Shiota,\nShingo Tamaru, Hitoshi Kubota, Akio Fukushima, Shinji Yuas a, and Yoshishige Suzuki,\n“Large voltage-induced changes in the perpendicular magne tic anisotropy of an MgO-based\ntunnel junction with an ultrathin Fe layer,” Phys. Rev. Appl ied5, 044006 (2016).\n[8] M. Endo, S. Kanai, S. Ikeda, F. Matsukura, and H. Ohno, “El ectric-field effects on\nthickness dependent magnetic anisotropy of sputtered MgO/ Co40Fe40B20/Ta structures,”\nAppl. Phys. Lett. 96, 212503 (2010).\n[9] Yoichi Shiota, Takayuki Nozaki, Fr´ ed´ eric Bonell, Shi nichi Murakami, Teruya Shinjo, and\nYoshishige Suzuki, “Induction of coherent magnetization s witching in a few atomic layers of\nFeCo using voltage pulses,” Nat. Mater. 11, 39–43 (2012).\n[10] Yoichi Shiota, Shinji Miwa, Takayuki Nozaki, Fr´ ed´ er ic Bonell, Norikazu Mizuochi, Teruya\nShinjo, Hitoshi Kubota, Shinji Yuasa, and Yoshishige Suzuk i, “Pulse voltage-induced dynamic\nmagnetization switching in magnetic tunneling junctions w ith high resistance-area product,”\nAppl. Phys. Lett. 101, 102406 (2012).\n[11] S. Kanai, M. Yamanouchi, S. Ikeda, Y. Nakatani, F. Matsu kura, and H. Ohno, “Electric field-\ninduced magnetization reversal in a perpendicular-anisot ropy CoFeB-MgO magnetic tunnel\njunction,” Appl. Phys. Lett. 101, 122403 (2012).\n[12] Yoichi Shiota, Takayuki Nozaki, Shingo Tamaru, Kay Yak ushiji, Hitoshi Kubota, Akio\nFukushima, Shinji Yuasa, and Yoshishige Suzuki, “Evaluati on of write error rate for voltage-\ndriven dynamic magnetization switching in magnetic tunnel junctions with perpendicular\nmagnetization,” Appl. Phys. Express 9, 013001 (2016).\n[13] C. Grezes, F. Ebrahimi, J. G. Alzate, X. Cai, J. A. Katine , J. Langer, B. Ocker,\nP. Khalili Amiri, and K. L. Wang, “Ultra-low switching energ y and scaling in electric-\n20field-controlled nanoscale magnetic tunnel junctions with high resistance-area product,”\nAppl. Phys. Lett. 108, 012403 (2016).\n[14] Yoichi Shiota, Takayuki Nozaki, Shingo Tamaru, Kay Yak ushiji, Hitoshi Kubota, Akio\nFukushima, Shinji Yuasa, and Yoshishige Suzuki, “Reductio n in write error rate of\nvoltage-driven dynamic magnetization switching by improv ing thermal stability factor,”\nAppl. Phys. Lett. 111, 022408 (2017).\n[15] J. C. Slonczewski, “Current-driven excitation of magn etic multilayers,” J. Magn. Magn. Mater.\n159, L1–L7 (1996).\n[16] L. Berger, “Emission of spin waves by a magnetic multila yer traversed by a current,”\nPhys. Rev. B 54, 9353–9358 (1996).\n[17] E. B. Myers, D. C. Ralph, J. A. Katine, R. N. Louie, and R. A . Buhrman, “Current-induced\nswitching of domains in magnetic multilayer devices,” Scie nce285, 867–870 (1999).\n[18] H.B.G. Casimir, J. Smit, U. Enz, J.F. Fast, H.P.J. Wijn, E.W. Gorter, A.J.W. Duyvesteyn,\nJ.D. Fast, and J.J. de Jong, “Rapport sur quelques recherche s dans le domaine du magn´ etisme\naux laboratoires philips,” J. Phys. Radium 20, 360–373 (1959).\n[19] D. Apalkov and W.H. Butler, “US patent 8,780,665,” US Pa tent 8,780,665 (2014).\n[20] Rie Matsumoto, Hiroko Arai, Shinji Yuasa, and Hiroshi I mamura, “Spin-transfer-\ntorque switching in a spin-valve nanopillar with a conicall y magnetized free layer,”\nAppl. Phys. Express 8, 063007 (2015).\n[21] Rie Matsumoto, Hiroko Arai, Shinji Yuasa, and Hiroshi I mamura, “Theoretical analysis of\nthermally activated spin-transfer-torque switching in a c onically magnetized nanomagnet,”\nPhys. Rev. B 92, 140409 (2015).\n[22] Jeong-Won Lee, Jong-Ryul Jeong, Sung-Chul Shin, Jongg eol Kim, and Sang-Koog Kim,\n“Spin-reorientation transitions in ultrathin Co films on Pt (111) and Pd(111) single-crystal\nsubstrates,” Phys. Rev. B 66, 172409 (2002).\n[23] Holger Stillrich, Christian Menk, Robert Fromter, and Hans Peter Oepen, “Magnetic\nanisotropy and the cone state in Co/Pt multilayer films,” J. A ppl. Phys. 105, 07C308 (2009).\n[24] Kyung-Woong Park, June-Young Park, Seung-heon Chris B aek, Dae-Hoon Kim, Soo-Man Seo,\nSung-Woong Chung, and Byong-Guk Park, “Electric field contr ol of magnetic anisotropy in\nthe easy cone state of Ta/Pt/CoFeB/MgO structures,” Appl. P hys. Lett. 109, 012405 (2016).\n[25] J. M. Shaw, H. T. Nembach, M. Weiler, T. J. Silva, M. Schoe n, J. Z. Sun, and D. C.\n21Worledge, “Perpendicular magnetic anisotropy and easy con e state in Ta/Co 60Fe20B20/MgO,”\nIEEE Magnetics Letters 6, 3500404 (2015).\n[26] Yu Fu, I. Barsukov, Jing Li, A. M. Gon¸ calves, C. C. Kuo, M . Farle, and I. N. Krivoro-\ntov, “Temperature dependence of perpendicular magnetic an isotropy in CoFeB thin films,”\nAppl. Phys. Lett. 108, 142403 (2016).\n[27] A. A. Timopheev, R. Sousa, M. Chshiev, H. T. Nguyen, and B . Dieny,\n“Second order anisotropy contribution in perpendicular ma gnetic tunnel junctions,”\nSci. Rep. 6, 26877 (2016).\n[28] Mark D. Stiles and Jacques Miltat, “Spin-transfer torq ue and dynamics,” in\nSpin Dynamics in Confined Magnetic Structures III , Topics in Applied Physics, Vol.\n101, edited by Burkard Hillebrands and Andr´ e Thiaville (Sp ringer Berlin Heidelberg, 2006)\npp. 225–308.\n[29] R. Matsumoto and H. Imamura, “Critical current density of a spin-torque oscilla-\ntor with an in-plane magnetized free layer and an out-of-pla ne magnetized polarizer,”\nAIP Advances 6, 125033 (2016).\n[30] M. Beleggia, M. De Graef, Y. T. Millev, D. A. Goode, and G. Rowlands, “Demagnetization\nfactors for elliptic cylinders,” J. Phys. D: Appl. Phys. 38, 3333 (2005).\n[31] Shun Kanai, Martin Gajek, D. C. Worledge, Fumihiro Mats ukura, and Hideo Ohno, “Electric\nfield-induced ferromagnetic resonance in a CoFeB/MgO magne tic tunnel junction under dc\nbias voltages,” Appl. Phys. Lett. 105, 242409 (2014).\n[32] A. Okada, S. Kanai, M. Yamanouchi, S. Ikeda, F. Matsukur a, and H. Ohno, “Electric-field\neffects on magnetic anisotropy and damping constant in Ta/CoF eB/MgO investigated by\nferromagnetic resonance,” Appl. Phys. Lett. 105, 052415 (2014).\n[33] Takayuki Nozaki, Hiroko Arai, Kay Yakushiji, Shingo Ta maru, Hitoshi Kubota, Hiroshi\nImamura, Akio Fukushima, and Shinji Yuasa, “Magnetization switching assisted by high-\nfrequency-voltage-induced ferromagnetic resonance,” Ap pl. Phys. Express 7, 073002 (2014).\n[34] Witold Skowro´ nski, Takayuki Nozaki, Yoichi Shiota, S hingo Tamaru, Yakushiji Kay,\nHitoshi Kubota, Akio Fukushima, Shinji Yuasa, and Yoshishi ge Suzuki, “Perpendic-\nular magnetic anisotropy of Ir/CoFeB/MgO trilayer system t uned by electric fields,”\nAppl. Phys. Express 8, 053003 (2015).\n[35] Xiang Li, Kevin Fitzell, Di Wu, C. Ty Karaba, Abraham Bud itama, Guoqiang Yu,\n22Kin L. Wong, Nicholas Altieri, Cecile Grezes, Nicholas Kiou ssis, Sarah Tolbert, Zongzhi\nZhang, Jane P. Chang, Pedram Khalili Amiri, and Kang L. Wang, “Enhancement of\nvoltage-controlled magnetic anisotropy through precise c ontrol of Mg insertion thickness at\nCoFeB|MgO interface,” Appl. Phys. Lett. 110, 052401 (2017).\n23"    },    {        "title": "1408.0992v1.Tunable_asymmetric_magnetoimpedance_effect_in_ferromagnetic_NiFe_Cu_Co_films.pdf",        "content": "arXiv:1408.0992v1  [cond-mat.mtrl-sci]  5 Aug 2014Tunable asymmetric magnetoimpedance effect in ferromagnet ic NiFe/Cu/Co films\nE. F. Silva,1M. Gamino,2A. M. H. de Andrade,2M. A. Corrˆ ea,1M. V´ azquez,3and F. Bohn1,∗\n1Departamento de F´ ısica Te´ orica e Experimental,\nUniversidade Federal do Rio Grande do Norte, 59078-900 Nata l, RN, Brazil\n2Instituto de F´ ısica, Universidade Federal do Rio Grande de Sul, 91501-970 Porto Alegre, RS, Brazil\n3Instituto de Ciencia de Materiales de Madrid, CSIC, 28049 Ma drid, Spain\n(Dated: January 1, 2018)\nWe investigate the magnetization dynamics through the magn etoimpedance effect in ferromag-\nnetic NiFe/Cu/Co films. We observe that the magnetoimpedanc e response is dependent on the\nthickness of the non-magnetic Cu spacer material, a fact ass ociated to the kind of the magnetic\ninteraction between the ferromagnetic layers. Thus, we pre sent an experimental study on asym-\nmetric magnetoimpedance in ferromagnetic films with biphas e magnetic behavior and explore the\npossibility of tuning the linear region of the magnetoimped ance curves around zero magnetic field\nby varying the thickness of the non-magnetic spacer materia l, and probe current frequency. We\ndiscuss the experimental magnetoimpedance results in term s of the different mechanisms governing\nthe magnetization dynamics at distinct frequency ranges, q uasi-static magnetic properties, thickness\nof the non-magnetic spacer material, and the kind of the magn etic interaction between the ferro-\nmagnetic layers. The results place ferromagnetic films with biphase magnetic behavior exhibiting\nasymmetric magnetoimpedance effect as a very attractive can didate for application as probe element\nin the development of auto-biased linear magnetic field sens ors.\nPACS numbers: 75.40.Gb, 75.30.Gw, 75.60.-d\nKeywords: Magnetic systems, Magnetization dynamics, Magn etoimpedance effect, Ferromagnetic films\nThe magnetoimpedance effect (MI), known as the\nchange of the real and imaginary components of elec-\ntrical impedance of a ferromagnetic conductor caused by\nthe action of an external static magnetic field, is com-\nmonly employed as a tool to investigate ferromagnetic\nmaterials. For a general review on the effect, we sug-\ngest the Ref.1. In recent years, the interest for this phe-\nnomenon has grown considerably not only for its contri-\nbution to the understanding of fundamental physics as-\nsociated to magnetization dynamics2, but also due to the\npossibility of application of materials exhibiting magne-\ntoimpedance as probe element in sensor devices for low-\nfield detection3. In this sense, the sensitivity and linear-\nity as a function of the magnetic field are the most im-\nportantparametersinthepracticalapplicationofmagne-\ntoimpedance effect for magnetic sensors4. Experiments\nhave been carried out in numerous magnetic systems,\nincluding ribbons5–7, sheets8, wires9–13, and, magnetic\nfilms14–16,16,17,17–23. However, although soft magnetic\nmaterials are highly sensitive to small field variations at\nlow magnetic fields, due to magnetization process most\nof them essentially have nonlinear MI behavior around\nzero magnetic field, which prevents a simple straightfor-\nward derivation of an appropriate signal for sensor appli-\ncations23,24.\nThe shift ofthe sensoroperational regionand the lead-\ning of the linear MI behavior at around zero magnetic\nfield can be obtained primarily by applying a bias field or\nan electrical current to the ordinary MI element24. How-\never, this approach proved to be disadvantageous from\nthe practical and technological point of view, mainly due\nto energetic consumption. Recently, it has been shown\nthat materialsexhibiting asymmetricmagnetoimpedance\n(AMI) effect arise as promising alternative with poten-tial of application, opening possibilities for the use of this\nkind of materials for the development of auto-biased lin-\nearmagnetic field sensors. For thesematerials, the asym-\nmetric effects are obtained by inducing an asymmetric\nstatic magnetic configuration, usually done by magneto-\nstatic interactions13,24,25or exchange bias4,16,23,26,27.\nFor ferromagnetic films, the primary AMI results have\nbeen measured for exchange biased multilayers16,23,27.\nTheory and experiment agree well for MI curves shifted\nby the exchange bias field, following the main features\nof the magnetization curve, as well as it is verified that\nthe linear region of AMI curves can be tuned to around\nzero just by modifying the angle between applied mag-\nnetic field and exchange bias field, or changing the probe\ncurrent frequency. On the other hand, another promis-\ning possibility of AMI material resides in films presenting\nbiphase magnetic behavior, with hard and soft ferromag-\nnetic phases intermediated by a non-magnetic layer act-\ning together.\nIn this work, we investigate the magnetoimpedance ef-\nfectin ferromagneticNiFe/Cu/Cofilms. We observethat\nthe MI response is dependent on the thickness of the\nnon-magnetic Cu spacer material, a fact associated to\nthe kind of the magnetic interaction between the ferro-\nmagnetic layers. Here we show that the linear region\nof the asymmetric magnetoimpedance curves in these\nfilms is experimentally tunable by varying the thickness\nof the non-magnetic spacer material, and probe current\nfrequency. The results place ferromagnetic films with\nbiphase magnetic behavior exhibiting asymmetric mag-\nnetoimpedance effect as a very attractive candidate for\napplication as probe element in the development of auto-\nbiased linear magnetic field sensors.\nFor this study, we produce Ni 81Fe19(252\nnm)/Cu( tCu)/Co(50 nm) ferromagnetic films, with\ntCu= 0, 1.5, 3, 5, 7, and 10 nm. The films are\ndeposited by magnetron sputtering from targets of\nnominally identical compositions onto glass substrates,\nwith dimensions of 8 ×4 mm2. A buffer Ta layer is\ndeposited before the NiFe layer to reduce the roughness\nof the substrate, as well as a cap Ta layer is inserted\nafter the Co layer in order to avoid oxidation of the\nsample. The deposition is carried out with the following\nparameters: base vacuum of 10−8Torr, deposition\npressure of 2 .0 mTorr with a 99 .99% pure Ar at 32 sccm\nconstant flow, and DC source with power of 150 W for\nthe deposition of the NiFe and Co layers, while 100 W\nfor the Cu and Ta layers. During the deposition, the\nsubstrate rotates at constant speed to improve the film\nuniformity, and a constant magnetic field of 2 kOe is\napplied perpendicularly to the main axis of the substrate\nin order to induce a magnetic anisotropy and define\nan easy magnetization axis. X-ray diffraction results,\nnot shown here, calibrate the deposition rates and\nverify the Co(111) and NiFe(111) preferential growth\nof all films. Magnetization curves are obtained with a\nvibrating sample magnetometer, measured along and\nperpendicular to the main axis of the films, to verify the\nquasi-static magnetic behavior. Magnetization dynamics\nis investigated through MI measurements obtained using\na RF-impedance analyzer Agilent model E4991, with\nE4991Atest head connected to a microstrip in which\nthe sample is the central conductor. Longitudinal MI\nmeasurements are performed by acquiring the real R\nand imaginary Xparts of the impedance Zover a wide\nrange of frequencies, from 0 .1 GHz up to 3 .0 GHz, with\n0 dBm (1 mW) constant power applied to the sample,\ncharacterizing the linear regime of driving signal, and\nmagnetic field varying between ±300 Oe, applied along\nthe main axis of the sample. Detailed information\non the MI experiment is found in Refs.21,22. In order\nto quantify the sensitivity and MI performance as a\nfunction of the frequency, we calculate the magnitude\nof the impedance change at the low field range ±6 Oe\nusing the expression23\n|∆Z|\n|∆H|=|Z(H= 6Oe)−Z(H=−6Oe)|\n12.(1)\nHere, we consider the absolute value of ∆ Z, since the\nimpedance around zero field can present positive or neg-\native slopes, depending on the sample and measurement\nfrequency. In particular, it is verified that |∆Z|/|∆H|is\nroughly constant at least for a reasonable low field range.\nFigure 1 shows the quasi-static magnetization curves\nfor selected films, measured with the external in-plane\nmagnetic field applied along and perpendicular to the\nmain axis. When analyzed as a function of the the thick-\nness of the non-magnetic Cu spacer material, it is ob-\nserved an evolution of the shape of the magnetization\ncurves, indicating the existence of a critical thickness\nrange,∼3 nm, which splits the films in groups according\nthe magnetic behavior. For films with tCubelow 3 nm,the NiFe and Co layers are ferromagnetically coupled.\nThe angular dependence of the magnetization curves in-\ndicatesanuniaxialin-planemagneticanisotropy,induced\nby the magnetic field applied during the deposition pro-\ncess, and oriented perpendicularly to the main axis. De-\nspitethe similarmagneticbehavior, thefilm with 1 .5nm-\nthick Cu layer (not shown) has slightly higher coercive\nfield if compared to the one for the film without spacer\nmaterial, possibly associated to the increase of the whole\nsample disorder due to the non-formation of a regular\ncomplete Cu layer. The film with tCu= 3 nm, within\nthe critical Cu thickness range, presents an intermediate\nmagnetic behavior, with smaller magnetic permeability,\ncharacterized by the first evidences of a small plateau\nin the measurement perpendicular to the main axis, and\nthe appearance of magnetization regions associated to\ndistinct anisotropy constants of the NiFe and Co layers.\nFilms with tCuabove the critical thickness range exhibit\na biphase magnetic behavior. The two-stage magnetiza-\ntion process is characterized by the magnetization rever-\nsion of the soft NiFe layer at low magnetic field, followed\nbythereversionofthehardColayerathigherfield. None\nsubstantial difference between the magnetization curves\nmeasured for films with tCu>3 nm is verified. In prin-\nciple, the biphase magnetic behavior suggests that the\nferromagnetic layers are uncoupled. The easy magneti-\nzation axis remains perpendicular to the main axis of the\nsubstrate, as expected. The weaker anisotropy induction\nand increase of hysteretic losses are primarily related to\nthe roughness of the interfaces and lack of homogeneity\nof the Cu layer arisen as its thickness is raised18.\nIt is well-known that quasi-static magnetic proper-\nties play a fundamental role in the dynamic magnetic\nresponse and are reflected in the MI behavior23. The\nshape and amplitude of the magnetoimpedance curves\nare strongly dependent on the orientation of the applied\nmagnetic field and accurrent with respect to the mag-\nnetic anisotropies, magnitude of the external magnetic\nfield, and probe current frequency, as well as are di-\nrectly related to the main mechanisms responsible for\nthe transversemagnetic permeability changes: skin effect\nand ferromagnetic resonance (FMR) effect23,28,29. How-\never, magnetoimpedance effect can also provide a further\ninsights on the nature of the interactions governing the\nmagnetization dynamics and energy terms affecting the\ntransverse magnetic permeability.\nRegarding the MI results, Fig. 2 shows the MI curves,\nat the selected frequency of 0 .75 GHz, for the films with\ndifferent thicknesses tCuof the non-magnetic Cu spacer\nmaterial. All samples exhibit a double peak behavior for\nthe whole frequency range, a signature of the perpen-\ndicular alignment of the external magnetic field and ac\ncurrent with the easy magnetization axis. An interesting\nfeature related to the MI behavior of the NiFe/Cu/Co\nfilms resides in the amplitude and position of the peaks\nwith the thickness of the non-magnetic Cu spacer mate-\nrial, and the probe current frequency.\nFilms with tCu<3 nm present the well-known sym-3\n−100 −50 0 50 100\nH (Oe)−1.0−0.50.00.51.0M/Ms0 nm\n⟂\n∥\n−100 −50 0 50 100\nH (Oe)−1.0−0.50.00.51.0M/Ms3 nm\n−100 −50 0 50 100\nH (Oe)−1.0−0.50.00.51.0M/Ms7 nm\nFIG. 1: Representative normalized quasi-static magnetiza tion curves for selected NiFe/Cu/Co films with different thic knesses\nof the non-magnetic Cu spacer material, obtained with the in -plane magnetic field applied along ( /bardbl) and perpendicular ( ⊥) to\nthe main axis. Films with tCubelow 3 nm present behavior similar to that verified for the fil m withtCu= 1.5 nm, while the\nfilms with tCuabove the critical thickness range have behavior identical to the one observed for the film with tCu= 7 nm. The\nfilm with tCu= 3 nm is within the critical Cu thickness range and have an int ermediate behavior.\nmetric magnetoimpedance behavior for anisotropic sys-\ntems. The MI curves have the double peak behavior,\nsymmetrical at aroud H= 0, with peaks with roughly\nthe same amplitude. For frequencies up to ∼0.85 GHz,\nthe position of the peaks remains unchanged close to the\nanisotropyfield, indicatingthattheskineffectisthemain\nresponsibleby the changesoftransversemagnetic perme-\nability governing the magnetization dynamics. For fre-\nquencies above this value, not presented here, besides the\nskin effect, the FMR effect also becomes an important\nmechanism responsible for variations of the MI effect, a\nfact evidenced by the displacement of the peaks position\ntoward higher fields as the frequency is increased. The\ncontribution of the FMR effect to Zis also verified us-\ning the method described by Barandiar´ an et al.30, and\npreviously employed by our group17.\nOn the other hand, films with tCu≥3 nm present\nnoticeable asymmetric magnetoimpedance effect. The\nasymmetric behavior is assigned by two characteristic\nfeatures: shift of the MI curve in field, depicted by the\nasymmetric position of the peaks, and asymmetry in\nshape, evidenced by the difference of amplitude of the\npeaks. Figure 3 shows the evolution of the MI curves, at\nselected frequencies between 0 .5 GHz and 3 .0 GHz, for\nthe film with tCu= 7 nm, as an example of the exper-\nimental result obtained for the ferromagnetic films with\ntCuabove 3 nm. Here, it is important to notice that\nthe presented MI behavior is acquired when the field\ngoes from negative to positive values. However, the MI\ncurves are acquired over a complete magnetization loop\nand present hysteretic behavior. In particular, when the\nfield goes from positive to negative values, the MI behav-\nior is reversed.\nFrom Fig. 3, regarding the position of the peaks, since\nthe skin effect commands the dynamical behavior, the\npeaks remain invariable at the low frequency range. For\nthis film, the peak at negative field values is located at\n∼ −4 Oe, while the one at positive fields is at ∼+30\nOe. For the other films with tCu>3 nm and biphase\nmagnetic behavior, the peak at positive field is placed\nat similar value, although the location of the peak at−300 −150 0 150 300\nH (Oe)4.204.224.244.264.28Z (Ω)0 nm\n−300 −150 0 150 300\nH (Oe)4.294.314.334.35Z (Ω)1.5 nm\n−300 −150 0 150 300\nH (Oe)4.924.944.964.98Z (Ω)3 nm\n−300 −150 0 150 300\nH (Oe)5.035.065.095.125.15Z (Ω)5 nm\n−300 −150 0 150 300\nH (Oe)4.284.314.344.374.40Z (Ω)7 nm\n−300 −150 0 150 300\nH (Oe)5.325.355.385.415.44Z (Ω)10 nm\nFIG. 2: The MI curves at frequency of 0 .75 GHz measured\nfor the films with different thicknesses of the non-magnetic\nCu spacer material tCu. The MI curves are acquired over a\ncomplete magnetization loop and present hysteretic behavi or.\nHere, we show just part of the curve, when the field goes from\nnegative to positive values, to make easier the visualizati on\nof the whole MI behavior.\nnegative field present dependence with tCu, as will be\ndiscussed. With respect to the amplitude of the peaks\nat low frequencies, for all films with tCu>3 nm, the\nMI behavior exhibits an asymmetric two-peak behavior,\nwith the peak at negative field being with higher am-\nplitude than the peak at positive field. As a signature\nof the emergence of the FMR effect, the displacement of\nthe peak at negative field begins at ∼0.6 GHz, while the4\n−300 −150 0 150 300\nH (Oe)3.883.903.923.943.96Z (Ω)0.50 GHz\n−300 −150 0 150 300\nH (Oe)4.284.314.344.374.40Z (Ω)0.75 GHz\n−300 −150 0 150 300\nH (Oe)4.924.954.985.015.04Z (Ω)1.0 GHz\n−300 −150 0 150 300\nH (Oe)6.526.566.606.646.68Z (Ω)1.5 GHz\n−300 −150 0 150 300\nH (Oe)8.638.718.798.87Z (Ω)2.0 GHz\n−300 −150 0 150 300\nH (Oe)14.714.915.115.3Z (Ω)3.0 GHz\nFIG. 3: Evolution of the experimental MI curves for selected\nfrequencies for the ferromagnetic biphase fil with tCu= 7\nnm. Similar results are obtained for all the ferromagnetic\nfilms with tCuabove 3 nm and biphase magnetic behavior.\nWe show just part of the curve, when the field goes from\nnegative to positive values.\nposition of peak at positive field starts changing at ∼1.1\nGHz. Above ∼1.5 GHz, strong skin and FMR effects are\nresponsible by the MI variations. At this high frequency\nrange, the asymmetry still remains in the portion of the\nimpedance curve around the anisotropy fields. However,\nthe displacement of the peaks toward higher fields sup-\npress the impedance peak asymmetry, in position and\namplitude, resulting is symmetric peaks around H= 0\nwith same amplitude. For the film with tCu= 3 nm with\nintermediate magnetic behavior, similar features are ob-\nserved respectively at ∼0.75 GHz, ∼1.1 GHz, and ∼2.0\nGHz.\nThe most striking finding in the dynamic magnetic re-\nsponse resides in the asymmetry of the MI curves mea-\nsured for the films with biphase magnetic behavior. It\nis important to notice that the magnetoimpedance re-\nsponse is nearly linear for low magnetic field values, and\nthe shape of the Zcurves depends on the thickness of\nthe non-magnetic Cu spacer material and probe current\nfrequency. As a consequence, the best response can be\ntuned by playing with both parameters. Figure 4 shows\nthe frequency spectrum of impedance variations between\n±6 Oe, as defined by Eq. (1), for each film, indicating\nthe sensitivity around zero field.\nFrom the figure, we verify that the films split in dif-\nferent groups according the sensitivity around zero field.0.0 0.5 1.0 1.5 2.0 2.5 3.0\nf (GHz)0246810|∆Z|/|∆H| (mΩ/Oe) 10 nm\n 7 nm\n 5 nm\n 3 nm\n 1.5 nm\n 0 nm\nFIG. 4: Frequency spectrum of impedance variations between\n±6 Oe for the films with different thicknesses of the non-\nmagnetic Cu spacer material tCu, indicating the sensitivity\naround zero field. Notice the kind of saturation effect ob-\nserved as the tCuincreases above 3 nm, related to the ampli-\ntude and frequency at which the maximum impedance change\nis reached.\nIt is important to notice that each one is related to a\ngiven magnetic behavior, verified through the magneti-\nzation curves. Films with tCu<3 nm have the largest\nsensitivity valuesat ∼1.0GHz, the film with tCu= 3 nm\nat∼0.9 GHz, while the ones with tCu>3 nm at∼0.75\nGHz. For all of them, the sensitivity peak is found to\nbe at frequencies just after the FMR effect starts ap-\npearing, and while the MI peaks are still placed close to\nthe anisotropy fields. The highest sensitivity is observed\nfor the films with tCu>3 nm, reaching ∼8 mΩ/Oe,\nand seems to be insensitive to the thickness of the non-\nmagnetic spacer material. In this situation, the AMI\ncurves have nearly linear behavior at low magnetic field\nvalues, and the slope of the linear region at nearly zero\nfield is negative, due to the shape of the MI curve.\nOur results raise an interesting issue on the behav-\nior of the peaks in the magnetoimpedance effect and the\nenergy terms affecting the transverse magnetic perme-\nability. Generally, our films consist of two ferromagnetic\nlayers,withdistinctanisotropyfieldstrengths, intermedi-\nated by non-magnetic spacer material. We interpret our\nexperimental data as a result caused by the competition\nbetween two types of magnetic interactions between fer-\nromagnetic layers: exchange coupling between touching\nferromagnetic phases, and long-range dipolarlike or mag-\nnetostatic coupling31. In particular, they are strongly\ndependent on the thickness of the spacer material, and\nthe action of each one affect in different ways the MI\nbehavior.\nIf both ferromagneticlayersarequasi-saturated, where\nthere are no walls with wall stray fields, the coupling\nshould adjust the magnetizationofthe twoferromagnetic\nlayers parallel to each other. For tCu<3 nm, the strong\ncoupling is caused by the exchange interaction between5\ntouching ferromagnetic layers and through pinholes in\nthe non-magnetic spacer, and the whole sample behaves\nas a single ferromagnetic layer18. In this sense, we con-\nfirm the expected symmetric magnetoimpedance behav-\nior of single anisotropic systems.\nFortCu>3 nm, the Cu layer is completelly filled18,\nand the nature of the coupling is magnetostatic. If\nthe ferromagnetic layers were completely uncoupled, one\ncould expect multiple peak MI behavior, associated to\nthe anisotropy fields of each different layers, around ±30\nOe and±10 Oe. This behavior is not verified here, in-\ndicating that the AMI can not be explained assuming\nindependent reversal of the NiFe and Co layers. Thus,\nthe asymmetry arises as a result of the magnetostatic\ncoupling between the ferromagnetic layers. The origin\nof the magnetostatic coupling is ascribed to the hard Co\nmagnetic phasein terms ofan effective biasfield, induced\nby divergences of magnetization mainly due to roughness\nin the interfaces and limits of the sample24, that must be\ntaken into account as a contributorto the transverseper-\nmeability. The field penetrates the non-magnetic spacer\nlayer and results in a torque on the magnetization of\nthe opposite layer. It is important to point out that the\nanisotropyfieldofthe hardColayerisconsiderablelarger\nthan the soft NiFe layer, the reason why this asymmetric\nbehavior is not verified in traditional multilayers.\nInthissense, themainfeaturesoftheasymmetricmag-\nnetoimpedance verified in films with biphase magnetic\nbehavior can be explained through the effective interac-\ntion between the ferromagnetic layers. The influence on\nthe soft NiFe layer is dependent on the magnetic state of\nthe hard Co layer, as well as on the thickness of the Cu\nlayer spacer. The difference of amplitude of the peaks is\nunderstood from the magnetic saturated state in terms\nof the orientation of the two layers. The peak at nega-\ntive field is higher than that in positive field, since the\nmagnetization of the soft NiFe layer is parallel to the\nmagnetization of the hard Co layer and consequently to\nthe magnetostatic field, as well as to the external field.\nSince the magnetization of the NiFe layer is reverted as\nthe field is increased,the senseofthis magnetizationwith\nrespect to the magnetostatic field is modified, and this\nform closes the magnetic flux, resulting a lower peak24.\nSimilar dependance with the orientation of magnetiza-\ntions has alreadybeen verified in field-annealed Co-based\namorphous ribbons4. When the MI measurement is ana-\nlyzed for decreasingmagnetic field, the reverted behavior\nis observed, with the higher and smaller peaks at posi-\ntive and negative fields, respectively, since the sense of\nthe magnetization of the hard Co layer is the opposite.\nBy employing MI measurements, it is possible to es-\ntimate the effective coupling strenght between the NiFe\nand Co layers. This can be done by considering the lo-\ncation of the peaks in the MI curves at the low frequen-\ncies. Figure 5 shows the magnetic field values in which\nthe impedance peaksarelocated, for differentthicknesses\nof the non-magnetic Cu spacer material, at the low fre-\nquency range. The position of the impedance peak at0.0 2.5 5.0 7.5 10.0\nThickness (nm)010203040|Hpeak | (Oe)Peak at negative H\nPeak at positive H\nFIG. 5: Magnetic field values in which the impedance peaks\nare located, at0 .5 GHz, for thefilmswith differentthicknesses\nof the non-magnetic Cu spacer material tCu. Notice that the\nfilms with tCu≤1.5 nm present double peak behavior, sym-\nmetrical at aroud H= 0.\nnegative field presents a noticeable dependence with tCu.\nIn particular, it is verified a reduction of the field value\nwhere the peak is located as tCuis increased, corroborat-\ning the assumption of a magnetostatic origin of the cou-\npling between the ferromagnetic layers. In this sense, we\ninterpret the reduction as an indication of the decrease\nof the bias field intensity acting on the soft NiFe layer\nas the Cu thickness is increased. On the other hand, the\npeak atpositive field is located at ∼30Oe, except for the\nsample without the non-magnetic spacer material. The\nconstancy in the peak location, irrespective of tCuand\nfield value of the Co reversion for each sample, suggests\nthat this value corresponds to a intrinsic feature of the\nferromagnetic Co layer, since it presents similar thick-\nnesses for all samples. Thereby, we understand it as the\nmagnitude of the bias field inducedby the hard Co layer.\nIn conclusion, we have investigated the magne-\ntoimpedance effect in ferromagnetic NiFe/Cu/Co films\nand observed the dependence of the MI curves, in par-\nticular, amplitude and position of the peaks, with the\nthickness of the non-magnetic Cu spacer material. We\nhave verified that the MI response of these films can be\ntaylored by the kind of magnetic interaction between the\nferromagnetic layers. In this sense, the coupling between\nthe layers is usefull to develop materials with asymmet-\nric MI. From the results, we have observed the crossover\nbetween two distinct magnetic behavior, associated to\ndistict kind of the magnetic behavior between the ferro-\nmagnetic layers, exchange interaction and magnetostatic\ncoupling, as the Cu thickness is altered crossing through\ntCu= 3 nm. Thus, we have tuned the linear regionof the\nasymmetric magnetoimpedance curves around zero mag-\nnetic field by varying the thickness of the non-magnetic\nspacer material, and probe current frequency. The high-\nest sensitivity is observed for the films with tCu>3 nm,6\nreaching ∼8 mΩ/Oe, and seems to be insensitive to the\nthickness of the non-magnetic spacer material. These\nresults extend the possibilities for application of ferro-\nmagnetic films with asymmetric magnetoimpedance as\nprobe element for the development of auto-biased linear\nmagnetic field sensors, placing films with biphase mag-\nnetic behavior as promissing candidates to optimize the\nMI performance.\nAcknowledgments\nThe authors thank Vivian Montardo Escobar for the\nfruitful discussions. The research is supported by theBrazilian agencies CNPq (Grants No. 471302/2013-9,\nNo. 310761/2011-5, No. 555620/2010-7), CAPES, and\nFAPERN (Grant Pronem No. 03 /2012). M.A.C. and\nF.B. acknowledge financial support of the INCT of Space\nStudies.\n∗Electronic address: felipebohn@gmail.com\n1M. Knobel, M. V´ azquez, and L. Kraus, Handbook of Mag-\nnetic Materials (Elsevier Science B. V., Amsterdam, 2003),\n(ed. K. H. Buschow)Ch. 5, 1 – 69.\n2A. Yelon, D. Menard, M. Britel, and P. Ciureanu, Appl.\nPhys. Lett. 69, 3084 (1996).\n3H. Chiriac, M. Tibu, A.-E. Moga, and D. D. Herea, J.\nMagn. Magn. Mater. 293, 671 (2005).\n4C. G. Kim, K. J. Jang, D. Y. Kim, and S. S. Yoon, Appl.\nPhys. Lett. 75, 2114 (1999).\n5R. L. Sommer and C. L. Chien, Phys. Rev. B 53, R5982\n(1996).\n6K. R. Pirota, L. Kraus, M. Knobel, P. G. Pagliuso, and\nC. Rettori, Phys. Rev. B 60, 6685 (1999).\n7F. Amalou and M. A. M. Gijs, J. Appl. Phys. 95, 1364\n(2004).\n8M. Carara, A. Gundel, M. N. Baibich, and R. L. Sommer,\nJ. Appl. Phys 84, 3792 (1998).\n9R. S. Beach and A. E. Berkowitz, Appl. Phys. Lett. 64,\n3652 (1994).\n10J. Vel´ azquez, M. V´ azquez, D.-X. Chen, and A. Hernando,\nPhys. Rev. B 50, 16737 (1994).\n11D. P. Makhnovskiy, L. V. Panina, and D. J. Mapps, Appl.\nPhys. Lett. 77, 121 (2000).\n12M. V´ azquez, M. Knobel, M. L. Sanchez, R. Valenzuela,\nand A. P. Zhukov, Sens. Actuators A 59, 20 (1997).\n13G. V. Kurlyandskaya et al., J. Appl.Phys. 87, 4822 (2000).\n14R. L. Sommer and C. L. Chien, Appl. Phys. Lett. 67, 3346\n(1995).\n15A. D. C. Viegas et al., J. Appl. Phys. 101, 033908 (2007).16R. B. da Silva et al., Appl. Phys. Lett. 94, 042501 (2009).\n17M. A. Corrˆ ea et al., J. Phys. D: Appl. Phys. 43, 295004\n(2010).\n18M. S. Marques et al., Thin Solid Films 520, 2173 (2012).\n19G. Kurlyandskaya et al., J. Magn. Magn. Mater. 242-245 ,\n291 (2002).\n20G. V. Kurlyandskaya et al., J. Phys.: Cond. Mat. 16, 6561\n(2004).\n21M. A. Corrˆ ea et al., J. Phys. D: Appl. Phys. 41, 175003\n(2008).\n22A. M. H. de Andrade, M. A. Corra, A. D. C. Viegas,\nF. Bohn, and R. L. Sommer, J. Appl. Phys. 115, 103908\n(2014).\n23R. B. da Silva et al., Appl. Phys. Lett. 104, 102405 (2014).\n24J. Torrej´ on, M. V´ azquez, and L. V. Panina, J. Appl. Phys.\n105, 033911 (2009).\n25J. Sinnecker, A. de Araujo, R. Piccin, M. Knobel, and\nM. V´ azquez, J. Magn. Magn. Mater. 295, 121 (2005).\n26N. A. Buznikov, C. Kim, C.-O. Kim, and S.-S. Yoon, Appl.\nPhys. Lett. 85, 3507 (2004).\n27C. Garc´ ıa, J. M. Florez, P. Vargas, and C. A. Ross, Appl.\nPhys. Lett. 96, 232501 (2010).\n28R. L. Sommer and C. L. Chien, Appl. Phys. Lett. 67, 857\n(1995).\n29M. A. Corrˆ ea et al., J. Appl. Phys. 110, 093914 (2011).\n30J. M. Barandiaran, A. Garcia-Arribas, and D. de Cos, J.\nAppl. Phys. 99, 103904 (2006).\n31T. C. Schulthess and W. H. Butler, J. Appl. Phys. 87,\n5759 (2000)."    },    {        "title": "2111.08188v1.Dynamic_alignment_and_plasmoid_formation_in_relativistic_magnetohydrodynamic_turbulence.pdf",        "content": "Draft version November 17, 2021\nTypeset using L ATEXtwocolumn style in AASTeX63\nDynamic alignment and plasmoid formation in relativistic magnetohydrodynamic turbulence\nAlexander Chernoglazov,1, 2Bart Ripperda\n ,2, 3,\u0003and Alexander Philippov\n2\n1Department of Physics, University of New Hampshire, 9 Library Way, Durham NH 03824, USA\n2Center for Computational Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA\n3Department of Astrophysical Sciences, Peyton Hall, Princeton University, Princeton, NJ 08544, USA\n(Revised ...; Accepted ...)\nSubmitted to ApJL\nABSTRACT\nWe present high resolution 2D and 3D simulations of magnetized decaying turbulence in relativistic\nresistive magneto-hydrodynamics. The simulations show dynamic formation of large scale intermittent\nlong-lived current sheets being disrupted by the tearing instability into plasmoid chains. These current\nsheets are locations of enhanced magnetic \feld dissipation and heating of the plasma. We \fnd magnetic\nenergy spectra/k\u00003=2, together with strongly pronounced dynamic alignment of Elsasser \felds and\nof velocity and magnetic \felds, for strong guide-\feld turbulence, whereas we retrieve spectra /k\u00005=3\nfor the case of a weak guide-\feld.\nKeywords: Magnetohydrodynamics | Plasma Astrophysics | Relativistic Fluid Dynamics\n1.INTRODUCTION\nTurbulence provides a route for the energy cascade\nand dissipation in a wide range of astrophysical plas-\nmas. This is relevant for astrophysical systems like black\nhole accretion disk-jet systems (e.g., Ripperda et al.\n2020, 2021; Mahlmann et al. 2020), magnetar magne-\ntospheres (Beloborodov 2020) and pulsar wind nebu-\nlae (e.g., Lyubarsky 1992; Begelman 1998). These as-\ntrophysical systems are typically relativistic, meaning\nthat the magnetization \u001b=B2=(4\u0019!\u001ac2)\u00151, whereB\nis the magnetic \feld strength, \u001ais the plasma den-\nsity, and!is the relativistic enthalpy density, indicat-\ning that the magnetic energy density is larger than the\nplasma energy density. This results in an Alfv\u0013 en speed\nvA=p\n\u001b=(\u001b+ 1)cthat is close to the speed of light c.\nMost turbulence studies have been in the realm of non-\nrelativistic magnetohydrodynamics (MHD) when the\nAlfv\u0013 en speed, vA, is much lower than the speed of light,\nc. Iroshnikov (1963); Kraichnan (1965) showed that the\nenergy cascade from large to small scales is caused by\nthe mutual shear of counter-propagating Alfv\u0013 en waves.\nCorresponding author: Alexander Chernoglazov\nalexander.chernoglazov@gmail.com\n\u0003Joint Princeton/Flatiron Postdoctoral FellowThirty years later Goldreich & Sridhar (1995, 1997) sug-\ngested that turbulent systems are in the critical balance\nregime meaning that an eddy is signi\fcantly deformed\nduring one Alfv\u0013 en-crossing time. This also means that\nthe turbulent eddies are elongated along the background\nmagnetic \feld. The \frst steps towards a theory of rel-\nativistic turbulence were taken recently by Chandran\net al. (2018), and they demonstrated that the relativis-\ntic picture is very similar to the Newtonian limit (more\ndetails are presented in Section 2). Boldyrev (2005,\n2006) suggested that turbulent eddies are anisotropic\nin all three directions: they are elongated along the\nguide magnetic \feld and have two di\u000berent sizes in the\nguide \feld-perpendicular plane. The ratio of these two\nsizes is called dynamic alignment angle. These eddies\nare progressively more elongated at smaller scales. Re-\ncent theories (e.g., Boldyrev & Loureiro 2017; Mallet\net al. 2017) proposed that the elongated eddies at small\nenough scale become unstable to the tearing instability,\ncausing a steepening of the turbulence spectrum.\nIn their recent paper, Dong et al. 2018 demonstrated\nthe formation of reconnecting current sheets in two-\ndimensional (2D) decaying non-relativistic turbulence.\nThey also demonstrated the formation of a turbulence\nspectrum and dynamic alignment in agreement with\nBoldyrev's theory. It is however as of yet uncleararXiv:2111.08188v1  [astro-ph.HE]  16 Nov 20212 Chernoglazov, A. et al.\nwhether these \fndings persist in the case of realistic\nthree-dimensional (3D) turbulence. In 3D MHD, de-\nspite prominent current sheet formation (Zhdankin et al.\n2013), it remains unclear whether reconnection can oc-\ncur in the fast regime, when the dissipation e\u000eciency\nis independent of resistivity. This regime is associated\nwith the formation of plasmoid chains, resulting in a\nuniversal reconnection rate of order 0 :01 (Bhattacharjee\net al. 2009; Uzdensky et al. 2010).\nPlasmoid-mediated reconnection in relativistic plas-\nmas can accelerate particles to non-thermal energies\n(e.g., Sironi & Spitkovsky 2014; Guo et al. 2014; Werner\net al. 2015), responsible for the high-energy emission\nin many environments of compact objects (e.g., Cerutti\net al. 2015; Beloborodov 2017). Recent studies of rela-\ntivistic turbulence in collisionless plasmas have shown\ne\u000ecient particle acceleration (Zhdankin et al. 2017;\nComisso & Sironi 2018) and the formation of reconnect-\ning current sheets, which are important for the process\nof initial particle acceleration (Comisso & Sironi 2018)\nboth in 2D and 3D. The high-energy power-law tail of\nthe distribution function has been shown to get steeper\nquickly for smaller ratios of the turbulent component of\nthe \feld to the guide \feld at the outer scale, \u000eB=B 0.\nThis observation further motivates the exploration of\ncurrent sheet properties at moderate \u000eB\u0018B0, when\nparticle acceleration is e\u000ecient.\nThe highly magnetized relativistic ( vA\u0019c) MHD\nlimit has been largely unexplored, and it is unclear\nwhether dynamic alignment forms in this regime, and\nwhether it plays an important role for the current sheet\nformation for situations where \u000eB\u0018B0. Neither any\npresence of dynamic alignment nor plasmoid unstable\ncurrent sheets were shown in the \frst relativistic ideal\nMHD simulations by Zrake & MacFadyen (2012).\nIn this Letter we present numerical relativistic resis-\ntive MHD simulations of decaying turbulence in highly-\nmagnetized plasma both in 2D and 3D. We demonstrate\nthat dynamic alignment forms both in 2D and 3D. We\nshow intermittent long-lived current sheets form natu-\nrally in the turbulence and become plasmoid-unstable.\n2.THEORETICAL OVERVIEW\nThe study of non-relativistic turbulence is usually\ndone with a reduced MHD approach. This method em-\nploys a few assumptions: a uniform, strong, in com-\nparison to the perturbation \u000eB, guide \feld B0and in-\ncompressibility of the \row ( cs!1 , wherecsis the\nsound speed). Under these assumptions, the only waves\nof interest are perpendicularly-polarized Alfv\u0013 en waves,\npropagating along the guide \feld. The reduced form ofMHD equations in this limit reads (Elsasser 1950):\n@z\u0006\n@t\u0007vA\u0001rz\u0006=\u0000z\u0007\u0001r?z\u0006\u0000rP=\u001a 0;r\u0001z\u0006= 0;\n(1)\nwherePis the total pressure, and z\u0006=\u000ev\u0006\u000eB=p4\u0019\u001a0\nare the Elsasser \felds, representing counter-propagating\nAlfv\u0013 en waves.\nThe ideal relativistic MHD equations consist of mass\nand stress-energy conservation laws, and the induction\nequation for the magnetic \feld evolution:\n@\u0017(\u001au\u0017) = 0; @\u0017T\u0016\u0017= 0; @\u0017(b\u0016u\u0017\u0000b\u0017u\u0016) = 0:(2)\nHere\u0016; \u0017 are 4-dimensional space-time indices, such\nthatu\u0016= (\u0000;\u0000v) is the four-velocity vector, \u0000 is\nLorentz factor, and T\u0016\u0017is the stress-energy tensor\nT\u0016\u0017=Eu\u0016u\u0017+\u0012\nP+b2\n2\u0013\n\u0011\u0016\u0017\u0000b\u0016b\u0017(3)\nwithE=\u001a!c2+b2and!= 1 + (\r=(\r\u00001))P=\u001a is the\nrelativistic enthalpy, \u0011\u0016\u0017= diagf\u00001;1;1;1g, the \rat-\nspacetime Minkowski metric. b\u0016is the magnetic \feld\nfour-vector\nb\u0016=1p\n4\u0019\u0012\n\u0000(v\u0001B);Bi\n\u0000+ \u0000(B\u0001v)vi\nc2\u0013\n; (4)\nandb2=b\u0016b\u0016. Introducing the relativistic Elsasser\n\felds\nz\u0016\n\u0006=u\u0016\u0006b\u0016=p\nE (5)\nand modi\fed pressure term, \u0005 = (2 P+b2)=(2E), one can\nrewrite the relativistic MHD equations in the Elsasser-\ntype form: (Chandran et al. 2018; TenBarge et al. 2021)\n@\u0017(z\u0016\n\u0006z\u0017\n\u0007+\u0005\u0011\u0016\u0017)+\u00123\n4z\u0016\n\u0006z\u0017\n\u0007+1\n4z\u0016\n\u0007z\u0017\n\u0006+ \u0005\u0011\u0016\u0017\u0013@\u0017E\nE= 0:\n(6)\nIn contrast to the non-relativistic case, one cannot for-\nmally introduce an incompressible limit in relativistic\nMHD wherein there is a maximum speed of propagation,\nc. The \fnite speed of light prevents easy elimination\nof the fast magnetosonic modes (Takamoto & Lazarian\n2017). However, it is still possible to order them out\nin the highly anisotropic limit, k?\u001dkjj, which implies\n!F\u0018kc\u001d!A\u0018kjjc, where!Fand!Aare the fre-\nquencies of the fast magnetosonic and Alfv\u0013 en modes,\ncorrespondingly (TenBarge et al. 2021). Here, we are\ninterested in the highly-magnetized limit, where the hot\nmagnetization parameter, \u001b, is large:\n\u001b=hB2i\n4\u0019\u001ac2!\u001d1: (7)Relativistic Turbulence 3\nThis implies b2\u001dP, or \u0005 = 1=2, and the elimination\nof the slow magnetosonic mode. The second term of the\nrelativistic Elsasser variable z\u0016\n\u0006is a unit vector in the\ndirection of the four-magnetic \feld vector as E\u0019p\nb2in\nthis limit.\nTenBarge et al. (2021) discusses the Elsasser-type\nequations of the relativistic MHD in the highly\nanisotropic limit, k?\u001dkk, constructed in the aver-\nage \ruid rest frame huii= 0 (Chandran et al. 2018).\nIn three-vector form the result is particularly similar to\nthe reduced non-relativistic MHD equations:\n@\u000ez\u0006\n@t\u0007vA\u0001r\u000ez\u0006=\u0000\u000ez\u0007\u0001r?\u000ez\u0006\u0000r?\u000e\u0005:(8)\n\u000ez\u0006=\u000ev\u0006\u000eB?pE0; \u000e\u0005 =\u00002\u000eP+\u000e\u001a\n2E0+2P0+\u001a0\n2E0\u000eE\nE0:\nDue to the close resemblance between the Newtonian\nand relativistic set of reduced MHD equations, once\nthe anisotropic cascade reaches su\u000eciently small scales,\nwherek?\u001dkkis satis\fed, one can expect that rel-\nativistic MHD turbulence is statistically similar to its\nNewtonian counterpart. In the case of interest, \u001b\u001d1,\nthe last term of (8) is negligible, and the Alfv\u0013 en speed\nis close to the speed of light, vA\u0019c, such that the\nequations are particularly simple:\n@\u000ez\u0006\n@t\u0007rjj\u000ez\u0006=\u0000\u000ez\u0007\u0001r?\u000ez\u0006: (9)\nThe applicability of equation (9) is limited to regimes\nwhere\u000eP\u001c\u000eB2. However, when plasma is heated\nto relativistic temperatures, e.g., in reconnection layers,\nthis assumption is not justi\fed, at least locally. Since\nthere is no a formally incompressible limit in relativis-\ntic systems, and our interest in systems with \u000eB\u0018B0,\nwhich is particularly challenging to explore analytically,\nwe turn to numerical simulations to con\frm these ex-\npectations.\nAn important feature of the highly magnetized MHD\nturbulence is the overall dominance of the magnetic and\nelectric \feld \ructuations, \u000eEB, over the kinetic energy,\n\u000eEkin. This can be seen from the following relations\nfor a single Alfv\u0013 en wave in the relativistic regime ( \u001b\u001d\n1;vA\u0018c):\n\u000eE\u0018\u000ev\ncB0\u0018vA\nc\u000eB\u0018\u000eB)\u000ev\u0018c\u000eB\nB0<c\n\u000eEkin\u0018\u001a\u000ev2\u0018\u000eB2\u001ac2\nB2\n0\u0018\u000eB21\n\u001bcold\u001c\u000eEB;\nwhere the inequality is used,\n\u001bcold=B2\n4\u0019\u001a0c2>B2\n4\u0019\u001a0c2!=\u001b: (10)Current sheets are important dissipative structures in\nmagnetized turbulence, and it is useful to compare their\nbehavior in non-relativistic and relativistic regimes. In\na near-stationary current sheet, the reconnection rate\nis the ratio of the in\row velocity to the out\row veloc-\nityvin=vout(Parker 1957; Sweet 1958). If the plasma\ncan be assumed incompressible, it then follows that\nvin=vout=\u000e=L where\u000eis the thickness and Lis the\nlength of the current sheet. The thickness of a Sweet-\nParker current sheet is determined from continuity of the\nresistive and ideal electric \felds as \u000e=\u0011=vin, where\u0011is\nthe resistivity. The out\row speed can be approximated\nas the Alfv\u0013 en speed vout\u0018vA, which in a relativistic\nplasma isvA\u0018c. This results in a relativistic recon-\nnection rate of vin=vout\u0018vin=c\u0018\u0011=(Lvin), i.e., a result\nidentical to the non-relativistic case (Lyubarsky 2005).\nThe reconnection rate in a Sweet-Parker sheet then\nscales as\u0018S\u00001=2, whereS=LvA=\u0011is the Lundquist\nnumber. For large Lundquist numbers, typical in astro-\nphysical sources, reconnection is mediated by the plas-\nmoid instability, which in non-relativistic settings gets\ntriggered at Scrit\u0015104, leading to a saturation of the\nreconnection rate at \u00190:01 (Loureiro et al. 2007; Bhat-\ntacharjee et al. 2009). It was shown semi-analtyically\nand numerically, by solving the full set of resistive rela-\ntivistic MHD equations, that this result holds for highly\nmagnetized relativistic plasmas (Del Zanna et al. 2016;\nRipperda et al. 2019).\nSince magnetic \feld \ructuations dominate over kinetic\nenergy, resistive dissipation dominates over viscous dissi-\npation in the inertial range of turbulence. We show that\nresistive dissipation in highly-magnetized MHD plasmas\nis also dominant in the exhausts of reconnection layers.\nOne can see this by comparing the rate of resistive en-\nergy dissipation\u0018\u0011B2=4\u0019\u000e2\nexhaust , to the rate of viscous\ndissipation\u00182\u0017Ekin=\u000e2\nexhaust , where\u0017is the viscosity,\nEkinis the kinetic energy in the exhaust region, B2=8\u0019is\nmagnetic energy and \u000eexhaust is the typical width scale in\nthe exhaust region. The ratio of resistive to viscous dis-\nsipation rates is then \u0011=\u0017\u0001(B2=Ekin)\u0018(B2=Ekin)1. In a\nnon-relativistic plasma, this ratio is always of order O(1)\nsinceEkin\u0018\u001av2\nAandv2\nA\u0018B2=\u001a. However, in highly\nmagnetized relativistic plasma, Ekin\u0018\u001a\rexhaustc2\u0018\n\u001ap\u001bc2(Lyubarsky 2005) while B2\u0018\u001a\u001bc2, and hence\nthe ratio between resistive and viscous dissipation rates\n1Our argument is applicable for the case of a scalar viscosity \u0017,\ni.e., independent of the gas pressure which is high in the exhaust\nregion. Whether this assumption is accurate for the e\u000bective vis-\ncosity of a collisionless relativistically hot plasma is an interesting\nquestion for further investigation.4 Chernoglazov, A. et al.\nis proportional top\u001b\u001d1 such that resistive dissipation\ndominates.\n3.NUMERICAL METHOD AND SETUP\nWe solve the set of special relativistic resistive MHD\n(SRRMHD) equations with the Black Hole Accretion\nCode ( BHAC , Porth et al. 2017; Olivares et al. 2019) and\nan Implicit-Explicit (IMEX) time stepping scheme to\nevolve the sti\u000b resistive Ohm's law (Ripperda et al. 2019;\nRipperda et al. 2019). We employ a constant and uni-\nform resistivity \u0011, which provides the simplest prescrip-\ntion to allow resolved magnetic reconnection.\nThe SRRMHD equations are numerically evolved in\na periodic domain of size L2in 2D andL2\u0002Lzin 3D.\nWe initialize an out-of-plane (in the ^ z-direction) guide\nmagnetic \feld and an in-plane ( x\u0000y) magnetic \feld\nperturbation \u000eB?:\n\u000eBx=NX\nm=1NX\nn=1\fmnnsin(kmx+\u001emn) cos(kny+'mn);\n\u000eBy=\u0000NX\nm=1NX\nn=1\fmnmcos(kmx+\u001emn) sin(kny+'mn);\nwhere\fmn= 2\u000eB?=(Np\nm2+n2),km= 2\u0019m=L , and\n\u001emn; 'mnare random phases. We set N= 8 initial\nwaves in each direction for 2D runs (64 initial modes)\nandN= 4 for 3D runs, in order to allow for a larger in-\nertial range in 3D simulations. The outer (or energy con-\ntaining) scale is then l?=L0=8 for 2D simulations and\nl?=L0=4 for 3D. The turbulence at smaller scales forms\nself-consistently via energy cascading. In 3D we modu-\nlate\u000eB?with two modes/sin(klz+ mnl), where mnl\nis also a random phase. The normalization coe\u000ecient is\nthen\fmnl= 2p\n2\u000eB=(NpNzp\nm2+n2). We initialize\nthe plasma at rest, with velocity \feld v= 0, and with\na uniform gas pressure p0and rest mass density \u001a0. We\nset an adiabatic index \r= 4=3, assuming an ideal rela-\ntivistic gas. Similar initial conditions for the magnetic\n\feld were employed in relativistic particle-in-cell (PIC)\n(Comisso & Sironi 2018; N attil a & Beloborodov 2020)\nturbulence simulations. For all simulations we set L= 1.\nIn order to characterize the strength of both the guide\nand the in-plane magnetic \feld we introduce two mag-\nnetization parameters\n\u001b0=hB2\nzi\n4\u0019\u001a0c2!; \u000e\u001b =h\u000eB2\n?i\n4\u0019\u001a0c2!; (11)\nA summary of the performed runs is given in Table\n1. We employ an elongated box with L= 1;Lz= 3\nfor run 3D[d] to enforce the critical balance condition\n\u000eB?=L\u0019B0=Lzat the outer scale.\nWe set the resistivity to either \u0011= 10\u00005;10\u00006in\nthe 2D setup, which corresponds to Lundquist numbersS\u0019104;105for the largest current sheets of length\nLcs\u00190:1, andvA=c\u00191. The simulation with \u0011= 10\u00006\nis well-above the critical Lundquist number limit Scrit,\nwhile the simulation with \u0011= 10\u00005is approximately at\nthe limitS\u0019Scrit. Potentially, current sheets can be-\ncome plasmoid unstable at a smaller critical Lundquist\nnumber in a turbulent \row (Loureiro et al. 2009). We\nexplore whether this e\u000bect is signi\fcant in 2D relativistic\nturbulence with our \u0011= 10\u00005simulation.\nIn order to ensure that the resistive length scales are\nresolved, and results are converged with numerical res-\nolution, we develop a novel adaptive mesh re\fnement\n(AMR) strategy (see Appendix A). We benchmark our\n2D results with a short simulation (until t= 0:5L=c)\non a uniform grid, with a resolution of 655362. In 3D\nit is impossible to fully converge due to numerical limi-\ntations, and instead we employ the highest feasible res-\nolution that allows to capture the development of the\nplasmoid instability in the longest current sheets. One\nhigh-resolution run is performed with 32003grid points\nto probe the formation of plasmoid chains. We addition-\nally present a study with di\u000berent values of the magne-\ntization parameter at a resolution of 20483grid points.\nThe SRRMHD algorithm relies on viscosity \u0017at the\ngrid level, such that the magnetic Prandtl number\nPrm=\u0017=\u0011\u001c1 for 2D simulations, and Pr m.1\nfor 3D simulations with a marginally resolved resistive\nscale, assuming resistive and viscous scales are similar\nand governed by the \fnite grid resolution. This choice\nis further motivated by the fact that viscous e\u000bects are\nsubdominant for highly-magnetized plasmas, as we have\ndemonstrated in Section 2.\n4.RESULTS\nWe present the results of simulations which we run\nuntilt= 2L=c, such that the turbulence is fully de-\nveloped and settles to a quasi-steady state. For the\ncase of decaying turbulence considered in this Let-\nTable 1. Summary of simulation parameters.\nSim Res \u0011 \u001b 0\u000e\u001b L;L zGrid\n2D[a] 65536210\u000065 5 12AMR\n2D[b] 65536210\u000055 5 12AMR\n2D[c] 32768210\u000065 5 12AMR\n2D[d] 32768210\u000061 1 12AMR\n2D[e] 32768210\u000061 5 12AMR\n2D[f] 65536210\u000065 5 12Uni\n3D[a] 3200310\u000061 5 12\u00021 Uni\n3D[b] 2048310\u000065 5 12\u00021 Uni\n3D[c] 2048310\u000061 1 12\u00021 Uni\n3D[d] 2048310\u000069 1 12\u00023 UniRelativistic Turbulence 5\nter, we de\fne a quasi-steady state when the spec-\ntral slope is constant for at least one outer scale\neddy turnover time, \u0018L=c, while the total energy\nEB=R\nB(x)2=8\u0019dx=R\nBk(k?)\u0001B\u0003\nk(k?)=8\u0019dkdissi-\npates. Here, Bkis the amplitude of the Fourier mode\nof the magnetic \feld with wavenumber k. It takes\n\u0001t\u00190:3L=cfor the energy to cascade from the initial\nlowkmodes to the resistive scale. At \u0001 t\u00191\u00002L=c\nthe spectrum \rattens until it reaches a quasi-steady\nstate. This behavior of the power spectrum is illus-\ntrated in the attached video2. In order to test conver-\ngence of the simulation, we compare spectra of magnetic\nenergy,E(k)dk=P\nk2dkBk\u0001B\u0003\nk=8\u0019, for di\u000berent reso-\nlutions: if the onset of the inertial range cuto\u000b does not\nchange with increasing resolution (the vertical lines in\nFigure 5c), i.e., if the cuto\u000b is determined by the resolved\nresistive scale, the simulation is considered converged.\nMore details about the AMR strategy and convergence\ntests are presented in the Appendix A.\nIn Figure 1a we present the distribution of the out-\nof-plane electric current density jz\u0018(r\u0002B)zfor a\n2D simulation with an e\u000bective resolution of 655362\ngrid points, 2D[a], and 3 AMR levels, for a resistivity\n\u0011= 10\u00006. Here, we set magnetizations \u001b0=\u000e\u001b= 5,\nequivalent to a total magnetization \u001b= 10. Very long\ncurrent sheets emerge at the interfaces of large merg-\ning eddies. The length of a current sheet is mainly\nde\fned by the size of the largest eddies present in the\nsystem. Estimating the length of these current sheets as\nLsheet=L\u00190:1 and accounting for the relativistic Alfv\u0013 en\nspeedvA=c=p\n\u001b=(\u001b+ 1)\u00191, we \fnd the Lundquist\nnumber to be S\u0019105\u001dScrit= 104. These current\nsheets are plasmoid-unstable and break up into current\nsheets of smaller length scales, such that their Lundquist\nnumbersSlocal\u0019Scrit. This results in a maximum\nnumber of\u001810 plasmoids, which is consistent with re-\nsults shown in Figure 1a. We also perform simulations\nfor\u001b0=\u000e\u001b= 1 and\u001b0= 1;\u000e\u001b= 5 and resistivity\n\u0011= 10\u00006. Plasmoid-unstable current sheets form ubiq-\nuitously for all of these settings. For resistivity \u0011= 10\u00005\nwe observe only few plasmoids (for the longest current\nsheets) in the whole domain indicating that the critical\nLundquist number Scrit\u0019104holds for the plasmoid\ninstability in current sheets in a 2D turbulent \row. By\nvarying numerical resolution, we \fnd that the onset of\nthe plasmoid instability occurs at lower resolution for\nthe cases with weaker guide \feld, \u001b0\u0014\u000e\u001b, motivating\nour choice to perform our highest resolution 3D simula-\ntion for\u001b0=\u000e\u001b= 1=5 (run 3D[a]).\n2Direct link: https://youtu.be/n7SZigrJ9kk2D and 3D weak guide \feld (3D[a]) simulations show\npronounced reconnection-mediated mergers of smaller\neddies. This process has also been recently observed\nin simulations of merging non-helical \rux tubes (Zhou\net al. 2020). Our very long 2D simulations with a\n(smaller) resolution of 32002demonstrate that the ter-\nminal state of the turbulence has two large eddies of\nopposite magnetic helicityR\nA\u0001Bdxremaining in the\nsimulation box. 3D simulations show similar behavior.\nIn order to identify current sheets, we choose a threshold\nin the current density, \u0018, and consider a point xto be\nin the current sheet if jz(x)> \u0018j rms, wherejrmsis the\nroot-mean-square of the electric current jzin the domain\n(Zhdankin et al. 2013). The long current sheets have an\nintermittent nature and occupy about 0.2-0.5% of the\ndomain in 2D, yet they are responsible for 20 \u000025% of\nthe magnetic \feld dissipation, /\u0011j2, for\u0011= 10\u00006. Our\nresults are insensitive to the exact value of \u0018, as long\nas\u0018&5. For a larger value of resistivity, \u0011= 10\u00005,\nonly few plasmoids form in the whole simulation box\n(see Figure 1b). Comparing to the \u0011= 10\u00006case, the\ncurrent sheets for \u0011= 10\u00005are thicker and, hence, have\na lower current density amplitude. In this case we \fnd\nonly 10% of the dissipation to happen in the localized\ncurrent sheets.\nThe anisotropic properties of the turbulence can\nbe quanti\fed by measuring the dynamic alignment\nangle of eddies in the plane perpendicular to the\nguide \feld (Boldyrev 2005, 2006). We employ\na Monte-Carlo method to compute dynamic align-\nment angle as a function of a point-separating\nvector. Following the method proposed by Ma-\nson et al. (2006); Perez et al. (2012), we com-\npute two structure functions S1\n1(l) =h\u000ev?(l)\u0002B?(l)i,\nS1\n2(l) =hj\u000ev?(l)jjB?(l)ji, where\u000ev?(l) and\u000eB?(l) are\nincrements of the velocity and the magnetic \feld perpen-\ndicular to the local guide \feld at scale l. The alignment\nangle is de\fned as\n\u0012(l)\u0011\u0012v;B(l)\u0011l\n\u0018=S1\n1\nS1\n2; (12)\nwhereland\u0018are sizes of the eddy in the guide \feld-\nperpendicular plane. Note that both S1\n1andS1\n2are \frst\norder structure functions Sn(l) =hjf(r+l)\u0000f(r)jni,\nand one can de\fne the alignment angle for any nas\n\u0012\u0019(Sn\n1(l)=Sn\n2(l))1=n. It turns out that the slope of the\nalignment angle function is dependent on the order of\nthe structure function n(Mallet et al. 2016) revealing\nthe intermittent nature of dynamic alignment (Frisch\n1995). We present more details on the intermittency of\ndynamic alignment in Appendix C.6 Chernoglazov, A. et al.\nFigure 1. 2D SRRMHD runs of highly-magnetized decaying turbulence. The top row shows snapshots of the out-of-plane\nnormalized electric current at t= 1 for a) simulation 2D[a], with the resistivity value \u0011= 10\u00006corresponding to the typical\nLundquist number S\u0019105for the longest current sheets; b) simulation 2D[b], \u0011= 10\u00005,S\u0019104. Insets show zooms into\nthe snapshot of simulation 2D[a], highlighting plasmoid-unstable current sheets. The bottom row shows statistical properties\nof the 2D turbulence: c) the spectrum of the normalized magnetic and kinetic (multiplied by 100) energies and d) the dynamic\nalignment angle at di\u000berent times during the simulation t= 0:5;t= 1:0;t= 2:0, in simulations 2D[a,b] and alignment angle\nfor Elsasser \feld, \u0012z+;z\u0000, for 2D[a] at t= 2:0. The results of a uniform grid simulation 2D[f] at t= 0:5 are presented to show\nnumerical convergence of the AMR criteria.\nThe slope of the dynamic alignment angle as a func-\ntion of the size of eddies is tightly connected to the\npower-law of the magnetic energy spectrum, PB(k),\nwhich is predicted to be k\u00003=2\n?for turbulence that is\nanisotropic in the plane perpendicular to the guide \feld\n(Boldyrev 2006). By introducing a non-linear time, \u001cc\n(Boldyrev 2005), we can relate the power spectrum with\nthe alignment angle:\n8\n<\n:\u000eB2\nl\n\u001cc\u0018\"\n\u001cc\u0018l\n\u000eBlsin\u0012;(13)\nE(k?)\u0018\u000eB2\nlk\u00001\n?\u0018\"2=3k\u00005=3\n?sin\u0012\u00002=3;\nwhere\"is the energy cascading rate, \u000eBlis the in-\ncrement of the magnetic \feld at a scale lin the\nplane perpendicular to the guide \feld. For sin \u0012(l)\u0018\nl1=4\u0018k\u00001=4\n?, it reproduces Boldyrev's spectrum\nE(k?)\u0018k\u00003=2\n?. In the case of no alignment beingpresent,\u0012(l)\u0018const, it reduces to the Goldreich & Srid-\nhar (1995) spectrum, E(k?)\u0018k\u00005=3\n?.\nIn Figure 1c we show the magnetic power spectrum\nin the steady state and multiply the result by k3=2\n?, to\nmake the di\u000berence between the power law indices \u00003=2\nand\u00005=3 more pronounced. Figure 1c clearly demon-\nstrates that the spectrum is closer to k\u00003=2\n?in the in-\nertial range. We de\fne the steady state of decaying\nturbulence when the spectral slope is constant in time,\natt\u00150:5L=c, allowing us to compare our results with\nsteady state theory. Note that the total energy \u000fB(t) =R\n(B2\u0000Bj2\nk=0)=8\u0019dV decreases in time while the nor-\nmalized spectrum ~E(k)dk=P\nk2dkBk\u0001B\u0003\nk=8\u0019\u000fb, which\nwe present in all spectrum plots, is constant in time. It is\nworth mentioning that in non-relativistic reduced MHD\nsimulations one typically analyzes the spectrum of the\ntotal kinetic and magnetic energy (Perez et al. 2012). In\nour highly magnetized relativistic simulations however,Relativistic Turbulence 7\n\u001a(\r\u00001)k(k?)\u001cB2\nk(k?), and we con\frmed that the\ncontribution of the kinetic energy is negligible for both\n2D and 3D simulations (see spectra in Figures 1c and\n2c). To preserve the kinetic to magnetic \feld energy\nratio, we also normalize the kinetic energy by \u000fb(t). In\nagreement with the k\u00003=2\n?power spectrum, the v\u0000B\ndynamic alignment (Figure 1d) demonstrates a perfect\nmatch with Boldyrev's prediction, \u0012(l)\u0018l1=4, at the in-\ntermediate scales, lres.l.lmax, wherelmax=L0=8 is\nde\fned by the number of modes in the initial conditions,\nandlresis de\fned by the resistive scale.\nChandran et al. (2015) proposed that mutual shear\nof counter-propagating Elsasser \felds \u000ez\u0006is responsible\nfor the dynamic alignment. They predict that these two\n\felds create a progressively decreasing alignment angle,\nwhile the slope becomes \ratter. To test this hypothesis,\nwe measure the alignment angle between two Elsasser\n\felds:\n\u0012z+;z\u0000=h\u000ez+\n?\u0002\u000ez\u0000\n?i\nhj\u000ez+\n?jj\u000ez\u0000\n?ji: (14)\nStraightforward application of the non-relativistic El-\nsasser \feld expression, \u000ez\u0006=\u000ev\u0006\u000eB=p4\u0019\u001a, results\nin\u000ez+\n?\u0002\u000ez\u0000\n?\u0018\u000ev\u0002\u000eB, whilej\u000ez+\n?jj\u000ez\u0000\n?j \u0018 j\u000eBj2,\ngiving that their ratio \u0012z+;z\u0000\u0018\u000ev=\u000eB\u001c1 in highly\nmagnetized plasma. However, one should use the rela-\ntivistic formulation of Elsasser \felds (5) in this regime,\nwhere uandb=p\nEcan be comparable. The dynamic\nalignment angle between the relativistic Elsasser \felds\nis \ratter than l0:25att= 2, for\u0011= 10\u00006(Figure 1d).\nThe average slope is close to the l0:1result, as predicted\nby Chandran et al. (2015), although it displays an un-\nexpected break at intermediate scales.\nThe smallest averaged dynamic alignment angle,\n\u0012v\u0000B, in the simulation with \u0011= 10\u00006is 0:175, and\nit is approximately constant for small scales. Devia-\ntions from Boldyrev's scaling l0:25are visible at scales\nwhere resistive e\u000bects become important. Note that this\nis also where the inertial range of the spectra ends. The\nplasmoid-unstable current sheets we observe in the sim-\nulation possess much smaller alignment angles \u0012\u00190:01,\nin accordance with Loureiro et al. (2007). The pres-\nence of such current sheets with alignment angles of an\norder of magnitude smaller than the minimal averaged\nalignment angle that we \fnd, implies the intermittent\nnature of these sheets (Dong et al. 2018). Formation\nof intermittent plasmoid-unstable current sheets can be\nresponsible for a steepening of the spectrum at the end\nof the inertial range, which we observe in the range\nk?\u0019300\u00001200 att= 1 in Figure 1c. However, we\nassume that the scale separation in our simulations is\nnot enough to robustly con\frm the k\u000011=5\n?prediction by\nBoldyrev & Loureiro (2017) and Mallet et al. (2017) forthe non-relativistic reconnection-mediated regime. We\nalso do not observe the increase of the alignment angle\nat small scales lcorresponding to wave-vectors k?in the\nsteepening range, as predicted in Boldyrev & Loureiro\n(2017).\nSince the onset of the plasmoid instability occurs at\nlower resolution in 2D simulations if a weaker guide\n\feld is assumed, we run a 3D simulation (3D[a]) with\n\u001b0= 1,\u000e\u001b= 5, and highest resolution of 32003grid\npoints. For 2D simulations we con\frm that full plas-\nmoid chains form for smaller values of \u000eB?=B0as well,\nbut higher resolutions are required to resolve the insta-\nbility. We refer to the case with initial \u000eB?=B0=p\n5=1\n(run 3D[a]) as a weak guide \feld, and to the case with\ninitial\u000eB?=B0= 1=3 (run 3D[d]) as a strong guide \feld.\nWe note that by t= 1\u00002, when we analyze the simu-\nlations, the turbulent component of the \feld decayed to\n\u000eB?=B0.1 (3D[a]) and \u000eB?=B0\u00180:2 (3D[d]).\nFor the strong guide \feld case, the energy cascade\nis developing mainly in k??^z, and the full 3D anal-\nysis can be reduced to a 2D analysis in a set of x\u0000y\nplanes (e.g., Perez et al. 2012). For simplicity, in the case\nof the weak guide \feld we also compute the spectrum\nfor wavevectors k?perpendicular to B0using the same\nmethod (a more accurate calculation would use struc-\nture functions which take into account a locally varying\nguide \feld, Cho & Vishniac 2000). In order to provide\na statistically signi\fcant result, we average the 2D spec-\ntrum and dynamic alignment angle in the set of x\u0000y\nplanes taken at various z. We con\frm that the spectrum\nand the alignment angles are independent of the choice\nof the sampling planes if Nplanes&Nz=3, whereNzis\nthe number of grid points in the direction along z.\nWe consider the turbulence at t= 2 to be in a steady\nstate, i.e., the dynamic alignment is fully formed (see\nFigure 2). We con\frm the steady state shape of the\ndynamic alignment angle function beyond t= 2 with\nlonger simulations at a lower numerical resolution, 20483\n(runs 3D[b], 3D[c]). The slope of the v\u0000Balignment\nangle is close to the predicted l0:25for the smaller ed-\ndies and is less pronounced for eddies of the system size\nscale (see Figure 2d). In simulation 3D[a] with the ini-\ntially weaker guide \feld \u000eB?=B0=p\n5=1, att\u00191, the\nalignment angle curve is signi\fcantly shallower, consis-\ntent with the steady state in driven non-relativistic tur-\nbulence at \u000eB?=B\u00181 (Mason et al. 2006). At this\ntime, the strength of turbulent \ructuations is similar\nto the strength of the guide \feld, hj\u000eB?ji \u0019 hj Bzji.\nFurther dissipation of the magnetic energy leads to\nhj\u000eB?ji\u0019 0:7hjBzjiatt= 2, and a steeper alignment\nangle curve. The spectrum of the turbulence develops\nsimultaneously with the dynamic alignment.8 Chernoglazov, A. et al.\nFigure 2. 3D SRRMHD runs of highly-magnetized decaying turbulence. The top row shows snapshots of the out-of-plane\nnormalized electric current jzfor run 3D[a] at a) t= 1:0 and b)t= 2:0. Insets of both \fgures show zooms into plasmoid-unstable\ncurrent sheets. The bottom row shows statistical properties of the 3D turbulence: c) the spectrum of normalized magnetic and\nkinetic (multiplied by 103) energies and d) the dynamic alignment angles \u0012for runs 3D[a] (solid lines) and 3D[d] (dashed lines)\nat two di\u000berent times t= 1:0; t= 2:0, and alignment angles for the Elsasser \felds, \u0012z+;z\u0000, att= 2 for runs 3D[a], 3D[d].\nThe slope of the z+\u0000z\u0000dynamic alignment angle,\n\u0012z+;z\u0000(l), is comparable to \u0012(l) for the strong guide \feld\n(run 3D[d], t= 2,hj\u000eB?ji=hjBzji\u0019 0:2). For the weak\nguide \feld (3D[a], t= 2,hj\u000eB?ji=hjBzji \u0019 0:7), the\nz+\u0000z\u0000alignment is very weakly pronounced. At the\nsame time, the slope of the energy spectrum of 3D[a] is\ncloser to\u00005=3 as predicted by Goldreich-Sridhar theory\nwith no dynamic alignment. It could be considered as\nan indication that the dynamic alignment of Elsasser\n\felds\u000ez+; \u000ez\u0000rather than the one of v;Breduces the\nnon-linearity.\n3D simulations show less pronounced boundaries of\nlarge-scale eddies, but the intermittent large current\nsheets are still present in the system with the weak guide\n\feld. Figure 2a and the linked video3demonstrate the\ndistribution of the electric current jzin the planes per-\npendicular to the guide \feld, Bz, att= 1, and Figure 2b\n3Direct link: https://youtu.be/nY3F4bnTtEMand the accompanying video4show the same at t= 2.\nSimilarly to the 2D results, intense current sheets occupy\nup to 4\u00005% of the total volume of the domain5and lead\nto 20% of the total dissipation of the magnetic energy.\nIntermittent long current sheets are clearly plasmoid-\nunstable as shown by the insets in Figure 2. A few ini-\ntial eddies are still clearly seen at t= 1, but many long\nintermittent current sheets are una\u000bected by the choice\nof the initial conditions. At t= 2 no visible features are\nassociated with the initial conditions (see Figure 2b).\nOverall, the structure of the electric current in the 3D\nweak guide \feld simulation looks similar to the one in\n2D (Figure 1a): there is a number of well-pronounced\nlong, plasmoid unstable current sheets formed at the\nouter scale. Their formation is likely associated with\nthe mergers and subsequent reconnection of large coher-\n4Direct link: https://youtu.be/8CRiWAZg Bo\n5The larger \flling fraction in 3D simulations is potentially at-\ntributed to the fact that for a similar value of resistivity,\n\u0011= 10\u00006, the widths of current sheets are not fully converged.Relativistic Turbulence 9\nFigure 3. 3D SRRMHD simulation of highly-magnetized\ndecaying turbulence, run 3D[d]. The color shows the out-of-\nplane component of the electric current jzin the snapshots at\nt= 2:0, whenh\u000eBi=hBzi= 0:2. The insets show zooms into\nindividual current sheets which indicate plasmoid formation.\nThe streamlines in the insets show the in-plane magnetic \feld\nlines. The current sheets in the middle and bottom insets do\nnot show a perfect anti-parallel \feld geometry because the\nlocal guide \feld is tilted with respect to the plane of the\nsnapshot.\nent structures (Hosking & Schekochihin 2020). Unlike in\nthe weak guide \feld regime, the strong guide \feld simu-\nlation 3D[d] shows the statistical properties of \\aligned\"\ncritically-balanced turbulence: the k\u00003=2\n?spectrum and\na pronounced dynamic alignment (dashed lines in Fig-\nure 2c and d). The spatial distribution of the electric\ncurrent is more uniform in this case (see Figure 3). The\nabsence of very long current sheets is consistent with the\nobservation of a very few plasmoids in the simulation\n(see insets of Figure 3). A possible explanation can be\nfound in the small ratio of the length, Lsheet\u00180:05, for\nthe sheets shown in the insets of Figure 3, to the width\nof these sheets, which at our resolution is still limited by\nthe numerical di\u000busion. We anticipate that the plasmoid\ninstability can be more reliably captured at much higher\nspatial resolution: for the typical length, Lsheet\u00180:05,\nand the width-to-length ratio \u0012\u00190:01, one requires\n(N\u000e=(Lsheet\u0012))3\u0019100003grid points, where N\u000e\u00195\ncells is the minimally desired resolution per width of the\nplasmoid-unstable current sheet.\nThe structure of a representative current sheet for the\nweak guide \feld simulation 3D[a] is presented in Figure\n4. The volume render represents the current density\namplitude, and solid black lines show selected magnetic\n\feld lines. The lower threshold for the volume rendering\nis chosen to be\u00192jrms, in order to remove the upstream\nregions without signi\fcant current. The initial (seed)points for the integration of magnetic \feld lines are set\ninside two randomly chosen plasmoids. Wrapped, helical\nmagnetic \feld is responsible for the large current density\ninside the plasmoids. The helical structure allows longer\nplasmoids (or, \rux tubes) to be kink-unstable if their\nlength is large enough to exceed the Kruskal-Shafranov\nstability limit. This instability likely limits the life time\nof plasmoids in current sheets and their axial extension.\nA zoom into the 3D structure of a plasmoid is shown in\nFigure 4b.\nAcceleration of the \row from the X-point of a Sweet-\nParker current sheet up to Alfv\u0013 enic speed creates a\nvelocity shear, which may become unstable to the\nKelvin-Helmholtz instability (KHI). The analytical non-\nrelativistic stability criterion \u0001 u.vA(Loureiro et al.\n2013) suggests that the strong upstream magnetic \feld\ncan lead to the stabilization of the KHI for the veloc-\nity shear \u0001 uand the Alfv\u0013 en speed vAdetermined by\nthe upstream magnetic \feld strength. A similar crite-\nrion was derived for a simpli\fed model with Bjjvin a\nfully relativistic case (Osmanov et al. 2008). Thus, we\nexpect current sheets in highly-magnetized turbulence\nto be stabilized by the strong upstream magnetic \feld.\nTo con\frm this prediction, we conduct localized numeri-\ncal experiments with conditions inferred from turbulence\nsimulations (see Appendix B for the description of the\nsetups) that con\frm that long plasmoid-unstable cur-\nrent sheets are Kelvin-Helmholtz-stable both in 2D and\n3D simulations.\n5.CONCLUSIONS\nIn this Letter we present the \frst 2D and 3D numer-\nical SRRMHD simulations of highly magnetized decay-\ning turbulence. We calculate statistical properties of\nthe turbulence, by analyzing a quasi steady state at two\nAlfv\u0013 en-crossing times of the simulation box. We show\nthat the spectrum of magnetic energy in both cases is\nclose to Boldyrev's spectrum, k\u00003=2\n?, and the v\u0000Bdy-\nnamic alignment angle follows an l1=4dependence. De-\nspite the dynamic alignment angle of vandB\felds\nin 2D is perfectly following Boldyrev's prediction, its\nformation cannot be explained by the uncertainty prin-\nciple originally employed by Boldyrev (2006). On the\nother hand, intermittent structures are vastly present in\nthe simulations, favoring the theory of mutual shear-\ning of Elsasser \felds by Chandran et al. (2015): an\nin-depth analysis of this approach is presented in Ap-\npendix C. We demonstrate that long-lived intermittent\ncurrent sheets form dynamically throughout the evolu-\ntion. These sheets are plasmoid unstable and KH-stable.\nThey occupy a very small fraction of the numerical do-10 Chernoglazov, A. et al.\nFigure 4. 3D volume rendering of the current density in a representative long current sheet in simulation 3D[a] at t= 1. Color\nshows the amplitude of the current density, and thick black lines show magnetic \feld lines near plasmoids. a) Structure of a\nsheet. The red-blue line presents the slice across the current sheet shown in Figure 6, di\u000berent colors of the line represent the\ndi\u000berent sides of the current sheet. b) Zoom into the structure of the plasmoid.\nmain but provide a signi\fcant fraction of the total mag-\nnetic \feld dissipation.\nIn our simulations we only employ explicit resistivity\nwhile viscosity is dictated by the \fnite grid resolution.\nWe expect that the magnetic energy dominates the ki-\nnetic energy at all scales, and dissipation is governed\nby resistivity. It will be useful to perform simulations\nwith explicit viscosity and \fxed magnetic Prandtl num-\nber Pr min the future studies, and to consider the trans-\nrelativistic regime, \u001b\u00181. These studies can be applied\nto turbulence in the accretion disk-jet boundary with\nmoderate magnetization (Ripperda et al. 2020).\nIn order to study the properties of intermittent cur-\nrent sheets in a statistical steady state, it is important to\nstudy driven turbulence in highly magnetized plasmas\n\u001b\u001d1. High magnetization leads to e\u000ecient heating\nof the plasma due to the dissipation of magnetic energy\nand a signi\fcant drop of \u001b. To mediate the e\u000bect of run-\naway heating, radiative cooling of the plasma should be\nincorporated in the simulations (Zhdankin et al. 2021).\nThe limitation of computational resources does not\nallow to reach numerical resolutions signi\fcantly above\n100003in the nearby future. This is too low to reach\nalignment angles substantially below \u0012\u00180:1 at the\nsmallest scales. On the other hand, the intriguing sim-\nilarity of statistical properties of 2D and 3D turbu-\nlence in our simulations makes it interesting to per-\nform even higher resolution simulations of 2D turbu-lence. The most signi\fcant milestone will be a resolution\nof\u0018(108)2which allows progressively elongated eddies\nto reach an alignment angle \u0012\u00180:01 corresponding to\nthe plasmoid instability of these eddies. The steepen-\ning of the turbulence spectrum due to the onset of the\nplasmoid instability in intermittent current sheets (or\ndue to the linear tearing instability in elongated eddies,\nBoldyrev & Loureiro 2017) can be measured reliably at\nresolutions of\u0018(106)2, realistically attainable in the\nnearby future, in particular with the AMR criterion we\npropose here.\n6.ACKNOWLEDGEMENTS\nWe acknowledge useful discussions with Lev Arza-\nmasskiy, Amitava Bhattacharjee, Benjamin Chandran,\nLuca Comisso, Mikhail Medvedev, Joonas N attil a, Ja-\nson TenBarge, James Stone, and help in navigating\nthrough 3D-visualization by Hayk Hakobyan. The au-\nthors acknowledge insightful comments by the anony-\nmous referee which helped to signi\fcantly improve the\nmanuscript. The computational resources and services\nused in this work were provided by facilities supported\nby the Scienti\fc Computing Core at the Flatiron Insti-\ntute, a division of the Simons Foundation; and by the\nVSC (Flemish Supercomputer Center), funded by the\nResearch Foundation Flanders (FWO) and the Flem-\nish Government { department EWI. A.C. gratefully ac-\nknowledges support and hospitality from the SimonsRelativistic Turbulence 11\nFigure 5. Analysis of numerical convergence of 2D simualations with numerical resolution. a) An example of the structure of\nthe re\fned grid close to a plasmoid-unstable current sheet. b) Coverage of the numerical domain by blocks of di\u000berent re\fnement\nlevels during the simulations. c) Resolution study for 2D simulations with uniform 163842;327682;655362and AMR 655362,\nwhich shows the comparison of magnetic energy spectra. Vertical dashed lines show the end of the inertial range for simulations\nwith 163842(left line) and 327682=655362(right line) grid points.\nFoundation through the pre-doctoral program at the\nCenter for Computational Astrophysics, Flatiron Insti-\ntute. B.R. is supported by a Joint Princeton/Fellowship\nPostdoctoral Fellowship. A.P. acknowledges support bythe National Science Foundation under Grant No. AST-\n1910248.\nSoftware :BHAC , (Porth et al. 2017; Olivares et al.\n2019; Ripperda et al. 2019), Python (Oliphant 2007;\nMillman & Aivazis 2011), NumPy (van der Walt et al.\n2011), Matplotlib (Hunter 2007), Mayavi (Ramachan-\ndran & Varoquaux 2011)\nAPPENDIX\nA.ADAPTIVE MESH REFINEMENT AND CONVERGENCE STUDY\nFor the fully resolved and converged 2D simulations we present the adaptive mesh re\fnement (AMR) criterion\nwe designed to accelerate the simulations and to simultaneously capture the main properties of the turbulence and\ndissipative structures. The main principle is that the largest eddies are resolved by many cells at low resolution. To\ncapture the physics at smallest scales, one needs to re\fne the resolution in the smallest eddies, capturing both velocity\nand magnetic \feld gradients. We de\fne the characteristic sizes of the eddies as\nlv=j(r\u0002v)zjq\nv2x+v2y; lB=j(r\u0002B)zjq\nB2x+B2y: (A1)\nThe re\fnement routine is called if the size of anyof the two quantities is less than a threshold value: lv;B<\u000b\u0001x\nat the point. Coe\u000ecient \u000bis chosen to be such that the threshold scale is larger than the numerical resistive scale. In\nthe simulations we use \u000b= 8, which is larger than the numerical resistive scale in simulations 2D[a] and 2D[c], and\n\u0001xis the grid spacing at a given resolution. Coarsening of the grid in the numerical domain is only allowed if both\nquantities at a given grid point are larger than the threshold. Since the electric current density is roughly given by the\ngradient of the magnetic \feld, j\u0018r\u0002 B, regions of the large electric current density (indicating current sheets) are\nautomatically re\fned. Since the inverse cascade is very pronounced in 2D simulations, AMR shows very high e\u000eciency\nat early times, when the spectrum is being formed, and at later times, when small eddies merge in larger ones (see\nFigure 5a for \u0011= 10\u00006). Since the resistive scale is much larger for \u0011= 10\u00005, the coverage by the highest resolution\nlevel does not exceed 15% in this case.\nThe threshold value is tested for a resolution of 327682grid points by comparing spectra of the magnetic \feld energy\nfor uniform grid and AMR-enabled runs (where the e\u000bective resolution for the AMR runs indicates the total resolution12 Chernoglazov, A. et al.\nFigure 6. Analysis of the KH-stability of current sheets in turbulence. a) Slices across the current sheet in simulation 2D[a],\nused to extract the shear \row parameters. Arrows show directions of Bjj;B?, orvjj;v?. b) Behavior of the reconnecting in-plane\nand out-of-plane magnetic \feld components and parallel velocity along the slice shown with a solid line in panel a. c) Similar\nquantities along the slice across the current sheet in 3D simulation 3D[a].\nif the whole domain were re\fned to the highest AMR level) at the same moment in time (see Figure 5b). Interestingly,\nthe most accurate spectra are produced by simulations where the re\fnement algorithm is called only every 50 \u0000100\ntime-steps, most likely due to less numerical noise being generated during the re\fning and coarsening of the grid and\nre-interpolation. The frequency of the re\fnement calls is de\fned to ensure that the \fnest structures are always located\ninside the re\fned grid block during their motion in the domain. For the bulk velocity of the \ruid u\u00190:1c, and CFL\nnumber 0:4, an element of the \ruid travels about 20 cells between two calls of the re\fnement, while the minimum size\nof a re\fned grid is 322cells.\nThis AMR strategy does not seem to be e\u000bective in 3D simulations due to the overall low grid resolution, compared\nto the extreme resolutions employed in 2D: AMR automatically chooses the resolution needed to resolve all the features\nin the block. Since the size of all features in the \row is rather de\fned by the numerical resolution than by an explicit\nresistivity, AMR re\fnes the whole domain up to the highest available resolution. It is impossible to \fnd a reasonable\nthreshold\u000bfor the 2D counterpart of the highest resolution 3D run with 32003grid points: any chosen \u000beither\ntruncates the inertial range of the spectrum or re\fnes the whole domain shortly after the start of the simulation.\nFigure 5c demonstrates that a base resolution of &320002grid points is needed to fully resolve the resistive scale for\n\u0011= 10\u00006and keep the inertial range of the turbulence una\u000bected by the resolution. In order to demonstrate this, we\ncompare spectra for resolutions with 163842;327682;655362points.\nB.KELVIN-HELMHOLTZ STABILITY OF CURRENT SHEETS\nIn order to study the stability of the magnetized shear \row in our plasmoid-unstable current sheets in 2D, we\ncalculate the value of the in-plane reconnecting magnetic \feld component Bjjand the out-of-plane component Bzas\nwell as velocity \feld components vjj;vzfor each of the three slices shown in Figure 6a by green lines. For all these slices\nwe \fndjB?j\u001cjBjjjandjv?j\u001cjvjjj, wherejjrepresents the direction parallel to the current sheet at a given point in\nthe slice (the arrows in Figure 6a indicate parallel and perpendicular directions, and the z-direction is out-of-plane).\nWe show the typical behavior of these parameters in Figure 6b, which implies that the \row satis\fes the non-relativistic\nstability criterion j\u000evj<j\u000eBj.\nFor each slice across the current sheet, we run a local simulation of the shear \row, with one \row having parameters\n(\u001a;Bjj;Bz;vjj;vz) given by the upstream of the current sheet, and its counter-\row having parameters from the interior\nof the current sheet, particularly, Bjj= 0. Zero parallel magnetic \feld in one of the two interacting \rows prohibits\nreconnection at their interface in these experiments, but allows to study the KHI. Plasma pressure is adjusted to\nmaintain the force balance across the interface of the \rows.\nWe run simulations with a resolution of 40962grid points for 20 light-crossing times along the sheet, which exceeds\nthe life time of intermittent current sheets in the full 2D turbulence simulation. For the intermediate slice (shown\nby the solid line in Figure 6a), we run an AMR-enabled simulation with an e\u000bective resolution of 327682grid points,\nwhich resolves all the scales up to the resistive scale for a resistivity \u0011= 10\u00006(see Appendix A for the resolution study).Relativistic Turbulence 13\nFigure 7. Intermittency of the turbulence: The two top rows show the conditional probability distribution function (PDF) of\nthe dynamic alignment angle for a \fxed amplitude of the Elsasser \feld \u000ez+=\u000ez+\nland at a \fxed scale l(the top row, l= 0:1,\ncorresponds to the low values of k?, approximately at the beginning of the intertial range; the middle row, l= 0:01,- corresponds\nto the high- k?end of the inertial range), measured at t= 2. Here, \u000ez+\nl= exphln\u000ez+jliis the geometrical mean of \u000ez+at a\n\fxed scalel. The bottom row shows the PDF of \u000ez+=\u000ez+\nlfor 30 logarithmically distributed scales between l= 0:9L(red line)\ntol= 0:01L(blue line).\nThis \fnest grid covers the whole interface of the \rows at any moment of the simulation. In all of these experiments\nwe do not observe any instability growth. This implies that the in-plane magnetic \feld in the upstream of the current\nsheet is capable of preserving KH-stability in both the upstream and downstream of the current sheet.\nIn order to explore the KH-stability of the 3D current sheet, we select slices between the two plasmoids (red-blue\nline in Figure 4), and perform test simulations in a 3D setup with a geometry similar to the one described above for\n2D simulations. This slice is also normal to the surface of the current sheet at the point of their intersection. We test\nthe sheet's stability by running a 3D simulation with a resolution of 10243grid points for 20 light-crossing times along\nthe shear interface, with parameters corresponding to the slice shown in Figure 4c.14 Chernoglazov, A. et al.\nC.AMPLITUDE-DEPENDENT INTERMITTENCY OF THE TURBULENCE.\nSince the lack of scale invariance is the most prominent sign of intermittency, we focus on the scale dependence\nof the Elsasser \feld increments, \u000ez+. The results presented in this Appendix are similar for \u000ez\u0000as we expect the\nturbulence to be balanced, \u000ez+\u0018\u000ez\u0000. As shown by Mallet et al. (2015), the scale invariance can be characterized by\nthe similarity of the conditional probability distribution functions (PDFs) P(\u000ez+jl) of\u000ez+at di\u000berent scales l. We\ncomputed these PDFs for 2D (run 2D[a]) and 3D (runs 3D[a] and 3D[d]) simulations. To measure the PDF, we use\na set of 30 logarithmically spaced scales fligfrom 0:01Lto 0:9L. The smallest scale l1, corresponding to k?\u0019100,\nlies deeply in the inertial range of the energy spectrum, while the largest scale of the considered set, l30, is located\nat the outer scale of the turbulence. As the bottom row of Figure 7 shows, the smaller-scale eddies (darker lines)\nhave higher probability to reach large normalized amplitudes of \u000ez+. The \rattening of the PDFs at smaller scales can\nbe attributed to the sheet-like structures emerging at these scales. For 2D and weak guide \feld 3D simulations, the\npresence of long current sheets can also explain a \ratter tail of the PDF at the larger scales, while the PDF for the\nstrong guide \feld 3D case has an abrupt cuto\u000b at the high \u000ez+for the same eddy sizes. We normalized \u000ez+(l) by a\ngeometrical mean of \u000ez+= exphln\u000ez+jliat a given scale l, as it is less sensitive to outliers than an arithmetical mean.\nThe intermittent, scale-dependent, nature of the dynamic alignment can also be shown by measuring the PDF of\nthe dynamic alignment angle at given scales, as considered by Dong et al. (2018). We are, however, also interested in\ntesting the assumption of Chandran et al. (2015) that large \u000ez+rotates\u000ez\u0000into alignment, while balanced collisions\n\u000ez+\u0018\u000ez\u0000\u0018\u000ez\u0006are not aligned. This anti-correlation of the alignment angle with the amplitude of \u000ez+contradicts\nthe intuitive explanation of the dynamic alignment by an uncertainty principle. To test this, we measure the conditional\nPDF of the dynamic alignment angle P(\u0012jl;\u000ez+) for a given scale land the amplitude of the Elsasser \feld \u000ez+=\u000ez+.\nThe middle and top rows of Figure 7 show that the the prediction is matched perfectly for strong guide \feld 3D\nturbulence: the larger \u000ez+, the more aligned \u000ez+and\u000ez\u0000are. For 2D and weak guide \feld 3D turbulence there is\na deviation from this prediction at the outer scale: while the statement holds for intermediate amplitudes of \u000ez+, at\nhigh amplitudes eddies become uncorrelated again. The most powerful increments \u000ez+are associated to current sheets\nand plasmoids, and one can expect that circular plasmoids have an alignment angle (the ratio of two length scales of\nthe eddy)\u0012\u00181 that can explain decorrelation of the alignment angle at high \u000ez+=\u000ez+.\nREFERENCES\nBegelman, M. C. 1998, ApJ, 493, 291.\nhttps://arxiv.org/abs/astro-ph/9708142\nBeloborodov, A. M. 2017, ApJ, 850, 141,\ndoi: 10.3847/1538-4357/aa8f4f\nBeloborodov, A. M. 2020, Emission of magnetar bursts and\nprecursors of neutron star mergers.\nhttps://arxiv.org/abs/2011.07310\nBhattacharjee, A., Huang, Y.-M., Yang, H., & Rogers, B.\n2009, Physics of Plasmas, 16, 112102\nBoldyrev, S. 2005, ApJL, 626, L37.\nhttps://arxiv.org/abs/astro-ph/0503053\n|. 2006, PhRvL, 96, 115002.\nhttps://arxiv.org/abs/astro-ph/0511290\nBoldyrev, S., & Loureiro, N. F. 2017, ApJ, 844, 125.\nhttps://arxiv.org/abs/1706.07139\nCerutti, B., Philippov, A., Parfrey, K., & Spitkovsky, A.\n2015, MNRAS, 448, 606, doi: 10.1093/mnras/stv042\nChandran, B. D. G., Foucart, F., & Tchekhovskoy, A. 2018,\nJournal of Plasma Physics, 84, 905840310.\nhttps://arxiv.org/abs/1707.06216\nChandran, B. D. G., Schekochihin, A. A., & Mallet, A.\n2015, ApJ, 807, 39. https://arxiv.org/abs/1403.6354Cho, J., & Vishniac, E. T. 2000, ApJ, 539, 273,\ndoi: 10.1086/309213\nComisso, L., & Sironi, L. 2018, PhRvL, 121, 255101.\nhttps://arxiv.org/abs/1809.01168\nDel Zanna, L., Papini, E., Landi, S., Bugli, M., &\nBucciantini, N. 2016, MNRAS, 460, 3753,\ndoi: 10.1093/mnras/stw1242\nDong, C., Wang, L., Huang, Y.-M., Comisso, L., &\nBhattacharjee, A. 2018, PhRvL, 121, 165101.\nhttps://arxiv.org/abs/1804.07361\nElsasser, W. M. 1950, Physical Review, 79, 183,\ndoi: 10.1103/PhysRev.79.183\nFrisch, U. 1995, Turbulence: The Legacy of A. N.\nKolmogorov (Cambridge University Press),\ndoi: 10.1017/CBO9781139170666\nGoldreich, P., & Sridhar, S. 1995, ApJ, 438, 763\n|. 1997, ApJ, 485, 680.\nhttps://arxiv.org/abs/astro-ph/9612243\nGuo, F., Li, H., Daughton, W., & Liu, Y.-H. 2014, PhRvL,\n113\nHosking, D. N., & Schekochihin, A. A. 2020, arXiv e-prints,\narXiv:2012.01393. https://arxiv.org/abs/2012.01393Relativistic Turbulence 15\nHunter, J. D. 2007, Computing in Science Engineering, 9,\n90, doi: 10.1109/MCSE.2007.55\nIroshnikov, P. S. 1963, AZh, 40, 742\nKraichnan, R. H. 1965, Physics of Fluids, 8, 1385\nLoureiro, N. F., Schekochihin, A. A., & Cowley, S. C. 2007,\nPhysics of Plasmas, 14, 100703.\nhttps://arxiv.org/abs/astro-ph/0703631\nLoureiro, N. F., Schekochihin, A. A., & Uzdensky, D. A.\n2013, PhRvE, 87, 013102.\nhttps://arxiv.org/abs/1208.0966\nLoureiro, N. F., Uzdensky, D. A., Schekochihin, A. A.,\nCowley, S. C., & Yousef, T. A. 2009, MNRAS, 399, L146,\ndoi: 10.1111/j.1745-3933.2009.00742.x\nLyubarsky, Y. E. 1992, Soviet Astronomy Letters, 18, 356\n|. 2005, MNRAS, 358, 113,\ndoi: 10.1111/j.1365-2966.2005.08767.x\nMahlmann, J. F., Levinson, A., & Aloy, M. A. 2020,\nMNRAS, 494, 4203. https://arxiv.org/abs/2001.03171\nMallet, A., Schekochihin, A. A., & Chandran, B. D. G.\n2015, MNRAS, 449, L77, doi: 10.1093/mnrasl/slv021\n|. 2017, MNRAS, 468, 4862.\nhttps://arxiv.org/abs/1612.07604\nMallet, A., Schekochihin, A. A., Chandran, B. D. G., et al.\n2016, MNRAS, 459, 2130, doi: 10.1093/mnras/stw802\nMason, J., Cattaneo, F., & Boldyrev, S. 2006, PhRvL, 97,\n255002. https://arxiv.org/abs/astro-ph/0602382\nMillman, K., & Aivazis, M. 2011, Computing in Science &\nEngineering, 13, 9 , doi: 10.1109/MCSE.2011.36\nN attil a, J., & Beloborodov, A. M. 2020, arXiv e-prints,\narXiv:2012.03043. https://arxiv.org/abs/2012.03043\nOliphant, T. E. 2007, Computing in Science Engineering, 9,\n10, doi: 10.1109/MCSE.2007.58\nOlivares, H., Porth, O., Davelaar, J., et al. 2019, A&A, 629,\nA61. https://arxiv.org/abs/1906.10795\nOsmanov, Z., Mignone, A., Massaglia, S., Bodo, G., &\nFerrari, A. 2008, A&A, 490, 493.\nhttps://arxiv.org/abs/0802.2607\nParker, E. N. 1957, J. Geophys. Res., 62, 509,\ndoi: 10.1029/JZ062i004p00509\nPerez, J. C., Mason, J., Boldyrev, S., & Cattaneo, F. 2012,\nPhysical Review X, 2, 041005.\nhttps://arxiv.org/abs/1209.2011Porth, O., Olivares, H., Mizuno, Y., et al. 2017,\nComputational Astrophysics and Cosmology, 4, 1.\nhttps://arxiv.org/abs/1611.09720\nRamachandran, P., & Varoquaux, G. 2011, Computing in\nScience & Engineering, 13, 40 ,\ndoi: 10.1109/MCSE.2011.35\nRipperda, B., Bacchini, F., & Philippov, A. A. 2020, ApJ,\n900, 100. https://arxiv.org/abs/2003.04330\nRipperda, B., Liska, M., Chatterjee, K., et al. 2021, arXiv\ne-prints, arXiv:2109.15115.\nhttps://arxiv.org/abs/2109.15115\nRipperda, B., Porth, O., Sironi, L., & Keppens, R. 2019,\nMonthly Notices of the Royal Astronomical Society, 485,\n299{314. http://dx.doi.org/10.1093/mnras/stz387\nRipperda, B., Bacchini, F., Porth, O., et al. 2019, ApJS,\n244, 10. https://arxiv.org/abs/1907.07197\nSironi, L., & Spitkovsky, A. 2014, ApJL, 783, L21.\nhttps://arxiv.org/abs/1401.5471\nSweet, P. A. 1958, in Electromagnetic Phenomena in\nCosmical Physics, ed. B. Lehnert, Vol. 6, 123\nTakamoto, M., & Lazarian, A. 2017, MNRAS, 472, 4542,\ndoi: 10.1093/mnras/stx2292\nTenBarge, J. M., Ripperda, B., Chernoglazov, A., et al.\n2021, arXiv e-prints, arXiv:2105.01146.\nhttps://arxiv.org/abs/2105.01146\nUzdensky, D. A., Loureiro, N. F., & Schekochihin, A. A.\n2010, PhRvL, 105, 235002,\ndoi: 10.1103/PhysRevLett.105.235002\nvan der Walt, S., Colbert, S. C., & Varoquaux, G. 2011,\nComputing in Science and Engineering, 13, 22,\ndoi: 10.1109/MCSE.2011.37\nWerner, G. R., Uzdensky, D. A., Cerutti, B., Nalewajko,\nK., & Begelman, M. C. 2015, ApJ, 816, L8\nZhdankin, V., Uzdensky, D. A., & Kunz, M. W. 2021, ApJ,\n908, 71. https://arxiv.org/abs/2007.12050\nZhdankin, V., Uzdensky, D. A., Perez, J. C., & Boldyrev,\nS. 2013, ApJ, 771, 124. https://arxiv.org/abs/1302.1460\nZhdankin, V., Werner, G. R., Uzdensky, D. A., &\nBegelman, M. C. 2017, PhRvL, 118, 055103,\ndoi: 10.1103/PhysRevLett.118.055103\nZhou, M., Loureiro, N. F., & Uzdensky, D. A. 2020,\nJournal of Plasma Physics, 86, 535860401.\nhttps://arxiv.org/abs/2001.07291\nZrake, J., & MacFadyen, A. I. 2012, ApJ, 744, 32.\nhttps://arxiv.org/abs/1108.1991"    },    {        "title": "1205.5679v1.Dynamics_of_the_Chiral_Magnetic_Effect_in_a_weak_magnetic_field.pdf",        "content": "Dynamics of the Chiral Magnetic E \u000bect in a weak magnetic field\nHarmen J. Warringa\u0003\nFrankfurt Institute for Advanced Studies (FIAS), Frankfurt am Main, Germany\nand Institut f¨ ur Theoretische Physik, Goethe-Universit¨ at Frankfurt am Main, Germany\n(Dated: September 25, 2018)\nWe investigate the real-time dynamics of the chiral magnetic e \u000bect in quantum electrodynamics (QED)\nand quantum chromodynamics (QCD). We consider a field configuration of parallel (chromo)electric and\n(chromo)magnetic fields with a weak perpendicular electromagnetic magnetic field. The chiral magnetic e \u000bect\ninduces an electromagnetic current along this perpendicular magnetic field, which we will compute using linear\nresponse theory. We discuss specific results for a homogeneous sudden switch-on and a pulsed (chromo)electric\nfield in a static and homogeneous (chromo)magnetic field. Our methodology can be easily extended to more\ngeneral situations. The results are useful for investigating the chiral magnetic e \u000bect with heavy ion collisions and\nwith lasers that create strong electromagnetic fields. As a side result we obtain the rate of chirality production\nfor massive fermions in parallel electric and magnetic fields that are static and homogeneous.\nI. INTRODUCTION\nIn quantum electrodynamics (QED) and quantum chromo-\ndynamics (QCD) certain gauge field configurations will in-\nduce chirality as follows from the axial Ward identity [1, 2].\nThese gauge field configurations have a net topological charge\ndensity, which means that the contraction of the field strength\ntensor with its dual is nonvanishing. An example of such\ngauge field configuration is one in which the (chromo)electric\nand (chromo)magnetic fields are parallel. For massless parti-\ncles, chirality is the asymmetry between the number of parti-\ncles plus antiparticles with right-handed helicity and the num-\nber of particles plus antiparticles with left-handed helicity.\nHere right-handed helicity means that spin and momentum\nare parallel, whereas left-handed helicity means they are anti-\nparallel. If a system of charged fermions possesses a net chi-\nrality, then an applied (electromagnetic) magnetic field will\ninduce an electromagnetic current in the direction of the mag-\nnetic field [3]. This is called the chiral magnetic e \u000bect. In\nmagnetic fields that are large enough to fully polarize the\nfermions, one can easily convince oneself that the magni-\ntude of this current is equal to the chirality times the absolute\ncharge of the fermion [4].\nTo improve the understanding of the chiral magnetic e \u000bect\nit is important to obtain the magnitude of the induced elec-\ntromagnetic current in di \u000berent situations. In many studies of\nthe chiral magnetic e \u000bect the chirality is introduced by hand\nthrough a chiral chemical potential \u00165[3, 5, 6]. However, in\norder to get a full understanding of the chiral magnetic e \u000bect,\none should include the dynamics that leads to a net chiral-\nity. Such dynamical studies have been performed in Euclidean\nspace-time using lattice QCD [7] and with analytic methods,\nboth in Euclidean [8, 9] and Minkowski [10] space-time.\nIn this article we will study the real-time dynamics of the\nchiral magnetic e \u000bect with an analytic method. We will con-\nsider a specific gauge field configuration of parallel homoge-\nneous (chromo)electric and (chromo)magnetic fields that are\npointing in the z-direction. These parallel fields are the source\n\u0003Electronic address: warringa@th.physik.uni-frankfurt.de\nzy\njy=?\nEzBy\n/BulletBzFIG. 1: Field configuration in which the chiral magnetic e \u000bect is\nstudied in this article. The electric field along the z-direction ( Ez) is\nhomogeneous and can have arbitrary time dependence. The magnetic\nfield along the z-direction is static and homogeneous. The goal is\nto compute the current density jyalong the magnetic field in the y-\ndirection ( By).\nof the chirality. The (chromo)magnetic field is assumed to be\nconstant in time, but the (chromo)electric field can have any\ntime dependence. Perpendicular to these fields we will con-\nsider an electromagnetic magnetic field with arbitrary time de-\npendence that points in the y-direction. We have sketched the\nsituation in Fig. 1. The goal of this article is to obtain the\ncurrent density in the y-direction, which is due to the chiral\nmagnetic e \u000bect.\nThe chiral magnetic e \u000bect will take place unsuppressed if\nthe fields have a magnitude that is at least of order of the\nfermion mass squared. In the near future, one hopes to cre-\nate such enormous electromagnetic fields with high intensity\nlasers (see e.g. [11]). If one could engineer a field configura-\ntion like the one of Fig. 1 with these lasers, one could falsify\nthe chiral magnetic e \u000bect in QED in a controlled environment.\nIn heavy ion collisions a similar configuration built out of flux-\ntubes of parallel chromoelectric and chromomagnetic fields is\ninduced [12]. Perpendicular to these color fields a large elec-\ntromagnetic magnetic field is generated through the currents\ncreated by the colliding charged ions [4, 13].\nIn QCD the probability of generating gauge field configu-\nrations with positive and negative topological charge is equal\n(assuming\u0012=0). Hence in heavy ion collisions, the cur-\nrent along the perpendicular field vanishes on average. How-\never, because of fluctuations in the initial state [12], and the\npresence of sphalerons [14] the topological charge fluctuates.\nHence the current along the perpendicular magnetic field willarXiv:1205.5679v1  [hep-th]  25 May 20122\nfluctuate as well. That will lead to di \u000berences in net electric\ncharge between the two sides of the reaction plane (the plane\nin which the beam-axis and the impact parameter lie) on an\nevent-by-event basis [15]. This e \u000bect can be analyzed experi-\nmentally through a charge correlation study [16]. Charge cor-\nrelations have indeed been observed at both the Relativistic\nHeavy Ion Collider (RHIC) [17] and the Large Hadron Col-\nlider (LHC) [18]. However, currently it is not clear if these\ncorrelations are caused by the chiral magnetic e \u000bect or by an-\nother mechanism [19]. A possible observation of the chiral\nmagnetic e \u000bect in QCD can teach us about the relevance of\nconfigurations with topological charge.\nTheoretically, the field configuration displayed in Fig. 1 is\ninteresting, because when all fields are static and homoge-\nneous it is possible to obtain a full analytic expression for\nthe induced current density along the perpendicular magnetic\nfield [10]. The current density behaves exactly according to\nthe qualitative expectations of the chiral magnetic e \u000bect. For\nexample, for a large perpendicular magnetic field the magni-\ntude of the current was found to be equal to the chirality times\nthe absolute value of the charge of the fermion. Furthermore,\nthe current density is suppressed by the fermion mass, and\nvanishes if there is no chirality production.\nThe calculation in Ref. [10] was build upon a nice argument\nbased on Lorentz transformations and the particle production\nrate in parallel, homogeneous and static fields. However, this\nargument cannot be applied to more general situations that are\nuseful to investigate. For example, in heavy ion collisions the\nmagnetic field has a strong time dependence. Furthermore,\nwith high-intensity lasers one will create inhomogeneous and\ntime-dependent fields. And for purely academic reasons it is\nof interest to analyze the real-time dynamics of the chiral mag-\nnetic e \u000bect in a spherical symmetric background gluon field\nthat carries one unit of topological charge, and is a solution of\nthe Yang-Mills equations.\nTo obtain the induced current density along the magnetic\nfield in a general situation one can in principle just solve the\nDirac equation numerically, and construct the current from the\nsolutions. However, this might be quite di \u000ecult in practice. In\norder to make progress, we will start a program to investigate\nthe chiral magnetic e \u000bect using linear response theory. We\nwill expand to first order in the electromagnetic field that is the\nsource of the perpendicular magnetic field. In this way we can\ninvestigate the chiral magnetic e \u000bect for weak perpendicular\nmagnetic fields that have arbitrary space and time dependence.\nIn practical terms we will confirm in this article the results of\nRef. [10] in the weak magnetic field limit. Furthermore we\nwill analyze the chiral magnetic e \u000bect in pulsed electric fields.\nMost of the calculations in this article will be performed\nin QED. From the QED results it is easy to obtain the results\nfor QCD as in Ref. [10] and as explained at the end of the\nintroduction. We hope that our results will inspire the reader\nto perform other studies of the chiral magnetic e \u000bect using\nlinear response theory.\nThis article is organized as follows. In Sec. II we will con-\nstruct the fermion propagator of QED in a background of par-\nallel electric and magnetic fields. To support Ans ¨atze made\nin Ref. [10], we compute the electromagnetic current densityfor massive fermions analytically in Sec. III. In this section\nwe will also obtain an analytic expression for the rate of chi-\nrality production for massive fermions in parallel electric and\nmagnetic fields. In Sec. IV we will then apply linear response\ntheory to obtain the induced current density along a magnetic\nfield that is perpendicular to these background fields. We will\ndiscuss our numerical results in Sec. V and show among other\nthings that we can reproduce the results of Ref. [10] in the\nlimit of weak magnetic fields. The conclusions of this work\ncan be found in Sec. VI.\nSome details of this work can be found in the appendices. In\nAppendix A we give several relations involving the wavefunc-\ntions in a magnetic field. Finally in Appendix B we discuss\nthe wavefunctions in a sudden switch-on electric field.\nThe notational conventions we will use in this article are\nsummarized below. We also explain the relation between the\nQED and the QCD setup below.\nNotation\nWe will use as a metric g\u0016\u0017=diag(1;\u00001;\u00001;\u00001). We write\np=p\u0016=(p0;p) to denote a 4-vector. A component of a four-\nvector will be written with either numbers or with an italic\nsub- or superscript. For example, pz=p3=\u0000pz=\u0000p3.\nWe will use roman subscripts to indicate a component of a 3-\nvector, i.e. p=(px;py;pz)=(px;py;pz). Hence pz=pz=\n\u0000pz. Likewise we write x=(xx;xy;xz) which we most often\nsimplify to x=(x;y;z).\nRelation between QCD and QED setup\nAs in Ref. [10] we consider an Abelianized gluon field con-\nfiguration of the form Aa\n\u0016=A\u0016nain the QCD setup. Here\nnadenotes a direction in the adjoint space. Since the elec-\ntromagnetic current density, the object we wish to compute,\nis gauge invariant, we can perform a gauge transformation to\nrotate nato the a=3 direction. From the covariant deriva-\ntiveD\u0016=@\u0016\u0000igA\u0016t3, it then follows that the red quarks\ne\u000bectively experience an electromagnetic field of magnitude\nqA\u0016=\u0000gA\u0016=2 whereas the green quarks experience an e \u000bec-\ntive electromagnetic field with size qA\u0016=gA\u0016=2. The blue\nquarks are not a \u000bected by the field. Here qis the charge of the\nfermion and gis the QCD coupling constant. The current den-\nsity along the perpendicular magnetic field in the QCD setup\ncan thus be obtained from the QED result, yielding\njy;QCD=2X\nfjy;QED(qEz=gEz=2;qBz=gBz=2;\nq=qf;m=mf):(1)\nHere fis a sum over quark flavors, mis the mass, and\n(Ez)Ezand (Bz)Bzdenote respectively the z-component of the\n(chromo)electric and (chromo)magnetic field.\nThere is an important di \u000berence between the QED and\nQCD setup. In QED the electromagnetic current has also a\ncomponent in the z-direction due to acceleration of charges3\nalong the electric field. In the QCD setup however, this contri-\nbution to the electromagnetic current vanishes due to a cancel-\nlation from the contribution of red and green quarks. The elec-\ntromagnetic current in the QCD setup has thus only a compo-\nnent in the y-direction due to the chiral magnetic e \u000bect.\nII. FERMION PROPAGATOR IN ELECTROMAGNETIC\nBACKGROUND\nThe goal of this section is to obtain a convenient expres-\nsion for the fermion propagator in a background of electric\nand magnetic fields both pointing in the z-direction. We will\nconsider a homogeneous static magnetic field, B=Bzez. The\nelectric field is assumed to be homogeneous as well with an\narbitrary time-dependence, E=E(t)ez. The corresponding\ngauge fields will be chosen as: A0=0 andA=(0;Bzx;Az(t))\nwith lim t!\u00001 Az(t)=0 so that E(t)=\u0000@tAz(t).\nIn this section we will first compute the Dirac spinors in\nthis electromagnetic background. Using these spinors we will\nconstruct the quantum field for fermions, from which we will\nderive the di \u000berent fermion propagators.\nA. Dirac spinors\nThe spinors  (x) can be obtained from the Dirac equation\nwhich reads\n\u0010\ni\r\u0016D\u0016\u0000m\u0011\n (x)=0; (2)\nwhere the covariant derivative D\u0016=@\u0016+iqA\u0016with qthe elec-\ntric charge of the fermion.\nTo solve the Dirac equation in the background given above\nwe will apply the Ritus method [20]. This method amounts to\nmaking the following ansatz for the particle spinors ( +) with\nmomentum pand antiparticle spinors ( \u0000) with momentum \u0000p\n +\nps(x)=F+\np(t)Gp(x)eipyy+ipzzus( ˜p+); (3)\n \u0000\nps(x)=F\u0000\np(t)Gp(x)eipyy+ipzzvs( ˜p\u0000): (4)\nHere the four-vector ˜ p\u0016\n\u0006=(\u0014;0;\u0007\u0015;0), and us( ˜p+) and vs( ˜p\u0000)\ndenote the usual particle and antiparticle spinors of the free\nDirac equation with momentum ˜ p\u0006. The free Dirac spinors\nsatisfy the following equations\n(/˜p+\u0000m)us( ˜p+)=0;(/˜p\u0000+m)vs( ˜p\u0000)=0: (5)\nFurthermore, \u0014and\u0015are constants that depend on p, and will\nbe determined below. Finally, F\u0006\np(t) is a 4\u00024 matrix that has\nto commute with \r1and\r2, and Gp(x) a 4\u00024 matrix that\nshould commute with \r0and\r3.\nBy inserting the Ritus ansatz into the Dirac equation it fol-\nlows that this ansatz is indeed a general solution if the matrices\nF\u0006\np(t) and Gp(x) obey the following two equations\nh\ni\r0@t\u0000\r3(pz\u0000qAz(t))i\nF\u0006\np(t)=\u0006\u0014F\u0006\np(t)\r0; (6)\nh\ni\r1@x\u0000\r2(py\u0000qBzx)i\nGp(x)=\u0015Gp(x)\r2: (7)We will solve these two equations explicitly below.\nBy multiplying Eq. (5) from the left with ( /˜p\u0006\u0006m) it fol-\nlows that ˜ p2\n\u0006=m2, so that\u00142=\u00152+m2. We will normalize\nthe free Dirac spinors as in Ref. [21]. In that convention the\npolarization sums, which are the only non-trivial property of\nthe free spinors required in this article, read\nX\ns=\u0006us(p)¯us(p)=(/p+m); (8)\nX\ns=\u0006vs(p)¯vs(p)=(/p\u0000m): (9)\nIn the remaining part of this subsection, we will determine\nthe matrices F\u0006\np(t) and Gp(x), and the values of \u0014and\u0015ex-\nplicitly. Without loss of generality both F\u0006\np(t) and Gp(x) can\nbe taken diagonal.\n1. Computation of G p(x)\nFrom Eq. (7) it follows that Gp(x) is a function of x \u0000\npy=qBz. Therefore let us write the diagonal elements of Gp(x)\nasga(x\u0000py=qBz) with a=1:::4. By multiplying Eq. (7)\nfrom the left with \r2one finds that\n(@x+qBzx)g1(x)=\u0015g4(x); (10)\n(@x\u0000qBzx)g4(x)=\u0000\u0015g1(x): (11)\nSince the other two components satisfy the same set of equa-\ntions, g2(x) is proportional to g4(x) and g3(x) is proportional to\ng1(x). In order to ensure that Gp(x) commutes with \r0and\r3\nas required, we have to take g2(x)=g4(x) and g1(x)=g3(x).\nBy combining Eqs. (10) and (11) one finds that ga(x) satis-\nfies the following equation\n\u0010\n\u0000@2\nx+q2B2\nzx2\u0000sqB z\u0011\nga(x)=\u00152ga(x); (12)\nwith s=1 for a=1;3 and s=\u00001 for a=2;4. Equation\n(12) is the eigenvalue equation of a harmonic oscillator with\nangular frequency jqBzj. The eigenvalues are given by \u00152=\n(2n+1)jqBzj\u0000sqB zwith quantum number n=0;1;2;:::. The\ncorresponding eigenfunctions \u001en(x) read\n\u001en(x)=1p\n2nn! jqBzj\n\u0019!1=4\nHn(p\njqBzjx) exp \n\u00001\n2jqBzjx2!\n;\n(13)\nhere Hn(z) denotes a Hermite polynomial of degree n. For\nconvenience we define \u001e\u00001(x)=0. The eigenfunctions \u001en(x)\nare normalized as\nZ1\n\u00001dx\u001en(x)\u001em(x)=\u000enm; (14)\nand satisfy the completeness relation\n1X\nn=0\u001en(x)\u001en(x0)=\u000e(x\u0000x0): (15)\nLet us now write \u00152=2kjqBzjwith k=0;1;2;:::. The full\nsolution to Eq. (12) can be found by using that g1(x) and g4(x)4\nmust have the same value of \u00152. This gives for qBz>0,\ng1(x)=c1\u001ek(x) and g4(x)=c4\u001ek\u00001(x). Choosing the nor-\nmalization such that c1=1, it follows from Eq. (10) that\nc4=c1=1. In this way one obtains also the sign of \u0015, giving\n\u0015=sgn(qBz)p\n2jqBzjk. As long as qBz>0 one then finds\nga(x)=\u001ek(x) for a=1;3 and ga(x)=\u001ek\u00001(x) for a=2;4.\nIfqBz<0 the solution is ga(x)=\u001ek\u00001(x) for a=1;3 and\nga(x)=\u001ek(x) for a=2;4.\nThe diagonal matrix G(x) can now be written in the follow-\ning compact notation\nGp(x)=X\ns=\u0006gps(x)Ps\ng (16)\nwhere we introduced gp+(x)=\u001ek(x\u0000py\nqBz) and gp\u0000(x)=\n\u001ek\u00001(x\u0000py\nqBz). The projection operators Ps\ngare given by\nPs\ng=(1+issgn(qBz)\r1\r2)=2.\n2. Computation of F\u0006\np(t)\nLet us write the diagonal elements of the matrix F\u0006\np(t) as\nf\u0006\na(t) with a=1:::4. Multiplying Eq. (6) from the left with\n\r0yields\n\u0002i@t+pz\u0000qAz(t)\u0003f\u0006\n1(t)=\u0006\u0014f\u0006\n3(t); (17)\u0002i@t\u0000pz+qAz(t)\u0003f\u0006\n3(t)=\u0006\u0014f\u0006\n1(t): (18)\nSince the other two components satisfy the same set of equa-\ntions it follows that f\u0006\n2(t)/f\u0006\n3(t) and f\u0006\n4(t)/f\u0006\n1(t). In order\nto ensure that F\u0006\np(t) commutes with \r1and\r2we have to take\nf\u0006\n2(t)=f\u0006\n3(t) and f\u0006\n4(t)=f\u0006\n1(t). We can therefore write\nF\u0006\np(t)=X\ns=\u0006f\u0006\nps(t)Ps\nf; (19)\nwhere f\u0006\np+(t)=f\u0006\n1(t),f\u0006\np\u0000(t)=f\u0006\n3(t) and the projection opera-\ntorsPs\nf=(1+s\r3\r0)=2.\nBy taking the complex conjugate of Eqs. (17) and (18) it\ncan be seen that f\u0000\nps(t) satisfies the same two di \u000berential equa-\ntions as f+\np\u0000s(t)\u0003. Hence both functions are proportional to\neach other. Since it is natural to normalize the particle and\nantiparticle spinors in the same way, we take\nf\u0000\nps(t)=f+\np\u0000s(t)\u0003: (20)\nWe will use this relation throughout to express our final results\nin terms of particle wavefunctions only.\nFrom Eqs. (17) and (18) it also follows that the sum\njf\u0006\np+(t)j2+jf\u0006\np\u0000(t)j2is independent of time. As it turns out it\nwill be convenient to normalize this combination as\njf\u0006\np+(t)j2+jf\u0006\np\u0000(t)j2=2: (21)\nIn the next subsection we will check that the normalizations\nwe have made are consistent, by verifying that the quantum\nfields satisfy the canonical anti-commutation relations.\nIn general the wavefunctions f\u0006\nps(t) can only be obtained\nnumerically. There are only a few cases in which analytic\nsolutions are known. We will now discuss a few of them.For vanishing electromagnetic field, Az(t)=0, combining\nEqs. (17) and (18) gives\nh\n\u0000@2\nt\u0000p2\nzi\nf\u0006\nps(t)=\u00142f\u0006\nps(t); (22)\nwith s=\u0006. The solution of the last equation is a linear combi-\nnation of phase factors exp( \u0006ip0t), where p0=p\n\u00142+p2z. The\ntwo di \u000berent solutions correspond to particles and antiparti-\ncles, hence f\u0006\ns(t)=c\u0006\nsexp(\u0007ip0t) where c\u0006\nsis a normalization\nconstant. From Eq. (17) it follows that the ratio of the normal-\nization constants is c\u0006\n+=c\u0006\n\u0000=\u0014=(p0\u0006pz). Applying Eq. (21)\ngives ( c\u0006\n+)2+(c\u0006\n\u0000)2=2, so that\nf\u0006\nps(t)=rp0\u0007spz\np0exp(\u0007ip0t): (23)\nIf\u0014=0, the two di \u000berential equations for f\u0006\nps(t) decouple.\nIn that case Eqs. (17) and (18) can be integrated straightfor-\nwardly, yielding\nf\u0006\nps(t)\f\f\f\u0014=0=p\n2 exp\"\n\u0007ijpzjt\u0000isZt\n\u00001dt0qAz(t0)#\n\u0012(\u0007spz):\n(24)\nIn a sudden switch-on electric field the wavefunctions f\u0006\nps(t)\nare known analytically [22]. We review the calculation in Ap-\npendix B. It is also possible to obtain an analytic solution in a\npulsed field of the form Ez(t)=Ez=cosh2(t=\u001c) [23].\nB. Quantum field\nThe Dirac field in the electromagnetic background is given\nby\n\t(x)=X\ns=\u0006X\np1p2\u0014ph\nbps +\nps(x)+dy\n\u0000ps \u0000\nps(x)i\n; (25)\nwhere we introducedP\np\u0011P1\nk=0Rdpy\n2\u0019Rdpz\n2\u0019and\u0014p=\u0014. Here\n \u0006\nps(x) are the particle ( +) and antiparticle ( \u0000) spinors in the\nbackground field, which are given explicitly in Eqs. (3) and\n(4). The operators bpsanddpsdenote respectively the annihi-\nlation operators for particles and antiparticles with momentum\npand spin sin a background magnetic field. The creation op-\nerator for antiparticles in Eq. (25) has negative momentum,\nreflecting the fact that in our notation  \u0000\nps(x) denotes an an-\ntiparticle spinor with momentum \u0000p.\nThe creation and annihilation operators satisfy the follow-\ning anti-commutation relations\nfbps;by\np0s0g=fdps;dy\np0s0g\n=(2\u0019)2\u000ekk0\u000e(py\u0000p0\ny)\u000e(pz\u0000p0\nz)\u000ess0:(26)\nAll other anti-commutation relations vanish.\nTo check that all normalization conditions are consistent we\nwill verify that the quantum field given in Eq. (25) satisfies the\ncanonical equal-time anti-commutation relation, which reads\nf\ta(t;x);\ty\nb(t;x0)g=\u000eab\u000e3(x\u0000x0): (27)5\nUsing the explicit expression of the quantum field gives after\nusing the properties of the creation and annihilation operators\nf\t(t;x);\ty(t;x0)g=X\nu;s=\u0006X\np1\n2\u0014p u\nps(t;x) u\nps(t;x0)y:(28)\nInserting the explicit solution for the spinors, and summing\nover spins by applying Eqs. (8) and (9) yields\nf\t(t;x);\ty(t;x0)g=X\nu=\u0006X\np1\n2\u0014peipy(y\u0000y0)+ipz(z\u0000z0)\n\u0002n\nuFu\np(t)\r0Fu\np(t)yh\nmGp(x)Gp(x0)\u0000\u0015Gp(x)\r2Gp(x0)i\n+\u0014pFu\np(t)Fu\np(t)yGp(x)Gp(x0)o\n:(29)\nThe last equation can be simplified by inserting the explicit\nexpression for F\u0006\np(t). In this way we find that\n1\n2X\nu=\u0006Fu\np(t)Fu\np(t)y=114; (30)\nX\nu=\u0006uFu\np(t)\r0Fu\np(t)y=0: (31)\nTo obtain the last two equations, we have used that by com-\nbining Eqs. (20) and (21) it can be shown that\njf+\nps(t)j2+jf\u0000\nps(t)j2=2; (32)\nf+\nps(t)f+\np\u0000s(t)\u0003=f\u0000\nps(t)f\u0000\np\u0000s(t)\u0003: (33)\nInserting Eqs. (30) and (31) into Eq. (29) yields\nf\t(t;x);\ty(t;x0)g=X\npeipy(y\u0000y0)+ipz(z\u0000z0)Gp(x)Gp(x0):(34)\nBy using the explicit expression for Gp(x) and applying the\ncompleteness relation Eq. (15), the canonical anticommuta-\ntion relation, Eq. (27), follows directly. Hence the normaliza-\ntions we have chosen are consistent.\nC. Propagator\nLet us introduce the following definitions for the two-point\ncorrelation functions\nS+\nab(x;x0)\u0011 h0j\ta(x)¯\tb(x0)j0i; (35)\nS\u0000\nab(x;x0)\u0011 h0j¯\tb(x0)\ta(x)j0i: (36)\nHerej0idenotes the in-vacuum, which in this article is the\nvacuum before the electric field has been switched on. The\ndi\u000berent propagators (retarded, advanced, Feynman) can be\nfound by the appropriate linear combinations of the two-point\nfunctions. By applying Eq. (25) it follows that the two-point\ncorrelation functions expressed in terms of Dirac spinors read\nS\u0006\nab(x;x0)=X\ns=\u0006X\np1\n2\u0014ph\n \u0006\nps(x)i\nah\n \u0006\nps(x0)y\r0i\nb: (37)By inserting the explicit expressions for the Dirac spinors, and\nsumming over spins, the two-point correlation functions be-\ncome\nS\u0006(x;x0)=X\np1\n2\u0014peipy(y\u0000y0)+ipz(z\u0000z0)\n\u0002F\u0006\np(t)Gp(x)(/˜p\u0006\u0006m)\r0F\u0006\np(t0)y\r0Gp(x0):(38)\nTo evaluate the current density and related quantities, one\nhas to contract a two-point correlation function with an arbi-\ntrary combination of gamma matrices denoted by \u0000. These\nquantities can be expressed in terms of S\u0006(x;x0) in the fol-\nlowing charge symmetric way,\nh0j¯\t(t;x0)\u0000\t(t;x)j0i=\u00001\n2X\nu=\u0006utr\u0002Su(t;x;t;x0)\u0000\u0003\n+1\n2trh\n\r0\u0000i\n\u000e(x\u0000x0);(39)\nFor the current density j\u0016we have \u0000 = q\r\u0016, for the chirality\nn5we have \u0000 =\r0\r5, and for the pseudoscalar condensate\n\u0000 =i\r5. Only for \u0000 =\r0the second trace in Eq. (39) does not\nvanish.\nIII. CURRENT AND AXIAL ANOMALY IN PARALLEL\nELECTRIC AND MAGNETIC FIELD\nWe will now compute the induced current density, chiral-\nity density and pseudoscalar condensate in the background of\nparallel homogeneous time-dependent electric and static mag-\nnetic fields. To compute these quantities we need to evaluate\nthe trace of S\u0006(x;x0)\u0000as follows from Eq. (39). This trace can\nbe easily performed using a symbolic manipulation program\nsuch as Mathematica. Inserting the explicit forms of F\u0006\np(t) and\nGp(x) gives\ntrh\nS\u0006(x;x0)\r3i\n=\u00001\n2X\npX\nr;s=\u0006eipy(y\u0000y0)+ipz(z\u0000z0)\n\u0002s f\u0006\nps(t)f\u0006\nps(t0)\u0003gpr(x)gpr(x0);(40)\ntrh\nS\u0006(x;x0)\r0\r5i\n=\u00001\n2X\npX\nr;s=\u0006eipy(y\u0000y0)+ipz(z\u0000z0)\n\u0002sgn(qBz)s f\u0006\nps(t)f\u0006\nps(t0)\u0003rgpr(x)gpr(x0);(41)\ntrh\nS\u0006(x;x0)\r5i\n=\u0007X\npX\nr;s=\u0006m\n2\u0014peipy(y\u0000y0)+ipz(z\u0000z0)\n\u0002sgn(qBz)s f\u0006\nps(t)f\u0006\np\u0000s(t0)\u0003rgpr(x)gpr(x0):(42)\nWe have to evaluate these correlators at equal time ( t=t0)\nwith the x and y-components of the direction vectors equal\n(x=x0and y =y0). For reasons we explain below, we will\nkeep z and z0di\u000berent and introduce \u0001\u0011z\u0000z0.6\nIn this situation we can perform the pyintegration using the\northogonality properties of the functions gps(x). This yields\n\u00001\n2X\nu=\u0006utrh\nSu(t;\u0001)\r3i\n=\njqBzj\n4\u00191X\nk=0\u000bkZdpz\n2\u0019eipz\u0001X\ns=\u0006sjf+\nps(t)j2;(43)\n\u00001\n2X\nu=\u0006utrh\nSu(t;\u0001)\r0\r5i\n=\nqBz\n4\u0019Zdpz\n2\u0019eipz\u0001X\ns=\u0006sjf+\nps(t)j2\f\f\f\fk=0;(44)\n\u00001\n2X\nu=\u0006utrh\nSu(t;\u0001)\r5i\n=\nsgn(m)qBz\n4\u0019Zdpz\n2\u0019eipz\u0001X\ns=\u0006s f+\nps(t)f+\np\u0000s(t)\u0003\f\f\f\fk=0:(45)\nwhere\u000bk=1 for k=0 and\u000bk=2 for k>0. We have\nmade use of Eqs. (20) and (21) to express all results in terms\nof particle wavefunctions f+\nps(t).\nThe quantities we will compute can be obtained from\nEqs. (43)-(45) in the limit \u0001!0. Naively putting \u0001 = 0 will\nnot give a well defined result, due to the presence of ultraviolet\ndivergences. Therefore we have to regularize our expressions.\nFor consistency, this regularization should be performed in a\ngauge invariant way.\nA natural way to achieve this is by using the point-split reg-\nularization [24]. Instead of evaluating the two-point functions\nat z=z0one computes them at z \u0000z0\u0011\u0001and integrates them\nover a distribution h(\u0001). This distribution has to be normal-\nized to unity, and should be sharply peaked around \u0001 =0. We\nwill choose h(\u0001)=exp(\u0000\u00012=4\u000f)=(2p\u0019\u000f) and take the limit\n\u000f!0.\nIn order to maintain gauge invariance, the correlators\nS\u0006(t;\u0001) have to be augmented by a gauge link link Ucon-\nnecting zwith z0. This gauge link is for both S+(t;\u0001) and\nS\u0000(t;\u0001) given by\nU(t;\u0001)=exp\"\niqZz\nz0dx\u0016A\u0016(x)#\n=exp\u0002\u0000iqAz(t)\u0001\u0003:(46)\nSummarizing, the full point-split regularization prescription\nreads\ntrr\u0002S\u0006(t)\u0000\u0003=lim\n\u000f!0Z\nd\u0001h(\u0001)U(t;\u0001) tr\u0002S\u0006(t;\u0001)\u0000\u0003;(47)\nhere the subscript rstands for regularized.\nWe will now apply this regularization prescription to\nEqs. (43)-(45) to evaluate the current density, chirality den-\nsity and pseudoscalar condensate. The integral over \u0001can be\nperformed exactly giving\njz(t)=lim\n\u000f!0qjqBzj\n2\u00191X\nk=0\u000bkZdpz\n2\u0019e\u0000\u000f[pz\u0000qAz(t)]2\n\u0002h\njf+\np+(t)j2\u00001i\n;(48)n5(t)=lim\n\u000f!0qBz\n2\u0019Zdpz\n2\u0019e\u0000\u000f[pz\u0000qAz(t)]2\n\u0002h\njf+\np+(t)j2\nk=0\u00001i\n;(49)\n\u0011(t)=lim\n\u000f!0i sgn( m)qBz\n4\u0019Zdpz\n2\u0019e\u0000\u000f[pz\u0000qAz(t)]2\n\u0002X\ns=\u0006s f+\nps(t)f+\np\u0000s(t)\u0003\f\f\f\fk=0;(50)\nwhere we used that from Eq. (21) it follows thatP\nsjf+\nps(t)j2=\n2(jf+\np+(t)j2\u00001). In the absence of a regulator and in the limit\nof vanishing magnetic field, Eq. (48) agrees with the results\nobtained in Ref. [25].\nWe can now verify the axial anomaly relation. In the mass-\nless limit one can show that jf+\np+(t)j2\nk=0\u00001=\u0000sgn(pz). We\ncan now perform the pzintegral in Eq. (49) in the limit \u000f!0\ngiving\nn5(t)=\u0000q2\n2\u00192BzAz(t): (51)\nTaking the derivative with respect to time in the last equation\ngives the axial anomaly relation for massless particles in par-\nallel electric and magnetic fields\ndn5(t)\ndt=q2\n2\u00192BzEz(t): (52)\nIf the fermions are massive the anomaly relation contains an\nadditional term proportional to the pseudoscalar condensate.\nTo obtain the anomaly relation in this case we will perform\nthe time derivative on the chirality given in Eq. (49) explicitly.\nThe limiting procedure \u000f!0 is equivalent to performing\nthe integration over pzin an interval symmetric around pz=\nqAz(t). Therefore\nn5(t)=lim\n\u0003!1qBz\n2\u0019Z\u0003+qAz(t)\n\u0000\u0003+qAz(t)dpz\n2\u0019h\njf+\np+(t)j2\nk=0\u00001i\n: (53)\nThe time derivative of n5(t) contains a part arising from the\nintegration boundaries and a part from the derivative on the\nwave-functions. It follows directly from Eq. (17) that\n@tjfp+(t)j2=i\u0014pX\ns=\u0006s f+\nps(t)f+\np\u0000s(t)\u0003: (54)\nAlso we will use that \u0014pjk=0=jmj. Furthermore, for large pz\none can neglect qAz(t), and from Eq. (23) it can be shown that\nlim pz!1jfp+(t)j2=0 and lim pz!\u00001jfp+(t)j2=2. By using\nEq. (50) and applying the time derivative to the integration\nboundaries it follows that\ndn5(t)\ndt=2m\u0011(t)+q2\n2\u00192BzEz(t): (55)\nwhich is exactly the anomaly relation in the presence of mass.\nLet us now consider a sudden-switch on electric field of the\nform E(t)=Ez\u0012(t). We discuss the functions f\u0006\nps(t) in this7\nfield in Appendix B. We can only compute the induced cur-\nrent density and the chirality density analytically in the large t\nlimit. Applying Eq. (B13) we find that for large t\ndjz(t)\ndt=qjqBzjqEz\n2\u00192e\u0000\u0019m2\njqEzj1X\nk=0\u000bke\u00002\u0019kjqBz\nqEzj\n=qjqBzjqEz\n2\u00192coth(jBz\nEzj\u0019)e\u0000\u0019m2\njqEzj; (56)\ndn5(t)\ndt=q2EzBz\n2\u00192e\u0000\u0019m2\njqEzj: (57)\nSince these results are obtained for large tin a sudden-switch\non electric field, they are exact in a constant electric field.\nThe result Eq. (56) was derived analytically in a di \u000berent\nway in Ref. [29]. It was also proved to be correct numerically\nin Ref. [30]. The result is easy to understand starting from\nthe production rate of fermion antifermion pairs in parallel\nhomogeneous electric and magnetic fields [10]. That rate per\nunit volume equals [22, 31] (see also [32–34])\n\u0000 =q2EzBz\n4\u00192coth Bz\nEz\u0019!\nexp \n\u0000m2\u0019\njqEzj!\n: (58)\nForBz=0 this rate reduces to the pair production rate in a ho-\nmogeneous electric field, that was first obtained by Schwinger\n[1]. The production of pairs gives rise to a homogeneous cur-\nrent density that has to point in the z-direction because of sym-\nmetry reasons. The particles are accelerated continuously by\nthe electric field. Therefore, at some point they will reach (al-\nmost) the speed of light. Hence, every time a pair is created\nthe current will eventually grow by twice the charge of the\nfermion. So therefore the rate of change of the current density\nis given by@tjz=2q\u0000sgn(qEz). Inserting Eq. (58) we see that\nwe exactly recover Eq. (56). One can also use this argument\nin the opposite order, in order to derive the pair production\nrate from the calculation of the current density.\nThe result Eq. (57) generalizes the well-known production\nrate of chirality in parallel electric and magnetic fields for\nmassless fermions to massive fermions. We are unaware of\nan earlier derivation of this result. The mass suppresses the\nproduction of chirality. By combining Eq. (57) with Eq. (55)\nwe find that the pseudoscalar condensate in static, homoge-\nnous and parallel electric and magnetic fields equals\n\u0011(t)=q2EzBz\n4\u00192m\u0012\ne\u0000\u0019m2\njqEzj\u00001\u0013\n: (59)\nIV . LINEAR RESPONSE TO MAGNETIC FIELD\nIn the previous section we have considered a time-\ndependent electric field and a constant magnetic field that\nwere both pointing in the z-direction. To this field config-\nuration we will now add a time-dependent magnetic field in\nthe y-direction, denoted by By(t). This magnetic field will be\naccompanied by a perpendicular electric field as can be seen\nfrom Faraday’s law, r\u0002E=\u0000@B(t)=@t. The additional mag-\nnetic field will induce a current density in the y-direction. Inthis section we will compute this current density to first order\ninBy(t) using linear response theory. In the next section we\nwill use this result to study the chiral magnetic e \u000bect.\nLet us write the full electromagnetic field as A\u0016(x)=\n¯A\u0016(x)+˜A\u0016(x). Here ¯A\u0016(x) denotes the background field, con-\nsisting of the electric and magnetic fields pointing in the z-\ndirection. The field ˜A\u0016(x) denotes the perturbation on this\nbackground, which in this case is the magnetic field in the y-\ndirection with its corresponding perpendicular electric field.\nFrom linear response theory it follows that to first order in\n˜A\u0016(x) the induced current density in the electromagnetic field\nA\u0016(x) equals j\u0016(x)=j\u0016\nA(x)=j\u0016\n¯A(x)+\u000ej\u0016\n¯A;˜A(x) where\n\u000ej\u0016\n¯A;˜A(x)=Z\nd4x0\u0005\u0016\u0017\nR(x;x0)˜A\u0017(x0): (60)\nHere the retarded current-current correlator (or equivalently\nphoton polarization tensor) in the background field ¯A\u0016(x) is\ngiven by \u0005\u0016\u0017\nR(x;x0)= \u0005\u0016\u0017\nC(x;x0)\u0012(t\u0000t0) with\n\u0005\u0016\u0017\nC(x;x0)=\u0000iq2h0jh¯\t(x)\r\u0016\t(x);¯\t(x0)\r\u0017\t(x0)i\nj0i:(61)\nUsing Eqs. (25) and (37) we can express \u0005\u0016\u0017\nC(x;x0) as\n\u0005\u0016\u0017\nC(x;x0)=\u0000iq2X\nu=\u0006utr\u0002\r\u0016Su(x;x0)\r\u0017S\u0000u(x0;x)\u0003;(62)\nwhere S\u0006(x;x0) is the two-point correlation function in the\nbackground field, given explicitly in Eq. (38).\nSince the background electric and magnetic fields are both\npointing in the z-direction, they cannot solely induce a current\nin the y-direction. As a result jy\n¯A(t)=0. Hence the induced\ncurrent density in the y-direction can only arise from the per-\nturbation and is therefore of the following form\njy(t)=Zt\n\u00001dt0H(t;t0)By(t0): (63)\nwhere H(t;t0) can be obtained from \u0005\u0016\u0017\nC(x;x0) as we will ex-\nplain in more detail below. The photon polarization tensor\nin an electric plus magnetic background has been studied by\nother authors before in di \u000berent contexts [26]. Furthermore,\nthe photon polarization tensor in a purely magnetic back-\nground has been studied in detail in several works [27], for\nrecent analyses and applications we refer to [5, 28].\nIn the following subsections we will compute the func-\ntionH(t;t0) in two cases, labeled by A and B. In case A we\nwill take the only non-vanishing component of the perturba-\ntion field to be ˜Az(x)=\u0000By(t)x. In case B the only non-\nvanishing component is chosen as ˜Ax(x)=By(t)z. These\ntwo cases lead to the same magnetic field in the y-direction,\nBy(t), but give rise to di \u000berent perpendicular electric fields.\nThe only non-vanishing component of the additional electric\nfield is in the first case Ez=@tBy(t)x, and in the second case\nEx=\u0000@tBy(t)z. By taking the average of the two cases, one\nobtains a more symmetric electric field, which is circular in\nthe x-z plane.\nIfBy(t) is constant in time, the perpendicular electric field\nvanishes and the two cases are gauge equivalent. However,8\nwe do not know how to implement a constant magnetic field\nexactly in a practical numerical calculation. In the numerical\nevaluation of Eq. (63) one might replace the lower integration\nbound by a finite time ta. But then one e \u000bectively deals with\na sudden switch-on perpendicular field of the following form\nBy(t)=(\n0 if t<ta;\nByift\u0015ta:(64)\nThis sudden switch-on leads to large perpendicular electric\nfields, and as a result the current density in case A and B can\nbe di \u000berent. To instead implement a magnetic field that is\ne\u000bectively constant in time we have to switch it on slowly. In\nour numerical calculations we will take a magnetic field of the\nfollowing form\nBy(t)=8>>><>>>:0 if t<ta;\nBy(t\u0000ta)=(tb\u0000ta) ifta\u0014t\u0014tb;\nBy ift>tb:(65)\nIf we choose tb\u0000talarge enough and tbsmall enough before\nthe important physical e \u000bects happen, the magnetic field is ef-\nfectively constant. In that situation the perpendicular electric\nfields are small and the induced current density in case A and\nB should approximately have the same magnitude. We will\nuse this feature to test our methodology.\nA. ˜Az(x)=\u0000By(t)x\nIn the case that ˜Az(x)=\u0000By(t)x we obtain\nH(t;t0)=Z\nd3x0x0\u000523\nC(x;x0): (66)\nAfter inserting the explicit expression for S\u0006(x;x0), taking the\ntrace, and integration over y, z, p0\nyandp0\nzwe find\nH(t;t0)=\u0000q2X\np1X\nk0=0Z1\n\u00001dx0x0\n\u0002\"\u0015p\n\u0014pVp;p0(t;t0)Wp;p0(x;x0)#\np0y=py;p0z=pz;(67)\nwhere the functions VandWare given by\nVp;p0(t;t0)=ImX\ns=\u0006s f+\nps(t)f+\np0s(t)f+\np\u0000s(t0)\u0003f+\np0s(t0)\u0003;(68)\nWp;p0(x;x0)=X\ns=\u0006gp\u0000s(x)gp0s(x)gps(x0)gp0s(x0):(69)\nThe expression for H(t;t0) can be simplified by performing\nthe integration over x0followed by integration over py. Using\nthe relations from Appendix A it can be shown that\nZ1\n\u00001dpy\n2\u0019Z1\n\u00001dx0x0Wp;p0(x;x0)\f\f\f\fp0y=py\n=p\njqBzj\n2\u0019p\n2p\nk\u00022\u000ek;k0\u0000\u000ek\u00001;k0\u0000\u000ek+1;k0\u0003:(70)Inserting Eq. (70) into Eq. (67) yields\nH(t;t0)=q2\n2\u00191X\nk=1Z1\n\u00001dpz\n2\u0019!k\u0002\n\u00022Vk;k(t;t0)\u0000Vk;k\u00001(t;t0)\u0000Vk;k+1(t;t0)\u0003\npz=p0z;(71)\nwhere\n!k=jqBzjkp\n2jqBzjk+m2: (72)\nTo speed up the numerical computation it is convenient to take\nthe wavefunctions with the same momenta together in the in-\ntegrand. For this reason we rewrite Eq. (71) into\nH(t;t0)=\u0000q2\n2\u0019Z1\n\u00001dpz\n2\u0019!1V1;0(t;t0)+q2\n2\u00191X\nk=1Z1\n\u00001dpz\n2\u0019\u0002\n\u00022!kVk;k(t;t0)\u0000!k+1Vk+1;k(t;t0)\u0000!kVk;k+1(t;t0)\u0003\npz=p0z:(73)\nB. ˜Ax(x)=By(t)z\nIf˜Ax(x)=By(t)z we obtain\nH(t;t0)=\u0000Z\nd3x0z0\u000521\nC(x;x0): (74)\nInserting the explicit expression for the two-point function and\nperforming the trace, we find that\n\u000521\nC(x;x0)=iq2\n2X\np;p0ei(py\u0000p0\ny)(y\u0000y0)+i(pz\u0000p0\nz)(z\u0000z0)\u0002\n(\nWA\np;p0(x;x0)\"\nVA\np;p0(t;t0)+m2\n\u0014p\u0014p0VB\np;p0(t;t0)#\n+\u0015p\u0015p0\n\u0014p\u0014p0WB\np;p0(x;x0)VB\np;p0(t;t0))\n;(75)\nwhere we have defined the following functions\nVA\np;p0(t;t0)=ImX\ns=\u0006f+\nps(t)f+\np0s(t)f+\nps(t0)\u0003f+\np0s(t0)\u0003;(76)\nVB\np;p0(t;t0)=ImX\ns=\u0006f+\nps(t)f+\np0s(t)f+\np\u0000s(t0)\u0003f+\np0\u0000s(t0)\u0003;(77)\nWA\np;p0(x;x0)=X\ns=\u0006sgps(x)gp0\u0000s(x)gps(x0)gp0\u0000s(x0);(78)\nWB\np;p0(x;x0)=X\ns=\u0006sgps(x)gp0\u0000s(x)gp\u0000s(x0)gp0s(x0):(79)\nTo simplify H(t;t0) we can make use of the following rela-\ntion\nZdpz\n2\u0019Zdp0\nz\n2\u0019Z\ndz0z0ei(pz\u0000p0\nz)(z\u0000z0)f(pz;p0\nz)=\n\u0000iZd ¯pz\n2\u0019\"@\n@heizhf( ¯pz+h=2;¯pz\u0000h=2)#\nh=0:(80)9\nFurthermore, using the relations from Appendix A it follows\nthat\nZ1\n\u00001dpy\n2\u0019Z1\n\u00001dx0WA\np;p0(x;x0)\f\f\f\fp0y=py\n=jqBzj\n2\u0019\u0000\u000ek+1;k0\u0000\u000ek\u00001;k0\u0001;(81)\nZ1\n\u00001dpy\n2\u0019Z1\n\u00001dx0WB\np;p0(x;x0)\f\f\f\fp0y=py=0: (82)\nUsing the last three equations above we find\nH(t;t0)=\u0000q2jqBzj\n2\u00191X\nk=0Zd ¯pz\n2\u0019@\n@hh\nVA\nk;k+1(t;t0)\n+m2\n\u0014k\u0014k+1VB\nk;k+1(t;t0)i\n;(83)\nwhere pz=¯pz+h=2 and p0\nz=¯pz\u0000h=2.\nC. Numerical procedure\nWe now will discuss the details of the numerical evalua-\ntion of the current density along the perpendicular magnetic\nfield. Firstly, we obtain the wavefunctions f+\nps(t) numeri-\ncally through solving Eqs. (17) and (18) using a Runge-Kutta\nmethod implemented in Matlab. We make sure that we ob-\ntain f+\nps(t) at equally spaced time steps. The next step is to\nconstruct the integrand of Eqs. (73) and (83) for di \u000berent t0.\nThe derivative in the integrand of Eq. (83) is computed us-\ning finite di \u000berences. We then perform the t0integration in\nEq. (63) using the trapezoidal rule with a lower integration\ncuto\u000b. Thereafter we perform the pzintegral, in an interval\nsymmetric around pz=0, using the trapezoidal rule. The up-\nper and lower cuto \u000bare taken so large that varying them does\nnot change the results. The last step is to perform the sum over\nk. The results are dominated by the small kvalues, therefore\nwe sum over kuntil we reach convergence. Typically one only\nhas to sum over a few values of kto obtain an accurate answer.\nV . RESULTS\nWe will now study the current density generated by the chi-\nral magnetic e \u000bect in parallel electric and magnetic fields with\na perpendicular magnetic field as in Fig. 1. We will compute\nthis current density numerically using the linear response re-\nlation Eq. (63). Since the calculation is based on linear re-\nsponse, our results will be valid for perpendicular magnetic\nfields small compared to the parallel electric and magnetic\nfields.\nThe full electromagnetic current density has two compo-\nnents. Firstly, it has a component along the perpendicular\nmagnetic field due to the chiral magnetic e \u000bect. Secondly, it\nhas a component in the longitudinal direction along the elec-\ntric and magnetic field due to pair production. This compo-\nnent does not vanish for weak perpendicular magnetic fields,and can to first order be computed using Eq. (48). As ex-\nplained in the introduction the longitudinal component van-\nishes in the QCD setup due to a cancellation of the contribu-\ntion of red and green quarks.\nThe electromagnetic current density will generate fields\nthemselves which can modify the dynamics. This backreac-\ntion can be safely neglected as long as the fields induced by the\ncurrents stay small compared to the background fields. Such\nregime can always be reached by considering times short after\nthe switch-on of the background fields. In this article we will\nnot consider this backreaction and leave its study for future\nwork.\nWe will present results for the chiral magnetic e \u000bect in a\nsudden switch-on electric field and a pulsed electric field be-\nlow. We consider a perpendicular magnetic field that is ef-\nfectively constant in time as in Eq. (65). The formalism we\nhave developed in this article allows one to analyze the chiral\nmagnetic e \u000bect in other settings as well.\nWe have performed all numerical calculations in an e \u000bec-\ntively constant magnetic field using both gauge field choice\nA and B. These choices are approximately gauge equivalent,\nand the accuracy of the approximation can be improved by\nswitching on the e \u000bectively constant magnetic field slower.\nThe calculations performed using gauge field choice A and B\nare independent and hence can be used to test our methodol-\nogy. In the numerical calculations in an e \u000bectively constant\nmagnetic field we have found excellent agreement between\nthe results obtained with choice A and B.\nA. Sudden switch-on electric field\nHere we will consider the chiral magnetic e \u000bect in a\nsudden-switch on electric field of the following form E(t)=\nEz\u0012(t). The corresponding gauge field reads Az(t)=\u0000Ez\u0012(t)t.\nThe perpendicular magnetic field is taken to be e \u000bectively\nconstant. In Ref. [10] the current density in the y-direction\nwas computed exactly for t\u001d0. For small Byandt\u001d0 the\nrate of current density generation equals [10]\n@tjy=q2By\n2\u00192jqEzjB2\nz\nB2z+E2zcoth Bz\nEz\u0019!\nexp \n\u0000m2\u0019\njqEzj!\n: (84)\nIn order to cancel the rapid oscillations in the current den-\nsity arising from the sudden-switch on of the electric field\nwe will investigate the running average of the current density,\nhere defined as\nhjy(t)i=Zt+c=p\njqEzj\nt\u0000c=p\njqEzjdt0jy(t0): (85)\nWe display the results of the numerical computation of the\nrunning average of the current density using c=1 in Fig. 2 for\ndi\u000berent values of ˜ m=m=p\njqEzjandBz=Ez=1. The linear\nresponse calculation shows that after the switch-on the current\nquickly grows linear with time. A fermion mass suppresses\nthe production of chirality as can be seen from Eq. (57). This\nexplains why the current density is smaller for particles with\na larger mass.10\n20151050-50.5\n0.4\n0.3\n0.2\n0.1\n0˜m=0.5˜m=0.2˜m=0\nt/radicalbig\n|qEz|/angbracketleftjy/angbracketright\nq2By/radicalbig\n|qEz|\nFIG. 2: Running average of the current density generated by the chi-\nral magnetic e \u000bect in an electric field suddenly switched on at t=0.\nHere Bz=Ez=1.\n10864200.06\n0.04\n0.02\n0˜m=0.5˜m=0.2˜m=0\nqBz/qEz∂t/angbracketleftjy/angbracketright\nq3EzBy\nFIG. 3: Rate of current density generation due to the chiral magnetic\ne\u000bect at late times in a sudden-switch on electric field as a function of\nBz=Ezfor di \u000berent masses. The lines denote the small Bylimit of the\nexact result; points indicate the results of the numerical calculations\nusing linear response.\nIn Fig. 3 we compare our numerical results for the rate of\ncurrent density generation to the small Bylimit of the exact\nresult, given in Eq. (84). It can be seen that we find excellent\nagreement between the results obtained using linear response\nand the small Bylimit of the exact result. Thus our linear\nresponse approach has passed a critical test. It implies that\nthe study of the dynamics of the chiral magnetic e \u000bect using\nlinear response can be performed successfully. Alternatively,\nour results can be seen as an independent verification of the\nresults obtained in Ref. [10].\nIt can be seen that the rate of current density generation in-\ncreases if Bzis enlarged. This is natural, since the amount of\nchirality production is increased. But at the same time enlarg-\ningBzdecreases the degree of polarization of the fermions in\nthe y-direction. The combination of these two e \u000bects results\nin the saturation of the rate of current generation for large Bz.\n50403020100-100.15\n0.1\n0.05\n0Bz/Ez=10Bz/Ez=1\nt/radicalbig\n|qEz|jy\nq2By/radicalbig\n|qEz|FIG. 4: Current density due to the chiral magnetic e \u000bect in a pulsed\nelectric field, as a function of time. Here \u001c=1=p\njqEzjand ˜m=0.\n50403020100-100.06\n0.04\n0.02\n0Bz/Ez=10Bz/Ez=1\nt/radicalbig\n|qEz|jy\nq2By/radicalbig\n|qEz|\nFIG. 5: Same as in Fig. 4 but now for ˜ m=0:5.\nB. Pulsed electric field\nWe will now study the chiral magnetic e \u000bect in a pulsed\nelectric field that has the form Ez(t)=Ez=cosh2(t=\u001c). The\ncorresponding gauge field reads Az(t)=\u0000Ez[1+tanh( t=\u001c)]\u001c.\nThe perpendicular magnetic field is taken to be e \u000bectively\nconstant.\nIn Figs. 4 and 5 we display the current density generated\nby the chiral magnetic e \u000bect as a function of time for \u001c=\n1=p\njqEzjand respectively ˜ m=m=p\njqEzj=0 and ˜ m=0:5.\nIt can be seen that the current density rises quickly around\nt=0. This is because only then an electric field is present\nso that chirality will be produced. For large tthe current does\nnot longer grow because there is no production of chirality\nanymore. For large values of Bz=Ezwe find that the current\ndensity becomes approximately constant in time if ˜ m=0. If\nthe fermions are massive the current density exhibits a slowly\ndamped sinusoidal oscillation for large values of Bz=Ez. The\nmass also suppresses the magnitude of the current.\nWe observe that for smaller values of Bz=Ezthe current11\n5432100.6\n0.4\n0.2\n0˜m=0.5˜m=0.2˜m=0\nτ/radicalbig\n|qEz|/angbracketleftjy/angbracketright\nq2By/radicalbig\n|qEz|\nFIG. 6: Average current density at late times in a pulsed electric field\nas a function of \u001cfor di \u000berent ˜ m.\n2 1.51 0.501\n0.8\n0.6\n0.4\n0.2\n0exp(−π˜m1.4)\n˜mf(˜m)\nFIG. 7: The function f( ˜m) which describes the mass dependence of\nthe average current density at late times in a pulsed electric field.\nPoints are numerical results, solid line is a fit.\ndensity oscillates around the behavior of the current density\nat large Bz=Ez. Therefore the running average of the current\ndensity seems to be independent of Bz=Ez. To investigate the\ndependence on \u001cwe have displayed the running average of the\ncurrent density at late times in Fig. 6 for Bz=Ez=10. We find\nthat the running average increases linear with \u001c. Through ob-\nservation of our numerical results we find that for all values of\nBz=Ezthe running average of the current density at late times\nis summarized by the following formula\nhjyi=q2jqj\u001c\n\u00192ByEzf( ˜m)sgn( Bz); (86)\nwhere we have displayed f( ˜m) for di \u000berent values of ˜ min\nFig. 7. We find a reasonable fit to our data with the function\nf( ˜m)=exp(\u0000\u0019˜m1:4).VI. CONCLUSIONS\nIn this article we have investigated the real-time dynam-\nics of the chiral magnetic e \u000bect using linear response theory.\nWe have considered a field configuration in which a homoge-\nneous (chromo)electric field with arbitrary time-dependence\nlies parallel to a homogeneous and static (chromo)magnetic\nfield. These parallel fields are the source of the chirality. To\nthis field configuration we have added a perpendicular homo-\ngeneous and static magnetic field. We have computed the in-\nduced current density along this perpendicular magnetic field\nexplicitly for a sudden switch-on and a pulsed electric field.\nIn the sudden switch-on electric field we have obtained ex-\ncellent agreement with an earlier independent analytic com-\nputation of the current density. In the pulsed electric field we\ncould summarize the induced current density that we have ob-\ntained numerically with a simple analytic formula.\nThe main purpose of this article was to demonstrate the dy-\nnamics of the chiral magnetic e \u000bect using linear response the-\nory. We hope that our results will be extended to other inter-\nesting field configurations in the future. For example, in heavy\nion collisions it would be important to answer the question\nwhether there is enough time for the chiral magnetic e \u000bect to\noccur in the quickly decaying magnetic field. This question\ncould be addressed using our methodology. The chiral mag-\nnetic e \u000bect could also be investigated with lasers that create\nstrong electromagnetic fields. For that purpose it would be\nimportant to extend the results to field configurations that are\nas close to the experimental situation as possible.\nAs a side result of our work we have obtained a derivation\nof the induced current density in static homogeneous parallel\nelectric and magnetic fields. We have also obtained an ana-\nlytic formula for the chirality production for massive fermions\nin static homogeneous parallel electric and magnetic fields.\nAcknowledgments\nI would like to thank Gerald Dunne, Kenji Fukushima,\nDmitri Kharzeev, Larry McLerran and Vladimir Skokov for\ndiscussions. The work of H.J.W. was supported by the Ex-\ntreme Matter Institute (EMMI) and by the Alexander von\nHumboldt Foundation.\nAppendix A: Relations involving the function gps(x)\nTo evaluate these integrals we will make use of the follow-\ning four identities which directly follow from the properties\nof the Hermite polynomials,\nZ1\n\u00001dx0x0gps(x0)gp0s(x0)\f\f\f\f\f\np0y=py=py\nqBz\u000ek;k0\u00001\u0000\u000ek;0\u000es\u0000\u0001\n+1p\n2jqBzj\u0010p\nk\u000ek\u0000s;k0+p\nk+s\u000ek+s;k0\u0011\n;(A1)12\n1\nqBzZ1\n\u00001dpy\n2\u0019pygp\u0000s(x)gp0s(x)\f\f\f\f\f\fp0y=py=jqBzj\n2\u0019\"\nx\u000ek\u0000s;k0\n\u00001p\n2jqBzj\u0010p\nk\u0000s\u000ek\u00002s;k0+p\nk\u000ek;k0\u0011#\n;(A2)\nZ1\n\u00001dpy\n2\u0019gp\u0000s(x)gp0s(x)\f\f\f\f\f\fp0y=py=jqBzj\n2\u0019\u000ek\u0000s;k0: (A3)\nZ1\n\u00001dxgps(x)gp0\u0000s(x)\f\f\f\f\f\np0y=py=\u000ek+s;k0: (A4)\nAppendix B: Wave functions in a sudden switch-on electric field\nWe will review the explicit solutions for the wave functions\nf\u0006\nps(t) for a sudden switch-on electric field of the form E(t)=\nEz\u0012(t) [22]. Then we will evaluate an integral that is necessary\nfor computing the induced current and chirality production.\nFor the sudden switch-on electric field we have Az(t)=\n\u0000Ezt\u0012(t). For t<0 we can use the wave functions in van-\nishing electromagnetic field, given in Eq. (23). If t>0 the\nelectric field is not longer vanishing. It follows from combin-\ning Eqs. (17) and (18) that f\u0006\nps(t) then satisfies\n2666664\u0000@2\nt\u0000q2E2\nz \nt+pz\nqEz!2\n+isqE z3777775f\u0006\nps(t)=\u00142f\u0006\nps(t):(B1)\nEq. (B1) is an eigenvalue equation for a particle in an upside-\ndown harmonic potential. There are no bound states, so \u0014\nis not quantized. The solution is a linear combination of\nparabolic cylinder functions D\u0017(z),\nf\u0006\nps(t)=\u000b\u0006\nsD\u0017s(\u0018)+\f\u0006\nsD\u0017s(\u0000\u0018); (B2)\nwhere\u0017s=\u0000(ssgn(qEz)+1+i\u00142=jqEzj)=2 and\u0018=p\n2jqEzjei\u0019=4(t+pz=qEz).\nIn the conventional normalization the parabolic cylinder\nfunctions are given explicitly by the following integrals\nD\u0017(z)=1\n\u0000(\u0000\u0017)e\u00001\n4z2Z1\n0dt t\u0000\u0017\u00001e\u0000zt\u00001\n2t2; (B3)\nfor Re(\u0017)<0, and\nD\u0017(z)=r\n2\n\u0019e1\n4z2Z1\n0dt t\u0017cos\u0010\nzt\u00001\n2\u0019\u0017\u0011\ne\u00001\n2t2(B4)\nfor Re(\u0017)>\u00001. Using these relations one can show that\nD\u0017(0)=2\u0017=2p\u0019=\u0000[(1\u0000\u0017)=2]. Asymptotically the parabolic\ncylinder functions behave as follows\nlim\njzj!1D\u0017(z)=z\u0017e\u00001\n4z2\u0000c\u0017p\n2\u0019\n\u0000(\u0000\u0017)z\u0000\u0017\u00001e1\n4z2; (B5)\nwith c\u0017=0 forjarg(z)j<3\u0019\n4,c\u0017=exp(i\u0019\u0017) for\u0019\n4<jarg(z)j<\n5\u0019\n4, and for c\u0017=exp(\u0000i\u0019\u0017) for\u00005\u0019\n4<jarg(z)j<\u0000\u0019\n4. Theseasymptotic functions describe the real parabolic cylinder func-\ntions very good if jzj>2j\u0017j.\nThe constants \u000b\u0006\nsand\f\u0006\nsof Eq. (B2) can be found by re-\nquiring continuity of f\u0006\nps(t) and its derivative at t=0. This\ngives the following two equations\n\u000b\u0006\ns=1\nWrp0\u0007spz\np0h\n\u0000ei\u0019=4p\n2jqEzjD0\n\u0017s(\u0000\u00180)\n\u0006ip0D\u0017s(\u0000\u00180)i\n; (B6)\n\f\u0006\ns=1\nWrp0\u0007spz\np0h\n\u0000ei\u0019=4p\n2jqEzjD0\n\u0017s(\u00180)\n\u0007ip0D\u0017s(\u00180)i\n; (B7)\nwhere\u00180=p\n2ei\u0019=4pz=jqEzj1=2sgn(qEz) and Wdenotes the\nWronskian of the two independent solutions presented in\nEq. (B2). Applying Abel’s di \u000berential equation identity to\nEq. (B1) shows that the Wronskian is independent of \u0018. Hence\nwithout loss of generality we can evaluate Wat\u0018=0,\nwhich yields W=2ei\u0019=4p\n\u0019jqEzj=\u0000(\u0000\u0017s). Using the relation\nD0\n\u0017(z)=1\n2zD\u0017(z)\u0000D\u0017+1(z) we can simplify Eqs. (B6) and (B7)\nto\n\u000b\u0006\ns=\u0000(\u0000\u0017s)p\n2\u0019rp0\u0007spz\np0h\nD\u0017s+1(\u0000\u00180)\n+ei\u0019=4\np\n2jqEzj\u0000pzsgn(qEz)\u0006p0\u0001D\u0017s(\u0000\u00180)i\n;(B8)\n\f\u0006\ns=\u0000(\u0000\u0017s)p\n2\u0019rp0\u0007spz\np0h\nD\u0017s+1(\u00180)\n\u0000ei\u0019=4\np\n2jqEzj\u0000pzsgn(qEz)\u0006p0\u0001D\u0017s(\u00180)i\n: (B9)\nUsing the asymptotic expansion of the parabolic cylin-\nder functions, Eq. (B4), it is possible to obtain a very good\napproximation for the function f\u0006\nps(t) away from the points\npz=\u0000qEztandpz=0. Let us now for a moment choose\nqEz>0 and takep\njqEzjt\u001d1. In order to obtain the induced\ncurrent we need to evaluate jf+\np+(t)j2, for which we find after\ntaking the dominating terms in the asymptotic expansion and\napproximating p0byjpzjthe following\njf+\np+(t)j2\u00198>>>><>>>>:2 for pz.\u0000qEzt\u0000\u0001;\n2e\u0000\u0019\u00142\njqEzj+g(\u0010) for\u0000qEzt+ \u0001.pz.\u0000\u0001;\n0 for pz&\u0001:\n(B10)\nwhere \u0001 =2jqEzj1=2j1+i\u00142\n2jqEzjj,\u0010=p\njqEzj(t+pz=qEz), and\ng(\u0010)=2p\u00191\n\u0010\u0014\ne\u0000\u0019\u00142\n4jqEzj\u0000e\u00005\u0019\u00142\n4jqEzj\u0015\n\u0002Re\u0014\n\u0000\u0010\n1\u0000i\u00142\n2jqEzj\u0011\nei\u00102+i\u00142\njqEzjlogjp\n2\u0010j+i\u0019=4\u0015\n:(B11)\nAs follows from Eq. (48) we have to evaluate the following\nintegral to obtain the induced current\nI(\u0014;t)=lim\n\u000f!0Z1\n\u00001dpz\n2\u0019e\u0000\u000f(pz+qEzt)2h\njf+\np+(t)j2\u00001i\n: (B12)13\nIn general this integral can only be evaluated numerically.\nBut using Eq. (B10) we can obtain I(\u0014;t) exactly for large t.\nFirstly one realizes that taking the limit \u000f!0 implies that we\nhave to integratejf+\np+(t)j2\u00001 over pzin an interval symmetric\naround pz=\u0000qEzt. The contribution to the integral in the re-\ngion where the approximation Eq. (B10) breaks down can be\nbounded from below and above by a time-independent con-\nstant. The contributions for pz<\u00002qEztandpz>0 will can-\ncel against each other. The non-vanishing contribution comes\nfrom the intermediate region in which \u00002qEzt<pz<0. The\nintegral over the rapidly oscillating function g(\u0010) is subdomi-\nnant in the large tlimit. The dominant contribution in the largetlimit is qEztcoming from the interval \u00002qEzt<pz<\u0000qEzt\nandqEzt[2 exp(\u0000\u0019\u00142=jqEzj)\u00001] from the interval \u0000qEzt<\npz<0. Ading both contributions yields\nlim\nt!11\ntI(\u0014;t)=qEz\n\u0019e\u0000\u0019\u00142\njqEzj: (B13)\nBy repeating the analysis for qEz<0 one can verify that the\nsign of Eq. (B13) is correct, so Eq. (B13) holds for all values\nofqEz.\n[1] J. S. Schwinger, Phys. Rev. 82(1951) 664.\n[2] S. L. Adler, Phys. Rev. 177, 2426 (1969); J. S. Bell and\nR. Jackiw, Nuovo Cim. A60, 47 (1969).\n[3] A. Vilenkin, Phys. Rev. D 22, 3080 (1980).\n[4] D. E. Kharzeev, L. D. McLerran and H. J. Warringa, Nucl. Phys.\nA803, 227 (2008).\n[5] A. Y . Alekseev, V . V . Cheianov and J. Frohlich, Phys.\nRev. Lett. 81, 3503 (1998); K. Fukushima, D. E. Kharzeev\nand H. J. Warringa, Phys. Rev. D 78, 074033 (2008);\nD. E. Kharzeev and H. J. Warringa, Phys. Rev. D 80, 034028\n(2009); H. -U. Yee, JHEP 0911 , 085 (2009); A. Rebhan,\nA. Schmitt and S. A. Stricker, JHEP 1001 , 026 (2010);\nV . A. Rubakov, arXiv:1005.1888 [hep-ph]; A. Gynther,\nK. Landsteiner, F. Pena-Benitez and A. Rebhan, JHEP 1102 ,\n110 (2011); V . D. Orlovsky and V . I. Shevchenko, Phys. Rev. D\n82, 094032 (2010); L. Brits and J. Charbonneau, Phys. Rev. D\n83, 126013 (2011); D. Hou, H. Liu and H. -c. Ren, JHEP 1105 ,\n046 (2011); C. Hoyos, T. Nishioka and A. O’Bannon, JHEP\n1110 , 084 (2011); A. Yamamoto, Phys. Rev. D 84, 114504\n(2011); D. T. Son and N. Yamamoto, arXiv:1203.2697 [cond-\nmat.mes-hall].\n[6] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Nucl. Phys.\nA836, 311 (2010).\n[7] P. V . Buividovich, M. N. Chernodub, E. V . Luschevskaya and\nM. I. Polikarpov, Phys. Rev. D 80, 054503 (2009); M. Abram-\nczyk, T. Blum, G. Petropoulos and R. Zhou, arXiv:0911.1348\n[hep-lat].\n[8] S. i. Nam, Phys. Rev. D 80, 114025 (2009); S. i. Nam, Phys.\nRev. D 82, 045017 (2010).\n[9] G. Basar, G. V . Dunne and D. E. Kharzeev, Phys. Rev. D 85,\n045026 (2012).\n[10] K. Fukushima, D. E. Kharzeev and H. J. Warringa, Phys. Rev.\nLett. 104, 212001 (2010).\n[11] H. Gies, Eur. Phys. J. D 55, 311 (2009); M. Marklund and\nJ. Lundin, Eur. Phys. J. D 55, 319 (2009); G. V . Dunne, Eur.\nPhys. J. D 55, 327 (2009).\n[12] D. Kharzeev, A. Krasnitz and R. Venugopalan, Phys. Lett. B\n545, 298 (2002); T. Lappi and L. McLerran, Nucl. Phys. A 772,\n200 (2006);\n[13] H. Minakata and B. Muller, Phys. Lett. B 377, 135 (1996);\nV . Skokov, A. Y . Illarionov and V . Toneev, Int. J. Mod. Phys.\nA24, 5925 (2009); V . V oronyuk, V . D. Toneev, W. Cassing,\nE. L. Bratkovskaya, V . P. Konchakovski and S. A. V oloshin,\nPhys. Rev. C 83, 054911 (2011); W. -T. Deng and X. -G. Huang,\nPhys. Rev. C 85, 044907 (2012).\n[14] G. D. Moore and M. Tassler, JHEP 1102 , 105 (2011).[15] D. Kharzeev, Phys. Lett. B 633, 260 (2006); D. Kharzeev and\nA. Zhitnitsky, Nucl. Phys. A 797, 67 (2007).\n[16] S. A. V oloshin, Phys. Rev. C 70, 057901 (2004).\n[17] B. I. Abelev et al. [STAR Collaboration], Phys. Rev. Lett. 103,\n251601 (2009); B. I. Abelev et al. [STAR Collaboration], Phys.\nRev. C 81, 054908 (2010).\n[18] P. Christakoglou, J. Phys. G 38, 124165 (2011).\n[19] A. Bzdak, V . Koch and J. Liao, Phys. Rev. C 81, 031901 (2010);\nM. Asakawa, A. Majumder and B. Muller, Phys. Rev. C 81,\n064912 (2010); J. Liao, V . Koch and A. Bzdak, Phys. Rev. C\n82, 054902 (2010); H. Petersen, T. Renk and S. A. Bass, Phys.\nRev. C 83, 014916 (2011); A. Bzdak, V . Koch and J. Liao, Phys.\nRev. C 83, 014905 (2011); B. Muller and A. Schafer, Phys. Rev.\nC82, 057902 (2010).\n[20] V . I. Ritus, Annals Phys. 69, 555 (1972); V . I. Ritus, Sov. Phys.\nJETP 48, 788 (1978) [Zh. Eksp. Teor. Fiz. 75, 1560 (1978)].\n[21] M. E. Peskin and D. V . Schroeder. An introduction to Quantum\nField Theory , (Perseus Books 1995).\n[22] A. I. Nikishov, Zh. Eksp. Teor. Fiz. 57(1969) 1210.\n[23] N. B. Narozhnyi and A. I. Nikishov, Yad. Fiz. 11, 1072 (1970)\n[Sov. J. Nucl. Phys. 11, 596 (1970)].\n[24] J. Ambjorn, J. Greensite and C. Peterson, Nucl. Phys. B 221,\n381 (1983).\n[25] Y . Kluger, J. M. Eisenberg, B. Svetitsky, F. Cooper and E. Mot-\ntola, Phys. Rev. D 45, 4659 (1992).\n[26] L. F. Urrutia, Phys. Rev. D 17, 1977 (1978); V . P. Barashev,\nA. E. Shabad and S. M. Shvartsman, Sov. J. Nucl. Phys. 43,\n617 (1986) [Yad. Fiz. 43, 964 (1986)]; C. Schubert, Nucl. Phys.\nB585, 407 (2000);\n[27] S. L. Adler, Annals Phys. 67(1971) 599; W. -y. Tsai, Phys. Rev.\nD10, 2699 (1974); A. E. Shabad, Annals Phys. 90, 166 (1975).\n[28] K. Fukushima, Phys. Rev. D 83, 111501 (2011); E. J. Ferrer,\nV . de la Incera and A. Sanchez, Phys. Rev. Lett. 107, 041602\n(2011); H. Gies and L. Roessler, Phys. Rev. D 84, 065035\n(2011); F. Karbstein, L. Roessler, B. Dobrich and H. Gies,\narXiv:1111.5984 [hep-ph]; N. Sadooghi and F. Taghinavaz,\narXiv:1203.5634 [hep-ph].\n[29] S. P. Gavrilov and D. M. Gitman, Phys. Rev. D 78, 045017\n(2008).\n[30] N. Tanji, Annals Phys. 324, 1691 (2009).\n[31] F. V . Bunkin and I. I. Tugov, Sov. Phys. Dokl. 14(1970) 678.\n[32] G. V . Dunne, arXiv:hep-th /0406216.\n[33] S. P. Kim and D. N. Page, Phys. Rev. D 73(2006) 065020.\n[34] T. D. Cohen and D. A. McGady, Phys. Rev. D 78(2008)\n036008."    },    {        "title": "1210.3668v1.Full_disk_nonlinear_force_free_field_extrapolation_of_SDO_HMI_and_SOLIS_VSM_magnetograms.pdf",        "content": "Astronomy & Astrophysics manuscript no. aa c\rESO 2018\nNovember 7, 2018\nFull-disk nonlinear force-free field extrapolation of SDO/HMI and\nSOLIS/VSM magnetograms\nTilaye Tadesse1;2, T. Wiegelmann3, B. Inhester3, P. MacNeice4, A. Pevtsov5, and X. Sun6\n1Department of Physics, Drexel University, Philadelphia, PA 19104-2875, USA\ne-mail: tasfaw@einstein.physics.drexel.edu\n2Addis Ababa University, Institute of Geophysics, Space Science, and Astronomy, Po.Box 1176, Addis Ababa, Ethiopia\ne-mail: tilaye.tadesse@gmail.com\n3Max-Planck-Institut f ¨ur Sonnensystemforschung, Max-Planck-Strasse 2, 37191 Katlenburg-Lindau, Germany\n4NASA /GSFC, Greenbelt, MD, U.S.A.\n5National Solar Observatory, Sunspot, NM 88349, U.S.A.\n6W.W. Hansen Experimental Physics Laboratory, Stanford University, Stanford, CA 94305, USA\nPreprint online version: November 7, 2018\nABSTRACT\nContext. The magnetic field configuration is essential to understand solar explosive phenomena such as flares and coronal mass\nejections. To overcome the unavailability of coronal magnetic field measurements, photospheric magnetic field vector data can be\nused to reconstruct the coronal field. A complication of this approach is that the measured photospheric magnetic field is not force-\nfree and that one has to apply a preprocessing routine in order to achieve boundary conditions suitable for the force-free modelling.\nFurthermore the nonlinear force-free extrapolation code should take uncertainties into account in the photospheric field data which\noccur due to noise, incomplete inversions or azimuth ambiguity removing techniques.\nAims. Extrapolation codes in cartesian geometry for modelling the magnetic field in the corona do not take the curvature of the\nSun’s surface into account and can only be applied to relatively small areas, e.g., a single active region. Here we apply a method for\nnonlinear force-free coronal magnetic field modelling and preprocessing of photospheric vector magnetograms in spherical geometry\nusing the optimization procedure to full disk vector magnetograms. We compare the analysis of the photospheric magnetic field\nand subsequent force-free modeling based on full-disk vector maps from Helioseismic and Magnetic Imager (HMI) on board solar\ndynamics observatory (SDO) and Vector Spectromagnetograph (VSM) of the Synoptic Optical Long-term Investigations of the Sun\n(SOLIS).\nMethods. We use Helioseismic and Magnetic Imager and Vector Spectromagnetograph photospheric magnetic field measurements to\nmodel the force-free coronal field above multiple solar active regions, assuming magnetic forces to dominate. We solve the nonlinear\nforce-free field equations by minimizing a functional in spherical coordinates over a full disk excluding the poles. After searching\nfor the optimum modeling parameters for the particular data sets, we compare the resulting nonlinear force-free model fields. We\ncompare quantities like the total magnetic energy content and free magnetic energy , the longitudinal distribution of the magnetic\npressure and surface electric current density using our spherical geometry extrapolation code.\nResults. The magnetic field lines obtained from nonlinear force-free extrapolation based on Helioseismic and Magnetic Imager and\nVector Spectromagnetograph data have good agreement. However, the nonlinear force-free extrapolation based on Helioseismic and\nMagnetic Imager data have more contents of total magnetic energy, free magnetic energy, the longitudinal distribution of the magnetic\npressure and surface electric current density compared to the one from Vector Spectromagnetograph data.\nConclusions.\nKey words. Magnetic fields – Sun: corona – Sun: photosphere – methods: numerical\n1. Introduction\nThe magnetic fields configuration is essential for us to under-\nstand solar explosive phenomena such as flares and coronal mass\nejections. The corona has been the subject of extensive modeling\nfor decades, but these e \u000borts have been hampered by our lim-\nited ability to determine the corona’s three-dimensional struc-\nture (Schrijver & Title 2011; Sandman & Aschwanden 2011).\nSince the corona is optically thin, direct measurements of these\nmagnetic fields are very di \u000ecult to implement, and the present\nobservations for the magnetic fields based on the spectropolari-\nmetric method (the Zeeman and the Hanle e \u000bects) are limited to\nlow layers of solar atmosphere (photosphere and chromosphere).\nThe problem of measuring the coronal field and its embedded\nelectrical currents thus leads us to use numerical modelling to in-fer the field strength in the higher layers of the solar atmosphere\nfrom the measured photospheric field. Due to the low value of\nthe plasma \f(the ratio of gas pressure to magnetic pressure),\nthe solar corona is magnetically dominated (Gary 2001). To de-\nscribe the equilibrium structure of the static coronal magnetic\nfield when non-magnetic forces are negligible, the force-free as-\nsumption is appropriate:\n(r\u0002B)\u0002B=0 (1)\nr\u0001B=0 (2)\nsubject to the boundary condition\nB=Bobs on photosphere (3)\nwhere Bis the magnetic field and Bobsis measured vector field\non the photosphere. Equation (1) states that the Lorentz forcearXiv:1210.3668v1  [astro-ph.SR]  13 Oct 20122 T. Tadesse et al.: Full-disk NLFFF extrapolation of SDO /HMI and SOLIS /VSM data\nvanishes (as a consequence of JkB, where Jis the elec-\ntric current density) and Equation (2) describes the absence of\nmagnetic monopoles. Based on the above assumption, the coro-\nnal magnetic field is modelled with nonlinear force-free field\n(NLFFF) extrapolation (Inhester & Wiegelmann 2006; Valori\net al. 2005; Wiegelmann 2004; Wheatland 2004; Wheatland &\nR´egnier 2009; Tadesse et al. 2009; Wheatland & Leka 2011;\nAmari & Aly 2010; Wiegelmann et al. 2012; Jiang & Feng\n2012). From a mathematical point of view appropriate boundary\ncondition for force-free modeling are the vertical magnetic field\nBnand the vertical current Jnprescribed only for one polarity of\nBn(Amari et al. 1997, 1999, 2006). A direct use of these bound-\nary conditions is implemented in Grad-Rubin codes (Amari et al.\n1999). Wheatland & R ´egnier (2009) and Wheatland & Leka\n(2011) implemented the use of B+\nnandB\u0000\nnsolution together with\nan error approximation to derive consistent solutions. Using the\nthree components of Bas boundary condition requires consistent\nmagnetograms, as outlined in Aly (1989). We use preprocessing\nand relaxation of the boundary condition to derive these consis-\ntent data on the boundary.\nAs an alternative to real measurement, nonlinear force-free\nfield (NLFFF) models are thought to be viable tools for investi-\ngating the structure, dynamics, and evolution of the coronae of\nsolar active regions. It has been found that NLFFF models are\nsuccessful in application to analytic test cases (Schrijver et al.\n2006; Metcalf et al. 2008), but they are less successful in ap-\nplication to real solar data. However, NLFFF models have been\nadopted to study various magnetic field structures and proper-\nties in the solar atmosphere. For instance, R ´egnier et al. (2002);\nR´egnier & Amari (2004); Canou et al. (2009); Canou & Amari\n(2010); Guo et al. (2010); Valori et al. (2012) have substantially\nstudied various magnetic field structures and properties using\ntheir respective NLFFF model codes. Di \u000berent NLFFF models\nhave been found to have markedly di \u000berent field line configu-\nrations and to provide widely varying estimates of the magnetic\nfree energy in the coronal volume, when applied to solar data\n(DeRosa et al. 2009). The main reasons for that problem are\n(1) the forces acting on the field within the photosphere, (2) the\nuncertainties on vector-field measurements, particularly on the\ntransverse component, and (3) the large domain that needs to be\nmodelled to capture the connections of an active region to its\nsurroundings(Tadesse et al. 2011, 2012a). In this study, we have\nconsidered those three points explicitly into account. However,\ncaution must still be needed while assessing results from this\nmodeling. This is because many aspects of the specific approach\nto modeling used in this work, such as the use of preprocessed\nboundary data, the missing boundary data, and the departure of\nthe model fields from the observed boundary fields may influ-\nence the results.\nIn this work, we use full-disk SDO /HMI and SOLIS /VSM\nphotospheric magnetic field measurements to model the NLFFF\ncoronal field above multiple solar active regions. Comparison of\nvector magnetograms for one particular active region observed\nwith two di \u000berent instruments from SOLIS and HMI and their\ncorresponding force-free models have been studied by Thalmann\net al. (2012, submitted to AJ) in Cartesian coordinates. We use a\nlarger computational domain which accommodates most of the\nconnectivity within the coronal region. We use a spherical ver-\nsion of the optimization procedure that has been implemented\nin Tadesse et al. (2011). We compare quantities like the total\nmagnetic energy content and free magnetic energy and the lon-\ngitudinal distribution of the magnetic pressure in the HMI and\nVSM-based model volumes in spherical geometry. We relate the\nappearing di \u000berences to the photospheric quantities such as themagnetic fluxes and electric currents but also show the extent of\nagreement of NLFFF extrapolations from di \u000berent data sources.\n2. Instrumentation and data set\n2.1. Solar Dynamics Observatory(SDO) - Helioseismic and\nMagnetic Imager(HMI)\nThe Helioseismic and Magnetic Imager (HMI; Schou et al.\n2012) is part of the Solar Dynamics Observatory (SDO) and ob-\nserves the full Sun at six wavelengths and full Stokes profile in\nthe Fe I 617.3 nm spectral line. HMI consists of a refracting\ntelescope, a polarization selector, an image stabilization system,\na narrow band tunable filter and two 4096 pixel CCD cameras\nwith mechanical shutters and control electronics. Photospheric\nline-of-sight LOS and vector magnetograms are retrieved from\nfiltergrams with a plate scale of 0.5 arc-second. From filter-\ngrams averaged over about ten minutes, Stokes parameters are\nderived and inverted using the Milne-Eddington (ME) inversion\nalgorithm of Borrero et al. (2011) (the filling factor is held at\nunity). Within automatically identified regions of strong mag-\nnetic fluxes (Turmon et al. 2010), the full disk inversion data\nare from the second HMI vector data release (JSOC data series\nhmi.ME 720s e15w1332). The 180-degree azimuthal ambiguity\nin the strong field region is resolved using the Minimum Energy\nAlgorithm (Metcalf 1994; Metcalf et al. 2006; Leka et al. 2009),\ntaken from the AR patches in the second release also (data series\nhmi.B 720s e15w1332 cutout). For the weak field region where\nnoise dominates, we adopt a radial-acute angle method to re-\nsolve the azimuthal ambiguity. The weak field region is defined\nas where field strength is below 200 G at disk center, 400 G on\nthe limb, and varies linearly in between. The noise level is \u001910G\nand\u0019100G for the longitudinal and transverse magnetic field,\nrespectively.\n2.2. Synoptic Optical Long-term Investigations of the\nSun(SOLIS) - Vector-SpectroMagnetograph(VSM\nThe Vector Spectromagnetograph (VSM; see Jones et al. 2002)\nis part of the Synoptic Optical Long-term Investigations of the\nSun (SOLIS) synoptic facility (SOLIS; see Keller et al. 2003).\nVSM is a full disk Stokes polarimeter. As part of daily synoptic\nobservations, it takes four di \u000berent observations in three spec-\ntral lines: Stokes I(intensity), V(circular polarization), Q, and\nU(linear polarization) in photospheric spectral lines Fe I 630.15\nnm and Fe I 630.25 nm , Stokes IandVin Fe I 630.15 nm and\nFe I 630.25 nm, similar observations in chromospheric spectral\nline Ca II 854.2 nm, and Stokes Iin the He I 1083.0 nm line and\nthe near by Si I spectral line. Observations of I,Q,U, and Vare\nused to construct full disk vector magnetograms, while I\u0000Vob-\nservations are employed to create separate full disk longitudinal\nmagnetograms in the photosphere and the chromosphere. The\nvector data are provided with a plate scale of one arc-second.\nThe lower limits for the noise levels are a few Gauss in the lon-\ngitudinal and 70G in the transverse field measurements.\nQuick-look (QL) vector magnetograms were created based\non an algorithm by Auer et al. (1977). Beginning January 2012,\nQL vector magnetograms are created using weak-field approx-\nimation (Ronan et al. 1987). The algorithm uses the Milne-\nEddington model of solar atmosphere, which assumes that the\nmagnetic field is uniform (no gradients) through the layer of\nspectral line formation (Unno 1956). It also assumes symmetric\nline profiles, disregards magneto-optical e \u000bects (e.g., Faraday\nrotation), and does not distinguish the contributions of mag-T. Tadesse et al.: Full-disk NLFFF extrapolation of SDO /HMI and SOLIS /VSM data 3\nnetic and non-magnetic components in spectral line profiles (i.e.,\nmagnetic filling factor is set to unity). A complete inversion of\nthe spectral data is performed later using a technique developed\nby Skumanich & Lites (1987). This latter inversion (called ME\nmagnetogram) also employs Milne-Eddington model of atmo-\nsphere, but solves for magneto-optical e \u000bects and determines the\nmagnetic filling factor i.e., (the fractional contribution of mag-\nnetic and non-magnetic components to each pixel). The ME in-\nversion is only performed for pixels with spectral line profiles\nabove the noise level. For pixels below the polarimetric noise\nthreshold, the magnetic field parameters are set to zero.\nFrom the measurements, the azimuths of transverse mag-\nnetic field can be determined with 180-degree ambiguity. This\nambiguity is resolved using the non-potential field calculation\n(NPFC; see Georgoulis 2005). The NPFC method was selected\non the basis of a comparative investigation of several methods\nfor 180-degree ambiguity resolution (Metcalf et al. 2006). Both\nQL and ME magnetograms can be used for potential and /or\nforce-free field extrapolation. However, in strong fields inside\nsunspots, the QL field strengths may exhibit an erroneous de-\ncrease inside the sunspot umbra due to so-called magnetic satu-\nration. For this study, we choose to use fully inverted ME mag-\nnetograms.\n3. Method\nPhotospheric field measurements are often subject to measure-\nment errors. In addition to this, there are finite non-magnetic\nforces which make the data inconsistent as a boundary for a\nforce-free field in the corona. In order to deal with these un-\ncertainties, one has to: 1.) preprocess the surface measurements\nin order to make them compatible with a force-free field and 2.)\nkeep a balance between the force-free constraint and deviation\nfrom the photospheric field measurements. Both methods con-\ntain free parameters, which have to be optimized for use with\ndata from SOLIS /VSM and SDO /HMI.\n3.1. Preprocessing of HMI and VSM data\nTo serve as suitable lower boundary condition for a force-free\nmodeling, vector magnetograms have to be approximately flux\nbalanced and on average a net tangential force acting on the\nboundary and shear stresses along axes lying on the boundary\nhave to reduce to zero. We use dimensionless parameters, \u000ff lux,\n\u000ff orce and\u000ftorque , to quantify such properties(Wiegelmann et al.\n2006; Tadesse et al. 2009; Aly 1989; Molodensky 1969). Even\nif we choose a su \u000eciently flux balanced region ( \u000ff lux), we find\nthat the force-free conditions \u000ff orce\u001c1 and\u000ftorque\u001c1 are not\nusually fulfilled for measured vector magnetograms. In order to\nfulfill those conditions, we use preprocessing method as imple-\nmented in Wiegelmann et al. (2006). The preprocessing scheme\nof Tadesse et al. (2009) involves minimizing a two-dimensional\nfunctional of quadratic form in spherical geometry similar to\nB=argmin (Lp);\nLp=\u00161L1+\u00162L2+\u00163L3+\u00164L4; (4)\nwhere Bis the preprocessed surface magnetic field from the in-\nput observed field Bobs. Each of the constraints Lnis weighted by\nan as yet undetermined factor \u0016n. The first term ( n=1) corre-\nsponds to the force-balance condition, the next ( n=2) to the\ntorque-free condition, and the last term ( n=4) controls the\nsmoothing. The explicit form of L1,L2, and L4can be found\nin Tadesse et al. (2009). The term ( n=3) controls the di \u000berence\nbetween measured and preprocessed vector fields..3.2. Optimization principle\nWe solve the force-free equations (1) and (2) by optimization\nprinciple, as proposed by Wheatland et al. (2000) and general-\nized by Wiegelmann (2004) for cartesian geometry. The method\nminimizes a joint measure of the normalized Lorentz forces and\nthe divergence of the field throughout the volume of interest,\nV. Throughout this minimization, the photospheric boundary of\nthe model field Bis matched exactly to the observed Bobsand\npossibly preprocessed magnetogram values B. Here, we use the\noptimization approach for functional ( L!) in spherical geome-\ntry (Wiegelmann 2007; Tadesse et al. 2009) along with the new\nmethod, which instead of an exact match enforces a minimal de-\nviation between the photospheric boundary of the model field B\nand the magnetogram field Bobsby adding an appropriate sur-\nface integral term Lphoto (Wiegelmann & Inhester 2010; Tadesse\net al. 2011). These terms are given by\nB=argmin (L!)\nL!=Lf+Ld+\u0017Lphoto (5)\nLf=Z\nV!f(r;\u0012;\u001e)B\u00002\f\f\f(r\u0002B)\u0002B\f\f\f2r2sin\u0012drd\u0012d\u001e\nLd=Z\nV!d(r;\u0012;\u001e)\f\f\fr\u0001B\f\f\f2r2sin\u0012drd\u0012d\u001e\nLphoto=Z\nS\u0000B\u0000Bobs\u0001\u0001W(\u0012;\u001e)\u0001\u0000B\u0000Bobs\u0001r2sin\u0012d\u0012d\u001e\nwhere LfandLdmeasure how well the force-free Eqs. (1)\nand divergence-free (2) conditions are fulfilled, respectively,\nand both!f(r;\u0012;\u001e) and!d(r;\u0012;\u001e) are weighting functions. The\nweighting functions !fand!dinLfandLdin Eq. (5) are cho-\nsen to be unity within the inner physical domain V0and decline\nwith a cosine profile in the bu \u000ber boundary region (Wiegelmann\n2004; Tadesse et al. 2009, 2012b). They reach a zero value at\nthe boundary of the outer volume V. The distance between the\nboundaries of V0andVis chosen to be nd=10 grid points wide.\nThe third integral, Lphoto, is the surface integral over the photo-\nsphere which allows us to relax the field on the photosphere to-\nwards force-free solution without too much deviation from the\noriginal surface field data.\nW(\u0012;\u001e) is a space-dependent diagonal matrix the element of\nwhich are inverse proportional to the estimated squared mea-\nsurement error of the respective field component. In principle\none could compute Wfrom the measurement noise and er-\nrors obtained from the inversion of measured Stokes profiles\nto field components. Until these quantities become available,\na reasonable assumption is that the magnetic field is measured\nin strong field regions more accurately than in the weak field\nand that the error in the photospheric transverse field is at\nleast one order of magnitude higher as the line-of-sight com-\nponent. Appropriate choices to optimize \u0017andWfor use with\nSDO/HMI(Wiegelmann et al. 2012) and SOLIS /VSM(Tadesse\net al. 2011) magnetograms have been investigated. For a de-\ntailed description of the current code implementation, we refer\nto Wiegelmann & Inhester (2010) and Tadesse et al. (2011).\n4. Results\nWithin this work, we use the full disk data from SOLIS /VSM\nand SDO /HMI instruments obsrved on November 09 2011\naround 17:45UT. During this observation there were four ac-\ntive regions (ARs 11338, 11339, 11341 and 11342) along with4 T. Tadesse et al.: Full-disk NLFFF extrapolation of SDO /HMI and SOLIS /VSM data\nFig. 1. Magnetic vector maps of VSM and HMI on part of the lower boundary. The color coding shows Bron the photosphere and\nthe white arrow indicates the transverse components of the field. The vertical and horizontal axes show latitude, \u0012and longitude, \u001e\non the photosphere respectively.\nother smaller sunspots spreading on the disk. To accommodate\nthe connectivity between those ARs and their surroundings, we\nadopt a non uniform spherical grid r,\u0012,\u001ewith nr=225,\nn\u0012=375, n\u001e=425 grid points in the direction of radius,\nlatitude, and longitude, respectively, with the field of view of\n[rmin=1R\f:rmax=2R\f]\u0002[\u0012min=\u000050\u000e:\u0012max=50\u000e]\u0002[\u001emin=\n90\u000e:\u001emax=270\u000e]. Given the twice as large plate scale of VSM,\nwe bin the HMI vector maps to the resolution of VSM in or-\nder to compare the photospheric magnetic field and subsequent\nforce free modeling. To deal with vector magnetogram data be-\ning inconsistent with the force-free assumption, we use a pre-\nprocessing routine in spherical geometry, which derives suitable\nboundary conditions for force-free modeling from the measured\nphotospheric data. Applying this procedure to both SDO /HMI\nand SOLIS /VSM reduces \u000ff orce and\u000ftorque further significantly.\nThe two quantities are very well below unity after preprocess-\ning, which gives us some confidence that the data might serve\nas suitable boundary condition for a force-free modeling. Doing\nthis, we do not intend to suppress the existing forces in the pho-\ntosphere. Instead, we try to approximate the magnetic field at\na chromospheric level where magnetic forces are expected to\nbe much smaller than in the layers below. Both vector magne-\ntograms are almost flux balanced and the field of view was large\nenough to cover the full-disk. The unsigned magnetic flux of lon-\ngitudinal surface magnetic field from HMI is 1.57 times that ofVSM magnetogram. This is in agreement with recent compara-\ntive study by Pietarila et al. (2012), who found that the factor to\nconvert SOLIS /VSM to SDO /HMI increases with flux density\nfrom about 1 (weak fields) to about 1.5 (strong fields) for the\nline-of-sight full disk magnetograms. HMI inverts weak field re-\ngions, however, for VSM zeros are assigned to pixels where the\nmeasured polarization signal is too weak to perform a reliable\ninversion. Disregarding these zero-pixels, about 20% of the total\nnumber of pixels in the HMI and VSM full disk vector maps are\nremaining for comparison. HMI is found to detect most trans-\nverse field.\nWe used a standard preprocessing parameter set \u00161=\u00162=1\nand\u00163=0:001, which are similar to the values calculated from\nvector data used in previous studies(Wiegelmann et al. 2012) for\nHMI data in Cartesian coordinates. Table 1 lists the values of\ndimensionless parameters for the used HMI and VSM data-sets.\nIn this study, we have found that the optimal value for smoothing\nparameter is \u00164=0:05 for full-disk HMI data. These parameters\ncontrol the amount of force-freeness, torque-freeness, nearness\nto the actually observed data and smoothing, respectively. As\nthe result of parameter study, Tadesse et al. (2011) have found\n\u00161=\u00162=1,\u00163=0:03 and\u00164=0:45 as optimal for full-disk\nVSM data.\nThe preprocessing influences the structure of the magnetic\nvector data. It does not only smooths Bt(transverse field) butT. Tadesse et al.: Full-disk NLFFF extrapolation of SDO /HMI and SOLIS /VSM data 5\nTable 1. Flux-balance, force and torque free parameters of\nSOLIS /VSM and SDO /MHI full disk magnetograms.\nData set \u000ff lux\u000ff orce\u000ftorque\nHMI observed \u00000:0621 0:1305 0:1773\nHMI preprocessed \u00000:0313 0:0001 0:0002\nSOLIS observed \u00000:0857 0:4571 0:2947\nSOLIS preprocessed \u00000:0460 0:0015 0:0007\nalso alters its values in order to reduce the net force and torque.\nThe change in Btis more pronounced than the radial compo-\nnent Br(radial field) since Btis measured with lower accuracy\nthan the longitudinal magnetic field. Figure 1 shows the prepro-\ncessed and observed surface vector magnetic field obtained from\nSDO/HMI and SOLIS /VSM magnetograms. To identify the sim-\nilarity of vector components from HMI and VSM on the bottom\nsurface, we calculate their pixel-wise correlations before and af-\nter preprocessing. The correlation were calculated from\nCvec=P\nivi\u0001ui\u0010P\nijvij2P\nijuij2\u00111=2; (6)\nwhere vianduiare the vectors at each grid point ion the bottom\nsurface. If the vector fields are identical, then Cvec=1; if vi?ui\n, then Cvec=0. Table 2 shows the correlation ( Cvec) of the 2D\nsurface magnetic field vectors of observed and preprocessed data\nfrom HMI and VSM for the radial and transverse components.\nThe vector correlation between Btin the preprocessed HMI and\nVSM surface vector maps is clearly more closer to unity than the\ncorresponding surface vector maps without preprocessing. There\nis no such di \u000berence in correlations between Brbefore and af-\nter preprocessing. This is to be expected since the preprocessing\nscheme only smooths the longitudinal field while it smooths and\nalters the transverse field. The mean value of the changes due\nto preprocessing in the longitudinal field is 10\u00003G and for the\ntransverse field on the order of 10 G, i. e., well within the mea-\nsurement uncertainty of the HMI and VSM.\nBefore we perform nonlinear force-free extrapolations we use\nthe preprocessed radial component Brof the VSM and HMI-\ndata to compute the corresponding potential fields using spher-\nical harmonic expansion for initializing our code. We imple-\nment the new term Lphoto in Eq. (5) to work with boundary\ndata of di \u000berent noise levels and qualities or even neglect some\ndata points completely. SOLIS /VSM provides full-disk vector-\nmagnetograms, but for some individual pixels the inversion from\nline profiles to field values may not have been successfully in-\nverted and field data there will be missing for these pixels. Since\nthe old code without the Lphoto term requires complete boundary\ninformation, it cannot be applied to this set of SOLIS /VSM data.\nIn our new code, these data gaps are treated by setting W=0 for\nthese pixels in Eq. (5) (Wiegelmann & Inhester 2010; Tadesse\net al. 2011). For those pixels, for which Bobswas successfully\ninverted, we allow deviations between the model field Band the\ninput fields observed Bobssurface field using Eq. (5), so that the\nmodel field can be iterated closer to a force-free solution even if\nthe observations are inconsistent.\nThe improved optimization scheme allows us to relax the\nmagnetic field also on the lower boundary. The relaxation of the\nlower boundary introduces a further modification of the vector\ndata, in addition to that by the preprocessing applied before. The\nmean modification of the longitudinal field due to the relaxation\nFig. 2. Mask function for magnetic vector field distribution on\nfull disk from (a) VSM and (b) HMI. The vertical and horizon-\ntal axes show latitude, \u0012and longitude, \u001eon the photosphere\nrespectively.\nof the lower boundary is 10\u00004G and absolute values are on the\norder of 1 G. The mean changes of the transverse field are on the\norder of 10G and absolute values can be several 100 G. Given the\nnoise levels of HMI and VSM measurements of the longitudinal\n(\u001910G and a few G, respectively) and transverse field ( \u0019100G\nand&70G, respectively), the modifications are on the order of\nthe measurement error.\nFor nonlinear force-free fields we minimize the functional\nEq. (5). In order to control the speed with which the lower\nboundary is injected during the extrapolation, we vary the\nLangrangian multiplier \u0017. Unless an exact error computation be-\ncomes available from inversion and ambiguity removal of the\nphotospheric magnetic field vector, a reasonable assumption is\nthat the field is measured more accurately in strong field regions\nand one can carry out computations with the mask /Btand\n/B2\nt. We choose a mask function of W=\u0000Bt=max(Bt)\u00012, which\ngives more weight to strong field regions than to weak ones as\ninvestigated in Wiegelmann et al. (2012). Figure 2 shows sur-\nface distribution of mask function Wfor VSM and HMI full-disk\ndata. For strong field regions Wis close to unity and decline to\nzero in weaker field regions. We vary the Langrangian multiplier\n\u0017between 0:1 and 0:0001 to investigate the optimal parameter\nfor HMI full-disk data. To evaluate how well the force-free and\ndivergence-free condition are satisfied for di \u000berent Langrangian\nmultiplier\u0017, we monitor a number of expressions, such as Lf,Ld6 T. Tadesse et al.: Full-disk NLFFF extrapolation of SDO /HMI and SOLIS /VSM data\nTable 2. The correlations between the components of surface\nfields from HMI and VSM data.\nv u Cvec\nNo preprocessing ( BHMI)r(BVS M)r0:947\nNo preprocessing ( BHMI)t(BVS M)t0:893\nPreprocessed ( BHMI)r(BVS M)r0:965\nPreprocessed ( BHMI)t(BVS M)t0:951\nTable 3. Evaluation of force-free field models from preprocessed\nHMI data. The first column names the model and in column 2\nshows the used Langrangian multipliers. Columns 3-5 show dif-\nferent force-free consistency evaluations. Column 6 shows the\nratio of NLFFF energy density to the corresponding potential\nenergy density and column 7 the computing time.\n\u0017 Lf Ld sin\u00001(\u001bi)E=Epot Time\n0:1 21:7 13:4 25:8\u000e1:06 2h:17 min\n0:05 19:8 10:7 18:1\u000e1:12 3h:31 min\n0:001 2:9 1:5 4:8\u000e1:22 4h:39 min\n0:005 5:2 3:9 8:9\u000e1:23 11h:47 min\n0:0001 7:7 4:3 10:2\u000e1:26 48h:53 min\nTable 4. The magnetic energy associated with extrapolated\nNLFFF field configurations from full disk SDO /HMI and\nSOLIS /VSM data.\nModel Enl f f(1033erg)Efree(1033erg)\nSOLIS /VSM 8 :609 1 :375\nSDO/HMI 8 :913 1 :607\nand\n\u001bj=\u0010X\nijJi\u0002Bij\nBi\u0011\n=X\niJi; (7)\nwhere\u001bjis the sine of the current weighted average angle be-\ntween the magnetic field Band electric current J.\nFor a su \u000ecient small Lagrangian multiplier \u0017=0:001 we\nfound that the resulting coronal fields are force and divergence\nfree compared to other values as shown in Table 3. The weighted\nangle between the magnetic field and electric current is about 5\u000e\nfor\u0017=0:001. Injecting the boundary faster by choosing a higher\nLagrangian multiplier ( \u0017=0:1) speeds up the computation, but\nthe residual forces are higher and current and field are not well\naligned as investigated by Wiegelmann et al. (2012) for single\nAR.\nTo understand the physics of solar flares, including the lo-\ncal reorganization of the magnetic field and the acceleration of\nenergetic particles, one has to estimate the free magnetic en-\nergy available for these phenomena (R ´egnier & Priest 2007a;\nAschwanden 2008; Schrijver 2009). This is the free energy that\ncan be converted into kinetic and thermal energy. From the en-\nergy budget and the observed magnetic activity in the active re-\ngion, R ´egnier & Priest (2007b) and Thalmann et al. (2008) in-\nvestigated the free energy above the minimum-energy state for\nthe flare process. We estimate the free magnetic energy to be\nthe di \u000berence between the extrapolated force-free fields and the\npotential field with the same normal boundary conditions in the\nphotosphere. We therefore estimate the upper limit to the free\nmagnetic energy associated with coronal currents of the form\nEfree=1\n8\u0019Z\nV\u0010\nB2\nnl f f\u0000B2\npot\u0011\nr2sin\u0012drd\u0012d\u001e; (8)where BpotandBnl f frepresent the potential and NLFFF mag-\nnetic field,respectively. The magnetic energy densities asso-\nciated with the potential field configurations from SDO /HMI\nand SOLIS /VSM data are found to be 7 :306\u00021033ergand\n7:234\u00021033erg, respectively. This has to be expected as the un-\nsigned magnetic flux of longitudinal surface magnetic field from\nHMI is greater than that of VSM magnetogram. The magnetic\nenergy of NLFFF obtained from HMI data is greater that the\none obtained from VSM data as shown in Table 4. This is due\nto the fact that HMI data has more longitudinal unsigned mag-\nnetic flux and detects more transverse field than VSM. To study\nthe influence of the use of preprocessed boundary data from the\nobserved boundary fields on the estimation of free-magnetic en-\nergy , we have computed the magnetic energy associated with\nthe potential field and NLFFF configurations from the origi-\nnal SDO /HMI data without preprocessing and with preprocess-\ning. The case for SOLIS /VSM has been studied by Tadesse\net al. (2012a). As preprocessing procedure filters out small scale\nsurface field fluctuations, the magnetic energy associated with\nNLFFF obtained from preprocessed SDO /HMI boundary data is\nsmaller than the one without preprocessing. Obviously, the po-\ntential field energies of boundary data with and without prepro-\ncessing are close in value, since the potential field calculation\nmakes use of the radial magnetic field component which is not\na\u000bected too much by preprocessing procedure. The computed\nmagnetic energy from SDO /HMI original data without prepro-\ncessing is about 9 :067\u00021033erg, which is about 1 :7% higher\nthan the one obtained from preprocessed and modified observa-\ntional HMI boundary data. However, this energy does not cor-\nrespond to the nonlinear force-free magnetic field solution since\nthe original boundary data without preprocessing is not a con-\nsistent boundary condition for NLFFF modeling(Tadesse et al.\n2012a).\nWe investigate the magnetic field configurations of the VSM\nand HMI models by comparing the vector components. We cal-\nculate the vector correlation (using Equation (6) in the com-\nputational volume) of the potential fields and the NLFFF fields.\nThe average vector correlation between the potential fields based\non the HMI and VSM data is 0.97. The average vector correla-\ntion between the NLFFF fields of HMI and VSM data is 0.94.\nFigure 3 a. and b. show the surface radial magnetic field com-\nponent observed by HMI instrument on November 09 2011 and\nthe corresponding AIA (Atmospheric Imaging Assembly) image\nin 171Å, respectively. The magnetic field lines obtained from\nnonlinear force-free extrapolation based on HMI and VSM data\nhave good correlations as shown in Figure 3 c. and d., with\nthe foot points of the field lines from the two magnetograms\nare identical. However, there are some di \u000berences. For example,\nextrapolated field lines from SDO /HMI magnetogram (Figure\n3d) do not show transequatorial loops connecting trailing polar-\nity of NOAA AR 11339 (west of central meridian in northern\nhemisphere) and trailing polarity of AR11338 (southern hemi-\nsphere). This transequatorial loop is well represented by NLFFF\nextrapolation based on SOLIS /VSM. This di \u000berence can be at-\ntributed to presence of a patch of weak fields between two ac-\ntive regions (in SDO /HMI data). With this weak field patch,\nSDO/HMI model tends to close field lines originating in trail-\ning polarity of AR11339, while SOLIS /VSM model extends\nthem to AR11338. Both extrapolations indicate loops connecting\nAR11339 and AR11342 (East of central meridian in Northern\nhemisphere). Although SDO /AIA image (Figure 3b) does not\nshow coronal loops connecting these two active region, such\nloops are clearly visible in images taken by X-ray Telescope\non Hinode. These loops appear to fit better field lines fromT. Tadesse et al.: Full-disk NLFFF extrapolation of SDO /HMI and SOLIS /VSM data 7\nFig. 3. a) SDO /HMI and b) AIA images and their respective selected magnetic field lines plots reconstructed from c) SOLIS and\nd) HMI magnetograms using nonlinear force-free modelling. The color coding shows Bron the photosphere. Yellow field lines\nrepresent closed field lines, while field lines changing in color from yellow to brown (from bottom to the top) represent the open\nones. The gray area indicates the region where magnetic field values are close to zero.\nSOLIS /VSM model. Despite a relatively good visual agreement\nin extrapolated fields, the models show some notable disagree-\nment in derived magnetic energy. Thus, for example, the esti-\nmated free magnetic energy obtained from SDO /HMI is 14:4%\nhigher than that of SOLIS /VSM. This is due to the fact that\nHMI data includes small scale magnetic fields measurements.\nWe study the magnetic pressure, pm, in a longitudinal cross-\nsection at about \u0012=20\u000eas shown in Figure 4. The overall\npattern of pmappears to be the same when calculated from the\nHMI and VSM NLFFF model volume. In this study we found\nthat the magnetic pressure of NLFFF model field from HMI is\ngreater than that of VSM for same locations in the cross section.\nThis is expected, since the magnetic pressure is proportional to\nthe magnetic energy density as the magnetic energy of NLFFF\nmodel field from HMI is larger than that of VSM. The surface\nradial ( Jr) and transverse ( Jt) electric current densities of the\nNLFFF field models based on HMI and VSM data are shown\nin in Figure 5. The value of the total radial surface electric cur-\nrent density flux of the NLFFF field models based on HMI is\ngreater that than that of VSM. It agrees with fact that the HMI\ninstrument measures more transverse magnetic field than that ofVSM instrument. The transverse surface electric current density\nof the NLFFF field model based on HMI spreads more around\nthe active regions than that of VSM as shown in Figure 5. This\ncould reflect the fact (Pietarila et al. 2012, see) that the scal-\ning factor between SOLIS /VSM and SDO /HMI is di \u000berent for\nweak and strong fluxes. This di \u000berence is scaling factor may act\nas a weighting function when comparing electric currents de-\nrived from two models. In addition, the vector correlations of\nthe radial and transverse surface electric current densities of the\nNLFFF field models based on HMI and VSM are 0.96 and 0.88,\nrespectively. This indicate that there is more pronounced dis-\ncrepancy in transverse electric current densities than radial one.\n5. Conclusion and outlook\nWe have investigated the coronal magnetic field associated with\nfull solar disk on November 09 2011 by analysing SDO /HMI\nand SOLIS /VSM data. We carried out nonlinear force-free coro-\nnal field extrapolations of a full disk magnetograms. The vec-\ntor magnetogram is almost perfectly flux balanced and the field\nof view was large enough to cover all the weak field surround-8 T. Tadesse et al.: Full-disk NLFFF extrapolation of SDO /HMI and SOLIS /VSM data\nFig. 4. Magnetic pressure pmin the longitudinal cross-section at\n\u0012=20\u000efor the (a) VSM and (b) HMI. The vertical and horizon-\ntal axes show radial distance in solar radius and longitude, \u001eon\nthe photosphere, respectively.\ning the active regions. Both conditions are necessary in order to\ncarry out meaningful force-free computations. We have used the\noptimization method for the reconstruction of nonlinear force-\nfree coronal magnetic fields in spherical geometry (Wiegelmann\n2007; Tadesse et al. 2009) to campare the final NLFFF model\nfield solution from HMI and VSM full disk data.\nWe have found that the optimal value for smoothing param-\neter is\u00164=0:05 for full-disk HMI data for the purpose of pre-\nprocessing. We conclude that the choice \u0017=0:001 is the optimal\nchoices for HMI full disk data set for our new code as investi-\ngated in Wiegelmann et al. (2012).\nThe magnetic field lines obtained from nonlinear force-free\nextrapolation based on HMI and VSM data have good corre-\nlations. However, the models show some disagreement on the\nestimated relative free magnetic energy which can be released\nduring explosive events and surface electric current density.\nReconstructed magnetic field based on SDO /HMI data have\nmore contents of total magnetic energy, free magnetic energy,\nthe longitudinal distribution of the magnetic pressure and surface\nelectric current density compared to SOLIS /VSM data. Since\nthe disagreement in free energy can be attributed to presence\nof weaker transverse fields in SDO /HMI measurements, it is not\nclear how important is the found (14.4%) di \u000berence in free mag-\nFig. 5. Vector plot of the radial component of electric current\ndensity and vector field plot of transverse component of electric\ncurrent density with white arrows. The colour coding shows Jr\non the photosphere. The vertical and horizontal axes show lati-\ntude,\u0012and longitude, \u001eon the photosphere respectively.\nnetic energy for flare and CME processes originating in magnetic\nfields higher in the corona. This aspect deserves a separate study.\nAcknowledgements. The authors thank the anonymous referee for helpful com-\nments. Data are courtesy of NASA /SDO and the AIA and HMI science teams.\nSOLIS /VSM vector magnetograms are produced cooperatively by NSF /NSO\nand NASA /LWS. The National Solar Observatory (NSO) is operated by the\nAssociation of Universities for Research in Astronomy, Inc., under cooperative\nagreement with the National Science Foundation. This work was supported by\nNASA grant NNX07AU64G and the work of T. Wiegelmann was supported by\nDLR-grant 50 OC 453 0501.\nReferences\nAly, J. J. 1989, Sol. Phys., 120, 19\nAmari, T. & Aly, J. 2010, A&A, 522, A52 +\nAmari, T., Aly, J. J., Luciani, J. F., Boulmezaoud, T. Z., & Mikic, Z. 1997, Sol.\nPhys., 174, 129\nAmari, T., Boulmezaoud, T. Z., & Aly, J. J. 2006, A&A, 446, 691\nAmari, T., Boulmezaoud, T. Z., & Mikic, Z. 1999, A&A, 350, 1051\nAschwanden, M. J. 2008, Journal of Astrophysics and Astronomy, 29, 115\nAuer, L. H., House, L. L., & Heasley, J. N. 1977, Sol. Phys., 55, 47\nBorrero, J. M., Tomczyk, S., Kubo, M., et al. 2011, Sol. Phys., 273, 267\nCanou, A. & Amari, T. 2010, ApJ, 715, 1566\nCanou, A., Amari, T., Bommier, V ., et al. 2009, ApJL, 693, L27\nDeRosa, M. L., Schrijver, C. J., Barnes, G., et al. 2009, ApJ, 696, 1780\nGary, G. A. 2001, Sol. Phys., 203, 71\nGeorgoulis, M. K. 2005, ApJL, 629, L69T. Tadesse et al.: Full-disk NLFFF extrapolation of SDO /HMI and SOLIS /VSM data 9\nGuo, Y ., Ding, M. D., Schmieder, B., et al. 2010, ApJL, 725, L38\nInhester, B. & Wiegelmann, T. 2006, Sol. Phys., 235, 201\nJiang, C. & Feng, X. 2012, ApJ, 749, 135\nJones, H. P., Harvey, J. W., Henney, C. J., Hill, F., & Keller, U. C. 2002, ESA\nSP, 505, 15\nKeller, U. C., Harvey, J. W., & Giampapa, M. S. 2003, 4853, 194\nLeka, K. D., Barnes, G., Crouch, A. D., et al. 2009, Sol. Phys., 260, 83\nMetcalf, T. R. 1994, Sol. Phys., 155, 235\nMetcalf, T. R., De Rosa, M. L., Schrijver, C. J., et al. 2008, Sol. Phys., 247, 269\nMetcalf, T. R., Leka, K. D., Barnes, G., et al. 2006, Sol. Phys., 237, 267\nMolodensky, M. M. 1969, Soviet Astron.-AJ, 12, 585\nPietarila, A., Bertello, D., Harvey, F., & Pevtsov, A. 2012, Sol. Phys.\nR´egnier, S. & Amari, T. 2004, 425, 345\nR´egnier, S., Amari, T., & Kersal ´e, E. 2002, A&A, 392, 1119\nR´egnier, S. & Priest, E. R. 2007a, A&A, 468, 701\nR´egnier, S. & Priest, E. R. 2007b, ApJ, 669, L53\nRonan, R. S., Mickey, D. L., & Orrall, F. Q. 1987, Sol. Phys., 113, 353\nSandman, A. W. & Aschwanden, M. J. 2011, solphys, 270, 503\nSchou, J., Scherrer, P. H., Bush, R. I., et al. 2012, Sol. Phys., 275, 229\nSchrijver, C. J. 2009, Adv. Space Res., 43, 739\nSchrijver, C. J., Derosa, M. L., Metcalf, T. R., et al. 2006, Sol. Phys., 235, 161\nSchrijver, C. J. & Title, A. M. 2011, J. Geophys. Res., 116, A04108\nSkumanich, A. & Lites, B. W. 1987, ApJ, 322, 473\nTadesse, T., Wiegelmann, T., & Inhester, B. 2009, A&A, 508, 421\nTadesse, T., Wiegelmann, T., Inhester, B., & Pevtsov, A. 2011, A&A, 527, A30 +\nTadesse, T., Wiegelmann, T., Inhester, B., & Pevtsov, A. 2012a, Sol. Phys., 60\nTadesse, T., Wiegelmann, T., Inhester, B., & Pevtsov, A. 2012b, Sol. Phys., 277,\n119\nThalmann, J. K., Pietarila, A., Sun, X., & Wiegelmann, T. 2012, Astronomical J.\nThalmann, J. K., Wiegelmann, T., & Raouafi, N.-E. 2008, A&A, 488, L71\nTurmon, M., Jones, H. P., Malanushenko, O. V ., & Pap, J. M. 2010, Sol. Phys.,\n262, 277\nUnno, W. 1956, Publ. Astron. Soc. Japan, 8, 108\nValori, G., Green, L. M., D ´emoulin, P., et al. 2012, Sol. Phys., 278, 73\nValori, G., Kliem, B., & Keppens, R. 2005, A&A, 433, 335\nWheatland, M. S. 2004, Sol. Phys., 222, 247\nWheatland, M. S. & Leka, K. D. 2011, ApJ, 728, 112\nWheatland, M. S. & R ´egnier, S. 2009, ApJ, 700, L88\nWheatland, M. S., Sturrock, P. A., & Roumeliotis, G. 2000, ApJ, 540, 1150\nWiegelmann, T. 2004, Sol. Phys., 219, 87\nWiegelmann, T. 2007, Sol. Phys., 240, 227\nWiegelmann, T. & Inhester, B. 2010, A&A, 516, A107 +\nWiegelmann, T., Inhester, B., & Sakurai, T. 2006, Sol. Phys., 233, 215\nWiegelmann, T., Thalmann, J. K., Inhester, B., et al. 2012, Sol. Phys., 67"    },    {        "title": "2008.03119v1.Anisotropic_Ultrafast_Spin_Dynamics_in_Epitaxial_Cobalt.pdf",        "content": "Anisotropic Ultrafast Spin Dynamics in Epitaxial Cobalt\nVivek Unikandanunni,1Rajasekhar Medapalli,2, 3Eric E. Fullerton,2\nKarel Carva,4Peter M. Oppeneer,5and Stefano Bonetti1, 6,∗\n1Department of Physics, Stockholm University, SE-10691 Stockholm, Sweden\n2Center for Memory and Recording Research, University of California San Diego, San Diego, CA 92093, USA\n3Department of Physics, School of Sciences, National Institute of Technology, Andhra Pradesh-534102, India\n4Department of Condensed Matter Physics, Faculty of Mathematics and Physics,\nCharles University, Ke Karlovu 5, CZ 121 16 Prague, Czech Republic\n5Department of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-75120 Uppsala, Sweden\n6Department of Molecular Sciences and Nanosystems,\nCa’ Foscari University of Venice, 30172 Venice, Italy\nWe investigate the ultrafast spin dynamics in an epitaxial hcp(1 ¯100) cobalt thin film. By perform-\ning pump-probe magneto-optical measurements with the magnetization along either the easy or hard\nmagnetic axis, we determine the demagnetization and recovery times for the two axes. We observe a\n35% slower dynamics along the easy magnetization axis, which we attribute to magneto-crystalline\nanisotropy of the electron-phonon coupling, supported by our ab initio calculations. This points\ntowards an unambiguous and previously undisclosed role of anisotropic electron-lattice coupling in\nultrafast magnetism.\nUltrafast quenching of magnetic order at sub-\npicosecond time scales triggered by femtosecond laser\npulses and its subsequent recovery were observed in a\nferromagnetic nickel thin film in the pioneering experi-\nment by Beaurepaire et al. [1]. Since then, many experi-\nments have confirmed the occurence of this phenomenon\nin metallic thin-film ferromagnets [2–19]. Significant the-\noretical progress [20–26] has been made towards finding\nthe fundamental microscopic mechanisms able to explain\nhow angular momentum is lost and recovered at these\nultrafast time scales, orders of magnitude faster than ex-\npected, e.g. by the textbook Landau-Lifshitz theory [27].\nDespite a two-decade long quest, a complete understand-\ning of the phenomenon is still lacking.\nThe role of the lattice in ultrafast magnetism has been\ndiscussed since the early years following the pioneering\nexperiment. The Elliott-Yafet-type spin-flip scattering\nwas put forward as a possible mechanism through which\nangular momentum can be transferred from the spin sys-\ntem to the lattice, although the efficiency of this mech-\nanism has been debated [22, 23, 28–30]. Surprisingly,\nonly very few experimental studies [31–35] have investi-\ngated epitaxial systems, where the crystalline structure\nof the sample can be properly modeled. Recently, us-\ning femtosecond x-ray diffraction, it was observed that a\nfemtosecond optical pulse can trigger ultrafast coherent\nterahertz longitudinal acoustic phonons (up to 4 THz) in\nan epitaxial iron thin film [34], disproving the common\nassumption that the lattice cannot respond coherently on\nultrafast time scales. Even more recently, another ultra-\nfast x-ray experiment on a similar iron film [35] suggested\nthe possibility of the ultrafast version of the Einstein-\nde Haas experiment, where the demagnetization of the\nmaterial is compensated by a coherent mechanical ro-\ntation of the body, in this case, driven by generation\n∗stefano.bonetti@fysik.su.seof transverse acoustic phonons at terahertz frequencies.\nNotwithstanding, the unambiguous detection of the in-\nvolvement of the lattice structure in ultrafast magnetiza-\ntion dynamics is still to be achieved.\nIn this Letter, we investigate a different model sys-\ntem, an epitaxial hcp(1 ¯100) cobalt thin film, which we\nprobe with a femtosecond optical pump-probe setup. Us-\ning the time-resolved magneto-optical Kerr effect (TR-\nMOKE), we measure the demagnetization and recov-\nery of the sample magnetization on the femto- and pi-\ncosecond time scales. The magnetization is set along\nthe easy or hard magnetization axis, which corresponds\nto two orthonormal lattice directions in the thin film\nplane. Surprisingly, there do not yet exist systematic\nstudies on possible anisotropic ultrafast spin dynamics in\nepitaxial model systems with strong magnetocrystalline\nanisotropy, which would allow to pin-down the role of\nthe orientation-dependent electron-phonon coupling. In\nthe following, we show that our measurements are able\nto reveal distinct magnetization dynamics coupled to the\nanisotropic lattice structure, even without the atomic res-\nolution given by an x-ray probe, and that is consistently\nexplained by ab initio calculations of the anisotropic\nelectron-phonon interaction.\nA 15-nm-thick epitaxial hcp(1 ¯100)-cobalt thin film\nwas grown as Co[1 ¯100] on a MgO(110) substrate and a\nCr(211) seed layer. The Co layer was capped with a\n3-nm-thick Pt layer. The easy axis of magnetization is\nalong thec−axis [0001] and lies in the plane of the film, as\nshown in Fig. 1(a). The hard axis of magnetization [11 ¯20]\nis perpendicular to it and also in the sample plane. This\nstrong in-plane magnetic anisotropy of the film enabled\nus to measure the ultrafast demagnetization along two\ndifferent crystalline orientations by a simple in-plane ro-\ntation of the sample, as shown schematically in Fig. 1(b).\nMagneto-optical loops along the easy (orange) and hard\n(blue) magnetization axes measured with the longitudi-\nnal MOKE are shown in Fig. 1(c). The loops are qual-arXiv:2008.03119v1  [cond-mat.mes-hall]  7 Aug 20202\n(a)\npump(400 nm)probe(800 nm)Hextb(b)b\n(c)Hext\n[0001][11%00][112%0][0001][112%0][112%0][0001]\n[0001][112%0]\nFIG. 1. (a) Crystallographic directions of the hcp Co film and\ngeometry of the MOKE setup. (b) Relative orientation of the\nsample with respect to the externally applied magnetic field\nHextparallel to (left panel) the [0001] easy magnetization axis\nand (right panel) to the [11 ¯20] hard magnetization axis. (c)\nMagnetization loops along the easy (orange) and hard (blue)\naxes measured using the longitudinal MOKE.\nitatively similar to the vibrating sample magnetometer\ndata presented in the Supplemental Material (SM) [36].\nThe probing configuration and laser setup are the same\nas for the time-resolved data shown below. In this way,\nthe time-resolved data can be directly normalized with\nrespect to the magnetization loops.\nThe pump-probe experiments were performed with an\namplified Ti:Sapphire laser, with a temporal resolution of\napproximately 40 fs, repetition rate of 1 kHz and a cen-\ntral wavelength of 800 nm. As pump, we used the 400-nm\noptical pulses generated by frequency doubling the fun-\ndamental of the laser, using a β-barium borate crystal.\nAs probe, we used the fundamental of the laser at 800\nnm. The pump was incident at an angle θpump≈10 de-\ngrees and the probe at an angle θ2≈55 degrees, as shown\nin Fig. 1(a). In this configuration, we are optimized to\nmeasure the longitudinal MOKE, proportional to the in-\nplane component of the magnetization [37]. In addition,\nthe different pump and probe energies allow us to sup-\npress coherent optical artefacts due to the formation of\ntransient gratings in the film [38].\nFigures 2(a)-(b) show the TR-MOKE measurements\nperformed at selected fluences for the easy and hard\nmagnetization axes. The delay traces are calculated as\nthe difference of the delay traces recorded using mag-\nnetic fields of equal magnitude but opposite sign, and\nthen normalized by the amplitude of the magnetization\nloops shown in Fig. 1(c). The applied magnetic fields\nwere±400 mT and ±1000 mT for the easy and, respec-\ntively, hard magnetization axis, enough to reach satu-\nration. These values were chosen to result in the same\neffective field, as demonstrated by the measurement of\nthe sample FMR, see SM [36]. We have however checked\nthat none of the observations reported below depend on\nthe magnitude of the external field, given that the sample\nis saturated. Plotting the difference of opposite fields al-\n20\n15\n10\n5\n0K/K0 (%)\n(a)\nEasy axis - [0001]\n1 mJ/cm2\n1.5 mJ/cm2\n2 mJ/cm2\n0 1 2 3 4 5\nTime(ps)20\n15\n10\n5\n0K/K0 (%)\n(b)\nHard axis - [1120]\n1 mJ/cm2\n1.5 mJ/cm2\n2 mJ/cm2FIG. 2. Transient magnetization dynamics in the hcp Co sam-\nple measured along the (a) easy and (b) hard magnetization\naxes. The pump is s-polarized and the probe is p-polarized.\nThe calculated fluence absorbed by the film is given in the\nlegend.\nlows for isolating the pure transient magnetic signal, and\nremove the contribution from transient reflectivity signal.\nThe transient reflectivity is shown in the SM [36] and has\na maximum relative change of 0.2% for the highest flu-\nence. This value has to be compared with the maximum\nrelative change in Kerr rotation of about 20%. The much\nlarger variation indicates that the Kerr signal is measur-\ning genuine magnetization dynamics and not optical arti-\nfacts, i.e. ∆ θK(t)/θK0= ∆M(t)/M0[39]. Furthermore,\nwe checked the dynamical Kerr rotation and ellipticity\nof all combinations of sandppump and probe polar-\nizations [31, 40, 41], and the MOKE response stayed the\nsame in shape and amplitude, within the experimental\nuncertainty.\nThe demagnetization curves shown in Fig. 2 illustrate3\n0.0 0.5 1.0 1.5 2.0 2.5\nAbsorbed fluence (mJ/cm2)02004006008001000Fast recovery time, R1 (fs)\nEasy axis [0001]\nHard axis [1120]\nFIG. 3. Fluence dependence of the fast recovery time τR1of\nthe magnetization for (orange) easy and (blue symbols) hard\nmagnetization axes. Solid lines are linear fits to the data,\nimposing the crossing of the origin of the plot.\nthe typical response observed in this experiment: a rapid\nquench of the magnetization on a time scale of the order\nof∼100 fs followed by a fast recovery on the time scale\nof∼1 ps, and finally a much slower recovery on the 10\nto 100 ps time scales. The figure also shows that the\nmaximum demagnetization increases monotonically with\nthe absorbed fluence. In order to accurately determine\nthe demagnetization and recovery time constants, we fit-\nted the ultrafast demagnetization data with the equa-\ntion given in Ref. [42]. In those equations, the dynamics\nis described by the decay time τmfor the ultrafast de-\nmagnetization, τR1for the fast recovery, and τR2for the\nslow recovery, which we can extract by a careful fitting\nprocedure described in the SM [36].\nFor the three fluences presented in Fig. 2, there is only\nnegligible difference (i.e. within the error bars) in the de-\nmagnetization amplitude at each fluence for the two dif-\nferent magnetization orientations. The change of demag-\nnetization time constant τmfor these fluences is below\nthe resolution of our measurement [9], and we obtained\nthe best fit with τm= 130 fs for all these measurements.\nBy using this value for τm, we could reliably extract the\nfast recovery time constant τR1and the slow recovery\ntime constant τR2for all the measurements. We also note\nthat the slow recovery time τR2is a coarse approximation\nof the dynamics, and which excludes the full response of\nthe magnetization including the ferromagnetic resonance.\nHence, we do not discuss it further in the following.\nWe instead focus on the fast recovery time τR1. Fig. 3\nshows the extracted τR1for both orientations as a func-\ntion of the absorbed fluence. The fast recovery time in-\ncreases with increasing fluence, consistent with previous\nreports [9, 43, 44]. In addition, we also observe that\n0.0 0.5 1.0 1.5\nTime(ps)40\n30\n20\n10\n0K/K0 (%)\n0.6\n0.4\n0.2\n0.0\nR/R0 (%)\nK/K0 - [0001]\nK/K0 - [1120]\nR/R0 - [0001]\nR/R0 - [1120]\nFIG. 4. Transient reflectivity (open symbols) and ultrafast\nmagnetization dynamics (solid symbols) measured with the\nexternally applied magnetic field parallel to the easy or hard\naxis of magnetization in epitaxial hcp cobalt, for an absorbed\nfluence of approximately 4 mJ/cm2. The dotted lines show\nthe time point of maximum demagnetization.\nthe fast recovery of the magnetization along the hard\naxis orientation is always faster than for the easy axis\norientation. The data can be fitted assuming a linear\ndependence and forcing the fit to go through the origin.\nThe slopes of the lines are approximately 360 fs cm2/mJ\nfor the easy magnetization axis, and 240 fs cm2/mJ for\nthe hard magnetization axis. Hence, and this is one of\nthe main experimental findings of this work, the ultra-\nfast magnetization dynamics in hcp Co recovers system-\natically faster along the hard magnetization axis than\nalong the easy axis. In the absorbed fluence range up to\n2 mJ/cm2, the ratio of the easy/hard slopes is approxi-\nmately 1.5, i.e.the recovery is approximately 50% slower\nalong the easy magnetization axis. One could argue\nthat, if this is due to an intrinsic non-equilibrium spin-\nscattering mechanism within the material, one would ex-\npect not only the magnetization recovery to be faster,\nbut also the quenching. However, the expected change\nof the demagnetization time constant at these fluences\nis comparable with the resolution of our measurement,\nand such difference may be not measurable within our\nexperimental resolution.\nIn order test this hypothesis, we looked at slower de-\nmagnetization by increasing the maximum absorbed flu-\nence by a factor two, i.e. 4 mJ/cm2, close to the sam-\nple damage threshold, observed as a permanent change\nin sample reflectivity and magneto-optical signal. The\ndata recorded at such fluence is shown in Fig. 4 for the\ntwo magnetization axes, where we also show the corre-\nsponding transient reflectivity. With the demagnetiza-\ntion process slowed down, we can now resolve the differ-4\nent time constants of the quenching, with τm,hard ≈130\nfs andτm,easy ≈180 fs, showing that indeed the hard\nmagnetization axis has an overall faster dynamics than\nthe easy axis. We also note that in this measurement\nτm,easy/τm,hard ≈τR1,easy/τR1,hard≈1.38, suggesting\nthat the same microscopic mechanism is governing the\nquenching and relaxation processes. This ratio is slightly\nsmaller than in the lower fluence cases, but it confirms\nthat the dynamics along the easy magnetization axis is\nstill substantially slower than the one along the hard axis.\nThis can be due to a saturation regime approaching, or\nsimply to sample non-uniformity. In order to check the\nrobustness of our results, we performed several measure-\nments weeks apart, with the sample and the setup both\ndismounted and mounted again, and the absorbed flu-\nence varied between 1 and 2 mJ/cm2, far away from the\ndamage threshold. Over 10 independent measurements\nfor each axis, we found that the magnetization dynamics\nalong the easy axis recovers always slower than the hard\naxis one, by 35 ±19%. A summary of this data is shown\nin the SM [36].\nWe now turn to the transient reflectivity data shown\nin Fig. 4. Along both the [0001] (easy) and [11 ¯20] (hard)\naxes, there is a measurable delay between the maximum\nchange in reflectivity and the maximum change in mag-\nnetization, by approximately 250 fs for the easy magne-\ntization axis, and 150 fs for the hard magnetization axis.\nThis is similar to what was observed in Ref. [6] for bcc\niron pumped with 800 nm pulses and probed at shorter\nwavelengths (500 −540 nm). As stated in that work, this\nindicates that the spin dynamics follows the onset of a\nnon-equilibrium electronic distribution.\nIn order to explain the magneto-crystalline anisotropy\nof the de- and re-magnetization time constants, we con-\nsider first the overall picture for energy transfer from the\nlaser-excited conduction electrons. The conduction elec-\ntrons thermalize within about 100 fs and transfer their\nenergy to the cold phonons due to electron-phonon cou-\npling. On a time scale longer than the initial electron\nthermalization, this energy transfer process is reasonably\nwell described by the two-temperature model [45]. The\nrate of increase of the lattice temperature is given by\nCph∂Tph/∂t=G(Te−Tph) whereTe(Tph) is the electron\n(lattice) temperature, Cphis the lattice heat capacity,\nandGis the electron-phonon coupling constant [46, 47].\nThis quantity can be computed ab initio ; it is given by\n[46]\nG= 2πg(/epsilon1F)/planckover2pi1kB/integraldisplay∞\n0dΩα2F(Ω) Ω, (1)\nwhereg(/epsilon1F) is the electronic density of states at the\nFermi level /epsilon1F, andα2F(Ω) is the Eliashberg function,\nwith Ω the phonon frequency variable (for explicit ex-\npressions, see [26, 30]).\nTo investigate the dependence of Gon the magne-\ntization axis, we have calculated the phonon spectra\nand electron-phonon matrix elements of hcp Co self-\nconsistently, using the ELK full-potential code [48]. Notethat the spin-orbit interaction was included, which allows\nus to examine the influence of the magnetization axis on\nG. We find a significant magneto-crystalline dependence\nofG: for magnetization along [0001] the calculated cou-\npling wasG= 1.8×1018Wm−3K−1, while for magneti-\nzation along [11 ¯20] it was 2 .8×1018Wm−3K−1. Hence,\nthe electron-phonon coupling for Malong the hard axis\nis about 50 % larger as for Malong the easy axis.\nAnalyzing next where the magneto-crystalline\nanisotropy in Gcomes from, we found that the dif-\nferences between the phonon dynamical matrices\nand phonon spectra computed for the hard and easy\nmagetization axes are small (see the SM [36] for the\ncomputed phonon spectra). However, we found that the\nelectron-phonon interaction is much more sensitive to\nthe magnetization direction, due to small shifts of the\nenergy levels near the Fermi energy induced by spin-orbit\ncoupling. The biggest changes in the electron-phonon\ninteraction were obtained for high energy phonons.\nThe implication of the larger electron-phonon coupling\nGfor the hard magnetization axis is a stronger transfer\nof energy from hot electrons to cool phonons, and thus\na faster electron cooling, leading to a faster remagneti-\nzation, which is fully consistent with the magnetization\ndynamics measured for t >0.5 ps. The high magneto-\ncrystalline anisotropy of the remagnetization rate corrob-\norates that the recovery is driven by a mechanism that\ndepends strongly on spin-orbit interaction, as the Elliott-\nYafet electron-phonon spin-flip scattering.\nAn accurate description of the demagnetization dy-\nnamics in the first few hundred fs is however a more\ncomplex issue. Note that it is observed only at very high\nfluences. Considering an electron-phonon picture, the\ntransfer of spin angular momentum from the electrons to\nthe phonons is given by the Elliott-Yafet electron-phonon\nspin-flip scattering [9], which is exactly described by the\nspin-flip Eliashberg function (see [26, 30]). This quantity\nhas a very similar spectral dependence as the conven-\ntionalα2F(Ω) but it is about 40 times smaller [26]. It has\nnevertheless the same magneto-crystalline anisotropy as\nthe common α2F. The electron-phonon spin-flip scatter-\ning forMalong the hard axis is thus larger, which would\nimply a faster magnetization decay in the first few hun-\ndred fs, consistent with our measurements. It needs to be\nemphasized, though, that in this time interval there will\nbe nonthermal electron populations that depend on the\nused fluence and several nonequilibrium processes that\ncan be involved, which would limit the validity of the\ntwo-temperature model as well as of the here-used quasi-\nequilibrium electron-phonon scattering description. We\ncan therefore only conclude that the right trend is given\non the very short time scale.\nIn conclusion, we performed ultrafast magneto-optical\npump-probe experiments on epitaxial hcp cobalt, in or-\nder to measure the magnetization dynamics along the\neasy and hard magnetization axes. We observed a sys-\ntematic 35% slower quenching and relaxation dynamics\nalong the easy magnetization axis. Our ab initio calcula-5\ntions reveal a large magneto-crystalline anisotropy in the\nelectron-lattice coupling and the Elliott-Yafet spin-flip\nscattering, which explains the observed anisotropic mag-\nnetization dynamics. Our study furthermore introduces a\nnovel approach to probe, using wavelengths in the optical\nrange, the role of the lattice anisotropy in ultrafast mag-\nnetism. We envision that future experiments that mimic\nour approach will be able to explore other crystalline ma-\nterials with well-defined lattice structures. The investi-\ngation of model systems, as opposed to polycrystalline\nones, allows moreover for theoretical models to be tested\nto a greater accuracy. We anticipate that such studies\nmay give important hints towards completely solving the\nquestion of the dissipation of angular momentum at ul-\ntrafast time scales, which is yet not settled after more\nthan two decades of research.We gratefully acknowledge B. Wehinger for useful\ndiscussion. V.U. and S.B. acknowledge support from\nthe European Research Council, Starting Grant 715452\nMAGNETIC-SPEED-LIMIT. R.M. and E.E.F. were\nsupported by U.S. Department of Energy, Office of\nScience, Office of Basic Energy Sciences, under Con-\ntracts No. DE-SC0018237. P.M.O. acknowledges sup-\nport from the Swedish Research Council (VR), the K.\nand A. Wallenberg Foundation (Grant No. 2015.0060),\nthe CRC/TRR 227 “Ultrafast Spin Dynamics, and the\nSwedish National Infrastructure for Computing (SNIC).\nK.C. acknowledges support from the Czech Science Foun-\ndation (Grant No. 18-07172S) and The Ministry of Ed-\nucation, Youth and Sports Large Infrastructures for\nResearch, Experimental Development and Innovations\nproject “e-Infrastructure CZ LM2018140.\n[1] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y.\nBigot, Physical Review Letters 76, 4250 (1996).\n[2] J. Hohlfeld, E. Matthias, R. Knorren, and K. Benne-\nmann, Physical Review Letters 78, 4861 (1997).\n[3] C. Stamm, T. Kachel, N. Pontius, R. Mitzner, T. Quast,\nK. Holldack, S. Khan, C. Lupulescu, E. F. Aziz, M. Wi-\netstruk, et al. , Nature materials 6, 740 (2007).\n[4] B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa, and\nW. J. M. De Jonge, Physical Review Letters 95, 267207\n(2005).\n[5] F. Dalla Longa, J. T. Kohlhepp, W. J. M. De Jonge, and\nB. Koopmans, Physical Review B 75, 224431 (2007).\n[6] E. Carpene, E. Mancini, C. Dallera, M. Brenna, E. Pup-\npin, and S. De Silvestri, Physical Review B 78, 174422\n(2008).\n[7] G. Malinowski, F. Dalla Longa, J. H. H. Rietjens, P. V.\nPaluskar, R. Huijink, H. J. M. Swagten, and B. Koop-\nmans, Nature Physics 4, 855 (2008).\n[8] C. Boeglin, E. Beaurepaire, V. Halt´ e, V. L´ opez-Flores,\nC. Stamm, N. Pontius, H. A. D¨ urr, and J.-Y. Bigot,\nNature 465, 458 (2010).\n[9] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf,\nM. F¨ ahnle, T. Roth, M. Cinchetti, and M. Aeschlimann,\nNature materials 9, 259 (2010).\n[10] C. La-O-Vorakiat, E. Turgut, C. A. Teale, H. C. Kapteyn,\nM. M. Murnane, S. Mathias, M. Aeschlimann, C. M.\nSchneider, J. M. Shaw, H. T. Nembach, et al. , Physical\nReview X 2, 011005 (2012).\n[11] A. Kirilyuk, A. V. Kimel, and T. Rasing, Reviews of\nModern Physics 82, 2731 (2010).\n[12] I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius,\nH. A. D¨ urr, T. A. Ostler, J. Barker, R. F. L. Evans, R. W.\nChantrell, et al. , Nature 472, 205 (2011).\n[13] T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell,\nU. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui, L. B.\nP. J. Le Guyader, E. Mengotti, L. J. Heyderman, et al. ,\nNature communications 3, 1 (2012).\n[14] S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka,\nE. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E.\nSiemens, S. Eich, et al. , Proceedings of the National\nAcademy of Sciences 109, 4792 (2012).\n[15] D. Rudolf, C. La-O-Vorakiat, M. Battiato, R. Adam,J. M. Shaw, E. Turgut, P. Maldonado, S. Mathias,\nP. Grychtol, H. T. Nembach, et al. , Nature communi-\ncations 3, 1038 (2012).\n[16] E. Turgut, J. M. Shaw, P. Grychtol, H. T. Nembach,\nD. Rudolf, R. Adam, M. Aeschlimann, C. M. Schneider,\nT. J. Silva, M. M. Murnane, et al. , Physical Review Let-\nters110, 197201 (2013).\n[17] S. Mangin, M. Gottwald, C. Lambert, D. Steil, V. Uhl´ ıˇ r,\nL. Pang, M. Hehn, S. Alebrand, M. Cinchetti, G. Mali-\nnowski, et al. , Nature materials 13, 286 (2014).\n[18] C.-H. Lambert, S. Mangin, B. C. S. Varaprasad, Y. Taka-\nhashi, M. Hehn, M. Cinchetti, G. Malinowski, K. Hono,\nY. Fainman, M. Aeschlimann, et al. , Science 345, 1337\n(2014).\n[19] E. Turgut, D. Zusin, D. Legut, K. Carva, R. Knut, J. M.\nShaw, C. Chen, Z. Tao, H. T. Nembach, T. J. Silva,\nS. Mathias, M. Aeschlimann, P. M. Oppeneer, H. C.\nKapteyn, M. M. Murnane, and P. Grychtol, Phys. Rev.\nB94, 220408 (2016).\n[20] G. Zhang and W. H¨ ubner, Physical Review Letters 85,\n3025 (2000).\n[21] M. Djordjevic and M. M¨ unzenberg, Physical Review B\n75, 012404 (2007).\n[22] D. Steiauf and M. F¨ ahnle, Physical Review B 79, 140401\n(2009).\n[23] M. Krauß, T. Roth, S. Alebrand, D. Steil, M. Cinchetti,\nM. Aeschlimann, and H. C. Schneider, Physical Review\nB80, 180407 (2009).\n[24] M. Battiato, K. Carva, and P. M. Oppeneer, Physical\nReview Letters 105, 027203 (2010).\n[25] R. Chimata, A. Bergman, L. Bergqvist, B. Sanyal, and\nO. Eriksson, Physical Review Letters 109, 157201 (2012).\n[26] K. Carva, M. Battiato, D. Legut, and P. M. Oppeneer,\nPhysical Review B 87, 184425 (2013).\n[27] L. D. Landau, Collected papers of LD Landau (Pergamon,\n1965).\n[28] J.-Y. Bigot, M. Vomir, and E. Beaurepaire, Nature\nPhysics 5, 515 (2009).\n[29] D. Steiauf, C. Illg, and M. F¨ ahnle, Journal of Magnetism\nand Magnetic Materials 322, L5 (2010).\n[30] K. Carva, M. Battiato, and P. M. Oppeneer, Physical\nReview Letters 107, 207201 (2011).6\n[31] B. Koopmans, M. Van Kampen, J. T. Kohlhepp, and\nW. J. M. De Jonge, Physical Review Letters 85, 844\n(2000).\n[32] M. Vomir, L. Andrade, L. Guidoni, E. Beaurepaire, and\nJ.-Y. Bigot, Physical Review Letters 94, 237601 (2005).\n[33] J.-Y. Bigot, M. Vomir, L. Andrade, and E. Beaurepaire,\nChemical physics 318, 137 (2005).\n[34] T. Henighan, M. Trigo, S. Bonetti, P. Granitzka,\nD. Higley, Z. Chen, M. P. Jiang, R. Kukreja, A. Gray,\nA. H. Reid, et al. , Physical Review B 93, 220301 (2016).\n[35] C. Dornes, Y. Acremann, M. Savoini, M. Kubli, M. J.\nNeugebauer, E. Abreu, L. Huber, G. Lantz, C. A. Vaz,\nH. Lemke, et al. , Nature 565, 209 (2019).\n[36] See Supplemental Material for (i) Fitting procedure\n(ii) Static characterization of the sample: crystalline,\nmagneto-crystalline and magneto-optical measurements\n(iii) Dynamic characterization: Transient reflectivity dy-\nnamics and FMR, and (iv) ab initio calculation of the\nphonon spectra, which includes Refs. [9, 42, 49].\n[37] A. K. Zvezdin and V. A. Kotov, Modern magnetooptics\nand magnetooptical materials (CRC Press, 1997).\n[38] C.-W. Luo, Y. Wang, F. Chen, H. Shih, and\nT. Kobayashi, Optics express 17, 11321 (2009).\n[39] R. P. Prasankumar and A. J. Taylor, Optical techniques\nfor solid-state materials characterization (CRC Press,\n2016).\n[40] L. Guidoni, E. Beaurepaire, and J.-Y. Bigot, Physical\nReview Letters 89, 017401 (2002).[41] E. Carpene, F. Boschini, H. Hedayat, C. Piovera,\nC. Dallera, E. Puppin, M. Mansurova, M. M¨ unzenberg,\nX. Zhang, and A. Gupta, Physical Review B 87, 174437\n(2013).\n[42] F. Dalla Longa, Laser-induced magnetization dynamics:\nan ultrafast journey among spins and light pulses , Ph.D.\nthesis, TU Eindhoven (2008).\n[43] U. Atxitia, O. Chubykalo-Fesenko, J. Walowski,\nA. Mann, and M. M¨ unzenberg, Physical Review B 81,\n174401 (2010).\n[44] C. von Korff Schmising, M. Giovannella, D. Weder,\nS. Schaffert, J. L. Webb, and S. Eisebitt, New Journal\nof Physics 17, 033047 (2015).\n[45] S. I. Anisimov, B. L. Kapeliovich, and T. L. Perel-man,\nSov. Phys. JETP 39, 375 (1974).\n[46] G. Grimvall, The Electron-Phonon Interaction in Metals\n(North-Holland Publishing Company, 1981).\n[47] P. B. Allen, Phys. Rev. Lett. 59, 1460 (1987).\n[48] We used the full-potential linear augmented plane wave\ncode ELK (http://elk.sourceforge.net/). The electron-\nphonon matrix elements are calculated self-consistently\nfor each phonon wave vector qinside a supercell com-\nmensurate with q. A 4×4×4 mesh of phonon qpoints\nwas used.\n[49] B. D. Cullity and C. D. Graham, Introduction to mag-\nnetic materials (John Wiley & Sons, 2011).Supplemental Material\nAnisotropic Ultrafast Spin Dynamics in Epitaxial Cobalt\nVivek Unikandanunni,1Rajasekhar Medapalli,2, 3Eric E. Fullerton,2\nKarel Carva,4Peter M. Oppeneer,5and Stefano Bonetti1, 6,∗\n1Department of Physics, Stockholm University, SE-10691 Stockholm, Sweden\n2Center for Memory and Recording Research, University of California San Diego, San Diego, CA 92093, USA\n3Department of Physics, School of Sciences, National Institute of Technology, Andhra Pradesh-534102, India\n4Department of Condensed Matter Physics, Faculty of Mathematics and Physics,\nCharles University, Ke Karlovu 5, CZ 121 16 Prague, Czech Republic\n5Department of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-75120 Uppsala, Sweden\n6Department of Molecular Sciences and Nanosystems,\nCa’ Foscari University of Venice, 30172 Venice, Italy\n1. Fitting procedure\nIn this section, we describe the detailed fitting procedure, and report all the extracted values. We stress again\nthat a careful fitting with properly chosen conditions is key to obtain meaningful parameters in cases like this, where\nthe number of free parameters is larger than those needed to fit the data. In order to accurately determine the\ndemagnetization and recovery time constants, we fitted the ultrafast demagnetization data with the equation given\nin Ref. [1], namely\n∆M(t)\nM0=/parenleftBigg\nA1τR1−A2τm\nτR1−τme−t/τm−τR1(A1−A2)\nτR1−τme−t/τR1−A2/radicalbig\nt/τR2+ 1/parenrightBigg\n/circleasteriskG(t), (1)\nwhere each parameter is defined as follows. ∆ M(t)/M 0is the relative change in magnetization. Here, ∆ M(t) is\nthe pump-induced change in magnetization measured with the time-resolved magneto-optical Kerr effect (MOKE),\nwhileM0is proportional to the saturation magnetization, measured as the maximum Kerr rotation when the external\nmagnetic field applied to the sample is large enough to saturate the sample. τmis the demagnetization time constant,\nτR1andτR2are the fast and, respectively, the slow recovery time constants. A1andA2are adimensional constants\nrelated to the magnitude of the demagnetization. The entire expression within the round brackets is convoluted with\na Gaussian-shaped temporal profile G(t) which accounts for the finite duration of the probe pulse.\nThere are totally five free fitting parameters (A 1, A 2,τm,τR1andτR2) in Eq. (1), since the width of the Gaussian\nfunction is fixed and set equal to the experimentally measured optical autocorrelation of the laser pulse, in our case\n40 fs. We stress that a fit to the data performed using this equation with all the parameters running free, can\nproduce reasonable χ2values but at the same time return recovery time constants that are clearly at odds with the\nexperimental evidence. This problem can be avoided by reducing the number of free parameters, considering the\nconstraints imposed by the realistic physical conditions. We first noticed that τmis extracted reliably even when all\nparameters are free running, and we noticed that a value τm= 130 fs was able to accurately fit all the measurements\ndone in the low to medium fluence regime ( ≤2 mJ/cm2). This fact is consistent with the observation of Ref. [2],\nwhere they estimated, in this fluence range, a maximum change of τmof less than 40 fs, which is also our experimental\nresolution. For this fluence range, we could hence fix τm= 130 fs, and bring the number of free parameters to four.\nWe allowed again the fit to run free with now four parameters and looked at the adimensional amplitudes A 1\nand A 2. The parameters A 1is related to the maximum demagnetization amplitude, but not the demagnetization\namplitude itself, whereas the parameter A 2is the demagnetization amplitude after the fast recovery. With the four\nfree parameters, we extracted the values of A 1and A 2, and we checked that the extracted value of A 2corresponded\nto the observed demagnetization amplitude after the fast recovery. After this step, we allowed A 1and A 2to vary\nonly within the error of the measurement, substantially fixing them.\nFinally, with only two free parameters left, namely τR1andτR2, we could get a robust and reliable fit of the\nexperimental data using Eq. (1) consistent with the experimental observations. All the parameters extracted are\nreported in Table I.\n∗stefano.bonetti@fysik.su.searXiv:2008.03119v1  [cond-mat.mes-hall]  7 Aug 20202\nThe robustness in the determination of τR1for easy and hard magnetization axes was checked with ten repeated\nmeasurements by varying the pump fluence in the range 1 −2 mJ/cm2for various regions of the samples the sample.\nThese measurements are summarized in Fig. 1(a) in the form of an histogram plots and ordered by increasing τR1\nfor the hard magnetization axis. Figure 1(b) shows a normalized Gaussian distribution of these recovery times for\nthe easy and hard axes orientations, whose half width half maximum is the standard deviation of the ten repeated\nmeasurements. This plot is included to show schematically that even by completely neglecting the fluence dependence,\nthe extracted time constants for the hard magnetization axis cluster around a substantially lower value than the time\nconstants for the easy magnetization axis. This strongly prove the robustness of our results. We calculated the ratio\nof easy axis to hard axis recovery time to be 1.35, and the standard deviation of the ratio to be 0.19, as reported in\nthe main text. We also performed measurements with four different combinations of pump and probe polarization.\nIrrespective of these combinations, we obtained the same trend in the results as the ones reported here.\nTABLE I. Table Extracted fit values using Eq. (1) for selected fluence values.\nAmplitude (%) Relaxation time\nFluence (mJ/cm2) Easy axis / Hard axis A 1 A2 τm(fs)τR1(fs)τR2(ps)\n1 Easy -22 -5.5 130 350 15\nHard -21 -5.5 130 260 10\n1.5 Easy -26 -7 130 520 17\nHard -27 -8 130 360 12\n2 Easy -33 -10 130 750 20\nHard -34 -10 130 480 13\n4 Easy -60 -18 180 730 23\nHard -59 -17 130 540 18\n020040060080010001200\n12345678910Measurement Number(a)Fast recovery time, !\"#(fs)\n(b)(b)\nFIG. 1. (a) Fast recovery time constant τR1of easy axis and hard axis for ten different measurements, with the absorbed fluence\nvarying between 1 mJ/cm2to 2 mJ/cm2. (b) Gaussian distributions of the fast recovery time constants for the measurements\ngiven in panel (a). The distribution for the easy axis is centered around 780 fs with a standard deviation of 175 fs whereas that\nfor the hard axis is centered around 550 fs with a standard deviation of 140 fs.3\n2. Structural and magnetic characterization of the sample\nIn this section, we present the measurements used to characterize the crystalline structure, magneto-crystalline\nanisotropy and magneto-optical properties of the sample. The sample stack consists of 3 nm of platinum as the cap\nlayer, 15 nm of hcp-cobalt and 5 nm of chromium as the seed layer with MgO as the substrate.\nFig. 2 shows the out-of-plane and the in-plane structural characterization using X-Ray Diffraction (XRD). Panel\n(a) of the figure shows that the c-axis is in the sample plane, while panel (b) is the rocking curve. Figure 3 shows the\nsaturation magnetization for the easy and hard axes orientations, measured using a Vibrating Sample Magnetometer\n(VSM). A bias field of approximately ±50 mT is required to saturate the sample along easy axis, whereas a field of\n±800 mT is required to saturate it along the hard axis direction. The MOKE magnetometry measurements for the\neasy and hard axes orientations are given in Fig. 4. We used p-polarized, 800 nm low-intensity femtosecond optical\npulses for this measurement. They have the same qualitative shape of the magnetization loops measured using the\nVSM. The Kerr rotation corresponding to the saturation magnetization is 0.7 mrad. This value was used as M 0to\ncalculate the relative change of magnetization in Eq. ((1)).\n(11\"00)(a)\n(b)\ntheta/2theta (deg.)Omega (deg.)counts\ncounts\nFIG. 2. XRD characterization of the sample. (a) 2 θscan for out of plane characterization. (b) Rocking curve of cobalt layer\nfor in plane characterization. The FWHM of the rocking curve is about 1.5 degree.\n(a)\n(b)\nMagnetic moment X 10-4 (emu)Magnetic moment X 10-4 (emu)\n(a)\n(b)\nMagnetic moment X 10-4 (emu)Magnetic moment X 10-4 (emu)Magnetic Moment X 10-4 (emu) (a)(b)\nFIG. 3. Magnetic characterization of the sample using a VSM. Magnetization loops measured in the plane of the sample along\n(a) the easy axis of magnetization [0001] and (b) the hard axis of magnetization [11 ¯20].4\n150\n 100\n 50\n 0 50 100 150\nHext (mT)1.0\n0.5\n0.00.51.0K (mrad)\n(a)\nEasy axis - [0001]\n2000\n 1000\n 0 1000 2000\nHext (mT)\n(b)\nHard axis - [1120]\nFIG. 4. Characterization of the sample using the magneto-optical Kerr effect. Kerr rotations measured in the plane of the\nsample along (a) the easy axis of magnetization [0001] and (b) the hard axis of magnetization [11 ¯20].5\n3. Pump-probe measurements: additional data\nFig. 5 shows the pump-induced reflectivity change in the sample. We measured the change in reflectivity of p-\npolarized probe following the pump excitation. The maximum observed change of reflectivity is less than 0.2% when\npumped with a pulse of fluence 2 mJ/cm2, whereas the relative change in the magneto-optical response was two orders\nof magnitude larger.\nWe also observed and characterized the FMR of the sample. Fig. 6(a) shows FMR oscillations for easy and hard\naxes orientations with a bias field of 1 T. The observed FMR frequency of the hard axis is 34 GHz and that of the\neasy axis is 55 GHz. This difference in FMR frequency between easy and hard axes is understood in terms of the\ndifferent effective fields along the two directions, caused by the magneto-crystalline anisotropy of cobalt, which is\nestimated to be approximately 0.6 T [3]. In order to check the consistency of our arguments, we repeated the easy\naxis measurement with a lower saturation bias field of 0.4 T, which in addition to the anisotropy field would give 1 T,\nand thus it is expected to give approximately the same FMR frequency as for the hard axis saturated at 1 T. Figure\n6(b) shows the comparison of these two measurements, and indeed the extracted FMR frequency is approximately\nthe same (34 GHz) for both orientations. We also repeated the measurements with various bias fields, all above the\nsaturation field for the respective axis, and observed that none of our results (in particular the extraction of the fast\nrecovery time constant τR1) are sensitive to the bias field value.\n0 1 2 3 4 5\nTime(ps)0.20\n0.15\n0.10\n0.05\n0.00R/R0 (%)\n(a)\nEasy axis - [0001]\n1 mJ/cm2\n1.5 mJ/cm2\n2 mJ/cm2\n0 1 2 3 4 5\nTime(ps)\n(b)\nHard axis - [1120]\n1 mJ/cm2\n1.5 mJ/cm2\n2 mJ/cm2\nFIG. 5. Transient reflectivity dynamics measured along the (a) easy [0001] and (b) hard [11 ¯20] magnetization axes. The pump\niss-polarized and the probe is p-polarized. The calculated fluence absorbed by the film is given in the legend.\n0 10 20 30 40 50 60\nTime(ps)0.4\n0.2\n0.00.20.4K/K (%)\n(a)\nhard,1T=34GHz\neasy,1T=55GHz\neasy axis - 1T\nhard axis - 1T\n0 10 20 30 40 50 60\nTime(ps)\neasy,0.4T=34GHz\n(b)\neasy axis - 0.4 T\nhard axis - 1T\nFIG. 6. FMR measurements for easy and hard axes of magnetization when (a) external bias field of 1T for both directions,\nand (b) when the easy axis is saturated with 0.4 T and the hard axis with 1 T bias fields.6\n4. Calculated phonon dispersions\nThe phonon dispersions of hcp Co were calculated ab initio , using the first-principles formalism outlined in the\nmain text. In Fig. 7 we show the computed phonon dispersions along high-symmetry lines in the hcp Brillouin zone,\nfor the magnetization either along the hard or the easy magnetization axis.\nAL M Γ A H K Γ 0102030Energy (meV)M || [0001]M || [1120]\n_\nFIG. 7. Ab initio calculated phonon dispersions of hcp Co, for magnetization Meither along the easy magnetization axis\n([0001]) or along the hard axis ([11 ¯20]).\n[1] F. Dalla Longa, Laser-induced magnetization dynamics: an ultrafast journey among spins and light pulses , Ph.D. thesis, TU\nEindhoven (2008).\n[2] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. F¨ ahnle, T. Roth, M. Cinchetti, and M. Aeschlimann, Nature\nmaterials 9, 259 (2010).\n[3] B. D. Cullity and C. D. Graham, Introduction to magnetic materials (John Wiley & Sons, 2011)."    },    {        "title": "1312.3799v2.Quantum_states__symmetry_and_dynamics_in_degenerate_spin_s_1_magnets.pdf",        "content": "arXiv:1312.3799v2  [cond-mat.stat-mech]  14 Jan 2014Quantum states, symmetry and dynamics in degenerate spin s=1 magnets\nM.Y. Kovalevsky, A.V. Glushchenko\nKharkov Institute of Physics and Technology,\nAcademicheskaya 1, Kharkov, 61108, Ukraine,\ne-mail: mikov51@mail.ru\nAbstract\nThe article deals with spin s=1 magnets. The symmetry conditions for normal and\ndegenerate equilibrium states are defined and types of magnetic or dering found out.\nFor each type of symmetry breaking the structure of sourcein th e Gibbs statistical op-\nerator has been obtained and additional thermodynamic paramete rs introduced. The\nalgebra of Poisson bracket for magnetic degrees of freedom has b een established and\nnonlinear dynamic equations have been derived. Using the models of t he exchange\ninteraction, we have calculated the spectra of collective excitation s for two degener-\nate states whose order parameters have different signature und er the time reversal\ntransformation.\nkeywords: quantum states, spin 1, symmetry, dynamics, spec tra.\n1 Introduction\nCurrently, there is an increasing interest in studies of hig h spin magnets, which have the\nspin s≥1. These studies are relevant because of theoretical and exp erimental work on the\nphysics of quasi-crystalline structures created on the bas is of technology of optical lattices\n[1, 2]. The capability to control geometrical parameters of the lattice and the intensity of\nthe inter particle interaction makes them attractive when s tudying collective properties\nof quantum objects. The additional stimulus is associated w ith the Bose-Einstein con-\ndensation of neutral atoms with a non-zero spin [3, 4]. The av ailable data on quadrupole\nmagnetic states [5, 6, 7] reveal the finiteness of the applica bility of traditional physical con-\ncepts of magnetism to high-spin systems. In the papers [8, 9, 10, 11] it was investigated\nthe equilibrium states of the spin s=1 magnets and considere d models of Hamiltonian\nwith a strong biquadratic interaction. Based on this Hamilt onian, phase states of low-\ndimensional magnets have been analyzed and the possibility of nematic magnetic states\nhas been predicted. Non-equilibrium processes in normal st ates of the spin s=1 magnets\nhavebeenanalyzedin[12,13,14]. Theauthorsofthesepaper susedaset ofdynamicvalues\ncorresponding to pure quantum states. In [15, 16], dynamic e quations for magnetic values\ncharacterizing mixed states have been obtained. In papers [ 17, 18] relaxation processes\nof normal states of magnets with the spin s=1 have been studie d. In [17] the structure\nof dissipative fluxes in the dynamic equations has been estab lished for the case of the\nSU(3) symmetry of Hamiltonian. In [18] the nature of collect ive excitations damping has\nbeen found and the importance of the effect of the magnetic symm etry on the relaxation\nmechanism has been noted. The description of degenerate mag netic states leads to the\nexpansion of magnetic degrees of freedom. The uniaxial spon taneous symmetry breaking\ncorresponds to the antiferromagnetic case and the magnetic phase of superfluid He3-A,\nand the biaxial symmetry breaking is observed in spin glasse s, the superfluid phase of\nHe3-B [19].\nThe expected new physical phenomena in spin 1 magnets are mai nly due to three\nfactors. With the increase of the particle spin, the set of va lues required for a macro-\nscopically complete description of ordered magnets states is expanded. The diversity of\nsymmetry properties of ζεηthe magnetic exchange interaction with s ≥1 leads to a more\n1complicated structure of the equilibrium states and non-eq uilibrium dynamic processes.\nThese magnets have several types of symmetry breaking of equ ilibrium states due to the\ndifferent properties of the order parameters under time rever sal transformation.\nThe structure of the paper is as follows: in section 2, using t he concept of quasiav-\nerages, we have discussed the properties of equilibrium sta tes with spontaneously broken\nsymmetry and introduced additional thermodynamic paramet ers. In section 3, we have\nfound subalgebras of the Poisson brackets characterizing n ormal and degenerate states.\nIn section 4, it has been obtained dynamic equations for two t ypes of degenerate states of\nmagnets and calculated spectra of collective excitations.\n2 Symmetry of normal and degenerate equilibrium states\nLet us consider mixed quantum equilibrium states of the magn ets, the particles of which\nhavespins=1, andformulatetheirsymmetryproperties. Int heinvestigated media, normal\nequilibrium states have the SO(3) or SU(3) symmetry and are d escribed by the Gibbs\nstatistical operator\nˆˆw=exp(Ω−Yaˆˆγa). (1)\nWe have denoted the second quantization operators byˆˆAto distinguish them from finite-\ndimensional matrices. In case of the SO(3) symmetry, the exc hange HamiltonianˆˆHand\nthe spin operatorˆˆSα≡ −iεαβγ/integraltext\nd3xˆˆψ+\nβ(x)ˆˆψγ(x) are integrals of motion ˆˆγa=ˆˆH,ˆˆSa,\n(a= 0,α), acting in the Hilbert space. Hereˆˆψ+\nβ,ˆˆψγare field creation and annihilation\noperators of particles with the spin s=1. Thermodynamic for cesYαconjugate of integrals\nof motion are Y−1\n0≡T– temperature and −Yα/Y0≡hα– effective magnetic field. The\nthermodynamic potential Ω is determined from the normaliza tion condition of the Gibbs\nstatistical operator Sp ˆˆw= 1. The operation of taking the trace in the Hilbert space is\ndenoted by Sp to distinguish it from the similar one used for fi nite-dimensional matrices.\nThe Hamiltonian and the normal equilibrium state satisfy th e symmetry conditions [20]\n/bracketleftBigˆˆH,ˆˆSα/bracketrightBig\n= 0,/bracketleftBig\nˆˆw,ˆˆΣα(Y)/bracketrightBig\n= 0. (2)\nThe operator of the generalized spin momentˆˆΣα(Y) is given by\nˆˆΣα(Y)≡ˆˆSα+SY\nα,SY\nα≡ −iεαβγYβ∂\n∂Yγ\nand it acts in both the Hilbert space and space of the thermody namic force Y. It satisfies\nthe commutation relations\ni/bracketleftBigˆˆΣα(Y),ˆˆΣβ(Y)/bracketrightBig\n=−εαβγˆˆΣγ(Y), i/bracketleftbig\nsY\nα,Yβ/bracketrightbig\n=εαβγYγ. (3)\nThe coincidence of the properties of the SO(3) symmetry of th e Hamiltonian and the\nnormal equilibrium state should be understood in terms of re lation (2). Formulas (2),\n(3) show that the equilibrium state is invariant under unita ry transformations of the spin\nrotationˆˆU(θ,Y) =exp(iθαˆˆΣα(Y)),ˆˆU(θ,Y)ˆˆwˆˆU+(θ,Y) =ˆˆw, whereθis the transforma-\ntion parameter. In this state, the spin sα(Y) = Spˆˆw(Y)ˆˆsαis collinear to the vector Y\nand its value in terms of the thermodynamic potential densit yω(Y) = lim V→∞Ω/Vis\ngiven bysα(Y) = 2Yα∂ω(Y)/∂Y2. The spin density in the equilibrium state tends to zero\nsα(Y)→0 atY→0. This case is similar to the SO(3) symmetric paramagnetic s tate of\nspin s=1/2 magnets.\n2TheGibbsstatistical operatorofnormalequilibriumstate s ofmagnetswithSU(3)sym-\nmetries is also given by (1). The set of additive integrals of motionˆˆγa≡/parenleftBigˆˆH,ˆˆGαβ/parenrightBig\n,(a=\n0,αβ) contains a non-Hermitian matrix operator [21].\nˆˆGαβ=/integraldisplay\nd3x/parenleftBigˆˆψ+\nα(x)ˆˆψβ(x)−δαβˆˆψ+\nγ(x)ˆˆψγ(x)/3/parenrightBig\n≡ˆˆQαβ+i\n2εαβγˆˆSγ.(4)\nIts symmetric part is the quadrupole operator, and the antis ymmetric part expressed in\nterms of the spin operator. Due to the definition (4) and the co mmutation relations of\nsecond quantization Bose operators, the following relatio n are true\n/bracketleftBigˆˆGαβˆˆGµν/bracketrightBig\n=ˆˆGανδβµ−ˆˆGµβδαν. (5)\nProperties of the SU(3) symmetry of the Hamiltonian and the e quilibrium state are\nsimilar to the formulas (2):\n/bracketleftBigˆˆH,ˆˆGαβ/bracketrightBig\n= 0,/bracketleftBig\nˆˆw,ˆˆGαβ/parenleftBig\nˆY/parenrightBig/bracketrightBig\n= 0, (6)\nwhere we introduced the operator\nˆˆGαβ/parenleftBig\nˆY/parenrightBig\n=ˆˆGαβ+GˆY\nαβ, GˆY\nαβ≡Yαλ∂\n∂Yβλ−Yλβ∂\n∂Yλα. (7)\nThermodynamic forces Yαβare conjugate of additive integrals of motionˆˆGαβ. For the\noperators (7) the following relations are true\n/bracketleftBigˆˆGαβ/parenleftBig\nˆY/parenrightBig\n,ˆˆGµν/parenleftBig\nˆY/parenrightBig/bracketrightBig\n=ˆˆGαν/parenleftBig\nˆY/parenrightBig\nδβµ−ˆˆGµβ/parenleftBig\nˆY/parenrightBig\nδαν,/bracketleftBig\nGˆY\nαβ,Yµν/bracketrightBig\n=Yανδβµ−Yµβδαν.(8)\nUsingtheformulas (7)and(8), it iseasy toseethat theequil ibriumstate isinvariant under\nunitary transformationˆˆU/parenleftBig\nˆθ,ˆY/parenrightBig\n= expiθαβˆˆGβα/parenleftBig\nˆY/parenrightBig\n:ˆˆU/parenleftBig\nˆθ,ˆY/parenrightBig\nˆˆwˆˆU+/parenleftBig\nˆθ,ˆY/parenrightBig\n=ˆˆw. To\nmeet hermiticity condition of statistical operator and uni tarity condition for the operator\nˆˆU/parenleftBig\nˆθ,ˆY/parenrightBig\nthe thermodynamic parameters satisfy relations Y∗\nαβ=Yβα,θ∗\nαβ=θβα. Let us\npresent the equilibrium value of the matrix gαβ=Spˆˆwˆˆgαβin terms of the thermodynamic\npotential:gαβ=∂ω/∂Y βα. If the matrix tends to zero gαβ→0 atYαβ→0, then this case\ncorresponds to the SU(3) symmetric paramagnetic state of th e matter.\nThe symmetry conditions (2) and (6) of the Gibbs statistical operator at degenerate\nequilibrium states are not true. Degeneracy of the state lea ds to an additional dependence\nof this operator from the parameters of the unitary transfor mationˆˆU(θ,Y), orˆˆU/parenleftBig\nˆθ,ˆY/parenrightBig\n.\nWe study such equilibrium states using the concept of quasia verages [22]. In accordance\nwith it, let us define the equilibrium statistical operator b y relation\nˆˆwν≡exp/parenleftBig\nΩν−Yaˆˆγa−νY0ˆˆF/parenrightBig\n. (9)\nThe sourceˆˆF≡/integraltext\nd3x/parenleftBig\nfa(x)ˆˆ△a(x)+h.c./parenrightBig\n, breaking the symmetry of an equilibrium\nstate, is the linear functional of the order parameter opera torˆˆ△a(x). Herefa(x) is a\nfunction conjugate of the order parameter operator, which d efines its equilibrium value in\nterms of quasiaverages △a(x) =/angbracketleftBigˆˆ△a(x)/angbracketrightBig\n≡limv→0limV→∞Spˆˆwνˆˆ△a(x). The quasiaver-\nages depend on structure the sourceˆˆFfunction. Assignment of a source structure allows\n3describe magnets with different natures of symmetry breaking of equilibrium states. Let\nus note that states with the spontaneously broken symmetry a re possible in magnets,\nwhere there are several magnetic sublattices. In this case, operators of magnetic values\nwill acquire an index of the magnetic sublattice (n):ˆˆGαβ→ˆˆG(n)\nαβ. In the multisublattice\ncase, instead of commutation relations (5), we obtain\n/bracketleftBigˆˆG(n)\nαβ,ˆˆG(m)\nµν/bracketrightBig\n=/parenleftBigˆˆG(n)\nανδβµ−ˆˆG(n)\nµβδαν/parenrightBig\nδmn. (10)\nThepropertyoftheSO(3)symmetryoftheHamiltonian isgive n by(2), wheretheoperator\nˆˆSα≡/summationtext\nmˆˆS(m)\nαhas the physical significance of the complete spin moment. In the formula\n(6), the operatorˆˆGαβ≡/summationtext\nmˆˆG(m)\nαβshall be understood as the operatorˆˆGαβ. Let us\nintroduce the order parameter operators by the relationˆˆ△αβ(x)≡dmˆˆg(m)\nαβ(x), wheredm\nare some constants, which do not simultaneously become zero or one. Formula (10) leads\nto the relation /bracketleftBigˆˆGαβ,ˆˆ△µν(x)/bracketrightBig\n=ˆˆ△αν(x)δβµ−ˆˆ△µβ(x)δαν. (11)\nGiven the connection (11) we find\ni/bracketleftBigˆˆSλ,ˆˆ△µν(x)/bracketrightBig\n=ˆˆ△αν(x)εαµλ−ˆˆ△µβ(x)ενβλ. (12)\nThis implies the following formulas\ni/bracketleftBigˆˆSλ,ˆˆ△(s)\nµν(x)/bracketrightBig\n=ˆˆ△(s)\nαν(x)εαµλ+ˆˆ△(s)\nαµ(x)εανλ, i/bracketleftBigˆˆSα,ˆˆ△β(x)/bracketrightBig\n=−εαβλˆˆ△(s)\nλ(x).(13)\nHere we introduced the symmetric and antisymmetric parts in the order parameter oper-\natorˆˆ△αβ≡ˆˆ△(s)\nαβ−iεαβγˆˆ△γ/2.\nLet us consider the source breaking the equilibrium state sy mmetry given as\nˆˆF(θ) =/integraldisplay\nd3x/parenleftBig\nξαˆˆU(θ)ˆˆ△α(x)ˆˆU+(θ)+h.c./parenrightBig\n. (14)\nHereˆˆU(θ)≡ˆˆU+(θ,Y= 0). Constant complex vector ξ=ξ1+iξ2settles the Cartesian\ncoordinate system in the spin space. Relations (13) lead to t he transformation law of the\norder parameter operator\nˆˆU(θ)ˆˆ△α(x)ˆˆU+(θ) =Rαβ(θ)ˆˆ△β(x). (15)\nThe orthogonal rotation matrix Rαβ(θ) is associated with the spin rotation parameter θ\nby the formula Rαβ(θ)≡(exp(εθ))αβ,εαβγθγ≡(εθ)αβ. The source (14), considering (15),\nis given by\nˆˆF/parenleftBig\nˆR/parenrightBig\n=/integraldisplay\nd3x/parenleftBig\nξαRαβ(θ)ˆˆ△β(x)+h.c./parenrightBig\n(16)\nand corresponds to the biaxial nature of the SO(3) symmetry b reaking.\nNow, let us consider the source (14) with a real vector ξ≡ξ1:\nˆˆF(n) =/integraldisplay\nd3xξ1αˆˆU(θ)ˆˆ△1α(x)ˆˆU+(θ) =/integraldisplay\nd3xnβ(θ)ˆˆ△1β(x). (17)\nIn this case, it has been a violation of SO(3) symmetry. The Gi bbs statistical op-\neratorˆˆw(Y,n) additionally depends on the spin anisotropy unit vector (a ntiferromag-\nnetic vector) nβ(θ)≡ξ1αRαβ(θ). The density of the thermodynamic potential of the\n4Gibbs statistical operator (9) with the source (17) is a func tion of two scalar invari-\nantsω= limv→0limV→∞Ων/V=ω(Y,n) =ω(Y2,Yn). A general case, when n/ne}ationslash= 0\nandY/ne}ationslash= 0 characterize the ferrimagnetic ordering. The special ca sen/ne}ationslash= 0,Y= 0 at\nlimY→0∂ω/∂Yn→0correspondstotheantiferromagneticordering. Iflim Y→0∂ω/∂Yn/ne}ationslash=\n0, atn/ne}ationslash= 0,Y= 0 the ferromagnetic ordering [23] is realized. Let us consi der the source\nbreaking the SO(3) symmetry of an equilibrium state given by\nˆˆF(θ) =/integraltext\nd3xξαβˆˆU(θ)ˆˆ△βα(x)ˆˆU+(θ).\nThe real matrix ξαβis symmetric and traceless: ξαβ=ξβα,ξαα= 0. It settles the\nanisotropyofmagneticdegreesoffreedominanequilibrium state. Theunitarytransforma-\ntionˆˆU(θ)transformstheorderparameteroperatorˆˆU(θ)ˆˆ△αβ(x)ˆˆU+(θ) =Rαλ(θ)Rβγ(θ)ˆˆ△λγ(x).\nTherefore,\nˆˆF(ˆm) =/integraldisplay\nd3xξβαRαλ(θ)Rβγ(θ)ˆˆ△λγ(x) =/integraldisplay\nd3xmγλ(θ)ˆˆ△λγ(x). (18)\nHeremγλ(θ) is a real, symmetric and traceless matrix. In this case, the Gibbs equilibrium\nstatistical operatordependsonthermodynamicforcesandt hesymmetricmatrix: ˆˆw(ˆY,ˆm).\nFinally, let us consider the case where a complete spontaneo us breaking of the SU(3)\nsymmetry occurs. Because of (11), the source in the Gibbs ope rator given by\nˆˆF(ˆa) =/integraldisplay\nd3xξαβˆˆU(ˆθ)ˆˆ△βα(x)ˆˆU+(ˆθ) =/integraldisplay\nd3xaαβ(ˆθ)ˆˆ△βα(x), (19)\nwhereˆξ+=ˆξ, ˆa/parenleftBig\nˆθ/parenrightBig\n≡ˆD/parenleftBig\nˆθ/parenrightBig\nˆξˆD−1/parenleftBig\nˆθ/parenrightBig\nandˆD/parenleftBig\nˆθ/parenrightBig\n≡exp/parenleftBig\n−i/parenleftBig\nˆθ/parenrightBig/parenrightBig\n. Sources (16)-(19)\ncharacterize various ways of symmetry breaking of an equili brium state. In the studied\nmagnets, there are two types of normal equilibrium states wi th the SO(3) and the SU(3)\nsymmetry and four types of degenerate states. One of them has the SU(3) symmetry\nbroken, and other three – SO(3) symmetry broken.\nUndertimereversal transformationˆˆT, thespinˆˆSγandtheorderparameterˆˆ△γchanges\nsign:ˆˆTˆˆSαˆˆT+=−ˆˆS∗\nα,ˆˆTˆˆ△αˆˆT+=−ˆˆ△∗\nα. The asterisk ”*” denotes complex conjugation.\nFor the Hamiltonian, quadrupole operator and order paramet er operatorˆˆ△(s)\nαfollowing\nrelations are true:ˆˆTˆˆHˆˆT+=ˆˆH∗,ˆˆTˆˆQαβˆˆT+=ˆˆQ∗\nαβ,ˆˆTˆˆ△(s)\nαβˆˆT+=ˆˆ△(s)∗\nαβ. Using this relations\nand taking into account (17)-(19) we can find the transformat ion law for the Gibbs statis-\ntical operators (1),(9). This allows one to find some thermod ynamic values in equilibrium\nstates.\n3 Subalgebras of the Poisson brackets of physical quantitie s\nand types of magnetic states\nIn accordance with the approach [16], for the construction o f Hamiltonian mechanics let\nus introduce Hermitian 3 ×3 matrices (ˆ a= ˆa+,ˆb=ˆb+), which are canonically conjugate\nvariables of spin s=1 magnets. This means that the following Poisson brackets are true\n{bαβ(x),bµν(x′)}= 0,{aαβ(x),aµν(x′)}= 0,\n{bαβ(x),aµν(x′)}=−δανδβµδ(x−x′).(20)\nIn terms of these matrices, we introduce the Hermitian matri x\nˆg(x)≡i/floorleftBig\nˆb(x),ˆa(x)/floorrightBig\n, (21)\n5which has the physical significance of the density of the SU(3 ) symmetry generator. Using\nthe formulas (21) and (20), we get the Poisson bracket algebr a for this variable:\ni/braceleftbig\ngαβ(x),gγρ(x′)/bracerightbig\n= (gγβ(x)δαρ−gαρ(x)δγβ)δ(x−x′). (22)\nFormulas (20) and (21) allow us to obtain the Poisson bracket of matrices ˆ a(x) and ˆg(x)\ni/braceleftbig\naαβ(x),gγρ(x′)/bracerightbig\n= (aγβ(x)δαρ−aαρ(x)δγβ)δ(x−x′). (23)\nDue to (23) {trˆa(x),gγρ(x′)}= 0, therefore it can be further assumed that trˆa= 0, so\nthat the matrix ˆ acontains eight independent variables.\nMagnetic degrees of freedom of spin s=1 magnets consist of th e spin vector sα(x) and\nthe quadrupolematrix qαβ(x), which are associated with the matrix gαβ(x) by the relation\ngαβ(x)≡qαβ(x)−iεαβγsγ(x)/2. (24)\nThese variables completely characterize the normal states of the studied magnets. The\nquadrupole matrix qαβis real, symmetric and traceless tensor: qαβ=qβα,qαα= 0. In\naddition to these variables, in degenerate case, the state i s also characterized by the matrix\nˆa. It is clear that for vector sα(x), the following Poisson brackets are true due to (24),\n(22)/braceleftbig\nsα(x),sβ(x′)/bracerightbig\n=δ(x−x′)εαβγsγ(x). (25)\nFor the variables sα(x),qαβ(x), we get\n/braceleftbig\nsα(x),qβγ(x′)/bracerightbig\n=δ(x−x′)(εαβρqργ(x)+εαγρqρβ(x)),/braceleftbig\nqαβ(x),qµν(x′)/bracerightbig\n=δ(x−x′)sγ(x)(εγανδβµ+εγβµδαν+εγβνδαµ+εγαµδβν)/4.(26)\nBy a similar way, let us connect the Hermitian matrix ˆ awith physical values\naαβ(x)≡mαβ(x)−iεαβγnγ(x)/2.\nVectornhas a physical significance of an antiferromagnetic vector. The tensor ˆ whas the\nsignificance of a T-even order parameter of the nematic order ing. Because of (23), we\nobtain the following Poisson brackets\n/braceleftbig\nsα(x),nβ(x′)/bracerightbig\n=δ(x−x′)εαβγnγ(x),/braceleftbig\nnα(x),qβγ(x′)/bracerightbig\n=δ(x−x′)(εαβρmργ(x)+εαγρmρβ(x)),/braceleftbig\nsα(x),mβγ(x′)/bracerightbig\n=δ(x−x′)(εαγρmβρ(x)+εαβρmγρ(x)),/braceleftbig\nmαβ(x),qµν(x′)/bracerightbig\n=δ(x−x′)nγ(x)(εανγδβµ+εβµγδαν+εβνγδαµ+εαµγδβν)/4.(27)\nFormulas(25)-(27)reveal thesubalgebrasofthePoissonbr ackets andallowustodetermine\nthedynamicsofmagnets withthespins=1forall thetypesofo rdering. Letuscharacterize\neach of them in detail:\nCase 1: The minimal subalgebra of the Poisson brackets (25) c ontains only the spin\nvector. The Hamiltonian formalism leads to Landau-Lifshit z equation [24] describing\ns=1/2 magnets.\nCase2: Themagneticdegrees offreedomconsistofthespinde nsityandthequadrupole\nmatrix. The dynamic equations of this kind of states have bee n obtained and analyzed in\npapers [15, 16].\nCase 3: The set of magnetic dynamic values consists of the spi n density and the vector\nof spin anisotropy, for which a closed subalgebra of the Pois son brackets (25),(27) is valid.\nThis case describes uniaxial T-odd SO(3) symmetry breaking with respect to rotations in\nthe spin space. The dynamics of such magnets is equivalent to the antiferromagnet.\n6Case 4: Themagnetic degrees of freedom consist of thespin de nsity and the orthogonal\nmatrix of rotation ˆR. To determine the Poisson brackets of the last variable with the spin\ndensity, let us note that the arbitrary orthogonal matrix ca n be expressed in terms of a\nreal anti-symmetric matrix ˆR≡(1+ˆη)(1−ˆη)−1. Let us define the matrix ηαβin terms of\nthe matrix aαβby relation ηαβ≡i(aαβ−aβα). By further using the formula (23), we get\n/braceleftbig\nRαβ(x′),sλ(x)/bracerightbig\n=δ(x−x′)(ελγαRγβ(x)−ελβγRαγ(x)). (28)\nFormulas (25),(28) are the set of Poisson brackets for the ca se of biaxial T-odd breaking\nof SO(3) symmetry.\nCase 5: The spin vector and the tensor mαβ(x) form a subalgebra of the Poisson\nbrackets (25),(27). It is a physically new case of T-even bre aking of the SO(3) symmetry,\nwhich is absent in spin s=1/2 magnets.\nCase 6: A set of magnetic values consists of Hermitian matric es ˆaand ˆg. Formulas\n(22),(23) allow us to obtain dynamics equations of spin s=1 m agnets in condition of\ncomplete breaking of SU(3) symmetry. However, in view of inc onvenience, we do not\nconsider them in this paper.\n4 Dynamic equations and excitation spectra of degenerate\nstates\nThe main interaction in magnets has an exchange nature. The c onsideration of dynamic\nprocesses requires the formulation of conservation laws in the differential form, taking into\naccount theHamiltonian symmetry. Thecondition of theSO(3 ) symmetry of the exchange\nenergy density is given by\n{Sα,e(x)}= 0. (29)\nThe exchange energy density has the form e=ehom+einhom. Here the homogeneous part\nof the energy density depends on the spin density and the vari ables associated with the\nbroken symmetry. For simplicity, we consider the contribut ion to the inhomogeneous part\nof the energy only in the form of gradients matrices ˆRor ˆm.\nCase 4. The Poisson bracket (25),(28) and the symmetry condi tion (29) lead to the\nfollowing dynamics equations\n˙sα=−∇kεαβγ/parenleftbigg∂e\n∂∇kRβλRγλ+∂e\n∂∇kRλβRλγ/parenrightbigg\n,\n˙Rαβ= (ερβγRαρ+εαγρRρβ)δH\nδsγ.(30)\nWe construct the model expression of the exchange energy den sity for spins s=1 magnets\nfrom the Casimir invariant of the Poisson bracket (25) and Ca simir invariants for an\nexpanded set of Poisson brackets (25),(28). They are R1≡trˆR,R≡trˆR2= 4cos2θ−\n1,R3≡trˆR3. Since these invariants are connected, we choose Ras a single independent\nvariable. We shall form the exchange energy model so that its homogeneous part had a\nspecific sign, and the inhomogeneous part is a positive. We ch oose the energy density as\nfollows [25]:\nehom=−1\n2As2−1\n2BR2+1\n4Es4+1\n4FR4++1\n2Js2R2,\neinhom=1\n2D(∇kRαβ)2+1\n2C∇ks2.(31)\n7HereA,B,E,F,J are effective exchange integrals of the homogeneous magnetic interac-\ntion andDis the exchange integral of inhomogeneous interaction. The stability of the\nequilibrium state in case C=0 1) s0= 0,R0= 0 is provided by inequalities A<0,B <0.\nThe Goldstone wave spectrum is linear ω=k√\n−2AD|sinθ0|. 2) The state s0= 0,R2\n0=\nB/Fis stable, if F >0,B >0,BN > FA . The Goldstone wave spectrum is linear:\nω=k/radicalbig\n2D(NB/F−A)|sinθ0|; the wave propagates transversely with respect to the axis\nθ0/|θ0|. 3) The ferromagnetic state s2\n0=A/E,R 0= 0 is stable, if: A >0,E >0,AN >\nBE. The spin wave spectrum is linear ω= 2k/radicalbig\n(s2\n0−(s0,θ0/|θ0|)2)DE|sinθ0|and the\nwave propagates transversely to the direction s0×θ0/|θ0|; 4) The equilibrium state s2\n0=\n(AF−NB)/(FE−N2),R2\n0= (ER−AN)/(FE−N2) is stable, if E >0,AF >BN,EF >\nN2,BE >AN . Thespectrumislinear ω= 2k/radicalbig\nDE(AF−NB)/(FE−N2)|sinθ0||sinϕ0|,\nwhereϕ0is angle between the spin vector sand the vector θ.\nCase 5. Considering formulas (25),(27) and the symmetry con dition (29) of the energy\ndensitye=e(s,ˆm,∇ˆm), we get equations\n˙sα=−2∇kεαβγ/parenleftbigg∂e\n∂∇kmβλmγλ/parenrightbigg\n,˙mβγ=−(εαγρmρβ+εαβρmργ)hα.(32)\nHerehα=δH/δs α. The solution of (32) at the equilibrium point leads to the re lations: 1)\nhα= 0,ˆm- const; 2)hα=hnαandmαβ=m(eαeβ−1/3δαβ), uniaxial case; 3) hα=hlα,\nmαβ=m(nαnβ−fαfβ), biaxial case. Vectors f,n,l=f×nare the orthonormal frame in\nspin space.\nLet us choose the exchange energy density model in the form of (31) with substitutions\n(trˆR2)→trˆm2and (∇Rαβ)2→(∇mαβ)2. It is clear that for this energy model, the\nfollowing equilibriumstates arepossible: 1) s0= 0,m0= 0–theparamagnetic equilibrium\nstate is stable, if: A <0,B <0; there is no real part of spectrum. 2) The solution\ns2\n0=A/E,m 0= 0 is a stable ferromagnetic equilibrium state, if: E >0,EB < JA,A >\n0. The spin wave spectrum is quadratic ω=Cs0k2; 3)s0= 0,m2\n0= 3B/2F– the\nquadrupole equilibrium state (spin nematic) is stable, if: B >0,F >0,AF < JB .\nThe quadrupole wave spectrum is given by ω=k/radicalbig\n6DB(−FA+JB+FCk2)/F. 4)\nSolutionss2\n0= (AF−BJ)/(EF−J2),m2\n0= 3(BE−AJ)/2(EF−J2), describe the\nstable equilibrium state, if BE >AJ,EF >J2,AF >BJ,E > 0. The spectrum is linear\nω= 2/radicalbig\n6DE(BE−AJ)(AF−BJ)k|sinψ|/(EF−J2), whereψis angle between the spin\nvectors0and the matrix axis ˆ m0.\nThe analysis of the symmetry of the equilibrium magnetic sta tes shows that along\nwith the two types of normal states with SO(3) or SU(3) symmet ry, there are three types\nof degenerate states: two T-odd types of SO(3) symmetry brea king (uniaxial and biaxial\nvector order parameter) and one T-even state (quadrupole or der parameter). To date,\nnot found experimental confirmation of SU(3) symmetry of the equilibrium state in spin\n1 magnets. In our work we have shown the possibility of manife station of the quadrupole\ndegree of freedom in terms of T-even SO(3) symmetry breaking of the equilibrium state\nof such magnets for which the spectra of magnetic excitation s are found.\nReferences\n[1] M. Lewenstein, A. Sanpera, V. Ahufinger, B. Damski, et al. , Advances in Physics 56\n(2007) 243.\n[2] I. Bloch, J. Dalibard, W. Zwerger, Rev. Mod. Phys. 80 (200 8) 885.\n[3] Ming-Shien Chang, Qishu Qin, Wenxian Zhang, et al., Natu re physics, 1 (2005) 111.\n8[4] R.Barnett, A.Turner, E. Demler, arXiv:cond-mat/06072 53v4 [cond-mat.str-el] 7 Nov\n(2006).\n[5] T.Matsumura, S.Nakamura, T.Goto, H. Shida, T.Suzuki, P hysicaB,223-224 (1996)\n385.\n[6] D. Hall, Z. Fisk, R. Goodrich, Phys. Rev. B 62 (2000) 84.\n[7] S. Demishev, A. Semeno, A. Bogach, et.al. Physica B: Cond ensed Matter, 378 (2006)\n602.\n[8] M. Nauciel-Bloch, G. Sarma, A. Castets, Phys. Rev.B 5 (19 72) 4603.\n[9] N. Papanicolaou, Nuclear Physics B, 305, (1988) 367.\n[10] G. Fath, J. Solyom, Phys. Rev. B, 51 (1995) 3620.\n[11] A.F. Andreev, I.A. Grishchuck, Sov. Phys. JETP, 60 (198 4) 267.\n[12] V.M. Loktev and V.S. Ostrovsky, Fiz. Nizk. Temp. 20 (199 4) 983.\n[13] B.A. Ivanov, A.K. Kolezhuk, Phys. Rev. B, 052401. (2003 ).\n[14] Kh.Kh. Muminov, ArXiv: 1206.1415v2\n[15] J.Bernatska, P. Holod, J. Phys. A: Math. Theor. 42 (2009 ) 075401.\n[16] M.Y. Kovalevsky, Tran Quang Vuong, Physics Letters A 37 4 (2010) 3676 .\n[17] M.Y. Kovalevsky, Theoretical and mathematical physic s 168 (2011) 245.\n[18] V.G. Bar’yakhtar, V.I. Butrim, A.K. Kolezhuk, B.A. Iva nov, Phys. Rev. B 87 (2013)\n224407.\n[19] D. Vollhardt, P. Wolfle, The superfluidphases of helium 3 Ed. F. Taylor. London-New\nYork-Philadelphia 1990.\n[20] N.N. Bogoliubov (jr.), M.Y. Kovalevsky, A.M. Kurbatov , S.V. Peletminsky, A.N.\nTarasov, Sov. Phys. Usp. 32 (1989): 1041. in Russian: Usp. Fi z. Nauk 159 (1989):\n585.\n[21] L.I. Plimak, C. Weib, R. Walser, W.P. Schleich, Optics C ommunications 264 (2006)\n311.\n[22] N.N. Bogolubov, N.N. Bogolubov (Jr.), Introduction to Quantum Statistical Mechan-\nics, World Scientific Publishing Company, Singapore, 2009.\n[23] N.N. Bogoliubov (jr.), M.Y. Kovalevsky, Ukrainian Jou rnal of Physics 50 (2005) 104.\n[24] L.D. Landau, E.M. Lifshits, Phys. Z. Sov. 8 (1935) 155.\n[25] P.M. Chaikin, T.C. Lubensky, Principles of condensed m atter physics (Cambridge\nUniversity Press, Cambridge, UK, 2003.)\n9"    },    {        "title": "1202.5873v1.Fast_magnetization_reversal_of_nanoclusters_in_resonator.pdf",        "content": "arXiv:1202.5873v1  [cond-mat.mes-hall]  27 Feb 2012Fast magnetization reversal of nanoclusters in\nresonator\nV.I. Yukalov1,aand E.P. Yukalova2\n1Bogolubov Laboratory of Theoretical Physics,\nJoint Institute for Nuclear Research, Dubna 141980, Russia\n2Laboratory of Information Technologies,\nJoint Institute for Nuclear Research, Dubna 141980, Russia\nAbstract\nAn effective method for ultrafast magnetization reversal of n anoclusters is sug-\ngested. The method is based on coupling a nanocluster to a res onant electric circuit.\nThis coupling causes the appearance of a magnetic feedback fi eld acting on the clus-\nter, which drastically shortens the magnetization reversa l time. The influence of the\nresonator properties, nanocluster parameters, and extern al fields on the magnetiza-\ntion dynamics and reversal time is analyzed. The magnetizat ion reversal time can be\nmade many orders shorter than the natural relaxation time. T he reversal is studied for\nboth the cases of a single nanocluster as well as for the syste m of many nanoclusters\ninteracting through dipole forces.\nPACS numbers : 75.75.Jn, 75.40.Gb, 75.50.Tt, 75.60.Jk,\naElectronic mail: yukalov@theor.jinr.ru\n11 Introduction\nThe effect of magnetization reversal in nanomaterials is of consider able importance for vari-\nous magneto-electronic devices, magnetic recording and storage , and other information pro-\ncessing techniques. The standard way of recording an information bit is to reverse the\nmagnetization by applying a magnetic field antiparallel to the magnetiz ation. From another\nside, the nanoparticle magnetic moment has to be sufficiently stable, which can be achieved\nby theuse of materialswith highmagnetic anisotropy. But thelatter complicates theprocess\nof magnetization reversal. In order to resolve the contradiction b etween these two require-\nments, different methods of magnetization reversal have been su ggested, as can be inferred\nfrom the review articles [1,2].\nMagnetization reversals caused by thermal fluctuations and phon on-assisted quantum\ntunneling are rather slow processes at low temperatures, below th e blocking temperature,\nwhere nanoclusters exhibit stable magnetization [1,2,3-10]. To make t he reversal faster, sev-\neral methods have been suggested. Thus, one can employ transv erse magnetic constant fields\n[11] or short pulses [12-17], transverse microwave alternating field s at magnetic resonance\nfrequency [18-24], and optical laser pulses [25].\nIn the present paper, we suggest another method for achieving a n ultrafast magnetization\nreversal of nanoclusters. The method is based on coupling the con sidered nanocluster with\na resonator by placing the cluster into the magnetic coil of an electr ic circuit. Then the\nmotion of the cluster magnetization produces a magnetic feedback field acting on the cluster.\nThis feedback mechanism essentially accelerates the magnetization reversal. Actually, the\neffect of the accelerated thermalization of nuclear magnets was pr oposed by Purcell [26] and\nconsidered, using the classical Bloch equations, by Blombergen and Pound [27]. Here we\nstudy the magnetization reversal by employing quantum microscop ic Hamiltonians typical\nof strongly anisotropic nanoclusters possessing large spins.\nWe study magnetic clusters with the effective sizes shorter than th e exchange length of\natoms composing them, when the magnetic cluster is in a single-domain state and its magne-\ntization can be represented by a large total spin. Such clusters ar e necessarily of nanosizes,\nwhich explains their name of nanoclusters. To avoid complications, du e to distributions\nof particle sizes and shapes, we consider the magnetization dynamic s of similar nanoclus-\nters. There are two essentially different cases. One is the magnetiz ation reversal of a single\nnanocluster. And the other is the magnetization dynamics in an ense mble of nanoclusters\ninteracting through dipolar forces. We study both these limiting cas es.\n2 Spin Dynamics of Magnetic Nanoclusters\nLet us, first, consider a single nanocluster. The typical Hamiltonian of a nanocluster, with\nthe total spin S, can be written in the form\nˆH=−µ0B·S−D(Sz)2+D2(Sx)2, (1)\nwhereµ0=−/planckover2pi1γSisthecluster magneticmoment, γS≈2µB//planckover2pi1isthegyromagnetic ratio, µB,\nBohr magneton, Bis the total magnetic field acting on the cluster, DandD2are anisotropy\nconstants. ThisHamiltonianremindstheclassical Stoner-Wohlfart hmodel [28,29]. However,\nwe start with the microscopic quantum Hamiltonian (1), where the sp in vector is treated as\nan operator. This will allow us to explicitly define all system parameter s and to take into\naccount quantum effects that can be important for the dynamics o f nanocluster assemblies.\n2One usually represents the cluster energy in a reduced form [1,2] wit h the anisotropy\nparameters related to DandD2as\nK=DS2\nV1, K 2=D2S2\nV1, (2)\nwithV1being the single-cluster volume and S, the cluster spin value. The second-order\nmagnetic anisotropy is caused by magnetocrystalline anisotropy, s hape anisotropy, and sur-\nface anisotropy [1,2]. Sometimes, one includes the fourth-order an d sixth-order anisotropy.\nHowever such higher-order anisotropy terms are usually much sma ller than the second-order\nterms, so their inclusion does not essentially change the overall pict ure.\nThe total magnetic field is the sum\nB=B0ez+Hex+B1ey (3)\nof an external constant magnetic field, generally having the longitu dinal,B0, and transverse,\nB1, components, and of the resonator feedback field Hdirected along the axis of the coil,\nwhere the cluster is inserted to. The field B0is directed opposite to the initial cluster magne-\ntization. The magnetic field, created by the coil, is described by the K irchhoff equation. The\nlatter, as is known, defines electric current generated in the coil b y varying magnetization.\nIn turn, the current produces magnetic field along the coil axis. Th e resulting equation for\nthe generated magnetic field H=H(t) can be written [30,31] in the form\ndH\ndt+2γH+ω2/integraldisplayt\n0H(t′)dt′=−4πηdmx\ndt, (4)\nin which ωis the circuit natural frequency, γis the circuit attenuation, η=V1/Vresis the\nfilling factor, with Vresbeing the volume of the resonant coil, and\nmx≡µ0\nV1/an}bracketle{tSx/an}bracketri}ht (5)\nis the average magnetization density, corresponding to the mean s pin in the direction of the\ncoil axis. Thus, moving spins produce magnetic field that acts back o n spins, accelerating\ntheir motion.\nTowritedowntheequationsofmotionforthespinoperators, itisco nvenient tointroduce\nseveral notations. We define the Zeeman frequency\nω0≡ −µ0\n/planckover2pi1B0=γSB0, (6)\nthe transverse frequency\nω1≡ −µ0\n/planckover2pi1B1=γSB1, (7)\nand the effective transverse field acting on the spin,\nf≡ −i\n/planckover2pi1µ0H+ω1. (8)\nThen the Heisenberg equations of motion for the spin operators S±=Sx±Sytake the form\ndS−\ndt=−iω0S−+fSz+iD\n/planckover2pi1/parenleftbig\nS−Sz+SzS−/parenrightbig\n+iD2\n/planckover2pi1(SxSz+SzSx),(9)\n3plus its Hermitian conjugate. And the equation for the longitudinal s pin becomes\ndSz\ndt=−1\n2/parenleftbig\nf∗S−+S+f/parenrightbig\n+D2\n/planckover2pi1(SxSy+SySx). (10)\nThe measurable quantities are the average spin components, such as the reduced trans-\nverse component\nu≡1\nS/an}bracketle{tS−/an}bracketri}ht (11)\nand the reduced longitudinal magnetization\ns≡1\nS/an}bracketle{tSz/an}bracketri}ht. (12)\nAveraging the equations of motion (9) and (10), we use the decoup ling\n/an}bracketle{tSαSβ+SβSα/an}bracketri}ht=/parenleftbigg\n2−1\nS/parenrightbigg\n/an}bracketle{tSα/an}bracketri}ht/an}bracketle{tSβ/an}bracketri}ht(α/ne}ationslash=β), (13)\nwhich preserves the correct behavior for all spins. Thus, for spin one-half it gives exactly\nzero, because of the anticommutativity of different Pauli matrices , and it becomes asymp-\ntotically exact for large spins [32-34]. We also take into account that nanoclusters possess a\nlongitudinal relaxation rate γ1that is due to phonon-assisted tunneling and to spin-phonon\ninteractions of the nanocluster with its surrounding.\nLet us introduce the longitudinal anisotropy frequency\nωD≡(2S−1)D\n/planckover2pi1, (14)\nthe transverse anisotropy frequency\nω2≡(2S−1)D2\n/planckover2pi1, (15)\nthe effective anisotropy frequency\nωA≡ωD+1\n2ω2, (16)\nand the effective rotation frequency\nΩ≡ω0−ωAs . (17)\nAlso, let us define the effective field\nF≡f+i\n2ω2u∗=−i\n/planckover2pi1µ0H+ω1+i\n2ω2u∗. (18)\nThen averaging Eqs. (9) and (10) yields the equations for the tran sverse spin component,\ndu\ndt=−iΩu+Fs , (19)\nplus its complex conjugate, and for the spin magnetization,\nds\ndt=−1\n2(u∗F+F∗u)−γ1(s−ζ), (20)\nwhereζis the equilibrium magnetization of a cluster.\nThere exists a large variety of different nanoclusters [1,2,6,35-38], b ecause of which the\nsystem parameters can take values in a wide range. Concrete exam ples will be discussed in\nthe concluding section.\n43 Approximate Analysis of Spin Dynamics\nThe equations of motion (19) and (20) are derived for arbitrary clu ster parameters. Their\nsolution can be done numerically. But before we pass to numerical inv estigation, it is useful\nto give an approximate qualitative analysis allowing for the better und erstanding of physics\ninvolved. For this purpose, we shall assume that some of the param eters are small as com-\npared to the Zeeman and resonator frequencies.\nFirst, it is possible to show [32-34] that the feedback equation (4) c an be represented as\nthe integral equation\nH=−4πη/integraldisplayt\n0G(t−t′) ˙mx(t′)dt′, (21)\nin which the transfer function is\nG(t) =/bracketleftBig\ncos(/tildewideωt)−γ\n/tildewideωsin(/tildewideωt)/bracketrightBig\ne−γt,\nwith the shifted frequency\n/tildewideω≡/radicalbig\nω2−γ2,\nand the source is\n˙mx=µ0S\n2V1d\ndt(u∗+u).\nTheresonatorfeedbackfieldefficiently actsontheclusteronlywhe ntheresonatornatural\nfrequency is tuned close to the Zeeman frequency, so that the de tuning be small, satisfying\nthe resonance condition /vextendsingle/vextendsingle/vextendsingle/vextendsingle∆\nω/vextendsingle/vextendsingle/vextendsingle/vextendsingle≪1 (∆ ≡ω−ω0). (22)\nThe effective rotation frequency (17) should also be close to ω, which requires that the\nanisotropy frequency (16) be also sufficiently small,\n/vextendsingle/vextendsingle/vextendsingleωA\nω/vextendsingle/vextendsingle/vextendsingle≪1. (23)\nWe assume that all attenuation parameters are smaller than ω, such that\nγ\nω≪1,γ1\nω≪1. (24)\nAnd let us introduce the parameter playing the role of the feedback rate\nγ0≡πηµ2\n0S\n/planckover2pi1V1=πµ2\n0S\n/planckover2pi1Vres, (25)\nwhich characterizes the attenuation caused by the coupling betwe en the cluster and res-\nonator.\nUnder these conditions, the solution to the integral equation (21) can be found by an\niterative procedure [32-34], which here gives\nµ0H=i/planckover2pi1(αu−α∗u∗), (26)\nwhere the coupling function , for small detuning, such that |∆|< γ, reads as\nα=γγ0Ω\nγ2+∆2/parenleftbig\n1−e−γt/parenrightbig\n. (27)\n5Substituting form (26) into Eqs. (19) and (20) results in the equat ions\ndu\ndt=−iΩu+αsu+ω1s−s(/tildewideαu)∗,\ndw\ndt= 2αsw+ω1(u∗+u)s−2sRe/parenleftbig\n/tildewideαu2/parenrightbig\n,\nds\ndt=−αw−ω1\n2(u∗+u)−γ1(s−ζ)+Re/parenleftbig\n/tildewideαu2/parenrightbig\n, (28)\nin which\nw≡ |u|2,/tildewideα≡α+i\n2ω2. (29)\nAccording to inequalities (22) to (24), the variables wandscan be treated as slowly\nvarying in time, while uas a fast variable. The existence of two time scales allows us to\ninvokethescaleseparationapproach[30,31,39,40]thatisavarianto ftheaveragingtechniques\n[41]. Then we solve the equation for u, keeping there wandsas quasi-integrals of motion,\nwhich gives\nu=−iω1s\nΩ+iαs+/parenleftbigg\nu0+ω1s\nΩ+iαs/parenrightbigg\nexp{−i(Ω+iαs)t}, (30)\nwhereu0≡u(0). This solution is to be substituted into the equations for the slow variables\nwands, with averaging their right-hand sides over time, thus, obtaining th e equations for\nthe guiding centers of wands.\nThe scale separation approach is a very powerful method for dealin g with a system of\nmany interacting elements, such as the assemblies of spins [30-34] o r of quantum dots [42].\nHowever, it requires the use of limitations on the range of the syste m parameters, such as\ninequalities (22)to(24). While, inourcase, wehavederived Eqs. (19 )and(20)thatarevalid\nfor arbitrary parameter values. The latter equations are not diffic ult to solve numerically.\nBut the present approximate consideration is useful for underst anding the following physical\npoints.\nThe source of creating the resonator feedback field (21) is the mo tion of the cluster spin,\nwhich generates the magnetic field (26) acting back on the spin. The back action is absent\nat the initial time. The coupling of the cluster spin and the resonator increases with time\naccording to the behavior of the coupling function (27). The trans verse spin component (30)\noscillates with time with the rotation frequency (17). The initial push to the spin oscillation\nis done by the transverse field B1entering through the transverse frequency (7). This also\nrequires that the initial magnetization (12) be nonzero. Generally, there is one more source\ntriggering the initial spin motion. These are quantum spin fluctuation s that start the spin\nmotion even in the absence of a transverse field, as has been shown for multi-spin systems\n[30-33]. Such quantum spin fluctuations areespecially important for particles with low spins.\nThe input of quantum fluctuations for large spins S≫1 diminishes as 1 /S, in agreement\nwith Eq. (13). Therefore, in the case of a single nanocluster, thes e fluctuations can be\nneglected as soon as the transverse field B1is present. The transverse field, that can be\neasily regulated, serves as a convenient triggering mechanism for in itiating spin motion.\n4 Numerical Solution of Evolution Equations\nNowwegobacktothegeneralevolutionequations(19)and(20). F ornumericalinvestigation,\nit is appropriate to pass to dimensionless quantities. To this end, we d efine the dimensionless\n6feedback field\nh≡ −µ0H\n/planckover2pi1γ0=γSH\nγ0. (31)\nThe effective forces (8) and (18), respectively, become\nf=iγ0h+ω1, F=iγ0h+i\n2ω2u∗+ω1. (32)\nInstead of the complex variable (11), let us use the real componen ts\nx≡1\nS/an}bracketle{tSx/an}bracketri}ht, y≡1\nS/an}bracketle{tSy/an}bracketri}ht, (33)\nwhen\nu=1\nS/an}bracketle{tS−/an}bracketri}ht=x−iy .\nAnd in the following, let us measure time in units of 1 /γ0.\nWith these notations, Eqs. (19) and (20) yield the equations for th e spin components:\ndx\ndt=−ω0y+ωDys+ω1s , (34)\ndy\ndt=ω0x−(ωD+ω2)xs−hs , (35)\nds\ndt=hy−ω1x+ω2xy−γ1(s−ζ). (36)\nThe feedback equation (4) can be rewritten in the form\nd2h\ndt2+2γdh\ndt+ω2h= 4d2x\ndt2. (37)\nWe assume that at the starting moment of time the cluster magnetiz ation is polarized along\nthe axisz, which implies the initial spin components\nx0≡x(0) = 0, y 0≡y(0) = 0, s 0≡s(0) = 1. (38)\nAnd for the initial feedback field we set\nh0≡h(0) = 0,˙h0≡˙h(0) = 0, (39)\nwhere the overdot signifies time derivative.\nThe overall time evolution is described by Eqs. (34) to (37), with the initial conditions\n(38) and (39). These equations possess the stable stationary so lution\nx∗= 0, y∗=ω1ζ\nω0−ωDζ, s∗=ζ , h∗=˙h∗= 0, (40)\nthat is reached in the relaxation time T1= 1/γ1. The attenuation γ1can be defined by the\nArrhenius law\nγ1=γAexp/parenleftbigg\n−EA\nkBT/parenrightbigg\n,\n7whereEA=/planckover2pi1ωASis the anisotropy barrier. At low temperatures, below the blocking\ntemperature, the attenuation γ1is exponentially suppressed, so that the relaxation time T1\nis very long. However the magnetization reversal can be ultrafast due to the action of the\nresonator feedback field. In our calculations, we set γ1= 10−3(in units of γ0) and take the\ninitial conditions (38) and (39).\nFigure 1 shows the solutions to the evolution equations (34) to (37) , with the initial\nconditions (38) and (39), as functions of time (in units of 1 /γ0) for typical parameters\nexpressed in units of γ0. The resonator feedback field realizes the magnetization reversa l\nthat can be many orders shorter than the relaxation time T1. Comparing the behavior in\nthe presence of the resonator and in its absence, we see that in th e time scale of 1 /γ0the\ninfluence of γ1is not noticeable at all.\nFigure 2 demonstrates the influence of the resonator attenuatio n on the magnetization\nreversal. Too short γmeans a large ringing time 1 /γ, when the resonator several times\nexchanges the energy with the cluster, which leads to the oscillation s of the magnetization.\nThe optimal value of the resonator attenuation is γ= 1, when the resonator attenuation γ\ncoincides with the attenuation γ0characterizing the coupling between the resonator and the\ncluster.\nFigure 3 illustrates the role of the Zeeman frequency. The larger th e latter, the shorter\nthe reversal time. However too large ω0leads to many oscillations of magnetization, which\nis not good, if one aims at the stable reversal.\nFigure 4 shows the influence of the transverse field B1entering through the transverse\nfrequency ω1. The larger the latter, the shorter the reversal time. But too lar geω1results\nin multiple oscillations, which may be inconvenient for practical purpos es.\nFigure 5 demonstrates the role of magnetic anisotropy. The anisot ropy frequencies that\nare smaller than the Zeeman frequency do not strongly influence th e reversal. But if the\nanisotropy frequencies are larger than the Zeeman frequency, t hen the reversal is blocked.\nFigure6showstheimportanceoftheresonancecondition. Larged etuningfromresonance\nmakes the magnetization reversal slower. The larger the detuning , the slower the reversal.\nThe reversal is the fastest when the resonance condition ω0=ωis valid.\nThese figures demonstrate that the coupling to a resonant electr ic circuit results in the\nultrafast magnetizationreversal of asingle nanocluster, ascomp ared to itsnatural relaxation\ntime caused by thermal fluctuations and phonon-assisted tunnelin g. Estimates for typical\nnanoclusters will be given in the concluding section.\n5 System of Nanoclusters with Dipolar Interactions\nIn the previous sections, we have treated the magnetization reve rsal of a single nanocluster.\nAn important question is whether such an ultrafast magnetization r eversal could be achieved\nfor an ensemble of nanoclusters. The basic difference of the latter case from that of a\nsingle cluster is the existence of strong dipolar interactions betwee n the nanoclusters. These\ndipolar interactions completely suppress coherent spin motion in dep hasing time T2, so that\ncollective spin rotation, without a resonant feedback, becomes imp ossible [32-34,43]. One\nshould not confuse the real dipolar interactions between spins with the effective interactions\nthrough photon exchange of resonant atoms radiating at optical frequencies. The dipolar\nspin interactions dephase spin motion, while the atomic interactions t hrough the photon\nexchange, vice versa, collectivize atomic radiation [44]. Spin motion, o f course, also produces\n8electromagneticradiationthat, however, isextremely weakandca nnever collectivize spinsin\ntimeshorter thanthedephasing time[33,34,43,44]. Self-organizedco herent atomicradiation,\ncalled superradiance, is the Dicke effect [45]. The principally different c ollectivization of spin\nmotion by means of a resonator feedback field is what is termed the P urcell effect [26]. The\nDicke effect for spin systems is impossible, so that the self-organize d coherent spin motion\nis admissible only through the Purcell effect [32-34,43,44], which neces sarily requires the\ncoupling of the spin system with a resonator.\nLet us consider a system of Nnanoclusters in volume V, with the density ρ≡N/V. The\nsystem Hamiltonian reads as\nˆH=N/summationdisplay\nj=1ˆHj+1\n2N/summationdisplay\ni/negationslash=jˆHij. (41)\nHere the first term is the sum of the single-cluster Hamiltonians\nˆHj=−µ0B·Sj−D(Sz\nj)2+D2(Sx\nj)2, (42)\nwith the cluster spin operators Sjand the index j= 1,2,...,Nenumerating the nan-\noclusters. Aiming at studying the role of the dipolar interactions, we keep in mind similar\nnanoclusters with close anisotropy parameters and spins S. The dipolar interactions are\ncharacterized by the Hamiltonian parts\nˆHij=/summationdisplay\nαβDαβ\nijSα\niSβ\nj, (43)\nwith the dipolar tensor\nDαβ\nij=µ2\n0\nr3\nij/parenleftBig\nδαβ−3nα\nijnβ\nij/parenrightBig\n, (44)\nin which\nrij≡ |rij|,nij≡rij\nrij,rij=ri−rj.\nThe resonator feedback field is described by the same Eq. (4), but with\nmx=µ0\nVN/summationdisplay\nj=1/an}bracketle{tSx\nj/an}bracketri}ht. (45)\nTo write the equations of motion in a compact form, we introduce sev eral notations. The\ndipolar terms are combined into the variables\nξ0\ni≡1\n/planckover2pi1N/summationdisplay\nj(/negationslash=i)/parenleftbig\naijSz\nj+c∗\nijS−\nj+cijS+\nj/parenrightbig\n, ξ i≡i\n/planckover2pi1N/summationdisplay\nj(/negationslash=i)/parenleftbigg\n2cijSz\nj−1\n2aijS−\nj+2bijS+\nj/parenrightbigg\n,\n(46)\nwhere\naij≡Dzz\nij, b ij≡1\n4/parenleftbig\nDxx\nij−Dyy\nij−2iDxy\nij/parenrightbig\n, c ij≡1\n2/parenleftbig\nDxx\nij−iDyz\nij/parenrightbig\n.(47)\n9In the evolution equations, as is well known, there arise pair spin cor relators that need to\nbe decoupled for obtaining a closed system of differential equations . For different clusters,\nwe use the semiclassical decoupling\n/an}bracketle{tSα\niSβ\nj/an}bracketri}ht=/an}bracketle{tSα\ni/an}bracketri}ht/an}bracketle{tSβ\nj/an}bracketri}ht(i/ne}ationslash=j), (48)\nsupplemented by the account of quantum spin correlations yielding t he appearance of the\ndephasing term γ2. And the spin correlators for different components of the same clu ster,\nsimilar to Eq. (13), are decoupled as\n/an}bracketle{tSα\njSβ\nj+Sβ\njSα\nj/an}bracketri}ht=/parenleftbigg\n2−1\nS/parenrightbigg\n/an}bracketle{tSα\nj/an}bracketri}ht/an}bracketle{tSβ\nj/an}bracketri}ht(α/ne}ationslash=β), (49)\nin order to retain the correct limiting expressions for spin one-half a nd large spins [32-34].\nThe angle brackets, as earlier, imply statistical averaging over the initial statistical operator.\nThis type of spin decoupling will lead to the appearance of the expres sions\nξ0≡1\nNN/summationdisplay\nj=1/an}bracketle{tξ0\nj/an}bracketri}ht, ξ≡1\nNN/summationdisplay\nj=1/an}bracketle{tξj/an}bracketri}ht. (50)\nAs we see, the consideration of an ensemble of magnetic nanocluste rs with dipolar in-\nteractions is essentially more complicated than that of a single nanoc luster, treated in the\nprevious sections.\n6 Magnetization Reversal in Ensemble of Nanoclusters\nThe quantities of interest are the average spin components, for w hich it is convenient to\nintroduce, instead of Eqs. (11) and (12), the relative transvers e component\nu≡1\nSNN/summationdisplay\nj=1/an}bracketle{tS−\nj/an}bracketri}ht (51)\nand the relative longitudinal component\ns≡1\nSNN/summationdisplay\nj=1/an}bracketle{tSz\nj/an}bracketri}ht. (52)\nThe effective force acting on a cluster spin, instead of Eq. (18), no w is\nF=−i\n/planckover2pi1µ0H+ω1+i\n2ω2u∗+ξ . (53)\nWriting down the equations of motion for the spin operators and ave raging them [46], we\ncome to the evolution equations for the mean spin components (51) and (52) in the form\ndu\ndt=−i(Ω+ξ0−iΓ2)u+Fs ,ds\ndt=−1\n2(u∗F+F∗u)−γ1(s−ζ),(54)\nwhere\nΓ2=γ2/parenleftbig\n1−s2/parenrightbig\n, γ 2=ρµ2\n0S\n/planckover2pi1=ρ/planckover2pi1γ2\nSS . (55)\n10Equations (54) for a nanocluster system replace Eqs. (19) and (2 0) for a single cluster.\nThese equations are to be considered together with Eq. (4) for th e resonator feedback field,\nwith\nmx=1\n2ρµ0S(u∗+u). (56)\nInstead of the feedback rate (25) for a single cluster, for Nclusters, we now have\nγ0(N) =πNµ2\n0S\n/planckover2pi1Vres=πηρ/planckover2pi1γ2\nSS . (57)\nAnd instead of the coupling function (27), we get\nα(N) =gγ2(1−As)/parenleftbig\n1−e−γt/parenrightbig\n, (58)\nwith the dimensionless coupling parameter\ng≡γω0γ0(N)\nγ2(γ2+∆2)(59)\nand the effective anisotropy parameter\nA≡ωA\nω0=2ωD+ω2\n2ω0. (60)\nNotice that, for πη∼1, the feedback rate (57) is of order of γ2. Under good resonance, with\na small detuning ∆ ∼0, the coupling parameter (59) is g∼ω0/γ.\nThe system of equations (54) and (21) can be solved numerically, eit her directly, as has\nbeendoneformagneticmolecules[47],orinvoking theaveragingtech niques [39,40],whenfast\noscillations are averaged out, so that the resulting curves are smo othed. Both these methods\ngive close results. The averaging techniques provide more physically transparent description\nof the initial stage of spin motion, showing that this motion starts wit h stochastic spin\nfluctuations caused by nonsecular terms of dipolar interactions. T hus, dipolar interactions\nplay at the initial stage the positive role of a triggering mechanism initia ting spin motion,\nwhile at the later stage they play the negative role, by destroying th e coherence of spin\nrotation.\nIn Fig. 7, we show the results of numerical calculations, involving the averaging tech-\nniques, for the time dependence of the reduced spin polarization (5 2) for different system\nparameters. Time is measured in units of γ−1\n2and all frequencies, in units of γ2. The realistic\nvalue for the anisotropy parameter (60) is taken as A= 0.1. It is worth emphasizing that\nforγ2> ω1, dipolar interactions initiate spin dynamics so that ω1plays a minor role. In\norder to stress the triggering role of the dipolar interactions, the transverse frequency is set\nto zero,ω1= 0. Analogously to Eqs. (38), the initial conditions are\nw0≡w(0) = 0, s 0≡s(0) = 1,\nwherew=|u|2=x2+y2. The figure shows that ultrafast magnetization reversal happen s\nalso in a system of nanoclusters interacting through dipolar forces . Even more interesting is\nthe fact that the dipolar interactions play the role of a triggering me chanism starting spin\ndynamics. The magnetization reversal is realized during the time of o rder 1/gγ2=T2/g,\nwhich, for g >1 is shorter that the dephasing time T2= 1/γ2. Hence it is feasible to find the\nsystem parameters, when dipolar interactions do not disturb the c oherence of spin motion,\nprovided the sample is coupled to a resonator. Without the latter, t he Purcell effect does\nnot exist and the ultrafast magnetization reversal is impossible.\n117 Discussion\nWehavesuggested amethodforrealizing anultrafastmagnetizatio nreversal ofnanoclusters.\nThe possibility of such a fast reversal is important for a number of a pplications, e.g., for\nthe functioning of various magneto-electronic devices, spintronic s, magnetic recording and\nstorage, andotherinformationprocessingtechniques [48-50]. Th eideaofthemethodisbased\nonthecouplingofthenanoclusterwitharesonantelectriccircuit. T hisiseasilyachievableby\nplacing thenanoclusters inside amagneticcoil. Then themotionofthe nanocluster magnetic\nmoment produces electric current in the circuit, which creates mag netic field acting back on\nthe cluster magnetization. This feedback field of the resonator ac celerates the magnetization\nreversal. The reversal time can be made many orders shorter tha n the natural relaxation\ntime.\nFirst, we have considered a single nanocluster, which makes it possib le to avoid compli-\ncations due to distributions of particle sizes, shapes, spin values, a nd so on, which could arise\nin the case of an assembly of many nanoclusters. In the latter case , the basic complication\nis the necessity of taking into account dipole interactions between t he clusters. All these\nadditional problems are avoided when dealing with a single cluster.\nThe case of an ensemble of nanoclusters, interacting through dipo lar forces is also an-\nalyzed. The ultrafast magnetization reversal is feasible for this ca se as well. The reversal\noccurs during the time shorter than the dipole dephasing time, beca use of which dipolar\ninteractions do not destroy coherent spin motion that is responsib le for the ultrafast rever-\nsal. Even more, dipolar interactions are useful at the initial stage, when they trigger spin\ndynamics.\nTo give the feeling of typical values for the characteristic paramet ers, let us make esti-\nmates for some nanoclusters. Actually, the family of magnetic nano clusters is very wide and\nthese can display rather different properties [1-3,51-54]. To be con crete, let us keep in mind\nthe values typical of Co, Fe, and Ni nanoclusters. The coherence radius for these clusters,\nbelow which they are in a single-domain state and can display coherent rotation of magne-\ntization is Rcoh∼10 nm. The standardly formed clusters have the radii R∼1−3 nm. The\ncorresponding cluster volume is V1∼10−20cm3. A cluster contains about N1∼103atoms.\nThe atomic density in a cluster is ρ1∼1023cm−3. The cluster spin is proportional to the\nnumber of atoms in the cluster, hence S∼103, that is, the magnetic moment is of the order\n103µB. The magnetic anisotropy parameters (2) are K1∼K2∼106erg/cm3. The fourth-\norder anisotropy is much smaller, K4∼105erg/cm3. This, for the anisotropy parameters of\nHamiltonian (1), gives D∼D2∼10−20erg. And for the fourth-order anisotropy, this would\nmakeD4∼10−27erg.\nThese values, for the anisotropy frequencies (14) to (16) yield ωD∼ω2∼ωA∼1010\nHz. This corresponds to the anisotropy field BA≡ωA/γS∼103G. The Zeeman frequency,\nfor the magnetic field B0∼1 T isω0∼1011Hz. Note that the present day facilities allow\nfor the generation of magnetic fields as high as about 100 T [55]. The f eedback rate (25) is\nγ0∼1010s−1. This rate provides the reversal time trev∼1/γ0∼10−10s.\nThe blocking temperature, below which thermally activated reversa ls are exponentially\nsuppressed is TB∼10−40 K. The typical prefactor in the Arrhenius law is γA∼109−1011\ns−1. The anisotropy energy barrier in the Arrhenius law is EA∼10−14erg. This gives the\nanisotropy temperature EA/kB∼100 K. The resulting relaxation time T1≡1/γ1, below\nthe blocking temperature, is rather long. Thus, even at the tempe ratureT= 10 K, we have\nT1∼10−5s. At the temperature T= 5 K, one has T1∼0.1 s. And for temperature T= 1\n12K, the thermal reversal time is astronomically large, being T1∼1034s. But, coupling the\nnanocluster to a resonator, produces a very short reversal tim etrev∼10−10s, independently\nof the value T1. The reversal time could be made shorter by choosing the appropr iate types\nof nanoclusters and resonator properties.\nAs has been explained above, the spin relaxation in the system of nan oclusters, which\nwould be caused by the photon exchange between different spins is n egligible [33,34,43,44].\nThe corresponding radiation width is\nγrad=2ω3µ2\n0S\n3/planckover2pi1c3=2ω3\n3ρc3γ2.\nFor the typical density of nanoclusters ρ∼1020cm−3, the spin of a cluster S∼103, and\nfrequency ω∼1011Hz, we get γrad∼10−8s−1, which is much smaller than the dipolar\nwidthγ2∼1010s−1. Therefore the relaxation time, due to the photon exchange betw een\nspins,trad= 1/γrad∼108s∼10 years, is enormously larger than the dipolar dephasing\ntimeT2∼10−10s. This confirms that the photon exchange mechanism plays no role in the\nspin relaxation, that is, the Dicke effect for spin systems does not e xist. But the ultrafast\nmagnetization reversal is completely due to the Purcell effect.\nThe typical density of an ensemble of clusters is ρ∼1020cm−3. With the natural dipolar\nwidthγ2∼1010s−1, the dephasing time is T2∼10−10s. As is seen in Fig. 7, the reversal\ntime can be an order shorter than the dephasing time, being trev∼10−11s.\nConcluding, by coupling nanoclusters to a resonant electric circuit, it is possible to realize\nultrafast magnetization reversal for single nanoclusters as well a s for assemblies of nanoclus-\nters. For such nanoclusters as Fe, Co, or Ni, the reversal time ca n be made as short as 10−11\ns.\nAcknowledgement\nThe authors acknowledge financial support from the Russian Foun dation for Basic Re-\nsearch.\n13References\n[1] W. Wernsdorfer, Adv. Chem. Phys. 118, 99 (2001).\n[2] J. Ferr´ e, Topics Appl. Phys. 83, 127 (2002).\n[3] B. Barbara, L. Thomas, F. Lionti, I. Chiorescu, and A. Sulpice, J . Magn. Magn. Mater.\n200, 167 (1999).\n[4] W. Wernsdorfer, E.B. Orozco, B. Barbara, A. Benoit, and D. Ma illy, Physica B 280,\n264 (2000).\n[5] R.H. Koch, G. Grinstein, G.A. Keefe, Y. Lu, P.L. Throuilloud, and W.J . Gallagher,\nPhys. Rev. Lett. 84, 5419 (2000).\n[6] S. Takagi, Macroscopic Quantum Tunneling (Cambridge University, Cambridge, 2002).\n[7] G.F. Goya, F.C. Fonseca, R.F. Jardim, R. Muccillo, N.L. Carreno, E. Longo, and E.R.\nLeite, J. Appl. Phys. 93, 6531 (2003).\n[8] F. Luis, J. Bartolom´ e, F. Petroff, L.M. Garcia, A. Vaures, and J . Carrey, J. Appl. Phys.\n93, 7032 (2003).\n[9] F. Luis, F. Petroff, and J. Bartolom´ e, J. Phys. Condens. Matt er16, 5109 (2004).\n[10] S. Krause, G. Herzog, T. Stapelfeldt, L. Berbil-Bautista, M. B ode, E.Y. Vedmedenko,\nand R. Wiesendanger, Phys. Rev. Lett. 103, 127202 (2009).\n[11] Y.B. Bazaliy and A. Stankiewicz, Appl. Phys. Lett. 98, 142501 (2011).\n[12] H.C. Siegmann, E.L. Garwin, C.Y. Prescott, J. Heidmann, D. Maur i, D. Weller, R.\nAllenspach, W. Weber, J. Magn. Magn. Mater. 151, 8 (1995).\n[13] O.S. Kolotov, A.P. Krasnozhon, and V.A. Pogozhev, Phys. Solid S tate40, 277 (1998).\n[14] C.H. Back, R. Allenspach, W. Weber, S.S. Parkin, D. Weller, E.L. Ga rwin, and H.C.\nSiegmann, Science 285, 864 (1999).\n[15] H.W. Schumacher, C. Chappert, R.C. Sousa, P.P. Freitas, and J . Miltat, Phys. Rev.\nLett.90, 017204 (2003).\n[16] S.J. Gamble, M.H. Burkhardt, A. Kashuba, R. Allenspach, S.S. Pa rkin, H.C. Siegmann,\nand J. St¨ ohr, Phys. Rev. Lett. 102, 217201 (2009).\n[17] A. Sukhov and J. Berakdar, Phys. Rev. B 79, 134433 (2009).\n[18] C. Thirion, W. Wernsdorfer, and D. Mailly, Nature Mater. 2, 524 (2003).\n[19] S.I. Denisov, T.V. Lyutyy, P. H¨ anggi, and K.N. Throhidou, Phys . Rev. B 74, 104406\n(2006).\n[20] P.M. Pimentel, B. Leven, B. Hillebrands, and H. Grimm, J. Appl. Phy s.102, 063913\n(2007).\n14[21] C. Nistor, K. Sun, Z. Wang, M. Wu, C. Mathieu, and M. Hadley, Ap pl. Phys. Lett. 95,\n012504 (2009).\n[22] S. Li, B. Livshitz, H.N. Bertram, E.E. Fullerton, and V. Lomakin, J . Appl. Phys. 105,\n07B909 (2009).\n[23] T. Yoshioka, T. Nozaki, T. Seki, M. Shiraishi, T. Shinjo, Y. Suzuk i, and Y. Uehara,\nAppl. Phys. Express 3, 013002 (2010).\n[24] S.V. Titov, P.M. D´ ejardin, H.E. Mrabti, and Y.P. Kalmykov, Phys. Rev. B82, 100413\n(2010).\n[25] C.D. Stanciu, F. Hansteen, A.V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T.\nRasing, Phys. Rev. Lett. 99, 047601 (2007).\n[26] E.M. Purcell, Phys. Rev. 69, 681 (1946).\n[27] N. Bloembergen and R.V. Pound, Phys. Rev. 95, 8 (1954).\n[28] E.C. Stoner and E.P. Wohlfarth, Phil. Trans. Roy. Soc. London A 240, 599 (1948).\n[29] A. Thiaville, J. Magn. Magn. Mater. 182, 5 (1998).\n[30] V.I. Yukalov, Phys. Rev. Lett. 75, 3000 (1995).\n[31] V.I. Yukalov, Phys. Rev. B 53, 9232 (1996).\n[32] V.I. Yukalov, Laser Phys. 12, 1089 (2002).\n[33] V.I. Yukalov and E.P. Yukalova, Phys. Part. Nucl. 35, 348 (2004).\n[34] V.I. Yukalov, Phys. Rev. B 71, 184432 (2005).\n[35] D.J. Sellmyer and R. Skomski, eds. Advanced Magnetic Nanostructures (Springer,\nBerlin, 2006).\n[36] J.P. Liu, E. Fullerton, O. Guttfleisch, and D.J. Sellmyer, eds. Nanoscale Magnetic Ma-\nterials and Applications (Springer, Berlin, 2009).\n[37] J.S. Miller and D. Gatteschi, Chem. Soc. Rev. 40, 3065 (2011).\n[38] J.S. Miller, Chem. Soc. Rev. 40, 3266 (2011).\n[39] V.I. Yukalov, Laser Phys. 3, 870 (1993).\n[40] V.I. Yukalov and E.P. Yukalova, Phys. Part. Nucl. 31, 561 (2000).\n[41] N.N. Bogolubov and Y.A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear\nOscillations (Gordon and Breach, New York, 1961).\n[42] V.I. Yukalov and E.P. Yukalova, Phys. Rev. B 81, 075308 (2010).\n[43] V.I. Yukalov and E.P. Yukalova, Laser Phys. Lett. 2, 302 (2005).\n15[44] V.I. Yukalov, Laser Phys. Lett. 2, 356 (2005).\n[45] R.H. Dicke, Phys. Rev. 93, 99 (1954).\n[46] V.I. Yukalov and E.P. Yukalova, Laser Phys. Lett. 8, 804 (2011).\n[47] V.I. Yukalov, V.K. Henner, and P.V. Kharebov, Phys. Rev. B 77, 134427 (2008).\n[48] F. Troiani and M. Affronte, Chem. Soc. Rev. 40, 3119 (2011).\n[49] S. Sanvito, Chem. Soc. Rev. 40, 3336 (2011).\n[50] V.I. Yukalov and D. Sornette, Laser Phys. Lett. 6, 833 (2009).\n[51] R.H. Kodama, J. Magn. Magn. Mater. 200, 359 (1999).\n[52] G.C. Hadjipanayis, J. Magn. Magn. Mater. 200, 373 (1999).\n[53] M. Jamet, W. Wernsdorfer, C. Thirion, V. Dupuis, P. M´ elinon, A . P´ erez, and D. Maily,\nPhys. Rev. B 69, 024401 (2004).\n[54] G.F. Goya and M.P. Morales, J. Metast. Nanocryst. Mater. 20, 673 (2004).\n[55] M. Motokawa, Rep. Prog. Phys. 67, 1995 (2004).\n16Figure Captions\nFig. 1. Solutions to the evolution equations: (a) x=x(t); (b)y=y(t); (c)s=s(t);\n(d)h=h(t) for the parameters ω0=ω= 10,ωD=ω1=ω2= 1,γ= 1,γ1= 10−3. The\nparameters are measured in units of γ0and time, in units of 1 /γ0. To emphasize the role of\nthe resonator feedback field, the solutions in the presence of the resonator (solid lines) are\ncompared with those for the case of no resonator (dashed lines).\nFig. 2. Role of the resonator attenuation. Magnetization as a function o f time for the\nparameters ω0=ω= 10,ωD=ω1=ω2= 1, and varying resonator attenuation: (a) γ= 0.1;\n(b)γ= 1 (solid line) and γ= 10 (dashed line).\nFig. 3. Role of the Zeeman frequency. Magnetization as a function of time for the\nparameters ωD=ω1=ω2= 1,γ= 1, and varying Zeeman frequency: (a) ω0=ω= 1 (solid\nline) and ω0=ω= 10 (dashed line); (b) ω0=ω= 100.\nFig. 4. Role of the triggering field. Magnetization as a function of time for t he parame-\ntersω0=ω= 10,ωD=ω2= 1, and varying triggering field: (a) ω1= 0.001 (solid line) and\nω1= 1 (dashed line); (b) ω1= 10.\nFig. 5. Role of the anisotropy. Magnetization as a function of time for the parameters\nω0=ω= 10,ω1= 1,γ= 1, and varying anisotropy frequencies ωD=ω2= 1 (dashed-doted\nline);ωD=ω2= 10 (solid line); ωD=ω2= 15 (dashed line).\nFig. 6. Role of the resonance. Magnetization as a function of time for the parameters\nω0= 10,ωD=ω1=ω2= 1,γ= 1, and varying detuning from the resonance, with ω= 1\n(solid line); ω= 10 (dashed-dotted line); ω= 20 (dashed line).\nFig. 7. Magnetization reversal in the system of nanoclusters with dipolar interactions.\nThe system parameters are γ1= 10−3, ∆ = 0, A= 0.1,ω1= 0. Time is measured in units\nofT2≡1/γ2= 10−10s, and the frequencies, in units of γ2. Other parameters are: γ= 10,\nω0=ω= 1000,g= 100 (solid line); γ= 1,ω0=ω= 100,g= 100 (dashed-dotted line);\nγ= 10,ω0=ω= 100,g= 10 (dashed line). The shown functions of time are: (a) coherence\nintensity w(t); (b) reduced magnetization s(t).\n170 2 4 6 8 10−1−0.8−0.6−0.4−0.200.20.40.60.81\ntx(t)\nh ≠ 0\nh = 0(a)\n0 2 4 6 8 10−1−0.8−0.6−0.4−0.200.20.40.60.81\nty(t)\nh = 0\nh ≠ 0(b)\n012345678−1−0.8−0.6−0.4−0.200.20.40.60.81s(t)\nth ≠ 0h = 0(c)\n0 2 4 6 8 10−10−8−6−4−202468\nth(t)\n(d)\nFigure 1: Solutions to the evolution equations: (a) x=x(t); (b)y=y(t); (c)s=s(t);\n(d)h=h(t) for the parameters ω0=ω= 10,ωD=ω1=ω2= 1,γ= 1,γ1= 10−3. The\nparameters are measured in units of γ0and time, in units of 1 /γ0. To emphasize the role of\nthe resonator feedback field, the solutions in the presence of the resonator (solid lines) are\ncompared with those for the case of no resonator (dashed lines).\n180 2 4 6 8 10−1−0.8−0.6−0.4−0.200.20.40.60.81\nts(t)\nγ = 0.1(a)\n0 2 4 6 8 10−1−0.8−0.6−0.4−0.200.20.40.60.81\nts(t)\nγ = 10\nγ = 1(b)\nFigure 2: Role of the resonator attenuation. Magnetization as a fu nction of time for the\nparameters ω0=ω= 10,ωD=ω1=ω2= 1 and varying resonator attenuation: (a) γ= 0.1;\n(b)γ= 1 (solid line) and γ= 10 (dashed line).\n0 1 2 3 4 5 6−1−0.8−0.6−0.4−0.200.20.40.60.81\nts(t)\nω0 = ω = 10ω0 = ω = 1(a)\n0 1 2 3 4 5 6−1−0.8−0.6−0.4−0.200.20.40.60.81\nts(t)\nω0 = ω = 100(b)\nFigure 3: Role of the Zeeman frequency. Magnetization as a functio n of time for the param-\netersωD=ω1=ω2= 1,γ= 1 and varying Zeeman frequency: (a) ω0=ω= 1 (solid line)\nandω0=ω= 10 (dashed line); (b) ω0=ω= 100.\n19012345678−1−0.8−0.6−0.4−0.200.20.40.60.81\nts(t)\nω1 = 0.001\nω1 = 1(a)\n012345678−1−0.8−0.6−0.4−0.200.20.40.60.81\nts(t)\nω1 = 10(b)\nFigure 4: Role of the triggering field. Magnetization as a function of t ime for the parameters\nω0=ω= 10,ωD=ω2= 1, and varying triggering field: (a) ω1= 0.001 (solid line) and\nω1= 1 (dashed line); (b) ω1= 10.\n012345678−1−0.8−0.6−0.4−0.200.20.40.60.81\nts(t)\nωD = ω2 = 10ωD = ω2 = 15\nωD = ω2 = 1\nFigure 5: Role of the anisotropy. Magnetization as a function of time for the parameters\nω0=ω= 10,ω1= 1,γ= 1, and varying anisotropy frequencies ωD=ω2= 1 (dashed-doted\nline);ωD=ω2= 10 (solid line); ωD=ω2= 15 (dashed line).\n20012345678−1−0.8−0.6−0.4−0.200.20.40.60.81\nts(t)\nω = 1\nω = 10ω = 20\nFigure 6: Role of the resonance. Magnetization as a function of time for the parameters\nω0= 10,ωD=ω1=ω2= 1,γ= 1, and varying detuning from the resonance, with ω= 1\n(solid line); ω= 10 (dashed-dotted line); ω= 20 (dashed line).\n00.1 0.2 0.3 0.4 0.5 0.600.20.40.60.81\ntw(t)(a)\n0 0.2 0.4 0.6 0.8 1−1−0.8−0.6−0.4−0.200.20.40.60.81\nts(t)(b)\nFigure 7: Magnetization reversal in the system of nanoclusters wit h dipolar interactions.\nThe system parameters are γ1= 10−3, ∆ = 0, A= 0.1,ω1= 0. Time is measured in units\nofT2≡1/γ2= 10−10s, and the frequencies, in units of γ2. Other parameters are: γ= 10,\nω0=ω= 1000,g= 100 (solid line); γ= 1,ω0=ω= 100,g= 100 (dashed-dotted line);\nγ= 10,ω0=ω= 100,g= 10 (dashed line). The shown functions of time are: (a) coherence\nintensity w(t); (b) reduced magnetization s(t).\n21"    },    {        "title": "1910.01218v1.Magnetic_Scattering_Chapter.pdf",        "content": "1 \n Magnetic Scattering  \n \nJeffrey W. Lynn1 and Bernhard Keimer2 \n1NIST Center for Neutron Research  \nNational Institute of Standards and Technology  \nGaithersburg, MD 20899- 6102 (USA)  \n  \n2Max -Planck Institute  for Solid State Research  \nHeisenbergstrasse 1  \nD-70569 Stuttgart (Germany)  \n \nin  \n \nHandbook of Magnetism  \nedited by Michael Coey and Stuart Parkin  \n  2 \n  \nContents  \nI. Introduction  ......................................................................................................................................... 3 \nII. Magnetic Neutron Diffraction Technique  ..................................................................................... 3 \n a.  Polarized Neutron Techniques  \n b.  Polarized Neutron Reflectometry  \nIII. Resonant Magnetic X -ray Diffraction Technique  ........................................................................ 8 \nIV.       Dynamics  ........................................................................................................................................ 11 \n          a.  Inelastic Neutron Scattering Technique  .................................................................................... 11 \n         b.  Resonant Inelastic X -ray Scattering Technique  ........................................................................ 12 \nV.     Magnetic Diffraction Examples with Neutrons  .......................................................................... 13 \nVI. Magnetic Diffraction Examples with X -rays  .............................................................................. 16 \nVII.  Spin Dynamics with Neutrons  ...................................................................................................... 18 \nVIII.     Spin Dynamics with RIXS ............................................................................................................ 20 \nIX. Facilities  and online information  .................................................................................................  21 \nX. Summary and Future Directions  ..................................................................................................... 22 \nXI. Acknowledgments  ......................................................................................................................... 22 \nReferences  \n \n  3 \n I. Introduction \nScattering measurements provide essential information about the intrinsic electronic \ninteractions in magnetic material s.  Traditionally neutron scattering has been the unique magnetic \nscattering tool,  [1] [2] but that situation has changed recently following  remarkable advances in \nresonant x -ray scattering techniques.  [3] Both techniques collect data as functions of the energy \nand the momentum transf erred from the spin system to the neutron or photon beam. The resulting \nfive-dimensional data sets serve as powerful probes of magnetic materials. Elastic scattering \nelucidates the magnetic configuration, direction of the spins, symmetry of the magnetic state, \nspatial distribution of the magnetization density, and dependence of the order parameter on \nthermodynamic fields such as temperature, pressure, magnetic and electric fields.  Inelastic \nscattering determines the energies of the fundamental excitations , which can be used to elucidate  \nthe nature and strength of the exchange interactions.   \nThe information provided by x -ray and neutron scattering is  largely complementary.  [4] For \ninstance, resonant elastic x -ray scattering can be used to measure magneti c order parameters in an \nelement -specific manner  by tuning the x -ray energy to an electronic transition , and t he enormous \nphoton flux at modern synchrotron facilities allows  measurement s of very small samples and films \nas thin  as a single unit cell . [5] [6] The magnetic dynamics of such samples can be explored with \nResonant Inelastic x-ray Scattering (RIXS),  [7] and pump- probe techniques open new avenues of \nresearch into non -equilibrium phenomena.  Elastic neutron scattering, on the other hand, provides \nquantitative information about the magnitude  of magnetic order parameters that is difficult to \nobtain with x-rays, and the energy resolution offered by inelastic magnetic neutron scattering is \norders -of-magnitude finer  than can be currently achieved in RIXS experiments .  Both techniques \ncan also measure the lattice structure and dynamics, as reviewed elsewhere in this volume , and \nthose cross section s can be uniquely distinguished from magnetic scattering by polarization \ntechniques. [8, 9]    \nIn this chapter, we describe how to employ neutron and x -ray scattering to explore the \nmagnetism of materials, paying particular attention to the complementarity of both techniques.   \n \nII. Magnetic Neutron Diffraction Technique  \n \nMagnetic neutron scattering originates from the neutron’s magnetic dipole moment . As a spin -\n½ particle , the neutron carries a magnetic dipole moment of - 1.913 nuclear magnetons that \ninteracts with the unpaired electrons in the sample, either through the dipole moment associated \nwith an electron’s spin or via the orbital motion of the elec tron.  The strength of this magnetic \ninteraction is comparable to the neutron- nuclear interaction . The  magnetic scattering cross -section  \nreveal s the magnetic structure and dynamics of materials over wide ranges of length scale and \nenergy.  Magnetic neutron scattering plays a central role in determining and understanding the \nmicroscopic properties of a vast variety of magnetic systems  – from the fundamental nature, \nsymmetry, and dynamics of magnetically ordered materials to elucidating the magnetic \ncharacteristics essential in technological applications.   4 \n  \nOne traditional role of magnetic neutron scattering has been the measurement of magnetic \nBragg intensities in the magnetically ordered regime.  Such measurements can be used to \ndetermine the spatial arrangement and directions of the atomic magnetic moments, the atomic \nmagnetization density of the individual atoms in the material, and the value of the ordered moments \nas a function of external  parameters such as temperature, pressure, and applied  magne tic or electric \nfields.  These types of measurements can be carried out on single crystals, powders, thin films, and \nartificially grown multilayers, and often the information collected can be obtained by no other \nexperimental technique.  For magnetic pheno mena that occur over length scales that are large \ncompared to atomic distances, the technique of magnetic Small Angle Neutron Scattering (SANS) \ncan be applied.  This is an ideal technique to explore domain structures, long wavelength \noscillatory magnetic s tates, vortex structures in superconductors, skyrmions, nanomagnets, and \nother spatial variations of the magnetization density on length scales from 1 to 1000 nm.  Another \nspecialized technique is neutron reflectometry, which can be used to investigate the  magnetization \nprofile in the near -surface regime of single crystals, as well as the magnetization density of thin \nfilms and multilayers.  This particular technique has enjoyed dramatic growth during the last \ndecade or so  due to the rapid advancement of at omic -layer  deposition capabilities.  \n \nThe cross -section for magnetic Bragg scattering can be written as  [10]  \n() ()22\n22\n)(2hkl M B hkl M FAmceC I g ggθγM\n\n\n=        (1) \n \nwhere IM is the integrated intensity for the magnetic Bragg reflection located at the reciprocal \nlattice vector g hkl, the neutron- electron coupling constant in parentheses is − 0.27× 10-12 cm, C is an \ninstrumental constant which includes the resolution of the measur ement, A(θB) is an angular factor \nwhich depends on the method of measurement (sample angular rotation, θ :2θ scan, etc.), and M g \nis the multiplicity of the reflection (for a powder sample).  The magnetic structure factor F M(g) is \ngiven in the general case by  [1] [11]   \n \n() ()ej jW\njN\njig\nhkl M e F−\n=⋅\n\n× × =∑ gg g grˆ ˆ\n1M (2) \n \nwhere gˆ is a unit vector in the direction of the reciprocal lattice vector g hkl, Mj(ghkl) is the vector \nform factor of the jth ion located at rj in the unit cell, Wj is the Debye -Waller factor that accounts \nfor the thermal vibrations of the jth ion, and the sum is over all (magnetic) atoms in the unit cell.  \nThe triple cross product originates from the vector nature of the dipole -dipole interaction of the \nneutron with the electron.  A quantitative calculation of M j(g) in the general case involves \nevaluating matrix elements of the form ±+±⋅O SeiRg2 , where S  is the electron  spin operator, O  \nis the symmetrized orbital operator introduced by [12] , and ± represents the angular momentum \nstate.  This can be quite a complicated angular -momentum computation involving all the electron 5 \n orbitals in the unit cell, but has the simple result that only the components of the magnetic moment \nthat are perpendicular to g hkl (or more generally the wave vector K ) contribute to the scattering.  \nOften the atomic spin density is collinear , by which we mean that at each point in the spatial extent \nof the electron’s probability distribution, the atomic magnetization density points in the same \ndirection .  In this case the direction of M j(g) does not depend on g, and the form factor is just a \nscalar function, f (g), which is simply related to the Fourier transform of the magnetization density.  \nThe free -ion form factors have bee n tabulated for essentially all the magnetic elements  [see, for \nexample, https://www.ill.eu/sites/ccsl/ffacts/ffachtml.html ].  Note that for x -ray scattering the \nform factor for charge scattering corresponds to the Fourier transform of the total charge density \nof all the electrons, while in the magnetic neutron case it is the transform of the “magnetic” \nelectrons  only, which are the electrons whose spins are unpaired.  Recalling t hat a Fourier \ntransform inverts the relative size of objects, the magnetic form factor typically decreases much \nmore rapidly with |ghkl| than for the case of x -ray charge scattering  since the unpaired electrons are \nusually the outermost ones of the ion.  T his dependence of the scattering intensity on f (g) is a \nconvenient way to distinguish magnetic cross -sections from nuclear cross -sections, where the \nequivalent of the form factor is just a constant since the nucleus ( ≈10-5 Å) looks like a point particle \nto a thermal/cold neutron ( see below the nuclear coherent scattering amplitude b in Eq.  (6)).  \nIf in addition to the magnetization density being collinear, the magnetic moments in the \nordered state point along a unique direct ion (i.e. the magnetic structure is a ferromagnet, or a \nsimple + -  + - type antiferromagnet), then the square of the magnetic structure factor simplifies to  \n \n \n() ()22^^21∑−⋅−=⋅\nji\nj M ee f FjW\njz\njj\ngrg\ng g µηη  (3) \n \nwhere η^ denotes the (common) direction of the ordered moments and ηj the sign of the moment \n(±1), µz\nj is the average value of the ordered moment in thermodynamic equilibrium at ( T, H, P, \n…), and the orientation factor 2^^\n1⋅−ηg  represents an average over all possible domains.  If \nthe magnetic moments are the same type, then this expression further simplifies to  \n \n() ()2\n22 2^^21 ∑−⋅⋅−=\njW i\njz\nM ee f Fj jrgg g g η µη     .      (4) \n \nWe see from these expressions that neutrons can be used to determin e several important quantities; \nthe location of magnetic atoms in the unit cell and the spatial distribution of their magnetic \nelectrons;  the dependence of < µz> on temperature, field, pressure, or other thermodynamic \nvariables, which is directly related to the order parameter for the phase transition (e.g. the sublattice 6 \n magnetization).  Often the preferred magnetic axis η^ can also be determ ined from the relative \nintensities.  Finally, the scattering can be put on an absolute scale by internal comparison with the \nnuclear Bragg intensities I N from the same sample, given by  \n \n() ()2)( g g FN B N A C I θτM=          (5) \nwith  \n()2\n2∑−⋅=\njW i\nj N eeb Fj jrgg     .          (6) \nHere b j is the coherent nuclear scattering amplitude for the jth atom in the unit cell, and the sum is \nover al l atoms in the unit cell.  Typically  the nuclear structure is known accurately and F N can be \ncalculated, whereby the saturated value of the magnetic moment in Bohr magnetons can be \nobtained. \n \n There are several ways that magnetic Bragg scattering can be distinguished from the \nnuclear scattering from the structure.  Above the magnetic order ing temperature all Bragg peaks \nare nuclear (structural) in origin, while as the temperature drops below the ordering temperature \nthe intensities of the magnetic Bragg peaks rapidly develop, and for unpolarized neutrons the \nnuclear and magnetic intensities  simply add.  If these new Bragg peaks occur at positions that are \ndistinct from the nuclear reflections, then it is straightforward to distinguish magnetic from nuclear \nscattering.  In the case of a ferromagnet, however, or for some antiferromagnets which contain two \nor more magnetic atoms in the chemical unit cell, these Bragg peaks can occur at the same position.  \nOne standard technique for identifying the magnetic Bragg scattering is to make one diffraction \nmeasurement in the paramagnetic state well abo ve the ordering temperature, and another in the \nordered state at the lowest temperature possible, and then subtract the two sets of data.  In the \nparamagnetic state the (free ion) diffuse magnetic scattering is given by [5,6]   \n \n()Kfp Ieff ParamceC2 22\n22\n2 32\n\n\n\n=γ        (7) \n \nwhere peff is the effective magnetic moment (= g[J(J+1)]1/2 for a free ion).  This is a magnetic \nincoherent cross -section, and the only angular dependence is through the magnetic form factor \nf(K).  Hence this scattering looks like “background”.  There is a sum rule on the magnetic scattering \nin the system, though, and in the ordered state  this diffuse scattering shifts into the coherent \nmagnetic Bragg peaks  and magnetic excitations .  A subtraction of the high temperature data (Eq. \n(7)) from the data obtained at low temperature (Eq. (1)) will then yield the magnetic  Bragg peaks, \non top of a deficit (negative) of scattering away from the Bragg peaks due to the disappearance of \nthe diffuse paramagnetic scattering in the ordered state.  On the other hand, all the nuclear cross -\nsections usually do not change significantl y with temperature  (apart from the Debye -Waller factor \ne-2W), and hence drop out in the subtraction.  A related subtraction technique is to apply a large 7 \n magnetic field in the paramagnetic state, to induce a net (ferromagnetic -like) moment.  The zero \nfield (nuclear) diffraction pattern can then be subtracted from the high -field pattern to obtain the \ninduced- moment diffraction pattern. \n \na. Polarized Neutron  Technique s \n \n When the neutron beam that impinges on a sample has a well -defined polarization state, \nthen the nuclear and magnetic scattering that originates from the sample interferes coherently, in \ncontrast to being separate cross -sections like Eq. (1) and Eq. (5) whe re magnetic and nuclear \nintensities just add.  Polarized neutron diffraction measurements with polarization analysis of the \nscattered neutrons can be used to establish unambiguously which peaks are magnetic, which are \nnuclear, and more generally to separat e the magnetic and nuclear scattering at Bragg positions \nwhere there are both nuclear and magnetic contributions. The standard polarization analysis \ntechnique is straightforward in principle [9]  [13].  Nuclear coherent Bragg scattering never causes \na reversal, or spin -flip, of the neutron spin direction upon scattering.  Thus the nuclear peaks will \nonly be observed in the non- spin- flip scattering geometry.  We denote this configuration as (+ +), \nwhere the incident spin of the neutron is ‘ up’ spin and remains in the up state after scattering.  Non -\nspin- flip scattering also occurs if the incident neutron is in the ‘ down’  state, and remains in the \ndown state after scatter ing (denoted ( − −)).  The magnetic cross -sections, on the other hand, depend \non the relative orientation of the neutron polarization P  and the reciprocal lattice vector g.  In the \nconfiguration where P ⊥g, typically half the magnetic Bragg scattering involves a reversal of the \nneutron spin (denoted by ( − +) or (+ − )), and half does not ; the details depend on the specific \nHamiltonian describing the magnetism.  Thus for an isotropic Heisenberg -type model the magnetic \ncontribution to the reflec tion consists of the spin- flip (− +) and non -spin- flip (+ +) intensities of  \nequal intensity.  For the case where P g, all the magnetic scattering is spin -flip.  Hence for a pure \nmagnetic Bragg reflection where (S x,Sy,Sz) are active, the spin- flip scattering should be twice as \nstrong as for the P ⊥g configuration. \n The arrangement of having P g or P⊥g provides an experimental simplification and hence \ndata that are straightforward to interpret.  More generally, however, P  and g can have any relative \nangle.  This more general technique of neutron polarimetry is more difficult to realize \nexperimentally an d can complicate the interpretation of  the data, but can provide additional details \nabout the magnetic structure that cannot be obtained otherwise . [14]    \nb. Polarized Neutron Reflectometry  \n \nIf neutrons are incident on a surface at (very small) grazing angles the scattering can be \ncast in the form of a neutron ‘optical potential’ , analogous to photons in optical fibers.  For most \nmaterials the waveleng th-dependent index of refraction for neutr ons (and x -rays) , n, is slightly less \nthen unity, so that at suffi ciently small angles of incidence the scattering can be described by the \none-dimensional Schrödinger equation and the neutrons undergo total external reflection —the \nbasis for neutron guides .  For a simple material with a net magnetization , interference between \nnuclear and magnetic scattering leads to the following expression for  n [10]  [15]  : 8 \n  2/1\n22 2\n21\n\n\n\n\n\n\n\n\n\n\n\n±−=±µγ\nπλ\nmcebN n        (8) \nwhere N is the number density of the material and < µ> is the average moment.  The magnetic form \nfactor is unity since we are scattering at very small angles.  Note that Nb  is the nuclear scattering \nlength density for the material, and the magnetic term is the magnetic scattering length density.  \nThe critical angle below which we have mirror reflection is given by  \n2/1\n22 22/1\n22 2\n2 2arcsin\n\n\n\n\n\n\n\n\n\n\n\n± ≅\n\n\n\n\n\n\n\n\n\n\n\n± = µγ\nπλµγ\nπλθmcebNmcebNC     (9) \nwhere ± denotes the two polarization states of the neutron.  Above the critical angle the neutrons  \npenetrate the surface, and Fourier transforming the scattering provides a quantitative measure of \nthe structural profile and magnetic  profile of the material .  For thin films and multilayers  the layers, \nsubstrate, and front and back surfaces produce interference effects that provide a standard and very \npowerful technique for determining the properties of a wide variety of magnetic materials. [16] \n[17] \n \nIII. Resonant M agnetic X -ray Diffraction  Technique  \n \nMagnetic x -ray scattering was first demonstrated off resonance, that is, with photons that were not \ntuned to any absorption edge of the material under study.  However, the non- resonant magnetic x -\nray scattering cross section is so small that this technique is not useful for magnetic structure \ndetermination.  Magnetic x -ray scattering has only risen to prominence when synchrotron radiation \nenabled experim ents with photons tuned to x -ray absorption edges, where the resonant cross \nsection can be enhanced by several orders of magnitude.  [5] [6] The enhancement is greatest when \nthe partially occupied valence shell is reached by an electric dipole- allowed transition, that is, at  \nthe L2,3-absorption edges of transition metals  with valence d- electrons , and at the M 4,5-absorption \nedges of lanthanides or  actinides  with valence f -electrons . Magnetic x -ray scattering is then \nactivated by the strong core- hole spin- orbit coupling in the intermediate state , prior to reemission \nof the photon. \n \nFrom an instrumental perspective , one can group magnetic x -ray scattering experiments into three \ncategories, depending on the photon energy E required to reach the respective absorption edges, \nnamely soft ( E < 1 keV), intermediate (1 ≤ E ≤ 5 keV) and hard ( E > 5 keV). Whereas soft x -ray \nexperiments use gratings to monochromate the synchrotron radiation, intermediate and hard x -ray \nexperiments are performed with single -crystal monochromators. Because of air absorption, soft \nand intermediate x -ray experiments are carried out under vacuum conditions. The soft and \nintermediate x -ray ranges comprise the L -edges of 3d (4d) metals and the M -edges of 4f - (5f-) \nelectron systems, respectively. Experiments at the dipole -active L -edges of 5d metals are carried \nout with hard x -rays, as are experiments at the K -absorption edges of d -electron systems and L -\nabsorption edges of  f-electron systems where the resonant en hancement of the magnetic cross \nsection is weaker.  9 \n  \nUnlike neutron scattering, resonant magnetic x -ray scattering experiments require photons with a \nspecific energy , so that only the direction and not the magnitude of the photon momentum  is \nadjustable . Momentum conservation yields kinematic constraints that are particularly severe for \nsoft x -ray experiments on the important class of 3d metal compounds , where simple \nantiferro magnetic Bragg reflections characteristic of a doubled crystallographic unit cell cannot be \nreached in many cases (Fig. 1 [7]). Magnetic order with larger periodicities (and correspondingly \nshorter reciprocal lattice vectors) can be studied by resonant x -ray diffraction, but dynamical \ndiffraction effects can be important (see the example below). For resonant x -ray diffraction with \nintermediate and hard x -rays (Fig. 1), these constraints do not apply.  \n \nIn contrast to magnetic neutron scattering  which is generally straightforward to interpret , a \ncomplete quantitative calculation of the magnetic x -ray scattering cross section requires numerical \nelectronic structure calculations that describe the many -body correlations in the intermediate state. \nIn many cases, however, one is interested in the magnetic moment orientation, which can be \nextracted from the dependence of the scattered intens ity on the photon polarization without \nreference to such calculations. In spherical symmetry, the scattering tensor can be expressed in the \nfollowing way:  [18]  \n ()()⋅−⋅⋅ +⋅× +⋅ =∗ ∗ ∗ ∗\no i j o j i j o i o i j M M E M E E EF εε εεσεεσεεσ31)( )( )( )()2( )1( )0( (10)  \nwhere Mj is the magnetization vector of the ion j, εi and εo are the polarization vectors of the \nincoming and outgoing photons, and σ(0), σ(1), and σ(2) are proportional to the x -ray absorption \n(XAS), x -ray magnetic circular dichroism (XMCD), and x -ray magnetic linear dichroism (XMLD) \ntensors , respectively.  Additional terms arise from the crystal field, but they tend to be small for \ncollinear spin structures , as long as M  points along a high- symmetry direction of the crystal lattice. \n[18] \nTo separate magnetic scattering from charge scattering  (first term in Eq. 10) , magnetic x -ray \nscattering experiments can be carried out in crossed linear polarization. With the caveats \nmentioned above, the  intensity of a magnetic Bragg reflection  of a collinear antiferromagnet  at the \nreciprocal lattice vector g  can then be written as  \n2\n)1()(∑ ⋅× =∗ ⋅\njj o i jrigM E e Ijεεσ         (11) \nwhere the summation r uns over the magnetic unit cell . To determine the spin structure of a given \nmaterial, one commonly uses the so- called “azimuthal scan” where the momentum transfer g is \nkept fixed, and the sample is rotated such that the orientation of  M varies relative to the photon \npolarization vectors. In this way, simple  spin structure s can be determined based on  a single Bragg \nreflection.   10 \n Even for simple spin structures, however, it is important to keep in mind that the spectral functions  \nσ(E) are tensors with properties that may be strongly influenced by the symmetry of the crystal \nlattice.  If the site symmetry is tetragonal, for instance, the XAS spectra for light polarized in the \nxy-plane and along the z -axis, σ(0)xy  and σ(0)z, are generally different – a phenomenon known as \n“natural linear dichroism”. σ(1) and σ(2) are also generally anisotropic.  \nThe deviations from spherical symmetry are particularly prominent in situations where orbital \norder is present. An elementary example is  the Cu2+ ion with electron configuration 3d9 (i.e., a \nsingle hole in the d- electron shell).  [18] Materials based on Cu2+ usually exhibit Jahn- Teller \ndistortions that lift the degeneracy between d- orbitals of x2-y2 and 3z2-r2 symmetry. The lobes of \nthese orbitals are extended in the xy- plane and along the z -axis, respectively . For instance, t he \ncuprate high-temperatur e superconductors  exhibit a tetragonal structure with hole s in the x2-y2 \norbital. In this case, the electric dipole selection rules prohibit excitation of a 2p core electron into \nthe valence shell with z -polarized light, so that σ(0)z = 0 whereas σ(0)xy ≠ 0. The selection rule  \ncompletely changes  the az imuthal scans,  as observed  in resonant elastic scattering experiments on \ncopper -oxide compounds . [19] This example illustrates the important influence of orbital order on \nazimuthal scans in magnetic x -ray scattering. Proper consideration of the crystal symmetry is \nespecially important for experiments performed with polarized incident light, but without \npolarization analysis of the  scattered beam, because magnetic and charge scattering may then both \ncontribute to the detected signal.  \nThe photon energy dependence of the scattering tensor σ(E) contains a lot of additional \ninformation , some of which can be extracted without extensive model calculations.  In particular, \nthe large enhancement of the scattering intensity  at the absorption edges of magnetic metal atoms \ngives rise to the element sensitivity of magnetic x -ray scattering, which is particularly useful for \nmultinary co mpounds and for magnetic multilayers with different magnetic species. In principle, \nresonant magnetic x -ray scattering is also sensitive to the valence state of metal ions, which can \nbe inferred from the maximum of σ (E). Resonant scattering experiments on mixed -valent \ncompounds have indeed been reported.  [20]  However, the analysis and quantitative interpretation \nof such experiments require careful consideration of the multiplets in the intermediate state.  \nIn the discussion so far, we have not considered the spin- orbit coupling in the valence shell, which \nis generally weak for 3d metal compounds. In 4f  and 5f electron systems, however, the spin- orbit \ncoupling is so strong that it dominates the interatomic exchange interactions, so that models of \nsuch compounds are ba sed on firmly locked spin and orbital angular momenta. In 4d and 5d \nelectron systems, on the other hand, the intra -atomic spin -orbit coupling turns out to be comparable \nto other important energy scales including the on -site Coulomb interactions and the inter -atomic \nexchange coupling. Comparative magnetic x -ray diffraction experiments at the L 2 and L 3 \nabsorption edges have recently proven to be a powerful probe of the s pin-orbit compos ition of the \nground state wave function in such materials.  [19]  \n \n 11 \n IV. Dynamics  \na. Inelastic Neutron Scattering Technique  \n \nNeutrons can also scatter inelastical ly, to reveal the magnetic fluctuation spectrum of a material \nover wide ranges of energy ( ≈10-8→1 eV) and over the entire Brillouin zone. Neutron scattering \nplays a truly unique role in that it is the only technique that can directly determine the complete \nmagnetic excitation spectrum, whether it is in the form of the dispersion relations for spin wave \nexcitations, wave -vector and energy dependence of critical fluctuations, crystal field excitations, \nmagnetic excitons, or moment/valence fluctuations.  In the present overview we will discuss some \nof these possibilities.  \n \nAs an example, consider identical spins S localized on a simple cubic lattice, with a coupling given \nby -JSi⋅Sj where J is the Heisenberg  exchange interaction between neighbors separated by the \ndistance a.  The collective excit ations are magnons [ref. Chap ter on Spin Waves] .  If we have J>0  \nso that the lowest energy configuration is where the spins are parallel (a ferromagnet), then the \nmagnon dispersion along the edge of the cube (the [100] direction) is given by  \n \nE(q)= 8 JS[sin2(qa/2)]   .    .    (12) \n \nAt each wave vector q  a neutron can either create a magnon at ( q, E) with a concomitant change \nof momentum and loss of energy  of the neutron, or conversely destroy a magnon with a gain in \nenergy.  The observed change in momentum and energy for the neutron can then be used to map \nthe magnon dispersion relation.  Neutron scattering is particularly well suited for such inelastic \nscattering studies since neutrons typically have energies that are comparable to the energies of \nexcitations in the solid, and theref ore the neutron energy changes are large and easily measured.   \nAdditional information about the nature of the excitations can be obtained by polarized inelastic \nneutron scattering techniques, which are finding increasing use.  The cross section for spin w ave \nscattering from a simple Heisenberg ferromagnet is given by [1]  [13]  [9] \n \n()( )()()GqKq\nGq,q − +\n\n\n= ∑\n\n\n\nΩ±\n   δδπ γσ\nEE nVS\nkkgfmce\ndd\n21212\n2')(2 223\n22\n22\n \n \n× \n\n\n\n••••+^ ^ ^2^ ^\n2 1 ηK KPηK     (13) \n \nwhere n q is the Bose thermal population factor and ^\nη is a unit vector in the direction of the spins.  \nGenerally s pin wave scattering is represented by the familiar raising and lowering operators S± = \nSx ± iSy, which cause a reversal of the neutron spin when the magnon is created or destroyed.  \nThese “spin -flip” cross -sections are denoted by (+ − ) and (− +).  If the neutron polarization P  is 12 \n parallel to the momentum transfer K , PK, then the spin angular momentum is conserved (as there \nis no orbital contribution in this case).  In this experimental geometry, Eq. (13) shows us that we \ncan only create a spin wave in the ( − +) configuration, which at the same time causes the total \nmagnetization of the sample to decrease by one unit (1 µ B for a spin -only system ).  Alternatively, \nwe can destroy a spin wave only in the (+ − ) configuration, while increasing the magnetization by \none unit.  This gives us a unique way to unambiguously identify the spin wave scattering, and \npolarized beam techniques in general can be used to distinguish magnetic from nuclear scattering \nin a manner similar to the case of Bragg scattering.  \nFinally, we note that the magnetic Bragg scattering is comparable in strength to the overall \nmagnetic inelastic scattering.  However, all the Bragg scattering is located at a single point in \nreciprocal space, while the inelastic scattering  is distributed throughout the three dimensional \nBrillouin zone.  Hence when actually making inelastic measurements to determine the dispersion \nof the excitations one can only observe a small portion of the dispersion surface at any one time, \nand thus the observed inelastic scattering  is typically two to three orders of magnitude less intense \nthan the Bragg peaks.  Consequently , these are much more time consuming measurements, and \nlarger samples are needed to offset the reduction in intensity.  Of course, a successful determination \nof the dispersion relations yields a complete determination of the fundamental magnetic \ninteractions in the solid.  \n \nb. Resonant Inelastic X -ray Scattering Techniqu e \nThe mechanism underlying magnetic resonant inelastic x -ray scattering (RIXS) is analogous to the \none for resonant elastic scattering discussed in Section III and depicted in Fig. 1. A photon tuned \nto a dipol e-allowed transition promotes a core electron into the pa rtially occupied valence s hell. In \nthe intermediate state, the core- hole spin- orbit coupling induces an electronic  spin- flip, so that  the \nre-emitted photon leaves a magnetically excited state behind. Single magnetic excitations are then \nobservable in crossed polarization, analogous  to elastic magnetic scattering (Eq. 10 ). [7] In this \nsense, the relationship between elastic and inelastic resonant x -ray scattering is analogous to the \none between elastic and inelastic neutron  scattering.  Another useful analogy is optical Raman \nscattering, where single magnetic excitations at q = 0 can be activated by the spin -orbit coupling \nin the intermediate state [21]  which is, however, usually much weaker than the core -hole spin-\norbit coupling in RIXS. A more com mon Raman scattering experiment  addresses  bi-magnon \nexcitations that do not involve an electronic spin- flip. Such experiments are also possible with \nRIXS in parallel polarization geometry. As in optical Raman scattering, however, they only \ndetermine the Brillouin -zone averaged spectrum of magnetic excitations. The unique advantage of \nsingle -magnon RIXS is that the full magnon dispersion can be determined even for single crystal s \nof micrometer dimensions , or for  atomically thin films and heterostructures.  \nFrom an instrumental perspective, RIXS experiments on magnetic excitations  are challenging \nbecause the energy of the photons required to induce the atomic dipole  transition ( E = 0.4- 1 keV \nfor 3d metal L -edges) l argely  exceeds the typical energy of magnons in solids. A breakthrough \nwas achieved in 2009, when the resolving power of soft x -ray RIXS instrumentation passed the \nthreshold of E/ΔE ≈ 10000. This enabled the first RIXS observation of  high-energy magnons in \nundoped layered cuprates , which exhibit an exc eptionally large bandwidth of ≈300 meV. [22] 13 \n Shortly thereafter, high -energy paramagnons were also observed by RIXS in superconducting \ncuprates [23] [24] and in iron -based high- temperature superconductors at the Fe L 2,3 edges.  [25]  \nKinematical constraints analogous to those in resonant elastic scattering restrict these experiments \nto a fraction of the Brillouin zone that does not include the magnetic ordering wave vectors  of the \nrespective parent compounds. The kinematical constraint s are even more severe for RIXS \nexperiments of bi -magnon excitations in metal oxides at the oxygen K -edge (1 s-2p, 0.5 eV).  [26]  \nParallel advances in RIXS instrumentation for hard x -rays allowed the observation of single \nmagnons in antiferromagnetically ordered iridium oxides with 5d electron systems . [27] The larger \nresonance energies of the 2p-5d transition, with  correspondingly larger photon wave vectors , allow \nthe detection of magnons over the entire Brillouin zone. Instrumentation for RIXS at the L -\nabsorption edges of 4d metals and M -edges of actinides at intermediate photon energies (2.5 ≤ E \n≤ 5 eV) ha s only recently been developed . [28]  \nIn contrast to inelastic neutron scattering, the theoretical description of RIXS is still under \ndevelopment, and several open questions are actively debated  in the literature. These include the \nseparation of spin excitations from orbital excitations in mul ti-orbital systems, and from charge \nexcitations in metallic systems. This challenge is particularly severe in the iron pnictides, which \nare metals with multiple Fermi surfaces originating from different Fe d- orbitals. A complete \nresolution of this problem will likely require a transition to full polarization analysis in RIXS, so \nthat the  different excitation channels  can be separated completely . The first experiments using \nRIXS polarimeters have already been reported.  [29] Another open issue is the influence of the \ncore- hole potential in the RIXS intermediate state of the valence electron system in metallic \nsystems , where the core- hole lifetime may be comparable to intrinsic time scale s of the valence \nelectrons .  \n \nV.  Magnetic Diffraction Examples with Neutrons  \n \nAs an example of magnetic powder diffraction, the scattering from a sample of Na 5/8MnO 2 is \nshown in Fig. 2  [30].  This material exhibits Mn3+ and Mn4+ charge stripes and vacancy ordering \nof the Na  subsystem , which results in a rather complicated low -temperature magnetic structure \nthat can be determined from this pattern.  Of course, Rietveld refinement s for the crystallographic \nstructure can be performed from the full patterns at both high and low  temperatures to determine \nthe full crystal structure;  lattice parameters, atomic positions in the unit cell, site occupancies, etc. , \nas well as the value of the ordered moment.  The inset shows the temperature  dependence of the \nmagnetic peak intensity, which we see from Eq. (4) is the square of the sublattice magnetization —\nthe order parameter of the magnetic phase transition .  Note that we can identify the magnetic \nscattering through its temperature dependence,  as mag netic Bragg peaks vanish above the Néel \ntemperature where long range magnetic order occurs.  Note also that  the magnetic intensities \nbecome weak at high scattering angles as f (g) falls off with increasing scattering angle.  14 \n A more elegant way to iden tify magnetic scattering is to employ the neutron polarization \ntechnique , particularly if the material has a crystallographic rearrangement or distortion associated \nwith the magnetic transition.  It is more involved and time -consuming experimentally, but yields \nan unambiguous identification and separation of magnetic and nuclear Bragg peaks.  Figure 3  \nshows the polarized beam results for two peaks of polycrystalline YBa 2Fe3O8. [31]  The top section \nof the figure shows the data for the P ⊥g configuration. The peak on the left has the identical \nintensity for both spin- flip and non- spin- flip scattering, and hence we conclude that this scattering \nis purely magnetic in origin.  The peak on the right has strong intensity for (+ +), while the inte nsity \nfor (- +) is smaller by the instrumental flipping ratio.  Hence this peak is a pure nuclear reflection.  \nThe center row shows the same peaks for the  P||g configuration, while the bottom row shows the \nsubtraction of the P ⊥g spin-flip scattering from the P||g spin- flip scattering.  In this subtraction \nprocedure instrumental background, as well as all nuclear scattering cross sections, cancel, \nisolating the magnetic scattering.  We see that there is magnetic intensity only for the low angle \nposition, whi le no intensity survives for the peak on the right, unambiguously establishing that the \none peak is purely magnetic and the other purely nuclear.  These data also demonstrate that all \nthree components of the angular momentum contribute to the magnetic scat tering. This simple \nexample demons trates how the technique works;  obviously it plays a more critical role in cases \nwhere it is not clear from other means what is the origin of the peaks, such as in regimes where \nthe magnetic and nuclear peaks overlap, or i n situations where the magnetic transition is \naccompanied by a structural distortion  where the structural peaks change significantly in intensity . \nWhen investigating the magnetic structures of new materials, it is generally best to first carry \nout powder diffraction experiments to establish the basic properties of the magnetic structure, \nassuming of course that the ordered moment is large enough to observe the magnetic Bragg peaks.  \nOnce the basics are established, on the other hand, measurements on a singl e crystal can provide \nmuch  higher quality and more detailed information about the magnetic properties.  Figure 4  shows \na map of the scattering intensity in the ( h,k,0) scattering plane at 22 K for a single crystal of the \nmultiferroic Co 3TeO 6, which orders antiferromagnetically at 26 K. [32]  The crystal structure is \nmonoclinic, and we see four satellite magnetic peaks around each (integer) structural peak, \nindicating that the initial magnetic structure is incommensurate in both the h and k (and l as well, \nit turns out [33] ) directions .  With further decrease of temperature a series of additional transitions \nare observed , detail that would be difficult to determine with a powder.  At lower temperature , \nseparate commensurate peaks develop, then there is a lock -in transition along k  that includes a \nferroelectric order parameter, and then finally a transition into the ground state with both \ncommensurate magnetic order and incommensurate order along h, k , and l . [33] [34]  \nThe magnetic superconductor ErNi 2B2C goes superconducting at T C = 11 K, and then develops \nincommensurate antiferro magnetic order below T M= 6 K as shown in Fig. 5 . [35]  The wave vector \nfor the ordering is (h,0,0) with h ≈ 0.55, with the spin direction transverse, along (0,y ,0).  Initially \nthe magnetic order exhibits a simple sinusoidal spin -density -wave (SDW) that is transversely \npolarized, as shown in the  bottom of the  figure.  As the amplitude of the SDW increases, third, \nfifth, and higher -order wave vector peaks develop as the wave squares  up.  This is the expected 15 \n behavior since for localized moments entropy mandates that a simple spin density wave cannot be \nthe ground state magnetic structure.  \nFor any SDW  structure, only odd- order peaks will have non- zero intensity due t o time-reversal \nsymmetry , because on average the net magnetization is zero.  Below 2.3 K we see that a new set \nof even-order peaks is found along the ( h,0,0) direction of ErNi 2B2C.  One possibility is that the \neven -order peaks are due to a structural distortion , a charge- density wave (CDW)  that follows the \nSDW  due to a magnetoelastic interaction.  Hence  the even -order peaks would be structural peaks \nand the odd- order peaks magnetic.  In t he present material, however, a net magnetization develops \nin the superconducting state  in the magnetic ground state , so that the even -order peaks could be \nstructural, magnetic, or both. To establish the nature of these peaks unambiguously  polarized \nneutron diffraction was used , as shown in Fig. 6.  The data are measured in the ( h,0,l) scattering \nplane, with k  then perpendicular to the scattering plane.  For P ||g the spins are perpendicular to the \nscattering plane and hence perpendicular to P and then the magnetic scattering is all spin -flip.  \nNote that the polarization dependence of the cross sections is quite different than the YBa 2Fe3O8 \nexample above, emphasizing that the spin- flip and non- spin- flip magnetic cross sections depend \non the details of the magnetic structure.  The structural scattering is always non -spin- flip.  The data \nshow that both odd- order and even -order  are purely magnetic  in this system .   \nFor antiferromagnets there is no net magnetization produced by the magnetic ordering.  When \nthe sublattice magnetizations are not compensated and there is a net magnetization, on the other \nhand, the superconductivity must respond to and try to screen this magnetization.  If the internally \ngenerated field is below HC1 then the supercurrents will exactly compensate the net magnetization \nand the total field will be zero.  If the field exceeds H C2 then the superconductivity will be \nextinguished as happens in ma terials such as ErRh 4B4 and HoMo 6S8. [36]  Between these two  \ncases , vortices are expected to be spontaneously generated, and this possibility can be i nvestigated \nwith SANS.  Figure 7  shows SANS data from  a single crystal of ErNi 2B2C. [37]  The inset presents \nthe image on the two -dimensional SANS detector, where K =0 is in the center.  We see the expected  \nhexagonal pattern of scattering from the vortex lattice.  Below the ferromagnetic transition \nadditional vortices spontaneously form due to the internally generated magnetic field, which adds \nto the applied field. To accommodate the additional vortices the y rearrange themselves with a \nsmaller lattice parameter for the vortex lattice, which is reflected by the peak of the vortex \nscattering moving to larger K . [38]  \nThe above examples demonstrate scattering from long range magnetic  order where the \nmagnetic diffraction consists of resolution -limited Bragg peaks .  But that is not always the case , \nand some of the best examples occur where competing magnetic interactions lead to frust ration \nand suppress the order or  prevent it completely.  Arguably the best example of  a frustrated lattice \noccurs in the cubic rare -earth  (R) pyrochlore (R2Ti2O7) systems  where the R  ions occupy corner -\nsharing tetrahedra. [39]  For R = Ho, Dy, for example, the single -ion anisotropy restricts the \nmoments to point along diagonal [111] directions, along lines that intersect the center of each \ntetrahedron.  The ground state turns out to be with two of the moments pointing into each \ntetrah edron and two pointing out.  But you don’t know which two are in and which two are out, \nexactly like the hydrogen bonding in ice where two H  move into the oxygen in the center of the 16 \n tetrahedron and bond and two move out , resulting in a macroscopic degener acy that violates the \nthird law of thermodynamics.  The first measurement of the ground state correlations was carried \nout for Ho 2Ti2O7, where the observed scattering  from the correlated moments  agreed quite well \nwith simulations.  [40]  An interesting simplification occurs for a field applied along the [111] \ndirection, which isolates the layers and forms  two-dimensional ‘kagom é spin- ice’.  The scattering \nfor this case is shown in Fig. 8 for Dy 2Ti2O7, which shows the broad distributions of diffuse \nmagnetic scattering that are in excellent agreement with Monte Carlo simulations.  [41]  \nThe ground state properties are not the only remarkable property of spin- ice, as the magnetic \nexcitations are equally fascinating.  Theory showed that these excitations, which simply consist of \nflipping one of the spins in a tetrahedron so that you have three pointing out and one pointing in \n(and in the adjacent tetrahedron three point in and one out), correspond to the creation of a \nmagnetic monopole and anti -monopole. [42]  The subsequent motion of these particles is  governed \nby the Coulomb Hamiltonian for magnetic charges, and this scenario was subsequently confirmed \nby neutron scattering measurements.  [43]  [41]  [44] \nAdvances in  thin film de position methods  have facilitated the synthesis of complex \nheterostructures with atomic layer accuracy, which has enabled investigators to control the \nmagnetic properties by tailoring the exchange interactions within and between layers.  These \ncapabilities combined with advance s in experimental reflectometry techniques  have made neutron \nscattering an essenti al tool to elucidate the atomic depth profile and magnetization density of thin \nfilms and multilayers.  An interesting example is the multilayer oxide heterostructure consisting \nof the (approximately cubic) antiferromagnets  LaMnO 3 and SrMnO 3, grown on a Sr TiO 3 substrate.  \nThe structural indices of refraction for these two materials are almost identical, rendering the \nstructural scattering practically invisible.   Occasionally an extra layer of LaMnO 3 was deposited \nto dope the interface, which produced an eff ective composition of La 0.44Sr0.56MnO 3, which is in \nthe ferromagnetic regime.  Figure 9  shows the non- spin- flip polarized neutron reflectivity data in \nthe two polarization states, R++ and R--, that are sensitive to the ferromagnetism.  The resulting \nmagnetic depth profile reveals that the magnetic modulation is quite large, varying from 0.7 µB to \n2.2 µB, and that its  period corresponds precisely to  the LMO superlattice structure.  [45]  High angle \ndiffraction data on the epitaxial multilayer confirmed the canted modulated spin structure of the \nsuperlattice.  \n \nVI. Magnetic Diffraction Examples with X -rays \n \nAs an example of resonant magnetic x -ray scattering, we first highlight experiments on the \nantiferromagnet Sr 2IrO 4 with hard x -rays tuned to the Ir L 2,3 edges  [46]. The crystal structure of \nSr2IrO 4 is composed of IrO 2 square lattices , closely similar to La 2CuO4, the parent compound of a \nprominent family of high -temperature superconductors.  Prior to the x -ray experiments, m agnetic \nsusceptibility measurements had suggested antiferromagnetic order with a Néel temperature of  \n240 K, but neutron diffraction experiments had proven difficult because of the large neutron \nabsorption cross section of Ir, and because large single crystals could not be grown. The hard x -17 \n ray data on a crystal of sub- millimeter dimensions show multiple magnetic Bragg reflections that \ncan be analyzed by refining the Bragg intensities according to Eq. (11) in a manner entirely \nanalogous to m agnetic neutron diffraction. The analysis revealed a canted  antiferromagnetic \nstructure in the IrO 2 planes , with alternating stacking in the direction perpendicular to the planes.  \n \nThe photon energy dependence of the resonant magnetic x -ray scattering cross section yields \nadditional information about the magnetic ground state of Sr 2IrO 4 that would be difficult to obtain \nwith neutron diffraction, even under ideal conditions. The Ir va lence electrons occupy 5d orbitals \nof xy, xz, and yz  symmetry. For materials with 3d valence electrons, the crystal field lifts the \ndegeneracy between these orbitals and quenches the orbital magnetization. In the 5d electron shell, \nhowever, the strong intra -atomic spin -orbit coupling can generate  complex admixtures of these \norbitals in the ground- state wave function, which correspond to a nonzero orbital magnetic  \nmoment. This, in turn, affects the matrix elements for the photon- induced transitions from the spin-\norbit split 2p  shell into the 5d shell such that the diffraction intensities at the L 2 and L 3 edges ( 2p1/2-\n5d and 2p3/2-5d, respectively) can become different. The strong disparity of the diffraction \nintensities observed experimentally (Fig. 10 ) [46] indicates that the orbital magnetization is largely \nunquenched, and t hat the spin and orbital components of the magnetic order parameter in Sr 2IrO 4 \nare of comparable magnitude. Similar observations have been made for other iridates. M odels of \nmagnetism in the iridates are therefore commonly expressed in terms of the total angular \nmomentum, J eff =S+L . For Sr 2IrO 4, Jeff = ½ in the ground state.  \n \nThe large resonant scattering cross section, combined with the hi gh photon flux at synchrotron \nbeamlines and the focusing capability of advance d x-ray instrumentation, allow magnetic x -ray \nscattering experiments with beam dimensions well below typical magnetic domain sizes. Figure \n11 provides an example of such an experiment  on the layered antiferromagnet La 0.96Sr2.04Mn 2O7, \nwhich comprises alternately stacked sheets of ferromagnetically aligned Mn spins  [47]. The (001) \nmagnetic Bragg reflection of this spin array can be reached with photons tuned to the Mn L 3-edge.  \nThe data shown in Fig. 11 were taken with a beam of 3 00 nm diameter. T hey reveal that the \ndiffracted intensity varies on a characteristi c length scale of several microns. A detailed analysis \nshows that the intensity variation results from domains with different spin directions, which \ndiffract photons with different scattering amplitude due to the photon polarization dependence of \nthe scatt ering cross section (Eq . (11)). In another study, domains with different helicities in a spiral \nmagnet were imaged by resonant diffraction with circularly polarized x -rays. The spatial resolution \nand imaging capabilities of magnetic x -ray scattering methods are expected to dev elop rapidl y \nwith the advent of coherent x -ray beams at fourth -generation synchrotron sources.  \n \nIn analogy  to neutron reflectometry, polarized magnetic x -ray reflectometry has recently \ndeveloped into a powerful, element -sensitive probe of complex oxide thin films, heterostructures , \nand superlattices . As an example, we discuss  resonant x-ray diffraction data on R NiO 3-based films \nand superlattices (where R denote s a lanthanide atom). R NiO 3 perovskites exhibit  a Mott metal-\ninsulator transition as a function of the radius of the R  cation, which modulates the Ni -O-Ni bond \nangle. Recent work has shown that the metal -insulator transition can also be controlled by epitaxial 18 \n strain and by spatial confinement of the c onduction electron system. Antiferromagnetism with \nordering vector g  = (¼, ¼, ¼) develops in the Mott -insulating phase. Fig. 12( top) shows azimuthal \nscans at the corresponding magnetic Bragg reflection taken with photons tuned to the Ni L 3-edge  \n[48]. The data analysis demonstrates that the magnetic order is non -collinear, with Ni spins \nforming a spiral propagati ng along the (111) direction of the perovskite unit cell. The polarization \nplane of the spiral can be contr olled by epitaxial strain.   \n \nFig. 12(bottom ) shows a contour map of the resonant scattering intensity from a  LaNiO 3-LaAlO 3 \nsuperlattice as a function of the azimuthal angle and the momentum transfer perpendicular to  the \nsuperlattice  plane.  [49] Strong modifications of the azimuthal- angle dependence of the intensity \noccur particularly under grazing- incidence or grazing -exit conditions, where t he incident or \nscattered beams are strongly refracted at the external and internal interfaces of the superlattice. \nThese data illustrate the possibly important influence of dynamical effects in resonant soft x -ray \ndiffraction from thin -film structures, which go beyond the kinematic approxi mation usually \nemployed in the analysis of such data.  \n \nVery recently, x -ray free -electron lasers have enabled time- resolved resonant magnetic diffraction \nexperiments capable of imaging the real -time dynamics of magnetic order  under non- equilibrium \nconditions . As an illustration of th is emerging capability, Fig.  13 shows the time evolution of the \ng = (¼, ¼, ¼) antiferromagnetic Bragg peak of a NdNiO 3 film following a THz pump pulse exciting \nan infrared -active phonon mode  of the LaAlO 3 substrate  [50].  As the phonon propagates from the \nsubstrate through the film , it obliterates the antiferromagnetic order in its wake on a picosecond \ntime scale. The mechanism underlying this “non -thermal  melting” phenomenon may involve \ntransient distortions of the NiO 6 octahedra, which weaken  the magnetic exchange interactions \nbetween Ni spins.  \n \nVII. Spin Dynamics with Neutrons  \nThere are many types of magnetic excitations and fluctuations that can be measured  with \nneutron scattering techniques, such as magnons, spinons, critical fluctuations, crystal field \nexcitations, magnetic excitons, and moment/valence fluctuations.  We start with classic magnons \nin an isotropic ferromagnet , where  the excitations are gapless and the dispersion relation  is given \nby Eq. (12).  Figure 14 (left) shows a measurement for La 0.67Ca0.33MnO 3, which is a colossal \nmagnetoresistive (CMR ) material.  [51]  The data reveal two magnon peaks  at a given  wave vector , \none in energy gain where the neutron destroys a magnon and gains energy, and one in energy loss  \nwhere a magnon is created .  This is a small q  (long wavelength) excitation, and in fact this sample \nis polycrystalline rather than single crystal, and the data were collected around the (0,0,0) \nreciprocal lattice position.  Such measurements are restricted in wave vector and energy, and are \nonly viable for isotropic ferromagnets; otherwise the excitation s fall outside the accessible \nexperimental window  dictated by momentum and energy conservation.  If there is a question of \nwhether these excitations are magnons or phonons, t he polarized  beam technique can be employed \nas shown in Fig. 14(right)  for the prototypical isotropic ferromagnet  amorphous Fe 86B14. [52].  \nThese data were taken with the neutron polarization P  parallel to the momentum transfer K  (PK).  \nIn this configuration magnons require the neutron spin direction to reverse (spin- flip), while 19 \n phonons can only be observed in the non- spin- flip configuration.  For magnons  we should be able \nto create a spin wave only in the ( − +) configuration  where t he incident neutron moment is \nantiparallel to the magnetization; the scattered neutron moment is then parallel to the \nmagnetization direction , and the magnetization is decreased by one unit by the creation of the \nmagnon.  On the energy gain side the proces s is reversed and we destroy a magnon  only in the (+ \n−) configuration.  This is prec isely what we see in the data; for the ( − +) configuration the spin \nwaves can only be observed for neutron energy loss scattering (E > 0), while for the (+ − ) \nconfiguration spin waves can only be observed in neutron energy gain (E < 0).  This behavior of \nthe scattering uniquely identifies  these excitations  as magnons . \nExpanding the sine in Eq. (12) we see that the small- q dispersion relation can be written as E sw \n= D(T)q2, where D  is the spin wave “stiffness” constant.  The general form of the spin wave \ndispersion relation is the same for all isotropic ferromagnets, a requirement of the (assumed) \nperfect rotational symmetry  of the magnetic system, while the numerical value of D  depends on \nthe details of the magnetic interactions and the nature of the magnetism.  The small- q dispersion \nrelation can be readily measured , as shown in Fig. 15(left)  for a single crystal of La 0.85Sr0.15MnO 3, \nand D(T) obtained. [53]  The effect of  temperature is to soften the average exchange interaction as \nthe magnetization decreases, and hence the magnons renormalize to lower energies with increasing \ntemperature as also  shown Fig. 15.  With single crystals the dispersion curves can be determine d \nin different directions and throughout the Brillouin zone , as shown in Fig. 15(right)  for a number \nof perovskite CMR systems. [54]  Such measurements enable to determine in detail all  exchange \ninteractions, rather than just the long wavelength (average) behavior.  Any gap(s) in the excitation \nspectrum can also be directly measured.  \nIn addition to the magnon energies, t he lifetimes  of the excitations  can also be determined by \nextracting the intrinsic widths of the excitations, both in the ground state for itinerant electron \nsystems, and as a function of temperature.   An example of the li newidths in the ground state are \nshown for La 0.85Sr0.15MnO 3 in Fig. 16. [53]  In the simplest localized -spin model negligible  \nintrinsic spin wave linewidths would be expected at low temperatures, while we see here that the \nobserved linewidths are substantial  at all measured wave vectors and highly anisotropic, indicating \nthat an itinerant electron type of model is a more appropr iate description for this system.   In \nparticular , the linewidths become very large at large wave vectors .  These substantial linewidths \nare easy to measure with conventional instrumentation.  I nsulating magnets, on the other hand, \ngenerally have much sm aller linewidths and require much higher instr umental resolution to \nmeasure.  Figure 16(right)  shows the measured linewidths for the prototype insulating \nantiferromagnet  Rb2MnF 4. [55]  Here the spin- echo triple -axis technique has been employed , \nwhich has extraordinarily good (µeV) resolution.  The theoretically calculated linewidths from \nspin- wave th eory are shown by the solid curves at a series of temperatures, and are in quantitative \nagreement with the data.   \nOne area where neutron scattering has played an essential role is elucidating the spin dynamics \nof the high temperature superconductors, first for the copper  oxide systems  [56] and more recently \nfor the iron -based superconductors. [57]  The magnetic excitations in these classes of materials \nextend to quite high energies —as high as ≈0.5 eV —making the measurements particularly \nchallenging since the magnetic form factor requires that the magnitude of K  must be kept small, \nnecessitating  quite high incident ene rgy neutrons.  These requirement s are well matched to the \ntime-of-flight capabilities of spallation neutron facilities where high energy neutrons are plentiful.  20 \n To illustrate the basic technique, consider the excitations from BaFe 2As2, which is one of the \nantiferromagnetic ‘parent’ materials of the iron -based superconductors.  The antiferromagnetic \nordering temperature T N = 138 K, which corresponds to a thermal energy of  just ≈12 meV  (1 meV \n 11.605 K) .  Yet  we see from Fig. 17  that the magnons extend up to 200 meV, an order -of-\nmagnitude  higher energies than the ordering temperature represents , indicating that the system has \na substantial component of low -dimensional character . [58]  The in-plane dispersion relations are \nalso quite anisotropi c, even though the orthorhombic distortion away from tetragonal symmetry \n(that accompanies the magnetic order) is small.  Another very interesting aspect of the magnetic \nexcitations is that they have quite large linewidths at high energies, indicating that the magnetic \nelectrons are itinerant in nature.  Indeed,  the iron d-bands where the magnetism originates cross \nthe Fermi energy —the definition of itineracy.   \nOur final neutron example concerns the spin dynamics of one -dimensional  (1D)  magnets , \nwhich  (together with  2D magnets) have played a special role  in developing a fundamental \nunderstanding of quantum magnetic systems .  This is because they are theoretically  more tractable \nand therefore enable a deeper comparison with experiment .  They also entail the emergence of new \ntypes of cooperative states and their associated excitations.  Arguably the most  interesting case is \nfor the spin one -half antiferromagnet chai n, where quantum effects are maximal, represented by \nmaterials such as KCuF 3 [59] and CuSO 4⋅5D 2O [60] which have enjoyed a long and int eresting \nhistory of  investigations.   The ground state turns out to be an entangled macroscopic singlet , but \nwhere the two -spin correlation function decay s only algebraically, rendering long  lengths of the \nchain to be correlated antiferromagnetically.  The fundamental excitations of such a 1D system are \nspinons  in the se (isolated)  spin chain s, which can be considered to a first approximation as moving \ndomain wall s.  Measurements of the dynamic structure factor for CuSO 4⋅5D 2O are shown in Fig. \n18. [60]  Spinons carry fractional spin, and hence these fractionalized excitations can only be \ncreated in pairs in the scattering process .  Thus  the lower energy part of the spectrum  corresponds \nto two -spinon excitations  and has t he appearance of a  simple  antifer romagnetic spin wave \ndispersion relation.  However, only 71 % of the spectral weight is contained in this two -spinon \ncomponent , with essentially  all the remainder bein g accounted for by the four -spinon contribution.  \nPrecise calculations of the dynamic str ucture factor for two -spinon and four -spinon scattering are \nalso show n in Fig. 18, which account for essentially the entire measured spectral weight , and are \nin excellent agreement with the measurements.  [60]    \n \nVIII.  Spin Dynamics with RIXS  \nThe set of materials investigated by high -resolution RIXS is thus far limited to magnets with \ncharacteristic exchange interactions of the order of 100 meV. A milestone was set by early \nexperiments on La2CuO 4, the antiferromagnetic, Mott- insulating e nd member of a family  of high-\ntemperature superconductors, which exhibit s an exceptionally large magnon bandwidth of ≈ 300 \nmeV. A RIXS spectrometer with  energy resolution of ΔE ≈ 100 meV proved to be  capable of \nseparating  these excitations from the elastic line over a substantial fraction  of the Brillouin zone  \n(Fig. 19) . [23, 22]  Comparison with prior inelastic neutron scattering data on the same materials \ndemonstrated that the  RIXS excitation features  indeed originate from single antiferromagnetic \nmagnons.  21 \n RIXS experiments have also revealed the persistence of high -energy paramagnon excitations in \nhighly doped, superconducting cuprates. Based on the polarization dependence of t he scattering \ncross section  at specific scattering geometries , they can be separated from charge excitations , as \nshown in Fig. 20 for YBa 2Cu3O6+x. [29]  The measurements are complementary to inelastic neutron \nscattering experiments, which have much higher energy resolution and can therefore access spin \nexcitations with energies from 1 -100 meV, comparable to the superconducting energy gap. The \nRIXS measurements, on the other hand, are more sensitive to high -energy excitations , which can \nalso be investigated with high energy neutrons from spallation sources. The photon energy \ndependence of the RIXS intensity yields additional insight into the nature of these excitations. \nWhereas the spin excitation energy is indep endent of photon energy, as expected for collective \nmodes, the spectral weight of the charge excitations shifts upon detuning the photon energy away \nfrom the L -edge resonance, signaling a broad excitation continuum. This supports models that treat \ncollecti ve spin excitations as mediators of unconventional superconductivity.   \nHard x -ray RIXS experiments on the layered iridates have revealed magnon dispersions \nremarkably similar to those of the cuprates  – a finding that has fueled predictions of \nunconventional superconductivity in the iridates. In addition to the usual low -energ y magnon \nbranches emanating from the antiferromagnetic Bragg reflections , these experiments have also \nrevealed weakly dispersive  “spin -orbit exciton” modes corresponding to spin excitations from the \nJeff = ½ ground state into the J eff = 3/2 excited state (Fig. 21). [27] Since the dispersion of these \nmodes is controlled by the combination of the intra -atomic spin -orbit coupling, the crystalline \nelectric field, and the inter -atomic exchange interactions, RIXS experiments are an incisive probe \nof the low -energy electronic structure of these materials.   \nFinally, to illustrate  the diversity of inelastic x-ray scattering methods applied to magnetism, we \nhighlight results of an  x-ray emission spectroscopy study of i ron arsenide superconductors  of \ncomposition Ca 1-xRxFe2As2 (where R = rare earth).  [61] The goal of this experiment was to \nelucidate the origin of a pressure -induced structural phase transition from an antiferromagnetic to \na nonmagnetic state that is associated with a large volume reduction.  [62] [63] To measure the \nlocal magnetic moment of the Fe  ions independent of any interatomic correlations , x-ray photons \nwere  tuned to the Fe K -absorption edge (1s -3d), and the spectrum of emitted x -rays was monitored \naround the  dipole -active  Kβ emission line ( 2p-1s). A local moment on the Fe site induces a splitting \nof this line (inset of Fig. 22) whose size depends on the moment amplitude. These experiments led \nto the discovery of a pressure induced spin- state transition  from a high- spin to a low -spin \nconfiguration of the Fe atoms. The lower volume of the low -spin Fe atoms  explains the vol ume \ncollapse in the nonmagnetic phase  at high pressures.  \n \n \nIX. Facilities  and Online Information  \n A list of current neutron scattering facilities around the world can be found at \n(http://e n.wikipedia.org/wiki/Neutron_research_facility ). Numerical values of the free- ion 22 \n magnetic form factors for neutrons can be obtained at \nhttps://www.ill.eu/sites/ccsl/ffacts/ffachtml.html .  Values of the coherent nuclear scattering \namplitudes and other nuclear cross -sections can be found at http://www.ncnr.nist.gov/resources/n-\nlengths/ . \n \n A list of current x -ray scattering facilities can be found at  \n(http://en.wikipedia.org/wiki/Lis t_of_synchrotron_radiation_facilities ).   \n \nValues for characteristic x -ray energies and a guide to the literature on x -ray form factors can be \nfound at http://xdb.lbl.gov/ . \n \nEnergy Units:  Traditionally  magnetic excitations are  quoted in units of meV but sometimes \nauthors use THz, particularly for phonons in older literature.  Raman and IR experimenters often \nuse cm-1.  1 meV   0.24180 THz   8.0655 cm-1  11.605 K. \n \nFor a wavelength λ  = 1.54 Å the photon energy is 8.05 keV, the electron energy is 63.4 eV, and \nfor a neutron the energy is 34.5 meV. \n \nX. Summary and Future Directions  \nIn this review we have discussed the basic characteristics of magnetic neutron and x -ray \nscattering and provided a number of experimental examples of how these techniques can be \nemployed.  Neutron scattering is a rather mature technique which has the advantage of being  a \nweakly interacting probe that does not affect the properties of the sample.  The source of neutrons \nhas traditionally been steady state reactor based facilities, but this has now been complemented by \nthe newer, pulsed spallation neutron source facilities  which can offer higher peak flux than steady -\nstate reactors .  Both types of sources have many different types of spectrometers that enable \nmagnetic investigations over many orders -of-magnitude in both spatial and time domains.  In \naddition to new sources and new types of sources, many of the advancements in ne utron technique s \nover the years ha ve com e from developments in how to tailor and manipulate neutrons, vast arrays \nof detectors, and the software to analyze and visualize the data , and this progress continues \nunabated .  New sources and new instrumentation currently are being planned and developed, with \nthe anticipation that measurement capabilities will be greatly increased together with an increased \nquantity and scale of data acquired.  \n Resonant x -ray scattering is a much newer technique, with high brightness that allows \nmeasurements of small bulk samples, thin films, and multilayers.  It also has the advantage of \nbeing element specific as the resonance is tuned to an absorption edge.  Tremendous progress in \nmeasurement capabilities has been realized in the last few years, both with magnetic diffr action \nand magnetic inelastic scattering.  In contrast to neutron scattering which is on a solid theoretical \nfoundation, the theoretical understanding and interpretation of magnetic x -ray scattering is \nundergoing considerable development, which should lead to improved interpretation of \nexperimental data and exciting new capabilities.  \nThe future of both techniques is brilliant, pun intended. 23 \n  \nAcknowledgments  \nWe thank our many colleagues who have collaborated with us on our own projects, where \nwe have taken a number  of the examples for our own convenience and familiarity with the \nsystems.   \n \nReferences  \n \n[1]  S. W. Lovesey, Theory of neutr on scattering from condensed matter, Oxford: Clarendon Press - \nOxford, 1984.  \n[2]  J. W. Lynn, \"Magnetic Neutron Scattering,\" J. Applied Physics, vol. 75, p. 6806, 1994.  \n[3]  M. Blume, \"Magnetic Scattering of X -rays,\" J. Appl. Phys., vol. 57, p. 3615, 1985.  \n[4]  E. Balcar and S. Lovesey, Theory of Magnetic Neutron and Photon Scattering, Oxford: Clarendon \nPress, 1989.  \n[5]  J. Hill and D. McMorrow, \"Resonant Exchange Scattering: Polarization Dependence and Correlation \nFunctions,\" Acta Cryst A, vol. 52, pp. 236 -  244, 1996.  \n[6]  T. Matsumura, H. Nakao and Y. Murakami, \"Resonant x -ray scattering experiments on the ordering \nof electronic degrees of freedom,\" J. Phys. Soc. Jpn., vol. 82, p. 021007, 2013.  \n[7]  L. Ament, M. van Veenendaal, T. P. Devereaux, J. P. Hill and J. van den Brink, \"Resonant inelastic s -\nray scattering studies of elementary excitations,\" Rev. Mod. Phys., vol. 83, p. 705, 2011.  \n[8]  P. D. Hatton, R. D. Johnson, S. R. Bland, C. Mazzoli, T. A. W. Beale, C. -H. Du and S. B. Wilkins, \n\"Magnetic structure determination using polarised resonant X -ray scattering,\" J. Magnetism & \nMagnetic Materials, vol. 321, p. 810 , 2009.  \n[9]  R. M. Moon, T. Riste and W. C. Koehler, \"Polarization Analysis,\" Phys. Rev., vol. 181, pp. 920 - 931, \n1969.  \n[10]  G. E. Bacon, Neutron Diffraction, Third ed., Oxford: Oxford University Press, 1975.  \n[11]  M. Blume, \"Orbital Contribution o f Form Factor of Ni++,\" Phys. Rev., vol. 124, p. 96, 1961.  \n[12]  G. T. Trammell, Phys.. Rev., vol. 92, p. 1287, 1953.  \n[13]  G. W. Williams, Polarized Neutrons, Oxford: Clarendon Press - Oxford, 1988.  24 \n [14]  A. Poole, E. Lelievre -Berna and A. S. Wills, \"General Refinement Strategy for Magneitc Structures \nusing Spherical Neutron Polarimetry and Representation Analysis,\" Physica B, vol. 404, p. 2535, \n2009.  \n[15]  H. Zhang and J. W. Lynn, \"Analytic Calculation of Polarized Neutron Reflectivity from \nSuperconductors,\" Phys. Rev. B, vol. 48, p. 15893, 1993.  \n[16]  J. Daillant and A. Gibaud, Eds., X -ray and Neutron Reflectivity: Principles and Applications, Berlin: \nSpringer, 2009.  \n[17]  B. P. Toperverg and H. Zabel, \"Neutron Scattering in Nanomagnetism,\" in Experimental Methods in \nthe Physical Sciences: Neutron Scattering - Magnetic and Quantum Phenomena, vol. 48, F. \nFernandez- Alonso and D. L. Price, Eds., London, Elsivier, 2015, p. 339.  \n[18]  M. W. Haverkort, N. Hollmann, I. P. Krug and A. Tanaka, \"Symmetry analysis of magneto -optical \neffects: The case of x- ray diffraction and absorption at the transition metal L2,3 edge,\" Phys. Rev. \nB, vol. 82, p. 094403, 2010.  \n[19]  P. Leininger, M. Rahlenbeck, M. Raichle, B. Rohnenbuck, A. Maljuk, C. T. Lin, B. Keimer, E. \nWeschke, E. Schierle, S. Seki, Y. Tokura and J. W. Freeland, \"Electronic structure, magnetic, and \ndielectric properties of the edge- sharing copper oxide chain compo und NaCu2O2,\" Phys. Rev. b, \nvol. 81, p. 085111, 2010.  \n[20]  D. B. McWhan, E. D. Isaacs, P. Carra, S. M. Shapiro, B. T. Thole and S. Hoshino, \"Resonant magnetic \ns-ray scattering from mixed -valence TmSe,\" Phys. Rev. B, vol. 47, p. 8630, 1993.  \n[21]  P. A.  Fleury and R. Loudon, \"Scattering of Light by One - and Two -Magnon Excitations,\" Phy. Rev., \nvol. 166, pp. 514 - 530, 1966.  \n[22]  L. Braicovich, L. J. P. Ament, V. Bisogni, F. Forte, C. Aruta, G. Balestrino, N. B. Brookes, G. M. De \nLuca, P. G. Medaglia, F . M. Granozio, M. Radovic, M. Salluzzo, J. van den Brink and G. Ghiringhelli, \n\"Dispersion of Magnetic Excitations in the Cuprate La2CuO4 and CaCuO2 Compounds Measured \nUsing Resonant X -Ray Scattering,\" Phys. Rev. Lett. , vol. 102, p. 167401, 2009.  \n[23]  L. Braicovich, J. van den Brink, V. Bisogni, M. Moretti Sala, L. J. P. Ament, N. B. Brookes, G. M. \nDeLuca, M. Salluzzo, T. Schmitt, V. N. Strocov and G. Ghiringhelli, \"Magnetic Excitations and Phase \nSeparation in the Underdoped La2 -xSrxCuO4 Superconductor m easured by Resonant Inelastic x -\nray scattering,\" Phys. Rev. Lett., vol. 104, p. 077002, 2010.  \n[24]  M. L. Tacon, G. Ghiringhelli, J. Chaloupka, M. Moretti Sala, V. Hinkov, M. W. Haverkort, M. Minola, \nM. Bakr, K. J. Zhou, S. Blanco -Canosa, C. Monney, Y. T . Song, G. L. Sun, C. T. Lin, G. DeLuca, M. \nSalluzzo, G. Khaliullin, L. Braicovich, T. Schmitt and B. Keimer, \"Intense paramagnon excitations in a \nlarge family of high -temperature superconductors,\" Nature Physics, vol. 7, pp. 725 -730, 2011.  25 \n [25]  K.-J. Zhou, Y. -B. Huang, C. Monney, X. Dai, V. N. Strocov, N. -L. Wang, Z. -G. Chen, C. Zhang, P. Dai, L. \nPatthey, J. van den Brink, H. Ding and T. Schmitt, \"Persistent high -energy spin excitations in iron -\npnictide superconductors,\" Nature Communications, vol. 4, p . 1470, 2013.  \n[26]  V. Bisogni, L. Simonelli, L. Ament, F. Forte, M. Moretti Sala, M. Minola, S. Huotari, J. van den Brink, \nG. Ghiringhelli, N. Brookes and L. Braicovich, \"Bimagnon studies in cuprates with resonant inelastic \nx-ray scattering at the O K edge. I. Assessment on La2CuO4 and comparison with the excitation at \nCu L3 and Cu K edges,\" Phys. Rev. B, vol. 85, p. 214527, 2012.  \n[27]  J. Kim, M. Daghofer, A. H. Said, T. Gog, J. van den Brink, G. Khaliullin and B. J. Kim, \"Excitonic \nquasiparticles in a spin -orbit Mott insu lator,\" Nature Communications, vol. 5, p. 4453, 2014.  \n[28]  H. Suzuki, H. Gretarsson, H. Ishikawa, K. Ueda, Z. Yang, H. Liu, H. Kim, D. Kukusta, A. Yaresko, M. \nMinola, J. Sears, S. Francoual, H. -C. Wille, J. Nuss, H. Takagi, B. Kim, G. Khaliullin, H. Yav as and B. \nKeimer, \"Spin waves and spin -state transitions in a ruthenate high -temperature antiferromagnet,\" \nNature Materials, vol. 18, p. 563, 2019.  \n[29]  M. Minola, G. Dellea, H. Gretarsson, Y. Y. Peng, Y. Lu, J. Porras, T. Loew, F. Yakhou, N. B. Brookes , Y. \nB. Huang, J. Pelliciari, T. Schmitt, G. Ghiringhelli, B. Keimer, L. Braicovich and M. Le Tacon, \n\"Collective Nature of Spin Excitations in Superconducting Cuprates Probed by Resonant Inelastic S -\nray Scattering,\" Phys. Rev. Lett., vol. 114, p. 217003, 2015.  \n[30]  X. Li, X. Ma, D. Su, L. Liu, R. Chisnell, S. P. Ong, H. Chen, A. Toumar, J. -C. Idrobo, Y. Lei, J. Bai, F. \nWang, J. W. Lynn, Y. S. Lee and G. Ceder, \"Direct Visualization of the Jahn -Teller Effect Coupled to \nNa Ordering in  Na5/8MnO2,\" Nature Materials, vol. 13, p. 586, 2014.  \n[31]  Q. Huang, P. Karen, V. L. Karen, A. Kjekshus, A. D. Migdall, N. Rosov, J. W. Lynn and A. Santoro, \n\"Neutron Powder Diffraction Study of the Nuclear and Magnetic Structures of YBa2Fe3O8 at Room \nTemperature,\" Phys. Rev. B, vol. 45, p. 9611, 1992.  \n[32]  W.-H. Li, C. -W. Wang, D. Hsu, C. -H. Lee, C. -M. Wu, C. -C. Chou, H. -D. Yang, Y. Zhao, S. Chang, J. W. \nLynn and H. Berger, \"Interplay between the magnetic and electric degree -of-freedoms in \nmultiferroics Co3TeO6,\" Phys. Rev. B, vol. 85, p. 094431, 2012.  \n[33]  C.-H. Lee, C. -W. Wang, Y. Zhao, W. -H. J. Li, A. Harris, K. Rule, H. -D. Yang and H. Berger, \"Complex \nmagnetic incommensurability and electronic charge transfer through the ferroelectric transition in \nmultiferroic Co3TeO6,\" Scientific Reports, vol. 7, p. 643 7, 2017.  \n[34]  C.-M. Wang, C. -H. Lee, C. -Y. Li, C. -M. Wu, W. -H. Li, C. -C. Chou, H. -D. Yang, J. W. Lynn, Q. Huang, A. \nB. Harris and H. Berger, \"Complex Magnetic Couplings in Co3TeO6,\" Phys. Rev. B, vol. 88, p. \n184427, 2013.  \n[35]  S.-M. Choi, J. W. Lynn,  D. Lopez, P. L. Gammel, P. C. Canfield and S. L. Bud'ko, \"Direct Observation \nof Spontaneous Weak -Ferromagnetism in the Superconductor ErNi2B2C,\" Phys. Rev. Lett., vol. 87, \np. 107001, 2001.  26 \n [36]  W. Thomlinson, J. W. Lynn, G. Shirane and D. E. Moncton, N eutron Scattering Studies of Magnetic \nOrdering in Ternary Superconductors, Vol. 34, Chap. 8 ed., M. B. M. a. O. Fischer, Ed., New York: \nSpringer -Verlat, 1983.  \n[37]  S.-M. Choi, J. W. Lynn, D. Lopez, P. L. Gammel, C. M. Varma, P. C. Canfield and S. L. Bud 'ko, \n\"Ferromagnetism and Spontaneous Vortex Formation in Superconducting ErNi2B2C,\" NCNR annual \nreport 2001, http://www.ncnr.nist.gov/AnnualReport/FY2001_pdf/, 2001.  \n[38]  H. Kawano -Furukawa, Y. Ishida, F. Yano, R. Nagatomo, A. Noda, T. Nagara, S. Ohira -Kawamura, C. \nKobayashi, Yoshizawa, K. Littrell, B. L. Winn, N. Furakawa and H. Takeya, \"Creation of vortices by \nferromagnetic order in ErNi2B2C,\" Physica C, vol. 470, p. 5716, 2010.  \n[39]  J. S. Gardner, M. J. P. Gingras and J. E. Greedan, Rev. Mod. Phys.,  vol. 82, p. 53, 2010.  \n[40]  S. T. Bramwell, M. J. Harris, B. C. den Hertog, M. J. P. Gingras, J. S. Gardner, D. F. McMorrow, A. R. \nWildes, A. L. Cornelius, D. M. Champion, R. G. Melko and T. Fennell, \"Spin Correlations in \nHo2Ti2O7: A Dipolar Spin Ice System,\" Phys. Rev. Lett., vol. 8 7, p. 047205, 2001.  \n[41]  H. Kadowaki, N. Doi, Y. Aoki, Y. Tabata, T. J. Sato, J. W. Lynn, K. Matsuhira and Z. Hiroi, \n\"Observation of Magnetic Monopoles in Spin Ice,\" J. Phys. Soc. Japan, vol. 78, p. 103706, 2009.  \n[42]  C. Costelnovo, R. Moessner and S . L. Sondhi, \"Magnetic Monopoles in Spin Ice,\" Nature, vol. 451, p. \n42, 2008.  \n[43]  D. J. P. Morris, D. A. Tennant, S. A. Grigera, B. Klemke, C. Castelnovo, R. Moessner, C. Cztemasty, \nM. Meissner, K. C. Rule, J. U. Hoffmann, K. Kiefer, S. Gerischer, D. Slobinsky and R. S. Perry, \"Dirac \nStrings and Magnetic Monopoles in the Spin Ice Dy2Ti2 O7,\" Science, vol. 326, p. 411, 2009.  \n[44]  T. Fennell, P. P. Deen, A. R. Wildes, K. Schmalzl, D. Prabhakaran, A. T. Boothroyd, R. J. Aldus, D. F. \nMcMorrow and S. T. Bramwell, \"Magnetic Coulomb Phase in the Spin Ice Ho2Ti2O7,\" Science, vol. \n326, p. 415, 2009.  \n[45]  T. S. Santos, B. J. Kirby, S. Kumar, S. J. May, J. A. Borchers, B. B. Maranville, J. Zarestky, S. G. E. Te \nVelthuis, J. van den Brink and A. Bhattacharya, \"Delta Doping of Ferromagnetism in \nAntiferromagnetic Manganite Superlattices,\" Phys. Re v. Lett., vol. 107, p. 167202, 2011.  \n[46]  B. J. Kim, H. Ohsumi, T. Komesu, S. Sakai, T. Morita, H. Takagi and T. Arima, \"Phase sensitive \nobservation of a spin -orbital Mott state in Sr2IrO4,\" Science, vol. 323, p. 1329, 2009.  \n[47]  M. Garcia -Fernandez, S. B. Wilkins, M. Lu, L. Qing'an, K. E. Gray, H. Zheng, J. F. Mitchell and D. \nKhomskii, \"Antiferromagnetic domain structure in a bilayer manganite,\" Phys. Rev. B, vol. 88, p. \n075134, 2013.  \n[48]  A. Frano, E. Schierle, M. W. Haverkort , Y. Lu, M. Wu, S. Blanco -Canosa, U. Nwankwo, A. V. Boris, P. \nWochner, G. Cristiani, H. U. Habermeier, G. Logvenov, V. Hinkov, E. Benckiser, E. Weschke and B. 27 \n Keimer, \"Orbital control on noncollinear magneti order in Nickel Oxide Heterostructures,\" Phys. \nRev. Lett., vol. 111, p. 106804, 2013.  \n[49]  S. Macke, J. E. Hamann -Borrero, R. J. Green, B. Keimer, G. A. Sawatzky and M. W. Haverkort, \n\"Dynamical Effects in Resonant X -ray Diffraction,\" Phys. Rev. Lett., vol. 117, p. 115501, 2016.  \n[50]  M. Forst, A. C aviglia, R. Scherwitzl, R. Mankowsky, P. Zubko, V. Khanna, H. Bromberger, S. B. \nWilkins, Y. D. Chuang, W. S. Lee, W. F. Schlotter, J. J. Tumer, G. L. Dakovski, M. P. Minitti, J. \nRobinson, S. R. Clark, D. Jaksch, J. M. Triscone, J. P. Hill, S. S. Dhesi and A. Cavallen, \"Spatially \nresolved ultrafast dynamics initiated at a complex oxide interface,\" Nature Materials, vol. 14, p. \n883, 2015.  \n[51]  J. W. Lynn, R. W. Erwin, J. A. Borchers, Q. Huang, A. Santoro, J. Peng and Z. Li, \"Unconventional \nFerromagnetic Transition In La1 -xCaxMnO3,\" Phys. Rev. Lett., vol. 76, p. 4046, 1996.  \n[52]  J. W. Lynn, N. Rosov and G. Fish, \"Polarization Analysis of the Magnetic Excitations in Invar and \nNon -Invar Amorphous Ferromagnets,\" J. Appl. Phys., pp. 5369 -  5371, 1993.  \n[53]  L. Vasiliu -Doloc, J. W. Lynn, A. H. Moudden, A. M. de Leon -Guevara and A. Revcolevschi, \"Structure \nand spin dynamics of La0.85Sr0.15MnO3,\" Phys. Rev. B, vol. 58, p. 14913, 1998.  \n[54]  F. Ye, P. Dai, J. A. Fernandez -Baca, H. Sha, J. W. Lynn, H. Kawano -Furukawa, Y. Tomioka, Y. Tokura \nand J. Zhang, \"Evolution of Spin -Wave Excitations in Ferromagnetic Metallic Manganites,\" Phys. \nRev. Lett., vol. 96, p. 047204, 2006.  \n[55]  S. P. Bayrakci, D. A. Tennant, P. Leinninger, T. Keller, C. R. Gibson, S. D. Wilson, K. Habicht, R. J. \nBirgeneau and B. Keimer, \"Lifetimes of Antiferromagnetic Magnons in Two and Three Dimensions: \nExperiment, Theory, and Numerics,\" Phys. Rev. Lett., vol. 111, p. 017204, 2006.  \n[56]  M. Fujita, H. Hiraka, M. Matsuda, M. Matsuura, J. M. Tranquada, S. Wakimoto, G. Xu and K. \nYamada, \"Progress in Neutron Scattering Studies of Spin Excitations in High -Tc Cuprates,\" J. Phys. \nSoc. Japan, vol. 81, p. 011007, 2012.  \n[57]  M. D. Lumsden and A. D. Christianson, \"Magnetism in Fe -based superconductors,\" J. Phys. \nCondens. Matter, vol. 22, p. 203203, 2010.  \n[58]  L. W. Harriger, H. Q. Luo, M. S. Liu, C. Frost, J. P. Hu, M. R. Norman and P. Dai, \"Nematic spin fluid \nin the tetragonal phase of BaFe2As2 ,\" Phys. Rev. B, vol. 84, p. 054544, 2011.  \n[59]  B. Lake, D. A. Tennant, J. -S. Caux, T. Barthel, U. Schollwock, S. E. Nagler and C. D. Frost, \n\"Multispinon Continua at Zero and Finite Temperature in a Near- Ideal Heisenberg Chain,\" Phys. \nRev. Lett., vol. 1 11, p. 137205, 2013.  \n[60]  M. Mourigal, M. Enderle, A. Klöpperpieper, J. -S. Caux, A. Stunault and H. M. Ronnow, \"Fractional \nspinon excitations in the quantum Heisenberg antiferromagnetic chain,\" Nature Physics, vol. 9, p. \n435, 2013.  28 \n [61]  H. Gretarsson, S. R. Saha, T. Drye, J. Palione, J. Kim, D. Casa, T. Gog, W. Wu, S. R. Julian and Y. -J. \nKim, \"Spin state transition in the Fe pnictides,\" Phys. Rev. Lett., vol. 110, p. 047003, 2013.  \n[62]  A. Kreyssig, M. Green, Y. lee, G. Samolyuk, P. Zaj del, J. Lynn, S. Bud'ko, M. Torikachvili, N. Ni, S. \nNandi, J. Leao, S. Poulton, D. Argyriou, B. Harmon, R. McQueeney, P. Canfield and A. Goldman, \n\"Pressure -induced volume -collapsed tetragonal phase of CaFe2As2 as seen via neutron \nscattering,\" Phys. Rev. B,  vol. 78, p. 184517, 2008.  \n[63]  S. Saha, N. Butch, T. Drye, J. Magill, S. Ziemak, K. Kirshenbaum, Z. P. Y. J. Lynn and J. Paglione, \n\"Structural collapse and superconductivity in rare -earth -doped CaFe2As2,\" Phys. Rev. B, vol. 85, p. \n024525, 2012.  \n \n \n  29 \n Figures  \n \n \n \n \n \n \n \n \n \n \nFigure   1. (top) Energy/density -of-states diagram illustrating RIXS with photons near the \nL-absorption edges of 3d  (green) and 4d (blue) metals. (bottom) Reciprocal lattice of \northorhombic perovskite antiferromagnets, with structural (black) and magnetic (red) Bragg \nreflections. Circles indicate the maximal coverage of RIXS with photons at the Cu (green) \nand Ru (blue) L 2,3-edges.  (top panel adapted from [7], © American Physical Society 2011).  30 \n  \nFigure 2 .  Magnetic diffraction pattern for Na 5/8MnO 2, obtained by subtracti ng the \ncrystallographic diffraction pattern obtained at 100 K, above the antiferromagnetic phase \ntransition, from the data at 2.5 K in the magnetic ground state.  The structural scattering cancels \nin the subtraction if there is no significant change when the  sample magnetically orders.  The \ninset shows the temperature dependence of the intensity of the strongest magnetic peak, and \nreveals a transition temperature of ≈60 K ( adapted from  [30] , ©Spinger Nature 2014).  \n31 \n  \nFigure 3.  Polarized neutron diffraction on polycrystalline YBa 2Fe3O8.  The top portion of the \nfigure is for P⊥g, where the open circles show the non- spin- flip scattering and the filled circles \nare in the spin -flip configuration.  The low angle peak has equal intensity for both cross sections, \nand thus is identified as a pure magnetic reflection, while the ratio of the (+ +) to ( - +) scattering \nfor the high angle peak is just the instrumental flipping ratio.  Hence this is a pure nuclear \nreflection.  The center portion of the figure is for P||g, and the bottom portion is the subtraction \nof the spin- flip data for the P ⊥g configuration from the spin- flip data for P||g.  Note that in the \nsubtraction procedure all background and nuclear cross sections cancel, thereby  isolating the \nmagnetic scattering.  (reprinted by  permission from  [31] , © American Physical Society 1992 ). \n32 \n  \nFigure 4 .  Neutron diffraction intensity map observed in the ( h, k, 0) scattering plane of a single \ncrystal of the multiferroic Co 3TeO 6.  The temperature is 22 K, just below the antiferromagnetic \nphase transition at T N=26 K.  The nuclear Bragg peaks at integer positions are accompanied by \nfour satellite magnetic reflections, indicating the development of incommensurate (ICM) \nmagnetic order.  Note that the ordering wave vector is incommensurate in both h and k .  No \nenerg y analyzer was used for these measurements so that the data are energy -integrated, and \nthere is clear diffuse scattering surrounding the ICM peaks at this temperature originating from  \ninelastic magnetic excitations ( adapted from  [32] , © American Physical Society 2012 ). \n \n33 \n  \n \n \nFigure 5 .  (top) Unpolarized neutron diffraction measurements along the ( h,0,0) direction at 1.3 \nK, 2.4 K, and 4.58 K  of a single crystal of ErNi 2B2C.  At 10 K no peaks are observed in this \nwave vector range.  The data have been offset along the intensity axis for clarity. Above the \nweak ferromagnetic transition at 2.3 K the fundamental incommensurate peak is observed at \nh=0.55, along with higher odd- order harmonics.  Below the ferromagnetic transition a new set of \neven -order harmonics develops, indicated by the arrows.  (bottom) Schematic of the i nitial \ntransversely polarized spin- density -wave, and ground state square -wave ( adapted from  [35] , © \nAmerican Physical Society 2001 ). \n \n34 \n  \nFig. 6.  Polarized neutron diffraction measurements on a single crystal of ErNi 2B2C showing \nboth the odd order (5th) and even- order (16th) harmonics for the P||g configuration.  The solid \ncircles ( - , +) and solid triangles (+ , -) are spin -flip scattering, while the open circles (+ , +) and \nopen triangles ( - , -) are non -spin- flip scattering.  The data demonstrate that both types of \nreflections are magnetic in or igin, with the moment direction along the b axis  (adapted from  [35] , \n© American Physical Society 2001 ). \n \n35 \n  \n \nFigure 7 .  Radially averaged small angle neutron scattering i ntensity of the vortex scattering in \nErNi 2B2C vs. wave vector K  at 85  mT, above and below the weak ferromagnetic transition.  The \nshift in the peak position demonstrates that additional vortices spontaneously form as the \nmacroscopic magnetization develops at low temperatures.  The temperature dependence shows \nthat this spontaneous vortex formation is directly related to the weak ferromagnetic transition.  \nThe inset shows vortex Bragg peaks on the two- dimensional SANS detector;  K  = 0 is in the \ncenter.  (adapted from  [37] ). \n  \n36 \n  \nFigure 8.  (A ) Neutron measurements of the diffuse magnetic scattering in the kagomé spin- ice \ncompound Dy 2Ti2O7 at T=0.43 K and B=0.5 T.  The sharp structural Bragg peaks, such as (2,-\n2,0), are contained within one pixel and have been removed from the plot.  (C ) Monte Carlo \nsimulations of the expected scattering in this  kagomé spin- ice state.   The overall features are in \nexcellent agreement with the data ( adapted from  [41] , © The Physical Society of Japan 2009 ). \n  \n37 \n  \nFigure 9 .  (a) Non -spin- flip polarized neutron reflectivity data R++ (red) and R- -(blue) on a \nLaMnO 3/SrMnO 3 multilayer, measured in a 675 mT  field at 120 K.  The inset shows a schematic \nof the superlattice.  (b) Magnetic depth profile  determined  by the fit to the data.  Location  of the \nLaMnO3 (pink)  and SrMnO3 (green)  layers  are shown.  (c)  Spin- flip intensity,  showing  the \nantiferromagnetic peak and satellite peak.   Inset  shows the non- spin- flip scattering in the same \nrange ( adapted from  [45] , © American Physical Society 2011 ). \n38 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 1 0.  Photon energy dependence of the (1, 0, 22) magnetic Bragg reflection at the  L3-\n(left) and L2-edges  (right) of Sr 2IrO 4. The black lines show the x -ray absorption spectra for \ncomparison ( reprinted with permission from  [46] , © American Associati on for the \nAdvancement of Science 2009). 39 \n  \n \n \n \n \n \n \n  \nFigure 11.  Map of the resonant elastic x -ray scattering intensity at the (0, 0, 1) magnetic \nBragg reflection of La 0.96Sr2.04Mn 2O7 at the Mn L 3-edge. The data indicate domains where \nthe Mn spins point in different directions in the MnO 2 layers ( reprinted with  permission \nfrom  [47] , © American Physical Society 2013).  40 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 12. (top left) Ni L 3-edge scans through the (¼, ¼, ¼) magnetic Bragg reflection of \nLaNiO 3-LaAlO 3 superlattices with different numbers of consecutive unit cells. The absence \nof the magnetic Bragg peak in superlattices with 3 or more LaNiO 3 layers indicates that the \nmagnetic order in the 2x2 superlattice is induced by spatial confinement of the conduction \nelectrons. ( reprinted with  permission from  [48]) (top right) Azimuthal angle dependence of \nthe (¼, ¼, ¼) magn etic Bragg peak of nickelate thin films and superlattices with simulations \nthat rule out collinear (CM) and favor non- collinear (NCM) magnetism. (bottom) Simulated \ncontour map of the scattering intensity of 2x2 LaNiO 3-LaAlO 3 superlattice as functions of \nazimuthal angle and momentum transfer perpendicular the to superlattice plane, \ndemonstrating the importance of dynamical diffraction effects ( reprinted with  permission \nfrom [49] , © American Physical Society 2016).  41 \n  \n  \nFigure 13.  (a) Schematic illustration of the demagnetization proce ss of a NdNiO 3 film \ntriggered by a coherently excited photon of the LaAlO 3 substrate. (b) Depth profile of the  \n(¼, ¼, ¼) resonant magnetic Bragg peak  intensity at different time delays between the \nphonon pump pulse and the resonant x -ray diffraction probe measurement. ( reprinted with  \npermission from  [50] , ©Springer Nature 2015).  42 \n  \nFigure 14 .  Spin waves in isotropic ferromagnets.  (left) Energy scan at a wave vector q  of 0.07 \nÅ-1 for La 0.7Ca0.3MnO 3, (published with permission from  [51] , © American Physical Society \n1996)  showing the spin waves in energy gain (E<0) and energy loss (E>0).  (Right) polarized \nbeam energy scan on the Fe86B14 amorphous ferromagnet at a fixed wave vector of 0.09 Å-1, with \nthe neutron polarization parallel to q .  In this configuration spin angular  momentum is conserved, \nand the neutron can only create an excitation (E>0) if its moment is initially antiparallel to the \nmagnetization, and can only destroy a spin wave (E<0) when its moment is parallel  (reprinted \nwith permission from  [52] , © American Institute of Physics 1996) . \n  \n43 \n  \n \n \n \nFigure 15 .  (a) Low energy spin wave dispersion relations at two different temperatures.  The \ndispersion relation follows a quadratic dependence expected for a ferromagnet, which defines the \nspin stiffness D , and no significant gap in the excitation spectrum is observed indicating an \nisotropic system.  D (T) is shown in (b), which follows a power law behavior as the Curie \ntemperature is approached  (reprinted with  permission from  [53] , © American Physical Society \n1998).  (right)  Spin wave dispersion relations for a series of colossal magnetoresistive \nperovskite oxides  (reprinted with  permission from  [54] , © American Physical Society 2006 ). \n  \n44 \n  \n \nFigure 16 .  (left) Intrinsic spin wave linewidths for the ground state magnetic excitations in \nLa0.85Sr0.15MnO 3.  The linewidths are quite anisotropic, and are significant at small wave vectors \nbut become very large at large q ( reprinted with  permission from  [53] , © American Physical \nSociety 1998).   (Right) magnon linewidths as a function of temperature for a series of q’s in the \ninsulating antiferromagnet Rb 2MnF 4, measured using the high resolution spin- echo triple -axis \ntechnique  (reprinted with  permission from  [55] , © American Physical Society 2006 ).  The solid \ncurves are calculations using spin wave theory.  \n  \n45 \n  \n \n \n \nFig. 17.  Left:  Constant -energy cuts of the magnetic excitations in BaFe 2As2 at a series of \nenergies.  The solid curves are the fits to the spin wave model.  (Right) Spin wave dispersion \nalong the (1, K ) direction as determined by energy and Q cuts of the raw data.  The solid line is a \nHeisenberg model calculation using anisotropic exchange couplings SJ 1a = 59.2 ± 2.0, SJ 1b = \n−9.2 ± 1.2, SJ 2 = 13.6 ± 1.0, SJc = 1.8 ± 0.3 meV determined by fitting the full cross section. \nThe dotted line is a Heisenberg model calculation assuming isotropic exchange coupling SJ 1a = \nSJ1b = 18.3 ± 1.4, SJ 2 = 28.7 ± 0.5, and SJc = 1.8 meV  (adapted from  [58] , © American \nPhysical Society 2011 ). \n \n46 \n  \n \nFig. 18.  Intensity color maps of the experimental inelastic neutron scattering spectrum measured \nalong the Cu chain  in CuSO 4⋅5D 2O are shown in the left, compared with the theoretical two - and \nfour-spinon dynamic structure factor  (reprinted with  permission from  [60] , © Springer Nature \n2013) . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n47 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 19.  (left) RIXS profile of La 2CuO 4 taken at the Cu L 3 edge with ≈100 meV energy \nresolution. The spectrum can be decomposed into elastic (A), single magnon (B), multiple \nmagnon (C) and optical phonon (D) components . The inset shows the x -ray absorption \nspectrum  near the Cu L 3 edge, the arr ow marks the energy of the incident photons. (right) \nSingle magnon dispersion determined by RIXS (blue dots), compared to inelastic  neutron \nscattering data ( dashed line)  (reprinted with  permission from  [23] , © American Physical \nSociety 2010).  48 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 20.  Photon energy dependence of the RIXS intensity ((a),(b) ) for undoped \nantiferromagnetic YBa 2Cu3O6.1 and ((c), (d) ) superconducting  Ca-substituted YBa 2Cu3O7 in \npolarization geometries that predominantly select spin ( a),(c) and charge (b),(d) excitations. \nThe horizontal dashed lines highlight the energy i ndependence of the magnetic pe ak \nposition, while the dashed green line is a guide to the e ye underlining the fluorescence \nbehavior of the continuum of charge excitations from the doped holes  (reprinted with  \npermission from  [29] , © American Physical Society 2015).  49 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 21.  (top)  The spin –orbital level scheme of Sr 2IrO 4. The spin -orbit coupling λ splits \nthe d -electron manifold into J eff=1/2 and 3/2 multiplets. The crystal field Δ lifts the \ndegeneracy of the J eff=3/2  multiplet. O range (blue) colors in the images of the orbitals \nrepre sent spin up (down) projections. (bottom) Dispersion of magnons and spin- orbit \nexcitons (marked with QP for “quasiparticle”) extracted from RIXS data at the Ir L -edge \n(reprinted with  permission from  [27] , © Springer Nature 2014).  50 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure 22.  Fe K β emission spectra of Ca 1-xRxFe2As2 with R = Nd, and difference spectrum \nwith FeCrAs where Fe is in a nonmagnetic spin -0 state. The difference spectrum indicates a \nsplitting of the emission line due to a local magnetic moment on the Fe site (inset) \n(reprinted with  permission from  [61] , © American Physical Society 2013).  "    },    {        "title": "1302.1964v1.Investigation_on_the_critical_dynamics_of_real_magnetics_models_by_computational_physics_methods.pdf",        "content": "Investigation on the critical dynamics of real magnetics models by computational \nphysics methods  \nA.K. Murtazaev, V.A. Mutailamov \nInstitute of Physics, Dagestan Scientific Center, Russian Academy of Sciences, Makhachkala, 367003 Russia \ne-mail: vadim.mut@mail.ru  \n \nThe critical dynamics of classical 3D Heisenberg m odel and complex model of the real antiferromagnetic \nCr2O3 is investigated with use of the method of molecular dynamics. The dynamic criti cal exponent z are determined for \nthese models on the basis of the theory dynamic finite-size scaling.   \nDespite of considerable successes reached to the present time in investigations of the critical \ndynamics of spin systems, the quantitative study of this problem still is one of actual problems of a \nmodern statistical physics [1]. Some theoretical approaches available in this field, developed \nindependent from each other and are based on completely different ideas. Among these approaches \nwe shall mark the mode-coupling theory and theory of  the dynamic scaling [1,2]. These theories \nhave allowed qualitatively correctly  explaining many experimental fact s. It is possible to receive \nquantitative results for simple model systems only fr om the theoretical appro aches with use of the \nrenormalization group theory and ε-expansion theory [2,3]. \nRecently, the various approaches based on numer ical methods are wide ly applied to study a \ndynamic critical behavior [4]. Note that usi ng of numerical experiment methods allows \ninvestigating of complex real syst em models. The existing theoretica l approaches do not allow it to \nmake because of a strong dependence of dynamic cr itical phenomena from details of a Hamiltonian \nand type of potential. A method of the molecular dynamics together with the Monte-Carlo method \nis used in one of such approaches . So in papers [5,6] th e approach was used for an investigation of \nthe critical dynamics of so me simple magnetic models. \nThe main point of this method consists in th e following. The system is brought to a state of \nthe thermodynamic equilibrium by the Monte-Carlo method. Then the system of differential equations of spins motion is solved. Using th e dynamic finite-size scaling theory [7] and a \nprocedure mentioned in [5,6] can be determ ine a value of the dynamic critical exponent z. As \napprobation we explored the Heisenberg ferroma gnet model with the linear dimensions from L=6 \nup to L=17. For this model the Hamiltonian and the system of equations of spins motion with \nk=x,y,z have the following view \n , ,21\n,\n\n\n  \njk\njk\nik\ni\njiji ij s JstsssJ H   .1 , is Jstt  \nThe solution of equations of spins motion allo ws to receive space-displaced, time-displaced \ncorrelation function \n.0 ,k\nrk\nrkStS trrC \n  \nUsing space-time Fourier transf orm of correlation functions \n      \n   cutoff\ncutofft\ntk\nrrk dttrrCti rrqi qS\n \n2, exp exp ,\n,   \nand define a charac teristic frequency ωm from the condition \n                                                 \n  Published in Journal of Magnetism and Magnetic Materials, Vol. 258-259, 2003, pp. 48-50.  2  , ,21, \n\n  \nd qS d qSk km\nm  \nit is possible to define dependenc e of a characteristic frequency ωm versus the linear \ndimensions of system L. From the dynamic finite-size scaling theory follows, that \nqLfL Lqz\nm, . \nSo it is possible to define value of the dynami c critical exponent from the above mentioned \nequation maintaining a relation qL=const  for all systems. \nThe value obtained of the dynamic critical exponent z is 2.49±0.09, that agrees good both \nwith the theoretical value for three-dimensional isotropic ferromagnets ( z=5/2) and results obtained \nin [6]. Fig.1 represents the dependence of a char acteristic frequency versus the system sizes for a \ncase qL=2π. \nA study of the critical dynamics of the real magnetic materials represents the essential \ninterest at the present stage of investigations of phase transitions and critical phenomena. This \ninterest is conditioned by the ab sence of clear and unique view a bout the dynamic critical behavior \nof the real spin systems. In particular, there is a number of questions on a definition of the \nuniversality classes of the dynamic critical behavior. \nNow we carry out investigations  on the dynamic critical prope rties of models of complex \nreal magnetic materials. In particular, the model of antiferromagnetic Cr 2O3 are studied. \nThe Hamiltonian for Cr 2O3 model have the following view \n  ,1 ,21\n21 2\n0\n,2\n,1         \niz\ni\nlklk\njiji D J J H  \nwhere, according to the experimental data of [8], J1 and J2 are the parameters of the \ninteraction of each spin with one nearest neighbor and three ne arest neighbors, respectively \n(J2=0.45 J1, J1<0, J2<0). The various relativistic interactions were fixed by the effective single-ion \nanisotropy D0>0. We considered the followi ng ratio between the anisitropy D0 and exchange J1 \n, 105.24\n1 0JD  \ncorresponding to real Cr 2O3 samples [9]. The temperature was the temperarure of phase \ntransition: T=Tc=0.466. This value of T c we obtained from our previous  investigations. Earlier we \ncarefully investigated the stat ic critical properties of Cr 2O3 model with a calculation of statical \ncritical exponents [10, 11]. \nThe result for Cr 2O3 model represent the value of z are z=1.59±0.19. We note that this value \nof dynamic critical index agrees with th e theoretical estimates for isotropic ( z=d/2, G model [2]) \nantiferromagnets. \nComplete investigations of models of real fe rro- and antiferromagnetic materials will allow \nto determine features of cri tical behaviour of models with  weak additional interactions. \n \nThe investigation is supported by the Russ ian Foundation for Basic Research (projects №01-02-16103 and \n№02-02-06873-MAC) and by grant of a Commission of RAS for Supporting Young Scientists.  3 \n[1] I.K. Kamilov, Kh.K. Aliev, Usp. Fiz. Nauk 168 (1998)  953 [Phys. Usp. 41 (1998) 865]. \n[2] P.C. Hohenberg, B.I. Halperin, Rev. of Mod. Phys. 49 (1977) 435. \n[3] K.G. Wilson, J. Kogut, Phys. Rep. 12C (1974) 75. \n[4] D.P. Landau, Physica A 205 (1994) 41. \n[5] A.Bunker, K.Chen, D.P.Landau, Phys. Rev. B 54 (1996) 9259. \n[6] D.P. Landau and M. Krech, J. Phys: Condens. Matter 11 (1999) 179. \n[7] I.K. Kamilov, A.K. Murtazaev, Kh.K. Aliev, Usp. Fiz. Nauk 169 (1999) 773   [Phys. Usp. 42 \n(1999) 689]. \n[8] E.J. Samuelson, M.T. Hutchings, G. Shirane, Physica 48 (1970) 13. \n[9] M. Marinelli, F. Mercuri, U. Zammit, R. Pizzo ferrato, F. Scudieri, D. Dadarlat, Phys. Rev. B. \n49 (1994) 9523. \n[10] A.K. Murtazaev, I.K. Kamilov, Kh.K. Aliev, J. Magn. Magn. Mater. 204 (1999) 151. \n[11] A.K. Murtazaev, Fiz. Nizk. Temper. 25 (1999) 469. \n \n \n \n  \nFig.1. Dependence of a characteristic fre quency versus the system sizes for a case \nqL=2π for classical 3D Heisenberg ferromagnet model. "    },    {        "title": "1610.09628v2.Cosmic_ray_acceleration_via_magnetic_reconnection_of_magnetic_islands_flux_ropes.pdf",        "content": "arXiv:1610.09628v2  [physics.space-ph]  17 Nov 2016Version xx as of May 16, 2018\nPrimary authors: Anil Raghav\nTo be submitted to Arxiv\nComment to raghavanil1984@gmail.com\n,\nCosmic ray acceleration via magnetic reconnection of magne tic islands/flux-ropes\nAnil Raghav*1and Zubair Shaikh1\n1University Department of Physics, University of Mumbai,\nVidyanagari, Santacruz (E), Mumbai-400098, India\n(Dated: May 16, 2018)\nThe dynamicprocesses ofmagnetic reconnection andturbule ncecausemagnetic islands/flux-ropes\ngeneration. The in-situ observations suggest that the coal escence or/and contraction of magnetic\nislands are responsible to the charged particle accelerati on (keV to MeV energy range). Numerical\nsimulations also support this acceleration mechanism. How ever, the most fundamental question\nraise here is, does this mechanism contribute to the cosmic r ays acceleration? To answer this, we\nreport,in-situevidence of flux-ropes formation, their magnetic reconnect ion and its manifestation\nas cosmic ray (GeV charged particle) acceleration in interp lanetary counterpart of coronal mass\nejection(ICME). Further, we propose that cosmic ray (high a nd/or ultra-high energy) acceleration\nby Fermi mechanism is valid not only through stochastic refle ctions of particles from the shock\nboundaries butalso through theboundaries of contractingm agnetic islands or/and their merging via\nmagnetic re-connection. This has significant implications on cosmic ray origin and their acceleration\nprocess.\nPACS numbers:\nINTRODUCTION\nThe origin of cosmic rays and its acceleration mecha-\nnism is the most fundamental problem in cosmic ray re-\nsearch. The relation between dimensions of particle con-\nfinement cavity and their energy (gyro-radius) indicate\ncosmic ray source region in broad perspective e.g. helio-\nsphere (Solar (including anomalous) cosmic ray), galaxy\n(galactic cosmic ray), and outside galaxy (extragalac-\ntic cosmic ray) etc. The present understanding of cos-\nmic ray acceleration and the recent observations have\nled to a search for sources within the spacial structures\ne.g. the active galacticnuclei, supernovaeremnants, neu-\ntron stars etc. could be the possible sources of ultra\nhigh energy galactic or extra-galactic cosmic rays. How-\never, how these sources contribute to cosmic ray accel-\neration is poorly understood. Fermi first order (diffusive\nshock)andsecondorder(randomcollisionswithinterstel-\nlar clouds with characteristic velocity) acceleration pro-\ncess is considered as the primal cosmic ray acceleration\nmechanism. Besides this, magnetic reconnection (strong\nlocalized electric field) is also thought to be contributed\nin cosmic ray acceleration [1].\nMagnetic reconnection is a fundamental process which\nrearranges the magnetic field-line configuration, i.e. pro-\nduces magnetic islands/structures[2, 3]. Further, it is re-\nsponsible for energy conversion in magnetized astrophys-\nical and laboratory plasmas and contributes in particle\nacceleration [4–6].To understand the magnetic reconnec-\ntion process and the corresponding particle acceleration,researchers have hunted for magnetic structures in solar\ncorona and space. Recent numerical simulation suggest\nthat the magnetic islands originate from the dynamical\nprocesses of magnetic reconnection and turbulence [7–9].\nMoreover, observational studies also identified magnetic\nisland formation regions. The literature suggest that the\nSun isanaturalgeneratorofmagneticislands, e.g. CME.\nThe regular solar wind interaction produces small scale\nmagnetic islands as well. The occurrence of magnetic\nislands also observed near helio-spheric current sheets,\nmagnetopause,Earthsmagnetotailetc. The magneticre-\nconnection process of these small-scaleislands/flux-ropes\ngive rise to an anti-reconnection electric field that can\naccelerate charged particles and further leads to merging\nor contraction of island. The particle acceleration is also\npossible through the contraction of islands. The trapped\nparticlesexperiencemultiplereflectionsfromthe strongly\ncurved field of contracting island gaining energy during\neachreflectionviaeitherFermifirst-orderorsecond-order\nmechanism [1, 10].The supporting evidences for particle\nacceleration (from keV to MeV energy range) are ob-\nserved at various places in interplanetary space (where\nthe magnetic islands observed) [10–15]. The recent nu-\nmerical simulation based on this physical mechanism of\ncharged particle acceleration strongly support observa-\ntions [11, 16–18]. However, does this physical mecha-\nnism contribute in cosmic ray acceleration processes is\nthe fundamental question. Here we report, in-situ evi-\ndence of flux-rope formation in turbulent ICME shock-\nsheath, their reconnection signature and possible indica-2\ntion of cosmic rays (GeV energy range) acceleration.\nDATA\nWe have selected turbulent ICME shock-sheath [19–\n21] which crossed the Advanced Composition Explorer\n(ACE) satellite, Wind satellite and the Earthon Septem-\nber 24-25, 1998 for this study. To understand the spa-\ntial properties of interplanetary space during the tran-\nsit, we have used 1-minute time resolution OMNI data.\nThe OMNI data (time corrected to the Earth bow-shock\nnose) includes total interplanetary magnetic field (IMF\nBT) and its three components ( Bx,By,Bz) in GSE coor-\ndinate system, solar wind speed, plasma beta, plasma\ntemperature and plasma density. We have also used\nfive minute time resolution neutron flux data taken from\nthe NMDB database ( www.nmdb.eu ) to investigate cos-\nmic rays response to this ICME shock-sheath transit.\nThe cosmic ray data processing method used here are\nbrieflydiscussedinRaghavetal(2016)[22]. Generally,A\n2D-hodogram analysis is widely used in magnetospheric\nphysics and express impressive visualization of rotating\nIMF within the magnetic island events [10, 14]. The\nobservation of semicircle/circle arc in one of the planes\nBx−ByorBz−ByorBz−Bxduring island crossing\nmanifest as rotation of IMF. However, we will not have\ninformation of time evolution of rotating plane in 2D-\nhodogram method. Therefore, 4D-hodogram method is\nperformed, in which 1 second time resolution ACE satel-\nlite data ( Bx,By&Bzin GSE coordinate system) is\nused. Beside this, to cross verify the signature of mag-\nnetic reconnection of flux-ropes electron density, velocity\nx-component (in GSE-coordinate system), electron flux\n(5 kev and 20 keV) and ion flux (0.14 keV, 4 keV and 19\nkeV) data from Wind satellite are used.\nOBSERVATIONS AND DISCUSSION\nThe variations of comic ray flux and interplanetary\nparameter during the selected event is shown in Figure\n(1). The ICME boundaries are defined using Richardson\n& Cane (2010) [21] and presented as dashed blue verti-\ncal lines. The ICME shock-sheath region is divided in 4\nparts for better investigation using dashed magenta ver-\ntical lines. Figure 2 shows 4 hodograms for all selected\nregions of the Figure 1.\nThe sharp enhancement in total IMF BTand solar\nwind suggest the onset of interplanetary shock-front at\nthe Earth’s Bow-shock nose. In region 1, random fluc-\ntuations are seen in total IMF including all its compo-\nnents. We have also observed sudden increase in plasma\ntemperature and density. Moreover, the top left panel\nhodogram (for region 1) of Figure 2, shows fuzzy signa-\nture of semicircle but highly dominated by random fluc-\nFIG. 1: ICME crossing event occurred on September 24-\n25, 1998. The figure has four panels, top most panel shows\ntemporal variation of normalized neutron flux with their re-\nspective band of rigidities. The 2ndpanel show interplane-\ntary magnetic field ( BTotalandBX,BY,BZ-component). The\n3rdpanel from top illustrates solar wind speed and plasma\nbeta variations. The bottom panel depicts proton density an d\nplasma temperature variation.\nSignature of \nreconnection\nFIG. 2: The four 4D hodograms for selected ICME shock\nsheath region. Each hodogram shows Bz−Byas front plane\nof 3D projection and Bxvariations are shown as in-out direc-\ntions. The temporal variations of event crossing is shown as\ncolor-bar. The top left and right panel hodogram illustrate\nregion 1 and region 2 and the bottom left and right panel\nhodograms depict region 3 and region 4 of Figure 1 respec-\ntively.3\nFIG. 3: IMF and Particle flux variation during ICME tran-\nsit occurred on September 25, 1998. The figure has four pan-\nels, top most panel shows temporal variation of interplanet ary\nmagnetic field ( BTotalandBX,BY,BZ-component). The 2nd\npanel show electron density and its velocity component in x d i-\nrection. The 3rdpanel from top illustrates 5 keV and 20 keV\nelectron flux variation. The bottom panel depicts 0.14 keV, 4\nkeV and 19 keV ion flux variations.\ntuations. These observations could be ascribed as the\nresidence of turbulence in shock-front. Compression and\nplasma heating in shock-front could be the cause of this\nobserved turbulence.\nIn region 2, especially the faint red-brown shaded part\ndemonstrates clear rotation in BzandBycomponents\nof IMF, however total IMF is gradually increasing and\nthen decreasing. The plasma temperature/density shows\ndecrease/increase, and solar wind shows steady varia-\ntions during the transit of red-brown shaded region. The\ntop right panel hodogram (for region 2) in Figure 2,\nshows explicit visualization of semicircle with varying\ncircle arc. This structure further continued in remain-\ning two hodograms (bottom left and right). These are\nclear evidences of magnetic island/flux-rope formation in\nshock sheath. It is important to note that the red-brown\nshaded region is the only possible signature of flux-rope\nis seen from Figure 1. However, 4D hodogram depicts\nmuchbettervisualizationoftheflux-ropeformingregions\nand extend its boundaries to region 3. During region 3\ntransit, IMF B and its components remain steady and\nIMF wasmainly directed in Y-direction i.e dawn to dusk.\nThe solar wind speed, plasma beta and plasma density\nshow steady variations, only plasma temperature gradu-\nallydecreasestoambient value. In region4 speciallyblue\nFIG. 4: The combined 4D hodograms for region 2 to 4 as\nshown in Figure 1. The hodogram shows Bz−Byas front\nplane of 3D projection and temporal variations are shown as\nin-out directions as well as color-bar. The sub-figure (A)\nshows the blue top layer of semicircle of 4D hodogarm in\nBz−Byplane. The magenta circle is simulated by assuming\n[BZ= 23∗sin(2πt/T)∗rand()] and [BY= 23∗cos(2πt/T)∗\nrand()].\nshaded region shows sharp transition in all IMF com-\nponents. The plasma density, plasma beta and cosmic\nray flux depict corresponding enhancement. Basically,\nflux-rope of the shock-sheath region and ICME flux-rope\nboundary lie in this region. The left bottom hodogram\n(for region3), shows sharpswitch in the oscillatingorien-\ntation (check brown circle data) which further continued\nin region 4. We have also noted similar oscillation trip-\nping at the onset of ICME flux-rope (in bottom right\nhodogram for region 4). This could be the indication\nof combination of two different flux-ropes (shock-sheath\nand ICME) via magnetic reconnection. To support the\nobservedsignatureofmagneticreconnectionelectronand\nion flux data are investigated. Figure 3 shows temporal\nvariation of electron density, its velocity in x direction,\nelectron and ion flux from wind satellite. Moreover, all\nstudied parameters (except IMF which used for reference\nbetween Figure 1 and 3) explicitly depicts enhancement\nin pink shaded region of Figure 3. These observations\nare clear indication of magnetic reconnection and corre-\nsponding charged particle acceleration.\nTo understand origin and evolution of observed flux-\nrope within shock-sheath, the complete overview of mag-\nnetic flux-ropes formed in shock-sheath region (from re-\ngion 2 to 4 in Figure 1)) is presented as 4D hodogram in4\nFigure 4. Initially, the arclength of circle is decreasingto\nminimum and then again increasing with slowly contin-\nuous oscillation of the arc-circle plane (see the circle arc\nwith different color, on-line only). Moreover, to estimate\nthe phase difference between BZandBY, the top circle\narc layer is selected and shown as the sub-figure (A) in\nFigure4. TheIMFcomponents( BY&BZ)aresimulated\nusing equations as [ BZ= 23∗sin(2πt/1600)∗rand()]\nand [BY= 22∗cos(2πt/1600)∗rand()]. Here, tvaries\nfrom 1 to 1600and rand() is computer generatedrandom\nnumber which varies from 0 to 1. The simulated data is\npresented as magenta circle in sub-figure (A). The sim-\nulated data is clearly fitted with the observed top circle\narc layer data. This proves that the BZandBYare hav-\ning phase angle π/2. Further, in region 2 of figure 1,\nwe have observed BZandBYare rotating and finally\ntotal IMF is oriented along the y-direction. Now, if we\nassume toroid shape of flux-rope, in which BZandBY\npresents poloidal and toroidal field direction respectively\nas shown in Figure 5 and the ACE spacecraft crossingdi-\nrection is not parallel to the axis of the flux-rope. Then\nobservationssuggestthat the different circle arcobserved\nin hodogram are the different layers of flux-rope. From\ntop to inside, at every layer, the poloidal component of\nmagnetic field slowly aligning with the toroidal compo-\nnent. This can be seen as the oscillation of arc plane\nin Figure 4. Finally BZpoloidal component reach to\nits minimum whereas the total IMF of flux-rope is rep-\nresented by toroidal component i.e. BY. This ascribed\nthattheplasmainshock-sheathregionminimizetheirpo-\ntential energy. This physical process is known as plasma\nrelaxation(self-organizationofa plasma)by magneticre-\nconnection [25]. Figure 5 (artistic illustration) exhibits\nthe updated visualization of ICME propagation in in-\nterplanetary space. The ICME flux-rope with turbulent\nshock-sheath is the earlier hypothesis. In this work, we\nadvancesthis traditionalhypothesisand demonstratethe\nfirst in-situ evidence of flux-rope formation other than\nICME flux-rope in shock-sheath region. The flux-rope\ncould be originated from the interaction of ICME with\nambient solar wind or fragmentation of ICME flux-rope\ndue to solar wind interaction [10].\nBeside this, the enhancement of cosmic rays flux (in\nall observed energy band) is unambiguously observed in\nred and blue shaded region of Figure 1. In addition,\nin Figure 2, the top right hodogram shows the indica-\ntion of rotating magnetic island formation and bottom\nleft hodogram demonstrates the signature of magnetic\nreconnection during respective time. The enhancement\nin red shaded region could be understood on the basis of\ngradual decrease in toroidal and poloidal field strength\nof flux-rope during corresponding period. Moreover, the\nenhancement in charged particle flux is generally inter-\npreted as the particle energization i.e. acceleration [14].\nThe simulation studies suggest that the particle accel-\neration is possible when particle gyro-radius is smallerthan the magnetic islands scale length [26]. That means\nthe accelerated particle energy range is depends on the\ndimensions of magnetic island. In present case, the di-\nmensionofobservedmagneticislandisestimatedapprox-\nimately 3 ∗107kmusing the time duration of magnetic\nisland crossing and average solar wind speed. The esti-\nmated dimension is much higher than the order of cosmic\nray (1-10 GeV energy range observed by neutron moni-\ntors) gyro-radius. Therefore, the enhancement in cosmic\nray flux in red shaded region also could be the outcome\nof acceleration through a stochastic repeated reflections\n(first-orderFermi mechanism (in the case ofcompressible\ncontraction) or a second-order Fermi mechanism (if the\ncontractionis incompressible))from contractingormerg-\ning flux-rope [10, 14]. However, enhancement in cosmic\nray flux in blue shaded region could be ascribed as ex-\nplicit evidence of cosmic ray acceleration via magnetic\nreconnection of shock-sheath flux-rope and ICME flux-\nrope.\nCONCLUSIONS\nA charged particle acceleration via contraction and/or\nmerging of magnetic islands through magnetic reconnec-\ntion is observationally evident in keV to MeV energy\nrange and strongly supported by numerical simulations.\nThepresentworkextendthisacceleratingprocesstoGeV\nenergy range charged particles i.e. cosmic rays. On the\nbasis of dimensional argument including field strength\ninside the magnetic island and the particle energy, we\npropose that the same processes should contribute sig-\nnificantly in high energy and/or ultra-high energy cos-\nmic ray acceleration. This further imply that not only\nthe active core region but also transient disturbances in\nthe spatially extended region of active core-regionare re-\nsponsible for cosmic ray acceleration. For example, after\nsupernova expansion, the hot thermal material generates\nshocksinto the undisturbed interstellarmedium [1]. This\nphysical scenario is similar to the ICME shock in helio-\nsphere. The transient disturbances could give rise to tur-\nbulent and dynamic conditions leads to magnetic islands\nformation. Moreover, the interactions of these magnetic\nstructures through magnetic reconnection induce merg-\ning and contraction of islands, ultimately contributes to\nthe charge particle (cosmic ray) acceleration.\nIn summary, we conclude that the cosmic rays accel-\neration by Fermi mechanism is valid not only through\nstochastic reflections of particles from the shock bound-\naries but also through merging and/or contraction of\nmagnetic islands via magnetic reconnection. This may\nprovide some insight into the origin of ultra-high energy\ncosmic rays. The ultra-high energy cosmic rays may not\nbe one-step process but a multi-step one. The region re-\nsponsible for final acceleration may be thought of as the\norigin of these ultra-high energy cosmic rays. .5\nFIG. 5:The updated schematic diagram (artistic illustration) of I CME in interplanetary space.\nAcknowledgment\nWe acknowledge the NMDB database (www.nmdb.eu)\nfounded under the European Union’s FP7 programme\n(contract no. 213007). We are also thankful to all neu-\ntron monitor observatories listed on website. We are\nthankful to CDAWeb and ACE science center for making\ninterplanetary data available. We are thankful to De-\npartment of Physics (Autonomous), University of Mum-\nbai, for providing us facilities for fulfillment of this work.\nAuthorswouldalsoliketothankA.Bhaskar,S.Kasthuri-\nrangan, N. Bijewar and M. Gadyali for valuable discus-\nsion.\n[1] Kulsrud, Russell M. Plasma physics for astrophysics,\n(chapter -12). Princeton: Princeton University Press,\nfirst Indian sub-continent edition, 2010, ISBN-978-81-\n224-3133-9.\n[2] ML Cartwright and MB Moldwin. Heliospheric evolution\nof solar wind small-scale magnetic flux ropes. Journal of\nGeophysical Research: Space Physics , 115(A8), 2010.\n[3] MB Moldwin, S Ford, R Lepping, J Slavin, and A Sz-\nabo. Small-scale magnetic flux ropes in the solar wind.\nGeophysical research letters , 27(1):57–60, 2000.\n[4] Masaaki Yamada, Russell Kulsrud, and Hantao Ji.\nMagnetic reconnection. Reviews of Modern Physics ,82(1):603, 2010.\n[5] PP Kronberg, SA Colgate, H Li, and QW Dufton. Giant\nradio galaxies and cosmic-ray acceleration. The Astro-\nphysical Journal Letters , 604(2):L77, 2004.\n[6] Yang Ren, Masaaki Yamada, Stefan Gerhardt, Hantao\nJi, Russell Kulsrud, and Aleksey Kuritsyn. Experimen-\ntal verification of the hall effect during magnetic recon-\nnection in a laboratory plasma. Physical review letters ,\n95(5):055003, 2005.\n[7] Stefano Markidis, Pierre Henri, Giovanni Lapenta, An-\ndrey Divin, Martin Goldman, DavidNewman, andErwin\nLaure. Kinetic simulations of plasmoid chain dynamics.\nPhysics of Plasmas (1994-present) , 20(8):082105, 2013.\n[8] Liping Yang, Lei Zhang, Jiansen He, Chuanyi Tu,\nLinghua Wang, Eckart Marsch, Xin Wang, Shaohua\nZhang, and Xueshang Feng. The formation of rota-\ntional discontinuities in compressive three-dimensional\nmhd turbulence. The Astrophysical Journal , 809(2):155,\n2015.\n[9] A Greco, S Servidio, WH Matthaeus, and P Dmitruk.\nIntermittent structures and magnetic discontinuities on\nsmall scales inmhdsimulations andsolar wind. Planetary\nand Space Science , 58(14):1895–1899, 2010.\n[10] Olga V Khabarova, Gary P Zank, Gang Li, Olga E Ma-\nlandraki, Jakobus A le Roux, and Gary M Webb. Small-\nscale magnetic islands in the solar wind and their role in\nparticle acceleration. ii. particle energization inside m ag-\nnetically confined cavities. The Astrophysical Journal ,\n827(2):122, 2016.\n[11] JF Drake, M Swisdak, H Che, and MA Shay. Electron\nacceleration from contracting magnetic islands during re-6\nconnection. Nature, 443(7111):553–556, 2006.\n[12] Maha Ashour-Abdalla, Mostafa El-Alaoui, Melvyn L\nGoldstein, Meng Zhou, David Schriver, Robert Richard,\nRaymond Walker, Margaret G Kivelson, and Kyoung-\nJoo Hwang. Observations and simulations of non-local\nacceleration of electrons in magnetotail magnetic recon-\nnection events. Nature Physics , 7(4):360–365, 2011.\n[13] L-J Chen, A Bhattacharjee, PA Puhl-Quinn, H Yang,\nN Bessho, S Imada, S M¨ uhlbachler, PW Daly, B Lefeb-\nvre, Yuri Khotyaintsev, et al. Observation of ener-\ngetic electrons within magnetic islands. Nature Physics ,\n4(1):19–23, 2008.\n[14] O Khabarova, GP Zank, G Li, JA le Roux, GM Webb,\nA Dosch, and OE Malandraki. Small-scale magnetic is-\nlands in the solar wind and their role in particle accelera-\ntion.i. dynamicsofmagnetic islands neartheheliospheric\ncurrent sheet. The Astrophysical Journal , 808(2):181,\n2015.\n[15] Goetz Paschmann, I Papamastorakis, N Sckopke,\nG Haerendel, BU Sonnerup, SJ Bame, JR Asbridge,\nJT Gosling, CT Russel, and RC Elphic. Plasma accel-\neration at the earth’s magnetopause-evidence for recon-\nnection. Nature, 282:243–246, 1979.\n[16] Nicolas H Bian and Eduard P Kontar. Stochastic\nacceleration by multi-island contraction during turbu-\nlent magnetic reconnection. Physical review letters ,\n110(15):151101, 2013.\n[17] JA Le Roux, GP Zank, GM Webb, and O Khabarova.\nA kinetic transport theory for particle acceleration and\ntransport in regions of multiple contracting and recon-\nnecting inertial-scale flux ropes. The Astrophysical Jour-\nnal, 801(2):112, 2015.\n[18] GP Zank, JA Le Roux, GM Webb, A Dosch, and\nO Khabarova. Particle acceleration via reconnection pro-cesses in the supersonic solar wind. The Astrophysical\nJournal, 797(1):28, 2014.\n[19] D Venkatesan, AG Ananth, Suresh Pillai, et al. Signifi-\ncance oftheturbulentsheathfollowing theinterplanetary\nshocks in producing forbush decreases. Solar physics ,\n137(2):345–353, 1992.\n[20] Thomas H Zurbuchen and Ian G Richardson. In-situ\nsolar wind andmagnetic fieldsignatures ofinterplanetary\ncoronal mass ejections. Space Science Reviews , 123(1-\n3):31–43, 2006.\n[21] IG Richardson and HV Cane. Near-earth interplanetary\ncoronal mass ejections during solar cycle 23 (1996–2009):\nCatalog and summary of properties. Solar Physics ,\n264(1):189–237, 2010.\n[22] Anil Raghav, Zubair Shaikh, Ankush Bhaskar, Gauri\nDatar, and Geeta Vichare. Does local structure within\nshock-sheath and magnetic cloud affects cosmic ray de-\ncrease?arXiv preprint arXiv:1608.03772 , 2016.\n[23] M Øieroset, TD Phan, M Fujimoto, RPLin, and RPLep-\nping. In situ detection of collisionless reconnection in th e\nearth’s magnetotail. Nature, 412(6845):414–417, 2001.\n[24] Alessandro Retin` o, David Sundkvist, Andris Vaivads,\nF Mozer, Mats Andr´ e, and CJ Owen. In situ evidence\nof magnetic reconnection in turbulent plasma. Nature\nPhysics, 3(4):236–238, 2007.\n[25] JBj Taylor. Relaxation and magnetic reconnection in\nplasmas. Reviews of Modern Physics , 58(3):741, 1986.\n[26] Jakobus A le Roux, Gary M Webb, Gary P Zank, and\nOlga Khabarova. Energetic ion acceleration by small-\nscale solar wind flux ropes. In Journal of Physics: Con-\nference Series , volume 642, page 012015. IOPPublishing,\n2015."    },    {        "title": "1804.05818v1.Triple_Exponential_Relaxation_Dynamics_in_a_Metallacrown_Based___Dy__III_Cu__II__5___3d_4f_Single_Molecule_Magnet.pdf",        "content": "Triple Exponential Relaxation Dynamics in a Metallacrown-Based {DyIIICuII5} 3d-4f Single-Molecule Magnet Quan-Wen Li,[a] Rui-Chen Wan,[a] Jin Wang,[a] Yan-Cong Chen,[a] Jun-Liang Liu,[a]* Daniel Reta,[b] Nicholas F. Chilton,[b]* Zhen-Xing Wang,[c] and Ming-Liang Tong*[a]  [a] Dr. Q.-W. Li, R.-C. Wan, J. Wang, Dr. Y.-C. Chen, Dr. J.-L. Liu, Prof. Dr. M.-L. Tong Key Laboratory of Bioinorganic and Synthetic Chemistry of Ministry of Education, School of Chemistry, Sun Yat-Sen University, Guangzhou 510275 (P. R. China) E-mail: liujliang5@mail.sysu.edu.cn (J.L.L.); tongml@mail.sysu.edu.cn (M.L.T.) [b] Dr. D. Reta, Dr. N. F. Chilton School of Chemistry, The University of Manchester, Oxford Road, Manchester M13 9PL (UK) E-mail: nicholas.chilton@manchester.ac.uk (N.F.C.) [c] Dr. Z.-X Wang Wuhan National High Magnetic Center, Huazhong University of Science and Technology, Wuhan 430074 ( P. R. China)  Abstract: The interplay of strong single-ion anisotropy and magnetic interactions often give rise to novel magnetic behavior and can provide additional routes for controlling magnetization dynamics. However, novel effects arising from interactions between lanthanide and transition-metal ions are nowadays rarely observed. Herein, a {DyIIICuII5} 3d-4f single-molecule magnet (SMM) is constructed as a rigid and planar [15-MC-5] metallacrown (MC), where the DyIII ion is trapped in the central pseudo-D5h pocket. A strong axial crystal field (CF) imbues the DyIII ion with large Ising-type magnetic anisotropy, and we are able to observe and model the magnetic interactions between the CuII-CuII and DyIII-CuII pairs. Butterfly-shaped magnetic hysteresis shows clear steps at ±0.4 T, coincident with level crossings in our model exchange Hamiltonian between the {CuII5} and DyIII spin systems. Most intriguingly, this air-stable SMM exhibits three distinct regimes in its magnetic relaxation dynamics, all clearly displaying an exponential dependence on temperature. Single-molecule magnets (SMMs)1,2 have attracted intensive attention as potential materials for ultra-high-density magnetic information storage and quantum information processing.3-5 Recently there have been some significant breakthroughs on dysprosium single-ion magnets (SIMs), resulting in large effective energy barriers (Ueff) and high blocking temperatures (TB).6-16 There have also been many nice d-f SMMs reported such as {MII2DyIII} (M = Fe, Co)17,18, {MIII2LnIII2} (M = Cr, Co; Ln = Dy, etc.)19,20, {UVxMnIIy} (x = 12, y = 6 or x = 1, y = 2)21,22 and {CoII2DyIII2}23, showing interesting magnetic dynamics such as exchange-biased magnetic hysteresis. The main benefit of heterometallic d-f SMMs over homometallic species is the ability to exploit hierarchical energy scales such as the crystal field (CF), and d-d/d-f magnetic exchange interactions, which can introduce a higher density of magnetic states to provide additional routes for tuning magnetic relaxation pathways. It has become apparent in recent years that high-performance DyIII SIMs should be constructed with strong axial CF potentials and high-order symmetry axes (e.g. D4d, D5h and Cn>6), in order to suppress quantum tunneling of magnetization (QTM) by reducing mixing between opposing projections of the magnetic moment.6 However, many of the recent developed species are coordinatively unsaturated and therefore usually highly reactive in air;9,11-16 such inherent instability is a problem for progressing the field from academic research to industrial applicability, and therefore chemical stability under ambient conditions must also be a core goal for the community.24 Therefore, we are keen to exploit a new strategy for designing 3d-4f SMMs that are stable under ambient conditions by marrying high pseudo-symmetry at the 4f ion with 3d-4f magnetic exchange. Considering the success of pentagonal bipyramidal lanthanide SIMs,6-10 one strategy to achieve our goal is to place a 4f ion into a pseudo-D5h-symmetric pocket, with 3d ions located at the periphery. This design typifies what is known as a metallacrown (MC), that can be assembled into various congeners to afford different geometries by tuning ditopic ligands and transition metal ions.25 Inspired by this concept, our aim was to employ a [15-MC-5] congener to build a ~D5h symmetric pocket for the lanthanide ion, and to also foster magnetic exchange between the 3d ions in the MC ring with the 4f ion in the center. In addition to the high-order pseudo-symmetry axis, the axial ligands should be rich in electron density to provide a strong axial CF and stabilize the largest magnetic states of our target central ion, DyIII.26,27 Suitable ligands include phenoxides, phosphine oxides, alkoxides, carbanions and cyclopentadienyls, which have been proven to generate excellent SMMs.6-16 Among these, phenoxide ligands stand out as being relatively stable in both air and polar solvents, which is beneficial for synthesizing, investigating and utilizing molecular materials under mild conditions. \n Figure 1. (a) Side view of the {DyIIICuII5} metallacrown; (b) Top view of the {DyIIICuII5} core; (c) Coordination environment of the DyIII ion. H atoms are omitted for clarity. Color codes: Dy, teal; Cu, pale blue; N, blue; O, red; C, light grey. The golden sticks show the [15-MC-5] metallacrown and the green pentagonal area show the planar coordination environment of DyIII ion. \nWe have utilized quinaldichydroxamic acid (H2quinha) and CuII to build a [15-MC-5] ring that encapsulates a single DyIII ion in the presence of salicylaldehyde (Hsal) and pyridine (see Supporting Information for synthetic details). Single-crystal X-ray diffraction reveals a hexanuclear {DyIIICuII5} complex, [DyCu5(quinha)5(sal)2(py)5][CF3SO3]·py·4H2O (1), that crystallizes in the P-1 space group (Table S1) with the molecular structure shown in Figure 1. Five quinha2− ligands bridge five CuII ions end to end, forming a [15-MCquinha,Cu(II)-5] ring with a -[Cu-N-O]- repeat unit. The [15-MC-5] ring equatorially encapsulates a DyIII ion with five hydroximate oxygen donor atoms, while two phenoxide groups from the salicylaldehydes coordinate axially. The average equatorial and axial Dy-O distances are 2.413(3) and 2.198(3) Å, respectively. The equatorial O-Dy-O angles vary from 70.74(10)° to 70.33(10)° while the axial one is 177.33(13) (Table S2 and S3). This indicates that the DyIII possesses pentagonal bipyramidal geometry (Figure 1c), and is further confirmed by CShM analysis28 with gives a small value of 0.244 for this geometry (Table S4). On the other hand, each CuII ion on the MC ring is five-coordinate (square pyramidal, Table S5) with two quinha2− ligands providing two oxygen atoms and two nitrogen atoms to form the equatorial plane, and one axial pyridine. The adjacent intramolecular Cu···Cu and Cu···Dy distances are 4.595-4.643 Å and 3.904-3.964 Å, respectively. Nearest-neighbor {DyCu5}+ cores are well separated by triflate anions and solvent molecules, with the shortest intermolecular Dy···Dy distance being ca. 11 Å. To obtain complementary insight into the magnetic behavior of 1, we synthesized its diamagnetic 4f analogue {YIIICuII5}, formulated as [YCu5(quinha)5(sal)2(py)5][CF3SO3]·py·4H2O (2); all attempts to obtain the diamagnetic 3d analogue {DyIIIZnII5} failed. The magnetic susceptibilities of 1 and 2 were measured on polycrystalline samples under a 1 kOe direct-current (dc) field (Figure 2). Upon cooling, the χMT of 2 decreases from 1.58 cm3 K mol−1 at 300 K to 0.45 cm3 K mol−1 at 2 K, which is as expected for a spin-ground state of S = 1/2 (0.45 cm3 K mol-1, g = 2.2), indicating antiferromagnetic interactions between adjacent CuII ions. For 1, the χMT value at 300 K of 15.1 cm3 K mol−1 is close to the sum of one Curie-like DyIII ion (14.17 cm3 K mol−1, 6H15/2, gJ = 4/3) and the room-temperature χMT value of 2 (1.58 cm3 K mol-1, sum of 15.75 cm3 K mol−1). On cooling, the χMT of 1 shows a gradual decline, owing to a combination of the CF splitting of the 6H15/2 ground multiplet of DyIII and the antiferromagnetic interaction among CuII ions. Below 15 K, there is a clear rise in χMT, indicating the presence of ferromagnetic interactions between the DyIII and CuII ions. At lower temperatures still, the χMT product drops; this could be due to magnetic blocking, which is suggested by the divergence of zero-field-cooled/field-cooled (ZFC/FC) magnetic susceptibilities below 6.5 K (Figure 2, inset), or to subtleties of the low-energy exchange manifold. \n Figure 2. The χMT products measured under a 1 kOe dc field for 1 (red circles) and 2 (green circles). The solid lines are corresponding to the best fits described in the text. Inset: the ZFC(blue)/FC(orange) magnetic susceptibilities for 1 under a 2 kOe dc field sweeping at 2 K/min in warming mode. To interpret the temperature dependence of the magnetic susceptibility of 1, we first proceed to interpret that of the orbitally non-degenerate analogue 2. As no zero-field splitting is possible for the local CuII S = 1/2 states, the magnetic properties of 2 can be reliably described using the well-known Heisenberg-Dirac-van Vleck (HDVV) and Zeeman Hamiltonians (Equation 1). To minimize overparameterization, electron paramagnetic resonance (EPR) on a powder sample of 2 is employed to determine gCu^ = 2.03(2) and gCu|| = 2.25(2) for the S = 1/2 ground state (Figure S13); given all CuII sites have their anisotropy axes roughly parallel, we take these values also for the local CuII g-values. The molecular structure of 2 suggests two types of CuII-CuII magnetic interactions, those where neighboring CuII ions have syn-syn or syn-anti arrangements of the coordinated pyridines (Figure 1 and Scheme 1); two approximate groups of magnetic interactions is also supported by DFT calculations (see ESI). Therefore, we fit the dc magnetic data of 2 with the PHI program29 using a two-J model (Figure 2 and Scheme 1). 𝑯=−2𝐽&'(&'𝑺&'*𝑺&'++𝑺&'+𝑺&'-+𝑺&'.𝑺&'/−2𝐽′&'(&'𝑺&'-𝑺&'.+𝑺&'*𝑺&'/+𝜇2𝑔&'𝑺&'45678𝐵  (1) \n Scheme 1. The illustration of magnetic exchange couplings for {YCu5} and {DyCu5}. The best-fit of the data gives JCu−Cu = −44(2) cm−1 and J’Cu−Cu = −68(2) cm−1 with gCu^ and gCu|| fixed from EPR; the ratio of the fitted exchange values is in good agreement with that extracted from broken-symmetry DFT calculations (Table S6 and S8). This results in an S = 1/2 ground state with a first excited S = 1/2 state at 51 cm−1 (Table S9).  To proceed to modelling the dc magnetic data for 1, we require a model for the CF states for the DyIII ion. Complete active space self-consistent field spin-orbit (CASSCF-SO) calculations (see ESI for details) on the DyIII site of 1 reveal that the ground and first excited Kramers doublets are well described by the |±15/2> and |±13/2> states, respectively, quantized along approximately the O11-O13 axis (Figure 1), separated by 314 cm−1 (Table S10). We model the DyIII-CuII interactions with a Lines-type exchange interaction between the true spin of DyIII, SDy = 5/2, and the SCu = 1/2 CuII ions;30 the CF parameters for DyIII are fixed from CASSCF-SO (Table S11), along with the CuII parameters from the fitted data for 2, and thus there are only two fitting parameters viz. gJ and JDy−Cu (Equation 2). The experimental dc magnetic data are well described with JDy−Cu = +0.88(5) cm−1 and gJ = 1.30(1) (Figures 2 and S3), the latter being only slightly lower than the free-ion value of 4/3, presumably accounting for the formation of molecular orbitals,31 in contrast to the pure free-ion basis functions used in PHI (indeed, CASSCF-SO predicts gJ = 1.32). 𝑯=−2𝐽&'(&'𝑺&'*𝑺&'++𝑺&'+𝑺&'-+𝑺&'.𝑺&'/−2𝐽′&'(&'𝑺&'-𝑺&'.+𝑺&'*𝑺&'/−2𝐽:;(&'𝑺:;𝑺&'45678+𝜇2𝑔&'𝑺&'45678+𝑔<𝑱𝐵+𝐵>?𝑶>?>?7(>>7A,C,D   (2)  As the exchange interaction between DyIII and CuII is so much smaller than the DyIII CF, JCu−Cu and J’Cu−Cu parameters, the low-lying electronic states for 1 are well described as simple product functions between the DyIII and {CuII5} spin systems (Table S12). Although the overall molecule is non-Kramers, many of the resulting states are pseudo-doublets; the low-lying states are sequential excitations of the {CuII5} spin system (Table S9) coupled to the ground |±15/2> state of DyIII. In zero dc field, the low-lying pseudo-doublets arising from the DyIII−CuII exchange interactions are linear combinations of the total angular momentum along the z axis Mz. Upon application of a small magnetic field along the z-axis, the degeneracy of the pseudo-doublets is lost and the linear combinations are resolved into components of ±Mz (Figure 3). The dc magnetization data for 1 exhibit butterfly-shaped hysteresis loops at low temperatures (Figure 4) that remain open up to 12 K at a sweep rate of 0.02 T/s. The hysteresis loops are closed at zero-field due to fast relaxation from the superimposed Mz states, but rapidly open upon application of the field. There is an obvious step in both sweep directions at around 0.4 T (Figure 4b), consistent with a level crossing predicted at ca. 0.48 T from our exchange model (Figure 4c) between the ground and the first excited doublet arising from the DyIII−CuII exchange bias; this provides further validation of our exchange model. \n Figure 3. Simulated energy diagram of {DyCu5}, showing the expectation value of the total angular momentum projection onto the z-axis (<Mz>) for each state. Simulations use the CF parameters obtained from ab initio calculations and other parameters extracted from fitting the magnetic and EPR data, as employed to model that magnetic susceptibility in Figure 2, using Equation 2. The external dc field applied along the z-axis is 0.5 kOe. \n Figure 4. (a) Magnetic hysteresis loops for 1. The data were continuously collected at time intervals of 1 s with the field sweeping rate of 0.02 T/s at various temperatures. (b) 1st derivative of magnetization vs. applied field. (c) Zeeman splitting of the four lowest magnetic states along the z-axis, calculated using Equation 2. Given that 1 shows magnetic hysteresis, we have studied its magnetic relaxation properties. Alternating-current (ac) measurements show clear temperature- and frequency-dependent out-of-phase signals (Figure S4-6), from which we extract the relaxation times (τ) for 1 by fitting with the generalized Debye model (Figure S7). Under zero dc field, an Arrhenius plot reveals two linear regions (Figure 6, red circles in green and red zones), characteristic of exponential temperature dependencies. [Analogous ac measurements for the yttrium analogue 2 do not show any signs of slow magnetic relaxation under zero field, revealing that this behavior cannot owe to the {CuII5} ring alone.] Applying an optimized field of 2 kOe (Figure S6), the resulting relaxation times also exhibit two linear regions (Figure 6, blue circles in green and blue zones). Both datasets in the high temperature region (64–45 K) are nearly identical, while those in the low temperature region (<45 K) are very different, revealing three different slopes in all. We note that these relaxation profiles do not show power-law dependencies (as they all curve on a log-log plot of τ−1 vs. T, Figure S9) and thus cannot be ascribed to Raman-like processes (Figure S10). Given the large change in the relaxation profile under an applied dc field, we were curious about the relaxation dynamics under an intermediate dc field between 0 and 2 kOe. With an applied dc field of 500 Oe, we curiously observe three unique domains in the relaxation dynamics (Figures 5 and S8), which also all show exponential temperature dependencies (Figure 6, green circles). At high temperatures, the relaxation times are nearly identical to the 0 and 2 kOe data (green zone), and below 45 K the slope becomes smaller and appears very similar to that under 2 kOe (7–45 K, blue zone). On further lowering the temperature the slope decreases again (2.5–6 K, red zone), and is parallel to that observed under zero dc field (2.5–45 K, red zone). Similarly to the 0 and 2 kOe data, these three domains are linear on an Arrhenius plot and are curved on a log-log plot of τ−1 vs. T (Figures 6 and S9) and thus clearly have an exponential, and not a power-law, dependence on temperature. After observing three distinct domains in the 500 Oe relaxation data, we were curious if the dynamics under a 2 kOe field would also show a 3rd domain at lower temperatures. Indeed, dc relaxation experiments at 2 kOe (Figure S12) yield relaxation times that show a low slope between 2.5–5 K (Figure 6, blue diamonds in red zone), similar to the zero dc field (2.5–45 K, red zone) and 500 Oe (2.5–6 K, red zone) data. The presence of three unique exponential processes contrasts to the more common observation of either one or two processes combined with other relaxation mechanisms,6-12 or the different case of multiple relaxation times under the same conditions. 13,16,32-35 All three datasets can be fitted simultaneously with a sum of three exponential terms (Equation 3), with three common exponents (Ueff) and field-dependent pre-exponential factors (τ0). We note pre-emptively that the phenomenological Ueff are not necessarily \nattributed to traditional Orbach process. Fitting the data gives Ueff(1) = 623(22) cm−1 (896(31) K), Ueff(2) = 38.7(5) cm−1 (55.7(7) K), and Ueff(3) = 5.2(2) cm−1 (7.5(3) K), with τ0 values in Table 1.  τ−1 = τ0(1)−1exp(−Ueff(1)/kBT) + τ0(2)−1exp(−Ueff(2)/kBT) + τ0(3)−1exp(−Ueff(3)/kBT)      (3)  \n Figure 5. The temperature-dependent ac susceptibilities for 1 under a 0.5 kOe dc field with the frequency of 1−1488 Hz. The solid lines are guide for eyes. \n Figure 6. The temperature-dependent relaxation times for 1 under zero dc field (red circles), 0.5 kOe dc field (green circles) and 2 kOe dc field (blue circles from ac magnetic susceptibility and blue diamond from dc magnetization decay as shown in Figure S12). The blue, red and green lines are Arrhenius law fits for 0, 0.5 and 2 kOe dc field (high [green], intermediate [blue] and low [red] temperature region), respectively. Table 1. Effective energy barriers (Ueff) and pre-exponential factors (τ0) for complex 1.  0 Oe 0.5 kOe 2 kOe Ueff(1) [cm−1] 623(22) Ueff(2) [cm−1] - 38.7(5) Ueff(3) [cm−1] 5.2(2) τ0(1) [s] 2.1(11)×10−11 2.4(12)×10−11 2.7(15)×10−11 τ0(2) [s] - 1.63(8)×10−4 1.53(9)×10−3 τ0(3) [s] 9.4(6)×10−5 9.0(8)×10−2 2.0(2)×101 The best-fit parameters (Table 1) show negligible field dependence for the pre-exponential factors of the 1st term, τ0(1), as expected given the data are practically coincident. However, we observe a marked field dependence for the other two, τ0(2) and τ0(3). The large exponent (Ueff(1) = 623(22) cm−1) for the 1st process is believed to be due to a true Orbach relaxation process involving the CF states of the DyIII ion, and relaxation likely occurs via a set of highly mixed states at 510−611 cm−1 (Figure 3a). To the best of our knowledge, Ueff(1) = 623 cm−1 is the largest energy barrier for any reported d-f SMM, eclipsing Ueff = 416 cm−1 for a {CoII2DyIII} complex.18 For the 2nd and 3rd relaxation processes we note that Ueff(3) = 5.2 and Ueff(2) = 38.7 cm−1 are similar to the 1st and 2nd excited states determined from our exchange model of 1 (4.4 and 50.6 cm−1, respectively; Figure 3b and Table S12), and therefore these relaxation mechanisms could conceivably involve combined DyIII and CuII spin-flip processes. However, we do not endorse such an assertion without reservation, and undoubtedly a thorough temperature- and field-dependent study is necessary to determine the nature of these two relaxation processes. \nBy utilizing a [15-MC-5] metallacrown and salicylaldehyde ligand, we have prepared a unique 3d-4f SMM with ferromagnetic 3d-4f interactions. In conjunction with DFT and CASSCF-SO calculations, we have modelled the static magnetic properties of the molecule, providing a plausible origin for the steps in the magnetic hysteresis at 0.4 T. Intriguingly, we observe three different exponential relaxation processes, the largest of which has Ueff(1) = 623(22) cm−1 (a record for d-f SMMs), that likely arises from the DyIII single-ion anisotropy. The other two exponential processes have strong field dependence, and as yet we do not understand their origins. However, we suggest that the interplay between the hierarchical energy scales of the DyIII anisotropy, CuII−CuII exchange and DyIII−CuII exchange are central to these dynamics. Unraveling the nature, origins and implications of these three distinct relaxation domains will be the subject of an upcoming full study. Acknowledgements This work was supported by the “973 Project” (2014CB845602), the NSFC (Grant nos 21620102002, 21773316, 21371183 and 91422302, 21701198), the Fundamental Research Funds for the Central Universities (Grant 17lgjc13 and 17lgpy81), China Postdoctoral Science Foundation (National Postdoctoral Program for Innovative Talents, Grant BX201700295), the Ramsay Memorial Trust (fellowship to NFC), the EPSRC (EP/P002560/1) and The University of Manchester.  [1] R. Sessoli, D. Gatteschi, A. Caneschi, M. A. Novak, Nature 1993, 365, 141-143. [2] D. Gatteschi, R. Sessoli, R. Villain, Molecular Nanomagnets, Oxford University Press, New York, 2006. [3] M. N. Leuenberger, D. Loss, Nature 2001, 410, 789-793. [4] M. Mannini, F. Pineider, P. Sainctavit, C. Danieli, E. Otero, C. Sciancalepore, A. M. Talarico, M.-A. Arrio, A. Cornia, D. Gatteschi, R. Sessoli, Nat. Mater. 2009, 8, 194-197. [5] S. Thiele, F. Balestro, R. Ballou, S. Klyatskaya, M. Ruben, W. Wernsdorfer, Science 2014, 344, 1135-1138. [6] J.-L. Liu, Y.-C. Chen, Y.-Z. Zheng, W.-Q. Lin, L. Ungur, W. Wernsdorfer, L. F. Chibotaru, M.-L. Tong, Chem. Sci. 2013, 4, 3310-3316. [7] Y.-C. Chen, J.-L. Liu, L. Ungur, J. Liu, Q.-W. Li, L.-F. Wang, Z.-P. Ni, L. F. Chibotaru, X.-M. Chen, M.-L. Tong, J. Am. Chem. Soc. 2016, 138, 2829-2837. [8] J. Liu, Y.-C. Chen, J.-L. Liu, V. Vieru, L. Ungur, J.-H. Jia, L. F. Chibotaru, Y. Lan, W. Wernsdorfer, S. Gao, X.-M. Chen, M.-L. Tong, J. Am. Chem. Soc. 2016, 138, 5441-5450. [9] Y.-S. Ding, N. F. Chilton, R. E. Winpenny, Y.-Z. Zheng, Angew. Chem. Int. Ed. 2016, 55, 16071-16074. [10] S. K. Gupta, T. Rajeshkumar, G. Rajaraman, R. Murugavel, Chem. Sci. 2016, 7, 5181-5191. [11] F.-S. Guo, B. Day, Y.-C. Chen, M.-L. Tong, A. Mansikamäkki, R. Layfield, Angew. Chem. Int. Ed. 2017, 56, 11445-11449. [12] C. A. P. Goodwin, F. Ortu, D. Reta, N. F. Chilton, D. P. Mills, Nature 2017, 548, 439-442. [13] S.-D. Jiang, B.-W. Wang, H.-L. Sun, Z.-M. Wang, S. Gao, J. Am. Chem. Soc. 2011, 133, 4730-4733. [14] K. R. Meihaus, J. R. Long, J. Am. Chem. Soc. 2013, 135, 17952-17957. [15] L. Ungur, J. J. Le Roy, I. Korobkov, M Murugesu, L. F. Chibotaru, Angew. Chem. Int. Ed. 2014, 53, 4413-4417. [16] M. Gregson, N. F. Chilton, A.-M. Ariciu, F. Tuna, I. F. Crowe, W. Lewis, A. J. Blake, D. Collison, E. J. L. McInnes, R. E. P. Winpenny, S. T. Liddle, Chem. Sci. 2016, 7, 155-165. [17] J.-L. Liu, J.-Y. Wu, Y.-C. Chen, V. Mereacre, A. K. Powell, L. Ungur, L. F. Chibotaru, X.-M. Chen, M.-L. Tong, Angew. Chem. Int. Ed. 2014, 53, 12966-12970. [18] J.-L. Liu, J.-Y. Wu, G.-Z. Huang, Y.-C. Chen, J.-H. Jia, L. Ungur, L. F. Chibotaru, X.-M. Chen, M.-L. Tong, Sci. Rep. 2015, 5, 16621. [19] S. K. Langley, N. F. Chilton, L. Ungur, B. Moubaraki, L. F. Chibotaru, K. S. Murray, Inorg. Chem. 2012, 51, 11873-11881. [20] S. K. Langley, D. P. Wielechowski, V. Vieru, N. F. Chilton, B. Moubaraki, B. F. Abrahams, L. F. Chibotaru, K. S. Murray, Angew. Chem. Int. Ed. 2013, 52, 12014-12019. [21] V. Mougel, L. Chatelain, J. Pécaut, R. Caciuffo, E. Colineau, J.-C. Griveau, M. Mazzanti, Nat. Chem. 2012, 4, 1011-1017. [22] L. Chatelain, J. P. S. Walsh, J. Pécaut, F. Tuna, M. Mazzanti, Angew. Chem. Int. Ed. 2014, 53, 13434-13438. [23] K. C. Mondal, A. Sundt, Y. Lan, G. E. Kostakis, O. Waldmann, L. Ungur, L. F. Chibotaru, C. E. Anson, A. K. Powell, Angew. Chem. Int. Ed. 2012, 51, 7550-7554. [24] R. Sessoli, Nature 2017, 548, 400-401. [25] G. Mezei, C. M. Zaleski, V. L. Pecoraro, Chem. Rev. 2007, 107, 4933-5003. [26] J. D. Rinehart, J. R. Long, Chem. Sci. 2011, 2, 2078-2085. [27] N. F. Chilton, D. Collison, E. J. L. McInnes, R. E. P. Winpenny, A. Soncini, Nat. Commun. 2013, 4, 2551. [28] S. Alvarez, P. Alemany, D. Casanova, J. Cirera, M. Llunell, D. Avnir, Coord. Chem. Rev. 2005, 249, 1693-1708. [29] N. F. Chilton, R. P. Anderson, L. D. Turner, A. Soncini, K. S. Murray, J. Comput. Chem. 2013, 34, 1164-1175. [30] M. E. Lines, J. Chem. Phys. 1971, 55, 2977-2984. [31] L. Ungur, L. F. Chibotaru, Phys. Chem. Chem. Phys. 2011, 13, 20086-20090. [32] Y.-N. Guo, G.-F. Xu, P. Gamez, L. Zhao, S.-Y. Lin, R. Deng, J. Tang, H.-J. Zhang, J. Am. Chem. Soc. 2010, 132, 8538-8539. [33] I. J. Hewitt, J. Tang, N. T. Madhu, C. E. Anson, Y. Lan, J. Luzon, M. Etienne, R. Sessoli, A. K. Powell, Angew. Chem. Int. Ed. 2010, 49, 6352-6256. [34] M. Jeletic, P.-H. Lin, J. J. Le Roy, I. Korobkov, S. I. Gorelsky, M. Murugesu, J. Am. Chem. Soc. 2011, 133, 19286-19289. [35] J. Ruiz, A. J. Mota, A. Rodriquez-Diéguez, S. Titos, J. M. Herrera, E. Ruiz, E. Cremades, J. P. Costes, E. Colacio, Chem. Commun. 2012, 48, 7916-7918. \t Supporting Information Table of Contents S1. Experimental Section…………………………………………………………………………………….……….2 S2. Crystal Data and Structures………………………………………………………………………………..……3 S3. Magnetic Characterization…………………………………………………………………………………..…...6 S4. Theoretical Calculations………………………………………………………………………………….………10 S1. Experimental Section Materials and Physical Measurements. The ligand H2quinHA was synthesized as literature described.1 Metal salts and other reagents were commercially available and used as received without further purification. The C, H, and N elemental analyses were carried out with an Elementar Vario-EL CHNS elemental analyzer. Powder X-ray diffraction (PXRD) patterns were performed on Bruker D8 Advance Diffratometer (Cu-Kα, λ = 1.54056 Å). Thermogravimetric analysis (TGA) was carried out on a NETZSCH TG209F3 thermogravimetric analyzer. Magnetic susceptibility measurements were performed with a Quantum Design MPMS-XL7 SQUID. The ZFC-FC (0.2 T, 2 K/min), hysteresis (0.02 T/s) were measured on a Quantum Design PPMS. Polycrystalline samples were embedded in vaseline to prevent torqueing. Data were corrected for the diamagnetic contribution calculated from Pascal constants. Single Crystal X-ray Crystallography. Diffraction data were collected on a Bruker D8 QUEST diffractometer with Mo-Kα radiation (λ = 0.71073 Å) for complexes 1 and 2 at 120(2) K. The Data indexing and integration were carried out using a Bruker Smart program. The structures were solved by direct methods, and all non-hydrogen atoms were refined anisotropically by least-squares on F2 using the SHELXTL program suite.2 Anisotropic thermal parameters were assigned to all non-hydrogen atoms. The hydrogen atoms attached to carbon, nitrogen and oxygen atoms were placed in idealised positions and refined using a riding model to the atom to which they were attached. The SQUEEZE program of PLATON was employed to deal with the disordered solvent molecules.3 The solvent accessible volume and the number of residual electrons per unit cell is 330 Å3, 89e and 342 Å3, 87e for 1 and 2, respectively. The elemental analysis and thermogravimetric analysis were applied to determine the disordered solvent molecules as three water molecules, which are consistent with the analysis results of SQUEEZE. CCDC 1563751 (1) and 1563752 (2) contain the supplementary crystallographic data for this paper. These data can be obtained free of charge via https://www.ccdc.cam.ac.uk/structures/. Synthesis. [DyCu5(quinha)5(sal)2(py)5](CF3SO3)·py·4H2O (1): A mixture of H2quinHA (0.125 mmol, 23.5 mg) and Cu(CF3SO3)2 (0.125 mmol, 45.2 mg) were dissolved in 5 mL MeOH. Under stirring, Dy(CF3SO3)3 (0.025 mmol, 15.2 mg) was added into the solution. Then solution was transferred to a Teflon container and kept at 75 °C in the oven for 12 hr. Green microcrystals (Yield: 30 mg) were collected as precursor for the next steps. The precursor (20 mg) was dissolved in 5 mL MeOH and 1 mL pyridine. Under stirring, salicylaldehyde (Hsal, 0.2 mmol, 24.4 mg) was added. Then, the solution was stirred for 2 hr. After that, the solution was filtered and kept silent for evaporation. Green crystals (Yield, 15 mg, 38 % based on Dy) suitable for X-ray analysis were obtained in two weeks. Elemental analysis calcd (%) for DyCu5C95H78N16O21SF3; C: 48.57, H: 3.35, N: 9.54; found (%): C: 48.68, H: 3.57, N: 9.36. [YCu5(quinha)5(sal)2(py)5](CF3SO3)·py·4H2O (2): the procedure was the same as that employed for complex 1, except that Dy(CF3SO3)3 was replaced by Y(CF3SO3)3 (Yield 13.4 mg, 34% based on Y). Green crystals were obtained in two weeks by slow evaporation of the solution. Elemental analysis calcd (%) for YCu5C95H78N16O21SF3; C: 50.15, H: 3.46, N: 9.85; found (%): C: 49.98, H: 3.61, N: 9.72. \t\tS2. Crystal Data and Structures \n Figure S1. Experimental and simulated X-ray powder diffraction (XRPD) patterns for 1 (a) and 2 (b).   \n Figure S2. Thermogravimetric analysis of 1 (a) and 2 (b). The red dash lines show the stage of escaping of solvent molecules. \t\t\t\t\t\t\t\t\t\t\t\t \nTable S1. Crystallographic data and structural refinements for 1 and 2. Compound 1 2 Formula DyCu5C95H78N16O21SF3 YCu5C95H78N16O21SF3 Formula weight 2348.99 2275.40 Temperature / K 120(2) 120(2) Crystal system Triclinic Triclinic Space group P-1 P-1 a / Å 13.3601(5) 13.3721(5) b / Å 18.4387(8) 18.3699(7) c / Å 21.1192(8) 21.2229(8) α / ° 108.139(1) 107.9971(13) β / ° 106.261(1) 106.4516(13) γ / ° 92.988(1) 92.7421(13) V / Å3 4690.6(3) 4702.6(3) Z 2 2 ρcalcd. (g/cm3) 1.663 1.607 μ (mm-1) 2.009 1.830 F (000) 2364 2310 Reflections collected 67814 81364 Independent reflections 21351 21536 GOF on F2 1.052 1.027 R1, wR2 [I ≥ 2σ(I), squeeze]a 0.0486, 0.1122 0.0544, 0.1448 R1, wR2 (all data, squeeze) 0.0724, 0.1247 0.0739, 0.1584 CCDC No. 1563751 1563752 a R1 = ∑||Fo| − |Fc||/∑|Fo|. wR2 = [∑w(Fo2 – Fc2)2/∑w(Fo2)2]1/2.   Table S2. Selected bonds lengths [Å] and angles [°] for 1 and 2. 1 2 Dy1-O1 2.449(3) Y1-O1 2.441(3) Dy1-O3 2.408(3)  Y1-O3 2.398(3) Dy1-O5 2.405(3) Y1-O5 2.390(3) Dy1-O7 2.408(3) Y1-O7 2.392(2) Dy1-O9 2.394(3) Y1-O9 2.382(3) Dy1-O11 2.197(3) Y1-O11 2.195(3) Dy1-O13 2.199(3) Y1-O13 2.196(3) O1-Dy1-O3 70.74(10) O1-Y1-O3 70.83(9) O3-Dy1-O5 73.33(10) O3-Y1-O5 73.25(9) O5-Dy1-O7 71.78(10) O5-Y1-O7 71.80(9) O7-Dy1-O9 72.21(10) O7-Y1-O9 72.27(9) O9-Dy1-O1 72.06(10) O9-Y1-O1 71.97(9) O11-Dy1-O13 177.33(13) O11-Y1-O13 177.48(12)    Table S3. Comparison of equatorial and axial Dy-O bonds in pentagonal bipyramidal Dy-based SMMs. Compounds Dy–Oequatorial bonds (Å) Dy–Oaxial bonds (Å) Oaxial–Dy–Oaxial angle (o) Ref 1 2.394-2.449 2.197, 2.199 177.33 This work [Zn2Dy(L1)2(MeOH)]NO3·3MeOH·H2O 2.366-2.427 2.195, 2.221 168.6 1 [Fe2Dy(L2)2(H2O)]ClO4·2H2O 2.324-2.492 2.190, 2.193 169.2 2 [Co2Dy(L1)2(H2O)]NO3·3H2O 2.355-2.420 2.175, 2.198 169.7 3 [Co2Dy(L1)2(H2O)]NO3 2.326-2.508 2.171, 2.175 169.8 3 [Dy(Cy3PO)2(H2O)5]Cl3·(Cy3PO)·H2O·EtOH 2.327-2.380 2.217, 2.221 175.79 4 [Dy(Cy3PO)2(H2O)5]Br3·2(Cy3PO)·2H2O·2EtOH 2.336-2.365 2.189, 2.210 179.04 4 [Dy(L3)2(H2O)5]I3·(L3)2·H2O 2.355-2.375 2.203, 2.208 175.14 5 [Dy(bbpen)Cl] / 2.166 154.3 6 [Dy(bbpen)Br] / 2.163 155.8 6 [Dy(OtBu)2(py)5](BPh4) / 2.110, 2.114 178.91 7 * L1 = 2,2′,2″-(((nitrilotris(ethane-2,1-diyl))tris(azanediyl))tris(methylene))tris-(4-bromophenol); L2 = 2,2′,2″-(((nitrilotris(ethane-2,1-diyl))tris(azanediyl))tris(methylene))tris-(4-chlorophenol)); L3 = tBuPO(NHiPr)2; H2bbpen = N,N′-bis(2-hydroxybenzyl)-N,N′-bis(2-methylpyridyl)ethylenediamine.  Table S4. Continuous shape measures calculations (CShM) for rare-earth ions in 1 and 2. Complex HP-7 (D7h) HPY-7 (C6v) PBPY-7 (D5h) COC-7 (C3v) CTPR-7 (C2v) JPBPY-7 (D5h) JETPY-7 (C3v) Dy in 1 33.352 24.487 0.244 7.845 5.996 2.304 23.844 Y in 2 33.316 24.555 0.233 7.826 5.999 2.357 23.885 HP-7 = Heptagon; HPY-7 = Hexagonal pyramid; PBPY-7 = Pentagonal bipyramid; COC-7 = Capped octahedron; CTPR-7 =Capped trigonal prism; JPBPY-7 = Johnson pentagonal bipyramid J13; JETPY-7 = Johnson elongated triangular pyramid J7.  Table S5. Continuous Shape Measures (CShM) calculations for CuII ions in 1 and 2.  PP-5 vOC-5 TBPY-5 SPY-5 JTBPY-5 complex 1      Cu1 25.838 2.174 5.377 1.350 8.893 Cu2 24.102 2.395 5.959 1.705 8.349 Cu3 28.548 2.005 4.214 0.886 6.767 Cu4 26.464 2.048 5.410 1.226 8.930 Cu5 26.113 1.821 6.003 1.307 8.242 complex 2      Cu1 25.676 2.265 5.413 1.387 9.026 Cu2 24.420 2.339 5.868 1.630 8.248 Cu3 28.686 2.001 4.218 0.859 6.764 Cu4 26.424 2.070 5.378 1.231 8.970 Cu5 26.065 1.828 5.898 1.299 8.127 *PP-5 = Pentagon; vOC-5 = Vacant octahedron; TBPY-5 = Trigonal bipyramid; SPY-5 = Spherical square pyramid; JTBPY-5 = Johnson Trigonal. \t\tS3. Magnetic Characterization \n Figure S3. Variable-field magnetization data for 1 (a) and 2 (b). The solid lines are simulation from the fitted parameters for 1 and the best fit for 2 by using PHI program. Data were collected from 0−7 T in steady fields.  \n Figure S4. (a) Frequency dependence of the in-phase product (χ′M) and out-of-phase (χ″M) at 10 K under different applied fields (0−4 kOe) for 1. The solid lines are guides for the eyes. (b) Field-dependent relaxation times for 1 at 10 K.  \n Figure S5. (a) The temperature dependence of the in-phase product (χ′MT) and out-of-phase (χ′′M) for 1 at zero dc field with the ac frequency of 11−1488 Hz. (b) The frequency dependence of the in-phase product (χ′MT) and out-of-phase (χ′′M) for 1 at zero dc field with the temperature of 2.5−66 K. The solid lines are guides for the eyes.  \n Figure S6. (a) The temperature dependence of the in-phase product (χ′MT) and out-of-phase (χ′′M) for 1 at 2 kOe dc field with the ac frequency of 0.03−1488 Hz. (b) The frequency dependence of the in-phase product (χ′MT) and out-of-phase (χ′′M) for 1 at 2 kOe dc field with the temperature of 3−68 K. The solid lines are guides for the eyes.  \n Figure S7. Cole-Cole plots for the ac susceptibilities under 0 and 2 kOe dc field for 1 (Left: full-temperature range; Right: high-temperature range). The solid lines are the best fit for the generalized Debye model.   \n Figure S8. (a) The frequency dependence of the in-phase product (χ′MT) and out-of-phase (χ′′M) for 1 at 0.5 kOe dc field with the temperature of 2.5−68 K. The solid lines are guides for the eyes. (b,c) Cole-Cole plots for the ac susceptibilities under 0.5 kOe dc field (b: full-temperature range; c: high-temperature range). The solid lines are the best fit for the generalized Debye model.  \n Figure S9. The temperature-dependent relaxation times (log-log scale plot of τ−1 vs. T) for 1 under indicated magnetic fields. The solid lines are fits with equation 3 for 0, 0.5 and 2 kOe dc field, respectively.  \n Figure S10. The temperature-dependent relaxation times for 1. The data are fitted by a sum of Orbach + Raman processes, and show a serious deviation.  \n Figure S11. (a) The plots of relaxation times of 1 under zero dc field. The black solid line represents a sum of two Arrhenius law, giving Ueff(1) = 623 cm−1, τ0(1) = 2.1 × 10−11 s (blue dotted line) and Ueff(3) = 5.2 cm−1, τ0(3) = 9.4 × 10−5 s (blue dashed line). We are not currently certain on the origin of the bump highlighted in dark brown, and these data have been omitted from the fit in order to correctly estimate Ueff(1) and Ueff(3).  \n Figure S12. Various temperature magnetization decay and the best fit curves at 2 kOe.  \n Figure S13. HF-EPR spectra of 2 at 4.2 K and various frequencies. The spectra are offset in proportion to the frequency. Black dots represent the resonance fields for each spectrum. Solid lines are the linear fit to each resonance branch. The obtained g-values are g|| = 2.25(2) and g^ = 2.03(2).\t\t\nS4. Theoretical Calculations Broken-symmetry DFT calculations. To obtain an estimate of the magnetic coupling between the copper centres, we performed DFT calculations on the experimental crystal structure of YCu5 with no optimisation in Gaussian09d,4 using the well-known B3LYP5 hybrid functional and the following basis sets: LANL2DZ6 pseudopotential for yttrium, cc-pVTZ7 for copper and cc-pVDZ8 for all other atoms. We assume the isotropic Heissenberg-Van Vleck-Dirac (HDVV) model spin Hamiltonian to be a good model, and proceed to map the expectation energy values of the different ferro- (FM) and antiferromagnetic (AFM) solutions to the exact non-relativistic, time-independent Hamiltonian. This is formally the same as employing the eigenstates of the simplified Ising model spin Hamiltonian. The molecular structure suggests that one may expect two different magnetic coupling constants within the CuII5 pentagon, which can be represented by the following HDVV Hamiltonian, where 𝑖,𝑗 and 𝑘,𝑙 are the indexes of the syn-syn (3 pairs) and syn-anti (2 pairs) pairs of Cu ions, respectively, accounting for the orientation of the pyridine ligands. 𝐻J:KK=−2𝐽\t\t𝑺6\t·\t𝑺N6,N−2𝐽O\t\t𝑺>\t·\t𝑺P>,P\tMaking use of the set of functions presented in Table S6, one can obtain expressions for the HDVV expectation values in terms of the magnetic coupling constants (Table S7). Combining the calculated energy differences in Table S6 with the associated analytical expressions in Table S7, one can solve all set of linearly independent equations to obtain the different coupling constant values, as reported in Table S8. While there is some dispersion of 𝐽 and 𝐽O, our DFT calculations give 𝐽=−98(4) cm-1 and 𝐽O=−66(9) cm-1, supporting a two-J model for fitting the magnetic data. The ratio 𝐽𝐽O=1.50 from DFT is consistent with that found experimentally of 𝐽𝐽O=1.55.  Table S6. DFT computed energies and corresponding energy differences with respect to the ferromagnetic solution. The ordering of spins in the functions 𝑓 is the same as in Scheme 1. Labels are used to indicate which energy differences have been used in Table S8. 𝑓≡𝑆&'8,𝑆&'A,𝑆&'^,𝑆&'C,𝑆&'5 Energy (a.u.) ∆𝐸 (cm-1) Label 𝛼𝛼𝛼𝛼𝛼 -13545.64838       𝛽𝛼𝛼𝛼𝛼 -13545.64916 171.52 1 𝛼𝛽𝛼𝛼𝛼 -13545.64928 199.09 2 𝛼𝛼𝛽𝛼𝛼 -13545.64910 159.98 3     𝛼𝛽𝛼𝛽𝛼 -13545.64999 353.88 4 𝛼𝛽𝛼𝛼𝛽 -13545.64999 354.41 5 𝛽𝛼𝛼𝛽𝛼 -13545.64986 326.03 6 \tTable S7. Specification of FM and AFM HDVV expectation values and associated energy differences used to extract the two different magnetic coupling constants. Note that the ordering of spins in the functions 𝑓 is the same as in Scheme 1, and the one to one correspondence with the entries in Table S6. 𝑓  𝑓𝐻J:KK𝑓 𝐹𝑀𝐻J:KK𝐹𝑀−𝐴𝐹𝑀𝐻J:KK𝐴𝐹𝑀     𝛼𝛼𝛼𝛼𝛼 −12(2𝐽+3𝐽′)       𝛽𝛼𝛼𝛼𝛼 −12∙𝐽′ −𝐽−𝐽′ 𝛼𝛽𝛼𝛼𝛼 −12∙𝐽′ −𝐽−𝐽′ 𝛼𝛼𝛽𝛼𝛼 −12(2𝐽−𝐽′) −2∙𝐽′     𝛼𝛽𝛼𝛽𝛼 12(2𝐽+𝐽′) −2∙(𝐽+𝐽′) 𝛼𝛽𝛼𝛼𝛽 12(2𝐽+𝐽′) 𝛽𝛼𝛼𝛽𝛼 12(2𝐽+𝐽′) \t\t\tTable S8. Coupling constants extracted for the different pathways, using the different set of equations available, as labelled in Table S6. Labels of energy differences used 𝐽 (cm−1) 𝐽O (cm−1) 1 and 2 -100 -72 2 and 3 -100 -60 2 and 6 -100 -63 2 and 4 -100 -55 3 and 4 -97 -63 1 and 4 -91 -80  Table S9. Spin states and energies for {YCu5}. ST Energy (cm−1) 1/2 0 1/2 51 1/2 143 3/2 156 3/2 173 1/2 271 1/2 273 3/2 283 3/2 299 5/2 362  CASSCF-SO calculations We employed Molcas 8.09 to perform CASSCF-SO calculations on the experimental crystal structure of {DyCu5} where the CuII ions were substituted by diamagnetic ZnII ions. Basis sets from the ANO-RCC library10,11 were used with VTZP quality for the Dy atom, VDZP quality for the nearest oxygen atoms and VDZ for all other atoms. The two electron integrals were Cholesky decomposed with a threshold of 10-8 to save disk space and computational resources. The state-averaged CASSCF orbitals of the 21 sextets were optimised with the RASSCF module. These orbitals were then used to obtain the 224 quartets by means of a configuration-interaction-only (no orbital optimisation). These two sets of spin-free states were then used to construct and diagonalise the spin-orbit coupling Hamiltonian in the basis of 21 and 128 sextets and quartets, respectively, with the RASSI module. The crystal field decomposition of the ground J = 15/2 multiplet of the 6H term was performed with the SINGLE_ANISO module.12  Table S10. Crystal field splitting of the 6H15/2 multiplet for Dy in DyCu5, calculated with CASSCF-SO. Ab initio energy (cm-1) gx gy gz gz Angle (º) Crystal Field Energy (cm-1) Crystal Field Wavefunction 0 0.00 0.00 19.89 - 0 100%±152 314 0.00 0.00 17.10 1.3 313 100%±132 511 1.43 7.35 13.46 89.9 510 89%±12+5%∓12+4%∓32+2%±32 541 0.19 0.49 13.61 4.3 542 92%±112+6%±32+2%±12 556 1.86 4.07 4.62 21.6 556 65%±32+22%∓32+4%±112+3%±52+2%±12+1%∓12+1%∓52 621 0.72 1.82 6.82 6.3 622 91%±52+5%±72+3%±32+1%±92 657 0.23 0.39 12.84 25.8 656 88%±92+11%±72 677 0.21 0.39 13.42 45.9 676 84%±72+11%±92+5%±52       Table S11. CASSCF-SO calculated CF parameters (cm−1) k q B(k,q) 2 -2 0.94703622185356E-02 2 -1 0.30252465348580E-01 2 0 -0.30197369300000E+01 2 1 -0.26997414970545E+00 2 2 0.15815788570198E+00 4 -4 -0.25058550502222E-02 4 -3 0.43492341289488E-02 4 -2 0.43348325188428E-03 4 -1 -0.15352173492993E-02 4 0 -0.11725362007375E-01 4 1 0.30283268386897E-02 4 2 -0.13493839249758E-02 4 3 -0.84289430330379E-02 4 4 0.79918998501252E-03 6 -6 -0.76411284866855E-05 6 -5 -0.69517717456357E-04 6 -4 -0.10660970745767E-04 6 -3 0.17406687745382E-04 6 -2 -0.43435319915386E-06 6 -1 0.24789033577866E-06 6 0 0.27771888397897E-04 6 1 -0.57161468366960E-04 6 2 -0.47113012046145E-06 6 3 -0.43314822885444E-04 6 4 0.79152086655417E-06 6 5 -0.88315016262650E-04 6 6 0.78251082173440E-05  Table S12. Lowest lying 14 pseudo-doublets of {DyCu5}. Energy (cm-1) mS (Cu5) mJ (Dy) Mz 0.0 ±0.5 ±7.5 ±8 4.4 ∓0.5 ±7.5 ±7 50.6 ±0.5 ±7.5 ±8 55.0 ∓0.5 ±7.5 ±7 142.9 ±0.5 ±7.5 ±8 147.3 ∓0.5 ±7.5 ±7 152.0 ±1.5 ±7.5 ±9 156.3 ±0.5 ±7.5 ±8 160.7 ∓0.5 ±7.5 ±7 165.1 ∓1.5 ±7.5 ±6 168.7 ±1.5 ±7.5 ±9 173.1 ±0.5 ±7.5 ±8 177.5 ∓0.5 ±7.5 ±7 181.9 ∓1.5 ±7.5 ±6   References 1. Jankolovits J, Kampf JW, Pecoraro VL. Solvent Dependent Assembly of Lanthanide Metallacrowns Using Building Blocks with Incompatible Symmetry Preferences. Inorg Chem 53, 7534-7546 (2014). 2. Sheldrick GM. Crystal structure refinement with SHELXL. Acta Cryst. C71, 3-8 (2015). 3. Spek AL. PLATON SQUEEZE: a tool for the calculation of the disordered solvent contribution to the calculated structure factors. Acta Cryst C71, 9-18 (2015). 4. Frisch, MJ, Trucks, GW, Schlegel, HB, Scuseria, GE, Robb, MA, Cheeseman, JR, Scalmani, G, Barone, V, Mennucci, B, Petersson, GA, Nakatsuji, H, Caricato, M, Li, X, Hratchian, HP, Izmaylov, AF, Bloino, J, Zheng, G, Sonnenberg, JL, Hada, M, Ehara, M, Toyota, K, Fukuda, R, Hasegawa, J, Ishida, M, Nakajima, T, Honda, Y, Kitao, O, Nakai, H, Vreven, T, Montgomery, J, Peralta, JAJE, Ogliaro, F, Bearpark, M, Heyd, JJ, Brothers, E, Kudin, KN, Staroverov, VN, Kobayashi, R, Normand, J, Raghavachari, K, Rendell, A, Burant, JC, Iyengar, SS, Tomasi, J, Cossi, M, Rega, N, Millam, JM, Klene, M, Knox, JE, Cross, JB, Bakken, V, Adamo, C, Jaramillo, J, Gomperts, R, Stratmann, RE, Yazyev, O, Austin, AJ Cammi, R, Pomelli, C, Ochterski, JW, Martin, R,L, Morokuma, K, Zakrzewski, VG, Voth, GA, Salvador, P, Dannenberg, JJ, Dapprich, S, Daniels, AD, Farkas, Ö, Foresman, JB, Ortiz, JV, Cioslowski, J, Fox, DJ. (2009). 5. Becke AD. Density-functional thermochemistry. III. The role of exact exchange. J Chem Phys 98, 5648-5652 (1993). 6. Hay PJ, Wadt WR. Ab initio effective core potentials for molecular calculations. Potentials for the transition metal atoms Sc to Hg. J Chem Phys 82, 270-283 (1985). 7. Kendall RA, Jr. THD, Harrison RJ. Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions. J Chem Phys 96, 6796-6806 (1992). 8. Jr. THD. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J Chem Phys 90, 1007-1023 (1989). 9. Aquilante F, et al. Molcas 8: New capabilities for multiconfigurational quantum chemical calculations across the periodic table. J Comput Chem 37, 506-541 (2016). 10. Roos BO, Lindh R, Malmqvist P-Å, Veryazov V, Widmark P-O. Main Group Atoms and Dimers Studied with a New Relativistic ANO Basis Set. J Phys Chem A 108, 2851-2858 (2004). 11. Roos BO, Lindh R, Malmqvist P-Å, Veryazov V, Widmark P-O. New relativistic ANO basis sets for actinide atoms. Chem Phys Lett 409, 295-299 (2005). 12. Ungur L, Chibotaru LF. Ab Initio Crystal Field for Lanthanides. Chem Eur J 23, 3708-3718 (2017).  "    },    {        "title": "2211.08521v1.Nonlinear_chiral_photocurrent_in_parity_violating_magnetic_Weyl_semimetals.pdf",        "content": "Nonlinear chiral photocurrent in parity-violating magnetic Weyl semimetals\nShiva Heidari1,\u0003and Reza Asgari1, 2, 3,y\n1School of Physics, Institute for Research in Fundamental Sciences, IPM, Tehran, 19395-5531, Iran\n2School of Physics, University of New South Wales, Kensington, NSW 2052, Australia\n3ARC Centre of Excellence in Future Low-Energy Electronics Technologies, UNSW Node, Sydney 2052, Australia\nThe strong correlation between the non-trivial band topology and the magnetic texture makes\nmagnetic Weyl semimetals excellent candidates for the manipulation and detection of magnetization\ndynamics. The parity violation together with the Pauli blocking cause only one Weyl node to\ncontribute to the photocurrent response, which in turn a\u000bects the magnetic texture due to the spin\ntransfer torque . Utilizing the Landau-Lifshitz-Gilbert equation and the spin-transfer torque in non-\ncentrosymmetric Weyl magnets, we show that the chiral photocurrent rotates the magnetization\nfrom the easy caxis to the aorbaxis, which leads to an exotic current next to the photocurrent\nresponse. The chiral photocurrent is calculated in the context of quantum kinetic theory and it has a\nstrong resonance on the order of mA/W near the Weyl nodes, the magnitude of which is controlled by\nthe momentum relaxation time. Remarkably, we study the in\ruence of magnetic texture dynamics\non the topological nonlinear photocurrent response, including shift and injection currents along with\nthe new chiral photocurrent, and show that both the magnitude and the in-plane orientation of the\nchiral photocurrent are strongly correlated with the direction of the magnetic moments.\nI. INTRODUCTION\nThe magnetic topological materials provide an ideal\nplatform for rich and fundamental scienti\fc discover-\nies stemming from the interplay of topology and mag-\nnetism [1{3]. An important advance is the realization of\nmagnetic Weyl semimetals [4{6], which are distinguished\namong topological materials by the dynamic interplay\nof magnetic texture and topological band crossings [7{\n9]. The topological notion of Weyl nodes can be un-\nderstood as e\u000bective magnetic monopoles with opposite\ncharges in momentum space as the origin of the diverg-\ning Berry curvature [10], leading to the large intrinsic\nanomalous Hall conductivity [11{13]. The other quan-\ntum metrics as the geometric origin of the nonlinear pho-\ntocurrent response appear in inversion-asymmetric topo-\nlogical materials [14, 15]. The parity-violation in Weyl\nsystems leads to an inverted asymmetric transition of\nthe electron position and velocity in the nonlinear opti-\ncal response, resulting in the shift and injection currents\n[14, 16]. These nonlinear photocurrent conductivities are\nthoroughly governed by system symmetries that dictate\nthe divergent behavior of topological quantum geome-\ntries near the Weyl crossing points. The recently dis-\ncovered parity-violating magnetic Weyl semimetals are\nproposed as promising candidates to generate both shift\nand injection currents due to parity-time (PT) symmetry\nbreaking. In TaAs Weyl semimetals the non-liner chiral\nphotocurrent can be induced by a femtosecond circularly\npolarized (CP) pulse through the non-equilibrium chiral\nmagnetic e\u000bect [17].\nThe magnetic texture in magnetic materials can be\ncontrolled by a spin-polarized current without a magnetic\n\u0003shiva.heidari@ipm.ir\nyr.asgari@unsw.edu.au, asgari@ipm.ir\feld [18{22]. The tunable magnetization at the inter-\nface between a topological insulator and a ferromagnetic\ninsulator opens up an intriguing venue to discover the\nintimate relation between non-trivial band topology and\nmagnetic con\fguration [23{29]. The signi\fcant role of\nstrong spin-orbit coupling in magnetic Weyl semimetals\nhas attracted much attention in more e\u000ecient spintronic\napplications and magnetic dynamics detection [7, 30, 31].\nIn particular, the electrically induced structural phase\ntransition in domain walls in a magnetic Weyl semimetal\nis accompanied by transient nonlinear electrical signals\nJy/(r\u0002 ^M(x;t))E2\nx, which can be taken as evidence for\nthe magnetic dynamics [32]. Further, the nonlinear opti-\ncal response in non-centrosymmetric topological magnets\ncan be employed as a powerful tool for realizing the in-\ntrinsic connection among the optical response, quantum\ngeometry in momentum space and magnetic texture in\nreal space. Speci\fcally, in a Weyl magnet, the magnitude\nand direction of magnetization determine the spacing and\norientation of a pair of Weyl nodes of opposite chirality\nin momentum space. Therefore, the potential to control\nthe magnetization can cause a topological redistribution\nof Weyl nodes in k-space, leading to a remarkable change\nin the nonlinear shift and injection currents.\nIn this research, we investigate the optical manipula-\ntion of the magnetic texture in an inversion-asymmetric\nWeyl magnet. The strong spin-orbit coupling in such ma-\nterials induces a spin-transfer torque (STT) arising from\nthe optical transitions of a single Weyl node. Our key\nquestion here is how a luminous light can rotate the mag-\nnetization and how these magnetic dynamics then lead\nto the signi\fcant changes in the nonlinear photocurrent\nresponses. The Weyl nodes in an inversion-asymmetric\nWeyl semimetal do not have the same energy, then the\nlight-induced resonant transitions in one node are Pauli-\nblocked in another node. Therefore, in the frequency\nwindow facing a node, only speci\fc Weyl fermions with\nchirality\u001fcontribute to the interband transitions andarXiv:2211.08521v1  [cond-mat.mes-hall]  15 Nov 20222\ncorrespondingly generate a chiral photocurrent response\nJ5. Such a chiral current can interact with the magnetic\ntexture and lead to magnetic rotation. We show that the\ndirection of magnetization rotates from the initial caxis\nto the \fnal aorbaxis, leading to signi\fcant changes in\nmagnitude and orientation of the in-plane photocurrent\nresponses. Our research provides new insights into un-\nderstanding the role of the tunable magnetic direction in\nconstructing the nonlinear topological shift and injection\ncurrents.\nThe paper is structured as follows. In Sec. II we in-\ntroduce the features and symmetries of parity-violating\nmagnetic Weyl semimetals and provide some examples of\ntheoretically predicted and experimentally observed such\nmaterials. Sec. III predicts the light-induced magnetiza-\ntion rotation leading to an additional current owing to\nthe strong coupling between topological Weyl fermions\nand magnetic orientation. The general properties of the\nchiral photocurrent response are discussed in Sec. IV\nand Sec. V is devoted to the quantum kinetic theory\nof the second-order DC photocurrent, which stems from\nthe non-trivial quantum geometry of the band structure.\nThe nonlinear Drude response arises from the intraband\ntransitions, and the interband shift, gyration and injec-\ntion currents and the in\ruence of magnetic dynamics on\nthem are discussed in Sec. VI and Sec. VII. We conclude\nour \fndings in Sect. VIII.\nII. PARITY-VIOLATING MAGNETIC WEYL\nSEMIMETALS\nWeyl semimetals have been proposed and have\nemerged in band structures in which either time-reversal\n(TR) or inversion (I) symmetries breaks. It has been the-\noretically proposed and experimentally con\frmed that a\nlarge class of Weyl materials in RAlX (R=rare earths,\nAl, X=Si, Ge) realize the Weyl fermions, respecting or\nviolating TR symmetry or I symmetry depending on the\nchoice of rare earth components and can even be catego-\nrized into type I or type II Weyl semimetals [6, 9, 33].\nMore speci\fcally, (Pr,Ce)AlX can be ferromagnetic with\nan easy axis along the candadirections [34{38], while\nLaAlGe is nonmagnetic [39].\nIn this study, we focus on the ferromagnetic Weyl\nsemimetals, where the intrinsic magnetic texture breaks\nthe time-reversal symmetry below the Curie temperature\nTc, and also their peculiar lattice crystal breaks the in-\nversion symmetry (IS). Table. I has classi\fed some exam-\nples of proposed or realized Weyl semimetals according\nto TR, I and Lorentz symmetries (type I or type II) and\ntheir magnetic properties. The ferromagnetism in such\nmaterials arises from the ordering of the local moments of\nthe f-electrons. For example, in LaAlGe, a non-magnetic\nWeyl semimetal, the f-orbital in the electronic con\fgura-\ntion of the La atom is empty, citing the density of states\n(DOS) of the localized f-orbital found only in the con-\nduction band. On the other hand, in (Pr,Ce)AlGe, theTABLE I. Some distinguished examples of experimentally ob-\nserved or theoretically proposed magnetic and non-magnetic\nWeyl semimetals. The last two compounds denoted by\u0003,\nRAlX (R=(Pr, Ce), X=(Ge,Si)), represent three examples of\ntheoretically proposed and experimentally observed parity-\nviolating magnetic Weyl semimetals.\nCompounds TRS ISType Magnetic Properties\nAB [40{43]p\u0002 I Non-Magnetic\nA=(Ta,Nb),B=(As,P)\n(W,Mo)Te 2[44, 45]/Ta 3S2[46]p\u0002II Non-Magnetic\nLaAlGe [39]p\u0002II Non-Magnetic\nMn3A \u0002pII Non-Collinear\nA=(Sn [47],Ge [48]) Anti-Ferromagnetic\nCuMnSb \u0002\u0002II Collinear\n[10, 49, 50] Anti-Ferromagnetic\nYbMnBi 2\u0002pII Canted\n[51] Anti-Ferromagnetic\nAlternative Layers of Magnetically \u0002\u0002 I Magnetic Impurities\nDoped TI and NI as the Spacer [52] Order Ferromagnetically\nCo3Sn2S2[5]/HgCr 2Se4[53]\u0002pI Ferromagnetic\nPrAlSi [54, 55] \u0002pI Ferromagnetic\nPrAlX\u0003\u0002\u0002 I Ferromagnetic\nX=(Ge [6, 33],Si [56])\nCeAlGe\u0003[33, 38] \u0002\u0002II Ferromagnetic\nParity -violating\ntime-invariant WSM\n𝑘𝑧\n𝑘𝑥𝑘𝑧\n𝑘𝑥𝑀∥Ƹ𝑧Parity -violating \nmagnetic WSM\n𝒘𝟏 𝒘𝟐\n𝒘𝟑𝒘𝟒𝒘𝟏𝒘𝟐\n𝒘𝟑𝒘𝟒(a)\n𝒘𝟑\n𝑘𝑧𝐸\n𝒘𝟏\n𝒘𝟐𝒘𝟒𝑘𝑧𝐸\n𝒘𝟏𝒘𝟑\n𝒘𝟐 𝒘𝟒(b)\n𝑀∥Ƹ𝑧𝜎𝑥𝑦𝐴𝐻=0 𝜎𝑥𝑦𝐴𝐻≠0\nFIG. 1. (a) Left hand side: The minimum model for\na time-invariant inversion asymmetric Weyl semimetal with\nfour nodes. The blue and red circles represent the Weyl\nnodes with opposite chiralities. Right hand side: The mag-\nnetism along ^ zrearranges the Weyl nodes to violate TRS. The\ngreen lines are the Fermi arcs connecting two Weyl nodes.\n(b) Represents the corresponding schematics energy disper-\nsion of Weyl nodes in non-magnetic and ferromagnetic Weyl\nsemimetals. The solid lines denoted the Weyl nodes with\nkx;y>0 and the dashed lines are nodes with kx;y<0.\nf-orbital in the Pr (or Ce) atom contains two (or one)\nelectrons, leading to ferromagnetization in these two ma-\nterials [33]. In magnetic Weyl semimetals with inversion\nsymmetry, i.g., Co 3Sn2S2[5], HgCr 2Se4[53] and PrAlSi\n[54, 55], the magnetism of texture splits the Weyl nodes\nin k-space. However, the parity-violating magnetic Weyl\nsemimetals are generated by IS breaking, i.e., nodes are\nsplit by inversion symmetry breaking, and the magnetic\ntexture only recon\fgures the Weyl nodes along the mag-3\nnetic direction in order to break TR symmetry. Figure 1\nshows how magnetism rearranges the Weyl nodes in an\ninversion asymmetric crystal. The low-energy Hamilto-\nnian describing a minimum model near w 1and w 4nodes\nis given by [Appendix. A]\nH0=~vF(k?\u0000kw\n?)\u001c0\n\u001b?+~vF(kz\u001cz\u0000 (1)\n(kw\nz\u0000kM)\u001c0)\n\u001bz+u \u001cz\n\u001b0+\u0015 \u001cy\n\u001bz:\nThe last two terms are responsible for inversion symme-\ntry breaking, while other symmetries are preserved. The\nterm\u0015\u001cy\n\u001bzis the momentum-independent spin-orbit\ninteraction that split the degeneracy at every points ex-\ncept the Weyl crossings. This term is closely analogous\nto the Dresselhous spin-orbit interaction term allowed in\nthe absence of IS. The role of u\u001cz\n\u001b0is shifting two tips\nof Weyl cones in energy and breaks IS. The positions of\nthe nodes are given by k(w1;w3)= (\u0006kw\n?;\u0006kw\nz\u0000kM) and\nk(w2;w4)= (\u0007kw\n?;\u0006kw\nz+kM), where the vector kw=\n(kw\n?;kw\nz) denotes the node positions in time-invariant in-\nversion asymmetric WSM and kM= (JS=~vF)^Mis the\nmomentum separation due to magnetism [Fig. 1], where\nJis the ferromagnetic exchange interaction and Sis the\nmagnitude of the texture magnetic moment. In the time-\ninvariant WSMs [Fig. 1: Left], e.g. TaAs family, no\nlinear anomalous Hall e\u000bect is observed. On the other\nhand, parity-violating magnetic Weyl semimetals [Fig. 1:\nRight], i.e. PrAl(Ge,Si) and CeAlGe, the Chern number\nis nonzero in the regions between nodes that are displaced\nby texture magnetic moments. Therefore, the anomalous\nHall e\u000bect is expected to be present in the linear response\nto an electric \feld in parity-violating magnetic WSMs\n[11]. For PrAlGe with node spacing kw= 0:15\u0017A\u00003the\nintrinsic anomalous Hall conductivity is estimated to be\n\u001bAH= 738 \n\u00001cm\u00001[6].\nThe divergent behavior of the quantum geometry of\nthe electron wave function near the Weyl points plays an\nimportant role in the bulk photogalvanic e\u000bect in topo-\nlogical materials [14]. The presence of TR or parity-time\n(PT) symmetry helps us predict which of the nonlinear\nresponse elements may be zero [14, 57]. In the presence\nof the PT symmetry, the Berry curvature vanishes at any\npoint inkspace, so there is no way to get the Weyl phase.\nMnGeO 3is a 3D Dirac semimetal with both TR and I\nsymmetry broken while PT symmetry is preserved. How-\never, in parity-violating magnetic Weyl semimetals such\nas PrAl(Ge,Si) and CeAlGe, both TR and PT symme-\ntry are broken and both shift and injection currents are\nexpected to be present. In the following, we will discuss\ntwo impactful parameters in the non-linear photocurrent\nresponse in Weyl semimetals. The \frst is the magneti-\nzation direction and the second is the chemical poten-\ntial which determines the energy di\u000berence between the\nFermi surface and the singularity of quantum geometries.III. CHIRAL CURRENT-INDUCED MAGNETIC\nROTATION\nIn this section, we aim to calculate the light-\ninduced spin-transfer-torque (STT)Tethrough the non-\nequilibrium spin polarization of electrons h\u001b(r)i, which\nappears when one node contributes more than the other\nnode, leading to the chiral current J5=evFh\u001b(r)i. The\nchiral current can be induced as a non-equilibrium photo-\nresponse. Using the well-known Landau-Lifshitz-Gilbert\n(LLG) equation [58]\nd^M\ndt=\r0Be\u000b\u0002^M+\u000b^M\u0002d^M\ndt+Te; (2)\nthe dynamic behavior of the magnetic texture can be ex-\ntracted. The parameter \r0is the gyromagnetic ratio ,Be\u000b\nis an e\u000bective magnetic \feld, \u000bis the Gilbert orviscous\ndamping parameter , which is proportional to the energy\nloss rate [59], and Teis the STT describing the electronic\nbackground contribution to the magnetic texture dynam-\nics.\nThe STTTeon the right-hand side arises from the\nexchange interaction between itinerant electrons and the\nmagnetic moments and is given by Te=jkMj\ne\u001as^M\u0002J5;\nwhereJ5=P\n\u001f\u001fJ\u001fis the chiral stream. The wave vector\nkMdetermines the separation of the Weyl nodes in mo-\nmentum space due to magnetism, i.e. \u001fkM^Mz, which\nis given by kM=JS\n~vF, and\u001asis the number of lo-\ncal magnetic moments per unit volume. Therefore, the\nSTT arises when one node contributes more than the\nother node in inducing a torque on localized magnetic\nmoments. In other words, the non-equilibrium spin po-\nlarization of electrons is essential to rotate the magnetic\ntexture, so the total current J=P\n\u001fJ\u001fcan not induce\nSTT. Here, the main origin of such a nodal imbalance\nis the optical transitions of a single Weyl node while the\nother node is Pauli blocked. We note that because of\n^M(t= 0) = ^zthe STT only applies to the in-plane com-\nponents of the magnetic moments. Figure 2 represents\nthe time evolution of magnetic moments induced by the\nchiral current from the initial z-axis to the \fnal x- or\ny-axis, depending on the magnitude of the chiral pho-\ntocurrentJ0\n5;kand its orientation 'j\n0att= 0 [Appendix.\nB]. Using the LLG equation, and in the absence of a\nmagnetic \feld Be\u000b, we show that strong spin-orbit cou-\npling in magnetic Weyl semimetals together with nonlin-\near light-induced electronic spin polarization leads to an\nadditional nonlinear photocurrent response generated by\nmagnetic dynamics. In magnetic Weyl semimetals, the\nmagnetic texture in real space is inherent to the nodes\nposition in momentum space and exhibits a dynamical\ninterplay with it [32, 60]. This key feature leads to an\ne\u000bective U(1) axial vector A5= (JS=evF)^M, in the low\nenergy Hamiltonian [Appendix.A]. Hence, the dynami-\ncal behavior of A5gives rise to an axial electric \feld\nE5=\u0000JS\nevF_^M. Using Eq. (2), E5can be written as4\n[Appendix. B]\nE5=\u0000~jkMj2\ne2\u001asexp(\u000b^\u0002)(^M\u0002J5); (3)\nwhere exp(\u000b^\u0002)O= (1+\u000b^M\u0002O+\u000b2^M\u0002(^M\u0002O)+\u0001\u0001\u0001)\nand\u000bis the dissipation parameter and J5is the chiral\nphotocurrent response. We should note that the axial\nelectric \feld E5only exists in the dynamic regime be-\nfore approaching the steady state. Such a pseudo-electric\n\feld induces a longitudinal drift current \u000ej\u001f=\u001f\u000e\u001b\u001fE5\nfor each node \u001f[60]. The longitudinal conductivity \u000e\u001b\u001f\nis well de\fned if two Fermi surfaces of two nodes are far\nenough apart in momentum space that the scattering be-\ntween the nodes can be neglected. Then the longitudinal\nconductivity for Weyl semimetals can be estimated as\n\u000e\u001b\u001f=\u0000e2\u001cu2\n\u001f\n3\u00192~3vF'\u0000(104\u0000105) S/m; (4)\nwhere we choose u\u0006= 20 (or 50) meV (the energy sepa-\nration between nodes and Fermi surfaces), vF= 5\u0002105\nm/s and\u001c= 1 ps. Using the expression for the axial\nelectric \feld in Eq. (3), we may write the (axial) current\n\u000ej(5)as a consequence of the interplay between electronic\ndegree of freedom and the magnetic texture ^M\n\u000ej= exp(\u000b^\u0002)(^M\u0002J5)X\n\u001f\u0011\u001f;\n\u000ej5= exp(\u000b^\u0002)(^M\u0002J5)X\n\u001f\u001f\u0011\u001f(5)\nwhere\u0011\u001f=\u001cu2\n\u001fjkMj2\n3\u00192~2vF\u001asis a dimensionless quantity and\ndepends on the system-dependent or non-universal pa-\nrameters. Therefore, the light-induced magnetic dynam-\nics in magnetic Weyl semimetals can induce chiral current\n\u000ej5=\u000ej+\u0000\u000ej\u0000with components\n\u000ej5;x=\u0000\u0011(J5;y+\u000bJ5;x+O(\u000b2) +\u0001\u0001\u0001);\n\u000ej5;y=\u0011(J5;x\u0000\u000bJ5;y+O(\u000b2) +\u0001\u0001\u0001);(6)\nwhere\u0011=\u0011++\u0011\u0000, and\u0006stands to opposite chiralities.\nFigure 3(a) estimates the parameter \u0011using the quanti-\nties\u001c= 1 ps (the typical relaxation time in semimetals),\nvF= 5\u0002105m/s [61],kM= 0:15\u0017A\u00001,\u001as'1:5\u00021028\nm\u00003(volume of unit cell: V= 262:1\u0017A3for a ferromag-\nnetic Weyl semimetal candidate PrAlGe [11]). Figure\n3(b) represents the change in magnitude of the in-plane\nchiral current ( xandycomponents) at time t, i.e.,Jt\nk,\nin comparison to its magnitude at t= 0. If\u0011exceeds\nthe viscose damping parameter \u000b(\u0011 >\u000b ), the magneti-\nzation dynamics increase the magnitude of the in-plane\nchiral photocurrent response and vice versa in the inset.\nHaving used Eq. (6), we conclude that magnetic texture\ndynamics can also change the orientation of the in-plane\ncurrent density as\ntan'j\nt=tan'j\n0+\u0011\u0000\u000b\u0011tan'j\n0\n1\u0000\u0011tan'j\n0\u0000\u0011\u000b; (7)\n01 2 3 4 \n0\n1 \n-1.580-1.422-1.264-1.106-0.9480-0.7900-0.6320-0.4740-0.3160-0.15800.0000.15800.31600.47400.63200.79000.94801.1061.2641.4221.580\n0\nFIG. 2. (a): The light (propagates along the z-axis)-induced\nmagnetic texture evolution in a parity-violating magnetic\nWeyl semimetals. The magnetization direction rotates away\nfrom the initial c-axis to the \fnal a- orb-axis depending on the\ninitial chiral photocurrent magnitude J0\n5;kand its orientation\n'j\n0. In the intermediate stage, between the initial and equilib-\nrium states, an axial electric \feld E5is induced to the system.\nWe de\fnea=JS\ne~vF\u001as. (b): The density plot of the modi\fed\nin-plane photocurrent due to the magnetic texture dynamics.\n'0\njand't\njare the orientation of in-plane chiral photocurrent\nin the absence and presence of magnetic dynamics.\nwhere'j\n0= tan\u00001(J5;y=J5;x) and'j\nt= tan\u00001((J5;y+\n\u000eJ5;y)=(J5;x+\u000eJ5;x)). Here'j\ntand'j\n0are the polar angles\nbetweenxandycomponents of current with and with-\nout taking into account the magnetic texture dynamics\n[Fig. 2(b)]. According to Fig. 2 (b), the magnetic tex-\nture dynamics can reorient and even reverse the in-plane\nphotocurrent. Therefore, the magnetic texture dynamics\nmay lead to a signi\fcant change in the orientation of the\nin-plane chiral photocurrent.\nIt is worth noting that the above discussion is a gen-\neral result and can be applied for both linear and circular\npolarization of light, which can induce chiral current into\nthe system. The spin manipulation in topological mate-\nrials, particularly Weyl semimetlas, has received consid-\nerable attention due to its wide applications in spintronic5\n0204060801000.00.10.20.30.40.50\n.00.10.20.30.40.51.001.041.081.120\n.00 .1\n  \n \n \n \n \nFIG. 3. (a): The parameter \u0011with respect to \u0016 uwhere\n\u0016u=q\n(u2\n++u2\n\u0000)=2. (b): The e\u000bect of magnetic texture dy-\nnamics on the magnitude of chiral photocurrent. The modi\f-\ncation coe\u000ecient of chiral current due to magnetic dynamics\nat timetis given byjJt\n5;kj=p\n(1\u0000\u000b\u0011)2+\u00112jJ0\n5;kj. The\ninset demonstrates that in the case of \u000b > \u0011 , the relative\nmagnitude slightly lowers.\ndevices [8, 9, 31, 32, 62, 63]. In the next section, we will\ndiscuss the electronic contribution of chiral photocurrent\n(J(2)\n5) in the context of the non-linear quantum kinetic\ntheory, and show how the magnitude and direction of the\nin-plane components can be a\u000bected by the direction and\nmagnitude of the magnetic moments.\nIV. CHIRAL PHOTOCURRENT RESPONSE\nMotivated by recent measurements of the nonlinear op-\ntical response in transition metal monopnictides such as\nTaAs, TaP, NbAs and the Weyl semimetal RhSi and\nCoSi [64{66], we perform the chiral photocurrent re-\nsponse in magnetic Weyl semimetals.\nThe general form of the second-order response to the\nelectric \feld of light is de\fned as [67, 68]\nJ(2)\nl(!;!1;!2) =\nX\ni;jZd!1d!2\n(2\u0019)2\u001bl;ij(!;!1;!2)Ei(!1)Ej(!2);(8)\nwhere!=!1+!2. The DC-photocurrent response due\nto the irradiation of light with frequency \n is character-\nized in the condition of != 0 or!1=\u0000!2. We de\fne\n!1=\u0000!2= \n, then we will have Ei(\n)Ej(!\u0000\n) =\nEi(\n)Ej(\u0000\n) or!= 0 in our formalism. We assume\nthe electric \feld of incident light E(t) =E0^e, where\n^e= (^icos\u0012pcos \nt+^jsin\u0012psin \nt) is the unit polariza-\ntion vector in which \u0012p= 0(\u0006\u0019=4) denotes the linearly\n(circularly) polarized light.\nDepending on the nature of light polarization, we can\ndecompose the bulk photovoltaic e\u000bect (BPVE) into the\npiezoelectric-like response (Linearly-polarized (LP) pho-\ntocurrent), and gyrotropic response (Circularly-polarized(CP) photocurrent) [69, 70]:\nJ(2)\nl(!= 0) =\njE0j2X\ni;jZd\n2\u0019(\fL\nl;ij(\n)Re[eie\u0003\nj] +\fC\nlr(\n)\u0014r):(9)\nThe superscripts L and C represent the linear and circu-\nlar BPVE, respectively. The symmetry of \fL\nl;ij(\n) is the\nsame as in the piezoelectric tensor, i.e. \fL\nl;ij=\fL\nl;ji, and\nits form is symmetric under index permutation, then we\ncan use\fL\nl;ij=1\n2[\u001bl;ij+\u001bl;ji]. On the other hand, the vec-\ntor\u0014is de\fned as\u0014=ie\u0002e\u0003which is non-zero only for\ncircularly polarized light and \fC\nlr(\n) =i\u000fijr\u001bl;ijis an-\ntisymmetric under index permutation and is called the\ngyration tensor. The nonlinear Drude response JDr,the\nBerry curvature dipole termJBCD, the injection stream\nJInj, the shift currentJShand the gyration currentJGyr\nare classi\fed into the second-order photocurrent response\nin the DC limit [16]. With the exception of the Drude\nresponse, the other terms are directly governed by vari-\nous intrinsic geometric quantities of the band structure.\nQuantum geometries such as Berry curvature and the or-\nbital magnetic moment play key roles in linear electronic\nand optical transport e\u000bects [71{73]. In the nonlinear\nresponse, light-induced direct current or the generation\nof second harmonics, the other quantum geometries ap-\npear in the formalism [14, 57, 74, 75]. We note that\nthemetric connection (\u0000) and the symplectic connection\n(~\u0000) have contributions to the shift and gyration currents,\nand also the quantum geometric berry curvature (B)\nand quantum metric (G) appear in LP and CP injec-\ntion terms, respectively. Furthermore, the berry curva-\nture dipole (BCD) vanishes in an untilted parity violated\nmagnetic Weyl semimetals in which both T and PT sym-\nmetries are broken [75]. However, it has been shown that\na tilt parameter can induce BCD in a T symmetric (non-\nmagnetic) but non-centrosymmetric Weyl systems lead-\ning to the accordingly a nonlinear anomalous Hall e\u000bect\n[76, 77]. The BCD vanishes for an untilted and isotropic\nWeyl semimetal. In the following sections, we will discuss\nthat the nonlinear interband photocurrent increases near\nthe Weyl crossing points, which is directly attributed to\nthe divergence behavior of quantum geometries near the\nWeyl nodes, and we will also consider the e\u000bect of mag-\nnetic dynamics into the photocurrent response.\nV. THEORETICAL FRAMEWORK: QUANTUM\nKINETIC EQUATION\nThe quantum Liouville equation of density matrix in\nthe Bloch representation, i.e., jn;ki= exp(ik\u0001r)jun;ki,\nis given by\n@h\u001a(k;t)i\n@t+i\n~[H;h\u001a(k;t)i] +\u0014(h\u001a(k;t)i) = 0:(10)\nHere,H=H0+HEis the complete Hamiltonian, k\nthe crystal momentum, nthe band index, \u0014(h\u001a(k;t)i)6\nis that scatter integral and H0is the non-interacting\nHamiltonian. In the presence of a light-matter interac-\ntion, the e\u000bect of the electromagnetic \feld on the time-\ndependent perturbation can be mapped according to the\nelectric dipole approximation Hamiltonian in the linear\napproach, i.e. HE=er\u0001E(t), where the electric \feld\ncouples to the Hamiltonian via a dipole energy. The ma-\ntrix representation of position operator in the quantum\nframework would be\n[rk]nm=i\u000enm@k+Rnm(k); (11)\nwhereR(k) =P\na=x;y;zRa(k)ekis the Berry vector poten-\ntial with components [ Ra(k)]nm=ihun\nkj@kaum\nki. This\nterm leads to the topologically non-trivial transport phe-\nnomena such as the well-known Hall conductivity in sys-\ntems with broken TR symmetry. The velocity operator\nin the Heisenberg picture can be obtained as\n[vk]nm= [ _rk]nm=~\u00001(@k\u000fn)\u000enm+i~\u00001\u000fnmRnm(k);\n(12)\nwhere\u000fnm=\u000fn\u0000\u000fmis the di\u000berence of the eigenvalues\nof the unperturbed Hamiltonian H0. The second term\ncontributes to the o\u000b-diagonal components or the inter-\nband responses. If the magnitude of the electric \feld\njEjis small enough, it can be viewed as a perturbation\nof the Bloch-Hamiltonian operator. Then we may ex-\npand the density matrix \u001ain powers of the electric \feld\n\u001a=P\nN\u001a(N), where\u001a(N)is theNthcorrection to \u001a(0)due\nto the electric \feld. Then, the quantum kinetic equation\nin its recursive form would be\n@\u001a(N)\n(t)\n@t+i\n~[H0;\u001a(N)\n(t)] +\u0014nm(\u001a(N)\n(t)) =eE(t)\n~\u0001[Dk\u001a(N\u00001)\n(t)]:\n(13)\nHere,DkO=DO\nDk=rkO\u0000i[Rk;O] is de\fned as the\ncovariant derivative. The solution of Eq. (13). i.e., the\nNthorder correction to the density matrix, is given by\n\u001a(N)\nnm(!) =eZd\n2\u0019d!\nnmEi(\n) [Di\nk\u001a(N\u00001)(!\u0000\n)]nm;(14)\nwhered!\nnm= (~!\u0000\u000fnm+i~\u001c\u00001)\u00001, and for simplicity,\nwe estimate the scattering integral as \u0014nm(\u001a(N))'h\u001ai\n\u001c,\nand\u001cis assumed to be k-independent. The zeroth cor-\nrection is the Fermi-distribution function at zero fre-\nquency,\u001a(0)\nnm= 2\u0019\u000e(!)\u000enmf(\u000fkn), withf(\u000fkn) = (1 +\ne\f(\u000fkn\u0000\u0016))\u00001. The linear-order correction, yields\n\u001a(1)\nnm(!) = 2\u0019e d!\nnmEi(!)f@kf(0)(\u000fkn)\u000enm+iRnm\nkFnmg;\n(15)\nwhereFnm\u0011f(\u000fkn)\u0000f(\u000fkm) is de\fned as the di\u000berence\nbetween the occupation in bands nandmin equilib-\nrium. The \frst term in Eq. (15) is diagonal and captures\nthe intra-band Drude conductivity in metals or doped\nsemimetals with \fnite Fermi surface, and the second term\nis o\u000b-diagonal and obtains the inter-band optical transi-\ntions which in the case of 3D Dirac or Weyl semimetals\nis linear in !.Using the linear response in Eq. (15), the second cor-\nrection\u001a(2)would be\n\u001a(2)\nnm(!) =eZd\n2\u0019d!\nnmEi(\n) [Di\nk\u001a(1)(!\u0000\n)]nm\n=\u001a(2)\nI;nm+\u001a(2)\nII;nm+\u001a(2)\nIII;nm:(16)\nWe \fnd\u001a(1)\nnm(!\u0000\n) =ed!\u0000\nnmEi(!\u0000\n)[Di\nk\u001a(0)]nm, where\nwe have used Eq. (14) and \u001a(0)(!\u0000\n0\u0000\n) =\u001a(0)\u000e(!\u0000\n\n0\u0000\n). Then, we can decompose the diagonal and o\u000b-\ndiagonal parts of the second order correction of density\nmatrix\n[\u001a(2)\nI(!)]nm=\ne2Zd\n2\u0019d!\nnmd!\u0000\nnmEi(\n)Ej(!\u0000\n)@ki@kjf(\u000fkn)\u000enm;\n(17)\n[\u001a(2)\nII(!)]nm=\nie2Zd\n2\u0019d!\nnmEi(\n)Ej(!\u0000\n)f@ki(d!\u0000\nnmRj\nnmFnm)+\niX\nn0(Ri\nk;n0nRj\nk;n0md!\u0000\nn0mFn0m\u0000Rj\nk;nn0Ri\nk;n0md!\u0000\nnn0Fnn0)g;\n(18)\n[\u001a(2)\nIII(!)]nm=\nie2Zd\n2\u0019d!\nnmd!\u0000\nnnEi(\n)Ej(!\u0000\n)Ri\nk;nm@kjFnm:(19)\nThe summation over the repeating indices i;j=x;y;z\nis implicit. The terms \u001a(2)\nIand\u001a(2)\nIIIare determined by\nderivative of the Fermi distribution function, then it\nwould be non-zero for materials with a \fnite Fermi sur-\nface and di\u000berent velocities in bands mandn, e.g. metals\nor doped semi-metals. The other term \u001a(2)\nIIis also \fnite\nwithout a Fermi surface, so it can exist for insulators as\nwell as for metals or doped semimetals. The Nthorder\ncorrection to the light-induced current is obtained by\nJ(N)\ni(!) =\u0000eX\nn;mZdk\n(2\u0019)dvi\nk;nm\u001a(N)\nk;mn(!): (20)\nThe second-order response can be obtained by setting\nN= 2, and using the second-order correction to the den-\nsity matrix, i.e., \u001a(2)\nk;mn.\nVI. INTRA-BAND CONTRIBUTION:\nNON-LINEAR DRUDE RESPONSE\nThe intra-band transitions ( n=m) in the non-linear\nresponse, similar to the linear response, can be only cap-\ntured for a single band with \fnite Fermi surface. The\nterm\u001a(2)\nI;nnin Eq. (17) is responsible for this non-linear7\nFIG. 4. Non-linear Drude response of inversion asymmetric\nmagnetic Weyl semimetals as a function of light \n. The real\nand imaginary parts of the conductivity are demonstrated by\nblue and red lines, respectively. We set \u001c= 1 ps ( ~=\u001c= 0:65\nmeV), T=10 K ( kBT=0.86 meV), ~vF= 3:25\u0002103meV\u0017A,\n~vFkw\nx(y)= 140 meV, ~vFkw\nz= 210 meV,JS= 100 meV,\nu+= 50 meV, u\u0000= 20 meV.\nintra-band response. Therefore, the non-linear Drude\nconductivity tensor is given by\n\u001b(2;Dr)\nl;ij=\u0000e3X\nnZdk\n(2\u0019)dd0\nnnd\u0000\nnnvl\nk;nn@ki@kjf(\u000fkn):\n(21)\nWe note that the above conductivity is symmetric un-\nderi$j, therefore the non-linear Drude conductivity is\nclassi\fed as the LP-photocurrent response. The product\nofd0\nnnd\u0000\nnnis a complex function, so the above conduc-\ntivity tensor can be divided into the real and imaginary\nparts as\nRe[\u001b(2;Dr)\nl;ij] =\u0000e3\n~\u001c2\n1 + \n2\u001c2Tr[Ulij\nnnf(\u000fkn)];\nIm[\u001b(2;Dr)\nl;ij] =\u0000e3\n~\n\u001c3\n1 + \n2\u001c2Tr[Ulij\nnnf(\u000fkn)];(22)\nwhereUlij\nnn=hnj(@kl@ki@kjH0(k))jni, and Tr[O] =P\nk;nO. The ratio of the real part to the imaginary part\nis given byRe[\u001b(2;Dr)\nl;ij]\nIm[\u001b(2;Dr)\nl;ij]=1\n\n\u001c, so the current survives\nin the limit of \n \u001c\u001c1. Figure 4 shows the numerical\nresult for the nonlinear Drude response of an inversion-\nasymmetric magnetic Weyl semimetal with respect to\nthe light frequency \n. As the \fgure shows, the nonlin-\near Drude current is suppressed by dissipation when the\nlight period 1 =\n is shorter than the quasiparticle lifetime\n\u001c. Only the intraband response is determined by deriva-\ntives of the band energy or the group velocity and does\nnot depend on the quantum geometries. Also, the in-\ntraband current does not generate a chiral current, since\nboth nodes are activated at the same time. Therefore,the Drude current cannot rotate the magnetization. In\nthe following, we will discuss that the inter-band contri-\nbution to the chiral photocurrent can trigger the magne-\ntization dynamics, resulting in a remarkable impact on\nthe in-plane photocurrent response.\nVII. INTER-BAND RESPONSE\nThe contribution of \u001a(2)\nII;mn(Eq. (18)) in the photocur-\nrent response results in\nJ(2;II)\nl(!= 0) =\u0000e\n~X\nn;mZdk\n(2\u0019)dvl\nk;nm\u001a(2)\nII;mn:(23)\nWe can decompose Eq. (18) into two distinct terms; (1):\nthe \frst term and the second term with n0=n;m which\nwe denote it by \u001a(2)\nII;1and (2): The second term with\nn06=n;m, which we denote it by \u001a(2)\nII;2. Then, we have\n\u001a(2)\nII;1(!= 0)jn;m=\nie2Zd\n2\u0019d0\nnmEi(\n)Ej(\u0000\n)Di\nnm(d\u0000\nnmRj\nnmFnm);(24)\nand\n\u001a(2)\nII;2(!= 0)jn;m=\u0000e2X\nn06=n;mZd\n2\u0019d0\nnmEi(\n)Ej(\u0000\n)\n(Ri\nk;n0nRj\nk;n0md\u0000\nn0mFn0m\u0000Rj\nk;nn0Ri\nk;n0md\u0000\nnn0Fnn0);\n(25)\nwhereDi\nnm=@ki\u0000i(Ri\nnn\u0000Ri\nmm) is the U(1)-covariant\nderivative, andFnm=f(\u000fkn)\u0000f(\u000fkm) is again the dif-\nference in distribution function of state nandm. Worth\nnoting that the diagonal part of [ \u001a(2)\nII;1]nmwithn=m\nvanishes due toFnn= 0, while [ \u001a(2)\nII;2]nmexists for both\ndiagonal (n=m) and o\u000b-diagonal ( n6=m) parts. The\nfollowing subsections include the resultant photocurrent\nresponses given by second-order quantum kinetic formal-\nism.\nA. Shift current and Gyration Current\nThe o\u000b-diagonal part of the velocity (second term\nin Eq. (12)) together with \u001a(2)\nII;1and\u001a(2)\nII;2, lead to\nthe LP shift current, \fL;Sh\nl;ij, which originates from the\ninversion-assymmetric transition of electron position and\nis de\fnded when \u001c!1 [69, 74], and CP gyration cur-\nrent response , \fC;Gyr\nlr, satisfying the following expression\nJ(2;II)\nl=\njE0j2X\ni;jZd\n2\u0019~f\fL;Sh\nl;ij(\n)eie\u0003\nj+i\fC;Gyr\nlr(\n)(e\u0002e\u0003)rg:\n(26)8\n05 01 00150-200200\n5 01 00150-60060-10 1 -101\n-\n404\n-\n10 1 -101-\n0.0300.03\n-\n10 1 -101-\n404\n-\n10 1 -101\n-\n0.0500.05\n \n \n \n \nFIG. 5. The momentum-space distribution of quantum ge-\nometries for xxyandxxxcomponents of (a),(c): \u0000 vc(metric\nconnection) and (b),(d) ~\u0000xxy\nvc(symplectic connection), near the\nWeyl point with \u001f= +1. The subscript vandcdenote va-\nlence and conduction bands, respectively. (e): The frequency\ndependence of circular gyration response and (f): the linear\nshift current along the x- andy- direction where resonance\npeaks stem from node \u001f= +1 or\u001f=\u00001. (x(y);0): dark\nblue and red, while ( x(y);t): light blue and orange, denote\nthexorycomponents of photo-response before and during\nthe magnetic dynamics.\nWe set\u001c!1 ,T= 10 K (kBT=0.86 meV),\n~vF= 3:25\u0002103meV\u0017A,JS= 500 meV, u+= 50 meV,\nu\u0000= 20 meV.\nwhere\n\fL;Sh\nl;ij(\n) =\n\u0000e3\n~Tr[\u0000lij\nnmP1\n~\n\u0000\u000fnmFnm+\u0019~\u0000lij\nnm\u000e(~\n\u0000\u000fnm)Fnm];\n(27)\nand\n\fC;Gyr\nlr(\n) =\n\u0000\u000fijre3\n2~Tr[~\u0000lij\nnmP1\n~\n\u0000\u000fnmFnm\u0000\u0019\u0000lij\nnm\u000e(~\n\u0000\u000fnm)Fnm]:\n(28)Here, the third-rank metric connection \u0000jli\nnm, and sym-\nplectic connection ~\u0000jli\nnmcontrol the shift and gyration\ncurrents near the gap closing point, which are de\fned\nas\n\u0000lij\nnm= Re[[DlRi\nk]nmRj\nk;mn];\n~\u0000lij\nnm= Im[[DlRi\nk]nmRj\nk;mn]:(29)\nThe integration over 3D k-space can be decomposed\ninto an integration over energy and an integration\nover 2D surface with \fxed energy \u000f, i.e.,Rd3k\n(2\u0019)3=\nR1\n0d\u000f\n2\u0019~R\n\u000fd2\u001b\n(2\u0019)21\njvkj, wherevkis the group velocity.\nThe signi\fcant contribution arises when the 2D surface\nin thek-space surrounds a Weyl node. Worth noting\nthat the \frst term in Eq. 28 could manifest a Fermi-\nsurface e\u000bect after using the band-resolved Berry curva-\nture and conducting a partial derivative,P\nm~\u0000lij\nnmFnm=\niP\nm@lBij\nnmFnm=iBl\nn(@lFnm), where the Berry curva-\nture for the nthband is de\fned as Bl\nn=P\nn0\u000flij\n2Bij\nnn0=\ni\n2P\nn0\u000flij[Ri\nnn0Rj\nn0n\u0000Rj\nnn0Ri\nn0n]:\nFigure 5 demonstrates the di\u000berent behavior of xxx\nandxxycomponents of the metric and symplectic con-\nnection near the Weyl nodes. Such nonlinear topological\nphotoresponses arise from the divergence in the quan-\ntum geometries near the Weyl points. Therefore, only\noptical excitations with frequency windows facing the\nWeyl crossing points make a maximum contribution to\nthe nonlinear photoresponse. Since nodes with opposite\nchirality have di\u000berent energies in inversion-asymmetric\nWeyl semimetals, i.e. u+andu\u0000, for light frequencies\n2 min(u+;u\u0000)6\n<2 max(u+;u\u0000) andu+6=\u0000u\u0000\nonly one Weyl node contributes to the interband excita-\ntions. The other node plays a role for \n >2 max(u+;u\u0000).\nTherefore, the optical excitations with di\u000berent energies\nare activated at di\u000berent frequencies when the chemical\npotential is tuned so that it is not equidistant from two\nnodes, ie 2\u00166=u++u\u0000. Figures 5 (e) and (f) demon-\nstrate the linear shift and circular gyration currents in\nthe absence and presence of magnetic dynamics. The\ndynamics of magnetic textures leading to a remarkable\nchange in magnitude and sign of both the xandycom-\nponents of photocurrents obtained from quantum kinetic\ntheory, corresponding to an in-plane reaction orientation\nchange ( according to Fig. 2(b)).\nB. Injection Current\nThe diagonal part of the velocity vk;nn, together with\n\u001a(2)\nII;2, lead to the injection current in terms of the LP and\nCP photocurrent\nJ(2;Inj)\nl=jE0j2X\ni;jZd\n2\u0019f\fL;Inj\nl;ijeie\u0003\nj+i\fC;Inj\nlr(e\u0002e\u0003)rg;\n(30)9\n408 01 20010\n5 01 001 50-202-10 1 -1010\n510\n-\n10 1 -101\n0\n35\n \n \n \n \n \n \nFIG. 6. The momentum-space distribution of quantum ge-\nometries (a):Bz\nc(Berry curvature) and (b): Gxx\nvc(quantum\nmetric), near the Weyl point with \u001f= +1. The subscript v\nandcdenote valence and conduction bands, respectively. (c):\nThe quantized circular and (d): the linear injection responses\nas a function of frequency in the absence and presence of mag-\nnetic dynamics (non-zero E5). In (c), the circular injection\ncurrent is a quantized response and vary with the relaxation\ntime. The magnetization dynamics lead to an increase in uni-\nversal magnitude, which is denoted by a green solid line. In\n(d), the magnitude of x- andy- components change, resulting\nin the rotation of photocurrent. We set \u001c= 1 ps ( ~=\u001c= 0:65\nmeV), T=10 K ( kBT=0.86 meV), ~vF= 3:25\u0002103meV\u0017A,\nJS= 500 meV, u+= 50 meV, u\u0000= 20 meV.\nwhere\n\fL;Inj\nl;ij=\u0000\u0019\u001ce3\n~Tr[X\nn0\u0001l\nnn0Gij\nn0nFn0n\u000e(~\n\u0000\u000fnn0)];\n\fC;Inj\nlr=\u000fijr\u0019\u001ce3\n2~Tr[X\nn0\u0001l\nnn0Bij\nn0nFn0n\u000e(~\n\u0000\u000fnn0)];\n(31)\nwhereGij\nn0nis called the quantum metric and Bij\nn0nis the\nBerry curvature which are de\fned as\nGij\nn0n=1\n2[Ri\nnn0Rj\nn0n+Rj\nnn0Ri\nn0n];\nBij\nn0n=i[Ri\nnn0Rj\nn0n\u0000Rj\nnn0Ri\nn0n];(32)\n, respectively. The injection current clearly depends on\nthe velocity di\u000berence along the current response between\ntwo bands, topology of bands as well as the relaxation\ntime [78]. Similar to the previous linear shift and circu-\nlar gyration currents, the geometric singularities near the\nWeyl closing points signi\fcantly contribute to the light-\ninduced injection current. Figures 6(a) and (b) represent\nthe quantum metric Gxx\nvcand Berry curvature Bc\nznear a\nFIG. 7. The nonlinear optical response of the linear in-\njection, shift current responses (solid lines), circular gyration\nand injection currents (dashed line) as a function of light fre-\nquency \n. The linear shift and circular gyration currents\n(with\u001c!1 ) are multiplied by a factor of 20 due to the\ngraphical purpose. The current J=P\nl=x;y;zJlis the to-\ntal current where the \frst and second peaks arise from node\n\u001f= +1 and\u001f=\u00001, respectively. I0is the light intensity in\nunits of W/m2where we setjEj2= 2I2\n0=\u000f0cand\u000f0c=e2\n4\u0019\u000b~\nwith\u000b= 1=137. Other parameters are the same as Fig. 6.\nWeyl node with \u001f= +1. The circular injection current\n\fC;Inj\nlris a quantized and constant response that depends\non the fundamental and universal quantities like topolog-\nical charge of Weyl nodes \u001f, electric charge eand Planck\nconstanth:P\nl=x;y;z\fC,Inj\nlz=\u0000\u0019\u001ce3\nh2\u001f=\u0000\f0\u001f(A/V2)\n[79][Fig. 6(c)]. The magnetization dynamics can change\nthe magnitude of \fxzand\fyz, resulting in enhancement\nof universal magnitude for circular injection response for\nmagnetic Weyl semimetals [Green dashed line in Fig.\n6(c)].\nFig. 7 collects the results of the nonlinear optical re-\nsponses of both linearly and circularly polarized lights.\nAccordingly, the injection currents are much stronger\nthan the shift and gyration currents and then can be\nthe dominant optical response in the system. The pho-\ntovoltaic current exhibits a strong resonance in order of\nmA/W in the vicinity of the Weyl nodes, with a magni-\ntude controlled by the momentum relaxation time. Since\nthe Weyl nodes can be arranged by magnetic texture\ndirection, the nonlinear optical response could be used\nas the basis for a terahertz photo-detector. With some\nstraightforward algebra, we can show that the magnetic\ndynamics may not be able to notably modify the mag-\nnitude of total photocurrent jJj=pP\nljJlj2in Fig. 7,\nalthough it results in a change in both direction and mag-\nnitude of the in-plane photo-response [Eq. 6].\nFinally, we note that the Pauli blocking is symmetric\nin an untilted Weyl cone, which means that the \n = 2 u\u001f10\nfrequency window is symmetric. For tilted nodes, the\nPauli blocking becomes asymmetric. In other words, the\nfrequency window2u\u001f\n1+vt=vf<\n<2u\u001f\n\u0006(1\u0000vt=vf)becomes\nwider and more asymmetric for transitions between the\nbands. The character \u0006designates the inclined type I\nor type II. Furthermore, in a highly asymmetric type\nII Weyl semimetal a partial compensation between two\nnodes may occur when the tilt parameter satisfy the con-\nditionvt\nvf>1+u1=u2\n1\u0000u1=u2leading to the partially activation\nof two nodes simultaneously. The in\ruence of tilting\non the nonlinear photoresponse is extensively studied in\n[14, 74, 80].\nVIII. CONCLUSION\nThis work has investigated the signi\fcant correlation\nof nonlinear DC photocurrent with magnetic texture in\na parity-violating magnetic Weyl semimetal. The tun-\nable chemical potential and IS breaking lead to a chiral\nphotocurrent generated by interband transitions of Weyl\nfermions belonging to a node with chirality \u001fwhile the\nother node has not yet been activated due to the Pauli\nblocking. We have shown that this chiral photocurrentinduces STT that causes magnetic texture dynamics, re-\nsulting in magnetic texture rotation from the initial c-\naxis to the \fnal a- andb-axes. The momentum space\npositions of the Weyl nodes kware a\u000bected by the mag-\nnitude and direction of the magnetization, so any magne-\ntization dynamics can move the Weyl nodes in momen-\ntum space, giving them a time dependence of the form\nkw=M(t). Accordingly, the dynamic magnetic mo-\nments can be mapped to an axial electric \feld E5. In\nthe dynamic regime of magnetic texture, the presence of\nan axial electric \feld induces an additional in-plane cur-\nrent arising from the interplay between non-trivial band\ntopology in momentum space and magnetic texture in\nreal space. Our theory predicts that the in-plane orienta-\ntion of photocurrents in parity-violating magnetic Weyl\nsemimetals is strongly correlated with the direction of\nmagnetic texture moments.\nIX. ACKNOWLEDGMENTS\nR. A. acknowledges support from the Australian\nResearch Council Centre of Excellence in Future\nLow-Energy Electronics Technologies (project number\nCE170100039).\nAppendix A: Model Hamiltonian and Topological Characteristics\nThe low-energy Hamiltonian describing a 3D parity-violating magnetic Weyl semimetal with minimum model with\nw1and w 4nodes is given by\nH0=~vF\u001cz\n\u001b\u0001(k\u0000kw\n?)\u0000~vF\u001c0\n(\u001bzkw\nz\u0000kM\u001b\u0001^M(r;t)) +u \u001cz\n\u001b0+\u0015 \u001cy\n\u001bz; (A1)\nwherevFis the Fermi velocity without tilt, kw= (kw\n?;kw\nz) is the node coordinate in the k-space, the wave-vector\nkM=JS=~vFdescribes the shift of nodes due to the coupling between electrons and magnetic texture through the\nexchange interaction J, and the 2\u00022 Pauli matrices \u001band\u001crepresents the spin and orbital degree of freedom,\nrespectively. The last two terms are added to the Hamiltonian to violate inversion symmetry (IS), but respect all\nother symmetries. The term \u0015\u001cy\n\u001bzis the momentum-independent spin-orbit interaction that split the degeneracy\nat every points except the Weyl crossings. This term is closely analogous to the Dresselhous spin-orbit interaction\nterm allowed in the absence of IS. The role of u\u001cz\n\u001b0is shifting two tips of Weyl cones in energy and breaks IS.\nThe inversion operator changes the sign of the momentum and orbital degree of freedom, i.e., H(k)!\u001cxH(k)\u001cxand\n\u001cx\u001cy(z)\u001cx=\u0000\u001cy;(z). Using the canonical transformation, i.e., \u001bx;y!\u001cz\u001bx;y, and ^Mk^z, the above Hamiltonian is\nwritten as\nH0=~vF(k?\u0000kw\n?)\u001c0\n\u001b?+~vF(kz\u001cz\u0000(kw\nz\u0000kM)\u001c0)\n\u001bz+u \u001cz\n\u001b0+\u0015 \u001cy\n\u001bz; (A2)\nIn this representation the above Hamiltonian can be presented as a block diagonal Hamiltonian given by H\u001f\n0(k) =\n~vf(k?\u0000kw\n?)\u0001\u001b?+~vf(\u001fkz\u0000(kw\nz\u0000km))\u001bz+\u001fu\u001f\u001b0=fx\u001bx+fy\u001by+f\u001f;z\u001bz+\u001fu\u001f\u001b0, whereu\u001fdetermines the shift of\nnodes in energy due to the IS breaking. Then, the corresponding eigenvalues are given by \u000f\u001f\ntk=\u001fu\u001f+tq\nf2x+f2y+f2\u001f;z,\nwherefx=~vF(kx\u0000kw\nx),fy=~vF(ky\u0000kw\ny),f\u001f;z=~vF(kz\u0000\u001f(kw\nz\u0000kM)), andt=\u0006denotes the conduction and\nvalence bands, respectively, and \u001f=\u0006represents the chirality. The corresponding eigenstates of Eq. (A2) would\nbejn\u001f\ntik=1p\n20\n@p\n1 +f\u001f\nz=\u000f\u001f\ntk\ntei'kp\n1\u0000f\u001f\nz=\u000f\u001f\ntk1\nA;whereei'k=ei'\u0000kw\n?\nk?ei'win which'and'ware the polar angles of vectors\nkandkwin thekx\u0000kyplane, respectively. The Berry connection or Berry vector potential in the eigenstates11\n-20 2 -404-\n20 2 -404\nFIG. 8. The band dispersion of model Hamiltonian in Eq. (A1) along the k?=kw\n?line as a function of momentum kzalong\nthe Weyl nodes (a) without tilt vt=vf= 0 and (b): with tilted cones vt=vf= 1.\nrepresentation is given by [ Rk;a]nn0=ihnj@kan0i, wherea=x;y;z andn;n0=\u0006denotes the conduction and valence\nbands, respectively. The individual components of the Berry connection are given by\n[Rk;x]nn0= ~\u001b01\n2k?(sin'\u0000kw\n?\nk?sin'w)\u0000~\u001bz1\n2k?f\u001f\nz\n\u000f\u001f\nk(sin'\u0000kw\n?\nk?sin'w)\u0000\n~\u001by~vFf\u001f\nz\n2(\u000f\u001f\nk)2(cos'\u0000kw\n?\nk?cos'w)\u0000~\u001bx~kF\n2\u000f\u001f\nk(sin'\u0000kw\n?\nk?sin'w);\n[Rk;y]nn0=\u0000~\u001b01\n2k?(cos'\u0000kw\n?\nk?cos'w) + ~\u001bz1\n2k?f\u001f\nz\n\u000f\u001f\nk(cos'\u0000kw\n?\nk?cos'w)\u0000\n~\u001by~vFf\u001f\nz\n2(\u000f\u001f\nk)2(sin'\u0000kw\n?\nk?sin'w) + ~\u001bx~vF\n2\u000f\u001f\nk(cos'\u0000kw\n?\nk?cos'w);\n[Rk;z]nn0= ~\u001by~vFk?\n2(\u000f\u001f\nk)2@f\u001f\nz(kz)\n@kz;(A3)\nwhere ~\u001baare the Pauli matrices in the eigenstates basis to represent the matrix elements of [ Rk;a]nn0, i.e., [Rk;a]nn0=\u0012\nh+jRk;aj+i h+jRk;aj\u0000i\nh\u0000jRk;aj+i h\u0000jRk;aj\u0000i\u0013\n. These components of Berry vectors clearly depend on the speci\fc con\fguration of Weyl\nnodes in the Brillouin zone, which is determined by the magnetic-texture properties and lattice structure. The\nBerry vectors are the building blocks of the topological and geometrical features of the band structure which play the\nprominent and vital role in photocurrent response. The quantum geometrical quantities also depend on the topological\ncharge or chirality of each node \u001f.\nAppendix B: Light-induced magnetic texture dynamics\nIn the absence of a magnetic \feld, the magnetic dynamics based on Landau-Lifshitz-Gilbert (LLG) equation [58]\nwould be\nd^M\ndt=Te+\u000b^M\u0002d^M\ndt=Te+\u000b^M\u0002(Te+\u000b^M\u0002d^M\ndt) =\nTe+\u000b^M\u0002Te+\u000b2^M\u0002(^M\u0002Te) +:::= exp(\u000b^\u0002)Te(B1)\nwhere ^\u0002O=^M\u0002O and the spin-transfer torque is given by Te=jkMj\ne\u001as^M\u0002J5. For spin-transfer torque to be non-\nvanishing, the chiral current J5must be generated in the system. This chiral current can be induced in an inversion\nasymmetric WSM, since only one Weyl node will be activated due to the Pauli blocking in another node.\nThe magnetic dynamics can also be understood in terms of an axial electric \feld E5=\u0000~\nejkMj_^M=\n\u0000~\nejkMjexp(\u000b^\u0002)Te. Therefore, the axial electric \feld can be written as\nE5=\u0000~jkMj2\ne2\u001asexp(\u000b^\u0002)(^M\u0002J5); (B2)\nThe light-induced time evolution of magnetic texture up to the \frst order of \u000bis given by\nd^M\ndt=Te+\u000b^M\u0002Te=jkMj\ne\u001asf^M\u0002J5+\u000b^M\u0002(^M\u0002J5)g (B3)12\nleading to the following equations\nd^Mx\ndt=\u0000jkMj\ne\u001as(J0\n5;ksin'j\n0+\u000bJ0\n5;kcos'j\n0);\nd^My\ndt=jkMj\ne\u001as(J0\n5;kcos'j\n0\u0000\u000bJ0\n5;ksin'j\n0);\nd^Mz\ndt= 0:(B4)\nHere,'j\n0is the orientation of in-plane photo-current when ^M= ^z. The time evolution of magnetic moments is\nillustrated by Fig. 2(a).\nAppendix C: Inter-band photo-currents\n1. Shift and Gyration Responses\nThe o\u000b-diagonal part of the velocity (second term in Eq. (12)) together with \u001a(2)\nII;1and\u001a(2)\nII;2, leads to the conductivity\ntensor as following\n\u001b(2;II)\nl;ij;O=e3\n~Tr[d0\nmn\u000fnmRl\nk;nmDi\nmn(d\u0000\nmnRj\nk;mnFnm)]\n+ie3\n~Tr[d0\nmn\u000fnmRl\nk;nmX\nn06=m[fij\nn0]mn];(C1)\nwhere [fij\nn0]mn=Ri\nk;n0mRj\nk;n0nd\u0000\nn0nFn0n\u0000Rj\nk;mn0Ri\nk;n0nd\u0000\nmn0Fmn0, and Tr =P\nnR\n[dk] indicates both a matrix trace\nand an integration of momentum kover the Brillouin zone, and the summation over m(6=n) is implicit. The term\niRl\nk;nmP\nn0[fij\nn0]mnin Eq. (C1), can be written in a more compact form by using the dummy nature of indices,\niRl\nk;nmX\nn06=n6=m[fij\nn0]mn=iX\nn06=n6=mRl\nk;nmRi\nk;n0mRj\nk;n0nd\u0000\nn0nFn0n\u0000iX\nn0Rl\nk;nmRj\nk;mn0Ri\nk;n0nd\u0000\nmn0Fmn0\n=iX\nn06=n6=m(Rl\nk;nmRi\nk;n0m\u0000Ri\nk;nmRl\nk;mn0)Rj\nk;n0nd\u0000\nn0nFn0n:(C2)\nUsing the sum-rule [81]\nX\nmi(Rl\nk;nmRi\nk;n0m\u0000Ri\nk;nmRl\nk;mn0) = [DlRi\nk]nn0\u0000[DiRl\nk]nn0; (C3)\nthe second term of Eq. (C1) is recast as\nie3\n~Tr[d0\nmn\u000fnmRl\nk;nmX\nn06=m[fij\nn0]mn] =e3\n~Tr[d0\nmn\u000fnm([DlRi\nk]nm\u0000[DiRl\nk]nm)Rj\nk;mnd\u0000\nmnFmn]: (C4)\nConducting the partial derivative in the \frst term of Eq. (C1) and summing with Eq. (C4), we \fnd the following\nexpression for \u001b(2;II)\nl;ij;O\n\u001b(2;II)\nl;ij;O=e3\n2~Tr[d0\nmn\u000fnm[DlRi\nk]nmRj\nk;mnd\u0000\nmnFnm] + [(i;\u0000\n)$(j;\n)]: (C5)\nWe introduce the symmetric and anti-symmetric quantities as\nSl;ij\nnm= [DlRi\nk]nmRj\nk;mn+ [DlRj\nk]mnRi\nk;nm;\nAl;ij\nnm= [DlRi\nk]nmRj\nk;mn\u0000[DlRj\nk]mnRi\nk;nm:(C6)13\nWe also de\fne the third-rank geometric tensors such as the metric connection \u0000jli\nnm, and symplectic connection ~\u0000jli\nnm,\nwhich are de\fned as\n\u0000lij\nnm= Re[[DlRi\nk]nmRj\nk;mn];\n~\u0000lij\nnm= Im[[DlRi\nk]nmRj\nk;mn]:(C7)\nwhereDl\nnm=@kl\u0000i(Rl\nnn\u0000Rl\nmm). Then we can write Sl;ij\nnmandAl;ij\nnmin terms of the geometric quantities\nSl;ij\nnm= \u0000lij\nnm+i~\u0000lij\nnm+ (i$j);\nAl;ij\nnm= \u0000lij\nnm+i~\u0000lij\nnm\u0000(i$j):(C8)\nFor the inter-band transitions to dominate, the frequency of light must be larger than the energy di\u000berence between\nthe statesnandm, i.e., \n\u0015\u000fnm\u001d1=\u001c, therefore, the term \u000fnmd0\nmnd\u0000\nmncan be decomposed into the real and\nimaginary parts as\n\u000fnmd0\nmnd\u0000\nmn=\u0000fP1\n\n\u0000\u000fnm\u0000i\u0019\u000e(\n\u0000\u000fnm)g; (C9)\nwhere P stands for the principal value in k-integration. Then, the conductivity expression in Eq. (C5) is simpli\fed\nas\n\u001b(2;II)\nl;ij;O=\u0000e3\n2~Tr[Sl;ij\nnmP1\n\n\u0000\u000fnmFnm\u0000i\u0019Al;ij\nnm\u000e(~\n\u0000\u000fnm)Fnm]: (C10)\nThe above expression can be obtained in terms of LP and CP photocurrent response by using the general formula in\nEq. (9)\nJ(2;II)\nl=jE0j2X\ni;jZd\n2\u0019~f\fL;Sh\nl;ij(\n)eie\u0003\nj+i\fC;Gyr\nlr(\n)(e\u0002e\u0003)rg: (C11)\nwhere\n\fL;Sh\nl;ij(\n) =\u0000e3\n~Tr[\u0000lij\nnmP1\n\n\u0000\u000fnmFnm+\u0019~\u0000lij\nnm\u000e(~\n\u0000\u000fnm)Fnm];\n\fC;Gyr\nlr(\n) =\u0000\u000fijre3\n2~Tr[~\u0000lij\nnmP1\n\n\u0000\u000fnmFnm\u0000\u0019\u0000lij\nnm\u000e(~\n\u0000\u000fnm)Fnm]:(C12)\nThe LP photocurrent, \fL;Sh\nl;ij, is classi\fed into the Shift current [69], while the CP photo-response, \fC;Gyr\nlr, is classi\fed\ninto the Gyration current .\n2. Injection currents\nOn the other hand, the diagonal part of the velocity vk;nn together with \u001a(2)\nII;2leads to the following conductivity\ntensor\n\u001b(2;II)\nl;ij;D=e3\n2~Tr[d0\nmnvl\nk;nm\u000enmX\nn06=m[fij\nn0]mn] =\u0000i\u001ce3\n2~Tr[vl\nk;nnX\nn0\n(n06=n)[fij\nn0]nn]\n=\u0000i\u001ce3\n2~Tr[X\nn0\u0001l\nnn0Ri\nnn0Rj\nn0nd\u0000\nn0nFn0n] + [(i;\u0000\n)$(j;\n)]\n=\u0000i\u001ce3\n2~Tr[X\nn0\u0001l\nnn0Ri\nnn0Rj\nn0n(d\u0000\nn0n+d\nnn0)Fn0n](C13)\nwhere we have used [ fij\nn0]nn=Ri\nk;n0nRj\nk;n0nd\u0000\nn0nFn0n\u0000Rj\nk;nn0Ri\nk;n0nd\u0000\nnn0Fnn0, and \u0001l\nnn0=vl\nn\u0000vl\nn0is the group velocity\ndi\u000berence between the band nandn0, Tr =P\nnR\n[dk].14\nThe real and imaginary parts of the quantity Ri\nnn0Rj\nn0ncan be de\fned in terms of symmetric and antisymmetric\nquantum geometric quantities as [57]\nRi\nnn0Rj\nn0n=Gij\nn0n\u0000iBij\nn0n=2; (C14)\nwhereGij\nn0nis called the quantum metric andBij\nn0nis the Berry curvature which are de\fned, respectively\nGij\nn0n=1\n2[Ri\nnn0Rj\nn0n+Rj\nnn0Ri\nn0n];\nBij\nn0n=i[Ri\nnn0Rj\nn0n\u0000Rj\nnn0Ri\nn0n]:(C15)\nThe quantityBij\nn0nis related to the Berry curvature for the nthband as\nBl\nn=X\nn0\u000flij\n2Bij\nnn0=i\n2X\nn0\u000flij[Ri\nnn0Rj\nn0n\u0000Rj\nnn0Ri\nn0n]: (C16)\nUsing the fact that d\u0000\nn0n+d\nnn0=\u00002i\u000e(~\n\u0000\u000fnn0), The conductivity \u001b(2;II)\nl;ij;Dis written as\n\u001b(2;II)\nl;ij;D=\u0000\u0019\u001ce3\n~Tr[X\nn0\u0001l\nnn0Fn0n(Gij\nn0n\u0000i\n2Bij\nn0n)\u000e(~\n\u0000\u000fnn0)] (C17)\nThe term contains Gij\nn0n(Bij\nn0n) satis\fes the symmetric (anti-symmetric) condition under permutation i$j, then it\nis classi\fed into LP (CP) current, known as the injection current arising from the longitudinal velocity injection.\nThen, the injection current response in terms of general expression for LP and CP photocurrent would be\nJ(2;Inj)\nl=jE0j2X\ni;jZd\n2\u0019f\fL;Inj\nl;ijeie\u0003\nj+i\fC;Inj\nlr(e\u0002e\u0003)rg; (C18)\nwhere\n\fL;Inj\nl;ij=\u0000\u0019\u001ce3\n~Tr[X\nn0\u0001l\nnn0Gij\nn0nFn0n\u000e(~\n\u0000\u000fnn0)];\n\fC;Inj\nlr=\u000fijr\u0019\u001ce3\n2~Tr[X\nn0\u0001l\nnn0Bij\nn0nFn0n\u000e(~\n\u0000\u000fnn0)]:(C19)\n[1] Y. Tokura, K. Yasuda, and A. Tsukazaki, Nature Re-\nviews Physics 1, 126 (2019), URL https://doi.org/10.\n1038%2Fs42254-018-0011-5 .\n[2] P. Wang, J. Ge, J. Li, Y. Liu, Y. Xu, and J. Wang, The\nInnovation 2, 100098 (2021), URL https://doi.org/10.\n1016%2Fj.xinn.2021.100098 .\n[3] Y. Fan and K. L. Wang, SPIN 06, 1640001 (2016), URL\nhttps://doi.org/10.1142%2Fs2010324716400014 .\n[4] D. F. Liu, A. J. Liang, E. K. Liu, Q. N. Xu, Y. W. Li,\nC. Chen, D. Pei, W. J. Shi, S. K. Mo, P. Dudin, et al.,\nScience 365, 1282 (2019), URL https://doi.org/10.\n1126%2Fscience.aav2873 .\n[5] N. Morali, R. Batabyal, P. K. Nag, E. Liu, Q. Xu, Y. Sun,\nB. Yan, C. Felser, N. Avraham, and H. Beidenkopf, Sci-\nence365, 1286 (2019), URL https://doi.org/10.1126%\n2Fscience.aav2334 .\n[6] D. S. Sanchez, G. Chang, I. Belopolski, H. Lu, J.-X. Yin,\nN. Alidoust, X. Xu, T. A. Cochran, X. Zhang, Y. Bian,\net al., Nature Communications 11(2020), URL https://doi.org/10.1038%2Fs41467-020-16879-1 .\n[7] S. Howlader, R. Ramachandran, Y. Singh, and G. Sheet,\nJournal of Physics: Condensed Matter 33, 075801\n(2020), URL https://doi.org/10.1088%2F1361-648x%\n2Fabc4d1 .\n[8] C. Lee, P. Vir, K. Manna, C. Shekhar, J. E. Moore, M. A.\nKastner, C. Felser, and J. Orenstein, Nature Commu-\nnications 13(2022), URL https://doi.org/10.1038%\n2Fs41467-022-30460-y .\n[9] D. Destraz, L. Das, S. S. Tsirkin, Y. Xu, T. Neupert,\nJ. Chang, A. Schilling, A. G. Grushin, J. Kohlbrecher,\nL. Keller, et al., npj Quantum Materials 5(2020), URL\nhttps://doi.org/10.1038%2Fs41535-019-0207-7 .\n[10] K. Manna, Y. Sun, L. Muechler, J. K ubler, and C. Felser,\nNature Reviews Materials 3, 244 (2018), URL https:\n//doi.org/10.1038%2Fs41578-018-0036-5 .\n[11] B. Meng, H. Wu, Y. Qiu, C. Wang, Y. Liu, Z. Xia,\nS. Yuan, H. Chang, and Z. Tian, APL Materials 7,\n051110 (2019), URL https://doi.org/10.1063%2F1.15\n5090795 .\n[12] J. K ubler and C. Felser, EPL (Europhysics Letters)\n108, 67001 (2014), URL https://doi.org/10.1209%\n2F0295-5075%2F108%2F67001 .\n[13] E. Liu, Y. Sun, N. Kumar, L. Muechler, A. Sun, L. Jiao,\nS.-Y. Yang, D. Liu, A. Liang, Q. Xu, et al., Nature\nPhysics 14, 1125 (2018), URL https://doi.org/10.\n1038%2Fs41567-018-0234-5 .\n[14] J. Ahn, G.-Y. Guo, and N. Nagaosa, Phys. Rev. X\n10, 041041 (2020), URL https://link.aps.org/doi/\n10.1103/PhysRevX.10.041041 .\n[15] N. Nagaosa, T. Morimoto, and Y. Tokura, Nature Re-\nviews Materials 5, 621 (2020).\n[16] H. Watanabe and Y. Yanase, Phys. Rev. X 11,\n011001 (2021), URL https://link.aps.org/doi/10.\n1103/PhysRevX.11.011001 .\n[17] K. Taguchi, T. Imaeda, M. Sato, and Y. Tanaka, Phys.\nRev. B 93, 201202 (2016), URL https://link.aps.org/\ndoi/10.1103/PhysRevB.93.201202 .\n[18] L. Berger, Phys. Rev. B 54, 9353 (1996), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.54.9353 .\n[19] M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang,\nM. Seck, V. Tsoi, and P. Wyder, Phys. Rev. Lett.\n80, 4281 (1998), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.80.4281 .\n[20] W. H. Rippard, M. R. Pufall, S. Kaka, S. E.\nRussek, and T. J. Silva, Phys. Rev. Lett. 92,\n027201 (2004), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.92.027201 .\n[21] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Emley,\nR. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph, Na-\nture425, 380 (2003), URL https://doi.org/10.1038%\n2Fnature01967 .\n[22] I. N. Krivorotov, N. C. Emley, J. C. Sankey, S. I.\nKiselev, D. C. Ralph, and R. A. Buhrman, Sci-\nence307, 228 (2005), URL https://doi.org/10.1126%\n2Fscience.1105722 .\n[23] I. Garate and M. Franz, Phys. Rev. Lett. 104,\n146802 (2010), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.104.146802 .\n[24] T. Yokoyama, Y. Tanaka, and N. Nagaosa, Phys. Rev.\nB81, 121401 (2010), URL https://link.aps.org/doi/\n10.1103/PhysRevB.81.121401 .\n[25] F. S. Nogueira and I. Eremin, Phys. Rev. Lett. 109,\n237203 (2012), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.109.237203 .\n[26] K. Nomura and N. Nagaosa, Phys. Rev. B 82,\n161401 (2010), URL https://link.aps.org/doi/10.\n1103/PhysRevB.82.161401 .\n[27] Y. Tserkovnyak and D. Loss, Phys. Rev. Lett. 108,\n187201 (2012), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.108.187201 .\n[28] Y. Ferreiros and A. Cortijo, Phys. Rev. B 89,\n024413 (2014), URL https://link.aps.org/doi/10.\n1103/PhysRevB.89.024413 .\n[29] J. Linder, Phys. Rev. B 90, 041412 (2014), URL https:\n//link.aps.org/doi/10.1103/PhysRevB.90.041412 .\n[30] Y. Sun, C. Lee, H.-Y. Yang, D. H. Torchinsky,\nF. Tafti, and J. Orenstein, Phys. Rev. B 104,\n235119 (2021), URL https://link.aps.org/doi/10.\n1103/PhysRevB.104.235119 .\n[31] T. Suzuki, L. Savary, J.-P. Liu, J. W. Lynn, L. Balents,\nand J. G. Checkelsky, Science 365, 377 (2019), URL\nhttps://doi.org/10.1126%2Fscience.aat0348 .[32] S. Heidari, R. Asgari, and D. Culcer, arXiv preprint\narXiv:2210.01150 (2022), URL https://arxiv.org/abs/\n2210.01150 .\n[33] G. Chang, B. Singh, S.-Y. Xu, G. Bian, S.-\nM. Huang, C.-H. Hsu, I. Belopolski, N. Alidoust,\nD. S. Sanchez, H. Zheng, et al., Phys. Rev. B 97,\n041104 (2018), URL https://link.aps.org/doi/10.\n1103/PhysRevB.97.041104 .\n[34] S. Bobev, P. H. Tobash, V. Fritsch, J. D. Thompson,\nM. F. Hundley, J. L. Sarrao, and Z. Fisk, Journal of\nSolid State Chemistry 178, 2091 (2005), ISSN 0022-\n4596, URL https://www.sciencedirect.com/science/\narticle/pii/S002245960500174X .\n[35] H. Flandorfer, D. Kaczorowski, J. Gr obner, P. Rogl,\nR. Wouters, C. Godart, and A. Kostikas, Journal of\nSolid State Chemistry 137, 191 (1998), ISSN 0022-\n4596, URL https://www.sciencedirect.com/science/\narticle/pii/S002245969797660X .\n[36] E. Gladyshevskii, N. Nakonechna, K. Cenzual, R. Glady-\nshevskii, and J.-L. Jorda, Journal of Alloys and\nCompounds 296, 265 (2000), ISSN 0925-8388, URL\nhttps://www.sciencedirect.com/science/article/\npii/S0925838899005393 .\n[37] S. Dhar and S. Pattalwar, Journal of Magnetism\nand Magnetic Materials 152, 22 (1996), ISSN 0304-\n8853, URL https://www.sciencedirect.com/science/\narticle/pii/0304885395004386 .\n[38] P. Puphal, V. Pomjakushin, N. Kanazawa, V. Uk-\nleev, D. J. Gawryluk, J. Ma, M. Naamneh, N. C.\nPlumb, L. Keller, R. Cubitt, et al., Physical Review\nLetters 124 (2020), URL https://doi.org/10.1103%\n2Fphysrevlett.124.017202 .\n[39] S.-Y. Xu, N. Alidoust, G. Chang, H. Lu, B. Singh, I. Be-\nlopolski, D. S. Sanchez, X. Zhang, G. Bian, H. Zheng,\net al., Science Advances 3, e1603266 (2017), URL https:\n//doi.org/10.1126%2Fsciadv.1603266 .\n[40] B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao,\nJ. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen,\net al., Phys. Rev. X 5, 031013 (2015), URL https://\nlink.aps.org/doi/10.1103/PhysRevX.5.031013 .\n[41] Y. Sun, S.-C. Wu, and B. Yan, Phys. Rev. B 92,\n115428 (2015), URL https://link.aps.org/doi/10.\n1103/PhysRevB.92.115428 .\n[42] C.-C. Lee, S.-Y. Xu, S.-M. Huang, D. S. Sanchez,\nI. Belopolski, G. Chang, G. Bian, N. Alidoust,\nH. Zheng, M. Neupane, et al., Phys. Rev. B 92,\n235104 (2015), URL https://link.aps.org/doi/10.\n1103/PhysRevB.92.235104 .\n[43] G. Chang, S.-Y. Xu, H. Zheng, C.-C. Lee, S.-M.\nHuang, I. Belopolski, D. S. Sanchez, G. Bian, N. Ali-\ndoust, T.-R. Chang, et al., Phys. Rev. Lett. 116,\n066601 (2016), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.116.066601 .\n[44] I. Belopolski, S.-Y. Xu, Y. Ishida, X. Pan, P. Yu, D. S.\nSanchez, H. Zheng, M. Neupane, N. Alidoust, G. Chang,\net al., Phys. Rev. B 94, 085127 (2016), URL https://\nlink.aps.org/doi/10.1103/PhysRevB.94.085127 .\n[45] B. Feng, Y.-H. Chan, Y. Feng, R.-Y. Liu, M.-Y. Chou,\nK. Kuroda, K. Yaji, A. Harasawa, P. Moras, A. Barinov,\net al., Phys. Rev. B 94, 195134 (2016), URL https://\nlink.aps.org/doi/10.1103/PhysRevB.94.195134 .\n[46] G. Chang, S.-Y. Xu, D. S. Sanchez, S.-M. Huang, C.-\nC. Lee, T.-R. Chang, G. Bian, H. Zheng, I. Belopolski,\nN. Alidoust, et al., Science Advances 2, e1600295 (2016),16\nURL https://doi.org/10.1126%2Fsciadv.1600295 .\n[47] K. Kuroda, T. Tomita, M.-T. Suzuki, C. Bareille,\nA. A. Nugroho, P. Goswami, M. Ochi, M. Ikhlas,\nM. Nakayama, S. Akebi, et al., Nature Materials 16, 1090\n(2017), URL https://doi.org/10.1038%2Fnmat4987 .\n[48] A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel,\nA. C. Komarek, C. Shekhar, N. Kumar, W. Schnelle,\nJ. K ubler, et al., Science Advances 2(2016), URL https:\n//doi.org/10.1126%2Fsciadv.1501870 .\n[49] K. Endo, T. Ohoyama, and R. Kimura, Journal of the\nPhysical Society of Japan 25, 907 (1968), URL https:\n//doi.org/10.1143%2Fjpsj.25.907 .\n[50] A. Regnat, A. Bauer, A. Senyshyn, M. Meven,\nK. Hradil, P. Jorba, K. Nemkovski, B. Pedersen,\nR. Georgii, S. Gottlieb-Sch onmeyer, et al., Physical Re-\nview Materials 2(2018), URL https://doi.org/10.\n1103%2Fphysrevmaterials.2.054413 .\n[51] S. Borisenko, D. Evtushinsky, Q. Gibson, A. Yaresko,\nK. Koepernik, T. Kim, M. Ali, J. van den Brink,\nM. Hoesch, A. Fedorov, et al., Nature Communi-\ncations 10(2019), URL https://doi.org/10.1038%\n2Fs41467-019-11393-5 .\n[52] A. A. Burkov and L. Balents, Phys. Rev. Lett. 107,\n127205 (2011), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.107.127205 .\n[53] G. Xu, H. Weng, Z. Wang, X. Dai, and Z. Fang, Phys.\nRev. Lett. 107, 186806 (2011), URL https://link.aps.\norg/doi/10.1103/PhysRevLett.107.186806 .\n[54] M. Lyu, J. Xiang, Z. Mi, H. Zhao, Z. Wang, E. Liu,\nG. Chen, Z. Ren, G. Li, and P. Sun, Phys. Rev. B\n102, 085143 (2020), URL https://link.aps.org/doi/\n10.1103/PhysRevB.102.085143 .\n[55] T. Ng, Y. Luo, J. Yuan, Y. Wu, H. Yang, and L. Shen,\nPhys. Rev. B 104, 014412 (2021), URL https://link.\naps.org/doi/10.1103/PhysRevB.104.014412 .\n[56] A. P. Sakhya, C.-Y. Huang, G. Dhakal, X.-J. Gao,\nS. Regmi, X. Yao, R. Smith, M. Sprague, B. Singh,\nH. Lin, et al., arXiv preprint arXiv:2203.05440 (2022).\n[57] P. Bhalla, K. Das, D. Culcer, and A. Agarwal, arXiv\npreprint arXiv:2108.04082 (2021).\n[58] L. Landau and E. Lifshitz, Sov. Phys 8, 153 (1935).\n[59] A. Malozemo\u000b and J. C. Slonczewski, Magnetic domain\nwalls in bubble materials: advances in materials and de-\nvice research , vol. 1 (Academic press, 2016).\n[60] Y. Araki, Annalen der Physik 532, 1900287 (2019), URL\nhttps://doi.org/10.1002%2Fandp.201900287 .\n[61] J. Behrends, A. G. Grushin, T. Ojanen, and J. H. Bardar-\nson, Physical Review B 93(2016), URL https://doi.\norg/10.1103%2Fphysrevb.93.075114 .\n[62] J. Hannukainen, A. Cortijo, J. H. Bardarson, and Y. Fer-\nreiros, SciPost Physics 10(2021), URL https://doi.\norg/10.21468%2Fscipostphys.10.5.102 .\n[63] A. Brataas, A. D. Kent, and H. Ohno, Nature Mate-\nrials11, 372 (2012), URL https://doi.org/10.1038%2Fnmat3311 .\n[64] L. Wu, S. Patankar, T. Morimoto, N. L. Nair, E. Thewalt,\nA. Little, J. G. Analytis, J. E. Moore, and J. Orenstein,\nNature Physics 13, 350 (2017).\n[65] D. Rees, K. Manna, B. Lu, T. Morimoto, H. Borrmann,\nC. Felser, J. Moore, D. H. Torchinsky, and J. Orenstein,\nScience advances 6, eaba0509 (2020).\n[66] Z. Ni, K. Wang, Y. Zhang, O. Pozo, B. Xu, X. Han,\nK. Manna, J. Paglione, C. Felser, A. G. Grushin, et al.,\nNature communications 12, 1 (2021).\n[67] B. I. Sturman, V. M. Fridkin, and J. Bradley, The photo-\nvoltaic and photorefractive e\u000bects in noncentrosymmetric\nmaterials (Routledge, 2021).\n[68] R. W. Boyd, Nonlinear optics (Elsevier Science and Tech-\nnology; Academic Press, 2020).\n[69] B. I. Sturman, Physics-Uspekhi 63, 407 (2020), URL\nhttps://doi.org/10.3367%2Fufne.2019.06.038578 .\n[70] J. Orenstein, J. Moore, T. Morimoto, D. Torchinsky,\nJ. Harter, and D. Hsieh, Annual Review of Condensed\nMatter Physics 12, 247 (2021).\n[71] N. Nagaosa, J. Sinova, S. Onoda, A. H. Mac-\nDonald, and N. P. Ong, Rev. Mod. Phys. 82,\n1539 (2010), URL https://link.aps.org/doi/10.\n1103/RevModPhys.82.1539 .\n[72] D. T. Son and N. Yamamoto, Phys. Rev. Lett. 109,\n181602 (2012), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.109.181602 .\n[73] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys.\n82, 1959 (2010), URL https://link.aps.org/doi/10.\n1103/RevModPhys.82.1959 .\n[74] Y. Gao, S. A. Yang, and Q. Niu, Phys. Rev. Lett.\n112, 166601 (2014), URL https://link.aps.org/doi/\n10.1103/PhysRevLett.112.166601 .\n[75] I. Sodemann and L. Fu, Phys. Rev. Lett. 115,\n216806 (2015), URL https://link.aps.org/doi/10.\n1103/PhysRevLett.115.216806 .\n[76] C. Zeng, S. Nandy, and S. Tewari, Phys. Rev. B 103,\n245119 (2021), URL https://link.aps.org/doi/10.\n1103/PhysRevB.103.245119 .\n[77] Y. Zhang, Y. Sun, and B. Yan, Phys. Rev. B 97,\n041101 (2018), URL https://link.aps.org/doi/10.\n1103/PhysRevB.97.041101 .\n[78] F. Mahmoudi and R. Asgari, Phys. Rev. B 105,\n085403 (2022), URL https://link.aps.org/doi/10.\n1103/PhysRevB.105.085403 .\n[79] F. de Juan, A. G. Grushin, T. Morimoto, and J. E.\nMoore, Nature Communications 8(2017), URL https:\n//doi.org/10.1038%2Fncomms15995 .\n[80] Q. Ma, S.-Y. Xu, C.-K. Chan, C.-L. Zhang, G. Chang,\nY. Lin, W. Xie, T. Palacios, H. Lin, S. Jia, et al., Na-\nture Physics 13, 842 (2017), URL https://doi.org/10.\n1038%2Fnphys4146 .\n[81] C. Aversa and J. E. Sipe, Phys. Rev. B 52,\n14636 (1995), URL https://link.aps.org/doi/10.\n1103/PhysRevB.52.14636 ."    },    {        "title": "1105.4359v1.Generalization_of_radiative_jet_energy_loss_to_non_zero_magnetic_mass.pdf",        "content": "arXiv:1105.4359v1  [nucl-th]  22 May 2011Generalization of radiative jet energy loss to non-zero mag netic mass\nMagdalena Djordjevic1and Marko Djordjevic2\n1Institute of Physics Belgrade, University of Belgrade, Ser bia\n2Faculty of Biology, University of Belgrade, Serbia\n(Dated: October 29, 2018)\nReliable predictions for jet quenching in ultra-relativis tic heavy ion collisions require accurate\ncomputation of radiative energy loss. With this goal, an ene rgy loss formalism in a realistic finite\nsize dynamical QCD medium was recently developed. While thi s formalism assumes zero magnetic\nmass - in accordance with the one-loop perturbative calcula tions - different non-perturbative ap-\nproaches report a non-zero magnetic mass at RHIC and LHC. We h ere generalize the energy loss\nto consistently include a possibility for existence of non- zero magnetic screening. We also present\nhow the inclusion of finite magnetic mass changes the energy l oss results. Our analysis indicates a\nfundamental constraint on magnetic to electric mass ratio.\nPACS numbers: 25.75.-q, 25.75.Nq, 12.38.Mh, 12.38.Qk\nINTRODUCTION\nHeavyflavorsuppressionis consideredto be a powerful\ntool to study the properties of a QCD medium created\nin ultra-relativistic heavy ion collisions [1]. The suppres-\nsion results from the energy loss of high energy partons\nmoving through the plasma [2]. Therefore, reliable com-\nputations of heavy quark energy loss are essential for the\nreliable predictions of jet suppression. In [3, 4], we de-\nveloped a theoretical formalism for the calculation of the\nfirst order in opacity radiative energy loss in a dynam-\nical QCD medium (see also a viewpoint in [5]). That\nstudy models radiative energy loss in a realistic finite\nsizeQCD medium with dynamical constituents, there-\nfore removing a major approximation of static scattering\ncenters present in previous calculations (see e.g. [6–12]).\nThe dynamical energy loss formalism [3, 4] is based\non HTL perturbative QCD, which requires zero mag-\nnetic mass. However, different non-perturbative ap-\nproaches [13–16] suggest a non-zero magnetic mass at\nRHIC and LHC. This, therefore, arises a question if fi-\nnite magnetic mass can be consistently included in the\ndynamical energy loss calculations, and how this inclu-\nsion would modify the energy loss results.\nRADIATIVE ENERGY LOSS IN A DYNAMICAL\nQCD MEDIUM\nIn [3, 4], we used finite temperature field theory (HTL\napproximation) and calculated the radiative energy loss\nin a finite size dynamical QCD medium. The obtained\nexpression for the energy loss is given by Eq. (1):\n∆Erad\nE=CRαs\nπL\nλdyn/integraldisplay\ndxd2k\nπd2q\nπv(q)f(k,q,x),(1)\nwheref(k,q,x) = 2/parenleftBigg\n1−sin(k+q)2+χ\nxE+L\n(k+q)2+χ\nxE+L/parenrightBigg\n×(k+q)\n(k+q)2+χ/parenleftbigg(k+q)\n(k+q)2+χ−k\nk2+χ/parenrightbigg\n.(2)\nIn Eqs. (1) and (2), Lis the length of the finite size\ndynamical QCD medium and Eis the jet energy. k\nis transverse momentum of radiated gluon, while qis\ntransverse momentum of the exchanged (virtual) gluon.\nαs=g2\n4πis coupling constant and CR=4\n3.v(q) is\nthe effective crossection in dynamical QCD medium and\nλ−1\ndyn≡C2(G)αsT= 3αsT(C2(G) = 3) is defined as“dy-\nnamical mean free path” (see also [17]). χ≡M2x2+m2\ng,\nwherexis the longitudinal momentum fraction of the\nheavy quark carried away by the emitted gluon, Mis the\nmass of the heavy quark, mg=µE/√\n2 is the effective\nmass for gluons with hard momenta k > T[18], and µE\nis the Debye mass. We assume constant coupling g. Fur-\nthermore, we note that in Eq. (1) effective crossection\nv(q) represents the interaction between the jet and ex-\nchanged gluon, while f(k,q,x) represents the interaction\nbetween the jet and radiated gluon [3, 4].\nThe goal of this section is to start from the above ex-\npression,andgeneralizeit toinclude theexistenceofnon-\nzeromagneticmass[13–16]. Toproceed, wenotethat the\ninclusion ofmagneticmass modifies the gluon selfenergy,\nand therefore our goal is to study how modified self en-\nergy of radiated and exchanged gluons change the energy\nloss result. We also note that from [3], it is straightfor-\nward to show that non-zero magnetic mass does not al-\nter the factorization ( v(q)f(k,q,x)) in the integrand of\nEq.(1), duetothe factthatthe factorizationdoesnotde-\npend on specific form of self energy. Since v(q) depends\nonly on the exchanged gluon self energy, while f(k,q,x)\ndepends only on a radiative gluon self energy, we below\nseparately study how the inclusion of magnetic mass will\nmodifyv(q) andf(k,q,x).2\nModification of the effective crossection due to\nmagnetic screening. The effective crossection v(q)\ncan be written in the following form\nv(q) =vL(q)−vT(q), (3)\nwherevT(q) (vL(q)) is transverse (longitudinal) contri-\nbution to the effective crossection, given by [3, 19]\nvT,L(q) =1\nq2+ReΠ T,L(∞)−1\nq2+ReΠ T,L(0),(4)\nwhere Π Tand Π Lare gluon self energies. While in [3, 4]\nthe derivation of the effective crossection was made\nthrough a hard thermal loop for the self-energy Π, one\nshould note that the crossection does not depend on spe-\ncific form of gluon self energy [19]. That is, the expres-\nsion is valid for any self-energy satisfying the following\nassumptions [19]:\n1. Π depends only on x≡k0/k\n2. ImΠ(x= 0) = 0\n3. ImΠ(x) = 0 ifx≥1\n4. ReΠ(x)≥0 ifx≥1,\nwhich are reasonable approximations for any system of\nwell defined quasiparticles.\nTherefore, we see that the result given by Eq. (4)\ndepends only on four numbers: ReΠ T,L(∞) and\nReΠT,L(0); due to this, we don’t need to know the full\ngluonpropagatortogeneralizethe effectivecrossectionto\nthe case of finite magnetic screening. The first two num-\nbers arethe massesofthe longitudinaland the transverse\ngluons at zero momentum (so called plasmon masses).\nTheseareshowntobeequalduetoSlavnor-Tayloridenti-\nties [20–22]. Physically, this property means that there is\nno way to distinguish transverse and longitudinal modes\nfor a particle at rest [19]. Therefore, we need only to\nintroduce one plasmon mass:\nReΠT(∞) = ReΠ L(∞)≡µ2\npl. (5)\nThe second two quantities are squares of the screening\nmasses for the transverse and longitudinal static gluon\nexchanges. The longitudinal (electric) screening mass is\nthe familiar Debye mass:\nµ2\nE≡ReΠL(0) (6)\nIn the HTL approximation, there is no screening for the\ntransverse static gluons, but this is not expected to hold\ngenerally. The corresponding screening mass is the mag-\nnetic mass, and is denoted\nµ2\nM≡ReΠT(0). (7)The general expressions for the transverse and longitu-\ndinal contributions to the effective crossections vT,L(q)\nthen become\nvL,T(q) =1\n(q2+µ2\npl)−1\n(q2+µ2\nE,M)(8)\nAfter replacing the expressions for vL,T(q) from Eq. (8)\ninto Eq. (3), we finally obtain the expression for the ef-\nfective crossection in the case of non-zero magnetic mass:\nv(q) =µ2\nE−µ2\nM\n(q2+µ2\nM)(q2+µ2\nE). (9)\nNote that v(q) in Eq. (9) does not depend on plasmon\nmass. In other words, all the dependence on the plasmon\nmass drops out in this expression. This seems reasonable\ngiven that v(q) involves only space-like gluon exchanges\n(see [3, 4, 17]), while the plasmon mass is a property of\ntime-like gluons [19]. Therefore, we only need to know\nthe two screening masses µEandµM, in order to gener-\nalize the effective crossection to non-zero magnetic mass.\nModification of f(k,q,x)due to magnetic\nscreening. As we discussed above, the introduction of\nthe magnetic mass leads to the modification of the ex-\nchanged and radiated gluon self energy. In this subsec-\ntion, westudy howthe introduction ofthe magneticmass\nin the radiated gluon self energy modifies the radiative\nenergy loss.\nTo proceed with this study, we note that all radiative\nenergyloss calculations [3, 6–12, 23] areperformed by as-\nsumingvalidityofthe softgluon( ω≪E) andsoftrescat-\ntering (ω≫ |k| ∼ |q| ∼q0,qz) approximations. Within\nthese approximations, we showed that in the finite tem-\nperature QCD medium radiated gluons have similar dis-\npersionrelationasinthevacuum, withthedifferencethat\nthe gluons now acquire a “mass” [18]. We also showed\nthat the gluon mass in the medium is approximately\nequal to the value of gluon self energy at x= 1 [18, 24]\n(so called asymptotic mass m∞=/radicalbig\nΠT(x= 1)).\nTherefore, analogously to the previous section, we see\nthat the dependence of the f(k,q,x) on gluon self energy\nreduces to just a single number: Π T(x= 1), which is\ndefinedasasquareofgluonmass mg. Duetothis,instead\nof knowing the full gluon propagator, we only need to\nknow how mgchanges in order to obtain how f(k,q,x)\nis modified in the case of non-zero magnetic mass.\nIn principle, gluon mass may change with the in-\ntroduction of non-zero magnetic screening, but (to our\nknowledge) no study up to now addressed how non-\nperturbativecalculationswould modify the gluonasymp-\ntotic mass. Consequently, our approach in the next sec-\ntion is to introduce an ansatz in order to numerically\ninvestigate how perturbations of mg, for a magnitude\ncorrespondingto magnetic mass, change radiativeenergy\nloss results.3\nModification of the energy loss expression due\nto magnetic screening. After replacing the effective\ncrossection v(q) (see Eq. (9)) into Eq. (1), the total en-\nergy loss becomes\n∆Erad\nE=CRαs\nπL\nλdyn(µ2\nE−µ2\nM)/integraldisplay\ndxd2k\nπd2q\nπ\n×1\n(q2+µ2\nM)(q2+µ2\nE)f(k,q,x),(10)\nwheref(k,q,x) is given by Eq. (2). Note that in Eq. (2),\nχ≡M2x2+M2\ng, where the gluon mass Mgcan now be\ndifferent from mg=µE/√\n2 (see previous subsection).\nA constraint on the magnetic mass range. We\nfirst discuss one interesting qualitative observation, that\ncomes directly from Eq. (10): Since integrand in Eq. (10)\nis positive definite, if magnetic mass becomes larger\nthan electric mass, the net energy loss becomes nega-\ntive. Therefore, if magnetic mass is larger than electric\nmass, the quark jet would, overall, start to gain (instead\nof lose) energy in this type of plasma. The origin for\nthis effect can be traced from Eq. (8): if the magnetic\nmass is larger than electric mass, the energy gain from\nmagnetic contribution becomes so large, that it, overall,\nleads to the total energy gain of the jet. One should\nnote that such a gain would involve transfer of energy\nof disordered motion of plasma constituents, to energy\nof ordered jet motion. Such transfer of “low” to “high”\nquality energy would be in a violation of the second law\nof thermodynamics. From this follows that it is impos-\nsible to create a plasma with magnetic mass larger than\nelectric, which places a fundamental limit on magnetic\nmassrange. Indeed, inanagreementwith thislimit, vari-\nous non-perturbative approaches [13–16] suggest that, at\nRHIC and LHC, 0 .4< µM/µE<0.6.\nNUMERICAL RESULTS\nIn this section, we numerically study how the inclu-\nsion of non-zero magnetic mass modifies the energy loss\nresults. To address this, we consider a quark-gluon\nplasma of temperature T=225MeV, with nf=2.5 effec-\ntive light quark flavors and strong interaction strength\nαs=0.3, as representative of average conditions encoun-\ntered in Au+Au collisions at RHIC. For the light quark\njets we assume that their mass is dominated by the\nthermal mass M=µ/√\n6, where µ=gT/radicalbig\n1+Nf/6≈0.5\nGeV is the Debye screening mass. The charm mass is\ntaken to be M=1.2GeV, and for the bottom mass we\nuseM=4.75GeV. To simulate (average) conditions in\nPb+Pb collisions at the LHC, we use the temperature of\nthe medium of T=400MeV.\nWe firstinvestigatehowpossiblechangesofgluonmass\n(i.e.f(k,q,x)) due to non-zero magnetic screening maychange radiative energy loss (see the previous section).\nToinvestigatethis,weintroduceanansatzthatbothelec-\ntric and magnetic masses equally contribute to gluon self\nenergy at x= 1 (i.e. Mg=/radicalBig\nµ2\nE+µ2\nM\n2). With this ansatz,\nwhich changes the gluon mass for a magnitude compa-\nrable to magnetic mass, we obtain a negligible change in\nradiative energy loss (data not shown). For simplicity,\nwe will therefore further assume that the gluon mass of\nradiated gluon remains the same as in [3, 18], i.e. that\nMg=µE/√\n2. Consequently, in the rest of this section,\nwe numerically study how the inclusion of magnetic mass\ninto the effective crossection modifies the energy loss re-\nsults compared to the results presented in [3].\nEnergylossdependence onthe magneticmassis shown\nin Fig. 1 for RHIC and LHC case. As expected, we see\nthat energy loss decreases with the increase in magnetic\nmass. Note that, when magnetic masses becomes larger\nthan electric mass, the net energy loss becomes negative,\nas discussed in the previous section. In Fig. 2, we show\nmomentum dependence of fractional energy loss, where\nwe concentrate on the range 0 .4< µM/µE<0.6, as\nsuggested by various non-perturbative approaches [13–\n16]. We see that finite magnetic mass reduces the energy\nloss in dynamical QCD medium by 25% to 50%.\nWe note that, contrary to what one may naively ex-\npect, majority of the energy loss decrease does not come\nfrom the introduction of magnetic screening in the de-\nnominator of the effective crossection (see Eq. (9)). In\nfact, the major decreasein the energy lossactually comes\nfrom the presence of the magnetic mass in the numera-\ntor of the energy loss expression (numerical results not\nshown). For example, for the ratio µM/µE= 0.5, 25%\ndecrease in the energy loss comes from the presence of\nthe magnetic mass in the numerator, while only 14% de-\ncrease comes from the presence of magnetic screening in\nthe denominator of the effective crossection. The reason\nbehind this is that introduction of magnetic screening\nin the denominator of effective crossection does not reg-\nulate the logarithmic divergence, as might be expected\nfrom Eq. (10). This is because this divergence is already\nnaturally regulated in Eq. (1), where all the relevant di-\nagrams are taken into account [3, 4].\nSUMMARY\nThis paper generalizes dynamical energy loss formal-\nism to non-zero magnetic screening. While the introduc-\ntionofmagneticmassintoanyperturbativecalculationis\ninherently phenomenological, the presented inclusion of\nthe effects of modified gluon self energy into our radia-\ntive energy loss formalism is of general validity as long as\na well defined quasiparticle system is assumed. Analysis\nof the finite magnetic mass effects leads to a constraint\nthat it is impossible to create a plasma with magnetic4\n0 0.5 1 1.5 200.20.4\n/Minus0.2\n/Minus0.4\nΜM/Slash1ΜE/CapDeltaErad/Slash1Eu,d\nc\nb\n0 0.5 1 1.5 200.51\n/Minus0.5\n/Minus1\nΜM/Slash1ΜE/CapDeltaErad/Slash1Eu,d,c\nbFIG. 1: Fractional radiative energy loss\nis shown as a function of magnetic\nand electric mass ratio. Assumed path\nlength is L= 5fm and initial jet energy\nis 10 (50) GeV for a left (right) panel.\nFull, dashed anddot-dashedcurves cor-\nrespond to light, charm and bottom\nquark respectively. Note that for left\n(right) panel, we assume RHIC (LHC)\nconditions, with a medium of tempera-\ntureT= 225 (400) MeV.\n5 10 15 20 2500.20.4\nE/LParen1GeV/RParen1/CapDeltaErad\nERHIC\n010020030040050000.20.40.60.81\nE/LParen1GeV/RParen1/CapDeltaErad\nELHCFIG. 2: Fractional radiative energy loss for an\nassumed path length L= 5fm as a function\nof momentum for charm quarks. Left (right)\npanel corresponds to RHIC (LHC) conditions.\nFull curve corresponds to the case when magnetic\nmass is zero. Gray band corresponds to the en-\nergy loss when magnetic mass is non-zero (i.e.\n0.4< µM/µE<0.6). Upper (lower) boundary\nof the band corresponds to the case µM/µE= 0.4\n(µM/µE= 0.6).\nmass larger than electric. Results presented in this pa-\nper allow including non-zero magnetic screening into jet\nsuppression calculations, and open a possibility for more\naccurate mapping of QGP properties.\nAcknowledgments: Valuable discussions with Joseph\nKapusta, Antony Rebhan, and Miklos Gyulassy are\ngratefully acknowledged. This work is supported by\nMarie Curie International Reintegration Grant within\nthe 7thEuropean Community Framework Programme\n(PIRG08-GA-2010-276913) and by the Ministry of Sci-\nence and Technological Development of the Republic of\nSerbia, under projects No. ON171004 and ON173052.\nMarko Djordjevic is supported in part by Marie Curie\nInternational Reintegration Grant within the 7thEuro-\npean Community Framework Programme (PIRG08-GA-\n2010-276996).\n[1] N. Brambilla et al., Preprint hep-ph/0412158 (2004).\n[2] M. Guylassy, I. Vitev, X. N. Wang and B. W. Zhang, in\nQuark Gluon Plasma 3, edited by R. C. Hwa and X. N.\nWang, p. 123 (World Scientific, Singapore, 2003)\n[3] M. Djordjevic, Phys. Rev. C 80, 064909 (2009).\n[4] M. Djordjevic and U. Heinz, Phys. Rev. Lett. 101,\n022302 (2008).\n[5] M. Gyulassy, Physics 2, 107 (2009).\n[6] M. Gyulassy, P. Levai and I. Vitev, Nucl. Phys. B 594,\n371 (2001).\n[7] M. Gyulassy and X. N. Wang, Nucl. Phys. B 420, 583\n(1994); X. N. Wang, M. Gyulassy and M. Plumer, Phys.\nRev. D51(1995) 3436.[8] U.A. Wiedemann, Nucl. Phys. B 588, 303 (2000); and\nNucl. Phys. B 582, 409 (2000).\n[9] E. Wang and X. N. Wang, Phys. Rev. Lett. 87142301,\n(2001). X. N. Wang and X. F. Guo, Nucl. Phys. A 696,\n788 (2001); X. F. Guo and X. N. Wang, Phys. Rev. Lett.\n853591, (2000).\n[10] N. Armesto, C. A. Salgado and U. A. Wiedemann, Phys.\nRev. D69, 114003 (2004).\n[11] M. Djordjevic and M. Gyulassy, Phys. Lett. B 560, 37\n(2003); and Nucl. Phys. A 733, 265 (2004).\n[12] M. Djordjevic, Phys. Rev. C, Phys. Rev. C 73, 044912\n(2006).\n[13] Yu. Maezawa et al. [WHOT-QCD Collaboration], Phys.\nRev. D81091501 (2010); Yu. Maezawa et al. [WHOT-\nQCD Collaboration], PoS Lattice 194 (2008).\n[14] A. Nakamura, T. Saito and S. Sakai, Phys. Rev. D 69,\n014506 (2004).\n[15] A. Hart, M. Laine and O. Philipsen, Nucl. Phys. B 586,\n443 (2000).\n[16] D. Bak, A. Karch and L. G. Yaffe, JHEP 0708, 049\n(2007).\n[17] M. Djordjevic and U. Heinz, Phys. Rev. C 77, 024905\n(2008).\n[18] M. Djordjevic, M. Gyulassy, Phys. Rev. C 68, 034914\n(2003).\n[19] P. Aurenche, F. Gelis and H. Zaraket, JHEP 0205, 043\n(2002).\n[20] R.L. Kobes, G. Kunstatter, A. Rebhan, Phys. Rev. Lett.\n64, 2992 (1990).\n[21] E. Braaten, R.D. Pisarski, Phys. Rev. D 42, 2156 (1990).\n[22] M. Dirks, A. Niegawa, K. Okano, Phys. Lett. B 461, 131\n(1999).\n[23] P. Arnold, G. D. Moore, L. G. Yaffe, JHEP 0111, 057\n(2001); JHEP 0206, 030 (2002); JHEP 0301, 030(2003).\n[24] A. Rebhan, Lect. Notes Phys. 583, 161 (2002);"    },    {        "title": "2401.01807v1.Local_distortion_driven_magnetic_phase_switching_in_pyrochlore__Yb_2_Ti__1_x_Sn_x__2O_7_.pdf",        "content": "Local distortion driven magnetic phase switching in pyrochlore Yb 2(Ti 1−xSnx)2O7\nYuanpeng Zhang1, Zhiling Dun2, Yunqi Cai2, Chengkun Xing3, Qi Cui2, Naveen Kumar\nChogondahalli Muniraju4,5,6, Qiang Zhang1, Yongqing Li2, Jinguang Cheng2,†∗, Haidong Zhou3,‡∗\n1Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA\n2Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China\n3Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA\n4Forschungszentrum J¨ ulich GmbH, J¨ ulich Centre for Neutron Science (JCNS),\nForschungszentrum J¨ ulich, D-52425 J¨ ulich, Germany\n5Chemical and Engineering Materials Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA\n6Institute of Physics, Bijeniˇ cka cesta 46, 10000, Zagreb, Croatia\n(Dated: January 4, 2024)\nWhile it is commonly accepted that the disorder induced by magnetic ion doping in quantum\nmagnets usually generates a rugged free-energy landscape resulting in slow or glassy spin dynamics,\nthe disorder/distortion effects associated with non-magnetic ion sites doping are still illusive. Here,\nusing AC susceptibility measurements, we show that the mixture of Sn/Ti on the non-magnetic ion\nsites of pyrochlore Yb 2(Ti1−xSnx)2O7induces an antiferromagnetic ground state despite both parent\ncompounds, Yb 2Ti2O7, and Yb 2Sn2O7, order ferromagnetically. Local structure studies through\nneutron total scattering reveals the local distortion in the non-magnetic ion sites and its strong\ncorrelation with the magnetic phase switching. Our study, for the first time, demonstrates the local\ndistortion as induced by the non-magnetic ion site mixture could be a new path to achieve magnetic\nphase switching, which has been traditionally obtained by external stimuli such as temperature,\nmagnetic field, pressure, strain, light etc.\nThe complex interplay between various degree of free-\ndom in frustrated quantum systems enables rich physics\ninvolving complex ground states and exotic excitations\n[1, 2]. The balancing between those weak interaction\nterms in the Hamiltonian (long-range dipolar interac-\ntions, exchange interaction beyond the nearest neighbor-\ning, site disorder, lattice distortions, etc.) and the frus-\ntration could determine the coordination in the phase\ndiagram [3–6]. Exploring such a complex interplay thus\nbecomes critical in terms of understanding the formation\nof those exotic states such as the quantum spin liquid\n(QSL) state where the long range magnetic ordering is\ncompletely suppressed by frustration and the quantum\nfluctuation yields strong dynamics even at zero temper-\nature. In practice, the QSL state of matter (e.g., the\nnon-Abelian anyons in 2D QSL) hosts promising appli-\ncation in error-proof quantum computation [7–9] and also\nis believed to be closely related to high temperature su-\nperconductivity [10–12].\nFor the study of frustrated quantum systems, of criti-\ncal concern is the underlying lattice characteristics, such\nas the chemical order/disorder, lattice/sub-lattice dis-\ntortion, etc. For example, the impact of various types\nof disorder has been attracting notable attention – this\nis especially true for the disorder effect with respect to\nthe magnetic species, as they are directly (predominantly\nthrough near neighbor interaction) associated with the\nmagnetic Hamiltonian. Depending on the level of disor-\nder, it can either perturb spontaneous symmetry break-\n∗ †jgcheng@iphy.ac.cn\n‡hzhou10@utk.eduing [13] or promote magnetic ordering and break a con-\ntinuous degeneracy via an order by disorder mechanism\n[14]. Specifically concerning the pyrochlore systems, on\nwhich the current report is to focus, there have been ex-\ntensive studies over the magnetic species relevant dis-\norder effects, including those diluted [15–19] or stuffed\n[20–23] pyrochlores. Following similar pathway, there\nhave been some recently emerging studies on the disor-\nder effect of the non-magnetic-only species on the non-\nmagnetic site. Several celebrated examples recently on\nthis topic are YbMgGaO 4(YMGO) [24–28], Sr 3CuTa 2O9\n(SCTO) [29], Sr 2Cu (Te 1−xWx)O6(SCTWO) [30–32],\nSrLaCuSbO 6(SLCSO) & SrLaCuNbO 6(SLCNO) [33],\nand Lu 3Sb3Mn2O14(LSMO) [34]. As compared to the\nmagnetic species involved chemical disorder, first, the\nnon-magnetic-only disorder has to take effect through the\nmagnetic interactions beyond the nearest-neighbor. For\nexample, in SCTWO, substituting W for Te alters the\nmagnetic interactions from the strong nearest-neighbor\ntype to the strong next-nearest-neighbor type [31], result-\ning in strong exchange interaction disorder that is absent\nin parent compounds Sr 2CuTeO 6[35] and Sr 2CuWO 6\n[36, 37]. Theoretically, the scenario could be described\nas the random-singlet (RS) state [30, 32, 38], in which\nthe randomness in a quantum magnet can induce spin-\nsinglet dimers of varying strengths with a spatially ran-\ndom manner and therefore account for the spin liquid like\nbehaviors. Second, for non-magnetic-only order/disorder\nsystems, it is relatively easier to separate out the lattice\ndistortion effect from the chemical order/disorder effect,\nespecially for those systems in which the magnetic inter-\nactions beyond the nearest neighbor are non-critical.\nEqually important as the order/disorder effect, the lo-\ncal lattice distortion also plays manifold critical roles inarXiv:2401.01807v1  [cond-mat.str-el]  3 Jan 20242\npromoting different ground states and quantum excita-\ntions. Both the crystal field environment and the deli-\ncate balance between the anisotropic exchanges could be\ntuned by the local distortions [39–41], and such an in-\nterplay could potentially induce accidental degeneracy in\nthe vicinity of the phase boundary and thus could lead\nto the emergence of a QSL [42]. Another typical and\nintricate scenario involving local distortion is for those\nsystems presenting strong spin-orbital coupling – on one\nside, the local distortion is potentially destroying the de-\ngeneracy of energy levels and thus suppressing the quan-\ntum fluctuation that is necessary for the formation of\nQSL states. On the other, the local symmetry break-\ning may lead to orbital quenching and therefore suppress\nthe spin-orbital coupling, which is then beneficial for the\nQSL states formation, since the spin-orbital coupling is\nsusceptible of lifting the degeneracy of energy levels and\nthus is detrimental for the QSL states. As such, sev-\neral typical examples were showing the dominant effect\nof suppressing the quantum fluctuations and its winning\nover the orbital quenching effect to give exotic quantum\nstates were reported for the NiRh 2O4 system [43–46].\nSuch impacts of local distortion, along with other poten-\ntial effects such as the formation of local dimer or trimer\nclusters [47–49], induction of fast spin fluctuations [50],\nreduction of the effective dimension of magnetic coupling\n[51], and among others, infers the importance of directly\nprobing the local distortion and constructing a clear pic-\nture of the interplay with magnetic coupling.\nWith this regard, extensive existing studies focus on\nthe impact of local distortion on the scheme of exchange\ninteractions, and among other aspects such as electron\nlocalization and charge density wave. Typically for QSL\ncandidate systems, various studies reported the direct\nprobe of the local environment and its link to the exotic\nQSL states, using NMR [52, 53], Electron Spin Resonance\n(ESR) [54], X-ray Absorption Spectroscopy (XAS) [55],\nand X-ray/neutron total scattering [56, 57]. We real-\nized enormous such research in the herbertsmithite and\nbarlowite QSL candidate systems. However, for the py-\nrochlore systems, direct experimental probing of the local\nstructural variation and its link to the magnetic coupling\nis still lacking, though, excessive theoretical work has\nprovided clear indication for the impact of local struc-\ntural distortion, via the spin-lattice coupling effect, upon\nthe magnetic states and spin ordering [58–64]. To the\nbest of our knowledge, there is only limited experimen-\ntal work focusing on the spin-lattice coupling effect in\npyrochlore systems from the local perspective, like the\nreport by P. M. Thygesen, et al. demonstrating the lo-\ncal orbital dimerization of Jahn-Teller active Mo4+ions\ninstead of random compositional or site disorder drives\nthe spin-glass state in Y 2Mo2O7[65]. More relevant ex-\nperimental efforts, though still limited, were mainly from\nthe average long-range structure perspective [20, 66–68].\nHowever, while the local distortion could be extracted\nfrom conventional Bragg diffraction, the local and short-\nrange probe could provide unique pathway and differentangle to such explorations, as is already demonstrated\nby the work on the herbertsmithite and barlowite QSL\ncandidate systems [52–57]. In this report, we were try-\ning to utilize the neutron total scattering measurement,\nwhich incorporates both the Bragg peaks and the dif-\nfuse scattering signal, to study the spin-lattice coupling\nin the Sn-doped Yb 2Ti2O7(YTSO). For such a system,\nit is believed only the nearest neighbor coupling is domi-\nnant in the magnetic Hamiltonian [6, 69–72]. Meanwhile,\nthe doping in our studied YTSO system is only on the\nnon-magnetic site, which infers the chemical disorder ef-\nfect in the YTSO system could be singled out from the\npotentially existing distortion effect. Through our mod-\neling for the neutron total scattering data together with\nmagnetic susceptibility measurements, a magnetic phase\nswitching could be clearly identified, as proposed to be\ninduced by the local distortion associated with the Sn-\ndoping on the non-magnetic Ti sites.\nFIG. 1. Average structure of Yb 2(Ti 1−xSnx)2O7and\nYb 2(Ti 1−xGex)2O7.a.Neutron powder diffraction pattern\n(red circles) for Yb 2(Ti0.6Sn0.4)2O7measured at 300 K with\nthe central wavelength of 1.5 ˚A on POWGEN diffractometer.\nThe solid black line is the Rietveld refinement using FullProf\n[73]. Solid blue line at the bottom of the panel shows the\ndifference curve. The Bragg peaks are marked with green\nmarkers. b.The doping concentration dependence of the\nlattice parameter. c.The doping level dependence of the\nratio between the two different Yb-O bond lengths and the\nYb-O2-Yb angle.\nNeutron diffraction. For comparison, we synthe-\nsized both Yb 2(Ti1−xSnx)2O7and Yb 2(Ti1−xGex)2O7\nsamples and used neutron powder diffraction (NPD) to\ncharacterize their lattice structures. Fig. 1(a) shows the\nrefinement for the NPD data of Yb 2(Ti0.6Sn0.4)2O7mea-\nsured at room temperature using the POWGEN diffrac-\ntometer. The data could be well fitted by the Fd¯3m\npyrochlore structure. The NPD data for several other Sn3\nand Ge doped samples was also refined (not shown here),\nwhich all exhibits pure pyrochlore structure. As sum-\nmarized in Fig. 1(c), the lattice parameter adecreases\nfrom Yb 2Sn2O7to Yb 2Ti2O7and then Yb 2Ge2O7for\nall doped samples. This is reasonable since the lat-\ntice parameter is strongly related to the ironic radius\nof the (Sn/Ti/Ge) site, and therefore the Sn sample\nhas the largest lattice parameter, Ti sample the sec-\nond, and Ge sample the smallest. We further used\nρ=dYb-O2/dYb-O1and the Yb-O2-Yb angle to char-\nacterize the axial distortion of the YbO 8polyhedra, here\ndYb-O1represents the bond length for the 6 longer Yb-\nO1 bonds in the plane perpendicular to the ⟨111⟩axis and\ndYb-O2represents the bond length for the two shorter\nYb-O2 bonds along the ⟨111⟩axis. As shown in Fig. 1(d),\nagain, both of them decrease linearly, without abrupt\nchange.\nAC susceptibility. Fig. 2 shows the AC suscep-\ntibility measured at different DC magnetic fields for\nYb2(Ti1−xSnx)2O7and Yb 2(Ti1−xGex)2O7. For all sam-\nples, the data at zero field exhibits a peak, which repre-\nsents the long range magnetic ordering at T*, and such a\npeak shows an obvious shift under applied DC fields. As\ndemonstrated in Ref. [74], the field dependence of AC\nsusceptibility can be used as a convenient tool to iden-\ntify the nature of a long-range magnetic ordering, i.e.,\nthe ferromagnetic (FM) ordering temperature will shift\nto higher temperatures with increasing DC field due to\nthe contribution of domain magnetization while the anti-\nferromagnetic (AFM) ordering temperature will show an\nopposite DC field dependence.\nThe field dependence of T* for each doping level was\nsummarized in Fig. 3(a). The data shows (i) for\nYb2Ti2O7and Yb 2Sn2O7, the T* increases with in-\ncreasing field, which is consistent with the fact that\nboth samples have a splayed ferromagnetic (SF) ground\nstate; (ii) for Yb 2Ge2O7, the T* decreases with increas-\ning field, which is consistent with its AFM ground state\n[74–76]; (iii) for all Sn- and Ge-doped samples except\nYb2(Ti0.8Sn0.2)2O7, the T* decreases first with increas-\ning field and then increases while the field surpasses a\ncritical value Hc. This indicates that as soon as Sn and\nGe are doped, certain volume of AFM phase is intro-\nduced. This AFM order should be in long range na-\nture since it dominates the bulk magnetism at low fields.\nWith H > H c, the sample comes back to ferromagnetic\nor is fully polarized; (iv) for Yb 2(Ti0.8Sn0.2)2O7, it has\nferromagnetic ground state since its T* monotonically\nincreases with increasing field.\nMagnetic Phase Diagram. Accordingly, a magnetic\nphase diagram of T* and Hcfor Yb 2(Ti1−xSnx)2O7and\nYb2(Ti1−xGex)2O7is summarized in Fig. 3(b). For\nGe-doped samples, both T* and Hcmonotonically in-\ncrease with increasing Ge-doping level. On the other\nhand, for Sn-doped samples, (i) while the T* generally\ndecreases with increasing Sn-doping level, it exhibits a\ndome around x= 0.5; (ii) the Hcfirst increases with in-\ncreasing Sn-doping level, peaks at x= 0.5, and thereafterdecreases.\nThe evolution of T* and Hcin the Ge-doped sam-\nples is expected. The magnetic ground states of Yb-\npyrochlores are determined by the ratio among the\nanisotropic exchange interactions [75, 77, 78]. Un-\nlike Yb 2Ti2O7, Yb 2Ge2O7orders antiferromagnetically\nin the Γ 5manifold [76]. From the point view of\nchemical pressure effect, with increasing the Ge-doping\nlevel in Yb 2(Ti1−xGex)2O7, the lattice parameter de-\ncreases monotonically and gradually tunes the balance of\nanisotropic exchange interactions that drives the system\ntowards the AFM Γ 5phase from the SF phase [75, 76].\nAlternatively, if we assume that the phase coexistence in\nGe-doped samples is similar to that of Yb 2Ti2O7, then\ntheHccould be scaled to the volume fraction of the AFM\nphase since the larger the Hcis, the more difficult to po-\nlarize the system. Therefore, the evolution of Hcin Fig.\n3(b) means a monotonic increase in volume fraction of\nthe AFM phase in Yb 2(Ti1−xGex)2O7, consistent with\nthe SF and AFM phases in Yb 2Ti2O7and Yb 2Ge2O7,\nrespectively.\nHowever, although we cannot determinate whether it\nis the ψ2orψ3phase of the Γ 5manifold, the appear-\nance of the long range AFM order in Yb 2(Ti1−xSnx)2O7\nis surprising. Since both Yb 2Ti2O7and Yb 2Sn2O7have\nthe SF ground state, an AFM ground state should not be\nexpected for Sn-doped samples from the view of chem-\nical pressure effects. Even if there is still magnetic\nphase coexistence [79], it is puzzling to observe this non-\nmonotonic change of the volume fraction of AFM phase\n(or,Hc) in Yb 2(Ti1−xSnx)2O7.\nTotal Scattering. The total scattering signal con-\ntains both the Bragg peaks and the diffuse scattering\ncontribution. The diffuse scattering part, which is usu-\nally taken as the background and thus subtracted off\nin conventional Bragg peaks analysis, in fact provides\nunique access to the local structure. Although both the\nBragg peaks and total scattering data would yield infor-\nmation about local distortions, the former is based on the\ndistance between average positions whereas the latter is\nbased on the average of distances ensemble . Therefore,\nthe local probe via total scattering data could provide\naccess to the local structural variation that is inaccessi-\nble through the conventional Bragg analysis. Here, we\ncollected neutron total scattering data on POWGEN for\nthe series of Yb 2(Ti1−xSnx)2O7samples and performed\nthe reverse Monte Carlo (RMC) [80, 81] modeling [82] to\nextract the local structure. A 10 ×10×10 supercell con-\ntaining more than 100, 000 atoms could be constructed\nfrom the RMC modeling and the following statistical cal-\nculation could be performed to extract the key structural\naspects. Typically, here we focused on the various bond\nangles involving both the Yb, Ti/Sn, and O atoms – from\nthe RMC resulted configuration, a statistical distribution\nof various triplet angles could be obtained [83]. Further,\nthe width of the triplet angles distribution could be ex-\ntracted through a Gaussian peak fitting, which indicates\nthe dispersiveness of the bond angle and thus infers the4\nFIG. 2. AC susceptibility .a.Real part of AC susceptibility measured under different DC magnetic fields in the arbitrary\nunit (a.u.) for Yb 2(Ti1−xSnx)2O7with x= 0.2, 0.4, 0.6, and 0.8. b.Similar data for Yb 2(Ti1−xGex)2O7with x= 0.2, 0.4, 0.6,\nand 0.8. The used AC frequency is 3317 Hz for the Sn-doped system and 331 Hz for the Ge-doped system with a magnitude\nof 5 Oe. Dashed arrows indicate the evolution of the peak’s position with increasing DC fields.\nFIG. 3. Magnetic phase diagram. a. The field\ndependence of the magnetic ordering temperature T∗for\nYb2(Ti1−xSnx)2O7and Yb 2(Ti1−xGex)2O7with different\ndoping concentrations. b.Magnetic phase diagram as a\nfunction of the doping concentration, x, temperature, T, and\nthe external DC field, µ0H. The red and blue regions repre-\nsent the FM and AFM phases, respectively. Data points for\nYb2Ti2O7, Yb 2Sn2O7, and Yb 2Ge2O7are from Dun et al.\n[74].\nsignificance of the local distortion. The results are pre-\nsented in Fig. 4. No abrupt change as the function of the\ndoping level could be observed except for the Ti&Sn-O-\nTi&Sn triplet (see the inset of Fig. 4). This indicates the\nnon-magnetic-site involved local distortion is enhanced as\nthe result of the doping. Assuming the nearest neighbor\nmagnetic interaction is dominant in the YTSO system,\nFIG. 4. Local distortion as characterized by various\nlocal bond angles, as extracted from the RMC re-\nsulted configuration . The inset is an illustration for the\nlocal geometry involving the critical triplet angle – Ti&Sn-O-\nTi&Sn. Yb →cyan, Ti&Sn →blue, O→red. Error bar was\nestimated via running 10 repeated RMC modeling followed\nby the same bond angle calculation. The actual error bar is\nsmaller than the symbol for all the data points as presented\nand here we just take the symbol size as the representative\nerror bar level.\nthe result here infers the potential link between the mag-5\nnetic phase switching and the local distortion. As of our\nknowledge, this is the first experimental evidence demon-\nstrating the magnetic phase switching in pyrochlore sys-\ntems that is potentially coupled with the local structural\ndistortion beyond the compositional randomness or site\ndisorder. Our finding here falls in line with various theo-\nretical studies for the spin-lattice coupling in pyrochlore\nsystems. For example, through many-body quantum-\nchemical calculations, N. Bogdanov et al. showed the\nmagnetic interactions and ordering in Cd 2Os2O7are cru-\ncially dependent on the local geometrical features [62]. H.\nShinaoka, et al. showed the coupling of spin-glass tran-\nsitions to local lattice distortions on pyrochlore lattices\nvia Monte Carlo simulation [59]. Also using Monte Carlo\nsimulation, K. Aoyama et al. revealed a lattice distor-\ntion induced spin ordering in the breathing pyrochlore\nlattice[58]. Based on such theoretical results, we believe\nthe non-magnetic site mixing in the YTSO system is im-\nposing effect upon the magnetic coupling through the\ninduced local lattice distortion, which then further on-\nsets the magnetic phase switching at a certain level of\nmixing. According to a recent report by A. Scheie et al. ,\nit was revealed that the QSL physics in Yb 2Ti2O7is fun-\ndamentally the FM-AFM phase competition in the low\ntemperature regime [79]. With this regard, the FM-AFM\nphase switching and its link to the local lattice distortion\nis of strong relevance and infers that the non-magnetic\nsite doping and the accordingly induced local structural\nvariation should be paid serious attention for the study\nof QSL physics in pyrochlore systems.\nAs a final remark, we want to emphasize that the con-\nclusion we arrived at here about there being a strong\ncorrelation between the local distortion and magnetic\nphase switching is partly based on the assumption that\nthe magnetic coupling beyond the nearest neighbor is not\ntaking critical effect in the YTSO system – in fact, such\nan assumption was indeed adopted in various previous\nreports as mentioned earlier [6, 69–72]. Without such\nan assumption, the induced randomness in the Hamilto-\nnian as the result of chemical disorder and its effect uponthe magnetic ordering cannot be rigorously excluded. As\nsuch, future studies are needed – theoretically, magnetic\ninteraction beyond the nearest neighbor needs to be in-\nspected and its effect on the magnetic phase diagram\nshould be studied.\nACKNOWLEDGMENTS\nWe thank Martin Mourigal and Itamar Kimchi for\nhelpful discussion. J.G.C. is supported by the Na-\ntional Natural Science Foundation of China (12025408,\n11874400, 11921004), the Key Research Program of Fron-\ntier Sciences of CAS ( QYZDB-SSW-SLH013), the CAS\nInterdisciplinary Innovation Team (JCTD-2019-01) and\nLujiaxi international group funding of K. C. Wong Ed-\nucation Foundation (GJTD-2020-01). The work at the\nUniversity of Tennessee is supported by the U. S. Depart-\nment of Energy under grant No. DE-SC0020254. Part\nof the research conducted at SNS was sponsored by the\nScientific User Facilities Division, Office of Basic Energy\nSciences, US Department of Energy. The following fund-\ning is acknowledged: US Department of Energy, Office\nof Science (contract No. DE-AC05-00OR22725). This\nresearch used resources of the Computeand Data Envi-\nronment for Science (CADES) at the Oak Ridge National\nLaboratory,which is supported by the Office of Science of\nthe U.S. Department of Energy under Contract No. DE-\nAC05-00OR22725. This research used resources of the\nCompute and Data Environment for Science (CADES) at\nthe Oak Ridge National Laboratory, which is supported\nby the Office of Science of the U.S. Department of En-\nergy under Contract No. DE-AC05-00OR22725. This\nresearch used resources of the National Energy Research\nScientific Computing Center (NERSC), a U.S. Depart-\nment of Energy Office of Science User Facility located\nat Lawrence Berkeley National Laboratory, operated un-\nder Contract No. DE-AC02-05CH11231 using NERSC\naward ERCAP0024340. We thank Dr. Emil S. Bozin for\nthe discussion about the local structure studies.\n[1] J. S. Gardner, M. J. P. Gingras, and J. E. Greedan, Rev.\nMod. Phys. 82, 53 (2010).\n[2] J. Shamblin, Z. Dun, M. Lee, S. Johnston, E. S. Choi,\nK. Page, Y. Qiu, and H. Zhou, Phys. Rev. B 96, 174418\n(2017).\n[3] K. A. Ross, Phd thesis, McMaster University (2012).\n[4] K. A. Ross, J. P. C. Ruff, C. P. Adams, J. S. Gardner,\nH. A. Dabkowska, Y. Qiu, J. R. D. Copley, and B. D.\nGaulin, Phys. Rev. Lett. 103, 227202 (2009).\n[5] Z. Lu, L. Ge, G. Wang, M. Russina, G. G¨ unther, C. R.\ndela Cruz, R. Sinclair, H. D. Zhou, and J. Ma, Phys. Rev.\nB98, 094412 (2018).\n[6] R. Applegate, N. R. Hayre, R. R. P. Singh, T. Lin,\nA. G. R. Day, and M. J. P. Gingras, Phys. Rev. Lett.\n109, 097205 (2012).[7] M. Banerjee, M. Heiblum, V. Umansky, D. E. Feldman,\nY. Oreg, and A. Stern, Nature 559, 205 (2018).\n[8] C. Nayak, S. H. Simon, A. Stern, M. Freedman, and\nS. Das Sarma, Rev. Mod. Phys. 80, 1083 (2008).\n[9] R. Willett, J. P. Eisenstein, H. L. St¨ ormer, D. C. Tsui,\nA. C. Gossard, and J. H. English, Phys. Rev. Lett. 59,\n1776 (1987).\n[10] J. R. Chamorro, T. M. McQueen, and T. T. Tran, Chem-\nical Reviews 121, 2898 (2021).\n[11] L. Balents, Nature 464, 199 (2010).\n[12] P. W. Anderson, Science 235, 1196 (1987).\n[13] J. Villain, Zeitschrift f¨ ur Physik B Condensed Matter 33,\n31 (1979).\n[14] C. L. Henley, Phys. Rev. Lett. 62, 2056 (1989).\n[15] A. Keren, J. S. Gardner, G. Ehlers, A. Fukaya, E. Segal,6\nand Y. J. Uemura, Phys. Rev. Lett. 92, 107204 (2004).\n[16] Sheetal, A. Ali, S. Rajput, Y. Singh, T. Maitra, and C. S.\nYadav, Journal of Physics: Condensed Matter 32, 365804\n(2020).\n[17] S. K. Takahashi, J. Wang, A. Arsenault, T. Imai,\nM. Abramchuk, F. Tafti, and P. M. Singer, Phys. Rev. X\n9, 031047 (2019).\n[18] V. Dantas and E. C. Andrade, Phys. Rev. Lett. 129,\n037204 (2022).\n[19] W.-H. Kao, J. Knolle, G. B. Hal´ asz, R. Moessner, and\nN. B. Perkins, Phys. Rev. X 11, 011034 (2021).\n[20] K. A. Ross, T. Proffen, H. A. Dabkowska, J. A. Quilliam,\nL. R. Yaraskavitch, J. B. Kycia, and B. D. Gaulin, Phys.\nRev. B 86, 174424 (2012).\n[21] D. F. Bowman, E. Cemal, T. Lehner, A. R. Wildes,\nL. Mangin-Thro, G. J. Nilsen, M. J. Gutmann, D. J.\nVoneshen, D. Prabhakaran, A. T. Boothroyd, D. G.\nPorter, C. Castelnovo, K. Refson, and J. P. Goff, Nature\nCommunications 10, 637 (2019).\n[22] G. C. Lau, R. S. Freitas, B. G. Ueland, B. D. Muegge,\nE. L. Duncan, P. Schiffer, and R. J. Cava, Nature Physics\n2, 249 (2006).\n[23] J. S. Gardner, G. Ehlers, P. Fouquet, B. Farago, and\nJ. R. Stewart, Journal of Physics: Condensed Matter 23,\n164220 (2011).\n[24] Y. Li, D. Adroja, P. K. Biswas, P. J. Baker, Q. Zhang,\nJ. Liu, A. A. Tsirlin, P. Gegenwart, and Q. Zhang, Phys.\nRev. Lett. 117, 097201 (2016).\n[25] Y. Li, S. Bachus, B. Liu, I. Radelytskyi, A. Bertin,\nA. Schneidewind, Y. Tokiwa, A. A. Tsirlin, and P. Gegen-\nwart, Phys. Rev. Lett. 122, 137201 (2019).\n[26] Y. Shen, Y.-D. Li, H. Wo, Y. Li, S. Shen, B. Pan,\nQ. Wang, H. C. Walker, P. Steffens, M. Boehm, Y. Hao,\nD. L. Quintero-Castro, L. W. Harriger, M. D. Frontzek,\nL. Hao, S. Meng, Q. Zhang, G. Chen, and J. Zhao, Na-\nture540, 559 (2016).\n[27] J. A. M. Paddison, M. Daum, Z. Dun, G. Ehlers, Y. Liu,\nM. B. Stone, H. Zhou, and M. Mourigal, Nature Physics\n13, 117 (2017).\n[28] Y. Shen, Y.-D. Li, H. C. Walker, P. Steffens, M. Boehm,\nX. Zhang, S. Shen, H. Wo, G. Chen, and J. Zhao, Nature\nCommunications 9, 4138 (2018).\n[29] B. Sana, M. Barik, S. Lee, U. Jena, M. Baenitz,\nJ. Sichelschmidt, S. Luther, H. Kuehne, K. Sethupathi,\nM. S. R. Rao, K. Y. Choi, and P. Khuntia, (2023),\narXiv:2304.13116 [cond-mat.str-el].\n[30] W. Hong, L. Liu, C. Liu, X. Ma, A. Koda, X. Li, J. Song,\nW. Yang, J. Yang, P. Cheng, H. Zhang, W. Bao, X. Ma,\nD. Chen, K. Sun, W. Guo, H. Luo, A. W. Sandvik, and\nS. Li, Phys. Rev. Lett. 126, 037201 (2021).\n[31] E. Fogh, O. Mustonen, P. Babkevich, V. M. Katukuri,\nH. C. Walker, L. Mangin-Thro, M. Karppinen, S. Ward,\nB. Normand, and H. M. Rønnow, Phys. Rev. B 105,\n184410 (2022).\n[32] H.-D. Ren, T.-Y. Xiong, H.-Q. Wu, D. N. Sheng, and\nS.-S. Gong, Phys. Rev. B 107, L020407 (2023).\n[33] M. Watanabe, N. Kurita, H. Tanaka, W. Ueno, K. Mat-\nsui, T. Goto, and M. Hagihala, Phys. Rev. B 105, 054414\n(2022).\n[34] C. Lee, S. Lee, H.-S. Kim, S. Kittaka, Y. Kohama,\nT. Sakakibara, K. H. Lee, J. van Tol, D. I. Gorbunov,\nS.-H. Do, S. Yoon, A. Berlie, and K.-Y. Choi, Phys. Rev.\nB107, 214404 (2023).\n[35] T. Koga, N. Kurita, M. Avdeev, S. Danilkin, T. J. Sato,and H. Tanaka, Phys. Rev. B 93, 054426 (2016).\n[36] S. Vasala, M. Avdeev, S. Danilkin, O. Chmaissem, and\nM. Karppinen, Journal of Physics: Condensed Matter\n26, 496001 (2014).\n[37] S. Vasala, H. Saadaoui, E. Morenzoni, O. Chmaissem,\nT.-S. Chan, J.-M. Chen, Y.-Y. Hsu, H. Yamauchi, and\nM. Karppinen, Phys. Rev. B 89, 134419 (2014).\n[38] I. Kimchi, A. Nahum, and T. Senthil, Phys. Rev. X 8,\n031028 (2018).\n[39] M. R. Norman, Rev. Mod. Phys. 88, 041002 (2016).\n[40] R. Yadav, N. A. Bogdanov, V. M. Katukuri, S. Nishi-\nmoto, J. van den Brink, and L. Hozoi, Scientific Reports\n6, 37925 (2016).\n[41] C. Balz, B. Lake, J. Reuther, H. Luetkens,\nR. Sch¨ onemann, T. Herrmannsd¨ orfer, Y. Singh, A. T. M.\nNazmul Islam, E. M. Wheeler, J. A. Rodriguez-Rivera,\nT. Guidi, G. G. Simeoni, C. Baines, and H. Ryll, Nature\nPhysics 12, 942 (2016).\n[42] E. Kermarrec, J. Gaudet, K. Fritsch, R. Khasanov,\nZ. Guguchia, C. Ritter, K. A. Ross, H. A. Dabkowska,\nand B. D. Gaulin, Nature Communications 8, 14810\n(2017).\n[43] J. R. Chamorro, L. Ge, J. Flynn, M. A. Subramanian,\nM. Mourigal, and T. M. McQueen, Phys. Rev. Mater. 2,\n034404 (2018).\n[44] F. L. Buessen, M. Hering, J. Reuther, and S. Trebst,\nPhys. Rev. Lett. 120, 057201 (2018).\n[45] S. Das, D. Nafday, T. Saha-Dasgupta, and\nA. Paramekanti, Phys. Rev. B 100, 140408 (2019).\n[46] G. Chen, Phys. Rev. B 96, 020412 (2017).\n[47] A. Paul, C.-M. Chung, T. Birol, and H. J. Changlani,\nPhys. Rev. Lett. 124, 167203 (2020).\n[48] Z. A. Kelly, T. T. Tran, and T. M. McQueen, Inorganic\nChemistry 58, 11941 (2019).\n[49] R. Flint and P. A. Lee, Phys. Rev. Lett. 111, 217201\n(2013).\n[50] P. Bonville, I. Mirebeau, A. Gukasov, S. Petit, and\nJ. Robert, Phys. Rev. B 84, 184409 (2011).\n[51] B. Zhang, P. J. Baker, Y. Zhang, D. Wang, Z. Wang,\nS. Su, D. Zhu, and F. L. Pratt, Journal of the American\nChemical Society 140, 122 (2018).\n[52] C. M. Pasco, B. A. Trump, T. T. Tran, Z. A. Kelly,\nC. Hoffmann, I. Heinmaa, R. Stern, and T. M. McQueen,\nPhys. Rev. Mater. 2, 044406 (2018).\n[53] P. Malavi, S. Pal, D. V. S. Muthu, S. Sahoo, S. Karmakar,\nand A. K. Sood, Phys. Rev. B 101, 214402 (2020).\n[54] A. Zorko, M. Herak, M. Gomilˇ sek, J. van Tol,\nM. Vel´ azquez, P. Khuntia, F. Bert, and P. Mendels, Phys.\nRev. Lett. 118, 017202 (2017).\n[55] R. W. Smaha, I. Boukahil, C. J. Titus, J. M. Jiang, J. P.\nSheckelton, W. He, J. Wen, J. Vinson, S. G. Wang, Y.-S.\nChen, S. J. Teat, T. P. Devereaux, C. Das Pemmaraju,\nand Y. S. Lee, Phys. Rev. Mater. 4, 124406 (2020).\n[56] D. P. Shoemaker, R. Seshadri, M. Tachibana, and A. L.\nHector, Phys. Rev. B 84, 064117 (2011).\n[57] E. S. Bozin, M. Abeykoon, S. Conradson, G. Baldinozzi,\nP. Sutar, and D. Mihailovic, (2023), arXiv:2301.05670\n[cond-mat.str-el].\n[58] K. Aoyama and H. Kawamura, Phys. Rev. B 99, 144406\n(2019).\n[59] H. Shinaoka, Y. Tomita, and Y. Motome, Phys. Rev.\nLett. 107, 047204 (2011).\n[60] T. Haku, K. Kimura, Y. Matsumoto, M. Soda, M. Sera,\nD. Yu, R. A. Mole, T. Takeuchi, S. Nakatsuji, Y. Kono,7\nT. Sakakibara, L.-J. Chang, and T. Masuda, Phys. Rev.\nB93, 220407 (2016).\n[61] Y.-P. Huang, G. Chen, and M. Hermele, Phys. Rev. Lett.\n112, 167203 (2014).\n[62] L. Savary and L. Balents, Phys. Rev. Lett. 118, 087203\n(2017).\n[63] N. A. Bogdanov, R. Maurice, I. Rousochatzakis,\nJ. van den Brink, and L. Hozoi, Phys. Rev. Lett. 110,\n127206 (2013).\n[64] E. Berg, E. Altman, and A. Auerbach, Phys. Rev. Lett.\n90, 147204 (2003).\n[65] P. M. M. Thygesen, J. A. M. Paddison, R. Zhang, K. A.\nBeyer, K. W. Chapman, H. Y. Playford, M. G. Tucker,\nD. A. Keen, M. A. Hayward, and A. L. Goodwin, Phys.\nRev. Lett. 118, 067201 (2017).\n[66] X. Li, W. M. Li, K. Matsubayashi, Y. Sato, C. Q. Jin,\nY. Uwatoko, T. Kawae, A. M. Hallas, C. R. Wiebe, A. M.\nArevalo-Lopez, J. P. Attfield, J. S. Gardner, R. S. Freitas,\nH. D. Zhou, and J.-G. Cheng, Phys. Rev. B 89, 064409\n(2014).\n[67] S. Gao, A. F. May, M.-H. Du, J. A. M. Paddison,\nH. S. Arachchige, G. Pokharel, C. dela Cruz, Q. Zhang,\nG. Ehlers, D. S. Parker, D. G. Mandrus, M. B. Stone, and\nA. D. Christianson, Phys. Rev. B 103, 214418 (2021).\n[68] Z. Y. Zhao, S. Calder, A. A. Aczel, M. A. McGuire, B. C.\nSales, D. G. Mandrus, G. Chen, N. Trivedi, H. D. Zhou,\nand J.-Q. Yan, Phys. Rev. B 93, 134426 (2016).\n[69] J. G. Rau, R. Moessner, and P. A. McClarty, Phys. Rev.\nB100, 104423 (2019).\n[70] K. A. Ross, L. Savary, B. D. Gaulin, and L. Balents,\nPhys. Rev. X 1, 021002 (2011).\n[71] N. R. Hayre, K. A. Ross, R. Applegate, T. Lin, R. R. P.\nSingh, B. D. Gaulin, and M. J. P. Gingras, Phys. Rev. B\n87, 184423 (2013).\n[72] J. D. Thompson, P. A. McClarty, H. M. Rønnow, L. P.\nRegnault, A. Sorge, and M. J. P. Gingras, Phys. Rev.\nLett. 106, 187202 (2011).\n[73] J. Rodr´ ıguez-Carvajal, Recent advances in magneticstructure determination by neutron powder diffraction,\nPhys. B: Condens. Matter 192, 55 (1993).\n[74] Z. L. Dun, M. Lee, E. S. Choi, A. M. Hallas, C. R. Wiebe,\nJ. S. Gardner, E. Arrighi, R. S. Freitas, A. M. Arevalo-\nLopez, J. P. Attfield, H. D. Zhou, and J. G. Cheng, Phys.\nRev. B 89, 064401 (2014).\n[75] Z. L. Dun, X. Li, R. S. Freitas, E. Arrighi, C. R.\nDela Cruz, M. Lee, E. S. Choi, H. B. Cao, H. J. Sil-\nverstein, C. R. Wiebe, J. G. Cheng, and H. D. Zhou,\nPhys. Rev. B 92, 140407 (2015).\n[76] A. M. Hallas, J. Gaudet, M. N. Wilson, T. J. Munsie,\nA. A. Aczel, M. B. Stone, R. S. Freitas, A. M. Arevalo-\nLopez, J. P. Attfield, M. Tachibana, C. R. Wiebe, G. M.\nLuke, and B. D. Gaulin, Phys. Rev. B 93, 104405 (2016).\n[77] A. W. C. Wong, Z. Hao, and M. J. P. Gingras, Phys.\nRev. B 88, 144402 (2013).\n[78] M. Taillefumier, O. Benton, H. Yan, L. D. C. Jaubert,\nand N. Shannon, Phys. Rev. X 7, 041057 (2017).\n[79] A. Scheie, J. Kindervater, S. Zhang, H. J. Changlani,\nG. Sala, G. Ehlers, A. Heinemann, G. S. Tucker, S. M.\nKoohpayeh, and C. Broholm, Proc. Natl. Acad. Sci. 117,\n27245 (2020).\n[80] Y. Zhang, M. Eremenko, V. Krayzman, M. G. Tucker,\nand I. Levin, Journal of Applied Crystallography 53,\n1509 (2020).\n[81] M. G. Tucker, D. A. Keen, M. T. Dove, A. L. Goodwin,\nand Q. Hui, Journal of Physics: Condensed Matter 19,\n335218 (2007).\n[82] See Supplemental Material at [URL to be inserted] for the\ntechnical details about the reverse Monte Carlo modeling.\n[83] See Supplemental Material at [URL to be inserted] for\nthe statistical distribution plots for the various triplet\nangles.\n[84] X. Rao, G. Hussain, Q. Huang, W. J. Chu, N. Li,\nX. Zhao, Z. Dun, E. S. Choi, T. Asaba, L. Chen, L. Li,\nX. Y. Yue, N. N. Wang, J.-G. Cheng, Y. H. Gao, Y. Shen,\nJ. Zhao, G. Chen, H. D. Zhou, and X. F. Sun, Nature\nCommunications 12, 4949 (2021)."    },    {        "title": "2112.00961v2.Hamilton_Jacobi_Equations_of_Nonholonomic_Magnetic_Hamiltonian_Systems.pdf",        "content": "arXiv:2112.00961v2  [math.SG]  15 Jun 2022Hamilton-Jacobi Equations for Nonholonomic Magnetic\nHamiltonian Systems\nHong Wang\nSchool of Mathematical Sciences and LPMC,\nNankai University, Tianjin 300071, P.R.China\nE-mail: hongwang@nankai.edu.cn\nIn Memory of Great Geometer Shiing Shen Chern\nJune 15, 2022\nAbstract: In order to describe the impact of different geometric structu res and constraints for\nthe dynamics of a Hamiltonian system, in this paper, for a mag netic Hamiltonian system defined by\na magnetic symplectic form, we first drive precisely the geom etric constraint conditions of magnetic\nsymplectic form for the magnetic Hamiltonian vector field. w hich are called the Type I and Type II\nof Hamilton-Jacobi equation. Secondly, for the magnetic Ha miltonian system with nonholonomic\nconstraint, we first define a distributional magnetic Hamilt onian system, then derive its two types\nof Hamilton-Jacobi equation. Moreover, we generalize the a bove results to nonholonomic reducible\nmagnetic Hamiltonian system with symmetry. We define a nonho lonomic reduced distributional\nmagnetic Hamiltonian system, and prove two types of Hamilto n-Jacobi theorem. These research\nwork reveal the deeply internal relationships of the magnet ic symplectic structure, nonholonomic\nconstraint, the distributional two-form, and the dynamica l vector field of the nonholonomic mag-\nnetic Hamiltonian system.\nKeywords: Hamilton-Jacobi equation, magnetic Hamiltonian system, n onholonomic\nconstraint distributional magnetic Hamiltonian system, n onholonomic reduction.\nAMS Classification: 70H20, 70F25, 53D20.\nContents\n1 Introduction 1\n2 Nonholonomic Magnetic Hamiltonian System 4\n3 Hamilton-Jacobi Equation of Magnetic Hamiltonian System 7\n4 Hamilton-Jacobi Equation for Distributional Magnetic Hamiltonian System 12\n5 Nonholonomic Reduced Distributional Magnetic Hamiltonian System 17\n1 Introduction\nIt is well-known that Hamilton-Jacobi theory is an importan t research subject in mathematics and\nanalytical mechanics, see Abraham and Marsden [ 1], Arnold [ 2] and Marsden and Ratiu [ 19], and\n1the Hamilton-Jacobi equation is also fundamental in the stu dyof the quantum-classical relationship\nin quantization, and it also plays an important role in the st udy of stochastic dynamical systems,\nsee Woodhouse [ 33], Ge and Marsden [ 10], and L´ azaro-Cam´ ı and Ortega [ 12]. For these reasons it\nis described as a useful tool in the study of Hamiltonian syst em theory, and has been extensively\ndeveloped in past many years and become one of the most active subjects in the study of modern\napplied mathematics and analytical mechanics.\nJust as we have known that Hamilton-Jacobi theory from the va riational point of view is\noriginally developed by Jacobi in 1866, which state that the integral of Lagrangian of a mechanical\nsystem along the solution of its Euler-Lagrange equation sa tisfies the Hamilton-Jacobi equation.\nThe classical description of this problem from the generati ng function and the geometrical point of\nview is given by Abraham and Marsden in [ 1] as follows: Let Qbe a smooth manifold and TQthe\ntangent bundle, T∗Qthe cotangent bundle with a canonical symplectic form ωand the projection\nπQ:T∗Q→Qinduces the map TπQ:TT∗Q→TQ.\nTheorem 1.1 Assume that the triple (T∗Q,ω,H)is a Hamiltonian system with Hamiltonian vec-\ntor field XH, andW:Q→Ris a given generating function. Then the following two assert ions\nare equivalent:\n(i)For every curve σ:R→Qsatisfying ˙σ(t) =TπQ(XH(dW(σ(t)))),∀t∈R, thendW·σis an\nintegral curve of the Hamiltonian vector field XH.\n(ii)Wsatisfies the Hamilton-Jacobi equation H(qi,∂W\n∂qi) =E,whereEis a constant.\nFrom the proof of the above theorem given in Abraham and Marsd en [1], we know that the\nassertion (i) with equivalent to Hamilton-Jacobi equation (ii) by the generating function, gives a\ngeometric constraint condition of the canonical symplecti c form on the cotangent bundle T∗Qfor\nHamiltonian vector field of the system. Thus, the Hamilton-J acobi equation reveals the deeply\ninternal relationships of the generating function, the can onical symplectic form and the dynamical\nvector field of a Hamiltonian system.\nNow, it is a natural problem how to generalize Theorem 1.1 to fi t the nonholonomic systems\nand their reduced systems. Note that if take that γ=dWin the above Theorem 1.1, then γis\na closed one-form on Q, and the equation d(H·dW) = 0 is equivalent to the Hamilton-Jacobi\nequation H(qi,∂W\n∂qi) =E, where Eis a constant, which is called the classical Hamilton-Jacob i\nequation. This result is used the formulation of a geometric version of Hamilton-Jacobi theorem\nfor Hamiltonian system, see Cari˜ nena et al [ 5,6]. Moreover, note that Theorem 1.1 is also gen-\neralized in the context of time-dependent Hamiltonian syst em by Marsden and Ratiu in [ 19], and\nthe Hamilton-Jacobi equation may be regarded as a nonlinear partial differential equation for some\ngenerating function S. Thus, the problem is become how to choose a time-dependent c anonical\ntransformation Ψ : T∗Q×R→T∗Q×R,which transforms the dynamical vector field of a time-\ndependent Hamiltonian system to equilibrium, such that the generating function Sof Ψ satisfies\nthe time-dependent Hamilton-Jacobi equation. In particul ar, for the time-independent Hamilto-\nnian system, ones may look for a symplectic map as the canonic al transformation. This work offers\nan important idea that one can use the dynamical vector field o f a Hamiltonian system to describe\nHamilton-Jacobi equation. In consequence, if assume that γ:Q→T∗Qis a closed one-form on\nQ, and define that Xγ\nH=TπQ·XH·γ, whereXHis the dynamical vector field of Hamiltonian\nsystem ( T∗Q,ω,H), then the fact that Xγ\nHandXHareγ-related, that is, Tγ·Xγ\nH=XH·γis\nequivalent that d(H·γ) = 0,which is given in Cari˜ nena et al [ 5,6]. Motivated by the above re-\nsearch work, Wang in [ 28] prove an important lemma, which is a modification for the cor responding\nresult of Abraham and Marsden in [ 1], such that we can derive precisely the geometric constrain t\nconditions of the regular reduced symplectic forms for the d ynamical vector fields of a regular re-\nducible Hamiltonian system on the cotangent bundle of a confi guration manifold, which are called\n2the Type I and Type II of Hamilton-Jacobi equation, because t hey are the development of the\nabove classical Hamilton-Jacobi equation given by Theorem 1.1, see Abraham and Marsden [ 1] and\nWang [28]. Moreover, Le´ on and Wang in [ 14] generalize the above results to the nonholonomic\nHamiltonian system and the nonholonomic reducible Hamilto nian system on a cotangent bundle,\nby using the distributional Hamiltonian system and the redu ced distributional Hamiltonian system.\nIn order to describe the impact of different geometric structu res and constraints for the dynam-\nics of a Hamiltonian system, in the following we first conside r the magnetic Hamiltonian system.\nDefine a magnetic symplectic form ωB=ω−π∗\nQB,and theπ∗\nQBis called a magnetic term on T∗Q,\nwhereωis the usual canonical symplectic form on T∗Q, andBis the closed two-form on Q, and\nthe map π∗\nQ:T∗Q→T∗T∗Q. A magnetic Hamiltonian system is a Hamiltonian system defin ed\nby the magnetic symplectic form, which is a canonical Hamilt onian system coupling the action of\na magnetic field B. Under the impact of magnetic term π∗\nQB, the magnetic symplectic form ωB,\nin general, is not the canonical symplectic form on T∗Q, we cannot prove the Hamilton-Jacobi\ntheorem for the magnetic Hamiltonian system just like same a s the above Theorem 1.1. We have\nto look for a new way. In this paper, we first drive precisely th e geometric constraint conditions of\nmagnetic symplectic form for the magnetic Hamiltonian vect or field. These conditions are called\nthe Type I and Type II of Hamilton-Jacobi equation, which are the development of the Type I and\nType II of Hamilton-Jacobi equation for a Hamiltonian syste m given in Wang [ 28].\nNext, we consider the magnetic Hamiltonian system with nonh olonomic constraint, which is\ncalled the nonholonomic magnetic Hamiltonian system. In me chanics, it is very often that many\nsystems have constraints, and usually, under the restricti on given by nonholonomic constraint, in\ngeneral, the dynamical vector field of a nonholonomic magnet ic Hamiltonian system may not be\nHamiltonian. Thus, we can not describe the Hamilton-Jacobi equations for nonholonomic magnetic\nHamiltonian system from the viewpoint of generating functi on as in the classical Hamiltonian case,\nthat is, we cannot prove the Hamilton-Jacobi theorem for the nonholonomic magnetic Hamilto-\nnian system, just like same as the above Theorem 1.1. In this p aper, by analyzing carefully the\nstructure for the nonholonomic dynamical vector field, we fir st give a geometric formulation of the\ndistributional magnetic Hamiltonian system for the nonhol onomic magnetic Hamiltonian system,\nwhich is determined by a non-degenerate distributional two -form induced from the magnetic sym-\nplectic form. The distributional magnetic Hamiltonian sys tem is not Hamiltonian, however, it is\na dynamical system closely related to a magnetic Hamiltonia n system. Then we drive precisely\ntwo types of Hamilton-Jacobi equation for the distribution al magnetic Hamiltonian system, which\nare the development of the Type I and Type II of Hamilton-Jaco bi equation for a distributional\nHamiltonian system given in Le´ on and Wang [ 14].\nThirdly, it is a natural problem to consider the nonholonomi c magnetic Hamiltonian system\nwith symmetry. In this paper, we generalize the above result s to nonholonomic reducible mag-\nnetic Hamiltonian system with symmetry. By using the method of nonholonomic reduction given\nin Bates and ´Sniatycki [ 3], and analyzing carefully the structure for the nonholonom ic reduced\ndynamical vector field, we first give a geometric formulation of the nonholonomic reduced distri-\nbutional magnetic Hamiltonian system. Since the nonholono mic reduced distributional magnetic\nHamiltonian system is not yet Hamiltonian, but, it is a dynam ical system closely related to a mag-\nnetic Hamiltonian system. Then we can derive precisely the g eometric constraint conditions of the\nnon-degenerate, and nonholonomic reduced distributional two-form for the nonholonomic reducible\ndynamical vector field, that is, the two types of Hamilton-Ja cobi equation for the nonholonomic\nreduced distributional magnetic Hamiltonian system. Thes e research work reveal the deeply inter-\nnal relationships of the magnetic symplectic structure, no nholonomic constraint, the induced (resp.\nreduced) distributional two-forms, and the dynamical vect or fields of the nonholonomic magnetic\n3Hamiltonian system.\nThe paper is organized as follows. In section 2 we first give so me definitions and basic facts\nabout the magnetic Hamiltonian system, the nonholonomic co nstraint, the nonholonomic magnetic\nHamiltonian system andthedistributional magnetic Hamilt onian system, whichwill beusedinsub-\nsequent sections. In section 3, for a magnetic Hamiltonian s ystem defined by a magnetic symplectic\nform, we first drive precisely the geometric constraint cond itions of magnetic symplectic form for\nthe magnetic Hamiltonian vector field. which are called the T ype I and Type II of Hamilton-Jacobi\nequation. In section 4, we derive two types of Hamilton-Jaco bi equation for a distributional mag-\nnetic Hamiltonian system, by the analysis and calculation i n detail. The nonholonomic reducible\nmagnetic Hamiltonian system with symmetry is considered in section 5, and derive precisely the\ngeometric constraint conditions of the non-degenerate, an d nonholonomic reduced distributional\ntwo-form for the nonholonomic reducible dynamical vector fi eld, that is, the two types of Hamilton-\nJacobi equations for the nonholonomic reduced distributio nal magnetic Hamiltonian system. These\nresearch work develop Hamilton-Jacobi theory for the nonho lonomic magnetic Hamiltonian system,\nas well as with symmetry, and make us have much deeper underst anding and recognition for the\nstructures of the nonholonomic magnetic Hamiltonian syste ms.\n2 Nonholonomic Magnetic Hamiltonian System\nIn this section we first give some definitions and basic facts a bout the magnetic Hamiltonian\nsystem, the nonholonomic constraint and the nonholonomic m agnetic Hamiltonian system. More-\nover, by analyzing carefully the structure for the nonholon omic dynamical vector field, we give a\ngeometric formulation of distributional magnetic Hamilto nian system, which is determined by a\nnon-degenerate distributional two-form induced from the m agnetic symplectic form. All of them\nwill be used in subsequent sections.\nLetQbe ann-dimensional smooth manifold and TQthe tangent bundle, T∗Qthe cotangent\nbundle with a canonical symplectic form ωand the projection πQ:T∗Q→Qinduces the map π∗\nQ:\nT∗Q→T∗T∗Q.We consider the magnetic symplectic form ωB=ω−π∗\nQB,whereωis the canonical\nsymplectic form on T∗Q, andBis the closed two-form on Q, and the π∗\nQBis called a magnetic term\nonT∗Q. A magnetic Hamiltonian system is a triple ( T∗Q,ωB,H), which is a Hamiltonian system\ndefined by the magnetic symplectic form ωB, that is, a canonical Hamiltonian system coupling the\naction of a magnetic field B. For a given Hamiltonian H, the dynamical vector field XB\nH, which is\ncalled the magnetic Hamiltonian vector field, satisfies the m agnetic Hamilton’s equation, that is,\niXB\nHωB=dH. In canonical cotangent bundle coordinates, for any q∈Q,(q,p)∈T∗Q,we have\nthat\nω=n/summationdisplay\ni=1dqi∧dpi, B=n/summationdisplay\ni,j=1Bijdqi∧dqj,dB= 0,\nωB=ω−π∗\nQB=n/summationdisplay\ni=1dqi∧dpi−n/summationdisplay\ni,j=1Bijdqi∧dqj,\nand the magnetic Hamiltonian vector field XB\nHwith respect to the magnetic symplectic form ωB\ncan be expressed that\nXB\nH=n/summationdisplay\ni=1(∂H\n∂pi∂\n∂qi−∂H\n∂qi∂\n∂pi)−n/summationdisplay\ni,j=1Bij∂H\n∂pj∂\n∂pi.\nSee Marsden et al. [ 17].\n4In order to describe the nonholonomic magnetic Hamiltonian system, in the following we first\ngive the completeness and regularity conditions for nonhol onomic constraints of a mechanical sys-\ntem, see Le´ on and Wang [ 14]. In fact, in order to describe the dynamics of a nonholonomi c\nmechanical system, we need some restriction conditions for nonholonomic constraints of the sys-\ntem. At first, we note that the set of Hamiltonian vector fields forms a Lie algebra with respect\nto the Lie bracket, since X{f,g}=−[Xf,Xg].But, the Lie bracket operator, in general, may not\nbe closed on the restriction of a nonholonomic constraint. T hus, we have to give the following\ncompleteness condition for nonholonomic constraints of a s ystem.\nD-completeness LetQbe a smooth manifold and TQits tangent bundle. A distribution\nD ⊂TQis said to be completely nonholonomic (or bracket-generating) if Dalong with all of\nits iterated Lie brackets [ D,D],[D,[D,D]],···,spans the tangent bundle TQ. Moreover, we con-\nsider anonholonomicmechanical system on Q, whichis given bya Lagrangian function L:TQ→R\nsubject to constraints determined by a nonholonomic distri butionD ⊂TQon the configuration\nmanifold Q. Then the nonholonomic system is said to be completely nonholonomic , if the\ndistribution D ⊂TQdetermined by the nonholonomic constraints is completely n onholonomic.\nD-regularity In the following we always assume that Qis a smooth manifold with coordi-\nnates (qi), andTQits tangent bundle with coordinates ( qi,˙qi), andT∗Qits cotangent bundle\nwith coordinates ( qi,pj), which are the canonical cotangent coordinates of T∗Qandω=dqi∧dpi\nis canonical symplectic form on T∗Q. If the Lagrangian L:TQ→Ris hyperregular, that is,\nthe Hessian matrix ( ∂2L/∂˙qi∂˙qj) is nondegenerate everywhere, then the Legendre transform ation\nFL:TQ→T∗Qis a diffeomorphism. In this case the Hamiltonian H:T∗Q→Ris given by\nH(q,p) = ˙q·p−L(q,˙q) with Hamiltonian vector field XH, which is defined by the Hamilton’s\nequation iXHω=dH, andM=FL(D) is a constraint submanifold in T∗Q. In particular, for the\nnonholonomic constraint D ⊂TQ, the Lagrangian Lis said to be D-regular , if the restriction of\nHessian matrix ( ∂2L/∂˙qi∂˙qj) onDis nondegenerate everywhere. Moreover, anonholonomic sys tem\nis said to be D-regular , if its Lagrangian LisD-regular . Note that the restriction of a positive\ndefinite symmetric bilinear form to a subspace is also positi ve definite, and hence nondegenerate.\nThus, for a simple nonholonomic mechanical system, that is, whose Lagrangian is the total kinetic\nenergy minus potential energy, it is D-regular automatically.\nA nonholonomic magnetic Hamiltonian system is a 4-tuple ( T∗Q,ωB,D,H), which is a mag-\nnetic Hamiltonian system with a D-completely and D-regularly nonholonomic constraint D ⊂TQ.\nUnder the restriction given by constraint, in general, the d ynamical vector field of a nonholonomic\nmagnetic Hamiltonian system may not be magnetic Hamiltonia n, however the system is a dynam-\nical system closely related to a magnetic Hamiltonian syste m. In the following we shall derive\na distributional magnetic Hamiltonian system of the nonhol onomic magnetic Hamiltonian system\n(T∗Q,ωB,D,H), by analyzing carefully the structure for the nonholonomi c dynamical vector field\nsimilar to the method in Le´ on and Wang [ 14]. It is worthy of noting that the leading distributional\nHamiltonian system is also called a semi-Hamiltonian syste m in Patrick [ 23].\nWe consider that the constraint submanifold M=FL(D)⊂T∗QandiM:M →T∗Qis the\ninclusion, the symplectic form ωB\nM=i∗\nMωB, is induced from the magnetic symplectic form ωBon\nT∗Q. We define the distribution Fas the pre-image of the nonholonomic constraints Dfor the\nmapTπQ:TT∗Q→TQ, that is, F= (TπQ)−1(D)⊂TT∗Q,which is a distribution along M, and\nF◦:={α∈T∗T∗Q|< α,v > = 0,∀v∈TT∗Q}is the annihilator of FinT∗T∗Q|M. We consider\nthe following nonholonomic constraints condition\n(iXωB−dH)∈ F◦, X∈TM, (2.1)\n5from Cantrijn et al. [ 4], we know that there exists an unique nonholonomic vector fie ldXnsatisfy-\ning the above condition (2 .1), if the admissibility condition dim M= rankFand the compatibility\ncondition TM∩F⊥={0}hold, where F⊥denotes the magnetic symplectic orthogonal of Fwith\nrespect to the magnetic symplectic form ωBonT∗Q. In particular, when we consider the Whitney\nsum decomposition T(T∗Q)|M=TM⊕F⊥and the canonical projection P:T(T∗Q)|M→TM,\nthen we have that Xn=P(XB\nH).\nFrom the condition (2.1) we know that the nonholonomic vecto r field, in general, may not\nbe magnetic Hamiltonian, because of the restriction of nonh olonomic constraints. But, we hope\nto study the dynamical vector field of nonholonomic magnetic Hamiltonian system by using the\nsimilar method of studying magnetic Hamiltonian vector fiel d. From Le´ on and Wang [ 14] and\nBates and ´Sniatycki [ 3], by using the similar method, we can define the distribution K=F ∩TM.\nandK⊥=F⊥∩TM,whereK⊥denotes the magnetic symplectic orthogonal of Kwith respect\nto the magnetic symplectic form ωB, and the admissibility condition dim M= rankFand the\ncompatibility condition TM∩F⊥={0}hold, then we know that the restriction of the symplectic\nformωB\nMonT∗Mfibrewise to the distribution K, that is, ωB\nK=τK·ωB\nMis non-degenerate, where\nτKis the restriction map to distribution K. It is worthy of noting that ωB\nKis not a true two-form on\na manifold, so it does not make sense to speak about it being cl osed. We call ωB\nKas a distributional\ntwo-form to avoid any confusion. Because ωB\nKis non-degenerate as a bilinear form on each fibre of\nK, there exists a vector field XB\nKonMwhich takes values in the constraint distribution K, such\nthat the distributional magnetic Hamiltonian equation\niXB\nKωB\nK=dHK (2.2)\nholds, where dHKis the restriction of dHMtoK, and the function HKsatisfiesdHK=τK·dHM,\nandHM=τM·His the restriction of HtoM. Moreover, from the distributional magnetic Hamil-\ntonian equation (2.2), we have that XB\nK=τK·XB\nH.Then the triple ( K,ωB\nK,HK) is a distributional\nmagnetic Hamiltonian system of the nonholonomic magnetic H amiltonian system ( T∗Q,ωB,D,H).\nThus, the geometric formulation of the distributional magn etic Hamiltonian system may be sum-\nmarized as follows.\nDefinition 2.1 (Distributional Magnetic Hamiltonian System) Assume that the 4-tuple (T∗Q,ωB,\nD,H)is aD-completely and D-regularly nonholonomic magnetic Hamiltonian system, whe re the\nmagnetic symplectic form ωB=ω−π∗\nQBonT∗Q, andωis the canonical symplectic form on T∗Q\nandBis a closed two-form on Q, andD ⊂TQis aD-completely and D-regularly nonholonomic\nconstraint of the system. If there exist a distribution K, an associated non-degenerate distribu-\ntional two-form ωB\nKinduced by the magnetic symplectic form ωBand a vector field XB\nKon the\nconstraint submanifold M=FL(D)⊂T∗Q, such that the distributional magnetic Hamiltonian\nequation iXB\nKωB\nK=dHKholds, where dHKis the restriction of dHMtoKand the function HK\nsatisfies dHK=τK·dHMas defined above, then the triple (K,ωB\nK,HK)is called a distributional\nmagnetic Hamiltonian system of the nonholonomic magnetic H amiltonian system (T∗Q,ωB,D,H),\nandXB\nKis called a nonholonomic dynamical vector field of the distri butional magnetic Hamiltonian\nsystem(K,ωB\nK,HK). Under the above circumstances, we refer to (T∗Q,ωB,D,H)as a nonholo-\nnomic magnetic Hamiltonian system with an associated distr ibutional magnetic Hamiltonian system\n(K,ωB\nK,HK).\nMoreover, in section 5, we consider the nonholonomic magnet ic Hamiltonian system with sym-\nmetry. Byusingthesimilarmethodfornonholonomicreducti ongiveninBatesand ´Sniatycki [ 3]and\nLe´ on and Wang [ 14], and analyzing carefully the structure for the nonholonom ic reduced dynam-\nical vector field, we also give a geometric formulation of the nonholonomic reduced distributional\nmagnetic Hamiltonian system.\n63 Hamilton-Jacobi Equation of Magnetic Hamiltonian System\nIn order to describe the impact of different geometric structu res and constraints for the Hamilton-\nJacobi theory, in this paper, we shall give two types of Hamil ton-Jacobi equations for the magnetic\nHamiltonian system, the distributional magnetic Hamilton ian system and the nonholonomic re-\nduced distributional magnetic Hamiltonian system.\nIn this section, we first derive precisely the geometric cons traint conditions of the magnetic\nsymplectic form for the dynamical vector field of a magnetic H amiltonian system, that is, Type I\nand Type II of Hamilton-Jacobi equation for the magnetic Ham iltonian system. In order to do this,\nin the following we first give some important notions and prov e a key lemma, which is an important\ntool for the proofs of two types of Hamilton-Jacobi theorem f or the magnetic Hamiltonian system.\nDenote by Ωi(Q) the set of all i-forms on Q,i= 1,2.For any γ∈Ω1(Q), q∈Q,then\nγ(q)∈T∗\nqQ,and we can define a map γ:Q→T∗Q, q→(q,γ(q)).Hence we say often that the\nmapγ:Q→T∗Qis an one-form on Q. If the one-form γis closed, then dγ(x,y) = 0,∀x,y∈TQ.\nNote that for any v,w∈TT∗Q,we have that dγ(TπQ(v),TπQ(w)) =π∗(dγ)(v,w) is a two-form\non the cotangent bundle T∗Q, whereπ∗:T∗Q→T∗T∗Q.Thus, in the following we can give a\nweaker notion.\nDefinition 3.1 The one-form γis called to be closed with respect to TπQ:TT∗Q→TQ,if for\nanyv,w∈TT∗Q,we have that dγ(TπQ(v),TπQ(w)) = 0.\nFor the one-form γ:Q→T∗Q,dγis a two-form on Q. Assume that Bis a closed two-\nform on Q, we say that the γsatisfies condition dγ=−B, if for any x,y∈TQ, we have that\n(dγ+B)(x,y) = 0.In the following we can give a new notion.\nDefinition 3.2 Assume that γ:Q→T∗Qis an one-form on Q, we say that the γsatisfies\ncondition that dγ=−Bwith respect to TπQ:TT∗Q→TQ,if for any v,w∈TT∗Q,we have that\n(dγ+B)(TπQ(v),TπQ(w)) = 0.\nFrom the above definition we know that, if γsatisfies condition dγ=−B, then it must satisfy\ncondition dγ=−Bwith respect to TπQ:TT∗Q→TQ.Conversely, if γsatisfies condition\ndγ=−Bwith respect to TπQ:TT∗Q→TQ,then it may not satisfy condition dγ=−B. We can\nprove a general result as follows, which states that the noti on thatγsatisfies condition dγ=−B\nwith respect to TπQ:TT∗Q→TQ,is not equivalent to the notion that γsatisfies condition\ndγ=−B.\nProposition 3.3 Assume that γ:Q→T∗Qis an one-form on Qand it doesn’t satisfy condition\ndγ=−B. we define the set N, which is a subset of TQ, such that the one-form γonNsatisfies the\ncondition that for any x,y∈N,(dγ+B)(x,y)/ne}ationslash= 0.Denote by Ker(TπQ) ={u∈TT∗Q|TπQ(u) =\n0},andTγ:TQ→TT∗Qis the tangent map of γ:Q→T∗Q.IfTγ(N)⊂Ker(TπQ),thenγ\nsatisfies condition dγ=−Bwith respect to TπQ:TT∗Q→TQ.\nProof: In fact, for any v,w∈TT∗Q,ifTπQ(v)/∈N,orTπQ(w))/∈N,then by the defini-\ntion ofN, we know that ( dγ+B)(TπQ(v),TπQ(w)) = 0; If TπQ(v)∈N,andTπQ(w))∈N,\nfrom the condition Tγ(N)⊂Ker(TπQ),we know that TπQ·Tγ·TπQ(v) =TπQ(v) = 0,\nandTπQ·Tγ·TπQ(w) =TπQ(w) = 0,where we have used the relation πQ·γ·πQ=πQ,\nand hence ( dγ+B)(TπQ(v),TπQ(w)) = 0.Thus, for any v,w∈TT∗Q,we have always that\n(dγ+B)(TπQ(v),TπQ(w)) = 0,that is,γsatisfies condition dγ=−Bwith respect to TπQ:\nTT∗Q→TQ./squaresolid\n7From the above Definition 3.1 and Definition 3.2, we know that, whenB= 0, the notion that,\nγsatisfies condition dγ=−Bwith respect to TπQ:TT∗Q→TQ,become the notion that γis\nclosed with respect to TπQ:TT∗Q→TQ.Now, we can prove the following lemma, which is a\ngeneralization of a corresponding to lemma given by Wang [ 28], and the lemma is a very important\ntool for our research.\nLemma 3.4 Assume that γ:Q→T∗Qis an one-form on Q, andλ=γ·πQ:T∗Q→T∗Q.For\nthe magnetic symplectic form ωB=ω−π∗\nQBonT∗Q, whereωis the canonical symplectic form on\nT∗Q, andBis a closed two-form on Q, then we have that the following two assertions hold.\n(i)For any v,w∈TT∗Q, λ∗ωB(v,w) =−(dγ+B)(TπQ(v), TπQ(w));\n(ii)For any v,w∈TT∗Q, ωB(Tλ·v,w) =ωB(v,w−Tλ·w)−(dγ+B)(TπQ(v), TπQ(w)).\nProof:We first prove the assertion (i). Since ωis the canonical symplectic form on T∗Q, we know\nthat there is an unique canonical one-form θ, such that ω=−dθ.From the Proposition 3.2.11 in\nAbraham and Marsden [ 1], we have that for the one-form γ:Q→T∗Q, γ∗θ=γ.Then we can\nobtain that for any x,y∈TQ,\nγ∗ω(x,y) =γ∗(−dθ)(x,y) =−d(γ∗θ)(x,y) =−dγ(x,y).\nNote that λ=γ·πQ:T∗Q→T∗Q,andλ∗=π∗\nQ·γ∗:T∗T∗Q→T∗T∗Q,then we have that for\nanyv,w∈TT∗Q,\nλ∗ω(v,w) =λ∗(−dθ)(v,w) =−d(λ∗θ)(v,w) =−d(π∗\nQ·γ∗θ)(v,w)\n=−d(π∗\nQ·γ)(v,w) =−dγ(TπQ(v), TπQ(w)).\nHence, we have that\nλ∗ωB(v,w) =λ∗ω(v,w)−λ∗·π∗\nQB(v,w)\n=−dγ(TπQ(v), TπQ(w))−(πQ·γ·πQ)∗B(v,w)\n=−dγ(TπQ(v), TπQ(w))−π∗\nQB(v,w)\n=−(dγ+B)(TπQ(v), TπQ(w)),\nwhere we have used the relation πQ·γ·πQ=πQ.It follows that the assertion (i) holds.\nNext, we prove the assertion (ii). For any v,w∈TT∗Q,note that v−T(γ·πQ)·vis vertical,\nbecause\nTπQ(v−T(γ·πQ)·v) =TπQ(v)−T(πQ·γ·πQ)·v=TπQ(v)−TπQ(v) = 0,\nThus,ω(v−T(γ·πQ)·v,w−T(γ·πQ)·w) = 0,and hence,\nω(T(γ·πQ)·v, w) =ω(v, w−T(γ·πQ)·w)+ω(T(γ·πQ)·v, T(γ·πQ)·w).\nHowever, the second term on the right-hand side is given by\nω(T(γ·πQ)·v, T(γ·πQ)·w) =γ∗ω(TπQ(v), TπQ(w)) =−dγ(TπQ(v), TπQ(w)),\nIt follows that\nω(Tλ·v,w) =ω(T(γ·πQ)·v, w)\n=ω(v, w−T(γ·πQ)·w)−dγ(TπQ(v), TπQ(w))\n=ω(v,w−Tλ·w)−dγ(TπQ(v), TπQ(w)).\n8Hence, we have that\nωB(Tλ·v,w) =ω(Tλ·v,w)−π∗\nQB(Tλ·v,w)\n=ω(v,w−Tλ·w)−dγ(TπQ(v), TπQ(w))−B(TπQ·Tλ·v, TπQ(w))\n=ωB(v,w−Tλ·w)+π∗\nQB(v,w−Tλ·w)\n−dγ(TπQ(v), TπQ(w))−B(T(πQ·λ)·v, TπQ(w))\n=ωB(v,w−Tλ·w)+π∗\nQB(v,w)−B(TπQ(v), TπQ·Tλ·w)\n−dγ(TπQ(v), TπQ(w))−B(T(πQ·γ·πQ)·v, TπQ(w))\n=ωB(v,w−Tλ·w)+π∗\nQB(v,w)−B(TπQ(v), T(πQ·λ)·w)\n−dγ(TπQ(v), TπQ(w))−B(TπQ(v), TπQ(w))\n=ωB(v,w−Tλ·w)+π∗\nQB(v,w)−B(TπQ(v), TπQ(w))−(dγ+B)(TπQ(v), TπQ(w))\n=ωB(v,w−Tλ·w)−(dγ+B)(TπQ(v), TπQ(w)).\nThus, the assertion (ii) holds. /squaresolid\nSince a magnetic Hamiltonian system is a Hamiltonian system defined by the magnetic sym-\nplectic form, and it is a canonical Hamiltonian system coupl ing the action of a magnetic field B.\nUsually, under the impact of magnetic term π∗\nQB, the magnetic symplectic form ωB=ω−π∗\nQB,\nin general, is not the canonical symplectic form ωonT∗Q, we cannot prove the Hamilton-Jacobi\ntheorem for the magnetic Hamiltonian system just like same a s the above Theorem 1.1. But, in\nthe following we can give precisely the geometric constrain t conditions of magnetic symplectic form\nfor the dynamical vector field of the magnetic Hamiltonian sy stem, that is, Type I and Type II\nof Hamilton-Jacobi equation for the magnetic Hamiltonian s ystem. At first, for a given magnetic\nHamiltonian system ( T∗Q,ωB,H) onT∗Q, by using the above Lemma 3.4, magnetic symplectic\nformωBand the magnetic Hamiltonian vector field XB\nH, we can derive precisely the following type\nI of Hamilton-Jacobi equation for the magnetic Hamiltonian system (T∗Q,ωB,H).\nTheorem 3.5 (Type I of Hamilton-Jacobi Theorem for a Magnetic Hamiltonia n System) For a\ngiven magnetic Hamiltonian system (T∗Q,ωB,H)with the magnetic symplectic form ωB=ω−π∗\nQB\nonT∗Q, whereωis the canonical symplectic form on T∗QandBis a closed two-form on Q, assume\nthatγ:Q→T∗Qis an one-form on Q, andXγ=TπQ·XB\nH·γ, whereXB\nHis the dynamical\nvector field of the magnetic Hamiltonian system (T∗Q,ωB,H), that is, the magnetic Hamiltonian\nvector field. If the one-form γ:Q→T∗Qsatisfies the condition that dγ=−Bwith respect to\nTπQ:TT∗Q→TQ,thenγis a solution of the equation Tγ·Xγ=XB\nH·γ.The equation is called\nthe Type I of Hamilton-Jacobi equation for the magnetic Hami ltonian system (T∗Q,ωB,H). Here\nthe maps involved in the theorem are shown in the following Di agram-1.\nT∗QπQ/d47/d47Q\nXγ\n/d15/d15γ/d47/d47T∗Q\nXB\nH/d15/d15\nT(T∗Q)TQTγ/d111/d111 T(T∗Q)TπQ/d111/d111\nDiagram-1\nProof: If we take that v=XB\nH·γ∈TT∗Q,and for any w∈TT∗Q, Tπ Q(w)/ne}ationslash= 0,from Lemma\n3.4(ii) and dγ=−Bwithrespectto TπQ:TT∗Q→TQ,thatis, (dγ+B)(TπQ·XB\nH·γ, TπQ·w) = 0,\n9we have that\nωB(Tγ·Xγ, w) =ωB(Tγ·TπQ·XB\nH·γ, w) =ωB(T(γ·πQ)·XB\nH·γ, w)\n=ωB(XB\nH·γ, w−T(γ·πQ)·w)−(dγ+B)(TπQ·XB\nH·γ, TπQ·w)\n=ωB(XB\nH·γ, w)−ωB(XB\nH·γ, Tλ·w).\nHence, we have that\nωB(Tγ·Xγ, w)−ωB(XB\nH·γ, w) =−ωB(XB\nH·γ, Tλ·w). (3.1)\nIfγsatisfies the equation Tγ·Xγ=XB\nH·γ,from Lemma 3.4(i) we know that the right side of (3.1)\nbecomes that\nωB(XB\nH·γ, Tλ·w) =ωB(Tγ·Xγ, Tλ·w)\n=ωB(Tγ·TπQ·XB\nH·γ, Tλ·w)\n=ωB(Tλ·XB\nH·γ, Tλ·w)\n=λ∗ωB(XB\nH·γ, w)\n=−(dγ+B)(TπQ·XB\nH·γ, TπQ·w) = 0,\nsinceγ:Q→T∗Qsatisfies the condition that dγ=−Bwith respect to TπQ:TT∗Q→TQ.\nBut, because the magnetic symplectic form ωBis non-degenerate, the left side of (3.1) equals zero,\nonly when γsatisfies the equation Tγ·Xγ=XB\nH·γ.Thus, if the one-form γ:Q→T∗Qsatisfies\nthe condition that dγ=−Bwith respect to TπQ:TT∗Q→TQ,thenγmust be a solution of\nthe Type I of Hamilton-Jacobi equation Tγ·Xγ=XB\nH·γ,for the magnetic Hamiltonian system\n(T∗Q,ωB,H)./squaresolid\nIt is worthy of noting that, when B= 0, in this case the magnetic symplectic form ωBis just the\ncanonical symplectic form ωonT∗Q, and the magnetic Hamiltonian system ( T∗Q,ωB,H) becomes\nthe Hamiltonian system ( T∗Q,ω,H) with the canonical symplectic form ω, and the condition that\nthe one-form γ:Q→T∗Qsatisfies the condition, dγ=−Bwith respect to TπQ:TT∗Q→TQ,\nbecomes the condition that γis closed with respect to TπQ:TT∗Q→TQ.Thus, from above\nTheorem 3.5, we can obtain Theorem 2.5 in Wang [ 28], that is. the Type I of Hamilton-Jacobi\ntheorem for a Hamiltonian system. On the other hand, from the proof of Theorem 2.5 in Wang [ 28],\nwe know that if an one-form γ:Q→T∗Qis not closed on Qwith respect to TπQ:TT∗Q→TQ,\nthenγis not a solution of the Type I of Hamilton-Jacobi equation Tγ·Xγ=XH·γ.But, note\nthat, ifγ:Q→T∗Qis not closed on Qwith respect to TπQ:TT∗Q→TQ,that is, there exist\nv,w∈TT∗Q,such that dγ(TπQ(v), TπQ(w))/ne}ationslash= 0,and hence γis not yet closed on Q. However,\nbecause d·dγ=d2γ= 0,and hence the dγis a closed two-form on Q. Thus, we can construct\na magnetic symplectic form on T∗Q, that is, ωB=ω+π∗\nQ(dγ) =ω−π∗\nQB,whereB=−dγ,\nandωis the canonical symplectic form on T∗Q, andπ∗\nQ:T∗Q→T∗T∗Q. Moreover, we hope to\nlook for a new magnetic Hamiltonian system, such that γis a solution of the Type I of Hamilton-\nJacobi equation for the new magnetic Hamiltonian system. In fact, for a given Hamiltonian system\n(T∗Q,ω,H) with the canonical symplectic form ωonT∗Q, andγ:Q→T∗Qis an one-form on\nQ, and it is not closed with respect to TπQ:TT∗Q→TQ.Then we can construct a magnetic\nsymplectic form on T∗Q,ωB=ω+π∗\nQ(dγ),whereB=−dγ,and a magnetic Hamiltonian system\n(T∗Q,ωB,H), its dynamical vector field is given by XB\nH, which satisfies the magnetic Hamiltonian\nequation, that is, iXB\nHωB=dH. In this case, for any x,y∈TQ,we have that ( dγ+B)(x,y) = 0,\nand hence for any v,w∈TT∗Q,we have ( dγ+B)(TπQ(v),TπQ(w)) = 0,that is, the one-form\nγ:Q→T∗Qsatisfies the condition, dγ=−Bwith respect to TπQ:TT∗Q→TQ.Thus, by using\nLemma 3.4 and the magnetic Hamiltonian vector field XB\nH, from Theorem 3.5 we can obtain the\nfollowing Theorem 3.6.\n10Theorem 3.6 For a given Hamiltonian system (T∗Q,ω,H)with the canonical symplectic form ω\nonT∗Q, and assume that the one-form γ:Q→T∗Qis not closed with respect to TπQ:TT∗Q→\nTQ.Then one can construct a magnetic symplectic form on T∗Q, that is, ωB=ω+π∗\nQ(dγ),where\nB=−dγ,and a magnetic Hamiltonian system (T∗Q,ωB,H). Denote Xγ=TπQ·XB\nH·γ, where\nXB\nHis the dynamical vector field of the magnetic Hamiltonian sys tem(T∗Q,ωB,H). Thenγis a\nsolution of the Type I of Hamilton-Jacobi equation Tγ·Xγ=XB\nH·γ,for the magnetic Hamiltonian\nsystem(T∗Q,ωB,H).\nNext, for any symplectic map ε:T∗Q→T∗Qwith respect to the magnetic symplectic form\nωB, we can derive precisely the following Type II of Hamilton-J acobi equation for the magnetic\nHamiltonian system ( T∗Q,ωB,H). For convenience, the maps involved in the following theor em\nand its proof are shown in Diagram-2.\nT∗Qε /d47/d47T∗Q\nXB\nH·ε/d15/d15Xε\n/d36/d36❍❍❍❍❍❍❍❍❍πQ/d47/d47Qγ/d47/d47T∗Q\nXB\nH/d15/d15\nT(T∗Q)TQTγ/d111/d111 T(T∗Q)TπQ/d111/d111\nDiagram-2\nTheorem 3.7 (Type II of Hamilton-Jacobi Theorem for a Magnetic Hamiltonia n System) For the\nmagnetic Hamiltonian system (T∗Q,ωB,H)with the magnetic symplectic form ωB=ω−π∗\nQBon\nT∗Q, whereωis the canonical symplectic form on T∗QandBis a closed two-form on Q, assume\nthatγ:Q→T∗Qis an one-form on Q, andλ=γ·πQ:T∗Q→T∗Q, and for any symplectic map\nε:T∗Q→T∗Qwith respect to ωB, denote by Xε=TπQ·XB\nH·ε, whereXB\nHis the dynamical vector\nfield of the magnetic Hamiltonian system (T∗Q,ωB,H), that is, the magnetic Hamiltonian vector\nfield. Then εis a solution of the equation Tε·XB\nH·ε=Tλ·XB\nH·ε,if and only if it is a solution of\nthe equation Tγ·Xε=XB\nH·ε,whereXB\nH·ε∈TT∗Qis the magnetic Hamiltonian vector field of the\nfunction H·ε:T∗Q→R. The equation Tγ·Xε=XB\nH·ε,is called the Type II of Hamilton-Jacobi\nequation for the magnetic Hamiltonian system (T∗Q,ωB,H).\nProof: If we take that v=XB\nH·ε∈TT∗Q,and for any w∈TT∗Q, Tλ(w)/ne}ationslash= 0,from Lemma 3.4\nwe have that\nωB(Tγ·Xε, w) =ωB(Tγ·TπQ·XB\nH·ε, w) =ωB(T(γ·πQ)·XB\nH·ε, w)\n=ωB(XB\nH·ε, w−T(γ·πQ)·w)−(dγ+B)(TπQ(XB\nH·ε), TπQ(w))\n=ωB(XB\nH·ε, w)−ωB(XB\nH·ε, Tλ·w)+λ∗ωB(XB\nH·ε, w)\n=ωB(XB\nH·ε, w)−ωB(XB\nH·ε, Tλ·w)+ωB(Tλ·XB\nH·ε, Tλ·w).\nBecause ε:T∗Q→T∗Qis symplectic with respect to ωB, and hence XB\nH·ε=Tε·XB\nH·ε,alongε.\nFrom the above arguments, we can obtain that\nωB(Tγ·Xε, w)−ωB(XB\nH·ε, w)\n=−ωB(XB\nH·ε, Tλ·w)+ωB(Tλ·XB\nH·ε, Tλ·w)\n=−ωB(Tε·XB\nH·ε, Tλ·w)+ωB(Tλ·XB\nH·ε, Tλ·w)\n=ωB(Tλ·XB\nH·ε−Tε·XB\nH·ε, Tλ·w).\nBecause the magnetic symplectic form ωBis non-degenerate, it follows that Tγ·Xε=XB\nH·ε,is\nequivalent to Tε·XB\nH·ε=Tλ·XB\nH·ε. Thus,εis a solution of the equation Tε·XB\nH·ε=Tλ·XB\nH·ε,\nif and only if it is a solution of the Type II of Hamilton-Jacob i equation Tγ·Xε=XB\nH·ε./squaresolid\n11Remark 3.8 It is worthy of noting that, the Type I of Hamilton-Jacobi equa tionTγ·Xγ=XB\nH·γ,is\nthe equation of the differential one-form γ; and the Type II of Hamilton-Jacobi equation Tγ·Xε=\nXB\nH·ε,is the equation of the symplectic diffeomorphism map ε. When B= 0, in this case the\nmagnetic symplectic form ωBis just the canonical symplectic form ωonT∗Q, and the magnetic\nHamiltonian system is just the canonical Hamiltonian syste m itself. From the above Type I and Type\nII of Hamilton-Jacobi theorems, that is, Theorem 3.5 and Theore m 3.7, we can get the Theorem\n2.5 and Theorem 2.6 in Wang [ 28]. It shows that Theorem 3.5 and Theorem 3.7 can be regarded\nas an extension of two types of Hamilton-Jacobi theorem for H amiltonian system given in [ 28] to\nthat for the magnetic Hamiltonian system.\n4 Hamilton-Jacobi Equation for Distributional Magnetic Ha mil-\ntonian System\nIn this section we shall derive precisely the geometric cons traint conditions of the induced distribu-\ntional two-form for thenonholonomicdynamical vector field of distributional magnetic Hamiltonian\nsystem, that is, the two types of Hamilton-Jacobi equation f or the distributional magnetic Hamil-\ntonian system. In order to do this, in the following we first gi ve some important notions and prove\na key lemma, which is an important tool for the proofs of two ty pes of Hamilton-Jacobi theorem\nfor the distributional magnetic Hamiltonian system.\nAssumethat D ⊂TQis aD-regularly nonholonomic constraint, and the constraint su bmanifold\nM=FL(D)⊂T∗Q, the distribution F= (TπQ)−1(D)⊂TT∗Q,andγ:Q→T∗Qis an one-form\nonQ, andBis a closed two-form on Q, in the following we first introduce two weaker notions.\nDefinition 4.1 (i)The one-form γis called to be closed on Dwith respect to TπQ:TT∗Q→TQ,\nif for any v,w∈TT∗Q,andTπQ(v), TπQ(w)∈ D,we have that dγ(TπQ(v),TπQ(w)) = 0;\n(ii)The one-form γ:Q→T∗Qis called that satisfies condition that dγ=−BonDwith respect\ntoTπQ:TT∗Q→TQ,if for any v,w∈TT∗Q,andTπQ(v), TπQ(w)∈ D,we have that (dγ+\nB)(TπQ(v),TπQ(w)) = 0.\nFrom the above Definition 4.1, we know that, when B= 0, the notion that, γsatisfies condition\nthatdγ=−BonDwith respect to TπQ:TT∗Q→TQ,become the notion that γis closed on\nDwith respect to TπQ:TT∗Q→TQ.On the other hand, it is worthy of noting that the notion\nthatγsatisfies condition that dγ=−BonDwith respect to TπQ:TT∗Q→TQ,is weaker than\nthe notion that γsatisfies condition dγ=−BonD,that is, ( dγ+B)(x,y) = 0,∀x,y∈ D. In\nfact, ifγsatisfies condition dγ=−BonD, then it must satisfy condition that dγ=−BonD\nwith respect to TπQ:TT∗Q→TQ.Conversely, if γsatisfies condition that dγ=−BonDwith\nrespect to TπQ:TT∗Q→TQ,then it may not satisfy condition dγ=−BonD. We can prove a\ngeneral result as follows, which states that the notion that , theγsatisfies condition that dγ=−B\nonDwith respect to TπQ:TT∗Q→TQ,is not equivalent to the notion that γsatisfies condition\ndγ=−BonD.\nProposition 4.2 Assume that γ:Q→T∗Qis an one-form on Qand it doesn’t satisfy condition\ndγ=−BonD. We define the set N, which is a subset of TQ, such that the one-form γon\nNsatisfies the condition that for any x,y∈N,(dγ+B)(x,y)/ne}ationslash= 0.DenoteKer(TπQ) ={u∈\nTT∗Q|TπQ(u) = 0},andTγ:TQ→TT∗Q.IfTγ(N)⊂Ker(TπQ),thenγsatisfies condition\ndγ=−Bwith respect to TπQ:TT∗Q→TQ.and hence γsatisfies condition dγ=−BonDwith\nrespect to TπQ:TT∗Q→TQ.\n12Proof: If theγ:Q→T∗Qdoesn’t satisfy condition dγ=−BonD, then it doesn’t yet satisfy\ncondition dγ=−B. From the proof of Lemma 3.3, for any v,w∈TT∗Q,we have always that\n(dγ+B)(TπQ(v),TπQ(w)) = 0.In particular, for any v,w∈TT∗Q,andTπQ(v), TπQ(w)∈ D,\nwe have ( dγ+B)(TπQ(v),TπQ(w)) = 0.that is,γsatisfies condition that dγ=−BonDwith\nrespect to TπQ:TT∗Q→TQ./squaresolid\nNow, we prove the following Lemma 4.3. It is worthy of noting t hat this lemma and Lemma\n3.4 given in §3 are the important tool for the proofs of the two types of Hami lton-Jacobi theorems\nfor the distributional magnetic Hamiltonian system and the nonholonomic reduced distributional\nmagnetic Hamiltonian system.\nLemma 4.3 Assume that γ:Q→T∗Qis an one-form on Q, andλ=γ·πQ:T∗Q→T∗Q,\nandωis the canonical symplectic form on T∗Q, andωB=ω−π∗\nQBis the magnetic symplectic\nform on T∗Q. If the Lagrangian LisD-regular, and Im (γ)⊂ M=FL(D),then we have that\nXB\nH·γ∈ Falongγ, andXB\nH·λ∈ Falongλ, that is, TπQ(XB\nH·γ(q))∈ Dq,∀q∈Q, and\nTπQ(XB\nH·λ(q,p))∈ Dq,∀q∈Q,(q,p)∈T∗Q.Moreover, if a symplectic map ε:T∗Q→T∗Qwith\nrespect to the magnetic symplectic form ωBsatisfies the condition ε(M)⊂ M,then we have that\nXB\nH·ε∈ Falongε.\nProof:Under the canonical cotangent bundle coordinates, for any q∈Q,(q,p)∈T∗Q,we have\nthat\nXB\nH·γ(q) = (n/summationdisplay\ni=1(∂H\n∂pi∂\n∂qi−∂H\n∂qi∂\n∂pi)−n/summationdisplay\ni,j=1Bij∂H\n∂pj∂\n∂pi)γ(q).\nand\nXB\nH·λ(q,p) = (n/summationdisplay\ni=1(∂H\n∂pi∂\n∂qi−∂H\n∂qi∂\n∂pi)−n/summationdisplay\ni,j=1Bij∂H\n∂pj∂\n∂pi)γ·πQ(q,p).\nThen,\nTπQ(XB\nH·γ(q)) =TπQ(XB\nH·λ(q,p)) =n/summationdisplay\ni=1(∂H\n∂pi∂\n∂qi)γ(q)∈TqQ.\nSince Im( γ)⊂ M,andγ(q)∈ M(q,p)=FL(Dq),from the Lagrangian LisD-regular, and FLis a\ndiffeomorphism, then there exists a point ( q, vq)∈ Dq,such that FL(q, vq) =γ(q).Thus,\nTπQ(XB\nH·γ(q)) =TπQ(XB\nH·λ(q,p)) =FL(q, vq)n/summationdisplay\ni=1(∂H\n∂pi∂\n∂qi)∈ Dq,\nit follows that XB\nH·γ∈ Falongγ, andXB\nH·λ∈ Falongλ. Moreover, for the symplectic map\nε:T∗Q→T∗Qwith respect to the magnetic symplectic form ωB, we have that\nXB\nH·ε(q,p) = (n/summationdisplay\ni=1(∂H\n∂pi∂\n∂qi−∂H\n∂qi∂\n∂pi)−n/summationdisplay\ni,j=1Bij∂H\n∂pj∂\n∂pi)ε(q,p).\nIfεsatisfies the condition ε(M)⊂ M,then for any ( q,p)∈ M(q,p), we have that ε(q,p)∈ M(q,p),\nand there exists a point ( q, vq)∈ Dq,such that FL(q, vq) =ε(q,p).Thus,\nTπQ(XB\nH·ε(q,p)) =n/summationdisplay\ni=1(∂H\n∂pi∂\n∂qi)ε(q,p) =FL(q, vq)n/summationdisplay\ni=1(∂H\n∂pi∂\n∂qi)∈ Dq,\nit follows that XB\nH·ε∈ Falongε./squaresolid\n13We note that for a nonholonomic magnetic Hamiltonian system , under the restriction given\nby nonholonomic constraint, in general, the dynamical vect or field of a nonholonomic magnetic\nHamiltonian system may not be Hamiltonian. On the other hand , since the distributional mag-\nnetic Hamiltonian system is determined by a non-degenerate distributional two-form induced from\nthe magnetic symplectic form, but, the non-degenerate dist ributional two-form is not a ”true two-\nform” on a manifold, and hence the leading distributional ma gnetic Hamiltonian system can not\nbe Hamiltonian. Thus, we can not describe the Hamilton-Jaco bi equations for the nonholonomic\nmagnetic Hamiltonian system from the viewpoint of generati ng function as in the classical Hamil-\ntonian case, that is, we cannot prove the Hamilton-Jacobi th eorem for the nonholonomic magnetic\nHamiltonian system, just like same as the above Theorem 1.1. Since the distributional magnetic\nHamiltonian system is a dynamical system closely related to a magnetic Hamiltonian system, in\nthe following by using Lemma 3.4, Lemma 4.3, and the non-dege nerate distributional two-form ωB\nK\nand the nonholonomic dynamical vector field XB\nKgiven in §2 for the distributional magnetic Hamil-\ntonian system, we can derive precisely the geometric constr aint conditions of the non-degenerate\ndistributional two-form ωB\nKfor the nonholonomic dynamical vector field XB\nK, that is, the two types\nof Hamilton-Jacobi equation for the distributional magnet ic Hamiltonian system ( K,ωB\nK,HK). At\nfirst, we prove the following Type I of Hamilton-Jacobi theor em for the distributional magnetic\nHamiltonian system.\nTheorem 4.4 (Type I of Hamilton-Jacobi Theorem for the Distributional Ma gnetic Hamiltonian\nSystem) For the nonholonomic magnetic Hamiltonian system (T∗Q,ωB,D,H)with an associated\ndistributional magnetic Hamiltonian system (K,ωB\nK,HK), assume that γ:Q→T∗Qis an one-form\nonQ, andXγ=TπQ·XB\nH·γ, whereXB\nHis the magnetic Hamiltonian vector field of the associated\nunconstrained magnetic Hamiltonian system (T∗Q,ωB,H). Moreover, assume that Im (γ)⊂ M=\nFL(D),and Im(Tγ)⊂ K.If the one-form γ:Q→T∗Qsatisfies the condition, dγ=−BonD\nwith respect to TπQ:TT∗Q→TQ,thenγis a solution of the equation Tγ·Xγ=XB\nK·γ.Here\nXB\nKis the nonholonomic dynamical vector field of the distributi onal magnetic Hamiltonian system\n(K,ωB\nK,HK). The equation Tγ·Xγ=XB\nK·γis called the Type I of Hamilton-Jacobi equation for the\ndistributional magnetic Hamiltonian system (K,ωB\nK,HK). Here the maps involved in the theorem\nare shown in the following Diagram-3.\nM\nXB\nK/d15/d15iM/d47/d47T∗Q\nXB\nH/d15/d15πQ/d47/d47Q\nXγ\n/d15/d15γ/d47/d47T∗Q\nXB\nH/d15/d15\nKT(T∗Q)τK/d111/d111 TQTγ/d111/d111 T(T∗Q)TπQ/d111/d111\nDiagram-3\nProof: At first, we note that Im( γ)⊂ M,and Im(Tγ)⊂ K,in this case, ωB\nK·τK=τK·ωB\nM=\nτK·i∗\nM·ωB,along Im( Tγ). Moreover, from the distributional magnetic Hamiltonian equation (2.2),\nwe have that XB\nK=τK·XB\nH,andτK·XB\nH·γ=XB\nK·γ. Thus, using the non-degenerate distributional\ntwo-form ωB\nK, from Lemma 3.4(ii) and Lemma 4.3, if we take that v=XB\nH·γ∈ F,and for any\n14w∈ F, Tλ(w)/ne}ationslash= 0,andτK·w/ne}ationslash= 0,then we have that\nωB\nK(Tγ·Xγ, τK·w) =ωB\nK(τK·Tγ·Xγ, τK·w)\n=τK·i∗\nM·ωB(Tγ·TπQ·XB\nH·γ, w) =τK·i∗\nM·ωB(T(γ·πQ)·XB\nH·γ, w)\n=τK·i∗\nM·(ωB(XB\nH·γ, w−T(γ·πQ)·w)−(dγ+B)(TπQ·XB\nH·γ, TπQ·w))\n=τK·i∗\nM·ωB(XB\nH·γ, w)−τK·i∗\nM·ωB(XB\nH·γ, T(γ·πQ)·w)\n−τK·i∗\nM·(dγ+B)(TπQ·XB\nH·γ, TπQ·w)\n=ωB\nK(τK·XB\nH·γ, τK·w)−ωB\nK(τK·XB\nH·γ, τK·T(γ·πQ)·w)\n−τK·i∗\nM·(dγ+B)(TπQ·XB\nH·γ, TπQ·w)\n=ωB\nK(XB\nK·γ, τK·w)−ωB\nK(XB\nK·γ, τK·Tγ·TπQ(w))\n−τK·i∗\nM·(dγ+B)(TπQ·XB\nH·γ, TπQ·w),\nwhere we have used that τK·Tγ=Tγ,since Im( Tγ)⊂ K,andτK·XB\nH·γ=XB\nK·γ∈ K.Note that\nXB\nH·γ, w∈ F,andTπQ(XB\nH·γ), TπQ(w)∈ D.If the one-form γ:Q→T∗Qsatisfies the condition,\ndγ=−BonDwith respect to TπQ:TT∗Q→TQ,then (dγ+B)(TπQ·XB\nH·γ, TπQ·w) = 0,\nand hence\nτK·i∗\nM·(dγ+B)(TπQ(XB\nH·γ), TπQ(w)) = 0,\nThus, we have that\nωB\nK(Tγ·Xγ, τK·w)−ωB\nK(XB\nK·γ, τK·w) =−ωB\nK(XB\nK·γ, τK·Tγ·TπQ(w)).(4.1)\nIfγsatisfies the equation Tγ·Xγ=XB\nK·γ,from Lemma 3.4(i) we know that the right side of (4.1)\nbecomes that\n−ωB\nK(XB\nK·γ, τK·Tγ·TπQ(w)) =−ωB\nK(Tγ·Xγ, τK·Tγ·TπQ(w))\n=−ωB\nK(τK·Tγ·Xγ, τK·Tγ·TπQ(w))\n=−τK·i∗\nM·ωB(Tγ·TπQ(XB\nH·γ), Tγ·TπQ(w))\n=−τK·i∗\nM·λ∗ωB(XB\nH·γ, w)\n=τK·i∗\nM·(dγ+B)(TπQ·XB\nH·γ, TπQ·w) = 0.\nBecause the distributional two-form ωB\nKis non-degenerate, the left side of (4.1) equals zero, only\nwhenγsatisfies the equation Tγ·Xγ=XB\nK·γ.Thus, if the one-form γ:Q→T∗Qsatisfies the\ncondition that dγ=−BonDwith respect to TπQ:TT∗Q→TQ,thenγmust be a solution of the\nType I of Hamilton-Jacobi equation Tγ·Xγ=XB\nK·γ,for the distributional magnetic Hamiltonian\nsystem (K,ωB\nK,HK)./squaresolid\nIt is worthy of noting that, when B= 0, in this case the magnetic symplectic form ωBis\njust the canonical symplectic form ωonT∗Q, and the nonholonomic magnetic Hamiltonian system\n(T∗Q,ωB,D,H) becomes the nonholonomic Hamiltonian system ( T∗Q,ω,D,H) with the canonical\nsymplectic form ω, and the distributional magnetic Hamiltonian system ( K,ωB\nK,HK) becomes the\ndistributional Hamiltonian system ( K,ωK,HK), and the condition that the one-form γ:Q→T∗Q\nsatisfies the condition that dγ=−BonDwith respect to TπQ:TT∗Q→TQ,becomes that\nγis closed on Dwith respect to TπQ:TT∗Q→TQ.Thus, from above Theorem 4.4, we can\nobtain Theorem 3.5 in Le´ on and Wang [ 14], that is. the Type I of Hamilton-Jacobi theorem for the\ndistributional Hamiltonian system. On the other hand, from the proofs of Theorem 3.5 in Le´ on and\nWang [14], we know that, if the one-form γ:Q→T∗Qis not closed on Dwith respect to TπQ:\nTT∗Q→TQ,then the γis not yet closed on D, that is, dγ(x,y)/ne}ationslash= 0,∀x,y∈ D, and hence γis not\n15yet closed on Q. However, in this case, we note that d·dγ=d2γ= 0,and hence the dγis a closed\ntwo-form on Q. Thus, we can construct a magnetic symplectic form on T∗Q,ωB=ω+π∗\nQ(dγ),\nwhereB=−dγ,. Moreover, we can also construct a nonholonomic magnetic Ha miltonian system\n(T∗Q,ωB,D,H)withanassociateddistributionalmagneticHamiltonians ystem(K,ωB\nK,HK), which\nsatisfies the distributional magnetic Hamiltonian equatio n (2.2), iXB\nKωB\nK=dHK. In this case,\nthe one-form γ:Q→T∗Qsatisfies also the condition that dγ=−BonDwith respect to\nTπQ:TT∗Q→TQ,by using Lemma 3.4, Lemma 4.3, and the magnetic Hamiltonian v ector field\nXB\nH, from Theorem 4.4 we can obtain the following Theorem 4.5.\nTheorem 4.5 For a given nonholonomic Hamiltonian system (T∗Q,ω,D,H)with the canonical\nsymplectic form ωonT∗QandD-completely and D-regularly nonholonomic constraint D ⊂TQ,\nand assume that the one-form γ:Q→T∗Qis not closed on Dwith respect to TπQ:TT∗Q→TQ.\nThen one can construct a magnetic symplectic form on T∗Q,ωB=ω+π∗\nQ(dγ),whereB=−dγ,\nand a nonholonomic magnetic Hamiltonian system (T∗Q,ωB,D,H)with an associated distribu-\ntional magnetic Hamiltonian system (K,ωB\nK,HK). Denote Xγ=TπQ·XB\nH·γ, whereXB\nHis the\ndynamical vector field of the magnetic Hamiltonian system (T∗Q,ωB,H). Thenγis a solution of\nthe Type I of Hamilton-Jacobi equation Tγ·Xγ=XB\nK·γ,for the distributional magnetic Hamilto-\nnian system (K,ωB\nK,HK).\nNext, for any symplectic map ε:T∗Q→T∗Qwith respect to the magnetic symplectic form\nωB, we can prove the following Type II of Hamilton-Jacobi theor em for the distributional magnetic\nHamiltonian system. For convenience, the maps involved in t he following theorem and its proof\nare shown in Diagram-4.\nM\nXB\nK/d15/d15iM/d47/d47T∗Q\nXB\nH·ε/d15/d15Xε\n/d36/d36❍❍❍❍❍❍❍❍❍πQ/d47/d47Qγ/d47/d47T∗Q\nXB\nH/d15/d15\nKT(T∗Q)τK/d111/d111 TQTγ/d111/d111 T(T∗Q)TπQ/d111/d111\nDiagram-4\nTheorem 4.6 (Type II of Hamilton-Jacobi Theorem for a Distributional Magn etic Hamiltonian\nSystem) For the nonholonomic magnetic Hamiltonian system (T∗Q,ωB,D,H)with an associated\ndistributional magnetic Hamiltonian system (K,ωB\nK,HK), assume that γ:Q→T∗Qis an one-form\nonQ, andλ=γ·πQ:T∗Q→T∗Q,and for any symplectic map ε:T∗Q→T∗Qwith respect to the\nmagnetic symplectic form ωB, denote by Xε=TπQ·XB\nH·ε, whereXB\nHis the dynamical vector field\nof the magnetic Hamiltonian system (T∗Q,ωB,H). Moreover, assume that Im (γ)⊂ M=FL(D),\nandε(M)⊂ M,and Im(Tγ)⊂ K.Thenεis a solution of the equation τK·Tε(XB\nH·ε) =Tλ·XB\nH·ε,\nif and only if it is a solution of the equation Tγ·Xε=XB\nK·ε. HereXB\nH·εis the magnetic\nHamiltonian vector field of the function H·ε:T∗Q→R,andXB\nKis the dynamical vector field of\nthe distributional magnetic Hamiltonian system (K,ωB\nK,HK). The equation Tγ·Xε=XB\nK·ε,is\ncalled the Type II of Hamilton-Jacobi equation for the distri butional magnetic Hamiltonian system\n(K,ωB\nK,HK).\nProof: In the same way, we note that Im( γ)⊂ M,and Im(Tγ)⊂ K,in this case, ωB\nK·τK=\nτK·ωB\nM=τK·i∗\nM·ωB,along Im( Tγ). Moreover, from the distributional magnetic Hamiltonian\nequation (2.2), we have that XB\nK=τK·XB\nH,andτK·XB\nH·ε=XB\nK·ε. Note that ε(M)⊂ M,and\nTπQ(XB\nH·ε(q,p))∈ Dq,∀q∈Q,(q,p)∈ M(⊂T∗Q),and hence XB\nH·ε∈ Falongε. Thus, using\nthe non-degenerate distributional two-form ωB\nK, from Lemma 3.4 and Lemma 4.3, if we take that\n16v=XB\nH·ε∈ F,and for any w∈ F, Tλ(w)/ne}ationslash= 0,andτK·w/ne}ationslash= 0,then we have that\nωB\nK(Tγ·Xε, τK·w) =ωB\nK(τK·Tγ·Xε, τK·w)\n=τK·i∗\nM·ωB(Tγ·TπQ·XB\nH·ε, w) =τK·i∗\nM·ωB(T(γ·πQ)·XB\nH·ε, w)\n=τK·i∗\nM·(ωB(XB\nH·ε, w−T(γ·πQ)·w)−(dγ+B)(TπQ(XB\nH·ε), TπQ(w)))\n=τK·i∗\nM·ωB(XB\nH·ε, w)−τK·i∗\nM·ωB(XB\nH·ε, Tλ·w)\n−τK·i∗\nM·(dγ+B)(TπQ(XB\nH·ε), TπQ(w))\n=ωB\nK(τK·XB\nH·ε, τK·w)−ωB\nK(τK·XB\nH·ε, τK·Tλ·w)\n+τK·i∗\nM·λ∗ωB(XB\nH·ε, w)\n=ωB\nK(XB\nK·ε, τK·w)−ωB\nK(τK·XB\nH·ε, Tλ·w)+ωB\nK(Tλ·XB\nH·ε, Tλ·w),\nwhere we have used that τK·Tγ=Tγ, τK·Tλ=Tλ,andτK·XB\nH·ε=XB\nK·ε,since Im( Tγ)⊂ K.\nNote that ε:T∗Q→T∗Qis symplectic with respect to the magnetic symplectic form ωB, and\nXB\nH·ε=Tε·XB\nH·ε,alongε, and hence τK·XB\nH·ε=τK·Tε·XB\nH·ε,alongε. Then we have that\nωB\nK(Tγ·Xε, τK·w)−ωB\nK(XB\nK·ε, τK·w)\n=−ωB\nK(τK·XB\nH·ε, Tλ·w)+ωB\nK(Tλ·XB\nH·ε, Tλ·w)\n=ωB\nK(Tλ·XB\nH·ε−τK·Tε·XB\nH·ε, Tλ·w).\nBecause the induced distributional two-form ωB\nKis non-degenerate, it follows that the equation\nTγ·Xε=XB\nK·ε,is equivalent to the equation τK·Tε·XB\nH·ε=Tλ·XB\nH·ε. Thus,εis a solution\nof the equation τK·Tε·XB\nH·ε=Tλ·XB\nH·ε,if and only if it is a solution of the Type II of\nHamilton-Jacobi equation Tγ·Xε=XB\nK·ε./squaresolid\nRemark 4.7 It is worthy of noting that, the Type I of Hamilton-Jacobi equa tionTγ·Xγ=XB\nK·γ,is\nthe equation of the differential one-form γ; and the Type II of Hamilton-Jacobi equation Tγ·Xε=\nXB\nK·ε,is the equation of the symplectic diffeomorphism map ε. If the nonholonomic magnetic\nHamiltonian system we considered has not any constrains, in this case, the distributional magnetic\nHamiltonian system is just the magnetic Hamiltonian system itself. From the above Type I and Type\nII of Hamilton-Jacobi theorems, that is, Theorem 4.4 and Theore m 4.6, we can get the Theorem\n3.5 and Theorem 3.7. It shows that Theorem 4.4 and Theorem 4.6 can b e regarded as an extension\nof two types of Hamilton-Jacobi theorem for the magnetic Ham iltonian system to the system with\nnonholonomic context. On the other hand, when B= 0, in this case the magnetic symplectic form\nωBis just the canonical symplectic form ωonT∗Q, and the distributional magnetic Hamiltonian\nsystem is just the distributional Hamiltonian system itsel f. From the above Type I and Type II of\nHamilton-Jacobi theorems, that is, Theorem 4.4 and Theorem 4. 6, we can get the Theorem 3.5 and\nTheorem 3.6 given by Le´ on and Wang in [ 14]. It shows that Theorem 4.4 and Theorem 4.6 can be\nregarded as an extension of two types of Hamilton-Jacobi the orem for the distributional Hamiltonian\nsystem to that for the distributional magnetic Hamiltonian system.\n5 Nonholonomic Reduced Distributional Magnetic Hamiltoni an\nSystem\nItiswell-known that thereductiontheoryforthemechanica l systemwithsymmetryis animportant\nsubject and it is widely studied in the theory of mathematics and mechanics, as well as applica-\ntions; see Abraham and Marsden [ 1], Arnold [ 2], Libermann and Marle [ 15], Marsden [ 16], Marsden\net al. [17,18], Marsden and Ratiu [ 19], Marsden and Weinstein [ 21], and Ortega and Ratiu [ 22]\nand so on, for more details and development. In particular, t he reduction of nonholonomically\n17constrained mechanical systems is also very important subj ect in geometric mechanics, and it is\nregarded as a useful tool for simplifying and studying concr ete nonholonomic systems, see Bates\nand´Sniatycki [ 3], Cantrijn et al. [ 4], Cendra et al. [ 7], Cushman et al. [ 8] and [9], Koiller [ 11], Le´ on\nand Rodrigues [ 13] and Le´ on and Wang [ 14] and so on.\nIn this section, we shall consider the nonholonomic reducti on and Hamilton-Jacobi theory of a\nnonholonomic magnetic Hamiltonian system with symmetry. A t first, we give the definition of a\nnonholonomic magnetic Hamiltonian system with symmetry. T hen, by using the similar method\nin Le´ on and Wang [ 14] and Bates and ´Sniatycki [ 3]. and by analyzing carefully the dynamics and\nstructure of the nonholonomic magnetic Hamiltonian system with symmetry, we give a geometric\nformulation of the nonholonomic reduced distributional ma gnetic Hamiltonian system, Moreover,\nwe derive precisely the geometric constraint conditions of the non-degenerate, and nonholonomic\nreduced distributional two-form for the nonholonomic redu cible dynamical vector field, that is,\nthe two types of Hamilton-Jacobi equation for the nonholono mic reduced distributional magnetic\nHamiltonian system, which are an extension of the above two t ypes of Hamilton-Jacobi equation for\nthe distributional magnetic Hamiltonian system given in se ction 4 under nonholonomic reduction.\nAssume that the Lie group Gacts smoothly on the manifold Qby the left, and we also consider\nthe natural lifted actions on TQandT∗Q, and assume that the cotangent lifted action on T∗Qis\nfree, proper and symplectic with respect to the magnetic sym plectic form ωB=ω−π∗\nQBonT∗Q,\nwhereωis the canonical symplectic form on T∗QandBis a closed two-form on Q. The orbit space\nT∗Q/Gis a smooth manifold and the canonical projection π/G:T∗Q→T∗Q/Gis a surjective\nsubmersion. Assume that H:T∗Q→Ris aG-invariant Hamiltonian, and the D-completely and\nD-regularly nonholonomic constraint D ⊂TQis aG-invariant distribution, that is, the tangent of\ngroup action maps DqtoDgqfor anyq∈Q. A nonholonomic magnetic Hamiltonian system with\nsymmetry is 5-tuple ( T∗Q,G,ωB,D,H), which is a magnetic Hamiltonian system with symmetry\nandG-invariant nonholonomic constraint D.\nIn the following we first consider the nonholonomic reductio n of a nonholonomic magnetic\nHamiltonian system with symmetry ( T∗Q,G,ωB,D,H). Note that the Legendre transformation\nFL:TQ→T∗Qis a fiber-preserving map, and D ⊂TQisG-invariant for the tangent lifted left\naction ΦT:G×TQ→TQ,then the constraint submanifold M=FL(D)⊂T∗QisG-invariant\nfor the cotangent lifted left action ΦT∗:G×T∗Q→T∗Q. For the nonholonomic magnetic\nHamiltonian system with symmetry ( T∗Q,G,ωB,D,H), in the sameway, we definethe distribution\nF, which is the pre-image of the nonholonomic constraints Dfor the map TπQ:TT∗Q→TQ,\nthat is,F= (TπQ)−1(D), and the distribution K=F ∩TM. Moreover, we can also define the\ndistributional magnetic two-form ωB\nK, which is induced from the magnetic symplectic form ωBon\nT∗Q, that is, ωB\nK=τK·ωB\nM,andωB\nM=i∗\nMωB. If the admissibility condition dim M= rankF\nand the compatibility condition TM ∩ F⊥={0}hold, then ωB\nKis non-degenerate as a bilinear\nform on each fibre of K, there exists a vector field XB\nKonMwhich takes values in the constraint\ndistribution K, such that for the function HK, the following distributional magnetic Hamiltonian\nequation holds, that is,\niXB\nKωB\nK=dHK, (5.1)\nwhere the function HKsatisfiesdHK=τK·dHM, andHM=τM·His the restriction of HtoM,\nand from the equation (5.1), we have that XB\nK=τK·XB\nH.\nIn the following we define that the quotient space ¯M=M/Gof theG-orbit in Mis a smooth\nmanifold with projection π/G:M → ¯M(⊂T∗Q/G),which is a surjective submersion. The\nreduced magnetic symplectic form ωB¯M=π∗\n/G·ωB\nMon¯Mis induced from the magnetic symplectic\n18formωB\nM=i∗\nMωBonM. SinceGis the symmetry group of the system ( T∗Q,G,ωB,D,H), all\nintrinsically defined vector fields and distributions are pu shed down to ¯M. In particular, the vector\nfieldXB\nMonMis pusheddown to a vector field XB¯M=Tπ/G·XB\nM, and the distribution Kis pushed\ndown to a distribution Tπ/G·Kon¯M, and the Hamiltonian His pushed down to h¯M, such that\nh¯M·π/G=τM·H. However, ωB\nKneed not to be pushed down to a distributional two-form define d\nonTπ/G·K, despite of the fact that ωB\nKisG-invariant. This is because there may be infinitesimal\nsymmetry ηKthat lies in M, such that iηKωB\nK/ne}ationslash= 0. From Bates and ´Sniatycki [ 3], we know that in\norder to eliminate this difficulty, ωB\nKis restricted to a sub-distribution UofKdefined by\nU={u∈ K |ωB\nK(u,v) = 0,∀v∈ V ∩K} ,\nwhereVis the distribution on Mtangent to the orbits of GinMand it is spanned by the infinites-\nimal symmetries. Clearly, UandVare both G-invariant, project down to ¯MandTπ/G·V= 0, and\ndefine the distribution ¯Kby¯K=Tπ/G·U. Moreover, we take that ωB\nU=τU·ωB\nMis the restriction\nof the induced magnetic symplectic form ωB\nMonT∗Mfibrewise to the distribution U, whereτU\nis the restriction map to distribution U, and the ωB\nUis pushed down to a distributional magnetic\ntwo-form ωB¯Kon¯K, such that π∗\n/GωB¯K=ωB\nU. It is worthy of noting that the distributional magnetic\ntwo-form ωB¯Kis not a true two-form on a manifold, so it does not make sense t o speak about it being\nclosed. Thus, it is called the nonholonomic reduced distrib utional magnetic two-form to avoid any\nconfusion.\nFrom the above construction we know that, if the admissibili ty condition dim ¯M= rank¯Fand\nthe compatibility condition T¯M ∩¯F⊥={0}hold, where ¯F⊥denotes the symplectic orthogonal\nof¯Fwith respect to the reduced magnetic symplectic form ωB¯M, then the nonholonomic reduced\ndistributional magnetic two-form ωB¯Kis non-degenerate as a bilinear form on each fibre of ¯K, and\nhence there exists a vector field XB¯Kon¯Mwhich takes values in the constraint distribution ¯K, such\nthat the nonholonomic reduced distributional magnetic Ham iltonian equation holds, that is,\niXB\n¯KωB¯K=dh¯K, (5.2)\nwheredh¯Kis the restriction of dh¯Mto¯Kand the function h¯Ksatisfies dh¯K=τ¯K·dh¯M, and\nh¯M·π/G=HMandHMis the restriction of the Hamiltonian function HtoM. In addition,\nfrom the distributional magnetic Hamiltonian equation (2. 2),iXB\nKωB\nK=dHK,we have that XB\nK=\nτK·XB\nH,and from the nonholonomic reduced distributional magnetic Hamiltonian equation (5.2),\niXB\n¯KωB¯K=dh¯K, we have that XB¯K=τ¯K·XB\nh¯K,whereXB\nh¯Kis the magnetic Hamiltonian vector field\nof the function h¯Kwith respect to the reduced magnetic symplectic form ωB¯M, and the vector fields\nXB\nKandXB¯Kareπ/G-related, that is, XB¯K·π/G=Tπ/G·XB\nK.Thus, the geometrical formulation of a\nnonholonomic reduced distributional magnetic Hamiltonia n system may be summarized as follows.\nDefinition 5.1 (Nonholonomic Reduced Distributional magnetic Hamiltoni an System) Assume\nthat the 5-tuple (T∗Q,G,ωB,D,H)is a nonholonomic magnetic Hamiltonian system with sym-\nmetry, where ωBis the magnetic symplectic form on T∗Q, andD ⊂TQis aD-completely and\nD-regularly nonholonomic constraint of the system, and DandHare both G-invariant. If there\nexists a nonholonomic reduced distribution ¯K, an associated non-degenerate and nonholonomic re-\nduced distributional two-form ωB¯Kand a vector field XB¯Kon the reduced constraint submanifold\n¯M=M/G,whereM=FL(D)⊂T∗Q, such that the nonholonomic reduced distributional mag-\nnetic Hamiltonian equation iXB\n¯KωB¯K=dh¯Kholds, where dh¯Kis the restriction of dh¯Mto¯Kand\nthe function h¯Ksatisfies dh¯K=τ¯K·dh¯Mandh¯M·π/G=HMas defined above. Then the triple\n(¯K,ωB¯K,h¯K)is called a nonholonomic reduced distributional magnetic H amiltonian system of the\n19nonholonomic magnetic Hamiltonian system with symmetry (T∗Q,G,ωB,D,H), andXB¯Kis called\na nonholonomic reduced dynamical vector field. of the system (¯K,ωB¯K,h¯K). Under the above circum-\nstances, we refer to (T∗Q,G,ωB,D,H)as a nonholonomic reducible magnetic Hamiltonian system\nwith the associated distributional magnetic Hamiltonian s ystem(K,ωB\nK,HK)and nonholonomic re-\nduced distributional magnetic Hamiltonian system (¯K,ωB¯K,h¯K).\nSince the non-degenerate and nonholonomic reduced distrib utional two-form ωB¯Kis not a ”true\ntwo-form” on a manifold, and it is not symplectic, and hence t he nonholonomic reduced distribu-\ntional magnetic Hamiltonian system ( ¯K,ωB¯K,h¯K) may not be yet a Hamiltonian system, and may\nhave no generating function, and hence we can not describe th e Hamilton-Jacobi equation for the\nnonholonomicreduceddistributionalmagneticHamiltonia n systemjustlikeasinTheorem1.1. But,\nsince the nonholonomic reduced distributional magnetic Ha miltonian system is a dynamical sys-\ntem closely related to a magnetic Hamiltonian system, for a g iven nonholonomic reduciblemagnetic\nHamiltonian system ( T∗Q,G,ωB,D,H) with the associated distributional magnetic Hamiltonian\nsystem ( K,ωB\nK,HK) and the nonholonomic reduced distributional magnetic Ham iltonian system\n(¯K,ωB¯K,h¯K), by using Lemma 3.4 and Lemma 4.3, we can also derive precise ly the geometric con-\nstraint conditions of thenonholonomic reduceddistributi onal two-form ωB¯Kforthe dynamical vector\nfieldXB¯K, that is, the two types of Hamilton-Jacobi equation for the n onholonomic reduced dis-\ntributional magnetic Hamiltonian system ( ¯K,ωB¯K,h¯K). At first, using the fact that the one-form\nγ:Q→T∗Qsatisfies the condition that dγ=−BonDwith respect to TπQ:TT∗Q→TQ,\nIm(γ)⊂ M,and it is G-invariant, as well as Im( Tγ)⊂ K,we can prove the Type I of Hamilton-\nJacobi theorem for the nonholonomic reduced distributiona l magnetic Hamiltonian system. For\nconvenience, the maps involved in the following theorem and its proof are shown in Diagram-5.\nM\nXB\nK\n/d15/d15iM/d47/d47T∗Q\nXB\nH/d15/d15πQ/d47/d47Q\nXγ\n/d15/d15γ/d47/d47T∗Q\nXB\nH/d15/d15π/G/d47/d47T∗Q/G\nXB\nh¯M/d15/d15¯Mi¯M /d111/d111\nXB\n¯K/d15/d15\nKT(T∗Q)τK/d111/d111 TQTγ/d111/d111 T(T∗Q)TπQ/d111/d111Tπ/G/d47/d47T(T∗Q/G)τ¯K /d47/d47¯K\nDiagram-5\nTheorem 5.2 (Type I of Hamilton-Jacobi Theorem for a Nonholonomic Reduce d Distributional\nmagnetic Hamiltonian System) For a given nonholonomic redu cible magnetic Hamiltonian system\n(T∗Q,G,ωB,D,H)with the associated distributional magnetic Hamiltonian s ystem(K,ωB\nK,HK)\nand the nonholonomic reduced distributional magnetic Hami ltonian system (¯K,ωB¯K,h¯K), assume\nthatγ:Q→T∗Qis an one-form on Q, andXγ=TπQ·XB\nH·γ, whereXB\nHis the magnetic\nHamiltonian vector field of the corresponding unconstraine d magnetic Hamiltonian system with\nsymmetry (T∗Q,G,ωB,H). Moreover, assume that Im (γ)⊂ M,and it is G-invariant, Im (Tγ)⊂\nK,and¯γ=π/G(γ) :Q→T∗Q/G.If the one-form γ:Q→T∗Qsatisfies the condition that\ndγ=−BonDwith respect to TπQ:TT∗Q→TQ,then¯γis a solution of the equation T¯γ·\nXγ=XB¯K·¯γ.HereXB¯Kis the dynamical vector field of the nonholonomic reduced dis tributional\nmagnetic Hamiltonian system (¯K,ωB¯K,h¯K). The equation T¯γ·Xγ=XB¯K·¯γ,is called the Type\nI of Hamilton-Jacobi equation for the nonholonomic reduced distributional magnetic Hamiltonian\nsystem(¯K,ωB¯K,h¯K).\nProof: At first, from Theorem 4.4, we know that γis a solution of the Hamilton-Jacobi equation\nTγ·Xγ=XB\nK·γ.Next, we note that Im( γ)⊂ M,and it is G-invariant, Im( Tγ)⊂ K,and hence\nIm(T¯γ)⊂¯K,in this case, π∗\n/G·ωB¯K·τ¯K=τU·ωB\nM=τU·i∗\nM·ωB,along Im( T¯γ). From the\ndistributional magnetic Hamiltonian equation (2.2), we ha ve thatXB\nK=τK·XB\nH,andτK·XB\nH·γ=\nXB\nK·γ. Because the vector fields XB\nKandXB¯Kareπ/G-related, Tπ/G(XB\nK) =XB¯K·π/G, and hence\n20τ¯K·Tπ/G(XB\nK·γ) =τ¯K·(Tπ/G(XB\nK))·(γ) =τ¯K·(XB¯K·π/G)·(γ) =τ¯K·XB¯K·π/G(γ) =XB¯K·¯γ.\nThus, using the non-degenerate, nonholonomic reduced dist ributional two-form ωB¯K, from Lemma\n3.4(ii) and Lemma 4.3, if we take that v=XB\nH·γ∈ F,and for any w∈ F, Tλ(w)/ne}ationslash= 0,and\nτ¯K·Tπ/G·w/ne}ationslash= 0,then we have that\nωB¯K(T¯γ·Xγ, τ¯K·Tπ/G·w) =ωB¯K(τ¯K·T(π/G·γ)·Xγ, τ¯K·Tπ/G·w)\n=π∗\n/G·ωB¯K·τ¯K(Tγ·Xγ, w) =τU·i∗\nM·ωB(Tγ·TπQ·XB\nH·γ, w)\n=τU·i∗\nM·ωB(T(γ·πQ)·XB\nH·γ, w)\n=τU·i∗\nM·(ωB(XB\nH·γ, w−T(γ·πQ)·w)−(dγ+B)(TπQ(XB\nH·γ), TπQ(w)))\n=τU·i∗\nM·ωB(XB\nH·γ, w)−τU·i∗\nM·ωB(XB\nH·γ, T(γ·πQ)·w)\n−τU·i∗\nM·(dγ+B)(TπQ(XB\nH·γ), TπQ(w))\n=π∗\n/G·ωB¯K·τ¯K(XB\nH·γ, w)−π∗\n/G·ωB¯K·τ¯K(XB\nH·γ, T(γ·πQ)·w)\n−τU·i∗\nM·(dγ+B)(TπQ(XB\nH·γ), TπQ(w))\n=ωB¯K(τ¯K·Tπ/G(XB\nH·γ), τ¯K·Tπ/G·w)−ωB¯K(τ¯K·Tπ/G(XB\nH·γ), τ¯K·T(π/G·γ)·TπQ(w))\n−τU·i∗\nM·(dγ+B)(TπQ(XB\nH·γ), TπQ(w))\n=ωB¯K(τ¯K·Tπ/G(XB\nH)·π/G(γ), τ¯K·Tπ/G·w)−ωB¯K(τ¯K·Tπ/G(XB\nH)·π/G(γ), τ¯K·T¯γ·TπQ(w))\n−τU·i∗\nM·(dγ+B)(TπQ(XB\nH·γ), TπQ(w))\n=ωB¯K(XB¯K·¯γ, τ¯K·Tπ/G·w)−ωB¯K(XB¯K·¯γ, T¯γ·TπQ(w))\n−τU·i∗\nM·(dγ+B)(TπQ(XB\nH·γ), TπQ(w)),\nwherewehaveusedthat τ¯K·Tπ/G(XB\nH·γ) =τ¯K·XB¯K·¯γ=XB¯K·¯γ,andτ¯K·T¯γ=T¯γ,sinceIm( T¯γ)⊂¯K.\nNote that XB\nH·γ, w∈ F,andTπQ(XB\nH·γ), TπQ(w)∈ D.If the one-form γ:Q→T∗Qsatisfies\nthe condition that dγ=−BonDwith respect to TπQ:TT∗Q→TQ,then we have that\n(dγ+B)(TπQ(XB\nH·γ), TπQ(w)) = 0,and hence\nτU·i∗\nM·(dγ+B)(TπQ(XB\nH·γ), TπQ(w)) = 0,\nThus, we have that\nωB¯K(T¯γ·Xγ, τ¯K·Tπ/G·w)−ωB¯K(XB¯K·¯γ, τ¯K·Tπ/G·w) =−ωB¯K(XB¯K·¯γ, T¯γ·TπQ(w)).(5.3)\nIf ¯γsatisfies the equation T¯γ·Xγ=XB¯K·¯γ,from Lemma 3.4(i) we know that the right side of (5.3)\nbecomes that\n−ωB¯K(XB¯K·¯γ, T¯γ·TπQ(w)) =−ωB¯K·τ¯K(T¯γ·Xγ, T¯γ·TπQ(w))\n=−¯γ∗ωB¯K·τ¯K(TπQ·XB\nH·γ, TπQ(w))\n=−γ∗·π∗\n/G·ωB¯K·τ¯K(TπQ·XB\nH·γ, TπQ(w))\n=−γ∗·τU·i∗\nM·ωB(TπQ(XB\nH·γ), TπQ(w))\n=−τU·i∗\nM·γ∗ωB(TπQ(XB\nH·γ), TπQ(w))\n=τU·i∗\nM·(dγ+B)(TπQ(XB\nH·γ), TπQ(w)) = 0,\nwhereγ∗·τU·i∗\nM·ωB=τU·i∗\nM·γ∗·ωB,because Im( γ)⊂ M.But, since the nonholonomic reduced\ndistributional two-form ωB¯Kis non-degenerate, the left side of (5.3) equals zero, only w hen ¯γsatisfies\nthe equation T¯γ·Xγ=XB¯K·¯γ.Thus, if the one-form γ:Q→T∗Qsatisfies the condition that\ndγ=−BonDwith respect to TπQ:TT∗Q→TQ,then ¯γmust be a solution of the Type I of\n21Hamilton-Jacobi equation T¯γ·Xγ=XB¯K·¯γ./squaresolid\nNext, for any G-invariant symplectic map ε:T∗Q→T∗Qwith respect to ωB, we can prove\nthe following Type II of Hamilton-Jacobi theorem for the non holonomic reduced distributional\nmagnetic Hamiltonian system. For convenience, the maps inv olved in the following theorem and\nits proof are shown in Diagram-6.\nM\nXB\nK\n/d15/d15iM/d47/d47T∗Q\nXB\nH·ε/d15/d15Xε\n/d35/d35❍❍❍❍❍❍❍❍❍πQ/d47/d47Qγ/d47/d47T∗Q\nXB\nH/d15/d15π/G/d47/d47T∗Q/G\nXB\nh¯M/d15/d15¯Mi¯M/d111/d111\nXB\n¯K/d15/d15\nKT(T∗Q)τK/d111/d111 TQTγ/d111/d111 T(T∗Q)TπQ/d111/d111Tπ/G/d47/d47T(T∗Q/G)τ¯K /d47/d47¯K\nDiagram-6\nTheorem 5.3 (Type II of Hamilton-Jacobi Theorem for a Nonholonomic Reduce d Distributional\nmagnetic Hamiltonian System) For a given nonholonomic redu cible magnetic Hamiltonian system\n(T∗Q,G,ωB,D,H)with the associated distributional magnetic Hamiltonian s ystem(K,ωB\nK,HK)\nand the nonholonomic reduced distributional magnetic Hami ltonian system (¯K,ωB¯K,h¯K), assume\nthatγ:Q→T∗Qis an one-form on Q, andλ=γ·πQ:T∗Q→T∗Q,and for any G-invariant\nsymplectic map ε:T∗Q→T∗Qwith respect to ωB, denote by Xε=TπQ·XB\nH·ε, whereXB\nHis\nthe magnetic Hamiltonian vector field of the corresponding u nconstrained magnetic Hamiltonian\nsystem with symmetry (T∗Q,G,ωB,H). Moreover, assume that Im (γ)⊂ M,and it is G-invariant,\nε(M)⊂ M, Im(Tγ)⊂ K,and¯γ=π/G(γ) :Q→T∗Q/G, and¯λ=π/G(λ) :T∗Q→T∗Q/G,and\n¯ε=π/G(ε) :T∗Q→T∗Q/G.Thenεand¯εsatisfy the equation τ¯K·T¯ε·XB\nh¯K·¯ε=T¯λ·XB\nH·ε,if and\nonly if they satisfy the equation T¯γ·Xε=XB¯K·¯ε.HereXB\nh¯K·¯εis the magnetic Hamiltonian vector\nfield of the function h¯K·¯ε:T∗Q→R,andXB¯Kis the dynamical vector field of the nonholonomic\nreduced distributional magnetic Hamiltonian system (¯K,ωB¯K,h¯K). The equation T¯γ·Xε=XB¯K·¯ε,is\ncalled the Type II of Hamilton-Jacobi equation for the nonhol onomic reduced distributional magnetic\nHamiltonian system (¯K,ωB¯K,h¯K).\nProof: In the same way, we note that Im( γ)⊂ M,and it is G-invariant, Im( Tγ)⊂ K,and hence\nIm(T¯γ)⊂¯K,in this case, π∗\n/G·ωB¯K·τ¯K=τU·ωB\nM=τU·i∗\nM·ωB,along Im( T¯γ). Moreover, from\nthe distributional magnetic Hamiltonian equation (2.2), w e have that XB\nK=τK·XB\nH.Note that\nε(M)⊂ M,andTπQ(XB\nH·ε(q,p))∈ Dq,∀q∈Q,(q,p)∈ M(⊂T∗Q),and hence XB\nH·ε∈ F\nalongε. Because the vector fields XB\nKandXB¯Kareπ/G-related, Tπ/G(XB\nK) =XB¯K·π/G, and hence\nτ¯K·Tπ/G(XB\nK·ε) =τ¯K·(Tπ/G(XB\nK))·(ε) =τ¯K·(XB¯K·π/G)·(ε) =τ¯K·XB¯K·π/G(ε) =XB¯K·¯ε.Thus, using\nthe non-degenerate and nonholonomic reduced distribution al two-form ωB¯K, from Lemma 3.4 and\nLemma 4.3, if we take that v=XB\nH·ε∈ F,and for any w∈ F, Tλ(w)/ne}ationslash= 0,andτ¯K·Tπ/G·w/ne}ationslash= 0,\n22then we have that\nωB¯K(T¯γ·Xε, τ¯K·Tπ/G·w) =ωB¯K(τ¯K·T(π/G·γ)·Xε, τ¯K·Tπ/G·w)\n=π∗\n/G·ωB¯K·τ¯K(Tγ·Xε, w) =τU·i∗\nM·ωB(Tγ·Xε, w)\n=τU·i∗\nM·ωB(T(γ·πQ)·XB\nH·ε, w)\n=τU·i∗\nM·(ωB(XB\nH·ε, w−T(γ·πQ)·w)−(dγ+B)(TπQ(XB\nH·ε), TπQ(w)))\n=τU·i∗\nM·ωB(XB\nH·ε, w)−τU·i∗\nM·ωB(XB\nH·ε, Tλ·w)\n−τU·i∗\nM·(dγ+B)(TπQ(XB\nH·ε), TπQ(w))\n=π∗\n/G·ωB¯K·τ¯K(XB\nH·ε, w)−π∗\n/G·ωB¯K·τ¯K(XB\nH·ε, Tλ·w)+τU·i∗\nM·λ∗ωB(XB\nH·ε, w)\n=ωB¯K(τ¯K·Tπ/G(XB\nH·ε), τ¯K·Tπ/G·w)−ωB¯K(τ¯K·Tπ/G(XB\nH·ε), τ¯K·T(π/G·λ)·w)\n+π∗\n/G·ωB¯K·τ¯K(Tλ·XB\nH·ε, Tλ·w)\n=ωB¯K(τ¯K·Tπ/G(XB\nH)·π/G(ε), τ¯K·Tπ/G·w)−ωB¯K(τ¯K·Tπ/G(XB\nH)·π/G(ε), τ¯K·T¯λ·w)\n+ωB¯K(τ¯K·Tπ/G·Tλ·XB\nH·ε, τ¯K·Tπ/G·Tλ·w)\n=ωB¯K(XB¯K·¯ε, τ¯K·Tπ/G·w)−ωB¯K(XB¯K·¯ε, T¯λ·w)+ωB¯K(T¯λ·XB\nH·ε, T¯λ·w),\nwhere we have used that τ¯K·Tπ/G(XB\nH·ε) =τ¯K(XB¯K)·¯ε=XB¯K·¯ε,andτ¯K·Tπ/G·Tλ=T¯λ,since\nIm(T¯γ)⊂¯K.From the nonholonomic reduced distributional magnetic Ham iltonian equation (5.3),\niXB\n¯KωB¯K=dh¯K,we have that XB¯K=τ¯K·XB\nh¯K,whereXB\nh¯Kis the magnetic Hamiltonian vector field\nof the function h¯K:¯M(⊂T∗Q/G)→R.Note that ε:T∗Q→T∗Qis symplectic with respect to\nωB, and ¯ε=π/G(ε) :T∗Q→T∗Q/Gis also symplectic along ¯ ε, and hence XB\nh¯K·¯ε=T¯ε·XB\nh¯K·¯ε,\nalong ¯ε, and hence XB¯K·¯ε=τ¯K·XB\nh¯K·¯ε=τ¯K·T¯ε·XB\nh¯K·¯ε,along ¯ε. Then we have that\nωB¯K(T¯γ·Xε, τ¯K·Tπ/G·w)−ωB¯K(XB¯K·¯ε, τ¯K·Tπ/G·w)\n=−ωB¯K(XB¯K·¯ε, T¯λ·w)+ωB¯K(T¯λ·XB\nH·ε, T¯λ·w)\n=ωB¯K(T¯λ·XB\nH·ε−τ¯K·T¯ε·XB\nh¯K·¯ε, T¯λ·w).\nBecause the nonholonomic reduced distributional two-form ωB¯Kis non-degenerate, it follows that\nthe equation T¯γ·Xε=XB¯K·¯ε,is equivalent to the equation T¯λ·XB\nH·ε=τ¯K·T¯ε·XB\nh¯K·¯ε.Thus,ε\nand ¯εsatisfy the equation T¯λ·XB\nH·ε=τ¯K·T¯ε·XB\nh¯K·¯ε,if and only if they satisfy the Type II of\nHamilton-Jacobi equation T¯γ·Xε=XB¯K·¯ε./squaresolid\nFor a given nonholonomic reducible magnetic Hamiltonian sy stem (T∗Q,G,ωB,D,H) with the\nassociated distributional magnetic Hamiltonian system ( K,ωB\nK,HK) and the nonholonomic reduced\ndistributional magnetic Hamiltonian system ( ¯K,ωB¯K,h¯K), we know that the nonholonomic dynami-\ncal vector field XB\nKand the nonholonomic reduced dynamical vector field XB¯Kareπ/G-related, that\nis,XB¯K·π/G=Tπ/G·XB\nK.Then we can prove the following Theorem 5.4, which states the relation-\nship between the solutions of Type II of Hamilton-Jacobi equ ations and nonholonomic reduction.\nTheorem 5.4 For agiven nonholonomic reducible magnetic Hamiltonian sy stem(T∗Q,G,ωB,D,H)\nwith the associated distributional magnetic Hamiltonian s ystem(K,ωB\nK,HK)and the nonholonomic\nreduced distributional magnetic Hamiltonian system (¯K,ωB¯K,h¯K), assume that γ:Q→T∗Qis\nan one-form on Q, andλ=γ·πQ:T∗Q→T∗Q,andε:T∗Q→T∗Qis aG-invariant\nsymplectic map with respect to ωB. Moreover, assume that Im (γ)⊂ M,and it is G-invariant,\nε(M)⊂ M, Im(Tγ)⊂ K,and¯γ=π/G(γ) :Q→T∗Q/G, and¯λ=π/G(λ) :T∗Q→T∗Q/G,\nand¯ε=π/G(ε) :T∗Q→T∗Q/G.Thenεis a solution of the Type II of Hamilton-Jacobi equation,\n23Tγ·Xε=XB\nK·ε,for the distributional magnetic Hamiltonian system (K,ωB\nK,HK), if and only if\nεand¯εsatisfy the Type II of Hamilton-Jacobi equation T¯γ·Xε=XB¯K·¯ε,for the nonholonomic\nreduced distributional magnetic Hamiltonian system (¯K,ωB¯K,h¯K).\nProof: Note that Im( γ)⊂ M,and it is G-invariant, Im( Tγ)⊂ K,and hence Im( T¯γ)⊂¯K,in this\ncase,π∗\n/G·ωB¯K·τ¯K=τU·ωB\nM=τU·i∗\nM·ωB,along Im( T¯γ), andτ¯K·T¯γ=T¯γ, τ¯K·XB¯K=XB¯K.Since\nnonholonomicvectorfield XB\nKandthevectorfield XB¯Kareπ/G-related, thatis, XB¯K·π/G=Tπ/G·XB\nK,\nusing the non-degenerate and nonholonomic reduced distrib utional two-form ωB¯K, we have that\nωB¯K(T¯γ·Xε−XB¯K·¯ε, τ¯K·Tπ/G·w)\n=ωB¯K(T¯γ·Xε, τ¯K·Tπ/G·w)−ωB¯K(XB¯K·¯ε, τ¯K·Tπ/G·w)\n=ωB¯K(τ¯K·T¯γ·Xε, τ¯K·Tπ/G·w)−ωB¯K(τ¯K·XB¯K·π/G·ε, τ¯K·Tπ/G·w)\n=ωB¯K·τ¯K(Tπ/G·Tγ·Xε, Tπ/G·w)−ωB¯K·τ¯K(Tπ/G·XB\nK·ε, Tπ/G·w)\n=π∗\n/G·ωB¯K·τ¯K(Tγ·Xε, w)−π∗\n/G·ωB¯K·τ¯K(XB\nK·ε, w)\n=τU·i∗\nM·ωB(Tγ·Xε, w)−τU·i∗\nM·ωB(XB\nK·ε, w).\nIn the case we considered that τU·i∗\nM·ωB=τK·i∗\nM·ωB=ωB\nK·τK,andτK·Tγ=Tγ, τK·XB\nK=XB\nK,\nsince Im( γ)⊂ M,and Im(Tγ)⊂ K.Thus, we have that\nωB¯K(T¯γ·Xε−XB¯K·¯ε, τ¯K·Tπ/G·w)\n=ωB\nK·τK(Tγ·Xε, w)−ωB\nK·τK(XB\nK·ε, w)\n=ωB\nK(τK·Tγ·Xε, τK·w)−ωB\nK(τK·XB\nK·ε, τK·w)\n=ωB\nK(Tγ·Xε−XB\nK·ε, τK·w).\nBecause the distributional two-form ωB\nKand the nonholonomic reduced distributional two-form ωB¯K\nare both non-degenerate, it follows that the equation T¯γ·Xε=XB¯K·¯ε,is equivalent to the equation\nTγ·Xε=XB\nK·ε.Thus,εis a solution of the Type II of Hamilton-Jacobi equation Tγ·Xε=XB\nK·ε,\nfor the distributional magnetic Hamiltonian system ( K,ωB\nK,HK), if and only if εand ¯εsatisfy the\nType II of Hamilton-Jacobi equation T¯γ·Xε=XB¯K·¯ε,for the nonholonomic reduced distributional\nmagnetic Hamiltonian system ( ¯K,ωB¯K,h¯K)./squaresolid\nRemark 5.5 It is worthy of noting that, the Type I of Hamilton-Jacobi equa tionT¯γ·Xγ=XB¯K·¯γ,\nis the equation of the nonholonomic reduced differential one -form¯γ; and the Type II of Hamilton-\nJacobi equation T¯γ·Xε=XB¯K·¯ε,is the equation of the symplectic diffeomorphism map εand the\nnonholonomic reduced symplectic diffeomorphism map ¯ε. When B= 0, in this case the magnetic\nsymplectic form ωBis just the canonical symplectic form ωonT∗Q, and the nonholonomic reducible\nmagnetic Hamiltonian system is just the nonholonomic reduc ible Hamiltonian system itself, and the\nnonholonomic reduced distributional magnetic Hamiltonia n system is just the nonholonomic reduced\ndistributional Hamiltonian system. From the above Type I an d Type II of Hamilton-Jacobi theorems,\nthat is, Theorem 5.2 and Theorem 5.3, we can get the Theorem 4.2 an d Theorem 4.3 given in Le´ on\nand Wang [ 14]. It shows that Theorem 5.2 and Theorem 5.3 can be regarded as an e xtension of two\ntypes of Hamilton-Jacobi theorem for the nonholonomic redu ced distributional Hamiltonian system\nto that for the nonholonomic reduced distributional magnet ic Hamiltonian system.\nIn order to describe the impact of different geometric structu res and constraints for the dynam-\nics of a Hamiltonian system, in this paper, we study the Hamil ton-Jacobi theory for the magnetic\nHamiltonian system, the nonholonomic magnetic Hamiltonia n system and the nonholonomic re-\nducible magnetic Hamiltonian system on a cotangent bundle, by using the distributional magnetic\n24Hamiltonian system and the nonholonomic reduced distribut ional magnetic Hamiltonian system,\nwhicharethedevelopment oftheHamilton-Jacobi theoryfor thenonholonomicHamiltonian system\nand the nonholonomic reducible Hamiltonian system given in Le´ on and Wang [ 14]. These research\nworks reveal from the geometrical point of view the internal relationships of the magnetic symplec-\ntic form, nonholonomic constraint, non-degenerate distri butional two form and dynamical vector\nfields of a nonholonomic magnetic Hamiltonian system and the nonholonomic reducible magnetic\nHamiltonian system. It is worthy of noting that, Marsden et a l. in [20] set up the regular reduction\ntheory of regular controlled Hamiltonian systems on a sympl ectic fiber bundle, by using momen-\ntum map and the associated reduced symplectic forms, and fro m the viewpoint of completeness\nof Marsden-Weinstein symplectic reduction, and some devel opments around the above work are\ngiven in Wang and Zhang [ 32], Ratiu and Wang [ 24], and Wang [ 25–28]. Since the Hamilton-Jacobi\ntheory is developed based on the Hamiltonian picture of dyna mics, it is natural idea to extend\nthe Hamilton-Jacobi theory to the (regular) controlled (ma gnetic) Hamiltonian systems and their\na variety of reduced systems, and it is also possible to descr ibe the relationship between the RCH-\nequivalence for controlled Hamiltonian systems and the sol utions of corresponding Hamilton-Jacobi\nequations, see Wang [ 29–31] for more details. Thus, our next topic is how to set up and dev elop the\nnonholonomic reduction and Hamilton-Jacobi theory for the nonholonomic controlled (magnetic)\nHamiltonian systems and the distributional controlled (ma gnetic) Hamiltonian systems, by ana-\nlyzing carefully the geometrical and topological structur es of the phase spaces of these systems. It\nis the key thought of the researches of geometrical mechanic s of the professor Jerrold E. Marsden\nto explore and reveal the deeply internal relationship betw een the geometrical structure of phase\nspace and the dynamical vector field of a mechanical system. I t is also our goal of pursuing and\ninheriting. In addition, we note also that there have been a l ot of beautiful results of reduction\ntheory of Hamiltonian systems in celestial mechanics, hydr odynamics and plasma physics. Thus,\nit is an important topic to study the application of reductio n theory and Hamilton-Jacobi theory\nof the systems in celestial mechanics, hydrodynamics and pl asma physics. These are our goals in\nfuture research.\nAcknowledgments: The year of 2021 is S.S. Chern’s year of Nankai University in C hina, for the\n110th anniversary of the birth of Professor S.S. Chern. Shii ng Shen Chern (1911-10-28—2004-12-\n03) is a great geometer and a model of excellent scientists. A ll of the differential geometry theory\nand methods for us used in the research of geometrical mechan ics, are studied from his books and\nhis research papers. He is a good example of us learning from h im forevermore.\nReferences\n[1] Abraham R., Marsden J.E., Foundations of Mechanics, sec ond ed., Addison-Wesley, Reading,\nMA, (1978).\n[2] Arnold V.I., Mathematical Methods of Classical Mechani cs, second ed., in: Graduate Texts in\nMathematics, vol. 60, Springer-Verlag, (1989).\n[3] Bates L. and ´Sniatycki J., Nonholonomic reduction, Rep. Math. Phys. 32, 9 9-115(1993).\n[4] Cantrijn F., de Le´ on M., Marrero J.C. and Martin de Diego D., Reduction of constrained\nsystems with symmetries, J. Math. Phys., 40(2), 795-820(19 99).\n[5] Cari˜ nena J.F., Gr` acia X., Marmo G., Mart´ ınez E., Mu˜ n oz-Lecanda M. and Rom´ an-Roy N.,\nGeometric Hamilton-Jacobi theory, Int. J. Geom. Methods Mo d. Phys. 3, 1417-1458(2006).\n25[6] Cari˜ nenaJ.F., Gr` acia X., MarmoG., Mart´ ınez E., Mu˜ n oz-LecandaM.andRom´ an-Roy N., Ge-\nometric Hamilton-Jacobi theory for nonholonomic dynamica l systems, Int. J. Geom. Methods\nMod. Phys. 7, 431-454(2010).\n[7] Cendra H., Marsden J.E. and Ratiu T.S., Geometric mechan ics, Lagrangian reduction and\nnonholonomic systems, In ”Mathematics Unlimited 2001 and B eyond” (eds. B. Engquist and\nW. Schmid), Springer-Verlag, New York, 221-273(2001).\n[8] Cushman R., Duistermaat H. and ´Sniatycki J., Geometry of Nonholonomic Constrained Sys-\ntems, Advanced series in nonlinear dynamics, 26, (2010).\n[9] Cushman R., Kemppainen D., ´Sniatycki J. and Bates L., Geometry of nonholonomic con-\nstraints, Rep. Math. Phys. 36(2/3), 275-286(1995).\n[10] Ge Z. and Marsden J.E., Lie-Poisson integrators and Lie -Poisson Hamilton-Jacobi theory,\nPhys. Lett. A, 133, 134-139(1988).\n[11] Koiller J., Reduction of some classical non-holonomic systems with symmetry, Arch. Rational\nMech. Anal. 118, 113-148(1992).\n[12] L´ azaro-Cam´ ı J-A and Ortega J-P, The stochastic Hamil ton-Jacobi equation, J. Geom. Mech.\n1, 295-315(2009).\n[13] Le´ on M. and Rodrigues P.R., Methods of Differential Geom etry in Analytical Mechanics,\nNorth-Holland, Amsterdam, (1989).\n[14] Le´ on M. and Wang H., Hamilton-Jacobi equations for non holonomic reducible Hamiltonian\nsystems on a cotangent bundle, (arXiv: 1508.07548, a revise d version).\n[15] Libermann P. and Marle C.M., Symplectic Geometry and An alytical Mechanics, Kluwer Aca-\ndemic Publishers, (1987).\n[16] Marsden J.E., Lectures on Mechanics, in: London Mathem atical Society Lecture Notes Series,\nvol. 174, Cambridge University Press, (1992).\n[17] Marsden J.E., Misiolek G., Ortega J.P., Perlmutter M. a nd Ratiu T.S., Hamiltonian Reduction\nby Stages, in: Lecture Notes in Mathematics, vol. 1913, Spri nger, (2007).\n[18] MarsdenJ.E., Montgomery R.andRatiuT.S., Reduction, SymmetryandPhasesinMechanics,\nin: Memoirs of the American Mathematical Society, vol. 88, A merican Mathematical Society,\nProvidence, Rhode Island, (1990).\n[19] Marsden J.E. and Ratiu T.S., Introduction to Mechanics and Symmetry, second ed., in: Texts\nin Applied Mathematics, vol. 17, Springer-Verlag, New York , (1999).\n[20] Marsden J.E., Wang H. and Zhang Z.X., Regular reduction of controlled Hamiltonian sys-\ntem with symplectic structure and symmetry, Diff. Geom. Appl. , 33(3), 13-45(2014), (arXiv:\n1202.3564).\n[21] MarsdenJ.E.andWeinstein A., Reductionofsymplectic manifoldswithsymmetry, Rep.Math.\nPhys. 5, 121-130(1974).\n[22] Ortega J-P and Ratiu T.S., Momentum Maps and Hamiltonia n Reduction, in: Progress in\nMathematics, vol. 222, Birkh¨ auser, (2004).\n[23] Patrick G.W., Variational development of the semi-sym plectic geometry of nonholonomic me-\nchanics, Rep. Math. Phys. 59, 145-184(2007).\n26[24] Ratiu T.S. and Wang H., Poisson reduction by controllab ility distribution for a controlled\nHamiltonian system, (arXiv: 1312.7047).\n[25] Wang H. Some developments of reduction theory for contr olled Hamiltonian system with sym-\nmetry ( in Chinese), Sci Sin Math, 2018, 48, 1-12.\n[26] Wang H. Reductions of controlled Hamiltonian system wi th symmetry, In: Symmetries and\nGroups in Contemporary Physics, ( Bai C M. Gazeau J P. and Ge M L . eds), World Scientific,\n2013, 639-642.\n[27] Wang H., Regular reduction of a controlled magnetic Ham iltonian system with symmetry of\nthe Heisenberg group, (arXiv: 1506.03640, a revised versio n).\n[28] Wang H., Hamilton-Jacobi theorems for regular reducib le Hamiltonian systems on a cotangent\nbundle, Jour. Geom. Phys., 11982-102, (2017).\n[29] Wang H., Hamilton-Jacobi equations for a regular contr olled Hamiltonian system and its re-\nducedsystems, (arXiv: 1305.3457, arevised version), Toap pearin Acta Mathematica Scientia,\nEnglish Series, 2022.\n[30] Wang H., Dynamical equations of the controlled rigid sp acecraft with a rotor, (arXiv:\n2005.02221).\n[31] Wang H., Symmetric reduction and Hamilton-Jacobi equa tions for the controlled underwater\nvehicle-rotor system, ( arXiv: 1310.3014, a revised versio n ).\n[32] Wang H. and Zhang Z.X., Optimal reduction of controlled Hamiltonian system with Poisson\nstructure and symmetry, Jour. Geom. Phys., 62 (5), 953-975( 2012).\n[33] Woodhouse N.M.J., Geometric Quantization, second ed. , Clarendon Press, Oxford, (1992).\n27"    },    {        "title": "1611.03541v2.Magnetism_and_ultra_fast_magnetization_dynamics_of_Co_and_CoMn_alloys_at_finite_temperature.pdf",        "content": "Magnetism and ultra-fast magnetization dynamics of Co and Co-Mn alloys at \fnite\ntemperature\nR. Chimata,1,\u0003E. K. Delczeg-Czirjak,1A. Szilva,1R. Almeida,1, 2Y.O. Kvashnin,1M. Pereiro,1\nS. Mankovsky,3H. Ebert,3D. Thonig,1B. Sanyal,1A. B. Klautau,2and O. Eriksson1\n1Department of Physics and Astronomy, Material Theory, University Uppsala, SE-75120 Uppsala, Sweden\n2Faculdade de F\u0013 \u0010sica, Universidade Federal do Par\u0013 a, Bel\u0013 em, PA, Brazil\n3Department of Chemistry, University of Munich,\nButenandtstrasse 5-13, D-81377 Munich, Germany\n(Dated: August 27, 2018)\nTemperature-dependent magnetic experiments like pump-probe measurements generated by a pulsed\nlaser have become a crucial technique for switching the magnetization in the picosecond time scale.\nApart from having practical implications on the magnetic storage technology, the research \feld\nof ultrafast magnetization poses also fundamental physical questions. To correctly describe the\ntime evolution of the atomic magnetic moments under the in\ruence of a temperature-dependent\nlaser pulse, it remains crucial to know if the magnetic material under investigation has magnetic\nexcitation spectrum that is more or less dependent on the magnetic con\fguration, e.g. as re\rected\nby the temperature dependence of the exchange interactions. In this article, we demonstrate from\n\frst-principles theory that the magnetic excitation spectra in Co with fcc, bcc and hcp structures\nare nearly identical in a wide range of non-collinear magnetic con\fgurations. This is a curious\nresult of a balance between the size of the magnetic moments and the strength of the Heisenberg\nexchange interactions, that in themselves vary with con\fguration, but put together in an e\u000bective\nspin Hamiltonian results in a con\fguration independent e\u000bective model. We have used such a\nHamiltonian, together with ab-initio calculated damping parameters, to investigate the magnon\ndispersion relationship as well as the ultrafast magnetisation dynamics of Co and Co-rich CoMn\nalloys.\nPACS numbers: 75.75.+a, 73.22.-f, 75.10.-b\nI. INTRODUCTION\nUltrafast magnetism, with relevant time-scales being\nan order of a few pico seconds, has become an intense\nresearch \feld. The motivation may be found both in\nfundamental aspects as well as practical implications of\nthese phenomena. Most of the information stored techno-\nlogically is done in a magnetic medium. Hence, the pos-\nsibility to write and retrieve information in a magnetic\nmaterial at a high speed and with low energy consump-\ntion has obvious societal implications. For this reason,\nultrafast magnetization dynamics has naturally become\nan intense research \feld. The experiment by Beaure-\npaire and co-workers1represents a breakthrough experi-\nment, with several experimental studies that followed.2{8\nHowever, despite several years of intense investigations, a\nmicroscopic understanding of the processes of ultra-fast\nmagnetization dynamics is far from been established.\nThe most common experimental technique is by pump-\nprobe, where an optical laser pulse excites the electron\nsub-system. The excited electrons become thermalized\nquickly9, and the thermal energy of the electron sub-\nsystem is transferred to the spin- and lattice sub-systems.\nThis de\fnes three thermal reservoirs, and typically the\nthree reservoirs reach thermal equilibrium after some 10-\n20 pico seconds. The time evolution of the temperatures\nof these reservoirs may be quanti\fed by the so called three\ntemperature model.1,10,11It should be noted that in the\n\frst few pico seconds of a pump-probe experiment, thematerial is not in thermal equilibrium between the three\nreservoirs, but after su\u000eciently long time after the pump\npulse, the temperature is the same in the di\u000berent sub-\nsystems.\nOn the theoretical side it has been argued that atom-\nistic spin-dynamics simulations should be relevant over\na time-scale of pico seconds and longer.12The argument\nhere is that a description of atomistic moments is rel-\nevant, and that these moments evolve in time with a\nspin-temperature that is given by the three-temperature\nmodel.13In this description, the magnetic moments and\nall parameters of an e\u000bective spin-Hamiltonian are eval-\nuated from \frst principles theory. Coupled to the equa-\ntions of motion of the atomic spins,14this allows for nu-\nmerical results of the time evolution of the magnetic mo-\nments, forming an ab-initio theory that does not rely\non experimental results as input. The dominating pa-\nrameters of such a spin-Hamiltonian are the size of the\natomic moments coupled to the inter-atomic exchange\ninteraction.15To mention an example of the fruitfulness\nof this approach we note that the \frst experimental re-\nsult of fcc Ni, was reproduced by such simulations with\ngood accuracy.17\nRecently, it was shown that the inter-atomic exchange\ninteractions of bcc Fe has a distinct temperature de-\npendence, and only good agreement with experimen-\ntal room temperature magnon excitations was achieved\nwhen the exchange parameters were evaluated at room\ntemperature.18This demonstrates that if a too broadarXiv:1611.03541v2  [cond-mat.mtrl-sci]  24 Apr 20172\ntemperature interval needs to be covered, bcc Fe is not\nan ideal Heisenberg system, and that the normal con-\ncepts of a Heisenberg Hamiltonian, e.g. magnons, can\nstill be considered although con\fguration dependent ex-\nchange parameters must be evaluated and used. This\nputs high demands if a three-temperature model is at-\ntempted to be used to reproduce an experimental pump-\nprobe experiment, since at each time-step the exchange\nparameters (and magnetic moment) should in principle\nbe recalculated, in order to take the changing tempera-\nture of the spin-system into account. If a material is a\ngood Heisenberg system or not, i.e. if the exchange pa-\nrameters are independent on the temperature or not, is\ndi\u000ecult to stipulate before a \frst principles calculation of\nthe con\fgurational dependent exchange parameters have\nbeen made, but several systems have by now been sug-\ngested to have exchange parameters that depend more or\nless strongly on temperature. Hence, it seems that there\nare indeed very few materials that are good Heisenberg\nsystems. In this article we demonstrate that Co, in fcc,\nbcc and hcp form, is rather unique in this sense, at least\namong the elemental metals, displaying the features of a\nHeisenberg magnet in a wide range of magnetic con\fgu-\nrations. As we shall see below, this comes with a twist,\nsince both the values of the magnetic moments and the\nstrength of the Heisenberg exchange depend on con\fgu-\nration, but put together in a spin Hamiltonian they form\na model that curiously is con\fguration independent. We\nalso investigate the magnetization dynamics of this sys-\ntem, and compare it to a Co-Mn alloy in the bcc and bct\nstructures.\nThe article is organized as follows. In Sec. II we present\nthe theoretical tools used to describe the ground state and\ndynamical properties. Sec. III contains the numerical\ndetails of the calculations. Results and discussions are\npresented in Sec. IV. Finally, we give conclusions in Sec.\nV.\nII. METHODS\nIn order to investigate ultrafast demagnetization dy-\nnamics of Co and Co-Mn alloys, we combined \frst princi-\nples electronic structure calculations with atomistic spin\ndynamics simulations. These methods are described be-\nlow.\nA. Electronic structure calculations\nThe ground state electronic structure and magnetic\nproperties of the studied materials are obtained via den-\nsity functional theory (DFT) calculations. The Kohn-\nSham equations are solved within the Korringa-Kohn-\nRostoker (KKR) Green's function formalism as imple-\nmented in the spin-polarized relativistic KKR (SPR-\nKKR) package20, and within linear mu\u000en-tin orbital\nmethod21in atomic sphere approximation (LMTO-ASA). We used both real-space (RS)22and reciprocal-\nspace23realisations of LMTO-ASA. The relativistic\ne\u000bects are considered by solving the fully relativis-\ntic Dirac equation24. The substitutional disorder is\ntreated by making use of the coherent potential ap-\nproximation (CPA)25. The high temperature paramag-\nnetic phase was modelled by disordered local moment\n(DLM) approximation26,27in combination with CPA.\nHere the Co-Mn binary systems were treated as quater-\nnary (Co\"\n0:5Co#\n0:5)1\u0000x(Mn\"\n0:5Mn#\n0:5)xalloys, with a ran-\ndom mixture of the two magnetic orientations of Co and\nMn. DLM approach is believed to accurately describe the\nhigh temperature paramagnetic phase26, therefore we ap-\nply this tool, in case of alloys, to evaluate whether the\nlocal magnetic moments and magnetic coupling constants\nare sensitive to the temperature induced \ructuations or\nnot.\nThe interatomic exchange interactions, Jij, are\ncalculated via the Liechtenstein-Katsnelson-Antropov-\nGubanov formalism (LKAG)15as implemented in the\nSPR-KKR and the RS-LMTO-ASA codes and its ex-\ntension to non-collinear spin arrangement18, (see Section\nII B). The site and element resolved Gilbert damping pa-\nrameters (\u000b,\u000bCo,\u000bMn) are calculated based on the linear\nresponse formalism28, (see Section II C).\nB. Calculation of the interatomic Jijexchange\nparameters\nInteratomic magnetic exchange interaction parame-\nters,Jij, are calculated from \frst principles. For collinear\natomic spin alignment, the method of in\fnitesimal spin\nrotation was derived almost thirty years ago15. The en-\nergy (grand potential) variation can be calculated when\nthe atomic spin is rotated by a small angle simultane-\nously, and mapped onto a bilinear spin model:\nH=\u00001\n2X\ni6=jJij~ ei\u0001~ ej; (1)\nwhere the unit vector ~ ei(~ ej) denotes the direction of\nthe spin at site i(j). Although it might seem trivial,\nwe write also this spin-Hamiltonian in the more common\nform, that explicitly describes the coupling of atomic spin\nmoments,~ m:\nH=\u00001\n2X\ni6=j~Jij~ mi\u0001~ mj: (2)\nFor the discussion of our results (below) it becomes rele-\nvant to make a distinction between Eq. (1) and (2), and\nthe fact that Jijand ~Jijdi\u000ber only by a factor ~ mi=mi~ ei.\nThe LKAG interatomic exchange formula can be writ-\nten asJij=A\"#\nij, where the symbol \"and#refer to the\nup and down spin channels, respectively, while\nA\u000b\f\nij=1\n\u0019EFZ\n\u00001d\"Im TrL\u0010\npiT\u000b\nijpjT\f\nji\u0011\n: (3)3\nFor collinear spin con\fguration the corresponding T\"\nijand\nT#\nijmatrices denote the component of the scattering path\noperator (SPO), \u001cij, in the two spin channels between site\niandjwhilepiandpjstand for the (spin-part) of the\ninverse of the one-site scattering matrix15. In order to\ntreat alloys or alloy analogy models, the defect atom \u0016\nis created at site iby a defect matrix D\u0016\ni. This de-\nfect matrix is considered in the e\u000bective CPA medium\n~\u001ci\u0016;j\u0017 =D\u0016\ni\u001cCPA\nijD\u0017\nj16. Hence, the scattering path oper-\nator ~\u001ci\u0016;j\u0017 replaces related components of the scattering\npath operator in Eq. (3).\nIn non-collinear spin arrangement, the SPO matrix el-\nements can be grouped into a charge and spin part with\nthe help of the two times two unit matrix and the Pauli\nmatrices. Hence, one can de\fne the exchange matrix A\u000b\f\nij\nwhere indices \u000band\frun over 0,x,yorz. By using trace\nproperties, the general symmetry relation A\u000b\f\nij=A\f\u000b\nji\nwas found. In the absence of spin-orbit coupling we can\nwrite that\u0000\nT\u000b\nij\u0001T=T\u000b\nji, which implies that A\u000b\f\nij=A\f\u000b\nij,\ni.e., the A-matrix is symmetric. The grand potential\n(pairwise) variation is proportional to the variation of\nthe integrated density of states, which is determined by\nusing the Lloyd formula29. This leads to the expression\n\u000eEij=\u00002Jnc\nij\u000e~ ei\u000e~ ej\u00004X\n\u0016;\u0017=x;y;z\u000ee\u0016\niA\u0016\u0017\nij\u000ee\u0017\nj (4)\nwhen the spin-orbit interaction is not considered, where\nJnc\nij=A00\nij\u0000Axx\nij\u0000Ayy\nij\u0000Azz\nij: (5)\nAt low temperature, where the degree of non-collinearity\nbetween atomic spins is small (a regime we denote the\nquasi collinear regime) the second term of Eq. (4) does\nnot give a signi\fcant contribution, hence we will here\nresolveJnc\nijfor di\u000berent systems, which can be mapped\nonto Eq. (1), i.e. onto a Heisenberg model when the\ncalculated exchange parameters are spin con\fguration-\nindependent. Note that in the exact collinear limit (e.g.\nin ferromagnetic ground state) Eq. (5) reduces to the\nexpression A00\nij\u0000Azz\nij, and it can be shown that this is\nequal to the LKAG formula given by A\"#\nij.\nC. Element and site resolved damping parameters\nWithin the present work, the Gilbert damping pa-\nrameter is calculated on the basis of the linear response\nformalism28. The approach used derives from a represen-\ntation of the electronic structure in terms of the Green\nfunctionsG+(E) that in turn is determined by means of\nthe multiple scattering formalism30. The diagonal ele-\nments\u0016=x;y;z of the Gilbert damping tensor can be\nwritten as28:\n\u000b\u0016\u0016=g\n\u0019mtotX\njTr\nT\u0016\n0~\u001c0jT\u0016\nj~\u001cj0\u000b\nc; (6)\nwhere the e\u000bective g-factor g= 2(1 +morb=mspin) and\ntotal magnetic moment mtot=mspin+morbare givenby the spin and orbital moments, mspinandmorb, re-\nspectively, ascribed to a unit cell. Eq. (6) gives \u000b\u0016\u0016for\nthe atomic cell at lattice site 0 and implies a summation\nover contributions from all sites indexed by jincluding\nj= 0. The elements of the matrix ~ \u001c0jare given by\n~\u001c\u0003\u00030\n0j=1\n2i(\u001c\u0003\u00030\n0j\u0000\u001c\u00030\u0003\n0j) where\u001c0jis the so-called SPO\nmatrix28evaluated for the Fermi energy, EF. Finally, the\nmatrixT\u0016\njis represented by the matrix elements\nT\u0016;\u00030\u0003\nj =Z\nd3r(Z\u00030\nj(~ r))\u0002[\f\u001b\u0016Bxc(~ r)]Z\u0003\nj(~ r);(7)\nof the torque operator ^T\u0016=\f(~ \u001b\u0002^mz)\u0016Bxc(~ r)31. Here,\nZ\u0003\nj(~ r) is a regular solution to the single-site Dirac equa-\ntion for the Fermi energy EFlabeled by the combined\nquantum numbers \u0003 (\u0003 = ( \u0014;\u0016)), with\u0014and\u0016being\nthe spin-orbit and magnetic quantum numbers32.\nTo calculate the con\fgurational average indicated by\nthe bracketsh:::ic, in the case of disordered alloys, the\nCPA alloy theory is used. This is done using the scheme\ndeveloped by Butler33in the context of electrical con-\nductivity, that splits the summation in Eq. (6) into a site\ndiagonal part,hT0\u0016~\u001c00T\u0016\n0~\u001c00ic, and a site o\u000b-diagonal\npart,P\nj6=0\nT\u0016\n0~\u001c0jT\u0016\nj~\u001cj0\u000b\nc, respectively. Dealing with\nthe second term one has to account in particular for\nthe so-called \\in-scattering processes\" that deals with\nvertex corrections of crucial importance for the Gilbert\ndamping34.\nAs indicated above, Eq. (6) gives in the case of a unit\ncell involving in the case of an alloy several atomic types\na value for \u000b\u0016\u0016that is averaged over these types. In the\ncase of a system consisting only of magnetic components,\ni.e. none of its components has an induced magnetic mo-\nment, one may also introduce a type-projected damping\nparameter\u000b\u0016\u0016\nt. As the average for the site diagonal as\nwell as site o\u000b-diagonal contributions to \u000b\u0016\u0016involve a\nsum over the types twith the type-speci\fc contribution\nweighted by the corresponding concentration xt33one is\nled in a natural way to the expression:\n\u000b\u0016\u0016\nt=gt\n\u0019mtTrT\u0016\n0hD\n\u001c00T\u0016\n0\u001c00E\nton0\n+X\nj6=0X\nt0onjxt0D\n~\u001c0jT\u0016\nj~\u001cj0E\nton0;t0onji\n(8)\nwithtandt0denoting the atomic types at the lattice po-\nsitions 0 and j, respectively. Here, we use a type-speci\fc\ng-factorgtand magnetic moment mtgiven by the cor-\nresponding spin and orbital moments, mt\nspinandmt\norb,\nrespectively. The resulting de\fnition for the element pro-\njected Gilbert damping \u000b\u0016\u0016\ntleads now to an average for\nthe unit cell according to: \u000b\u0016\u0016=P\ntxt\u000b\u0016\u0016\nt. Because of\nthe normalizing factor g=mtotused in Eq. (6) this expres-\nsion will lead in general to results slightly deviating from\nthat based on Eq. (6).\nThe calculations of the Gilbert damping parameter for\n\fnite temperature presented below have been done us-\ning the so called alloy analogy model28. This approach4\nis based on the adiabatic approximation assuming ran-\ndom temperature dependent displacements of the atoms\nfrom their equilibrium positions. Using a discrete set of\ndisplacements with each displacement treated as an alloy\ncomponent, the problem of calculating the thermal aver-\nage for a given temperature Tis reduced to the problem\nof calculating the con\fgurational average as done for sub-\nstitutional alloys28.\nD. Atomistic spin dynamics\nThe temperature dependent evolution of spins are cal-\nculated from atomistic spin-dynamics (ASD) simulation\nat di\u000berent temperatures using the framework of Landau-\nLifshitz-Gilbert (LLG) formalism. The temporal evolu-\ntion of an atomic moment in LLG formalism is given\nby35,\nd~ mi(t)\ndt=\u0000\r\n(1 +\u000b2)\u0010\n~ mi(t)\u0002~Bi(t)+ (9)\n\u000b\nmi~ mi(t)\u0002(~ mi(t)\u0002~Bi(t))\u0013\n;\nwhere\ris the gyromagnetic ratio, \u000brepresents the di-\nmensionless Gilbert damping constant and ~ mistands\nfor an individual atomic moment on site i. Note that\n~ mi=mi~ eiwheremiis the magnitude of the magnetic\nmoment (at site i). The e\u000bective magnetic \feld is rep-\nresented by ~Bi=\u0000@H\n@~ mi+~bi, whereHis given by Eq.\n(2) and~biis a time evolved stochastic magnetic \feld\nwhich depends on the spin temperature from the two-\ntemperature (2T) model36.\nThe analytical expression of the two temperature\nmodel reads as,\nTs=T0+ (10)\n(TP\u0000T0)\u0002(1\u0000exp(\u0000t=\u001cinitial ))\u0002exp(\u0000t=\u001cfinal)+\n(TF\u0000T0)\u0002(1\u0000exp(\u0000t=\u001cfinal))\nwhereTsis the spin temperature, T0is initial tempera-\nture of the system, TPis the peak temperature after the\nlaser pulse is applied and TFis the \fnal temperature.\n\u001cinitial and\u001c\fnalare exponential parameters. The calcu-\nlated spin temperature from Eq. (10), is explicitly passed\ninto LLG equation via the stochastic magnetic \feld ~bi\nin Eq. (9), which takes into account thermal \ructuations\nof the system and the strength of the stochastic \feld is\nde\fned as D=\u000bkBTs\n\rm,kBis the Boltzmann constant.\nAlloying Co is treated by spatial random disorder of the\nMn dopant.\nThe dynamical structure factor, which describes the\nmagnon dispersion relation, is obtained from the Fourier\ntransform of space and time displaced correlation func-\ntion\nC\u0016(r;r0;t) =hm\u0016\nr(t)m\u0016\nr0(0)i\u0000hm\u0016\nr(t)ihm\u0016\nr0(0)i;(11)where the ensemble average is represented in the angular\nbrackets and \u0016=x;y;z is the Cartesian component, and\nits Fourier transform is written as,\nS\u0016(q;!) =1p\n2\u0019NX\nr;r0eiq\u0001(r\u0000r0)Z1\n\u00001dtei!tC\u0016(r;r0;t);\n(12)\nwhere qand!are the momentum and energy transfer,\nrespectively. Nis the number of terms in the summation.\nTo estimate the Curie temperatures we used the fourth\norder size dependent Binder cumulant19, which is de\fned\nas,\nUL= 1\u0000hM4iL\n3hM2i2\nL; (13)\nwhereMis the total or average magnetization. h:::iis\nthe ensemble and time average. Binder cumulants ex-\nploits the critical point and critical exponents in a phase\ntransition from the crossing point of magnetization curves\nfor di\u000berent sizes Lof the system.\nIII. NUMERICAL DETAILS\nThe Perdew, Burke and Ernzerhof (PBE)37version of\nthe generalized gradient approximation is used to de-\nscribe the exchange-correlation potential. The spin po-\nlarized scalar relativistic full-potential (SR-FP) mode20\nis used to calculate the total energies as a function of\nvolume (E(V)) and the total ( M) and element resolved\n(mCo,mMn) magnetic moments. For the exchange inte-\ngral (Jij) and damping parameter ( \u000b,\u000bCo,\u000bMn) calcula-\ntions, the potential is described within the atomic sphere\napproximation (ASA) using the scalar-relativistic (SR)\nand relativistic (R) mode, respectively20,34. The basis set\nconsisted of s,p,dandforbitals (lmax=3). The num-\nber of kpoints was set to \u0019300,\u0019500 and\u00191000000 for\nthe calculation of the ground state properties, density of\nstates (magnetic exchange integrals) and site and element\nresolved damping parameters, respectively. Equilibrium\nlattice constants are obtained by \ftting E(V) curves with\na Morse type of equation of state38. The exchange con-\nstants are calculated up to 12 nearest neighbour shells.\nWe performed ASD simulations by using the UppASD\nsoftware39,40for the Co-based systems with a size of 20\n\u000220\u000220 unit cells and also with periodic boundary\nconditions. Here we used calculated exchange constants\nand averaging over 16 ensembles.\nIV. RESULTS AND DISCUSSION\nA. Static properties of Co and Co-Mn alloys\n1. Electronic structure and magnetic properties\nThe estimated theoretical lattice parameters ( a) are5\nTABLE I. Theoretically estimated lattice parameters ( a), total (M) and element ( mCo,mMn) projected magnetic moments,\nCurie temperatures ( TC), total and element projected densities of states at the Fermi level (DOS( EF)) and Gilbert damping\nparameters ( \u000b) calculated for bcc, fcc and hcp Co as well as for Co 1\u0000xMnx(x= 0:1;0:15;0:2;0:3) alloys in the bcc and bct\ncrystallographic phase. The experimental Curie temperatures, lattice constant, magnetic moment are shown in brackets and\nthe DLM results are shown in parentheses. The magnetic moments are calculated using experimental lattice parameters.\nSystem a/c(\u0017A) M(\u0016B/atom) TC(K) DOS(EF) (states/Ry) \u000b\nbcc Co 2.85[2.8348] 1.70[1.7749,1.5050] 1280 25.3(31.448) 0.0091(0.011)\nfcc Co 3.58[3.5451] 1.62[1.68]541311[139265] 16.8(29.170) 0.0057(0.009)\nhcp Co 2.48/4.04[2.50/4.05]661.59[1.52]551306[138864] 12.8(28.68) 0.0030(0.019)\nSystem M m ComMn total Co Mn \u000b \u000b 1(Co)\u000b2(Mn)\nCo0:90Mn0:10672.86 1.89 1.77(1.51) 2.99(2.75) 1280(1054) 20.1 21.6 7.1 0.0072 0.0083 0.0013\nCo0:85Mn0:15672.87 1.96 1.78 2.98 1248 19.5 21.2 9.1 0.0066 0.0081 0.0015\nCo0:80Mn0:20 2.88 2.03 1.79 2.97 1129 18.3 20.2 11.0 0.0058 0.0076 0.0016\nCo0:70Mn0:30 2.89 2.13 1.80(1.46) 2.89(2.72) 1050(833) 16.0 17.5 12.7 0.0045 0.0061 0.0022\nCo0:90Mn0:10bct 2.83 [2.7168] 1.79 1.69 2.73 1235[121568]19.1 20.2 9.0 0.0080 0.0090 0.0025\nCo0:70Mn0:30bct 2.87 [2.9068] 2.11 1.80 2.85 1054 [84268]16.8 17.6 14.9 0.0047 0.0062 0.0025\nlisted in Table I for bcc, fcc, and hcp Co as well as\nCo1\u0000xMnxalloys in the bcc and bct crystallographic\nphase, calculated using the PBE exchange-correlation\nfunctional. The local density approximation (LDA)\ncalculations underestimates the lattice parameter with\nabout 2% when compared to PBE. The presented PBE\nvalues for pure Co are in good agreement with the ex-\nperimental data found for bcc Co grown in GaAs surface\n(2.82 \u0017A)41and for fcc Co/Cu \flm (3.54 \u0017A)42, as well as\nwith the results of the previous DFT simulations43. The\nCo-Mn alloys can be grown on a GaAs surface in bcc44,45\nor bct46,47crystal structure. The theoretical lattice con-\nstantaof bcc alloys increases with Mn addition, which is\nconsistent with the larger atomic radius for Mn compared\nto Co. The estimated lattice constants for x=0.3 (see Ta-\nble I) andx=0.4 (2.89 \u0017A) is in line with the experimen-\ntal lattice parameters data reported for x=0.32 (0.4)44,45\nwhich is 2.9 (2.89) \u0017A. The in-plane lattice parameter for\nthe bct phase of Co 0:90Mn0:10and Co 0:70Mn0:30alloys are\ntaken from experiments, Refs. [46] and [47], respectively,\nwhile the out-of-plane lattice parameters have been opti-\nmised theoretically (see Table I).\nCalculated densities of states (DOS) for pure Co in bcc,\nfcc and hcp crystal structure are presented in Fig. 1. For\nthese crystal structures, the 3 dmajority spin channel is\nfully occupied, resulting in a low DOS at the Fermi level\nDOS\"(EF), while DOS#(EF) lies near a peak in the 3 d\nDOS. The energy split between the majority and minor-\nity channels leads to a magnetic moment of 1.73 \u0016Bfor the\nbcc lattice in good agreement with the previous theoret-\nical data43,49. The experimental value for the magnetic\nmoments of Co in the bcc structure are estimated from\nCo \flms grown on GaAs50,52. The average value is given\nas 1.4\u0016Bbut in the centre of the \flm (50 \u0017A) the esti-\nmated experimental value for the Co magnetic moment\nin the bcc structure is \u00181.7\u0016B52which is in good agree-\nment with the theoretically estimated value. For fcc Co,the calculated magnetic moment is in agreement with the\npreviously published theoretical value of 1.64 \u0016B43,53and\nin decent agreement with the experimental value of 1.68\n\u0016B54. Finally the magnetic moment of hcp Co is in good\nagreement with the reported experimental55and theoret-\nical data43.\nIn both bcc and bct phases of Co-Mn alloys the\nDOS\"(EF) of Co is small due to the full occupation of\nthe 3dmajority channels, and the shift in the occupa-\ntion of the majority and minority channels results in a\nmagnetic moment higher than that in pure bcc Co and it\nis found to increase with increased Mn content (see Ta-\nble I). The Mn 3 dmajority band is not fully occupied,\nand the 3dminority band contains less states than in\ncase of Co, due to the reduced number of electrons for\nMn. The exchange splitting results in a higher magnetic\nmoment for Mn than for Co (see Table I). As Table I\nshows for Co-Mn alloys the coupling between Co and Mn\nmoments is ferromagnetic, with a large moment on both\natoms. The results for the bcc structure give larger mo-\nments compared to data for the bct structure.\nThe smallest magnetic moments are given by the SR-\nFP mode. The SR-ASA (R spin) moments are in average\n0.6 (0.5) % larger compared to the SR-FP moments. We\n\fnd the same trends for SR-ASA and R spin moments\nas a function of composition and structure as in the case\nof SR-FP moments. The orbital moment of Co is 0.085\n\u0016Bin the bcc phase. Its variation among di\u000berent crystal\nstructures and alloying is within 7% and follows the same\ntrend as for the spin moments. The orbital moment of\nMn in bcc Co 0:9Mn0:1is 0.018\u0016B. This value decrease to\n0.016\u0016Bforx=0.3 Mn content in the bcc phase. The or-\nbital moment of Mn in the bct phase is smaller compared\nto its value in the bcc phase for the corresponding com-\nposition. Local magnetic moments of Co and Mn in the\nDLM phase of Co 1\u0000xMnxforx= 0:1 and 0:3 are also\npresented in Table I. Here we \fnd that mCoandmMn6\n-8 -7 -6 -5 -4 -3 -2 -1 0 1 2\n-2 -2-1 -10 01 12 23 3\nbcc\nfcc\nhcp\n-2-1012DOS (states/eV)\nCo\nMn-2-1012\n-8 -6 -4 -2 0 2\nE-EF (eV)-2-1012\n-8 -6 -4 -2 0 2-2-1012Co\nbct x = 0.1bcc x = 0.3 bcc x = 0.1\nbct x = 0.3\nFIG. 1. (Color online) Density of states (DOS) per atom of\nbcc Co (blue), fcc Co (red), hcp Co (green) (upper panel)\nand bcc and bct Co 1\u0000xMnx(lower panel). DOS for Co in\nCo1\u0000xMnxalloys is labeled by full line while DOS of Mn is\nrepresented by dashed line. Dotted line illustrates the Fermi-\nenergy.\nare reduced with 15% and 20%, respectively, in the DLM\nphase compared to that of FM solution.\nAll entries in Table I show that the DLM con\fgura-\ntion results in lower magnetic moments than for the FM\ncon\fguration. To analyse this further we calculated the\nsize of the magnetic moment of a supercell of 16 atoms\nin a bcc lattice, in which only the central atom had its\nmagnetic moment rotated away from the z-axis with an\nangle\u0012. The rotated moment is denoted as miand the\nrest of the spins are labeled mj. The self-consistently ob-\ntained values of miandmjare shown for each value of \u0012\nin Fig. 2. One can see that once \u0012increases, the magni-\ntude of the moment mitends to decrease. We repeated\nthe same calculations for bcc Fe and obtained qualita-\ntively the same behaviour. Thus, the results of Fig. 2 are\nconsistent with the data in Table I, and seem to re\rect a\nquite general phenomenon that the rotation of a moment\nin a system with predominant FM interactions leads to\nthe decrease of its length. This fact can be understood\non the basis of a simple model, containing the energy\nof longitudinal spin variation (containing even powers of\nmagnetization, as appropriate for a Landau expansion)\nand a nearest-neighbour exchange coupling ( J1). In the\ncase of the single spin rotation in the ferromagnetic back-\nground, one obtains:\nE=\u0000\u000b1m2\n1\u0000\u000b2M2+\f1m4\n1+\f2M4\u0000~J1~ mi\u0001~M;(14)\nwhereM=PN\nj=1mjrepresents the macro-spin formed\nby allNnearest neighbour spins from the FM back-\nground. The parameters \u000b1,\u000b2,\f1and\f2are phe-\nnomenological constants originating from the local ex-\nchange interactions. The energy penalty stemming from\n0 20 40 60\nθ (°)1.41.51.61.71.8Magnetic moment  (µΒ)\nmj ≠ i\nmiFIG. 2. Calculated magnetic moment as a function of rotation\nof a single spin with an angle \u0012, in the FM background of bcc\nCo.\nthe Heisenberg term when rotating the moment miwith\nan angle\u0012will be given by:\nE(\u0012)\u0000E(0) = ~J1miM(1\u0000cos(\u0012)): (15)\nIt is straightforward to show that, if the magnitude of mi\nis allowed to change, the system will try to minimize the\nenergy costs of the single spin rotation by decreasing its\nlength. In principle a reduction of Mwould also reduce\nthe energy cost of rotating a single spin, but the Landau\nparameters describing this change are not in favor of this\nscenario. Finding the minima of the energy with respect\ntomileads to the solution of non-linear equation, which\ncan be solved numerically. The numerical results con\frm\nthat with an increase of \u0012, the value of micorresponding\nto the minimum of the energy goes down.\nThus, for the case of a single spin rotation in bcc Co,\nwe have shown that the magnitude of the magnetic mo-\nments unavoidably depends on the magnetic con\fgura-\ntion. In order to quantify how sensitive the magnetic\nexcitations to inter-atomic non-collinearity, we have per-\nformed a series of spin spiral calculations56for the same\nstructure. Spin spiral states are characterized by the\npropagation direction ( ~ q) and the cone angle \n between\nthe magnetization and ~ qvectors. Note that spin spi-\nrals with in\fnitesimal \n would correspond to the actual\nmagnon excitation. The bottom panel of Fig. 3 shows\nthe self-consistently obtained value of the magnetic mo-\nment in all di\u000berent spin spiral states. Just as in the\ncase of the single spin rotation (Fig. 2), the magnetic\nmoment experiences a variation when ~ qis changed. In\nthe top panel of Fig. 3 we show the relative energies of\nthe spin spirals calculated for various ~ qand \n values.\nOnyaxis we plot E~ q\u0000Eq=0=sin2(\n), which is supposed\nto be \n-independent for a truly Heisenberg magnet (see\ne.g. Ref. 57). It is clearly seen from Fig. 3 that, despite\nthe changes in the magnetic moment values, all curves lie7\n0.00.10.20.3(Eq - Eq=0) / sin2(Ω) (eV)\nΓ N P Γ H0.00.40.81.21.6Moment ( µB)\nΩ=5ο\n30o\n45o\n60o\n90o\nFIG. 3. Calculated relative spin spiral energies for di\u000berent\nvalues of cone angle \n along with the self-consistent values of\nthe magnetic moment as a function of the propagation vector\n~ q.\nnearly on top of each other, if \n lies within the range of 5\nto 45 degrees. At larger \n angles, most of spin spiral en-\nergies are still very close to each other and the largest dif-\nferences appear for ~ q-vectors along \u0000-H direction. Hence,\none can see that Co is a remarkable system, which is char-\nacterized by a con\fguration-independent magnetic exci-\ntations in a wide range of magnetic states. From Fig. 3\nwe estimated that the Heisenberg Hamiltonian (Eq. (1))\nis perfectly valid up to a critical value of the angle be-\ntween the nearest-neighbouring spins of about 90 degrees.\nQuite importantly, the results indicate an intriguing in-\nterplay between the strength of the ~Jij's and the magni-\ntude of~ mi's, which tend to balance each other resulting\nin a con\fguration-independent Jij's.\n2. Temperature dependent exchange interactions Jij\nThe calculated exchange interactions, Jij, for all Co-\nbased systems at T= 0 K are plotted in Fig. 4. The\nCo-Co interactions have positive values showing a fer-\nromagnetic coupling between Co atoms. In the Co-Mn\nalloys, the Mn atoms are ferromagneticaly coupled to the\nCo atoms and favour antiparallel coupling to the nearest\nMn atoms, while the Co-Co interactions are ferromag-\nnetic. AllJij's decay fast with distance. Increased Mn\ncontent is found to enhance all interactions, which can\nbe explained by an increase of mCoandmMn, since ac-\n051015hcp Cobcc Co\nfcc Co (DLM)\nhcp Co(DLM)\nbcc Co(DLM)\n0.51 1.5 2 2.5 0\n51015fcc Co\n-20\n-1001020Jij (meV)-20\n-10010200.5\n1 1.5 2 -20-10010200.5\n1 1.5 2 2.5 d \n(a units)-20-1001020Co0.1Mn0.9Co0.7Mn0.3Co\n0.1Mn0.9Co0.7Mn0.3FIG. 4. (Color online) Exchange integrals ( Jij) from FM con-\n\fguration for fcc, hcp and bcc Co (up triangle) are repre-\nsented in solid lines, DLM (with dashed lines) and bcc and\nbct Co 1\u0000xMnxalloys plotted as a function of distance d(in\nalattice parameter units). JCo\u0000Coare labeled by \flled red\ntriangles,JCo\u0000Mnby \flled black squares and JMn\u0000Mnby \flled\ngreen circles. The exchange parameters are obtained for the\nFM state.\ncording to Eq. (1), the magnitude of the moments is ef-\nfectively contained within the Jij's. The results obtained\nfor the di\u000berent phases of elemental Co are in overall\ngood agreement with prior DFT studies, the di\u000berences\ncoming from the employed computational methods.58{62\nTo continue the analysis of exchange interactions in\nthese systems we investigated whether hcp, fcc, and bcc\nCo have Heisenberg exchange parameters that are con\fg-\nuration (temperature) dependent. To this end we deter-\nminedJnc\nijde\fned by Eq. (5). Note that the second term\nin Eq. (4) did not give a signi\fcant contribution in Co\nsystems we considered here. We compare these results\nto those of bcc Fe which has been already shown to have\nJij's that are con\fguration dependent, and hence not to\nbe a perfect Heisenberg system63. Note that ~ eiin Eq. (4)\ndenotes the direction of the spin at site i, which can be\nformulated as ei=e(\u0012i;\u001ei) where\u0012iand\u001eiare the polar\nand azimuthal angles of the spin direction, respectively.\nThe most simple non-collinear spin con\fguration may be\nthe case when one spin in a ferromagnetic background\nis being rotated by a \fnite angle \u0012. The dashed lines in\nFig. 5 show the Jnc\nij's for the nearest neighbour couplings\nin bcc Co and bcc Fe. We \fnd that bcc Fe is more con-\n\fguration dependent than bcc Co, i.e., bcc Co is closer\nto a \\perfect\" Heisenberg system. However, Fig. 2 shows\nthat this story is somewhat more complex, since for single\nsite rotation the magnetic moment changes signi\fcantly\nwith angle of rotation. As was already demonstrated for\nthe case of spin spirals (Fig. 3), there is no contradiction,\nsinceJijvalue is de\fned in such a way that it contains8\nthe magnetic moment value in itself. Thus, we witness\nonce again that bcc Co re\rects the physics intrinsic to\nan ideal Heisenberg magnet despite the sensitivity of its\nmagnetic moment to the environment.\n010 20 30 40 50 60 θ\n05101520Jnnnc(meV)Fe bcc 1NN\nCo bcc 1NN\nFe bcc 1NN\nCo bcc 1NN\nFIG. 5. (Color online) Solid lines: Non-collinear exchange\ncouplingJnc\nijde\fned by Eq. (5) for \frst neighbour spin pairs\nin bcc Fe and Co when one spin is \fxed and the spin directions\nat its \frst neighbour sites are rotated by \u0012and\u001e. Dashed\nlines: Non-collinear exchange coupling Jnc\nijde\fned by Eq. (5)\nfor \frst neighbour spin pairs in bcc Fe and Co when one spin\nis rotated by \u0012and\u001e. The azimuthal angle \u001eis set by a\nrandom number generator.\nA more realistic spin con\fguration can be constructed\nwhen one spin is \fxed at, say, site i, and where the spin\ndirections at its \frst neighbour sites are rotated by \u0012and\n\u001e. These results are shown by the solid lines in Fig. 5,\nmodelling a \fnite temperature disordered background.\nFor such a con\fguration we also calculated the Jnc\nij's for\nneighbours, with varying distance. Fig. 6 shows all the\ninteratomic exchange coupling parameters for the \frst six\nnearest neighbour shells in bcc Fe, fcc Co, bcc Co and hcp\nCo, respectively. As can be seen, all the Co phases seem\nto have an excitation spectrum that is close to an ideal\nHeisenberg system, but again it is due to a decrease of\nthe individual moments and an increase of the Heisenberg\nexchange interaction, ~Jij, as given by Eq. (2).\nThe \fndings presented above motivate that atomistic\nspin dynamics simulations can be made from Heisenberg\nexchange parameters from collinear ferromagnetic phases\nof bcc, fcc and hcp Co, if the de\fnition of Eq. (1) is used\nfor the energy excitations. As mentioned above, of all\nsystems investigated here elemental Co stands out to be\nunique in this regard.\nTheJijof Co-Mn alloys are evaluated only in the DLM\nphase (not shown) and only for alloy composition x= 0:1\nand 0:3 , since the aforementioned approach is cumber-\nsome to apply for random alloys due to methodological\nreasons. The nearest neighbour JCo\u0000Co,JCo\u0000Mnand\nJMn\u0000Mninteractions are reduced compared to the value\n-2381318-5\n051015201NN\n2NN\n3NN\n4NN\n5NN\n6NN\n0\n5 1015 20 25 -50510150\n5 1015 20 25 30 -5051015 Jijnc(meV)a)b) c)\nd) bcc Fefcc Co bcc Co\nhcp Co θ\nFIG. 6. (Color online) Non-collinear exchange coupling Jnc\nij\nde\fned by Eq. (5) for the \frst six neighbour spin pairs in\nbcc Fe when one spin is \fxed and the spin directions at its\n\frst neighbour sites are rotated by \u0012and\u001e. Similar for the\n\frst six neighbour spin pairs in fcc, bcc and hcp Co systems\nreceptively. In these \fgures, the azimuthal angle \u001eis set by a\nrandom number generator.\nof the pure element by 21%, 26%, 7%, respectively, for\nx= 0:1 and 35%, 24%, 38%, respectively, for x= 0:3 .\nThis variations are in the same order as those obtained for\nCo in the DLM phase. Hence, Co and doped Co tend to\nbehave like a \"bad\" Heiseberg system close to the phase\ntransition temperature. In the low temperature regime,\nhowever, exchange parameter of Mn are likely to change\n(not shown here), similar to Fe, giving evidence for a non-\nHeisenberg behavior of CoMn even at low temperature.\n3. Curie temperatures\nThe calculated and experimental values of the Curie\ntemperatures of Co-based systems are presented in Ta-\nble I. The calculated Curie temperature values of bcc, fcc\nand hcp Co are in a rather good agreement with exper-\nimental data. The measured Curie temperatures of fer-\nromagnetic Co 1\u0000xMnxalloys decreases linearly with an\nincrease of concentration of Mn and becomes zero around\n0.4. The calculated Curie temperature of Co 1\u0000xMnxal-\nloys decrease linearly up to 30% of Mn and the TC's\nare somewhat overestimated when compared with exper-\nimental values. We address this discrepancy to slight\nvariations of the Heisenberg parameter with Mn doping\nat the phase transition temperature. This \fnding is sup-\nported by the reduced TCvalues calculated in DLM con-\n\fguration for Co 0:9Mn0:1and Co 0:7Mn0:3(see Table I).\nSince the di\u000berence in the phase transition temperature\nbetween the FM and the DLM phase are small com-\npared to the maximal temperature obtained for the sim-\nulated pump-probe experiment, we make the approxima-9\ntion that a small amount of Mn in Co-Mn alloys will\nnot change the temperature independence of the Heisen-\nberg exchange parameters. Hence, the extracted FM\ncollinear parameters are used in simulated pump-probe\nexperiments of bcc, fcc and Co, as well as bcc and bct\nCo-Mn alloys, as detailed below.\n4. Damping parameters\nResults for the Gilbert damping parameter of pure Co\nas well as for Co 1\u0000xMnxalloys calculated at T= 324 K\nare presented in Table I together with the DOS( EF). In\nthe case of pure bcc Co, this temperature corresponds\napproximately to the minimum of the \u000b(T) curve, that\nindicates the cross-over of the contributions due to the in-\ntraband (dominating at low temperature) and the inter-\nband (dominating at high temperature) electron scatter-\ning events.69In the case of Co 1\u0000xMnxalloys, on the other\nhand, the interband spin-\rip scattering events are respon-\nsible for magnetization dissipation in the whole tempera-\nture regime, similar to the case of Cu impurities in Ni28.\nWhen the temperature increases above room temperature\n(not shown here), the thermal lattice vibrations lead for\nCo as well as Co 1\u0000xMnxalloys to an increase of \u000b(T).\nAs can be seen in Table I, an increase of the Mn concen-\ntrationxfor bcc Co 1\u0000xMnxresults in a decrease of the\nGilbert damping, which correlates well with a decrease\nof the total DOS( EF). For the atom-resolved damping\nparameters we also \fnd its correlation with the value of\nthe DOS(EF)/atom. With increasing Mn concentration\nin these alloys, the damping parameter of Mn spins and\nDOS(EF)/atom both tend to increase, while the oppo-\nsite is found for the Co spins. Comparing the Gilbert\ndamping for the bcc and bct phases of Co 0:90Mn0:10\n(Co0:70Mn0:30), we \fnd that the damping reaches higher\nvalues in bct phase than in bcc for both sublattices, which\nis in contrast to the DOS( EF)/atom. This point also in-\ndicates that other e\u000bects also play role in determining\nthe damping parameter, and that the correlation between\ndamping and DOS is strongest for alloys within the same\ncrystal structure.\nB. Dynamical properties\n1. Dispersion relations\nThe calculated dynamical structure factor or magnon\nspectrumSz(q;!) of fcc, bcc and hcp Co along the high\nsymmetry directions of the Brillouin zone are presented in\nFig. 7. These simulations used the exchange parameters\nreported in Fig. 4 and the Gilbert damping of 0.005. The\nfcc Co magnon dispersion along \u0000-X high-symmetry path\nis in good agreement with experimental magnon data71.\nThe hcp Co magnon dispersion along \u0000-M and \u0000-A high-\nsymmetry path agrees also quite well with experimental\nmagnon data already reported in Refs. [72 and 73]. Also,the results in Fig. 7 are consistent with results published\nby Etz et al.70.\n/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s54/s48/s48/s55/s48/s48/s56/s48/s48/s57/s48/s48\n/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s54/s48/s48/s55/s48/s48/s56/s48/s48/s57/s48/s48\n/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48/s54/s48/s48/s55/s48/s48/s56/s48/s48/s57/s48/s48\n/s75\n/s101/s120/s112/s46/s32/s66/s97/s108/s97/s115/s104/s111/s118/s32/s50/s48/s48/s57/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41/s67/s111/s32/s102/s99/s99/s97/s41\n/s77 /s65 /s72 /s76/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41/s97/s41 /s97/s41/s97/s41 /s97/s41\n/s97/s41 /s97/s41\n/s67/s111/s32/s98/s99/s99/s98/s41\n/s72 /s78 /s80\n/s65/s101/s120/s112/s46/s32/s80/s101/s114/s114/s105/s110/s103/s32/s49/s57/s57/s53\n/s101/s120/s112/s46/s32/s83/s104/s105/s114/s97/s110/s101/s32/s49/s57/s54/s56/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s41/s67/s111/s32/s104/s99/s112/s99/s41/s76 /s88 /s87 /s75\nFIG. 7. (Color online) Magnon dispersion relation of fcc, bcc\nand hcp Cobalt along high symmetry directions of the Bril-\nlouin zone. Dots represent experimental measurements from\nRef. [71]. All simulations were done assuming a negligibly low\ntemperature and in FM state.\n2. Ultrafast magnetization under laser \ruence\nIn the following section we present ultrafast magnetiza-\ntion dynamics of fcc, bcc and hcp Co as well as bcc and\nbct Co-Mn alloys under the in\ruence of a femtosecond\nlaser pulse. The results are obtained by the integration\nof the atomistic LLG equation in combination with the\nanalytical two temperature model74, and are plotted in\nFig. 8. The simulated laser pulse results in a temperature\npro\fle that initially starts at room temperature as initial\ntemperature, T0and reaches its maximum at TP=1500K10\nand \fnally relaxes to TF= 450K via several scattering\nprocesses (see top panel of Fig. 8). The exponential pa-\nrameters,\u001cinitial = 1\u000110\u000014and\u001c\fnal= 3\u000110\u000012are\nused in the 2TM model. Damping values taken from\nthe \frst principles theory are considered in the simula-\ntions, called, \u000b, but also arti\fcially enhanced values of\nthe damping are used to investigate how the damping\nin\ruence the ultrafast magnetization. The enhancement\nconsidered is 5 and 10 times larger, i.e. 5 \u0002\u000band 10\u0002\u000b.\nThe magnetization decreases rapidly to a minimum of\nabout 44% to 84% of the total magnetization in fcc Co,\n42% to 84% in bcc Co and 20% to 57% in hcp Co for\ndi\u000berent damping values, as shown in Fig. 8.\n0.20.40.60.81α=0.0091 5·\nα10·\nα0\n0.20.40.60.81α=0.0057 5·\nα10·\nα0.3\n0.6 0.9 1.2 1.5 1.8 00.20.40.60.81α=0.0030\n5·\nα10·\nα30060090012001500Ts (K)T\nSM/M0t (ps)\nbcc Cofcc Co\nhcp Co\nFIG. 8. (Color online)Time dependent normalized average\nmagnetization (M/M 0) dynamics of Co systems after ultra-\nshort laser irradiation (constant \ruency) for di\u000berent damping\nvalues with FM state. Top panel shows time dependent spin\ntemperature. The exchange parameters are obtained from\nSPR-KKR calculations. Where M 0is magentization at 300K\nAs Fig. 8 shows, the quenching of magnetization in-\ncreases with the increase of damping parameter. This\nhighlights the fact that both the demagnetisation time\nand the reduction of the magnetic moment in laser in-\nduced demagnetisation measurements depend critically\nupon the damping parameter. Furthermore, Fig. 9 shows\nthat for systems with more than one magnetic sublattice,\nthe magnetisation dynamics may be di\u000berent and that\nthese sublattices therefore display di\u000berent demagneti-\nsation times. The demagnetization times are calculated\nusing double exponential \ftting function, as described in\nRef. [75] and \u001cmare listed in Table II. We would like to\navoid a detailed comparison between the obtained theo-\nretical and experimental demagnetization times because\nthe theoretical values are calculated for single crystal\nphases while the measurements are made for polycrys-talline samples, potentially with several crystallographic\nphases present. However the gross features of the num-\nbers listed in Table II may be comparable to experimental\ndata11. To end this section, in Table II, it is shown that\nthe demagnetization time is reduced for increasing value\nof the damping parameter in agreement with the \fnding\npublished in Refs. [76 and 77].\nTABLE II. Element speci\fc demagnetization times for Co-\nbased systems. \u000b1refers to Co and \u000b2refers to Mn. Experi-\nmental data from polycrystalline samples presented in brack-\nets.\nSystem Co Mn\n\u001cde(ps) \u001cde(ps)\nfcc Co (\u000b= 0:0057) 0.255 [0.15-0.25]11\nbcc Co (\u000b= 0:0091) 0.250[0.15-0.25]11\nhcp Co (\u000b= 0:0030) 0.300[0.15-0.25]11\nCo0:90Mn0:10(\u000b1 = 0:0083;\u000b2 = 0:0013) 0.219 0.6\nCo0:90Mn0:10(\u000b= 0:0072) 0.22 0.375\nCo0:70Mn0:30(\u000b1 = 0:0061;\u000b2 = 0:0022) 0.23 1.0\nCo0:70Mn0:30(\u000b= 0:0045) 0.5 0.65\nCo0:90Mn0:10bct (\u000b1 = 0:0090;\u000b2 = 0:0025) 0.215 0.56\nCo0:90Mn0:10bct (\u000b= 0:0080) 0.225 0.375\nCo0:70Mn0:30bct (\u000b1 = 0:0062;\u000b2 = 0:0025) 0.22 0.7\nCo0:70Mn0:30bct (\u000b= 0:0047) 0.45 0.6\nIn Fig. 10 we show the distribution of azimuthal an-\ngles (\u0012) of the atomic spins during the demagnetisation\nprocess, for bcc Co. Note that at each time the distri-\nbution of the \u0012angles is found to follow essentially a\nBoltzmann distribution function. This is not an obvious\nresult since the atomic spins are in out-of-equilibrium sit-\nuation. We have also calculated the angles between the\nnearest-neighboring spins (not shown) and found that at\neach time step these angles do not exceed 90 degrees. As\nwe have shown above (Fig. 3), the magnetic excitation\nenergies in bcc Co are represented accurately in this in-\nterval of angles using Eq. (1) (see also Fig. 5), which lends\ncredence to the approach adopted here to study ultrafast\nmagnetisation dynamics.\nNext we analyze the Co-Mn alloys in more detail, and\nwe focus on Co 0:90Mn0:10and Co 0:70Mn0:30in the bcc\nand bct structures, respectively. Element-speci\fc damp-\ning parameters ( \u000b1(Co),\u000b2(Mn)) of Co-Mn alloys are\nused to investigate the angular momentum exchange be-\ntween the sublattices in the spin dynamics simulations.\nResults are presented in Fig. 9. The calculated element-\nspeci\fc de- and re-magnetization of Co-Mn alloys show\na variety of possible situations that can be encountered\nin laser experiments on alloys with two magnetic sublat-\ntices. The demagnetization of Co precedes that of Mn\nby 0.15-0.6 ps due to the low damping value of Mn. The\nincrease of damping parameters on both sublattices by\n5 to 10 times in all considered alloys reduces the rela-\ntive di\u000berence between the sublattice magnetism in the\ndemagentization phase. Furthermore, our results show11\n00.20.40.60.81Co, α1=0.0083Mn, \nα2=0.0013Co, 5·α1Mn, 5·\nα20\n0.5 1.0 1.5 00.20.40.60.81Co, 10· α1 Mn, 10·\nα2 0\n0.5 1.0 1.5 2.0 Co, α1=0.0072Mn, \nα2=0.0072t (ps)\nM/M\n0\nCo0.90Mn0.10\n00.20.40.60.81Co, α1=0.0061Mn, \nα2=0.0022Co, 5·α1Mn, 5·\nα20\n0.5 1.0 1.5 00.20.40.60.81Co, 10· α1 Mn, 10·\nα2 0\n0.5 1.0 1.5 2.0 Co, α1=0.0045Mn, \nα2=0.0045t (ps)\nM/M\n0\nCo0.70Mn0.30\n00.20.40.60.81Co, α1=0.0090Mn, \nα2=0.0025Co, 5·α1Mn, 5·\nα20\n0.5 1.0 1.5 00.20.40.60.81Co, 10· α1 Mn, 10·\nα2 0\n0.5 1.0 1.5 2.0 Co, α1=0.0080Mn, \nα2=0.0080t (ps)\nM/M\n0\nCo0.90Mn0.10(bct)\n00.20.40.60.81Co, α1=0.0062Mn, \nα2=0.0025Co, 5·α1Mn, 5·\nα20\n0.5 1.0 1.5 00.20.40.60.81Co, 10· α1 Mn, 10·\nα2 0\n0.5 1.0 1.5 2.0 Co, α1=0.0047Mn, \nα2=0.0047t (ps)\nM/M\n0\nCo0.70Mn0.30(bct)\nFIG. 9. (Color online) Time evolution of the normalized average magnetization (M/M 0) of Co 0:90Mn0:10and Co 0:85Mn0:15\nand alloys under the in\ruence of a thermal heat pulse for di\u000berent sublattice damping parameters with FM state. The black\nand red lines represent the Co and Mn sublattices respectively. The damping, \u000b1 and\u000b2, refer to Co and Mn sublattices,\nrespectively. The exchange parameters are obtained from SPR-KKR calculations.12\n0-10\n10-20\n20-30\n30-40\n40-50\n50-6060-7070-8080-90\n90-100\n100-110\n110-120\n120-130\n130-140\n140-150\n150-160\n160-170\n170-180\nFIG. 10. (Color online) The time evolution of the distribution\nof the azimuthal angle \u0012, for the di\u000berent atomic spins for\nbcc Co. Note that the distribution is shown in intervals 0-10\ndegrees, 10-20 degrees etc. where each interval is shown as a\nbar with a speci\fc colour. The distribution is shown for t=0\nps, 0.25 ps, 0.5 s, 1.0 ps and 5.0 ps after the laser pulse starts\nto heat up the sample.\nthat the quenching of magnetization on both sublattices\nincreases with the increased damping parameter. For the\nlowest damping value of Mn in bcc Co 0:70Mn0:30, the Mn\ndemagentization time is \u00184 times slower than that of the\ndemagentization of Co. The results of Fig. 9 show that a\nlarge asymmetry in the damping parameter in multicom-\nponent magnets is a good parameter to use when one\nwants to identify systems with very di\u000berent behavior\nof the demagnetization in ultrafast pump probe experi-\nments. We propose that this is a parameter that should\nbe explored when one tries to identify alloys and com-\npounds in which the element speci\fc magnetisation dy-\nnamics is drastically di\u000berent. We hope these results can\nmotivate further experimental studies.\nV. CONCLUSION\nWith the extension of LKAG ab initio interatomic ex-\nchange calculation method for non-collinear spin systems,\nwe have analysed the exchange interactions and magnetic\nmoments of Co with fcc, bcc and hcp crystal structures.We found that elemental Co is unique in that it has ex-\ncitations energies that re\rect an almost perfect Heisen-\nberg system in a rather wide range of angles between the\nspins. This has a signi\fcant importance in the correct\ndescription of the time evolution of the atomic magnetic\nmoments under the in\ruence of a temperature-dependent\nlaser pulse. Note that in contrast to Co, bcc Fe shows\nsigni\fcant spin con\fguration-dependence, for any de\fni-\ntion of the spin-Hamiltonian, as it was shown here and\nin previous studies18.\nMn spins, on the contrary, exhibit strongly non-\nHeisenberg behavior already for small degree of inter-\natomic non-collinearity. A relatively small amount of Mn\ndopants, as in the case of the alloys studied here, is not\nexpected to drastically alter the system's properties. We\nnote, however, that high Mn concentration will de\fnitely\nlead to the breakdown of the Heisenberg picture and we\nplan to investigate it in detail in future.\nThe calculated structural and magnetic properties of\nCo-rich Co-Mn alloys are compared with experimental\ndata. We \fnd that they are in very good agreement\nwith observations. The calculated TC's reproduce well\nthe measured values and shows a linear decrease as a\nfunction of increasing Mn content, in line with the ex-\nperiments. The magnon dispersion curves of fcc, bcc and\nhcp Co are plotted along the high symmetry directions of\nthe Brillouin zone and they are indeed in good agreement\nwith experimental data, where comparison can be made.\nWe have also addressed the temporal behavior of the\nmagnetism of Co in the bcc, fcc and hcp structures as\nwell as Co-Mn alloys, after a laser excitation. Ultrafast\nmagnetization dynamics of these Co systems were stud-\nied for di\u000berent damping parameters, and it was found\nthat the demagnetization behaviour depends critically on\nthe damping parameter as well as the strength of the\nexchange interaction represented by di\u000berent concentra-\ntions of Mn. This becomes especially interesting for Co-\nMn alloys that have very di\u000berent values of the damping\nparameters and exchange interactions of the constitute,\nwhich lead to drastically di\u000berent magnetization dynam-\nics of the Co and Mn sublattices.\nVI. ACKNOWLEDGEMENTS\nThe authors thank to Tom Silva, Justin Shaw, Mar-\ntin Shoen and A.N. Yaresko (MPI Stuttgart) for fruitful\ndiscussions. We acknowledge \fnancial support from the\nSwedish Research Council. O.E Acknowledged support\nfrom KAW (projects 2013.0020 and 2012.0031). O.E.\nand E. K. D.-Cz. acknowledges STandUP for \fnan-\ncial support. The calculations were performed at NSC\n(Link oping University, Sweden) under a SNAC project.\n\u0003raghuveer.chimata@physics.uu.se1E. Beaurepaire, J. C. Merle, A. Daunois, and J.Y. Bigot,\nPhys. Rev. Lett., 76, 4250, (1996).13\n2A. Kimel, A. Kirilyuk, A.Tsvetkov, R Pisarev, and T. Ras-\ning, Nature, 429, 850 (2004).\n3A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev,\nA. Kirilyuk, and T. Rasing Nat. Phys., 5, 727 (2009).\n4J. Y. Bigot, L. Guidoni, E. Beaurepaire, and P. N. Saeta,\nPhys. Rev. Lett. 93, 077401 (2004).\n5I. Tudosa, C. Stamm, A. B Kashuba, F. King, H. C. Sieg-\nmann, J Stohr, G. Ju, B. Lu, and D. Weller, Nature, 428,\n831 (2004).\n6J. Y. Bigot, M. Vomir, and E. Beaurepaire, Nat. Phys., 5,\n515 (2009).\n7K. Vahaplar, A. M. Kalashnikova, A. V. Kimel, S. Ger-\nlach, D. Hinzke, U. Nowak, R. Chantrell, A. Tsukamoto,\nA. Itoh, A. Kirilyuk, and T. Rasing Phys. Rev. B, 85,\n104402 (2012).\n8C. Stamm, T. Kachel, N. Pontius, R. Mitzner, T. Quast,\nK. Holldack, S. Khan, C. Lupulescu, E. F. Aziz, M. Wi-\netstruk, H. A. D urr, and W. Eberhardt, Nature Mater. 6,\n740, (2007).\n9H. S. Rhie, H. A. D urr, and W. Eberhardt, Phys. Rev.\nLett. 90, 247201 (2003).\n10B. Koopmans, H. Kicken, M. van Kampen, and de. Jonge,\nW J. Magn. Magn. Mater. 286, 271 (2005).\n11B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf,\nM. Fahnle, T. Roth, M. Cinchetti, and M. Aeschlimann,\nNat. Mater. 9, 259 (2010).\n12O. Eriksson, A. Bergman, L. Bergqvist, J. Hellsvik (un-\npublished).\n13R. Chimata, A. Bergman, L. Bergqvist, B. Sanyal, and O.\nEriksson Phys. Rev. Lett. 109, 157201 (2012).\n14V. P. Antropov, M. I. Katsnelson, B. N. Harmon, M. van\nSchilfgaarde, and D. Kusnezov Phys. Rev. B 54, 1019\n(1996).\n15A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, and\nV. A. Gubanov, J. Magn. Magn. Mater. 67, 65 (1987).\n16D. B ottcher, A. Ernst, and J. Henk. J. Magn. Magn. Mater.\n324610, 2012.\n17R. F. L. Evans, U. Atxitia, and R. W. Chantrell, Phys.\nRev. B 91, 144425 (2015).\n18A. Szilva, M. Costa, A. Bergman, L. Szunyogh, L. Nord-\nstrom, and O. Eriksson, Phys. Rev. Lett. 111, 127204\n(2013)\n19K. Binder, Z. Phys. B: Condensed Matter 43, 119 (1981);\nPhys. Rev. Lett. 47, 693 (1981).\n20The Munich SPR-KKR package, version 6.3, H. Ebert\net al. , http://ebert.cup.uni-muenchen.de/SPRKKR; H.\nEbert, D. K odderitzsch, and J. Min\u0013 ar: Rep. Prog. Phys.\n74, 096501 (2011).\n21O. K. Andersen, Phys. Rev. B 12, 3060 (1975).\n22P. R. Peduto, S. Frota-Pessoa and M. S. Methfessel, Phys.\nRev. B 44, 13283 (1991).\n23V. Antonov, B. Harmon, and A. Yaresko, Electronic Struc-\nture and Magneto-Optical Properties of Solids (Kluwer\nAcademic,Dordrecht, 2004).\n24P. Weinberger, and R. Podloucky, Phys. Rev. B 22, 645\n(1980); J. Staunton, B. L Gy or\u000by, and P. Weinberger, J.\nPhys. F: Met. Phys. 10, 2665 (1980).\n25P. Soven, Phys. Rev. 156, 809 (1967).\n26A. J. Pindor, J. Staunton, G. M. Stocks, H. Winter, J.\nPhys. F Metal Phys. 13, 979 (1983).\n27J. Staunton, B. Gyor\u000by, A. Pindor, G. Stocks, H. Winter,\nJ. Magn. Magn. Mater. 45, 15 (1984).\n28H. Ebert, S. Mankovsky, D. K odderitzsch, and P. J. Kelly\nPhys. Rev. Lett. 107, 066603 (2011).29P. Lloyd, Proc. Phy. 90, 207 (1967).\n30H. Ebert, D. K odderitzsch, and J. Min\u0013 ar, Rep. Prog. Phys.\n74, 096501 (2011).\n31H. Ebert and S. Mankovsky Phys. Rev. B 79, 045209\n(2009).\n32https://openlibrary.org/works/OL3517103W/\nRelativistic_electron_theory\n33W. H. Butler Phys. Rev. B 31, 3260 (1985).\n34S. Mankovsky, D. K odderitzsch, G. Woltersdorf, and H.\nEbert Phys. Rev. B 87, 014430 (2013).\n35V. P. Antropov, M. I. Katsnelson, B. N. Harmon, M. van\nSchilfgaarde, and D. Kusnezov, Phys. Rev. B 54, 1019\n(1996).\n36U. Bovensiepen, J. Phys. Cond. Matter 19, 083201 (2007).\n37J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett.\n77, 3865 (1996).\n38V. L. Moruzzi, J. F. Janak, and K. Schwarz, Phys. Rev. B\n37, 790 (1988).\n39B. Skubic, J. Hellsvik, L. Nordstrom, and O. Eriksson, J.\nPhys. Cond. Matter 20, 315203 (2008).\n40O.Eriksson, A.Bergman, L.Bergqvist and J.Hellsvik,\nAtomistic Spin Dynamics: Foundations and Applications,\nOxford Univ. Press (2017).\n41Y. U. Idzerda, W. T. Elam, B. T. Jonker, and G. A. Prinz,\nPhys. Rev. Lett. 62, 2480 (1989).\n42G. R. Harp, R. F. C. Farrow, D. Weller, T. A. Rabedeau,\nand R. F. Marks, Phys. Rev. B 48, 17538 (1993).\n43G. Y. Guo, and H. H. Wang, Chin. J. Phys. 38, 949 (2000).\n44G. S. Dong, Y. Z. Wu, C. Jing, Y. Chen, W. R. Zhu, X. F.\nJin, H. W. Zhao, M. Lu, H. R. Zhai, X. L. Shen, L. Li, J.\nCryst. Growth 187, 444 (1998).\n45D. Wu, G. L. Liu, C. Jing, Y. Z. Wu, D. Loison, G. S.\nDong, X. F. Jin, and Ding-Sheng Wang, Phys. Rev. B 63,\n214403 (2001).\n46L. Zhang, D. Basiaga, J. R. O \u0013Brien, D. Heiman, J. Appl.\nPhys. 98, 106101 (2005).\n47H. W. Zhao, Y. Chen, W. R. Zhu, G. S. Dong, X. F. Jin,\nM. Lu, and H. R. Zhai, J. Appl. Phys. 81, 2036 (1997).\n48G. A. Prinz Phys. Rev. Lett. 54, 1051 (1985).\n49A. D\u0013 \u0010az-Ortiz, R. Drautz, M. F ahnle, H. Dosch, and J. M.\nSanchez Phys. Rev. B 73, 224208, (2006).\n50G. A. Prinz, Phys. Rev. Lett. 54, 1051 (1985).\n51American Institute of Physics Handbook (3rd ed.),\nMcGraw-Hill, New York (1970) Table 9a-2\n52J. A. C. Bland, R. D. Bateson, P. C. Reidi, R. G. C. Gra-\nham, H. J. Lauter, J. Penfold, and C. Shackleton, J. Appl.\nPhys. 69, 4989 (1991).\n53O. Eriksson, B. Johansson, R. C. Albers, A. M. Boring and\nM.S.S Brooks, Phys. Rev. B 42, 2707 (1990).\n54X. Liu, M. M. Steiner, R. Sooryakumar, G. A. Prinz, R. F.\nC. Farrow, and G. Harp Phys. Rev. B 53, 12166 (1996).\n55http://materials.springer.com/bp/docs/\n978-3-540-39667-3\n56L. M. Sandratskii, Journal of Physics: Condensed Matter\n3, 8565 (1991).\n57S. V. Halilov, H. Eschrig, A.Y. Perlov and P.M. Oppeneer,\nPhys. Rev. B 58, 293 (1998).\n58M. van Schilfgaarde and V. P. Antropov, J. Appl. Phys.\n85, 4827 (1999).\n59M. Pajda, J. Kudrnovsk\u0013 y, I. Turek, V. Drchal, and P.\nBruno Phys. Rev. B 64, 174402 (2001).\n60I. Turek, J. Kudrnovsk\u0013 y, V. Drchal, P. Bruno, and S.\nBl ugel, Phys. Status Solidi B 236, 318 (2003).14\n61M. Lezai\u0013 c, P. Mavropoulos, and S. Bl ugel, Appl. Phys.\nLett. 90, 082504 (2007).\n62Y. O. Kvashnin, W. Sun, I. Di Marco, and O. Eriksson\nPhys. Rev. B 92, 134422 (2015).\n63Y. O. Kvashnin, R. Cardias, A. Szilva, I. Di Marco, M. I.\nKatsnelson, A. I. Lichtenstein, L. Nordstr om, A. B. Klau-\ntau, and O. Eriksson Phys. Rev. Lett. 116, 217202 (2016).\n64J. Crangle, Philos. Mag. 46, 499 (1955).\n65Vega, A. and Nolting, W. (1996), Phys. Stat. Sol. (b), 193:\n177-187.\n66C. M. Singal and T. P. Das Phys. Rev. B 16, 5068, 1977\n67M. Acet, C. John, E.F. Wassermann J. Appl. Phys. 70\n(1991), p. 6556.\n68L. Zhang, D. Basiaga, J. R. OBrien, and D. J. Heiman, J.\nAppl. Phys. 98, 106101 (2005).\n69K. Gilmore, Y. U. Idzerda, and M. D. Stiles Phys. Rev.\nLett. 99, 027204 (2007).\n70Corina Etz, Lars Bergqvist, Anders Bergman, Andrea\nTaroni and Olle Eriksson J. Physics. Cond. Matter. 27,\n243202 (2015)71P. Buczek, A. Ernst, and L. M. Sandratskii Phys. Rev. B\n84, 174418 (2011).; T. Balashov, Ph.D. thesis, Physikalis-\nches Institut, Universitat Karlsruhe (TH), 2009.\n72G. Shirane, V. J. Minkiewicz, and R. Nathans, J. Appl.\nPhys. 39, 383 (1968).\n73T. Perring, A. Taylor, and G. Squires, Physica B 213-214,\n348(1995).\n74R. Chimata, L. Isaeva, K. Kadas, A. Bergman, B. Sanyal,\nJ. H. Mentink, M. I. Katsnelson, T. Rasing, A. Kirilyuk,\nA. Kimel, O. Eriksson and M. Pereiro, Phys. Rev. B 92,\n094411 (2015).\n75S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka,\nE. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E.\nSiemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschli-\nmann, M. M. Murnane and H. C. Kapteyn, Proc. Natl.\nAcad. Sci. U.S.A. 109, 4792 (2012).\n76C. L. S. Kantner, M. C. Langner, W. Siemons, J. L. Blok,\nG. Koster, A. J. H. M. Rijnders, R. Ramesh, and J. Oren-\nstein Phys. Rev. B 83, 134432 (2011).\n77B. Koopmans, J. J. M. Ruigrok, F. DallaLonga, and W. J.\nM. de Jonge, Phys. Rev. Lett. 95, 267207 (2005)."    },    {        "title": "1312.7786v1.Dynamic_behaviors_of_the_hexagonal_Ising_nanowire.pdf",        "content": "1 \n Dynamic behavior s of the hexagonal  Ising nano wire  \n \nMehmet Ertaş1, * and Yusuf Kocakaplan2 \n1Department of Physics, Erciyes University, 38039 Kayseri, Turkey  \n2Graduate School of Natural and Applied Sciences, Erciyes University, 38039 Kayseri, Turkey  \n \nAbstract  \nBy utilizing the effective -field theory based on the Glauber -type stochastic dynamics, t he \ndynamic behavior s of the hexagonal Ising nanowire  (HIN) system  in the presence of a time \ndependent magnetic field are obtained. The time variations of average order parameters and  the \nthermal behavior of the dynamic order parameters are studied to analyze the nature of transitions \nand to obtain the dynamic phase transition points. The dynamic phase diagrams are introduced  in \nthe plane of the reduced temp erature v ersus magnetic field amplitude . The dynamic phase \ndiagrams exhibit coexistence phase region , several ordered phases and critical point  as well  as a \nreentrant behavior.   \n \nKeywords: Magnetic nanostructured materials ; Hexagonal  Ising nanowire ; Dynamic  phase \ndiagram ; Effective -field theory ; Glauber -type stochastic dynamics  \n \n \n1. Introduction  \nRecent years, magnetic nanoparticle systems such as nanowire, nanotube, nanofilms and \nnanorods  have been receiving considerable attention by the experimental and  theoretical \nresearchers  [1-7]. Because, compared with those in bulk materials, these magnetic nanomaterials \nhave many peculiar physical properties [8]. Another reason is that they have important potential \ntechnological applications in various areas, e.g. they can be used for medical applications [9, 10] , \nenvironmental  remediation [11], permanent magnets [12], sensors [13], magnetic recording \n                                                           \n*Corresponding author.  \nTel: + 90 352 2076666 # 33134  \nE-mail address: mehmetertas@erciyes.edu.tr  (Mehmet Ertaş)  \n Revised Manuscript2 \n media [14 –16], nonlinear optics [17], biotechnology [18 , 19] and also used for bio -separation [20, \n21].  \nOn the othe r hand , many researchers have used  the spin-1/2 Ising system  to study equilibrium \nproperties of magnetic nanostructured materials. An early attempt to examine nanoparticles was \ndone by Kaneyoshi [22]. Kaneyoshi investigated the phase diagrams of a ferroele ctric \nnanoparticle described with the transverse Ising model by using the mean -field theory (MFT) and \nthe effective -field theory (EFT) corresponding to the Zernike approximation. Kaneyoshi also [23] \nstudied, via the MFT, the thermal variations of longitudi nal and transverse magnetizations for a \nferroelectric nanoparticle system by the transverse Ising model. MFT and EFT were used to study \nthe magnetizations [24] and phase diagrams [25] of a transverse Ising nanowire and found that \nthe equilibrium behavior o f the system is strongly affected by the surface situations. Kaneyoshi \n[26] also examined magnetic properties, such as the thermal behavior of total magnetization, \ncompensation temperature, phase diagrams, total susceptibility and inverse total susceptibil ity, of \na cylindrical spin -1/2 Ising nanowire (or nanotube) in detail within the EFT. He investigated \nphase diagrams and magnetizations in the cylindrical nanowire system with a diluted surface \ndescribed by the transverse spin -1/2 Ising model [27] via the EFT. Keskin et al.  [28] studied the \nhysteresis behaviors of the cylindrical Ising nanowire at temperatures below, around, and above \nthe critical temperature within EFT with correlations. For ferromagnetic and antiferromagnetic \ninteractions between the shel l and the core, they obtained the hysteresis curves for different \nreduced temperatures. The effects of the randomly distributed magnetic field on the phase \ndiagrams of a spin -1/2 Ising nanowire have studied by Akıncı [29] via the EFT. For the \nferroelectric  and anti -ferroelectric interfacial coupling, the hysteresis behaviors of the nanotube in \nwhich consisting of a ferroelectric core of spin -1/2 surrounded by a ferroelectric shell of spin -1/2 \nhave examined by Zaim et al.  with the transverse Ising model [30] . They obtained a number of \ncharacteristic behaviors such as the existence of triple hysteresis loops for appropriate values of \nthe system parameters. By using the EFT based on the probability distribution technique, Bouhou \net al.  [31] investigated the mag netization, susceptibility, and hysteresis loops of a magnetic \nnanowire. In particular, the effects of the exchange interaction between core/shell and the \nexternal fields on the magnetization and the susceptibility were examined. They also described \nthe di luted magnetic nanowire by the spin -1/2 Ising model and examined magnetic properties \n[32]. By using the EFT, Yüksel et al.  [33] examined the effects of bond dilution on the magnetic 3 \n properties of a cylindrical Ising nanowire. Very recently, Kaneyoshi studi ed nanoscaled thin \nfilms by the spin -1/2 Ising model and presented phase diagrams [34] and the thermal behavior of \ntotal magnetization and reentrant phenomena in a transverse Ising nanowire (or nanotube) with a \ndiluted surface [35].  \nWe should also mention  that although the spin -1/2 Ising systems have used to investigate the \nequilibrium properties of magnetic nanostructured materials, there have been only  a few works \nthat the spin -1/2 Ising systems used to investigate dynamic magnetic properties of nanostru ctured \nmaterials. Wang et al.  [36] studied the dynamic properties of  the phase diagram in cylindrical \nferroelectric nanotubes, especially they showed effects structure factors of the ferroelectric \nnanotubes in phase diagrams.  Deviren and Keskin [37] inves tigated thermal behavior of dynamic \nmagnetizations, hysteresis loop areas and cerrelations of a cylindrical Ising nanotube in a time \ndependent magnetic field within the EFT based on Glauber -type stochastic dynamics [38], \nnamely dynamic effective -field theo ry (DEFT). Yüksel et al.  [39] analyzed the nonequilibrium \nphase transition properties and hysteretic behavior of a ferromagnetic core -shell nanoparticle in \nthe presence of a oscillating magnetic field by means of the MC simulations. Deviren et al.  [40-\n42] investigated dynamic phase transitions temperature of cylindrical Ising nanowire [40], \ntransverse cylindrical Ising nanowire [41] and the dynamic phase diagrams of the kinetic \ncylindrical Ising nanotube [42] using DEFT , in detail.  \nAs far as we kn ow the dynamics behaviors of  the HIN system have not been studied yet. To \novercome this deficiency in the literature, t herefore, the aim of this paper is to investigate the \ndynamic phase transition temperatures and dynamic phase diagrams of the spin-1/2 HI N system \nin an oscillating magnetic field within the DEFT. We employ the Glauber -type stochastic \ndynamics to construct the dynamic effective -field equation for the average magnetizations. We \ninvestigate the stationary solutions of the set of coupled dynami c effective -field equations and \nthis investigation leads us to find the phases in the system. In order to characterize the nature \n(continuous and discontinuous) of the phase transitions and obtain the DPT points, the thermal \nbehavior of the dynamic order p arameters have also studied. Finally, the dynamic phase diagrams \nin the plane of the reduced temperature versus magnetic field amplitude have presented.  4 \n The organization of the remaining part of this paper is as follows. In Section 2, the model are \ndefined  and given briefly the formulation of the spin-1/2 HIN system . In Section 3, we present \nthe numerical results and discussions. Finally, Section 4 contains the summary and conclusions.  \n2. Model and formulation  \nThe schematic presentation  of the HIN system with core/shell structure is illustrated in Fig. 1; \nhence the blue and red spheres indicate magnetic atoms at the surface shell and core, respectively.  \nEach site of the HIN system is occupied by a spin -1/2 magnetic atoms and each magnetic atom is \nconnected to the two nearest -neighbor magnetic atoms on the above and below sections along the \nwire. The Hamiltonian of the system is given by  \nS i j C m n 1 i m i m\nij mn im i mH J S S J S S J S S h(t) S S ,        \n     (1)  \nwhere \nij , \nmn  and \n<im> stand for the summations over all pairs of neighboring spins at \nthe shell surface, core and between shell surface and core, respectively. \niS is the Pauli spin \noperator and it takes value \n1/ 2\n . \nh(t)  is a time -dependent external oscillating magnetic field \nthat is given by\n0 h(t) = h  sin( t)  and \n0h  and \n = 2     are the amplitude and the angular \nfrequency of the oscillating field, respectively. The \nSJ  and \nCJ  are the interaction parameters \nbetween two nearest -neighbor magnetic atoms at the surface shell and core, res pectively, and \n1J  \nis the interaction parameters between two nearest -neighbor magnetic atoms at the surface shell \nand the core. In order to clear up of the surface effects on the physical properties of the system, \nSJ\nsurface interaction parameter is often defined as \nSCJJS = 1+  [26, 28, 41, 42].  \nIn the  framework of the EFT with correlations by using the van der Waerden identity  [43] we \ncan easily obtain \nCm magnetization at the core and \nSm magnetization in the shell surface for the \nHIN system depicted in Fig. 1 as follow:  \n \n  26\nC C i C 1 j 1 C x=0\ni=1 j=1m = cosh(J ) S sinh(J ) cosh(J ) S sinh(J ) F x+h           \n,   (2a) 5 \n \n  41\nS S i S 1 j 1 S x=0\ni=1 j=1m = cosh(J ) S sinh(J ) cosh(J ) S sinh(J ) F x+h           ,   (2b)  \nwhere \n/x  is the differential operator. The \nCS(x h) and (x h)FF  functions are defined as \nfollow;  \n  \nCS11tanh β x + h\n22F (x h) F (x h) .\n          (3) \nIn Eq. (3), \nB1/ k T where \nBk  is the Boltzmann constant  and T is absolute temperature. \nMoreover, the total magnetization of per site in the HIN system can be obtain via \nT CS 1/ 7 6 m m m \n. Also, for the following discussions, at this point let us define the r \nparameter as \n1C rJ / J .  \nIn order to obtain the dynamic effective -field theory (DEFT) equations of motion for the \nCm\nmagnetization at the core and \nSm magnetization in the shell surface, we apply the Glauber -type \nstochastic process [38] ba sed on the master equation as follows;   \n  C26C\nC C C 1 S 1 Cx0dm= - m +\ndtcosh(J ) m sinh(J ) cosh(J ) m sinh(J ) F x h ,      \n           \n(4a) \n  4\nS S S S 1 C 1 Sx0S.dm= - m + cosh(J ) m sinh(J ) cosh(J ) m sinh(J ) F x hdt      \n               \n(4b) \nBy expanding the right -hand sides of Eq. (4), and by applying the \nexp( )F(x) F(x )  \nexpression, we can obtain many coefficients. Due to their being so long and complicated, these \ncoefficients will not be given here. The solution of Eq. (4) gives the phases in the system, and \nthese will be presented and discussed in the next section  \n3. Numerical results and discussions  \n3.1. Phases in this system  6 \n In this subsection, at first the time variations of the average shell and core magnetizations \nare examined to obtain the phases in the  system. In order to investigated  the behaviors of time \nvariations of the average magnetizations, the stationary solutions of the set of coupled Eqs. ( 4a)-\n(4b) dynamic effective -field equations have been studied for  different interaction parameter \nvalues.  The stationary solutions of solutions  of these equations will be a periodic function of   \nwith period 2 ; that is, \nSSmξ+2π = m ξ  and \nCCmξ+2π =m ξ .  Furthermore, they can be \none of the three  types according to whether they have or do not have the properties  \n      \nSSmξ+π = -m ξ ,             (5a) \nand \n      \nCCmξ+π = - m ξ .            (5b) \nwhere  \nξ= t . The first type of solution satisfies Eqs. ( 5a) and ( 5b) which is known as  a \nsymmetric solution which corresponds to a paramagnetic phase . In th e solution, average \nmagnetizations delayed with respect to the external magnetic field. The second type of s olution \ndoes not satisfy Eqs. (5 a) and ( 5b), and is called a non-symmetric solution that corresponds to a \nferromagnetic (f) or antiferromagnetic (af)  solution. In this case, average magnetizations do not \nfollow the external magnetic field any more, but instead of oscillating around zero value.  The \nthird type of solution, which satisfies Eq. (5a) but does not satisfy Eq. (5 b), corresponds to a \nnonmagnetic solution (nm).  These facts are seen obviously by solving Eqs. ( 4a) and ( 4b) utilizing \nthe Adams -Moulton predictor -corrector method for a given set of parameters and initial values, \nand obtained fundamental phases are presented in Fig. 2. Fig. 2  (a)-(d) illustrate paramagnetic, \nferromagnetic, antiferromagnetic and nonmagnetic fundamental phases  for different interaction \nparameters and initial values , respectivelly.  In these figures each curve shows different one initial \nvalues, nam ely m C = 0.5 and m S = 0.5 or  mC = 0.5 and m S = -0.5, and  m C = -0.5 and m S = 0.5 \netc. In Fig. 2(a), average magnetizations  are equal to each other and  oscillate  around  zero value (\nCSmξ = m ξ  =0\n). Hence, the system shows symetric solut ion, namely paramagnetic phase. In \nFig. 2(b) , average  shell and core  magnetizations are equal to each other and oscillate around \nCSmξ = m ξ  =0.5\n and the system illustrates  ferromagnetic phase. In Fig. 2(c), as shell \nmagnetization oscillates around 0.5 value, core magnetization oscillates around -0.5 value  and 7 \n system illustrates antiferromagnetic phase.  In Fig. 2(d), core magnetization oscillates around the \nzero value and is delayed with respect to the external magnetic field and shell magnetization  does \nnot follow the external magnetic field anymore, but instead of oscillating around a zero value, it \noscillates around a nonzero value , \nSmξ =0.5.  These  four solutions do n ot depend on the initial \nvalues, seen in Fig. 2(a) -(d) explicitl y. In Fig. 2(e), we have two solutions, namely nm and p \nphases or solutions coexist in the system and in this case, the solution depend on the initial \nvalues.   \n3.2. Dynamic phase transition points  \nThe dynamic shell and core magnetizations or dynamic order parameters as the time -averaged \nmagnetization over a period of the oscillating magnetic field are given as    \n                                 \n2\nSS\n01M = m ( )d2π\n                                                                (6a) \n                                 \n2\nCC\n01M = m ( )d .2π\n                                                                (6b) \nBy combination  of the Adams -Moulton predictor corrector with Romberg integration numerical \nmethods, we solve Eqs. (6a) and (6b) and examine the thermal behavior of M S and M C for \ndifferent values of Hamiltonian parameters.  The thermal behaviors of M S and M C gives the \ndynami c phase transition (DPT) point and the type of the dynamic phase transition.  The obtained \nnumerical results of Eqs. (6a) and (6b) are seen in Fig s. 3(a)-(d). In Fig. 3, TC and T t show the \ncritical or the second -order phase transitions and the first -order phase transition temperatures, \nrespectively.  Fig. 3(a) shows the behavior of dynamic  shell M S and dynamic core magnetization \nMC as a function of temperature for\n=2 , \nr 1.0= ,\n0.0S=  and h 0 = 0.1  values.  At zero \ntemperature, M S = M C = 0.5 and wit h the increase of temperature they  decrease to zero \ncontinuously ; thus the system undergoes a second order phase transition from the ferromagnetic \nphase to the paramagnetic phase at T C = 1.25.  For\n=2 , \nr 0.5= ,\n0.6S=  and h 0 = 0.12  \nvalues , the dynamic behavior of M S and M C was obtained in Fig. 3(b) . In this figure,  MS and M C \ntake 0.5 and -0.5 values at zero temperature , and they exhibits a discontinuous jump  to zero from \nthese values . Hence, the system undergoes a first -order phase transition from the 8 \n antiferromagnetic phase to the paramagnetic phase at Tt = 0.6.  Moreover, Fig s. 3(c) and (d) are \nplotted for\n=2 , \nr 0.1= , \n3.0S= , h0 = 6.05 and different initial values, namely Fig. 3(c) for \nMC = 0.5, M S = 0.5 or M C = 0.5, M S = -0.5 and Fig 3( d) for M C = 0.0, M S = 0.0 values. In Fig. \n3(c), a s the temperature values increase, M C always becomes zero. In contrast, M S take the 0.5 \nvalue at zero temperature and with the increase of temperature, it undergoes a first -order phase \ntransition at T t = 0.1.  Above T t, MS and MC always  equal to zero . On the other hand, the initial \nvalues of M S and M C are taken 0.0, they are equal to 0.0 at zero temperature and with the  increase  \nof temperature values, and they always become zero as seen in Fig. 3(d).  Therefore, the system \nundergoes a first -order phase transition from the nm + p mixed phase to the paramagnetic phase \nat T t = 0.1.  This fact is seen in the phase diagram of Fig. 4(e) for h 0 = 6.05, explicitly.  On the \nother hand, in the past two decades, both experimental (see [44 -48] and references therein) and \ntheoretical (see [49 -52] and references therein) investigations of the nonequilibrium critical \nphenomena, especially the DPT, have received a great deal of attention due to the reason that \nbesides the scientific interests the study of DPT can also inspired new methods in materials and \nmanufacturing process and processing as well as in nanotechnology [53].  \n       3.3. Dynamic phase diagrams  \nNow, we can o btain  the dynamic phase diagrams of the system. The dynamic phase diagrams \nare represented in the (h, T) planes for diverse values of the Hamiltonian parameters as seen in \nFig. 4.  In Fig. 4, the solid and dashed lines stand for the second - and first -order phase  transition \nlines, respectively;  the dynamic tricritical point  (TCP)  is represented by a filled circle. The \ndynamic phase diagrams contain the p, f, af, nm fundamental phases and the nm + p mixed phase . \nThe system illustrates the TCP in Figs. 4(c) -4(h), bu t the system does not exhibit TCP in Figs. \n4(a) and 4(b). In Fig. 4(a), we can see that the system always undergoes a second -order phase \ntransition and contain dynamic zero temperature (Z) point for the \nr 0.3=  and\n0.95S=  values. \nFig. 4(b) is obtained for \nr 4.0=  and\n0.5S=  values. It is similar to Fig. 4(a), except that the \nreentrant behavior is also observed in Fig. 4(b). With the temperature increase, the system passes \nfrom the p  phase to the f phase, and then back to the p phase again.  We should also mention that \nseveral weakly frustrated ferromagnets, such as in manganite LaSr 2Mn 2O7 by electron and x -ray \ndiffraction, in the bulk bicrystals of the oxide superconductor BaPb 1-xBixO3 and Eu xSr1-xS and \namorphous -Fe1-xMn x, demonstrate the reentrant phenomena [ 54]. Fig. 4(c) and 4(d) are calculated 9 \n for \nr 1.0=  and\n0.0S= , and \nr 1.0=  and \n0.0S= , respectively. The overall structure of Fig. 4(c) \nand (d) is similar. Both of them contain one TCP. But, Fig. 4(c) include the f and p phases while \nFig. 4(d) comprise the af and p phases. Fig. 4(e) is plotted for \nr 0.1=  and\n3.0S= , and different \ninitial values. We can clearly see that it contains the second - and first -order phase transitions, the \nTCP, the Z special point and the f, p, nm and nm  + p phases.  For  \nr 0.2=  and\n5.0S= , besides  \nthe Z special point, nm and p phase and the TCP, Fig. 4(f) is also include reentrant behavior. Fig. \n4(g) is similar to Fig. 4(f), except that the critical temperature is small.  Finally, Fig. 4(h) is \nobtained for \nr 0.5=  and\n0.6S=  and it contain two TCP, and the af and p phases.  \n4. Summary and conclusions  \nThe dynamic phase transition points  (DPTs)  and dynamic phase diagrams of the HIN system \nunder a time oscillating longitudinal magnetic field were investigated using the EFT with \ncorrelations. By utilizing the Glauber -type stochastic process, t he EFT equations of motion for \nthe average shell and core magnetizations are obtained for the HIN sys tem. Our results show that \nthe dynamic phase dia grams contain the p , f, af and nm fundamental phases and the nm  + p \nmixed phase as well as special Z and TC P points. We should also mention that  both experimental \nand theoretical investigations of the nonequi librium critical phenomena, especially the DPT, have \nreceived a great deal of attention due to the reason that besides the scientific interests the study of \nDPT can also inspired new methods in materials and manufacturing process and processing as \nwell as in nanotechnology. Moreover, o ur results also show the reentrant behaviors  and several \nweakly frustrated ferromagnets, such as in manganite LaSr 2Mn 2O7 by electron and x -ray \ndiffraction, in the bulk bicrystals of the oxide superconductor BaPb 1-xBixO3 and Eu xSr1-xS and \namorphous -Fe1-xMn x, demonstrate the reentrant phenomena. Finaly, although the equilibrium \nphase transitions of the nanosystems have been conspicuously studied, the dynamic or \nnonequilibrium properties of these systems have not been investigated  considerably. Hence, we \nhope that our work contributes to close this shortcoming in the literature.  \nReferences  \n[1] R. Skomski, J. Phys.: Condens. Matter 15 (2003) R841.  \n     X. Zou, G. Xiao, Phys. Rev. B 77 (2008) 054417.  \n[2] W. Jiang, X -X. Li, l -M. Liu, Physica E 53 (2013) 29.  \n[3] W. Jiang, H -Y. Guan, Z. Wang, A -B. Guo, Physica B 407 (2012) 378.  10 \n [4] W. Jiang, F. Zhang, X -X. Li, H -Y. Guan, A -B. Guo, Z. Wang, Physica E 47 (2013) 95.  \n[5] L -M. Liu, W. Jiang, Z. Wang, H. Y. Guan, A -B. Guo, J. Magn. Magn. Mater . 324 (2012) \n4034.  \n[6] W. Jiang, L -M. Liu, X -X. Li, Q. Deng, H. -Y. Guan, F. Zhang, A -b. Guo, Physica B 407 \n(2012) 3933.  \n[7] W. Jiang, X -X. Li, L -M. Liu, J -N. Chen, F. Zhang, J. Magn. Magn. Mater. 353 (2014) 90.  \n[8] A.E. Berkowitz, R.H. Kodama, S.A. Makhlou f, F.T. Parker, F.E. Spada, E.J. McNiff Jr., S. \nFoner, J. Magn. Magn. Mater. 196 (1999) 591.  \n[9] A.P.Y. Wong, M. H. W. Chan, Phys. Rev. Lett. 65 (1990) 2567.  \n[10] C. Alexiou, A. Schmidt, R. Klein, P. Hullin, C. Bergemann, W. Arnold, J. Magn. Magn. \nMater. 2 52 (2002) 363.  \n[11] D.W. Elliott, W. -X. Zhang, Environ. Sci. Technol. 35 (2001) 4922.  \n[12] H. Zeng, J. Li, J.P. Liu, Z.L. Wang, S. Sun, Nature 420 (2002) 395.  \n[13] G.V. Kurlyandskaya, M.L. Sanchez, B. Hernando, V.M. Prida, P. Gorria, M. Tejedor , Appl. \nPhys. Lett. 82 (2003) 3053.  \n[14] J.E. Wegrowe, D. Kelly, Y. Jaccard, Ph. Guittienne, J.Ph. Ansermet, Eur. Phys. Lett. 45 \n(1999) 626.  \n[15] A. Fert, L. Piraux, J. Magn. Magn. Mater. 200 (1999) 338.  \n[16] R.H. Kodama, J. Magn. Magn. Mater. 200 (1999) 3 59. \n[17] S. Nie, S.R. Emory, Science 275 (1997) 1102.  \n[18] S.J. Son, J. Reichet, B. He, M. Schuchman, S.B. Lee, J. Am. Chem. Soc. 127 (2005) 7316.  \n[19] D. Lee, R.E. Cohen, M.F. Rubner, Langmuir 23 (2007) 123.  \n[20] A. Hultgren, M. Tanase, C.S. Chen, D.H. Re ich, IEEE Trans. Magn. 40 (2004) 2988.  \n[21] A. Hultgren, M. Tanase, E.J. Felton, K. Bhadriraju, A.K. Salem, C.S. Chen, D.H. Reich, \nBiotechnol. Prog. 21 (2005) 509.  \n[22] T. Kaneyoshi, Phys. Stat. Sol. (b) 242 (2005) 2948.  \n[23] T. Kaneyoshi, J. Magn. Magn. M ater. 321 (2009) 3430;  \n        T. Kaneyoshi, Solid State Commun. 152 (2012) 883.  \n[24] T. Kaneyoshi, J. Magn. Magn. Mater. 322 (2010) 3014.  \n[25] T. Kaneyoshi, J. Magn. Magn. Mater. 322 (2010) 3410.  \n[26] T. Kaneyoshi, Phys. Stat. Sol. (b) 248 (2011) 250;  \n        T. Kaneyoshi, Physica A 390 (2011) 3697;  \n      T. Kaneyoshi, J. Magn. Magn. Mater. 323 ( 2011 ) 1145;  \n      T. Kaneyoshi, J. Magn. Magn. Mater. 323 (2011) 2483;  \n      T. Kaneyoshi, Solid State Commun. 151 (2011) 1528.  \n[27] T. Kaneyoshi, Physica A 39 1 (2012) 3616.  \n[28] M. Keskin, N. Şarlı, B. Deviren, Solid State Commun. 151 (2011) 1025.  \n[29] Ü. Akıncı, J. Magn. Magn. Mater. 324 (2012) 3951;  \n        Ü. Akıncı, J. Magn. Magn. Mater. 324 (2012) 4237.   \n[30] A. Zaim, M. Kerouad, M. Boughrara, A. Ainane , J.J. de Miguel, J. Supercond. Novel Magn. \n25 (2012) 2407.  \n[31] S. Bouhou, I. Essaoudi, A. Ainane, M. Saber, F. Dujardin, J. J. de Miguel J. Magn. Magn. \nMater. 324 (2012) 2434.  \n[32] S. Bouhou, I. Essaoudi, A. Ainane, M. Saber, R. Ahuca and F. Dujardin F J . Magn. Magn. \nMater. 336 (2013) 75;  \n      S. Bouhou, I. Essaoudi, A. Ainane, F. Dujardin, R. Ahuca and M. Saber M J Supercond Nov \nMagn. 26 (2013) 201.  \n[33] Y. Yüksel, Ü. Akıncı, H. Polat, Phys. Stat. Sol. (b) 250 (2013) 196.  11 \n [34] T. Kaneyoshi, Physica B 4 08 (2013) 126.  \n[35] T. Kaneyoshi, J. Magn. Magn. Mater. 339 (2013) 151.  \n[36] C. Wang, Z. Z. Lu, W. X. Yuan, S. Y. Kwok and B. H. Teng Phys. Lett. A 375 (2011) 3405.  \n[37] B. Deviren, M. Keskin, Phys. Lett. A 376 (2012) 1011.  \n[38] R. J. Glauber J. Math. Phys. 4 ( 1963 ) 294.  \n[39] Y. Yüksel, E. Vatansever and H. Polat, J. Phys.: Condens. Matter 24 (2012) 436004.  \n[40] B. Deviren, E. Kantar and M. Keskin, J. Magn. Magn. Mater  324 (2012) 2163.  \n[41] B. Deviren, M. Ertaş and M. Keskin Phys. Scr. 85 (2012) 055001.  \n[42] B. Deviren, Y. Şener and M. Keskin Physica A 392 (2013) 3969.  \n[43] H. B. Callen Phys. Lett. 4 (1963) 161;  \n        M. Suzuki Phys. Lett. 19 (1965) 267;  \n        T. Balcerzak J. Magn. Magn. Mater. 246 (2002) 213.  \n[44] N. Gedik, D.S. Yang, G. Logvenov , I. Bozovic, A.H. Zewail, Science 316 (2007) 425.  \n[45] D.T. Robb, Y.H. Xu, O. Hellwing, J. McCord, A. Berger, M.A. Novotny, P.A. Rikvold, \nPhys. Rev. B 78 (2008) 134422.  \n[46] J.J. Arregi, O. Idigoras, P. Vavassori, A. Berger, Appl. Phys. Lett. 100 (2012) 262403.  \n[47] A. Berger, O. Idigoras, P. Vavassori, Phys. Rev. Lett. 111 (2013) 190602.  \n[48] L. Chan, H. Li, B. Feng, Z. Ding, J. Qui, P. Cheng, K. Wu, S. Meng, Phys. Rev. Lett. 110 \n(2013) 085504.  \n[49] M. Ertaş, Y. Kocakaplan, M. Keskin, J. Magn. Magn. Mate r. 348 (2013) 113.  \n[50] E. Vatansever, Ü. Akıncı, Y. Yüksel  and H. Polat , J. Magn. Magn. Mater. 329 (2013) 14.  \n[51] M. Ertaş, B. Deviren, M. Keskin, Phys. Rev. E 86 (2012) 051110.  \n[52] B. Deviren, M. Keskin, J. Magn. Magn. Mater. 324 (2012) 1051.  \n[53] M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65 (1993) 851.  \n[54] J.Q. Li, Y. Matsui, T. Kimura, Y. Tokura, Phys. Rev. B 57 (1998) R3205; T. Sata, T. \nYamaguchi, K. Matsusaki, J. Membr. Sci. 100 (1995) 229; K. Binder, A.P. Young, Rev. \nModern Phys. 58 (1986)  801. \n \n \nList of the Figure Captions  \n \nFig. 1. (Color online) Schematic presentation of hexagonal Ising nanowire. The blue and red \nspheres indicate magnetic atoms at the surface shell and core, respectively.  \n \nFig. 2.  (Color online) Time variations of the core and shell magnetizations (m C and m S):  \n(a) Paramagnetic phase (p), \nr 0.3= , \n0.95S= , h0 = 0.78 , and T = 0.6.  \n(b) Ferromagnetic phase (f), \nr 4.0= , \n0.5S= , h0 = 1.5, and T = 1.5.  \n(c) Antiferromagnetic phase (af), \nr 1.0= , \n0.0S= , h0 = 1.20, and T = 0.8.  \n(d) Nonmagnetic phase (nm), \nr 0.2= , \n5.0S= , h0 = 0.4, and T = 0.1.  \n(e) Mixed phase (nm+p), \nr 0.1= , \n3.0S= , h0 = 6.0, and T = 0.1.  12 \n  \nFig. 3. (Color online) Thermal behaviors  of the  dynamic core and shell  magnetization s with the \nvarious values of r and \nS . \n   (a) r = 1.0, \n0.0S= , and h0 = 0.1 . \n   (b) r = - 0.5, \n0.96S= , and h 0 = 0.12.  \n   (c) and (d) r = 0.1, \n3.0S= , and h 0 = 6.05.  \n             \nFig. 4. The dynamic phase diagrams in (h -T) plane of the hexagonal Ising nanowire.  Dashed and \nsolid lines show  the first - and second -order phase transitions, respectively. The \ntricritical points are indicated with filled ci rcles . \n   (a) r = 0.3 and \n0.95S= ;  (b) r =  4.0 and \n0.5S= ;  (c) r = 1.0 and \n0.0S= ; \n   (d) r = -1.0 and \n0.0S= ;  (e) r = 0.1 and \n3.0S= ;  (f) r = 0.2 and \n5.0S= ; \n   (g) r = -0.5 and \n3.0S= ;  (h) r = - 0.5 and \n0.6S= . \n\n\n\n\n\n\n\n                                            Fig. 1 \nFigure 1 \n \n     Fig. 2  0 2 04 06 08 0mC (), mS ()\n-0.50-0.250.000.250.50Paramagnetic phase( a )\n0 20 40 60 80 100 120 140mC (), mS ()\n-0.50-0.250.000.250.50\nFerromagnetic phase\n0 50 100 150 200 250 300mC (), mS ()\n-0.50-0.250.000.250.50\n0 20 40 60 80 100 120 140 160mC (), mS ()\n-0.50-0.250.000.250.50( b )\nAntiferromagnetic phase( c )\nNonmagnetic phase( d )mC = mS\nmC = mS\nmC mS\nmC\n0 2 04 06 08 0 1 0 0-0.50-0.250.000.250.50( e )\nmC (), mS ()\nmC, mSmC, mS\nnm + p mixed phase mSFigure 2T0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4MC, MS\n0.00.10.20.30.40.5\nT0.0 0.2 0.4 0.6 0.8MC, MS\n-0.50-0.250.000.250.50\nMCMS\nMC= MSMC\nMS\nT0.00 0.04 0.08 0.12 0.16MC, MS\n0.000.250.50MS\nMC= MSMCTC\nTt\nTt\nT0.00 0.04 0.08 0.12 0.16-0.05-0.030.000.030.050.08\nMC= MS( a )\n( b )\n( c ) ( d )\n\nFig. 3 Figure 30.0 0.1 0.2 0.3 0.4h\n0.00.20.40.60.8Z\nfp( a )\n0.0 0.5 1.0 1.5 2.00.01.02.03.04.0\nZ\nfp( b )\n0.0 0.2 0.4 0.6 0.8 1.0 1.2h\n0.00.51.01.52.02.5\nfp\n0.0 0.2 0.4 0.6 0.8 1.0 1.20.00.51.01.52.02.5\nafp( c ) ( d )\n0.0 0.5 1.0 1.5 2.0 2.5 3.0h\n0.01.02.03.04.05.06.0\nfnmp\nZnm + p( e )\n0.00 0.05 0.10 0.15 0.200.00.10.20.30.40.50.6nmp Z( f )\nT0.00 0.04 0.08 0.12 0.16h\n0.00.10.20.3nmpZ( g )\nT0.0 0.1 0.2 0.3 0.4 0.5 0.60.00.20.40.60.81.01.2\nafp( h )\n\n  Fig. 4 Figure 4"    },    {        "title": "0805.2210v1.Dynamical_Casimir_effect_for_magnons_in_a_spinor_Bose_Einstein_condensate.pdf",        "content": "arXiv:0805.2210v1  [cond-mat.other]  15 May 2008Dynamical Casimir effect for magnons in a spinor Bose-Einste in condensate\nHiroki Saito1and Hiroyuki Hyuga2\n1Department of Applied Physics and Chemistry, The Universit y of Electro-Communications, Tokyo 182-8585, Japan\n2Department of Physics, Keio University, Yokohama 223-8522 , Japan\n(Dated: October 27, 2018)\nMagnon excitation in a spinor Bose-Einstein condensate by a driven magnetic field is shown\nto have a close analogy with the dynamical Casimir effect. A ti me-dependent external magnetic\nfield amplifies quantum fluctuations in the magnetic ground st ate of the condensate, leading to\nmagnetization of the system. The magnetization occurs in a d irection perpendicular to the magnetic\nfieldbreakingtherotation symmetry. This phenomenonis num erically demonstratedandtheexcited\nquantum field is shown to be squeezed.\nPACS numbers: 03.75.Mn, 03.70.+k, 42.50.Lc, 42.50.Dv\nI. INTRODUCTION\nVacuum fluctuations play an important role in a vari-\nety of situations in quantum physics. For instance, the\nstatic Casimir effect [1] originates from vacuum fluctu-\nations in the electromagnetic field and in the electronic\nstates in matter. If the definition of the vacuum state\ndepends on time due to a time-dependent external con-\ndition and the system cannot follow the instantaneous\nvacuum state adiabatically, the vacuum fluctuation ma-\nterializes as real particles [2, 3, 4, 5]. This phenomenon\nis called the nonstationary or dynamical Casimir effect\n(DCE).\nThe DCE has been extensively studied [6], especially\nfor photons and the massless scalar field. When the mir-\nror of an optical cavity is rapidly moved [7] or the di-\nelectric constant of the matter in a cavity is rapidly al-\ntered [8, 9, 10, 11], photons are created in the cavity\neven if the initial state of the electromagnetic field is\nin the vacuum state. The generated photons are in a\nsqueezed state [12], which modifies the Casimir force ex-\nerted on the mirrors [13]. The finite temperature correc-\ntion to the DCE [14] and decoherence via the DCE [15]\nhave also been studied. Since there is no intrinsic Hamil-\ntonian in the original formulation of the moving mirror\nproblem [4], the effective Hamiltonian approach has been\ndeveloped [16, 17, 18, 19]. We have derived an effective\nHamiltonian for the moving mirror problem by quantiz-\ning both the electromagnetic field and the polarization\nfield in the mirrors [20].\nHowever,photoncreationbythe DCEhasnotyetbeen\nexperimentally observed. This is because the mirror of\na cavity must be vibrated at a frequency of the order of\nGHz (at least for a microwave cavity) in order to reso-\nnantly amplify the photon field [21]. Recently, the INFN\ngroup [22] proposed a method to detect the DCE using\na semiconductor layer illuminated by laser pulses, which\nenables the rapid displacement of the position of the cav-\nity mirror. The proposed experiment using this method,\nhowever, has not been completed to date.\nIn the present paper, we propose a novel system to\nrealize the DCE: a Bose Einstein condensate (BEC) of\nan ultracold atomic gas with spin degrees of freedom.In this system, magnons are created through the DCE.\nThe time-dependent external condition that causes the\nDCE corresponds to the time-dependent magnetic field\napplied to the BEC. Unlike in the case of photons, the\ntypical energy of magnons in a BEC for a magnetic field\nof∼1 G is∼h×100 Hz (h: Planck constant). Magnetic\nfieldmodulation atthisfrequencyisexperimentallyfeasi-\nble. Continuous amplification of magnons by an oscillat-\ning magnetic field leads to magnetization of the system,\nwhich can be observed by in situmeasurements [23].\nIn a broad sense, quasiparticle excitations in a nonsta-\ntionary BEC may be regarded as the DCE. For exam-\nple, a time dependent trapping potential [24], a rapid in-\ncrease in the interatomic interaction [25], and collapse of\na BEC by an attractive interaction [26] generate Bogoli-\nubov quasiparticles. In these cases, the BEC itself is also\nexcited and its shape depends on time, whose dynam-\nics is described by the mean-field Gross-Pitaevskii (GP)\nequation. However, it is difficult to distinguish the quasi-\nparticle excitation from the mean-field excitation, and\ntherefore these systems are unsuitable for demonstrating\nthe DCE. In contrast, in our model, the time-dependent\nmagnetic field excites only the vacuum fluctuation in the\ninitial quasiparticle vacuum state, giving an ideal testing\nground for the DCE.\nThis paper is organized as follows. Section II formu-\nlates the problem. Section III discusses the relation of\nthe magnon excitation in the present system with the\nDCE. Section IV numerically demonstrates the proposed\nphenomena using the GP equation with quantum fluctu-\nations. Section V provides discussion and conclusions.\nII. FORMULATION OF THE PROBLEM\nA. Hamiltonian for the system\nWe consider spin-1 bosonic atoms with mass Mcon-\nfined in an optical trapping potential U(r). The single-\nparticle part of the Hamiltonian without a magnetic field2\nis given by\nˆH0=1/summationdisplay\nm=−1/integraldisplay\ndrˆψ†\nm(r)/bracketleftbigg\n−¯h2\n2M∇2+U(r)/bracketrightbigg\nˆψm(r),(1)\nwhereˆψm(r) is the field operator that annihilates an\natom with spin magnetic quantum number m=−1,0,1\nat the position r.\nThe interatomic interaction for ultracold spin-1 atoms\nis described by the s-wave scattering lengths a0anda2,\nwhere the subscripts 0 and 2 indicate the total spin of\ntwo colliding atoms. The interaction Hamiltonian can\nbe written in spin-independent and spin-dependent parts\nas [27, 28]\nˆHint=/integraldisplay\ndr/bracketleftBig\nc0: ˆρ2(r) : +c1:ˆF(r)·ˆF(r) :/bracketrightBig\n,(2)\nwhere the symbol :: denotes the normal ordering and\nˆρ(r) =1/summationdisplay\nm=−1ˆψ†\nm(r)ˆψm(r), (3)\nˆF(r) =/summationdisplay\nm,m′ˆψ†\nm(r)(f)m,m′ˆψm′(r),(4)\nwithf= (fx,fy,fz) being the vector of the spin-1 3 ×3\nmatrices. The interaction coefficients c0andc1in Eq. (2)\nare given by\nc0=4π¯h2\nMa0+2a2\n3, (5)\nc1=4π¯h2\nMa2−a0\n3. (6)\nWe restrict ourselves to the case of the hyperfine spin\nF= 1 of an alkali atom with nuclear spin I= 3/2 and\nelectron spin S= 1/2 (e.g.,23Na and87Rb). Because of\nthe hyperfine coupling between the nuclear and electron\nspins and their different magnetic moments, the Zeeman\nenergy is a nonlinear function of B[29]. Taking the first\nand second order terms, the Hamiltonian becomes\nˆHB=/integraldisplay\ndr/bracketleftBig\np1B(t)ˆFz+p2B2(t)/parenleftBig\nˆψ†\n1ˆψ1+ˆψ†\n−1ˆψ−1/parenrightBig/bracketrightBig\n,\n(7)\nwherep1andp2are the linear and quadratic Zeeman\ncoefficients, respectively, andweassumethat theuniform\nmagnetic field B(t) is applied in the zdirection. We\ndefine the quadratic Zeeman energy as\nq(t)≡p2B2(t), (8)\nwhich is positive for the F= 1 hyperfine state.\nThus, the total Hamiltonian has the form\nˆH=ˆH0+ˆHB+ˆHint. (9)\nSince the linear Zeeman term in Eq. (7) commutes with\nthe other part of the Hamiltonian and only rotates the\nspin uniformly around the zaxis, we neglect the linear\nZeeman term henceforth.B. Bogoliubov approximation\nThe initial state considered in the present paper is the\nground state for an initial value of the quadratic Zeeman\nenergy,q=q(0), under the restriction\n/integraldisplay\ndr∝angb∇acketleftˆFz(r)∝angb∇acket∇ight= 0. (10)\nInSec. Vwediscusshowtogeneratethisstate. Thespin-\nindependentinteractioncoefficient c0mustbepositivefor\nthe existence of the ground state. Either for c1>0 or\nforc1<0 andq>∼2|c1|˜ρ, where ˜ρis a typical atomic\ndensity, almost all atoms are in the m= 0 state for this\ninitial state [30]. We therefore employ the Bogoliubov\napproximation by setting\nˆψ0(r) =e−iµt/¯h/bracketleftBig\nΨ0(r)+ˆφ0(r)/bracketrightBig\n,(11)\nˆψ±1(r) =e−iµt/¯hˆφ±1(r), (12)\nwhere Ψ 0(r) is a real function that minimizes the energy\nfunctional,\nE0=/integraldisplay\ndr/bracketleftbigg\nΨ0/parenleftbigg\n−¯h2\n2M∇2+U/parenrightbigg\nΨ0+c0\n2Ψ4\n0/bracketrightbigg\n,(13)\nµis the chemical potential,\nµ=/integraldisplay\ndr/bracketleftbigg\nΨ0/parenleftbigg\n−¯h2\n2M∇2+U/parenrightbigg\nΨ0+c0Ψ4\n0/bracketrightbigg\n,(14)\nandˆφm(r) are the fluctuation operators. The wave func-\ntion Ψ 0(r) isnormalizedas/integraltext\ndrΨ2\n0(r) =NwithNbeing\nthe number of atoms. The Heisenberg equations of mo-\ntion forˆφmare then written as [27]\ni¯h∂ˆφ0\n∂t=/parenleftbigg\n−¯h2\n2M∇2+U−µ/parenrightbigg\nˆφ0+c0Ψ2\n0/parenleftBig\n2ˆφ0+ˆφ†\n0/parenrightBig\n,\n(15)\ni¯h∂ˆφ±1\n∂t=/parenleftbigg\n−¯h2\n2M∇2+U+q+c0Ψ2\n0−µ/parenrightbigg\nˆφ±1\n+c1Ψ2\n0/parenleftBig\nˆφ±1+ˆφ†\n∓1/parenrightBig\n, (16)\nwhere we neglect the second and third orders of\nˆφm. Equation (15) is identical with that of a scalar\nBEC, which gives the Bogoliubov spectrum E(0)\nk=/radicalbig\nεk(εk+2c0Ψ2\n0) for a uniform system, where εk=\n¯h2k2/(2M).\nIfqis constantintime, wecansolveEq.(16) bysetting\nˆφ±1=/summationdisplay\nλ/bracketleftBig\nuλ(r)ˆb±,λe−iE(1)\nλt/¯h+v∗\nλ(r)ˆb†\n∓,λeiE(1)\nλt/¯h/bracketrightBig\n,\n(17)\nwhereˆb±,λare bosonic operators for quasiparticles and\nuλ,vλ, andE(1)\nλare determined by the Bogoliubov-de3\nGenne equations,\n/parenleftbigg\n−¯h2\n2M∇2+U+q+c0Ψ2\n0+c1Ψ2\n0−µ/parenrightbigg\nuλ+c1Ψ2\n0vλ\n=E(1)\nλuλ, (18)\n/parenleftbigg\n−¯h2\n2M∇2+U+q+c0Ψ2\n0+c1Ψ2\n0−µ/parenrightbigg\nvλ+c1Ψ2\n0uλ\n=−E(1)\nλvλ. (19)\nThe quasiparticles created by ˆb†\n±,λare excitations of m=\n±1 states, which we call magnons.\nFor a uniform system, Eqs. (18) and (19) are solved to\ngive\nuk=eik·r\n√\n2V/radicalBigg\nεk+q+c1Ψ2\n0\nE(1)\nk+1, (20)\nvk=−eik·r\n√\n2V/radicalBigg\nεk+q+c1Ψ2\n0\nE(1)\nk−1,(21)\nwhereVis the volume of the system and\nE(1)\nk=/radicalBig\n(εk+q)(εk+q+2c1Ψ2\n0).(22)\nWhen all Bogoliubov energies are real, the system is dy-\nnamically stable. If c1<0 andq <2|c1|Ψ2\n0, Eq. (22)\nis imaginary for long wavelengths and the system is\ndynamically unstable against spontaneous magnetiza-\ntion [31, 32]. In the present paper, we consider such\nparameters that the system is dynamically stable.\nC. Effective Hamiltonian for oscillating q\nHere we assume that the quadratic Zeemanenergy q(t)\noscillates as\nq(t) =q0(1+δqsinΩt) (23)\nthroughtheoscillationofthestrengthoftheappliedmag-\nnetic field. For δq= 0, the Bogoliubov Hamiltonian for\nmagnons is given by\nˆHq=q0\nmag=/summationdisplay\nλE(1)\nλ/parenleftBig\nˆb†\n+,λˆb+,λ+ˆb†\n−,λˆb−,λ/parenrightBig\n,(24)\nwhereE(1)\nλ,ˆb+,λ, andˆb−,λare energies and annihilation\noperators of magnons defined at q=q0. The magnon\nHamiltonian for δq∝negationslash= 0 is then written as\nˆHq\nmag=ˆHq=q0mag+q0δqsinΩt/integraldisplay\ndr/parenleftBig\nˆφ†\n1ˆφ1+ˆφ†\n−1ˆφ−1/parenrightBig\n.\n(25)When the last term in Eq. (25) is treated as the pertur-\nbation, its interaction representation becomes\nq0δqsinΩt/summationdisplay\nλλ′/integraldisplay\ndr/bracketleftBig\nei(E(1)\nλ−E(1)\nλ′)t/¯h(u∗\nλuλ′+v∗\nλvλ′)\n×/parenleftBig\nˆb†\n+,λˆb+,λ′+ˆb†\n−,λˆb−,λ′/parenrightBig\n+e−i(E(1)\nλ+E(1)\nλ′)t/¯h(uλvλ′+vλuλ′)ˆb+,λˆb−,λ′\n+H.c.+const./bracketrightBig\n, (26)\nwhere H.c.denotes the Hermitian conjugate of the pre-\nceding term. If the frequency Ω is resonant with ( E(1)\nλ+\nE(1)\nλ′)/¯hfor someλandλ′, the terms proportional to\nˆb+,λˆb−,λ′andˆb†\n+,λˆb†\n−,λ′become dominant.\nWhen the system is uniform and Ω is resonant with\n2E(1)\nk/¯h, the Hamiltonian in Eq. (26) reduces to\nˆHeff=−iq0δqV/summationdisplay\nk′\n|ukvk|/parenleftBig\nˆb+,kˆb−,−k−ˆb†\n+,kˆb†\n−,−k/parenrightBig\n,\n(27)\nwhere/summationtext\nk′denotes that the summation is taken for\n¯h2k2/(2M)≃2E(1)\nk. The form of Eq. (27) suggests that\nthe magnon state develops into the squeezed state by the\noscillation of q, which will be confirmed in Sec. IVB nu-\nmerically.\nIII. RELATION WITH THE DYNAMICAL\nCASIMIR EFFECT\nWe now discuss the relation of the present system to\nthe usual DCE.\nFirst let us review the DCE for an electromagnetic\nfield. The initial state is the vacuum state for some static\nconfigurations of mirrors and dielectrics. The initial ex-\npectation values of the electric and magnetic fields there-\nfore vanish: ∝angb∇acketleftˆE∝angb∇acket∇ight=∝angb∇acketleftˆH∝angb∇acket∇ight= 0. These operators obey the\nHeisenberg equations of motion,\n∂\n∂tε(r,t)ˆE(r,t) =∇׈H(r,t), (28)\n∂\n∂tˆH(r,t) =−∇×ε(r,t)ˆE(r,t),(29)\nwhereεis a dielectric constant, and the electric field op-\neratorˆEmust vanish at the mirrors. From Eqs. (28)\nand (29), ∝angb∇acketleftˆE∝angb∇acket∇ightand∝angb∇acketleftˆH∝angb∇acket∇ightalways remain zero as in classi-\ncal electrodynamics, whereas the vacuum fluctuation of\nthe electromagnetic field can be amplified, leading to the\ncreation of photons.\nFor the present spinor BEC system, the initial state\nis assumed to be the ground state satisfying Eq. (10).\nSince the Hamiltonian (9) has spin-rotation symmetry\naroundthe zaxis, the expectation value ofthe transverse\nmagnetization for the initial state vanishes: ∝angb∇acketleftˆF+∝angb∇acket∇ight= 0,4\nwhereˆF+=ˆFx+iˆFy. In the Bogoliubov approxima-\ntion, using Eqs. (11) and (12), the magnetization opera-\ntorˆF+=√\n2(ˆψ†\n1ˆψ0+ˆψ†\n0ˆψ−1) reduces to\nˆF+≃√\n2Ψ0/parenleftBig\nˆφ†\n1+ˆφ−1/parenrightBig\n. (30)\nFrom Eq. (16), the Heisenberg equations of motion for\nˆF+are obtained as\n¯h∂ˆF+\n∂t=/bracketleftbigg\n−¯h2\n2M∇2+V+q(t)+c0Ψ2\n0−µ/bracketrightbigg\nˆΠ+,(31)\n¯h∂ˆΠ+\n∂t=−/bracketleftbigg\n−¯h2\n2M∇2+V+q(t)\n+(c0+2c1)Ψ2\n0−µ/bracketrightbigg\nˆF+, (32)\nwhere we define\nˆΠ+=−i√\n2Ψ0/parenleftBig\nˆφ†\n1−ˆφ−1/parenrightBig\n. (33)\nThe expectation value of this operator also vanishes,\n∝angb∇acketleftˆΠ+∝angb∇acket∇ight= 0, for the initial state. We find from Eqs. (31)\nand (32) that ∝angb∇acketleftˆF+∝angb∇acket∇ightand∝angb∇acketleftˆΠ+∝angb∇acket∇ightalways remain zero. On\nthe other hand, their quantum fluctuations can be am-\nplified due to the temporal variation of q(t), leading to\nthe creation of Bogoliubov quasiparticles, i.e., magnons.\nThus, the present situation is similar to that of the DCE\nin the electromagnetic field in that the amplification of\nquantum fluctuations and (quasi)particle creations occur\nwhile the expectation values of the fields remain constant\n(∝angb∇acketleftˆF+∝angb∇acket∇ight=∝angb∇acketleftˆΠ+∝angb∇acket∇ight= 0).\nWhat kind of excitation phenomenon can we regard\nas the DCE? For example, can the excitation of rip-\nples on water in a vibrating bucket be attributed to\nthe DCE? The answer is partially yes, because ampli-\nfication of the quantum fluctuation on the water surface\ndoes occur, which creates ripplons. However, this quan-\ntum excitation is overwhelmed by the classical excitation\nof the water surface and identification of the quasipar-\nticle excitations should therefore be extremely difficult.\nThus, the suitable conditionforstudying the DCE isthat\nonly the quantum fluctuation is excited and the classi-\ncal field, i.e., the expectation value of the relevant quan-\ntum field, remains constant during the temporal vari-\nation of external parameters. For the electromagnetic\nfield,∝angb∇acketleftˆE(r,t)∝angb∇acket∇ight=∝angb∇acketleftˆH(r,t)∝angb∇acket∇ight= 0, even when the position\nof the mirror and ε(r,t) of matter are changed. The\npresent spinor BEC system is also suitable for studying\nthe DCE, since ∝angb∇acketleftˆF+(r,t)∝angb∇acket∇ight=∝angb∇acketleftˆΠ+(r,t)∝angb∇acket∇ight= 0 holds during\nthe change of q(t).\nIV. NUMERICAL ANALYSIS\nA. Mean-field theory with quantum fluctuations\nIn this section, we numerically demonstrate the DCE\nof magnons in a spinor BEC. Since performing a fullquantum many-body simulation is difficult, we employ a\nmean-field approximation taking into account the initial\nquantum fluctuations. Recently, this method wasused to\npredict spin vortex formation through the Kibble-Zurek\nmechanism in magnetization of a spinor BEC [33].\nThe mean-field GP equations for a spin-1 BEC are\ngiven by\ni¯h∂ψ±1\n∂t=/bracketleftbigg\n−¯h2\n2M∇2+V+q(t)+c0ρ/bracketrightbigg\nψ±1\n+c1/parenleftbigg1√\n2F∓ψ0±Fzψ±1/parenrightbigg\n,(34)\ni¯h∂ψ0\n∂t=/parenleftbigg\n−¯h2\n2M∇2+V+c0ρ/parenrightbigg\nψ0\n+c1√\n2(F+ψ1+F−ψ−1), (35)\nwhereψm(r,t) are the macroscopic wave functions and\nρ,Fz, andF±=Fx±iFyare defined by the forms in\nEqs. (3) and (4) in which ˆψmare replaced by ψm. The\ninitial wave function for ψ0is the ground state solution\nΨ0of Eq. (13), which is a stationary solution of Eq. (35)\nforψ±1= 0. Although ∝angb∇acketleftˆF±∝angb∇acket∇ight= 0 for the exact many-\nbody initial state, we do not set ψ±1= 0, since the right-\nhand side of Eq. (34) vanishes and no time evolution is\nobtained.\nWe include a small initial noise in ψ±1to reproduce\nvacuum fluctuation in magnetization within the Bogoli-\nubov approximation. To this end, we consider the mag-\nnetic correlation function for the Bogoliubov vacuum,\n∝angb∇acketleftˆF+(r)ˆF−(r′)∝angb∇acket∇ight ≃2Ψ0(r)Ψ0(r′)/summationdisplay\nλfλ(r)f∗\nλ(r′),(36)\nwherefλ(r)≡uλ(r)+vλ(r) and we have used Eqs. (17)\nand(30). Tofind theappropriatemean-fieldinitial states\nofψ±1, we assume the form\nψ±1(r) =/summationdisplay\nλ/bracketleftbig\nuλ(r)b±,λ+v∗\nλ(r)b∗\n∓,λ/bracketrightbig\n,(37)\nwhereb±,λare random numbers whose probability dis-\ntribution is determined below. Substituting Eq. (37) into\nF+(r)F−(r′) and taking the average with respect to the\nprobability distribution, we obtain\n∝angb∇acketleftF+(r)F−(r′)∝angb∇acket∇ightavg= 2Ψ 0(r)Ψ0(r′)\n×/summationdisplay\nλ,λ′/bracketleftbig\nf∗\nλ(r)fλ′(r′)∝angb∇acketleftb∗\n+,λb+,λ′∝angb∇acket∇ightavg\n+fλ(r)f∗\nλ′(r′)∝angb∇acketleftb−,λb∗\n−,λ′∝angb∇acket∇ightavg\n+fλ(r)fλ′(r′)∝angb∇acketleftb−,λb+,λ′∝angb∇acket∇ightavg\n+f∗\nλ(r)f∗\nλ′(r′)∝angb∇acketleftb∗\n+,λb∗\n−,λ′∝angb∇acket∇ightavg/bracketrightbig\n.(38)\nUsing the fact that both ( u,v) and (u∗,v∗) are solutions\nofEqs. (18) and (19) and assumingthat the randomvari-\nablesb+,λandb−,λobey the same distribution, we find5\nthat Eq. (38) coincides with Eq. (36) by a probability\ndistribution satisfying\n∝angb∇acketleftb∗\n±,λb±,λ′∝angb∇acket∇ightavg=1\n2δλ,λ′,∝angb∇acketleftb±,λb∓,λ′∝angb∇acket∇ightavg= 0.(39)\nWe also assume ∝angb∇acketleftb±,λb±,λ′∝angb∇acket∇ightavg=∝angb∇acketleftb±,λb∗\n∓,λ′∝angb∇acket∇ightavg= 0.\nThus, the initial mean-field states (37) with probabil-\nity distributions obeying Eq. (39) reproduce the same\nmagnetic correlation as for the Bogoliubov approxima-\ntion. Similarly, we can show that the time evolution of\n∝angb∇acketleftF+(r)F−(r′)∝angb∇acket∇ightavgalso agrees with that of ∝angb∇acketleftˆF+(r)ˆF−(r′)∝angb∇acket∇ight\nat the level of the Bogoliubov approximation.\nB. Numerical results\nWe numerically solve Eqs. (34) and (35) for the time-\ndependent magnetic field that gives a sinusoidal oscil-\nlation of the quadratic Zeeman energy as in Eq. (23).\nFor simplicity, we consider a uniform two-dimensional\n(2D) system with a periodic boundary condition. We as-\nsume that 107 23Na atoms are in a 100 µm×100µm\nspace with atomic density n= 2.8×1014cm−3. In a\nrecent experiment [35], a2−a0∝c1for anF= 123Na\natomwaspreciselymeasuredtobe about 2.47Bohrradii.\nThe system is therefore dynamically stable according to\nEq. (22). The initial state is ψ0=√nandψ±1are given\nby Eqs. (37) and (39) in which the complex random vari-\nablesb±,kare assumed to follow the Gaussian distribu-\ntionP(b) =/radicalbig\n2/πe−2|b|2. The initial random variables\nb±,kare cut off for 2 π/k<∼1.5µm. The strength of the\nmagnetic field at t= 0 is 500 mG, which corresponds to\nq0≃h×69 Hz≃1.02c1n.\nFigure 1 (a) shows time evolution of the average\nsquared transverse magnetization [34],\nGT=/integraltext\ndr|F+(r)|2\n/integraltext\ndrρ2(r), (40)\nand longitudinal magnetization,\nGL=/integraltext\ndr|Fz(r)|2\n/integraltext\ndrρ2(r), (41)\nfor Ω = 2/radicalbig\nq0(q0+2c1n)≃2π×237 Hz and δq= 0.2.\nThis frequency is resonant with 2 E(1)\nk=0/¯h, and thek≃0\nmodes are expected to be excited according to the effec-\ntive Hamiltonian (27). From Fig. 1 (a), we find that the\ntime constant for the exponential growth of GTis≃22\nms.\nSolving the Heisenberg equation for the effective\nHamiltonian with k= 0,\nˆH0\neff=−iξ0/parenleftBig\nˆb+,0ˆb−,0−ˆb†\n+,0ˆb†\n−,0/parenrightBig\n,(42)\nwe obtain\nˆb±,0(t) =ˆb±,0e−iE(1)\n0t/¯hcoshξ0t+ˆb†\n∓,0eiE(1)\n0t/¯hsinhξ0t,\n(43)00.10.2\n02004006008001000GT, GL\nt [ms]GT\nGL(a)\nt [ms]\n0\n285\n319\n800(b)|F+| / n F z / n arg F+\n0 1 −π π\n -0.5 0.5\nFIG. 1: (Color) (a) Time evolution of the average squared\ntransverse magnetization GT(red curve) and longitudinal\nmagnetization GL(blue curve) for n= 2.8×1014cm−3,\nN= 107, Ω = 2p\nq0(q0+2c1n)≃2π×237 Hz, and δq= 0.2.\nThe inset magnifies the dashed square. (b) Profiles of |F+|/n,\nargF+, andFz/natt= 0, 285, 319, and 800 ms. The field\nof view is 100 µm×100µm. The red circles indicate the\ntopological defects.\nwhereξ0≡q0δq|u0v0|V. Substitution of this solution\ninto Eq. (30) with\nˆφ±1(r,t) =/summationdisplay\nk/bracketleftBig\nuk(r)ˆb±,k(t)+v∗\nk(r)ˆb†\n±,k(t)/bracketrightBig\n(44)6\ngives\nˆF+(t)≃√\n2n/summationdisplay\nke−ik·r(|uk|+|vk|)/bracketleftBig\nˆb†\n+,k(t)+ˆb−,−k(t)/bracketrightBig\n≃√\n2n(|u0|+|v0|)\n×/parenleftBig\nˆb†\n+,0eiE(1)\n0t/¯h+ˆb−,0e−iE(1)\n0t/¯h/parenrightBig\neξ0t,(45)\nwhere in the second line we take only the k= 0 compo-\nnent because of the exponential factor eξ0t. The average\nsquared transverse magnetization GTthus grows expo-\nnentially with a time constant 1 /(2ξ0)≃20 ms, which is\nin good agreement with the numerical results of 22 ms.\nThe spatial profile of the magnetization is shown in\nFig. 1 (b). The transverse magnetization first grows\nwith long wavelength [the second row of Fig. 1 (b)], in\nwhich we can see a few topological defects (red circles).\nThese spin vortices are generated through the Kibble-\nZurek mechanism [33, 36]. The transverse magnetization\nthen exhibits an interesting concentric pattern [the third\nrow of Fig. 1 (b)], which may be due to the nonlinearity.\nThe magnetization finally breaks into complicated frag-\nments and the longitudinal magnetization also begins to\ngrow [the fourth row of Fig. 1 (b)].\nWe now discuss symmetry breaking in the DCE. The\noriginal Hamiltonian (9) commutes with/integraltextˆFzdrand the\nsystem has spin-rotation symmetry around the zaxis.\nIn reality, however, local magnetization in the x-ydirec-\ntions occurs in a ferromagnetic BEC breaking the spin-\nrotation symmetry spontaneously [34], and we expect\nthat the symmetry breaking also occurs in the present\nsystem. This is because the exact quantum state with\nspin-rotation symmetry is the macroscopic superposition\nof magnetized states in the x-ydirections, which is there-\nfore fragile against external perturbations. Such symme-\ntry breaking phenomena should also occur in the DCE\nof the electromagnetic field. For example, in the 1D\nmoving mirror problem [7], there is rotation symmetry\naround the axis of the 1D cavity and the generated pho-\nton state is a superposition of all polarizations, giving\n∝angb∇acketleftˆE∝angb∇acket∇ight=∝angb∇acketleftˆB∝angb∇acket∇ight= 0. When we measure the local electromag-\nnetic field, however, we obtain a nonzero value as a result\nof the symmetry breaking.\nFigure 2 shows the magnitude of the Fourier transform\nofF+,\n/vextendsingle/vextendsingle/vextendsingle˜F+(k)/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\ndrF+(r)eik·r\n√\nV/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (46)\nInitially only the k≃0 modes grow as shown in Fig. 2\n(a). Then, the momentum distribution becomes broad\nas shown in Fig. 2 (b). Specific wave numbers can be\nselectively excited using larger Ω. Figure 2 (c) shows\nthe result for Ω = 2 π×358 Hz, which is resonant with\n2E(1)\nk/¯hfork= 0.5µm−1. We can see the ring at this\nwave number.\nFrom the form of the effective Hamiltonian in Eq. (27),\ngenerated magnons are expected to have quantum corre-\n(a) (b) (c)\narb. scale0 1-10\n-11ky\nkx[µm  ]-1[µm  ]-1\nFIG. 2: (Color) Magnitude of the Fourier transform of F+at\n(a)t= 285 ms and (b) t= 800 ms. The parameters are the\nsame as in Fig. 1. (c) Magnitude of the Fourier transform of\nF+att= 500 ms for Ω = 2 π×358 Hz. The other parameters\nare the same as in Fig. 1.\nlations. Defining new operators,\nˆB+,k=1√\n2/parenleftBig\nˆb+,k+ˆb−,k/parenrightBig\n, (47)\nˆB−,k=1√\n2/parenleftBig\nˆb+,k−ˆb−,k/parenrightBig\n, (48)\nwe can rewrite Eq. (27) as\nˆHeff=−i\n2/summationdisplay\nk′\nξk/parenleftBig\nˆB2\n+,k−ˆB†2\n+,k−ˆB2\n−,k+ˆB†2\n−,k/parenrightBig\n,(49)\nwhereξk≡q0δq|ukvk|V. We can clearly see that this\nHamiltonian generates the squeezed state.\nFigure3(a)plotsthe valuesof B+,0= (b+,0+b−,0)/√\n2\natt= 0 and 30 ms. The distribution of B+,0corresponds\nto the quantum fluctuation in Eq. (47) with k= 0, which\nis expected to be squeezed. The values of b±,0are ob-\ntained from b±,k=√\nV[|uk|˜ψ±1(k)− |vk|˜ψ∗\n∓1(k)] with\n˜ψ±1(k) being the Fourier transform of ψ±1(r). Each\npoint in Fig. 3 (a) corresponds to a single simulation run\nand 1000 simulations are performed. Figure 3 (b) shows\nthe variance of Re( B+,0e−iθ),\nV(θ) =∝angb∇acketleft[Re(B+,0e−iθ)]2∝angb∇acket∇ightavg−∝angb∇acketleftRe(B+,0e−iθ)∝angb∇acket∇ight2\navg.(50)\nFortheinitialstate, the varianceisisotropic, V(θ) = 1/4,\nfromEq.(39). Thesmalldeviationofthesolidcurvefrom\n1/4 in Fig. 3 (b) is the statistical error. At t= 30 ms,\nthe distribution of B+,0is clearly squeezed, and the max-\nimum and minimum values of V(θ) are 1.09 and 0.057.\nFor the present parameters, e2ξt≃4.38 att= 30 ms,\nwhich is in good agreement with 1 .09/0.25≃0.25/0.057.\nSince 1.09×0.057≃1/42, the squeezed state is almost\nthe minimum uncertainty state.\nIn this section, we assumed the use of23Na atoms.\nSimilarresultscanbeobtainedalsofor F= 187Rbatoms\nwith a ferromagnetic interaction ( c1<0), whereq(t)\nmust always be larger than |c1|Ψ2\n0in order to suppress\nthe spontaneous magnetization.7\n00.20.40.60.811.2\n0 π/2 πV(θ)\nθt=0 mst=30 ms(a)\n(b)-4-2024\n-4-2024Im B+,0\nRe B+,0t=0 ms\n-4-2024\n-4-2024Im B+,0\nRe B+,0t=30 ms\nFIG. 3: (a) Distributions of B+,0= (b+,0+b−,0)/√\n2 att= 0\nand 30 ms obtained by 1000 simulation runs with different\ninitial states produced by random numbers. The parameters\nare the same as in Fig. 1. (b) Variance V(θ) of Re(B+,0e−iθ)\nfor the data in (a). The dotted line indicates V(θ) = 1/4.\nV. DISCUSSION AND CONCLUSIONS\nWe now discuss the possibility of experimental obser-\nvation of the proposed phenomena. The magnetization\nprofileF(r) as shown in Fig. 1 can be measured by\nspin-sensitive phase-contrastimaging in a nondestructive\nmanner [23, 34], from which GT,GL, and˜F(k) are ob-\ntained. The squeezing of the field as shown in Fig. 3 can\nalso be observed. From Eq. (45), we can measure both\nb∗\n+,k+b−,−kandb∗\n+,k−b−,−k, and therefore we obtain\nB±,k. In orderto assurethat the observedmagnetization\nis definitely due to amplification of the vacuum fluctua-tion, we must prepare the appropriate initial state. In\nprinciple, the initial magnon vacuum state can be pre-\npared as follows. First, a BEC is prepared in the m= 0\nstate at a sufficiently strong magnetic field ( q≫c1Ψ2\n0),\nin which the m= 0 state is almost the ground state. The\nresidual atoms in the m=±1 states must be eliminated\ncompletely. Then, the magnetic field is adiabatically de-\ncreased to the desired strength, which gives a magnon\nvacuum state satisfying Eq. (10).\nIn conclusion, we have studied the magnon excitation\nin a spinor BEC by a driven external magnetic field and\nhave demonstrated the close analogy of this phenomenon\nwith the DCE. The present system is suitable for study-\ning the DCE of quasiparticles, since the time-dependent\nmagnetic field applied to the Bogoliubov ground state\namplifies only the vacuum fluctuation, keeping the “clas-\nsical fields” constant in the Bogoliubov approximation.\nWe numerically demonstrated magnon excitation in a\nspinor BEC using the mean-field GP equation, in which\nthevacuumfluctuationistakenintoaccountbytheinitial\nrandom noise. We have shown that the oscillating exter-\nnal magnetic field resonantly amplifies the vacuum fluc-\ntuation, leading to magnetization of the system (Fig. 1).\nThe Fourier transform of the excited field reveals that\nthe specific wave number kcan be selectively amplified\n(Fig. 2). The excited quantum field is squeezed (Fig. 3)\nas in the DCE of photons. The growth of magnetization\nin Fig. 1 (a) and the degree of squeezing in Fig. 3 can be\nwell described by the effective Hamiltonian in Eq. (27).\nA spinor BEC is thus a good testing ground for the\nDCE and our proposal is feasible with current experi-\nmental techniques. Study of the DCE of quasiparticles\nmay serve as a stepping stone to the observation of the\nDCE of photons.\nAcknowledgments\nThis work was supported by the Ministry of Educa-\ntion, Culture, Sports, Science and Technology of Japan\n(Grants-in-Aid for Scientific Research, No. 17071005and\nNo. 20540388) and by the Matsuo Foundation.\n[1] H. B. G. Casimir, Proc. K. Ned. Akad. Wet. 51, 793\n(1948).\n[2] Y. Takahashi and H. Umezawa, Nuovo Cimento 6, 1324\n(1957).\n[3] L. Parker, Phys. Rev. Lett. 21, 562 (1968).\n[4] G. T. Moore, J. Math. Phys. 11, 2679 (1970).\n[5] S. A. Fulling and P. C. W. Davies, Proc. R. Soc. London\nA348, 393 (1976).\n[6] For review, see, V. V. Dodonov, Adv. Chem. Phys. 119,\n309 (2001); V. V. Dodonov and A. V. Dodonov, J. Phys.:\nConf. Ser. 99, 012006 (2008).\n[7] See, e.g., M. Castagnino and R. Ferraro, Ann. Phys.(N.Y.)154, 1 (1984); V. V. Dodonov, A. B. Klimov,\nand D. E. Nikonov, J. Math. Phys. 34, 2742 (1993); C.\nK. Law, Phys. Rev. Lett. 73, 1931 (1994).\n[8] E. Yablonovitch, Phys. Rev. Lett. 62, 1742 (1989).\n[9] V. V. Dodonov, A. B. Klimov, and D. E. Nikonov, Phys.\nRev. A47, 4422 (1993).\n[10] H. Johnston andS. Sarkar, Phys.Rev.A 51, 4109 (1995).\n[11] H. Saito and H. Hyuga, J. Phys. Soc. Jpn. 65, 1139\n(1996);ibid.653513 (1996).\n[12] V. V. Dodonov, A. B. Klimov, and V. I. Man’ko, Phys.\nLett. A149, 225 (1990).\n[13] M. T. Jaekel and S. Reynaud, J. Phys. I France 2, 1498\n(1992).\n[14] G. Plunien, R. Sch¨ utzhold, and G. Soff, Phys. Rev. Lett.\n84, 1882 (2000).\n[15] D. A. R. Dalvit and P. A. Maia Neto, Phys. Rev. Lett.\n84, 798 (2000).\n[16] M. Razavy, Lett. Nuovo Cimento 37, 449 (1983).\n[17] G. Barton and C. Eberlein, Ann. Phys. (N.Y.) 227, 222\n(1993).\n[18] C. K. Law, Phys. Rev. A 49, 433 (1994).\n[19] J. Haro and E. Elizalde, Phys. Rev. Lett. 97, 130401\n(2006).\n[20] H. Saito and H. Hyuga, Phys. Rev. A 65, 053804 (2002).\n[21] V. V. Dodonov and A. B. Klimov, Phys. Rev. A 53, 2664\n(1996).\n[22] C. Braggio, G. Bressi, G. Carugno, C. Del Noce, G.\nGaleazzi, A. Lombardi, A. Palmieri, G. Ruoso, and D.\nZanello, Europhys. Lett. 70, 754 (2005).\n[23] J. M. Higbie, L. E. Sadler, S. Inouye, A. P. Chikkatur, S.\nR. Leslie, K. L. Moore, V. Savalli, and D. M. Stamper-\nKurn, Phys. Rev. Lett. 95, 050401 (2005).\n[24] Y. Castin and R. Dum, Phys. Rev. Lett. 79, 3553 (1997).\n[25] C. K. Law, P. T. Leung, and M. -C. Chu, Phys. Rev. A\n66, 033605 (2002).\n[26] E. A. Calzetta and B. L. Hu, Phys. Rev. A 68, 043625(2003).\n[27] T. -L. Ho, Phys. Rev. Lett. 81, 742 (1998).\n[28] T. Ohmi and K. Machida, J. Phys. Soc. Jpn. 67, 1822\n(1998).\n[29] See, e.g., C. J. Pethick and H. Smith, Bose-Einstein Con-\ndensation in Dilute Gases (Cambridge Univ. Press, Cam-\nbridge, 2002).\n[30] J. Stenger, S. Inouye, D. M. Stamper-Kurn, H. -J. Mies-\nner, A. P. Chikkatur, and W. Ketterle, Nature (London)\n396, 345 (1998).\n[31] H. Saito and M. Ueda, Phys. Rev. A 72, 023610 (2005).\n[32] H. Saito, Y. Kawaguchi, and M. Ueda, Phys. Rev. Lett.\n96, 065302 (2006); Phys. Rev. A 75, 013621 (2007).\n[33] H. Saito, Y. Kawaguchi, and M. Ueda, Phys. Rev. A 76,\n043613 (2007).\n[34] L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore ,\nand D. M. Stamper-Kurn, Nature (London) 443, 312\n(2006).\n[35] A. T. Black, E. Gomez, L. D. Turner, S. Jung, and P. D.\nLett, Phys. Rev. Lett. 99, 070403 (2007).\n[36] B. Damski and W. H. Zurek, Phys. Rev. Lett. 99, 130402\n(2007)."    },    {        "title": "2209.11574v1.Magnetic_interactions_in_orbital_dynamics.pdf",        "content": "Draft version September 26, 2022\nTypeset using L ATEX default style in AASTeX63\nMagnetic interactions in orbital dynamics\nBenjamin C. Bromley1and Scott J. Kenyon2\n1Department of Physics & Astronomy, University of Utah, 201 JFB, Salt Lake City, UT 84112\n2Smithsonian Astrophysical Observatory, 60 Garden Street, Cambridge, MA 02138\nABSTRACT\nThe magnetic \feld of a host star can impact the orbit of a stellar partner, planet, or asteroid if\nthe orbiting body is itself magnetic or electrically conducting. Here, we focus on the instantaneous\nmagnetic forces on an orbiting body in the limit where the dipole approximation describes its magnetic\nproperties as well as those of its stellar host. A permanent magnet in orbit about a star will be\ninexorably drawn toward the stellar host if the magnetic force is comparable to gravity due to the\nsteep radial dependence of the dipole-dipole interaction. While magnetic \felds in observed systems\nare much too weak to drive a merger event, we con\frm that they may be high enough in some close\ncompact binaries to cause measurable orbital precession. When the orbiting body is a conductor, the\nstellar \feld induces a time-varying magnetic dipole moment that leads to the possibility of eccentricity\npumping and resonance trapping. The challenge is that the orbiter must be close to the stellar host, so\nthat magnetic interactions must compete with tidal forces and the e\u000bects of intense stellar radiation.\nKeywords: planets and satellites: dynamical evolution | planets and satellites: formation | planets:\nMercury\n1.INTRODUCTION\nGravity, light, and wind guide the formation of planets around most stars. Yet stellar hosts can also in\ruence\nmaterial around them through magnetic interactions. A body with a permanent or induced magnetic moment can\nrespond to the local magnetic \feld tied to a magnetic host star. While these interactions are weak compared to\ngravity, cumulative e\u000bects of magnetic interactions can change the orbital energy of asteroids (Bromley & Kenyon\n2019), rocky planets (Kislyakova et al. 2017; Kislyakova et al. 2018; Kislyakova & Noack 2020; Noack et al. 2021), and\ngas giants (Laine et al. 2008; Laine & Lin 2012; Chang et al. 2012; Strugarek et al. 2017; Chyba & Hand 2021). Even\nstellar partners can in\ruence each other magnetically (Joss et al. 1979; Campbell 1983; Katz 1989, 2017; Mik\u0013 oczi 2021;\nBourgoin et al. 2022).\nMagnetic interactions can arise between a stellar host and any magnetic or electrically conducting body. A per-\nmanently magnetized body will feel a force in a \feld gradient, while free charges in a small, non-magnetic conductor\nmoving with respect to stellar magnetic \feld lines experience the Lorentz force. A large, conducting planet forms eddy\ncurrents in response to changes in the stellar magnetic \feld along its orbit (e.g., Gi\u000en et al. 2010; Nagel 2018); the\nLorentz forces on these eddy currents do a small amount of work that can a\u000bect orbital energy over time. A conductor\nin orbit through a static stellar magnetic \feld can inspiral as a result, with Ohmic losses in the eddy currents draining\norbital energy (Laine et al. 2008; Kislyakova et al. 2018; Bromley & Kenyon 2019). If the star and its magnetic dipole\nmoment are spinning, the orbiting body can migrate outward, tapping the energy of the rotating \feld. Even when\nthe net time-averaged force is negligible, as when a highly conductive asteroid e\u000eciently generates surface currents to\ne-mail: bromley@physics.utah.edu\ne-mail: skenyon@cfa.harvard.eduarXiv:2209.11574v1  [astro-ph.EP]  23 Sep 20222\noppose changes in the magnetic \rux, the instantaneous Lorentz force can be signi\fcant (Nagel 2018). The magnetic\ninteraction then generates an oscillatory driving force that can impact orbital dynamics.\nGoldreich & Lynden-Bell (1969) also considered the possibility that Lorentz forces drive current \rows in \rux tubes\nbetween a magnetic primary and a conducting body. In contrast to the induction mechanism just described, in which\na conducting planet acts as an AC transformer with an induced magnetic dipole, this unipolar interaction requires a\nmediating plasma to support the current connecting the primary and the conductor. The original focus of Goldreich &\nLynden-Bell (1969) on Jupiter-Io as a DC battery has since been broadened to include a wide range of systems (e.g.,\nLai 2012; Piro 2012).\nThe strength of the stellar magnetic \feld determines the importance of magnetic interactions compared with gravity.\nStars of nearly all spectral types harbor magnetic \felds (Donati & Landstreet 2009), during the T Tauri phase (e.g.\nLavail et al. 2017; Grankin 2021), while on the main sequence (Babcock 1958; Landstreet 1992; Johns-Krull & Valenti\n1996; Kochukhov 2021), and among compact evolved stars (e.g., Angel 1978; Valyavin 2015; Konar 2017). Extreme\ncases include: Babcock's Star, an Ap star with a \u001830 kG surface \feld (Babcock 1960); white dwarfs with \felds around\n109G (Caiazzo et al. 2021); and magnetars, with \feld strengths over 1014G (Olausen & Kaspi 2014). While dipole\n\feld strengths decrease as the inverse distance cubed, extremely magnetic stars produce \felds capable of in\ruencing\norbital motion at least within a few stellar radii. Around main sequence stars, stellar winds can buoy stellar magnetic\n\felds beyond this distance, supporting a more shallow, 1 =r2fall-o\u000b (e.g., Johnstone 2012, and references therein)\nAn orbiting body's material properties also determine the astrophysical relevance of its magnetic interactions\n(Kislyakova et al. 2017; Bromley & Kenyon 2019). For example, a conducting body responds to variations in the\nstellar magnetic \feld along its orbit by generating eddy currents to oppose changes in magnetic \rux. A poor con-\nductor produces only weak currents and barely perturbs the magnetic \feld. Ohmic losses and long-term changes to\nthe orbit are negligible. At higher conductivity, eddy currents are stronger; signi\fcant orbital evolution from Ohmic\nlosses is possible. In the limit of high conductivity, the orbiting body generates surface currents that e\u000eciently repel\nthe \rux changes entirely, yet there is little Ohmic dissipation or change in the orbital energy. In realistic astrophysical\nscenarios, planets consisting of modestly conducting silicates or water may interact magnetically in more interesting\nways than bodies made of insulators or superconductors (Bromley & Kenyon 2019).\nMagnetic interactions have observable astrophysical consequences (Joss et al. 1979). For example, a conducting\nasteroid around a magnetized white dwarf can be drawn inside the Roche radius (Bromley & Kenyon 2019), where it is\neventually accreted by the host star (Kenyon & Bromley 2017), contributing to the `pollution' of the star's atmosphere\nby metals (e.g., Zuckerman & Reid 1998; Zuckerman et al. 2010; Kepler et al. 2016; Farihi 2016). Kislyakova et al.\n(2017) suggested the possibility that Ohmic dissipation can lead to volcanism within rocky planets (see also Kislyakova\net al. 2018; Kislyakova & Noack 2020), an e\u000bect that might enable smaller, close-in bodies to contribute to stellar\npollution (Bromley & Kenyon 2019). More recently, Hogg et al. (2021) suggested that the pollution of white dwarfs\nmight be enhanced as small diamagnetic grains on tight orbits interact with the stellar magnetic \feld. Closer to home,\nmagnetic interactions with Jupiter and its major satellites have served as probes of the moons' internal structure (e.g.,\nKhurana et al. 1998; Zimmer et al. 2000; Kivelson et al. 2000).\nHere, we concentrate on interactions where the magnetic \feld is the sole intermediary between a magnetic star\nand an orbiting partner without a surrounding plasma to sustain a current loop. Magnetic forces between bodies\nare approximated by assigning or deriving dipole \felds for the stellar host and its partner. With this idealization,\nwe examine the stability of a stellar binary when each partner has a strong fossil \feld. We also derive the induced\nmagnetic dipole of planetary companions as they orbit through the stellar magnetic \feld. Highly conducting planets\nlike Mercury have dipoles that track the local stellar \feld but are antiparallel to it. A hypothetical planet composed\nof ferromagnetic material would produce a magnetic dipole moment that lines up with the instantaneous local \feld.\nEach of these con\fgurations has a potential impact on orbital dynamics.\nThis paper begins with a general description of orbital dynamics when dipole-dipole interactions are present alongside\ngravity ( §2), with the main focus on the stellar magnetic \feld. Then, in §3. we discuss the magnetic dipole moments\nof the stellar or planetary companions of a magnetic star, comparing fossil \felds, magnetic material, and conductors.\nWe give examples of speci\fc astrophysical scenarios in §4, and summarize in §5.3\nêzêrYZ\nXB𝝁\nFigure 1. Schematic of a body orbiting a magnetic star. The unit vector ^ ezis aligned with the star's magnetic dipole moment\n~ \u0016, while the unit vector ^ ergives the direction from the center of the star to the orbiting body. Field lines and a sample \feld\nvector (~B, in blue) are also indicated to give a sense of the spatial distribution of the magnetic \feld. The curved lines with\narrows (in purple) show the default sense of stellar spin and the motion of the body. Unless otherwise speci\fed, we assume that\nthe stellar spin and the angular moment of the orbiting body are aligned, as shown.\n2.ORBITAL DYNAMICS\nTo calculate orbital solutions for a body in motion around a magnetized star, we begin by assuming that the stellar\nmagnetic \feld is a dipole, characterized completely by its magnetic moment,1\n~ m?=4\u0019B?R3\n?\n\u00160^ez; (1)\nwhereR?is the star's radius, B?is the surface \feld strength at the dipolar equator, \u00160is the permeability of free\nspace, and unit vector ^ ezspeci\fes the orientation of the dipole. The instantaneous magnetic \feld exterior to the star\nis\n~B(~ r) =B?R3\n?\nr3[3^er(^ez\u0001^er)\u0000^ez] (2)\nwhere~ ris a position relative to the star's center of mass, while ^ eris the unit vector in the direction of ~ r. When\nthe magnetic moment is not aligned with the star's spin axis, it varies in time as the star rotates; the unit vector ^ ez\nchanges accordingly. For simplicity, and unless otherwise stated, we assume throughout that ~ rrefers to the orbiting\nbody's location and that it lies in a plane perpendicular to the star's spin axis. Figure 1 illustrates this geometry.\nA magnetized or conducting body orbiting through the stellar magnetic \feld feels a magnetic force. We work in the\nlimit where the body's size is small compared with the spatial variations in the stellar magnetic \feld. If the `body' is\na stellar companion, then our assumption is that the stars are separated by a distance that is large compared with the\nsecondary's radius. As with the host star, we assume that the body's magnetic properties are encoded by its magnetic\nmoment,~ mmag, the dipole moment of \feld that the body itself generates, either because of its own material properties\n1Throughout, we adopt the SI system for Maxwell's equations and derived quantities, following Bidinosti et al. (2007). When providing\ncharacteristic values of observed quantities, we use a mix of units to match typical uses in the literature.4\nor in response to the stellar \feld. Then, the instantaneous magnetic force that the body feels as it orbits the star is\n~FB=~r\u0010\n~ mmag\u0001~B\u0011\n=~ mmag\u0001~r~B (3)\nwhere the right-most equation treats the magnetic moment as independent of position, and ~r~B, with (~r~B)ij\u0011\ndBi=dxj, is the Jacobian of the stellar magnetic \feld. In a standard ( x;y;z ) rectilinear coordinate system with the\nz-axis tied to the stellar dipole moment,\n~B(~ r) =B?R3\n?\nr5[3xz^ex+ 3yz^ey+ (3z2\u0000r2)^ez] (4)\nwhere ^eiare unit vectors so that position vector ~ r=x^ex+y^ey+z^ez. Then, the Jacobian is\n~r~B= 3B?R3\n?\nr72\n64zr2\u00005x2z\u00005xyz xr2\u00005xz2\n\u00005xyz zr2\u00005y2z yr2\u00005yz2\nxr2\u00005xz2yr2\u00005yz23zr2\u00005z33\n75: (5)\nIf we can determine the magnetic moment of an orbiting body, we can calculate the instantaneous magnetic force on\nit, and thus derive an orbit solution. Because of our choice of simple orbital geometries, as well as magnetic moments\nof orbiters that are induced by the stellar \feld, we focus on this center-of-mass force and do not consider magnetic\ntorques.\nThe magnetic force in Equation (3) has a qualitatively simple description for an orbit in the plane perpendicular to\na \fxed stellar magnetic moment. Then, with z= 0, the Jacobian reduces to\n~r~B= 3B?R3\n?\nr4(^er^ez+ ^ez^er); (6)\nwhere dyadic notation is used to indicate the outer product of these unit vectors. If the magnetic moment of the\norbiting body is parallel or antiparallel to the stellar dipole, it feels a force in the radial direction from a gradient in\nthe strength of the stellar \feld. If the orbiting body's magnetic moment is radially directed, it experiences a vertical\nforce produced by a vertical \feld gradient associated with the curved \feld lines of the stellar dipole.\n2.1. Orbital stability\nTo derive how magnetic forces may impact orbits around a magnetic star, we assume an idealized con\fguration\nin which the orbiting body has a magnetic moment that is antiparallel to the stellar dipole and moves in the plane\nperpendicular to its magnetic moment vector. Then, the force on the body is purely radial, corresponding to a speci\fc\npotential energy\nUmag=\u00003\n3 +\r\u0018rB?R3\n?(Ms+M?)\nMsM?r3+\r; (7)\nwhereMsis the mass of the orbiter, and we have expressed the magnetic moment in the form\nmmag=\u0018rr\u0000\r; (8)\nthe power-law index \r= 0 corresponds to an orbiting body with a \fxed magnetic moment, while \r= 3 applies\nto a planet or asteroid whose magnetic moment is induced by the stellar magnetic \feld ( §2 and Appendix A). By\nincluding both the orbiter mass and the mass of the star, this potential could apply when the `orbiting body' is a\nstellar companion, perhaps the secondary partner in a magnetized binary system. The e\u000bective radial potential for\nreduced two-body motion is\nUe\u000b=L2\n2r2\u0000G(Ms+M?)\nr+Umag; (9)\nwhereLis the speci\fc angular momentum. The \frst derivative of Ue\u000bwith respect to r,\ndUe\u000b=dr=\u0000L2\nr3+3\u0018rB?R3\n?(Ms+M?)\nMsM?r4+\r+G(Ms+M?)\nr2; (10)5\nhas roots that correspond to circular orbits. The stability of these orbits is determined by the sign of the second\nderivative,\nd2Ue\u000b=dr2= 3L2\nr4\u00003(4 +\r)\u0018rB?R3\n?(Ms+M?)\nMsM?r5+\r\u00002G(Ms+M?)\nr3; (11)\nif this second derivative is negative, the orbit is unstable and will lead to inspiral. Setting both \frst and second\nderivatives of Ue\u000bto zero, we solve for the orbital distance that delimits stable and unstable circular orbits. This\nmagnetic `minimum stable circular orbit' is\nrmsco=\u00143(1 +\r)\u0018rB?R3\n?\nGMsM?\u00151=(2+\r)\n(12)\nUnsurprisingly, it will turn out that in astrophysical systems, the orbital radius rmscoformally takes on values that are\nmuch less than the physical radii of magnetic stars, so that no such minimum stable orbits exist around these objects\n(§4). Magnetically driven mergers are implausible.\n2.2. Orbit precession\nMagnetic forces on an asteroid, planet or stellar companion from a magnetic star are weak compared with gravity\n(§4). The instantaneous force in Equation (3) is a perturbation that has only a small impact on a body's orbit on\ndynamical time scales. However, it can lead to orbital precession and, for conducting bodies, a steady, long-term loss\nof energy that drives orbital evolution away from an orbit that corotates with the stellar magnetic \feld. Here, we\nassume the stellar \feld is \fxed in a frame tied to the star's rotation, although it need not be aligned with the spin\nangular momentum vector. Bourgoin et al. (2022) provide a more general analysis when both the orbiting body and\nthe stellar host have \fxed magnetic dipole moments.\nDespite the comparative weakness of magnetic forces in orbital dynamics, they contribute to secular evolution. For\nexample, the time-averaged potential associated with the magnetic interaction (e.g., Eq (7)), yields apsidal and nodal\nprecession rates, respectively,\n_$\u0019\u0000\u00141\n2\nr2d\ndr\u0012\nr2dUmag\ndr\u0013\f\f\f\f\nr=rcirc;z=0(13)\n_\u0013\u0019\u00141\n2\nzdUmag\ndz\f\f\f\f\nr=rcirc;z=0: (14)\nwherercircis the semimajor axis of the orbit and \n is the orbital frequency. When the magnetic dipoles are antiparallel\nto each other and oriented nearly perpendicular to the orbital plane, we \fnd\n_$\u00193(2 +\r)\n2p\nG\u0018rB?R3\n?pMs+M?\nMsM?r7=2+\r(15)\n_\u0013\u0019\u00003p\nG\u0018rB?R3\n?pMs+M?\nMsM?r7=2+\r: (16)\nWhen the orbiter's magnetic moment is \fxed ( \r= 0), the apsidal and nodal precession rates are equal in magnitude in\nthe limit of small eccentricity (cf. Bourgoin et al. 2022). If the orbiter's magnetic moment derives from magnetization\nor induction in the stellar \feld ( \r= 3), the nodal precession rate is slower than the apsidal precession rate.\n2.3. Loss or gain of orbital energy\nIf the orbiting body is a conductor, then magnetic interactions can also drive changes to the orbital energy as\nre\rected by its semimajor axis. Induced currents in a conducting asteroid or planet will slowly but steadily dissipate\nenergy from the system through Ohmic heating. Around a non-rotating star with a static magnetic \feld, the conductor\nwill steadily lose orbital energy and inspiral toward the star; if the star and its magnetic \feld rotate faster and in the\nsame sense as the conductor, the conductor's orbit will tend to push outward from the star (e.g., Laine et al. 2008;\nLaine & Lin 2012; Kislyakova et al. 2018; Kislyakova & Noack 2020, see also Bromley & Kenyon 2019).6\nThe secular change in the orbital energy of a conductor around a magnetic star is determined by the time-averaged\nmechanical power,\nPmech =\u0000D\n(~ mmag\u0001~r~B)\u0001~ vE\n; (17)\nwhere~ vis the conductor's velocity as measured in a reference frame tied to the star's magnetic dipole moment. While\nthe mechanical power accounts for the loss of kinetic energy in this frame, the conducting body will gain or lose orbital\nenergy. Equation (17) corresponds to energy that is converted to heat through Ohmic dissipation of currents induced\nby the stellar magnetic \feld. Thus, we can estimate the rate of change in orbital energy directly from the Ohmic\nheating rate,\nP\n=1\n2Z\nVdVj~Jj2\n\u001b(18)\nwherej~Jjis the maximum amplitude of the current density, \u001bis the conductivity, and the integral is over the volume\nof the conductor.\nAlternatively, the average power dissipated through Ohmic heating may be determined directly from the conductor's\nmagnetic moment. For example, if the stellar magnetic \feld (or one component of it) has an amplitude of jBj, and\nthe frequency of \feld variation experienced by the conductor from orbital motion and stellar rotation is !, then the\npower loss is\nP\n=1\n2!jBj=fmmagg; (19)\nwhere the complex magnetic moment magnitude keeps track of the phase di\u000berence between the induced currents and\nthe \feld oscillations (Appendix A). A purely real-valued magnetic moment is in phase with the \feld oscillations; the\naverage magnetic force then vanishes. A purely imaginary magnetic moment is 90\u000eout of phase with the \feld; then,\nthe magnetic force acts only in one direction, either opposing the orbital motion or aligned with it. In this formalism,\nthe magnetic force is just the real part of Equation (3). Following Laine et al. (2008), the sign of !determines whether\nthe body gains or loses orbital energy. For example, if the orbiting body overtakes slowly corotating stellar magnetic\n\feld lines,! >0, which generally leads to a loss of orbital energy. Rapidly corotating \feld lines sweep up the body,\nand!<0.\nEquation (19) is essentially the time derivative of the dipole's magnetic energy corresponding to a mode oscillating\nat frequency !. In the context of orbital dynamics, !is the synodic frequency of motion as the conducting body orbits\nthrough the stellar host's (possibly) rotating magnetic \feld. Harmonics of this frequency may also contribute to the\ntotal power loss.\nOrbital energy changes from Ohmic heating translates to a rate of change in orbital distance,\n_rcirc\nrcirc\u00192rcircP\nGMsM?: (20)\nwhere the power P\nis negative if a body is plowing through a static or slowly rotating magnetic \feld, and positive if\norbital energy is gained at the expense of the magnetic \feld's rotation (the Ohmic loss occurs in that rotating frame).\nEccentricity can also change; energy is transferred between the \feld and the conductor most strongly at periastron,\nleading to circularization or eccentricity pumping depending on whether the orbital period is longer than the star's\nrotation period.\nNon-conducting bodies, even if strongly magnetized, do not experience this type of orbital energy exchange. There\nare interesting potential dynamical e\u000bects nonetheless. We explore one example next.\n2.4. Resonance trapping\nAs a \fnal example of the impact of magnetic interactions on orbiting bodies, we note that the orbiter experiences\nan oscillatory driving force depending on the orbital motion relative to the rotation of the star. The frequency of the\n\feld variation in the orbit frame is !\u0011\n?\u0000\n, where \n ?is the stellar spin rate and \n is the orbital frequency; for\na planet or asteroid with an induced magnetic moment in a tidally locked frame, the driving force has a frequency of\n2!. A resonance occurs when\n\n?\u0019(\n(1\u00001=2n)\n (\n>\n?)\n(1 +n=2)\n (\n\u0014\n?)(n= 1;2;3;:::): (21)7\nThe precise resonance condition will be a\u000bected by apsidal precession (Eq (13)). When an object experiences slow\nchanges in orbital elements, speci\fcally semimajor axis, it can get trapped in a resonance, as in planetary systems\n(e.g., Malhotra 1996; Wyatt 2003) or galactic dynamics (e.g., Moreno et al. 2015). While the trapping is weak, it is\nplausible, as we demonstrate below for the case of a slowly varying stellar rotation period ( §4). The phenomenon leads\nto changes in semimajor axis, but also eccentricity pumping and the possibility of capture by the star.\nAll of the phenomena just described depend on the nature of the magnetic moments generated in the asteroids,\nplanets or stellar companions of a magnetic star. We consider this topic next.\n3.MAGNETIC MOMENTS\nMagnetic interactions between a star and an orbiting companion depend on the star's magnetic \feld and the magnetic\nmoment of the orbiting companion. In one simple scenario, the orbiter has a \fxed, `fossil' \feld with a magnetic moment\nthat is independent of the orbit. Alternatively, the star's \feld can induce magnetization (remanence) if the companion\nis composed of magnetic material. Finally, temporal variations in the local magnetic \feld from orbital motion and/or\nstellar rotation induce free currents that generate a magnetic moment in the companion. In this section, we examine\nall of these cases.\n3.1. An orbiting magnetic body\nThe orbiting body's magnetic moment encodes material properties, its fossil magnetization, and its response to the\nstellar magnetic \feld. In one simple case, when the orbiting body is a \fxed, permanent magnet, the magnetic moment,\n~ mmag, is constant in time. If the orbiter is a companion star with a fossil \feld, its magnetic moment may be speci\fed\nas in Equation (1). Other sources of the magnetic \feld are possible, too; younger stars in particular may have strong\ndynamos, e.g., Johnstone et al. 2021, and planetary \felds also may have a dynamical origin (e.g. Christensen 2010).\nIf the orbiter behaves as a permanent or hard magnet composed of ferromagnetic material with high coercivity, then\nthe magnetic moment has a strength of\n~ mmag=4\u0019R3\n3\u00160~Brem (permanent) ; (22)\nwhereRis the magnet's radius, and ~Bremis the \\remanant\" (or residual) \feld. For ferrite, the remanence can reach\nabout 3000 G. For comparison, a magnetic white dwarf has the equivalent of a residual \feld with a strength as high\nas 109G.\nWhen the body is a \\magnetizable\" low-conductivity, soft magnet, its magnetic moment depends on the local value\nof the stellar magnetic \feld. In a linear material, the response to an external stellar \feld is characterized by its\nmagnetic permeability, \u0016. If the stellar \feld penetrates into the bulk material unperturbed by induced currents, then\nthe magnetic moment is\n~ mmag=\u00004\u0019B?\n\u00160R3\n?\nr3R3\u0016\u0000\u00160\n\u0016+ 2\u00160^ez(magnetizable) ; (23)\nwhere thez-axis aligns with the stellar dipole. The physics is familiar from magnetostatics; bathed in an external\n\feld, a magnetic body will generate its own \feld in the same direction as the external one. This \feld is associated with\nbound currents that conceptually represent the underlying molecular or sub-atomic response to the external \feld.\nIn these examples, the magnetic moment of an orbiting body is either speci\fed independently of the stellar magnetic\n\feld, or is derived from the local value of that \feld. In either case, instantaneous forces are straightforward to calculate\nin an orbit integration code. When the orbiter is also a conductor, its magnetic moment can be more complicated to\nderive. We consider this situation next.\n3.2. An orbiting conductor\nThe magnetic moment of a conducting body is generated by currents induced as the orbiting body experiences\ntemporal variations in the stellar magnetic \feld. These variations come from orbital motion or rotation of the stellar\nmagnetic moment, or a combination of both. Toward obtaining orbit solutions, we typically assume that the force of8\ngravity dominates the orbital dynamics. Then, osculating Keplerian orbital elements provide the conductor's trajectory\nthrough the stellar magnetic \feld; the \feld variations it experiences along its orbit and Faraday's law of induction\ndetermine the induced currents and the magnetic moment associated with them.\nTo derive the magnetic moment, we assume that this body is perfectly spherical with radius R, and that its bulk\nproperties, including mass density \u001a, conductivity \u001b, and magnetic permeability \u0016, vary in the radial direction only.\nAs this sphere orbits the star, the stellar magnetic \feld it travels through varies in time t; we focus here on a single\noscillatory mode with frequency !, so that the magnetic \feld at the orbiter's location is ~B0exp(\u0000i!t), where~B0is\na constant vector that is approximately uniform over the volume of the conductor. We work in a regime where !is\nsmall enough that the wavelength of light propagating through the sphere at this frequency is larger than the sphere\nitself.\nThe induced current density and the stellar magnetic \feld are described by Faraday's law and Amp\u0012 ere's law,\n~r\u0002~J\n\u001b=\u0000i!~B;~r\u0002~B\n\u0016=~J; (24)\nrespectively, where we have adopted Ohm's law to write the electric \feld in Faraday's law in terms of current density\n(~E=~J=\u001b). These two equations, combined with the curl of either one, along with boundary conditions at the\nconducting surface and a regularity condition at the conductor's center, lead to a solution for the current density and\nthe magnetic \feld. Appendix A provides details (see Joss et al. 1979; Bidinosti et al. 2007; Laine et al. 2008; Gi\u000en\net al. 2010; Chang et al. 2012; Kislyakova et al. 2017; Bromley & Kenyon 2019, for example).\nThe solutions for ~Jshow trends that depend on the magnetic Reynolds number, de\fned as\nRm\u00112R2=\u000e2=\u0016\u001b!R2; (25)\nwhere\u000eis the 'skin depth', which characterizes how deep the stellar magnetic \feld permeates into a spherical conductor.\nIfRmis much less than unity, so that \u000e\u001dR, then the conductor is bathing in an unperturbed external \feld; weak\neddy currents form throughout. If Rmis large compared with \u00162=\u00162\n0, the magnetic \feld can only penetrate into a thin\nlayer of depth \u000e\u001cRon the conductor's surface, where strong currents are generated (the 'skin e\u000bect'). In a magnetic\nconductor, \u0016=\u0016 0>1, when 1 .Rm.\u00162=\u00162\n0, induced free currents are con\fned to the surface of the conductor, but\nthe magnetic \feld penetrates into the sphere's interior, supported by bound currents (Bromley & Kenyon 2019).\nFigure 2 provides an illustration. It shows examples of numerical calculations of the current density for homogeneous\nconductors, as well as for a sphere with a low-conductivity shell around a highly conductive core.\nThe magnetic moment of a conducting body follows directly from the current density:\n~ mmag=Z\ndV\u00141\n2~ r\u0002~J+(\u0016\u0000\u00160)\n\u0016~B\u0015\n(26)\nwhere~Jis the current density and ~B\u0018~r\u0002~J=\u001b is the magnetic \feld within the conductor. For a homogeneous\nmedium, Bidinosti et al. (2007) give\nmmag=2\u0019R3B0\n\u001602(\u0016\u0000\u00160)j0(kR) + (2\u0016+\u00160)j2(kR)\n(\u0016+ 2\u00160)j0(kR) + (\u0016\u0000\u00160)j2(kR); (27)\nwherek=pi\u0016\u001b! whilej0andj2are the zeroth- and second-order spherical Bessel functions. In the limits of low and\nhigh magnetic Reynolds number, the magnetic moment is\nj~ mmagj=8\n<\n:4\u0019B0R3\n\u0016+2\u00160\f\f\f\u0016\u0000\u00160\n\u00160+3i\u00162\u001b!R2\n10(\u0016+2\u00160)\f\f\f(Rm\u001c10)\n2\u0019B0R3\n\u00160(Rm\u001d10\u00162=\u00162\n0):(28)\nwhile the magnetic force (Eq. 3), formally becomes\n~FB=~B0\u0001\u0010\n~r~B\u0011\n\u00028\n<\n:2\u0019i\u001b!R5=15 (Rm\u001c10 and\u0016\u0019\u00160)\n\u00002\u0019R3=\u00160 (Rm\u001d10\u00162=\u00162\n0):(29)9\n0.0 0.2 0.4 0.6 0.8 1.0\nr (km)106\n104\n102\n100102| J | (A/m2)low , (1,1)\nhigh , (103,105)\nlow + high , core\nFigure 2. The peak magnitude of the current density as a function of radius inside kilometer-sized spheres of various materials\nas determined from numerical calculations. The blue-green curve corresponds to non-magnetic, low-conductivity material in a\njB0j= 1 T \feld varying sinusoidally on a 10-day cycle. A sphere made of magnetic, high-conductivity material is indicated in\npurple. These curves are superimposed on light gray curves from the theoretical predictions of Bidinosti et al. (2007). The legend\nshows the permeability in units of \u00160and the conductivity \u001bis in units of S/m. The remaining curve, in black, corresponds\nto a sphere with an outer shell of the low conductivity material surrounding an inner core of high conductivity and magnetic\npermeability. To avoid numerical errors, we smoothly transition between high and low values of \u0016and\u001bin a radial zone of\n0.1 km in thickness that separates these two materials.\nwhere the upper equation applies to non-magnetic material. The transition between these low and high Reynolds\nnumber regimes occurs near Rm\u001910 for non-magnetic material, while for magnetic material, there is an intermediate\nregime that extends from Rm\u001810 toRm\u001810\u00162=\u00162\n0, for which the magnet moment originates from a mixture of free\nand bound currents.\nThe power, likewise, has characteristic behavior in each of these regimes. From Equation (19),\nP\n\u00198\n>><\n>>:3\u0019B2\n0f\u00162\u001b!2R5=5(\u0016+ 2\u00160)2Rm\u001c10\n2\u0019B2\n0p\n\u001b!3=\u0016R410.Rm.10\u00162=\u00162\n0\n3\u0019B2\n0p\n\u0016!=2\u001bR2Rm\u001d10\u00162=\u00162\n0:(30)\nThe top and bottom expressions are exact in the limits shown, while the middle expression is only an approximate\nscaling relation. The bottom expression corrects a missing factor of 1 =p\n2 in Bromley & Kenyon (2019, Eq. (9)\ntherein).\nFigure 3 provides an illustration of the magnetic moment of homogenous spherical conducting bodies as a function\nof Reynolds number. It shows examples of non-magnetic and magnetic bodies. The power dissipation in Ohmic heat10\n1001021041061081010\nReynolds Number102\n101\n100Magnetic Moment ×0/2B0R3 \n/0=1\n/0=1000\nFigure 3. The magnetic moment of homogeneous conducting spheres in a sinusoidally varying magnetic \feld as a function of\nmagnetic Reynolds number. The magnetic moment is scaled to a dimensionless quantity (e.g., Eq. (28)). The gold curves are\nfor a non-magnetic sphere, while the blue curves correspond to a sphere made of magnetic material. In each case, the line types\ndistinguish the magnitude of the magnetic moment (solid curves) from the real (dashed) and imaginary (dotted) parts. Note\nthat the real part is negative for the non-magnetic sphere, while for the magnetic sphere, it is positive at low Reynolds (below\nRm\u0018\u00162=\u00162\n0\u0018106) and negative at high Reynolds number.\nis gleaned from the imaginary part of the each curve (Eq. (19)). Figure 4 gives examples of the instantaneous forces\nexperienced by theses bodies.\n3.3. Comparison with motional EMF\nA conducting planet traveling through a stellar magnetic \feld experiences a charge separation from the Lorentz\nforce independently of any induced currents. In a reference frame that moves with the planet, the stellar magnetic\n\feld lines \row past with some velocity ~ vrel, producing an electric \feld as a result of this motion:\n~E=\r~ vrel\u0002~B; (31)\nwhere the Lorentz factor \rmay be ignored in the context of planetary orbital dynamics. Assuming an instantaneous\nresponse from charges within the conducting planet's interior, the planet is polarized with electric dipole moment\n~ p= 4\u0019\u000f0R3~E: (32)11\n1001021041061081010\nReynolds Number106\n105\n104\n103\n102\n101\nAcceleration (cm/s2)\n/0=1\n/0=1000\nFigure 4. The acceleration of a conducting sphere traveling through a strong, spatially varying magnetic \feld as a function\nof magnetic Reynolds number. The \feld gradient exists only in the direction of travel. The Lorentz force is sinusoidal, with\nan average force that opposes the motion through the \feld, consistent with the power dissipated by Ohmic heating. The gold\ncurves show the acceleration of a non-magnetic sphere with constant conductivity; the solid curve is the average acceleration that\nopposed the direction of travel. The instantaneous acceleration has an oscillating part from the interplay between the induced\nmagnetic moment of the conductor and the magnetic \feld. The dotted line shows the maximum value of the instantaneous\nacceleration. The blue curves correspond to a magnetic sphere with \u0016= 1000\u00160.\nThe force on the conductor then is approximately\n~FE=~ p\u0001~r~E (33)\n= 4\u0019\u000f0R3\u0010\n~ vrel\u0002~B\u0011\n\u0001\u0010\n~ vrel\u0002~r~B\u0011\n(34)\n= 4\u0019\u000f0R3h\nv2\nrel~B\u0000(~ vrel\u0001~B)~ vreli\n\u0001~r~B (35)\nTo roughly compare this motional electromotive force (EMF) with the induction force, we use the ratio of magnitudes,\nFE\nFB\u0018\u000f0R3v2\nrel\u0014B2\nR3\u0014B2=\u00160=v2\nrel\u00160\u000f0=v2\nrel\nc2(\u0016\u001d\u00160orRm\u001d\u00162=\u00162\n0) (36)\n\u0018\u000f0R3v2\nrel\u0014B2\nR5\u001bvrel\u00142B2\u0018v2\nrel\u000f0\u00160\nR2\u00160\u001bv\u0014=v2\nrel\nc2Rm(Rm\u001c10 and\u0016\u0019\u00160); (37)\nwhere\u0014is a wave number that characterizes the \feld variations along the planet's trajectory. For example, if the planet\nis on a circular, polar orbit with respect to a static stellar magnetic dipole, \u0014\u00181=R. The product vrel\u0014thus represents\nthe frequency !of the \feld in the planet's frame of reference. This comparison suggests that electric polarization is\ncomparatively small when the magnetic Reynolds number is high, but that the polarization force can dominate over12\nthe magnetic force when Rmis low, as with bodies that are small or made of non-magnetic, insulating material. We\nfocus here only on scenarios where magnetic forces are more signi\fcant than those from motional EMF.\n4.ASTROPHYSICAL EXAMPLES\nTo apply the results of the previous sections, we consider three astrophysical examples: (i) bodies with fossil magnetic\n\felds, (ii) those that are magnetizable, as in soft-iron spheres, and (iii) conducting asteroids and planets. Our goal is\nnot to provide an exhaustive analysis, but to understand whether magnetic interactions plausibly lead to observable\nphenomena. We begin with fossil magnetic \felds, focusing on compact, magnetized binary stars.\n4.1. A compact stellar binary with fossil \felds\nMagnetic interactions between compact partners in a close binary pair may produce measurable dynamic e\u000bects\nduring a merger event or even earlier (Bourgoin et al. 2022). To assess the impact of magnetic interactions on orbits\nwell before merging, we assume the partners have 'fossil' (permanent) magnetic dipole moments oriented antiparallel\nto each other and perpendicular to the orbital plane. Then, from the formalism in §2, the force experienced by one\npartner (labeled 'a') by the other star (labeled 'b') is\n~FB=~ \u0016a\u0001~r~Bb (38)\n=\u000012\u0019BaBbR3\naR3\nb\n\u00160r4\u0014\u0012\n1\u00005z2\nr2\u0013\n^er+ 2z\nr^ez\u0015\n(~ \u0016ak\u0000~ \u0016b) (39)\nwhereris the binary separation, and the positive z-axis is in the direction of ~ \u0016b, the magnetic moment of star 'b'.\nWhenz= 0, the binary orbit is strictly in the plane perpendicular to the stellar magnetic moments.\nThis force law leads to the possibility of an inspiral orbit. Following the analysis in §2.1, adapted for a pair of\nidentical partners | white dwarfs or neutron stars | the magnetic minimum stable circular orbit is\nrmsco= 2\u00143\u0019\n\u00160GM2?\u00151=2B?R3\n?\nM?(40)\nWith typical radii and masses of white dwarfs and neutron stars, along with surface \feld strengths near the maximum\nobserved values ( R?;M?;B?) = (1 R\b;1 M\f;109G) and (10 km ;2 M\f;1015G) respectively, magnetic minimum stable\ncircular orbits have values that are roughly 0.1% of the stars' physical radii, formally placing these orbits deep within\nthe stellar interiors. Even in these extreme astrophysical systems the \feld strengths are a few orders of magnitude too\nsmall to drive mergers.\nAt observed \feld strengths and orbital con\fgurations, apsidal and nodal precession are potentially measurable.\nWhen the magnetic dipoles are perpendicular to the orbital plane and antiparallel to each other, apsidal and nodal\nprecession rates are\n_$=\u0000_\u0013\u001921=212\u0019B2\n?R6\n?\n\u00160G1=2M3=2\n?r7=2; (41)\nwhich can yield precession rates for identical white dwarfs (WDs) or neutron stars in close binaries that are signi\fcant\non dynamical time scales if the orbital separations are O(100) stellar radii or less. For example, with orbital and stellar\nparameters for the eclipsing WD-WD binary ZTF J153932.16+502738.8 (Burdge et al. 2019),\n_$=\u0000_\u0013\u00195:3\u0014Ba\n108G\u0015\u0014Bb\n108G\u0015\ndeg/yr (42)\nAlthough the apsidal precession rate can be formally high, tidal forces and gravitational radiation circularize the\nbinary orbit, preventing detection. Nodal precession, on the other hand, may occur if the fossil \felds are not exactly\nantiparallel or are not strictly perpendicular to the binary's orbital plane. The rate of precession is expected to be\ncomparable the value in Equation (42). Light curves monitored over a period of time might provide novel constraints\non the stellar magnetic \felds.13\n4.2. A magnetic body orbiting in the midplane of a \fxed stellar dipole\nNext, we consider a magnetizable asteroid or planet ( \u0016\u001d\u00160) orbiting in the midplane of the stellar dipole. If either\nthe sphere's radius or its conductivity are small, the Reynolds number is low and the magnetic \feld permeates the\nsphere. Then, the magnetic moment of the sphere is given by Equation (23). Close-in orbits are of most interest, since\nthe magnetic force falls o\u000b steeply with orbital distance ( r\u00007). The Roche radius gives an approximate lower limit to\nthis distance; it is approximately\nRRoche =K\u0012M?\n\u001a\u00131=3\n\u00190:85\u0014K\n0:8\u0015\u0014M?\n1 M\f\u00151=3\u0014\u001a\n5 g/cm3\u0015\u00001=3\nR\f (43)\nwhere\u001ais the orbiting body's mass density and Kis a constant near unity that depends on the material properties\n(e.g., Veras et al. 2017). Within this distance, tidal forces are destructive.\nApplying the stability analysis of §2.1, we equate the Roche radius to the magnetic minimum stable circular orbit\nradiusrmsco to \fnd the stellar magnetic \feld strength needed to destabilize circular orbits. We start with the real\npart of the magnetic moment in Equation (28), applicable to a magnetizable, low-conductivity body. Then, we use\n\u0018r=\u0016r\r, where\r= 3 in this case, in Equation (40). Finally, setting rmsco =RRoche , we can solve for the required\nmagnetic \feld:\nB?&R5=2\nRoche\n6R3?\u0014\u00160\u001aM?(\u0016+ 2\u00160)\n(\u0016\u0000\u00160)\u00151=2\n(44)\n\u0019K5=2M4=3\n?\n6\u001a1=3R3?\u0014\u00160(\u0016+ 2\u00160)\n(\u0016\u0000\u00160)\u00151=2\n: (45)\nFor magnetic permeability signi\fcantly above unity, as for most ferromagnetic materials, the minimum destabilizing\n\feld is\nB?&4:7\u0002106\u0014K\n0:8\u00155=2\u0014\u001a\n5 g/cm3\u0015\u00001=3\u0014M?\n1 M\f\u00154=3\u0014R?\n2 R\f\u0015\u00003\nG: (46)\nThis limit is two orders of magnitude higher than \feld strengths of \u00182 kG observed in T Tauri stars, to which the\n\fducial values of other parameters apply (e.g., Villebrun et al. 2019). Similar assessments for white dwarfs indicate\nthat their dipole \felds are also too weak by orders of magnitude to destabilize circular orbits of magnetized material\naround them. Increasing the material strength of the magnetic material, thereby reducing its Roche radius, lowers\nthe requirement on magnetic \feld strength. Yet even a decrease in the Roche radius by a factor of 10 demands a \feld\nstrength that is an order of magnitude higher than the most extreme values observed on white dwarfs. Neutron stars,\neven magnetars, have \feld strengths that are also too weak to destabilize these orbits. Finally, unstable orbits would\nhave to be so close to the stellar host that it would be too hot to retain ferromagnetic properties.\nOrbital precession, on the other hand, may occur at realistic \feld strengths. From §2.2, we \fnd apsidal and nodal\nprecession rates of\n_$\u0019\u00002:5_\u0013\u001945B2\n?R6\n?\n2\u001aG1=2M1=2\n?r13=2\u0016\u0000\u00160\n\u0016+ 2\u00160(47)\n\u00193:0\u0014\u001a\n5 g/cm3\u0015\u00001\u0014M?\n1 M\f\u00151=2\u0014B?\n3 kG\u00152\u0014R?\n2 R\f\u00156hr\n0:01 aui13=2\ndeg/yr (T Tauri) (48)\n\u00194:4\u0014\u001a\n5 g/cm3\u0015\u00001\u0014M?\n1 M\f\u00151=2\u0014B?\n109G\u00152\u0014R?\n1 R\b\u00156\u0014r\n0:2 R\f\u001513=2\ndeg/yr (WD) (49)\nwhere the middle equation is for a body on a surface-skimming orbit around a T Tauri star and the bottom equation\nis for an orbit around a white dwarf. In both cases, \u0016\u001d\u00160. The \fducial values for the white dwarf are extreme; the\nmagnetic \feld strength is near the maximum observed value, and the orbital distance is well below the Roche radius\nin Equation (43), indicating that the magnet must be small ( .100 km) so that it is held together not by self-gravity\nbut by its own material strength (see Brouwers et al. 2022, and references therein). Under these conditions orbital\nprecession is formally signi\fcant. These same conditions will also likely a\u000bect the orbiter's magnetic properties; high14\nequilibrium temperatures near the stellar host will reduce the magnetic permeability. While a magnetic asteroid may\nbe able to maintain internal magnetization when cool, unlike larger bodies with internal heating (see Kislyakova &\nNoack 2020, for example), it is unlikely to remain solid or magnetic while occupying an orbit so close to the stellar\nhost.\n4.3. Orbital dynamics at high Reynolds number: a conductor in a spinning stellar dipole\nWe next consider the high Reynolds number limit, where induced currents and the Lorentz force play a role in the\norbital dynamics of a conducting body. To explore this scenario, we assume that a rotating star has a magnetic dipole\nmoment at right angles to its spin axis, so that the moment vector rotates in a plane. We further assume that a\nconducting planet is on a low-eccentricity orbit in that same plane. When the planet is on a circular orbit, the planet's\nposition as a function of time tis\n~ r=rcirc[cos(\nt)^ex+ sin(\nt)^ey]; (50)\nwherercircis the mean orbital distance of the planet. At that distance, the stellar magnetic moment evolves according\nto\n~ m?=4\u0019B?R3\n?\n\u00160[cos(\n?t)^ex+ sin(\n?t)^ey] ; (51)\nin these expressions, both the stellar magnetic moment and the planet's orbit lie in the x\u0000yplane. At the planet's\nlocation, the magnetic \feld is\n~B\fx(t) =B?R3\n?\nr3\ncircf3 cos(!t) [cos(\nt)^ex+ sin(\nt)^ey]\u0000[cos(\n?t)^ex+ sin(\n?t)^ey]g (52)\n=B?R3\n?\n2r3\ncircf[3 cos(2\nt\u0000\n?t) + cos(\n ?t)] ^ex+ [3 sin(2\nt\u0000\n?t) + sin(\n?t)] ^eyg; (53)\nwhere!\u0011\n?\u0000\n. We next measure the magnetic \feld at the planet's location when the planet is tidally locked to\nthe star. We adopt a set of basis vectors that include ^ er(see Eq. (50)), a unit vector directed radially outward from\nthe host star, along with the unit vector\n^ev=\u0000sin(\nt)^ex+ cos(\nt)^ey; (54)\nthat is aligned with the planet's instantaneous direction of travel in the stellar rest frame. The local stellar magnetic\n\feld measured in this frame is\n~Block(t) = (~B\u0001^er)^er+ (~B\u0001^ev)^ev (55)\n=B?R3\n?\nr3\ncirc[2 cos(!t)^er\u0000sin(!t)^ev]: (56)\nIn the planet's tidally-locked frame, there are again two modes of oscillation, although now with the same frequency.\nA similar analysis yields the Jacobian of the \feld in the orbital plane:\n~r~Block=B?R3\n?\nr4\ncirc[\u00006 cos(!t)^er^er+ 2 sin(!t)^er^ev+ 3 sin(!t)^ev^er+ cos(!t)^ev^ev]lock; (57)\nwhere the subscript reminds that this expression applies in the tidally-locked frame.\nFinally, the induced magnetic dipole moment in the high-Reynolds number limit (Eq. A13) is real-valued (in phase\nwith the local stellar magnetic \feld), so that the acceleration experienced by the planet is\n~ aB=3B2\n?R6\n?\n4\u00160\u001ar7\ncircf[9 cos(2!t) + 15] ^er\u00003 sin(2!t)^evg; (58)\nan expression which, we caution, is valid only when the planet experiences \feld variations rapid enough so that\nRm\u001d1. As in Equation (19), a purely real magnetic moment means negligible Ohmic losses. Thus, there is no net15\n0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00\nspin-orbit frequency ratio0.0960.0970.0980.0990.1000.1010.1020.1030.104orbital distance (R)\nFigure 5. Simulated minimum and maximum radial excursion distances of a conducting asteroid as a function of the spin-orbit\nfrequency (\n ?=\n). The star is a 0.5 M \f, 1.5 R \bwhite dwarf with a 109G magnetic \feld that is oriented perpendicularly to\nthe star's spin axis. The asteroid (1 km radius, density of 5 g/cm3) is placed on an initially circular orbit at 0 :1 R\fin the\nplane of the rotating stellar dipole moment. The asteroid's orbit acquires some small eccentricity over hundreds of orbits, as\nillustrated by the minimum and maximum radial excursions indicated by pairs of black and gray points, vertically displaced\nfrom one another. Each pair corresponds to a single orbit integration. At these orbital distances, the eccentricty \\pumping\" is\nweak, and occurs only near a spin-orbit resonance ( \n ?=\n\u0019(1=2;1;3=2;2).\nchange of orbital energy. Furthermore, with a vanishing time-averaged acceleration, orbital precession from secular\ntheory is insigni\fcant.\nThe oscillatory driving force, with a magnitude comparable to that experienced by a ferromagnet, has dynamical\nconsequences nonetheless. To demonstrate, we adapted Orchestra , a hybrid n-body{coagulation code for planet\nformation (Bromley & Kenyon 2006, 2011; Kenyon & Bromley 2012, 2014), to include magnetic interactions. The\nnewOrchestra code integrates the equations of motion for a conducting planet, initially on a circular orbit at two\nstellar radii from its magnetic host, over 250 orbits. In a suite of trials, the spin rate of the star is varied, and we\nestimate the planet's radial excursions (e.g., Sutherland & Kratter 2019) as a function of the spin rate of the star.\nFigure 5 illustrates the outcome in a plot of the minimum and maximum radial excursions versus the ratio of the\nstar's spin frequency (\n ?) to the orbital frequency (\n). We \fnd that magnetic interactions can pump up the orbital\neccentricity, depending on the ratio frequencies of stellar spin and orbital rotation. The radial excursions are broadest\nnear resonances (\n ?: \n = 1:2, 1:1, 3:2, 2:1).\nFigure 6 provides a second illustration of how magnetic interactions can impact orbits. It shows results from\nsimulations of a conducting asteroid on a highly eccentric orbit about a white dwarf, with apoastron at 1 R \f, and\nperiastron at 3 R \b(two stellar radii). In some simulations, near spin-orbit resonances, the periastron distance evolves,\nbringing the asteroid close to the stellar surface.16\n0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00\nspin-orbit frequency ratio0.0140.0160.0180.0200.0220.0240.0260.028orbital distance (R)\nFigure 6. Simulated minimum radial excursion distances of a conducting asteroid as a function of the spin-orbit frequency\n(\n?=\n). The asteroid and host star have the same physical parameters as in the previous plot, Figure 5. Here, the asteroid is\ninitially on a highly eccentric orbit with periastron at two stellar radii and apoastron at 1 R \f. The eccentricity pumping near\nspin-orbit resonances is su\u000ecient to signi\fcantly reduce the periastron distance, allowing a close approach to the stellar surface\n(the horizontal line at the bottom of the plot).\nFor an asteroid on an eccentric orbit, tidal dissipation could play an important role in orbital evolution. However,\nwith peak losses only near closest approach to the star, the asteroid would tend to circularize near its original periastron\ndistance over centuries, much longer than our simulation time. For example, a 1% loss of orbital energy per orbit at\n0.5 au would mean that the asteroid would have to absorb O(1011) erg/g, which would vaporize it (assuming a speci\fc\nheat capacity of\u0018107erg/g/K). Our assumption here is that the asteroid is strong and rigid enough to survive each\nclose passage. In contrast to this secular evolution, the resonant e\u000bects from the magnetic interaction can draw the\nasteroid closer to the stellar host on dynamical time scales. Still, other in\ruences, including interactions with a gas\ndisk and collisions, may compete with or overwhelm the magnetic phenomenon described here (Brouwers et al. 2022).\nFigures 5 and 6 illustrate e\u000bects that would be di\u000ecult to \fnd in real astrophysical systems, even in the absence\nof other orbital perturbations. The host star must have a \feld strength that is near the peak of observed values and,\neven then, the conductor's orbit must be well within the formal Roche limit around the host. For a metallic object\nto survive at that location, we anticipate that the stellar host must be cool enough not to erode or evaporate the\nasteroid. Cool white dwarfs are known; an extreme example is DES J214756.46-403529.3 (Apps et al. 2021). Its Gaia\ncolors (G BP-GRP\u00182) suggest a surface temperature above 3000 K. Yet, if the asteroid lingered at two stellar radii,\nit would certainly melt, losing the material strength it would need to withstand the tidal forces there. The asteroid\ncould remain solid if it came close to the star only during periastron passage. Thus, the narrowest sliver of plausibility\nis allowed only by an unusually cold, highly magnetized white dwarf with a strong, conducting asteroid on a very\neccentric orbit.17\n0 20 40 60 80 100 120\ntime (years)0.200.250.300.350.400.450.500.55orbital distance (R)\nFigure 7. The radial excursions of a conducting asteroid trapped in a spin-orbit resonance. The white dwarf ( M?= 0:5 M\f,\nR?= 1:5 R\b, andB?= 109G) hosts an asteroid ( R= 1 km,\u001a= 5 g/cm3) on an initially circular orbit at 0 :2 R\f, as in\nFigure 5. This plot shows the evolution of the minimum and maximum radial excursions over a single orbit as the star's spin is\nslowly decreased. The spin-orbit frequency ratio is locked at 3:2 after the location of that resonance crosses the asteroid's initial\npath.\nSpin-orbit interactions can drive orbital evolution if a conducting asteroid or planet becomes trapped in a resonance.\nFor example, a young star with a strong magnetic \feld and rapid spin may experience spin-down, causing resonance\nlocations to sweep outward. Conducting bodies in the sweep zone may get caught and pushed outward. In simulations,\nwe \fnd they experience eccentricity pumping as well. Figure 7 provides an illustration of this phenomenon for an\nasteroid around a white dwarf. The spin-down rate is rapid | the star's spin drops by a factor of almost two in a\nmatter of a century. In astrophysical systems, inferred spin-down rates are much slower (e.g., de Jager et al. 1994;\nJohnstone et al. 2015, 2021), with magnetars being the possible exception (e.g., Rea et al. 2010). Our choice is for\nnumerical convenience only; we observed no lower limit to the spin-down rate for resonant trapping. On the contrary,\ntrapping appears more robust for slower rates. Simulated conductors fell out of the resonant trap when spin-down was\nmore rapid than in Figure 7.\n4.4. Orbital evolution of a terrestrial planet in a spinning stellar dipole\nIn this \fnal example, we consider a more realistic composition for the orbiting body, modelled after the planet\nMercury. Our hypothetical planet consists of a crust, mantle, inner core and outer core, each with unique conducting\nand magnetic properties as speci\fed in Table 1. We assume spherical symmetry overall and apply the algorithm in\nAppendix A to determine the complex magnetic moment, ~ mmag. This calculation includes an assumed orbital distance\nand fundamental frequency of the oscillating \feld at the location of the planet, != \nk\u0000\n?; Figure 8 shows the\nmagnetic moment for a range of frequencies. We also assume that the planet is tidally locked | consistent with a18\nTable 1. Internal structure of Mercury.\nouter radius thickness density conductivity permeability\ncrust 2440 km 90 km 2.8 g/cm30.001 S/m 1 \u00160\nmantle 2350 360 3.2 3.0 1\nouter core 1990 1030 7.4 5 :0\u00021051\ninner core 960 { 7.8 7 :8\u00021051000\nNote |The radial distances and densities are representative of the results presented\nby Genova et al. (2019).\nclose-in orbit where the magnetic \feld of the star is strong | so that it has \fxed orientation in the rotating reference\nframe of the star.2Then, the planet's magnetic interactions involve two orthogonal magnetic \feld components, one\ndirected radially outward from the star, and the other perpendicular to it in the orbital plane (Eq. (55)). These\nprescriptions allow us to completely specify the magnetic force on a \\Mercury\" in a close-in orbit.\nWriting the planetary magnetic moment as\n~ mmag=jmmagje\u0000i\u001e^eB; (59)\nwhere\u001eis a phase angle, and ^ eBis a unit vector in the direction of the background magnetic \feld, we obtain the\nphysical \feld\n~ mmag,phys =jmmagjcos(!t+\u001e)^eB; (60)\nwhere time tis zero when the stellar magnetic dipole points at the planet's location. This equation carries all the\ninformation about how the magnetic dipole moment changes in time. Since the planet is tidally locked, there are two\n\felds contributing, with di\u000berent zero-points for time t, because the radial and tangential components of the stellar\ndipole peak at di\u000berent locations.\nThe orbital evolution of this hypothetical Mercury broadly depends on how quickly it orbits the host star compared\nto the stellar rotation (Laine et al. 2008). If the planet orbits more quickly than the star rotates, then it plows through\nthe stellar magnetic \feld, experiencing instantaneous Lorentz forces including a net drag force that causes inspiral.\nQuantitatively, we can see this behavior by focusing on the imaginary part of the magnetic moment, which is positive\nvalued, as it interacts with the radial component of the stellar magnetic \feld in the tidally locked frame of the planet.\nThe time dependence of the dipole moment goes as sin !t, which is 180 degrees out of phase with the gradient of the\n\feld in the direction of orbital motion. As in Equation (3), the planet feels a force in opposition to its motion.\nWhen the stellar rotation is faster than the planet's orbit ( ! < 0), the planet perceives the stellar magnetic \feld\nlines as coming at it from the direction opposite to its orbital motion. Then, there is a net force in the direction\nof travel, boosting its orbital energy and pushing it outward. In quantitative terms, the sign of !\rips the sense of\nthe sinusoidally varying magnetic moment, causing it to be in phase with the \feld gradient. This component of the\nmagnetic force is thus directed along the planet's velocity vector.\nWhen the planet corotates with the star, it experiences no change in the magnetic \rux in its tidally locked frame;\nno currents are induced. The \feld is free to permeate through the interior of the planet, interacting only with the\nmagnetic inner core. This con\fguration does not seem stable; if the planet drifts slightly inward, its new increased\norbital speed will allow it to overtake the magnetic \feld lines, drawing it closer to the host star. If the planet is moved\nbeyond the corotation radius, the stellar \feld lines overtake it, generating a torque that pushes out further still.\nA similar scenario plays out for binary stars with an important distinction. The Ohmic losses in a conductive\ncompanion as it interacts with the magnetic \feld of its rotating partner a\u000bect the partner's spin more than the orbital\nenergy (Joss et al. 1979; Campbell 1983). The torques from the magnetic interactions spin up the magnetic partner\n2Planetary spin would a\u000bect the local magnetic \rux calculations, and would be a source of energy in the Ohmic dissipation process. However,\nfor the close-in orbits considered here, tidal locking is assumed.19\n102\n101\n100101102\nSynodic Period (days)103\n102\n101\nMagnetic Moment ×0/2B0R3\nAbs\nRe ×(1)\nIm\nFigure 8. The complex magnetic moment of a hypothetical Mercury, based on the parameters in Table 1, as a function of the\nperiod of oscillation of the stellar magnetic \feld at the planet's location. For each value of the period of oscillation, a grid-based\ncalculation yields the complex magnetic moment vector. The grid consists of 105radial points; values of the conductivity on\nthis grid are smoothed with an approximately Gaussian window with a standard deviation equivalent to 5 km (the thinnest\nradial zone is the crust, with a depth of 90 km). As in the legend, the solid line is the absolute magnitude of the planet's\nmagnetic moment, the dashed line is the real part, times a factor of \u00001 since it is negative-valued (as an indicator of the phase\nshift between the applied and induced \felds), and the dotted line is the imaginary part. Only the imaginary part leads to a net\nchange in orbital energy as a result of Ohmic dissipation.\nif the conducting star is within the corotation radius, pushing toward synchronization. Similarly, a conducting star\nbeyond the corotation radius will spin down its magnetic partner, leading to synchronization.\nTo establish how the magnetic force might impact a Mercury-like planet ( R= 2440 km and M= 3:30\u00021026g) on\na close-in orbit, we consider a stellar host with magnetic dipole moment vector lying in the plane perpendicular to the\nstar's spin axis. The planet is on a circular orbit in that plane. The acceleration of the planet from interactions with\nthe stellar magnetic \feld scales as (Eq. (3))\namag\u0018mmag\nMB?R3\n?\nr4\ncirc\u0018!1:4\u001aB2\n?R6\n?\nr7\ncirc; (61)\nwhere!is the \feld variation frequency at the planet's location, and the power-law scaling is an approximation based\non the data in Figure 8. Adopting R?= 0:2 R\f,M?= 0:2 M\f,B?= 8 kG (near the most extreme \feld strengths\nobserved for M dwarfs, e.g., Shulyak et al. 2017, 2019; Kochukhov 2021), an orbital distance of rcirc= 0:002 au,\nand a frequency !of the same order as the planet's orbital frequency (2 \u0019d\u00001), the planet is accelerated by roughly\n10\u00008cm/s2as a result of magnetic interactions. This value is roughly \u001810\u000012times weaker than the magnitude of\nthe gravitational acceleration. Any orbital evolution as a result of magnetic interactions will be slow compared with\nthe dynamical time.20\nTable 2. Mercury around a variety of stellar hosts.\nstar type M?R?B?P?rcircrcirc=_rcirc\nT Tauri 1 M \f 2 R\f 3 kG 4 day 0.01 AU -150 Myr\nM dwarf 0.2 M \f0.2 R\f 8 kG 1 day 0.002 AU -270 Myr\nAp/Bp 3 M \f 3 R\f 30 kG 10 day 0.015 AU -3 Myr\nwhite dwarf 0.5 M \f 1 R\b 109G 1 day 0.003 AU -41 Myr\nneutron star 1.5 M \f 15 km 1015G 10 s 0.004 AU +3.9 Gyr\nNote |Stellar parameters are intended to be typical, except for the magnetic\n\feld strengths, which are characteristic of one the most extreme observed\nvalues for each type of star. For T Tauri stars see Johnstone et al. (2014);\nfor M dwarfs Saar & Linsky (1985); Johns-Krull & Valenti (1996), Johnstone\net al. (2014), and Kochukhov (2021). Babcock (1960) and Landstreet (1992)\ndiscuss Ap/Bp stars, and (Ferrario et al. 2020) summarizes observations of\nwhite dwarfs. (Olausen & Kaspi 2014) introduce a catalog of neutron stars\nand magnetars, while Younes et al. (2017) focuses on an extreme example.\nThe orbital distance of the Mercury-like planet is nearest the larger of the\nstellar radius or the Roche radius (Eq. (43)). The sign of the dynamical time\nindicates a growing (+) or shrinking ('-') planetary orbit.\nDespite this comparative weakness, the magnetic interaction yields a continuous drag force on the hypothetical\nMercury in a frame that corotates with the \feld lines. Over time, this force changes orbital energy; in terms of the\norbital distance (Eq. (20)),\ndrcirc\ndt\u00182=fmmaggB\nr2\ncirc\nGM?\u0018\n1:4\u001aB2\n?R6\n?\nr6\ncirc: (62)\nThe imaginary part of the magnetic moment associated with the average drag force is comparable to the magnitude of\nthe magnetic moment itself in the case of the Mercury analog. In the example of a Sun-like star with a 10 kG magnetic\n\feld, the decay rate is 0.3 au/Gyr, suggesting inspiral within tens of millions of years from a distance of a few Solar\nradii around a slowly rotating star. Table 2 provides estimates of the orbital evolution time scales for a Mercury-like\nplanet around a variety of stellar hosts. There, we use extreme values for the magnetic \feld strengths to illustrate\nwhat is possible, not what is typical.\nAs in Table 2, orbital evolution time scales (last column) are most rapid for a Mercury-like planet around a peculiar A\n(Ap) star or white dwarf. However, equilibrium temperatures at the close-in distances required for megayear dynamical\ntimes may well a\u000bect the assumed planetary structure for these calculations. Dynamical times around T Tauri stars,\neven at close-in distances, exceed the time that young stars spend in the T Tauri phase; as evolution proceeds, the\nmagnetic \feld strength shrinks along with the stellar radius making long-term orbital evolution through magnetic\ninteractions unlikely. Around a neutron star, a Mercury anaolog will be slowly pushed away from the Roche radius\non gigayear time scales. Perhaps the most promising astronomical scenario is orbital evolution around a cool dwarf,\nat least early on in its evolution when it is active. Equilibrium temperatures are comparatively low, even for close-\nin orbits, and surface \feld strengths can exceed a kilogauss. Then, a close-in planet will be drawn inward to tidal\ndestruction and accretion by the host star within a gigayear.\n5.SUMMARY\nThis work is an exploration of magnetic interactions between a host star and an orbiting companion. Previous\nwork (e.g., Laine et al. 2008) demonstrated that induced currents and associated Ohmic heating losses cause secular\nchanges in orbits, leading to inspiral in extreme cases. Here, we focus on instantaneous Lorentz forces that arise from\ninteractions between a magnetic or conducting body and the stellar magnetic \feld. The following list summarizes our\nmain \fndings:\n•The magnetic force between a star and an orbiting companion generally falls o\u000b more steeply than gravity,\nleading to the possibility of an unstable zone where mergers are inevitable. In astrophysical systems, the stellar21\nmagnetic \felds have dipole \feld strengths that are orders of magnitude too weak to produce this type of unstable\nzone. At lower, more realistic magnetic \feld strengths, binaries with white dwarfs and/or neutron stars with\nfossil magnetic \felds may cause measurable orbital precession (Bourgoin et al. 2022).\n•Conducting bodies in a time-varying stellar magnetic \feld develop eddy currents to suppress changes in magnetic\n\rux. In the dipole-dipole interaction picture presented here, the magnetic moment of the body, ~ mmag, has a\nstrength and phase that depend on the magnetic Reynolds number. At low Rm, the magnetic moment is 90\u000e\nout of phase with the local stellar magnetic \feld; the work done on an orbiting body moving through a spatially\nvarying magnetic \feld in this situation always opposes the motion (e.g., Gi\u000en et al. 2010). The amount of work\nrises with increasing Rm, as when the conductivity is higher, peaking at Rm\u00181. AsRmincreases beyond unity,\nthe phase shifts toward 180\u000e, and the force becomes oscillatory, with little work done. The amplitude of the\ninstantaneous force is strongest at high Reynolds number.\n•The dipole-dipole interaction picture provides an estimate of the work done on a conducting body moving through\na magnetic \feld that is consistent with calculations based on Ohmic dissipation (Joss et al. 1979; Campbell 1983;\nLaine et al. 2008; Laine & Lin 2012; Kislyakova et al. 2017; Kislyakova et al. 2018; Bromley & Kenyon 2019).\nThe dipole-dipole interaction view also allows for consideration of non-dissipative forces that also may impact\norbital dynamics.\n•When the magnetic Reynolds number of a conducting body is much greater than unity, the Ohmic losses are low.\nThen, the induced currents are in phase with the magnetic \feld itself; Lorentz forces are oscillatory. With these\ndriving forces at play, orbital precession, eccentricity pumping, and resonant trapping are possible in speci\fc\ncon\fgurations.\n•Orbiting bodies with high magnetic susceptibility (as when \u0016=\u0016 0\u001d1) and a magnetic Reynolds number that is\nmuch less than unity experience similar phenomena caused by oscillatory Lorentz forces.\n•The most promising astrophysical systems for the observation of dipolar magnetic interactions are compact mag-\nnetic binaries. Fossil \felds between close-in white dwarfs can cause signi\fcant orbital precession, a phenomenon\nalso identi\fed recently by Bourgoin et al. (2022). Nodal precession of misaligned \felds may detectable even in\ncircularized systems like ZTF J153932.16+502738.8 (Burdge et al. 2019), depending on the orientation of the\nstellar dipole moments.\nThese main conclusions are drawn on the basis of several simplifying assumptions. One is that the stellar magnetic\n\felds a pure dipoles, falling o\u000b as 1 =r3. This choice may underestimate the strength of the magnetic \feld around\nstars with winds (e.g., Johnstone 2012). Stellar winds can support and strengthen the \feld at larger radii, yielding\na substantially shallower fallo\u000b, 1 =r2. Depending on the azimuthal dependence of the wind-swept \feld, the e\u000bects\ndescribed here may be stronger at large radii Other plasma e\u000bects, including the formation of \rux tubes (e.g., Goldreich\n& Lynden-Bell 1969; Lai 2012), may modify the local magnetic \feld. These phenomena o\u000ber a potentially rich layer\nto the orbital dynamics that is well beyond what we describe here.\nACKNOWLEDGMENTS\nWe are grateful to referees for providing thoughtful comments that improved the content and presentation of this\nwork. We acknowledge generous allotments of computer time on the NASA `discover' cluster, provided by the NASA\nHigh-End Computing (HEC) Program through the NASA Center for Climate Simulation (NCCS). Guidance and\ncomments from M. Geller improved our presentation.\nSoftware: Scipy (Jones et al. 2001{)\nAPPENDIX22\nA.INDUCED CURRENTS IN CONDUCTING ASTEROIDS OR PLANETS\nIn this section, we consider the currents induced in a conducting sphere by an oscillating magnetic \feld. The sphere\nhas a radius R, is spherically symmetric with bulk properties, including mass density \u001a, conductivity \u001b, and magnetic\npermeability \u0016, that may vary with radius. The external magnetic \feld in which the sphere resides is spatially uniform,\nwith an amplitude that varies in time twith frequency !, so that~B0=B0exp(\u0000i!t)^ez, whereB0is a constant.\nThe changes in the magnetic \feld will induce currents in the conductor (Faraday's law), which will in turn generate a\nmagnetic \feld that opposes the changes (Lenz's law). When the frequency !is low (as quanti\fed below), the induced\ncurrents are weak; the external \feld penetrates throughout the conducting body. When the magnetic \feld oscillations\nare rapid, the induced currents are strong and can generate an induced \feld that e\u000bectively prevents the external\n\feld from penetrating into the bulk of the conductor | the skin e\u000bect. The key parameter in distinguishing these\nregimes is the magnetic Reynolds number, Rm\u0011\u0016\u001b!R2(Eq. (25)). For Rm\u001c1, the \feld varies slowly, and/or the\nconductivity and permeability are low; the external \feld bathes the entire conductor. When Rm\u001d1, the external\n\feld can only penetrate into a thin layer on the conductor's surface.\nWhile we consider a range of frequencies, we assume throughout that the wavelength of electromagnetic waves within\nthe sphere at frequency !is much larger than the depth of \feld penetration into the sphere. This limitation on high\nfrequencies allows us to ignore dielectric properties of the conductor (Bidinosti et al. 2007). We begin with the case of\na homogeneous, nearly-perfect conductor.\nA.1. A weakly conductive, non-magnetic sphere\nIn the limit of low magnetic Reynolds number ( Rm\u001c1), the external magnetic \feld is largely una\u000bected by the\nweak currents that it induces in the conducting body. We can use the `Physics II' formalism for Amp\u0012 ere's law to\nestimate the current density:Z\nrJ(r;\u0012)=\u001bd` =i\u0019!sin2(\u0012)r2B0;(r<R ) (A1)\nwhere we have adopted spherical coordinates centered on the conductor and have assumed that the current density J\nlies in the azimuthal direction. Thus\nJ(r) =i\n2\u001b!rsin(\u0012)B0; (A2)\nwhere the factor of iindicates that the current density lags the magnetic \feld by a phase of 90\u000e. The magnetic moment\nof the sphere is thus\n~ mmag=1\n2Z\nVdV~ r\u0002~J=2\u0019i\n15\u001b!R5~B0: (A3)\nThe power dissipated by the induced currents through Ohmic heating is\nP\n=Z\nVdV\f\f\f~J\f\f\f2\n\u001b=2\u0019\n15\u001b!2R5B2\n0: (A4)\nBy associating the frequency !with an orbital speed through a spatially varying magnetic \feld, we can deduce that\nOhmic heating comes at the expense of orbital energy (Gi\u000en et al. 2010; Kislyakova et al. 2017; Bromley & Kenyon\n2019).\nA.2. A highly conducting sphere\nIn the limit of high conductivity, the penetration depth of the magnetic \feld into a non-magnetic spherical body\nis small. We follow Joss et al. (1979), focusing on a small patch on the sphere's surface with an in\fnitesimal planar\nsurface element of area dAthat interacts with a uniform, applied magnetic \feld exterior to it. In the region beneath\nthe conducting surface, Faraday's law and Amp\u0012 ere's law are, respectively,\n~r\u0002~J=\u0000i!\u001b~B;~r\u0002~B=\u0016~J; (A5)23\nwhere~Jis the current density, and ~Bis the magnetic \feld, including both the applied and induced components; here,\n!is the driving frequency, \u001bis the conductivity and \u0016is the permeability inside the conductor, all of which we assume\nto be constant. We also have used the microscopic version of Ohm's law, ( ~J=\u001b~E), to write the electric \feld ~Ein\nterms of the free current density ~J. The combination of the two equations gives\n~r\u0002\u0010\n~r\u0002~J\u0011\n=i\u0016\u001b!~J: (A6)\nRecognizing the (local) planar symmetry of the problem, we overlay a Cartesian grid ( x;y;z ) on the surface of the\nconductor with z>0 specifying a vertical distance away from the conductor's surface. If we assume the magnetic \feld\nis parallel to the local surface, oriented along the x-axis, or simply just consider this component of the \feld, a solution\nemerges:\n~J=J0exp (\u0000kjzj)~ ey (A7)\nwhere\nk=p\ni\u0016\u001b! = (1 +i)p\n\u0016\u001b!= 2 (A8)\nandJ0is a constant. A boundary condition at the conducting surface is that the tangential \feld just beneath the\nsurface,B(\u0000)\nk=\u0016, equalsBk=\u00160, the tangential \feld above it. Then,\nJ0=kB+\nk\n\u00160; (A9)\nThe power dissipated per unit surface area is\ndP\ndA=Z\ndzjJj2\n\u001b=B2\nk\n4\u00162\n0p\n\u0016\u001b!= 2: (A10)\nZooming out to view the whole conductor as a sphere of radius R, we seek the tangential magnetic \feld at the\nconductor's surface, Bk, by assuming that the surface currents create a uniform magnetization within the sphere,\nwhile outside of the sphere, these currents generate a magnetic dipole \feld:\n~B(r<R ) =2\u00160\n4\u0019R3~ m+~B0 (A11)\n~B(r>R ) =\u00160\n4\u0019R3[3(~ m\u0001^r)^er\u0000~ m] +~B0; (A12)\nwhereRis the radius of the sphere, ~ mmagis the induced dipole moment, and B0is the strength of the \feld far from\nthe sphere.3Since the \feld inside the conductor is zero, the magnetic moment is\n~ mmag=\u00002\u0019R3\n\u00160~B0: (A13)\nand the tangential component of the \feld just outside of the conductor's surface is\nBk=3\n2sin(\u0012)B0 (A14)\nWith this result, integration over the sphere's surface yields the total dissipated power,\nP\n=Z\nd\u001eZ\nsin(\u0012)d\u0012dP\ndAsin2(\u0012) =3\u0019B2\n0\n\u00162\n0r\u0016!\n2\u001bR2: (A15)\nThis analysis is valid so long as the conductivity is high, and that the skin depth of magnetic di\u000busion into the\nsphere,\n\u000e=p\n2=\u0016\u001b!; (A16)\nis very small compared with R.\n3A derivation from potential theory is straightforward; here we rely on uniqueness and that fact that the induced \feld inside the sphere\nmust be uniform in order to cancel the ambient \feld. To match boundaries at the sphere's surface, the induced \feld outside of the sphere\nmust be a dipole.24\nA.3. Homogeneous material with arbitrary conductivity and permeability\nA similar treatment to the one above allows us to consider the more general case of arbitrary conductivity in a\nhomogeneous material where the magnetic \feld can permeate deep within the sphere. Following Bidinosti et al.\n(2007), the current density satis\fes the Faraday-Amp\u0012 ere-Ohm Equation (A6),\nr2~J+k2~J= 0: (A17)\nwhere ther2operator is the vector Laplacian, and we have made use of Gauss' law,\n~r\u0001~J= 0 (A18)\nin the absence of free charges. As in the previous example of an almost-perfectly conducting sphere, symmetry requires\nthe current density ~Jto be toroidal, with a dependence on polar angle \u0012that matches that of the background magnetic\n\feld tangent to the surface of the sphere (Eq. (A9)). Thus, we seek a solution of the form\n~J=f(r) sin\u0012^e\u001e (A19)\nin standard spherical coordinates ( r,\u0012,\u001e) that are aligned with the background magnetic \feld. Then, the azimuthal\ncomponent of Equation (A17) gives\n@2f\n@r2+2\nr@f\n@r+\u0012\nk2\u00002\nr2\u0013\nf= 0: (A20)\nThe solution is\nf(r) =Aj1(kr); (A21)\nwherejnis a spherical Bessel function of order n, and we have enforced the boundary condition that f(r) must be\nregular at the origin.\nWe next obtain the constant Aby matching boundary conditions across the spherical conductor's surface at radius\nR. From Faraday's law, the magnetic \feld components inside the sphere are\nBr=2A\u0016\nk2rji(kr) cos\u0012; B\u0012=\u0000A\u0016\nk2r[j1(kr) +krj0\n1(kr)] sin\u0012; (A22)\nand outside of the sphere ( r>R ),\nBr= (B0+ 2 ~m=r3) cos\u0012; B\u0012= (\u0000B0+ ~m=r3) sin\u0012; (A23)\nwhere ~mis a constant related to the induced \feld from the sphere, which we take to be a dipole (Bidinosti et al. 2007).\nThe boundary conditions at the sphere's surface are that the radial component of ~Bis continuous, as is the tangential\ncomponent of ~H=\u0016~B. These two conditions allow us to solve for the two unknowns, Aand ~m, where only the former\nis needed to specify the current density.\nSolving for A, and invoking identities relating spherical Bessel functions and their derivatives, Bidinosti et al. (2007,\nsee also Nagel 2018) \fnd that\nA=9B0\n2ki\u0016\u001b!\n(\u0016+ 2\u00160)j0(kR) + (\u0016\u0000\u00160)j2(kR)): (A24)\nWith Equations (A19 and (A21), this expression completes the solution for the current density J.\nThe magnetic moment associated with the induced current is\nmmag=2\u0019R3B0\n\u001602(\u0016\u0000\u00160)j0(kR) + (2\u0016+\u00160)j2(kR)\n(\u0016+ 2\u00160)j0(kR) + (\u0016\u0000\u00160)j2(kR)(A25)\n(Bidinosti et al. 2007).\nThe average dissipated power comes from Ohm's Law in microscopic form,\nP\n=1\n2Z\nVdVj~Jj2\n\u001b; (A26)25\nwhich we express most generally in the form\nP\n\u0011B2\n0\n2\u00160!4\u0019R3\n3F (A27)\n(Bromley & Kenyon 2019). For the homogeneous, linear material considered in this section,\nF==\u001a9\u0016[j0(kR) +j2(kR)]\n2[(\u0016+ 2\u00160)j0(kR) + (\u0016\u0000\u00160)j2(kR)]\u001b\n: (A28)\nThe dissipated power's behavior falls into several regimes, depending on Reynolds number Rmand the magnetic\npermeability, \u0016. In the \frst regime, the Reynolds number is low, with Rm\u001c10; the magnetic \feld permeates\nthroughout the conductor and is largely unperturbed by the induced currents. In a second regime, Rm\u001d\u00162=\u00162\n0\u001510;\nthen, surface currents are produced that expel the magnetic \feld | there is little penetration of the magnetic \feld\ninto the conductor. For magnetic materials, with \u0016>\u0016 0, there is an intermediate regime, 10 .Rm.\u00162=\u00162\n0, where\nthe \feld permeates the conducting material, but bound currents throughout the medium also respond to the \feld.\nTo get numerical values for quantities presented here, the Python SciPy.special module can evaluate Bessel\nfunctions with complex arguments. In some cases, the multiple-precision capabilities of the mpmath module are required.\nWe recommend using multiple-precision arithmetic for all equations in this section.\nA.4. Radially varying conductivity and permeability\nLaine et al. (2008) and Kislyakova et al. (2017, and references therein) describe numerical approaches to calculate\nthe power dissipated in a conducting sphere with radially varying conductivity in a time-varying magnetic \feld. Here,\nwe follow a similar approach, while also accommodating radial variations in magnetization. Our starting point is the\nFaraday-Amp\u0012 ere-Ohm equation (A20), now modi\fed to accommodate variable \u0016and\u001b:\n~r\u0002 \n1\n\u0016~r\u0002~J\n\u001b!\n=i!~J: (A29)\nA few choice vector identities applied to the \u001ecomponent of the above expression leads to\n@2f\n@r2+\u00122\nr+g1\u0013@f\n@r+\u0012\nk2\u00002\nr2+g1\nr+g2\u0013\nf (A30)\nwherej~Jj=fsin(\u0012), as above, and\ng1=\u00001\n\u0016@\u0016\n@r\u00002\n\u001b@\u001b\n@r(A31)\ng2=1\n\u0016\u001b@\u0016\n@r@\u001b\n@r+2\n\u001b2\u0012@\u001b\n@r\u00132\n\u00001\n\u001b@2\u001b\n@r2: (A32)\nWe solve Equation (A29) numerically by discretizing the radial domain: ~ r!frjgforj= 0;1;2;:::;N\u00001, where\nr0\u0018\u0001r,rj=j\u0001r+r0, andrN\u00001=Rare points on an equally-spaced mesh. Converting the derivatives in\nEquation (A30) to \fnite di\u000berences, the expression takes on a matrix form,\nA~f=~b (A33)\nschematically, the matrix Aand the two vectors ~fand~bare\nA=2\n66666641 0::: 0\na\u0000\n1d1a+\n10:::\n0a\u0000\n2d2a+\n2:::\n......\n0::: 0 0 13\n7777775;~f=2\n6666664f0\nfi\n...\nfN\u00002\nfN\u000013\n7777775and ~b=2\n6666664finner\n0\n...\n0\nfouter3\n7777775; (A34)26\nhere, the subscripts correspond to radial positions, so that fj=f(rj). The elements of the tridiagonal matrix are\na\u0006\nj= 1=\u0001r2\u0006(2=rj+g1;j)=2\u0001r\ndj=\u00002=\u0001r2+k2\nj\u00002=r2\nj+g1;j=rj+g2;j (A35)\nwheregi;jare from Equations (A31) and (A32), derived either analytically from \u0016(r) and\u001b, or with their numerical\n(\fnite-di\u000berence) derivatives.\nThe two remaining unde\fned constants in the matrix equation (A33) are the \frst and last elements of the RHS\nvector ~b,b0=finner andbN\u00001=fouter, corresponding to Dirichlet boundary conditions. We set finner = 0, since the\ncurrent density vanishes at the origin. (In practice, the innermost radial grid point is close to but not exactly at the\norigin, because of a coordinate singularity there.)\nTo obtainfouter, we follow this simple plan: We \frst assume a trial value for the current density at the outer surface\nwithft(R) = 1. We then solve the matrix equation\n~ft=A\u00001~bt: (A36)\nwherebt\nN\u00001= 1 and all other elements are zero. We derive the magnetic \feld at the surface such that key boundary\nconditions from Maxwell's equations are satis\fed. The tridiagonal form of Aallows for fast inversion compared with\na general matrix; in our Python implementation, we use the SciPy linalg package and the solve banded routine.\nThe \fnal step is to rescale the current density so that the derived background \feld matches the actual \feld, ~B0. The\n(rescaled) solution we are after is\n~f=C~ft; (A37)\nwhereCis a complex constant. To obtain C, we assume that the magnetic \feld has the form\nBr(R+;0) =B0+ 2BMandB\u0012(R+;\u0019=2) =\u0000BI+BM (A38)\nwhere the left equation is the purely radial \feld at the point just above the sphere's pole at z= +R, and the the right\nequation corresponds to the poloidal \feld just beyond the sphere's equator, and BMis the strength of the induced\ndipole \feld from the sphere just outside its equator at radius R+. From Faraday's law, the current density and \feld\njust inside the conductor's surface are related by\nBr(R\u0000;0) =2f(R\u0000)\ni\u001b!RandB\u0012(R\u0000;\u0019=2) =\u0000f(R\u0000) +Rf0(R\u0000)\ni\u001b!R: (A39)\nThe boundary conditions that the radial component of ~H\u0011~B=\u0016 and the tangential component of ~Bare continuous\nacross the boundary yield two independent equations with two unknowns. Eliminating BM, we \fnd\nB=1\n3\u0012\nBr(R\u0000;0)\u00002\u00160\n\u0016B\u0012(R\u0000;\u0019=2)\u0013\n=2\n3(\u0016+\u00160)f(R\u0000) +R\u00160f0(R\u0000)\ni\u0016\u001b!R; (A40)\nsincef=Cft, we infer that\nC=3i\u0016\u001b!RB 0\n2(\u0016+\u00160)f(R\u0000) + 2R\u00160f0(R\u0000): (A41)\nIn our numerical implementation, we choose R\u0000to berN\u00002, and evaluate the above expression with a \fnite di\u000berence\noperation:\nC=3i\u0016N\u00002\u001bN\u00002!rN\u00002B0\n2(\u0016N\u00002+\u00160)ft\nN\u00002+ 2rN\u00002\u00160(ft\nN\u00001\u0000ft\nN\u00003)=2\u0001r; (A42)\nwhich then multiplies all ft\njto yield the full set, ~f. This direct method is an alternative to iterative approaches of Laine\net al. (2008, Newton-Raphson-Kantorovich) and Kislyakova et al. (2017, recursive application of boundary conditions\non radial shells).\nREFERENCES\nAngel, J. R. P. 1978, ARA&A, 16, 487,\ndoi: 10.1146/annurev.aa.16.090178.002415Apps, K., Smart, R. L., & Silvotti, R. 2021, Research Notes\nof the AAS, 5, 229, doi: 10.3847/2515-5172/ac2df227\nBabcock, H. W. 1958, ApJS, 3, 141, doi: 10.1086/190035\n|. 1960, ApJ, 132, 521, doi: 10.1086/146960\nBidinosti, C., Chapple, E., & Hayden, M. 2007, Concepts in\nMagnetic Resonance Part B: Magnetic Resonance\nEngineering, 31B, 191,\ndoi: https://doi.org/10.1002/cmr.b.20090\nBourgoin, A., Le Poncin-La\ftte, C., Mathis, S., & Angonin,\nM. C. 2022, PhRvD, 105, 124042,\ndoi: 10.1103/PhysRevD.105.124042\nBromley, B. C., & Kenyon, S. J. 2006, AJ, 131, 2737,\ndoi: 10.1086/503280\n|. 2011, ApJ, 731, 101, doi: 10.1088/0004-637X/731/2/101\n|. 2019, ApJ, 876, 17, doi: 10.3847/1538-4357/ab12e9\nBrouwers, M. G., Bonsor, A., & Malamud, U. 2022,\nMNRAS, 509, 2404, doi: 10.1093/mnras/stab3009\nBurdge, K. B., Coughlin, M. W., Fuller, J., et al. 2019,\nNature, 571, 528, doi: 10.1038/s41586-019-1403-0\nCaiazzo, I., Burdge, K. B., Fuller, J., et al. 2021, Nature,\n595, 39, doi: 10.1038/s41586-021-03615-y\nCampbell, C. G. 1983, MNRAS, 205, 1031,\ndoi: 10.1093/mnras/205.4.1031\nChang, Y.-L., Bodenheimer, P. H., & Gu, P.-G. 2012, ApJ,\n757, 118, doi: 10.1088/0004-637X/757/2/118\nChristensen, U. R. 2010, in Heliophysics: Evolving Solar\nActivity and the Climates of Space and Earth, ed. C. J.\nSchrijver & G. L. Siscoe (Cambridge: Cambridge\nUniversity Press), 179{216\nChyba, C. F., & Hand, K. P. 2021, ApJL, 922, L38,\ndoi: 10.3847/2041-8213/ac399d\nde Jager, O. C., Meintjes, P. J., O'Donoghue, D., &\nRobinson, E. L. 1994, MNRAS, 267, 577,\ndoi: 10.1093/mnras/267.3.577\nDonati, J. F., & Landstreet, J. D. 2009, ARA&A, 47, 333,\ndoi: 10.1146/annurev-astro-082708-101833\nFarihi, J. 2016, NewAR, 71, 9,\ndoi: 10.1016/j.newar.2016.03.001\nFerrario, L., Wickramasinghe, D., & Kawka, A. 2020,\nAdvances in Space Research, 66, 1025,\ndoi: 10.1016/j.asr.2019.11.012\nGenova, A., Goossens, S., Mazarico, E., et al. 2019,\nGeophysical Research Letters, 46, 3625,\ndoi: https://doi.org/10.1029/2018GL081135\nGi\u000en, A., Shneider, M., Kalra, C. S., Ames, T. L., & Miles,\nR. B. 2010, arXiv e-prints, arXiv:1004.5412.\nhttps://arxiv.org/abs/1004.5412\nGoldreich, P., & Lynden-Bell, D. 1969, ApJ, 156, 59,\ndoi: 10.1086/149947\nGrankin, K. 2021, Acta Astrophysica Taurica, 2, 9,\ndoi: 10.31059/aat.vol2.iss1.pp9-20Hogg, M. A., Cutter, R., & Wynn, G. A. 2021, MNRAS,\n500, 2986, doi: 10.1093/mnras/staa3316\nJohns-Krull, C. M., & Valenti, J. A. 1996, ApJL, 459, L95,\ndoi: 10.1086/309954\nJohnstone, C. P. 2012, PhD thesis, Saint Andrews\nUniversity, UK\nJohnstone, C. P., Bartel, M., & G udel, M. 2021, A&A, 649,\nA96, doi: 10.1051/0004-6361/202038407\nJohnstone, C. P., G udel, M., Brott, I., & L uftinger, T.\n2015, A&A, 577, A28, doi: 10.1051/0004-6361/201425301\nJohnstone, C. P., Jardine, M., Gregory, S. G., Donati,\nJ. F., & Hussain, G. 2014, MNRAS, 437, 3202,\ndoi: 10.1093/mnras/stt2107\nJones, E., Oliphant, T., Peterson, P., et al. 2001{, SciPy:\nOpen source scienti\fc tools for Python.\nhttp://www.scipy.org/\nJoss, P. C., Katz, J. I., & Rappaport, S. 1979, ApJ, 230,\n176, doi: 10.1086/157074\nKatz, J. I. 1989, MNRAS, 239, 751,\ndoi: 10.1093/mnras/239.3.751\n|. 2017, ApJ, 835, 150, doi: 10.3847/1538-4357/835/2/150\nKenyon, S. J., & Bromley, B. C. 2012, AJ, 143, 63,\ndoi: 10.1088/0004-6256/143/3/63\n|. 2014, ApJ, 780, 4, doi: 10.1088/0004-637X/780/1/4\n|. 2017, ApJ, 850, 50, doi: 10.3847/1538-4357/aa9570\nKepler, S. O., Pelisoli, I., Koester, D., et al. 2016, MNRAS,\n455, 3413, doi: 10.1093/mnras/stv2526\nKhurana, K. K., Kivelson, M. G., Stevenson, D. J., et al.\n1998, Nature, 395, 777, doi: 10.1038/27394\nKislyakova, K., & Noack, L. 2020, A&A, 636, L10,\ndoi: 10.1051/0004-6361/202037924\nKislyakova, K. G., Fossati, L., Johnstone, C. P., et al. 2018,\nThe Astrophysical Journal, 858, 105,\ndoi: 10.3847/1538-4357/aabae4\nKislyakova, K. G., Noack, L., Johnstone, C. P., et al. 2017,\nNature Astronomy, 1, 878,\ndoi: 10.1038/s41550-017-0284-0\nKivelson, M. G., Khurana, K. K., Russell, C. T., et al. 2000,\nScience, 289, 1340, doi: 10.1126/science.289.5483.1340\nKochukhov, O. 2021, A&A Rv, 29, 1,\ndoi: 10.1007/s00159-020-00130-3\nKonar, S. 2017, Journal of Astrophysics and Astronomy, 38,\n47, doi: 10.1007/s12036-017-9467-4\nLai, D. 2012, ApJL, 757, L3,\ndoi: 10.1088/2041-8205/757/1/L3\nLaine, R. O., & Lin, D. N. C. 2012, ApJ, 745, 2,\ndoi: 10.1088/0004-637X/745/1/2\nLaine, R. O., Lin, D. N. C., & Dong, S. 2008, ApJ, 685,\n521, doi: 10.1086/58917728\nLandstreet, J. D. 1992, A&A Rv, 4, 35,\ndoi: 10.1007/BF00873569\nLavail, A., Kochukhov, O., Hussain, G. A. J., et al. 2017,\nA&A, 608, A77, doi: 10.1051/0004-6361/201731889\nMalhotra, R. 1996, AJ, 111, 504, doi: 10.1086/117802\nMik\u0013 oczi, B. 2021, arXiv e-prints, arXiv:2109.10722.\nhttps://arxiv.org/abs/2109.10722\nMoreno, E., Pichardo, B., & Schuster, W. J. 2015, MNRAS,\n451, 705, doi: 10.1093/mnras/stv962\nNagel, J. R. 2018, IEEE Antennas and Propagation\nMagazine, 60, 81, doi: 10.1109/MAP.2017.2774206\nNoack, L., Kislyakova, K. G., Johnstone, C. P., G udel, M.,\n& Fossati, L. 2021, A&A, 651, A103,\ndoi: 10.1051/0004-6361/202040176\nOlausen, S. A., & Kaspi, V. M. 2014, ApJS, 212, 6,\ndoi: 10.1088/0067-0049/212/1/6\nPiro, A. L. 2012, ApJ, 755, 80,\ndoi: 10.1088/0004-637X/755/1/80\nRea, N., Esposito, P., Turolla, R., et al. 2010, Science, 330,\n944, doi: 10.1126/science.1196088\nSaar, S. H., & Linsky, J. L. 1985, ApJL, 299, L47,\ndoi: 10.1086/184578\nShulyak, D., Reiners, A., Engeln, A., et al. 2017, Nature\nAstronomy, 1, 0184, doi: 10.1038/s41550-017-0184\nShulyak, D., Reiners, A., Nagel, E., et al. 2019, A&A, 626,\nA86, doi: 10.1051/0004-6361/201935315Strugarek, A., Bolmont, E., Mathis, S., et al. 2017, ApJL,\n847, L16, doi: 10.3847/2041-8213/aa8d70\nSutherland, A. P., & Kratter, K. M. 2019, MNRAS, 487,\n3288, doi: 10.1093/mnras/stz1503\nValyavin, G. 2015, in Astronomical Society of the Paci\fc\nConference Series, Vol. 494, Physics and Evolution of\nMagnetic and Related Stars, ed. Y. Y. Balega, I. I.\nRomanyuk, & D. O. Kudryavtsev, 107\nVeras, D., Carter, P. J., Leinhardt, Z. M., & G ansicke, B. T.\n2017, MNRAS, 465, 1008, doi: 10.1093/mnras/stw2748\nVillebrun, F., Alecian, E., Hussain, G., et al. 2019, A&A,\n622, A72, doi: 10.1051/0004-6361/201833545\nWyatt, M. C. 2003, ApJ, 598, 1321, doi: 10.1086/379064\nYounes, G., Baring, M. G., Kouveliotou, C., et al. 2017,\nApJ, 851, 17, doi: 10.3847/1538-4357/aa96fd\nZimmer, C., Khurana, K. K., & Kivelson, M. G. 2000,\nIcarus, 147, 329, doi: 10.1006/icar.2000.6456\nZuckerman, B., Melis, C., Klein, B., Koester, D., & Jura,\nM. 2010, ApJ, 722, 725,\ndoi: 10.1088/0004-637X/722/1/725\nZuckerman, B., & Reid, I. N. 1998, ApJL, 505, L143,\ndoi: 10.1086/311608"    },    {        "title": "1102.2847v1.Application_of_Resonance_Perturbation_Theory_to_Dynamics_of_Magnetization_in_Spin_Systems_Interacting_with_Local_and_Collective_Bosonic_Reservoirs.pdf",        "content": "arXiv:1102.2847v1  [quant-ph]  14 Feb 2011Application of Resonance Perturbation Theory to\nDynamics of Magnetization in Spin Systems Interacting\nwith Local and Collective Bosonic Reservoirs\nM. Merkli∗\nDept. of Mathematics and Statistics\nMemorial University of Newfoundland\nSt. John’s, NL, Canada A1C 5S7G.P. Berman†and A. Redondo‡\nTheoretical Division, MS B213\nLos Alamos National Laboratory\nLos Alamos, NM 87545, USA\nDecember 26, 2018\nLA-UR 11-00280\nAbstract\nWe apply our recently developed resonance perturbation theory t o describe the\ndynamics of magnetization in paramagnetic spin systems interacting simultane-\nouslywithlocalandcollectivebosonicenvironments. Wederiveexplicit expressions\nfortheevolutionofthereduceddensitymatrixelements. Thisallows ustocalculate\nexplicitly the dynamics of the macroscopic magnetization, including ch aracteristic\nrelaxation and dephasing time-scales. We demonstrate that collect ive effects (i)\ndo not influence the character of the relaxation processes but me rely renormalize\nthe relaxation times, and (ii) significantly modify the dephasing times, leading\nin some cases to a complicated (time inhomogeneous) dynamics of the transverse\nmagnetization, governed by an effective time-dependent magnetic field.\n1 Introduction\nWhen quantum systems interact with their environments the e ffects of relaxation and\ndecoherence occur [1–8]. In this paper we study relaxation a nd decoherence in quan-\ntummacroscopic systems of “effective” spinsinteracting sim ultaneously withbothlocal\nand collective thermal environments. By “effective” we mean t hat our approach can\nbe applied not only to magnetic spin systems, but also to many quantum systems with\ndiscrete energy levels, including recently widely discuss ed quantum bits (qubits) based\non superconducting Josephson junctions and SQUIDs [9–14]. We also would like to\nmention here the research on ephaptic coupling of cortical n eurons, when both local\nand collective electrical fields play a significant role in th e synchronization dynamics\nof neurons [15]. We assume that spins do not interact directl y among themselves, but\nonly through their interactions with collective (energy co nservingand energy exchange)\n∗Email: merkli@mun.ca, http://www.math.mun.ca/ ∼merkli/\n†Email: gpb@lanl.gov\n‡Email: redondo@lanl.govbosonic environments (“thermal baths”, “reservoirs”). Th e relaxation in these systems\nis caused by energy exchange between the environments and sp ins. The rate of relax-\nation is usually characterized by the spectral density of no ise of the reservoirs at the\ntransition frequency, ω, of spins in their effective magnetic field, and by the interac-\ntion constant between a spin and an environment [7,9,14]. Th e rate of decoherence\nusually has a more complicated dependence on the parameters of spins, their local en-\nvironments, and interaction constants [7–9,14,16]. In par ticular, low-frequency noise\n(1/fnoise) makes a significant contribution to the decoherence r ate [17–20] (see also\nreferences therein).\nUsually relaxation anddecoherence areunwanted effects, for example, in aquantum\ncomputer one must maintain quantum coherence for long times [5]. But dissipative\neffects can also be put to good use, for example, in magnetic res onance imaging (MRI)\n[21–23]. Indeed, in this case, different values of relaxation times (T1) for different\nsubstances (e.g. water and biological tissues) allow one to distinguish and visualize\npathological developments in tissues [21]. Dissipative effe cts can also be utilized, for\nexample, to analyze and classify the influence of many types o f defects and impurities,\nin order to improve the properties of materials [24].\nImproving our understanding of relaxation and decoherence processes is important\nfor many fields of science and for many applications. The main problem associated\nwith dissipative effects is that there are many different source s of noise and thermal\nfluctuations which lead to relaxation and decoherence. We me ntion only some of them,\nelectromagnetic and acoustic fluctuations (bosonic degree s of freedom), magnetic fluc-\ntuations (such as two-level systems in superconducting mat erials), charge defects, and\nnon-equilibrium quasiparticles. Generally, it is impossi ble to eliminate all sources of\nnoise, so some additional classification can be useful. For e xample, in [16] we demon-\nstrated that, for a system of Nspins interacting with bosonic environments, one can\nintroduce clusters of reduced density matrix elements in su ch a way that to a given\ncluster corresponds a decoherence rate describing the fadi ng of all matrix elements\nbelonging to it. When dealing with a quantum algorithm in qua ntum computation,\nthis could imply that the decay of some clusters is rapid, but – if the algorithm is\nbuilt mainly on the use of slower decaying clusters – that dec ay may not influence\nsignificantly the fidelity of the quantum protocol.\nIn this paper we are mainly interested in the effects produced b y simultaneous\ninfluence of both local and collective bosonic environments on the dynamics of a col-\nlective magnetization in a system of Nnon-interacting (paramagnetic) spins in a time-\nindependent magnetic field. The local and collective enviro nments include both energy\nconserving and energy exchange interactions with spins. Th is allows us to determine\nconditions of applicability of the Bloch equation for descr ibing the evolution of the\nmagnetization. We also consider two (and more) ensembles of spins with different pa-\nrameters and strengths of interactions with their environm ents. Using our approach\nbased on resonant perturbation theory [16], we derive expli cit expressions for the time\nevolution of the reduced density matrix elements and, conse quently, for the macro-\nscopic magnetization. We explicitly calculate the relevan t relaxation and decoherence\nrates. The obtained results are important for many applicat ions including MRI and\nfor studying collective effects in materials for superconduc ting qubits.\n2Main results of the paper\n•Single spin dynamics. We consider a microscopic, Hamiltonian model of Nspins\ninteracting with local and collective bosonic thermal rese rvoirs, via energy conserving\nand energy exchange interactions. In Theorem 2.1 we derive a rigorous expression\nfor the reduced density matrix of a single spin, consisting o f a main term describ-\ning relaxation and dephasing, plus a remainder term which is small in the couplings\nhomogeneously in time.\n•Single spin relaxation. We show that the single-spin relaxation rate is given by\nγrelax=1\n4coth(βω/2)/braceleftbig\nλ2Jgc(ω)+µ2Jgℓ(ω)/bracerightbig\n,\nwhereωisthespinfrequency, λandµarethestrengthsoftheenergyexchangecollective\nandlocal couplings, respectively, andwhere Jg(ω)isthereservoirspectraldensity. Only\nenergy-exchange couplings contribute to this rate, and the effect of the local and the\ncollective reservoirs are the same.\n•Single spin dephasing. We show that the single-spin dephasing rate is given by\nγdeph=1\n2γrelax+γcons+γ′,\nwhereγconsisacontribution stemmingonlyfromtheenergy conservingl ocal andcollec-\ntive interactions, determined by the spectral density of th e reservoir at zero frequency\n(see (3.3)). The contribution γ′encodes the effect on dephasing of a single spin due\nto all other spins. It is defined as follows. The time-depende nce of the single spin\noff-diagonal density matrix elements has a very complicated, not exponentially decay-\ning contribution coming from the collective coupling. The t ermγ′is defined to be the\nreciprocal of the time by which that quantity is reduced to ha lf its initial value.\nThe explicit expression of γ′is not simple (see (2.33), (2.31)). For small ratio\nrbetween the strengths of the collective to the local couplin gs we have γ′=O(r2)\n(independent of the number Nof spins). For large collective coupling we have γ′∼\nconst.γrelax, for a constant not depending on N.\n•Evolution of magnetization. We consider the spins in a homogeneous magnetic\nfield pointing in the z-direction. We show that the z-component of the total magne-\ntization vector relaxes to its equilibrium value at the sing le-spin relaxation rate γrelax.\nThis verifies the correctness of the usual Bloch equation (3. 13) for the z-component.\nThe equation for the transverse total magnetic field is given by a modified Bloch equa-\ntion (3.15), with a time-dependent dephasing time ( T2=T2(t)) and a time-dependent\neffective magnetic field. For large times, the coefficients in th e modified Bloch equation\napproach stationary values and give rise to the usual Bloch e quation with renormalized\nT2(∞) time and renormalized effective magnetic field. We show that\n1\nT2(∞)=1\n2γrelax+γcons+(N−1)γ′′,\nwhereγ′′≥0 is independent of N. For small ratio rbetween the strengths of the col-\nlective to the local couplings we have γ′′=O(r2). Consequently, if r∼N−1/2then the\n3collective coupling gives a non-vanishing renormalizatio n to the (asymptotic) T2time,\nwhile ifr∼N−1/2−ǫ(anyǫ>0) or smaller, then no collective effect is visible in the de-\nphasing. An interesting question is what happens for r∼N−1/2+ǫor larger. Then the\nexpression for T2(∞) suggests that the collective interaction may decrease the T2time\ndrastically for large N. However, this range of interaction parameters is not acces sible\nby our perturbation theory approach, and more work in this di rection is required. It\nis important to note here that in order to derive our rigorous result, Theorem 2.1, we\nneed a strong smallness condition on all coupling constants (see (2.11)). As explained\nin Section 5, we expect that our result should hold for collec tive coupling constants\nup to sizeO(N−1/2) and local coupling constants of size O(N0) (relative to the spin\nfrequency). However, for r=O(N−1/2+ǫ) we do not think that usual perturbation\ntheory can be applied, and a different approach should be taken .\nWe also examine the situation where we have two (or more) spec ies of spins, Aand\nB, each species coupled homogeneously to local and collectiv e reservoirs (with a single\ncollective reservoir for both species). We show that the z-component of the magne-\ntization of either species relaxes with single-spin relaxa tion time (associated to that\nspecies). The transverse magnetization dephases followin g a modified Bloch equation\nwith time-dependent T2-time and effective magnetic field. For large times, the T2-time\nof speciesAapproaches the limiting value\n1\nT2,A(∞)=1\n2γrelax,A+γcons,A+(NA−1)γA+NBγB,\nwhereNAandNBarethenumberofspinsineach class, and γA,γB≥0. For small ratio\nrA,rBof the collective and local coupling constants, we have γA=O(r2\nA),γB=O(r2\nB).\nThe total magnetization is the sum of that of species AandB. It is the sum of two\nterms decaying (relaxing and dephasing) at different rates, a nd so we cannot associate\nto it a total relaxation time or a total dephasing time.\nThe effects of collective interactions between effective spins and thermal environ-\nments, discussed in this paper, can represent a significant i nterest, for example, in\nNMR, MRI, and quantum computation. The presence of energy co nserving and energy\nexchange collective effects can be investigated experimenta lly, for example, in NMR\nexperiments by (i) creation and controlling of collective e ffects and (ii) analyzing relax-\nation and dephasing time-scales and time-dependencies of m agnetization as functions\nof characteristic parameters.\n42 Model, single spin dynamics\nWe consider Nnon interacting spins 1 /2 coupled to local and collective bosonic heat\nreservoirs. The full Hamiltonian is given by\nH=−/planckover2pi1N/summationdisplay\nn=1ωnSz\nn+N/summationdisplay\nn=1HRn+HR (2.1)\n+N/summationdisplay\nn=1λnSx\nn⊗φc(gc)+N/summationdisplay\nn=1κnSz\nn⊗φc(fc) (2.2)\n+N/summationdisplay\nn=1µnSx\nn⊗φn(gn)+N/summationdisplay\nn=1νnSz\nn⊗φn(fn). (2.3)\nBelow we use dimensionless variables and parameters. To do s o, we introduce a\ncharacteristic frequency, ω0, typically of the order of spin transition frequency. The\ntotal Hamiltonian, energies of spin states, and temperatur e are measured in units /planckover2pi1ω0.\nThe frequencies of spins, ωn>0, bosonic excitations, ω(k) =c|/vectork|(wherecis the\nspeed of light), the wave vectors of bosinic excitations are normalized by ω0/c, and all\nconstants of interactions are measured in units ω0. A dimemsionless time is defined as\nt→ω0t.\nIn (2.2), (2.3), ωn>0 is the frequency of spin n,\nSz=1\n2/bracketleftbigg1 0\n0−1/bracketrightbigg\nandSx=1\n2/bracketleftbigg0 1\n1 0/bracketrightbigg\n, (2.4)\nandSz,x\nndenotes the Sz,xof spinn.HRis the Hamiltonian of the bosonic collective\nreservoir,\nHR=/integraldisplay\nR3|k|a∗(k)a(k)d3k, (2.5)\nandHRnis that same Hamiltonian pertaining to the n-th individual reservoir. For a\nsquare-integrable form factor h(k),k∈R3,φ(h) is given by\nφ(h) =1√\n2/integraldisplay\nR3{h(k)a∗(k)+h(k)∗a(k)}d3k. (2.6)\nThe real numbers λn,κn,µn,νnare coupling constants, measuring the strengths of\nthe various interactions as follows:\nλnenergy exchange collective coupling\nκnenergy conserving collective coupling\nµnenergy exchange local coupling\nνnenergy conserving local coupling\nWe introduce the maximal size of all couplings,\nα:= max\nn{|κn|,|λn|,|µn|,|νn|}. (2.7)\n5The energies of the Nuncoupled spins are the eigenvalues of H/vectorS=−/summationtextN\nn=1ωnSz\nn,\ngiven by −1\n2/summationtextN\nn=1ωnσn, whereσn∈ {1,−1}. We denote by ϕσ=ϕσ1⊗···⊗ϕσNthe\ncorresponding eigenvector. Bohr energies (energy differences) are thus given by\ne(σ,τ) =−1\n2N/summationdisplay\nn=1ωn(σn−τn). (2.8)\nAssumptions.\n(A)We consider the spin frequencies {ωn}to be uncorrelated in the following sense:\nIfe(σ,τ) =e(σ′,τ′)thenσn−τn=σ′\nn−τ′\nnfor alln. (2.9)\nIn particular, we do not allow any of the ωnto be the equal. However, we can\ndescribe a homogeneous magnetic field within the constraint (2.9) by considering\na distribution ωn=ω+δωnfor some fluctuation δωnhaving, say, uniform distri-\nbution in some interval. Then assumption (A) is satisfied alm ost surely. In view\nof such fluctuations, relation (2.9) is reasonable from a phy sical point of view, and\nits mathematical advantage is that it breaks permutation sy mmetry and hence\nreduces the degeneracies of the energies e.\n(B)The smallest gap between different Bohr energies of the non-in teracting spin\nenergies (2.8) is\n∆ =1\n2min\nmn,m′n\n\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleN/summationdisplay\nj=1ωn(mn−m′\nn)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n\n\\{0}, (2.10)\nwheretheminimumistaken oversequences mn,m′\nn∈ {−2,0,2}. Asourapproach\nis based on perturbation theory of Bohr energy differences, th eir displacement\nunder interaction, which is of size N2α2, should be small relative to ∆,\nN2λ2< <∆. (2.11)\nForωn=ωconstant, we have ∆ = /planckover2pi1ω. Hence for a homogeneous magnetic field\nωn=ω+δωnwith small fluctuation /a\\}bracketle{tδωn/a\\}bracketri}ht/ω< <1, we have ∆ = ω+O(/a\\}bracketle{tδω/a\\}bracketri}ht), and\nther.h.s. of(2.11) isindependentof N. Condition(2.11) isaseriousrestriction on\nthe coupling strength for large systems (big N). Our analysis uses this condition\nin several technical estimates of remainder terms, stemmin g from perturbation\ntheory (see also [16]). However, it is seen from physical con siderations, presented\nin Section 5, that the true condition should read α2\ncN < <ωandαℓ< <ω, where\nαcandαℓare the sizes of collective and local coupling constants, an dωis the\ntypical frequency of a spin.\n(C)Regularity of form factors: denote by hany of the functions fc,gc,fn,gnin the\nHamiltonian H. Letr≥0, Σ∈S2be the spherical coordinates of R3. Then\nh(r,Σ) =rpe−rmh′(Σ), withp=−1/2 +n,n= 0,1,2,...andm= 1,2, and\nwhereh′is anyangularfunction. (Lessrestrictive requirementson harenecessary\n6only [16], but they are more technical to describe, so we rest rict our attention to\nhsatisfying this condition. This family of form factors cont ains the usual physical\nones [8].)\nGivene, (2.8), the number\nN0(e) ={n:σn=τnfor any (σ,τ) withe(σ,τ) =e} (2.12)\ndependson ealone, andthenumberofdifferent configurations ( σ,τ)with constant value\n(2.8) is 2N0(e). There are 2N0(e)elements/angbracketleftbig\nϕσ,ρ/vectorSϕτ/angbracketrightbig\nof the (reduced) density matrix of\nthe spins with fixed value (2.8). As shown in [16], these eleme nts evolve in time jointly,\nand independently of elements associated with any other val ue of (2.8).\nWe consider unentangled initial states\nρ0=ρS1⊗···⊗ρSN⊗ρR1⊗···⊗ρRN⊗ρR,\nwhereρSjare arbitrary single spin states, and ρRj,ρRare thermal equilibrium states\nof single reservoirs, all at temperature T= 1/β >0.\nThe reduced density matrix ρ(j)\ntof spinjis given by\nρ(j)\nt= Tr(j)e−itHρ0eitH,\nthe trace being taken over all spins n/\\e}atio\\slash=jand over all reservoirs.\nLetAjbe an observable of the j-th spin, and denote its dynamics by\n/a\\}bracketle{tAj/a\\}bracketri}htt= Trρ(j)\ntAj,\nwhere the trace is taken over the space of Sj. Our goal is to find a representation of\n/a\\}bracketle{tAj/a\\}bracketri}htt. For a square integrable form factor h(k) =h(|k|,Σ) (spherical coordinates of\nR3), thespectral density of the reservoir associated to his given by\nJh(ω) =πω2/integraldisplay\nS2|h(ω,Σ)|2dΣ.1(2.13)\nDecay rates are given by coupling constants squared times Jh(ω)coth(βω/2) at values\nωcorresponding to Bohr frequencies of the spin system. Energ y-conserving processes\nare associated with the Bohr frequency ω= 0, and since coth( βω/2)∼ω−1asω∼0,\nwe introduce\n/tildewideJh(0) = lim\nω→0+Jh(ω)\nω. (2.14)\n1LetCh(t) =1\n2[/angbracketleftφ(h)eitHRφ(h)e−itHR/angbracketrightβ+/angbracketlefteitHRφ(h)e−itHRφ(h)/angbracketrightβ] be the symmetrized correlation\nfunction of a reservoir in thermal equilibrium at temperatu reT= 1/β, withHRandφ(h) given in (2.5)\nand (2.6). The Fourier transform /hatwideCh(ω) =/integraltext∞\n0e−iωtC(t)dt,ω≥0, is related to the spectral density by\nRe/hatwideCh(ω) =Jh(ω)coth(βω/2).\n7Define the quantities\nbj=1\n4eβωj\neβωj−1/braceleftbig\nλ2\njJgc(ωj)+µ2\njJgj(ωj)/bracerightbig\n(2.15)\ncj= e−βωj(2.16)\nZβ,j= e−βωj/2+eβωj/2(2.17)\nXj=1\n8πP.V./integraldisplay\nRλ2\njJgc(|u|)+µ2\njJgj(|u|)\nu+ωjcoth(β|u|/2)du (2.18)\nYj=1\n8/braceleftbig\nλ2\njJgc(ωj)+µ2\njJgj(ωj)/bracerightbig\ncoth(βωj/2)+1\n2β/braceleftbig\nκ2\nj/tildewideJfc(0)+ν2\nj/tildewideJfj(0)/bracerightbig\n(2.19)\nWith this notation in place we have the following result.\nTheorem 2.1 (Dynamics of single spin) For any observable Ajof spinj,j=\n1,...,N, andt≥0, we have\n/a\\}bracketle{tAj/a\\}bracketri}htt=Z−1\nβ,jTr e−βHSjAj (2.20)\n+e−tbj(cj+1)/braceleftbigg\n[ρ(j)\n0]11−1\ne−βωj+1/bracerightbigg/parenleftbig\n[Aj]11−[Aj]22/parenrightbig\n(2.21)\n+eit(−ωj+Xj+iYj)Cj(N,t) [ρ(j)\n0]21[Aj]12 (2.22)\n+eit(−ωj+Xj+iYj)Cj(N,t) [ρ(j)\n0]21[Aj]21 (2.23)\n+O(α2). (2.24)\nThe quantity Cj(N,t) involves the interaction parameters and the initial condi tion of\nall spins other than j,\nCj(N,t) =/productdisplay\nl/ne}ationslash=j/braceleftbigg/bracketleftBig\neitz+\nl−eitz−\nl/bracketrightBig1+clαl\n1+clα2\nl/parenleftbig\nαl+[ρ(l)\n0]11(1−αl)/parenrightbig\n+eitz−\nl/bracerightbigg\n(2.25)\nz±\nl=1\n2/braceleftbigg\nibl(1+cl)±/radicalBig\n−b2\nl(1+cl)2+4al[al−ibl(1−cl)]/bracerightbigg\n(2.26)\nαl= 1+iz+\nl−al\nblcl(2.27)\nIn (2.26), the square root is the principal value (cut on the n egative real axis), bl,clare\ngiven in (2.15), (2.16), and\nal=−1\n2κ2\nlP.V./integraldisplay\nR3|fc(p)|2\n|p|d3p. (2.28)\nDiscussion of the factor Cj(N,t).\nClearlyCj(N,0) = 1. For vanishing energy-conserving collective couplin g,κl= 0 (all\nl), we have Cj(N,t) = 1 for all t≥0. This follows from al= 0,z+\nl= ibl(1+cl),z−\nl= 0\nandαl=−1/cl.\n8As soon as the collective energy-conserving coupling is swi tched on, κl/\\e}atio\\slash= 0, the\nanalysis of Cj(N,t) is difficult. The factors in the product (2.25) decay with rat e at\nleast\nγj= min{ℑz+\nj,ℑz−\nj}, (2.29)\nso for a homogeneous system (each factor the same) we have the estimate |C(N,t)| ≤\nCNe−γ(N−1)t, with\nγ= min\njγj. (2.30)\nOf course, Cjdoes not depend on janymore. This estimate says that Cdecays in time\nwith rateγ(N−1), but we have a prefactor dependingon N. We have the upperbound\nCN≤e(N−1)c′, for\nc′= ln/braceleftbigg\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1+cα\n1+cα2(α+[ρ0]11(1−α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle+1/bracerightbigg\n>0. (2.31)\nWe know that for t= 0 the true upper bound on |C(N,t)|corresponds to CN= 1, but\nthis does not mean at all that |C(N,t)| ≤e−γ(N−1)t. The estimate\n|C(N,t)| ≤e(N−1)[−γt+c′],withγ= minjγj, see (2.29) (2.32)\nshowsthat |C|decays to 1/2 (half of its initial value) nolater than at time γ−1[ln2\nN−1+c′].\nWe thus call\nγ′=γ/bracketleftbiggln2\nN−1+c′/bracketrightbigg−1\n(γ′≈γ/c′for largeN) (2.33)\nthe decay rate of of |C(N,t)|. Let us examine this decay rate in the two cases where\nr=|al|\nbl∼κ2\nl\nλ2\nl+µ2\nl(2.34)\nis either very small or very close to one. (2.34) very small co rresponds to the situation\nwhere the collective interactions ( κ≈λ) are much smaller than the local ones ( µ≈ν).\nThe situation where (2.34) is unity describes very large col lective coupling relative to\nthe local ones.\nFor small collective coupling ,r≈0, we obtain γ′∼const.rκ, where const .does not\ndepend on N. In the limit r→0 we getγ′= 0, which is the correct behaviour as we\nhave seen above ( C= 1 in this setting, no decay).\nFor large collective coupling ,r≈1, we obtain γ′∼const.b, where const .does not\ndepend onNandbis given in (2.15).\nComparison to exactly solvable model.\nIf the spins interact with the reservoirs only through energ y conserving channels, then\nλn=µn= 0 in (2.2), (2.3). This model is exactly solvable . By proceeding as in [16]\n(“Resonance theory of decoherence and thermalization”, pr oof of Proposition 7.4), one\nfindsthefollowing exact formula fortheevolution ofthereduceddensitymatrixelement\nof a single spin. For simplicity of notation, we take all κnto be constant κc(collective)\n9and allνnconstantνℓ(local). We also take all local form factors equal ( fℓ) and all\ncollective ones too ( fc). Then we find\n[ρ(j)\nt]21= [ρ(j)\n0]21e−iωjte−ν2\nℓΓℓ(t)−κ2\ncΓc(t)/summationdisplay\nσk,k/ne}ationslash=jN/productdisplay\nl,l/ne}ationslash=j[ρ(l)\n0]σlσle−2iσlκ2\ncSc(t).(2.35)\nThe sum is over σk=±1/2,k= 1,...,N,k/\\e}atio\\slash=j, whereσj= 1/2 corresponds to the\nenergy eigenstate ϕ1= [1 0]TofSz(see (2.4)). The decoherence functions and Lamb\nshift are given by\nΓ(t) =/integraldisplay\nR3|f(k)|2coth(β|k|/2)sin2(|k|t/2)\n|k|2d3k (2.36)\nS(t) =−1\n2/integraldisplay\nR3|f(k)|2|k|t−sin(|k|t)\n|k|2d3k. (2.37)\nOf course, the populations are time-independent in this mod el, [ρ(j)\nt]ll= [ρ(j)\n0]llfor all\nt≥0,l= 1,2. The time-dependence in the exponentials in (2.35) become s linear for\nlarge times, Γ( t)→t/tildewideJ(0) andκ2\ncS(t)→taast→ ∞, where /tildewideJ(0) andaare given in\n(2.14) and (2.28) with κlreplaced by κc(see [16]).\nIt is not hard to see that upon the replacements Γ( t)/ma√sto→t/tildewideJ(0) andκ2\ncS(t)/ma√sto→tathe\nexact formula (2.35) coincides precisely with expression ( 2.20)-(2.23) for As.t. [A]11=\n[A]22= [A]21= 0, [A]12= 1 (so that /a\\}bracketle{tA/a\\}bracketri}htt= [ρt]21). The factor Cj(N,t) =C(N,t) is\nthus identified with the sum of the product in (2.35). If all sp ins are initially in the\nsame state, characterized by the population probability 0 ≤p≤1 for the state with\nσ= 1/2, we obtain\nC(N,t) =/bracketleftbig\npe−iat+(1−p)eiat/bracketrightbigN−1,withagiven in (2.28).\nClearly|C(N,t)| ≤1foralltimesandall N. Also, forall n∈Z,wehave |C(N,nπa−1)|=\n1 and|C(N,(n+1\n2)πa−1)|=|1−2p|N−1≈0 forNlarge andp/\\e}atio\\slash= 0,1. Therefore the\nfactorC(N,t) oscillates in size between zero and one, with frequency |a|/πproportional\nto the square of the energy-conserving collective coupling κ2\nc.\n3 Evolution of single spins and of magnetization\n3.1 Single spin relaxation and dephasing times\nThe term on the r.h.s. of (2.20) is the equilibrium average at temperature T= 1/β.\nFrom (2.21) we obtain the relaxation rate of spin j, namelyγrelax,j=bj(cj+1). The\nsingle spin relaxation rate is\nγrelax,j= 1/τrelax,j=1\n4coth(βωj/2)/braceleftbig\nλ2\njJgc(ωj)+µ2\njJgj(ωj)/bracerightbig\n.(3.1)\nThe single-spin relaxation time depends on the local ( µj) and collective ( λj) couplings\nin the same manner: In the relaxation process, the collective reservoir acts as a local\nreservoir.\n10Next we consider the dephasing time determined by (2.22), (2 .23). There are two\ncontributions to the time decay. One comes from spin jitself and is given by Yj, the\nother one comes from all other spins than jand is given by Cj. One sees from (2.26)\nthatℑz±\nl≥0, and that min {ℑz+\nl,ℑz−\nl}= 0⇔albl= 0. It follows that if the energy\nconserving collective coupling and at least one of the energ y-exchange couplings (local\nor collective) do not vanish (so that albl/\\e}atio\\slash= 0), then we have (2.32) with γ >0. The\nsingle-spin dephasing rate is thus Yj+γ′which we can write as\nγdeph,j=1\n2γrelax,j+γcons,j+γ′, (3.2)\nwhere\nγcons,j=1\n2β/braceleftbig\nκ2\nj/tildewideJfc(0)+ν2\nj/tildewideJfj(0)/bracerightbig\n(3.3)\nis a contribution coming purely from the energy-conserving interactions, in which the\nlocal and collective couplings play the same role. The last t erm in expression (3.2) is\ndue to the presence of the N−1 spins other than the considered one. As we have\nseen after (2.34), if the collective coupling is small ( r≈0), thenγ′∼κ2r <<κ2and\nhence the last term in (3.2) is negligible. If the collective coupling is large ( r≈1), then\nγ′∼b∼γrelax,j.\nConclusions. •The single-spin relaxation rate is the sum of two contributi ons\nfrom the local and the collective energy-exchange interact ions (3.1). The collective\nterm has the same form as the local term, and the presence of al l other spins does not\ninfluence the single spin relaxation rate.\n•The single-spin dephasing rate has three contributions (3. 2). One is half the\nrelaxation rate (exchange interactions), one comes from en ergy conserving interactions\n(local and collective), and a third term which is due to the pr esence of all other spins.\nThat last term ( γ′) is negligible for small collective coupling, and renormal izes the\ndephasing rate for strong collective couplings by an amount independent of the number\nof spins.\n3.2 Evolution of magnetization\nLet\n/vectorS=\nSx\nSy\nSz\n\nbe the total magnetization vector, where Sx,y,z=/summationtextN\nj=1Sx,y,z\nj. It is convenient to\nintroduce the complex (non-hermitian) observable\nS−\nj=Sx\nj−iSy\nj.\nWe use Theorem 2.1 with Aj=Sx,y,z\njto obtain\n/angbracketleftbig\nSz\nj/angbracketrightbig\nt=1\n2tanh(βωj/2)[1−e−t/τrelax,j]+e−t/τrelax,j/angbracketleftbig\nSz\nj/angbracketrightbig\n0+O(α2),(3.4)\n/a\\}bracketle{tS−\nj/a\\}bracketri}htt= eit(−ωj+Xj+iYj)Cj(N,t)/a\\}bracketle{tS−\nj/a\\}bracketri}ht0+O(α2). (3.5)\n11Purely local coupling. In the absence of collective coupling ( λn= 0 =κn), the\nabove equations simplify to\n/angbracketleftbig\nSz\nj/angbracketrightbig\nt=1\n2tanh(βωj/2)[1−e−t/τrelax,j]+e−t/τrelax,j/angbracketleftbig\nSz\nj/angbracketrightbig\n0+O(α2),(3.6)\n/a\\}bracketle{tS−\nj/a\\}bracketri}htt= eit(−ωj+Xj+iYj)/a\\}bracketle{tS−\nj/a\\}bracketri}ht0+O(α2). (3.7)\nwhereτrelax,jis given by 1 /γrelax,j,. (3.1) with λj= 0,Xj,Yjare given in (2.18), (2.19)\nwithλj=µj= 0. The factor Cj(N,t) equals 1 (as discussed after (2.28)).\n3.2.1 Homogeneous magnetic field\nIn this section we derive the evolution of the magentization vector in a homogeneous\nmagnetic field, characterized by ωj=ω+δωjwithδωj→0 (see also assumption (B)\nafter (2.11)). This is the description of an elementary volu me of many spins sitting in\na magnetic field with gradient much smaller than the size of th e elementary volume.\nWe consider all spins initially in the same state. We take all local couplings to be\nthe same, i.e., gj=gℓand all collective couplings to be the same, fj=fc, and all\ncoupling constants independent of j. In this limit, we have in formulas (3.4), (3.5)\nωj=ω, X j=X, Y j=Y,\nwhereτrelax,X,Yare given in (3.1), (2.18), (2.19), with ωj,fj,gjand all coupling\nconstants replaced by their constant values, in particular ,\nγrelax= 1/τrelax=1\n4coth(βω/2)/braceleftbig\nλ2Jgc(ω)+µ2Jgℓ(ω)/bracerightbig\n. (3.8)\nFurthermore, we have Cj(N,t) =C(N,t), with (see (2.25))\nC(N,t) = [D(t)]N−1(3.9)\nD(t) =/bracketleftBig\neitz+−eitz−/bracketrightBig1+cα\n1+cα2/parenleftbig\nα+[ρ0]11(1−α)/parenrightbig\n+eitz−.(3.10)\nWe sum equations (3.4) and (3.5) over jto obtain (dropping the O(α2) terms)\n/a\\}bracketle{tSz/a\\}bracketri}htt=N\n2tanh(βω/2)[1−e−t/τrelax]+e−t/τrelax/a\\}bracketle{tSz/a\\}bracketri}ht0 (3.11)\n/a\\}bracketle{tS−/a\\}bracketri}htt= eit(−ω+X+iY)[D(t)]N−1/a\\}bracketle{tS−/a\\}bracketri}ht0. (3.12)\nIt is clear that (3.11) is the integrated version of the Bloch equation\nd\ndt/a\\}bracketle{tSz/a\\}bracketri}htt=−1\nτrelax/bracketleftbig/angbracketleftbig\nSz\nj/angbracketrightbig\nt−N\n2tanh(βω/2)/bracketrightbig\n(3.13)\ncorresponding to the homogeneous magnetic field /vectorB=Bz/vector ez=−ω/vector ez, with relaxation\ntime\nT1=τrelax= 1/γrelax,\n12see (3.8). The Bloch equation for the transverse magnetization would read\nd\ndt/a\\}bracketle{tS−/a\\}bracketri}htt=−1\nT2/a\\}bracketle{tS−/a\\}bracketri}htt+iBz/a\\}bracketle{tS−/a\\}bracketri}htt. (3.14)\nHowever the true evolution, (3.12), is not of this form. By di fferentiating (3.12) we\nobtaind\ndt/a\\}bracketle{tS−/a\\}bracketri}htt=−Γ(t)/a\\}bracketle{tS−/a\\}bracketri}htt+iB(t)/a\\}bracketle{tS−/a\\}bracketri}htt, (3.15)\nwith\nΓ(t) =1\n2γrelax+γcons−(N−1)Red\ndtlnD(t), (3.16)\nB(t) =−ω+X+(N−1)Imd\ndtlnD(t), (3.17)\nwhereγrelaxis the single-spin relaxation rate (3.1) and γconsis the single-spin dephasing\nrate due to the energy-conserving interactions (3.3).\nComparing (3.15) with (3.14) leads us to the identification o f atime-dependent\ndephasing time T2= 1/Γ(t) anda time-dependent effective magnetic field Bz=B(t).\nThe deviation of the true equation of evolution from the Bloc h equation is given by\nthe terms d /dtlnD(t) in (3.16), (3.17). We now estimate the size of this term for w eak\ncollective coupling, where ris small, see (2.34). It is not hard to see that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle˙D(t)\nD(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C|r|,lim\nt→∞˙D(t)\nD(t)= iz−= 4ibrtanh(βω/2)+O(r2), (3.18)\nfor a constant Cindependent of t(andN).\nConclusions. The Bloch equation for the total magnetization (homogeneou s\nmagnetic field) holds with relaxation time T1given by the single-spin relaxation rate\n(3.1) (no influence of the other spins). The total magnetizat ion dephases with a time-\ndependentT2-time,T2=T2(t). We have 1 /T2(t) =1\n2γrelax+γcons+(N−1)ReY′(t), see\nalso (3.1), (3.3). The time-dependent part Y′(t) stems from the collective interaction.\nFor weak collective interaction, rsmall (see (2.34)), we have |Y′(t)| ≤C|r|(all times).\nThe term ( N−1)Red\ndtD(t) in (3.16) is O(Nr2) for large times, see (3.18). If ris\nof the order 1 /√\nNthen this is of order one, and the collective interaction giv es time-\ndependent modification of the dephasing time T2with an asymptotically renormalized\nvalue\n1/T2(∞) =1\n2γrelax+γcons+(N−1)Imz−,Imz−=O(r2).\nIfris smaller than N−1−ǫ(anyǫ >0) then the collective interaction has no effect\nin (3.16), (3.17) and the Bloch equation for transversal rel axation holds with T2the\nsingle-spin dephasing time [ γrelax/2+γcons]−1. For larger collective interaction we may\nget large corrections to the Bloch equation, since the last t erms in (3.16), (3.17) may\nbecome large (big N). This regime does not enter the present perturbative setup , and\nmore work on this issue is needed.\nNote that in any event, since Im z−≥0, the collective interactions can only accel-\nerate the dephasing process.\n133.2.2 Multi-species inhomogeneity\nConsider the situation where Nspins are grouped into two (or more) classes Aand\nB. We describe the situation where within each class, the spin s are homogeneous. We\nhave two magnetic fields ωA,ωB, two sets of coupling constants ( λA,λBetc), two sets\nof form factors ( gc,A,gc,B,gA,gBetc). LetNAandNBbe the relative sizes,\nNA+NB=N.\nIf spinjbelongs to class A, then (2.25) becomes\nCj(N,t) = [DA(t)]NA−1[DB(t)]NB, (3.19)\nwithDA(t),DB(t) given as in (3.10) for species A,B. Let\n/vectorSA=/summationdisplay\njinclassA/vectorSj\nand, correspondingly, for the three components of this vect or. We sum (3.4) and (3.5)\nover all indices of spins belonging to class Ato obtain\n/a\\}bracketle{tSz\nA/a\\}bracketri}htt=NA\n2tanh(βωA/2)[1−e−t/τrelax,A]+e−t/τrelax,A/a\\}bracketle{tSz\nA/a\\}bracketri}ht0+O(α2) (3.20)\n/a\\}bracketle{tS−\nA/a\\}bracketri}htt= eit(−ωA+XA+iYA)[DA(t)]NA−1[DB(t)]NB/a\\}bracketle{tS−\nA/a\\}bracketri}ht0+O(α2). (3.21)\nHence class Arelaxes with single-spin relaxation time τrelax,Aaccording to the usual\nBloch equation (3.18). For the transverse magnetization we obtain again a modified\nBloch equation with time-dependent relaxation time and effec tive magnetic field,\nd\ndt/a\\}bracketle{tS−\nA/a\\}bracketri}htt=−ΓA(t)/a\\}bracketle{tS−\nA/a\\}bracketri}htt+iBA(t)/a\\}bracketle{tS−\nA/a\\}bracketri}htt+O(α2), (3.22)\nwith\nΓA(t) =1\n2γrelax,A+γcons,A−(NA−1)Red\ndtlnDA(t)−NBRed\ndtlnDB(t) (3.23)\nBA(t) =−ωA+XA+(NA−1)Imd\ndtlnDA(t)+NBImd\ndtDB(t). (3.24)\nAs in the previous paragraph, we see that for weak collective coupling and large times,\nΓA(t) converges to1\n2γrelax,A+γcons,A−(NA−1)Imz−\nA, andBA(t) converges to −ωA+\nXA+(NA−1)Imz−\nA. We thus obtain the (asymptotic) dephasing rate for species A,\nγdeph,A(∞) =1\n2γrelax,A+γcons,A+(NA−1)Imz−\nA+NBImz−\nB.2(3.25)\n2A straightforward generalization to sspeciesA1,...,A swith sizes NA1+···+NAs=Ngives the\ntransverse relaxation (i.e., dephasing) rates\nγdeph,Aj(∞) =1\n2γrelax,Aj+γcons,Aj+(NAj−1)ImzAj+/summationdisplay\nk/negationslash=jNAkImzAk,\nforj= 1,...,s. We conclude that the relaxation rate of each species is a sin gle-spin relaxation rate,\nwhile the dephasing contains collective effects. In particu lar, the dephasing rate of class Ajdepends\non all other classes.\n14Recall again that for small collective interaction, Im z−\nA,B=O(r2\nA,B), (2.34).\nConclusions. Thez-component of the total magnetization of each species A\nandBevolves according to the Bloch equation (3.13) with single- spin relaxation rates\nγrelax,Aandγrelax,B, (3.1).\nThe transverse total magnetization of species Aevolves according to a modified\nBloch equation (3.22) (similarly for B). The dephasing time becomes time-dependent\n(3.23), and takes the value T2,A(∞) = 1/γdeph,A(∞), (3.25) for large times and small\ncollective coupling.\nThe total magnetization is the sum of that of species AandB,/a\\}bracketle{tS/a\\}bracketri}htt=/a\\}bracketle{tSA/a\\}bracketri}htt+\n/a\\}bracketle{tSB/a\\}bracketri}htt. Thez-component relaxes as a sum of two exponentially decaying qu antities\nwith different rates (corresponding to AandB). Therefore we cannot associate to it a\ntotal a single decay rate.\nThe total transverse magnetization is the sum of that of spec iesAandB. Each\ncontribution evolves according to the modified Bloch equati on. For large times, the\ndephasing time approaches a renormalized constant value. B eing again a sum of two\nterms decaying at different rates, the total transverse magne tization does not have a\nsingle decay rate.\n4 Proof of Theorem 2.1\nWe setj= 1 in this proof (the case of general jis obtained merely by a change in\nnotation). Following the method developed in [16], the dyna mics ofA1is represented\nas\n/a\\}bracketle{tA1/a\\}bracketri}htt=/angbracketleftbig\nψ0,B1···BNeitKA1Ω/vectorS⊗Ω/vectorR/angbracketrightbig\n. (4.1)\nThe scalar product on the r.h.s. is that of the GNS Hilbert spa ce (“doubled space”).\nHere, Ω /vectorS= ΩS1⊗···⊗ΩSN, Ω/vectorR= ΩR1⊗···⊗ΩRN⊗ΩRand ΩSjis the trace state of Sj,/angbracketleftbig\nΩSj,AΩSj/angbracketrightbig\n=1\n2([A]11+[A]22), and Ω Rjare reservoir equilibrium states at temperature\nT= 1/β.\nTheBjare unique operators (in the commutant of the algebra of obse rvables of\nspinj) satisfying\nψSj=BjΩSj, (4.2)\nwhereψSjis the initial state of Sj.\nThe operator Kis the Liouville operator acting on all spins and all reservo irs,\nsatisfying\nKΩ/vectorS⊗Ω/vectorR= 0. (4.3)\nIts explicit form is easily written down (even though it is so mewhat lengthy, see [16])\nThe main property is the representation\nP/vectorReitKP/vectorR=/summationdisplay\ne,seitε(s)\neQ(s)\ne+O(α2e−γt), (4.4)\nwhereP/vectorR=|Ω/vectorR/a\\}bracketri}ht/a\\}bracketle{tΩ/vectorR|projects out all degrees of freedom of the reservoirs. The su m\nruns over all eof the form (2.8), i.e., eigenvalues of the operator\nL/vectorS=H/vectorS⊗1l/vectorS−1l/vectorS⊗H/vectorS(4.5)\n15acting on C2N⊗C2N(which are also the eigenvalues of Kwithα= 0). For each e\nfixed,sindexes its splitting into ε(s)\ne, 1≤s≤mult(e), as an eigenvalue of K, under\nthe perturbation (2.2) plus (2.3).3We haveε(s)\ne/\\e}atio\\slash=ε(s′)\neunlesss=s′. TheQ(s)\neare the\n(not orthogonal) spectral projections of K, andγ >0 satisfies 0 ≤ ℑε(s)\ne<2γ < T\n(temperature).\nWe now describe the perturbation expansion in αofε(s)\neandQ(s)\ne. Due to As-\nsumption (2.9) the eigenspace of L/vectorSassociated to an eigenvalue eis obtained as fol-\nlows. Associated to eare unique indices 1 ≤j1< j2<···< jN0(e)≤N(recall\n(2.12)) satisfying σj=τj⇔j∈ {j1,...,jN0(e)}for any (σ,τ) withe(σ,τ) =e. Let\n̺= (̺1,...,̺N0(e))∈ {−1,+1}N0(e)and define vectors in C2by\nξ̺j\nj=ξ±\nj=/bracketleftbigg1\nα±\nj/bracketrightbigg\n, (4.6)\n/tildewideξ̺j\nj=/tildewideξ±\nj=1\n1+cj[(α±\nj)∗]2/bracketleftbigg1\ncj(α±\nj)∗/bracketrightbigg\n, (4.7)\naccording to whether ̺j=±1. Here,cjis given in (2.16) and\nα±\nj= 1+iz±\nj−aj\nbjcj(4.8)\nwithaj,bj,z±\njfrom (2.28), (2.15) and (2.26).\nGivene,̺, set\nη(̺)\ne=ϕσ1,τ1⊗···⊗ξ̺1\nj1⊗···⊗ξ̺N0(e)\njN0(e)⊗···⊗ϕσN,τN, (4.9)\n/tildewideη(̺)\ne=ϕσ1,τ1⊗···⊗/tildewideξ̺1\nj1⊗···⊗/tildewideξ̺N0(e)\njN0(e)⊗···⊗ϕσN,τN, (4.10)\nwhereϕσk,τk=ϕσk⊗ϕτk∈C2, and where at locations jk, we replace ϕσk,τkbyξ(or\n/tildewideξ) with the appropriate value of ̺k.\nLethbe a form factor. We define\nGh(u) =/integraldisplay\nS2|h(u,Σ)|2dΣ,andγ+(h) = lim\nu→0+uGh(u).(4.11)\nLet{hn}and{αn}be form factors and coupling constants, respectively. For a n eigen-\n3To be more precise, one has to use a ‘spectral deformation’ Kθof the operator Kin this argument\n[16], but the deformation does not influence the physical res ults.\n16valueeas in (2.8), set\nxe({αn},{hn}) =−1\n8/summationdisplay\n{n:σn/ne}ationslash=τn}α2\nnσnP.V./integraldisplay\nRu2Ghn(|u|)\nu+ωncoth(β|u|/2)du(4.12)\nye({αn},{hn}) =π\n8/summationdisplay\n{n:σn/ne}ationslash=τn}α2\nn(ωn)2Ghn(ωn)coth(βωn/2), (4.13)\ny′\ne=π\n2β/summationdisplay\n{n:σn/ne}ationslash=τn}ν2\nnγ+(fn), (4.14)\ny′′\ne=π\n8βγ+(fc)[e0(e)]2, (4.15)\ne0(e) =/summationdisplay\n{n:σn/ne}ationslash=τn}κn(σn−τn). (4.16)\nNote that the indices over which the sums are taken are the sam e for any pair of spin\nconfigurations ( σ,τ) withe(σ,τ) =e. Furthermore, we define\nXe=xe({λn},gc)+xe({µn},{gn}), (4.17)\nYe=y′′\ne+y′\ne+ye({λn},gc)+ye({µn},{gn}). (4.18)\nThen we have:\nProposition 4.1 Suppose that the numbers e+δ(̺)\ne, where\nδ(̺)\ne=Xe+iYe+N0(e)/summationdisplay\nk=1z̺k\njk, (4.19)\nare distinct for all eand all̺(thezare given in (2.26)). Then, for nonzero, small α,\nthe eigenvalues of (the spectrally deformed) Kare all simple and have the expansion\nε(̺)\ne=e+δ(̺)\ne+O(α4) (4.20)\nwith corresponding eigenprojection\nQ(̺)\ne=|η(̺)\ne/a\\}bracketri}ht/a\\}bracketle{t/tildewideη(̺)\ne|+O(α2). (4.21)\nWe give a proof of the proposition in Section 4.1. Combining t he result of the Propo-\nsition with (4.1) and (4.4) gives\n/a\\}bracketle{tA1/a\\}bracketri}htt=/summationdisplay\ne/summationdisplay\n̺∈{±1}N0(e)eitε(̺)\ne/angbracketleftBig\nψS1···ψSN,B1···BN(|η(̺)\ne/a\\}bracketri}ht/a\\}bracketle{t/tildewideη(̺)\ne|)A1Ω/vectorS/angbracketrightBig\n+O(α2),\n(4.22)\nwith a remainder term uniformly bounded in t≥0. Since /tildewideη(̺)\nebelongs to the range of\nthe spectral projection P(L/vectorS=e), and since\nA1Ω/vectorS=P(LS2=···=LSN= 0)A1Ω/vectorS, (4.23)\n17only the terms e∈spec(L1) ={−ω1,0,0,ω1}in the sum in (4.22) contribute.\nLet us first consider e=−ω1. We haveN0(ω1) =N−1,\nη(̺)\n−ω1=ϕ+−⊗ξ̺2\n2⊗···⊗ξ̺N\nNand/tildewideη(̺)\n−ω1=ϕ+−⊗/tildewideξ̺2\n2⊗···⊗/tildewideξ̺N\nN.(4.24)\nThe term with e=−ω1in (4.22) equals\n/summationdisplay\n̺2,...,̺N∈{±1}eit[−ω1+X−ω1+iY−ω1+/summationtextN\nj=2z̺j\nj+O(α4)](4.25)\n×[ρ(1)\n0]21[A1]12N/productdisplay\nj=2/angbracketleftBig\nψSj,Bjξ̺j\nj/angbracketrightBig/angbracketleftBig\n/tildewideξ̺j\nj,ΩSj/angbracketrightBig\n(4.26)\n= eit(−ω1+X1+iY1+O(α4))C1(N,t)[ρ(1)\n0]21[A1]12, (4.27)\nwhere we set\nC1(N,t) =N/productdisplay\nj=2/bracketleftBig\neitz+\nj/angbracketleftBig\nψSj,Bjξ+\nj/angbracketrightBig/angbracketleftBig\n/tildewideξ+\nj,ΩSj/angbracketrightBig\n+eitz−\nj/angbracketleftBig\nψSj,Bjξ−\nj/angbracketrightBig/angbracketleftBig\n/tildewideξ−\nj,ΩSj/angbracketrightBig/bracketrightBig\n.(4.28)\nLet us analyze the factors of this product. Using (4.6) and (4 .7) we have (omitting the\nindexj)\n/a\\}bracketle{tψS,Bξ̺/a\\}bracketri}ht/angbracketleftBig\n/tildewideξ̺,ΩS/angbracketrightBig\n(4.29)\n=1\n1+c[α̺]2/a\\}bracketle{tψS,B(ϕ11+α̺ϕ22)/a\\}bracketri}ht/angbracketleftBig\nϕ11+c[α̺]∗ϕ22,2−1/2(ϕ11+ϕ22)/angbracketrightBig\n=1+cα̺\n1+c[α̺]2/angbracketleftBig\nψS,B2−1/2(ϕ11+α̺ϕ22)/angbracketrightBig\n=1+cα̺\n1+c[α̺]2/a\\}bracketle{tψS,B{|ϕ1/a\\}bracketri}ht/a\\}bracketle{tϕ1|⊗1l+α̺|ϕ2/a\\}bracketri}ht/a\\}bracketle{tϕ2|⊗1l}ΩS/a\\}bracketri}ht (4.30)\n=1+cα̺\n1+c[α̺]2/parenleftbig\n[ρ0]11+α̺[ρ0]22/parenrightbig\n. (4.31)\nIn the last step, we use that Bcommutes with |ϕj/a\\}bracketri}ht/a\\}bracketle{tϕj|⊗1l, and that BΩS=ψS. Next\nwe note the relation α+α−=−1/c, which can be derived readily, for instance from the\nfact that |ξ+/a\\}bracketri}ht/a\\}bracketle{t/tildewideξ+|+|ξ−/a\\}bracketri}ht/a\\}bracketle{t/tildewideξ−|= 1l. A short calculation then shows that\nζ:=1+cα+\n1+c(α+)2= 1−1+cα−\n1+c(α−)2,\nso that the factor in the product (4.28) becomes\n[ρ0]11/braceleftBig\neitz+ζ+eitz−(1−ζ)/bracerightBig\n+(1−[ρ0]11)/braceleftBig\neitz+α+ζ+eitz−α−(1−ζ)/bracerightBig\n.\nNext, collecting the terms proportional to [ ρ0]11and using\n(1−α−)(1−ζ) =−(1−α+)ζandα+ζ= 1−α−(1−ζ),\n18we obtain formula (2.25).\nOne can transfer the error term down from the exponent: with D=O(α4) we have\ne−tY1+tD−e−tY1= e−tY1/summationtext\nn≥1(tD)n\nn!and hence\n|e−tY1+tD−e−tY1| ≤e−tY1[et|D|−1]≤e−tY1t|D|et|D|(4.32)\n(mean value theorem). Now for |D| ≤Cα4andY1≥cα2>0, the r.h.s. can be\nbounded from above as follows: let ǫ>0, then for α≤cǫ/C, an upper bound is\nCtα4e−tα2c(1−ǫ)≤Cα2e−tα2c(1−2ǫ)sup\nx≥0xe−xcǫ=Cα2\necǫe−tα2c(1−2ǫ).\nThis gives that if Y1≥cα2>0, then for all ǫ>0 andαsmall enough,\ne−tY1+O(α4)= e−tY1+O(α2e−tcα2(1−ǫ)). (4.33)\nThe remainder depends on ǫ. Takingǫ= 1/2 andα<c/(2C), we get\ne−t[Y1+O(α4)]= e−tY1+O(α2) (4.34)\n(uniformly in t≥0). This gives the contribution (2.22).\nSimilarly to (4.27), one shows that the term in (4.22) with e=ω1equals (2.23). To\nderive this, one checks that under the change e/ma√sto→ −e, the exponent in (4.27) undergoes\na complex conjugation and a sign change, and C1turns into its complex conjugate.\nNext weconisder e= 0in (4.22). We have N0(0) =Nandobtain two contributions:\none associated with ε0= 0,η0= Ω/vectorS(see (4.3)), /tildewideη0=/tildewideξ+\n1⊗ ··· ⊗/tildewideξ+\nNand another\ncontribution with ε(−1,1,1,...)\n0 and\nη(−1,1,1,...)\n0 =ξ−\n1⊗ξ+\n2⊗···⊗ξ+\nNand/tildewideη(−1,1,1,...)\n0 =/tildewideξ−\n1⊗/tildewideξ+\n2⊗···⊗/tildewideξ+\nN.(4.35)\nIt is easily seen that these two contributions are (2.20) and (2.21). This completes the\nproof of Theorem 2.1 /squaresolid\n4.1 Level shift operators and proof of Proposition 4.1\nThe total Liouville operator has the form\nK=L0+W, (4.36)\nwhereL0=L/vectorS+L/vectorRis the free (non interacting) Liouville operator, and Wcontains\nall interactions and is of O(α2) (for more detail, see Appendix B of [16], Resonance\ntheory of decoherence and thermalization). To every eigenv alueeofL0we associate\nthe level shift operator\nΛe=−PeWPe(L0−e+i0+)−1PeWPe, (4.37)\nwherePeis the spectral projection of L0onto the eigenvalue e. The eigenvalues of Λ e\nare the second order (in α) corrections to the eigenvalues of Kunder the analytic (in\n19α) perturbation WofL0, see also Section 5 of [16], Resonance theory of decoherence\nand thermalization. Moreover, if ε(̺)\ne=e+δ(̺)\ne+O(α4) are the eigenvalues bifurcating\nout ofeforα/\\e}atio\\slash= 0 then the corresponding eigenprojections of K, to lowest order in α,\nare given by |η(̺)\ne/a\\}bracketri}ht/a\\}bracketle{t/tildewideη(̺)\ne|, where\nΛeη(̺)\ne=δ(̺)\neη(̺)\ne (4.38)\n[Λe]∗/tildewideη(̺)\ne= [δ(̺)\ne]∗/tildewideη(̺)\ne (4.39)\nand/angbracketleftBig\nη(̺)\ne,/tildewideη(̺)\ne/angbracketrightBig\n= 1. For more information on these facts, we refer to Section 6 of [16],\nResonance theory of decoherence and thermalization. We are assuming here that all\nenergiesδ(̺)\neare different, so that the corresponding eigenspaces are one- dimensional.\nThis is generically true in applications, but it is not neces sary for our strategy to work,\nsee e.g. Appendix A of [16], Dynamics of collective decohere nce and thermalization.\nBelow, we give the explicit form of the level shift operators associated with all\neigenvalues e(given by(2.8)). Each level shift operators splits into a su m\nΛe= Λcoll\ne+Λloc\ne (4.40)\nof two operators associated with the local and the collectiv e interactions. We find the\nspectrum and eigenvalues of the level shift operators. In vi ew of the explanations given\nat the beginning of this section, this gives a proof of Propos ition 4.1.\n4.1.1 Collective level shift operator\nLetebe an eigenvalue (2.8) and define\nrn=1\n4κne0(e)P.V./integraldisplay\nR3|fc(k)|2\n|k|d3k (4.41)\nProposition 4.2 (Collective LSO) The collective level shift operator associated to\neis given by\nΛcoll\ne=xe({λn},gc)+i[y′′\ne+ye({λn},gc)]+/summationdisplay\n{n:σn=τn}Mn\ncoll, (4.42)\nwhereMn\ncollacts onspan{ϕ++,ϕ−−}(doubled Hilbert space of n-th spin) as\nMn\ncoll= iπ\n4λ2\nn(ωn)2Ghn(ωn)\neβωn−1/bracketleftbigg1−1\n−eβωneβωn/bracketrightbigg\n−rn/bracketleftbigg1 0\n0−1/bracketrightbigg\n.(4.43)\nA proof of this proposition is obtained along the lines of Pro position 3.7 of [16] (Dy-\nnamics of collective decoherence and thermalization).\n204.1.2 Local level shift operator\nProposition 4.3 (Local LSO) The local level shift operator associated to eis given\nby\nΛloc\ne=xe({µn},{gn})+i[y′\ne+ye({µn},{gn})]+/summationdisplay\n{n:σn=τn}Mn\nloc,(4.44)\nwhereMn\nlocacts onspan{ϕ++,ϕ−−}(doubled Hilbert space of n-th spin) as\nMn\nloc= iπ\n4µ2\nn(ωn)2Ggn(ωn)\neβωn−1/bracketleftbigg1−1\n−eβωneβωn/bracketrightbigg\n. (4.45)\nA proof of this proposition is obtained along the lines of Pro position 5.1 of [16] (Reso-\nnance theory of decoherence and thermalization).\nRemark. The contributions to local and collective level shift opera tors coming\nfrom the energy-exchange interactions are the same.\n4.1.3 Proof of Proposition 4.1\nThe explicit forms of Λcoll\neand Λloc\negiven in Propositions 4.2 and 4.3, and relation\n(4.40) yield\nΛe=Xe+iYe+/summationdisplay\n{n:σn=τn}/braceleftbigg\nibn/bracketleftbiggcn−cn\n−1 1/bracketrightbigg\n−rn/bracketleftbigg1 0\n0−1/bracketrightbigg/bracerightbigg\n. (4.46)\nThe results (4.20), (4.19) and (4.9) for the eigenvalues and eigenvectors follow, and\nsimilarly for its adjoint.\nNote that for e= 0 we have e0(e) = 0, and the level shift operator becomes\nΛ0= iN/summationdisplay\nn=1bn/bracketleftbiggcn−cn\n−1 1/bracketrightbigg\n. (4.47)\nIt follows that z+\nn= 0 andz−\nn= ibn(cn+1), with\nξ+\nn=1√\n2/bracketleftbigg1\n1/bracketrightbigg\n, ξ−\nn=/bracketleftbiggcn\n−1/bracketrightbigg\n,/tildewideξ+\nn=√\n2\n1+cn/bracketleftbigg1\ncn/bracketrightbigg\n,/tildewideξ−\nn=1\n1+cn/bracketleftbigg1\n−1/bracketrightbigg\n.(4.48)\n5 Validity of perturbation expansion\nThe Heisenberg equations of motion corresponding to (2.1)- (2.3) are\n˙Sz\nn=−i[Sz\nn,H] =λcSy\nn⊗φc(gc)+µnSy\nn⊗φn(gn) (5.1)\n˙Sx\nn=ωnSy\nn−κnSy\nn⊗φc(fc)−νnSy\nn⊗φn(fn) (5.2)\n˙Sy\nn=−ωnSx\nn−λnSz\nn⊗φc(gc)+κnSx\nn⊗φc(fc)−νnSz\nn⊗φn(gn) (5.3)\n+νnSx\nn⊗φn(fn). (5.4)\n21For the local and collective annihilation operators an(k) andac(k) we have\ni˙an(k) = [an(k),H] =|k|an(k)+1√\n2µngn(k)Sx\nn+1√\n2νnfn(k)Sz\nn(5.5)\ni˙ac(k) =|k|ac(k)+1√\n2λngc(k)N/summationdisplay\nn=1Sx\nn+1√\n2κnfc(k)N/summationdisplay\nn=1Sz\nn. (5.6)\nThe latter two equations can be integrated,\nan(k,t) = e−i|k|tan(k,0)\n−1√\n2/integraldisplayt\n0e−i|k|(t−τ){µngn(k)Sx\nn(τ)+νnfn(k)Sz\nn(τ)}dτ (5.7)\nac(k,t) = e−i|k|tac(k,0)\n−1√\n2/integraldisplayt\n0e−i|k|(t−τ)/braceleftBigg\nλngc(k)N/summationdisplay\nn=1Sx\nn(τ)+κnfc(k)N/summationdisplay\nn=1Sz\nn(τ)/bracerightBigg\ndτ.(5.8)\nRemembering that φ(h) =1√\n2/integraltext\nR3{h(k)a∗(k)+h(k)a(k)}d3kwe insert (5.7), (5.8) into\n(5.2) to obtain\n˙Sx\nn(t) =ωnSy\nn(t)−κnSy\nn(t)φ(ei|k|tfc)\n+1\n2κnSy\nn(t)/integraldisplay\nR3/integraldisplayt\n0e−i|k|(t−τ)fc(k)/braceleftBigg\nλngc(k)N/summationdisplay\nn=1Sx\nn(τ)+κnfc(k)N/summationdisplay\nn=1Sz\nn(τ)/bracerightBigg\ndτd3k\n+h.c.\n+1\n2νnSy\nn(t)/integraldisplay\nR3/integraldisplayt\n0e−i|k|(t−τ)fn(k){µngn(k)Sx\nn(τ)+νnfn(k)Sz\nn(τ)}dτd3k\n+h.c. (5.9)\nLet us denote by h2(k) any of the products of functions of koccuring in the above\nintegrals (e.g. h2(k) =fc(k)gc(k) etc). Let us analyze the k-integrals in the last\nexpression for ˙Sx\nn. The product of form factors behaves like h2(k) =|k|p1+p2e−|k|/k0,\nwherepj=−1/2+nj,nj= 0,1,2,...(see Assumption (C)). So\n/integraldisplay\nR3ei|k|τh2(k)d3k∼/integraldisplay∞\n0r1+n1+n2eirτe−r/k0dr=∂1+n1+n2τ(−i)1+n1+n2\n1/r0+iτ,\nwhich decays at least as1\n1+τ2(worst case n1=n2= 0). Together with theboundedness\n|Sx,y,z\nn(τ)| ≤1/2 this implies that the integrals over τandkin (5.9) are bounded\nhomogeneously in t≥0, leading to\n˙Sx\nn(t) =ωnSy\nn(t)−κnSy\nn(t)φ(ei|k|tfc)+κnO(λnN+κnN)+νnO(µn+νn).\nFor the validity of perturbation theory homogeneously in t≥0, we should impose\nκ2\nnN,κnλnN < <ω nandν2\nn,νnµn< <ωn. Denoting by αcandαℓthe size of collective\nand local coupling parameters, we thus need α2\ncN < < ω ,αℓ< < ω, whereωis the\n(typical) Bohr frequency of the single spin.\n22Acknowledgements\nWe would like to thank M.A. Espy and P.L. Volegov for useful di scussions. This work\nwas carried out under the auspices of the National Nuclear Se curity Administration of\nthe U.S. Department of Energy at Los Alamos National Laborat ory under Contract\nNo. DE-AC52-06NA25396. Thisresearch byG.P. Berman wassup portedby theLDRD\nProgram at LANL and by the Office of the Director of National Int elligence (ODNI),\nand Intelligence Advanced Research Projects Activity (IAR PA). All statements of fact,\nopinion or conclusions contained herein are those of the aut hors and should not be\nconstrued as representing the official views or policies of IA RPA, the ODNI, or the\nU.S.M. M. Merkli acknowledges the support of NSERC under Dis covery Grant 205247,\nand of the Quantum Institute through the CNLS at LANL.\nReferences\n[1] W.H. Zurek, Environment-induced superselectionrules ,Phys. Rev. D ,26, 18621880\n(1982).\n[2] A.O. Caldeira and A.J. Leggett, Quantum tunnelling in a d issipative system, Ann.\nPhys.149, 374-456 (1983).\n[3] C.W. Gardiner and P. Zoller, Quantum Noise, Springer, 20 00.\n[4] U. Weist, Quantum Dissipative Systems, WSPC, 1999.\n[5] M.A. Nielsen and I.L. Chuang, Quantum Computation and Qu antum Information,\nCambridge University Press, 2000.\n[6] A. Shnirman, Yu. Makhlin, G. Sch¨ on, Noise and Decoheren ce in Quantum Two-\nLevel Systems, Physica Scripta ,T102, 147-154 (2002).\n[7] A.J. Leggett, S. Chakravarty, A.T. Dorsey, Matthew P.A. Fisher, Anupam Garg,\nand W. Zwerger, Dynamics of the dissipative two-state syste m,Rev. Mod. Phys. ,59,\n185 (1987).\n[8] G.M. Palma, K.-A. Suominen, and A.K. Ekert, Quantum comp uters and dissipa-\ntion,Proc. R. Soc. Lond. A ,452, 567-584 (1996).\n[9] Y. Makhlin, G. Sch¨ on, and A. Shnirman, Quantum-state en gineering with\nJosephson-junction devices, Rev. Mod. Phys. ,73, 380-400 (2001).\n[10] Y. Yu, S. Han, X. Chu, S.I Chu, and Z. Wang, Coherent tempo ral oscillations\nof macroscopic quantum states in a Josephson junction, SCIENCE ,296, 889-892\n(2002).\n[11] M.H. Devoret and J.M. Martinis, Implementing qubits wi th superconducting in-\ntegrated circuits, Quantum Information Processing ,3, 163-203 (2004).\n23[12] M. Steffen, M. Ansmann, R. McDermott, N. Katz, R.C. Bialcz ak, E. Lucero, M.\nNeeley, E.M. Weig, A.N. Cleland, and J.M. Martinis, State to mography of capaci-\ntively shunted phase qubits with high fidelity, Phys. Rev. Lett. 97050502-1-4 (2006).\n[13] N. Katz, M. Neeley, M. Ansmann, R.C. Bialczak, M. Hofhei nz, E. Lucero, and\nA. O’Connell, Reversal of the weak measurement of a quantum s tate in a supercon-\nducting phase qubit, Phys. Rev. Lett. ,101, 200401-1-4 (2008).\n[14] A.A. Clerk, M.H. Devoret, S.M. Girvin, Florian Marquar dt, and R.J.\nSchoelkopf, Introductiontoquantumnoise, measurementan damplification, Preprint\narXiv:0810.4729v1 [cond-mat] (2008).\n[15] C.A. Anastassiou, R. Perin, H. Markram, and C. Koch, Eph aptic coupling of\ncortical neurons, Nature Neuroscince ,14, 217-223 (2011).\n[16] M. Merkli, I.M. Sigal, G.P. Berman: Decoherence and the rmalization. Phys. Rev.\nLett.98no. 13, 130401, 4 pp (2007); Resonance theory of decoherence and ther-\nmalization. Ann. Phys. 323, 373-412 (2008); Dynamics of collective decoherence and\nthermalization. Ann. Phys. 323, no. 12, 3091-3112 (2008).\n[17] R.H. Koch, J. Clarke, W.M. Goubau, J.M. Martinis, C.M. P egrum and D.J. Har-\nlingen, Flicker (1 /f) noise in tunnel junction dc SQUIDS, J. Low Temp. Phys. ,51,\nno. 1-2, 207-224 (1983).\n[18] F. Yoshihara, K. Harrabi, A.O. Niskanen, Y. Nakamura, a nd J.S. Tsai, Decoher-\nence of flux qubits due to 1 /fflux noise, Phys. Rev. Lett. ,97, 167001-4 (2006).\n[19] R.C. Bialczak, R. McDermott, M. Ansmann, M. Hofheinz, N . Katz, E. Lucero, M.\nNeeley, A.D.O’Connell, H. Wang, A.N. Cleland, and J.M. Mart inis, 1/fflux noise\nin Josephson phase qubits, Phys. Rev. Lett. ,99, 187006-4 (2007).\n[20] A. Shnirman, I. Martin, and Y. Makhlin, 1 /fnoise and two-level systems in\nJosephson qubits, K. Scharnberg and S. Kruchinin (eds.), In : Electron Correlation\nin New Materials and Nanosystems, 343-356, Springer, 2007.\n[21] J. Clarke, M. Hatridge, and M. M¨ o βle, SQUID-detected magnetic resonance imag-\ning in microtesla fields, Annu. Rev. Biomed. Eng. ,9, 389-413 (2007).\n[22] M. Espy, M. Flynn, J. Gomez, C. Hanson, R. Kraus, P. Magne lind, K. Maskaly,\nA. Matlashov, S. Newman, T. Owens, M. Peters, H. Sandin, I. Sa vukov, L. Schultz,\nA. Urbaitis, P. Volegov and V. Zotev, Ultra-low-field MRI for the detection of liquid\nexplosives, Supercond. Sci. Technol. ,23, 034023-8 (2010) .\n[23] K.V.R. Chary and G. Govil, NMR in Biological Systems Fro m Molecules to Hu-\nmans, Springer, 2008.\n[24] R.W. Simmonds, K.M. Lang, D.A. Hite, S. Nam, D.P. Pappas , and J.M. Martinis,\nem Phys. Rev. Lett., 93, 077003-4 (2004).\n24"    },    {        "title": "1403.6427v1.Simulations_of_magnetic_nanoparticle_Brownian_motion.pdf",        "content": "Simulations of magnetic nanoparticle Brownian motion\nDaniel B. Reeves1and John B. Weaver1\nDepartment of physics and astronomy, Dartmouth College, Hanover, NH 03755,\nUSAa)\nMagnetic nanoparticles are useful in many medical applications because they interact with biology on a cellular\nlevel thus allowing microenvironmental investigation. An enhanced understanding of the dynamics of magnetic\nparticles may lead to advances in imaging directly in magnetic particle imaging (MPI) or through enhanced\nMRI contrast and is essential for nanoparticle sensing as in magnetic spectroscopy of Brownian motion (MSB).\nMoreover, therapeutic techniques like hyperthermia require information about particle dynamics for e\u000bective,\nsafe, and reliable use in the clinic. To that end, we have developed and validated a stochastic dynamical\nmodel of rotating Brownian nanoparticles from a Langevin equation approach. With no \feld, the relaxation\ntime toward equilibrium matches Einstein's model of Brownian motion. In a static \feld, the equilibrium\nmagnetization agrees with the Langevin function. For high frequency or low amplitude driving \felds, behavior\ncharacteristic of the linearized Debye approximation is reproduced. In a higher \feld regime where magnetic\nsaturation occurs, the magnetization and its harmonics compare well with the e\u000bective \feld model. On\nanother level, the model has been benchmarked against experimental results, successfully demonstrating that\nharmonics of the magnetization carry enough information to infer environmental parameters like viscosity\nand temperature.\nKeywords: Magnetic nanoparticles, Brownian stochastic simulations, biosensing\nI. INTRODUCTION TO BROWNIAN NANOPARTICLES\nThere is a wide range of possible uses for magnetic\nnanoparticles (MNPs) in medical applications. Novel\nmodalities image particles themselves as in magnetic\nparticle imaging (MPI)1and particles are also used\nas contrast agents in conventional magnetic resonance\nimaging2. Magnetic spectroscopy of Brownian mo-\ntion (MSB) uses particle dynamics to gather informa-\ntion about the microscopic environments the particles\ninhabit3. A therapeutic method called hyperthermia\ndamages unwanted cells (e.g in cancer treatment) by\ndepositing energy from particles that are unable to ro-\ntate in time with oscillating \felds due to viscous drag or\nanisotropic energy barriers4. In each of these techniques,\nit is necessary to understand the rotational dynamics of\nthe particles in magnetic \felds. Furthermore, simula-\ntion studies can be used by practitioners to choose ideal\nnanoparticle conditions and satisfy their speci\fc needs2.\nIn this work we examine Brownian rotation where par-\nticles physically rotate and experience viscous drag when\nplaced in an oscillating \feld. The physics of Brownian ro-\ntation is governed by the average time it takes a particle\nto relax to an equilibrium value given its surroundings.\nThe Brownian time constant \u001cBis based accordingly on\nthe viscosity \u0011and temperature Tof the suspending \ruid,\nthe hydrodynamic particle volume V, and Boltzmann's\nconstantkB5,6:\n\u001cB=3\u0011V\nkBT: (1)\nBrownian rotation is the dominant physical process when\nthe particles are thermally blocked. This means that the\na)Electronic mail: dbr@Dartmouth.edu.energy barrier for internal magnetic moment relaxation is\ngreater than the thermal energy of the system. If the par-\nticle is not thermally blocked, the electronic spins within\nsingle magnetic domains can rotate in unison, switching\nthe magnetization of the particle. This spin rearrange-\nment is referred to as N\u0013 eel relaxation, and as described\nin Ref.5has a time constant of \u001cN=\u001c0exp(KVc=kBT)\nwhereKis the magnetic anisotropy constant, Vcthe par-\nticle core volume, and \u001c0the N\u0013 eel attempt time charac-\nteristic of the material. Therefore, if a measurement is\ntaken on a scale shorter than the relaxation time, no N\u0013 eel\nspin \rips are expected on average.\nIt is important to note also that both relaxation pro-\ncesses occur without an applied \feld as the probabilities\nof random relaxation increase with heightened thermal\nactivity. In practice, because the particle volume is a\nde\fning element of the blocking energy, the two regimes\nare often separated by the size of the particles. There-\nfore, when we consider larger 100nm diameter particles\nlike those used in magnetic spectroscopy we are con\fdent\nthe dominant mechanism of relaxation is Brownian.\nII. STOCHASTIC MODELING FROM A BALANCE OF\nTORQUES\nIf we consider a dilute sample of MNPs we can imagine\neach as a separate isotropic dipole with a single magne-\ntization direction determined by an internal crystalline\nproperties. In a vacuum, the magnetic torque on such\na dipole is expressed as the cross product between its\nmagnetic moment and the \feld, T=m\u0002B. The mag-\nnetic moment can be found from the material saturation\nmagnetization multiplied by the magnetic core volume\nMsVcore. Nanoparticles are commonly dispersed in a\n\ruid so we must account for viscous drag during rota-arXiv:1403.6427v1  [cond-mat.mes-hall]  25 Mar 20142\ntion. To \frst approximation, this torque is proportional\nto the angular velocity of the particle with magnitude\ngiven by the Stokes-Einstein relation for small Reynold's\nnumber particles6. Written, the torque is T=\u00006\u0011V\n,\nwith\nthe angular velocity, Vthe hydrodynamic volume\nof the particle, and \u0011the \ruid's viscosity.\nAt the nanoscale we must also consider the powerful\nin\ruence of thermal e\u000bects caused by random collisions\nof nanoparticles with the minute molecules of the \ruid.\nIf we assume that the time between such interactions is\nmuch shorter than the reaction time of the particles, we\nare free to consider the statistical \ructuations Marko-\nvian { uncorrelated spatially or temporally. Accordingly\nwe also implement a white noise force Nin our model7.\nWhite noise is characterized theoretically by a \rat power\nspectrum in frequency so that in Fourier space the be-\nhavior is delta autocorrelated in time and space with zero\nmean value. Explicitly,\nhN(t)i= 0hNi(t)Nj(t0)i= 2D\u000eij\u000e(t\u0000t0):(2)\nThus the expectation value of our random force at any\ntime is zero, and no previous force a\u000bects the subsequent\nsteps. The Einstein-Smoluchowski di\u000busion constant D\nfor spherical rotations depends on both the thermal en-\nergy as well as the viscous drag D= 6kBT\u0011V. This\nmay be determined from the \ructuation dissipation the-\norem or the associated Fokker-Planck equation for the\ndynamics8. We now have the complete balance of torques\nT=m\u0002B\u00006\u0011V\n+p\n2DN: (3)\nThe acceleration term proportional to the moment of\ninertia can be ignored because we have already speci\fed a\nlow Reynolds number for the MNPs. This means that the\nfrictional drag forces are su\u000eciently intense as to prevent\ninertial rotation. With this assumption we can simplify\nconsiderably to a \frst order di\u000berential equation\n\n=1\n6\u0011V\u0010\nm\u0002B+p\n2DN\u0011\n: (4)\nWe can further clarify the expression by noting that\nthe magnetic moment vector will experience a tangential\nrate change determined by the perpendicular component\nof the angular velocity of the moment viz. dm=dt=\n\n\u0002m9. Using this in Eq. 4 we obtain the full stochas-\ntic Langevin equation governing rotational dynamics of\na magnetic Brownian particle suspended in \ruid and\nplaced in a magnetic \feld10,11:\ndm\ndt=1\n6\u0011V\u0010\nm\u0002B+p\n2DN\u0011\n\u0002m: (5)\nWe chose to make no further assumptions, instead re-\nsorting to stochastic numerical analysis. The construc-\ntion of the stochastic model is described below. Once\nprepared, we embarked on a series of simulations in order\nto benchmark against theories in various physically per-\ntinent regimes. This determines the strengths and weak-\nnesses of each analytical model, from the static Langevinfunction derived from Boltzmann statistics12, to the weak\n\feld Debye model13,14, and lastly the low frequency ef-\nfective \feld model15.\nIII. OTHER APPROACHES TO MODELING MNPS\nCurrently there is no closed form solution to the\nstochastic Langevin equation, yet insight can often be\nteased from the rotational di\u000busion equation, sometimes\nalso referred to as the Fokker-Planck equation. This ap-\nproach introduces a distribution function for nanoparti-\ncle magnetization that carries all the associated prob-\nability moments. In contrast, the stochastic method\nhas a more transparent di\u000berential equation, but re-\nquires solving for all the moments separately through\nrepeated trials. As in Felderhof and Jones' paper15\nthe orientation of a nanoparticle is presumed to satisfy\nthe Einstein-Smoluchowski equation for the distribution\nfunctionf(m;t), where the azimuthal symmetry of the\ndipole potential energy U=\u0000m\u0001Bresults in a simpli\fed\nrelation that is only a function of the polar angle f(\u0012;t),\n@f\n@t=D\u00141\nsin\u0012@\n@\u0012\u0012\nsin\u0012@f\n@\u0012+E(t) sin2\u0012f\u0013\u0015\n:(6)\nThe di\u000busion constant Dis the same as the stochastic\napproach above. The variable E(t) =mBsin (!t)=kBT\nis a ratio of magnetic to thermal energy with a perfect\nsinusoidal \feld. Eq. 6 could be used to examine the exact\ndistribution function over magnetic moment angles but\nhas not admitted an analytical solution.\nIn 1929, Peter Debye developed a \frst order approx-\nimation for small driving \felds that neglects non-linear\nbehavior, and for our purposes is particularly limited to\ncases with no magnetic saturation13. Linear response\ntheory as used in Debye's method leads to a susceptibil-\nity that is a combination of real (in phase) and complex\n(out of phase) components:\n\u001f=\u001f0+i\u001f00(7)\nso that the magnetization parallel to the driving \feld can\nbe expressed in terms of the Brownian relaxation time\n(Eq. 5), an approximation in and of itself combining mul-\ntiple relaxation mechanisms into one time constant. The\nDebye magnetization along the direction of the applied\n\feld is thus expressed\nM(t) =M0\n1 + (!\u001cB)2(cos!t+!\u001cBsin!t): (8)\nBoth perturbative methods16and a series expansion in\nLegendre polynomials17have been used to develop a\nmore complete solution to Eq. 6. A particularly use-\nful result, the so called e\u000bective \feld model, assumes\nthat the frequency of oscillation is low enough so that\nthe equilibrium distribution remains Maxwellian feq/\nexp(\u0000U=kBT). In low frequency regimes the quasi-static3\nformulation does produce accurate nonlinear dynamics\nfor the MNP magnetization with\ndM(t)\ndt=\u00002M(t)\n\u001cB\u0012\n1\u0000E(t)\n\u000be(t)\u0013\n(9)\nwhere again \u001cBis the Brownian relaxation time (Eq. 5),\nand at any moment in time, \u000be(t) =mB=kBTis the\nargument in the Langevin function,\nL(\u000b) = coth\u000b\u00001\n\u000b; (10)\nchanging in time but not necessarily equal to the per-\nfectly sinusoidal variation E(t) from Eq. 6. This slight\ndi\u000berence encodes the non-linearity into the magnetiza-\ntion to \frst approximation. In practice the model is used\nto \fnd the magnetization in the direction of the applied\n\feld by iterating the di\u000berential equation (Eq. 9) while\nat each time step inverting the Langevin function to \fnd\nthe proper e\u000bective \feld \u000be.\nAside from the two dynamical models we've seen, the\nLangevin function itself provides a benchmark for the\nmagnetization Meq=M0L(\u000b) in the static \feld case,\nprovided the system is allowed to reach equilibrium. The\nthree models presented will be compared to our stochas-\ntic model later in the work and the strengths and weak-\nnesses of each will be discussed.\nIV. NUMERICAL METHODS\nTo simulate the particle dynamics, we developed a\nMonte-Carlo scheme for MNPs in various \felds and en-\nvironmental conditions. To do so, the numerical recipe\nbelow (Eq. 11), was implemented to solve the stochastic\ndi\u000berential equation (Eq. 5) iteratively. Because at root\nwe must admit that we are approximating some colored\nnoise process as a white noise process, we take recourse to\nthe Stratanovich interpretation of stochastic integration.\nThe transformation also has practical bene\fts because\nthe Stratanovich calculus preserves the rules of ordinary\ndi\u000berential calculus (i.e. the chain rule)18. A \fctitious\ndrift term arises during the transformation from the It\u0016 o\ncalculus and hence we can employ a stochastic numerical\nintegration scheme which is transparently analogous to\ncommon methods. The simplest option in the stochas-\ntic realm, the Euler-Marayuma scheme (see for example\nRef.19Ch.7) is su\u000ecient because the slightly increased\naccuracy achieved through higher order methods is out-\nweighed by longer computation times. We interpret the\nwhite noise force accordingly as a Wiener process d W,\nthe derivative of a Markov process and also uncorrelated\nin varying space or time. It can be approximated numer-\nically as a Gaussian distribution scaled byp\n\u0001t19. This\nresults in the numerical algorithm\nm+ \u0001m=m\n6\u0011V(m\u0002H\u0002m\u00002kBTm) \u0001t\n+s\n2kBT\n6\u0011V\u0001tW\u0002mwheremis the magnetic moment magnitude and Wis a\nvector of random numbers with zero mean and unit stan-\ndard deviation in each cartesian direction. At this level of\nanalysis, single particle trajectories are practically mean-\ningless. The quantities of interest are instead statistical\nmoments, often requiring thousands to millions of sam-\nples. To examine equilibrium magnetization, simulation\nlengths were multiples of theoretical time constants. For\nsimulations using oscillating \felds, equilibrium dynamics\nwere determined by converging similarity between peri-\nods. The standard deviation between a cycle and its pre-\ndecessor divided by the amplitude of oscillation was used\na metric for convergence. Cycles were repeated until this\nvalue was less than the threshold of 1%. The constant nu-\nmerical parameters were chosen and varied around those\ntypical of MSB experiments. The mass magnetization\nwas taken from a nanoparticle supplier (Micromod) data\nsheet as 76 emu/g for 100nm diameter particles. The\nparticles are actually made up of many crystal domains,\nbut are treated as though the bulk properties mimic the\ncomponent properties and density is assumed constant\nso that the total magnetic moment can be written in\nits more common units of Joule/Tesla. The viscosity of\nthe solution is assumed to be \u00181cP, near that of water at\nroom temperature (293K). We pick realistic properties in\norder to see the correct physics in each regime of interest\nand this aids in the transition to quantitative comparison\nwith experiments.\nV. MODEL BENCHMARKING\nIn this section, we verify that the model predicts\nwell known physical results including a) the static \feld\nLangevin function, b) the high-frequency, low-amplitude\nDebye model, and c) the e\u000bective \feld model. The \frst\nbenchmark is that with no applied \feld, the relaxation to\nequilibrium is on a time scale determined by Einstein's\nformulation of Brownian motion (Eq. 5). If a sample of\nparticles is prepared to align in some direction but ex-\nperience no external force, their average magnetization\nquickly decays to zero. By \ftting to the normalized data\nan exponential decay of the form M=e\u0000t=\u001c, we obtain\nthe relaxation time. Increasing the viscosity intuitively\nincreases the time constant. Fig. 1 shows the stochastic\nmodel agrees with Einstein's theory with nanosecond pre-\ncision. It should be noted however that as the viscosity\nis decreased, and particularly at the limit where \u0011!0,\nEinstein's model becomes unphysical and the stochastic\nmodel predicts longer relaxation times.\nTo test the model's behavior in a static \feld we begin\nwith a sample of particles initially in the x-direction, and\nemploy a static \feld in the z-direction. The magnetiza-\ntion is allowed to evolve, and the value after \fve time con-\nstants is recorded. This gives a good estimate of the equi-\nlibrium magnetization and agrees with Langevin's theory\nas expected. Raising the viscosity does not change the\noutcome of the magnetization value, only delaying or ex-4\nFIG. 1. Two relaxation curves with associated relaxation\ntimes found from exponential \fts of the stochastic data. The\nexpected relaxation times are given by Einstein's equation for\nBrownian relaxation to be 379.42 \u0016s and 569.13 \u0016s respectively,\nwithin nanoseconds of the simulated data. The simulations\nused Eq. 11 at two viscosities. 106particles were averaged\nbeginning all in the z-direction, with no magnetic \feld.\npediting equilibrium. Increasing the temperature allows\nfor more thermal motion, and therefore lessens the total\nalignment with the \feld e\u000bectively decreasing the \fnal\nmagnetization value. The data are depicted in Fig. 2.\nMany models of MNPs benchmark against the equilib-\nrium and static case, but we have extended ours to os-\ncillating \felds in order to corroborate analytic approx-\nimations. Here we begin by considering an oscillating\n\feld of the form B=B0sin!tapplied with high fre-\nquency (!=2\u0019= 20kHz). In this case, the magnetization\ndoes not saturate and follows the behavior of the Debye\nModel (see Fig. 3).\nHowever, in a regime where particles come closer to\nsaturation, the magnetization obeys di\u000berent dynamics,\nincluding much more severe hysteretic e\u000bects, so that a\nsimple linearization method like Debye's can no longer be\napplied. Described previously, the e\u000bective \feld model\n(Eq. 9) uses an adaptive method to solve for the ef-\nfective thermal \feld from the inverse Langevin function\nand includes relaxation e\u000bects. Simulations demonstrate\nequivalence (Fig. 4) to the e\u000bective \feld model. Yet, the\nstochastic model extends past the range of the e\u000bective\n\feld model, which is only valid for low frequencies. Fur-\nthermore, our model replicates the correct behavior of\nsolely odd harmonics from symmetric distortion. This\nis an important result because magnetization harmonics\nare used exclusively in practical imaging and sensing20,21.\nFIG. 2. The stochastic model (Eq. 11, data points) is com-\npared to the Langevin function (Eq. 10, solid lines) at various\ntemperatures. The particles are initially aligned in the x-\ndirection and the \feld points in the z-direction. 104particles\nwere averaged, few enough to demonstrate that as tempera-\nture increases, the disorder in the system becomes much more\nprevalent.\nFIG. 3. In the low-\feld, high-frequency regime, the stochastic\nmodel (Eq. 11) agrees with the Debye model (Eq. 8). Shown\nis the 20th cycle using 105particles in a 20mT, 2MHz \feld.\nIn this simulation, the driving \feld is a cosine so the phase\nlag can be seen. The almost complete out of phase nature of\nthe curve can be understood by examining the Debye model.\nGiven a relaxation time of hundreds of microseconds and a\nfrequency of 20kHz, the out of phase component is an order\nof magnitude larger than the in phase signal.5\nFIG. 4. The Monte-Carlo code (Eq. 11) matches the e\u000bec-\ntive \feld model magnetization (Eq. 9) and harmonic structure\nwith a 20mT applied \feld of frequency 1kHz. The tempera-\nture is 300K and 105averages are used for a smooth curve.\nNote the applied \feld is sinusoidal and the phase-lag is evi-\ndent.\nVI. EXPERIMENTAL COMPARISON AND VALIDATION\nMagnetic particle spectroscopy can be used to dis-\ncover information about the environment surrounding\nnanoparticles. One of the \frst experiments to do this\nemployed MNPs as microscopic thermometers22. To do\nso, a scaling relationship between the temperature and\nthe magnetic \feld in the argument of the Langevin func-\ntion was exploited. The method makes use of the low\nfrequency, adiabatic limit assumption { that the system\nis assumed to always be in equilibrium and thus gov-\nerned by the Langevin function. Then if a control curve\nis developed for a known temperature, the scaling fac-\ntor necessary to map subsequent curves onto the control\ncan be used to quantify temperature. Note that ratios\nof higher harmonics are used for experimental reasons;\nto avoid dependence on particle number or systematic\nreceive coil errors. Because the simulations have vali-\ndated the Langevin function explicitly, we should not be\ntoo surprised to \fnd that the scaling of \feld and tem-\nperature in the Langevin argument is consistent when\nattempted at low frequencies. Yet, that it works for the\nharmonic ratios (see Fig. 5) is an important con\frma-\ntion. In the simulations, we have neglected any temper-\nature dependence on viscosity or magnetic moment, and\nthe adiabatic approximation fails above a few hundred\ncycles per second, at which point the harmonics do notcarry enough information to distinguish between temper-\natures.\nFIG. 5. A scaling argument can be made for temperature es-\ntimation of nanoparticles as in22. By sweeping through \feld\nstrengths and calculating the harmonic ratio for di\u000berent tem-\nperatures, we show that scaling the x-values by the ratio of\nthe control to new temperature shifts the data back on top\nof one another. Thus for an unknown temperature curve, a\nleast squares \ft can determine the scaling constant and the\ntemperature can be recovered. In this \fgure, a \feld of 200Hz\nwas applied at typical experimental values of 7.5-15mT. The\ntwo curves used 104particles at known temperatures of 293K\nand 303K respectively so in this case the scaling factor is 0.96.\nResults also agree with experiments that determine\nnanoparticle relaxation time due to viscosity shifts23.\nTaking ratios of harmonics at various frequencies, there\nis a characteristic shift in time which compensates for\nthe increasing relaxation time from an increased viscos-\nity. By scaling the frequency range by the appropriate\namount, the data can be shifted back onto a control\ncurve. Then by measuring the relaxation time through\nthe scaling value, obtained with a least-squares regres-\nsion, the nanoparticles can be used as viscometer. The\nsimulations are able to repeat the experimental method\n(Fig. 6). When scaled by viscosity, the curves align. Fur-\nthermore, by changing the speci\fc values of the nanopar-\nticles size and magnetic moment slightly from the values\ngiven by the manufacturer, the actual experimental data\nfrom Rauwerdink's paper can be reproduced.\nVII. CONCLUSIONS\nA Langevin-type stochastic di\u000berential equation for\nthe magnetization of non-interacting isotropic MNPs was6\nFIG. 6. A depiction of three frequency sweeps taking the har-\nmonic ratio of the \ffth to third harmonic as in23. The higher\nviscosity appears lower than the original, but can be scaled\nhorizontally (in frequency) to align with the old curve. This\ncompensates for viscous a\u000bects that are characterized by the\nrelaxation time. The 1.5cP curve is shown explicitly before\nand after shift, while the 2cP is only showed post scaling, to\njustify that the three curves all do align. The simulations\nused a 20mT, 1kHz \feld, 300K, and four runs of 104particles\neach.\ndeveloped and a numerical integration scheme was ap-\nplied to model the particle dynamics in various types\nof magnetic \felds. The simulations successfully mod-\neled the approach to equilibrium (through the classical\nEinstein relaxation time) and the eventual state of ther-\nmal equilibrium was validated by the Langevin function\nat various temperatures. In oscillating \felds, our ap-\nproach agreed with other standard analytical approxi-\nmations, surpassing the ability of both by achieving ac-\ncuracy through a wide frequency range. The agreement\nincluded the magnetization shape as well as it's harmonic\ncontent in a distorted regime. Finally, it was shown that\nthe model reproduces two experimental results from pre-\nvious work on MSB sensing: that the relaxation time\nshift due to viscosity changes can be compensated for by\na scaling in frequency so that nanoparticle solution vis-\ncosity can be measured, and that a similar method can\nbe used in the adiabatic limit to measure temperature by\nscaling \feld strength. One of the large bene\fts of using\nstochastic analysis is the transparency of the di\u000beren-\ntial equation. In the future, we will investigate further\nphysics including the potential of decoupling temperature\nand viscosity e\u000bects into separate relaxation times.VIII. ACKNOWLEDGEMENTS\nThis work was supported by an NIH-NCI grant\n1U54CA151662-01.\n1J. Weizenecker, B. Gleich, J. Rahmer, H. Dahnke, and J. Borgert,\nThree-dimensionoal real-time in vivo magnetic particle imaging,\nPhys. Med. Biol. 54, L1-L10 (2009).\n2Q.A. Pankhurst, J. Connolly, S.K. Jones and J. Dobson, Appli-\ncations of magnetic nanoparticles in biomedicine J. Phys. D 36,\nR167-R181 (2003).\n3J.B. Weaver, The use of magnetic nanoparticles in thermal ther-\napy monitoring and screening: Localization and imaging. J.\nAppl. Phys. 111(7) , 7B317-7B3173 (2012).\n4A.J. Giustini et al, Magnetic nanoparticle hyperthermia in cancer\ntreatment. Nano LIFE 01, 17 (2010).\n5W.F. Brown, Thermal \ructuations of a single domain particle.\nPhys. Rev. 130, 1677 (1963).\n6A. Einstein, Investigations of the Theory of Brownian Movement.\nMineola NY: Dover (1956).\n7D.S. Lemons, An Introduction to Stochastic Processes in Physics.\nBaltimore MA: John's Hopkins Univeristy Press (2002).\n8J.L. Garcia-Palacios, F.J. Lazaro, Langevin-dynamics study of\nthe dynamical properties of small magnetic particles. Phys. Rev.\nB58, 14937-14958 (1998).\n9Thornton and Marion. Classical Dynamics of Particles and Sys-\ntems (Ch10). Belmont CA: Brook/Cole (2004).\n10M. Raible and A. Engel, Langevin equation for the rotation of a\nsingle domain particle. Appl. Organometal. Chem. 18, (10) 536-\n541 (2004).\n11J. Weizenecker, B. Gleich, J. Rahmer, and J. Borgert, Particle\ndynamics of mono-domain particles in magnetic particle imag-\ning, in Proceedings of the 1st International Workshop on Mag-\nnetic Particle Imaging, Magnetic Nanoparticles, pp. 3-15, World\nScienti\fc (2010).\n12N.W. Ashcroft and N.D. Mermin, Solid State Physics (Ch31-33).\nNew York NY: Holt, Rinehart, and Winston (1976).\n13P. Debye, Polar Molecules. New York: Chemical Catalog Co.\n(1929).\n14M.I. Shliomis and V.I. Stepanov. Frequency dependence and long\ntime relaxation of the susceptibility of the magnetic \ruids. J.\nMagn. and Magn. Mater .122, No. 1-3, 176-181, (1993).\n15B.U. Felderhof and R.B. Jones, Nonlinear response of a dipolar\nsystem with rotational di\u000busion to an oscillating \feld. J. Phys.\nCondens. Matter 15, S1363 (2003).\n16A. Morita, and H. Watanabe, Nonlinear response and its behav-\nior in transient and stationary processes. Phys. Rev. A 35, 2690\n(1987).\n17Y.L. Raikher, V.I. Stepanov, and S.V. Burylov, Low-frequency\ndynamics of the orientational birefringence in a suspension of\ndipolar particles. J. Colloid Interface Sci. 144, 308 (1991).\n18B.K. Oksendal, Stochastic Di\u000berential Equations: An Introduc-\ntion with Applications. Springer, Berlin (2003).\n19T.C. Gard. Introduction to Stochastic Di\u000berential Equations.\nNY: Marcel Dekker Inc. (1998).\n20B. Gleich, and J. Weizenecker, Tomographic imaging using the\nnonlinear response of magnetic particles, Nature. 435, 1214-1217\n(2005).\n21A.M. Rauwerdink and J.B. Weaver, Viscous e\u000bects on nanoparti-\ncle magnetization harmonics, J. Magn. Magn. Mater .322, 609-\n613 (2010).\n22J.B. Weaver, A.M Rauwerdink, and E.W. Hansen, Magnetic\nnanoparticle temperature estimation. Med. Phys. 36, 5 (2009).\n23J.B. Weaver and E. Kuehlert. Measurement of magnetic nanopar-\nticle relaxation time. Med. Phys .39, 2765 (2012)."    },    {        "title": "1301.0421v1.Coherent_spin_dynamics_of_nanomolecules_and_magnetic_nanoclusters.pdf",        "content": "arXiv:1301.0421v1  [cond-mat.mes-hall]  3 Jan 2013Coherent spin dynamics of nanomolecules and\nmagnetic nanoclusters\nV.I. Yukalov1and E.P. Yukalova2\n1Bogolubov Laboratory of Theoretical Physics,\nJoint Institute for Nuclear Research, Dubna 141980, Russia\n2Laboratory of Information Technologies,\nJoint Institute for Nuclear Research, Dubna 141980, Russia\nAbstract\nSpin dynamics of nanomolecules and nanoclusters are analyz ed. The nanosizes of\nthese objects make it possible to consider them as single-do main magnets with a large\ntotal spin, where the motion of the spins of all atoms, compos ing a nanocluster, occurs\nin a coherent way. Another meaning of coherence in spin dynam ics is the coherent spin\nmotion of several nanomolecules or nanoclusters. Different a pproaches for treating\nspin dynamics are compared and the main mechanisms influenci ng the spin motion\nare studied. Spin dynamics of separate magnetic nanomolecu les and nanoclusters are\ninvestigated, as well as the spin dynamics of the ensembles o f these nano-objects.\n1 Introduction\nMagnetic nanomolecules and nanoclusters enjoy many similar proper ties because of which\nthe dynamics of their magnetization can be described by the same ty pe of equations. This\nis why, we consider both these nano-objects together. Of cours e, there is difference in their\nstructure and parameters which we shall take into account and ch aracterize them by the\nappropriate models. The detailed description of general physical p roperties and applications\nof different magnetic nanoparticles can be found in review articles [1- 9]. Here we briefly\nmention those of the properties and parameters that will be neces sary for the following\nconsideration.\nIt is worth stressing that there exist two types of magnetic nanop articles. One large\nclass consists of nanoclusters and nanomolecules, whose magnetic moments are formed by\nelectron spins. Another type includes nanomolecules that possess magnetic moments solely\ndue to polarized proton spins. Examples are propanediol C 3H8O2, butanol C 4H9OH, and\nammonia NH 3. In such nanomolecules, there is no any other magnetic moment exc ept that\ncaused by polarized protons. So, here the proton magnetic momen t is not a contribution,\nbut the main object.\nThe magnetic moment of an atom is composed of electron and proton moments, with the\nelectron magnetic moment µe=−geµBS=/planckover2pi1γeS=−µBand the proton magnetic moment\n1µp=gpµNS=/planckover2pi1γpS, wherege= 2 and gp= 5.586 are the electron and proton Land´ e\nfactors,µB=|e|/planckover2pi1/2meandµN=|e|/planckover2pi1/2mpare the Bohr and nuclear magnetons, γeandγp\nare the electron and proton gyromagnetic ratios. Since the proto n mass is larger than that\nof an electron, mp/me∼103, the proton magnetic moment is essentially smaller, µp/µe∼\n10−3. The electron and proton radii are re∼10−15cm andrp∼10−13cm, respectively. An\natom is called magnetic, when its total magnetic moment is nonzero. T he total spin of a\nmagnetic atom can be between 1/2 and S∼10, hence, its magnetic moment can be of order\n1µB−10µB. Atom radii are of order rA∼10−9−10−8cm. Examples of magnetic atoms are\nFe (Iron), Co (Cobalt), Ni (Nickel), Gd (Gadolinium), and Cr (Chrom ium).\nMagnetic nanomolecules are composed of many magnetic atoms an, a s is clear from\ntheir name, are of the nanometer size. An important property of a magnetic nanomolecule\nis that its total magnetic moment can be treated as being due to an e ffective total spin.\nGenerally, the molecule spin can be directed either up or down, with an energy barrier\nbetween these directions of order EA∼10−100 K. At high temperatures, above the blocking\ntemperature TB∼1−10 K, a magnetic molecule behaves as a superparamagnetic particle,\nwhosespinrandomlyoscillatesbetweentheupanddownpositions. Wh ilebelowtheblocking\ntemperature the spin is frozen in one of the directions.\nMagnetic nanoclusters are also made of magnetic atoms that are as sembled together in\na random way. This distinguishes them from magnetic molecules, wher e atoms are strictly\nconnected by chemical bonds. The sizes of nanoclusters can be in t he range between 1 nm\nand100nm, containingabout100 −105atoms. Thesevaluesdefinethe coherence radius Rcoh,\nbelow which a nanocluster is in a single-domain state and can be treate d as a large particle\nwith an effective spin. A cluster, with a size larger than Rcoh, separates into domains with\nopposite magnetizations. Similarly to magnetic molecules at low temper ature, the magnetic\nmoment of a nanocluster, below the blocking temperature TB∼10−100 K, is frozen in one\nof two possible directions. The effective spin of a nanocluster is form ed by electron spins\nand can be as large as S∼100−105.\nThe often considered nanoclusters are made of the magnetic atom s of Fe, Ni, and Co.\nThey can be made of oxides, such as NiO, Fe 2O3, NiFe 2O4or alloys, such as Nd 2Fe14B,\nPr2Fe14B,Tb2Fe14B,DyFe 14B,Pr2Co14B,Sm 1Fe11Ti1,Sm1Fe10V2,Sm2Fe17N23,Sm2Fe17C22,\nSm2Co17, Sm2Co5. To protect nanoclusters from oxidation, one coat them with grap hene or\nnoble metals, forming the double-component nanoclusters, such a s Fe-Au, Co-Au , Co-Ag,\nCo-Cu, Co-Pt, Co-Pd, Ni-Au, Ni-Ag, Ni-Pd, and Mn−-Au. The coating is done be means of\nchemical reactions or laser ablationtechniques. The nanoclusters are produced by employing\nthermal decomposition, microemulsion reactions, and thermal spr aying.\nMagnetic nanoclusters and nanomolecules find numerous application s, among which we\ncan mention magnetic chemistry, biomedical imaging, medical treatm ent, genetic engineer-\ning, waste cleaning, information storage, quantum computing, and creation of radiation\ndevices. Since both nanomolecules and nanoclusters possess many common properties and\ncan be considered as single particles with a large spin, we shall often t alk on nanoclusters,\nimplying that similar effects can be realized with both of them, molecules as well as clusters.\nThe use of these nano-objects requires the existence of two pro perties that contradict\neach other. From one side, to be able to keep memory, a cluster has to enjoy a stable state\nwith itsspin frozen inonedirection. But fromanother side, inorder t o beable to manipulate\nthe cluster magnetization, there should exist a way of suppressing the anisotropy. And it\nis necessary that the spin manipulation could be done sufficiently fast , so that the cluster\n2magnetization could be quickly reversed. Recall that thermal reve rsal is characterized by\nthe Arrhenius law giving the longitudinal relaxation time T1∼exp{EA/kBT}, whereEA\nis the anisotropy energy, so that, at temperatures below the bloc king temperature, the\nmagnetization is frozen.\nMagnetization reversal can be realized by different methods, by ap plying transverse con-\nstant or alternating magnetic fields and short magnetic field pulses [1 0]. To achieve fast\nreversal, one needs to find optimal values for the amplitude, frequ ency, and duration of such\nfield pulses.\nA very efficient method of achieving ultrafast magnetization revers al of magnetic nan-\noclusters has been suggested [11] by employing the acceleration eff ect caused by a resonator\nfeedback field. The efficiency of this method is due to self-optimizatio n of the spin motion\nproducing the resonator field acting back on the spins. Historically, this effect was described\nby Purcell [12] and considered by Bloembergen and Pound [13] using c lassical phenomeno-\nlogical equations. Such equations are not sufficient for describing d ifferent regimes of spin\nmotion. Microscopic theory of spin dynamics has been developed bein g applied to polarized\nproton spins of such molecules as propanediol C 3H8O2, butanol C 4H9OH, and ammonia NH 3\n(see review articles [4,14]) and to magnetic molecules [15-19].\nThe aim of the present paper is threefold. First, we concentrate o n the spin dynamics of\nnanoclusters, comparing the peculiarity of their spin motion with tha t of proton and molec-\nular spins. Second, we analyze the role of other effects, such as th e Nyquist-noise triggering\nand Dicke correlation, studying their influence on the spin dynamics o f nanoclusters. We\nshow that these effects are negligible as compared to the Purcell eff ect. And, third, we\ncompare different approaches to describing spin dynamics, demons trating the advantage of\nusing a microscopic approach based on quantum equations of motion .\n2 Phenomenological classical equations\nDynamics of the magnetic moment Mof a magnetic particle is usually described by the\nclassical equation\ndM\ndt=−|γS|M×Heff+R, (1)\nin which γSis the giromagnetic ratio of the particle with spin SandRis a relaxation term.\nThe effective magnetic field is given by the variational derivative Heff=−δE/δMof the\nparticle energy E. The length of the magnetic moment is conserved, when the right-h and\nside of the equation dM2/dt= 2M·Ris zero.\nChoosing the relaxation term in the form\nR=−α|γS|\nMM×(M×Heff), (2)\none gets the Landau-Lifshitz equation, where αis a dissipation parameter and M≡ |M|.\nUnder form (2), |M|is conserved. The equation was initially derived [20] for describing\nenergy dissipation in the process of magnetic domain wall motion inside bulk ferromagnetic\nmatter. Though it is often applied for treating the dynamics of ferr omagnetic particles [21].\nTaking the relaxation term as\nR=α\nMM×dM\ndt, (3)\n3one comes to the Gilbert equation [22]. This equation, up to a renotat ion of parameters, is\nequivalent to the Landau-Lifshitz equation. Hence, it has the same region of applicability,\nthough it is also used for describing the magnetization rotation of ma gnetic particles [10].\nAnother form of the relaxation term has been advanced by Bloch [23 ] as\nR=−Mx−M∗\nx\nT2ex−My−M∗\ny\nT2ey−Mz−M∗\nz\nT1ez, (4)\nwhereM∗isanequilibrium magnetization, eαareunit coordinatevectors, andtherelaxation\nparametersarecharacterized bythelongitudinalrelaxationtime T1andtransverse relaxation\ntimeT2. Thelatterisalsocalledthedephasingtime. Foranensembleof Nmagneticparticles\nwith a large average spin polarization\ns≡1\nSNN/summationdisplay\nj=1/an}bracketle{tSz\nj/an}bracketri}ht, (5)\nthe transverse term has to be renormalized [16,24] as 1 //tildewideT2= (1−s2)/T2.\nTheLandau-Lifshitsequationhasasingledissipationparameter αandpreserves spherical\nsymmetry, thus, describing isotropic magnetization rotation. Bec ause of these properties, it\nis appropriate for bulk macroscopic ferromagnetic matter with sph erical magnetic symmetry.\nIt may also be used for magnetic clusters, possessing this symmetr y, which, however, is a\nrather rare case.\nThe Bloch equation has two relaxation parameters, T1andT2. Therefore it can describe\nmoregeneral situation ofanisotropicrelaxation, which ismorerealis tic fortreating nanoclus-\nters in a medium or below the blocking temperature, when T2≪T1. The Bloch equations\nhave been employed for considering the electron and nuclear spin mo tion in a strongly coher-\nent regime [25-29] and for spin-polarized129Xe gas [30]. But these equations cannot describe\nthe whole process of spin relaxation starting from an incoherent qu antum stage, for which\na microscopic approach is necessary [29,31,32]. The initial stage of sp in relaxation is trig-\ngered by quantum spin fluctuations that can be identified with noneq uilibrium spin waves\n[4,16,31-33].\n3 Microscopic quantum approach\nIn a self-consistent quantum approach, we start with a microscop ic spin Hamiltonian ˆHthat\nis a functional of spin operators S. The evolution equations are given by the Heisenberg\nequations of motion\ni/planckover2pi1dS\ndt= [S,ˆH]. (6)\nThe advantage of using the quantum approach is in the following. Firs t, it takes into\naccount quantum effects that can be important for small clusters . Hence, it is more general.\nSecond, at the initial stage of free spin relaxation, quantum spin flu ctuations are of principal\nimportance, being the triggering mechanism for starting the spin mo tion. Third, being based\non an explicit spin Hamiltonian makes it possible to control the used app roximations and to\nhave well defined system parameters.\n4We assume that a magnetic cluster isinserted into anmagnetic coil, of nturns andlength\nl, of a resonant electric circuit characterized by resistance R, inductance L, and capacity C.\nThe coil axis is taken along the axis x. Moving magnetic moments induce in the coil the\nelectric current jdescribed by the Kirchhoff equation\nLdj\ndt+Rj+1\nC/integraldisplayt\n0j dt=−dΦ\ndt+Ef, (7)\nin which the magnetic flux Φ = 4 πnMx/clis formed by the mean transverse magnetization\nMx=µ0/summationtextN\nj=1/an}bracketle{tSx\nj/an}bracketri}ht, whereµ0≡/planckover2pi1γS. HereEfis an additional electromotive force, if any.\nThe resonator natural frequency and circuit damping, respectiv ely, are\nω=1√\nLC, γ=R\n2L. (8)\nThe coil current creates the magnetic field\nH=4πn\nclj (9)\nthat is the solution to the equation\ndH\ndt+2γH+ω2/integraldisplayt\n0H(t′)dt′=−4πηdmx\ndt, (10)\nwhereη≡V/Vcoilis the filling factor and\nmx≡Mx\nV=µ0\nVN/summationdisplay\nj=1/an}bracketle{tSx\nj/an}bracketri}ht\nis the transverse magnetization density. The external electromo tive force is omitted. The\nfieldHis the feedback field, created by moving spins and acting back on the m.\n4 Dynamics of a single nanocluster\nThe typical Hamiltonian of a nanocluster is\nˆH=−µ0B·S−D(Sz)2+D2(Sx)2+D4/bracketleftbig\n(Sx)2(Sy)2+(Sy)2(Sz)2+(Sz)2(Sx)2/bracketrightbig\n,(11)\nwhere the total magnetic field\nB=B0ez+B1ex+Hex (12)\nconsists of an external constant field B0, weak transverse anisotropy field B1, and the feed-\nback resonator field H. The anisotropy parameters D,D2,D4are defined by the particular\ntype of considered nanoclusters.\nThe main attention will be payed to the investigation of spin dynamics s tarting from a\nstrongly nonequilibrium initial state, where the magnetization is direc ted opposite to the\nconstant external magnetic field B0.\n5First, we study the influence of the thermal Nyquist noise of the co il in order to un-\nderstand whether it can trigger the spin motion in a nanocluster. Fo r the thermal-noise\nrelaxation time, we find\ntT=4γVcoil\n/planckover2pi1γ2\nSωtanh/parenleftbiggω\n2ωT/parenrightbigg\n, (13)\nwhereωT≡kBT//planckover2pi1isthethermal frequency defined bytemperature T. At lowtemperatures,\nbelow the blocking temperature, say at T= 1 K, we have ωT∼1012s−1. Then the thermal-\nnoise relaxation time is\ntT≃2γVcoil\n/planckover2pi1γ2\nSωT/parenleftbiggω\nωT≪1/parenrightbigg\n. (14)\nOn the other side, for the reversal time, caused by the resonato r feedback field, we have\ntrev≃Vcoil\nπ/planckover2pi1γ2\nSS. (15)\nThe ratio of the latter to the thermal time (13) is trev/tT∼ωT/2πγS. For the typical values\nT= 1 K,γ∼1010s−1, andS∼103, this ratio is small: trev/tT∼10−2. Therefore the\nthermal Nyquist noise does not play any role in the spin dynamics of a n anocluster.\nWe have accomplished numerical solution of the evolution equations f or nanocluster\nparameters typical of Fe, Ni, and Co nanoclusters. The Zeeman fr equency is taken as\nω0≡2µBB0//planckover2pi1∼1011s−1. For the feedback rate, we have γ0≡πη/planckover2pi1γ2\nSS/Vcoil∼1010\ns−1. The typical anisotropy parameters satisfy the relations D/(/planckover2pi1γ0)∼10−3,D2/(/planckover2pi1γ0)∼\n10−3,D4/(/planckover2pi1γ0)∼10−10. At the initial time, the spin is assumed to be directed along the axis\nz. The resonator natural frequency is taken to be in resonance wit h the Zeeman frequency\ndefined by the field B0. The behavior of the spin polarization (5) is shown in Fig. 1, where\nwe compare the spin motion in the presence of the resonator ( h/ne}ationslash= 0) and in the absence of\nthe latter ( h= 0). Clearly, without the resonator feedback field, the spin is block ed, while\nin the presence of the resonator, it reverses in short time trev∼10−10s.\n012345678−1−0.8−0.6−0.4−0.200.20.40.60.81s(t)\nth ≠ 0h = 0\nFigure 1: Spin reversal of a single nanocluster,\nwith parameters typical of nanoclusters made of\nFe, Ni, and Co.\n5 Dynamics of nanocluster assemblies\nThe ensemble of nanoclusters is described by the Hamiltonian\nˆH=/summationdisplay\niˆHi+1\n2/summationdisplay\ni/negationslash=jˆHij, (16)\n6where the indices i,j= 1,2,...,Nenumerate nanoclusters. The single nanocluster Hamil-\ntonians are\nˆHi=−µ0B·Si−D(Sz\ni)2+D2(Sx\ni)2+D4/bracketleftbig\n(Sx\ni)2(Sy\ni)2+(Sy\ni)2(Sz\ni)2+(Sz\ni)2(Sx\ni)2/bracketrightbig\n,(17)\nwith the total magnetic field\nB=B0ez+Hex. (18)\nThe interaction term takes into account the dipolar spin interaction s\nˆHij=/summationdisplay\nαβDαβ\nijSα\niSβ\nj, (19)\nthrough the dipolar tensor Dαβ\nij=µ2\n0/parenleftBig\nδαβ−3nα\nijnβ\nij/parenrightBig\n/r3\nij, in which rij≡ |rij|,nij≡rij/rij,\nandrij≡ri−rj.\nOne sometimes says that spin systems are similar to atomic systems, where transition\ndipoles are correlated by means of the photon exchange through t he common radiation field.\nThis correlation leads to coherent atomic radiation called the Dicke su perradiance [34]. One\nsays that moving spins also radiate electromagnetic field that could y ield the correlated spin\nmotion, in the same way as in the Dicke effect. To check whether this is so, we need to\ncompare the time trad, required for inducing spin correlations through the common radiat ion\nfield with the spin dephasing time T2. As the radiation time [35,36] for nanoclusters, we\nhave\ntrad=3c3\n2/planckover2pi1γ2\nSω3S, (20)\nwhile the spin dephasing time is\nT2=1\n/planckover2pi1ργ2\nSS. (21)\nFor the typical nanocluster density ρ∼1020cm−3andS∼103, the spin dephasing time is\nT2∼10−10s. While for the radiation time (20), with ω∼1011s−1, we have trad∼108s = 10\nyears. The ratio of times (20) and (21) is extremely large: trad/T2= 3c3ρ/(2ω3)∼1018. This\ntells us that the spin motion in no way can be correlated through elect romagnetic radiation.\nThat is, the Dicke effect has no relation to the coherent spin motion. But spins can be\ncorrelated only through the Purcell effect requiring the presence of a feedback field caused\nby a resonator.\nThe feedback rate due to the resonator is\nγ0=πηρ/planckover2pi1γ2\nSS . (22)\nThe reversal time for Ncorrelated nanoclusters becomes\ntrev=1\nγ0=Vcoil\nπ/planckover2pi1γ2\nSSN=t1\nrev\nN, (23)\nwheret1\nrevis the relaxation time (15) for a single nanocluster inside the same coil.\nWe solved the evolution equations for the nanocluster assemblies inv olving the scale\nseparation approach [4,14] that is a generalization of the Krylov-Bo golubov [37] averaging\nmethod. Four classes of spin objects have been investigated.\n7(i) Polarized nuclear materials, such as propanediol C 3H8O2, butanol C 4H9OH, and am-\nmonia NH 3, with the parameters: S= 1/2,ρ= 1022cm−3,T= 0.1K,B0∼104G,ω0∼\n108s−1,λ∼102cm,T1∼105s,T2∼10−5s,τ≡1/γ∼10−6s . Recall that in these\nnanomolecules the magnetization is due to polarized proton spins.\nThe following characteristic times are found: thermal-noise time tT∼1016s∼109years,\nradiation time trad∼1015s∼108years, and reversal time trev∼10−6s.\nTherefore, neither the Nyquist thermal noise nor the photon exc hange through the radi-\nated field play any role in the relaxation process. Spin dynamics, resu lting in the magnetiza-\ntion reversal, is completely due to the action of the resonator feed back field. As is explained\nabove, the same concerns nanomolecules and nanoclusters\n(ii)Nuclearpolarizedferromagnets, whereprotonspinsarepolariz edandinteractthrough\nhyperfine forces with electrons participating in forming ferromagn etic order. In such ma-\nterials, the electron subsystem plays the role of an additional reso nator enhancing effective\nnuclear correlations. Being interested in the motion of nuclear spins , under a fixed mean\nelectron magnetization, we find the reversal time trev∼10−9s.\n(iii) Molecular magnets, such as Mn 12and Fe 8, with the typical parameters: S= 10,ρ=\n1020−1021cm−3,TB= 1K,B0∼105G,ω0∼1013s−1,λ∼10−2cm,ωA≡EA//planckover2pi1∼1010−\n1012s−1,T1∼105−107s,T2∼10−10s. The reversal time is trev∼10−11s.\n(iv) Magnetic nanoclusters composed of Fe, Ni, and Co, at T= 1 K, with the typical\nparameters: S= 103,ρ= 1020cm−3,TB= 10−40K,B0∼104G = 1T,ω0∼1011s−1,λ∼\n1cm,T1∼1034s∼1027years,T2∼10−10s. The reversal time can be very small reaching the\nvaluetrev∼10−12s.\nIn the case of magnetic molecules and, especially, nanoclusters, be cause of their high\nspins, thesystemofmanyclusterscanproducequitestrongcohe rentradiationofthemaximal\nintensity\nImax∼2µ2\n0\n3c3S2ω4N2\ncoh, (24)\nwhereNcoh∼ρλ3is the number of clusters in a coherent packet. The intensity of rad iation\nof magnetic molecules, with Ncoh∼1014, is of order Imax∼105W. And for magnetic\nnanoclusters, with Ncoh∼1020, the radiation intensity can reach Imax∼1012W.\nThere can happen several regimes of spin dynamics depending on th e initial spin polar-\nization, the strength of a triggering pulse, and the effective couplin g parameter\ng≡γγ0ω0\nγ2(γ2\n2+∆2), (25)\nwhereγ2≡1/T2and ∆≡ω−ω0is the detuning from resonance. These regimes for\nnanoclusters can be classified analogously to those occurring for n uclear magnets [4,16]:\nincoherent free relaxation, weakly coherent free induction, weak ly coherent superradiance,\nstrongly coherent pure superradiance, strongly coherent trigg ered superradiance, pulsing\nsuperradiance, and punctuated superradiance [38].\nIt is important to stress that the existence of magnetic anisotrop y in magnetic nanoclus-\nters does not preclude the realization of fast spin reversal, provid ed the external magnetic\nfield is sufficiently strong. The influence of the anisotropy energy EAon the spin reversal\nof a nanocluster system is shown in Fig. 2, where A≡EA/ω0. This regime corresponds to\npure spin superradiance.\n80 0.05 0.1 0.15 0.2−1−0.500.51\nts(t)\nA = 0\nA = 0.5A = 1\nFigure 2: Influence of magnetic anisotropy on the\nspinreversal ofanensemble ofmanynanoclusters,\nwith the parameters typical of Fe, Ni, and Co.\nCoherent dynamics in the spin assemblies, formed by magnetic nanom olecules, have an\nimportant difference from the spin dynamics in an ensemble of magnet ic nanoclusters. Mag-\nnetic molecules are identical and form the systems with well organize d crystalline lattices.\nWhile magnetic nanoclusters vary in their shapes, sizes, and total s pins, which results in an\nessential nonuniform broadening. Computer simulations, accomplis hed together with V.K.\nHenner and P.V. Kharebov, demonstrate that this nonuniformity d oes not destroy coherent\nspin motion. A detailed analysis of the computer simulations, with nonu niform nanocluster\ndistributions, will be presented in a separate publication.\nIn conclusion, we have considered spin dynamics in magnetic nanomole cules and nan-\noclusters, starting from a strongly nonequilibrium state, with the m agnetization directed\nopposite to the applied external magnetic field. We have compared s everal methods of de-\nscribing the spin dynamics, showing that a microscopic approach, ba sed on the quantum\nequations of motion, is the most accurate. We also have analyzed th e influence of different\neffects on spin dynamics. The effects of the Nyquist-noise triggerin g and of Dicke correla-\ntions are found to be negligible for spin systems. This principally disting uishes spin systems\nfrom atomic systems or quantum dot systems [39], where correlatio ns, leading to coherent\nradiance, are caused by the Dicke effect of interactions through t he common radiation field.\nThe feedback field, developing in the resonator, reaches rather h igh values, of the order\nof the applied constant magnetic field. Such a strong feedback field suppresses the influence\nof mutual cluster interactions. Generally, in an ensemble of nanoclu sters of sufficiently high\ndensity, in addition to dipole interactions, there can appear exchan ge interactions [40] that\ncan influence equilibrium properties of nanoclusters. But in the cons idered case of strongly\nnonequilibrium spin dynamics, the exchange interactions are also sup pressed by the self-\norganized resonator feedback field.\nThis important conclusion can be formulated as follows: Coherent spin dynamics are\ncompletely governed by the Purcell effect that is caused by th e action of the resonator feedback\nfield.\nReferences\n[1] BarbaraB, ThomasL,Lionti F,Chiorescu I,andSulpiceA1999 J. Magn. Magn. Mater.\n200167\n[2] Wernsdorfer W 2001 Adv. Chem. Phys. 11899\n9[3] Ferre J 2002 Topics Appl. Phys. 83127\n[4] Yukalov V I and Yukalova E P 2004 Phys. Part. Nucl. 35348\n[5] Wang J and Zeng X C 2009 in: Nanoscale Magnetic Materials and Applications\n(Springer, Berlin), p. 35\n[6] Bedanta S and Kleemann W 2009 J. Phys. D 42013001\n[7] Berry C C 2009 J. Phys. D 42224003\n[8] Beveridge J S, Stephens J R, and Williams M E 2011 Annu. Rev. Anal. Chem. 4251\n[9] Hoang V V and Ganguli D 2012 Phys. Rep. 51881\n[10] Bauer M, Fassbender J, Hillebrands B, and Stamp R L 2000 Phys. Rev. B 613410\n[11] Yukalov V I and Yukalova E P 2012 J. Appl. Phys. 111023911\n[12] Purcell E M 1946 Phys. Rev. 69681\n[13] Bloembergen N and Pound R V 1954 Phys. Rev. 958\n[14] Yukalov V I and Yukalova E P 2000 Phys. Part. Nucl. 31561\n[15] Yukalov V I 2002 Laser Phys. 121089\n[16] Yukalov V I 2005 Phys. Rev. B 71184432\n[17] Yukalov V I, Henner V K, Kharebov P V and Yukalova E P 2008 Laser Phys. Lett. 5\n887\n[18] Yukalov V I, Henner V K and Kharebov P V 2008 Phys. Rev. B 77134427\n[19] Yukalov V I and Yukalova E P 2011 Laser Phys. Lett. 8804\n[20] Landau L D and Lifshitz E M 1935 Phys. Zeits. Sow. 8153\n[21] Henner V, Raikher Y and Kharebov P 2011 Phys. Rev. B 84144412\n[22] Gilbert T L 1956 PhD Thesis (Illinois Institute of Technology, Chica go)\n[23] Bloch F 1946 Phys Rev. 70460\n[24] Abragam A and Goldman M 1982 Nuclear Magnetism: Order and Disorder (Clarendon,\nOxford)\n[25] Yukalov V I 1992 Laser Phys. 2559\n[26] Fokina N P, Khutsishvili K O and Chkhaidze S G 1992 J. Exp. Theor. Phys. 1021013\n[27] Khutsishvili K O and Chkhaidze S G 1992 Physica B 17654\n[28] Fokina N P, Khutsishvili K O and Chkhaidze S G 1992 Physica B 179171\n10[29] Yukalov V I 1996 Nucl. Instrum. Methods Phys. Res. A 37345\n[30] Yoshimi A, Asahi K, Sakai K, Tsuda M, Yogo K, Ogawa H, Suzuki T and Nakagura M\n2002Phys. Lett. A 30413\n[31] Yukalov V I 1995 Phys. Rev. Lett. 753000\n[32] Yukalov V I 1996 Phys. Rev. B 539232\n[33] Rueckriegel A, Kreisel A and Kopietz P 2012 Phys. Rev. B 85054422\n[34] Dicke R H 1954 Phys. Rev. 9399\n[35] Yukalov V I and Yukalova E P 2005 Laser Phys. Lett. 2302\n[36] Yukalov V I 2005 Laser Phys. Lett. 2356\n[37] Bogolubov N N and Mitropolsky Y A 1961 Asymptotic Methods in the Theory of Non-\nlinear Oscillations (Gordon and Breach, New York)\n[38] Yukalov V I and Yukalova E P 2002 Phys. Rev. Lett. 88257601\n[39] Yukalov V I and Yukalova E P 2010 Phys. Rev. B 81075308\n[40] Kondratyev V N and Lutz H O 1998 Phys. Rev. Lett. 814508\n11"    },    {        "title": "1807.08120v1.Uniform_and_Nonuniform_Precession_of_a_Nanoparticle_with_Finite_Anisotropy_in_a_Liquid__Opportunities_and_Limitations_for_Magnetic_Fluid_Hyperthermia.pdf",        "content": "Uniform and Nonuniform Precession of a Nanoparticle with Finite Anisotropy in a\nLiquid: Opportunities and Limitations for Magnetic Fluid Hyperthermia\nT. V. Lyutyy\u0003, O. M. Hryshko, M. Yu. Yakovenko\nSumy State University, 2 Rimsky-Korsakov Street, UA-40007 Sumy, Ukraine\nAbstract\nWe focus on an in-depth study of the forced dynamics of a ferromagnetic single-domain uniaxial nanoparticle placed in\na viscous \ruid and driven by an external rotating magnetic \feld. The process of conversion of magnetic and mechanical\nenergies into heat is a physical basis for magnetic \ruid hyperthermia that is very promising for cancer treatment. The\ndynamical approximation allows us to establish the limits of the heating rate and understand the logic of selection of the\nsystem parameters to optimize the therapy. Based on the developed analytical and numerical tools, we analyze from a\nsingle viewpoint the synchronous and asynchronous rotation of the nanoparticle or/and its magnetization in the following\nthree cases. For the beginning, we actualize the features of the internal magnetic dynamics, when the nanoparticle body\nis supposed to be \fxed. Then, we study the rotation of the whole nanoparticle, when its magnetization is supposed to\nbe locked to the crystal lattice. And, \fnally, we realize the analysis of the coupled motion, when the internal magnetic\ndynamics is performed in the rotated nanoparticle body. In all these cases, we describe analytically the uniform mode, or\nsynchronous rotation along with an external \feld, while the nonuniform mode, or asynchronous rotation, is investigated\nnumerically.\nKeywords: Ferro\ruid, \fnite anisotropy, coupled dynamics, precessional mode, power loss\n1. Introduction\nFerro\ruid [1, 2] applications in biotechnologies and medicine\nare the key of stable interest to these media in recent years.\nHere, one should note targeted drug delivery [3, 4], biosen-\nsors, macro-molecule and virus separation [3, 5, 6], mag-\nnetic \ruid hyperthermia [3, 7, 8]. The latter is a promis-\ning method with principal advantages over conventional\nchemotherapy. First, hyperthermia is a local method and\nconcentrates its action on the injured tissue. Second, hy-\nperthermia operates with nontoxic matters and has com-\nparatively lower side e\u000bects. Here, treatment is realized\nthrough the local heating provided due to the absorption\nof an external alternating \feld by ferromagnetic nanopar-\nticles, which were injected and concentrated around the in-\njured tissue. There are three mechanisms of conversion of\nthe \feld energy into heat: 1) the rotation of the nanopar-\nticle body in a viscous carrier, 2) the damped precession of\nthe particle magnetization inside the nanoparticle, 3) the\nEddy currents inside the particle, which are induced by an\nexternal \feld. The latter is negligibly small, and further\nwe consider the \frst two cases.\nStrictly speaking, magnetic dynamics and mechanical\nmotion interfere with each other. However, in literature\nthe simpli\fed models are often used to describe the inter-\n\u0003Corresponding author\nEmail address: lyutyy@oeph.sumdu.edu.ua (T. V. Lyutyy)action of the nanoparticle with an alternating \feld. Sup-\nposing that the nanoparticle is immobilized, the problem\nof energy dissipation for the nanoparticle driven by a cir-\ncularly polarized \feld is considered both analytically and\nnumerically in [9, 10]. In the mentioned investigations,\nbased on the noise-free single-particle model, the exact ex-\npressions for the power loss were obtained in some speci\fc\ncases, and the numerical simulation has been performed\nfor the general case. The simulations of the nanoparti-\ncle ensemble, where only the magnetic damped precession\nproduces the losses, are reported in [11, 12]. Despite the\nresults obtained in the cited works have su\u000ecient scien-\nti\fc relevance, they do not exhaust the hyperthermia prob-\nlem, because the motion of the nanoparticle bodies in most\ncases has considerable in\ruence on losses.\nIn this regard, another approximation, which should\nbe considered, is the so-called the rigid dipole (or frozen\nmagnetic moment) model. Within this, the nanoparticle\nmagnetization is assumed to be \fxed to its anisotropy axis,\nand the nanoparticle performs the mechanical motion only.\nUsing the rigid dipole model, the response to an external\nalternating \feld and the power losses were treated in terms\nof the complex magnetic susceptibility [13, 14, 15, 16].\nThe detailed microscopic consideration of the stochastic\ndynamics of such nanoparticles was given in [15, 17], and\nthe in\ruence of the dipole interaction on the power loss was\nstudied in detail in [18]. Despite the rigid dipole model is\npartially valid for the system parameters, which are suit-\nPreprint submitted to Journal of Magnetism and Magnetic Materials July 24, 2018 arXiv:1807.08120v1  [cond-mat.mes-hall]  21 Jul 2018able for hyperthermia, even small deviations of the mag-\nnetization from the easy axis can lead to the signi\fcant\nchanges in the dissipation process.\nThe coupled dynamics of the nanoparticle body and its\nmagnetization are complex and cannot be described by a\nsimple superposition of the above mentioned types of mo-\ntion. The model equations were \frstly written in [19], but\nthe discussion of their explicit form has been continued\nuntil now [20, 21, 22, 23]. Special attention deserves also\nthe attempt to describe the energy absorption during the\nforced coupled motion presented in [24]. There, the power\nloss was obtained on the basis of the Lagrangian equa-\ntion in the noise-free and single-particle approximation.\nAfter this, the progress in the description of the energy\ndissipation of a viscously coupled nanoparticle with \fnite\nanisotropy driven by an alternating \feld was achieved in\n[25, 26, 27]. The further development of this model as-\nsumes the accounting of thermal noise \frstly. In this re-\ngard, several important results were obtained in [28, 29],\nbut the role of thermal \ructuations in the energy absorb-\ntion has not yet found.\nThe analysis performed suggests that even in the de-\nterministic, single-particle case, some issues remain un-\nclear that complicates further proceeding. Therefore, in\nthe present study we continue to develop the methodolog-\nical scheme, which is realized in [27]. To understand better\nthe role of both energy dissipation channels, we consider\nconsequently and on common footing the driven magnetic\ndynamics in the \fxed nanoparticle, the mechanical rota-\ntion of the nanoparticle with the \fxed magnetization, and,\n\fnally, the coupled motion of the nanoparticle with \f-\nnite anisotropy. We assume that the rotating \feld acts,\nand there are two types of motion, i.e. the uniform and\nnonuniform precession. The \frst type is treated analyti-\ncally, while for the latter the numerical description is de-\nmanded.\n2. Model and Basic Equations\nWe consider a uniform spherical single-domain ferro-\nmagnetic nanoparticle of radius R, magnetization M(jMj=\nM= const), and density \u001a. This particle performs the\nspherical motion (or motion with the \fxed center of mass)\nwith respect to a \ruid of viscosity \u0011. Then, we assume\nthat the nanoparticle is driven by the external circularly\npolarized \feld\nH(t) =exHcos(\nt) +ey%Hsin(\nt); (1)\nwhere ex,ey,ezare the unit vectors of the Cartesian\nframework, His the \feld amplitude, \n is the \feld fre-\nquency,tis the time, and %is the factor, which determines\nthe polarization type ( %=\u00061).\nThe magnetic energy of the nanoparticle is given by\nW=\u0000HaV\n2M(Mn)2\u0000VMH(t)\u0000VHzMez;(2)whereHais the magnitude of the uniaxial anisotropy, n\nis the unit vector de\fning the anisotropy axis direction,\nHzis the magnitude of the constant \feld. Taking into\naccount the action on the nanoparticle magnetization of\nits crystal lattice, the e\u000bective magnetic \feld acting on\nthe nanoparticle can be written as\nHe\u000b=\u0000V\u00001@W\n@M=HaM\u00001(Mn)n+H(t) +ezHz:(3)\nThe dynamics of the nanoparticle leads to the dissipa-\ntion of its energy W. In accordance with [10], the power\nloss is determined as Q= lim\u001c!1(1=\u001c)R\u001c\n0dtq, whereq=\n\u0000dW=dt is the instantaneous power loss. As follows from\nEq. (2),q=VHe\u000b\u0001dM=dt, and the reduced power loss\neQ=Q=(HaVM\n0) (\n 0is some characteristic frequency,\nwhich is determined on the model approach) is written in\nthe form\neQ= lim\ne\u001c!11\ne\u001cZe\u001c\n0d~the\u000b\u0001_m: (4)\nHere, he\u000b=He\u000b=Hais the reduced e\u000bective \feld, m=\nM=Mis the unit vector, which represents the direction\nof the nanoparticle magnetization, ~t= \n 0tis the reduced\ntime,e\u001c= \n 0\u001c. It is important to highlight that Eq. (4)\nis suitable for both the analytical and numerical further\ntreatment of the energy dissipation problem.\nAs stated above, for the analytical description there\nare three approaches to the nanoparticle dynamics: 1) the\nnanoparticle body is \fxed, and the magnetic moment can\nrotate about the anisotropy axis; 2) the magnetic moment\nis \fxed within the viscously rotated nanoparticle; 3) both\nthe nanoparticle body and its magnetic moment perform\nthe rotation. Let us describe all these approaches in detail.\n2.1. Internal magnetic dynamics\nIn the case of high frequencies, large enough parti-\ncles and/or carrier viscosities, and not so large anisotropy,\nthe nanoparticle body motion is negligible. Here, only\nthe magnetic moment dynamics should be considered. To\nthese purposes, the well-known Landau-Lifshitz-Gilbert equa-\ntion can be utilized\n_M=\u0000\rM\u0002He\u000b+\u000bM\u00001M\u0002_M: (5)\nHere,\r(>0) is the gyromagnetic ratio, \u000b(>0) is the di-\nmensionless damping parameter.\nIn the dimensionless form, Eq. (5) can be rewritten as\n_ m=\u0000\nrm\u0002he\u000b+\u000bm\u0002_ m: (6)\nUsing recursive substitution and taking into account the\nproperties of the vector product, it is easy to show that\nEq. (6) corresponds to\n(1 +\u000b2)_ m=\u0000\nrm\u0002he\u000b\u0000\u000b\nrm\u0002m\u0002he\u000b;(7)\nwhich is more convenient for the numerical treatment. Af-\nter standard transformations and accounting Eq. (1), one\n2can write the set of scalar equations with respect to the\npolar#and azimuthal 'angles of the vector m\n(1 +\u000b2)\n\u00001\nr_#=\u000bhcos#f+hf'\u0000\u000bsin#(cos#+hz);\n(1 +\u000b2)\n\u00001\nr_'=\u000bhcsc#f'\u0000hcot#f+cos#+hz;\n(8)\nwhereh=H=Ha,hz=Hz=Ha,\nf= cos'cos(\nt) +%sin'sin(\nt); (9)\nandf'=@f=@' .\n2.2. Motion of the nanoparticle body\nIn the case of strong anisotropy or weak coupling with\nthe environment, the internal magnetic dynamics can be\nnegligible. And here, the nanoparticle dynamics is de-\nscribed by the rigid dipole model, when the magnetization\nis supposed to be \fxed to the anisotropy axis. This model\nis introduced in [30] and has been successfully used up to\nnow. The main peculiarity of the analytical description is\nthe presence of two vector equations. The \frst equation,\nin fact, is the condition of the rigid body rotation, and the\nsecond one is the second Newton's law for the rotational\nmotion\n_n=!\u0002n;\nJ_!=VMn\u0002H\u00006\u0011V!: (10)\nHere,!is the nanoparticle angular velocity, J(= 8\u0019\u001aR5=15)\nis the nanoparticle moment of inertia, Vis the nanopar-\nticle volume, and dots over symbols represent derivatives\nwith respect to time. When the inertia momentum is too\nsmall and can be neglected, Eqs. (10) are transformed into\na simple form\n_n=\u0000\ncrn\u0002(n\u0002h); (11)\nwhere \ncr=MHa=(6\u0011) is the characteristic frequency of\nthe uniform mechanical rotation. After standard transfor-\nmations and accounting Eq. (1), one can write the set of\nscalar equations with respect to the polar \u0012and azimuthal\n\u001eangles of the vector n\n\n\u00001\ncr_\u0012=hcos\u0012cos(%\nt\u0000\u001e)\u0000hzsin\u0012;\n\n\u00001\ncr_\u001e=hsin\u00001\u0012sin(%\nt\u0000\u001e):(12)\n2.3. The coupled dynamics of the body and magnetic mo-\nment of the nanoparticle\nAs shown in detail in [22], the coupled magnetic dy-\nnamics and the mechanical motion cannot be described\nby a simple superposition of these two types of motion\nbecause of the signi\fcant changes in the basic equations.\nUltimately, it was stated that the coupled dynamics obeys\nthe following pair of coupled equations:\n_n=!\u0002n;\nJ_!=\r\u00001V_M+VM\u0002H\u00006\u0011V!; (13)_M=\u0000\rM\u0002Heff+\u000bM\u00001\u0010\nM\u0002_M\u0000!\u0002M\u0011\n:(14)\nIn the case when the inertia term in (13) is negligible,\nthis equation can be transformed into the more convenient\nform. Then, we transform the equation for the internal\nmagnetic dynamics (14) in order to separate the terms\ncontaining the time derivatives. As a result, we obtain\n\n\u00001\ncr_ n=_ m\u0002n=\nr+ (m\u0002h)\u0002n;\n(1 +\u000b2\n1)\n\u00001\nr1_ m=\u0000m\u0002h1\neff\u0000\u000b1m\u0002m\u0002h1\neff;\n(15)\nwhere\f=\u000bM= 6\r\u0011, \nr1= \nr=(1 +\f),\u000b1=\u000b=(1 +\f),\nh1\neff= (exhcos \nt+ey%hsin \nt) (1 +\f) + (mn)n:(16)\nAfter standard transformations and accounting Eq. (1), we\ncan write the set of scalar equations with respect to the\npolar\u0012and azimuthal \u001eangles of the vector n, as well as\nto the polar #and azimuthal 'angles of the vector m\n(1 +\u000b2\n1)\n\u00001\nr1_#=f1+\u000b1f2;\n(1 +\u000b2\n1)\n\u00001\nr1_'= sin\u00001#(\u000b1f1\u0000f2);\n\n\u00001\nr1_\u0012=\f\u000b\u00001(!ycos\u001e\u0000!xsin\u001e);\n\n\u00001\nr1_\u001e=\f\u000b\u00001\u0002\n!z\u0000cot\u0012(!ysin\u001e\n+!xcos\u001e)\u0003\n;(17)\nwhere\nf1=\u0002\nh(1 +\f) sin(%\nt\u0000\u001e)\u0000Fsin\u0012sin('\u0000\u001e)\u0003\n;\nf2= cos#\u0002\nh(1 +\f) cos(%\nt\u0000\u001e)\n+Fsin\u0012cos('\u0000\u001e)\u0003\n\u0000sin#\u0002\n(1 +\f)hz\n+Fcos\u0012\u0003\n;\nF= cos\u0012cos#+ cos('\u0000\u001e) sin\u0012sin#(=mn)\n!x=_#cos#cos'+ _'sin#sin'\n\u0000(1 +\f)\u0002\nhzsin#sin'+hcos#cos(\nt)\u0003\n;\n!y=_#cos#sin'+ _'sin#cos'\n\u0000(1 +\f)\u0002\nhzsin#cos'+hcos#sin(%\nt)\u0003\n;\n!z= (1 +\f)hsin(%\nt\u0000') sin#\u0000_#sin#:\n(18)\nWe want to underline here that the system Eqs. (17) along\nwith designations Eqs. (18) are appropriate for further nu-\nmerical treatment.\nTherefore, the model equations derived above allow us\nto perform the investigation of the precessional motion of\nthe nanoparticle induced by the external circularly polar-\nized \feld. The approach used neglects thermal \ructua-\ntions. Its validity is discussed in [10, 27]. The forced\nstochastic motion in the simpli\fed cases of the rigidly \fxed\nnanoparticle and the rigid dipole are considered in [31, 32]\nand in [17, 18], respectively. The stochastic motion in the\ncase of the coupled magnetic dynamics and the mechani-\ncal rotation is not completely studied yet. Some issues are\ndiscussed in [28, 29].\n33. Results and Discussion\n3.1. Internal magnetic dynamics\nIf the nanoparticle is supposed to be immobilized, there\nare two modes of the steady-state dynamics of munder\nthe action of the \feld of type Eq. (1) [33, 34, 35]. The\n\frst mode is the uniform rotation, which is performed syn-\nchronously with the external \feld. The second one is the\nnonuniform rotation, when the period of mdoes not co-\nincide with the period of H(t). From the analytical view-\npoint, the uniform mode is characterized by the constant\nprecession and lag angles, #1and'1, where'1='\u0000%\nt.\nAs follows from Eqs. (8), the precession angle satis\fes the\nequation[33, 31]\nh2=1\u0000cos2#1\ncos2#1\u0014\u0012\ncos#1+hz\u0000%e\n1 +\u000b2\u00132\n+\u0012\u000be\n cos#1\n1 +\u000b2\u00132\u0015\n;\n(19)\nand the lag angle is connected to the precession one as\nsin'1=\u0000%\u000be\nh(1 +\u000b2)sin#1: (20)\nHere \n 0= \nr,e\n = \n=\nr. After integration by parts of\nEq. (4), we obtain the general expression for the reduced\npower loss in the case of the periodic mode\neQ=\u000be\n2\n(1 +\u000b2)2sin2#1: (21)\nIn the nonuniform mode, the polar angle #of the vector\nmvaries periodically in time with a period, which does\nnot coincide with the \feld one. The similar oscillations\nare demonstrated by the azimuthal angle 'together with\nthe linear growth in time. This dynamics is accompanied\nby the power losses, which can be investigated only in the\nnumerical way. The di\u000berence scheme for the numerical\ncalculus of the power loss here is written as\neQ=1\nNNX\ni=1\u0014\nhxi(cos#icos'i\u0001#i\u0000sin#isin'i\u0001\u001ei)\n+hyi(cos#isin'i\u0001\u0012i+ sin#icos'i\u0001'i)\n\u0000hzisin#i\u0001#i\u0015\n; (22)\nwhereN=e\u001c=\u0001~t(e\u001c=\u001c\nrand is chosen as 105in the sim-\nulation) is the number of time steps on the external \feld\nperiod, \u0001 ~t(\u001c1=e\n) is the value of the time step within the\nnumerical calculation procedure, #i=#(~ti),'i='(~ti),\n\u0001#i=@#(~ti)\n@~t\u0001~t, \u0001'i=@'(~ti)\n@~t\u0001~t,hxi=hcos(%e\n~ti),hyi=\nhsin(%e\n~ti),hzi=hz+ cos#i.\nThe results of the series of simulations are illustrated\nin Fig. 1. For the uniform mode, these results are in ex-\ncellent agreement with those obtained from Eq. (21). The\nsharp changes of eQare associated with the changes in the\nprecession modes that is discussed in detail in [10, 34, 35].\nFigure 1: (Color online) Model of the \fxed nanoparticle: the most\ntypical dependencies of the power loss on the \feld frequencies for\ndi\u000berent \feld amplitudes. The values of the system parameters are\nthe following: \u000b= 0:1,%= +1,hz= 0. Triangle markers designate\nthe uniform precession in the \"up state\"; circle markers designate the\nuniform precession after the magnetization switching to the \"down\nstate\"; stars markers designate the nonuniform precession; square\nmarkers designate the uniform precession in the \"up state\" again.\nWhen the \feld amplitude is considered to be constant, the\nmost complicated case, which corresponds to the frequen-\ncies near the resonant one, is realized in the following way\n(see the curves for h= 0:21 andh= 0:35). For beginning,\nthe power loss increases with the \feld frequency within\nthe uniform mode, see the curves fractures with the trian-\ngle markers. Then, an abrupt increase in eQis caused by\nthe reorientation or switching to the \"down state\", see the\ncurves fractures with the circle markers. After that, the\nnonuniform mode starts to be generated that is testi\fed\nby a sharp increase in the eQ(e\n) curve, see star markers. It\nis important, the condition #<\u0019= 2 holds predominantly.\nFrom the view point of energy minimizing, this mode is\ngenerated in order to reduce the losses that is clear from\nthe \fgure. Finally, a further sharp increase in eQis the\nconsequence of switching to the uniform mode again, see\ncurves fractures with the squares.\n3.2. Motion of the nanoparticle body\nIf the nanoparticle magnetic moment is \fxed inside, the\nuniform and nonuniform precession modes can be also real-\nized. The \frst of them is the natural solution of Eqs. (12).\nThis mode is characterized by the constant lag angle \u001e1=\n\u001e\u0000%\ntand the constant angle of the precession cone \u00121.\nSubstituting these solutions into Eqs. (12), we derive the\nsystem of algebraic equations for the calculation of \u001e1and\n\u00121\ncos\u00121\u0010\ne\n2\u000bsin\u00121+hcos\u001e1\u0011\n=hzsin\u00121;\ne\n sin\u00121=hsin\u001e1:(23)\n4Here, \n 0= \ncrande\n = \n=\ncr. The average value of the\npower loss can be found easily in this case. The straight-\nforward calculations using Eqs. (23) and Eq. (4) yield\neQ=e\n2sin2\u00121: (24)\nTwo remarks are relevant here. First, Eq. (24) for a small\nangle of the precession cone coincides with the results ob-\ntained by Xi [24] in the linear approximation. And, second,\nwhen the static \feld is absent ( hz= 0), the relationships\n\u00121=\u0019=2, sin\u001e1=e\n=h, andeQ=e\n2are valid.\nTo describe the power loss behavior in the whole range\nof parameters and visualize the data, the numerical sim-\nulation is also demanded here. The di\u000berence scheme for\nthe numerical calculus of the power loss is written as\neQ=1\nNNX\ni=1\u0014\nhxi(cos\u0012icos\u001ei\u0001\u0012i\u0000sin\u0012isin\u001ei\u0001\u001ei)\n+hyi(cos\u0012isin\u001ei\u0001\u0012i+ sin\u0012icos\u001ei\u0001\u001ei)\n\u0000hzsin\u0012i\u0001\u0012i\u0015\n; (25)\nwhereN=e\u001c=\u0001~t(e\u001c=\u001c\ncrand is chosen as 105in the\nsimulation) is the number of time steps on the external\n\feld period, \u0001 ~t(\u001c1=e\n) is the value of the time step within\nthe numerical calculation procedure, \u0012i=\u0012(~ti),\u001ei=\u001e(~ti),\n\u0001\u0012i=@\u0012(~ti)\n@~t\u0001~t, \u0001\u001ei=@\u001e(~ti)\n@~t\u0001~t,hxi=hcos(%e\n~ti),hyi=\nhsin(%e\n~ti).\nAs follows from the analytical results discussed above,\nwhenh>e\n, the nanoparticle is rotated uniformly, and all\ncontributions into the power loss are due to this rotation.\nThis is con\frmed by the series of simulations, the results of\nwhich are shown in Fig. 2, see the triangle markers. At the\nsame time, when h<e\n andh\u0018e\n, the dynamics becomes\nnonuniform, see the star markers. Similar to the previous\ncase, here nperforms the rotation simultaneously with the\noscillations of a larger period. Since the nonuniform pre-\ncession is characterized by smaller instantaneous angular\nvelocity of the nanoparticle, a decrease in the power loss is\nobserved. It is expressed in a pronounced drop of eQ(e\n) for\nthe \fxed amplitude h(see Fig. 2). There are two features,\nwhich should be underlined in this regard. First, while the\n\feld frequency grows, the average angular velocity tends\nto zero, the oscillation frequency tends to e\n, and the oscil-\nlation amplitude tends to the saturated values predicted\nby Eq. (34) in [27]. Second, the resulting power loss in the\nnonuniform mode depends on the initial position of the\nnanoparticle.\n3.3. The coupled dynamics of the body and magnetic mo-\nment of the nanoparticle\nIn the case of synchronous precession of the vectors\nmandnwith the external circularly polarized \feld (see\nFig. 3), the stationary solution of the set of equations (13),\n(14) can be obtained in the form '=%\nt\u0000'1,#=#1,\n\u001e=%\nt\u0000\u001e1,\u0012=\u00121. To \fnd the unknown constants\nFigure 2: (Color online) Model of the \fxed magnetization: the most\ntypical dependencies of the power loss on the \feld frequencies for\ndi\u000berent \feld amplitudes. The values of the system parameters are\nthe following: \u0011= 0:05P,%= +1,hz= 0, and the initial condition\n\u00120= 0:01. Triangle markers designate the uniform precession; star\nmarkers designate the nonuniform precession.\n'1,\u001e1and#1,\u00121, we used the condition of absence of the\nmagnetic moment motion with respect to the nanoparticle\ncrystal lattice\n_m\u0000!\u0002m= 0: (26)\nAfter substitution of Eq. (26) into the second equation of\nEqs. (13) and neglecting the inertia term, we derive\n0 = \ncr(m\u0002n)(mn) +!: (27)\nThen, let us introduce the double-primed coordinate sys-\ntemx00y00z00, which is rotated along with the external \feld\nas follows from Fig. 3. In this new framework, the angular\nvelocity has a very simple form\n!00= (\u0000%\n sin\u00121;0;0): (28)\nSince Eq. (14) cannot be easily represented in the double-\nprimed system, we need to write the explicit form of all the\nvectors in the laboratory coordinate system. To perform\nthe necessary transformations, we should use the rotation\nmatrix\nC\u00001=0\n@cos\u00121cos \b 1\u0000sin \b 1sin\u00121cos \b 1\ncos\u00121sin \b 1cos \b 1sin\u00121sin \b 1\n\u0000sin\u00121 0 cos \u001211\nA;(29)\nwhere \b 1=%\nt\u0000\u001e1. Let us introduce the designation\nF1= sin\u00121sin#1cos(\u001e1\u0000'1) + cos\u00121cos#1 (30)\nand represent the vectors !,m, and nin the laboratory\nsystem using the rotation matrix Eq. (29). This allows us\n5Figure 3: Schematic representation of the behaviour of the vectors\nn,mand the coordinate systems used for the precession mode de-\nscription\nto straightforwardly obtain the set of algebraical equations\n\u0000sin#1sin'1\u0012%\n\nr\u0000hz\u0013\n= sin\u00121cos\u00121cos\u001e1%\n\ncr;\nhsin#1sin'1= sin2\u00121%\n\ncr;\nF1sin(#1\u0000\u00121) sin'1= sin#1\u0012%\n\nr\u0000hz\u0013\n;\nF1sin\u00121sin('1\u0000\u001e1) =hsin'1: (31)\nPerforming the direct integration of Eq. (4) and accounting\nthe representation of the vectors mandnin the spherical\ncoordinates\n_m= (\u0000%\n sin#1sin \t 1;%\n sin#1cos \t 1;0);(32)\nn= (sin\u00121cos \b 1;sin\u00121sin \b 1;cos\u00121); (33)\nwhere \t 1=%\nt\u0000'1, we obtain\neQ= 2%e\n sin#1sin'1: (34)\nHere,e\n = \n=\nr1. It is important to note that Eq. (34) is\nsimilar to the Eq. (21) bearing in mind Eq. (20).\nUsing Eq. (4) and the representation of the vectors\nmandnin the spherical coordinates, one can derive the\nexpression for the numerical calculation of the power loss.\nIt corresponds to the expression (25) with the following\ndi\u000berences:e\u001c=\u001c\nr1(is chosen as 105in the simulations),\nhxi=hcos(%e\n~ti) +Fisin\u0012icos\u001ei;\nhyi=hsin(%e\n~ti) +Fisin\u0012isin\u001ei;\nhzi=hz+Ficos\u0012i; (35)\nFi=F(~ti),\u0012i=\u0012(~ti),\u001ei=\u001e(~ti).\nAn additional degree of freedom inspires more inter-\nesting and complicated behaviour of the nanoparticle dy-\nnamics. First, like in the previous cases, a conventional\nnonuniform mode, which is characterised by oscillations of\nFigure 4: (Color online) Model of the nanoparticle in a \ruid with\n\fnite anisotropy: the most typical dependencies of the power loss\non the \feld frequencies for di\u000berent \feld amplitudes. The values\nof the system parameters are the following: \u000b= 0:1,\u0011= 0:006P,\nM= 338G,%= +1,hz= 0. The simulation in the chosen range\nof the \feld parameters is not sensitive to the initial conditions. Tri-\nangle markers designate the uniform precession in the \"up state\";\ncircle markers designate the uniform precession after switching to\nthe \"down state\"; star markers designate the nonuniform precession;\nsquare, hexagonal and down triangle markers designate the nonuni-\nform mode with the immobilized magnetic moment in the \"up state\";\ndiamond markers designate the nonuniform mode with the immobi-\nlized magnetic moment in the \"down state\".\nthe precession angles, is generated. We recall that the pe-\nriod of these oscillations does not coincide with the \feld\none. Second, like in the case of the \fxed nanoparticle, the\nswitching between two uniform modes, which is charac-\nterised by di\u000berent nanoparticle orientations, can occur.\nMoreover, another one interesting mode takes place. It is\ncharacterised by the immobilized magnetic moment, while\nthe nanoparticle body performs oscillations: the angles\n\u0012and\u001eof the vector nvary synchronously along with\nthe external \feld without a recognizable drift, while the\nangles#and'of the vector mremain practically con-\nstant. Such a type of motion is spread enough. It occurs\nin the \"up state\", \"down state\" and around the exter-\nnal \feld polarization plane. The transitions between the\ndiscussed types of motion are exhibited in the jumps in\nthe dependence eQ(e\n)). Ultimately, we conclude that the\nswitching between the uniform and nonuniform modes is\naccompanied by an abrupt increase in the power loss. The\nswitching between two uniform modes with the nanoparti-\ncle reorientation leads, at least, to one-order reduction of\nthe power loss. The most typical dependencies eQ(e\n) are\ndepicted in Fig. 4).\n4. Conclusions\nThe forced coupled dynamics of the nanoparticle body\nand its magnetization has been considered in the determin-\nistic approximation. The approach is based on the torque\n6equation and equation of damped precession of the mag-\nnetization, which are derived from the total momentum\nconservation law. Within this framework, two modes of\nmotion under the action of the circularly polarized \feld\nare described. While both the nanoparticle easy axis and\nits magnetic moment perform precession within the cones\nwith constant angles, we call this the uniform mode. While\nthe precession becomes unstable and the polar angle, at\nleast, for the easy axis undergoes the periodical changes,\nwe call this the nonuniform mode. To understand well the\nmechanisms of energy dissipation, we present the results\nfor the coupled motion together with the results for the\ncases of the rigidly \fxed nanoparticle inside the solid ma-\ntrix and the rigidly \fxed magnetization inside the mobile\nnanoparticle.\nSince the external \feld is supposed to be rotating, it\nis natural that the simplest mode of the forced motion\nis the uniform precession. It is characterised by the con-\nstant precession angles and lag angles, the values of which\ndepend on the system parameters. The algebraical equa-\ntions for these dependencies have been obtained for all the\ncases considered. The main properties of the solution of\nthe derived algebraical equations for the coupled dynam-\nics case are the following. First, the nanoparticle mag-\nnetic moment always constitutes a smaller angle with the\nexternal \feld than with the anisotropy axis. Second, the\nprecessional dynamics suggests the presence of some e\u000bec-\ntive \feld, which is perpendicular to the \feld polarization\nplane, depends on the \fled frequency and the polarization\ndirection. For high frequencies, this e\u000bective \feld is large\nenough and can hold the magnetic moment almost along\nthe direction, which is perpendicular to the polarization\nplane of the external \feld. The last fact was con\frmed\nnumerically. Finally, the expressions for the power loss\nhave been obtained for the case of viscous rotation of the\nnanoparticle with \fnite anisotropy and for the simpli\fed\ncases of the \fxed nanoparticle and the \fxed magnetiza-\ntion.\nThe nonuniform mode is a key issue of our investiga-\ntions and has been described numerically. This mode con-\nsists in the periodical changes of the polar angles with a\nperiod, which does not coincide with the \feld one. The\nactivation of the nonuniform precession mode is accompa-\nnied by the changes of the power loss. However, in the\ncase of the motion of the nanoparticle with the magne-\ntization \fxed inside, the nonuniform precession leads to\nthe power loss decrease. At the same time, in the case of\nthe magnetic dynamics inside the \fxed nanoparticle and\nin the case of the coupled motion of the nanoparticle with\n\fnite anisotropy in a viscous carrier, the nonuniform pre-\ncession is connected with a considerable increase in the\npower loss. The coupled motion exhibits another type\nof the nonuniform precession. It is characterised by the\nnanoparticle magnetization, which is stable in the labora-\ntory coordinates, and the nanoparticle body, which oscil-\nlates synchronously with the external \feld. This motion\nis realized for a wide enough range of parameters and caninclude several modes distinguished by discrete positions\nof the magnetization. The switching between them and\nthe modes of other types is also connected with abrupt\nmodi\fcations of the power loss.\nDespite the nonlinear e\u000bects in the coupled dynam-\nics still need to be investigated thoroughly, the reported\nresults allow us to state the following. The sharp tran-\nsitions between the di\u000berent precession modes is an im-\nportant phenomenon to control the heating process within\nhyperthermia. On the one hand, it can be used to se-\nlect the optimal parameters for therapy. On the other\nhand, we need to bear in mind it to prevent dangerous\noverheating. Although our approach does not take into\naccount the thermal agitation and the dipole interaction\nbetween the nanoparticles, its relevance is obvious. First,\nas it follows from the approximation of the \fxed magne-\ntization, the interaction and thermal noise decrease these\nvalues. Therefore, the deterministic approach establishes\nthe limit values of the power loss. Second, for the large\nenough nanoparticles ( \u001820nm) and comparatively intense\nexternal \felds, the regular component in the nanoparticle\ndynamics is dominant. Consequently, the deterministic\napproach gives the results close to the correct ones.\nAcknowledgment\nThe authors express appreciation to V. V. Reva for\nthe valuable help in the numerical simulations of the cou-\npled magnetic and mechanical motion. Moreover, the au-\nthors are grateful to the Ministry of Education and Sci-\nence of Ukraine for the \fnancial support under Grant\nNo. 0116U002622 and DAAD, the scholarship programme:\nResearch Stays for University Academics and Scientists,\n2018, Section: ST22, personal ref. no.: 91695699.\nReferences\nReferences\n[1] R. Rosensweig, Ferrohydrodynamics, Cambridge University\nPress, 1985.\n[2] M. I. Shliomis, Magnetic \ruids, Soviet Physics Uspekhi 17 (2)\n(1974) 153.\nURL http://stacks.iop.org/0038-5670/17/i=2/a=R02\n[3] Q. A. Pankhurst, J. Connolly, S. K. Jones, J. Dobson, Appli-\ncations of magnetic nanoparticles in biomedicine, Journal of\nPhysics D: Applied Physics 36 (13) (2003) R167.\nURL http://stacks.iop.org/0022-3727/36/i=13/a=201\n[4] O. Veiseh, J. W. Gunn, M. Zhang, Design and fabrication of\nmagnetic nanoparticles for targeted drug delivery and imag-\ning, Advanced Drug Delivery Reviews 62 (3) (2010) 284 { 304,\ntargeted Delivery Using Inorganic Nanosystem. doi:https:\n//doi.org/10.1016/j.addr.2009.11.002 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0169409X09003408\n[5] B. Tian, Z. Qiu, J. Ma, T. Z. G. de la Torre, C. Johansson,\nP. Svedlindh, M. Strmberg, Attomolar zika virus oligonucleotide\ndetection based on loop-mediated isothermal ampli\fcation and\nac susceptometry, Biosensors and Bioelectronics 86 (2016) 420\n{ 425. doi:https://doi.org/10.1016/j.bios.2016.06.085 .\nURL http://www.sciencedirect.com/science/article/pii/\nS095656631630625X\n7[6] B. Tian, T. Zardan Gomez de la Torre, M. Donolato, M. F.\nHansen, P. Svedlindh, M. Stromberg, Multi-scale magnetic\nnanoparticle based optomagnetic bioassay for sensitive dna and\nbacteria detection, Anal. Methods 8 (2016) 5009{5016. doi:\n10.1039/C6AY00721J .\nURL http://dx.doi.org/10.1039/C6AY00721J\n[7] A. Jordan, R. Scholz, P. Wust, H. Fhling, R. Felix, Magnetic\n\ruid hyperthermia (mfh): Cancer treatment with ac magnetic\n\feld induced excitation of biocompatible superparamagnetic\nnanoparticles, Journal of Magnetism and Magnetic Materials\n201 (1) (1999) 413 { 419. doi:http://dx.doi.org/10.1016/\nS0304-8853(99)00088-8 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0304885399000888\n[8] S. Dutz, R. Hergt, Magnetic particle hyperthermia: promising\ntumour therapy?, Nanotechnology 25 (45) (2014) 452001.\nURL http://stacks.iop.org/0957-4484/25/i=45/a=452001\n[9] P. F. de Ch^ atel, I. N\u0013 andori, J. Halk, S. M\u0013 esz\u0013 aros, K. Vad,\nMagnetic particle hyperthermia: N\u0013 eel relaxation in mag-\nnetic nanoparticles under circularly polarized \feld, Journal of\nPhysics: Condensed Matter 21 (12) (2009) 124202.\nURL http://stacks.iop.org/0953-8984/21/i=12/a=124202\n[10] T. V. Lyutyy, S. I. Denisov, A. Y. Peletskyi, C. Binns, En-\nergy dissipation in single-domain ferromagnetic nanoparticles:\nDynamical approach, Phys. Rev. B 91 (2015) 054425. doi:\n10.1103/PhysRevB.91.054425 .\nURL http://link.aps.org/doi/10.1103/PhysRevB.91.054425\n[11] C. Haase, U. Nowak, Role of dipole-dipole interactions for hy-\nperthermia heating of magnetic nanoparticle ensembles, Phys.\nRev. B 85 (2012) 045435. doi:10.1103/PhysRevB.85.045435 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.85.\n045435\n[12] G. T. Landi, Role of dipolar interaction in mag-\nnetic hyperthermia, Phys. Rev. B 89 (2014) 014403.\ndoi:10.1103/PhysRevB.89.014403 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.89.\n014403\n[13] C. Scherer, H.-G. Matuttis, Rotational dynamics of magnetic\nparticles in suspensions, Phys. Rev. E 63 (2000) 011504.\ndoi:10.1103/PhysRevE.63.011504 .\nURL https://link.aps.org/doi/10.1103/PhysRevE.63.\n011504\n[14] Y. L. Raikher, V. I. Stepanov, Power losses in a suspension of\nmagnetic dipoles under a rotating \feld, Phys. Rev. E 83 (2011)\n021401. doi:10.1103/PhysRevE.83.021401 .\nURL https://link.aps.org/doi/10.1103/PhysRevE.83.\n021401\n[15] B. U. Felderhof, R. B. Jones, Mean \feld theory of the nonlin-\near response of an interacting dipolar system with rotational\ndi\u000busion to an oscillating \feld, Journal of Physics: Condensed\nMatter 15 (23) (2003) 4011.\nURL http://stacks.iop.org/0953-8984/15/i=23/a=313\n[16] D. Soto-Aquino, C. Rinaldi, Nonlinear energy dissipation of\nmagnetic nanoparticles in oscillating magnetic \felds, Journal\nof Magnetism and Magnetic Materials 393 (2015) 46 { 55.\ndoi:https://doi.org/10.1016/j.jmmm.2015.05.009 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0304885315301335\n[17] T. V. Lyutyy, S. I. Denisov, V. V. Reva, Y. S. Bystrik,\nRotational properties of ferromagnetic nanoparticles driven by\na precessing magnetic \feld in a viscous \ruid, Phys. Rev. E 92\n(2015) 042312. doi:10.1103/PhysRevE.92.042312 .\nURL https://link.aps.org/doi/10.1103/PhysRevE.92.\n042312\n[18] T. V. Lyutyy, V. V. Reva, Energy dissipation of rigid dipoles\nin a viscous \ruid under the action of a time-periodic \feld: The\nin\ruence of thermal bath and dipole interaction, Phys. Rev. E\n97 (2018) 052611. doi:10.1103/PhysRevE.97.052611 .\nURL https://link.aps.org/doi/10.1103/PhysRevE.97.\n052611\n[19] A. O. Tsebers, Simultaneous rotational di\u000busion of the magneticmoment and the solid matrix of a single-domain ferromagnetic\nparticle, Magnetohydrodynamics 11 (3) (1975) 273{278.\nURL http://mhd.sal.lv/contents/1975/3/MG.11.3.2.R.html\n[20] H. Mamiya, B. Jeyadevan, Hyperthermic e\u000bects of dissipa-\ntive structures of magnetic nanoparticles in large alternating\nmagnetic \felds, Scienti\fc Reports 1 (2011) 157 EP, article.\ndoi:10.1038/srep00157 .\nURL http://dx.doi.org/10.1038/srep00157\n[21] N. A. Usov, B. Y. Liubimov, Dynamics of magnetic nanopar-\nticle in a viscous liquid: Application to magnetic nanoparti-\ncle hyperthermia, Journal of Applied Physics 112 (2) (2012)\n023901. arXiv:http://dx.doi.org/10.1063/1.4737126 ,doi:\n10.1063/1.4737126 .\nURL http://dx.doi.org/10.1063/1.4737126\n[22] K. D. Usadel, C. Usadel, Dynamics of magnetic single do-\nmain particles embedded in a viscous liquid, Journal of Applied\nPhysics 118 (23) (2015) 234303. arXiv:http://dx.doi.org/10.\n1063/1.4937919 ,doi:10.1063/1.4937919 .\nURL http://dx.doi.org/10.1063/1.4937919\n[23] N. Usov, B. Y. Liubimov, Magnetic nanoparticle motion in ex-\nternal magnetic \feld, Journal of Magnetism and Magnetic Ma-\nterials 385 (2015) 339 { 346. doi:http://dx.doi.org/10.1016/\nj.jmmm.2015.03.035 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0304885315002565\n[24] H. Xi, K.-Z. Gao, Y. Shi, S. Xue, Precessional dynamics of\nsingle-domain magnetic nanoparticles driven by small ac mag-\nnetic \felds, Journal of Physics D: Applied Physics 39 (22) (2006)\n4746.\nURL http://stacks.iop.org/0022-3727/39/i=22/a=002\n[25] H. Keshtgar, S. Streib, A. Kamra, Y. M. Blanter, G. E. W.\nBauer, Magnetomechanical coupling and ferromagnetic res-\nonance in magnetic nanoparticles, Phys. Rev. B 95 (2017)\n134447. doi:10.1103/PhysRevB.95.134447 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.95.\n134447\n[26] T. V. Lyutyy, O. M. Hryshko, A. A. Kovner, E. S. Denisova,\nPrecession of a \fne magnetic particle with \fnite anisotropy in\na viscous \ruid, J. Nano- Electron. Phys. 8 (4) (2016) 04086.\ndoi:10.21272/jnep.8(4(2)).04086 .\nURL https://dx.doi.org/10.21272/jnep.8(4(2)).04086\n[27] T. V. Lyutyy, O. M. Hryshko, A. A. Kovner, Power loss for\na periodically driven ferromagnetic nanoparticle in a viscous\n\ruid: The \fnite anisotropy aspects, Journal of Magnetism\nand Magnetic Materials 446 (Supplement C) (2018) 87 { 94.\ndoi:https://doi.org/10.1016/j.jmmm.2017.09.021 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0304885317310740\n[28] K. D. Usadel, Dynamics of magnetic nanoparticles in a viscous\n\ruid driven by rotating magnetic \felds, Phys. Rev. B 95 (2017)\n104430. doi:10.1103/PhysRevB.95.104430 .\nURL https://link.aps.org/doi/10.1103/PhysRevB.95.\n104430\n[29] J. Weizenecker, The fokker-planck equation for coupled brown-\nn\u0013 eel-rotation, Physics in Medicine and Biology 63 (3) (2018)\n035004.\nURL http://stacks.iop.org/0031-9155/63/i=3/a=035004\n[30] J. J. Newman, R. B. Yarbrough, Motions of a magnetic particle\nin a viscous medium, Journal of Applied Physics 39 (12) (1968)\n5566{5569. arXiv:https://doi.org/10.1063/1.1656014 ,doi:\n10.1063/1.1656014 .\nURL https://doi.org/10.1063/1.1656014\n[31] S. I. Denisov, T. V. Lyutyy, P. H anggi, Magnetization of\nnanoparticle systems in a rotating magnetic \feld, Phys. Rev.\nLett. 97 (2006) 227202. doi:10.1103/PhysRevLett.97.227202 .\nURL https://link.aps.org/doi/10.1103/PhysRevLett.97.\n227202\n[32] S. I. Denisov, A. Y. Polyakov, T. V. Lyutyy, Resonant sup-\npression of thermal stability of the nanoparticle magnetization\nby a rotating magnetic \feld, Phys. Rev. B 84 (2011) 174410.\ndoi:10.1103/PhysRevB.84.174410 .\n8URL https://link.aps.org/doi/10.1103/PhysRevB.84.\n174410\n[33] G. Bertotti, C. Serpico, I. D. Mayergoyz, Nonlinear magnetiza-\ntion dynamics under circularly polarized \feld, Phys. Rev. Lett.\n86 (2001) 724{727. doi:10.1103/PhysRevLett.86.724 .\nURL https://link.aps.org/doi/10.1103/PhysRevLett.86.\n724\n[34] T. V. Lyutyy, A. Y. Polyakov, A. V. Rot-Serov, C. Binns,\nSwitching properties of ferromagnetic nanoparticles driven by\na circularly polarized magnetic \feld, Journal of Physics: Con-\ndensed Matter 21 (39) (2009) 396002.\nURL http://stacks.iop.org/0953-8984/21/i=39/a=396002\n[35] S. Denisov, T. Lyutyy, C. Binns, P. Hnggi, Phase diagrams\nfor the precession states of the nanoparticle magnetization in\na rotating magnetic \feld, Journal of Magnetism and Magnetic\nMaterials 322 (9) (2010) 1360 { 1362, proceedings of the Joint\nEuropean Magnetic Symposia. doi:https://doi.org/10.1016/\nj.jmmm.2009.09.036 .\nURL http://www.sciencedirect.com/science/article/pii/\nS0304885309009330\n9"    },    {        "title": "2011.07277v1.Generic_Theory_of_the_Dynamic_Magnetic_Response_of_Ferrofluids.pdf",        "content": "arXiv:2011.07277v1  [cond-mat.soft]  14 Nov 2020Generic Theory of the Dynamic Magnetic Response of Ferroflui ds\nAngbo Fang\nSchool of Physics and Electronics, North China University o f Water Resources and Electric Power, Zhengzhou 450011, Chin a\n(Dated: November 17, 2020)\nFerrofluids belong to an important class of highly functiona l soft matter, benefiting from their\nmagnetically controllable physical properties. Therefor e, it is of central importance to quantitatively\npredict the dynamic magnetic response of ferrofluids. Tradi tional dynamic theories, however, are\noften restricted to the near-equilibrium regime and/or onl y apply to nearly ideal ferrofluids that are\nmonodisperse, dilute enough, and weakly interacting. In th is paper I develop a self-consistent and\nnonperturbative dynamical mean field theory for typical fer rofluids which are often polydisperse,\nconcentrated, and strongly interacting, possibly driven f ar from equilibrium. I obtain a general\nnonperturbative expression for the dynamic magnetic susce ptibility, quantitatively agreeing with\nthe spectra obtained from Brownian Dynamics simulations on both mono- and bidisperse samples.\nFurthermore, I derive a generic magnetization relaxation e quation (MRE) for both mono- and\npolydisperse ferrofluids by employing the projection opera tor technique in nonequlibrium statistical\nmechanics. This MRE is in simple closed form and independent of which model is employed to\napproximate the equilibrium magnetization curve. Existin g models can be recovered as low-order\napproximations of my generic and nonperturbative MRE. My th eory can play a key role in studying\nthe dynamics of ferrofluids and other polar fluids. It may also have substantial and immediate\nconsequences to various ferrofluid applications.\nI. INTRODUCTION\nFerrofluids have been studied extensively [1, 2] as a\nprototype of polar fluids and for their many applica-\ntions in industry and biomedicine. The number of fer-\nrofluid applications keeps growing, taking advantage of\ntheir response to applied magnetic fields which renders\ntheir physical (mechanical, optical, thermal, etc.) prop-\nerties controllable and tunable. Thus, it is crucial for\nus to quantitatively describe the dynamic magnetic re-\nsponse of ferrofluids, which on the macroscopic scale is\nvia a magnetization relaxation equation (MRE). While\nthe static properties of typical ferrofluids can be reliably\npredicted by existing models [3–9], however, there still\nlacks a quantitatively reliable dynamic theory.\nThedynamicmagneticresponseofferrofluidsismainly\ncomplicated by three factors: finite particle concentra-\ntion, often characterized by the hydrodynamic volume\nfraction φ; dipole-dipole interactions (DDI) character-\nized by the coupling constant λ; and the particle poly-\ndispersity. The earliest dynamic theory of dipolar flu-\nids was developed by Debye [10], based on the Smolu-\nchowski equation (SE) for an ensemble of noninteract-\ning dipolar particles (ideal ensemble) under a weak field.\nHenceDebye’stheoryonlyappliestodipolarfluidsonthe\nneighborhood of the unpolarized equilibrium state. The\nsame restriction applies to phenomenological MREs [12–\n14]obtainedfromlinearirreversiblethermodynamics. By\ntreating the relaxation coefficients as phenomenological\nstate-independent constants, the unjustified use of these\nMREs in far-from-equilibrium regimes can lead to re-\nsults even qualitatively wrong. In 1974, Martsenyuk,\nRaikher and Shliomis [11](MRSh) proposed an effective\nfield ansatz to derive the MRE for ideal ferrofluids, pos-\nsibly driven far away from equilibrium by a strong mag-\nnetic field or flow deformation. It can be regarded asa zeroth order approximation (with respect to φλ) to an\nexact theory and approximately applies to sufficiently di-\nlute and weakly interacting ferrofluids. However, while\nthere have been some progress in improving the MRSh\nmodel via perturbation approaches, the existing theories\nare still insufficient for describing dynamic response of\nreal ferrofluids which are often non-dilute, strongly in-\nteracting, and polydisperse.\nIn this paper I develop a generic nonperturbative the-\nory for the dynamic response of ferrofluids. It quantita-\ntively accounts for the effects of all three aforementioned\nimportant issues. My theory is neither restricted to the\nnear-equilibrium regime nor limited to weakly interact-\ning and dilute ferrofluids. While on the mesoscopic level\nit is in essence a dynamicalmean field theory(and can be\nderived in the framework of the dynamical density func-\ntional theory of classical fluids), it proves to be quanti-\ntatively reliable as long as particle clustering is insignif-\nicant (so that on the macroscopic level it is unnecessary\nto introduce new structural order parameterscoupling to\nmagnetization dynamics). This implies my theory well\ncaptures the time-averaged impact of inter-particle cor-\nrelations on magnetization dynamics on a slow time scale\neven though it does not track the much faster evolution\nof the inter-particle correlation functions themselves.\nA generic MRE is analytically derived based on the\nprojection operator method in non-equilibrium statisti-\ncal mechanics, for monodisperse as well as polydisperse\nferrofluids. For the first time, we obtain a MRE applica-\nble to polydisperse interacting ferrofluids driven far from\nequilibrium (the polydisperse versionofthe MRSh model\nis trivial, while other polydisperse theories are restricted\ntothe near-equilibriumregime). On the otherhand, such\na MRE, in simple and closed form, can recover all pre-\nvious MRE’s as low-order approximations. Remarkably,\neven for a new ferrofluid whose equilibrium magnetiza-2\ntion curve is only empirically known but not fitted into\nany existing analytical models, this MRE can be readily\nemployedtopredictitsmagnetizationdynamics, whether\nit is near or far from equilibrium. Importantly, I also ob-\ntain a simple, universal, and closed formula for the dy-\nnamicmagnetic susceptibility(DMS) forboth mono-and\npolydisperse ferrofluids. Furthermore, I have compared\nmy theoretical predictions with Brownian dynamics sim-\nulations. The quantitative agreements are superior to\nprevious models. The superiority increases as the fer-\nrofluid samples becomes more concentrated and/or more\nstrongly interacting (see Fig. 1). Especially, my theory\nis quantitatively accurate even for non-dilute interact-\ning ferrofluid samples driven far away from equilibrium\n(see Fig. 2), for which previous models may even become\nqualitatively unreliable.\nII. MONODISPERSE FERROFLUIDS\nTodevelopthetheory, Ifirstfocusonthemonodisperse\ncase. Consider Nidentical spherical particles (with hy-\ndrodynamic diameter d, carrying magnetic moment µ)\ndispersed in a solvent with viscosity ηs, occupying to-\ntal volume V, maintained at temperature T, and driven\nby a magnetic field H. The particle concentration is\nρ=N/Vand saturation magnetization is Ms=ρµ.\nTypically the relaxation of magnetic moments is domi-\nnated by Brownian mechanism. For an ideal ferrofluid\n(φ∼0,λ∼0), particle dynamics is characterized by the\nDebye or Brownian rotation time [10] τD=πηsd3/2kBT\nwithkBthe Boltzmann constant. Averaging the single-\nparticle SE for such an ideal ensemble using the effective\nfield ansatz leads to the celebrated MRSh equation [11]\nfor magnetization relaxation. However, for a ferrofluid\nwith finite φ=πρd3/6 andλ=µ0µ2/4πd3kBT(withµ0\nthe vacuum magnetic permeability thereafter made im-\nplicit), particles respond to external fields in a correlated\nway. Then we have to solve an intractable N-body prob-\nlem. Often we can approximate it by a single-particle SE\nwith an appropriate mean field potential (for a detailed\nderivation, see Ref. [15]).\nDenoting etheorientationalvectorforarepresentative\nparticle and W(e,t) its orientational distribution func-\ntion (ODF), the single-particle SE reads\n2τrdW\ndt=1\nkBTˆJWˆJ[kBTlnW−µe·H+Φint],(1)\nwithˆJ=e×∂/∂eandτrthe rotational self-diffusion\ntime which is in general larger than τDdue to hydro-\ndynamic interactions and DDI [16, 17]. Eq. (1) is sim-\nilar to the dynamical density functional theory of clas-\nsical fluids [18], if we identify the bracketed term as\nthe total chemical potential Φ( e,t) = Φ 0+ Φex+ Φint,\nwith Φ 0=kBTlnWthe ideal gas entropic contribution,\nΦex=−µe·Hthe dipole-field potential energy, and Φ int\ntheeffectiveone-bodypotentialduetoDDI. WithEq.(1)wecan in principle determine the evolutionofmagnetiza-\ntionM(t)≡ρµ/integraltexteW(e,t)de, though it is often difficult\nto obtain a MRE in closed form.\nIfHis a static field, Eq. (1) admits a stationary\nsolution, W0(e), determined by Φ = const. Then\nthe magnetic equation of state (MEOS) can be deter-\nmined, specifying the equilibrium magnetization M0as\na function of H. For example, for an ideal ferrofluid,\nΦ0+ Φex=constreproduces Langevin’s MEOS. For\nconvenience I define the scaled Langevin function by\n˜L(x) =MsL(µx/kBT), where L(x) = coth( x)−1/xis\nthe dimensionless Langevin function. Then Langevin’s\nMEOS is given by M0=˜L(H) and Langvin’s static ini-\ntial susceptibility is χL=ρµ2/3kBT≡8φλ. However,\ntaking interparticle correlations into account we would\nhaveM0=˜G(H), with˜Ga function different from ˜L.\nIn Eq. (1) Φ intdepends on the pair correlation func-\ntionthatusuallydecaysmuchfasterthan W. WithM(t)\nthe sole relevant slow variable, we may assume Φ int, or\nthe nonequilibrium mean dipolar field Hdddefined by\nΦint=−µe·Hdd, is a functional of M(t) alone. For ex-\nample, the Weiss model postulates Hdd=M(t)/3, from\nwhichaMREcanbederived. Tothe firstorderof χLitis\nequivalent to the MRE by Zubarev and Yushkov [19, 20]\nfor dilute and weakly interacting ferrofluids. Recently, I\nhave developed a dynamical second-order modified mean\nfield (MMF2) theory [21] in which a second-order ex-\npression for Hddis constructed. It applies to ferroflu-\nids with larger χLbeyond previous lower-order models.\nNow, a general inverse problem arises: Given a magne-\ntization curve ˜Gof nonperturbative or even empirical\nnature, what is the self-consistent form for Hdd?\nIt is a key observation that in all previous perturbative\ndynamic models [11, 20, 21] the mean dipolar field can\nbe obtained from the following universal form:\nHdd(t) =HL\ne(t)−He(t), (2)\nwhereHe(t) =˜G−1(M(t))ˆm(withˆm≡M(t)/M(t)) is\nthe thermodynamic effective field and I call HL\ne(t) =\n˜L−1(M(t))ˆmthe (auxiliary) Langevin effective field.\nWith Eq. (2) the equilibrium ODF is determined\nasW0(e)∝exp(e·HL/kBT), where HL=\n˜L−1(˜G(H))H/His the equilibrium Langevin field. Thus\nwe have M0=˜L(HL)≡˜G(H), reproducing the pre-\nscribed MEOS self-consistently.\nI comment that, while Eq. (2) initially arises as a con-\njecture based on previous perturbative models [21], it\ncan be derived from the dynamical density functional\ntheory of classical fluids with reasonable assumptions.\nIt should be distinguished from the Weiss mean field\nmodel (or other near-equilibrium models) which applies\nonly if the order parameter is sufficiently small (close\nto the unpolarized equilibrium state), for which we have\nHdd(t)∝M(t). In Eq. (2) the mean dipolar field is\nobtained as a nonlinear function of the instantaneous\nmagnetization. Its explicit form follows self-consistently\nfromthe prescribedMEOS,which holds ona slow(quasi-3\nequilibrium) time scale. Hence, the time-averaged effect\nof inter-particle correlations is embodied in my theory\nto all orders with respect to M(t)/Ms. In contrast, the\nMRSh model and the Weiss model only captures the cor-\nrelation effect to the zeroth and first orders, respectively.\nThis explains why my theory is more generic and ac-\ncurate, and quantitatively reliable even in the far-from-\nequilibrium regime. Furthermore, via Eq. (2), the mean\ndipolar field has a physically intuitive interpretation:\nit is the difference between the magnetic fields needed\nto prepare the same ferrofluid in a macroscopic quasi-\nequilibrium state characterized by M(t), when inter-\nparticle correlations are completely switch off or on, re-\nspectively.\nTo derive the MRE from Eqs. (1) and (2) I fol-\nlow Ref. [21] and employ the projection operator tech-\nnique [22, 23]. With M(t) the only relevant slow vari-\nable, we may discard memory effects and obtain\nτrdM\ndt=M\nHLe(H−He)/bardbl+1\n2/parenleftbigg\n3χL−M\nHLe/parenrightbigg\n(H−He)⊥,\n(3)\nwhere subscripts “ ∝bardbl” or “⊥” denotes the components of\nH−He(the thermodynamic driving force) parallel or\nperpendicular to M(t), respectively.\nEq. (3) is the generic nonperturbative MRE for\nmonodisperse ferrofluids. It is of canonical form [24, 25]\nin nonequilibrium thermodynamics. Interestingly, the\nstate-dependent transport coefficients are universally de-\ntermined by the ratio of M(t) toHL\ne=˜L−1(M(t)), its\nassociated instantaneous Langevin field, independent of\nthe MEOS. All previously obtained MRE’s, respectively\naccurate to the zeroth [11], first [19, 20], and second [21]\norder of χL, can be reproduced by substituting the cor-\nresponding MEOS into Eq. (3) to express Heas a func-\ntion ofM(t). Importantly, even if the MEOS is empir-\nically determined (hence immune to inaccuracies caused\nby model approximations), we can employ Eq. (3) to ac-\ncurately compute the magnetization dynamics.\nIn some situations H(t) consists of a large static part\nand a small time-dependent part. If µ|H−He| ≪kBT,\nthen we can linearize Eq. (3) and obtain the generalized\nDebye equation\ndM\ndt=−(M/bardbl−MH)\nτ/bardbl−M⊥\nτ⊥, (4)\nwhereMH=˜G(H(t))H/Hcorresponds to the magne-\ntization for a hypothetical quasi-equilibrium state pre-\npared by the instantaneous magnetic field, and M/bardbl,⊥\ndenotes the components of Mparallel or perpendicular\ntoH, respectively. τ/bardblandτ⊥arethe correspondingfield-\ndependent relaxation times:\nτ/bardbl(ξ)\nτr=dξL\ndξdlnL(ξL)\ndlnξL;τ⊥(ξ)\nτr=ξL\nξ2L(ξL)\nξL−L(ξL)(5)\nwithξ=µH/kBTandξL=µHL/kBT.Furthermore, if ξ≪1, the magnetization relaxation\nis restricted around the unpolarized equilibrium and my\nMRE reduces to a Debye-like equation [10]:\ndM\ndt=−M−χ0H\nτM;τM=χ0\nχLτr,(6)\nwhereχ0is the static initial susceptibility. In Eq. (6) an\nexact connection is established between the macroscopic\nmagnetization relaxation time, τM, and the microscopic\nrotational self-diffusion time, τr. Debye’s original equa-\ntion is for ideal polar fluids with χ0=χLandτM=τr.\nHowever, for strong ferrofluids we have χ0≫χLand\nhenceτM≫τr. Unfortunately, there is a widespread\nconfusion in which τMis improperly identified with τr\nand the latter is often approximated by τD. Overall this\ncould cause significantly inaccurate estimates of τMor\nparticle sizes, thereby leading to misinterpretations of\nrelevant experimental measurements.\nFor noninteracting ferrofluids, Debye’s theory predicts\ntheDMSas χD(ω) =χL/(1+iωτr). However,accounting\nfor interaction effects Eq. (6) leads to\nχ(ω) =χ0\n1+iωτM(7)\nas the generic DMS for monodisperse ferrofluids. Re-\nmarkably, but not obviously, Eq. (7) can be recast into\nthe following form:\nχ(ω) =χD(ω)\n1−gcχD(ω)/χL;gc≡1−χL\nχ0(8)\nwheregcis a measure of interparticle correlations. Set-\ntinggc= 0 we have χ(ω) =χD(ω). If we set χ0=\nχL(1 +χL/3) as in leading-order equilibrium perturba-\ntion theories [3, 4], Eq. (8) exactly reproduces the heuris-\ntic modified Weiss model proposed recently [26]. If this\nexpression is further subject to a second-order polyno-\nmial expansion, then it recoversthe DMS from the recent\nMMF1 theory [27, 28]. Thus my theory, of nonperturba-\ntive nature and in simple closed form, unifies all previous\nperturbative results.\nEq. (7) can also be directly obtained from the SE (1)\nwith the near-equilibrium approximation to Eq. (2):\nHdd= (χ−1\nL−χ−1\n0)M(t). (9)\nThis includes all-order(with respect to χL) contributions\nifχ0takes its exact value. On the other hand, Hddused\nin all previous DMS calculations [27, 29] are at best ac-\ncurate to the first order of χLand thus fail to describe\nferrofluids with either high concentration or strong DDI.\nI plot Figure 1 to compare the predicted DMS with the\nresults from Brownian Dynamics (BD) simulations and\nfrom the MMF1 theory [30]. Obviously, the latter be-\ncomes less and less accurate as φandλincrease. In\ncontrast, my predictions agree remarkably well with the\nsimulations for all samples.4\n/s49/s48/s45/s50\n/s49/s48/s45/s49\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52/s48/s46/s53\n/s32/s48/s61/s48/s46/s52/s54/s51\n/s77/s61/s49/s46/s49/s55\n/s32/s32\n/s32/s66/s68/s44/s32/s73/s109/s103\n/s32/s66/s68/s44/s32/s82/s101/s97/s108\n/s32/s77/s77/s70/s49\n/s32/s69/s113/s46/s32/s40/s55/s41/s40 /s41/s40/s97/s41/s32 /s61/s48/s46/s48/s53/s50/s44/s32 /s61/s49\n/s32/s32/s32/s32/s32\n/s76/s61/s48/s46/s52/s49/s57\n/s49/s48/s45/s50\n/s49/s48/s45/s49\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56\n/s48/s61/s49/s46/s55/s55\n/s77/s61/s49/s46/s52/s57/s40/s98/s41/s32 /s61/s48/s46/s49/s48/s53/s44/s32 /s61/s49/s46/s53\n/s32/s32/s32/s32/s32\n/s76/s61/s49/s46/s50/s54\n/s32/s32\n/s49/s48/s45/s50\n/s49/s48/s45/s49\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s40 /s41\n/s48/s61/s51/s46/s53/s49\n/s77/s61/s49/s46/s56/s49/s32/s40/s99/s41/s32 /s61/s48/s46/s50/s54/s50/s44/s32 /s61/s49\n/s32/s32/s32/s32/s32\n/s76/s61/s50/s46/s48/s57\n/s32/s32\n/s66/s49/s48/s45/s50\n/s49/s48/s45/s49\n/s49/s48/s48\n/s49/s48/s49\n/s49/s48/s50/s48/s49/s50/s51/s52\n/s48/s61/s51/s46/s56/s53\n/s77/s61/s50/s46/s54/s40/s100/s41/s32 /s61/s48/s46/s49/s48/s53/s44/s32 /s61/s50/s46/s53\n/s32/s32/s32/s32/s32\n/s76/s61/s50/s46/s48/s57\n/s32/s32\n/s66\nFIG. 1. The susceptibility spectra χ(ω) for monodisperse fer-\nrofluids with various φandλ. The symbols are from BD\nsimulations [30] and the dotted lines are from the dynamic\nMMF1 theory. The solid lines are from Eq. (7), with χ0and\nτM(in units of τB) specified for each sample.\nWhen a ferrofluid sample is put in a static field H0\nalongwithaweakprobingACfield ˜H(t), itsmagneticre-\nsponse can be described by Eq. (4). Unlike previous per-\nturbative models, my theory obtains the finite-field DMS\ninsimpleandclosedformforbothlongitudinal( ˜H∝bardblH0)\nand transverse ( ˜H⊥H0) setups. The peak frequen-\ncies of imaginary-part spectra are respectively given by\nτ−1\n/bardbl(ξ0) andτ−1\n⊥(ξ0), withξ0=µH0/kBT. Remarkably,\nmy theoretical predictions (supplemented with MEOS\nfrom MMF2 model) quantitatively agree with BD sim-\nulation results for all the studied ten samples [31], with\n6φ/π= 0.1,...,0.5 andλ= 1 or 2.5. Figure 2 only shows\nfield dependence of the static susceptibilities and peak\nfrequencies for some representative samples. For weakly\ninteracting samples ( λ= 1), fitting of peak frequencies\naccording to Eq. (5) leads to almost the same τras that\ndetermined from the zero-field spectra by Eq. (7). For\nstrongly interacting samples ( λ= 2.5),τrinferred from\nthe longitudinal and transverse spectra are apparently\ndifferent. Only the latter yields a τrclose to that from\nthe zero-field spectra (except for the most concentrated\nsample). Nevertheless, this properly indicates the forma-\ntion of transient chain-like structures along H0, which\nsubstantiallyinfluences relaxationofthe longitudinal but\nnot transversecomponentofmagnetization. Notably, my\npredictions are even quantitatively good for the sample\nwithφ= 0.262 and λ= 2.5 whose MEOS is no longer/s48 /s49 /s50 /s51 /s52 /s53/s48/s49/s50/s51/s52/s53\n/s40/s97/s41/s32 /s61/s48/s46/s49/s48/s53/s44/s32/s32 /s61/s49/s46/s48/s59\n/s32/s32/s32/s32/s32\n/s76/s61/s48/s46/s56/s51/s55/s56/s32\n/s32/s66/s68/s32/s40 /s124/s124 /s41\n/s32/s66/s68/s32/s40 /s41\n/s32/s84/s104/s101/s111/s114/s121/s32/s40 /s124/s124 /s41\n/s32/s84/s104/s101/s111/s114/s121/s32/s40 /s41/s32\n/s32/s80/s101/s97/s107/s32/s70/s114/s101/s113/s117/s101/s110/s99/s105/s101/s115/s32/s40\n/s66/s45/s49\n/s41\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s49/s50/s51/s52/s53\n/s40/s98/s41/s32 /s61/s48/s46/s50/s48/s57/s44/s32/s32 /s61/s49/s46/s48/s59\n/s32/s32/s32/s32/s32\n/s76/s61/s49/s46/s54/s55/s53/s53/s32\n/s32/s32\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s49/s50/s51/s52/s53\n/s48/s40/s99/s41/s32 /s61/s48/s46/s49/s53/s55/s44/s32/s32 /s61/s50/s46/s53/s59\n/s32/s32/s32/s32/s32\n/s76/s61/s51/s46/s49/s52/s49/s54/s32/s32/s32\n/s32/s32/s80/s101/s97/s107/s32/s70/s114/s101/s113/s117/s101/s110/s99/s105/s101/s115/s32/s40\n/s66/s45/s49\n/s41\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s49/s50/s51/s52/s53\n/s40/s100/s41/s32 /s61/s48/s46/s50/s54/s50/s44/s32/s32 /s61/s50/s46/s53/s59/s32/s32\n/s32/s32/s32/s32/s32\n/s76/s61/s53/s46/s50/s51/s54\n/s48\n/s32/s32\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s32/s48\n/s32/s32\n/s32 /s61/s48/s46/s50/s48/s57/s44/s32 /s61/s49\n/s32 /s61/s48/s46/s50/s54/s50/s44/s32 /s61/s49\n/s32 /s61/s48/s46/s50/s48/s57/s44/s32 /s61/s50/s46/s53\n/s32 /s61/s48/s46/s50/s54/s50/s44/s32 /s61/s50/s46/s53/s32/s32/s78/s111/s114/s109/s97/s108/s105/s122/s101/s100/s32/s83/s116/s97/s116/s105/s99/s32/s83/s117/s115/s99/s101/s112/s116/s105/s98/s105/s108/s105/s116/s121/s40/s101/s41/s32/s108/s111/s110/s103/s105/s116/s117/s100/s105/s110/s97/s108\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s48/s40/s102/s41/s32/s116/s114/s97/s110/s115/s118/s101/s114/s115/s101\n/s32/s32\nFIG. 2. Field dependence of the peak frequencies (a-d) of\nimaginary-part dynamic susceptibility spectra and the sta tic\nsusceptibilities (e-f) normalized by their zero-field coun ter-\nparts, for both the longitudinal ( /bardbl) and transverse ( ⊥) setups.\nSymbols are from BD simulations in Ref. [31] and lines are\nfrom my theoretical predictions. Different samples are dist in-\nguished by a pair of values for φandλ.\nadequately described by the MMF2 model. This may\nindicate the robustness of my theory.\nIII. POLYDISPERSE FERROFLUIDS\nA real ferrofluid is usually polydisperse and additional\ncomplications arise. For small particles the relaxation\nof magnetic moments can be dominated by N´ eel rather\nthan Brownian mechanism [32–34]. Large-size structural\nunits may form [33–36]. Here, I restrict myself to typ-\nical polydisperse ferrofluids in which the effects due to\nparticle aggregationand N´ eel relaxation are of minor im-\nportance.\nLet us assume nspecies of particles, with pk(k=\n1,...,n) the fraction of particles with magnetic core di-5\nameterdk. The magnetic moment carried by a k-particle\nisµk=πd3\nkMs/6. I useforfkto denote the population-\nweighted average of a species-specified quantity fk. If\nparticle interactions are negligible, the equilibrium mag-\nnetization is given by M0=G0(H), with G0(x) =/summationtext\nkpk˜Lk(x) and˜Lk(x) =ρµkL(µkx/kBT). Langevin’s\ninitial susceptibility is χL=χL\nk, withχL\nk=ρµ2\nk/3kBT.\nIn general, as suggested by modified mean field theo-\nries, we may write the polydisperse MEOS in the form of\nM0=G0(Λ(H)), with Λ a function to be specified. For\nexample, the polydisperse MMF2 model [6, 8] postulates\nΛ(x) =x+1\n3G0(x)+1\n144G0(x)G′\n0(x).\nOut of equilibrium, I denote Wk(e,t) the k-particle\nODF and Mk(t) its first-order moment multiplied by\nρµk. The total magnetization is M(t) =/summationtext\nkpkMk(t)\nand I denote by mits director. I assume Wk(e,t) still\nsatisfies SE (1), but with τrreplaced by τk, the k-particle\nrotational diffusion time, and Φ replaced by\nΦk(e,t) =kBTlnWk−µke·/bracketleftBig\nH+Hdd(t)/bracketrightBig\n,(10)\nthe k-particle chemical potential. In Eq. (10) the entropy\nduetospeciesmixingisneglectedand Hdd(t)isthemean\ndipolar field. Similar to Eq. (2), I postulate\nHdd(t) =HL\ne(t)−He(t), (11)\nwhereHe(t)≡Λ−1◦G−1\n0/parenleftbig\nM(t)/parenrightbig\nmandHL\ne(t)≡\nG−1\n0/parenleftbig\nM(t)/parenrightbig\nmare respectively the thermodynamic and\nLangevin effective field associated with M(t). Hence, a\nparticle, to whichever species it belongs, is subjected to\nthe same mean dipolar field (with inter-particle correla-\ntions time-averaged). In general, we expect Eq. (11) is\nonly a good approximation on a sufficiently slow time\nscale when M(t) is the sole relevant slow variable. How-\never, when the polydisperse ferrofluid remains close to\nthe unpolarized equilibrium state (for which Eq. (11) can\nbe linearized with respect to M(t)), it may also perform\nwell on a time scale faster than the time required to equi-\nlibrate all Mkin the far-from-equilibrium regime. The\nquantitative reliability of Eq. (11) should be checked a\nposterior .\nNow, treating {Mk;k= 1,...,n}as relevant slow vari-\nables, we can manipulate the k-particle SE using the pro-\njection operator technique and obtain\nτkdMk\ndt=Mk\nHLe(H−He)/bardbl+1\n2/bracketleftbigg\n3χL\nk−Mk\nHLe/bracketrightbigg\n(H−He)⊥\n+/integraldisplayt\n0dsAk(Mk(s),M(s),t,s)[H(s)−He(s)],(12)\nwhereon the righthand side the firstand secondline rep-\nresents the instantaneous response and memory effects,\nrespectively. The memory kernel Akis a second-order\ntensor and depends on all Mkin the past. Unlike the\nmonodisperse case with only one relevant slow variable,here we can not discard the memory effects. This is be-\ncause the relaxation times for different species can be\ndrasticallydifferentandtheirinteractionscannotbeade-\nquatelydescribed byinstantaneouscoarse-grainedforces.\nStill, on a time scale slow enough to wash out memory\neffects and validate the quasi-equilibrium approximation,\na polydisperse ferrofluid should also obey the principle\nof nonequilibrium thermodynamics. The rate of change\nofM(t) should be proportional to the instantaneous\nnonequilibrium thermodynamic force H(t)−He(t) with\ntransportcoefficientsdependingsolelyon M(t), irrespec-\ntive of whether the ferrofluid sample is monodisperse or\npolydisperse. Therefore, it is expected that the mem-\nory effects are to regulate the relaxation rates of different\nspeciessothatall Mk(t)becomesynchronized,rendering\nM(t) the sole and adequate slow variable characterizing\nthe instantaneous thermodynamic state. Hence we may\nabsorb the memory-effect term into τkdMk\ndtand write\nτBdMk\ndt=Mk\nHLe(H−He)/bardbl+1\n2/bracketleftbigg\n3χL\nk−Mk\nHLe/bracketrightbigg\n(H−He)⊥,\n(13)\nin which τBcould be a function of M.\nTaking the population-weighted average of Eq. (13)\nleads to the polydisperse MRE\nτBdM\ndt=M\nHLe(H−He)/bardbl+1\n2/bracketleftbigg\n3χL−M\nHLe/bracketrightbigg\n(H−He)⊥.\n(14)\nFor Eq. (14) to reduce to Eq. (3) in the monodisperse\nlimit,τBshouldbe stateindependent. Therefore, τBcan\nbe identified as the averaged rotational diffusion time.\nNear the unpolarized equilibrium Eq. (14) reduces to\na Debye-like equation and a DMS can be deduced:\nχs(ω) =χ0/(1+iωτBχ0/χL), (15)\nwithχ0the static susceptibility. Nevertheless, χs(ω) is\nonly a low-frequency approximation to the true DMS.\nThis is distinguished from monodisperse ferrofluids, for\nwhich the same DMS [Eq. (7)] is derived from either\nthe macroscopic MRE [Eq. (3)] or microscopic (strictly\nspeaking, mesoscopic) model [Eq.(9)]. This is because\nfor monodisperse ferrofluids there is a single microscopic\ntime scale and the magnetization is the unique slow vari-\nable. However,forpolydisperseferrofluids,therearemul-\ntiple microscopic time scales and multiple mutually cou-\npled slow variables. Except at a sufficiently slow time\nscale, memory effects can not be discarded, even in the\nlinear response regime. Hence we have to go back to the\nSE to obtain the polydisperse DMS.\nI denote the weak AC magnetic field as H(t) =\nH0exp(iωt), with ξ0\nk=µkH0/kBT≪1 fork=\n1,...,n. Under the linear response approximation we\nhaveMk(t) =χk(ω)H(t) andM(t) =χ(ω)H(t) with\nχ(ω) =/summationtext\nkpkχk(ω). To the first order of ξ0\nk, the mean\ndipolar field becomes\nHdd=/parenleftbig\nχ−1\nL−χ−1\n0/parenrightbig\nχ(ω)H0exp(iωt) (16)6\nand the k-particle ODF is\nWk(e,t) = 1+χk(ω)\nχk\nLµke·H0\nkBTexp(iωt).(17)\nPlugging Eqs. (16) and (17) into the SE’s, we obtain\n(1+iωτk)χk(ω) =χL\nk/bracketleftbig\n1+(χ−1\nL−χ−1\n0)χ(ω)/bracketrightbig\n(18)\nThe population-weighted average of Eq. (18) reads\nχ(ω) =χD(ω)\n1−gcχD(ω)/χL(19)\nwithgc= 1−χL/χ0the correlation factor and\nχD=/summationdisplay\nkpkχL\nk/(1+iωτk) (20)\nthe Debye’s DMS for noninteracting ploydisperse fer-\nrofluids. Remarkably, Eq. (19) is completely parallel to\nEq. (8) for the monodisperse DMS. In both cases, the\ncorrection to Debye’s DMS only depends on a single cor-\nrelationfactor gc. Thiscorrelationfactoronlydependson\nthe static initial susceptibility, which can be determined\neither empirically or by approximateequilibrium models.\nIf my theory is supplemented with χ0=χL(1+χL/3) as\npredictedby first-orderequilibriumperturbationtheories\nand neglect the difference between the τkandτBk(which\ndenotes the Brownian relaxation time for the k-particle\nin the infinitely dilute limit), then Eq. (19) reduces to\nthe heuristic modified Weiss model [26]. The latter was\nshownin goodagreementwith BDsimulationsforseveral\nbidisperse samples with χL≤3.2.\nTo demonstrate the accuracy of my theoretic predic-\ntion, I plot in Fig. 3 the imaginary part of the DMS\nfor a bidisperse ferrofluid sample. This sample consists\nof two species (k=1, 2) of particles, with their sizes (no\ncoating layer on every particle) and magnetic moments\ndesigned so that τB2/τB1= 10 and λ2= 2λ1= 2\n(withλk=µ0µ2\nk/4πd3\nkkBTcharacterizing the strength\nof DDI between k-particles). The number and volume\nfractions of small particles are 0 .744 and 0 .181, respec-\ntively. As clearly shown in Fig. 3, the heuristic modified\nWeiss model agrees well with BD simulations, but shows\nobservable deviations at both low and high frequencies.\nRemarkably, my prediction based on Eq. (19) is in nearly\nperfectagreementwithBDsimulationsinthewholestud-\nied frequency range. This is explained as follows. Ac-\ncording to my theory, the effects of inter-particle corre-\nlations are well separated into two parts. For the near-\nequilibrium behaviorofferrofluids, the static correlations\nare captured by the factor gc= 1−χL/χ0, whereas the\ndynamic correlations are manifested by the ratios of the\nmean relaxation time τkto the Brownian relaxation time\nτBk. In a manuscript being prepared by myself [37], it\nis proved that τkactually corresponds to the character-\nistic time describing the orientational self-diffusion of a\ntagged particle in the long time regime, during which it\nsamplesmanyconfigurationsofsurroundingparticles. As/s49/s48/s45/s50\n/s49/s48/s45/s49\n/s49/s48/s48\n/s49/s48/s49/s48/s46/s48/s48/s46/s51/s48/s46/s54/s48/s46/s57/s49/s46/s50/s49/s46/s53/s49/s46/s56\n/s32/s32/s73/s109/s91 /s40 /s41/s93\n/s66/s49/s32/s66/s68/s32/s115/s105/s109/s117/s108/s97/s116/s105/s111/s110/s115\n/s32/s77/s111/s100/s105/s102/s105/s101/s100/s32/s87/s101/s105/s115/s115/s32/s109/s111/s100/s101/s108\n/s32/s69/s113/s46/s40/s49/s57/s41\nFIG. 3. The imaginary part of DMS for a bidisperse fer-\nrofluid. The frequency is measured in terms of the inverse of\nτB1, which is the Brownian relaxation time for small parti-\ncles (for large particles τB2= 10τB1). Symbols are from BD\nsimulations for sample B in Ref. [26] (see Figure 5(b) in it).\nThe dashed blue line is from the modified Weiss model, while\nthe solid red line is from my theoretical prediction based on\nEq. (19). Relevant material parameters are provided in Ta-\nble I of Ref. [26]. By fitting Eq. (19) with the results from\nBD simulations, the values of τ1andτ2are determined as\nτ1= 1.137τB1andτ2= 1.078τB2, which seems quite reason-\nable [37].\na consequence of the adiabatic approximation implied in\nderiving the mean field SE (and other dynamical density\nfunctional theories), it actually describes the stochastic\ndynamics of a representative dressed particle and τk/τBk\ncharacterizes the integrated (renormalization) effects of\nthe neglected dynamic evolution of inter-particle correla-\ntions.\nWe usually have τk/τBk>1 due to hydrodynamic and\ndirect interactions. Based on my theory and the observa-\ntions from Fig. 3, the dynamic effect of inter-particle cor-\nrelations can cause enhanced dissipation at low frequen-\ncies but reduced dissipation at high frequencies. This\nmayhaveimplicationsinapplyingferrofluidsto magnetic\ntherapy. For the sample studied in Fig. 3, the modified\nWeissmodel wellcapturesthe effects ofstatic but not dy-\nnamic correlation, while Eq. (19) accountsfor both. This\nis why my theoretic prediction is even better. Neverthe-\nless, it is noted the DDI strength in all the bidispersed\nferrofluid samples studied in Ref. [26] is only moderately\nstrong. It remains of great interest to extensively inves-\ntigate polydisperse samples with much stronger DDI (for\nwhich the modified Weiss model is even insufficient to\ncapture the effect of static correlations) to check the ac-7\ncuracies of my theory. Furthermore, it is extremely diffi-\ncult (even for a monodisperse ferrofluid) to evaluate from\nfirst principles the value of τkdue to the many-body and\nnon-Markovian nature of the problem [37]. Nevertheless,\nthe sensitivity of the DMS on τkmay be exploited to\ndetermine it experimentally.\nFinally, I determine τB, the characteristic relaxation\ntime in the polydisperse MRE. By requiring χs(ω) to\nmatchχ(ω) to the first order of ω, we obtain\nτB=/summationdisplay\nkpkχL\nkτk/χL. (21)\nNotably, this expression is identical to the characteristic\ntime defined by Ivanov et. al. based on their MMF1 the-\nory [27], although they do not distinguish τkfromτBk.\nImportantly, by comparing χs(ω) withχ(ω), we may de-\ntermine the frequency cutoff or the critical time scale, τc,\nbeyond which the polydisperse MRE is no longer valid.\nOn a time scale faster than τc, we can not discard mem-\nory effects related to inter-species coupling. Then the\ninstantaneous total magnetization is no longer a well-\ndefined thermodynamic variable.IV. CONCLUSIONS\nIn conclusion, I have developed a nonperturbative and\nself-consistent dynamical mean field theory for mono-\nand polydisperse interacting ferrofluids. I obtain a\ngeneric magnetization relaxation equation and a general\nexpression for the dynamic magnetic susceptibility, both\nof which are in simple and closed form. My theory is ex-\npectedtoplayacrucialroleinstudyingferrofluiddynam-\nics, with important consequences on many areas of appli-\ncationssuchashyperthermia, drugdelivery,andmechan-\nical engineering. The general strategy presented here can\nbe extended to other soft matter systems to construct\ndynamical mean field theories and obtain equations of\nmotion for coarse-grained variables.\nACKNOWLEDGEMENTS\nI would like to thank Prof. Philip J. Camp for provid-\ning the data for Figure 1 and many helpful discussions.\nI appreciate the editor for his long waiting for my re-\nvised manuscript. I acknowledge the support from North\nChina University of Water Resources and Electric Power\nvia Grant No. 201803023.\n[1] R. E. Rosenzweig, Ferrohydrodynamics , Cambridge Uni-\nversity Press, London, 1985.\n[2] S. Odenbach (Ed.), Colloidal Magnetic Fluids: Basics,\nDevelopment and Application of Ferrofluids , Springer,\nBerlin Heidelberg, 2009.\n[3] Y. A. Buyevich and A. O. Ivanov, Physica A, 1992, 190,\n276.\n[4] A. F. Pshenichnikov, J. Magn. Magn. Mater., 1995, 145,\n319.\n[5] B. Huke, and M. L¨ ucke, Phys. Rev. E, 2000, 62, 6875.\n[6] A. O. Ivanov, and O. B. Kuznetsova, Phys. Rev. E, 2001,\n64, 041405.\n[7] B. Huke and M. L¨ ucke, Rep. Prog. Phys., 2004, 67, 1731.\n[8] A.O. Ivanov, S.S.Kantorovich, E. N.Reznikov, C. Holm,\nA. F. Pshenichnikov, A.V. Lebedev, A. Chremos and P.\nJ. Camp, Phys. Rev. E, 2007, 75, 061405.\n[9] A. Y. Solovyova, E. A. Elfimova, A. O. Ivanov and P. J.\nCamp, Phys. Rev. E, 2017, 96, 052609.\n[10] P. J. W. Debye, Polar Molecules , Chemical Catalog Com-\npany, New York, 1929.\n[11] M. A. Martsenyuk, Y. L. Raikher and M. I. Shliomis, J.\nExp. Theor. Phys., 1974, 38, 413.\n[12] H.W.M¨ uller andM.Liu, Phys.Rev.E,2001, 64, 061405.\n[13] B. U. Felderhof and H. J. Kroh, J. Chem. Phys., 1999,\n110, 7403.\n[14] M. I. Shliomis, Phys. Rev. E, 2001, 64, 063501.\n[15] A. Fang, “The Dynamical Mean Field Model for Inter-\nacting Ferrofluids: I. Derivations for both homogeneous\nand inhomogeneous cases”, in preparation.\n[16] B. Cichocki, M. L. Ekiel-Je` zewska and E. Wajnryb, J.\nChem. Phys., 1999, 111, 3265.[17] K. Makuch, M. Heinen, G. C. Abade and G. N¨ agele, Soft\nMatter, 2015, 11, 5313.\n[18] A. J. Archer and R. Evans, J. Chem. Phys., 2004, 121,\n4246.\n[19] A. Y. Zubarev and A. V. Yushkov, J. Exp. Theor. Phys.,\n1998,87, 484.\n[20] P. Ilg and S. Hess, Z. Naturforsch. A, 2003, 58, 589.\n[21] A. Fang, Phys. Fluids, 2019, 31, 122002.\n[22] R. Zwanzig, Phys. Rev., 1961, 124, 983.\n[23] H. Grabert, Projection Operator Techniques in Nonequi-\nlibrium Statistical Mechanics , Springer Verlag, Berlin,\n1982.\n[24] E. Blums, A. Cebers and M. M. Maiorov, Magnetic Flu-\nids, Walter de Gruyter, Berlin, 1997.\n[25] A. Fang, “Variational Approach to the Hydrodynamics\nof Interacting Ferrofluids”, submitted.\n[26] A. O. Ivanov and P. C. Camp, Phys. Rev. E, 2018, 98,\n050602(R).\n[27] A. O. Ivanov, V. S. Zverev and S. S. Kantorovich, Soft\nMatter, 2016, 12, 3507.\n[28] Thenear-equilibriumMMF1theorypresentedinRef.[27 ]\nis inequvalent to the model derived from our theory sup-\nplemented with the MMF1 EMOS in near-equilibrium\nregime. To obtain the mean dipolar potential acting on\na representative particle, their theory employs a Debye-\nLangevin equilibrium ODF for other particles. Hence in\ntheir approximation “other” particles are in a static con-\nfiguration and not on the same footing as the represen-\ntative particle.\n[29] B. U. Felderhof and R. B. Jones, J. Phys.: Condens.\nMatter, 2003, 15, 4011.8\n[30] J.O.Sindt,P.J.Camp, S.S.Kantorovich, E.A.Elfimova\nand A. O. Ivanov, Phys. Rev. E, 2016, 93, 063117.\n[31] T. M. Batrudinov, Y. E. Nekhoroshkova, E. I. Para-\nmonov, V. S. Zverev, E. A. Elfimova, A. O. Ivanov and\nP. J. Camp, Phys. Rev. E, 2018, 98, 052602.\n[32] Y. L. Raikher and M. I. Shliomis, Adv. Chem. Phys.,\n1994,87, 595.\n[33] B. H. Ern´ e, K. Butter, B. W. M. Kuipers and G. J.\nVroege, Langmuir, 2003, 19, 8218.\n[34] A. O. Ivanov, S. S. Kantorovich, V. S. Zverev, E. A.\nElfimova, A. V. Lebedev and A. F. Pshenichnikov, Phys.Chem. Chem. Phys., 2016, 18, 18342.\n[35] A.O. Ivanov, S. S. Kantorovich, E. A. Elfimova, V. S.\nZverev, J. O. Sindt and P. J. Camp, J. Magn. Magn.\nMater., 2017, 431, 141.\n[36] A. O. Ivanov, S. S. Kantorovich, V. S. Zverev, A. V.\nLebedev, A. F. Pshenichnikov and P. J. Camp, J. Magn.\nMagn. Mater., 2018, 459, 252.\n[37] A. Fang, “The Dynamical Mean Field Model for Inter-\nacting Ferrofluids: II. The proper relaxation time and\neffects of dynamic correlation”, in preparation."    },    {        "title": "1812.00581v1.Engineering_planar_transverse_domain_walls_in_biaxial_magnetic_nanostrips_by_tailoring_transverse_magnetic_fields_with_uniform_orientation.pdf",        "content": "Article\nEngineering planar transverse domain walls in biaxial\nmagnetic nanostrips by tailoring transverse magnetic\nfields with uniform orientation\nMingna Yu1, Mei Li2* and Jie Lu1,*\n1College of Physics, Hebei Advanced Thin Films Laboratory, Hebei Normal University, Shijiazhuang 050024,\nHebei, China; 1543916410@qq.com (M.Y.)\n2Physics Department, Shijiazhuang University, Shijiazhuang 050035, Hebei, China\n*Correspondence: limeijim@163.com; jlu@hebtu.edu.cn\nReceived: date; Accepted: date; Published: date\nAbstract: Designing and realizing various magnetization textures in magnetic nanostructures are\nessential for developing novel magnetic nanodevices in modern information industry. Among all\nthese textures, planar transverse domain walls (pTDWs) are the simplest and the most basic, which\nmake them popular in device physics. In this work, we report the engineering of pTDWs with\narbitrary tilting attitude in biaxial magnetic nanostrips by transverse magnetic field profiles with\nuniform orientation but tunable strength distribution. Both statics and axial-field-driven dynamics\nof these pTDWs are analytically investigated. It turns out that for statics these pTDWs are robust\nagain disturbances which are not too abrupt, while for dynamics it can be tailored to acquire higher\nvelocity than Walker’s ansatz predicts. These results should provide inspirations for designing\nmagnetic nanodevices with novel one-dimensional magnetization textures, such as 360\u000ewalls, or\neven two-dimensional ones, for example vortices, skyrmions, etc.\nKeywords: magnetic nanostrips; planar transverse domain walls; transverse magnetic fields\n1. Introduction\nArtificially prepared magnetic nanostructures have been forming the basic components of\nnanodevices in modern information industry for decades[ 1,2]. Various magnetization textures therein\nprovide the abundant choices of defining zeros and ones in binary world. Among them, domain\nwalls (DWs) are the most common ones which separate magnetic domains with interior magnetization\npointing to different directions[ 3–8]. In magnetic nanostrips with rectangular cross sections, numerical\ncalculations confirm that there exists a critical cross-section area[ 9,10]. Below (above) it, transverse\n(vortex) walls dominate. For nanodevices based on DW propagation along strip axis with high\nintegral level, strips are thin enough so that only transverse DWs (TDWs) appear. Their velocity\nunder external driving factors (magnetic fields, polarized electronic currents, etc.) determines the\nresponse time of nanodevices based on DW propagation. In the past decades, analytical, numerical\nand experimental investigations on TDW dynamics have been widely performed and commercialized\nto a great extent[ 11–27]. However, seeking ways to further increase TDW velocity, thus improve the\ndevices’ response performance, is always the pursuit of both physicists and engineers.\nBesides velocity, fine manipulations of DW structure are also essential for improving the device\nperformance. In the simplest case, a TDW with uniform azimuthal distribution, which is generally\ncalled a planar TDW (pTDW), is of the most importance. Historically the Walker ansatz[11] provides\nthe first example of pTDW, however its tilting attitude is fully controlled by the driving field or current\ndensity (in particular, lying within easy plane in the absence of external driving factors) thus can not\nbe freely adjusted. In the past decades, several strategies[ 28–31] have been proposed to suppress or at\nleast postpone the Walker breakdown thus makes TDWs preserve traveling-wave mode which has a\nhigh mobility (velocity versus driving field or current density). The nature of all these proposals is to\ndestroy the two-fold symmetry in the strip cross section, thus is equivalent to a transverse magneticarXiv:1812.00581v1  [cond-mat.mes-hall]  3 Dec 20182 of 14\nfield (TMF), no matter it’s built in or extra. In 2016, the “velocity-enhancement\" effect of uniform TMFs\n(UTMFs) on TDWs in biaxial nanostrips has been thoroughly investigated[ 32]. It turns out that UTMFs\ncan considerably boost TDWs’ propagation meanwhile inevitably leaving a twisting in their azimuthal\nplanes. However for applications in nanodevices with high density, the twisting is preferred to be\nerased to minimize magnetization frustrations and other stochastic fields. In 2017, optimized TMF\nprofiles with fixed strength and tunable orientation are proposed to realize pTDWs with arbitrary\ntilting attitude[ 33]. Dynamical analysis on these pTDWs reveals that they can propagate along strip\naxis with higher velocities than those without TMFs. However, there are several remaining problems:\nthe rigorous analytical pTDW profile (thus TMF distribution) is still lacking, the pTDW width can not\nbe fully controlled and the real experimental setup is challenging.\nIn this work, we engineer pTDWs with arbitrary tilting attitude in biaxial magnetic nanostrips\nby tailoring TMF profiles with uniform orientation but tunable strength distribution. For statics,\nthe well-tailored TMF profile manipulates pTDW with arbitrary tilting attitude, clear boundaries\nand controllable width. In particular, these pTDWs are robust again disturbances which are not too\nabrupt. For axial-field-driven dynamics with TMFs comoving, pTDWs will acquire higher velocity\nthan Walker’s ansatz predicts.\n2. Model and Preparations\n\u000b \f1zke\u000b \f2xke\nyemeTe\nIeT IA)\u000b \f,H ztA\n1HM Nanostrip \nFigure 1. Sketch of biaxial magnetic nanostrip under consideration. ( ex,ey,ez) is the global Cartesian\ncoordinate system in real space: ezis along strip axis, exis in the thickness direction and ey=ez\u0002ex.\nk1(k2)is the total magnetic anisotropy coefficient in easy (hard) axis. ( em,eq,ef) forms the local\nspherical coordinate system associated with the magnetization vector M(blue arrow with magnitude\nMs, polar angle qand azimuthal angle f). The total external field has two components: axial driving\nfield with magnitude H1and TMF with constant tilting attitude F?and tunable magnitude H?(z,t).\nWe consider a biaxial magnetic nanostrip with rectangular cross section, as depicted in Figure 1.\nThe zaxis is along strip axis, the xaxis is in the thickness direction and ey=ez\u0002ex. The magnetic\nenergy density functional of this strip can be written as,\nEtot[M,Hext] =\u0000m0M\u0001Hext\u0000k1\n2m0M2\nz+k2\n2m0M2\nx+J(rm)2, (1)\nin which m\u0011M/Mswith Msbeing the saturation magnetization. The magnetostatic energy density\nhas been described by quadratic terms of Mx,y,zvia three average demagnetization factors Dx,y,z[34]\nand thus been absorbed into k1,2ask1=k0\n1+ (Dy\u0000Dz)and k2=k0\n2+ (Dx\u0000Dy)[17,31,32], where\nk0\n1,2are the magnetic crystalline anisotropy coefficients. The external field Hexthas two components:\nthe axial driving field Hk\u0011H1ezand the TMF with general form\nH?=H?(z,t)\u0002\ncosF(z,t)ex+sinF(z,t)ey\u0003\n. (2)\nThe time evolution of M(r,t)is described by the Landau-Lifshitz-Gilbert (LLG) euqation[35] as3 of 14\n¶m\n¶t=\u0000gm\u0002Heff+am\u0002¶m\n¶t, (3)\nwhere aphenomenologically describes magnetic damping strength, g>0is the absolute value of\nelectron’s gyromagnetic ratio and Heff=\u0000(dEtot/dM)/m0is the effective field.\nWhen system temperature is far below Curie point, the saturation magnetization Msof magnetic\nmaterials can be viewed as constant. Thus M(r,t)is fully described by its polar angle q(r,t)\nand azimuthal angle f(r,t). In addition, for thin enough nanostrips (where TDWs dominate) the\ninhomogeneity in cross section can be ignored thus make them become quasi one-dimensional (1D)\nsystems ( r!z). Then reasonably one has (rm)2\u0011(rzm)2= (q0)2+sin2q(f0)2in which a\nprime means spatial derivative to z. After the transition from the global Cartesian coordinate system\n(ex,ey,ez) to the local spherical coordinate system ( em,eq,ef), the effective field Heffreads\nHeff=Hm\neffem+Hq\neffeq+Hf\neffef, (4a)\nHm\neff=H1cosq+H?(z,t)sinqcos[F?(z,t)\u0000f]+k1Ms\u0000Mssin2q\u0010\nk1+k2cos2f\u0011\n\u00002J\nm0Ms(q02+sin2qf02)2, (4b)\nHq\neff=\u0000H1sinq+H?(z,t)cosqcos[F?(z,t)\u0000f]\u0000Mssinqcosq\u0010\nk1+k2cos2f\u0011\n+2J\nm0Ms(q00\u0000sinqcosqf02)\u0011\u0000B , (4c)\nHf\neff=H?(z,t)sin[F?(z,t)\u0000f]+k2Mssinqsinfcosf+2J\nm0Ms1\nsinq\u0010\nsin2q\u0001f0\u00110\n\u0011A . (4d)\nPut it back into Eq. (3), the vectorial LLG equation turns to its scalar counterparts,\n(1+a2)˙q/g=A\u0000 aB, (5a)\n(1+a2)sinq˙f/g=B+aA, (5b)\nor equivalently\n˙q+asinq˙f=gA, (6a)\nsinq˙f\u0000a˙q=gB, (6b)\nwhere a dot means time derivative. These equations are all what we need for our work is this paper.\n3. Results\nIn this section, we present in details how to engineer pTDWs with arbitrary tilting attitude by\nproperly tailoring TMF profile along strip axis. As mentioned in Section 1, here we fix the TMF\norientation (thus F?(z,t)\u0011F0) and allow its strength tunable along strip axis, which is much\neasier to realize in real experiments. Both statics and axial-field-driven dynamics of pTDWs will be\nsystematically investigated.\n3.1. Statics\nFrom the roadmap of field-driven DW motion in nanostrips[ 17], in the absence of axial driving\nfields a TDW will finally evolve into its static configurations ( ˙q=˙f=0) under time-independent\nTMFs ( H?(z,t)\u0011H?(z)). For Eq. (6) this means A=B=0. In the absence of any TMF ( H?(z)\u00110),\nthe static TDW is a pTDW lying in easy plane with the well-known Walker’s profile[11],4 of 14\nq(z) =2 arctan ehz\u0000z0\nD0,f(z)\u0011np/2, (7)\nwhere D0\u0011p\n2J/(m0k1M2s)is the pTDW width, z0is the wall center, h= + 1(\u00001)denotes\nhead-to-head (tail-to-tail) pTDWs and n= + 1(\u00001)is the wall polarity (sign of hmyi). However,\nif we want to realize a static pTDW with arbitrary tilting attitude, i.e. f(z)\u0011fd, well-tailored\nposition-dependent TMF profile must be exerted.\n3.1.1. Boundary condition\nAs the first step, we need the boundary condition of this pTDW, which means the magnetization\norientation in the two domains at both ends of the strip. Without losing generality, our investigations\nare performed for head-to-head walls and 0<fd<p/2. In the two domains, the orientation of\nmagnetization should be uniform, meaning that the azimuthal angle satisfies f(z)\u0011fd, while the\npolar angle in the left (right) domain takes the value of qd(p\u0000qd). Meantime, the TMF strength\nshould be constant ( H?(z)!Hd\n?) in these two domains. Then A=B=0 becomes\nHd\n?sin(fd\u0000F0) =k2Mssinqdsinfdcosfd, (8a)\nHd\n?cos(F0\u0000fd) =Mssinqd(k1+k2cos2fd). (8b)\nThe solution to the above equation set provides the TMF profile in the two domains as\nF0=arctan\u0012k1\nk1+k2\u0001tanfd\u0013\n,Hd\n?=Hmax\n?\u0001sinqd, (9)\nwith\nHmax\n?=Msq\nk2\n1sin2fd+ (k1+k2)2cos2fd. (10)\nEq. (9) indicates that in both domains, TMF should be farther away from the easy plane than the\nmagnetization. Meanwhile, the existence condition of the pTDW ( qd6=p/2) requires that TMF\nstrength in domains has an upper limit,\nHd\n?<Hmax\n?. (11)\n3.1.2. Static pTDW profile\nNote that we have fixed TMF orientation to be F0, therefore in pTDW region A=B=0becomes\n0=H?(z)sin(F0\u0000f)+k2Mssinqsinfcosf+2J\nm0Ms1\nsinq\u0010\nsin2q\u0001f0\u00110\n, (12a)\n2J\nm0Msq00=\u0000H?(z)cosqcos(F0\u0000f)+Mssinqcosq\u0010\nk1+k2cos2f\u0011\n+2J\nm0Mssinqcosqf02. (12b)\nSince we are considering pTDWs with uniform tilting attitude f(z)\u0011fd, then the above equations\nbecome\nH?(z)sin(fd\u0000F0)=k2Mssinqsinfdcosfd, (13a)\n2J\nm0Msq00=\u0000H?(z)cosqcos(F0\u0000fd)+Mssinqcosq\u0010\nk1+k2cos2fd\u0011\n. (13b)\nCombing Eqs. (8a) and (13a), one has5 of 14\nH?(z) =Hd\n?\nsinqd\u0001sinq(z) =Hmax\n?\u0001sinq(z). (14)\nPutting it back into Eq. (13b) and considering Eq. (8b), it turns out that\n2J\nm0Msq00=sinqcosq\nsinqdh\nMssinqd(k1+k2cos2fd)\u0000Hd\n?cos(F0\u0000fd)i\n=0, (15)\nwhich means q(z)is linear in pTDW region,\nq(z) =C1+C2\u0001(z\u0000z0), (16)\nwhere z0is the pTDW center.\nze\nyexe\n:zM\ndcos T\ndcos T\u00100\nz\nz\n0z\n02z'\u001002z'\u000e\u000b \fzT\n\u000b \fzI\ndIdTdS T \u0010\n02SS\nFigure 2. Illustration of pTDW profile with arbitrary titling attitude fd, controllable width Dand linear\npolar angle distribution from qdtop\u0000qd. The color chart indicates the variation of Mzcomponent\nalong strip axis from cos qdin the left domain to \u0000cosqdin the right domain.\nIt is worth noting that in nearly all existing literatures, the boundary between “domains\" and\n“domain walls\" in nanostrips is not clear (or abrupt) since q(z)and f(z)and their derivatives are\nall continuous there. However, Eqs. (14) to (16) provide us an opportunity to realize a pTDW with\nclear boundary and tunable width, as depicted in Figure. 2. In summary, under the following TMF\ndistribution\nH?(z) =8\n><\n>:Hd\n?, z<z0\u0000D\n2\nHmax\n?\u0001sinn\nqd+p\u00002qd\nDh\nz\u0000\u0010\nz0\u0000D\n2\u0011io\n,z0\u0000D\n2<z<z0+D\n2\nHd\n?, z>z0+D\n2,F?(z)\u0011F0, (17)\na pTDW with the following profile will emerge in the nanostrip,6 of 14\nq0(z) =8\n><\n>:qd, z<z0\u0000D\n2\nqd+p\u00002qd\nDh\nz\u0000\u0010\nz0\u0000D\n2\u0011i\n,z0\u0000D\n2<z<z0+D\n2\np\u0000qd, z>z0+D\n2,f0(z)\u0011fd. (18)\nInterestingly, the above pTDW has the following features: (i) an arbitrary tilting attitude fd. (ii) a\nfully controllable width Dand (iii) two clear boundaries ( z0\u0006D/2) with the two adjacent domains.\nNote that the magnetization and TMF at z0\u0006D/2are both continuous, but rzmis not. This inevitably\nleads to a finite jump of exchange energy density right there.\nHowever, the pTDW has a critical width Dcunder which the entire strip has lower magnetic\nenergy compared with the single-domain state under the UTMF with strength Hd\n?and orientation F0.\nTo see this, we integrate EpTDW\ntot\u0000Edomain\ntot over the entire strip and thus\nDE=k1m0M2\ns\n2\u0001\u0014\n(D0)2(p\u00002qd)21\nD\u0000sin 2qd+ (p\u00002qd)cos 2 qd\n2(p\u00002qd)\u0012\n1+k2\nk1cos2fd\u0013\nD\u0015\n. (19)\nObviously, there exists a critical pTDW width\nDc\u0011D0\u0001\u0012\n1+k2\nk1cos2fd\u0013\u00001\n2\n\u0001k(qd),k(qd)\u0011s\n2(p\u00002qd)3\nsin 2qd+ (p\u00002qd)cos 2 qd. (20)\nAsHd\n?!Hmax\n?, by definingHd\n?\nHmax\n?=1\u0000ewe have qd=arcsinHd\n?\nHmax\n?\u0019p\n2\u0000p\n2e, thus sin2qd\u00192p\n2e,\ncos2qd\u0019\u0000 1+4eandp\u00002qd\u00192p\n2e. Putting all these approximations back into kin Eq. (20), we\nfinally get k!2which leads to a finite critical pTDW Dc. As a result, we can always make the pTDW\nenergetically preferred by setting D>Dc(thusDE<0).\n3.1.3. Stability analysis\nTo make the explorations on statics complete and self-consistent, we need to perform stability\nanalysis on the pTDW profile in Eq. (18). For simplicity, the variations on q(z)andf(z)are processed\nseparately. In the first step, f(z)\u0011f0is fixed (thus ˙f\u00110) and suppose the polar angle departs from\nits static profile as\nq=q0+dq. (21)\nPutting it back into Eq. (6b), by noting that ˙f0=0 and ˙q0=0, one has\nsinq˙f\u0000a˙q=gB)a\ng¶(dq)\n¶t=\u0000B. (22)\nOn the other hand, in pTDW region q0satisfies Eq. (13b). After performing series expansion on B\naround q0and preserving up to linear terms of dq, we finally get\na\ng¶(dq)\n¶t\u0019\u0014\n\u0000Mscos2q0(k1+k2cos2f0) +2J\nm0Ms(dq)00\ndq\u0015\n\u0001dq. (23)\nObviously, when\n\f\f\f\f(dq)00\ndq\f\f\f\f<cos2q0(1+k2cos2f0/k1)\n(D0)2, (24)\ndqfades out as times goes by. This implies that when the variation dqis not too abrupt, q0is stable. In\nfact, most variations satisfy this demand. For example, both tiny global translations along z\u0000axis and\nslight local variations proportional to z\u0000z0make (dq)00\u00110 thus assure the stability around q0.\nIn the second step, we keep q(z)\u0011q0and let the azimuthal angle varies as follows7 of 14\nf=f0+df. (25)\nSubstituting it into Eq. (6a), by recalling that ˙q0=0 and ˙f0=0, we have\n˙q+asinq˙f=gA)a\ng¶(df)\n¶t=A\nsinq0. (26)\nRemember in pTDW region q0andf0satisfy Eq. (13a). By performing series expansion on Aabout f0\nand at most keeping linear terms of dq, one has\na\ng¶(df)\n¶t\u0019\u0014\n\u0000Ms(k1+k2sin2f0) +2J\nm0Ms2 cot q0\u0001q0\n0\u0001(df)0+ (df)00\ndf\u0015\n\u0001df. (27)\nSimilarly, if dfdoes not varies too abruptly, that is\n\f\f\f\f2 cot q0\u0001q0\n0\u0001(df)0+ (df)00\ndf\f\f\f\f<1+k2sin2f0/k1\n(D0)2, (28)\nthe pTDW is stable around f0, which confirms the feasibility of engineering pTDWs in magnetic\nnanostrips. In particular, tiny global rotations around z\u0000axis or slight local twistings proportional to\nz\u0000z0will not drive pTDW away from its static profile shown in Eq. (18).\n3.1.4. Numerical confirmations\nTo confirm the above theoretical analysis, we perform numerical simulations using the OOMMF\nmicromagnetics package[ 36]. In our simulations, the nanostrip is 5 nm thick, 100 nm wide and 1 mm\nlong, which is quite common in real experiments. The three average demagnetization factors are:\nDx=0.00661366 ,Dy=0.07002950 and Dz=0.92335684 [34]. Magnetic parameters are as follows:\nMs=500kA/m, J=40\u000210\u000012J/m, K1=m0k0\n1M2\ns/2=200kJ/m3,K2=m0k0\n2M2\ns/2=50kJ/m3\nand a=0.1to speed up the simulation. Throughout the entire calculation, the strip is discretized\ninto 5\u00025\u00025nm3cells and all magnetic intensive quantities evaluated at each cell are the average of\ntheir continuous counterparts over the cell volume. In all figures, z0denotes the wall center which is\nthe algebraic average of the central positions ( f(z) =p/2) of each layer (row of cells with a certain\ny-coordinate). At last, the external TMF at each cell is the value from Eq. (17) at the cell center.\nWe aim to realize a pTDW with tilting attitude fd\u0011p/4and boundary condition qd\u0011p/6\nunder the TMF profile in Eq. (17). To do this, firstly simple algebra provides us D0=13.80 nm (14.14\nnm) when the demagnetization is (not) considered. Then the critical pTDW width Dc=35.66 nm (41.31\nnm) for each case. Therefore we set the pTDW width as D=100nm to assure energetic preference. We\nhave performed simulations for both cases in which magnetostatic effect is included or not. At each\ncase, a standard head-to-head Néel wall with width 20 nm is generated at the strip center beforehand.\nAfter it relaxes to its stable profile, a time-independent TMF described by Eq. (17) is exerted onto\neach calculation cell of this strip. The magnetization texture then begin to evolve accompanied by the\ndecreasing total magnetic energy due to the Gilbert damping process. We set the convergence strategy\nasjm\u0002Htotj/Ms<10\u00007, which is accurate enough. The results are plotted in Figure 3(a) and 3(b),\nrespectively.\nIn the simpler case, the pTDW profile under TMF distribution described in Eq. (17) with fd\u0011p/4,\nqd\u0011p/6andD=100nm in the absence of demagnetization is plotted in Figure 3(a). The solid\nblack and red lines are the analytical polar and azimuthal distributions from Eq. (18), respectively.\nThe open circles are numerical data from OOMMF simulation. Clearly the planar nature of wall\nis reproduced very well. For polar angle, the linear behavior near pTDW center is unambiguous.\nWhile the discontinuity in polar angle derivative at pTDW border ( z0\u000650 nm) is weakened due to\nthe inevitable “discretized sampling\" of TMF at calculation cells during numerical simulations. In\nsummary one may clearly see that the numerics and analytics fit very well.8 of 14\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s32/s32/s65/s110/s103/s108/s101/s115/s32/s40 /s41/s58/s32\n/s79/s79/s77/s77/s70\n/s58/s32\n/s84/s104/s101/s111\n/s58/s32\n/s79/s79/s77/s77/s70\n/s58/s32\n/s84/s104/s101/s111/s78/s111/s32/s100/s101/s109/s97/s103/s40/s97/s41\n/s45/s49/s48/s48 /s45/s53/s48 /s48 /s53/s48 /s49/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s32/s58/s32/s84/s111/s112\n/s58/s32/s67/s101/s110/s116/s114/s97/s108\n/s58/s32/s98/s111/s116/s116/s111/s109/s40/s98/s41\n/s87 /s105/s116/s104/s32/s100/s101/s109/s97/s103/s65/s110/s103/s108/s101/s115/s32/s40 /s41\n/s122/s45/s122\n/s48/s32/s40/s110/s109/s41\nFigure 3. Comparisons between analytical (solid lines) and numerical (hollow symbols) pTDW profiles\nunder TMF in Eq. (17) with fd\u0011p/4,qd\u0011p/6andD=100nm: (a) without demagnetization, (b)\nwith demagnetization. The magnetic parameters are as follows: Ms=500kA/m, J=40\u000210\u000012J/m,\nK1=m0k0\n1M2s/2=200 kJ/m3,K2=m0k0\n2M2s/2=50 kJ/m3anda=0.1.\nThen we switch on the magnetostatic interaction (demagnetization). Due to the complicated\ndipole-dipole interaction, the magnetization orientation in the strip cross section differs a little (not\ntoo much since the strip is thin enough). We then calculate the polar and azimuthal angles for three\ntypical layer (rows of cells with the same y\u0000coordinate): top, central and bottom. The resulting data\nare depicted in Figure 3(b) by different discrete hollow symbols: crosses, squares and triangles. It\nturns out that they overlap each other nicely and match the analytical profiles quite well. This not only\nreproves the validity of TMF in Eq. (17) for realizing pTDW in Eq. (18) under more complex situations,\nbut also shows once again the feasibility of simplifying magnetostatic energy by local quadratic terms\nin thin enough nanostrips.\n3.2. Axial-field-driven dynamics\nFrom the roadmap of field-driven DW dynamics[ 17], an axial magnetic field is crucial for driving\npTDWs to move along strip axis thus realizing bit-switchings in magnetic nanodevices based on them.\nWe focus on the traveling-wave mode of pTDWs in which their profile is generalized directly from Eq.\n(18) by allowing z0to depend on time meantime leaving the rest unchanged. To preserve the pTDW\nprofile, the TMF distribution is suggested to take the same form as in Eq. (17) but with the generalized\nz0, which means that TMF moves along with the pTDW sharing the same velocity. In this section,\nthe dynamics of these pTDWs are systematically investigated under two strategies: 1D collective\ncoordinate model (1D-CCM)[ 20] and 1D asymptotic expansion method (1D-AEM)[ 32,33,37,38]. As\nwill be shown below, they provide the same result which confirms the feasibility of both approaches.9 of 14\n3.2.1. 1D-CCM\nHistorically, 1D-CCM plays important role in the exploration of TDW dynamics for both\nfield-driven and current-driven cases. Generally it treats the center, tilting attitude and with of a\nDW as independent collective variables of the system Lagrangian or the resulting dynamical equations\n(i.e. LLG equation). The classical Walker ansatz (which is indeed a pTDW profile) in the absence of any\nTMFs is the first example and turns out to be the rigorous solution of LLG equation. In the presence of\nUTMFs, generally no rigorous solutions exist due to the mismatch between symmetries in different\nenergy terms. In most theoretical works, pTDWs with quasi-Walker profiles are often proposed to\nmimic the real complicated magnetization distribution. However, in Section 3.1 it has been shown that\nthe Walker ansatz is not the only choice that a pTDW can preceed. In this subsection, we provide the\npTDW velocity with comoving TMF profile in the framework of 1D-CCM.\nBefore the main context, we want to point out that to preserve the planar feature of these walls,\nthe strength of axial driving field should not be too high. To see this, we revisit the boundary condition\nin the two domains in the presence of axial driving field H1. Note that although in pTDW region, Heff\nis not parallel with m(otherwise the wall will not move), however in both domains it holds since\nmagnetization does not vary with time, hence A=B=0therein. After redefining the polar and\nazimuthal angles of magnetization in the left domain as ˜qdand ˜fd(p\u0000˜qdand ˜fdin the right domain),\none has\n0=Hd\n?sin(F0\u0000˜fd) +k2Mssin˜qdsin˜fdcos˜fd, (29a)\n0=H1sin˜qd\u0000Hd\n?cos˜qdcos(F0\u0000˜fd) +Mssin˜qdcos˜qd(k1+k2cos2˜fd), (29b)\nObviously, only when H1\u001cmin[Hd\n?,Ms]one has ˜qd\u0019qdand ˜fd\u0019fd. Then after the generalization\nof collective coordinate z0from constant to time-dependent, the pTDW in Eq. (18) is expected to move\nalong strip axis under the comoving TMF in Eq. (17) with the velocity equal to d z0/dt.\nTo determine wall velocity in traveling-wave mode, we perform time derivative of the pTDW\nprofile which gives\n˙q(z,t) =8\n><\n>:0, z<z0\u0000D\n2\n\u0000p\u00002qd\nD\u0001dz0\ndt,z0\u0000D\n2<z<z0+D\n2\n0, z>z0+D\n2,˙f(z,t)\u00110. (30)\nFrom Eq. (5b), the traveling-mode condition ˙f(z,t)\u00110leads toA=\u0000B/a. Putting back into Eq. (5a),\nit turns out that\u0000a˙q(z,t)/g=B. Substituting Eq. (30) into it, one has\na\ng\u0001p\u00002qd\nD\u0001dz0\ndt=H1sinq\u0000H?(z,t)cosqcos(F0\u0000f)+Mssinqcosq\u0010\nk1+k2cos2f\u0011\n\u00002J\nm0Msq00.\n(31)\nNote that the generalized TMF configuration and the resulting pTDW profile still satisfy Eq. (13b), thus\neliminate the last three terms in the right hand side of the above equation. Then after integrating Eq.\n(31) over the pTDW region, z2\u0010\nz0\u0000D\n2,z0+D\n2\u0011\n, and noting thatRz0+D/2\nz0\u0000D/21dz=D,Rz0+D/2\nz0\u0000D/2sinqdz=\n2Dcosqd/(p\u00002qd), we finally get\nVa\u0011dz0\ndt=gD\na\u0001w(qd)\u0001H1,w(qd)\u00112 cos qd\n(p\u00002qd)2. (32)\nNext we examine the asymptotic behavior of the boosting factor w(qd)when Hd\n?!Hmax\n?. Suppose\nagainHd\n?\nHmax\n?=1\u0000e, then cosqd\u0019p\n2eand p\u00002qd\u00192p\n2e. Putting them back into Eq. (32), we\nfinally have10 of 14\nw(qd)\u00191\n2p\n2e!+¥, (33)\nase!0+. This confirms the boosting effect of these TMFs on axial propagation of pTDWs.\nAt last, stability analysis to dynamical pTDW profile under comoving TMFs takes the same\nformat as static case and thus has been omitted for saving space. It turns out that for profile variations\nwhich are not too abrupt, the traveling-wave mode of pTDW is also stable. This is really important for\npotential commercial applications of these pTDWs.\n3.2.2. 1D-AEM\nNext we recalculate the pTDW velocity in traveling-wave mode with the help of 1D-AEM. In\nthis approach, the dynamical behavior of pTDWs is viewed as the response of their static profiles to\nexternal stimuli. Therefore it is the manifestation of linear response framework in nanomagnetism\nand should be suitable for exploring traveling-wave mode of pTDWs under small axial driving fields.\nNote that the TMF distribution in Eq. (17) indicates that at the pTDW center TMF strength reaches\nHmax\n?which is finite, thus we rescale the axial driving field and pTDW axial velocity simultaneously,\nH1=eh1,Vb=evb, (34)\nin which eis a dimensionless infinitesimal. This means a slight external stimulus ( H1) will lead to a\nweak response of the system, that is, a slow velocity ( Vb) of pTDW axial motion. We concentrate on\ntraveling-wave mode of pTDWs thus define the traveling coordinate\nx\u0011z\u0000Vbt=z\u0000evbt. (35)\nMeantime the TMF distribution takes the same one as in Eq. (17), except for the generalization of\nz!x. As a result, the real solution of pTDW can be expanded as follows,\nc(z,t) =c0(x) +ec1(x) +O(e2),c=q(f), (36)\nwhere q0(f0)denote the zeroth-order solutions and should be the static pTDW profile (will see later),\nwhile q1and f1are the coefficients of first-order corrections to zeroth-order solutions when H1is\npresent. Putting them into the LLG equation (6) and noting that ¶c/¶t= (\u0000evb)\u0001¶c/¶x, we have\n(\u0000evb)\u0001\u0012¶q0\n¶x+asinq0¶f0\n¶x\u0013\n+O(e2) =gA0+gA1\u0001e+O(e2), (37a)\n(\u0000evb)\u0001\u0012\nsinq0¶f0\n¶x\u0000a¶q0\n¶x\u0013\n+O(e2) =gB0+gB1\u0001e+O(e2), (37b)\nwith\nA0=H?(x)sin(F0\u0000f0) +k2Mssinq0sinf0cosf0+2J\nm0Ms\u0012\n2 cos q0¶q0\n¶x¶f0\n¶x+sinq0¶2f0\n¶x2\u0013\n, (38a)\nB0=\u0000H?(x)cosq0cos(F0\u0000f0)\u00002J\nm0Ms¶2q0\n¶x2+k1Mssinq0cosq0\"\n1+k2\nk1cos2f0+D2\n0\u0012¶f0\n¶x\u00132#\n,\n(38b)\nand11 of 14\nA1=Pq1+Qf1,\nP=k2Mscosq0sinf0cosf0+2J\nm0Ms\u0014\n2¶f0\n¶x\u0012\ncosq0¶\n¶x\u0000sinq0¶q0\n¶x\u0013\n+cosq0¶2f0\n¶x2\u0015\n,\nQ=\u0000H?(x)cos(F0\u0000f0) +k2Mssinq0cos 2 f0+2J\nm0Ms\u0012\n2 cos q0¶q0\n¶x¶\n¶x+sinq0¶2\n¶x2\u0013\n, (39)\nas well as\nB1=h1sinq0+Rq1+Sf1,\nR=H?(x)sinq0cos(F0\u0000f0)\u00002J\nm0Ms¶2\n¶x2+k1Mscos 2 q0\"\n1+k2\nk1cos2f0+D2\n0\u0012¶f0\n¶x\u00132#\n,\nS=\u0000H?(x)cosq0sin(F0\u0000f0) +k1Mssin 2q0\u0012\nD2\n0¶f0\n¶x¶\n¶x\u0000k2\nk1sinf0cosf0\u0013\n. (40)\nAt the zeroth order of e, Eq. (37) provides A0=B0=0. Combing with the definitions in Eq.\n(38), its solution is just the pTDW profile in Eq. (18) except for the substitution of z!x. This is not\nsurprising since zeroth-order solution describes the response of system under “zero\" stimulus which is\njust the static case.\nHowever to obtain the pTDW velocity, we need to proceed to the first order of e. In particular, we\nhave to deal with Rand Sto get the dependence of velocity ( vb) on axial driving field ( h1). By partially\ndifferentiatingB0=0 with respect to f0,Scan be simplified to\nS=D2\n0k1Mssin 2q0\u0012¶f0\n¶x¶\n¶x\u0000¶2f0\n¶x2\u0013\n\u00110 (41)\ndue to the planar nature of walls. On the other hand, the partial derivative of B0=0with respect to q0\nhelps to simplify Rto\nR=2J\nm0Ms\"\n\u0000¶2\n¶x2+\u0012¶q0\n¶x\u0013\u00001\u0012¶3q0\n¶x3\u0013#\n\u0011L, (42)\nwhich is the 1D self-adjoint Schrödinger operator appeared in previous works[ 32,33,37,38]. Then Eq.\n(40) rigorously turns to\nLq1=\u0000h1sinq0+ (\u0000vb)\u0001\u0012\n\u0000a¶q0\n¶x\u0013\n. (43)\nAgain the “Fredholm alternative\" requests the right hand side of the above equation to be orthogonal\nto the kernel of L(subspace expanded by ¶q0/¶x) for the existence of a solution q1, where the inner\nproduct in Sobolev space is defined as hf(x),g(x)i\u0011Rx=+¥\nx=\u0000¥f(x)\u0001g(x)dx. Noting thath¶q0\n¶x,sinq0i=\n2 cos qdandh¶q0\n¶x,¶q0\n¶xi= (p\u00002qd)2/D, we finally get\nVb\u0011dz0\ndt=gD\na\u00012 cos qd\n(p\u00002qd)2\u0001H1, (44)\nwhich is the same as Eq. (32) from 1D-CCM.\n4. Discussion\nIn Section 3.2 we point out that under axial driving fields, the pTDW velocity can be considerably\nincreased due to the divergent behavior of the boosting factor w(qd)when H?!Hmax\n?(see Eq. (33)).\nInterestingly, the contribution of pTDW width, i.e. D, is also an important boosting factor. From Eq.12 of 14\n(20) one has a finite critical pTDW width even when H?!Hmax\n?. Therefore to further increase the\npTDW velocity, broadening the pTDW width should also be effective.\nSecond, to realized pTDWs the “orientation-fixed\" strategy proposed here has several advantages\ncomparing with the “amplitude-fixed\" one introduced before[ 33]: (i) the wall width can be freely\ntuned. (ii) the rigorous pTDW profile and the corresponding TMF distribution can be explicitly written\nout. (iii) the asymptotic behavior of the boosting factor in axial-field-driven case can be analytically\nexplored. (iv) most importantly, the “orientation-fixed\" strategy is much easier to realize in real\nexperiments.\nFor example, the following procedure can be applied to realize a pTDW with center position z0,\nwidth D, tilting attitude fdand boundary condition qd(p\u0000qd). First a short and strong enough field or\ncurrent pulse is exerted to induce a wall around z0and after a transient process it finally becomes static\nin easy plane with Walker’s profile. Then a series of ferromagnetic scanning tunneling microscope\n(STM) tips are placed along the wire axis with fixed tilting attitude F0to produce a series of localized\nTMF pulsed. By arranging these tips with proper spacing and distance to strip, the envelope of these\npulses is tuned to be the TMF profile in Eq. (17). The resulting static wall profile is the pTDW shown in\nEq. (18). When driving by axial field, since the transient process prior to traveling-wave mode is short\n(picoseconds), the STM tips can be arrange to move at the velocity in Eq. (32) so as to synchronize with\nthe pTDW.\nAt last, our “orientation-fixed\" strategy can be generalized to the cases where pTDW motion is\ninduced by spin-polarized currents, spin waves or temperature gradient, etc. Similar discussions can\nbe performed to realized these pTDWs with clear boundaries. Magnetic nanostrips bearing with these\nwalls would serve as proving ground for developing new-generation nanodevices with fascinating\napplications.\n5. Conclusions\nIn this work, the “orientation-fixed\" TMF profiles are adopted to realize pTDW with arbitrary\ntilting attitude in biaxial magnetic nanostrips. After solving the LLG equation, unlike the classical\nWalker ansatz we obtain a pTDW with clear boundaries with adjacent domains and linear polar angle\ndistribution inside wall region. More interestingly, the wall width can be freely tuned for specific\nusages. With TMF profile synchronized along with, these pTDWs can propagate along strip axis with\nconsiderably high velocity (well above that from the Walker ansatz) when driven by axial magnetic\nfields. These results should provide new insights in developing fascinating new-generation magnetic\nnanodevices based on DW propagations in nanostrips.\nAuthor Contributions: Conceptualization, M.L. and J.L.; Methodology, M.L.; Validation, J.L.; Formal analysis,\nJ.L.; Investigation, M.Y. and M.L.; Writing—original draft preparation, M.Y.; Writing—review and editing, M.L.\nand J.L.; Supervision, J.L.; Project Administration, J.L.; Funding Acquisition, J.L.\nFunding: This research was funded by the National Natural Science Foundation of China (Grants No. 11374088).\nConflicts of Interest: The authors declare no conflict of interest.\nAbbreviations\nThe following abbreviations are used in this manuscript:\nDW Domain wall\nTDW Transverse DW\npTDW planar TDW\nTMF Transverse magnetic field\nUTMF Uniform TMF\nLLG Landau-Lifshitz-Gilbert\n1D one-dimensional\n1D-CCM 1D collective coordinate model\n1D-AEM 1D asymptotic expansion method13 of 14\nReferences\n1. Leeuw, F.H.D.; Doel, R.V .D.; Enz, U. Dynamic properties of magnetic domain walls and magnetic bubbles.\nReports on Progress in Physics 1980 ,43, 689, doi:10.1088/0034-4885/43/6/001.\n2. Tserkovnyak, Y.; Brataas, A.; Bauer, G.E.W.; Halperin, B.I. Nonlocal magnetization dynamics in ferromagnetic\nheterostructures. Reviews of Modern Physics 2005 ,77, 1375-1421, doi:10.1103/RevModPhys.77.1375.\n3. Allwood, D.A.; Xiong, G.; Faulkner, C.C.; Atkinson, D.; Petit, D.; Cowburn, R.P . Magnetic domain-wall logic.\nScience 2005 ,309, 1688-1692, doi:10.1126/science.1108813.\n4. Parkin, S.S.P .; Hayashi, M.; Thomas, L. Magnetic domain-wall racetrack memory. Science 2008 ,320, 190-194,\ndoi:10.1126/science.1145799.\n5. Hayashi, M.; Thomas, L.; Moriya, R.; Rettner, C.; Parkin, S.S.P . Current-controlled magnetic domain-wall\nnanowire shift register. Science 2008 ,320, 209-211, doi:10.1126/science.1154587.\n6. Franken, J.H.; Swagten, H.J.M.; Koopmans, B. Shift registers based on magnetic domain wall ratchets with\nperpendicular anisotropy. Nature Nanotechnology 2012 ,7, 499, doi:10.1038/nnano.2012.111.\n7. Münchenberger, J.; Reiss, G.; Thomas, A. A memristor based on current-induced domain-wall motion\nin a nanostructured giant magnetoresistance device. Journal of Applied Physics 2012 ,111, 07D303,\ndoi:10.1063/1.3671438.\n8. Parkin, S.; Yang, S.-H. Memory on the racetrack. Nature Nanotechnology 2015 ,10, 195,\ndoi:10.1038/nnano.2015.41.\n9. McMichael, R.D.; Donahue, M.J. Head to head domain wall structures in thin magnetic strips. IEEE\nTransactions on Magnetics 1997 ,33, 4167-4169, doi:10.1109/20.619698.\n10. Nakatani, Y.; Thiaville, A.; Miltat, J. Head-to-head domain walls in soft nano-strips: a refined phase diagram.\nJournal of Magnetism and Magnetic Materials 2005 ,290-291 , 750-753, doi:10.1016/j.jmmm.2004.11.355.\n11. Schryer, N.L.; Walker, L.R. The motion of 180odomain walls in uniform dc magnetic fields. Journal of Applied\nPhysics 1974 ,45, 5406-5421, doi:10.1063/1.1663252.\n12. Ono, T.; Miyajima, H.; Shigeto, K.; Mibu, K.; Hosoito, N.; Shinjo, T. Propagation of a magnetic domain wall\nin a submicrometer magnetic wire. Science 1999 ,284, 468, doi:10.1126/science.284.5413.468.\n13. Atkinson, D.; Allwood, D.A.; Xiong, G.; Cooke, M.D.; Faulkner, C.C.; Cowburn, R.P . Magnetic domain-wall\ndynamics in a submicrometre ferromagnetic structure. Nature Materials 2003 ,2, 85, doi:10.1038/nmat803.\n14. Beach, G.S.D.; Nistor, C.; Knutson, C.; Tsoi, M.; Erskine, J.L. Dynamics of field-driven domain-wall\npropagation in ferromagnetic nanowires. Nature Materials 2005 ,4, 741, doi:10.1038/nmat1477.\n15. Tretiakov, O.A.; Clarke, D.; Chern, G.-W.; Bazaliy, Y.B.; Tchernyshyov, O. Dynamics of domain walls in\nmagnetic nanostrips. Physical Review Letters 2008 ,100, 127204, doi:10.1103/PhysRevLett.100.127204.\n16. Yang, J.; Nistor, C.; Beach, G.S.D.; Erskine, J.L. Magnetic domain-wall velocity oscillations in permalloy\nnanowires. Physical Review B 2008 ,77, 014413, doi:10.1103/PhysRevB.77.014413.\n17. Wang, X.R.; Yan, P .; Lu, J. High-field domain wall propagation velocity in magnetic nanowires. Europhysics\nLetters 2009 ,86, 67001, doi:10.1209/0295-5075/86/67001.\n18. Wang, X.R.; Yan, P .; Lu, J.; He, C. Magnetic field driven domain-wall propagation in magnetic nanowires.\nAnnals of Physics 2009 ,324, 1815-1820, doi:10.1016/j.aop.2009.05.004.\n19. Sun, Z.Z.; Schliemann, J. Fast domain wall propagation under an optimal field pulse in magnetic nanowires.\nPhysical Review Letters 2010 ,104, 037206, doi:10.1103/PhysRevLett.104.037206.\n20. Shibata, J.; Tatara, G.; Kohno, H. A brief review of field- and current-driven domain-wall motion. Journal of\nPhysics D: Applied Physics 2011 ,44, 384004, doi:10.1088/0022-3727/44/38/384004.\n21. Berger, L. Emission of spin waves by a magnetic multilayer traversed by a current. Physical Review B 1996 ,\n54, 9353-9358, doi:10.1103/PhysRevB.54.9353.\n22. Slonczewski, J.C. Current-driven excitation of magnetic multilayers. Journal of Magnetism and Magnetic\nMaterials 1996 ,159, L1-L7, doi:10.1016/0304-8853(96)00062-5.\n23. Li, Z.; Zhang, S. Domain-wall dynamics and spin-wave excitations with spin-transfer torques. Physical\nReview Letters 2004 ,92, 207203, doi:10.1103/PhysRevLett.92.207203.\n24. Yamaguchi, A.; Ono, T.; Nasu, S.; Miyake, K.; Mibu, K.; Shinjo, T. Real-space observation of\ncurrent-driven domain wall motion in submicron magnetic wires. Physical Review Letters 2004 ,92, 077205,\ndoi:10.1103/PhysRevLett.92.077205.14 of 14\n25. Beach, G.S.D.; Knutson, C.; Nistor, C.; Tsoi, M.; Erskine, J.L. Nonlinear domain-wall velocity enhancement by\nspin-polarized electric current. Physical Review Letters 2006 ,97, 057203, doi:10.1103/PhysRevLett.97.057203.\n26. Hayashi, M.; Thomas, L.; Bazaliy, Y.B.; Rettner, C.; Moriya, R.; Jiang, X.; et al. Influence of current on\nfield-driven domain wall motion in permalloy nanowires from time resolved measurements of anisotropic\nmagnetoresistance. Physical Review Letters 2006 ,96, 197207, doi:10.1103/PhysRevLett.96.197207.\n27. Yan, P .; Wang, X.R. Optimal time-dependent current pattern for domain wall dynamics in nanowires. Applied\nPhysics Letters 2010 ,96, 162506, doi:10.1063/1.3413951.\n28. Nakatani, Y.; Thiaville, A.; Miltat, J. Faster magnetic walls in rough wires. Nature Materials 2003 ,2, 521,\ndoi:10.1038/nmat931.\n29. Lee, J.-Y.; Lee, K.-S.; Kim, S.-K. Remarkable enhancement of domain-wall velocity in magnetic nanostripes.\nApplied Physics Letters 2007 ,91, 122513, doi:10.1063/1.2789176.\n30. Bryan, M.T.; Schrefl, T.; Atkinson, D.; Allwood, D.A. Magnetic domain wall propagation in nanowires under\ntransverse magnetic fields. Journal of Applied Physics 2008 ,103, 073906, doi:10.1063/1.2887918.\n31. Lu, J.; Wang, X.R. Motion of transverse domain walls in thin magnetic nanostripes under transverse magnetic\nfields. Journal of Applied Physics 2010 ,107, 083915, doi:10.1063/1.3386468.\n32. Lu, J. Statics and field-driven dynamics of transverse domain walls in biaxial nanowires under uniform\ntransverse magnetic fields. Physical Review B 2016 ,93, 224406, doi:10.1103/PhysRevB.93.224406.\n33. Li, M.; Wang, J.; Lu, J. General planar transverse domain walls realized by optimized transverse magnetic\nfield pulses in magnetic biaxial nanowires. Scientific Reports 2017 ,7, 43065, doi:10.1038/srep43065.\n34. Aharoni, A. Demagnetizing factors for rectangular ferromagnetic prisms. Journal of Applied Physics 1998 ,83,\n3432-3434, doi:10.1063/1.367113.\n35. Gilbert, T.L. A phenomenological theory of damping in ferromagnetic materials. IEEE Transactions on\nMagnetics 2004 ,40, 3443-3449, doi:10.1109/TMAG.2004.836740.\n36. Donahue, M.J.; Porter, D.G. OOMMF User’s Guide, Version 1.0, Interagency Report NISTIR 6376 (National\nInstitute of Standards and Technology, Gaithersburg, MD, Sept 1999; http://math.nist.gov/oommf).\n37. Goussev, A.; Lund, R.G.; Robbins, J.M.; Slastikov, V .; Sonnenberg, C. Fast domain-wall\npropagation in uniaxial nanowires with transverse fields. Physical Review B 2013 ,88, 024425,\ndoi:10.1103/PhysRevB.88.024425.\n38. Goussev, A.; Lund, R.G.; Robbins, J.M.; Slastikov, V .; Sonnenberg, C. Domain wall motion in magnetic\nnanowires: an asymptotic approach. Proceedings of the Royal Society A: Mathematical, Physical and Engineering\nScience 2013 ,469, 20130308, doi:10.1098/rspa.2013.0308."    },    {        "title": "0909.1391v2.Memory__Aging_and_Spin_Glass_Nature__A_Study_of_NiO_Nanoparticles.pdf",        "content": "arXiv:0909.1391v2  [cond-mat.mes-hall]  9 Sep 2009\n/C5/CT/D1/D3/D6/DD /B8 /BT/CV/CX/D2/CV /CP/D2/CS /CB/D4/CX/D2 /BZ/D0/CP/D7/D7 /C6/CP/D8/D9/D6/CT/BM /BT /CB/D8/D9/CS/DD /D3/CU /C6/CX/C7 /C6/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7/CE/CX/CY/CP /DD /BU/CX/D7/CW /D8∗/CP/D2/CS /C3/BA/C8 /BA/CA/CP /CY/CT/CT/DA†/BW/CT/D4 /CP/D6/D8/D1/CT/D2/D8 /D3/CU /C8/CW/DD/D7/CX\r/D7/B8 /C1/D2/CS/CX/CP/D2 /C1/D2/D7/D8/CX/D8/D9/D8/CT /D3/CU /CC /CT \r/CW/D2/D3/D0/D3 /CV/DD /C3/CP/D2/D4/D9/D6 /BE/BC/BK/BC/BD/BI/B8 /C1/D2/CS/CX/CP/CF /CT /D6/CT/D4 /D3/D6/D8 /D7/D8/D9/CS/CX/CT/D7 /D3/D2 /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CS/DD/D2/CP/D1/CX\r/D7 /CX/D2 /C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /D3/CU /CP /DA /CT/D6/CP/CV/CT /D7/CX/DE/CT /BH /D2/D1/BA /CC /CT/D1/B9/D4 /CT/D6/CP/D8/D9/D6/CT /CP/D2/CS /D8/CX/D1/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /D3/CU /CS\r /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/B8 /DB /CP/CX/D8 /D8/CX/D1/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /D3/CU /D1/CP/CV/D2/CT/D8/CX\r /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2/B4/CP/CV/CX/D2/CV/B5 /CP/D2/CS /D1/CT/D1/D3/D6/DD /D4/CW/CT/D2/D3/D1/CT/D2/CP /CX/D2 /D8/CW/CT /CS\r /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D6/CT /D7/D8/D9/CS/CX/CT/CS /DB/CX/D8/CW /DA /CP/D6/CX/D3/D9/D7 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CP/D2/CS/AS/CT/D0/CS /D4/D6/D3/D8/D3 \r/D3/D0/D7/BA /CF /CT /D3/CQ/D7/CT/D6/DA /CT /D8/CW/CP/D8 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /D7/CW/D3 /DB/D7 /D1/CT/D1/D3/D6/DD /CP/D2/CS /CP/CV/CX/D2/CV /CX/D2 /AS/CT/D0/CS \r/D3 /D3/D0/CT/CS /CP/D2/CS /DE/CT/D6/D3 /AS/CT/D0/CS\r/D3 /D3/D0/CT/CS /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7/BA /CC/CW/CT/D7/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /D7/CW/D3 /DB /D8/CW/CP/D8 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /C6/CX/C7/D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /CX/D7 /D7/CX/D1/CX/D0/CP/D6 /D8/D3 /D7/D4/CX/D2 /CV/D0/CP/D7/D7/CT/D7/BA /CF /CT /CP/D6/CV/D9/CT /D8/CW/CP/D8 /D8/CW/CT /D7/D4/CX/D2 /CV/D0/CP/D7/D7 /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/D6/CX/CV/CX/D2/CP/D8/CT/D7 /CU/D6/D3/D1 /D8/CW/CT/CU/D6/CT/CT/DE/CX/D2/CV /D3/CU /D7/D4/CX/D2/D7 /CP/D8 /D8/CW/CT /D7/D9/D6/CU/CP\r/CT /D3/CU /D8/CW/CT /CX/D2/CS/CX/DA/CX/CS/D9/CP/D0 /D4/CP/D6/D8/CX\r/D0/CT/D7/BA/C8 /BT /BV/CB /D2 /D9/D1 /CQ /CT/D6/D7/BM /BJ/BH/BA/BH/BC/BA/CC/D8/B8 /BJ/BH/BA/BH/BC/BA/C4/CZ/B8 /BJ/BH/BA/BF/BC/BA/BV/D6/B8 /BJ/BH/BA/BG/BC/BA/BZ/CQ/C3/CT/DD/DB /D3/D6/CS/D7/BM /C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7/B8 /D1/CP/CV/D2/CT/D8/CX\r /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2/B8 /CP/CV/CX/D2/CV/B8 /D1/CT/D1/D3/D6/DD /CT/AR/CT\r/D8/D7/BA/C1/BA /C1/C6/CC/CA /C7/BW/CD/BV/CC/C1/C7/C6/CC/CW/CT /D7/D0/D3 /DB /CS/DD/D2/CP/D1/CX\r/D7 /D7/CW/D3 /DB/D2 /CQ /DD /D1/CP/CV/D2/CT/D8/CX\r /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7/CW/CP/D7 /CQ /CT/CT/D2 /CP/D2 /CP\r/D8/CX/DA /CT /CP/D6/CT/CP /D3/CU /D6/CT/D7/CT/CP/D6\r /CW /CU/D3/D6 /D8/CW/CT /D4/CP/D7/D8 /D8 /DB /D3/CS/CT\r/CP/CS/CT/D7 /CQ /CT\r/CP/D9/D7/CT /D3/CU /D2 /D9/D1/CT/D6/D3/D9/D7 /D8/CT\r /CW/D2/D3/D0/D3/CV/CX\r/CP/D0 /CP/D4/D4/D0/CX\r/CP/D8/CX/D3/D2/D7/CP/D7 /DB /CT/D0/D0 /CP/D7 /CU/D3/D6 /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS/CX/D2/CV /D8/CW/CT /D4/CW /DD/D7/CX\r/D7 /CQ /CT/CW/CX/D2/CS /D8/CW/CT /CT/DC/D3/D8/CX\r/D4/CW/CT/D2/D3/D1/CT/D2/CP /D3/CQ/D7/CT/D6/DA /CT/CS/BA\n/BD/BY /CT/D6/D6/D3 /CP/D2/CS /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /D2/CP/D2/D3/D4/CP/D6/B9/D8/CX\r/D0/CT/D7 /CW/CP /DA /CT /CQ /CT/CT/D2 /D7/D8/D9/CS/CX/CT/CS /D1/D3/D6/CT /D8/CW/CP/D2 /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r/D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /CQ /CT\r/CP/D9/D7/CT /D3/CU /D8/CW/CT/CX/D6 /D8/CT\r /CW/D2/D3/D0/D3/CV/CX\r/CP/D0 /D4 /D3/D8/CT/D2 /D8/CX/CP/D0 /CP/D7/D8/CW/CT/DD /CW/CP /DA /CT /CW/CX/CV/CW /D1/CP/CV/D2/CT/D8/CX\r /D1/D3/D1/CT/D2 /D8/D7/BA\n/BE/BT/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r/D1/CP/D8/CT/D6/CX/CP/D0/D7 /D7/CW/D3 /DB /CP /CS/D6/CP/D7/D8/CX\r \r /CW/CP/D2/CV/CT /CX/D2 /D8/CW/CT/CX/D6 /D1/CP/CV/D2/CT/D8/CX\r /D4/D6/D3/D4/B9/CT/D6/D8/CX/CT/D7 /DB/CW/CT/D2 /D8/CW/CT /D4/CP/D6/D8/CX\r/D0/CT /D7/CX/DE/CT /CV/D3 /CT/D7 /D8/D3 /D8/CW/CT /D2/CP/D2/D3 /D6/CT/CV/CX/D1/CT /CQ /CT/B9\r/CP/D9/D7/CT /D3/CU /D8/CW/CT /D9/D2\r/D3/D1/D4 /CT/D2/D7/CP/D8/CT/CS /D7/D4/CX/D2/D7 /CP/D8 /D8/CW/CT /D7/D9/D6/CU/CP\r/CT /DB/CW/CX\r /CW/CV/CX/DA /CT /D6/CX/D7/CT /D8/D3 /CP /D2/CT/D8 /D1/CP/CV/D2/CT/D8/CX\r /D1/D3/D1/CT/D2 /D8/BA /CC/CW/CX/D7 /D0/CT/CP/CS/D7 /D8/D3 /D1/CP/D2 /DD/CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV /D1/CP/CV/D2/CT/D8/CX\r /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /CT/BA/CV/BA /CP /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2 /CQ /CT/B9/D8 /DB /CT/CT/D2 /AS/CT/D0/CS \r/D3 /D3/D0/CT/CS /B4/BY /BV/B5 /CP/D2/CS /DE/CT/D6/D3 /AS/CT/D0/CS \r/D3 /D3/D0/CT/CS /B4/CI/BY /BV/B5 /D1/CP/CV/B9/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/B8 /CP /D4 /CT/CP/CZ /CX/D2 /CI/BY /BV /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/B8 /D7/D0/D3 /DB /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2/D3/CU /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/B8 /DB /CP/CX/D8 /D8/CX/D1/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /D3/CU /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /B4/CP/CV/CX/D2/CV/B5 /CP/D2/CS /D1/CT/D1/D3/D6/DD /CX/D2 /BY /BV /CP/D2/CS /CI/BY /BV /D1/CP/CV/D2/CT/B9/D8/CX/DE/CP/D8/CX/D3/D2 /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7/BA\n/BF /B8/BG/B8/BH/B8/BI/B8/BJ /B8/BK/B8/BL/B8/BD/BC/B8/BD/BD/C1/CU /D8/CW/CT /D4/CP/D6/D8/CX\r/D0/CT/D7 /CP/D6/CT/D2/D3/D2 /CX/D2 /D8/CT/D6/CP\r/D8/CX/D2/CV/B8 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CS/DD/D2/CP/D1/CX\r/D7 /CX/D7 /CS/CT/D7\r/D6/CX/CQ /CT/CS/CQ /DD /D7/D9/D4 /CT/D6/D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /CP/D7 /D4/D6/CT/CS/CX\r/D8/CT/CS /CQ /DD /C6/GH/CT/D0/B9/BU/D6/D3 /DB/D2 /D8/CW/CT/D3/D6/DD /BA\n/BD/BE /B8/BD/BF/C7/D2 /D8/CW/CT /D3/D8/CW/CT/D6 /CW/CP/D2/CS/B8 /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 \r/CP/D2/CV/CX/DA /CT /D6/CX/D7/CT /D8/D3 /CP /D7/D4/CX/D2 /CV/D0/CP/D7/D7 /D0/CX/CZ /CT /CQ /CT/CW/CP /DA/CX/D3/D6 /B4/D7/D9/D4 /CT/D6/D7/D4/CX/D2 /CV/D0/CP/D7/D7/B5/CX/D2 /CX/D2 /D8/CT/D6/CP\r/D8/CX/D2/CV /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7/BA\n/BF /B8/BH/B8/BI/B8/BJ/B8/BD/BG/C0/D3 /DB /CT/DA /CT/D6/B8 /D7/D4/CX/D2 /CV/D0/CP/D7/D7/CQ /CT/CW/CP /DA/CX/D3/D6 \r/CP/D2 /CP/D0/D7/D3 /CP/D6/CX/D7/CT /CX/D2 /D8/CW/CT /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /CS/D9/CT /D8/D3 /D7/D4/CX/D2/CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2 /CP/D8 /D8/CW/CT /D7/D9/D6/CU/CP\r/CT /D3/CU /CX/D2/CS/CX/DA/CX/CS/D9/CP/D0 /D4/CP/D6/D8/CX\r/D0/CT/D7/BA\n/BD/BD /B8/BD/BH/B8/BD/BI/B8/BD/BJ/BU/D9/D0/CZ /C6/CX\r /CZ /CT/D0 /D3 /DC/CX/CS/CT /B4/C6/CX/C7/B5 /CX/D7 /CZ/D2/D3 /DB/D2 /D8/D3 /CQ /CT /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/B9/D2/CT/D8/CX\r /DB/CX/D8/CW /CP /C6/GH/CT/D0 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT T/C6\n/D3/CU /BH/BE/BF /C3/BA /CC/CW/CT /D8/CT/D1/D4 /CT/D6/B9/CP/D8/D9/D6/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /D3/CU /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D3/CU /C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7/DB /CP/D7 /AS/D6/D7/D8 /D7/D8/D9/CS/CX/CT/CS /CX/D21956 /CQ /DD /CA/CX\r /CW/CP/D6/CS/D7/D3/D2 /CP/D2/CS /C5/CX/D0/D0/CX/CV/CP/D2 /CP/D2/CS/CP /D4 /CT/CP/CZ /CX/D2 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /DB /CP/D7 /CU/D3/D9/D2/CS /D1 /D9\r /CW/CQ /CT/D0/D3 /DB /D8/CW/CT /CQ/D9/D0/CZT/C6\n/BA\n/BD/BK/C1/D8 /DB /CP/D7 /D3/CQ/D7/CT/D6/DA /CT/CS /D8/CW/CP/D8 /D3/D2 /CS/CT\r/D6/CT/CP/D7/B9/CX/D2/CV /D8/CW/CT /D4/CP/D6/D8/CX\r/D0/CT /D7/CX/DE/CT/B8 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D2\r/D6/CT/CP/D7/CT/D7 /CP/D2/CS /D8/CW/CT/D4 /CT/CP/CZ /CX/D2 /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /D7/CW/CX/CU/D8/D7 /D8/D3 /D0/D3 /DB /CT/D6 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7/BA /C4/CP/D8/CT/D6/CX/D21961 /C6/GH/CT/D0 /D7/D9/CV/CV/CT/D7/D8/CT/CS /D8/CW/CP/D8 /D7/D1/CP/D0/D0 /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r/D4/CP/D6/D8/CX\r/D0/CT/D7 /D7/CW/D3/D9/D0/CS /CT/DC/CW/CX/CQ/CX/D8 /D7/D9/D4 /CT/D6/D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX/D7/D1 /CP/D2/CS /DB /CT/CP/CZ/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX/D7/D1/BA\n/BD/BL/CC/CW/CT /D3/CQ/D7/CT/D6/DA /CT/CS /D4/CP/D6/D8/CX\r/D0/CT /D1/D3/D1/CT/D2 /D8 /D3/CU /C6/CX/C7/D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /CX/D7 /CU/D3/D9/D2/CS /D8/D3 /CQ /CT /D1 /D9\r /CW /D0/CP/D6/CV/CT/D6 /D8/CW/CP/D2 /D8/CW/CP/D8 /D4/D6/CT/B9/CS/CX\r/D8/CT/CS /CQ /DD /D8/CW/CT /D8 /DB /D3 /D0/CP/D8/D8/CX\r/CT /D1/D3 /CS/CT/D0 /D3/CU /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/D7 /CP/D2/CS/CP /D1 /D9/D0/D8/CX /D7/D9/CQ/D0/CP/D8/D8/CX\r/CT /D1/D3 /CS/CT/D0 /CW/CP/D7 /CQ /CT/CT/D2 /D4/D6/D3/D4 /D3/D7/CT/CS /D8/D3 /CT/DC/D4/D0/CP/CX/D2/CX/D8 /CP/D2/CS /CP/D0/D7/D3 /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/CS /CW/CX/CV/CW \r/D3 /CT/D6\r/CX/DA/CX/D8/CX/CT/D7 /CP/D2/CS /D0/D3 /D3/D4 /D7/CW/CX/CU/D8/D7\n/CX/D2 /D8/CW/CT/D7/CT /D4/CP/D6/D8/CX\r/D0/CT/D7/BA\n/BE/BC /B8/BE/BD/CC/CW/CT/D6/CT /CW/CP /DA /CT /CQ /CT/CT/D2 /D7/D3/D1/CT /D6/CT/D4 /D3/D6/D8/D7 /D3/D2/D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /D3/CU /C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /DB/CW/CX\r /CW \r/D0/CP/CX/D1/D8/CW/CP/D8 /D8/CW/CT/DD /CP/D6/CT /D7/D9/D4 /CT/D6/D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r/BA\n/BE/BE /B8/BE/BF/B8/BE/BG/B8/BE/BH/B8/BE/BI /B8/BE/BJ/C0/D3 /DB /CT/DA /CT/D6/B8/D8/CW/CT/D6/CT /CP/D6/CT /CX/D7/D7/D9/CT/D7 /CX/D2 \r/D3/D2/D7/CX/CS/CT/D6/CX/D2/CV /D8/CW/CT/D1 /CP/D7 /D7/D9/D4 /CT/D6/D4/CP/D6/CP/D1/CP/CV/B9/D2/CT/D8/CX\r /CP/D7 /D8/CW/CT/CX/D6 /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 \r/CP/D2/D2/D3/D8 /CQ /CT /CS/CT/D7\r/D6/CX/CQ /CT/CS /CQ /DD /D8/CW/CT/D1/D3 /CS/CX/AS/CT/CS /C4/CP/D2/CV/CT/DA/CX/D2 /CU/D9/D2\r/D8/CX/D3/D2/BA\n/BE/BC/CC/CX/DB /CP/D6/CX /CT/D8 /CP/D0/BA /CW/CP /DA /CT /CS/D3/D2/CT/CP /CS/CT/D8/CP/CX/D0/CT/CS /D7/D8/D9/CS/DD /D3/D2 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /D3/CU /C6/CX/C7/D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /CP/D2/CS /CW/CP /DA /CT \r/D0/CP/CX/D1/CT/CS/B8 /D3/D2 /D8/CW/CT /CQ/CP/D7/CX/D7 /D3/CU /D7\r/CP/D0/B9/CX/D2/CV /CP/D6/CV/D9/D1/CT/D2 /D8/D7/B8 /D8/CW/CP/D8 /C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /D7/CW/D3 /DB /D7/D4/CX/D2 /CV/D0/CP/D7/D7/CQ /CT/CW/CP /DA/CX/D3/D6/BA\n/BD/BH/CC/CW/CT/DD /CW/CP /DA /CT /D4/D6/D3/D4 /D3/D7/CT/CS /D8/CW/CP/D8 /D8/CW/CT /D7/D9/D6/CU/CP\r/CT /D7/D4/CX/D2 /CS/CX/D7/B9/D3/D6/CS/CT/D6 /CP/D2/CS /CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2 /CV/CX/DA /CT /D6/CX/D7/CT /D8/D3 /D7/D9\r /CW /CQ /CT/CW/CP /DA/CX/D3/D6/BA /CF/CX/D2/CZ/D0/CT/D6/CT/D8 /CP/D0/BA /CW/CP /DA /CT /CS/D3/D2/CT /D1/CP/CV/D2/CT/D8/CX\r /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /D3/D2 /CQ /D3/D8/CW /CQ/CP/D6/CT/CP/D2/CS /D4 /D3/D0/DD/D1/CT/D6 /CS/CX/D7/D4 /CT/D6/D7/CT/CS /C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /D3/CU /D7/CX/DE/CT /BF /D2/D1/CP/D2/CS /CW/CP /DA /CT /CU/D3/D9/D2/CS /D8/CW/CP/D8 /D8/CW/CT/DD \r/CP/D2 /CQ /CT /D8/CW/D3/D9/CV/CW /D8 /D8/D3 /CQ /CT \r/D3/D2/B9/D7/CX/D7/D8/CX/D2/CV /D3/CU /CP/D2 /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r \r/D3/D6/CT /DB/CX/D8/CW /CP/D2 /D9/D2\r/D3/D1/D4 /CT/D2/B9/D7/CP/D8/CT/CS /D1/D3/D1/CT/D2 /D8 /CP/D2/CS /CP /CS/CX/D7/D3/D6/CS/CT/D6/CT/CS /D7/D9/D6/CU/CP\r/CT /D7/CW/CT/D0/D0/BA\n/BD/BI/CC/CW/CT/DD /CW/CP /DA /CT/D4/D6/D3/D4 /D3/D7/CT/CS /D8/CW/CP/D8 /D8/CW/CT /CX/D2 /D8/CT/D6/D4/CP/D6/D8/CX\r/D0/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 \r/CP/D2 /CX/D2\r/D6/CT/CP/D7/CT/D8/CW/CT /CT/AR/CT\r/D8/CX/DA /CT /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /CT/D2/CT/D6/CV/DD /D3/CU /D8/CW/CT \r/D3/D6/CT /D1/CP/CV/D2/CT/D8/CX\r /D1/D3/B9/D1/CT/D2 /D8/D7 /DB/CW/CX\r /CW /D6/CT/D7/D9/D0/D8/D7 /CX/D2 /D7/CW/CX/CU/D8/CX/D2/CV /D8/CW/CT /CU/D6/CT/CT/DE/CX/D2/CV /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7/D8/D3 /CW/CX/CV/CW/CT/D6 /DA /CP/D0/D9/CT/D7 /CP/D2/CS /CX/D2 /CT/D2/CW/CP/D2\r/CX/D2/CV /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT/D7/D4/CX/D2/D7 /CP/D8 /D8/CW/CT /D7/D9/D6/CU/CP\r/CT/BA /CC/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7/CX/D7 /CP/D0/D7/D3 /CU/D3/D9/D2/CS /D8/D3 /CS/CT/D4 /CT/D2/CS /D3/D2 /D8/CW/CT /D1/CT/D8/CW/D3 /CS /D3/CU /D4/D6/CT/D4/CP/D6/CP/D8/CX/D3/D2/B8/DB/CW/CT/D8/CW/CT/D6 /D8/CW/CT/DD /CP/D6/CT \r/D3/CP/D8/CT/CS /D3/D6 /D2/D3/D8/B8 /CP/D2/CS /D8/CW/CT /D2/CP/D8/D9/D6/CT /D3/CU /D8/CW/CT\r/D3/CP/D8/CX/D2/CV/BA\n/BD/BI /B8/BE/BE/B8/BE/BF/B8/BE/BK/B8/BE/BL/B8/BF/BC/B8/BF/BD/BT/CV/CX/D2/CV /CP/D2/CS /D1/CT/D1/D3/D6/DD /CT/AR/CT\r/D8/D7 /CW/CP /DA /CT /CQ /CT/CT/D2 /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CT/CS /CX/D2/D1/CP/D2 /DD /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT /D7/DD/D7/D8/CT/D1/D7 /D9/D7/CX/D2/CV /CP\r /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /CP/D2/CS/D0/D3 /DB /AS/CT/D0/CS /CS\r /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /DB/CX/D8/CW /DA /CP/D6/CX/D3/D9/D7/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CP/D2/CS /AS/CT/D0/CS /D4/D6/D3/D8/D3 \r/D3/D0/D7/BA\n/BH /B8/BI/B8/BJ /B8/BK/B8/BD/BC/B8/BD/BG/B8/BF/BE/B8/BF/BF/B8/BF/BG/B8/BF/BH/B8/BF/BI/B8/BF/BJ/C6/D3/D2/B9/CX/D2 /D8/CT/D6/CP\r/D8/CX/D2/CV /D4/CP/D6/D8/CX\r/D0/CT/D7 /CP/D6/CT /CT/DC/D4 /CT\r/D8/CT/CS /D8/D3 /D7/CW/D3 /DB /CP/CV/CX/D2/CV /CP/D2/CS/D1/CT/D1/D3/D6/DD /CT/AR/CT\r/D8/D7 /D3/D2/D0/DD /CX/D2 /BY /BV /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7/BA/CC/CW/CT/D7/CT /CT/AR/CT\r/D8/D7 /CW/CP /DA /CT /CQ /CT/CT/D2 /D3/CQ/D7/CT/D6/DA /CT/CS /CQ /DD /DA /CP/D6/CX/D3/D9/D7 /CP/D9/D8/CW/D3/D6/D7 /CP/D2/CS/D8/CW/CT/CX/D6 /CT/DC/D4/D0/CP/D2/CP/D8/CX/D3/D2 /CX/D7 /CQ/CP/D7/CT/CS /D3/D2 /CP /D7/CX/D1/D4/D0/CT /D7/D9/D4 /CT/D6/D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r/D1/D3 /CS/CT/D0 /DB/CW/CT/D6/CT /D3/D2/CT /CP/D7/D7/D9/D1/CT/D7 /CP /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D3/CU /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD/CT/D2/CT/D6/CV/DD /CQ/CP/D6/D6/CX/CT/D6/D7 /CP/D2/CS /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CS/D6/CX/DA /CT/D2 /CS/DD/D2/CP/D1/CX\r/D7/BA\n/BI /B8/BK/B8/BD/BC/B8/BF/BJ/BU/DD \r/D3/D2 /D8/D6/CP/D7/D8/B8 /CX/D2 /CX/D2 /D8/CT/D6/CP\r/D8/CX/D2/CV /D4/CP/D6/D8/CX\r/D0/CT/D7/B8 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/CS/DD/D2/CP/D1/CX\r/D7 /CX/D7 /D7/D4/CX/D2 /CV/D0/CP/D7/D7 /D0/CX/CZ /CT /CP/D2/CS /D7/D3 /CX/D8 /CX/D7 /CT/DC/D4 /CT\r/D8/CT/CS /D8/CW/CP/D8/D8/CW/CT/DD /DB /D3/D9/D0/CS /D7/CW/D3 /DB /CP/CV/CX/D2/CV /CP/D2/CS /D1/CT/D1/D3/D6/DD /CT/AR/CT\r/D8/D7 /CX/D2 /CQ /D3/D8/CW /BY /BV/CP/D2/CS /CI/BY /BV /D4/D6/D3/D8/D3 \r/D3/D0/D7 /D0/CX/CZ /CT /D7/D4/CX/D2 /CV/D0/CP/D7/D7/CT/D7/BA /C1/D2/CS/CT/CT/CS/B8 /D8/CW/CX/D7 /CX/D7 /D8/CW/CT\r/CP/D7/CT /CP/D2/CS /D1/D3 /CS/CT/D0/D7 /CQ/CP/D7/CT/CS /D3/D2 \r/CP/D2/D3/D2/CX\r/CP/D0 /D7/D4/CX/D2 /CV/D0/CP/D7/D7/CT/D7 /CW/CP /DA /CT/CQ /CT/CT/D2 /D9/D7/CT/CS /D8/D3 /CT/DC/D4/D0/CP/CX/D2 /D8/CW/CT/D7/CT /CT/AR/CT\r/D8/D7 /CX/D2 /D7/D9\r /CW /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7/DD/D7/D8/CT/D1/D7/BA\n/BH /B8/BI/CC/CW /D9/D7 /D8/CW/CT /D4/D6/CT/D7/CT/D2\r/CT /D3/CU /CP/CV/CX/D2/CV /CP/D2/CS /D1/CT/D1/D3/D6/DD /CX/D2/CI/BY /BV /D4/D6/D3/D8/D3 \r/D3/D0 /CX/D7 /D0/CX/CZ /CT /CP /D0/CX/D8/D1 /D9/D7 /D8/CT/D7/D8 /CU/D3/D6 /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D8/CX/D2/CV /D7/D4/CX/D2/CV/D0/CP/D7/D7/CT/D7 /CP/D2/CS /D7/D9/D4 /CT/D6/D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/D7/BA/BE/C5/D3/D7/D8 /D3/CU /D8/CW/CT /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /D7/D8/D9/CS/CX/CT/CS /CU/D3/D6 /CP/CV/CX/D2/CV /CP/D2/CS /D1/CT/D1/B9/D3/D6/DD /CT/AR/CT\r/D8/D7 /CP/D6/CT /CU/CT/D6/D6/D3 /D3/D6 /CU/CT/D6/D6/CX/D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CS /D8/CW/CT/D6/CT /CP/D6/CT /DA /CT/D6/DD/CU/CT/DB /D7/D8/D9/CS/CX/CT/D7 /D3/D2 /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7/BA /CF /CT /CU/CT/CT/D0/D8/CW/CP/D8 /CX/D8 /DB /D3/D9/D0/CS /CQ /CT /CX/D2 /D8/CT/D6/CT/D7/D8/CX/D2/CV /D8/D3 /D7/D8/D9/CS/DD /D8/CW/CT/D7/CT /CT/AR/CT\r/D8/D7 /CX/D2 /C6/CX/C7/D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7/B8 /CP/D2 /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D7/DD/D7/D8/CT/D1 /CX/D2 /DB/CW/CX\r /CW /D7/D9/D6/B9/CU/CP\r/CT /CT/AR/CT\r/D8/D7 /CP/D6/CT /CZ/D2/D3 /DB/D2 /D8/D3 /D4/D0/CP /DD /CP /D1/CP /CY/D3/D6 /D6/D3/D0/CT /CX/D2 /CS/CT/D8/CT/D6/D1/CX/D2/B9/CX/D2/CV /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /CQ /CT/CW/CP /DA/CX/D3/D6/BA /C1/D2 /CU/CP\r/D8/B8 /CX/D8 /CW/CP/D7 /CQ /CT/CT/D2 \r/D0/CP/CX/D1/CT/CS/D8/CW/CP/D8 /D8/CW/CT/D7/CT /D4/CP/D6/D8/CX\r/D0/CT/D7 /D7/CW/D3 /DB /D7/D4/CX/D2 /CV/D0/CP/D7/D7 /CQ /CT/CW/CP /DA/CX/D3/D6/BA\n/BD/BH /B8/BD/BI/C1/D2 /D8/CW/CX/D7/DB /D3/D6/CZ/B8 /DB /CT /D4/D6/CT/D7/CT/D2 /D8 /CP /CS/CT/D8/CP/CX/D0/CT/CS /D7/D8/D9/CS/DD /D3/D2 /CP/CV/CX/D2/CV /CP/D2/CS /D1/CT/D1/D3/D6/DD/CT/AR/CT\r/D8/D7 /CX/D2 /BH /D2/D1 /C6/CX/C7 /D4/CP/D6/D8/CX\r/D0/CT/D7 /DB/CX/D8/CW /DA /CP/D6/CX/D3/D9/D7 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/CP/D2/CS /AS/CT/D0/CS /D4/D6/D3/D8/D3 \r/D3/D0/D7 /CP/D2/CS /D8/D6/DD /D8/D3 /D7/CT/D8/D8/D0/CT /D8/CW/CT /CX/D7/D7/D9/CT /D3/CU /CX/D8/D7 /D7/D4/CX/D2/CV/D0/CP/D7/D7 /D2/CP/D8/D9/D6/CT/BA/C1 /C1/BA /BX/CG/C8/BX/CA/C1/C5/BX/C6/CC /BT/C4 /BW/BX/CC /BT/C1/C4/CB/C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /CP/D6/CT /D4/D6/CT/D4/CP/D6/CT/CS /CQ /DD /D8/CW/CT /D7/D3/D0 /CV/CT/D0/D1/CT/D8/CW/D3 /CS/BA\n/BD/BH /B8/BE/BC/B8/BE/BI/C6/CX\r /CZ /CT/D0 /CW /DD/CS/D6/D3 /DC/CX/CS/CT /D4/D6/CT\r/D9/D6/D7/D3/D6 /CX/D7 /D4/D6/CT\r/CX/D4/CX/B9/D8/CP/D8/CT/CS /CQ /DD /D6/CT/CP\r/D8/CX/D2/CV /CP/D5/D9/CT/D3/D9/D7 /D7/D3/D0/D9/D8/CX/D3/D2/D7 /D3/CU /D2/CX\r /CZ /CT/D0 /D2/CX/D8/D6/CP/D8/CT/B4/BL/BL/BA/BL/BL/BL/B1/B5/CP/D2/CS /D7/D3 /CS/CX/D9/D1 /CW /DD/CS/D6/D3 /DC/CX/CS/CT/B4/BL/BL/BA/BL/BL/B1/B5 /CP/D8 /D4/C0 /BP /BD/BE/B8/CP/D8 /D6/D3 /D3/D1 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/BA /CC/CW/CX/D7 /D4/D6/CT\r/CX/D4/CX/D8/CP/D8/CT /CX/D7 /DB /CP/D7/CW/CT/CS /D1/CP/D2 /DD/D8/CX/D1/CT/D7 /DB/CX/D8/CW /CS/CX/D7/D8/CX/D0/D0/CT/CS /DB /CP/D8/CT/D6 /D8/D3 /D6/CT/D1/D3 /DA /CT /D6/CT/D1/D2/CP/D2 /D8 /D2/CX/D8/D6/CP/D8/CT /CP/D2/CS/D7/D3 /CS/CX/D9/D1 /CX/D3/D2/D7/BA /C1/D8 /CX/D7 /D8/CW/CT/D2 /CS/D6/CX/CT/CS /CP/D8 /BD/BC/BC◦/BV /CU/D3/D6 /BI /CW/D3/D9/D6/D7 /D8/D3/CV/CT/D8 /CV/D6/CT/CT/D2 \r/D3/D0/D3/D6/CT/CS /D2/CX\r /CZ /CT/D0 /CW /DD/CS/D6/D3 /DC/CX/CS/CT /D4 /D3 /DB/CS/CT/D6/BA /C6/CX\r /CZ /CT/D0 /D3 /DC/CX/CS/CT/D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /CP/D6/CT /D4/D6/CT/D4/CP/D6/CT/CS /CQ /DD /CW/CT/CP/D8/CX/D2/CV /D2/CX\r /CZ /CT/D0 /CW /DD/CS/D6/D3 /DC/CX/CS/CT /CP/D8/BE/BH/BC◦/BV /CU/D3/D6 /D8/CW/D6/CT/CT /CW/D3/D9/D6/D7 /CX/D2 /AT/D3 /DB/CX/D2/CV /CW/CT/D0/CX/D9/D1 /CV/CP/D7/BA /CC/CW/CT /D7/CP/D1/B9/D4/D0/CT /CX/D7 \r /CW/CP/D6/CP\r/D8/CT/D6/CX/DE/CT/CS /CQ /DD /CG/B9/D6/CP /DD /CS/CX/AR/D6/CP\r/D8/CX/D3/D2 /B4/CG/CA/BW/B5 /D9/D7/CX/D2/CV /CP/CB/CT/CX/CU/CT/D6/D8 /CS/CX/AR/D6/CP\r/D8/D3/D1/CT/D8/CT/D6 /DB/CX/D8/CW /BV/D9 /C3α /D6/CP/CS/CX/CP/D8/CX/D3/D2/BA /CC/CW/CT /CP /DA /CT/D6/B9/CP/CV/CT /D4/CP/D6/D8/CX\r/D0/CT /D7/CX/DE/CT /CP/D7 /CS/CT/D8/CT/D6/D1/CX/D2/CT/CS /CQ /DD /CG/CA/BW /D9/D7/CX/D2/CV /D8/CW/CT /CB\r /CW/CT/D6/B9/D6/CT/D6 /CU/D3/D6/D1 /D9/D0/CP /CX/D7 /BH /D2/D1/BA /BT/D0/D0 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /CP/D6/CT/CS/D3/D2/CT /DB/CX/D8/CW /CP /CB/C9/CD/C1/BW /D1/CP/CV/D2/CT/D8/D3/D1/CT/D8/CT/D6 /B4/C9/D9/CP/D2 /D8/D9/D1 /BW/CT/D7/CX/CV/D2/B8/C5/C8/C5/CB /CG/C4/BH/B5/BA/C1 /C1 /C1/BA /CA/BX/CB/CD/C4 /CC/CB /BT/C6/BW /BW/C1/CB/BV/CD/CB/CB/C1/C7/C6/BT/BA /BT/CV/CX/D2/CV /BX/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/CC /CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /D3/CU /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /DB /CP/D7 /CS/D3/D2/CT/D9/D2/CS/CT/D6 /BY /BV /CP/D2/CS /CI/BY /BV /D4/D6/D3/D8/D3 \r/D3/D0/D7 /CP/D8 /CP /AS/CT/D0/CS /D3/CU /BD/BC/BC /C7/CT/BA /CB/CT/CT/BY/CX/CV/D9/D6/CT /BD/BA /CC/CW/CT/D6/CT /CX/D7 /CP /CQ/CX/CU/D9/D6\r/CP/D8/CX/D3/D2 /CX/D2 /BY /BV /CP/D2/CS /CI/BY /BV /D1/CP/CV/D2/CT/D8/CX/B9/DE/CP/D8/CX/D3/D2/D7 /DB/CW/CX\r /CW /D1/CP/D2/CX/CU/CT/D7/D8/D7 /CQ /CT/D0/D3 /DB /BE/BJ/BH /C3 /CP/D2/CS /D8/CW/CT /CI/BY /BV /D1/CP/CV/B9/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CW/CP/D7 /CP /CQ/D6/D3/CP/CS /D4 /CT/CP/CZ /CP/D8 /CP/CQ /D3/D9/D8 /BD/BK/BC /C3/BA /C1/D8 \r/CP/D2 /CQ /CT/D7/CT/CT/D2 /D8/CW/CP/D8 /D8/CW/CT /BY /BV /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D2\r/D6/CT/CP/D7/CT/D7 /DB/CX/D8/CW /CS/CT\r/D6/CT/CP/D7/CX/D2/CV/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CP/D4/D4/CP/D6/CT/D2 /D8/D0/DD /D8/CT/D2/CS/CX/D2/CV /D8/D3 /D7/CP/D8/D9/D6/CP/D8/CT/BA /CC/CX/D1/CT /CS/CT\r/CP /DD/D3/CU /D8/CW/CT/D6/D1/D3/D6/CT/D1/CP/D2/CT/D2 /D8 /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /B4/CC/CA/C5/B5 /DB /CP/D7 /CS/D3/D2/CT /CP/D8/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7 /BE/BH /C3/B8 /BH/BC /C3 /CP/D2/CS /BD/BC/BC /C3/BA /BY /D3/D6 /D8/CW/CT/D7/CT /D1/CT/CP/D7/D9/D6/CT/B9/D1/CT/D2 /D8/D7/B8 /DB /CT \r/D3 /D3/D0 /D8/CW/CT /D7/CP/D1/D4/D0/CT /CX/D2 /CP /AS/CT/D0/CS /D3/CU /BD/BC/BC /C7/CT /D8/D3 /D8/CW/CT /D8/CT/D1/B9/D4 /CT/D6/CP/D8/D9/D6/CT /D3/CU /CX/D2 /D8/CT/D6/CT/D7/D8 /CP/D2/CS /D8/CW/CT/D2 /D7/DB/CX/D8\r /CW /D3/AR /D8/CW/CT /AS/CT/D0/CS/BA /C6/D3 /DB/D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D7 /D1/CT/CP/D7/D9/D6/CT/CS /CP/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /D8/CX/D1/CT/BA /CB/CT/CT/CX/D2/D7/CT/D8 /D3/CU /BY/CX/CV/D9/D6/CT /BD/BA /C1/D8 \r/CP/D2 /CQ /CT /D3/CQ/D7/CT/D6/DA /CT/CS /D8/CW/CP/D8 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/B9/D8/CX/D3/D2 /CS/CT\r/CP /DD/D7 /D1/D3/D6/CT /D3/D6 /D0/CT/D7/D7 /D0/D3/CV/CP/D6/CX/D8/CW/D1/CX\r/CP/D0/D0/DD /BA /CC/CW/CX/D7 /CQ /CT/CW/CP /DA/CX/D3/D6/CX/D7 /CP \r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /D3/CU /CQ /D3/D8/CW /D7/D9/D4 /CT/D6/D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/D7 /CP/D2/CS /D7/D4/CX/D2/CV/D0/CP/D7/D7/CT/D7/BA /BT/D2 /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8 /D8/CW/CP/D8 \r/CP/D2 /CS/CX/D7/D8/CX/D2/CV/D9/CX/D7/CW /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT/CP/CQ /D3 /DA /CT /D8 /DB /D3 /D4 /D3/D7/D7/CX/CQ/CX/D0/CX/D8/CX/CT/D7 /CX/D7 /D8/CW/CT /DB /CP/CX/D8 /D8/CX/D1/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /D3/CU/D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /B4/CP/CV/CX/D2/CV/B5/BA /CF /CT \r/CP/D6/D6/CX/CT/CS /D3/D9/D8 /CP/CV/CX/D2/CV/CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CX/D2 /CQ /D3/D8/CW /BY /BV /CP/D2/CS /CI/BY /BV /D4/D6/D3/D8/D3 \r/D3/D0/D7 /CP/D7 /CU/D3/D0/D0/D3 /DB/D7/BM/BV/D3 /D3/D0 /D8/CW/CT /D7/CP/D1/D4/D0/CT /CX/D2 /CP /AS/CT/D0/CS /D3/CU /BD/BC/BC /C7/CT /CU/D3/D6 /BY /BV /B4/D3/D6 /CX/D2 /DE/CT/D6/D3/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s52/s56\n/s49/s48/s48 /s49/s48/s48/s48/s57/s50/s57/s54/s49/s48/s48/s77/s40/s116/s41/s32/s47/s32/s77/s40/s48/s41/s32/s105/s110/s32/s37\n/s84/s105/s109/s101/s40/s115/s101/s99/s41/s49/s48/s48/s32/s75/s53/s48/s32/s75/s50/s53/s32/s75/s77/s32/s40/s49/s48/s45/s50\n/s32/s101/s109/s117/s32/s47/s32/s103/s41\n/s84/s32/s40/s75/s41/s90/s70/s67/s70/s67/BY/CX/CV/D9/D6/CT /BD/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /CC /CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /D3/CU /D8/CW/CT /CS\r/D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D2 /CP /BD/BC/BC /C7/CT /AS/CT/D0/CS /CU/D3/D6 /CQ /D3/D8/CW /CI/BY /BV /CP/D2/CS /BY /BV /D4/D6/D3/D8/D3/B9\r/D3/D0/D7/BA /CC/CW/CT /CX/D2/D7/CT/D8 /D7/CW/D3 /DB/D7 /CS/CT\r/CP /DD /D3/CU /D8/CW/CT/D6/D1/D3/D6/CT/D1/CP/D2/CT/D2 /D8 /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/CP/D8 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7 /BE/BH /C3/B8 /BH/BC /C3 /CP/D2/CS /BD/BC/BC /C3/BA/AS/CT/D0/CS /CU/D3/D6 /CI/BY /BV/B5 /D8/D3 /D8/CW/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /D3/CU /CX/D2 /D8/CT/D6/CT/D7/D8/B8 /DB /CP/CX/D8 /CU/D3/D6 /CP/D7/D4 /CT\r/CX/AS/CT/CS /D8/CX/D1/CT /B4/DB /CP/CX/D8 /D8/CX/D1/CT/B5 /CP/D2/CS /D8/CW/CT/D2 /D7/DB/CX/D8\r /CW /D8/CW/CT /AS/CT/D0/CS /D3/AR/B4/D3/D6 /D3/D2 /CX/D2 \r/CP/D7/CT /D3/CU /CI/BY /BV/B5/BA /C6/D3 /DB /D6/CT\r/D3/D6/CS /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D7/CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /D8/CX/D1/CT/BA /CB/D9/D4 /CT/D6/D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/D7 /CP/D6/CT /CT/DC/D4 /CT\r/D8/CT/CS /D8/D3/D7/CW/D3 /DB /CP /DB /CT/CP/CZ /DB /CP/CX/D8 /D8/CX/D1/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /D3/CU /CC/CA/C5 /CP/D2/CS /D2/D3 /DB /CP/CX/D8/D8/CX/D1/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /CX/D2 /CI/BY /BV /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/BN /CX/D2 /D3/D8/CW/CT/D6 /DB /D3/D6/CS/D7/DB /CT/CP/CZ /BY /BV /CP/CV/CX/D2/CV /CP/D2/CS /D2/D3 /CI/BY /BV /CP/CV/CX/D2/CV/BA /CB/D4/CX/D2 /CV/D0/CP/D7/D7/CT/D7 /CP/D6/CT/B8 /CW/D3 /DB/B9/CT/DA /CT/D6/B8 /CZ/D2/D3 /DB/D2 /D8/D3 /D7/CW/D3 /DB /CQ /D3/D8/CW /BY /BV /CP/D2/CS /CI/BY /BV /CP/CV/CX/D2/CV/BA\n/BI /B8/BF/BK/BY/CX/CV/D9/D6/CT /BE/D7/CW/D3 /DB/D7 /D8/CW/CT /CS/CP/D8/CP /CU/D3/D6 /CP/CV/CX/D2/CV /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CX/D2 /BY /BV /CP/D2/CS /CI/BY /BV /D4/D6/D3/B9/D8/D3 \r/D3/D0/D7/BA /BT /D2/D3/D8/CX\r/CT/CP/CQ/D0/CT /DB /CP/CX/D8 /D8/CX/D1/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /CX/D2 /CQ /D3/D8/CW /BY /BV/CP/D2/CS /CI/BY /BV /D4/D6/D3/D8/D3 \r/D3/D0/D7 \r/CP/D2 /CQ /CT /D3/CQ/D7/CT/D6/DA /CT/CS /DB/CW/CX\r /CW /CX/D7 /CP/D2 /CT/DA/CX/CS/CT/D2\r/CT/CX/D2 /D7/D9/D4/D4 /D3/D6/D8 /D3/CU /D7/D4/CX/D2 /CV/D0/CP/D7/D7 /CQ /CT/CW/CP /DA/CX/D3/D6 /CX/D2 /C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7/BA/BU/BA /C5/CT/D1/D3/D6/DD /BX/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/CF /CT \r/CP/D6/D6/CX/CT/CS /D3/D9/D8 /D1/CT/D1/D3/D6/DD /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CX/D2 /CQ /D3/D8/CW /BY /BV /CP/D2/CS/CI/BY /BV /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7/BA /C1/D2 /D8/CW/CT /CI/BY /BV /D4/D6/D3/D8/D3 \r/D3/D0/B8/DB /CT /AS/D6/D7/D8 /D6/CT\r/D3/D6/CS /D8/CW/CT /CI/BY /BV /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /D7/D8/CP/D2/CS/CP/D6/CS/DB /CP /DD /CP/D2/CS \r/CP/D0/D0 /D8/CW/CX/D7 /CP/D7 /D8/CW/CT /D6/CT/CU/CT/D6/CT/D2\r/CT /CS/CP/D8/CP/BA /C6/D3 /DB /D8/CW/CT /D7/CP/D1/D4/D0/CT/CX/D7 \r/D3 /D3/D0/CT/CS /CX/D2 /DE/CT/D6/D3 /AS/CT/D0/CS /D8/D3 /BH /C3 /DB/CX/D8/CW /CP /D7/D8/D3/D4 /D3/CU /D3/D2/CT /CW/D3/D9/D6 /CP/D8/BD/BC/BC /C3/BA /BW/D9/D6/CX/D2/CV /D7/D9/CQ/D7/CT/D5/D9/CT/D2 /D8 /CW/CT/CP/D8/CX/D2/CV /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D7/D6/CT\r/D3/D6/CS/CT/CS /D9/D4 /D8/D3 /BF/BC/BC /C3/BA /C1/D2 /BY/CX/CV/D9/D6/CT /BF /DB /CT /D7/CW/D3 /DB /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2\r/CT/CX/D2 /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT /CI/BY /BV /CS/CP/D8/CP /DB/CX/D8/CW /D8/CW/CT /D7/D8/D3/D4/CP/D2/CS /D8/CW/CT /CI/BY /BV /D6/CT/CU/CT/D6/CT/D2\r/CT /CS/CP/D8/CP/BA /C1/D8 /CX/D7 \r/D0/CT/CP/D6 /D8/CW/CP/D8 /D8/CW/CT/D6/CT /CX/D7 /CP/CS/CX/D4 /CP/D8 /BD/BC/BC /C3/B8 /DB/CW/CT/D6/CT /D8/CW/CT /D7/D8/D3/D4 /DB /CP/D7 /D8/CP/CZ /CT/D2 /CS/D9/D6/CX/D2/CV /D8/CW/CT \r/D3 /D3/D0/CX/D2/CV/D4/D6/D3 \r/CT/D7/D7 /CT/D7/D8/CP/CQ/D0/CX/D7/CW/CX/D2/CV /D8/CW/CT /CI/BY /BV /D1/CT/D1/D3/D6/DD /CX/D2 /D8/CW/CT /D7/DD/D7/D8/CT/D1/BA /BY /D3/D6/CS/D3/CX/D2/CV /BY /BV /D1/CT/D1/D3/D6/DD /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7/B8 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D7 \r/D3 /D3/D0/CT/CS /CX/D2/D8/CW/CT /D4/D6/CT/D7/CT/D2\r/CT /D3/CU /CP /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /D8/D3 /BH /C3 /DB/CX/D8/CW /CX/D2 /D8/CT/D6/D1/CX/D8/D8/CT/D2 /D8/D7/D8/D3/D4/D7 /D3/CU /D3/D2/CT /CW/D3/D9/D6 /CP/D8 /BE/BH /C3/B8 /BH/BC /C3 /CP/D2/CS /BD/BC/BC /C3 /DB/CX/D8/CW /D8/CW/CT/AS/CT/D0/CS /D7/DB/CX/D8\r /CW/CT/CS /D3/AR /CS/D9/D6/CX/D2/CV /D8/CW/CT /D7/D8/D3/D4/D7/BA /CC/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/CX/D7 /D1/CT/CP/D7/D9/D6/CT/CS /DB/CW/CX/D0/CT \r/D3 /D3/D0/CX/D2/CV /CP/D2/CS /D8/CW/CT/D2 /CS/D9/D6/CX/D2/CV /D7/D9/CQ/D7/CT/D5/D9/CT/D2 /D8/CW/CT/CP/D8/CX/D2/CV/BA /CC/CW/CT /CS/CP/D8/CP /CX/D7 /D7/CW/D3 /DB/D2 /CX/D2 /D8/CW/CT /CX/D2/D7/CT/D8 /D3/CU /BY/CX/CV/D9/D6/CT /BF/BA /C1/D8\r/CP/D2 /CQ /CT /D3/CQ/D7/CT/D6/DA /CT/CS /D8/CW/CP/D8 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /D6/CT/D1/CT/D1 /CQ /CT/D6/D7 /D8/CW/CT /CW/CX/D7/D8/D3/D6/DD/BF\n/s49/s48 /s49/s48/s48 /s49/s48/s48/s48/s49/s46/s49/s48/s49/s46/s49/s53\n/s49/s48 /s49/s48/s48 /s49/s48/s48/s48/s52/s46/s53/s54/s52/s46/s54/s48\n/s116\n/s119/s61/s32/s48\n/s116\n/s119/s61/s32/s49/s104\n/s116\n/s119/s61/s32/s50/s104/s77/s32/s40/s49/s48/s45/s50\n/s32/s101/s109/s117/s32/s47/s32/s103/s32/s41\n/s84/s105/s109/s101/s40/s115/s101/s99/s41\n/s32/s116\n/s119/s61/s32/s48\n/s116\n/s119/s61/s32/s49/s104\n/s116\n/s119/s61/s32/s50/s104/s77/s32/s40/s32/s49/s48/s45/s50\n/s32/s101/s109/s117/s32/s47/s32/s103/s32/s41\n/s84/s105/s109/s101/s32/s40/s115/s101/s99/s41/s32/BY/CX/CV/D9/D6/CT /BE/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /CF /CP/CX/D8 /D8/CX/D1/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /D3/CU /CI/BY /BV /D1/CP/CV/B9/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D8 /BE/BH /C3/BA /C1/D2/D7/CT/D8 /D7/CW/D3 /DB/D7 /D8/CW/CT /DB /CP/CX/D8 /D8/CX/D1/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /D3/CU/CC/CA/C5 /CP/D8 /BE/BH /C3/BA\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s45/s51/s45/s50/s45/s49/s48\n/s48 /s53/s48 /s49/s48/s48/s53/s46/s52/s53/s46/s54/s53/s46/s56/s77/s32/s40/s49/s48/s45/s50\n/s32/s101/s109/s117/s47/s103/s41\n/s84/s32/s40/s75/s41/s82/s101/s102/s101/s114/s101/s110/s99/s101\n/s67/s111/s111/s108/s105/s110/s103\n/s72/s101/s97/s116/s105/s110/s103/s77/s32/s40/s32/s49/s48/s45/s52\n/s32/s101/s109/s117/s47/s103/s41\n/s84/s32/s40/s75/s41/BY/CX/CV/D9/D6/CT /BF/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /C5/CT/D1/D3/D6/DD /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CX/D2 /CI/BY /BV /D4/D6/D3/D8/D3/B9\r/D3/D0/BA /CC/CW/CT /CS/CX/AR/CT/D6/CT/D2\r/CT /CX/D2 /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /DB/CX/D8/CW /CP /D7/D8/D3/D4 /D3/CU /D3/D2/CT /CW/D3/D9/D6/CP/D8 /BD/BC/BC /C3 /CX/D2 /D8/CW/CT \r/D3 /D3/D0/CX/D2/CV /D4/D6/D3 \r/CT/D7/D7 /CP/D2/CS /D8/CW/CT /D6/CT/CU/CT/D6/CT/D2\r/CT /CS/CP/D8/CP/B8 /D4/D0/D3/D8/D8/CT/CS/CP/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/BA /C1/D2/D7/CT/D8/BM /C5/CT/D1/D3/D6/DD /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /CX/D2/BY /BV /D4/D6/D3/D8/D3 \r/D3/D0 /DB/CX/D8/CW /D7/D8/D3/D4/D7 /D3/CU /D3/D2/CT /CW/D3/D9/D6 /CS/D9/D6/CP/D8/CX/D3/D2 /CP/D8 /BD/BC/BC /C3/B8 /BH/BC /C3/B8/CP/D2/CS /BE/BH /C3/BA /CC/CW/CT /AS/CT/D0/CS /CX/D7 /D7/DB/CX/D8\r /CW/CT/CS /D3/AR /CS/D9/D6/CX/D2/CV /CT/CP\r /CW /D7/D8/D3/D4/BA/D3/CU /D8/CW/CT \r/D3 /D3/D0/CX/D2/CV /D4/D6/D3 \r/CT/D7/D7 /CP/D2/CS /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D8/CP/CZ /CT/D7 /CY/D9/D1/D4/D7\r/D0/D3/D7/CT /D8/D3 /D8/CW/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7 /DB/CW/CT/D6/CT /D8/CW/CT /D7/D8/D3/D4/D7 /DB /CT/D6/CT /D8/CP/CZ /CT/D2/BA/C5/CT/D1/D3/D6/DD /CX/D2 /BY /BV /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CW/CP/D7 /CQ /CT/CT/D2 /D3/CQ/D7/CT/D6/DA /CT/CS /CU/D3/D6/CQ /D3/D8/CW /CX/D2 /D8/CT/D6/CP\r/D8/CX/D2/CV /CP/D2/CS /D2/D3/D2 /CX/D2 /D8/CT/D6/CP\r/D8/CX/D2/CV /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /CP/D2/CS/CX/D8 /CW/CP/D7 /CQ /CT/CT/D2 /D7/CW/D3 /DB/D2 /D8/CW/CP/D8 /CP /CQ/D6/D3/CP/CS /CS/CX/D7/D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D3/CU /CT/D2/CT/D6/CV/DD/CQ/CP/D6/D6/CX/CT/D6/D7 /CX/D7 /D7/D9Ꜷ\r/CX/CT/D2 /D8 /D8/D3 /D4/D6/D3 /CS/D9\r/CT /D1/CT/D1/D3/D6/DD /CT/AR/CT\r/D8/D7 /CX/D2 /BY /BV/D4/D6/D3/D8/D3 \r/D3/D0/BA\n/BI/C0/D3 /DB /CT/DA /CT/D6 /D1/CT/D1/D3/D6/DD /CX/D2 /CI/BY /BV /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D7 /CP/CU/CT/CP/D8/D9/D6/CT /CX/D2/CW/CT/D6/CT/D2 /D8 /D8/D3 /D7/D4/CX/D2 /CV/D0/CP/D7/D7/CT/D7 /CP/D2/CS /CW/CP/D7 /D2/D3/D8 /CQ /CT/CT/D2 /D3/CQ/B9/D7/CT/D6/DA /CT/CS /CX/D2 /D7/D9/D4 /CT/D6/D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/D7/BA /CC/CW /D9/D7 /D8/CW/CT /D1/CT/D1/D3/D6/DD /D3/CQ/D7/CT/D6/DA /CT/CS/CX/D2 /CI/BY /BV /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /D4/D6/D3 /DA/CX/CS/CT/D7 \r/D3/D2\r/D0/D9/D7/CX/DA /CT\n/CT/DA/CX/CS/CT/D2\r/CT /CX/D2 /CU/CP /DA /D3/D6 /D3/CU /D8/CW/CT /D7/D4/CX/D2 /CV/D0/CP/D7/D7 /D2/CP/D8/D9/D6/CT /D3/CU /C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/B9/D8/CX\r/D0/CT/D7/BA /C0/D3 /DB /CT/DA /CT/D6/B8 /D8/CW/CT /DB/CX/CS/D8/CW /D3/CU /D8/CW/CT /CS/CX/D4 /CX/D2 /BY/CX/CV/D9/D6/CT /BF /CX/D7 /D6/CP/D8/CW/CT/D6/D0/CP/D6/CV/CT/B8 /CP/CQ /D3/D9/D8 /BD/BC/BC /C3/B8 /D8/CW/CT \r/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /AS/CV/D9/D6/CT /CU/D3/D6 \r/CP/D2/D3/D2/CX\r/CP/D0/D7/D4/CX/D2 /CV/D0/CP/D7/D7/CT/D7 /CQ /CT/CX/D2/CV /CP /CU/CT/DB /C3/CT/D0/DA/CX/D2/D7/BA\n/BF/BL/CC /D3 \r/D3/D1/D4/D0/CT/D1/CT/D2 /D8 /D8/CW/CT/D7/CT /D1/CT/D1/D3/D6/DD /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /DB /CT /CW/CP /DA /CT/CS/D3/D2/CT /D2/CT/CV/CP/D8/CX/DA /CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT \r/DD\r/D0/CX/D2/CV /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/D7 /DB/CX/D8/CW /AS/CT/D0/CS\r /CW/CP/D2/CV/CT /CX/D2 /CQ /D3/D8/CW /BY /BV /CP/D2/CS /CI/BY /BV /D4/D6/D3/D8/D3 \r/D3/D0/D7 /CP/D7 /D7/D9/CV/CV/CT/D7/D8/CT/CS /CQ /DD/CB/D9/D2 /CT/D8 /CP/D0/BA /CP/D2/CS /CP/CS/D3/D4/D8/CT/CS /CQ /DD /D1/CP/D2 /DD /CP/D9/D8/CW/D3/D6/D7/BA\n/BH /B8/BI/B8/BJ/B8/BK/B8/BL/C1/D2 /BY /BV/D4/D6/D3/D8/D3 \r/D3/D0/B8 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D7 \r/D3 /D3/D0/CT/CS /D8/D3 /BE/BH /C3 /CX/D2 /CP /AS/CT/D0/CS /D3/CU /BD/BC/BC /C7/CT/B8/D8/CW/CT /AS/CT/D0/CS /CX/D7 /D8/CW/CT/D2 /D7/DB/CX/D8\r /CW/CT/CS /D3/AR /CP/D2/CS /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D7/D6/CT\r/D3/D6/CS/CT/CS /CU/D3/D6 /CP /D8/CX/D1/CT /D4 /CT/D6/CX/D3 /CSt1\n/BA /CC/CW/CT/D2 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D7 \r/D3 /D3/D0/CT/CS/D8/D3 /BD/BH /C3/B8 /CP /AS/CT/D0/CS /D3/CU /BD/BC/BC /C7/CT /CX/D7 /CP/D4/D4/D0/CX/CT/CS /CP/D2/CS /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/CS/CP/D8/CP /CX/D7 /D8/CP/CZ /CT/D2 /CU/D3/D6 /CP /D4 /CT/D6/CX/D3 /CSt2\n/BA /CC /CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CX/D7 /D2/D3 /DB \r /CW/CP/D2/CV/CT/CS/CQ/CP\r /CZ /D8/D3 /BE/BH /C3/B8 /AS/CT/D0/CS /CX/D7 /D7/DB/CX/D8\r /CW/CT/CS /D3/AR /CP/D2/CS /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CX/D7/D6/CT\r/D3/D6/CS/CT/CS /CP/CV/CP/CX/D2 /CU/D3/D6 /CP /D4 /CT/D6/CX/D3 /CSt3\n/BA /C0/CT/D6/CTt1\n/BPt2\n/BPt3\n/BP /BE/BK/BC/BC/D7/CT\r/D3/D2/CS/D7/BA /CB/CT/CT /BY/CX/CV/D9/D6/CT /BG/B4/CP/B5/BA /C1/D8 \r/CP/D2 /CQ /CT /D7/CT/CT/D2 /D8/CW/CP/D8 /DB/CW/CT/D2 /D8/CW/CT/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CX/D7 /D6/CP/CX/D7/CT/CS /CQ/CP\r /CZ /D8/D3 /BE/BH /C3/B8 /D8/CW/CT /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /D7/D8/CP/D6/D8/D7/CP/D0/D1/D3/D7/D8 /CU/D6/D3/D1 /D8/CW/CT /D4 /D3/CX/D2 /D8 /CP/D8 /DB/CW/CX\r /CW /CX/D8 /DB /CP/D7 /D0/CT/CU/D8 /D3/AR /CX/D2 /D8/CW/CT /D4/D6/CT/DA/CX/B9/D3/D9/D7 /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /CP/D8 /BE/BH /C3/BA /C8/D0/CT/CP/D7/CT /D7/CT/CT /D8/CW/CT /CX/D2/D7/CT/D8 /D3/CU /BY/CX/CV/D9/D6/CT /BG /B4/CP/B5/BA/CC/CW/CX/D7 /D7/CW/D3 /DB/D7 /D8/CW/CP/D8 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CW/CP/D7 /CP /D1/CT/D1/D3/D6/DD /D3/CU /CP/D2 /CT/CP/D6/D0/CX/CT/D6/CP/CV/CX/D2/CV /CX/D2 /D7/D4/CX/D8/CT /D3/CU /CP/D2 /CX/D2 /D8/CT/D6/DA /CT/D2/CX/D2/CV /CP/CV/CX/D2/CV /CP/D8 /CP /D0/D3 /DB /CT/D6 /D8/CT/D1/D4 /CT/D6/B9/CP/D8/D9/D6/CT/BA /CF /CT /CW/CP /DA /CT /CP/D0/D7/D3 /CS/D3/D2/CT /D2/CT/CV/CP/D8/CX/DA /CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT \r/DD\r/D0/CX/D2/CV\n/s48 /s53/s48/s48/s48 /s49/s48/s48/s48/s48/s48/s54/s49/s50/s52/s56/s53/s52/s32\n/s32/s48 /s50/s48/s48/s48 /s52/s48/s48/s48/s52/s53/s46/s52/s52/s53/s46/s54/s52/s53/s46/s56/s52/s54/s46/s48\n/s116\n/s49/s116\n/s51\n/s116\n/s51/s116\n/s50/s72/s61/s48/s32/s79/s101\n/s84/s61/s50/s53/s75/s84/s61/s50/s53/s75/s84/s61/s49/s53/s75\n/s72/s61/s49/s48/s48/s32/s79/s101\n/s72/s61/s48/s32/s79/s101\n/s116\n/s49\n/s48 /s50/s52/s48/s48 /s52/s56/s48/s48/s49/s49/s46/s49/s49/s49/s46/s52/s49/s49/s46/s55\n/s116\n/s51/s116\n/s49/s116\n/s49/s116\n/s51/s72/s61/s49/s48/s48/s32/s79/s101/s84/s61/s50/s53/s32/s75\n/s84/s61/s50/s53/s32/s75\n/s84/s61/s32/s49/s53/s32/s75/s72/s61/s49/s48/s48/s32/s79/s101\n/s72/s61/s32/s48/s32/s79/s101/s116\n/s50/s77/s32/s40/s49/s48/s45/s51\n/s32/s101/s109/s117/s47/s103/s41\n/s40/s98/s41\n/s84/s105/s109/s101/s32/s40/s115/s101/s99/s41/s40/s97/s41/BY/CX/CV/D9/D6/CT /BG/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /C5/CP/CV/D2/CT/D8/CX\r /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /DB/CX/D8/CW /D2/CT/CV/CP/D8/CX/DA /CT/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT \r/DD\r/D0/CX/D2/CV /CP/D2/CS /CP /AS/CT/D0/CS \r /CW/CP/D2/CV/CT /CU/D3/D6 /B4/CP/B5 /BY /BV /D4/D6/D3/D8/D3 \r/D3/D0/BA/B4/CQ/B5 /CI/BY /BV /D4/D6/D3/D8/D3 \r/D3/D0/BA /CC/CW/CT /CX/D2/D7/CT/D8/D7 /D7/CW/D3 /DB /D8/CW/CP/D8 /D8/CW/CT /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /CS/D9/D6/CX/D2/CV/D8/CX/D1/CTt3\n/CX/D7 /CT/D7/D7/CT/D2 /D8/CX/CP/D0/D0/DD /D8/CW/CT \r/D3/D2 /D8/CX/D2 /D9/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /CS/D9/D6/B9/CX/D2/CVt1\n/B8 \r/D3/D2/AS/D6/D1/CX/D2/CV /D8/CW/CP/D8 /D8/CW/CT /D7/DD/D7/D8/CT/D1 /CW/CP/D7 /D8/CW/CT /D1/CT/D1/D3/D6/DD /D3/CU /CT/CP/D6/D0/CX/CT/D6/D6/CT/D0/CP/DC/CP/D8/CX/D3/D2/D7/BA/BG/CU/D3/D6 /CI/BY /BV /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /CX/D2 /CP /D7/CX/D1/CX/D0/CP/D6 /D1/CP/D2/D2/CT/D6/BA/CB/CT/CT /BG /B4/CQ/B5 /CP/D2/CS /CX/D8/D7 /CX/D2/D7/CT/D8/BA /CC/CW/CT /D6/CT/D7/D9/D0/D8/D7 /CP/CV/CP/CX/D2 \r/D3/D2/AS/D6/D1 /D8/CW/CT/CT/DC/CX/D7/D8/CT/D2\r/CT /D3/CU /D1/CT/D1/D3/D6/DD /CX/D2 /CI/BY /BV /D4/D6/D3/D8/D3 \r/D3/D0/BA/BV/BA /BW/CX/D7\r/D9/D7/D7/CX/D3/D2/CC/CW/CT /D4/D6/CT/D7/CT/D2\r/CT /D3/CU /CP/CV/CX/D2/CV /CP/D2/CS /D1/CT/D1/D3/D6/DD /CX/D2 /CI/BY /BV /D1/CP/CV/D2/CT/D8/CX/B9/DE/CP/D8/CX/D3/D2 /D3/CU /C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 \r/D3/D2/AS/D6/D1/D7 /D8/CW/CT/CX/D6 /D7/D4/CX/D2 /CV/D0/CP/D7/D7 /CQ /CT/B9/CW/CP /DA/CX/D3/D6/BA /CC/CW/CT/D6/CT /CW/CP /DA /CT /CQ /CT/CT/D2 /D7/D3/D1/CT /DB /D3/D6/CZ /D3/D2 /D3/D8/CW/CT/D6 /D2/CP/D2/D3/D4/CP/D6/D8/CX/B9\r/D0/CT /D7/DD/D7/D8/CT/D1/D7 /DB/CW/CT/D6/CT /CI/BY /BV /D1/CT/D1/D3/D6/DD /DB /CP/D7 /D3/CQ/D7/CT/D6/DA /CT/CS/BA\n/BG /B8/BH /B8/BI/B8/BF/BF/B8/BF/BG/B8/BF/BL/BT/D0/D0 /D8/CW/D3/D7/CT /DB /D3/D6/CZ/D7 /DB /CT/D6/CT /D3/D2 /CU/CT/D6/D6/CX /CP/D2/CS /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D1/CP/D8/CT/D6/CX/B9/CP/D0/D7 /CP/D2/CS /D8/CW/CT /CX/D2 /D8/CT/D6/D4/CP/D6/D8/CX\r/D0/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 /DB /CT/D6/CT /D7/CP/CX/CS /D8/D3 /CQ /CT /D6/CT/B9/D7/D4 /D3/D2/D7/CX/CQ/D0/CT /CU/D3/D6 /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/CS /CV/D0/CP/D7/D7/DD /CQ /CT/CW/CP /DA/CX/D3/D6/BA /CC/CW/CT /CS/CX/D4 /CX/D2/D8/CW/CT /CI/BY /BV /D1/CT/D1/D3/D6/DD /CX/D2 /D8/CW/CT /D4/D6/CT/D7/CT/D2 /D8 /DB /D3/D6/CZ /B4/BY/CX/CV/BA /BF/B5 /CX/D7 /D5/D9/CX/D8/CT/CQ/D6/D3/CP/CS \r/D3/D1/D4/CP/D6/CT/CS /D8/D3 /D8/CW/D3/D7/CT /D6/CT/D4 /D3/D6/D8/CT/CS /D3/D2 /D3/D8/CW/CT/D6 /D2/CP/D2/D3/D4/CP/D6/D8/CX/B9\r/D0/CT /D7/DD/D7/D8/CT/D1/D7/BA /CC/CW/CX/D7 /D7/D9/CV/CV/CT/D7/D8/D7 /D8/CW/CP/D8 /D8/CW/CT /D3/D6/CX/CV/CX/D2 /D3/CU /D7/D4/CX/D2 /CV/D0/CP/D7/D7/CQ /CT/CW/CP /DA/CX/D3/D6 /CX/D2 /C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /CX/D7/B8 /D4 /D3/D7/D7/CX/CQ/D0/DD /B8 /D2/D3/D8 /CX/D2 /D8/CT/D6/D4/CP/D6/B9/D8/CX\r/D0/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7/BA /C1/D2 /CU/CP\r/D8/B8 /D8/CW/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 /CQ /CT/D8 /DB /CT/CT/D2 /D8/CW/CT/D7/CT/D4/CP/D6/D8/CX\r/D0/CT/D7 /CP/D6/CT /DA /CT/D6/DD /DB /CT/CP/CZ /CP/D2/CS /CP/D6/CT /D2/D3/D8 /D7/D9Ꜷ\r/CX/CT/D2 /D8 /D8/D3 \r/CP/D9/D7/CT\r/D3/D0/D0/CT\r/D8/CX/DA /CT /CU/D6/CT/CT/DE/CX/D2/CV /D3/CU /D4/CP/D6/D8/CX\r/D0/CT /D1/D3/D1/CT/D2 /D8/D7 /CP/D8 /D7/D9\r /CW /CW/CX/CV/CW /D8/CT/D1/B9/D4 /CT/D6/CP/D8/D9/D6/CT/D7 /CP/D7 /CW/CP/D7 /CQ /CT/CT/D2 /CP/D6/CV/D9/CT/CS /CQ /DD /CC/CX/DB /CP/D6/CX /CT/D8 /CP/D0/BA\n/BD/BH/C0/D3 /DB /CT/DA /CT/D6/D8/CW/CT/D7/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 \r/CP/D2 /CT/D2/CW/CP/D2\r/CT /D8/CW/CT /CU/D6/D9/D7/D8/D6/CP/D8/CX/D3/D2 /D3/CU /D7/D4/CX/D2/D7/CP/D8 /D8/CW/CT /D7/D9/D6/CU/CP\r/CT /D3/CU /CX/D2/CS/CX/DA/CX/CS/D9/CP/D0 /D4/CP/D6/D8/CX\r/D0/CT/D7 /CP/D2/CS /D7/CW/CX/CU/D8 /D8/CW/CT /CU/D6/CT/CT/DE/B9/CX/D2/CV /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7 /D8/D3 /CW/CX/CV/CW/CT/D6 /DA /CP/D0/D9/CT/D7/BA\n/BD/BI/CC/CW/CT /CT/DC\r /CW/CP/D2/CV/CT /CQ/CX/CP/D7/CT/AR/CT\r/D8/D7 /D3/CQ/D7/CT/D6/DA /CT/CS /CX/D2 /C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /CX/D2/CS/CX\r/CP/D8/CT /D8/CW/CT /D4/D6/CT/D7/B9/CT/D2\r/CT /D3/CU /CQ /D3/D8/CW /CU/CT/D6/D6/D3 /CP/D2/CS /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7 /CP/D8\n/D8/CW/CT /D7/D9/D6/CU/CP\r/CT/B8 /DB/CW/CX\r /CW \r/CP/D2 /CU/D6/D9/D7/D8/D6/CP/D8/CT /D8/CW/CT /D7/D4/CX/D2/D7 /D0/CT/CP/CS/CX/D2/CV /D8/D3 /D7/D4/CX/D2/CV/D0/CP/D7/D7 /CQ /CT/CW/CP /DA/CX/D3/D6/BA\n/BG/BC /B8/BG/BD/CC/CW /D9/D7 /D8/CW/CT /D3/D6/CX/CV/CX/D2 /D3/CU /D7/D4/CX/D2 /CV/D0/CP/D7/D7 /D7/D8/CP/D8/CT /CX/D2/C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /D7/CT/CT/D1/D7 /D8/D3 /CQ /CT /D8/CW/CT /CU/D6/CT/CT/DE/CX/D2/CV /D3/CU /D7/D4/CX/D2/D7 /CP/D8 /D8/CW/CT/D7/D9/D6/CU/CP\r/CT /D3/CU /D8/CW/CT /CX/D2/CS/CX/DA/CX/CS/D9/CP/D0 /D4/CP/D6/D8/CX\r/D0/CT/D7/BA /CC/CW/CT /DB/CX/CS/CT /CS/CX/D4 /CX/D2 /CI/BY /BV/D1/CT/D1/D3/D6/DD /D3/CU /C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /CP/D7 \r/D3/D1/D4/CP/D6/CT/CS /D8/D3 \r/CP/D2/D3/D2/CX\r/CP/D0/D7/D4/CX/D2 /CV/D0/CP/D7/D7/CT/D7 \r/CP/D2 /D4 /D3/D7/D7/CX/CQ/D0/DD /CQ /CT /CP/D8/D8/D6/CX/CQ/D9/D8/CT/CS /D8/D3 /D8/CW/CT /AS/D2/CX/D8/CT /D7/CX/DE/CT/D3/CU /D8/CW/CT /D7/DD/D7/D8/CT/D1/BA/C1/CE/BA /BV/C7/C6/BV/C4/CD/CB/C1/C7/C6/CF /CT /CW/CP /DA /CT /CS/D3/D2/CT /CS\r /D1/CP/CV/D2/CT/D8/CX\r /D6/CT/D0/CP/DC/CP/D8/CX/D3/D2 /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7/D3/D2 /C6/CX/C7 /D2/CP/D2/D3/D4/CP/D6/D8/CX\r/D0/CT/D7 /DB/CX/D8/CW /DA /CP/D6/CX/D3/D9/D7 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CP/D2/CS /AS/CT/D0/CS/D4/D6/D3/D8/D3 \r/D3/D0/D7/BA /C7/D9/D6 /D6/CT/D7/D9/D0/D8/D7 /D7/CW/D3 /DB /D8/CW/CT /D4/D6/CT/D7/CT/D2\r/CT /D3/CU /CP/CV/CX/D2/CV /CP/D2/CS/D1/CT/D1/D3/D6/DD /CT/AR/CT\r/D8/D7 /CX/D2 /CQ /D3/D8/CW /BY /BV /CP/D2/CS /CI/BY /BV /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/D7/B8/D8/CW /D9/D7 /CT/D7/D8/CP/CQ/D0/CX/D7/CW/CX/D2/CV /D8/CW/CT /D7/D4/CX/D2 /CV/D0/CP/D7/D7 /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/CW/CT/D7/CT /D4/CP/D6/D8/CX/B9\r/D0/CT/D7/BA /CC/CW/CT /D3/D6/CX/CV/CX/D2 /D3/CU /D8/CW/CX/D7 /CQ /CT/CW/CP /DA/CX/D3/D6 /D7/CT/CT/D1/D7 /D8/D3 /CQ /CT /D7/D9/D6/CU/CP\r/CT /D7/D4/CX/D2/CU/D6/CT/CT/DE/CX/D2/CV /D3/CU /CX/D2/CS/CX/DA/CX/CS/D9/CP/D0 /D4/CP/D6/D8/CX\r/D0/CT/D7 /D6/CP/D8/CW/CT/D6 /D8/CW/CP/D2 /CX/D2 /D8/CT/D6/D4/CP/D6/D8/CX\r/D0/CT/CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/D7/BA/BT \r /CZ/D2/D3 /DB/D0/CT/CS/CV/D1/CT/D2 /D8/D7/CE/BU /D8/CW/CP/D2/CZ/D7 /D8/CW/CT /CD/D2/CX/DA /CT/D6/D7/CX/D8 /DD /BZ/D6/CP/D2 /D8/D7 /BV/D3/D1/D1/CX/D7/D7/CX/D3/D2 /D3/CU /C1/D2/CS/CX/CP/CU/D3/D6 /AS/D2/CP/D2\r/CX/CP/D0 /D7/D9/D4/D4 /D3/D6/D8/BA\n∗/BX/D0/CT\r/D8/D6/D3/D2/CX\r /CP/CS/CS/D6/CT/D7/D7/BM /DA/CX/CY/CP /DD/CQ\b/CX/CX/D8/CZ/BA/CP\r/BA/CX/D2\n†/BX/D0/CT\r/D8/D6/D3/D2/CX\r /CP/CS/CS/D6/CT/D7/D7/BM /CZ/D4/D6/CP /CY\b/CX/CX/D8/CZ/BA/CP\r/BA/CX/D2/BD/C2/BA /C4/BA /BW/D3/D6/D1/CP/D2/D2/B8 /BW/BA /BY/CX/D3/D6/CP/D2/CX/B8 /CP/D2/CS /BX/BA /CC /D6/D3/D2\r/B8 /BT /CS/DA /CP/D2\r/CT/D7 /CX/D2/BV/CW/CT/D1/CX\r/CP/D0 /C8/CW /DD/D7/CX\r/D7 /BL/BK /B8 /BE/BK/BF /B4/BD/BL/BL/BJ/B5/BA/BE/CB/D8/CT/CT/D2 /C5/GW/D6/D9/D4/B8 /BW/CP/D2/CX/CT/D0 /BX /C5/CP/CS/D7/CT/D2/B8 /BV/CP/D8/CW/D6/CX/D2/CT /BY /D6/CP/D2/CS/D7/CT/D2/B8 /BV/CW/D6/CX/D7/B9/D8/CX/CP/D2 /CA /C0 /BU/CP/CW/D0 /CP/D2/CS /C5/CX/CZ/CZ /CT/D0 /BY /C0/CP/D2/D7/CT/D2/B8 /C2/BA /C8/CW /DD/D7/BA/BM /BV/D3/D2/CS/CT/D2/D7/BA/C5/CP/D8/D8/CT/D6 /BD/BL /B8 /BE/BD/BF/BE/BC/BE /B4/BE/BC/BC/BJ/B5/BA/BF/CG/BA /BU/CP/D8/D0/D0/CT /CP/D2/CS /BT/BA /C4/CP/CQ/CP/D6/D8/CP/B8 /C2/BA /C8/CW /DD/D7/BA /BW /BF/BH /B8 /CA/BD/BH /B4/BE/BC/BC/BE/B5/BA/BG/C5/CP/D7/CP/D8/D7/D9/CV/D9 /CB/D9/DE/D9/CZ/CX/B8 /CB/CW/CP/D6/CQ/CP/D2/CX /C1/BA /BY /D9/D0/D0/CT/D1 /CP/D2/CS /C1/D8/D7/D9/CZ /D3 /CB/BA/CB/D9/DE/D9/CZ/CX/B8 /C8/CW /DD/D7/BA /CA/CT/DA/BA /BU /BJ/BL /B8 /BC/BE/BG/BG/BD/BK /B4/BE/BC/BC/BL/B5/BA/BH/CH/BA /CB/D9/D2/B8 /C5/BA /BU/BA /CB/CP/D0/CP/D1/D3/D2/B8 /C3/BA /BZ/CP/D6/D2/CX/CT/D6 /CP/D2/CS /CA/BA /CB/BA /BT /DA /CT/D6/CQ/CP\r /CZ/B8/C8/CW /DD/D7/BA /CA/CT/DA/BA /C4/CT/D8/D8/BA /BL/BD /B8 /BD/BI/BJ/BE/BC/BI /B4/BE/BC/BC/BF/B5/BA/BI/C5/BA /CB/CP/D7/CP/CZ/CX/B8 /C8 /BA /BX/BA /C2/D3/D2/D7/D7/D3/D2 /B8 /C0/BA /CC /CP/CZ /CP /DD /CP/D1/CP/B8 /CP/D2/CS /C0/BA /C5/CP/D1/CX/DD /CP/C8/CW /DD/D7/BA /CA/CT/DA/BA /BU /BJ/BD /B8 /BD/BC/BG/BG/BC/BH /B4/BE/BC/BC/BH/B5/BA/BJ/C5/CP/D0/CP /DD /BU/CP/D2/CS/DD /D3/D4/CP/CS/CW /DD /CP /DD /CP/D2/CS /CB/D9/D7/CW/CP/D2 /D8/CP /BW/CP/D8/D8/CP/CV/D9/D4/D8/CP/B8 /C8/CW /DD/D7/BA/CA/CT/DA/BA /BU /BJ/BG /B8 /BE/BD/BG/BG/BD/BC /B4/BE/BC/BC/BI/B5/BA/BK/BZ/BA /C5/BA /CC /D7/D3/CX/B8 /C4/BA /BX/BA /CF /CT/D2/CV/CT/D6/B8 /CD/BA /CB/CT/D2/CP/D6/CP/D8/D2/CT/B8 /CA/BA /C2/BA /CC /CP\r /CZ /CT/D8/D8/B8 /BX/BA/BV/BA /BU/D9\r/B8 /CA/BA /C6/CP/CX/CZ/B8 /C8 /BA /C8 /BA /CE /CP/CX/D7/CW/D2/CP /DA /CP /CP/D2/CS /CE/BA /C6/CP/CX/CZ/B8 /C8/CW /DD/D7/BA /CA/CT/DA/BA/BU /BJ/BE /B8 /BC/BD/BG/BG/BG/BH /B4/BE/BC/BC/BH/B5/BA/BL/CA/BA /C3/BA /CI/CW/CT/D2/CV/B8 /C0/D3/D2/CV/DB /CT/CX /BZ/D9/B8 /BU/CX/D2/CV /CG/D9/B8 /CP/D2/CS /CG/BA /CG/BA /CI/CW/CP/D2/CV/C8/CW /DD/D7/BA /CA/CT/DA/BA /BU /BJ/BE /B8 /BC/BD/BG/BG/BD/BI /B4/BE/BC/BC/BH/B5/BA/BD/BC/CB/BA /BV/CW/CP/CZ/D6/CP /DA /CT/D6/D8 /DD /B8 /C5/BA /BU/CP/D2/CS/DD /D3/D4/CP/CS/CW /DD /CP /DD /B8 /CB/BA /BV/CW/CP/D8/D8/CT/D6/CY/CT/CT/B8 /CB/BA/BW/CP/D8/D8/CP/CV/D9/D4/D8/CP/B8 /BT/BA /BY /D6/DD/CS/D1/CP/D2/B8 /CB/BA /CB/CT/D2/CV/D9/D4/D8/CP /CP/D2/CS /C8 /BA /BT/BA /CB/D6/CT/CT/D6/CP/D1/B8/C8/CW /DD/D7/BA /CA/CT/DA/BA /BU /BJ/BD /B8 /BC/BH/BG/BG/BC/BD /B4/BE/BC/BC/BH/B5/BA/BD/BD/BU/BA /C5/CP/D6/D8/CX/D2/CT/DE/B8 /CG/BA /C7/CQ/D6/CP/CS/D3/D6/D7/B8 /C4/BA /BU/CP/D0\r/CT/D0/D0/D7/B8 /BT/BA /CA/D3/D9/CP/D2/CT/D8 /CP/D2/CS /BV/BA/C5/D3/D2 /D8 /DD /B8 /C8/CW /DD/D7/BA /CA/CT/DA/BA /C4/CT/D8/D8/BA /BK/BC /B8 /BD/BK/BD /B4/BD/BL/BL/BK/B5/BA/BD/BE/C4/BA /C6/GH/CT/D0/B8 /BT/D2/D2/BA /BZ/CT/D3/D4/CW /DD/D7/BA /BV/BA/C6/BA/CA/BA/CB/BA /BH /B8 /BL/BL /B4/BD/BL/BG/BL/B5/BA/BD/BF/CF/BA /BY/BA /BU/D6/D3 /DB/D2 /C2/D6/BA /C8/CW /DD/D7/BA /CA/CT/DA/BA /BD/BF/BC /B8 /BD/BI/BJ/BJ /B4/BD/BL/BI/BF/B5/BA/BD/BG/CB/BA /CB/CP/CW/D3 /D3/B8 /C7/BA /C8 /CT/D8/D6/CP\r/CX\r /CP/D2/CS /CF/BA /C3/D0/CT/CT/D1/CP/D2/D2/B8 /C8/CW /DD/D7/BA/CA/CT/DA/BA/BU /BI/BJ /B8/BE/BD/BG/BG/BE/BE /B4/BE/BC/BC/BF/B5/BA\n/BD/BH/CB/BA /BW/BA /CC/CX/DB /CP/D6/CX /CP/D2/CS /C3/BA /C8 /BA /CA/CP /CY/CT/CT/DA/B8 /C8/CW /DD/D7/BA /CA/CT/DA/BA /BU /BJ/BE /B8 /BD/BC/BG/BG/BF/BF/B4/BE/BC/BC/BH/B5/BA/BD/BI/BX/BA /CF/CX/D2/CZ/D0/CT/D6/B8 /CA/BA /BW/BA /CI/DD/D7/D0/CT/D6/B8 /C5/BA /CE /CP/D7/D5/D9/CT/DE /C5/CP/D2/D7/CX/D0/D0/CP/B8 /BW/BA /BY/CX/D3/D6/CP/D2/CX/B8/BW/BA /CA/CX/D2/CP/D0/CS/CX/B8 /C5/BA /CE /CP/D7/CX/D0/CP/CZ /CP/CZ/CX /CP/D2/CS /C3 /C6 /CC /D6/D3/CW/CX/CS/D3/D9/B8 /C6/CP/D2/D3/D8/CT\r /CW/B9/D2/D3/D0/D3/CV/DD /BD/BL /B8 /BD/BK/BH/BJ/BC/BE /B4/BE/BC/BC/BK/B5/BN /BX/BA /CF/CX/D2/CZ/D0/CT/D6/B8 /CA/BA /BW/BA /CI/DD/D7/D0/CT/D6/B8 /C5/BA/CE /CP/D7/D5/D9/CT/DE /C5/CP/D2/D7/CX/D0/D0/CP /CP/D2/CS /BW/BA/BY/CX/D3/D6/CP/D2/CX /C8/CW /DD/D7/BA /CA/CT/DA/BA /BU /BJ/BE /B8 /BD/BF/BE/BG/BC/BL/B4/BE/BC/BC/BH/B5/BA/BD/BJ/CA/BA /C0/BA /C3 /D3 /CS/CP/D1/CP/B8 /BT/BA /BX/BA /BU/CT/D6/CZ /D3 /DB/CX/D8/DE/B8 /BX/BA /C2/BA /C5\r/C6/CX/AR/B8 /C2/D6/BA /CP/D2/CS /CB/BA/BY /D3/D2/CT/D6/B8 /C8/CW /DD/D7/BA /CA/CT/DA/BA /C4/CT/D8/D8/BA /BJ/BJ /B8 /BF/BL/BG /B4/BD/BL/BL/BI/B5/BA/BD/BK/C2/BA /CC/BA /CA/CX\r /CW/CP/D6/CS/D7/D3/D2 /CP/D2/CS /CF/BA /C7/BA /C5/CX/D0/D0/CX/CV/CP/D2/B8 /C8/CW /DD/D7/BA /CA/CT/DA/BA /BD/BC/BE/B8 /BD/BE/BK/BL/B4/BD/BL/BH/BI/B5/BA/BD/BL/C4/BA /C6/GH/CT/D0/B8 /CX/D2 /C4/D3 /DB /CC /CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /C8/CW /DD/D7/CX\r/D7/B8 /CT/CS/CX/D8/CT/CS /CQ /DD /BV/BA /BW/CT/DB/CX/D8/D8/CT/D8 /CP/D0/BA /B4/BZ/D3/D6/CS/CP/D2 /CP/D2/CS /BU/CT/CP\r /CW/B8 /C6/CT/DB /CH /D3/D6/CZ/B8 /BD/BL/BI/BE/B5 /B8 /D4/BA /BG/BD/BF/BA/BE/BC/CB/BA /BT/BA /C5/CP/CZ/CW/D0/D3/D9/CU/B8 /BY/BA /CC/BA /C8 /CP/D6/CZ /CT/D6/B8 /BY/BA /BX/BA /CB/D4/CP/CS/CP /CP/D2/CS /BT/BA /BX/BA/BU/CT/D6/CZ /D3 /DB/CX/D8/DE/B8 /C2/BA /BT/D4/D4/D0/BA /C8/CW /DD/D7/BA /BK/BD /B8 /BH/BH/BI/BD /B4/BD/BL/BL/BJ/B5/BA/BE/BD/CA/BA /C0/BA /C3 /D3 /CS/CP/D1/CP/B8 /CB/BA /BT/BA /C5/CP/CZ/CW/D0/D3/D9/CU /CP/D2/CS /BT/BA /BX/BA /BU/CT/D6/CZ /D3 /DB/CX/D8/DE/B8 /C8/CW /DD/D7/BA/CA/CT/DA/BA /C4/CT/D8/D8/BA /BJ/BL /B8 /BD/BF/BL/BF /B4/BD/BL/BL/BJ/B5/BA/BE/BE/C5/D3/D9/D1/CX/D8/CP /BZ/CW/D3/D7/CW/B8 /C3/CP/D2/CX/D7/CW/CZ /CP /BU/CX/D7/DB /CP/D7/B8 /BT/BA /CB/D9/D2/CS/CP/D6/CT/D7/CP/D2 /CP/D2/CS /BV/BA/C6/BA /CA/BA /CA/CP/D3/B8 /C2/BA /C5/CP/D8/CT/D6/BA /BV/CW/CT/D1/BA /BD/BI /B8 /BD/BC/BI /B4/BE/BC/BC/BI/B5/BA/BE/BF/C5/BA/CB/BA /CB/CT/CT/CW/D6/CP/B8 /C8 /BA /BW/D9/D8/D8/CP/B8 /C0/BA /CB/CW/CX/D1 /CP/D2/CS /BT/BA /C5/CP/D2/CX/DA /CP/D2/D2/CP/D2/B8 /CB/D3/D0/CX/CS/CB/D8/CP/D8/CT /BV/D3/D1/D1 /D9/D2/CX\r/CP/D8/CX/D3/D2/D7 /BD/BE/BL /B8 /BJ/BE/BD /B4/BE/BC/BC/BG/B5/BA/BE/BG/CH /D9/CZ /D3 /C1\r /CW/CX/DD /CP/D2/CP/CV/CX/B8 /C6/CP/D3/D8/D3 /CF /CP/CZ /CP/CQ/CP /DD /CP/D7/CW/CX/B8 /C2/D9/D2/CX\r /CW/CX/D6/D3 /CH /CP/D1/CP/DE/CP/CZ/CX/B8/CB/CP/D3/D6/CX /CH /CP/D1/CP/CS/CP/B8 /CH /D3/D7/CW/CX/CW/CX/CS/CT /C3/CX/D1/CX/D7/CW/CX/D1/CP/B8 /BX/D6/CX/CZ /D3 /C3 /D3/D1/CP/D8/D7/D9 /CP/D2/CS/C0/CX/D6/D3 /DD/D9/CZ/CX /CC /CP /CY/CX/D1/CP/B8 /C8/CW /DD/D7/CX\r/CP /BU /BF/BE/BL /B8 /BK/BI/BE /B4/BE/BC/BC/BF/B5/BA/BE/BH/C2/D3/D2/CV/D2/CP/D1 /C8 /CP/D6/CZ /CT/D8 /CP/D0/BA /BT /CS/DA/BA /C5/CP/D8/CT/D6/BA /BD/BJ /B8 /BG/BE/BL /B4/BE/BC/BC/BH/B5/BA/BE/BI/C2/BA /CC/BA /CA/CX\r /CW/CP/D6/CS/D7/D3/D2/B8 /BW/BA /C1/BA /CH/CX/CP/CV/CP/D7/B8 /BU/BA /CC /D9/D6/CZ/B8 /C3/BA /BY /D3/D6/D7/D8/CT/D6 /CP/D2/CS /C5/BA/CE/BA /CC /DB/CX/CV/CV/B8 /C2/BA /BT/D4/D4/D0/BA /C8/CW /DD/D7/BA /BJ/BC /B8 /BI/BL/BJ/BJ /B4/BD/BL/BL/BD/B5/BA/BE/BJ/BY /CP/D8/CT/D1/CT/CW /BW/CP /DA /CP/D6 /B8 /CI/CT/CX/D2/CP/CQ /BY /CT/D6/CT/D7/CW /D8/CT/CW /CP/D2/CS /C5/CP/D7/D3/D9/CS /CB/CP/D0/CP /DA /CP/D8/CX/B9/C6/CX/CP/D7/CP/D6/CX/B8 /C2/D3/D9/D6/D2/CP/D0 /D3/CU /BT/D0/D0/D3 /DD/D7 /CP/D2/CS /BV/D3/D1/D4 /D3/D9/D2/CS/D7 /BG/BJ/BI /B8 /BJ/BL/BJ/B4/BE/BC/BC/BL/B5/BA/BH/BE/BK/BY/BA /BU/D3 /CS/CZ /CT/D6/B8 /C5/BA/BY/BA /C0/CP/D2/D7/CT/D2/B8 /BV/BA /BU/CT/D2/CS/CT/D6 /C3 /D3 \r /CW /CP/D2/CS /CB/BA/C5/D3/D6/D9/D4/B8/C2/D3/D9/D6/D2/CP/D0 /D3/CU /C5/CP/CV/D2/CT/D8/CX/D7/D1 /CP/D2/CS /C5/CP/CV/D2/CT/D8/CX\r /C5/CP/D8/CT/D6/CX/CP/D0/D7 /BE/BE/BD /B8 /BF/BE/B4/BE/BC/BC/BJ/B5/BA/BE/BL/C5/BA /CB/BA /CB/CT/CT/CW/D6/CP/B8 /C0/BA /CB/CW/CX/D1/B8 /C8 /BA /BW/D9/D8/D8/CP/B8 /BT/BA /C5/CP/D2/CX/DA /CP/D2/D2/CP/D2 /CP/D2/CS /C2/BA/BU/D3/D2/CT/DA/CX\r /CW/B8 /C2/BA /BT/D4/D4/D0/BA /C8/CW /DD/D7/BA /BL/BJ /B8 /BD/BC/C2/BH/BC/BL /B4/BE/BC/BC/BH/B5/BA/BF/BC/C0/BA /CB/CW/CX/D1/B8 /BT/BA /C5/CP/D2/CX/DA /CP/D2/D2/CP/D2/B8 /C5/BA /CB/BA /CB/CT/CT/CW/D6/CP/B8 /C3/BA /C5/BA /CA/CT/CS/CS/DD /CP/D2/CS/BT/BA/C8/D9/D2/D2/D3 /D3/D7/CT/B8 /C2/BA /BT/D4/D4/D0/BA /C8/CW /DD/D7/BA /BL/BL /B8 /BC/BK/C9/BH/BC/BF /B4/BE/BC/BC/BI/B5/BA/BF/BD/CE/BA /CE/BA /C8/CX/D7/CW/CZ /D3/B8 /CB/BA /C4/BA /BZ/D2/CP/D8\r /CW/CT/D2/CZ /D3/B8 /CE/BA /CE/BA /CC /D7/CP/D4 /CT/D2/CZ /D3/B8 /CA/BA /C0/BA/C3 /D3 /CS/CP/D1/CP /CP/D2/CS /CB/CP/D0/CP/CW /BT/BA /C5/CP/CZ/CW/D0/D3/D9/CU/B8 /C2/BA /BT/D4/D4/D0/BA /C8/CW /DD/D7/BA /CE /D3/D0/BA /BL/BF /B8/BJ/BF/BK/BE /B4/BE/BC/BC/BF/B5/BA/BF/BE/C5/CX\r /CW/CP/CT/D0 /C7/D7/D8/CW/B8 /BW/CX/CS/CX/CT/D6 /C0/CT/D6/CX/D7/D7/D3/D2/B8 /C8 /CT/D6 /C6/D3/D6/CS/CQ/D0/CP/CS/B8 /C2/D3/D7/CT /BT/BA /BW/CT/CC /D3/D6/D3 /CP/D2/CS /C2/D3/D7/CT /C5/BA /CA/CX/DA /CT/CX/D6/D3/B8 /C2/D3/D9/D6/D2/CP/D0 /D3/CU /C5/CP/CV/D2/CT/D8/CX/D7/D1 /CP/D2/CS /C5/CP/CV/B9/D2/CT/D8/CX\r /C5/CP/D8/CT/D6/CX/CP/D0/D7 /BF/BD/BF /B8 /BF/BJ/BF /B4/BE/BC/BC/BJ/B5/BA/BF/BF/BV/BA /CA/CP /CY /CB/CP/D2/CZ /CP/D6/B8 /CB/BA /CE/CX/CY/CP /DD /CP/D2/CP/D2/CS/B8 /CB/CT/CT/D1/CP /CE /CT/D6/D1/CP /CP/D2/CS /C8 /BA /BT/BA /C2/D3 /DD /B8/CB/D3/D0/CX/CS /CB/D8/CP/D8/CT /BV/D3/D1/D1 /D9/D2/CX\r/CP/D8/CX/D3/D2/D7 /BD/BG/BD /B8 /BF/BC/BJ /B4/BE/BC/BC/BJ/B5/BA/BF/BG/BW/BA /C8 /CP/D6/CZ /CT/D6/B8 /BY/BA /C4/CP/CS/CX/CT/D9/B8 /BX/BA /CE/CX/D2\r/CT/D2 /D8/B8 /BZ/BA /C5/GH/D6/CX/CV/D9/CT/D8/B8 /BX/BA /BW/D9/CQ /D3/CX/D7/B8/CE/BA /BW/D9/D4/D9/CX/D7 /CP/D2/CS /CA/BA /C8 /CT/D6/DE/DD/D2/D7/CZ/CX/B8 /C2/BA /BT/D4/D4/D0/BA /C8/CW /DD/D7/BA /BL/BJ /B8 /BD/BC/BT/BH/BC/BE/B4/BE/BC/BC/BH/B5/BA/BF/BH/BW/BA /C8 /CP/D6/CZ /CT/D6/B8 /CE/BA /BW/D9/D4/D9/CX/D7/B8 /BY/BA /C4/CP/CS/CX/CT/D9/B8 /C2/BA /C8 /BA /BU/D3/D9\r /CW/CP/D9/CS/B8 /BX/BA\n/BW/D9/CQ /D3/CX/D7/B8 /CA/BA /C8 /CT/D6/DE/DD/D2/D7/CZ/CX /CP/D2/CS /BX/BA /CE/CX/D2\r/CT/D2 /D8/B8 /C8/CW /DD/D7/BA /CA/CT/DA/BA /BU /BJ/BJ /B8/BD/BC/BG/BG/BE/BK /B4/BE/BC/BC/BK/B5/BA/BF/BI/BZ/D9/CU/CT/CX /CI/CW/CP/D2/CV/B8 /C3/BA /C8 /D3/D8/DE/CV/CT/D6 /B8 /CB/CW/CT/D2/CV/D5/CX/CP/D2/CV /CI/CW/D3/D9/B8 /BT/BA /C5/H0\r /CZ/D0/CX\r /CW/B8/CH/CX\r/D3/D2/CV /C5/CP /CP/D2/CS /C2/BA /BY /CP/D7/D7/CQ /CT/D2/CS/CT/D6/B8 /C6/D9\r/D0/CT/CP/D6 /C1/D2/D7/D8/D6/D9/D1/CT/D2 /D8/D7 /CP/D2/CS/C5/CT/D8/CW/D3 /CS/D7 /CX/D2 /C8/CW /DD/D7/CX\r/D7 /CA/CT/D7/CT/CP/D6\r /CW /BU /BE/BI/BJ /B8 /BD/BH/BL/BI /B4/BE/BC/BC/BL/B5/BA/BF/BJ/CA/BA /C3/BA /CI/CW/CT/D2/CV/B8 /C0/D3/D2/CV/DB /CT/CX /BZ/D9 /BU/CX/D2/CV /CG/D9 /CP/D2/CS /CG/BA /CG/BA /CI/CW/CP/D2/CV/B8 /C8/CW /DD/D7/BA/CA/CT/DA/BA /BU /BJ/BE /B8 /BC/BD/BG/BG/BD/BI /B4/BE/BC/BC/BH/B5/BA/BF/BK/CC/BA /C2/D3/D2/D7/D7/D3/D2/B8 /C2/BA/C5/CP/D8/D8/D7/D7/D3/D2/B8 /BV/BA /BW/CY/D9/D6/CQ /CT/D6/CV/B8 /BY/BA/BT/BA/C3/CW/CP/D2/B8/C8 /BA/C6/D3/D6/CS/CQ/D0/CP/CS /CP/D2/CS /C8 /BA /CB/DA /CT/CS/D0/CX/D2/CS/CW/B8 /C8/CW /DD/D7/BA /CA/CT/DA/BA /C4/CT/D8/D8/BA /BJ/BH /B8 /BG/BD/BF/BK/B4/BD/BL/BL/BH/B5/BA/BF/BL/C8 /BA /BX/BA /C2/GU/D2/D7/D7/D3/D2/B8 /C0/BA /CH /D3/D7/CW/CX/D2/D3/B8 /C0/BA /C5/CP/D1/CX/DD /CP/B8 /CP/D2/CS /C0/BA /CC /CP/CZ /CP /DD /CP/D1/CP/B8/C8/CW /DD/D7/BA /CA/CT/DA/BA /BU /BJ/BD /B8 /BD/BC/BG/BG/BC/BG /B4/BE/BC/BC/BH/B5/BA/BG/BC/C2/BA /BU/BA /CH/CX/B8 /C2/BA /BW/CX/D2/CV/B8 /CH/BA /C8 /BA /BY /CT/D2/CV/B8 /BZ/BA /CF/BA /C8 /CT/D2/CV/B8 /BZ/BA /C5/BA /BV/CW/D3 /DB/B8 /CH/BA/C3/CP /DB /CP/DE/D3 /CT/B8 /BU/BA /C0/BA /C4/CX/D9/B8 /C2/BA /C0/BA /CH/CX/D2/B8 /CP/D2/CS /CB/BA /CC/CW/D3/D2/CV/D1/CT/CT/B8 /C8/CW /DD/D7/BA/CA/CT/DA/BA /BU /BJ/BI /B8 /BE/BE/BG/BG/BC/BE /B4/BE/BC/BC/BJ/B5/BA/BG/BD/CB/CP/D0/CP/CW /BT/BA /C5/CP/CZ/CW/D0/D3/D9/CU/B8 /C0/BA /BT/D0/B9/BT /D8/D8/CP/D6 /CP/D2/CS /CA/BA/C0/BA /C3 /D3 /CS/CP/D1/CP/B8 /CB/D3/D0/CX/CS/CB/D8/CP/D8/CT /BV/D3/D1/D1 /D9/D2/CX\r/CP/D8/CX/D3/D2/D7 /BD/BG/BH /B8 /BD /B4/BE/BC/BC/BK/B5/BA"    },    {        "title": "1702.04673v1.Dynamics_of_magnetic_nano_particles_in_a_viscous_fluid_driven_by_rotating_magnetic_fields.pdf",        "content": "Dynamics of magnetic nano particles in a viscous \ruid driven by rotating magnetic\n\felds\nKlaus D. Usadel\nTheoretische Physik, Universit at Duisburg-Essen, 47048 Duisburg, Germany\n(Dated: October 17, 2018)\nThe rotational dynamics of magnetic nano particles in rotating magnetic \felds in the presence\nof thermal noise is studied both theoretically and by performing numerical calculations. Kinetic\nequations for the dynamics of particles with uniaxial magnetic anisotropy are studied and the phase\nlag between the rotating magnetic moment and the driving \feld is obtained. It is shown that for\nlarge enough anisotropy energy the magnetic moment is locked to the anisotropy axis so that the\nparticle behaves like a rotating magnetic dipole. The corresponding rigid dipole model is analyzed\nboth numerically by solving the appropriate Fokker-Planck equation and analytically by applying\nan e\u000bective \feld method. In the special case of a rotating magnetic \feld applied analytic results\nare obtained in perfect agreement with numerical results based on the Fokker-Planck equation.\nThe analytic formulas derived are not restricted to small magnetic \felds or low frequencies and are\ntherefore important for applications. The illustrative numerical calculations presented are performed\nfor magnetic parameters typical for iron oxide.\nPACS numbers: 75.40.Gb, 75.40.Mg, 75.75.Jn, 75.60.Jk\nI. INTRODUCTION\nRecently, magnetic nanoparticles (MNPs) have been\nstudied for many biomedical applications such as mag-\nnetic particle imaging, separation of biological targets,\nimmunoassays, drug delivery, and hyperthermia treat-\nment [1{3]. In these applications the magnetic moment\nmay rotate within the particle with respect to some crys-\ntal axis - i.e. Ne\u0013 el relaxation - and move along with the\nparticle with respect to the liquid, i.e. Brownian relax-\nation. Hyperthermia, for instance, is based on the fact\nthat power is absorbed locally by the MNP when placed\nin an applied oscillating magnetic \feld. A high absorp-\ntion rate is achieved if both Ne\u0013 el relaxation and Brownian\nrelaxation contribute [4{6].\nOn the other hand if the anisotropy energy is large\ncompared to the thermal energy Ne\u0013 el relaxation becomes\nunimportant. In this limit the MNP can be considered as\na rigid body (RB) having a constant magnetic moment\n\frmly attached to it. In this rigid dipole model (RDM)\nthe external driving \feld will create a torque to the mag-\nnetic moment which is transferred to the RB. In a rotat-\ning magnetic \feld, for instance, the MNP will rotate as\nwell. Theoretically the dynamics of MNPs has been stud-\nied in recent years in very many papers for the case that\nthe particles are \fxed in space. The dynamics is then\nreduced to that of magnetic moments in external \felds\nfor which the stochastic Landau-Lifshitz-Gilbert (LLG)\nequation as introduced by Brown [7] is often used. This\napproach has gained increased interest recently because\nof its application to magnetic single-domain particles. In\nthese MNPs the magnetic moments within the particles\nare \frmly tied together making it possible to describe\nthe magnetic moment of the particle as one macro spin\nof constant length [8]. Its dynamics is expected to be well\ndescribed by a classical approach. The dynamics of these\nmacro spins is greatly in\ruenced by thermal \ructuationsoriginated from coupling of the spins to the surrounding\nmedium (the heat bath). Additionally, for mobile par-\nticles Brownian relaxation has to be taken into account\n[5, 9{16].\nRecently the dynamical properties of mobile particles\nhave been studied in some detail in connection with bio-\nlogical applications, i. e. the realization of homogeneous\nbiosensors [17{20] based on the response of a suspension\nof MNPs to an applied acmagnetic \feld. In these biolog-\nical applications a dilute suspension of MNPs is present\nthe physical properties of which are determined to a large\nextend by the magnetic and mechanical dynamics of in-\ndependent MNPs.\nQuite recently biosensors were proposed [21{23] which\nare based on the response of MNPs with Brownian dy-\nnamics to rotating magnetic \felds, i.e. to di\u000berences in\nthe phase lag between the rotating magnetic moment and\nthe driving \feld in order to determine the changes in\nhydrodynamic volume caused by analytes bound to the\nsurface of the MNPs. For this method to be successful,\nit is necessary to quantitatively clarify the dynamics of\nMNPs in rotating \felds. Theoretical studies on this prob-\nlem reported in Ref. [24] are based on an e\u000bective \feld\nmethod (EFM) [25, 26] for the dynamics of an ensem-\nble of MNPs treated as rigid dipoles placed in a viscous\nmedium. The EFM was derived from a Fokker-Planck\nequation (FPE) describing the rotational dynamics at \f-\nnite temperatures.\nResults obtained for the case that the applied \feld ro-\ntates were compared with measurements of the phase\nlag [23, 27] revealing a reasonable agreement between\nthe measured and calculated values in the low frequency\nregime and for small \feld amplitudes. However, signi\f-\ncant discrepancies were observed outside this regime. Nu-\nmerical calculations based directly on the FPE for the\nrigid dipole model [28] con\frmed these \fndings so that\na need for further theoretical work persists.arXiv:1702.04673v1  [physics.flu-dyn]  16 Jan 20172\nIn the present paper we therefore study the dynamics\nof mobile MNPs driven by a rotating magnetic \feld in\norder to contribute to an understanding of the physics\nunderlying the functionality of the class of biosensors\nmentioned [21{23]. The ensemble of MNPs is described\nby a set of kinetic equations proposed earlier [5] for the\ndynamics of mobile particles treating Brownian and Ne\u0013 el\ndynamics on equal footing.\nIn the \frst part of the paper we discuss brie\ry our\nkinetic equation [5] and present numerical results for\nthe dynamics of MNPs with \fnite uniaxial anisotropy\ndriven by rotating magnetic \felds. Evidence is given to\nthe expected blocking of the magnetic moment to the\nanisotropy axis for large enough anisotropy energies. The\nrelation between our kinetic equations and the RDM is\ndiscussed and it is shown how the RDM emerges from\nour kinetic equations.\nIn the second part of the paper we discuss in detail the\nRDM and sketch the derivation of the basic equations\nof the EFM [25, 26]. We show that for the special case\nof an applied rotating magnetic \feld these equations can\nbe solved without further assumptions. Analytic formu-\nlas are obtained for the phase lag and for the magnetic\nmoment and it is shown that these results are in nearly\nperfect agreement with results which follow from numer-\nical solutions of the FPE. The analytic formulas derived\nare not restricted to small magnetic \felds and/or fre-\nquencies and are therefore important for applications.\nII. DYNAMICS OF MAGNETIC NANO\nPARTICLES WITH FINITE ANISOTROPY\nA. Kinetic equations\nA nano particle considered in the present paper is mod-\neled as a uniformly magnetized spherical rigid body (RB)\nwith uniaxial magnetic anisotropy. The particle can ro-\ntate in a viscous medium. Its orientation in space is\ndescribed by a time dependent unit vector n(t) parallel\nto the anisotropy axis of the RB. This vector is \frmly at-\ntached to the RB so that its equation of motion is given\nby\ndn\ndt=!\u0002n (1)\nwhere!denotes the angular velocity of the RB. The\ndi\u000berentiation in Eq. (1) is performed in a coordinate\nsystem \fxed in space, the laboratory frame.\nThe magnetic moment \u0016of the NP with constant mag-\nnitude\u0016scan rotate in space along with the RB and\ncan also rotate relative to it depending primarily on the\nstrength of the anisotropy energy Dof the NP. We intro-\nduce e, an unit vector in the direction of the magnetic\nmoment, e=\u0016=\u0016s. For eand for the angular velocity !\nof the RB kinetic equations were proposed in [5] whichcan be written in compact form as\nde\ndt=!\u0002e\u0000\r\n1+\u000b2\u0010\ne\u0002(Be\u00001\n\r!)\n+\u000be\u0002(e\u0002(Be\u00001\n\r!))\u0011\n(2)\nand\n\u0002d!\ndt=\u0016s\n\rde\ndt+\u0016se\u0002(B+\u0010)\u0000\u0018!+\u000f: (3)\nThe e\u000bective \feld Beentering Eq.(2) consists of the ex-\nternal driving \feld Bwhich may be time dependent,\na term proportional to the anisotropy energy Dand a\nstochastic \feld \u0010relevant at elevated temperatures,\nBe=B+2D\n\u0016s(e\u0001n)n+\u0010: (4)\nThe quantity \u000bappearing in Eq.(2) is the dimen-\nsionless damping parameter usually used in the litera-\nture [29{31], \u0018denotes the friction coe\u000ecient usually ex-\npressed as\u0018= 6\u0011Vdwhere\u0011is the dynamical viscosity\nwhileVdwith radius rddenotes the hydrodynamic or to-\ntal volume of the particle, i.e. it is assumed that the\nparticle consists of a magnetic core region of volume Vm\nwith radius rmeventually covered by a nonmagnetic sur-\nfactant layer.\nFor the thermal \ructuations introduced above it is as-\nsumed as usual that they are Gaussian distributed with\nzero mean. Their correlators are chosen in such a way\nthat in equilibrium Boltzmann statistics is recovered.\nThis leads to\nh\u0010l(0)\u0010m(t)i\u0010=\u000el;m\u000e(t)2\u000bkBT=(\u0016s\r): (5)\nfor the magnetic \feld \ructuations [7] and\nh\u000fl(0)\u000fm(t)i\u000f=\u000el;m\u000e(t)2\u0018kBT: (6)\nfor the Brownian rotation [11, 32]. The angular brackets\ndenote averages over \u0010and\u000f, respectively, and landm\nlabel cartesian components of the \ructuating \felds.\nFor given \ructuating \felds \u000f(t) and\u0010(t) the quanti-\ntiesn(t) and e(t) as solutions of the stochastic equations\nare trajectories on the unit sphere because the equations\nof motion conserve the length of these vectors. Physical\nquantities of interest describing properties of an ensem-\nble of identical particles are obtained as averages over\nthese trajectories. The reduced magnetic moment at \f-\nnite temperatures, for instance, is given by\nm(t) =he(t)i\u0010;\u000f: (7)\nEqs. (1 - 6 ) constitute a closed set of kinetic equations\nfor the quantities n(t),e(t) and!(t) specifying the dy-\nnamics of the MNP. In deriving these equations we were\nguided by the requirement that for the isolated particle\nthe total angular momentum L+Shas to be conserved\nirrespectively of internal interactions within the MNP.3\nHere, L= \u0002!denotes the angular momentum and S\nthe spin momentum, S=\u0000\r\u00001\u0016, where\rdenotes the\ngyromagnetic ratio. For more details we refer to [5].\nThe kinetic equations proposed treat the dynamics of\nthe magnetic moment and the rotational motion of the\nNP on the same footing. In general, both processes are\ncoupled leading to a rather complex behavior. In the\nlimit of a large anisotropy energy, however, it is expected\nthat the magnetic moment is locked into a position par-\nallel to the anisotropy axis of the RB. This limiting case\nis the essence of the rigid dipole model.\nB. Numerical analysis: dependence on anisotropy\nenergy\nUsing our kinetic equations we investigate numeri-\ncally the dependence of phase lag and induced mag-\nnetic moment on the anisotropy energy considering the\nanisotropy constant K1as being adjustable. Other pa-\nrameters are chosen as typical for particles of iron oxides\n(magnetite). The magnetic moment \u0016sis expressed as\n\u0016s=MsVmwithMs= 4\u0001105A=m and the reduced vis-\ncosity ~\u0011is de\fned as ~ \u0011=\u0011=\u0011water with\u0011water = 10\u00003\nkg/ms. Results are expressed as function of K1=K10with\nK10= 104J=m3which orresponds to the anisotropy con-\nstant of magnetite. The rotating magnetic \feld is given\nby\nB=B0(cos(~!t)^x+ sin(~!t)^y) (8)\nwhere ~!= 2\u0019Fdenotes the frequency of the driving\n\feld. A cartesian coordinate system \fxed in space is\nused spanned by unit vectors ^x,^yand^z.\nThe initial conditions necessary for the numerical in-\ntegration of the kinetic equations are speci\fed as e(t=\n0) = n(t= 0) and!(t= 0) = 0 with randomly dis-\ntributed e(t= 0). For the integration of the stochastic\nequations Eqs.(1-3) methods well known by now from lit-\nerature are used [29{31]. In the stationary state which is\nreached after a time interval \u000etthe ensemble averaged\nquantitieshe(t)i\u0010;\u000fandhn(t)i\u0010;\u000fare monitored. Note\nthat\u000etdepends very much on parameters.\nIn the stationary state he(t)i\u0010;\u000frotates around ^zwith\nthe frequency of the driving \feld. A phase lag is observed\nbetween the direction of the rotating \feld and this rotat-\ning magnetic moment. The phase lag \u001eis obtained as the\nangel betweenhe(t)i\u0010;\u000fand the direction of the driving\n\feld averaged over about 1000 particles in the ensemble\nfollowed by an average over about 100 \feld cycles. Fig. 1\nshows the averaged phase lag as function of K1=K10for\nthree di\u000berent magnetic radii rmof the MNP, rm= 10\nnm (circles, black),14 nm (diamonds, blue) and 19 nm\n(squares, red).\nFor large anisotropy energies it is observed that the\nphase lag becomes independent of the anisotropy energy\nwhich indicates that the moment is locked to the RB,\ni.e. it behaves like a rotating rigid dipole. This is di-\nrectly con\frmed by the observation that hn(t)i\u0010;\u000fremains\n00.5 1 1.5 2 2.5 K\n1 / K1000.511.5φ\n00.5 1 1.5 2 2.5 K\n1 / K100.40.60.81mFIG. 1. (color online) Phase lag \u001e( upper panel ) and in-plane\nmagnetization m( lower panel ) versus reduced anisotropy\nparameterK1=K10for di\u000berent values of the magnetic radius\nrm:rm= 19 nm (circles, black), 14 nm (squares, blue), and\n10 nm (diamonds, blue). \u000b= 0:01, ~\u0011= 1:0,rd=rm= 2:5,\nF= 4 kHz,B= 5 mT. The horizontal lines indicate the\nvalues obtained within the RDM.\nnearly parallel to he(t)i\u0010;\u000f(not shown), i.e. magnetic mo-\nment and anisotropy axis rotate unisono.\nThe RDM is studied in detail in the next sections. Nu-\nmerical results will be obtained some of which are shown\nas dashed horizontal lines in Fig.(1). They represent\nphase lag and magnetic moment, respectively, calculated\nwithin the RDM. Obviously for large K1=K10 the values\nobtained from the RDM are approached asymptotically.\nA reduction of K1=K10, on the other hand, leads to a\nsharp drop in the phase lag accompanied by a signi\fcant\nincrease of m. In this limit the moment adjusts nearly\nparallel to the driving \feld and rotates with the \feld\nwhile the RBs stop rotating in a coherent fashion. This\nfollows from the time variation of hn(t)i\u0010;\u000f(not shown)\nconsisting of only small random variations around zero.\nIn the course of this work we kept the temperature\n\fxed atT= 300 K. A variation of Twill e\u000bect in par-\nticular the Ne\u0013 el relaxation time which depends on K1=T.\nWe therefore expect that the value of K1at which the\nlocking of the moment sets in will depend on T.\nThe increase of mwhen reducing the anisotropy en-4\nergy sets in for values of K1=K10for which the magnetic\nmoment unlocks from the RB moving towards the di-\nrection of the \feld. In the extreme limit K1= 0 the\nmagnetic moment is only subjected to a tiny frictional\ntorque resulting from the Gilbert - damping \u000b. Only for\nunrealistic high values of \u000bthis torque is large enough to\novercome the viscous damping necessary for the RB to\nrotate.\nC. Large anisotropy energy\nThe numerical results presented suggest that for large\nanisotropy energies our kinetic equations pass over to the\nRDM. This is supported by the following qualitative ar-\nguments. Consider a rotating frame \frmly attached to\nthe RB in which the anisotropy axis is \fxed. The time\nderivativede\ndtcan be split into two parts,\nde\ndt=!\u0002e+ (de\ndt)rf (9)\nwhere the second term in this equation is the time deriva-\ntive in the rotating frame. Comparison with Eq.(2) shows\nthat this time derivative is equal to the second term of\nEq.(2) so that\n(de\ndt)rf=\u0000\r\n1 +\u000b2\u0010\ne\u0002(Be\u00001\n\r!)\n+\u000be\u0002(e\u0002(Be\u00001\n\r!))\u0011\n; (10)\ndescribing the dynamics of a spin with a LLG-type\nequation with e\u000bective \feld Be\u00001\n\r!in the rotating\nframe. In this rotating frame the anisotropy axis is \fxed.\nTherefore ewill settle nearly instantaneously parallel to\nthe anisotropy axis and will stay there providing the\nanisotropy energy is much larger than the contributions\nfrom the driving \feld and from !and that the temper-\nature is such that thermal switching of edoes not take\nplace. This means that after a (short) initial time (de\ndt)rf\nwill go to zero and we are left with\nde\ndt=!\u0002e: (11)\nNote that this argument requires a \fnite Gilbert damping\n\u000b. If the damping parameter is strictly zero there will be\nno relaxation towards the anisotropy axis so that a rather\ncomplex precessional motion results.\nThus for large anisotropy energy we are lead to Eqs.\n(3) and (11) which constitute the basis of the RDM. Fur-\nther simpli\fcations are possible. First we note that in the\nequation of motion for !inertia e\u000bects can be ignored\nbecause of the small particle size combined with realistic\nvalues of\u0018so that one obtains\n!=1\n\u0018(\u0016s\n\rde\ndt+\u0016se\u0002(B+\u0010) +\u000f): (12)Eqs.(11-12) can be solved forde\ndtwith algebraic manip-\nulations known from the LLG-equation. The resulting\nequation forde\ndtdepends on the parameter \u0014,\n\u0014=\u0016s\n\u0018\r(13)\nwhich is extremely small. It is su\u000ece therefore to keep\nonly the leading terms with respect to \u0014resulting in the\nwell-known equation of motion for the dynamics of a\nmagnetic NP in the rigid dipole approximation,\nde\ndt=\u0000\u0018\u00001\u0016se\u0002(e\u0002B)\u0000\u0018\u00001e\u0002\u000f: (14)\nNote that the \ructuating \feld \u0010entering Eq.(12) can be\nshown to be negligible in lowest order in \u0014.\nIII. RIGID DIPOLE MODEL\nThe rigid dipole model became a matter of particular\ninterest quite recently because of its potential for biomed-\nical applications. The stochastic equation describing its\ndynamics at \fnite temperature, Eq.(14), can be studied\nnumerically. An alternative is using the FPE which cor-\nresponds to this stochastic process.\nA. Fokker-Planck equation for the RDM\nFor the RDM the probability density P(S;t) is de\fned\nas\nP(S;t) =<\u000e(S\u0000e(t))>\u000f (15)\nwhere the angular brackets denote an average over all\ntrajectories e(t) of Eq. (14).\nThe probability density P(S;t) de\fned is a solution of\nthe FPE obtained from Eq.(14). It reads\n@P(S;t)\n@t=r\u0001h\u0016s\n\u0018S\u0002(S\u0002B)\n\u00001\n2\u001cBS\u0002(S\u0002r)i\nP(S;t): (16)\nFor an elegant derivation of the FPE see for instance\nRef.[33]. The quantity \u001cBintroduced in Eq.(16) denotes\nthe Brownian relaxation time,\n\u001cB=3\u0011Vd\nkBT: (17)\nBecause the stochastic process conserves the length of e\nthe probability density is of general form\nP=\u000e(jSj\u00001)Q(S;t): (18)\nThe probability density P(S;t) contains all informa-\ntion about \ructuation averaged physical quantities. The5\naveraged reduced magnetic moment, for instance, is given\nby\nm(t) =Z\nd3SP(S;t)S; (19)\nthe \frst moment of the probability density, which is\nequivalent to the average over thermal \ructuations,\nEq.(7).\nIt is easy to see that the Boltzmann distribution P0\u0018\nexp(\u0016sS\u0001B0=kBT) is an equilibrium solution of Eq.(16).\nIn general, however, exact solutions of the FPE are not\nknown so that one has to rely on numerical solutions of\nthe FPE or on approximations.\nFor a numerical solution of the FPE the construction\nof a fast and robust algorithm has been described previ-\nously. The important point is to discretize the equation\nof motion for Q, Eq.(18), in such a way that the normal-\nization ofPis preserved independent of the mesh size.\nFor details the reader is referred to [34].\nAn approximate solution of the FPE, the e\u000bective \feld\nmethod, has been described in [25, 26] the basic steps of\nwhich are outlined in the next section.\nB. E\u000bective \feld method\nThe starting point is an ansatz for the probability den-\nsity as in the works before, [25, 26], assuming for P(S;t)\nan expression of the form of an equilibrium density with\nadjustable parameters,\nP=N\u00001\u000e(jSj\u00001) exp( A(t)\u0001S) (20)\nwith normalization factor Ngiven by\nN(t) = 4\u0019sinh(A(t))\nA(t)(21)\nandA(t) =p\n(A(t)\u0001A(t)).\nThe reduced magnetic moment resulting from this\nprobability density, i.e. the \frst moment of the prob-\nability density, is given by\nm(t) =L(A(t))A(t)\nA(t)(22)\nwhere\nL(A) =1\ntanh(A)\u00001\nA(23)\ndenotes the Langevin function. For the \frst moment of\nthe probability density we obtain from Eq.(16) after par-\ntial integration\nZ\nd3SS@P(S;t)\n@t=Z\nd3Sh\u0016s\n\u0018S\u0002(S\u0002B)\n\u00001\n2\u001cBS\u0002(S\u0002r)i\nP(S;t):(24)The e\u000bective \feld Ais determined by the requirement\nthat this equation is ful\flled for the probability density\nP(S;t) de\fned in Eq.(20) leading to\ndm\ndt=\u0000\u0016s\n\u0018\u0002\n(1\u00003m\nA) (B\u0001^ m)^ m+ (m\nA\u00001)B\u0003\n\u00001\n\u001cBm:\n(25)\n^ mdenotes a unit vector in the direction of m. Eqs.\n(22,23,25) are the basic equations of the EFM [25, 26].\nNote that for a time independent magnetic \feld B=\nB0the equilibrium solutiondm\ndt= 0 is parallel to B0so\nthat we obtain from Eq. (25) A=\u0016sB0\nkBT. For the reduced\nmagnetic moment we therefore obtain with Eq. (23)\nm=L(\u0016sB0\nkBT); (26)\ni.e. the equilibrium moment. This result is of course\nexpected because the equilibrium probability density is\nof the same form as that assumed in Eq.(20).\nIn the general case, however, because of the nonlinear\nrelation between m(t) andA(t) in Eq.(22), analytic solu-\ntions are not obvious. An exception is the special case of\na rotating magnetic \feld, Eq.(8), for which a stationary\nsolution of Eq.(25) can be found.\nC. Rotating magnetic \felds\nFor a rotating magnetic \feld a stationary solution ex-\nists with time independent amplitude m=p\n(m\u0001m)\nimplying a time independent A, Eq.(22).\nTo show this we note that for a time independent m\nwe have\nm\u0001dm\ndt= 0: (27)\nMultiplying Eq.(25) with mwe obtain\nB(t)\u0001m(t) =Amk BT\n\u0016s: (28)\nTherefore a time independent m(andA) requires a ro-\ntating magnetic moment mwith constant phase lag, i.e.\nmmust be of the form\nm=m(cos(~!t\u0000\u001e)^x+ sin(~!t\u0000\u001e)^y): (29)\nThis form indeed solves Eq.(25) and one \fnds after some\nalgebra for the phase lag\n\u001e= arctan(2 ~!\u001cB\nA=m\u00001) (30)\nand for the amplitude of the e\u000bective \feld\nA=\u0016sB0\nkBT(A=m\u00001)p\n(2~!\u001cB)2+ (A=m\u00001)2(31)6\nwith\nm=L(A): (32)\nEqs.(31-32) determine implicitly the quantities Aandm.\nWe note in passing that the lag angel \u001edetermines the\nenergy absorption of the MNPs by the rotating \feld. The\nabsorbed power can be obtained from [4, 5, 12]\nW=\u0000\u0016shedB\ndti (33)\nso that we obtain from Eqs.(8,29)\nW=\u0016s~!B0msin(\u001e): (34)\nIn the next section we will show that our analytic re-\nsults, Eq.(30-32), are in nearly perfect agreement with\nthose we obtained from numerical solutions of the FPE.\nDi\u000berent results for the case that a rotating \feld is ap-\nplied have been reported previously [24]. Those results\nare based on expansions of the EFM around the equilib-\nrium solution [25, 26]. The following expressions for the\nphase lag and for the e\u000bective \feld - called \u0018for clarity -\n010 20 30 40 F [kHz]\n00.511.5φ, m\n05 10 F [kHz]\n00.511.5φ, m\nFIG. 2. (color online) Phase lag \u001eand magnetic moment m\nversus frequency.\nParameters: ~ \u0011= 1:0;T= 300;rm= 15:0;B0= 5 mT.\nUpper panel: rd=rm= 1:5, lower panel rd=rm= 4:5.\nDots black and squares red: numerical solution of the FPE.\nSolid, blue: present paper, dashed-dotted, green: Ref.[24].were obtained [24]:\n\u001e= arctan(~!\u001cB2L(\u0018)\n\u0018\u0000L(\u0018)) (35)\nwith\n\u0018=\u0016sB0\nkBT(36)\nand\nm=L(\u0018)cos(\u001e): (37)\nThe e\u000bective \feld A, Eq.(31), is frequency dependent\nin general. For !!0 it becomes identical to \u0018, Eq.(36),\nso that in this limit the results obtained for the phase lag\ncoincide. Di\u000berences are obvious for \fnite frequencies.\nFinally we would like to mention that for small \felds\nthe analytic results presented agree with each other and\nthey also agree with results from linear response theory.\nThe reason is simply that for small \felds the quantity\nQ(S;t), Eq.(18), can be expanded as\nQ(S;t) =1+q(t)\u0001S+::: (38)\nwhere qis assumed to be linear in the \feld. Inserting this\nexpression into Eq.(16) and keeping only terms linear in\nB0we arrive at a di\u000berential equation for q(t) which\ncan be solved easily. It is important to note that this\nis an exact solution of the FPE (to linear order in B0)\ndescribing the stationary state.\nWe do not present details of this calculation here be-\ncause the results can also be obtained from an expansion\nof Eqs.(30-32) in linear order in B0. The reason for this\nis that for small e\u000bective \felds P(S;t), Eq.(20), agrees\nwith Eq.(38) in linear order in B0so that the results for\n\u001eandmobtained must coincide.\nD. Numerical analysis of the RDM\nResults obtained from a numerical solution of the FPE\nfor the RDM will be compared in the following with those\nobtained from our analytic results, Eqs.(30-32) and also\nwith those obtained in Ref. [24] (Eqs. (35-37)).\nFrom the numerical solution of the FPE in a rotat-\ning \feld the magnetic moment m(t) is calculated using\nEq.(19). After an initial time interval with length de-\npending again strongly on the parameters of the system\na stationary state with time independent phase lag is ob-\ntained. Results for the phase lag and the reduced mag-\nnetic moment in the stationary state are discussed in the\nfollowing.\nFig.(2) shows the phase lag \u001e(black circles and ascend-\ning curves) and the magnitude of the induced magnetic\nmoment,m, (red squares and descending curves) as func-\ntion of frequency for rd=rm= 1:5 (upper panel) and for\nrd=rm= 4:5 (lower panel). An increase of the dynamical7\n02 4 6 8 F [kHz]\n00.511.52φ, m\n02 4 6 8 F  [kHz]\n00.511.52φ, m\n01 2 3 4 F [kHz]\n00.511.5φ, m\nFIG. 3. (color online) Phase lag \u001eand magnetic moment m\nversus frequency.\nParameters: rd=rm= 2:5,rm= 19 nm.\nUpper panel: B= 2:5 mT, ~\u0011= 1:0,\nmiddle panel: B= 5 mT. ~\u0011= 1:0,\nlower panel: B= 5 mT. ~\u0011= 8:0.\nDots black and squares red: numerical solution of the FPE.\nSolid blue: present paper, dashed-dotted green: Ref.[24].\nradiusrdincreases the frictional torque which is compen-\nsated by the enormous increase of the lag angel observed\nat small frequencies (note the di\u000berent scales in the ab-\nscissa of these two graphs). This is consistent with the\nobservation that an increase of the phase lag leads to an\nincrease of the absorbed power according to Eq. (34).\nPhase lag and magnetic moment calculated from the\nnumerical solution of the FPE are in very good agree-\n24 6 r\nd/rm00.511.5φ, mFIG. 4. (color online) Phase lag \u001eand magnetic moment m\nversusrd=rm.\nParameters: ~ \u0011= 1:0;T= 300;rm= 19:0;B= 5:0 mT,F= 4\nkHz.\nDots black and squares red: numerical solution of the FPE.\nSolid blue: present paper, dashed-dotted green: Ref.[24].\nment with our analytic results obtained from the EFM\n(solid lines, blue) in the entire frequency region shown.\nResults obtained from [24] (dashed-dotted, green) devi-\nate from the numerically exact result especially for larger\nfrequencies.\nThe horizontal lines (solid, red) show the equilibrium\nmagnetic moment for a \feld B0, the amplitude of the\nrotating \feld. Obviously this result shows that for very\nsmall frequencies the system is in a state of quasi equi-\nlibrium in which the magnitude of the magnetic moment\nis close to its equilibrium value rotating slowly. With de-\ncreasing frequency the equilibrium value is approached\nasymptotically.\nThe results shown are representative in the sense that\nwe always found an extremely good agreement between\nthe results obtained from the FPE and those obtained\nfrom our analytic results based on the EFM.\nFig.(3) shows results for particles with an increased\nmagnetic radius, rm= 19 nm. An increase in the \feld\nstrength from B0= 2:5 mT (upper panel) to B0= 5 mT\n(middle panel) leads to a reduction in the lag angel and\nto a signi\fcant increase in the magnetic moment in the\nlow frequency region. This is explained by the increased\ndriving torque exerted by the larger magnetic \feld. An\nincreased frictional torque, on the other hand, leads to\nan increase of the lag angel and a decrease of mas can\nbe seen in the lower panel of Fig.(3) where the viscosity\nis enlarged by a factor of 8 (note again the di\u000berent scale\nin the abscissa of the \fgure in the lower panel).\nFinally, in Fig.(4) we show results for the dependence\nof the lag angel and the magnetic moment on the ratio\nof the hydrodynamic to the magnetic radius, rd=rm, im-\nportant for bioassay applications [22]. Remarkable is the\nsharp increase of \u001earoundrd=rm= 2:5. For larger values\nofrd=rmthe phase lag is nearly constant, i.e. insensitive\nto variations of rd.8\nIV. CONCLUSIONS\nOur numerical calculations based on the kinetic equa-\ntions for MNPs dissolved in a viscous liquid show that\nunder the in\ruence of a rotating magnetic \feld a tran-\nsition takes place from a state with magnetic moment\nlocked to the anisotropy axis of the MNP to a state with\nfree rotation of the moment depending on the anisotropy\nenergy. This scenario is expected physically and our\nresults support this picture quantitatively. From these\ninvestigations we can conclude that for MNPs at roomtemperature with magnetic parameters typical for iron\noxides (magnetite) the moment can be considered as be-\ning locked if the magnetic radius rmis larger than about\n12 nm, c.f. Fig.(1). Particles with a larger radius can be\ndescribed within the RDM.\nThe EFM which is based on the FPE for the RDM\nhas been reconsidered and it has been applied to the dy-\nnamics of MNP in rotating magnetic \felds. The analytic\nresults we obtain are shown to be in perfect agreement\nwith numerical solutions of the FPE in the range of pa-\nrameters studied. Therefore these results can be used\nwith success in applications avoiding lengthy calculations\non the basis of the FPE.\n[1] Q. M. Pankhurst, J. Connolly, S. K. Jones, and J. Dob-\nson, J. Phys. D: Appl. Phys., 36, R167 (2003).\n[2] Q. A. Pankhurst, N. K. T. Thanh, S. K. Jones, and J\nDobson, J. Phys. D 42, 224001 (2009).\n[3] Q. Li, Y. Xuan, and J. Wang, Exp. Therm. Fluid Sci.\n30, 109 (2005).\n[4] N. A. Usov, J. Appl. Phys. 107, 123909 (2010).\n[5] K. D. Usadel and C. Usadel, J. Appl. Phys. 118, 234303\n(2015).\n[6] L. Kafrouni and O. Savadogo, O. Prog Biomater (2016)\n5:147.doi:10.1007/s40204-016-0054-6\n[7] W. F. Brown, Phys. Rev. 130, 1677 (1963).\n[8] R. C. O'Handley, Modern magnetic materials: Principles\nand applications , (John Wiley & Sons, New York, 2000).\n[9] M. I. Shliomis, Sov. Phys. Usp. 17, 153 (1974).\n[10] M. I. Shliomis and V. I. Stepanov, Adv. Chem. Phys. 87,\n1 (1994).\n[11] W. T. Co\u000bey and Yu. P. Kalmykov, J. Magn. Magn.\nMater. 164, 133 (1996).\n[12] R. E. Rosensweig, J. Magn. Magn. Mater. 252, 370\n(2002).\n[13] M- Lahonian and A. A. Golneshan, IEEE Trans. Nano.\nBiosci. 10, 262 (2011).\n[14] G. Nedelcu, Dig. J. Nanomater. Bios., 3, 103 (2008).\n[15] N. A. Usov and B. Ya. Liubimov, J. Appl. Phys. 112,\n023901 (2012).\n[16] T. V. Lyutyy, S.I. Denisov, V.V. Reva, and Yu. S.\nBystrik, Phys. Rev, E 92, 042312 (2015).\n[17] J. Connolly and T.G. St Pierre, J. Magn. Magn. Mater.\n225, 156 (2001).\n[18] S. H. Chung, A. Ho\u000bmann, S. D. Bader, C. Liu, B. Kay,\nL. Makowski, and L. Chen, Appl. Phys. Lett. 85, 2971\n(2004).[19] K. Enpuku, T. Tanaka, T. Matsuda, F. Dang, N.\nEnomoto, J. Hojo, K. Yoshinaga, F. Ludwig, F. Gha\u000bari,\nE. Heim, and M. Schilling, J. Appl. Phys. 102, 054901\n(2007).\n[20] D. Eberbeck, C. Bergemann, F. Wiekhorst, U. Steinho\u000b,\nand L. Trahms, Nanobiotechnology 6, 4 (2008).\n[21] S. Schrittwieser, J. Schotter, T. Maier, R. Bruck, P.\nMuellner, N. Kataeva, K. Soulantika, F. Ludwig, A.\nHuetten, and H. Brueckl, Procedia Eng. 5, 1107 (2010).\n[22] J. Dieckho\u000b, S. Schrittwieser, J. Schotter, H. Remmer,\nM. Schilling, and F. Ludwig J. Magn. Magn. Mater. 380,\n205 (2015).\n[23] J. Dieckho\u000b, M. Schilling, and F. Ludwig, Appl. Phys.\nLett. 99, 112501 (2011).\n[24] M. I. Shliomis, in Ferro\ruids: Magnetically Controllable\nFluids and Their Applications , edited by S. Odenbach\n(Springer, New York, 2002), p.85.\n[25] M. A. Martsenyuk, Yu. L. Raikher, and M. I. Shliomis,\nSov. Phys. -JETP 38, 413 (1974).\n[26] Yu. L. Raikher and M. I Shliomis. Adv. Chem. Phys. 87,\n595 (1994).\n[27] J. Dieckho\u000b, D. Eberbeck, M. Schilling, and F. Ludwig,\nJ. Appl. Phys. 119, 043903 (2016).\n[28] T. Yoshida, K. Enpuku, J. Dieckho\u000b, M. Schilling, and\nF. Ludwig, J. Appl. Phys. 111, 053901 (2012).\n[29] J. L. Garcia { Palacios and F. Lazaro, Phys. Rev. B 58,\n14937 (1998).\n[30] K. D. Usadel, Phys. Rev. B 73, 212405 (2006).\n[31] U. Nowak, Ann. Rev. Comp. Phys, 9, 105 (2001).\n[32] W. T. Co\u000bey, Yu. P. Kalmykov, and J. T. Waldron, The\nLangevin Equation , 2nd ed. (World Scienti\fc, Singapore,\n2004).\n[33] D. A. Garanin, Phys. Rev. B 55, 3050 (1997).\n[34] K. D. Usadel, Phys. Rev. B 87, 174431 (2013)."    },    {        "title": "1106.2359v1.Quantum_Dynamics_of_a_Nanomagnet_driven_by_Spin_Polarized_Current.pdf",        "content": "arXiv:1106.2359v1  [cond-mat.mes-hall]  13 Jun 2011Quantum Dynamics of a Nanomagnet driven by Spin-Polarized C urrent\nYong Wang and L.J. Sham∗\nCenter for Advanced Nanoscience, Department of Physics,\nUniversity of California, San Diego, La Jolla, California 9 2093-0319, USA\nA quantumtheory of magnetization dynamics of ananomagnet a s asequence ofscatterings ofeach\nelectron spin with the macrospin state of the magnetization results in each encounter a probability\ndistribution of the magnetization recoil state associated with each outgoing state of the electron.\nThe quantum trajectory of the magnetization contains the av erage motion tending in the large spin\nlimit to thesemi-classical results of spin transfer torque and the fluctuations givingrise toa quantum\nmagnetization noise and an additional noise traceable to th e current noise.\nPACS numbers: 75.75.Jn,72.25.Mk,05.10.Gg\nIntroduction The spin transfer torque (STT) [1, 2] is\nboth a fundamental process in magnetization dynamics\nand important to spintronics in memory and informa-\ntion processing integration. The magnetization dynam-\nics driven by spin-polarized current via STT has made\ngreat progress by experiments and by a “semiclassical”\ntheory of treating the magnetization dynamics classically\nand the polarized current transport and the spin waves\nquantum mechanically (see a series of reviews introduced\nby Ref. [3]).\nRecent development in fast time-resolved measure-\nments [4–6] and coherent control[7] have made possi-\nble studies of magnetization dynamics and fluctuations\nnot masked by the inhomogeneity effects of the measure-\nment and the prospect of precision control. These recent\nexperiments including Ref. [8], by showing the stochas-\ntic nature of the magnetization dynamics at short time\nintervals, highlight three important aspects of the sub-\nject. First is the stochastic motion which may lead to\nan understanding of noise, a fundamental issue in mag-\nnetization dynamics [9]. Second, in interconnected sys-\ntems at the nanoscale, the particulate nature of the cur-\nrent electrons imprints shot noise on the magnetization\nthrough the quantum scattering process [10, 11]. Third,\nthe thermal fluctuations are important because, for ex-\nample, the thermal noise power is comparable to the mi-\ncrowave power generated by the magnetization preces-\nsion [12]. The measurements seem to be satisfactorily\ntreated by the semiclassical theory including micromag-\nnetics. However, the common treatment of angular mo-\nmentum transfer by the magnet as an immovable scat-\ntering potential is shown below by order of magnitude\nestimate to be inadequate for accurate computation of\nmean displacement and fluctuations in a nanomagnet.\nThe origin of the stochastic motion arises from the as-\nsumption ofrandomnessin either the currentorthe mag-\nnetization. The spin wavesseemgraftedonratherarising\nnaturally out of the vibration of the spins which consti-\ntute the moving magnetization. These points motivate\nus to question if inclusion of the quantum dynamics of\nthe magnetization may not only provide more accurate\nconnection between the current electrons and the magne-tization but alsoprovidequalitative understanding ofthe\nstochastic dynamics as well as bringing out the prospect\nof coherent motion for precision control. By a quantum\ntheory of the nanomagnet which gives its mean dynamics\nand fluctuations through scattering between the current\nelectrons and the movable magnet, we hope to illustrate\nthe fundamental aspect of our approach to STT. The de-\nvelopment of quantum optics after the laser operation\nwas understood by semiclassical theory provides perhaps\nan optimistic historical guide for the development of co-\nherent magnetization dynamics after the successes of the\nsemiclassical STT theory.\nConditional Scattering and Stochastic Schr¨ odinger\nEquation We take the magnetization of a single-domain\nnanomagnet to be represented by a coherent state of\nthe local spins [13] and view its dynamics in the spin-\npolarized current as a sequence of scattering by individ-\nual electrons. Since each scattering produces a number\nof outgoing macro-spin states, the serial scatterings cre-\nate a Monte Carlo trajectory of the macro-spin states\nwhich is governed by a stochastic Sch¨ odinger equation\n[14]. Repeated solutions form an ensemble of quantum\ntrajectories for the magnetization from which the mean\ntrajectory and the fluctuations can be obtained. Since\nthe outcomes of the macro-spin states are paired with\nthe scattered spin states of the current electron to form\nan entangled state, the scattered electron state may be\nviewed as conditioned on the corresponding macro-spin\nstate. After the first scattering, the electron recedes and\nthe magnetization is left in a reduced density matrix of\ntheoutgoingmacro-spinstates. Thesecondelectronfrom\nthe current may be viewed as the recipient of one of\nthe macro-spin states. This step is equivalent to mea-\nsurement or decoherence[15], and the magnetization un-\ndergoes a Brownian motion and not a quantum random\nwalk. The magnetization noise then has three sources,\nan intrinsic one due to the uncertainty of the quantum\nstate of the magnetization, two extrinsic ones due to the\nentangled state after each scattering and due to the par-\nticulate nature of each colliding electron with the coher-\nent macro-spin state. The last forms a channel for the\nmutual effects of the current noise and the magnetization2\nnoise.\nWe illustrate the quantum effects with a simple scat-\ntering model of the nanomagnet by a regular sequence\nof electrons and then layer the complexities of the real\ncurrent[16]. We study here the dynamics of the magneti-\nzation as an angular momentum coherent state |J,Θ,Φ/an}bracketri}ht\n[13]. Theeffect ofthe spinwavesassmalldeviationsfrom\nthecoherentstateisbeinginvestigated. Jisthequantum\nnumber of the total spin ˆJof the localized d-electrons in\nthe nanomagnet and is of the order of 106for typical\nnanomagnets. The angles Θ and Φ characterize the di-\nrection of the total spin of the nanomagnet. The current\nelectron has the incoming state |k,↑/an}bracketri}ht, its wave vector k\nimpinging normally on a ferromagnet film in the x= 0\nplane and its spin along the z-axis. The scattering po-\ntential,δ(x)[λ0ˆJ0+λˆ s·ˆJ], is a potential of fixed position\nbut includes a spin independent term indicated by the\nidentity operator ˆJ0of the total spin state space and the\nexchange term between the current electron spin and the\ntotal local spin. The δ-potential models the FM film of\nthickness of several nanometer, and the parameters λ0\nandλare the interaction strength between the current\nelectron and one d-electron, estimated from two simple\nexchange-split potentials.\nAfter a scattering event by one current electron, the\ntotal state of the system of the nanomagnet and the cur-\nrent electron is entangled[16]. In the Berger [2] basis\nstates{|±k,±/an}bracketri}ht}for the current electron, in which the z\naxis is rotated in the z-Jplane to the magnetization di-\nrection before the scattering, the conditional macro-spin\nstates associated with the outgoing single electron states\nare coherent states with an error of the order 1 /J2,\n|Ψ/an}bracketri}ht=4/summationdisplay\ni=1gi|J,Θi,Φi/an}bracketri}ht|ki,σi/an}bracketri}ht, (1)\nwhere Span( |ki,σi/an}bracketri}ht) = (|−k,+/an}bracketri}ht,|k,+/an}bracketri}ht,|−k,−/an}bracketri}ht,|k,−/an}bracketri}ht).\nThe Berger basis is therefore preferable to the original\nspin basis of the current electron of spin up and down\nalong the z-axis, and only the incoming electron state\nis shown in the original basis. The Berger basis may\nbe viewed as constituting a measurement basis for the\ncurrent electron with the outcomes associated with the\nprobabilities Gi=|gi|2,i= 1—4. A conditional state\nof the macro-spin state of the magnetization is deter-\nmined by the scattered current electron state, as shown\nbytheFeynmandiagramsin Fig.1. Notethat, afterscat-\ntering, the conditional macro-spin states associated with\nthe electron states |±k,−/an}bracketri}htsuffer only a phase changebe-\ncause the flip-flop terms /hatwides+/hatwideJ−+/hatwides−/hatwideJ+in the Heisenberg\nexchange interaction do not connect the state |k,+/an}bracketri}ht|J,J/an}bracketri}ht\nto|k,−/an}bracketri}ht|J,m/an}bracketri}htfor anym < J.\nThe evolution of the quantum state of magnetization\nfrom|J,Θ,Φ/an}bracketri}htto|J,Θi,Φi/an}bracketri}htstochastically by one spin-\npolarized current electron may be represented by a ro-\ntation,|J,Θ′,Φ′/an}bracketri}ht=e−iˆ n·ˆJϑ|J,Θ,Φ/an}bracketri}ht, with the rotationG1|J,Ĭr,ULj |-k, +Lj\n|J,Ĭ,Lj | k, LjG2| k, +Lj|J,Ĭt,WLj\n| k, Lj|J,Ĭ,LjG3|-k, -Lj|J,Ĭ,Lj\n|J,Ĭ,Lj | k, LjG4| k, -Lj|J,Ĭ,Lj\n| k, Lj|J,Ĭ,Lj\nFIG. 1. Four possible states of the nanomagnet in the Berger\nbasis{| ±k,±/an}bracketri}ht}after scattering. The straight lines repre-\nsent the spin-polarized electron and the wavy lines represe nt\nthe nanomagnet states. The macro-spin state |J,Θi,Φi/an}bracketri}htwith\nsufficesi=1—4 define the association with the electron spin\nstates in the Berger basis. Giare the conditional probabili-\nties.\naxisˆ n= (−sinϕ,cosϕ,0) and the rotation angle ϑas\nthe random variables with probability outcomes deter-\nmined by Eq. (1). Since the angle ϑis of the order of\n1/Jand the time interval τbetween two successive elec-\ntronsis smallcomparedwith the time scaleofmagnetiza-\ntion dynamics, the stochastic rotationofthe nanomagnet\nmay be treated as a continuous rotation governed by the\nstochastic Schr¨ odinger equation,\ni∂\n∂t|J,Θ,Φ/an}bracketri}ht=ωˆ n·ˆJ|J,Θ,Φ/an}bracketri}ht, (2)\nunder the effective magnetic field due to the polarized\ncurrent electrons, producing precession frequency, ω≡\nϑ/τabout the axis ˆ n.\nDynamics and Fluctuation of Magnetization, Electric\nCurrent and Current Noise To study the switching be-\nhavior of nanomagnet at zero magnetic field, we com-\nputed anensemble of500runs ofMonte Carlosteps, each\nwith sequential scatterings by Ne= 1.5×107electrons\nwith the macro-spin J= 106. To allow for the three di-\nmension nature of the electron wavevectordistribution in\na Fermi sphere of radius kFin the normal metal model,\nthe normal component to the interface kobeys the dis-\ntribution function f(k) = 2k/k2\nFfork∈[0,kF][16]. In\nour simulation, the wavevector kof each injected elec-\ntron is generated randomly according to f(k). The scat-\ntered state in each step is randomly selected according to\nthe probabilities Gi, associatedwith themacro-spinstate\n|J,Θi,Φi/an}bracketri}htand output electron state |ki,σi/an}bracketri}ht. The effect of\nSTT comes from the random encounter by each incom-\ning electron of a choice of the macro-spin states left by\nthe previous electron, giving rise of a random walk of the\nmagnetization. Fig. 2 shows the direction of the magne-\ntizationJaveraged over the ensemble of trajectories and\nits uncertainties in the Cartesian components ∆ Jα. 500\nruns were found to be sufficient to stabilize the ratio of\nthe uncertainties to the average.\nFig. 2(a) shows that the switching motion also con-\ntains a precession about the spin polarization direction\nof the injected electrons, which reproduced the semiclas-\nsical results of being driven by M×SandM×(M×S)\nfor the magnetization Mand current spin polarization S.3\n/s45/s49/s46/s48\n/s45/s48/s46/s53\n/s48/s46/s48\n/s48/s46/s53\n/s49/s46/s48/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s46/s48/s48/s48\n/s48/s46/s48/s48/s53\n/s48/s46/s48/s49/s48/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s49/s48\n/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s49/s48/s66/s67/s74\n/s90\n/s74/s89/s74\n/s88/s32/s40/s97/s41\n/s65/s66\n/s67\n/s65/s40/s98/s41\n/s32\n/s32\n/s32\n/s74\n/s90\n/s74/s89\n/s74\n/s88/s32\nFIG. 2. (color online). Time evolution of magnetization and\nits noise. (a) The average trajectory of the macro-spin stat e\nunder STT and its projections onto the Cartesian planes.\nThe Cartesian components Jα,α=x,y,zare normalized in\nunits ofJ. (b) The time evolution of the uncertainties of the\nmacro-spin componentsandits projections ontotheCartesi an\nplanes. ∆ Jαare normalized in units of J. The current elec-\ntron number is Ne= 1.5×107and the macro-spin J= 106.\nThe initial orientation is (Θ 0,Φ0) = (3,0.5) rad. The elec-\ntron wave vector obey the probability function f(k) = 2k/k2\nF\nfork∈[0,kF] withkF= 13.6 nm−1, and the spin polariza-\ntion vector is taken S= (0,0,1).λ0andλare determined\nby the spin-dependent potential ∆ += 1.3 V, ∆ −= 0.1 V,\nand layer thickness d= 3 nm. The switching time between\npointsAandCis estimated as 1 .3 ns for bias V= 1 mV and\ncross-section area A= 104nm2, by Eq. (3).\nIt is also found that the number of electrons needed for\nswitching is of the order of J. For typical experiments\n[17], the electric current is of the order of mA and the\nswitching time is of the order of ns, both consistent with\nthe electron number Newe used. Since the input pa-\nrameters to our model, namely the Heisenberg exchange\nenergy and the current, are of the order of the physical\nproperties of real systems, we see the ball-park agree-\nment of the switching time as an encouraging sign for the\nquantum approach so that the noise effects from the the-\nory may be worth being included with the thermal noise\nin interpretation of experimental measurements. With-\nout the thermal noise, the fluctuations ∆ Jα, shown in\nFig. 2(b) in time order from point AtoC, first increase\ntill the midpoint Band then decrease to C. The ini-\ntial state of nanomagnet is a pure angular momentum\ncoherent state |J,Θ,Φ/an}bracketri}htwith only intrinsic magnetization\nnoise (∆Jα∼√\nJ). Scattering by current electrons leads\nto stochastic motion which accounts for the rise of the\nextrinsicmagnetizationnoise. As all the trajectoriescon-\nvergeto the state |J,0,0/an}bracketri}ht, the magnetization noise finally\ndecreases to the intrinsic quantum fluctuation.\nThe injected current electron noise can come from ei-\nther its wave vector distribution f(k) or the uncertainty\nof its spin state. To identify their effects on the magne-\ntization dynamics of nanomagnet, we include them sep-\narately in the simulations and compared the case with\nno injected electric current noise. As shown in Fig. 3,\nthe inhomogeneous effect of khas less effect on the time-\nevolution of average value /an}bracketle{tΘ/an}bracketri}htand/an}bracketle{tΦ/an}bracketri}htand their fluc-\ntuations ∆Θ and ∆Φ; while the electron spin noise can/s48 /s49 /s50 /s51/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s49/s48/s48/s46/s48/s49/s53/s48 /s49 /s50 /s51/s48/s49/s50/s51/s52\n/s48 /s49 /s50 /s51/s48/s50/s52/s54/s56\n/s48 /s49 /s50 /s51/s48/s46/s48/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s49/s48/s48/s46/s48/s49/s53/s32/s32\n/s32/s32/s115/s105/s110/s103/s108/s101/s32/s107/s32/s32/s80/s32/s61/s32/s49/s46/s48\n/s32/s32/s102/s40/s107/s41/s44/s32/s32/s32/s32/s32/s32/s32/s32/s80/s32/s61/s32/s49/s46/s48\n/s32/s32/s115/s105/s110/s103/s108/s101/s32/s107/s44/s32/s80/s32/s61/s32/s48/s46/s53\n/s84/s105/s109/s101/s32/s91/s110/s115/s93/s40/s100/s41/s40/s99/s41/s40/s98/s41/s32/s32/s115/s105/s110/s103/s108/s101/s32/s107/s44/s32/s80/s32/s61/s32/s49/s46/s48\n/s32/s32/s102/s40/s107/s41/s44/s32/s32/s32/s32/s32/s32/s32/s32/s80/s32/s61/s32/s49/s46/s48\n/s32/s32/s115/s105/s110/s103/s108/s101/s32/s107/s44/s32/s80/s32/s61/s32/s48/s46/s53\n/s84/s105/s109/s101/s32/s91/s110/s115/s93/s32\n/s32/s32\n/s40/s97/s41\n/s32/s32\n/s32/s32/s115/s105/s110/s103/s108/s101/s32/s107/s32/s32/s80/s32/s61/s32/s49/s46/s48\n/s32/s32/s102/s40/s107/s41/s44/s32/s32/s32/s32/s32/s32/s32/s32/s80/s32/s61/s32/s49/s46/s48\n/s32/s32/s115/s105/s110/s103/s108/s101/s32/s107/s44/s32/s80/s32/s61/s32/s48/s46/s53\n/s84/s105/s109/s101/s32/s91/s110/s115/s93\n/s32/s32\n/s32/s32/s115/s105/s110/s103/s108/s101/s32/s107/s32/s32/s80/s32/s61/s32/s49/s46/s48\n/s32/s32/s102/s40/s107/s41/s44/s32/s32/s32/s32/s32/s32/s32/s32/s80/s32/s61/s32/s49/s46/s48\n/s32/s32/s115/s105/s110/s103/s108/s101/s32/s107/s44/s32/s80/s32/s61/s32/s48/s46/s53\n/s84/s105/s109/s101/s32/s91/s110/s115/s93\nFIG. 3. (color online). Effects of injected current electron\nnoise on the magnetization dynamics of nanomagnet. (a)\nTime evolution of the average value of Θ. (b) Time evolution\nof the average value of Φ. (c) Time evolution of the fluctua-\ntion of Θ. (d) Time evolution of the fluctuation of Φ. Three\ncases are considered here: the fully polarized electrons wi th a\nsingle wave vector 2 kF/3 (solid line); the fully polarized elec-\ntrons obey wave vector distribution function f(k) = 2k/k2\nF\nfork∈[0,kF] (dash line); partially polarized electrons with\na single wave vector 2 kF/3 (dotted line). The spin polar-\nized vector is S= (0,0,P). The current electron number is\nNe= 3.0×107, and the other simulation parameters are the\nsame as in Fig. 2.\nchange the magnetization dynamics significantly. The\nmagnetization switch time for spin polarization vector\nS= (0,0,0.5) is about twice of that for the fully-\npolarized electrons. The magnetization noise is also ob-\nviously enhanced, although its main contribution is still\nfrom the scattering process shown in Fig. 1.\nThe electric current Iand current shot noise Sin a\nnanomagnetic junction are given as [18]\nI=e2\n2π/planckover2pi1VNc/summationdisplay\nn=1Tn, S= 2ee2\n2π/planckover2pi1VNc/summationdisplay\nn=1Tn(1−Tn),(3)\nwhereTnis the transmission probability of the electron\nwith mixed spin states reduced from the entangled state\nof Eq. (1). Ncis the number of transport channels, esti-\nmated as Nc≃ Ak2\nF. The channel sums can be replaced,\nrespectively, by the ensemble average Ncδµand variance\nNc(∆δµ)2of the current electrons, where δµis a random\nvariable given the value 1 or 0 for a transmitted or re-\nflected electron. By Eq. (3), the time interval τbetween\ntwo successive injected electrons is given in terms of the\ncurrent as τ= 2π/planckover2pi1/NceV. Then the number of scatter-\ningeventsinthe simulationscanbe convertedtothe time\nscaleofmagnetizationswitch. Fig.4showsthecalculated\nelectroncurrent Iandcurrentnoise Sassociatedwiththe\nmagnetization dynamics in Fig. 3. The electric current\ndecreases during the magnetization switch, because the\nelectrons are scattered by spin-dependent potential bar-4\n/s48 /s49 /s50 /s51/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s48 /s49 /s50 /s51/s48/s46/s48/s53/s46/s48/s120/s49/s48/s45/s53/s49/s46/s48/s120/s49/s48/s45/s52/s49/s46/s53/s120/s49/s48/s45/s52/s50/s46/s48/s120/s49/s48/s45/s52\n/s115/s105/s110/s103/s108/s101/s32/s107/s44/s32/s80/s61/s48/s46/s53/s32/s115/s105/s110/s103/s108/s101/s32/s107/s44/s32/s80/s61/s49/s46/s48\n/s32/s102/s40/s107/s41/s44/s32/s32/s32/s32/s32/s32/s32/s32/s80/s61/s49/s46/s48\n/s32/s32/s67/s117/s114/s114/s101/s110/s116/s32/s40/s109/s65/s41\n/s84/s105/s109/s101/s32/s40/s110/s115/s41/s40/s97/s41\n/s115/s105/s110/s103/s108/s101/s32/s107/s44/s32/s80/s61/s48/s46/s53/s40/s98/s41\n/s32/s32/s83/s32/s40/s110/s65/s50\n/s47/s72/s122/s41\n/s84/s105/s109/s101/s32/s40/s110/s115/s41/s32/s115/s105/s110/s103/s108/s101/s32/s107/s44/s32/s80/s61/s49/s46/s48\n/s32/s102/s40/s107/s41/s44/s32/s32/s32/s32/s32/s32/s32/s32/s80/s61/s49/s46/s48/s32\nFIG. 4. (color online). Output electric current and current\nnoise. (a) Current during the switching of magnetization in\nFig. 3. (b) Current shot noise during the switching of mag-\nnetization in Fig. 3.\nriers ofthe nanomagnet, ∆ −= 0.1 V and ∆ += 1.3 V, at\nthe startingpoint andfinal point respectively. The calcu-\nlated current shot noise Srise and drop in sequence, with\nthe maximum amplitude corresponding to the transmis-\nsion probability T= 0.5 for electrons. The calculated\ncurrent amplitude and switching time are comparable to\nthe experiment results [17]. The amplitude of shot noise\nof the order of nA2/Hz is also consistent with the noise\nmeasurement[19]. We found that the change of current\nand current noise during the magnetization switch de-\ncrease with the reduction of current electron spin polar-\nization.\nThe quantum noise generated in magnetization and\nelectric current due to the scattering can be veiled by\nthe thermal noise in the system at high temperature.\nThe current shot noise will become important when\neV > κT[18, 19], where κis the Boltzmann constant\nandTis the temperature. For the bias V= 1 mV,\nthis gives T<11 K, and the shot noise was ob-\nserved experimentally at 12 K[19]. The magnetization\nthermal noise can be characterized by the correlator\n/an}bracketle{th(t)h(t′)/an}bracketri}ht= (2α0κT/γMV)δ(t−t′) for the random ef-\nfective field h(t)[10, 11, 20], where α0is the Gilbert\ndamping coefficient, γis the gyromagnetic ratio, Mis\nthe magnetization, and Vis the volume for the nano-\nmagnet. Then the thermal fluctuation for the rotation\nangleϑ(t) =γ/integraltextt+τ\nth(t′)dt′in the time interval τwill\nbe/an}bracketle{tδϑ2/an}bracketri}httherm= 2α0κTτ/J/planckover2pi1, here we used the relation\nγJ/planckover2pi1=MV. On the other hand, the quantum noise\nbased on Eq. (2) gives /an}bracketle{tδϑ2/an}bracketri}htquant∼ O(1/J2). Thus the\ncharacteristic temperature that the quantum noise will\nbe comparable to the thermal noise can be roughly esti-\nmated by the relation\nα0κT ∼/planckover2pi1ωs, (4)\nwhereωs= 1/Jτis a characteristic frequency related to\nthe switch time of the nanomagnet. For the nanomag-\nnet in Ref. [17], α0∼0.025, together with our simula-\ntion results above Jτ∼0.1 ns, the characteristic tem-\nperature Tis about 3 K. Relation (4) shows that the\nquantum magnetization noise will be more important forsmallerGilbertdampingcoefficient, smallernanomagnet,\nor shorter magnetization switch time.\nConclusion The quantum trajectory method we have\ndeveloped not only reproduced the semiclassical results\nof STT, but also revealed the generation of quantum\nnoiseand the noisetransferbetween the nanomagnetand\ncurrent. More quantum phenomena in the STT-related\nphysics are anticipated in further studies based on this\nquantum picture.\nAcknowledgement We acknowledge support of this\nwork by the U. S. Army Research Office under contract\nnumber ARO-MURI W911NF-08-2-0032 and thank W.\nYang, Y.J. Zhang, H. Suhl, and I.N. Krivorotov for valu-\nable discussions.\n∗lsham@ucsd.edu\n[1] J.C. Slonczewski, J. Magn. Magn. Mater. 159, L1(1996).\n[2] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[3] D.C. Ralph and M.D. Stiles, J. Magn. Magn. Mater. 320,\n1190 (2008).\n[4] T. Devolder, J. Hayakawa, K. Ito, H. Takahashi, S.Ikeda,\nP. Crozat, N. Zerounian, J.-V. Kim, C. Chappert, and H.\nOhno, Phys. Rev. Lett., 100, 057206 (2008).\n[5] H. Tomita, K. Konishi, T. Nozaki, H. Kubota, A.\nFukushima, K. Yakushiji, S. Yuasa, Y. Nakatani, T.\nShinjo, M. Shiraishi, and Y. Suzuki, Applied Physics Ex-\npress1, 061303 (2008).\n[6] Y.-T. Cui, G. Finocchio, C. Wang, J.A. Katine, R.A.\nBuhrman, andD.C.Ralph, Phys.Rev.Lett., 104, 097201\n(2010).\n[7] S. Garzon, L. Ye, R.A. Webb, T.M. Crawford, M. Cov-\nington, and S. Kaka, Phys. Rev. B 78, 180401 (2008).\n[8] X. Cheng, C.T. Boone, J. Zhu, and I.N. Krivorotov,\nPhys. Rev. Lett. 105, 047202 (2010).\n[9] H. Suhl, Relaxation processes in micromagnetics (Oxford\nUniversity Press, 2007).\n[10] J. Foros, A. Brataas, G. E. W. Bauer, and Y.\nTserkovnyak, Phys. Rev. B 79214407 (2009).\n[11] A. L. Chudnovskiy, J. Swiebodzinski, and A. Kamenev,\nPhys. Rev. Lett. 101, 066601 (2008).\n[12] A. Slavin and V. Tiberkevich, IEEE Transactions on\nMagnetics 45, 1875 (2009).\n[13] W.-M. Zhang, D.H. Feng, and R. Gilmore, Rev. Mod.\nPhys.62, 867 (1990).\n[14] H. J. Carmichael, An open systems approach to quantum\noptics(Springer, Berlin, 1993).\n[15] H. M. Wiseman and G. J. Milburn, Quantum Measure-\nment and control (Cambridge University Press, Cam-\nbridge, 2010).\n[16] See supplementary material for details.\n[17] I.N. Krivorotov, N.C. Emley, J.C. Sankey, S.I. Kise-\nlev, D.C. Ralph, and R.A. Buhrman, Science, 307, 228\n(2005).\n[18] Ya.M. Blanter and M. B¨ uttiker, Phys. Rep. 336, 1\n(2000).\n[19] K. Sekiguchi, T. Arakawa, Y. Yamauchi, K. Chida, M.\nYamada, H. Takahashi, D. Chiba, K. Kobayashi, and T.\nOno, Appl. Phys. Lett. 96, 252504 (2010).\n[20] W.F. Brown, Phys. Rev. 130, 1677 (1963)."    },    {        "title": "1912.06422v1.Magnetism_in_Massive_Stars.pdf",        "content": "Magnetism in Massive Stars\nKyle C. Augustson1\n1. AIM, CEA, CNRS, Universit\u0013 e Paris-Saclay, Universit\u0013 e Paris Diderot, Sarbonne Paris Cit\u0013 e, F-91191 Gif-sur-Yvette Cedex, France\nMassive stars are the drivers of star formation and galactic dynamics due to their\nrelatively short lives and explosive demises, thus impacting all of astrophysics.\nSince they are so impactful on their environments, through their winds on the\nmain sequence and their ultimate supernovae, it is crucial to understand how\nthey evolve. Recent photometric observations with space-based platforms such\nas CoRoT, K2, and now TESS have permitted access to their interior dynamics\nthrough asteroseismology, while ground-based spectropolarimetric measurements\nsuch as those of ESPaDOnS have given us a glimpse at their surface magnetic\n\felds. The dynamics of massive stars involve a vast range of scales. Extant\nmethods can either capture the long-term structural evolution or the short-term\ndynamics such as convection, magnetic dynamos, and waves due to computational\ncosts. Thus, many mysteries remain regarding the impact of such dynamics on\nstellar evolution, but they can have strong implications both for how they evolve\nand what they leave behind when they die. Some of these dynamics including\nrotation, tides, and magnetic \felds have been addressed in recent work, which is\nreviewed in this paper.\n1 Introduction\nMassive stars live fast and die young. Because of this fast lifecycle, massive stars\nhave been the primary drivers of galactic evolution and to some degree cosmological\nevolution from the the epoch of reionization and the formation of the \frst stars.\nDuring the main sequence, where stars spend most of their lives burning hydrogen\nin their cores, the strong winds of these hot stars impact their local environment\nand any stellar companions that may have formed nearby. When these stars die in\na brilliant supernova, their angular momentum, magnetic \felds, and heavy-element\nladen ionized ashes are redispersed into the local medium (e.g., Maeder, 2009; Langer,\n2012). This eventually leads to an enrichment of galaxy in elemental abundances\nand triggers new episodes of star formation, although with a modi\fed abundance\ndistribution that impacts the nature of the stars that form. But the precise statistical\nbehavior of this process is an important open question in galactic dynamics and\ncosmological evolution (e.g., Nomoto et al., 2013; Krumholz & Federrath, 2019),\none that future studies can help to address. Moreover, those explosive events, often\nleave behind a remnant: a white dwarf for lower mass stars, a neutron star for\nintermediate mass stars, and black holes for the more massive stars, whereas some\nof the most massive stars may completely disintegrate in a titanic pair-instability\ndriven explosion (e.g., Woosley et al., 2007; Groh et al., 2013). Indeed, one of the\nmysteries of these objects is why only a fraction of these white dwarf and neutron\nstar remnants have extremal surface magnetic \felds, whereas the remaining fraction\nhave comparatively weak magnetic \felds (e.g., Donati & Landstreet, 2009; M osta\net al., 2015; Kaspi & Beloborodov, 2017).\npta.edu.pl/proc/2021oct25/123 PTA Proceedings ?October 25, 2021 ?vol. 123?51arXiv:1912.06422v1  [astro-ph.SR]  13 Dec 2019Kyle C. Augustson\n344 F. Ligni`ereset al.\nFigure 4.(Colour online) This sketch summarizes then e wc o n t e x to fi n t e r m e d i a t e - m a s sm a g -netism set by the recent discoveries of the lower bound of Ap magnetism, the magnetic desertin longitudinal fieldBLand the Vega-like magnetism.been found in 70 % of the A-type Kepler stars, as expected in the presence of starspotsor other magnetic corotating features.4. On the origin of the magnetic dichotomyThe sketch of Fig. 4 illustrates the dichotomy between the Ap/Bp-like and Vega-likemagnetic fields of intermediate-mass stars. One can think of two different ways to explainthis dichotomy : the first one is to assume that the two types of magnetic fields havedifferent properties because they have been generated by two different processes. Anotherpossibility is that the observed fields have a common origin but during the evolution theirmagnetic field distribution split into two distinct families of low and high longitudinalfields.Braithwaite & Cantiello (2013) proposed that Vega-like stars are ”failed fossil” mag-netic stars meaning that their field, produced during star formation, is still decayingthus not truly fossil. Accordingly, the helicity of the initial field configuration must below enough to evolve towards a sub-gauss field amplitude at the age of Vega. By con-trast, the initial helicity of Ap/Bp-like magnetic stars has to be very high to produce in arelatively short time the observed fossil-like Ap/Bp magnetism. Braithwaite & Cantiello(2013) thus argue that two distinct generation mechanisms must be invoked to explainthese very different initial helicities. To account for Ap/Bp stars, they follow Ferrarioet al.(2009) and Tutukov& Fedorova (2010) who assume that these stars result fromthe merging of close binaries, their strong fields being produced by a powerful dynamoduring the merging phase.9CC#%\u000e\u0004\u0004))) 42 3$:58\u0019 \"$8\u00044\"$\u0019\u0004C\u0019$ % \u00019CC#%\u000e\u0004\u00045\": \"$8\u0004\u0006\u0005 \u0006\u0005\u0006\u000b\u00041\u0006\u000b\t\b\r\u0007\u0006\b\u0006\t\u0005\u0005\u0007\t\t\u0005\u0010\")!\u001f\"25\u00195\u00017$\" \u00019CC#%\u000e\u0004\u0004))) 42 3$:58\u0019 \"$8\u00044\"$\u0019 \u0001.0\u0001255$\u0019%%\u000e\u0001\n\u0005 \u0007\b\b \u0006\r\u0007 \u0006\u0005\u0006\u0002\u0001\"!\u0001\u0005\f\u0001/4C\u0001\u0007\u0005\u0006\r\u00012C\u0001\u0006\n\u000e\u0005\n\u000e\u0005\u000b\u0002\u0001%D3\u001e\u00194C\u0001C\"\u0001C9\u0019\u0001,2 3$:58\u0019\u0001,\"$\u0019\u0001C\u0019$ %\u0001\"7\u0001D%\u0019\u0002\u00012(2:\u001f23\u001f\u0019\u00012C\n(a)(b)(c)(d)\nFig. 1: (a) The magnetic desert, no detections of stars with magnetic \felds between about\n1G and 100 G (Ligni\u0012 eres et al., 2014). (b) Distribution of stars with detected magnetic\n\felds by stellar type (Wade et al., 2014). (c) Simulations of relaxed magnetic \felds for\na nearly uniform initial magnetic \feld amplitude (Braithwaite, 2008) and a similar obser-\nvation (Kochukhov et al., 2011). (d) Similar to (c) but for a more centrally concentrated\ninitial condition, with corresponding observation (Kochukhov et al., 2013).\nThroughout the evolution of massive stars, there are processes that can build in-\nternal magnetic \felds and transport angular momentum and chemical species (e.g.,\nMaeder, 2009; Mathis, 2013). The rotation and magnetic \felds of these stars drasti-\ncally impact both their evolution through modi\fed convective, transport processes,\nand mass loss and their environment through the strong winds associated with that\nmass loss and their tidal interactions with any companions (e.g., Ogilvie, 2014; Smith,\n2014). Since these dynamical processes a\u000bect the long-term evolution of such stars,\nthey must be modeled with high \fdelity in order to properly capture their impact\non many other astrophysical processes. Observationally calibrating these models\nis possible given the recent revolution in our knowledge of stellar dynamics pro-\nvided by the seismology of the interiors of the Sun and of stars (with SDO, CoRoT,\nKepler, K2, TESS, and BRITE) and through the ground-based spectropolarimetry\nthat characterizes stellar surface magnetic \felds (ESPaDOnS/CFHT, Narval/TBL,\nHARPSpol/ESO).\nMagnetism Spectropolarimetric campaigns by consortia such as MiMeS (Magnetism\nin Massive Stars) and BOB (B Fields in OB Stars) and LIFE (Large Impact of\nmagnetic Fields on the Evolution of hot stars) have been directed toward measuring\nmagnetic \felds on the surfaces of massive stars, some using Zeeman Doppler imaging\ntechniques. They report that only about 7% of O and B-type stars exhibit large-scale\nsurface magnetic \felds (See Figure 1; e.g., Donati & Landstreet, 2009; Wade et al.,\n2014; Fossati et al., 2015, 2016). As for interior magnetic \felds, strong magnetic\n\felds have potentially been detected deep in the cores of these stars through the\nsuppression of dipolar mixed oscillatory modes (Fuller et al., 2015; Stello et al.,\n2016). Such depressed modes are seen in those stars that had a convective core\nduring the main sequence, suggesting that they were indeed running a convective\ncore dynamo. Additionally, an asteroseismic method for detecting general magnetic\n\feld con\fgurations in the interiors of rapidly rotating stars has been developed for\ngravity waves and applied to Kepler stars to ascertain if the frequency shifts can be\ndetected (Prat et al., 2019), and for perturbative rotation and magnetic \felds for\ngeneral stellar waves in Augustson & Mathis (2018). Hence, the tools are in hand to\nassess data from ongoing and upcoming ground-based and space-based observational\n52?PTA Proceedings ?October 25, 2021 ?vol. 123 pta.edu.pl/proc/2021oct25/123Magnetism in Massive Stars\n(we chose this specific model to reproduce the conditions ofVega, a well-known A-type star). Using the numbers at thelocation in each layer of peak convective energy densityrv2c2,wefind Ra≈102, 103, and 107in the HCZ, HeICZ, andHeIICZ, respectively. The reason for the difference in Rabetween the layers is largely a consequence of the density: atlower density, the mean free path of the photons, which carryboth heat and momentum, is greater, and both thermal andkinetic diffusivity are higher. Indeed, at the photosphere, themean free path is comparable to the scale height. In addition,the scale height used in the numerator in Equation(4)is smallercloser to the surface.It is not well understood how stellar convection should lookat low Rayleigh numbers; there should perhaps be some kind ofviscous, nonturbulent motion, similar to what is observed inlaboratory experiments. Another uncertainty is our limitedinability to predict the temperature gradient and convectivevelocities accurately; we currently use MLT. Although standardin stellar evolution modeling, we see from laboratory experi-ments and numerical simulations, as well as the experience ofglider pilots, that MLT is based on an inaccurate physicalpicture. In reality, rising and fallingfluid parcels move past eachother rather than mixing, surviving over many scale heights;heating and cooling at the boundaries is crucial(see, e.g.,Öpik1950for an attempt to capture the horizontal exchange ofmatter between up- and downstreams in a 1D theory). We usethe standard MLT in our modeling; the numbers shouldtherefore be treated as the result of a dimensional analysis,and their dependence on the mixing length free parameteraMLTshould neither worry nor surprise us. In the literature, a numberof works have attempted to simulate these convective regionsusing multidimensional hydro simulations, although mostly forcool A stars with surface temperatures below 8500 K(see, e.g.,Kupka & Montgomery2002; Trampedach2004; Kochukhovet al.2007; Freytag et al.2012; Kupka & Muthsam2017). ForFigure 2.Normalized radial extension of core, surface, and subsurface convection zones for stars in the mass range 0.9–25M:. The models are extracted during themain sequence when the mass fraction of H at the stellar center is 0.5. The convective regions are associated with the ionization of H, He(HeIand HeII), and iron-peak elements(Fe). The stellar surfacer/R*=1 is defined as the location corresponding to optical depthτ=2/3.\nFigure 3.Number of main-sequence envelope convection regions in starsbetween 1.5 and 15M:on the HR diagram. Different colors show the numberof coexisting convective regions present in the stellar outer layers. The type ofconvection zones is annotated on the plot. The black outline shows modelswhere convection is present at the stellar surface.4The Astrophysical Journal,883:106(13pp), 2019 September 20 Cantiello & BraithwaiteM. Cantiello et al.: Sub-surface convection in hot stars 283Vini = 350 km/sVini = 250 km/sVini = 0 km/s1.5 HP\nFig. 4.Convective velocity in the FeCZ as function of radial distancefrom the stellar surface. The dotted line corresponds to a non-rotating20M⊙model atZ=0.02, while the dashed and solid lines refer to thesame model rotating at birth with 250 km s−1and 350 km s−1respec-tively. The values correspond to models having the same effective tem-perature (logT=4.339) and very similar luminosity (logL/L⊙=5.04for the non-rotating model and logL/L⊙=5.03 for the rotating ones).The gray band shows the upper 1.5 pressure scale heights of the FeCZ,which is the region considered for the computation of⟨/v1c⟩,c f .E q .(6).Convective velocities in the He convection zone are much lower than1k ms−1and are not visible in this plot.small scale or large scale clumping in massive star winds, mag-netic fields, and non-radial pulsations could be related to sub-surface convection. For each point, we first briefly discuss thetheoretical motivation, and then the corresponding observationalevidence.4.1. Microturbulence4.1.1. Theoretical considerationsThe convective cells in the upper part of a convection zone exciteacoustic and gravity waves that propagate outward. The genera-tion of sound waves by turbulent motions was first discussed byLighthill(1952)a n de x t e n d e dt oas t r a t i fi e da t m o s p h e r eb yStein(1967)a n dGoldreich & Kumar(1990). In a stratified medium,gravity acts as a restoring force and allows the excitation of grav-ity waves. For both acoustic and gravity waves, the most impor-tant parameter determining the emitted kinetic energy flux is thevelocity of the convective motions. This is why, in the follow-ing, we use the average convective velocity⟨/v1c⟩as the crucialparameter determining the efficiency of sub-surface convection.Goldreich & Kumar(1990)s h o w e dt h a tc o n v e c t i o ne x c i t e sacoustic and gravity waves, resulting in maximum emission forthose waves with horizontal wave vectorkh∼1/HP,cand angularfrequencyω∼/v1c/HP,c,w h e r en o w/v1candHP,care evaluated atthe top of the convective region. They calculated that the amountof convective kinetic energy flux going into acoustic and gravitywaves isFac∼FcM15/2c,(7)andFg∼FcMc,(8)respectively, where we takeFc∼ρc⟨/v1c⟩3andMcis the Machnumber in the upper part of the convective region. Since con-vection in our models is subsonic, gravity waves are expectedEnvelope convective zoneRadiative Layer\nRadiative LayerStellar surfaceClumpsAcoustic and gravity wavesMicroturbulence\nConvective ZoneBuoyant magnetic flux tubes\nFig. 5.Schematic representation of the physical processes connected tosub-surface convection. Acoustic and gravity waves emitted in the con-vective zone travel through the radiative layer and reach the surface, in-ducing density and velocity fluctuations. In this picture microturbulenceand clumping at the base of the wind are a consequence of the presenceof sub-surface convection. Buoyant magnetic flux tubes produced in theconvection zone could rise to the stellar surface.to extract more energy from the convective region than acousticwaves. These gravity waves can then propagate outward, reachthe surface and induce observable density and velocity fluctua-tions (Fig.5).The Brunt-Vaisäla frequency in the radiative layer abovethe FeCZ is about mHz. Molecular viscosity can only dampthe highest frequencies, while wavelengths that will be reso-nant with the scale length of the line forming region shouldnot be affected (see e.g.Lighthill 1967). This is the case forthe gravity waves stochastically excited by convective motions:they can easily propagate through the sub-surface radiative layer,steepening and becoming dissipative only in the region of lineformation.Again, multi-dimensional hydrodynamic simulations wouldbe the best way to compute the energy loss of these waves duringtheir propagation through the radiatively stable envelope abovethe FeCZ, but this is beyond what we can presently do. We can,however, obtain an upper limit to the expected velocity ampli-tudes at the stellar surface, where we only consider the energytransport through gravity waves. The kinetic energy per unit vol-ume associated with the surface velocity fluctuationsEsmustbe comparable to or lower than the kinetic energy density as-sociated with the waves near thes u b - s u r f a c ec o n v e c t i o nz o n e ,Eg∼Mcρc⟨/v1c⟩2,o rEg\nEs∼Mc/parenleftBiggρc\nρs/parenrightBigg/parenleftBigg⟨/v1c⟩\n/v1s/parenrightBigg2≥1,(9)whereρcis the density at the top of the convective region andρsis the surface density, and/v1sis the surface velocity amplitude. Inthis ratio we only consider energy density since the volume ofthe line forming region is comparable to the volume of the upperpart of the convective zone. Therefore, we expect/v1s≤⟨/v1c⟩/radicalbigg\nMcρc\nρs·(10)In our models with well developed FeCZs,/radicalbig\nMcρc/ρs≃1( o r -der of magnitude), and thus/v1sand⟨/v1c⟩should be on the sameorder of magnitude. It is difficult to estimate the typical corre-lation length of the induced velocity field at the stellar surface,but a plausible assumption is that it is about one photosphericpressure scale height,HP,s,g i v e nt h ep r o x i m i t yo ft h eF e C Zt o\nFig. 2: Left: Normalized radial extent of the core, surface, and subsurface convective zones\nfor stars between 0.9 and 25 M\f, corresponding to the main sequence structure of these\nstars. The convective regions are associated with the ionization of H, HeI, HeII, and the\niron group elements Fe. The stellar surface r=R\u0003is de\fned as the location where the\noptical depth \u001c= 2=3 (Cantiello & Braithwaite, 2019). Right: A sketch of the iron-bump\nconvection zone, showing a local box in the spherical domain with a portion of the inner\nradiative zone, FeCZ, and outer radiative photosphere where waves are excited and observed\nas macroturbulence (Cantiello et al., 2009).\ncampaigns for the internal structure and magnetic \felds of stars. However, both\nthe processes that lead to strong angular momentum transport and to convective\ndynamos remain di\u000ecult to universally parameterize so as to explain the observed\nproperties and the secular evolution of massive stars.\nConvection Throughout the evolution of massive stars, convection has profound ef-\nfects on both their stellar structure and on what we observe at the surface. In\nmain-sequence massive stars, the photosphere is in a stably-strati\fed region where\nradiation dominates the heat transport and convective motions are absent. Yet ob-\nservations show signi\fcant motions of unknown origin called macroturbulence at the\nstellar photosphere, with typically supersonic velocities of 20 \u000060 km s\u00001(Sundqvist\net al., 2013). A possible source of it may be a detached convection zone located well\nbelow the photosphere, where iron has a local maximum in its opacity (e.g., Cantiello\n& Braithwaite, 2011, 2019; Nagayama et al., 2019). These iron-bump convection\nzones host nearly sonic compressive convection, driving waves in the overlying re-\ngion (see Figure 2). However, there is a curious targe called \\Dash-2.\" This star is\nan outlier among the observed massive stars with an observed macroturbulence of\nonly 2:2 km s\u00001. It also has the strongest observed surface magnetic \feld of these\nstars, with a surface \feld of approximately 20 kG, versus typical \feld strengths of\n1 kG (Sundqvist et al., 2013). Dash-2's surface magnetic \felds are strong enough\nthat they rival the thermodynamic pressure in the iron-bump convection zone, po-\ntentially quenching the convection and thus the surface waves in Dash-2. In the\nother stars of Sundqvist et al. (2013), the \felds are too weak relative to the thermo-\ndynamic pressure. Hence, one puzzle to solve is the origin of macroturbulence and\nits link to near-surface convection and the in\ruence of magnetism. The iron-bump\nconvection zone is however only one of the convective regions, the deeper convec-\ntion zone and the seat of a global-scale dynamo is the convective core. This core\nconvection will drive internal waves and interact with the fossil magnetic \feld (e.g.,\nFeatherstone et al., 2009; Augustson et al., 2016). Such dynamo action will have\npta.edu.pl/proc/2021oct25/123 PTA Proceedings ?October 25, 2021 ?vol. 123?53Kyle C. Augustson\nFig. 3: (a) A Dedalus simulation of fully compressive convection below a stable region,\nreminiscent of both a convective core and the iron-bump convection zone, showing the\nconvective region, penetration into the stable region, and the excited waves in the stable\nregion as rendered in entropy \ructuations with cool regions in dark tones and warm regions\nin light tones (Courtesy of B. Brown). (b) Radial velocity of waves excited by a convective\ncore in a 3D simulation of a 15 M\fstar, showing a displacement proportional to the wave\namplitude (Andr\u0013 e, 2019). (c) An equatorial cut through the computational domain of the\n15M\fstar, showing the \fltered extraction of waves at two frequencies (Andr\u0013 e, 2019).\nimpacts later in the evolution of the star as the internal structure of the star freezes\nthe dynamo-generated \feld into a larger stable region, adding to the extant fossil\n\feld there.\nConvective Penetration Convective \rows cause mixing not only in regions of supera-\ndiabatic temperature gradients but in neighboring subadiabatic regions as well (see\nFigure 3(a)), as motions from the convective region contain su\u000ecient inertia to ex-\ntend into those regions before being buoyantly braked or turbulently eroded (e.g.,\nAugustson & Mathis, 2019; Korre et al., 2019). Thus, convective penetration and\nturbulence softens the transition between convectively stable and unstable regions,\nwith the consequence being that the di\u000berential rotation, opacity, compositional\nand thermodynamic gradients are modi\fed (e.g., Augustson et al., 2013; Brun et al.,\n2017; Pratt et al., 2017), while compositional gradients can drive further mixing\n(e.g., Garaud, 2018; Sengupta & Garaud, 2018). Such processes have an asteroseis-\nmic signature as has been observed in massive stars as well as lower mass stars (e.g.,\nAerts et al., 2003; Neiner et al., 2012, 2013; Moravveji et al., 2016; Pedersen et al.,\n2018). Indeed, the waves shown in Figure 3(b) and (c) depict the self-consistent\namplitude of waves excited by convection in the core. In stars with a convective\ncore, convective penetration can lead to a greater amount of time spent on stable\n54?PTA Proceedings ?October 25, 2021 ?vol. 123 pta.edu.pl/proc/2021oct25/123Magnetism in Massive Stars\nA&A 560, A29 (2013)\nFig. 15.Comparison of the cumulative distributions of projected rota-tional velocities of our work (VFTS-O sample – purple),Penny & Gies(blue),Huang & Gies(green), and the VFTS-B sample ofDufton et al.(2013) (red).The KP tests indicate that thePenny & Giesand VFTS B-stardistributions are statistically di↵erent, with a confidence levelbetter than 1%, while theHuang & Giesdistribution marginallyagrees with our O-star distribution (p⇠11%). ThatPenny &Giesdo not correct for macro-turbulence is a straightforwardexplanation for the absence of slow rotators in their sample. Theother three distributions agree well with respect to the fractionof extremely slow rotators. The fraction of VFTS O- and B-starsbelow our3esiniresolution limit (see Sect.3.6), for instance, isroughly similar.The distribution of3esiniof thePenny & Giessample peaksat the same projected rotational velocity as in our distribution.The lack of stars spinning faster than 300 km s\u00001in their sampleis intriguing, but may result from a selection e↵ect. Indeed theFUSE archives may not be representative of the population offast rotators in the LMC, as individual observing programs mayhave focused on stars most suitable for their respective scienceaims, possibly excluding fast rotators as these are notoriouslydi\u0000cult to analyze.The similarities between the O-star distribution in 30 Dorand the distribution of late-O and early-B Galactic stars ofHuang & Giessuggests a limited influence of metallicity. This isconsistent with our expectation that stellar winds do not play asignificant role in shaping the rotational velocity distributions inboth samples, because they are dominated by stars less massivethan 40M\u0000.The di↵erences with the VFTS B-star sample are strikingand lack a straightforward explanation. The B-type stars show abimodal population of very slow rotators and fast rotators, withfew stars rotating at rates that are typical of the low-velocity peakseen in the VFTS O-type stars. We return to this issue in Sect.5.4.5. Analytical representation of the3edistributionThe size of our sample is large enough to investigate the dis-tribution of intrinsic rotational velocities. By assuming that therotation axes are randomly distributed, we infer the probabilitydensity function of the rotational velocity distributionP(3e) fromFig. 16.Observed3esiniand Lucy-deconvolved3edistributions. Thedot-dashed line shows the estimates, after 4 iterations in the Lucy-deconvolution, of the probability density function for the projected rota-tional velocity distribution. The solid line shows the probability densityfunction of the actual rotational velocities.that of3esini. We adopt the iterative procedure ofLucy(1974),as applied inPaper Xfor the B-type stars in the VFTS, to esti-mate the pdf of the projected rotational velocityP(3esin i) and ofthe corresponding deprojected pdf velocityP(3e). As expected,P(3e) moves to higher velocities compared toP(3esin i) due tothe e↵ect of inclination. At3e\u0000300 km s\u00001,P(3e) presents smallscale fluctuations that probably result from small numbers in theobserved distribution. The two extremely fast rotators at3esini⇠>600 km s\u00001are excluded from the deconvolution for numericalstability reasons.We can approximate the deconvolved rotational velocity dis-tribution well by an analytical function with two components.We use a gamma distribution for the low-velocity peak and anormal distribution to model the high-velocity contribution:P(ve)⇡I\u0000g(ve;↵,\u0000)+INN(ve;µ,\u00002) (1)whereg(x;↵,\u0000)=\u0000↵\u0000(↵)x↵\u00001e\u0000\u0000x,(2)N⇣x;µ,\u00002⌘=1p2⇡\u0000e\u0000(x\u0000µ)2/2\u00002,(3)andI\u0000andINare the relative contributions of both distributionstoP(ve). The best representation, shown in Fig.17, is obtainedforP(ve)⇡0.43g(↵=4.82,\u0000=1/25)+0.67N✓µ=205 km s\u00001,\u00002=⇣190 km s\u00001⌘2◆.(4)The function is normalized to 0.99 to allow for including of anadditional 1% component to represent the two extremely fast ro-tators in our sample. One should note that the reliability of the fitfunction is limited by the sample size at extreme rotational ve-locities. This analytical representation of the intrinsic rotationalvelocity distribution may be valuable in stellar population syn-thesis models that account for rotational velocity distributions.A29, page 12 of16P. L. Dufton et al.: Rotational velocities of B-type stars in the Tarantula Nebula\nFig. 8.Histogram of observed normalisedvesinidistribution binnedto 40 km s−1.A l s os h o w na r et h ee s t i m a t e sa f t e r4i t e r a t i o n so ft h eprobability densities for the projected rotational velocity distribution,P(vesini)(˜φ4in Lucy’s notation – dot-dashed line) and for the rota-tional velocity,P(ve)(ψ4in Lucy’s notation– solid line).smaller scale structure. Also listed in this table is the cumula-tive distribution function (cdf) corresponding to this best esti-mate and because this is truncated atve≤600 km s−1,i td o e sn o treach unity. We recommend that:1. either the range up to the critical velocity is populated2. or that the cdf and estimate ofP(ve)a r er e n o r m a l i s e ddepending on the nature of the application.All the de-convolutions show a double peak in the equa-torial velocity distribution consistent with that observed in theprojected rotational velocity distribution. Approximately onequarter of the sample have a rotational velocity of less than100 km s−1and there appears to be another peak 250–300 km s−1although this is complicated by the varying degree of small scalestructure found in the different de-convolutions.We have attempted to fit the de-convolved distribution us-ing analytical functions. Initially two Gaussian profiles wereadopted but these were found to give a relatively poor fit to ourestimated distribution. Two Maxwellian functions as discussedbyZorec & Royer(2012)w e r ea l s oc o n s i d e r e db u tw o u l dn o thave reproduced the significant value ofP(ve)a sve→0. Wetherefore recommend that either the values listed in Table6or analytical fits appropriate to the specific applications areadopted.The most distinctive feature of thevesinidistribution is itsbi-modal nature, which is also present in the estimation of therotational velocity distribution, illustrated in Fig.8.A l t h o u g hour sample contains approximately 300 targets, it will still besubject to significant random sampling errors due to its finitesize. In Fig.3,t h ep o p u l a t i o no ft h et w o4 0k m s−1bins withthe lowest projected rotational velocities have values of approx-imately 40 corresponding to an estimated standard error of ap-proximately 6. Hence even decreasing the populations of boththese bins by twice this estimated error (note that if these binshad been overpopulated our estimate of the standard error wouldalso be too high) would not remove this bi-modal behaviour.Additionally Kolmogorov-Smirnov tests using uni-modal distri-butions (for example a single Gaussian fit to our estimated rota-tional velocity distribution) lead to very small probabilities thatthey were the parent populations. Hence we conclude that theprojected rotational velocity distribution (and hence the under-lying rotational velocity distribution) is bi-modal, although weTable 6.Estimates of the probability density of the rotational velocity,P(ve), and its cumulative distribution function, cdf.\nveP(ve)×103cdf\n02 . 2 0020 2.40 0.04640 2.45 0.09460 2.55 0.14480 2.25 0.192100 1.40 0.239120 1.00 0.253140 0.75 0.270160 1.05 0.288180 2.40 0.322200 3.00 0.377220 2.85 0.435240 2.75 0.492260 2.90 0.548280 3.20 0.609300 3.30 0.674320 3.20 0.739340 2.50 0.796360 2.00 0.841380 1.80 0.879400 1.50 0.912420 1.10 0.938440 0.75 0.956460 0.70 0.971480 0.50 0.983500 0.21 0.990520 0.09 0.993540 0.08 0.995560 0.06 0.996580 0.04 0.997\naccept that our sample size limits the information on the struc-ture of the two components.In Sect.3.5,w ed i s c u s s e dt h ep r o j e c t e dr o t a t i o n a lv e l o c i -ties in two other LMC samples. Those ofHunter et al.(2008b)showed some evidence for a bimodal distribution but that ofMartayan et al.(2006)a p p e a r e du n i m o d a l .H o w e v e rg i v e nt h edifferences in the samples in terms ofa g e ,f r a c t i o no ffi e l ds t a r sand undetected binaries, we do not consider this discrepancy tobe significant.5. Origin of bi-modal distributionThe bi-modal distribution of rotational velocities implies thatour stellar sample is composed of, at least, two different com-ponents. Which physical processes lead to the existence of thesetwo components? Why do most B-type main sequence stars ro-tate fast, while about a quarter or so rotate slowly? Whatever theanswers to these questions, they are not contained in the standardevolutionary theories of single stars, which predicts only smallchanges of the surface rotational velocity during core hydrogenburning (see, for example,Brott et al. 2011).The two components, which we find, could either corre-spond to different ages, different star formation conditions, oremerge as a consequence of differences in the evolution of stars(e.g., in close binaries). We emphasize again that age differencescan only lead to the difference in the mean rotation rate of thetwo components unless the standard stellar evolution picture iswrong or incomplete. Differences in age and star formation con-ditions could produce stellar components with different spatial orkinematic properties. We inspect our data in this respect in theA109, page 9 of12P. L. Dufton et al.: Rotational velocities of B-type stars in the Tarantula Nebula\nFig. 8.Histogram of observed normalisedvesinidistribution binnedto 40 km s−1.A l s os h o w na r et h ee s t i m a t e sa f t e r4i t e r a t i o n so ft h eprobability densities for the projected rotational velocity distribution,P(vesini)(˜φ4in Lucy’s notation – dot-dashed line) and for the rota-tional velocity,P(ve)(ψ4in Lucy’s notation– solid line).smaller scale structure. Also listed in this table is the cumula-tive distribution function (cdf) corresponding to this best esti-mate and because this is truncated atve≤600 km s−1,i td o e sn o treach unity. We recommend that:1. either the range up to the critical velocity is populated2. or that the cdf and estimate ofP(ve)a r er e n o r m a l i s e ddepending on the nature of the application.All the de-convolutions show a double peak in the equa-torial velocity distribution consistent with that observed in theprojected rotational velocity distribution. Approximately onequarter of the sample have a rotational velocity of less than100 km s−1and there appears to be another peak 250–300 km s−1although this is complicated by the varying degree of small scalestructure found in the different de-convolutions.We have attempted to fit the de-convolved distribution us-ing analytical functions. Initially two Gaussian profiles wereadopted but these were found to give a relatively poor fit to ourestimated distribution. Two Maxwellian functions as discussedbyZorec & Royer(2012)w e r ea l s oc o n s i d e r e db u tw o u l dn o thave reproduced the significant value ofP(ve)a sve→0. Wetherefore recommend that either the values listed in Table6or analytical fits appropriate to the specific applications areadopted.The most distinctive feature of thevesinidistribution is itsbi-modal nature, which is also present in the estimation of therotational velocity distribution, illustrated in Fig.8.A l t h o u g hour sample contains approximately 300 targets, it will still besubject to significant random sampling errors due to its finitesize. In Fig.3,t h ep o p u l a t i o no ft h et w o4 0k m s−1bins withthe lowest projected rotational velocities have values of approx-imately 40 corresponding to an estimated standard error of ap-proximately 6. Hence even decreasing the populations of boththese bins by twice this estimated error (note that if these binshad been overpopulated our estimate of the standard error wouldalso be too high) would not remove this bi-modal behaviour.Additionally Kolmogorov-Smirnov tests using uni-modal distri-butions (for example a single Gaussian fit to our estimated rota-tional velocity distribution) lead to very small probabilities thatthey were the parent populations. Hence we conclude that theprojected rotational velocity distribution (and hence the under-lying rotational velocity distribution) is bi-modal, although weTable 6.Estimates of the probability density of the rotational velocity,P(ve), and its cumulative distribution function, cdf.\nveP(ve)×103cdf\n02 . 2 0020 2.40 0.04640 2.45 0.09460 2.55 0.14480 2.25 0.192100 1.40 0.239120 1.00 0.253140 0.75 0.270160 1.05 0.288180 2.40 0.322200 3.00 0.377220 2.85 0.435240 2.75 0.492260 2.90 0.548280 3.20 0.609300 3.30 0.674320 3.20 0.739340 2.50 0.796360 2.00 0.841380 1.80 0.879400 1.50 0.912420 1.10 0.938440 0.75 0.956460 0.70 0.971480 0.50 0.983500 0.21 0.990520 0.09 0.993540 0.08 0.995560 0.06 0.996580 0.04 0.997\naccept that our sample size limits the information on the struc-ture of the two components.In Sect.3.5,w ed i s c u s s e dt h ep r o j e c t e dr o t a t i o n a lv e l o c i -ties in two other LMC samples. Those ofHunter et al.(2008b)showed some evidence for a bimodal distribution but that ofMartayan et al.(2006)a p p e a r e du n i m o d a l .H o w e v e rg i v e nt h edifferences in the samples in terms ofa g e ,f r a c t i o no ffi e l ds t a r sand undetected binaries, we do not consider this discrepancy tobe significant.5. Origin of bi-modal distributionThe bi-modal distribution of rotational velocities implies thatour stellar sample is composed of, at least, two different com-ponents. Which physical processes lead to the existence of thesetwo components? Why do most B-type main sequence stars ro-tate fast, while about a quarter or so rotate slowly? Whatever theanswers to these questions, they are not contained in the standardevolutionary theories of single stars, which predicts only smallchanges of the surface rotational velocity during core hydrogenburning (see, for example,Brott et al. 2011).The two components, which we find, could either corre-spond to different ages, different star formation conditions, oremerge as a consequence of differences in the evolution of stars(e.g., in close binaries). We emphasize again that age differencescan only lead to the difference in the mean rotation rate of thetwo components unless the standard stellar evolution picture iswrong or incomplete. Differences in age and star formation con-ditions could produce stellar components with different spatial orkinematic properties. We inspect our data in this respect in theA109, page 9 of12The Astrophysical Journal,7 6 4 : 1 6 6( 1 7 p p ) ,2 0 1 3F e b r u a r y2 0De Mink et al.amount of angular momentum exchanged during mass transferor merger, they do not affect the rotation rates of stars afterthese interactions. Although in future work adapting the initialspin distribution to reproduce the properties of suitable observedsamples may be considered, this is beyond our present scope.To compute the distribution of projected rotation ratesvsini,whereiis the inclination angle of the binary system, we assumethat the orientation of the binary orbits is random in space.Unless an observational campaign is designed to detect bina-ries, many companion stars will remain undetected. We assumethat only the rotation rate of the brightest star is measured inthis case. Therefore, when constructing the simulated distri-bution of rotation rates, we only include the most luminousmain-sequence star of each binary system.For our standard simulation we adopt a metallicity ofZ=0.008, which is appropriate for the LMC. This metallicity is alsoconsidered representative for star-forming regions at a redshift1–2, at the peak of star formation in the universe. The effect ofmetallicity is discussed in Section4.1.We derive the distribution of rotation rates for systems thatare brighter than 104and 105L⊙,r e s p e c t i v e l y .T op u tt h i si nperspective, in our models 104L⊙corresponds to the luminosityof a 8.5–12M⊙main-sequence star, depending on whether wetake the model at zero-age or at the end of the main sequence.Similarly, a luminosity cutoff of 105L⊙corresponds to starswith masses in excess of 20–28M⊙.E f f e c t i v e l y ,t h efi r s tg r o u pis dominated by early B-type stars and the second group byO-type stars. We refrain at this stage from applying other cutoffssuch as criteria based on temperature, color, or spectral type,since our predictions for the temperatures are less reliable thanthose for the luminosities.3. THE EVOLUTION OF THE ROTATIONAL VELOCITYFOR INDIVIDUAL SYSTEMSAs a star evolves, its rotational velocity is affected by variousprocesses, for example, as a result of angular momentum lossthrough stellar winds. While a single star can only lose angularmomentum as a result of mass loss, a star in a binary systemmay either lose or gain angular momentum as it interacts withits companion. Even when angular momentum is conserved, therotational velocity of a star can alter as a result of changes in thestellar interior. These effects are discussed in Section3.1.T h eeffect of binary interaction is discussed in Sections3.2and3.3.3.1. Effect of Changes in the Stellar StructureStars expand over the course of their main-sequence evolu-tion. Although one might expect intuitively that the rotationalvelocity decreases as the star expands, in practice this is notthe case. During the main sequence, contraction and conse-quent spin up of the core counteract the effect of the moderateexpansion of the envelope. In this context, it is instructive toinvestigate how the moment of inertia,I∼kR2, changes as thestar expands. Here,kdenotes the square of the effective gyra-tion radius, which depends on the internal density profile. If weapproximatek∼R−ξ,o re q u i v a l e n t l yξ≡−dlnkdlnR,(6)we find that the exponentξvaries only slightly during themain sequence with typical values of 1.5–2. In the case ofrigid rotation, we can now express how the rotation rateΩ, 0 200 400 600 800\n 2  4  6  8  10  20  40  60  80Initial mass (M)vK  (km s-1)t/tMS = 0.25t/t = 0.5t/tMS = 0.75SMzero-age main sequence (t = 0)central hydrogen exhaustion (t = tMS)0.5 tMS0.75 tMS0.25 tMS\nFigure 1.Evolution of the Keplerian rotational velocity for stars of differentinitial masses during their main-sequence evolution. Labels show the relativeage as a fraction of the main-sequence lifetimetMS. This diagram is constructedusing stellar radii from Pols et al. (1998)f o ram e t a l l i c i t yo fZ=0.008, usingthe prescription by Hurley et al. (2000). We note that the time-dependent radiifor stars more massive than about 40M⊙are very uncertain. In the most massivestars the decrease of the Keplerian velocity with time may well be more severethan shown here. See Section3.1for a discussion.(A color version of this figure is available in the online journal.)the rotational velocityvrot, and the ratio of the rotation rate andthe Keplerian rateω/ωKchange as the star expands,dlnΩdlnR=ξ−2,(7)dlnvrotdlnR=ξ−1,(8)dlnΩ/ΩKdlnR=ξ−12.(9)In other words, given typical values ofξ,w efi n dt h a tt h edecreasing gyration radius compensates for the expansion ofthe star such that the rotation rateΩdecreases only slightly asthe star evolves (Equation (7)), sincedlnΩ/dlnR≈−0.5−0.The rotational velocity at the equator,vrot,i n c r e a s e ss l i g h t l y(Equation (8)).Most interestingly, the last expression shows that stars nat-urally evolve toward the Keplerian limit (Equation (9)), if theamount of angular momentum loss is small and internal an-gular momentum transport between the core and envelope isefficient. An important implication of this is that when a starreaches the Keplerian rotation rate, it remains rotating near theKeplerian limit in the absence of an efficient angular momentumloss mechanism. We refer the reader to the excellent discussionby Ekstr¨om et al. (2008b), which describes this effect in detailedmodels of single stars that allow for differential rotation.Figure1shows the Keplerian rotational velocityvKas afunction of the initial mass of a star at different stages during themain sequence. In zero-age main-sequence stars the Keplerianvelocity increases monotonically with the initial stellar mass.The Keplerian velocity drops as stars evolve and expand. Thelargest change occurs during the final stages of main-sequenceevolution. The change in radius of more massive stars duringthe main sequence is larger, resulting in a more significant dropin the Keplerian velocity. As a result, the Keplerian rotationalvelocity at the end of the main sequence is around 400 km s−1with only a weak dependence on the stellar mass. Note that4\nFig. 4: Left: Observed bimodal distribution of vsinifor single B-type stars (Dufton et al.,\n2013). Center: Observed log-normal distribution of vsinifor single O-type stars (Ram\u0013 \u0010rez-\nAgudelo et al., 2013). Right: Theoretical Keplerian velocities of massive stars as a function\nof age and mass (de Mink et al., 2013).\nburning phases as fresh fuel is mixed into the burning region (e.g., Maeder, 2009;\nViallet et al., 2013; Jin et al., 2015). From the standpoint of stellar evolution, this is\nyet an open problem: to understand how the depth of penetration and the character\nof the convection in this region change with rotation, magnetism, and di\u000busion.\nRotation Observations using stellar spectra have shown that the majority of massive\nstars are fairly rapid rotators (e.g., Huang et al., 2010; de Mink et al., 2013). The\naverage projected equatorial velocity of these stars is about 150 km s\u00001on the main-\nsequence but have signi\fcant tails (Figure 4). The rotation rates of the core and\nradiative envelope of some B stars, notably \f-Cepheid variables, have been estimated\nusing asteroseismology, \fnding that most massive main sequence stars appear to be\nrigidly rotating (e.g., Aerts et al., 2017). Indeed, the small angular velocity contrasts\nfor observed massive stars may point to an e\u000bective angular momentum coupling be-\ntween the core and the envelope, possibly by gravity waves or magnetic \felds during\ntheir evolution. Stellar radiative regions are rotating and magnetized. Therefore,\ninternal gravity waves become magneto- gravito-inertial waves and the Coriolis ac-\nceleration and the Lorentz force cannot be treated apriori as perturbations. For\nexample, during the PMS of low-mass stars and in rapidly rotating massive stars,\nthe strati\fcation restoring force and the Coriolis acceleration can be of the same\norder of magnitude. In addition to the impact of rotation on convection and its\ndynamo processes, it also has an in\ruence in radiative zones that can lead to trans-\nport there through meridional \rows, shear turbulence, and internal waves (Mathis,\n2013). The large-scale meridional circulation occurring in stellar radiation zones is\noccurs due to the deformation of the star and its isothermal surfaces by the centrifu-\ngal acceleration. The radiative \rux is then no longer divergence-free and must be\nbalanced by heat advection, which is carried by the meridional \row. This \row can\nalso transport angular momentum and chemical species throughout the radiative\nenvelope. Shear turbulence can occur if the waves excited by convection become\nnonlinear and break or if there is a strong di\u000berential rotation that leads either to\nan magneto-rotational instability. Such turbulence has been successful in describing\nsome aspects of the dynamics in massive stars (Meynet & Maeder, 2000). Waves\nalso can transport energy, even if they are linear as they can propagate until they\nare dissipated through thermal di\u000busion. Thus, their transport is highly nonlocal.\nSuch internal waves are excited by the turbulent motions at convection to radiation\ntransitions in stellar interiors, namely the boundaries of convective envelopes.\nMultiplicity Recent surveys have shown that around 70-80% of massive stars are in\npta.edu.pl/proc/2021oct25/123 PTA Proceedings ?October 25, 2021 ?vol. 123?55Kyle C. Augustson\nAA51CH07-Duchene ARI 11 June 2013 19:5\n0.11.010.0Stellar mass (M  )0.00.51.01.5MF, CFVLM MGK A early BO\nFigure 1Dependency ofCF(companion frequency;red squares)a n dMF(multiplicity frequency;blue triangles)w i t hprimary mass for main-sequence stars and field very low-mass (VLM) objects. The horizontal error barsrepresent the approximate mass range for each population. For B and O stars, only companions down toq≈0.1 are included. The frequencies plotted here are the best-estimate numbers from Sections 3.1–3.5,also reported inTable 1.currently known has four components (Reid & Hawley 2005). Although it appears that the ratioof binary to higher order systems does not vary much for objects withM⋆≤1.5M⊙, it probablyrises significantly toward high-mass stars.5.1.2. Orbital period distribution.Figure 2presents the orbital period distribution for fieldobjects as a function of primary mass, based on results discussed in Section 3. To focus on ma-jor trends, the distributions are simplified as linear combinations of log-normal and power-lawdistributions. However, except for the well-characterized solar-type systems, the underlying truedistributions are likely more complicated.The distribution of orbital periods is unimodal for solar-type and lower mass stars, but both themedian separation and width of the distribution decrease sharply with decreasing stellar mass. Asa result, the frequency of companions in the 1–10-AU range (10–15%) does not vary significantlywith stellar mass forM⋆≤1.5M⊙, including substellar objects. The much lower overall multi-plicity of field BDs traces a marked deficit of wide companions as opposed to a uniform depletionover all separations.Intermediate- and high-mass stars have more complex distributions of orbital periods. A strongpeak at the shortest periods (logP≈0–1) is found in both populations with an amplitude thatincreases with stellar mass. VBs show a peak for intermediate-mass stars, whereas the situation isless clear for high-mass stars, for which a shallow power law is currently preferred. Interestingly,the frequency of companions in the 1–10 AU-range among intermediate-mass stars appears inreasonable agreement with that observed among solar-type stars.5.1.3. Mass ratio distribution.Although a simple power-law representation is imperfect for mostsamples, this formalism offers the most straightforward criterion to compare multiple systems ofvarious masses. As shown inFigure 3, the observed distribution of mass ratios is close to a flatwww.annualreviews.org•Stellar Multiplicity 289Changes may still occur before final publication online and in printAnnu. Rev. Astro. Astrophys. 2013.51. Downloaded from www.annualreviews.orgby Royal Melbourne Institute of Technology (RMIT) on 07/04/13. For personal use only.The Astrophysical Journal,7 6 4 : 1 6 6( 1 7 p p ) ,2 0 1 3F e b r u a r y2 0De Mink et al.\n05101520\n 1  2  4  6  8Age (Myr)Initial orbital period (d) 10  100  1000  10000 0 100 200 300 400 500 600\nmergersCase ABCase ABBCase ACase BCase C)a( elpmaxe)c( elpmaxe\nspin up by tidesspin down by tidesspun up secondaries vrot   (km s-1)spun up secondariesslow Case Amass transfer  200 km s-1\n 100 km s-1 100 km s-1 200 km s-1 500 km s-1 400 km s-1 300 km s-1\nFigure 3.Rotational velocity (color shading) as a function of initial orbital period and time for the brightest main-sequence star in a binary system. We adoptinitialmasses of 20 and 15M⊙, initial rotational velocities of 100 km s−1,a n dam e t a l l i c i t yo fZ=0.008. As the stars evolve along the main sequence their rotationalvelocity is altered by stellar winds, internal evolution, tides, and most notably mass accretion. The vertical dotted line indicates the maximum separation for whichthis system interacts by mass transfer. The examples shown in panels (a) and (c) of Figure2are part of this simulation.(A color version of this figure is available in the online journal.)Changes in the stellar structure.The effect of changes on thestellar structure as the star evolves can be observed, for example,in panel (a) of Figure2. During the first 9.5 Myr of the evolutionof this system both stars reside well within their Roche lobe andtheir evolution is similar to that of single stars. The rotationalvelocity of both stars remains roughly constant during this phase(cf. Section3.1). When the primary star approaches the end ofits main-sequence lifetime, its expansion accelerates. This leadsto the decrease of the rotational velocity that is visible in panel(a) at an age of 8–9.5 Myr. The expansion is also responsiblefor the decrease of the Keplerian rotational velocity with time.Stellar wind.The effect of angular momentum loss by stellarwinds is small during the major part of the main sequence inthese examples. The winds become stronger toward the endof the main sequence, which together with the expansion,contributes to the decrease in the rotational velocity of theprimary discussed above. However, spin-down by winds doesplay a significant role for the massive, rapidly rotating stars thatcan be produced as a result of mass transfer. This effect canbe seen most clearly in panel (c) for ages of 12–15 Myr. Thesecondary quickly spins down, reducing its rotational velocityby about a factor of 3.Tides.Tidal interaction tends to synchronize the rotation ofthe stars with the orbit in systems where the separation betweenthe stars is comparable to the stellar radii. The systems depictedin the upper and lower panels have initial separation on the orderof 120R⊙and 40R⊙, respectively; this is slightly less for thesystems on the right due to the smaller mass ratios. Tides donot play a significant role during the main sequence of the starsin the upper panels, but they are responsible for the gradualincrease in rotational velocity that can be seen in the lowerpanels for the primary star during the first 8 Myr. The orbitalperiod remains nearly constant during this phase. The primarystar is kept in corotation with the orbit as it expands, whichimplies that the rotational velocity gradually increases. In thisexample, the primary expands by just over a factor of two beforeit fills its Roche lobe, resulting in a rotational velocity in excessof 200 km s−1.In panel (c) of Figure2,t i d e sa r ea l s oi m p o r t a n tf o rt h esecondary star, during the first mass transfer phase, which startsat about 8 Myr. The secondary star spins up as it accretes massand angular momentum. However, the tides quickly spin the stardown to synchronous rotation. This system experiences a masstransfer phase, which lasts almost 4 Myr, during which both starsremain in synchronous rotation while the orbit gradually widens.Around 12 Myr a second rapid phase of mass transfer sets in,i.e., Case AB, as the primary star leaves the main sequenceand expands during hydrogen shell burning. As a result of thehigh-mass transfer rate and the fact that the orbit widens, tidesare no longer effective in preventing the accreting star fromspinning up.3.3. Effect of the Initial Separation and Mass RatioTo further illustrate the effect of the initial binary parameters,we depict in Figure3the evolution of the rotational velocity ofthe brightest main-sequence star in a binary system as a functionof the initial orbital period. For this example we adopted aninitial rotation rate of 100 km s−1,am e t a l l i c i t yo fZ=0.008,and initial masses of 20 and 15M⊙.T h ec o l o rs h a d i n gi n d i c a t e sthe equatorial rotational velocity of the brightest main-sequencestar in each system. Initially this is the primary star, but aftermass transfer the secondary becomes the brightest.Mass transfer with a main-sequence donor (Case A).Inshort-period systems the expansion of the primary star dur-ing its main-sequence evolution is sufficient to make it fill itsRoche lobe. These systems experience a phase of slow masstransfer that can last for several Myr. Mass transfer typicallyoccurs via direct impact onto the surface of the secondary.Tides keep both stars synchronized, which prevents the sec-ondary from reaching very high rotation rates. This phase endswhen the two stars come into contact and merge (for periodsP/lessorsimilar2 days) or when the primary star leaves the main sequence(for periodsP≈2–5 days), triggering a new mass transfer phase(Case AB). Both cases are expected to lead to the formation of amassive rapidly rotating star. In certain cases the rapidly rotatingstar experiences an additional spin-up phase, when the primaryfills its Roche lobe again as it expands during He shell burning(Case ABB). This effect is visible in Figure3around 14 Myrfor systems with orbital periods of 4–5 days.64586J. Vidal et al.\n(a)\n(b)Figure 7.(a) Three-dimensional snapshot of the magnetic field magnitude|B|at a given time. The rotation axis is alongz. (b) Time- and radius-averaged spectra of the magnetic energy as function of the spherical har-monic degreel. Simulation atEk=10−4,Pr=1,Pm=2, andϵ=0.2.predominantly toroidal, as expected from stability considerations innon-barotropic stars (Akg¨un et al.2013).Because of the complex time evolution, straightforward visual-izations of the instantaneous field are not illuminating. We show inFig.7(a) an instantaneous snapshot of the magnitude of the mag-netic field. The field is of rather small scale. We observe similaritieswith the temperature field shown in Fig.3(a). A description of thefield morphology is provided by the time-averaged spectrum of themagnetic field in Fig.7(b). The magnetic spectrum is dominated bycomponents of spherical harmonic degreesl≤10. It is maximumfor the dipolar component (l=1) and then slowly decays with apower lawE(B)∝l−0.04. The time-averaged spectrum, as well asthe instantaneous ones, are well resolved, proving that tidal motionsare able to drive a dipole-dominated dynamo in a stably stratifiedlayer.We show in Fig.8, the time-averaged magnetic field truncatedat spherical harmonics degreel=5, because higher degrees arenot observed (e.g. Donati & Landstreet2009;F a r e se ta l .2017).This time-averaged field is mostly dipolar (l=1) and axisymmetric(m=0). Non-axisymmetric components are averaged out becauseof the rapid spin. The time-averaged flow has a columnar structurealigned with the spin axis, as shown in Fig.8(b). These spin-aligned\nFigure 8.(a) Time-averaged radial magnetic field at the stellar surfaceand (b) time-averaged velocity magnitude in the equatorial plane and in ameridional plane. Simulations atEk=10−4,Pr=1,Pm=2, andϵ=0.2.Time-averaged fields computed fromt/tη=0t ot/tη=0.1 in Fig.6(b). Inboth figures, the spin axis is the verticalz-axis.structures are the global counterpart of the strong vortices almostinvariant along the rotation axis and filling the periodic boxes ofsimilar local simulations (Barker & Lithwick2013a,b). These flowsare expected in our stress-free computations with no viscous fric-tion at the boundary (Livermore et al.2016; Le Reun et al.2017).The emergence of such spin-aligned large-scale vortices are alsoobserved in rotating thermal convection (e.g. Guervilly et al.2014)and have been shown to be dynamos (Guervilly et al.2015).3.4 Tidal mixingWe have shown that the tidal instability is dynamo capable in oursimulations whenN0//Omega1s/lessorsimilar1 with a dynamo thresholdRmc≃3000. For stronger stratifications (N0//Omega1s≥10), we did not finddynamo action up toRm≃8000 in the simulations. Indeed, dynamoaction requires not only largeRm,b u ta l s oa d e q u a t e ,s u f fi c i e n t l ycomplex, flow structure (Kaiser & Busse2017). Here, we suspectMNRAS475,4579–4594 (2018)Downloaded from https://academic.oup.com/mnras/article-abstract/475/4/4579/4797183by Lancaster University useron 04 June 20184584J. Vidal et al.\nFigure 3.Three-dimensional views of the total temperatureT=T0+/Theta1in the non-linear regime of the tidal instability. Surfaces of constantTareshown in the equatorial plane and in a meridional plane. Simulations atEk=10−4,Pr=1, andϵ=0.2.kinetic energy of the global rotation. Three regimes are observed inthe simulations. Illustrative three-dimensional snapshots of the totaltemperature fieldT=T0+/Theta1in these regimes are shown in Fig.3.When 0≤N0//Omega1s/lessorsimilar1, the tidal instability flow is immune to thestable stratification, as in the linear growth phase. The instability isalmost four times critical in this range (ϵ/ϵc≃3.7) and the typicalReynolds number isRe=2000. The flow has a kinetic energy repre-senting about 5 per cent of the kinetic energy of the global rotation,consistent with the expected dimensional amplitudeϵ/Omega1sR∗in theneutral (N0=0) case (Barker & Lithwick2013a;G r a n n a ne ta l .2016; Barker2016). In Fig.3(a), the stratification seems to be wellmixed and eroded in the bulk (compare with Fig.1b), below a ther-Figure 4.Instantaneous fraction of poloidal to total kinetic energyEpol(u)/E(u), denotedFpol,a saf u n c t i o no fN0//Omega1s. Simulations atEk=10−4,Pr=1, andϵ=0.2mal boundary layer (due to our thermal boundary condition). Wenote that the fluid is no longer barotropic, since the instantaneousisolines ofTdo not coincide with the isopotentials anymore.When 1/lessorsimilarN0//Omega1s≤2, we observe a collapse of the kinetic energy.For these stratifications, the interplay between inertial and internalwaves reduces the saturation amplitude of the tidal instability. As aconsequence, we observe also a reduction in the mixing in Fig.3(b).The collapse when 1/lessorsimilarN0//Omega1s≤2 is due to a variation ofϵcthere,likely due to diffusion effects (see Appendix). This effect is notexpected in radiative stellar interiors, characterized by much lowerdiffusivities. Finally, whenN0//Omega1s≥2, the strong stratifications donot prevent the tidal instability. Instead the instability has an evenlarger amplitude, with a typical Reynolds numberRe=3000 and akinetic energy representing still about 5 per cent of the kinetic en-ergy of the basic flow, see Fig.2(b). This translates to a dimensionalflow amplitudeϵ/Omega1sR∗regardless of the strong stratification. Thetotal temperature field displayed in Fig.3(c) seems however hardlydisturbed by the instability, implying that the motions are mostlyconfined to spherical shells with almost no radial component. Thisis confirmed by the ratioFpolof poloidal kinetic energy to the totalkinetic energy, shown in Fig.4.F o rN0//Omega1s≤1,Fpolmostly liesbetween 0.3 and 0.4. WhenN0//Omega1s≥1, firstFpolseems to takevalues between 0.1 and 0.5, before dropping below 0.05 when thestratification is further increased in the rangeN0//Omega1≥10. Theselow values of the poloidal kinetic energy show that the flow hasconsistently a weak radial component whenN0//Omega1s≥10.3.2 Kinematic dynamosWe remove the Lorentz force (∇×B)×Bfrom the momentumequation (3a) to investigate kinematic dynamos. In this problem, weassess the dynamo capability of the non-linear tidal motions, withouta back reaction of the magnetic field on the flow. We introduce themagnetic Reynolds numberRm=PmRe,(11)withRethe Reynolds number previously introduced. If the structureof the tidal instability flow is suitable for dynamo action,Rmhasafi n i t ec r i t i c a lv a l u eRmcabove which the dynamo process starts,characterized by the growth of a magnetic field. Equivalently, thedynamo thresholdRmcis associated with a critical magnetic PrandtlnumberPmcfor a fixed value ofEk.We have considered several values of the magnetic Prandtl num-ber (1/lessorsimilarPm≤5), starting from random magnetic seeds, toMNRAS475,4579–4594 (2018)Downloaded from https://academic.oup.com/mnras/article-abstract/475/4/4579/4797183by Lancaster University useron 04 June 20184584J. Vidal et al.\nFigure 3.Three-dimensional views of the total temperatureT=T0+/Theta1in the non-linear regime of the tidal instability. Surfaces of constantTareshown in the equatorial plane and in a meridional plane. Simulations atEk=10−4,Pr=1, andϵ=0.2.kinetic energy of the global rotation. Three regimes are observed inthe simulations. Illustrative three-dimensional snapshots of the totaltemperature fieldT=T0+/Theta1in these regimes are shown in Fig.3.When 0≤N0//Omega1s/lessorsimilar1, the tidal instability flow is immune to thestable stratification, as in the linear growth phase. The instability isalmost four times critical in this range (ϵ/ϵc≃3.7) and the typicalReynolds number isRe=2000. The flow has a kinetic energy repre-senting about 5 per cent of the kinetic energy of the global rotation,consistent with the expected dimensional amplitudeϵ/Omega1sR∗in theneutral (N0=0) case (Barker & Lithwick2013a;G r a n n a ne ta l .2016; Barker2016). In Fig.3(a), the stratification seems to be wellmixed and eroded in the bulk (compare with Fig.1b), below a ther-Figure 4.Instantaneous fraction of poloidal to total kinetic energyEpol(u)/E(u), denotedFpol,a saf u n c t i o no fN0//Omega1s. Simulations atEk=10−4,Pr=1, andϵ=0.2mal boundary layer (due to our thermal boundary condition). Wenote that the fluid is no longer barotropic, since the instantaneousisolines ofTdo not coincide with the isopotentials anymore.When 1/lessorsimilarN0//Omega1s≤2, we observe a collapse of the kinetic energy.For these stratifications, the interplay between inertial and internalwaves reduces the saturation amplitude of the tidal instability. As aconsequence, we observe also a reduction in the mixing in Fig.3(b).The collapse when 1/lessorsimilarN0//Omega1s≤2 is due to a variation ofϵcthere,likely due to diffusion effects (see Appendix). This effect is notexpected in radiative stellar interiors, characterized by much lowerdiffusivities. Finally, whenN0//Omega1s≥2, the strong stratifications donot prevent the tidal instability. Instead the instability has an evenlarger amplitude, with a typical Reynolds numberRe=3000 and akinetic energy representing still about 5 per cent of the kinetic en-ergy of the basic flow, see Fig.2(b). This translates to a dimensionalflow amplitudeϵ/Omega1sR∗regardless of the strong stratification. Thetotal temperature field displayed in Fig.3(c) seems however hardlydisturbed by the instability, implying that the motions are mostlyconfined to spherical shells with almost no radial component. Thisis confirmed by the ratioFpolof poloidal kinetic energy to the totalkinetic energy, shown in Fig.4.F o rN0//Omega1s≤1,Fpolmostly liesbetween 0.3 and 0.4. WhenN0//Omega1s≥1, firstFpolseems to takevalues between 0.1 and 0.5, before dropping below 0.05 when thestratification is further increased in the rangeN0//Omega1≥10. Theselow values of the poloidal kinetic energy show that the flow hasconsistently a weak radial component whenN0//Omega1s≥10.3.2 Kinematic dynamosWe remove the Lorentz force (∇×B)×Bfrom the momentumequation (3a) to investigate kinematic dynamos. In this problem, weassess the dynamo capability of the non-linear tidal motions, withouta back reaction of the magnetic field on the flow. We introduce themagnetic Reynolds numberRm=PmRe,(11)withRethe Reynolds number previously introduced. If the structureof the tidal instability flow is suitable for dynamo action,Rmhasafi n i t ec r i t i c a lv a l u eRmcabove which the dynamo process starts,characterized by the growth of a magnetic field. Equivalently, thedynamo thresholdRmcis associated with a critical magnetic PrandtlnumberPmcfor a fixed value ofEk.We have considered several values of the magnetic Prandtl num-ber (1/lessorsimilarPm≤5), starting from random magnetic seeds, toMNRAS475,4579–4594 (2018)Downloaded from https://academic.oup.com/mnras/article-abstract/475/4/4579/4797183by Lancaster University useron 04 June 2018N/𝜴EP/E(a)(b)\n(c)(d)(e)\nFig. 5: (a) Average companion fraction (CF) and multiplicity fraction (MF) by spectral\ntype (Duch^ ene & Kraus, 2013). (b) Rotational evolution and outcomes of a massive binary\nsystem as function of initial orbital period (de Mink et al., 2013). (c) Ratio of poloidal\nmagnetic energy to total energy in simulations of tidally driven dynamos in a stable region,\n(d) magnetic \feld strength for N=\n = 0:5 and (e) its corresponding temperature \feld (Vidal\net al., 2018).\nmultiple star systems (Raghavan et al., 2010; Duch^ ene & Kraus, 2013). Given that\ntheir lives are short, they do not have much time to migrate far from their place\nof birth. Therefore, tidal interactions will be an important dynamical component\nfor many massive stars, which will impact their evolution, structure, and magnetic\n\felds. The equilibrium tides distort the shape of the star while it is the dynamical\ntides, or even nonlinear tides, that could lead to dynamo action if there is su\u000ecient\ncorrelation between the velocity \feld and the magnetic \feld (See Figure 5; Ogilvie,\n2014; Vidal et al., 2018, 2019). Such processes therefore should be accounted for\nin both 3D dynamical simulations of massive stars as well as in stellar evolution\nand structure computations. Indeed, as shown in Figure 5(c), the poloidal magnetic\nenergy generated through dynamo action induced by the tide can reach 20% of the\nequipartition value with the kinetic energy of the dynamical tide when the orbital\nfrequency greater than about 10% of the Brunt-V aisalla frequency. This suggests\nthat it is possible that tides could disrupt the fossil \felds formed during the pre-\nmain-sequence. Moreover, it implies that the tidal interaction between the disk and\nthe protostar is indeed quite dynamic even in the stably strati\fed regions of the\nprotostar.\n2 Evolution of Massive Star Magnetism\nThe formation of massive stars is quite di\u000berent from the slow accretion of low mass\nstars that can take 100 million years, with the formation time scale being compressed\nto a few tens or hundreds of thousands of years. What they do share is that whatever\n56?PTA Proceedings ?October 25, 2021 ?vol. 123 pta.edu.pl/proc/2021oct25/123Magnetism in Massive Stars\n64 C. Neineret al.\nFigure 1.Diagram showing the pre-main sequence (PMS) from the birthline to the ZAMS, withvarious evolutionary tracks from 1.2 to 8 M⊙shown with black solid lines. The PMS is dividedin 4 parts indicated with different grey shades (color version in on-line copy of this figure),showing 4 stages of the evolution of the structureo ft h es t a r sa n do ft h e i rf o s s i lm a g n e t i cfi e l d .2014; Braithwaite & Cantiello 2013). This kind of ultra-weak field has been observed inVega (Ligni`ereset al.2009; Petitet al.2010) and in a few Am stars (e.g., Petitet al.2011, see also Blazereet al., these proceedings).4. Lack of magnetic fields in hot binariesBinaMIcS (Binarity and Magnetic Interactions in various classes of Stars, Neineret al.2013; Alecianet al.2015) is an ongoing project that exploits binarity to yield new con-straints on the physical processes at work in hot and cool magnetic stars. It rests ontwo large programs of observations with the ESPaDOnS spectropolarimeter at CFHTin Hawaii and its twin Narval at TBL in France. BinaMIcS aims at studying the roleof magnetism during stellar formation, magnetospheric star-star (and star-planet) inter-actions, the impact of tidal flows on fossil and dynamo fields, its impact on mass andangular momentum transfer, etc.In the frame of BinaMIcS, a large survey of magnetism in hot spectroscopic binarysystems with 2 spectra (SB2) has been undertaken. Out of∼200 observed SB2, includingat least one star (and most of the time two stars) with spectral type O, B or A in eachsystem, none were found to host a magnetic field, while the detection threshold wassimilar to the one used in the MiMeS project on single hot stars. This lack of detectionsin∼400 stars with BinaMIcS, compared to the∼7% detection rate in∼500 single starswith MiMeS, is thus statistically significant: magnetism is less present in hot binariesthan in single hot stars.1(19\u001e12\u001e5\u00011C\u0001\u001bCC\"%\u000e\u0005\u0005)))\u000431\u001f2$9475\u0004!$7\u00053!$5\u0005C5$\u001f%\u0004\u0001\u001bCC\"%\u000e\u0005\u00054!9\u0004!$7\u0005\u0007\u0006\u0004\u0007\u0006\u0007\f\u0005/\u0007\f'\t\r\b\u0007\t\u0007\u000b\u0006\u0006'\u000b\b'\u0010!) \u001e!1454\u0001\u0019$!\u001f\u0001\u001bCC\"%\u000e\u0005\u0005)))\u000431\u001f2$9475\u0004!$7\u00053!$5\u0004\u0001,!\u001f\u001f9%%1$91C\u0001+\u0001\u001e\u0002, 5$795\u00011C!\u001f9#D5 \u0001! \u0001\b\r\u0001.!(\u0001\b\u0006\u0007\r\u00011C\u0001\u0007\u0007\u000e\b\u0007\u000e\t\r \u0001%D2:53C\u0001C!\u0001C\u001b5\u0001,1\u001f2$9475\u0001,!$5\u0001C5$\u001f%\u0001!\u0019\u0001D%5 \nFig. 6: A sketch of the evolution of a PMS massive star, showing the fully convective phase\n(1), the convective freeze-out (2), the radiative phase (3), and the ZAMS state with an\noblique axisymmetric relaxed magnetic \feld (4) (Neiner et al., 2015).\nangular momentum, chemical abundances, and magnetic \feld are present in the star\nforming region will initially shape the structure of the local patch of gas that if\nsu\u000eciently self-gravitating will eventually collapse into a disk that feeds a central\nobject. As gravitational energy is released while this central object contracts and it\nis fed with additional mass from the disk, parts of the protostar will be convectively\nunstable plasma (see Figure 6).\nThe stably strati\fed portions of the protostar will contain the geometrically am-\npli\fed magnetic \feld advected into it during its initial condensation. The convective\nportions on the other hand will be running an active magnetic dynamo wherein the\nrotation, di\u000berential rotation, and the buoyantly driven convection act together to\nbuild magnetic \felds that can be stronger than the originating \feld. The dynamo-\ngenerated \feld will link with the magnetic \feld in the stable regions of the star\nas well as with the magnetic \felds in the disk causing angular momentum transfer\nas well as potentially inciting instabilities (e.g., Romanova & Owocki, 2015). This\nformation process is extremely di\u000ecult both to observe and to theoretically describe\ngiven that the structure of the disk impacts the protostellar structure so strongly,\ne.g. it is unknown how the mass, angular momentum, and magnetic \feld are actually\nentrained into the protostar.\nMassive protostars begin fusing material in their cores before they \fnish forming\nleading to an even larger radiative \rux compared to their low-mass brethren that\nradiate only the gravitational energy. The radiation streaming from the massive\nprotostar's photosphere is su\u000ecient to blow away most of the material unless the\ndisk is some how screened from it. One way this can be circumvented is through\nthe magnetic collimation of polar jets of out\rowing gas and dust along which the\nradiation can preferentially stream. These jets can drive a circulation in the disk\nthat draws in more gas from its surroundings (e.g., Tan et al., 2014; Krumholz, 2015;\nRomanova & Owocki, 2015), replenishing the disk that feeds the star. This process\npta.edu.pl/proc/2021oct25/123 PTA Proceedings ?October 25, 2021 ?vol. 123?57Kyle C. Augustson\n351\nFigure 10.5: Radial velocity shown in an orthographic volumep r o j e c t i o nf o rC a s eM4,with the velocities outside the core scaled to 10 cm s−1and those in the core scaled to100 m s−1.D o w n fl o w s a r e d a r k , u p fl o w s l i g h t .G r a v i t y w a v e s a r e v i s i b l ein the radiativezone and columnar convective structures can be seen in the core. (b) Instantaneous velocitystreamlines in the convective core, cut at the equator in CaseM16.C o l o r i n d i c a t e s t h emagnitude of the velocity, with slow flows of about 10 m s−1in red and fast flows of about500 m s−1in blue. (c) Instantaneous velocity streamlines of single convective column, showingonly a weak linking between adjacent columns and a helicity reversal across the equator.Color indicates the direction of the vertical velocityvz,w i t hr e dp o s i t i v ea n db l u en e g a t i v e .filamentary nature of the flows within their cores and the highradial wavenumber gravitymodes that the convective overshooting excites in their radiative exteriors. It also servesto highlight the dramatically different amplitude of the modes excited in the two cases, thehydrodynamic case shows modes with a fair amount of enstrophy, whereas the magneticcase has much less. The reason for this becomes clear when oneconsiders Figures 10.4(c,f). CaseH4has a profound radial differential rotation with the deepestportions of the corerotating 50% more slowly than the frame rate, and 70% more slowly than the more rapidlyrotating flows near the overshooting region at 0.2R . T h u s t h e fl o w s c a r r y i n g e n e r g y i n t h i ssimulation are also very effective at transporting angular momentum out of the core. Themeridional profile of the angular velocity in this case is shown in Figure 10.4b, where aprominent Stewartson column is formed along the rotation axis. It is quite remarkable thatTeffLMass AccretionPMSMain SequenceSupergiantMass LossHHeCO-Ne-Mg-SSiFeHe+\nHPre-SupernovaThe Magnetic Furnace 7\nFigure 6.Time evolution of radial velocity and radial magnetic field in caseM16shown in a set of equatorial cuts covering the convective core (delineated bythe dashed circle) and a small portion of the radiative envelope. The tiny central circle is the inner domain cutout in our computational domain. (a)-(d) displayfour successive samplings of radial velocity vrthat are 20 days apart, with downflows blue and upflows red, and (e)-(h) show the accompanying radial magneticfieldsBr, with inwardly directed magnetic field in blue and outwardly pointing field in red. This interval is in the latest stage of the simulation where the dynamohas reached a statistically steady state. Clipping values for radial velocity and magnetic field are respectively±100ms-1and±105G.with the rotation axis. Such alignment is consistent with themotions being more constrained by the Coriolis force, hencebecoming quasi-2D in the spirit of the Taylor-Proudman the-orem (Pedlosky 1987). In particular, the rotationally alignedvertical structures present in the more rapidly rotating casesare akin to the Taylor columns expected for rapidly rotatingconvection, in that they are nearly invariant in the vertical di-rection. These columnar structures may be quite intricate, yethave some sense of continuity across the full core. Their num-ber increases with faster rotation, so that in caseM16roughlyeight to ten evolving vortical rolls are evident at any giventime. Indeed, following the work of Browning et al. (2004)and Miesch et al. (2006), it can be seen that the flows in themore rapidly rotating cases are quasi-geostrophic.The flows in the core generate internal gravity wavesthat then propagate following complex spiral-like ray pathsthrough the radiative envelope above it (e.g., Zahn et al. 1997;Rogers et al. 2013; Alvan et al. 2015). Such convectivecolumns formed in the core and the gravity waves that theydrive are apparent in Figure 3(a), where an orthographic pro-jection of the radial velocity in the domain is shown for caseM4. Since most of the gravity waves excited here have smallradial wavelengths, they damp fairly quickly due to a rel-atively large thermal diffusion (Zahn et al. 1997). A self-consistent simulation of such internal gravity waves that maybe pertinent to observations requires the capture of at leastthe entire resonant cavity encompassing the full radiative en-velope, which is not the case here. A detailed analysis ofthe gravity wave spectrum generated in this set of simulationswill not be presented here. The amplitude of the flows has twoscales in Figure 3(a), in the core the color table is clipped to±100ms-1in the core and to±0.1ms-1in the radiative zoneto emphasize the differing characteristics of the flow in theconvectively unstable and stable regions. Such columnar flowfeatures are further emphasized in Figure 3(b), which showsa volume rendering with a cutout of the radial velocity just inthe core.Turning now to the most rapidly rotating caseM16, thestructure of the radial velocity field is shown in Figure 5(a),and its time evolution in Figures 6(a)-(d). The columnar flowstructures involve fast flows interspersed with slower portions,thereby leading to a rich assembly of evolving folded sheetsand tubules with substructures. This is evidence of some as-pect of turbulent flow, yet they possess large-scale orderingthat imprints into the induced magnetic fields. Figures 5(b)and (c) show the magnetic field lines that accompany this in-stant in time, presenting both an equatorial cut through thefull core and a perspective view from just outside the coreboundary. The equatorial rendering reveals that the strongmagnetic fields extend across the core and show wrappingat larger radii consistent with the swirling sense of columnarflow cells viewed along their main axis. The choice here tocolor-code the field lines by the polarity of their radial mag-netic field component leads to a change in color from blue tored or vice versa as any given field line crosses the center ofthe core. The magnetic field amplitude reaches up to 2 MG asreported in Table 2. The helicoidal wrapping of the magneticfield lines close to the core-envelope boundary is evident inpanel (c) for which the field has been rendered at the sameinstant and in the same orientation as in panel (a). The dis-tinctive presence of the underlying columnar convection canbe seen in these magnetic field line structures. However, thispicture is somewhat confused by a number of magnetic fieldlines connecting across many cells. As the core edge is ap-proached, the magnetic field topology tends to become moretoroidal in character, with the field lines wrapping around thecore in the longitudinal direction. Some of these field linecharacteristics are sampled in panel (c).The time evolution of the flow structures associated withthe strongest upflows and downflows in the core occurs ap-proximately over a local convective overturning time, whichis about 134 days for caseM16. This can be appreciated in\nFig. 7: A sketch of the evolution of a massive star, showing 3D convection and its associated\ntransport and dynamo processes and the inclusion of 3D processes occurring in radiative\nregions being linked into the parameterized processes of main sequence evolution of a 2D\nstar. The 3D processes are depicted here as magnetic \feld lines in the equatorial plane\nwith strength indicated as 1 G (gray), 100 kG (blue), and 1 MG (red) for Task A, and as\nnormalized radial velocity, showing the wave \feld excited in the radiative envelope by the\nconvective core with down\rows in dark tones and up\rows in light tones (Augustson et al.,\n2016). The path on the left of the diagram illustrates the evolution of a rotating 40 M\f\nstar, from the accreting pre-main-sequence phase, to the main-sequence, to the post-main-\nsequence, and ultimately to its supernova.\ncontinues adding mass and angular momentum to the star until its radiation and\nwind output increases enough to overpower mass in\rowing from the disk, stalling\nthat \row and eventually eviscerating the disk.\nOnce the massive protostar has fully contracted, disconnected from the disk, and\nbegun its CNO-cycle fusing main-sequence life, its magnetic history is locked into\nits convectively stable regions as a fossil \feld that is connected to the convective\ndynamo of its core and into its radiation driven winds. The topological properties\nof this fossil \feld can impact whether or not the star has a magnetosphere. This in\nturn a\u000bects the star's mass loss rates and the rate of its spin down. Moreover it can\nimpact the nature of its near-surface convection zones and the waves it generates\nthat manifest as macroturbulence (Sundqvist et al., 2013; MacDonald & Petit, 2019).\nThus, understanding the properties of this fossil \feld is paramount.\n2.1 Fossil Magnetic Fields\nOne current puzzle regarding massive star magnetism is why do only about ten per-\ncent of such stars possess observable magnetic \felds. Could it be that 90 percent\nhave complex morphologies that do not lend themselves to spectropolarimetric de-\ntection? Or is it that there are con\fgurational instabilities that lead to only a subset\nof stable magnetic con\fgurations? The stability of magnetic \feld con\fgurations for\ncertain strati\fed \ruid domains have been considered both from a theoretical and a\nnumerical standpoint in the work of Duez & Mathis (2010) and Braithwaite & Spruit\n(2004); Duez et al. (2010), respectively. These magnetic \feld equilibria are valid for\n58?PTA Proceedings ?October 25, 2021 ?vol. 123 pta.edu.pl/proc/2021oct25/123Magnetism in Massive Stars\n−7−6−5−4−3−2−1 0 1\nlog10(1−t/tf)(Myr)02468101214m/M⊙M-S He C O-Ne-Mg-S Si\n012345678910\nlog10B(G)\nFig. 8: A magnetic Kippenhahn diagram showing the evolution of the equipartition mag-\nnetic \feld for a 15 M\fstar. The abscissa show the time remaining in Myr before the iron\ncore infall that occurs at tf. The burning phase of the core is indicated at the top of the\ndiagram.\na spherically-symmetric barotropic star, with the nonbarotropic component being\nhandled perturbatively assuming that the magnetic \feld is such that the magnetic\npressure is much less than the gas pressure everywhere in the domain considered.\nSuch magnetic equilibria are shown in Figure 1(c) and (d).\nOne crucial physical component that has been neglected so far is rotation for\nmost ZAMS massive stars will be rapidly rotating. Thus, if one considers rotation\nas discussed in Duez (2011) and Emeriau & Mathis (2015), and with the details\nforthcoming in Emeriau et al. (2020), there are additional ideal invariants in the\nsystem that permit the construction of self-consistent magnetic and mean veloc-\nity equilibria in the rotating frame. The underlying principle for \fnding magnetic\nequilibria in ideal magnetohydrodynamics is that the energy and dynamics must\nbe \fxed in time. For this to be true, the Lorentz force must be in balance with\nthe other forces in the system: the pressure, gravity, and the Coriolis force, form-\ning a magneto-rotational hydrodynamic equilibrium. Numerical simulations of such\nequilibria in rotating systems appear to yield relaxed states similar to those in the\nnonrotating system, but where slowly rotating systems tend toward misalignment\nof the magnetic \feld with respect to the rotation axis and more rapidly rotating\nsystems are aligned (Duez, 2011). However, the stability of such systems appears to\nbe quite sensitive to their initial distributions of magnetic helicity, which has already\nbeen seen in numerical simulations of magnetic \feld relaxation (e.g., Braithwaite,\n2008; Braithwaite & Spruit, 2017).\n2.2 Dynamo-generated Magnetic Fields\nIn convectively unstable regions, the buoyancy-driven plasma motions give rise to\nmagnetic induction through turbulent correlations that can generate both large and\nsmall scale magnetic \felds. Such \felds can link to the fossil \felds threading through\nsuch regions. In simulations of core convection, it has been found that superequipar-\npta.edu.pl/proc/2021oct25/123 PTA Proceedings ?October 25, 2021 ?vol. 123?59Kyle C. Augustson\ntition states can be achieved where the magnetic energy is greater than that con-\ntained in the convection itself, when the rate of rotation is su\u000eciently large and a the\nsystem enters into a magnetostrophic regime (Augustson et al., 2016). The structure\nof the magnetic \feld is shown in an equatorial slice of the convective core on the left\nhand side of Figure 7. Such states are possible because the Lorentz force that could\notherwise quench the \rows is mitigated by the \row and magnetic structures that are\nformed. Speci\fcally, the \row and magnetic structures are largely spatially separated\nin that the bulk of the magnetic energy exists in regions where there is little kinetic\nenergy and visa versa (e.g., Featherstone et al., 2009; Augustson et al., 2016). In\nthe regions where the two \felds overlap, new magnetic \feld is generated through\ninduction. The presence of a fossil \feld can modify this balance, enhancing certain\ncorrelations and leading to a greater level of superequipartition (Featherstone et al.,\n2009). Such magnetic \felds formed during the main sequence will slowly be added\nto the fossil magnetic \feld in the radiative envelope of these massive stars as the\nconvective core contracts. This process is continuous but occurs over evolutionary\ntime scales, whereas the new magnetic \feld con\fguration will relax on Alfv\u0013 enic time\nscales. Such continuous magnetic \feld relaxation could be at the origin of the de-\nclining prevalence of magnetic \felds during the main sequence evolution as shown\nin (Fossati et al., 2016).\nAs the star evolves past the main sequence, it moves directly to helium burn-\ning once the hydrogen has been exhausted in the core. During this phase, the core\ncontracts further and the density is larger, leading to greater kinetic energy in the\nconvective core. The superequipartition magnetic \felds now approximately contain\napproximately 100 times the energy compared to the main sequence. As the evo-\nlution continues, and carbon burning takes place, the density is once again very\nmuch larger and the superequipartition magnetic \felds will contain again 100 times\nthe energy of the helium burning phase. This pattern continues until the end-stage\nsilicon burning, where the magnetic \felds are of the order of 1010Gauss or more\ndepending upon the previous stages of burning. This evolutionary progression is\nshown in Figure 8 for a 15 M\fstar for magnetic \felds that are simply equipartition.\n3 Conclusions\nThere are still many puzzles to solve regarding the in\ruence of magnetic \felds on the\nevolution and structure of massive stars as well as on their winds and environment.\nNevertheless, as the numerous physical mechanisms at work are explored, a picture\ncan be constructed about both their births as well as their evolution toward their\ncataclysmic ends. Here we have explored recent work regarding the observed prop-\nerties of the magnetism of main-sequence massive stars, the in\ruence of convection\non their surface properties as well as on their core dynamo action, the impact of\nconvective penetration both on wave generation and in chemical mixing, and \fnally\nhow tides from multiple stellar companions can change the classical picture of single\nstar evolution. Later we have sketched the magnetic evolution of the interior of the\na massive star from the pre-main-sequence to the \fnal stages before its supernova.\nAcknowledgements. K. C. Augustson acknowledges support from the ERC SPIRE 647383\ngrant.\n60?PTA Proceedings ?October 25, 2021 ?vol. 123 pta.edu.pl/proc/2021oct25/123Magnetism in Massive Stars\nReferences\nAerts, C., Van Reeth, T., Tkachenko, A., ApJ 847, 1, L7 (2017)\nAerts, C., et al., Science 300, 5627, 1926 (2003)\nAndr\u0013 e, Q., Ondes gravito-inertielles dans les \u0013 etoiles et les plan\u0012 etes g\u0013 eantes, Ph.D. thesis,\nl'Universit\u0013 e de Paris (2019)\nAugustson, K. C., Brun, A. S., Toomre, J., ApJ 777, 2, 153 (2013)\nAugustson, K. C., Brun, A. S., Toomre, J., ApJ 829, 2, 92 (2016)\nAugustson, K. C., Mathis, S., in SF2A-2018: Proceedings of the Annual meeting of the\nFrench Society of Astronomy and Astrophysics, Di (2018)\nAugustson, K. C., Mathis, S., ApJ 874, 1, 83 (2019)\nBraithwaite, J., MNRAS 386, 4, 1947 (2008)\nBraithwaite, J., Spruit, H. C., Nature 431, 7010, 819 (2004)\nBraithwaite, J., Spruit, H. C., Royal Society Open Science 4, 2, 160271 (2017)\nBrun, A. S., et al., ApJ 836, 192 (2017)\nCantiello, M., Braithwaite, J., A&A 534, A140 (2011)\nCantiello, M., Braithwaite, J., ApJ 883, 1, 106 (2019)\nCantiello, M., et al., A&A 499, 1, 279 (2009)\nde Mink, S. E., et al., ApJ 764, 2, 166 (2013)\nDonati, J. F., Landstreet, J. D., ARA&A 47, 1, 333 (2009)\nDuch^ ene, G., Kraus, A., ARA&A 51, 1, 269 (2013)\nDuez, V., Astronomische Nachrichten 332, 983 (2011)\nDuez, V., Braithwaite, J., Mathis, S., ApJ 724, 1, L34 (2010)\nDuez, V., Mathis, S., A&A 517, A58 (2010)\nDufton, P. L., et al., A&A 550, A109 (2013)\nEmeriau, C., Mathis, S., in G. Meynet, C. Georgy, J. Groh, P. Stee (eds.) New Windows\non Massive Stars, IAU Symposium , volume 307, 373{374 (2015)\nEmeriau, C., Mathis, S., Augustson, K., A&A in prep. (2020)\nFeatherstone, N. A., Browning, M. K., Brun, A. S., Toomre, J., ApJ 705, 1, 1000 (2009)\nFossati, L., et al., A&A 582, A45 (2015)\nFossati, L., et al., A&A 592, A84 (2016)\nFuller, J., et al., Science 350, 6259, 423 (2015)\nGaraud, P., Annual Review of Fluid Mechanics 50, 1, 275 (2018)\nGroh, J. H., Meynet, G., Georgy, C., Ekstr om, S., A&A 558, A131 (2013)\nHuang, W., Gies, D. R., McSwain, M. V., ApJ 722, 1, 605 (2010)\nJin, J., Zhu, C., L u, G., PASJ 67, 19 (2015)\nKaspi, V. M., Beloborodov, A. M., ARA&A 55, 1, 261 (2017)\nKochukhov, O., Lundin, A., Romanyuk, I., Kudryavtsev, D., ApJ 726, 1, 24 (2011)\nKochukhov, O., Mantere, M. J., Hackman, T., Ilyin, I., A&A 550, A84 (2013)\nKorre, L., Garaud, P., Brummell, N. H., MNRAS 484, 1, 1220 (2019)\npta.edu.pl/proc/2021oct25/123 PTA Proceedings ?October 25, 2021 ?vol. 123?61Kyle C. Augustson\nKrumholz, M. R., The Formation of Very Massive Stars, Astrophysics and Space Science\nLibrary , volume 412, 43 (2015)\nKrumholz, M. R., Federrath, C., Frontiers in Astronomy and Space Sciences 6, 7 (2019)\nLanger, N., ARA&A 50, 107 (2012)\nLigni\u0012 eres, F., et al., in P. Petit, M. Jardine, H. C. Spruit (eds.) Magnetic Fields throughout\nStellar Evolution, IAU Symposium , volume 302, 338{347 (2014)\nMacDonald, J., Petit, V., MNRAS 487, 3, 3904 (2019)\nMaeder, A., Physics, Formation and Evolution of Rotating Stars (2009)\nMathis, S., Transport Processes in Stellar Interiors, volume 865, 23 (2013)\nMeynet, G., Maeder, A., A&A 361, 101 (2000)\nMoravveji, E., Townsend, R. H. D., Aerts, C., Mathis, S., ApJ 823, 2, 130 (2016)\nMosta, P., et al., Nature 528, 7582, 376 (2015)\nNagayama, T., et al., Phys. Rev. Lett. 122, 235001 (2019), URL https://link.aps.org/\ndoi/10.1103/PhysRevLett.122.235001\nNeiner, C., Mathis, S., Saio, H., Lee, U., in H. Shibahashi, A. E. Lynas-Gray (eds.) Progress\nin Physics of the Sun and Stars: A New Era in Helio- and Asteroseismology, Astronomical\nSociety of the Paci\fc Conference Series , volume 479, 319 (2013)\nNeiner, C., et al., A&A 546, A47 (2012)\nNeiner, C., et al., in K. N. Nagendra, S. Bagnulo, R. Centeno, M. Jes\u0013 us Mart\u0013 \u0010nez Gonz\u0013 alez\n(eds.) Polarimetry, IAU Symposium , volume 305, 61{66 (2015)\nNomoto, K., Kobayashi, C., Tominaga, N., ARA&A 51, 1, 457 (2013)\nOgilvie, G. I., ARA&A 52, 171 (2014)\nPedersen, M. G., Aerts, C., P\u0013 apics, P. I., Rogers, T. M., A&A 614, A128 (2018)\nPrat, V., et al., A&A 627, A64 (2019)\nPratt, J., et al., A&A 604, A125 (2017)\nRaghavan, D., et al., ApJS 190, 1, 1 (2010)\nRam\u0013 \u0010rez-Agudelo, O. H., et al., A&A 560, A29 (2013)\nRomanova, M. M., Owocki, S. P., Space Sci. Rev. 191, 1-4, 339 (2015)\nSengupta, S., Garaud, P., ApJ 862, 2, 136 (2018)\nSmith, N., ARA&A 52, 487 (2014)\nStello, D., et al., Nature 529, 7586, 364 (2016)\nSundqvist, J. O., et al., MNRAS 433, 3, 2497 (2013)\nTan, J. C., et al., in H. Beuther, R. S. Klessen, C. P. Dullemond, T. Henning (eds.)\nProtostars and Planets VI, 149 (2014)\nViallet, M., Meakin, C., Arnett, D., Moc\u0013 ak, M., ApJ 769, 1 (2013)\nVidal, J., C\u0013 ebron, D., Schae\u000ber, N., Hollerbach, R., MNRAS 475, 4, 4579 (2018)\nVidal, J., C\u0013 ebron, D., Ud-Doula, A., Alecian, E., arXiv e-prints arXiv:1902.10599 (2019)\nWade, G. A., et al., in P. Petit, M. Jardine, H. C. Spruit (eds.) Magnetic Fields throughout\nStellar Evolution, IAU Symposium , volume 302, 265{269 (2014)\nWoosley, S. E., Blinnikov, S., Heger, A., Nature 450, 7168, 390 (2007)\n62?PTA Proceedings ?October 25, 2021 ?vol. 123 pta.edu.pl/proc/2021oct25/123"    },    {        "title": "2307.15263v1.Reversible_magnetic_domain_reorientation_induced_by_magnetic_field_pulses_with_fixed_direction.pdf",        "content": "Reversible magnetic domain reorientation induced by magnetic field pulses with fixed direction\nXichao Zhang,1,∗Jing Xia,2,∗Oleg A. Tretiakov,3Guoping Zhao,4Yan Zhou,5\nMasahito Mochizuki,1,†Xiaoxi Liu,2,‡and Motohiko Ezawa6,§\n1Department of Applied Physics, Waseda University, Okubo, Shinjuku-ku, Tokyo 169-8555, Japan\n2Department of Electrical and Computer Engineering, Shinshu University, 4-17-1 Wakasato, Nagano 380-8553, Japan\n3School of Physics, The University of New South Wales, Sydney 2052, Australia\n4College of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, China\n5School of Science and Engineering, The Chinese University of Hong Kong, Shenzhen, Guangdong 518172, China\n6Department of Applied Physics, The University of Tokyo, 7-3-1 Hongo, Tokyo 113-8656, Japan\n(Dated: July 28, 2023)\nNanoscale magnetic domains with controllable configurations could be used for classical and quantum ap-\nplications, where the switching of magnetization configurations is an essential operation for information pro-\ncessing. Here, we report that the magnetic domain reorientation in a notched ferromagnetic nanotrack can be\nrealized and effectively controlled by applying uniform magnetic field pulses in a fixed in-plane direction per-\npendicular to the nanotrack. Our micromagnetic simulation results show that the configurations of magnetic\ndomains in the notched nanotrack can be switched between a head-to-head state and a tail-to-tail state in a re-\nversible manner driven by magnetic field pulses, while it is unnecessary to reverse the direction of the magnetic\nfield. Such a unique magnetic domain reorientation dynamics is found to depend on magnetic parameters and\nnanotrack geometries. The reorientation dynamics of magnetic domains also depends on the strength and length\nof the applied magnetic field pulse. In addition, we point out that the notches at the center of the nanotrack\nplay an important role for the stabilization of the head-to-head and tail-to-tail states during the magnetic do-\nmain reorientation. We also qualitatively explain the field-induced reorientation phenomenon with a simplified\ntwo-dimensional macrospin model. Our results may make it possible to build spintronic devices driven by a\nfixed magnetic field. Our findings may also motivate future studies to investigate the classical and quantum\napplications based on nanoscale magnetic domains.\nI. INTRODUCTION\nThe magnetization configurations in nanoscale ferromag-\nnetic tracks can be used to store binary information and per-\nforming storage functions [1–5]. For example, the magnetic\ndomain walls in a nanotrack with either in-plane or out-of-\nplane magnetization can be driven into motion by magnetic\nand electric means [6–22], which is essential for the racetrack-\ntype memory and relevant logic computing devices [1–5].\nSimilarly, chiral skyrmionic textures stabilized in nanotracks\nmay also be used for information storage and processing [23–\n36]. The racetrack-type memory applications based on mag-\nnetic domain walls could be driven by the magnetic field and\nelectric current [1–22], while the skyrmion-based one is usu-\nally driven by the electric current as the skyrmion cannot be\ndriven into translational motion by a uniform and constant\nmagnetic field [23–36].\nIn order to realize next-generation spintronic applications\nbased on nonvolatile magnetization configurations, it is still\nimportant to explore and understand the dynamics of magne-\ntization configurations induced by the most fundamental ex-\nternal stimuli, such as the magnetic field and electric current.\nThe dynamics of ferromagnetic domain walls in nanotracks\ncontrolled by magnetic field pulses has been investigated for\ndecades, however, most published reports are focused on the\n∗These authors contributed equally to this work.\n†Email: masa_mochizuki@waseda.jp\n‡Email: liu@cs.shinshu-u.ac.jp\n§Email: ezawa@ap.t.u-tokyo.ac.jpfield-induced or current-induced motion of domain walls [1–\n22]. In a racetrack-type memory device, the field-induced in-\nline motion of domain walls is a key operation for manipulat-\ning the encoded data on the nanotrack, however, domain walls\nwill be erased and re-created during the write-in and read-out\noperations [3–5]. In the same manner, the chiral skyrmionic\ntextures used in racetrack-type applications may also need to\nbe frequently created and deleted during the write-in and read-\nout operations [25, 26, 28, 30, 31, 33–35].\nA possible solution to avoid frequent creation and deletion\nof domain walls in magnetic memory and logic computing ap-\nplications is to utilize the magnetic domain reorientation and\nswitching dynamics in well-designed nanostructures [37–42].\nNamely, the data write-in and read-out process is based on\nthe manipulation of the domain configuration instead of the\ndomain wall position [37–42], because that each data bit is\nstored in a single isolated nanostructure or region so that it will\nbe unnecessary to frequently create, move, and delete domain\nwalls as the domain wall position can be fixed during the mag-\nnetic domain reorientation. However, the magnetic domain\nreorientation dynamics driven by magnetic field pulses in fer-\nromagnetic nanotracks is still elusive, however, the magnetic\ndomain reorientation dynamics could be vital for the design\nand development of future applications.\nIn this paper, we report the magnetic domain reorientation\ndynamics in a notched ferromagnetic nanotrack driven by uni-\nform in-plane magnetic field pulses perpendicular to the nan-\notrack, where the configurations of magnetic domains could\nbe reversibly switched by magnetic field pulses without re-\nversing the field direction. Our results could be useful for the\ndesign and development of novel information processing ap-arXiv:2307.15263v1  [cond-mat.mes-hall]  28 Jul 20232\nplications based on the manipulation of magnetic domains in\nnanoscale tracks with modified geometries.\nII. METHODS\nIn this work, we consider an ultra-thin ferromagnetic nan-\notrack with two square notches at the center of the upper and\nlower edges along the length direction, as shown in Fig. 4(a).\nThe default length of the nanotrack in the xdirection is equal\nto100nm, and the default width in the ydirection is set to\n10nm. The thickness of the nanotrack is fixed at 1nm. The\nsimulation is performed by using the functional micromag-\nnetics package Object Oriented MicroMagnetic Framework\n(OOMMF) [43]. The mesh size in all simulations is set to\n1×1×1nm3, which ensures good computational accuracy.\nThe open boundary conditions are applied in the xandydi-\nrections.\nWe assumed that the nanotrack is made of typical permal-\nloy, and the magnetic parameters are [44]: the saturation\nmagnetization MS= 860 kA m−1, the exchange constant\nA= 13 pJ m−1, and the magnetic anisotropy equals zero. In\nthe simulation, the magnetization dynamics in the permalloy\nnanotrack is controlled by the Landau-Lifshitz-Gilbert (LLG)\nequation [43],\n∂tm=−γ0m×heff+α(m×∂tm), (1)\nwheremis the reduced magnetization (i.e., m=M/M S),t\nis the time, γ0is the absolute gyromagnetic ratio, and αis the\nGilbert damping parameter with the default value being 0.3.\nheff=−1\nµ0MS·δε\nδmis the effective field, where µ0andεare\nthe vacuum permeability constant and average energy density,\nrespectively. The energy terms considered in the simulation\ninclude the ferromagnetic exchange energy, the applied mag-\nnetic field energy, and the demagnetization energy [43], as\nexpressed in the average energy density below\nε=A(∇m)2−µ0MS(m·Ha)−µ0MS\n2(m·Hd),(2)\nwhere Ais the ferromagnetic exchange constant, Hais the\napplied magnetic field, and Hdis the demagnetization field.\nNote that the magnetic anisotropy energy equals zero in our\nmodel as the permalloy nanotrack has no crystalline magnetic\nanisotropy [44], while the demagnetization effect may result\nin certain magnetic shape anisotropy.\nIII. RESULTS AND DISCUSSION\nA. A two-dimensional macrospin model\nBefore discussing the micromagnetic simulation results, we\nfirst analyze the magnetic field-induced switching of the in-\nplane magnetization in a nanotrack with a simplified two-\ndimensional macrospin model. As depicted in Fig. 1, we ne-\nglect the spatial variation of the magnetization within the left\nFIG. 1. Schematic illustration of a two-dimensional macrospin\nmodel describing the a single spin driven by a magnetic field pulse.\nThe angle between the spin vector direction and the +xdirection is\ndefined as θ. The magnetic field pulse is applied in the +ydirection.\nFIG. 2. The total energy of the macrospin system as a function of\nθforK= 1 andBy= 0−2.5.KandByare simplified param-\neters, which control the shape anisotropy magnitude and magnetic\nfield strength, respectively.\nhalf of the nanotrack and assume that the magnetization is de-\nscribed by a two-dimensional single macroscopic spin vector\nn= (nx, ny). We define the angle between the spin vector\ndirection and the +xdirection as θ, and therefore, the spin\nvector could be expressed as\nn= ˆxcosθ+ ˆysinθ. (3)\nTo mimic the demagnetization effect, we consider a uniaxial\nshape magnetic anisotropy with the xaxis being the easy axis.\nWe also consider a magnetic field pulse applied in the +ydi-\nrection. Thus, the energy of the system is given as\nE=−Kcos2θ−Bysinθ, (4)\nwhere KandByare simplified parameters that control the\nanisotropy magnitude and magnetic field strength, respec-\ntively. In Fig. 2, it can be seen that the total energy Ereaches\nits maximum value when θ= 90◦in the absence of the mag-\nnetic field (i.e., By= 0), while it reaches its minimum value\nwhen θ= 90◦upon the application of a strong magnetic field3\nFIG. 3. Field-induced reorientation of a single macrospin. θas func-\ntions of time for By= 1.0,2.5, and 3.0. The shape magnetic\nanisotropy strength is fixed at K= 0.5. The magnetic field is ap-\nplied during t= 0−500, as indicated by the gray background.\n(e.g., By= 2.5). However, it should be noted that the energy\nminimum may be found at θ= 0◦−90◦andθ= 90◦−180◦,\nwhich depends on the competition between the shape mag-\nnetic anisotropy Kand applied magnetic field By. In order\nto drive the switching of the spin, the strength of the applied\nmagnetic field pulse should be large enough to create an en-\nergy profile such as the curve for K= 1 andBy= 2.5in\nFig. 2, where θmust reach 90◦when the magnetic field is\napplied for certain time.\nTo qualitatively demonstrate the field-induced spin reorien-\ntation, we further obtain the Euler-Lagrange equation [45, 46]\nfor the two-dimensional macroscopic model from the La-\ngrangian\nL=−˙θ+Kcos2θ+Bysinθ, (5)\nand the Euler-Lagrange equation is given as\nd\ndt∂L\n∂˙θ−∂L\n∂θ=−∂\n∂˙θα\n2˙θ2, (6)\nwithαbeing the damping parameter. Thus, the LLG equation\nis obtained by polar coordinate θas\n˙θ=−2K\nαsinθcosθ+By\nαcosθ. (7)\nIn Fig. 3, we show three representative situations of the field-\ninduced spin dynamics by numerically solving Eq. (7) with\nthe simplified assumption that K=α= 0.5. As shown in\nFig. 3, when a relatively weak magnetic field pulse of By= 1\nis applied during t= 0−500and the system is relaxed un-\ntilt= 2000 , it can be seen that the spin vector first rotates\ntoward the +ydirection (i.e., θ→90◦) but finally relaxes\nto its initial direction (i.e., θ= 0◦). This situation suggests\nthat a weak magnetic field pulse may not be able to trigger a\nfull180◦reorientation of the magnetization in the nanotrack.\nWhen a stronger magnetic field pulse of By= 2.5is appliedduring t= 0−500, it shows that the spin vector is mag-\nnetized to the +ydirection soon upon the application of the\npulse (i.e., θ= 90◦). However, we note that θremains at\n90◦when the magnetic field pulse is turned off. The reason\nis due to the symmetry breaking problem, where the spin vec-\ntor direction gets stuck on an unstable equilibrium. Namely,\nthe spin vector is balanced at the maximum energy point and\ndoes not know which direction to rotate. In micromagnetic\nsimulations discussed below, the symmetry would be broken\ndue to the spin precession. Therefore, we apply a relatively\nstrong magnetic field pulse of By= 3 during t= 0−500,\nand meanwhile, consider a tiny perturbation of the magnetic\nfield direction to avoid the symmetry breaking problem in the\nmacrospin model. Hence, it can be seen from Fig. 3 that the\nspin vector first rotates to the +ydirection (i.e., θ= 90◦)\nwhen the magnetic field pulse is turned on, and then rotates to\nthe−xdirection when the magnetic field pulse is turned off.\nThis situation suggests that a strong magnetic field pulse could\nbe able to result in a full 180◦reorientation of the magnetiza-\ntion in the nanotrack. The prerequisite is that the magnetic\nfield pulse is strong and long enough to drive the spin to over-\ncome the energy barrier due to the shape magnetic anisotropy.\nB. Reversible reorientation phenomenon\nWe computationally demonstrate the typical reversible re-\norientation of magnetic domains in the notched nanotrack\nwith default geometry and material parameters, where the\nmagnetization dynamics is driven by in-plane magnetic field\npulses applied in the width direction of the nanotrack, i.e.,\npointing at the +ydirection. As shown in Fig. 4(a), the initial\nstate in the nanotrack at t= 0ps is a relaxed metastable head-\nto-head magnetization configuration, which is defined as the\nstate “0” as it could be used to carry the binary information bit\n“0”. Namely, the magnetization in the left half and right half\nof the nanotrack are almost aligned along the +x(red) and −x\n(blue) directions, respectively. We point out that the read-out\nof state “0” could be realized by placing two magnetic tun-\nnel junction (MTJ) reader sensors upon the left half and right\nhalf of the nanotrack. The two square notches at the center\nof the nanotrack are employed to stabilize and fix the position\nof a head-to-head or tail-to-tail domain wall at the nanotrack\ncenter. In this work, the default length and width of the notch\nare equal to 4nm and 2nm, respectively. The notches fabri-\ncated at nanotrack edges are usually used to pin domain walls\nand affect the domain wall dynamics [47–52]. The domain\nwall can exist in the nanotrack without the notches, however,\nit may be easily displaced or destroyed if it is not pinned by\nthe notches, which we will discuss later in this paper.\nWe apply a single magnetic field pulse in the +ydirection\nto drive the dynamics of magnetic domains in the nanotrack.\nThe magnetic field pulse strength is set to By= 100 mT, and\nthe pulse length is set to τ= 200 ps, which is applied during\nt= 20−220ps [see Fig. 4(b)]. It can be seen that the magneti-\nzation in the nanotrack rotate to the +ydirection driven by the\nmagnetic field pulse during t= 20−220ps, and the magne-\ntization continue to rotate mainly in the nanotrack plane after4\nFIG. 4. Field-induced reorientation of magnetic domains in a notched nanotrack leading to the switching from state “0” to state “1”. (a) Top-\nview snapshots showing the magnetization configurations in the notched nanotrack at selected times. The arrows represent the magnetization\ndirections. The in-plane magnetization component ( mx) is color coded: red is pointing at the +xdirection, blue is pointing at the −x\ndirection, and white is pointing at the ±ydirections. The initial state “0” is a metastable head-to-head magnetization configuration, while the\nfinal state “1” is a metastable tail-to-tail magnetization configuration. The switching from state “0” to state “1” is realized by the magnetization\nreorientation driven by a magnetic field pulse applied at the +ydirection. The field strength By= 100 mT, and the pulse length τ= 200 ps\n(i.e., applied during t= 20−220ps). The system is relaxed until t= 1000 ps. Here, α= 0.3,A= 13 pJ m−1, andMS= 860 kA m−1. (b)\nThe applied magnetic field strength Byas a function of time. (c) The reduced in-plane magnetization components for the left half ( mL\nx) and\nright half ( mR\nx) of the nanotrack as functions of time. (d) The reduced in-plane magnetization component myas a function of time. (e) The\nreduced out-of-plane magnetization component mzas a function of time. (f) The total energy as a function of time. (g) The exchange energy\nas a function of time. (h) The demagnetization energy as a function of time. (i) The applied magnetic field energy (i.e., Zeeman energy) as a\nfunction of time. The magnetic field pulse duration is indicated by the gray background.\nthe pulse application and finally relax to a tail-to-tail magneti-\nzation configuration, as shown in Fig. 4(a) at t= 1000 ps (see\nSupplemental Video 1 in Ref. 53). Such a tail-to-tail magneti-\nzation configuration is thus defined as the state “1” as it could\nbe used to carry the binary information bit “1”. Therefore,\nthe field-induced reorientation of magnetic domains leads to\na smooth switching from state “0” to state “1”, mimicking a\nunitary NOT operation that is fundamental to bit manipulationin spintronic information storage and processing devices.\nThe reorientation of the magnetic domains in the left half\nand right half of the nanotrack could be clearly indicated by\nthe reduced in-plane magnetization components for the left\nhalf ( mL\nx) and right half ( mR\nx) of the nanotrack, respectively\n[see Fig. 4(c)]. During the field-driven reorientation and re-\nlaxation, both the in-plane magnetization component my[see\nFig. 4(d)] and out-of-plane magnetization component mz[see5\nFIG. 5. Field-induced reorientation of magnetic domains in a notched nanotrack leading to the switching from state “1” to state “0”. (a) Top-\nview snapshots showing the magnetization configurations in the notched nanotrack at selected times. The arrows represent the magnetization\ndirections. The in-plane magnetization component ( mx) is color coded: red is pointing at the +xdirection, blue is pointing at the −xdirection,\nand white is pointing at the ±ydirections. The initial state “1” is a metastable tail-to-tail magnetization configuration, while the final state\n“0” is a metastable head-to-head magnetization configuration. The switching from state “1” to state “0” is realized by the magnetization\nreorientation driven by a magnetic field pulse applied at the +ydirection. The field strength By= 100 mT, and the pulse length τ= 200 ps\n(i.e., applied during t= 20−220ps). The system is relaxed until t= 1000 ps. Here, α= 0.3,A= 13 pJ m−1, andMS= 860 kA m−1. (b)\nThe applied magnetic field strength Byas a function of time. (c) The reduced in-plane magnetization components for the left half ( mL\nx) and\nright half ( mR\nx) of the nanotrack as functions of time. (d) The reduced in-plane magnetization component myas a function of time. (e) The\nreduced out-of-plane magnetization component mzas a function of time. (f) The total energy as a function of time. (g) The exchange energy\nas a function of time. (h) The demagnetization energy as a function of time. (i) The applied magnetic field energy (i.e., Zeeman energy) as a\nfunction of time. The magnetic field pulse duration is indicated by the gray background.\nFig. 4(e)] also vary with time, however, the variation ampli-\ntude of mzis much smaller as the magnetization lie in the x-y\nplane. The time-dependent total energy [see Fig. 4(f)], ex-\nchange energy [see Fig. 4(g)], demagnetization energy [see\nFig. 4(h)], and Zeeman energy [see Fig. 4(i)] are given in\nFig. 4. The total energy, exchange energy, and Zeeman en-\nergy decrease during the field-driven reorientation, while the\ndemagnetization energy increases during the reorientation pe-riod. When the magnetic field pulse is turned off at t= 220\nps, the total energy shows a sharp increase and then decreases\nto its initial value. The demagnetization energy also decreases\nto its initial value. Therefore, it can be seen that the forma-\ntion of the tail-to-tail magnetization configuration after the\npulse application is favored by the demagnetization effect, al-\nthough the exchange energy will slightly increase due to the\nformation of a domain wall structure between the two square6\nFIG. 6. Reversible reorientation of magnetic domains in a notched\nnanotrack leading to the switching between state “0” and state “1” in\na controlled manner. (a) The reduced in-plane magnetization com-\nponents for the left half ( mL\nx) and right half ( mR\nx) of the nanotrack as\nfunctions of time. (b) The total energy as a function of time. Here,\nα= 0.3,A= 13 pJ m−1, and MS= 860 kA m−1. The field\nstrength By= 100 mT. The pulse length τ= 200 ps, which is indi-\ncated by the gray background.\nnotches.\nWe also apply same in-plane magnetic field pulse to drive\nthe system with the state “1” being the initial state, namely, the\ninitial state is a tail-to-tail magnetization configuration iden-\ntical to the one obtained at t= 1000 ps in Fig. 4(a). The\ngeometric and material parameters are the same as that used\nin Fig. 4 and the magnetic field pulse with By= 100 mT and\nτ= 200 ps is also applied at the +ydirection. The results\nare given in Fig. 5, where we find that the magnetic domain\nreorientation induced by the magnetic field pulse could also\nlead to the smooth switching from state “1” to state “0”. This\nmeans that the switching between state “0” and state “1” is\nreversible due to the magnetic domain reorientation induced\nby magnetic field pulses applied at the +ydirection.\nHence, as shown in Fig. 6, we demonstrate the reversible\nswitching between state “0” and state “1” by applying a se-\nquence of magnetic field pulses to drive the nanotrack with an\ninitial state of head-to-head magnetization configuration (i.e.,\nstate “0”), where the direction of magnetic field pulses is fixed\nat the +ydirection (see Supplemental Video 2 in Ref. 53). In\nFig. 6, it can be seen that the reversible switching between\nstate “0” and state “1” can be reliably achieved provided that\nthe spacing between two adjacent pulses is long enough for\nthe system to fully relax into a metastable state.\nC. Geometry and parameter dependence\nIn Fig. 7, we further study the field-induced reorientation\ndynamics of magnetic domains for different strengths and\nlengths of the applied magnetic field pulse. The nanotrack\ngeometry and magnetic parameters are the same as that used\nin Fig. 4. The initial relaxed state is a head-to-head magneti-\nzation configuration referred as the state “0”. A single mag-\nFIG. 7. Switching phase diagram of a notched nanotrack driven by\na single magnetic field pulse for different pulse length τand field\nstrength By. The initial relaxed state before the pulse application is\nthe state “0” with a head-to-head magnetization configuration. The\nmagnetic field pulse is applied at the +ydirection. The switching\nfrom state “0” to state “1” is realized for certain values of τandBy.\nHere, α= 0.3,A= 13 pJ m−1, andMS= 860 kA m−1.\nnetic field pulse with strength Byand length τis applied in\nthe+ydirection and the system is relaxed after the pulse ap-\nplication until t= 1000 ps. It shows that the switching from\nstate “0” to state “1” could be realized only when the field\nstrength Byis larger than certain threshold value. For the\nmagnetic field pulse with a suitable strength that is enough to\ninduce the magnetic domain reorientation, only certain ranges\nof pulse lengths τcould result in the switching from state “0”\nto state “1”. When a larger Byis applied, both the maximum\nand minimum values of τrequired for the switching will be\nreduced as a general trend.\nIn Fig. 8, we explore the field-induced reorientation dynam-\nics of magnetic domains for different intrinsic magnetic mate-\nrial parameters. The nanotrack geometry and default magnetic\nparameters are the same as that used in Fig. 4. The initial re-\nlaxed state is a head-to-head magnetization configuration re-\nferred as the state “0”. A single magnetic field pulse with\nstrength By= 100 mT and length τ= 200 ps is applied in the\n+ydirection during t= 20−220ps. The system is relaxed af-\nter the pulse application until t= 1000 ps. It is found that the\ndamping parameter αmay obviously affect the magnetic do-\nmain reorientation dynamics [see Figs. 8(a) and 8(b)], where\nwe focus on the time-dependent reduced in-plane magnetiza-\ntion components for the left half mL\nxand right half mR\nxof the\nnanotrack. It shows that the field-induced switching from state\n“0” to state “1” could be realized when α= 0.25−0.35or\nwhen α= 0.05. For relatively larger α= 0.5, the state “0”\nis not switched after the pulse application, although the mag-7\nFIG. 8. Dynamics of magnetic domains driven by a single mag-\nnetic field pulse in notched nanotracks with different material pa-\nrameters. (a) Time-dependent reduced in-plane magnetization com-\nponent for the left half of the nanotrack mL\nxfor different damping\nparameter α. (b) Time-dependent reduced in-plane magnetization\ncomponent for the right half of the nanotrack mR\nxfor different α. (c)\nTime-dependent mL\nxfor different exchange constant A. (d) Time-\ndependent mR\nxfor different A. (e) Time-dependent mL\nxfor different\nsaturation magnetization MS. (f) Time-dependent mR\nxfor different\nMS. Here, the initial relaxed state before the pulse application is\nthe state “0” with a head-to-head magnetization configuration. The\nmagnetic field pulse is applied at the +ydirection. The field strength\nBy= 100 mT, and the pulse length τ= 200 ps (i.e., applied during\nt= 20−220ps as indicated by the gray background). The system\nis relaxed until t= 1000 ps.\nnetic domains are reorientated to the +ydirection by the field\npulse around t= 220 ps.\nAs shown in Figs. 8(c) and 8(d), the field-induced orien-\ntation of magnetic domains could lead to the a successful\nswitching from state “0” to state “1” for a wide range of ex-\nchange constant. Similarly, the switching from state “0” to\nstate “1” could also be realized for a wide range of satura-\ntion magnetization. These results suggest that the sensitivity\nof the field-induced reorientation of magnetic domains in the\ngiven nanotrack to damping parameter is higher than that to\nmagnetic parameters, indicating the importance of the mag-\nnetization damping precession on the outcome of switching.\nIn Fig. 9, we also investigate the effects of nanotrack geom-\netry on the field-induced reorientation dynamics of magnetic\ndomains as well as the switching between state “0” and state\n“1”. The magnetic parameters are the same as that used in\nFig. 4. The default nanotrack length is set as l= 100 nm, and\nthe default nanotrack width is set as w= 10 nm. We note that\nin this work, for the sake of simplicity, the length and width\nof the edge notch is fixed at 4nm and 2nm, respectively. The\ninitial relaxed state is a head-to-head magnetization configu-\nFIG. 9. Dynamics of magnetic domains driven by a single magnetic\nfield pulse in notched nanotracks with different geometric parame-\nters. (a) Time-dependent reduced in-plane magnetization component\nfor the left half of the nanotrack mL\nxfor different track length l. The\ntrack width is fixed at w= 10 nm. (b) Time-dependent reduced\nin-plane magnetization component for the right half of the nanotrack\nmR\nxfor different l. (c) Time-dependent mL\nxfor different track width\nw. The track length is fixed at l= 100 nm. (d) Time-dependent\nmR\nxfor different w. Here, the initial relaxed state before the pulse\napplication is the state “0” with a head-to-head magnetization con-\nfiguration. The magnetic field pulse is applied at the +ydirection.\nThe field strength By= 100 mT, and the pulse length τ= 200 ps\n(i.e., applied during t= 20−220ps as indicated by the gray back-\nground). The system is relaxed until t= 1000 ps.\nration referred as the state “0”. A single magnetic field pulse\nwith strength By= 100 mT and length τ= 200 ps is applied\nin the +ydirection during t= 20−220ps. The system is\nrelaxed after the pulse application until t= 1000 ps. It can\nbe seen from Figs. 9(a) and 9(b) that the field-induced reori-\nentation of magnetic domains and its consequential switching\nfrom state “0” to state “1” are only realized in nanotracks with\nl= 90−120nm and w= 10 nm. Also, by fixing the nan-\notrack length at l= 100 nm, the field-induced reorientation of\nmagnetic domains and its consequential switching from state\n“0” to state “1” are only realized for nanotracks with w= 10\nnm or w= 20 nm [see Figs. 9(c) and 9(d)]. The different\nshape of the nanotrack will result in different shape anisotropy\ndue to the demagnetization effect, which may favor different\nmagnetization configurations, and may also affect the field-\ninduced reorientation and switching.\nD. Effect of the notches\nFor the purpose of elucidating the effect of the notches on\nthe field-induced magnetic domain reorientation dynamics in\nthe nanotrack, we carry out three simulations with different\nnanotrack geometries to compare with each other. In the first\nsimulation, as shown in Fig. 10(a), the nanotrack geometry\nand magnetic parameters are the same as that used in Fig. 4.\nThe initial relaxed state is a head-to-head magnetization con-\nfiguration referred as the state “0”. A single magnetic field8\nFIG. 10. Field-induced reorientation of magnetic domains in nanotracks with and without the notches. (a) Top-view snapshots showing the\nmagnetization configurations in a default notched nanotrack at selected times. The arrows represent the magnetization directions. The in-plane\nmagnetization component ( mx) is color coded: red is pointing at the +xdirection, blue is pointing at the −xdirection, and white is pointing\nat the ±ydirections. The initial state “0” is a metastable head-to-head magnetization configuration, while the final state “1” is a metastable\ntail-to-tail magnetization configuration. The switching from state “0” to state “1” is realized by the magnetization reorientation driven by a\nmagnetic field pulse applied at the +ydirection with a tiny component in the +xdirection. The magnetic field B= (0.005,100,0)mT, and\nthe pulse length τ= 200 ps (i.e., applied during t= 20−220ps). The system is relaxed until t= 1000 ps. Here, α= 0.3,A= 13 pJ m−1,\nandMS= 860 kA m−1. The reference frame is the same as that in Fig. 4. (b) Top-view snapshots showing the magnetization configurations\nin a nanotrack without notches at selected times. The length and width of the nanotrack are the same as that in (a). All material and magnetic\nfield parameters are the same as that in (a). The initial state is a metastable head-to-head magnetization configuration, while the final state\nis a stable uniform magnetization configuration. (c) Top-view snapshots showing the magnetization configurations in two nanotracks without\nnotches at selected times. The system is the same as that in (a) but the area between the two notches is totally etched away. All material\nand magnetic field parameters are the same as that in (a). Treating the two nanotracks as a system, the initial state can be seen as a stable\nhead-to-head magnetization configuration, while the final state is identical to the initial state.\nFIG. 11. The reduced in-plane magnetization components for the left\nhalf (mL\nx) and right half ( mR\nx) of the nanotrack system as functions\nof time. (a) Time-dependent mL\nxandmR\nxfor the system given in\nFig. 10(a). (b) Time-dependent mL\nxandmR\nxfor the system given in\nFig. 10(b). (x) Time-dependent mL\nxandmR\nxfor the system given in\nFig. 10(c). The magnetic field pulse duration is indicated by the gray\nbackground.\npulse of B= (0.005,100,0)mT is applied to drive the mag-\nnetization dynamics in the nanotrack. Namely, the magnetic\nfield is applied mainly at the +ydirection but with a tiny com-\nponent in the +xdirection. The magnetic field is applied for\n200ps during t= 20−220ps, and then the system is relaxed\nafter the pulse application until t= 1000 ps. We note that\nwe consider a tiny component of magnetic field in the xdirec-\ntion to mimic an additional tiny perturbation in the field pulse,\nwhich would highlight the effect of the notches in enhancing\nthe stability of the domain wall configuration during the reori-\nentation. It can be seen that the applied magnetic field pulse\nFIG. 12. The total energies as a function of time for the three systems\ngiven in Fig. 10. The magnetic field pulse duration is indicated by\nthe gray background.\ndrives the reorientation of magnetic domains, and leads to the\nswitching from state “0” to state “1” after the relaxation (see\nSupplemental Video 3 in Ref. 53). The switching from state\n“0” to state “1” can also be seen from the time-dependent mL\nx\nandmR\nxof the system [see Fig. 11(a)].\nHowever, the snapshots selected at different times in\nFig. 10(a) show that during the magnetic domain reorienta-\ntion the domain wall structure at the center of the nanotrack\nfirst moves slightly toward the −xdirection (i.e., away from\nthe notches), and then moves toward the +xdirection (i.e.,\nback to the notches) [cf. Figs. 4(a) and 10(a)]. The domain\nwall motion toward the −xdirection is caused by the tiny +x\ncomponent of the applied magnetic field pulse, while its mo-\ntion toward the +xdirection is due to the attractive interaction9\nFIG. 13. Top-view snapshots showing the initially relaxed magneti-\nzation configurations in a default notched nanotrack with and without\nthe demagnetization effect. (a) An ideal head-to-head magnetization\nconfiguration given as the initial state before the relaxation. The ar-\nrows represent the magnetization directions. The in-plane magne-\ntization component ( mx) is color coded: red is pointing at the +x\ndirection, blue is pointing at the −xdirection, and white is point-\ning at the ±ydirections. (b) The relaxed state for the system with\nthe demagnetization effect is a metastable head-to-head magnetiza-\ntion configuration. (c) The relaxed state for the system without the\ndemagnetization effect is a uniform state magnetized along the +y\ndirection.\nbetween the domain wall and the notches.\nWe note that such a displacement of the domain wall from\nthe center of the nanotrack could result in the instability of the\nhead-to-head and tail-to-tail magnetization configurations in a\nclean nanotrack without any pinning effect. For example, as\nshown in Fig. 10(b), if we remove the notches from the center\nof the nanotrack while keeping other parameters unchanged,\nthe magnetic field pulse will result in the transition from the\ninitial head-to-head magnetization configuration to a uniform\nferromagnetic state by driving the domain wall structure out\nof the nanotrack (see Supplemental Video 4 in Ref. 53). The\ntransition is also indicated by the time-dependent mL\nxandmR\nx\nof the system [see Fig. 11(b)]. Here we point out that the\nreorientation could, in principle, be realized in a nanotrack\nwithout the notches under the exact same protocol used in\nFig. 4, i.e., B= (0,100,0)mT. However, this is only true\nwhen the head-to-head or tail-to-tail domain wall is initially\nplaced at the exact center of the nanotrack, and the applied\nmagnetic field is perfectly aligned perpendicularly to the nan-\notrack. Otherwise, the head-to-head or tail-to-tail domain wall\ncould easily be destroyed as demonstrated in Fig. 10(b).\nWe also apply a same magnetic field pulse of B=\n(0.005,100,0)mT to drive the system where the area between\ntwo notches is totally etched away, as shown in Fig. 10(c). It\ncan be seen that the initial and relaxed final states of the sys-\ntem remain unchanged after the application of the magnetic\nfield pulse (see Supplemental Video 5 in Ref. 53). The mag-\nnetization dynamics in the left and right parts of the system\nare also described by the time-dependent mL\nxandmR\nx, respec-\ntively [see Fig. 11(c)].\nIn Fig. 12, we show the time-dependent total energies for\nthe three simulations with different nanotrack geometries dis-\ncussed above. It shows that the nanotrack with two notchesat the upper and lower edges of the nanotrack center can\nrealize the field-induced switching between two metastable\nstates with the same energy (i.e., the head-to-head and tail-to-\ntail magnetization configurations). In the nanotrack without\nnotches, the final state is ground state with an energy lower\nthan that of the initial metastable state. In the nanotrack sys-\ntem with the area between two notches being totally etched\naway (i.e., a system of two shorter nanotracks), the initial\nand final states are ground states with the energy close to the\nground state in the longer nanotrack without notches. We also\npoint out that the demagnetization effect plays an important\nrole in the stabilization of the head-to-head or tail-to-tail mag-\nnetization configuration in our model. As shown in Fig. 13, if\nwe turn off the demagnetization in the simulation, the initial\nhead-to-head magnetization configuration cannot be relaxed\nand will evolve into an uniform state during the initial relax-\nation before the application of the magnetic field pulse. The\nresults discussed in this section suggest that the notches at the\nnanotrack center can pin the domain wall structure, and thus,\nenhance the stability of the head-to-head and tail-to-tail mag-\nnetization configurations, which are important for the realiza-\ntion of the fixed field-induced switching between state “0” and\nstate “1”.\nE. Implications for classical and quantum applications\nThe reversible switching between state “0” and state “1” in\nthe nanotrack without changing the driving field direction may\nhave important implications for both classical and quantum\nspintronic applications. For example, such a controlled re-\nversible single-bit flipping operation driven by magnetic field\napplied in a fixed in-plane direction could be used to build a\nmemory device, where information is stored in arrays of nan-\notracks with one bit per track, similar to the bit-patterned mag-\nnetic recording architecture [41].\nOn the other hand, recent studies have suggest that\nnanoscale magnetic textures, including domain walls [54],\nmerons [55], and skyrmions [56–58], can be used as build-\ning blocks in quantum computation. As the reversible switch-\ning between state “0” and state “1” in the given system could\nbe realized without reversing the sign of the applied in-plane\nmagnetic field, the system may also, in principle, make it pos-\nsible to build a quantum Pauli-X gate as long as the size of\nthe nanotrack is of a few nanometers [54] in real experiments.\nNamely, magnetic textures with dimensions down to a few\nnanometers could be quantum objects [54–60].\nNote that the Pauli-X gate is a quantum version of the clas-\nsical NOT gate, which transforms the pure state |0⟩to|1⟩\nand vice versa. Namely, the Pauli-X gate is an operation that\ncan be expressed by the Pauli matrix σxacting on a single\nqubit, for example, σx|0⟩=|0⟩⟨1|0⟩+|1⟩⟨0|0⟩=|1⟩and\nσx|1⟩=|0⟩⟨1|1⟩+|1⟩⟨0|1⟩=|0⟩, where σxmay correspond\nto an in-plane magnetic field with a fixed direction in real ex-\nperiments. It should be noted that it is not required to fix the\ndirection of the driving field in the operation of the classical\nNOT gate.\nBesides, for a classical system, the reversible switching be-10\nFIG. 14. Schematic illustrations showing possible experimental se-\ntups for the initialization, readout, and reset operations. (a) The ini-\ntialization and reset functions could be realized by using two mag-\nnetic tunnel junction (MTJ) elements with fixed layers placed upon\nthe left and right parts of the notched ferromagnetic nanotrack (i.e.,\nfree layers), respectively. (b) The readout operation could be real-\nized by using the above-mentioned two MTJ elements as the readout\nsensors.\ntween state “0” and state “1” induced by magnetic fields with a\nfixed direction may simplify the bit flipping operation as only\nthe strength of the magnetic field needs to be programmed.\nWe note that the magnetic field applied in the device plane\ncould be realized by using a micro coil covering the top and\nbottom surfaces of the nanotrack. One may also fabricate a\nnanotrack parallel and attached (i.e., underneath or beyond)\nto a flat current wire, so that the Oersted field in the in-plane\ndirection could be used to drive the magnetization dynamics\nin the nanotrack.\nIn Fig. 14, we also provide two schematic illustrations\nshowing possible experimental setups for the initialization,\nreadout, and reset operations. First, the initial head-to-head\nmagnetization configuration in the nanotrack could be created\nby using two magnetic tunnel junction (MTJ) elements placed\nupon the left and right parts of the notched nanotrack, where\nthe fixed layers in the two MTJ elements have opposite uni-\nform magnetization configurations as depicted in Fig. 14. By\ninjecting vertical spin currents through the two MTJ elements,\none should be able to create or reset the head-to-head mag-\nnetization configuration in the notched nanotrack. Besides,\nthe two MTJ elements could also be used as the readout sen-\nsors to detect the tail-to-tail magnetization configuration in the\nnotched nanotrack after the magnetic domain reorientation.\nIV . CONCLUSION\nIn conclusion, we have studied the reorientation of head-to-\nhead and tail-to-tail magnetic domains in a notched nanotrack\ndriven by in-plane magnetic field pulses, where the direction\nof magnetic field is fixed at the +ydirection, i.e., perpen-\ndicular to the length direction of the nanotrack. It is foundthat the field-induced reorientation of magnetic domains could\nlead to reversible switching between a head-to-head magneti-\nzation configuration and a tail-to-tail magnetization configu-\nration. The head-to-head and tail-to-tail magnetization config-\nurations in the nanotrack could be treated as binary bit states\n“0” and “1”, respectively. Thus, the reversible switching be-\ntween state “0” and state “1” could be induced by apply-\ning in-plane magnetic field pulses perpendicular to the nan-\notrack without changing the direction of the magnetic field.\nSuch a phenomenon is in stark contrast to typical field-driven\nand current-driven domain switching and domain wall motion,\nwhere the reversible switching and motion are obtained by re-\nversing the sign of driving force. The reorientation dynamics\nof magnetic domains depends on both the intrinsic magnetic\nparameters, the nanotrack geometry, and the applied magnetic\nfield pulse profile. The damping parameter and nanotrack\ngeometry may play important roles on the reorientation dy-\nnamics and the consequential switching between state “0” and\nstate “1”, however, the overall dynamic process (i.e., includ-\ning the field-driven reorientation and zero-field relaxation) is\nrobust to the variations of magnetic parameters, including the\nexchange constant and saturation magnetization. Namely, the\nreorientation operation is robust for a wide parameter range\nof exchange constant and saturation magnetization, although\nthe suitable parameter spaces for the reorientation are not ex-\ntremely wide for parameters such as the damping parameter.\nIt should be emphasized that the notches at the center of the\nupper and lower nanotrack edges could enhance the stability\nand reliability of the head-to-head and tail-to-tail magnetiza-\ntion configurations, especially in the presence of a perturba-\ntion in the driving field. Our results could be important for\nunderstanding the field-driven dynamics of magnetic domains\nin narrow nanotracks with notches, and may also provide a\nnew way for the design of novel spintronic devices. We also\nbelieve that our computational findings will stimulate more\ntheoretical and experimental efforts to explore novel magneti-\nzation dynamics in complex and artificial nanostructures.\nACKNOWLEDGMENTS\nX.Z. and M.M. acknowledge support by CREST, the\nJapan Science and Technology Agency (Grant No. JP-\nMJCR20T1). M.M. also acknowledges support by the Grants-\nin-Aid for Scientific Research from JSPS KAKENHI (Grant\nNo. JP20H00337). J.X. was a JSPS International Re-\nsearch Fellow supported by JSPS KAKENHI (Grant No.\nJP22F22061). O.A.T. acknowledges support by the Aus-\ntralian Research Council (Grant No. DP200101027), the Rus-\nsian Science Foundation (Grant No. 21-42-00035), the Co-\noperative Research Project Program at the Research Insti-\ntute of Electrical Communication, Tohoku University (Japan),\nand by the NCMAS grant. G.Z. acknowledges support by\nthe National Natural Science Foundation of China (Grants\nNo. 51771127, No. 51571126, and No. 51772004),\nand Central Government Funds of Guiding Local Scien-\ntific and Technological Development for Sichuan Province\n(Grant No. 2021ZYD0025). Y .Z. acknowledges sup-11\nport by the Guangdong Basic and Applied Basic Research\nFoundation (Grant No. 2021B1515120047), the Guang-\ndong Special Support Project (Grant No. 2019BT02X030),\nthe Shenzhen Fundamental Research Fund (Grant No.\nJCYJ20210324120213037), the Shenzhen Peacock Group\nPlan (Grant No. KQTD20180413181702403), the Pearl River\nRecruitment Program of Talents (Grant No. 2017GC010293),\nand the National Natural Science Foundation of China (Grant\nNo. 11974298). X.L. acknowledges support by the Grants-\nin-Aid for Scientific Research from JSPS KAKENHI (Grants\nNo. JP20F20363, No. JP21H01364, No. JP21K18872,and No. JP22F22061). M.E. acknowledges support by the\nGrants-in-Aid for Scientific Research from JSPS KAKENHI\n(Grant No. JP23H00171). M.E. also acknowledges support\nby CREST, JST (Grant No. JPMJCR20T2).\nM.E. and X.Z. conceived the idea. M.M. and X.L. coor-\ndinated the project. X.Z. and J.X. performed the computa-\ntional simulation and theoretical analysis supervised by M.M.,\nX.L., and M.E. The manuscript is drafted by X.Z. and J.X.,\nand revised with input from O.A.T., G.Z., Y .Z., M.M., X.L.,\nand M.E. All authors discussed the results and reviewed the\nmanuscript. X.Z. and J.X. contributed equally to this work.\n[1] C. Chappert, A. Fert, and F. Van Dau, Nat. Mater. 6, 813 (2007).\n[2] A. Brataas, A. Kent, and H. Ohno, Nat. Mater. 11, 372 (2012).\n[3] S. S. P. Parkin and S. H. Yang, Nat. Nanotech. 10, 195 (2015).\n[4] D. Kumar, T. Jin, S. Al Risi, R. Sbiaa, W. S. Lew, and S. N.\nPiramanayagam, IEEE Trans. Magn. 55, 1 (2019).\n[5] R. Bläsing, A. A. Khan, P. C. Filippou, C. Garg, F. Hameed, J.\nCastrillon, and S. S. P. Parkin, Proc. IEEE 108, 1303 (2020).\n[6] N. L. Schryer and L. R. Walker, J. Appl. Phys. 45, 5406 (1974).\n[7] M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno, Nature\n428, 539 (2004).\n[8] A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and T.\nShinjo, Phys. Rev. Lett. 92, 077205 (2004).\n[9] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004).\n[10] A. Thiaville, Y . Nakatani, J. Miltat, and Y . Suzuki, Europhys.\nLett. 69, 990 (2005).\n[11] D. C. Ralph and M. D. Stiles, J. Magn. Magn. Mater. 320, 1190\n(2008).\n[12] S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320, 190\n(2008).\n[13] M. Hayashi, L. Thomas, R. Moriya, C. Rettner, and S. S. P.\nParkin, Science 320, 209 (2008).\n[14] O. A. Tretiakov, D. Clarke, Gia-Wei Chern, Ya. B. Bazaliy, and\nO. Tchernyshyov, Phys. Rev. Lett. 100, 127204 (2008).\n[15] D. J. Clarke, O. A. Tretiakov, G.-W. Chern, Ya. B. Bazaliy, and\nO. Tchernyshyov, Phys. Rev. B 78, 134412 (2008).\n[16] O. A. Tretiakov, Y . Liu, and Ar. Abanov, Phys. Rev. Lett. 105,\n217203 (2010).\n[17] O. A. Tretiakov, Y . Liu, and Ar. Abanov, Phys. Rev. Lett. 108,\n247201 (2012).\n[18] A. Thiaville, S. Rohart, E. Jue, V . Cros, and A. Fert, Europhys.\nLett. 100, 57002 (2012).\n[19] S. Emori, U. Bauer, S.-M. Ahn, E. Martinez, and G. S. D.\nBeach, Nat. Mater. 12, 611 (2013).\n[20] S. H. Yang, K. S. Ryu, and S. S. P. Parkin, Nat. Nanotech. 10,\n221 (2015).\n[21] K. Sato and O. A. Tretiakov, Appl. Phys. Lett. 108, 122403\n(2016).\n[22] D. Kumar, J. P. Chan, and S. N. Piramanayagam, J. Appl. Phys.\n130, 213901 (2021).\n[23] N. Nagaosa and Y . Tokura, Nat. Nanotech. 8, 899 (2013).\n[24] M. Mochizuki and S. Seki, J. Phys.: Condens. Matter 27,\n503001 (2015).\n[25] G. Finocchio, F. Büttner, R. Tomasello, M. Carpentieri, and M.\nKläui, J. Phys. D: Appl. Phys. 49, 423001 (2016).\n[26] W. Kang, Y . Huang, X. Zhang, Y . Zhou, and W. Zhao, Proc.\nIEEE 104, 2040 (2016).\n[27] R. Wiesendanger, Nat. Rev. Mat. 1, 16044 (2016).\n[28] A. Fert, N. Reyren, and V . Cros, Nat. Rev. Mater. 2, 17031(2017).\n[29] W. Jiang, G. Chen, K. Liu, J. Zang, S. G. Velthuiste, and A.\nHoffmann, Phys. Rep. 704, 1 (2017).\n[30] K. Everschor-Sitte, J. Masell, R. M. Reeve, and M. Kläui, J.\nAppl. Phys. 124, 240901 (2018).\n[31] X. Zhang, Y . Zhou, K. M. Song, T.-E. Park, J. Xia, M. Ezawa,\nX. Liu, W. Zhao, G. Zhao, and S. Woo, J. Phys. Condens. Mat-\nter32, 143001 (2020).\n[32] C. Back, V . Cros, H. Ebert, K. Everschor-Sitte, A. Fert, M.\nGarst, T. Ma, S. Mankovsky, T. L. Monchesky, M. Mostovoy,\nN. Nagaosa, S. S. P. Parkin, C. Pfleiderer, N. Reyren, A. Rosch,\nY . Taguchi, Y . Tokura, K. von Bergmann, and J. Zang, J. Phys.\nD: Appl. Phys. 53, 363001 (2020).\n[33] B. Göbel, I. Mertig, and O. A. Tretiakov, Phys. Rep. 895, 1\n(2021).\n[34] S. Luo and L. You, APL Mater. 9, 050901 (2021).\n[35] C. H. Marrows and K. Zeissler, Appl. Phys. Lett. 119, 250502\n(2021).\n[36] C. Reichhardt, C. J. O. Reichhardt, and M. V . Milosevic, Rev.\nMod. Phys. 94, 035005 (2022).\n[37] D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit,\nand R. P. Cowburn, Science 309, 1688 (2005).\n[38] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D. Gan,\nM. Endo, S. Kanai, J. Hayakawa, F. Matsukura, and H. Ohno,\nNat. Mater. 9, 721 (2010).\n[39] I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V .\nCostache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and\nP. Gambardella, Nature 476, 189 (2011).\n[40] L. Liu, O. J. Lee, T. J. Gudmundsen, D. C. Ralph, and R. A.\nBuhrman, Phys. Rev. Lett. 109, 096602 (2012).\n[41] T. R. Albrecht et al., IEEE Trans. Magn. 51, 0800342 (2015).\n[42] S. Bhatti, R. Sbiaa, A. Hirohata, H. Ohno, S. Fukami, and S. N.\nPiramanayagam, Mater. Today 20, 530 (2017).\n[43] M. J. Donahue and D. G. Porter, “OOMMF User’s Guide, Ver-\nsion 1.0”, Interagency Report NO. NISTIR 6376 (National In-\nstitute of Standards and Technology, Gaithersburg, MD, 1999)\n[http://math.nist.gov/oommf/].\n[44] M.-Y . Im, P. Fischer, K. Yamada, T. Sato, S. Kasai, Y . Nakatani,\nand T. Ono, Nat. Commun. 3, 983 (2012).\n[45] H. Goldstein, Classical Mechanics 3rd Edition (Boston, MA:\nAddison-Wesley), 2002.\n[46] J. Shibata, G. Tatara, and H. Kohno, J. Phys. D: Appl. Phys. 44,\n384004 (2011).\n[47] M. Hayashi, L. Thomas, C. Rettner, R. Moriya, X. Jiang, and\nS. S. P. Parkin, Phys. Rev. Lett. 97, 207205 (2006).\n[48] D. Atkinson, D. S. Eastwood, and L. K. Bogart, Appl. Phys.\nLett. 92, 022510 (2008).\n[49] M.-Y . Im, L. Bocklage, P. Fischer, and G. Meier, Phys. Rev.12\nLett. 102, 147204 (2009).\n[50] J. Akerman, M. Muñoz, M. Maicas, and J. L. Prieto, Phys. Rev.\nB82, 064426 (2010).\n[51] H. Y . Yuan and X. R. Wang, Phys. Rev. B 89, 054423 (2014).\n[52] C. I. L. de Araujo, J. C. S. Gomes, D. Toscano, E. L. M. Paixão,\nP. Z. Coura, F. Sato, D. V . P. Massote, and S. A. Leonel, Appl.\nPhys. Lett. 114, 212403 (2019).\n[53] See Supplemental Material at [URL] for videos showing the\nfield-induced magnetization dynamics in different nanotrack\ngeometries.\n[54] J. Zou, S. Bosco, B. Pal, S. S. P. Parkin, J. Klinovaja, and D.\nLoss, arXiv:2212.12019 (2022).[55] J. Xia, X. Zhang, X. Liu, Y . Zhou, and M. Ezawa, Commun.\nMater. 3, 88 (2022).\n[56] C. Psaroudaki and C. Panagopoulos, Phys. Rev. Lett. 127,\n067201 (2021).\n[57] C. Psaroudaki and C. Panagopoulos, Phys. Rev. B 106, 104422\n(2022).\n[58] J. Xia, X. Zhang, X. Liu, Y . Zhou, and M. Ezawa, Phys. Rev.\nLett. 130, 106701 (2023).\n[59] V . Lohani, C. Hickey, J. Masell, and A. Rosch, Phys. Rev. X 9,\n041063 (2019).\n[60] P. Siegl, E. Y . Vedmedenko, M. Stier, M. Thorwart, and T.\nPosske, Phys. Rev. Research 4, 023111 (2022)."    },    {        "title": "1401.4518v1.Acceleration_of_magnetic_dipoles_by_the_sequence_of_current_turns.pdf",        "content": " 1          Acceleration of magnetic dipoles by the s equence of current turns \n \n                                                             S. N. Dolya \n \n         Joint Institute for Nuclear Research, Joli ot Curie str. 6, Dubna, Russia, 141980 \n \n \nAbstract \n \nAcceleration of magnetic dipoles is carried out by the running gradient of the magnetic field \nformed while sequent switching on the current turns . Magnetic dipoles, with a diameter of  \ndsh  = 60 mm and full length l tot  = 1m, are pre-accelerated by using the gas-dynamic  method to \nspeed V in  = 1 km / s, corresponding to the injection rate in to the main accelerator. To prevent \nthe turning of the dipoles by 180 degrees in the fi eld of the accelerating pulse and focus \nthem, the magnetic dipoles are accelerated inside t he titanium tube. The magnetic dipoles \nhave mass m = 2 kg and acquire the finite speed V fin = 5 km / s on the acceleration length \n L acc  = 300 m. \n \nIntroduction  \n  \n       There is a known [1], method to accelerate t he magnetic dipoles by the \ncurrent pulse moving in the space. Using a large nu mber of turns, in principle, \nallows one to reach a high finite speed of the magn etic dipoles. When the \nelectric current is flowing in the coils, there is the magnetic field which pulls \nthe magnetic dipole inside the coil with the curren t. After the magnetic dipole \npassing through the center of the coil, the magneti c field gradient changes its \nsign - due to this the magnetic dipole begins to ac celerate in the opposite \ndirection, i. e. inhibits. Therefore, to create a c ontinuous acceleration, the \ncurrent in the loop must quickly break off after th e magnetic dipole passing \nthrough the coil center. \n      \n   However, despite the apparent simplicity of this  method, its practical use is \naccompanied with serious difficulties. \n      \n    From the ferromagnetic materials being used for  magnetic dipoles the most \nsuitable is iron having a high specific magnetic mo ment and high Curie \ntemperature. The specific magnetic moment is the pr operty of the substance and \ncannot be increased. Furthermore, since the magneti c dipole must include a jet \nengine with the fuel supply and navigation devices,  the specific magnetic dipole \nmoment will be even less than that of the pure iron . That is why it is not \npossible to achieve a large finite speed of the mag netic dipole by using the \nacceleration method.  2 \n    We explain the details. The magnetic moment per  molecule in iron [2],  \npage 524, is of the n b = 2.219 Bohr magneton. The table value of the Bohr  \nmagneton power [2], page 31, mb = 9.27 * 10 -21  erg / Gs. Taking into account \nthat the atomic weight of iron is:  A Fe  = 56, we find that the magnetic moment \nper nucleon in iron is: mFe  ≈ 2 * 10 -10  eV / (Gs*nucleon) and this value  \nmFe  cannot be increased. Another principle disadvantag e of this device is that \nthe movement of the magnetic dipoles therein is uns table in the longitudinal \ndirection. The reason is that at the attraction of the magnet poles with opposite \nsigns there is phase instability [3]. \n        \n     Thus, if the magnetic dipole is slightly behin d the running current pulse \naccelerating it, then it will turn out to be in a l ower momentum field and, \nfinally, will be forever behind it. If the magnetic  dipole becomes too close to \nthe current pulse, it will fall into a stronger fie ld, being more and more attracted \nto the current pulse at the end. Finally, it will b e ahead of it and turn by  \n180 degrees. \n       \n     From the point of view of mutual positioning o f the accelerating running \npulse and the magnetic dipole, there is the only st able case when the pulse \npushes (not pulls) the magnetic dipole. This means that the region of the phase \n(longitudinal) stability is located on the front sl ope of the traveling pulse. In the \naccelerator technology it is called the principle o f phase stability [3]. \n      \n     The specific magnetic dipole moment can be, pr obably, increased (as \ncompared with iron) by applying a current of the su perconducting layer located \ninside the dipole. The magnetic field gradient, whi ch accelerates the dipole, can \nbe increased by superconductivity. The both ways le ad to increasing the force \naccelerating the dipole: F z = m * dB z / dz, where m - specific magnetic dipole \nmoment, dB z / dz - the magnetic field gradient. \n      \n     We assume that the initial speed of the dipole  is: V sh  = 1 km / s and it is \nachieved by the gas-dynamic acceleration. \n \n1. Opportunity of increasing the specific magnetic moment in the magnetic dipole \n       \n    The specific magnetic dipole moment can be incr eased (as compared with \niron), if to place the Nb 3Sn superconducting winding inside the dipole and le t \nthe ring current flow through it. \n        3    Let us calculate how much the specific magnetic  moment – the magnetic \nmoment per unit of the mass of the magnetic dipole,  will grow  if to put a layer \nof superconducting Nb3Sn with radius r cyl  = 3 cm and thickness δcyl  = 0.2 cm, in \nits cylindrical part with a length equal to l cy1 =40 cm. We assume the current \ndensity in the superconductor to be equal [2], page  312, to J ss  = 3 *10 5 A/cm 2. \nThen, linear density j ss  current (A / cm) in such a superconducting layer i s equal \nto: jss  (A / cm) = J sp  * δcyl  = 6 * 10 4 A / cm. Such linear current density on the \nsurface of the superconductor will create the magne tic field strength equal to  \nHss  (kGs) = 1.226 * j (A / cm ) ≈ 70 kGs, that does not contradict the \nopportunity of achieving the current density of J ss  ≈3 * 10 5 A/cm 2 [2],  \npage 312. The total current flowing in the supercon ducting layer is equal to   \nISC = j ss  *lcyl = 2.4 * 10 6 A, it will lead to the appearance of magnetic mome nt \nMss = I SC  * πrcyl 2 = 6.8 * 10 7A * cm2 or, in the CGS system,  \nMss  = 6.8 * 10 6 erg / Gs. \n \n     The total mass of the superconducting layer ca n be calculated from the \nfollowing: the density of Nb 3Sn superconductor is ρNb3Sn  = 8 g/cm3, the atomic \nmass: A = 400 and the total volume of superconducto r V sp = 150 cm3 contains  \nNNb3Sn  = 7.2 *10 26  nucleons. The specific magnetic moment, the magnet ic \nmoment per unit of the mass (nucleon), is then equa l to: Mss = m ss  /NNb3Sn  =  \n=0.94 * 10 -20  erg / (Gs *nucleon) = 5.9 * 10 -9 eV / (Gs * nucleon), that is \napproximately 30 times greater than in iron [2], pa ge 524. \n \n2. Ways to achieve the required parameters of accel eration \n \n     Let the mass of the superconductor in the magn etic dipole be  \nmNb3Sn  = 1.2 kg, the mass of the jet engine, fuel, naviga tion control devices is \nequal m Fuel = 0.8 kg, then the specific magnetic moment in the magnetic dipole \nwill be equal to: m0 = 3.5 * 10 -9 eV / (Gs *nucleon), that is approximately  \n17 times greater than in iron. The pulse duration o f the magnetic field can be \ndetermined from the following considerations. To pl ace the magnetic dipole on \nthe length of the pulse accelerating it, slowdown w avelength must be of the \norder of: λslow  = 4 m. The time period T 0 of the corresponding wave is \ndetermined from the following relationship: \n \n                                           λslow  = V sh  *T0,                                 ( 1) \n \nwhere we find that T 0 = 4 ms and the wave frequency corresponding to thi s \nperiod is equal to: f 0 = 250 Hz. \n  43. Selecting the thickness of the barrel wall of th e Gauss gun  \n       \n    The region of the phase stability in azimuthall y symmetric wave \ncorresponds to the region of the radial instability . The magnetic dipole will push \nitself from the pole of the same sign, but, first o f all, it will seek to turn around \nby 180 0 and be attracted by the opposite sign poles. To pr event the radial escape \nof the dipole and its reversal by 180 0 in the field of the pulse accelerating is \npossible if to place the dipole inside the titanium  tube whose inner diameter is \nthe same as the outer diameter of the dipole.  The titanium tube wall thickness \nmust be of such value to let the external magnetic field freely without distortion \npenetrate inside. It means that it should be much s maller than the skin-layer \ndepth in titanium. \n        \n     Electrical resistance of copper ρCu  = 1.67 *10 -6 Ohm * cm, titanium \nρTi  = 55 *10 -6 Ohm *cm, [2], page 305, the conductivity σ (dimension \nσ is 1 / s) are related with a specific resistance v alue: σ = 9 * 10 11  / ρ for copper \nthe conductivity is : σCu  = 5.4 * 10 17  1 / s, for titanium σTi = 3.23 * 10 16  1 / s. \nThis allows one to calculate the depth of the skin- layer and, thereby, to \ncalculate the possible thickness of the tube wall, where the magnetic dipole will \nbe accelerated. \n  \n    Let us find the thickness of the skin - layer f or titanium for frequency  \nf0 = 250 Hz. It can be calculated by the following fo rmula: \n \n                        δTi  = c/2 π (f 0σTi ) 1/2  = 1.68 cm.                                  (2) \n \n       This means that the wall thickness of the ti tanium tube ∆hTi , wherein the \nmagnetic dipole has to be accelerated, can be chose n to be equal to:  \n∆hTi  = 2 mm. \n \n4. Interaction of the dipole with the magnetic fiel d gradient  \n \n  For stable acceleration of magnetic dipoles it is  necessary to \"switch on” \nconsequently the magnetic coils according to the di pole move. The magnetic \nfield of the coil with a current can be written as follows: \n \n                          B z = I 0 * r 02 / [ 2 * (r 02 + z 2) 3/2 ],                               (3) \n \nwhere: I 0 - current in the loop, Ampere, r 0 - the radius of the coil with a current, \ncm, z - the distance from the coil plane to the obs ervation point.      5     In comparison of the multi section Gauss gun [ 1] the corresponding coil here \nis necessary “to be switched on\" after the magnetic  dipole passage through the \ncoil center but not to join the coil switched on in  advance. \n \n4.1. Acceleration of the magnetic dipole by the cur rent single coil  \n \n    We differentiate expression (3) with respect to  z and obtain a formula for the \nmagnetic field gradient: \n \n                      dB z / dz = ( 3/2 ) * r 02 * I 0 * z / (r 02 + z 2) 5/2 .                  (4) \n \nFrom this formula it is clear that the gradient fie ld is zero in the coil plane  \nat z = 0. \n \n     We assume that at a distance of the order of t he radius of the turn, the speed \nof the dipole varies slightly, i.e. it is possible to change variable z for V sh t. The \nspecific magnetic dipole moment increases while the  dipole passing through the \ncenter of the coil according to the law: \n \n                                      m = 2 m0 * z / l sh ,                                            (5) \n \nwhere m0 = 3.5 * 10 -9 eV / (Gs * nucleon), l sh  - length of the dipole . The \ncurrent in the coil after switching increases linea rly with time, according to the \nlaw: \n \n                                     I ≈ I 0 * (4t/T 0),                                                ( 6) \n \nwhere T 0 - time period of the slowdown wavelength. \n \nThe force influencing the dipole from the coil side  is: \n \n   Fz = m0 * (z / l sh ) * ( 3/2 ) * r 02 *z * I 0 * 4 (t/T 0) / (r 02 + z 2) 5/2 .      (7) \n \n   Substituting t for z / V sh  and integrating over z, we obtain an expression fo r \nthe energy gain rate while passing one loop by the magnetic dipole: \n \n∆W = ∫ F zdz =  \n= ∫ m0 * (z / l sh ) * ( 3/2 ) * r 02 *z * I 0 *4 (z/V sh T0) /(r 02 + z 2) 5/2 dz,   (8)                                                                        \n \nor  6                                                        l sh  / 2 \n     ∆W1 = 12 m0 * (r 02/l sh ) *(I 0/V sh T0) ∫ [z 3 / (r 02 + z 2) 5/2  ] dz.         (9) \n                                                           0 \n       \n      Integration in (9) should be performed till t he distance approximately equal \nto half of the length of the magnetic dipole: l sh  = 1m. The same order should be \nthe turn radius: r 0 = 1m. After the magnetic dipole passing a distance  l sh  / 2, its \nmagnetic moment does not any longer increase and th e dipole magnet will be \njust repelled by a coil with a current. \n       \n     The corresponding set of energy can then be wr itten as follows: \n                                                         r0 \n            ∆W2 = 6 m0 * r 02 *(I 0/V sh T0) ∫ [z 2 / (r 02 + z 2) 5/2 ] dz.           (10) \n                                                      lsh  / 2 \n                   \n     Substituting numerical values of  r 0 = 1m, l sh  = 1m, calculating the integrals \nand summing ∆W1 and ∆W2, we find that the energy acquired during the \npassage of one current loop by the magnetic dipole is equal to: \n \n                ∆W ≈ 12 m0 * (r 02/l sh ) * (I 0/V sh T0) * 5.5 * 10 -2.             (11) \n \n4.2. Acceleration of the magnetic dipole by consequ ence of current turns \n                                                                                                                            \n     We assume that the consequence of current turn s is as follows: per 1m there \nare 10 3 turns (10 3 / m), the current in each coil is assumed to be eq ual to:  \nI0 = 150 kA. Assuming m0 = 3.5 * 10 -9 eV / (Gs *nucleon) and averaging the \naction of the turns on the magnetic dipole with a c oefficient of ½, we finally \nobtain the formula for the energy gain rate of the magnetic dipole: \n \n                        ∆W = 4.33 * 10 -4 (eV / nucleon * m).                 (12) \n \n       Multiplying ∆W = 4.33 * 10-4 (eV / nucleon * m) by the length of  the \nacceleration L acc = 300 m, we find the finite energy of the magnetic dipoles: \nWfin  = 0.13 eV / nucleon, that corresponds to the finit e speed of the magnetic \ndipoles V fin  = 5 km / s. \n    \n     Figure 1 shows a diagram of the device.  7               \n  \n \nFig.1. (1) - gun, (2) - magnetic dipoles, (3) - cur rent coils, \n          (4) - titanium tube, (5) - pulse diaphrag m, (6) - pumping. \n \nConclusion  \n  \n          While increasing the diameter of the magnetic dipol e its total magnetic \nmoment grows as the area of its turn, i.e. proporti onally, as r 2. The dipole mass, \nat a constant current density increases as the peri meter of the loop, i.e., linearly \nwith the radius. Thus, the specific magnetic dipole  moment will grow linearly \nwith increasing of the turn radius. \n \nLiterature \n \n1. http://ru.wikipedia.org/wiki/ Пушка _Гаусса  \n \n2. Tables of physical quantities. Reference ed. Kik oin, \n     Moscow, Atomizdat, 1976 \n \n3. V. I. Veksler, Dokl. USSR Academy of Sciences, 1 944, v. 43,  \nIssue 8, p. 346, E. M. McMillan, Phys. Rev. 1945, v . 68, p. 143 \n \n \n \n "    },    {        "title": "1411.7505v1.The_critical_relaxation_of_the_model_of_iron_vanadium_magnetic_superlattice.pdf",        "content": "The critical relaxation of the model of  iron-vanadium magnetic superlattice1 \n \nVadim A. Mutailamov 1, Akai K. Murtazaev 1,2 \n1 Institute of Physics DSC RAS, 94 M. Yaragskii Str., Makhachkala, Russia, 367003. \nE-mail: vadim.mut@mail.ru \n2 Daghestan State University, 43a M. Gajiev Str., Makhachkala, Russia, 367025. \nE-mail: akai2005@mail.ru \n ABSTRACT \nThe critical relaxation of iron-vanadium magn etic superlattice in case of the equality \nbetween interlayer and intralayer exchange intera ctions is investigated. The dynamic and static \ncritical exponents of the model are calculated. A value of the cr itical temperature is evaluated. \n \nKeywords: A. Magnetic superlattices; D. Phase transitions; D. Critical phenomena; E. Short-\ntime dynamic. \n \n1. INTRODUCTION \nResearches on metallic nonmagnetic superalttices  consisting of alternat e atomic layers of \nmagnetic and non-magnetic materials are of great interest in the modern condensed-matter physics \n[1-3]. The possibility to control the fundament al properties of superl attices (magnetization, \ninterlayer exchange interaction, magnetoresistance,  and other characteristics)  by means of external \naction allows creating structures with predetermi ned parameters, what makes these materials unique \nobjects for the practical applicati on and theoretical investigation. \nSince the experimental researches of such sy stems meet with essential difficulties, the \ncomputational physics methods came to be successfu lly used for their study recently. In works [4-\n6], the static critical behavior of magnetic Fe\n2/V13 superlattices is invest igated, static critical \nexponents are calculated, and their dependence on correlation of intral ayer and interlayer exchange \ninteractions is studied. The critic al exponents of studied superlatti ce models are established to be \ndependent on a value of interlay er exchange interaction parameter. At the same time the scaling \ncorrelations between critical expon ents are carried out with extrem ely high precision. This situation \ndeviates from the modern theory of phase trans itions and critical phenome na. In this regard, the \ninvestigation of critical dynamics of these models arouses great interest that can be a principal for \nan explanation of difficulties appearing in th e exploration of static critical phenomena.  \n                                                 \nPublished in Journal of Magnetism and Magnetic Materials, Vol. 325, 2013, pp. 122-124.  2Recently, the critical dynamics of magnetic materials models ar e successfully studied using \na short-time dynamic method [7-9], where the cri tical relaxation of a ma gnetic model from non-\nequilibrium state into the equilibrium one is investigated within A model (Halperin and  Hohenberg \nclassification of universality classes of dynamic critical behavior [10]). Traditionally, it is \nconsidered that a universal scaling behavior exis ts in the state of thermodynamic equilibrium only. \nNevertheless, a universal scaling behavior for some dynamic systems is shown to be realized at \nearlier stages of their time evolution from hi gh-temperature disordered state into the state \ncorresponding to the phase transition temperature [ 11]. Such a behavior is realized after a certain \ntime period which is rather larger in a microsco pic sense, but remains small macroscopically. The \nsame picture is observed when the system evolvi ng from low-temperature ordered state [7-8].  \n \n2. INVESTIGATION METHOD \nUsing the renormalization group method, the author s [11] showed that far from equilibrium \npoint after microscopically small time period, a scaling form is realized for k-th moment of the \nmagnetization  \n) , , ,( ),,,(01 1 )(\n0)(0mbLb btb M b mLt Mx z k k k      ,    (1) \nwhere M(k) is k-th moment of the magnetization; t is the time; τ marks the reduced temperature; L \ndenots the linear size of the system; b is a scale coefficient; β and ν  are static critical exponents of \nmagnetization and correlation radius; z denotes a dynamic critical exponent; x0 denotes new \nindependent critical exponent defining the sc aling dimension of the initial magnetization m0. \nWhen starting from low-temperature ordered state ( m0 = 1) in the critical point ( τ = 0), \nassuming b = t 1/z in Eq.(1), for the systems with sufficiently large linear sizes the theory predicts an \npower-law behavior of the magneti zation in the short-time region  \nzc ttMc\n\n1, ~)(1.             (2) \nFinding the logarithm of both parts of  Eq. (2) and taking derivatives with respect to τ at \nτ = 0 we get the power law for the logarithmic derivative  \n zc t tMlcl\n 1, ~),( ln1 01 \n.      ( 3 )  \nFor Binder cummulant calculated by first and second moments of  magnetization, the finite-\nsize scaling theory gives dependence at τ = 0:  \nzdc tMMtUUc\nLU  , ~1)()(2)2(\n.      ( 4 )  \nSo, during one numerical experiment, the shor t-time dynamics method allows to determine \nthe values of three critical exponents β, ν, and z using correlations (2-4).  Moreover, the dependences  3(2) plotted at different temperature values perm its to detect a value Tc by their deviation from direct \nline in the log-log scale. \n 3. MODEL \nWe study the critical relaxation from low-temperature ordered state of Fe\n2/V13 superlattice \nby means of the short-time dynamics method. In mi croscopic model of the superlattice offered in \nworks [4-6], every atom of iron has four neares t neighbors from an adjacent iron layer. The iron \nlayers are shifted relative one anot her on half lattice constant along the x and y axes. The magnetic \nmoments of iron atom s are ordered in xy plane. The scheme of iron-va nadium sublattice is shown in \nFig.1. \n An interaction between the nearest neighbors wi thin layer has a ferromagnetic character and \nis determined by the exchange interaction parameter||J. The interlayer interaction Jbetween \nvanadium magnetic layers is tran sferred by the conduction electrons in the non-magne tic interlayer \nof vanadium (RKKI-interaction).  In the real subl attices, its value and sign can change depending on \na number of adsorbed hydrogen into the vanadium subsystem. Since an accu rate dependence of the \nRKKI-interaction is unknown, usua lly, when carrying out the numerical investigations the whole \nrange of interlayer in teraction values from||J J   to ||J J  is studied. In this work, we present \none special case of RK KI-interaction, when ||J J . \nAs in the experiment the distan ce between magnetic layers is substantially larger than the \ninteratomic distance, ever y atom interacts with the averaged mo ment of neighbor layers. A size of \naveraging region is a model parameter. Our investigations are made for limiting case, when every \natom interacts with only one nearest atom from neighbor layer. The study of  the static critical \nbehavior of magnetic superlattices [4-6] showed that  this approach describes a critical behavior of \nthese models to the best advantage. \nThus, the Hamiltonian of this model can be written as modified 3D XY-model [4-6] \n        \nkiy\nky\nix\nkx\ni\njiy\njy\nix\njx\ni SS SS J SS SS J H\n, ,||21\n21,    (5) \nwhere first sum takes into account  the direct exchange interactio n of each magnetic atom with \nnearest neighbors inside the la yer, and second denotes the R KKI-interaction with atoms of \nneighboring layers through non-magnetic interlayer; yx\niS,marks the components of the spin \nlocalized in a site i. \n    44. RESULTS \nWe study a system with the linear size L=64 containing 262144 spins in a case of equality \nbetween intralayer and interlayer exchange inter actions. Let`s note that in this case, according to \nEq. (5), the Hamiltonian of studied model is similar to the Hamiltonian of the classical 3D XY-model. The investigations are car ried out by Metropolis standard al gorithm of Monte-Carlo method. \nThe relaxation of the system is performed from initial fully ordered state with starting value of magnetization m\n0=1 during tmax=1000, where one Monte-Carlo step  per spin is taken as “time” \nunite. Relaxation dependencies are calculated up to 14 000 times, obt ained data are averaged among \nthemselves. \nThe critical temperatures are defined by the dependence of the magnetization on the time \nEq. (2), which, in a point of phase transition, must  be a straight line in log-log scale. The deviation \nof the straight line is estimated by the least-squares method.  The temperature, at which this \ndeviation is minimal, is taken as a critical. Fig.  2 presents the time dependence of magnetization at \nthree values of temperature around the phase trans ition point in log-log scal e (here and further all \nvalues are presented in arbitrary units). Estimate d value of the critical temperature in units of \nexchange integral kbT/J is Tc=1.752(1). The logarithmic derivativ e in a phase transition point is \ncalculated by the least-squares approximation by th ree time dependences of magnetization plotted at \ntemperatures T=1.742, T=1.752 and T=1.762. \nIn Fig. 3, the time dependence of Binder cummulants  UL is demonstrated in log-log scale at \nphase transition temperature. Th e analysis of curve shows that  exponential scaling behavior UL(t) is \nrealized from a time point t=100. \nTherefore, the overall least–squares approxi mation by Eq. (4) is carried out within time \nranges t=[100;1000]. This had resulte d in exponent value CU =1.54(3). According to Eq. (4) the \noverall value of dynamic critical exponent is z=1.95(3), what is close to the theoretical value \npredicted for anisotropic magnetics ( z=2, model A [10]). \nThe dependences of magnetization and magnetiza tion derivative on the time in log-log scale \nare presented in Fig. 4 and 5 respectively. The data are approximated within time ranges \nt=[100;1000] like Binder cummulants. The exponent values cl=0.29(3) and cl1=0.79(3) have \nresulted from approximation. The critical exponents of magnetization β=0.36(3) and correlation \nradius ν=0.65(3) are calculated by means of Eqs. (2-4).  \nIn works [4-6], where the static critical behavior of Fe 2/V13 superlattice model was studied \nby traditional equilibrium method, β=0.342(3) and ν=0.671(3) for the similar exponents. The theory \npredicts β=0.3485(3) and ν=0.67155(37) for classical XY- model [12]. As it is ev ident, our values of \nstatic critical exponents agree with results of both works.  \n  5REFERENCES  \n[1] B.Hjörvarsson, J.A.Dura, P.Isberg, et all, Phys. Rev. Lett. 79, 901 (1997). \n[2] V.Leiner, K.Westerholt, A.M.Blixt, H. Zabel, B.Hjörvarsson, Phys. Rev. Lett. 91, 37202 \n(2003). \n[3] V.Leiner, K.Westerholt, B.Hjörvarss on, H.Zabel, J. Phys. D: Appl. Phys. 35, 2377 (2002). \n[4] K.Sh.Khizriev, A.K.Murtazaev, V.M.Uzdin, Journal of Magnetism and Magnetic Materials \n300, e546 (2006). \n[5] A.K. Murtazaev, K.Sh. Khizriev, V.M. Uzdin, Bulletin of the Russian Academy of Sciences: \nPhysics 70, 695 (2006). \n[6] A.K. Murtazaev, Phys. Usp 49, 1092 (2006). \n[7] A.Jaster, J.Mainville, L.Schulke et al, E-Print arXiv: cond-matt/9808131 v1 (1998). \n[8] B.Zheng, Physica A 283, 80 (2000). \n[9] V.V.Prudnikov, P.V.Prudnikov, B.Zh eng et al, Prog. Theor. Phys. 117, 973 (2007). \n[10] P.C. Hohenberg, B.I. Halperin, Rev. Mod. Phys. 49, 435 (1977). \n[11] H.K.Janssen, B.Schaub, B.Schmittmanm, Z. Phys. B 73, 539 (1989). [12] M. Campostrini, M. Hasenbusch, A. Pelissetto, et al, Phys. Rev. B \n63, 214503 (2001).  6\n \nFig.1. The scheme of iron-vanadium Fe 2/V13 superlattice. Three of  the thirteen vanadium \nmonolayers presented for the clarity.   7\n \nFig.2. Time evolution of the magnetization at  three different temperatures around the phase \ntransition point.  8 \n \nFig.3. Time evolution of th e Binder cummulants at the ph ase transition temperature.  9\n \nFig.4. Time evolution of the magnetizati on at the phase transition temperature.  10\n \nFig.5. Time evolution of the magnetization de rivative at the phase transition temperature. \n \n  "    },    {        "title": "0806.4731v2.Dynamical_Regimes_Induced_by_Spin_Transfer_in_Magnetic_Nanopillars.pdf",        "content": "arXiv:0806.4731v2  [cond-mat.mtrl-sci]  16 Jul 2008Dynamical Regimes Induced by Spin Transfer in Magnetic Nano pillars.\nWeng Lee Lim, Andrew Higgins, and Sergei Urazhdin\nDepartment of Physics, West Virginia University, Morganto wn, WV 26506\nWe demonstrate the predicted out-of-plane precession indu ced by spin transfer in magnetic nanos-\ntructures with in-plane magnetic field. We show that other ma gnetic excitations have a significant\neffect on the stability of the out-of plane precession, makin g it extremely sensitive to the orienta-\ntion of the applied magnetic field. The data are supported wit h micromagnetic simulations. Our\nresults elucidate the relation between the excitation spec trum and the specific dynamical behaviors\nof nanoscale magnets.\nPACS numbers: 85.75.-d, 75.60.Jk, 75.70.Cn\nSpin transfer torque (ST) [1] exerted on nanomag-\nnets by spin-polarized current Ican induce dynamical\nstates not accessible by any other techniques, providing\na unique opportunity to test our understanding of nano-\nmagnetism [2]. At Ijust above the excitation threshold,\nST acting on the magnetic moment m1of a nanomagnet\nF1causes precession on an elliptically shaped orbit, de-\ntermined by a combination of the magnetic field Hand\nthe anisotropy of F 1. The nanomanget is usually a thin\nfilm whoseanisotropyisdominated bythe demagnetizing\nfield, with Hcommonly in the film plane. The precession\namplitude grows with I, resulting in an orbit known as\nthe clamshellmode (left inset in Fig.1). Good agreement\nbetween calculations and experiments has been achieved\nfor precessional dynamics in this regime [3, 4].\nHowever, fast operation of magnetic devices driven by\nST will likely be performed in the less explored high-\ncurrent regime. According to simulations [4, 5, 6], the\nextreme points of clamshell eventually merge, resulting\nin a crossover to the out-of-plane (OP) precession mode\nconsisting of either the lower or the upper half of the\nclamshell. Only a broad spectral feature indicative of\nthis mode has been seen [3], suggesting that micromag-\nnetic simulations may not be adequate for the highly ex-\ncited dynamical states of nanomagnets, or that current-\ninduced effects may not be fully described by the estab-\nlished ST mechanisms.\nST also affects the magnetic layer F 2used to polarize\nthe current, which can decrease the dynamical coherence\nand suppress the OP mode [7]. Here, we report observa-\ntion of the OP precession in devices where this effect was\nminimized by using an extended F 2. We identified and\nanalyzed the effects of varied direction and magnitude\nofH. The micromagnetic simulations support our inter-\npretation of the data, and provide insight into the mi-\ncroscopic origins of the observed behaviors. Some of the\nresults could not be reproduced by simulations, suggest-\ning that the understanding of current-induced dynamics\nin nanomagnets is still incomplete.\nMultilayers Cu(50)Py(20)Cu(5)Py(5)Au(20), where\nPy=Ni 80Fe20and thicknesses are in nm, were deposited\non oxidized silicon at room temperature (RT) by mag-netron sputtering at base pressure of 5 ×10−9Torr, in\n5 mTorr of purified Ar. F 1=Py(5) and about 5 nm-\nthick part of F2=Py(20) were patterned by Ar ion\nmilling through an evaporated Al mask with dimensions\nof 100 nm ×50 nm, followed by deposition of 30 nm of\nundoped Si without breaking the vacuum. This proce-\ndure avoids oxidation of the magnetic layers, which can\naffect the magnetic dynamics [8]. The mask was removed\nby a combination of ion milling with Ar beam nearlypar-\nallelto the samplesurface, andetching in aweaksolution\nof HF in water, followed by sputtering of a 200 nm thick\nCu top contact. We discuss data for one of three devices\nthat exhibited similar behaviors.\nAll measurements were performed at RT. The sample\nwas contacted by coaxial microwave probes, which were\nconnected through a bias tee to a current source, a lock-\nin amplifier, and a spectrum analyzer through a broad-\nband amplifier. To enable the detection of the preces-\nsional states by electronic spectroscopy, Hwas rotated\nin the sample plane by angle φ= 40◦with respect to\nthe nanopillar easy axis, unless specified otherwise. Pos-\nitiveIflowed upwards. The device was characterized by\nmagnetoresistive (MR) measurements of its response to\nHandI, yielding the parameters essential for modeling,\nsuch as the MR of 0 .21 Ω, the dipolar coupling field of\n200OecausedbythepartialpatterningofF 2, andtheco-\nercivityof175Oe. The latterwasconsistentwith Stoner-\nWohlfarth approximation, indicating uniform magnetic\nreversal. The measured microwave signals were adjusted\nfor the frequency-dependent gain of the amplifier and\nlosses in the cables and probes, determined with a cali-\nbrated microwave generator and a power meter.\nFig. 1 shows the dependence of the measured power\nspectraldensity(PSD) on H. Thespectraat H= 550Oe\nexhibit three harmonically related peaks at I <5 mA,\ncaused by the clamshell precession. The expected fun-\ndamental frequency of the OP mode is close to the\nfrequency of the clamshell’s second harmonic near the\ncrossover, since its trajectory is half that of clamshell.\nThe 550 Oe data in Fig. 1 exhibit a spectral feature at\nI >5 mA consistent with this relationship between the\ntwo modes. Below, we present measurements and micro-2\n/s52/s56/s49/s50\n/s72\n/s52/s56/s49/s50/s32/s73/s32/s40/s109/s65/s41/s32/s32/s80/s83/s68/s32/s40/s112/s87/s47/s77 /s72/s122/s41\n/s52/s56/s49/s50/s48/s46/s48/s52 /s48/s46/s49/s49 /s48/s46/s51/s53 /s49/s46/s49/s49 /s51/s46/s53/s48\n/s50 /s52 /s54 /s56 /s50 /s52 /s54 /s56/s52/s56/s49/s50\n/s102/s32/s40/s71/s72/s122/s41/s55/s48/s48/s32/s79/s101/s54/s53/s48/s32/s79/s101 /s45/s54/s53/s48/s32/s79/s101/s53/s53/s48/s32/s79/s101/s45/s32/s53/s53/s48/s32/s79/s101\n/s49/s32/s107/s79/s101 /s45/s49/s32/s107/s79/s101/s45/s55/s48/s48/s32/s79/s101\nFIG. 1: PSD vs. frequency fandI, at the labeled values of\nH. The logarithmic data scale brings out a break at 2 .93 GHz\nof about 50 fW/MHz due to the spectrum analyzer crossover.\nInset: clamshell trajectory of the magnetic moment, with H\nshown.\nmagnetic simulationsthat confirmour interpretationand\nelucidate other, more complex dynamical behaviors.\nAtH >550 Oe, the OP peak rapidly shifts to higher\nfrequency. Simultaneously, the high-current part of the\npeak splits from the lower part, and eventually merges\nwith the clamshell peaks ( H= 650 Oe and 700 Oe\ndata in Fig. 1). One can estimate the precession am-\nplitude based on the total microwave power under the\npeaks divided by I2(see also the description of sim-\nulations). The largest possible emitted power for hy-\npothetical oscillations between the parallel (P) and an-\ntiparallel (AP) configurations of the magnetic layers is\n37.7 pW/mA2for our sample. The power generated\nby clamshell precession at H= 700 Oe, I= 3.5 mA\nis 11.2 pW/mA2, and by the OP mode at the same H\nandI= 10 mA is 9 .9 pW/mA2. Both values correspond\nto the in-plane precession angle exceeding 90◦, providing\na strong evidence for our interpretation of the spectral\npeaks as large-angle precessional modes. If these spec-\ntral features were induced by inhomogeneous dynamicsrather than precession, they would result in significantly\nsmaller microwave power emission. The first clamshell\nharmonic exhibits the smallest FWHM of 30 MHz at\nH= 550 Oe, I= 3.5 mA. The peaks decrease in am-\nplitude and broaden with increasing H. These behaviors\nsuggest increasingly inhomogeneous dynamics, resulting\nin decoherence of precession. In contrast, the intensity of\nthe OP peak at I= 10 mA increases from 0 .86 pW/MHz\natH= 550 Oe to 2 .5 pW/MHz at 1 kOe, while the\nFWHMremainsapproximatelyconstantat140 ±15MHz.\nThe spectra are asymmetric with respect to the direc-\ntion ofH, as illustrated by the difference between the left\nand the right panels in Fig. 1. The clamshell peaks are\nconsistently broader at H <0 than at H >0. At large I,\nthey are replaced by incoherent noise rather than the OP\nmode. The direction of Hfor which the OP mode was\nobserved varied among the samples, indicating extrin-\nsic origin of asymmetry. At large H, the data became\nsimilar for positive and negative H. In particular, the\nH=±1 kOe data exhibit a sharp OP peak at I≥9 mA.\nTogaininsightintotheoriginofthebehaviorsshownin\nFig. 1, as a well as other results discussed below, we per-\nformed micromagnetic simulations with OOMMF open\ncode software [9]. The simulations included the current-\ninduced ST and Oersted field effects, but neglected ther-\nmal fluctuations. The dipolar field was accounted for\nby subtracting 200 Oe from the in-plane component of\nH. The cell size was 4 ×4×5 nm3. Reducing the size\nto 2×2×5 nm3did not significantly affect the results.\nThe parameter values available mostly from MR mea-\nsurements [10] were used: current polarization p= 0.7,\nPy exchange stiffness A= 1.3×10−6erg/cm, Gilbert\ndamping α= 0.03, and the ratio Λ = 1 .3 of the ST mag-\nnitudes near the AP and the P states. Saturation mag-\nnetization M= 750 emu/cm3of Py was determined by\nmagnetometry of a Py(5) film prepared under the same\nconditions as the nanopillar.\nThe spectra were calculated from the simulated time-\ndependent magnetization distribution of the nanopillar.\nThe calculated time-dependent resistance was R(t) =\nR0+ ∆R(t), where R0= (RP+RAP)/2, and ∆ R(t) =\n(RP−RAP)/angbracketlefts1·s2/angbracketright/2. Here, s1(t),s2(t) are the local\nnormalized magnetizations of F 1and F 2, and/angbracketleft/angbracketrightde-\nnotes averaging over the simulation grid. The ac volt-\nage on the input of the amplifier V(t) =I∆R(t)\n1+R0/50Ωwas\nobtained by assuming that a constant current Iis dis-\ntributed between a 50 Ω load and the sample resistance\nR(t). Fast Fourier transform (FFT) of V(t) over a pe-\nriod ofT= 16.4 ns with a 1 ps step was performed\nafter relaxation for 10 ns. The power spectral density\nwas determined by PSD(f) = 2V2(f)/(50Ω∆f), where\n∆f= 1/T, and a factor of 2 accounts for the negative- f\ncontribution to the FFT.\nOur calculations showed that a significant asymmetry\nof spectra with respect to the direction of Hcan be in-3\n/s50 /s52 /s54 /s56 /s49/s48/s50/s52/s54/s56/s49/s48\n/s102/s32/s40/s71/s72/s122/s41/s73/s40/s109/s65/s41/s97\n/s50 /s52 /s54 /s56 /s49/s48/s98\n/s102/s32/s40/s71/s72/s122/s41/s48/s46/s48/s49 /s48/s46/s48/s52 /s48/s46/s49/s57 /s48/s46/s56/s49 /s51/s46/s53/s48/s80/s83/s68/s32/s40/s112/s87/s47/s77 /s72/s122/s41\n/s72\n/s73/s73/s73 /s73/s73 /s73/s32/s99/s32\nFIG. 2: (a) Calculated PSD for H= 650 Oe. (b) same\nas (a), for H=−650 Oe. c, instantaneous magnetization\ndistribution for H=−650 Oe, I= 6 mA, captured in the\nI-II-III sequence with a 100 ps interval, showing the nucle-\nation, propagation, and annihilation of a vortex; the mo-\ntion of its core is marked with yellow dots. The intensity\nof blue(red) reflects the out-of-plane component of the mag-\nnetization above(below) the plane. Arrow shows the directi on\nofH.\nduced by the simultaneous effects of the current-induced\nOersted field and a modest asymmetry of the sample ge-\nometry. Fig. 2 shows results for a nanopillar approxi-\nmatedbytwosemi-ellipseswithminorsemi-axesof34nm\nand 18 nm, and a major axis of 104 nm. Coherent OP\nprecession was obtained at H >0, but was suppressed at\nH <0 by vortices and finite-wavelength spin-waves [11].\nThe vortices usually nucleated at the right upper edge\nof the nanopillar, and annihilated at the lower left edge\n(Fig. 2c). The chirality of the vortices coincided with the\ndirection of the Oersted field’s rotation, indicating that\nthe asymmetry of spectra is caused by the suppression or\nenhancement of vortex nucleation due to the interplay of\nsample shape asymmetry and the effect of Oersted field.\nSimulations could not reproduce severalfeatures of the\ndata for any reasonable variations of nanopillar shape,\ndistribution of current and its polarization, or Py stiff-\nness. Firstly, the simulated OP peak did not exhibit\nthe rapid shift and splitting with increasing Hseen in\ndata. Secondly, despite a significant asymmetry of the\ncalculated spectra, they did not reproduce the region\natI >5 mA where sharp OP peak was present for\nH= 500 Oe, but no dynamical features appeared at/s54 /s56 /s49/s48 /s49/s50 /s52 /s54 /s56 /s49/s48 /s49/s50/s49/s50/s51/s52/s53\n/s32/s73/s32/s40/s109/s65/s41/s97/s102/s32/s40/s71/s72/s122/s41/s98\n/s73/s40/s109/s65/s41\nFIG. 3: Frequency of the OP peak vsIforφ= 48◦(squares),\nφ= 40◦(open circles), φ= 24◦(triangles). (a) Measurements\nperformed at H= 550 Oe, (b) Calculations performed at\nH= 650 Oe, with the same notations.\nH=−550 Oe. The simulations also indicated that\nthe OP mode should exhibit multiple spectralharmonics,\nwhileonlyoneortwoharmonicscouldbeseenindata, re-\ngardless of the large amplitude of precession established\nfrom the analysis of peak intensity. These features may\nbe caused by additional effects of spin-polarized current\nneglected by the model, or by the dynamical states not\ndescribed by micromagnetic simulations.\nThedifferencesbetweendataandsimulationsalsoopen\nthe possibility that the high-current spectral feature in\nFig. 1is caused by dynamicsdifferent fromthe OPmode.\nCurrent-induced excitation of the polarizing layer F 2can\nlead to microwave peaks, appearing above the onset cur-\nrent determined by the ratio of the volumes of F 2and\nF1[3]. However, the effective volume of the extended\nlayer F 2in our samples far exceeds that of F 1, and thus\ncannot explain the onset current that at 550 Oe is only\n2.6 times larger than the onset of the clamshell preces-\nsion. The nanopillar shape imperfections can also re-\nsult in precession around a configuration intermediate\nbetweentheAPandPstates. However,thisintermediate\nstatewouldquicklybecomeunstableatincreased H, con-\ntrary to the high-field data in Fig. 1. Such a state would\nalso likely appear as an intermediate-resistance step not\nseen in our dc measurements of MR.\nBoth measurements and simulations showed that the\nOP precession is extremely sensitive to the orientation\nofH, when the latter was rotated in the film plane or\ntilted out of plane. Rotating Hin the plane changed the\ndependence of the OP mode frequency on I, Fig. 3(a).\nAtφ= 48◦, the peak exhibited a blue shift up to 6 .5 mA,\nabove which it gradually red shifted. At smaller values\nofφ, the peak broadened, decreased in intensity, and red\nshifted. The correlation between the width of the OP\npeak and the dependence of its frequency on Iis con-\nsistent with published simulations [2, 6]. Namely, blue\nshift is alwayspredicted in the macrospinapproximation,\nwhich is more applicable to narrow coherent peaks. In4\n/s50/s52/s54/s56/s49/s48/s49/s50\n/s99/s72/s32/s73/s32/s40/s109/s65/s41/s50/s52/s54/s56/s49/s48/s49/s50\n/s98/s72\n/s50 /s52 /s54 /s56 /s49/s48/s50/s52/s54/s56/s49/s48/s49/s50\n/s100/s72\n/s32/s102/s32/s40/s71/s72/s122/s41/s101\n/s102\n/s103\n/s50 /s52 /s54 /s56 /s49/s48/s104/s50/s52/s54/s56/s49/s48/s49/s50\n/s97/s72/s80/s83/s68/s32/s40/s112/s87/s47/s77 /s72/s122/s41\n/s48/s46/s48/s49 /s48/s46/s48/s52 /s48/s46/s49/s57 /s48/s46/s56/s49 /s51/s46/s53/s48\nFIG.4: PSDfor |H|= 900Oetiltedby45◦withrespecttothe\nsample plane, at φ= 40◦. Insets are side view schematics for\nthe orientation of Hwith respect tothe pillars, where positive\nin-plane direction of His to the right. (a)-(d) Measurements,\n(e)-(h) Simulations.\ncontrast, the peaks can red shift in micromagnetic simu-\nlations of the more inhomogeneous dynamics associated\nwith broader spectral peaks. The red shift originates\nfrom the decrease of the total magnetic moment of the\nnanopillar caused by the inhomogeneity. This interpreta-\ntion was supported by our simulations (Fig. 3(b)), where\ndecreasing φresulted in increasingly inhomogeneous OP\ndynamics. The OP mode red shifted with Iat small φ.\nAt larger φ, it blue shifted at small Iand red shifted at\nlargeI, in excellent overall agreement with the data. A\nsomewhat larger Hwas used in simulations to destabi-\nlize the static current-induced AP state at large I(see\ndiscussion of Fig. 1).\nForHtilted with respect to the sample plane, the dy-\nnamicsshoweddependence notonlyonthe signof H, but\nalso on its tilting direction (Fig. 4). The component of H\nperpendicular to the film plane was significantly smaller\nthan the demagnetizing field of 9 .4 kOe, resulting in only5◦tilting of m1when it was static. The OP mode ap-\npeared for both directions of Habove the film plane, and\nwas suppressed for Hbelow the film plane. The different\nresponse of dynamics to the opposite directions of tilt-\ning indicates a significant asymmetry of this state with\nrespect to the film plane. This asymmetry characteris-\ntic of the OP dynamics was reproduced by the simula-\ntions, asshowninpanels(e)-(h). Simulationsfor Htilted\nabove the sample plane showed predominantly coherent\nOP precession below the plane , in excellent agreement\nwith robust spectral features seen at large Iin the data\nofFigs. 4(b,c). We note that simulationswith in-plane H\nalso showed OP precession below the plane, determined\nby the relative orientations of Hand the nanopillar easy\naxis. In contrast, simulations with Htilted below the\nsample plane showed OP precession above the sample\nplane suppressed by dynamical inhomogeneities. These\nresults are consistent with the data.\nIn summary, coherent current-induced dynamics in\nmagnetic nanopillars was achieved by employing an ex-\ntended polarizing layer. A high-current spectral peak\nwas identified as the out-of-plane precession, whose de-\npendence on the orientation of Hwas in excellent agree-\nment with micromagnetic simulations. The dynamics ex-\nhibited a significant asymmetry with respect to the di-\nrection of H, attributed to a combination of nanopillar\nshape asymmetry and Oersted field.\nThis work was supported by the NSF Grant DMR-\n0747609 and a Cottrell Scholar award from the Research\nCorporation.\n[1] J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[2] S.I. Kiselev, J.C. Sankey, I.N. Krivorotov, N.C. Emley,\nR.J. Schoelkopf, R.A. Buhrman, D.C. Ralph, Nature\n425, 380 (2003).\n[3] I.N. Krivorotov, D.V. Berkov, N.L. Gorn, N.C. Emley,\nJ.C. Sankey, D.C. Ralph, and R.A. Buhrman, Phys. Rev.\nB 72, 064430 (2007).\n[4] D.V.Berkov, J. Miltat, J. Magn. Magn. Mater. 320, 1238\n(2008).\n[5] Z. Li, and S. Zhang, JournalPhys. Rev.B 680244042003.\n[6] D.V. Berkov and N.L. Gorn, Phys. Rev. B 72, 094401\n(2005).\n[7] S. Urazhdin, arXiv:0802.1560v1 (2008).\n[8] O. Ozatay, P.G. Gowtham, K.W. Tan, J.C. Read, K.A.\nMkhoyan, M.G. Thomas, G.D. Fuchs, P.M. Braganca,\nE.M. Ryan, K.V. Thadani, J. Silcox , D.C. Ralph , and\nR.A. Buhrman, Nature Mat. 7, 567 (2008).\n[9] M.J. Donahue and D.G. Porter, OOMMF User’s Guide ,\nNISTIR 6376 , NIST Gaithersburg, MD (1999).\n[10] J. Bass and W.P. Pratt Jr., J. Phys.: Condens. Matter\n19, 183201 (2007).\n[11] K.J. Lee, A. Deac, O. Redon, J.P. Noziers, and B. Dieny,\nNature Mat. 3, 877 (2004)."    },    {        "title": "2007.07618v2.Magnetic_Charge_Propagation_upon_a_3D_Artificial_Spin_ice.pdf",        "content": " 1 Magnetic Charge Propagation upon a 3D Artificial Spin -ice \nA. May1, M. Saccone2, A. van den Berg1, J. Askey1, M. Hunt1 and S. Ladak*1 \n1. School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA                              \nEmail: LadakS@cardiff.ac.uk  \n2. Physics Department, University of California, Santa Cruz, 1156 High Street, Santa Cruz, \nCA 95064, USA.  \n \nMagnetic charge propagation in spin -ice materials have yielded a paradigm -shift in \nscience, allowing the symmetry between electr icity and magnetism to be studied. Recent \nwork is now suggesting the spin -ice surface may be important in mediating the ordering \nand associated phase space in such materials. Here we detail a 3D artificial spin -ice, which \ncaptures the exact geometry of bul k systems, allowing magnetic charge dynamics to be \ndirectly visualized upon the surface. Using magnetic force microscopy, we observe vastly \ndifferent magnetic charge dynamics along two principal  directions. For a field applied \nalong the surface termination , local energetics force magnetic charges to nucleate over a \nlarger characteristic distance, reducing their magnetic Coulomb interaction and \nproducing uncorrelated monopoles. In contrast, applying a field transverse to the surface \ntermination yields highly  correlated monopole -antimonopole pairs. Detailed simulations \nsuggest it is the difference in effective chemical potentials that yields the striking \ndifferences in monopole transport.  \n \n \n  2 Introduction  \n \nThe concept of magnetic monopole transport within a co ndensed matter setting has captivated \nscientists, allowing established theory [ 1] to become an experimental realization [ 2-4] within \nthe bulk pyrochlore systems known as spin -ice [5]. In these three -dimensional (3D) systems, \nrare earth spins are located up on corner -sharing tetrahedra, and energy minimisation yields a \nlocal ordering principle known as the ice -rule, where two spins point into the centre of a \ntetrahedron and two spins point out. Representing each spin as a dimer, consisting of two equal \nand op posite magnetic charges ( ±𝑞), is a powerful means to understand the physics of spin -ice \n[5]. Using this description, known as the dumbbell model [ 1], the ice -rule is a result of charge \nminimisation, yielding a net magnetic charge of zero in the tetrahedra  centre ( 𝑄=∑𝑞!!=0). \nThen the simplest excitation within the manifold produces a pair of magnetic charges ( ∑𝑞!!=\n±2𝑞) which, once created, can propagate thermally and only at an energy cost equivalent to a \nmagnetic analogue of Coulomb’s law. The energy scale for the production of monopoles upon \nthe spin -ice lattice is controlled by the chemical potential ( µ), which is governed by properties \nintrinsic to the material such as lattice constant and magnetic moment [ 6]. Canonical spin -ice \nmaterials hav e a chemical potential that places them in a weakly correlated regime where only \na small fraction of bound monopole -antimonopole pairs are found. Recent theoretical work has \nstudied the ordering of magnetic charges upon cleaved spin -ice surfaces, perpendic ular to the \n[001] direction [7]. In such systems, the orphan bonds upon the surface are found to order in \neither a magnetic charge crystal or magnetic charge vacuum, depending upon the scales of \nexchange and dipolar energies [7]. Recent experimental work i s now hinting at the presence of \na surface -driven phase transition  [8] but the transport of magnetic charge across such surfaces \nhas not been considered previously.  \n \n  3 The arrangement of magnetic nanowires into two -dimensional lattices  has recently shown to \nbe a powerful means to explore the physics of frustration and associated emer gent physics. \nThese artificial spin -ice (ASI) systems [ 9-15], where each magnetic nanowire behaves as an \neffective Ising spin, have recently yielded an experimental realisation of the square ice model \n[16] and have also been used to study the thermal dynam ics of monopoles in the context of \nDebye -Hückel theory [ 17]. Controlled formation of magnetic charge is an exotic means to \nrealise advanced multistate memory devices. Such concepts have been shown in simple 2D \nlattices using magnetic force microscopy (MFM)  [18]. The extension of artificial spin -ice into \ntrue 3D lattices that capture the exact underlying geometry of bulk systems is paradigm -\nFigure 1: A 3D Artificial Spin -ice. (a) Schematic of a 3D artificial spin -ice system.  \nThe surface L1 layer (red) consists of an alternating sequence of coordination two and \ncoordination four vertices . Below this, the L2 (blue) and L3 (green) layers can be seen.  \nWithin these layers, only vertices of coordination four are present. The L4 lay er (grey) \nis the lower surface termination which again has an alternating sequence of coordination \ntwo and coordination four vertices. Inset: Cross -section of Ni 81Fe19 (grey) upon the \npolymer scaffold (yellow). ( b) A false colour scanning electron microsco py image of \nthe 3D artificial spin -ice lattice. Scale bar is 20µm. ( c) A false colour scanning electron \nmicroscopy image showing the L1 (red) and L2 (blue) sub -lattices, viewed at a 45°  tilt \nwith respect to the substrate plane. Scale bar is 1µm. ( d) Atomic force microscopy \nimage of the 3D artificial spin -ice system. Scale bar is 2µm.  Coordinate system for field \napplication is shown in top -right of image. (e) Possibilities for creating magnetic charge \nupon L1 . (f) The possible states and associated mag netic charge that can be realised at \nvertices of coordination two and coordination four.  \nB c\ne fA\nQ = +2qQ = +2q\n-4q -2q +2q +4qVertex Charge (Q)d\n ab\n 1.72µm\n0.00!\"\n!#!$\n!%x\ny\nz 4 shifting, allowing the exploration of ground state ordering and magnetic charge formation in \nthe bulk as well as upon t he surface. The production of 3DASI systems harbouring magnetic \ncharge also allows marriage with advanced racetrack device concepts [ 19,20] . \nIn this study, we use state -of-the-art 3D nanofabrication and processing in order to realise a \n3DASI in a diamond -bond 3D lattice geometry, producing an artificial experimental analogue \nof the originally conceived dumbbell model [1]. MFM is then harnessed to image the formation \nand propagation of magnetic charge upon the 3D nanowire lattice.  \nResults  \nFigure 1a shows a schematic of the 3DASI, which is composed of four distinct layers , labelled \nby colour. The system is fabricated by using two -photon lithography  [21-24] to define a \npolymer lattice in a diamond -bond geometry, upon which 50nm Ni 81Fe19 is evaporated (See \nmethods  for further details ). This yields NiFe nanowires within a diamond lattice geometry as \nshown previously [ 24]. Each nanowire has a crescent shaped cross -section  (Fig 1a inset) , is \nsingle domain  and exhibits Ising -like behaviour  [24]. The L1 layer, which is coloured red  in \nFig 1a,  is the  upper  surface termination  and consists of an alternating sequence of coordination \ntwo (bipods) and coordination four vertices (tetrapod s). The L2 and L3 layers, coloured blue \nand green respectively are ice -like with only vertices of coordination four. Finally, the L4 layer  \n(Grey)  is the lower surface  termination  of the lattice and consists  of vertices which alternate \nbetween coordination two and coordination four. \nThe overall array size is approximately 50  µm x 50 µm x 10 µm as seen in the scanning electron \nmicroscopy (SEM) image (Fig. 1b). Analysis of SEM data indicates the long -axis of L1  wires \nis orientated at θ = (33.11 ± 2.94)° from the substrate plane, matching within error the angle of \n35.25° which is expected for an idealised diamond -bond  geometry [5].  A higher magnification \nimage, clearly showing the L1 (red) and L2 layers  (blue) ca n be found in Fig 1c. The \ntopography of the upper three layers can be measured using atomic force microscopy (AFM)  5 as shown in Fig 1d. The coordinate system used to define field directions is also shown in Fig \n1d.  \nThe surface of the 3DASI lattice  provides interesting possibilities with respect to magnetic \ncharge creation and transport .  In Fig 1e we illustrate how magnetic charge propagates along \nthe L1 layer.  Starting with a saturated state, applying a magnetic field  above a critical value  \nalong the unit vector (1, -1,0) leads to the nucleation of a domain wall (DW) (Fig 1e top -left) \nwhich carries a mobile magnetic charge of magnitude 2q. When reaching the L1 -L2 junction \n(Fig 1e top -right), the effective vertex magnetic charge becomes Q=+2q. A further increment \nin magnetic field leads to the L1 -L2 junction emitting another DW (Fig 1e, bottom -left) and \nwhen this wire is fully switched a surface magnetic charge state of Q=+2q is realized (Fig 1e, \nFigure 2: Imaging the saturated states in a 3DASI. (a) An MFM image taken at \nremanence after application of saturating fields along the unit vectors (1, -1,0) and ( -1,-1,0). \nA coordination two, surface vertex is highlighted in pink, whilst a coordination four vertex \nat the intersection of L1 and L2 is highlight ed in red. The scale bar represents 2 µm. (b) \nMagnified example of the MFM contrast seen associated with L1 -L2 junctions as seen in \n(a). Arrows are coloured by the local in -plane magnetization components . (c) MFM image \ntaken at remanence after a further saturating field is now applied along unit vector ( -1,1,0). \n(d) Magnified example of the MFM contrast seen associated with L1 -L2 junctions as seen \nin (c). ( e) MFM image taken at remanence after a further saturating field is now applied \nalong unit v ectors (1, -1,0) and (1,1,0). ( f) Magnified example of the MFM contrast seen \nassociated with L1 -L2 junctions as seen in (e . \n b\na\nM\nL1\nL2\nc\nMe\nM\nd\n f\n 0.25 Deg\n -0.35 Deg\n2.0µm0.25 °\n-0.35 °\nA Ax\ny\nz 6 bottom -right). Note that a field applied in eithe r direction with a projection along the L2 sub -\nlattice produces only magnetic charges at four -way junctions (Fig S1). Overall, the 3DASI \nsurface can realise effective magnetic charge of ±4q, ±2q and 0 as summarised in Fig 1f.  \n \n \nImaging the magnetic config uration  of a 3DASI  \nMFM is a convenient method to deduce the magnetization configuration of the 3DASI during \nfield-driven experiments.  This imaging technique is sensitive to the second derivative of the \nstray field with respect to  z (d2Hz/dz2) which makes it ideal for imaging magnetic charge  [25] \nupon the 3DASI lattice. In the present study , we focus upon the field-driven transport of \nmagnetic charge upon the L1 and L2 layers.  The volume of the individual nanowires is \nsufficiently high that the 3DASI system is frozen  at room temperature  and thus thermal \nenergies are negligible when compared to the energy required to switch a wire.  \n \nIt is insightful to first study the simplest scenarios where each sub -lattice is saturated. Optical \nmagnetometry (see Fig  S2) indicates 30 mT is well above the saturating field for each sub -\nlattice. Figure 2a presents an MFM image , taken at remanence  following  a H = 30 mT in -plane \nmagnetic field, first applied  along unit vector (1, -1,0) and subsequently along unit vector ( -1,-\n1,0). Masks are placed over void regions to guide the eye to signal originating from L1 and L2. \nUnmasked data is provided in the supplementary information.  Every L1-L2 vertex within the \narray is seen to have identical contrast. A magnified example of th e contrast associated with an \nindividual L1 -L2 vertex  is also shown in Fig 2b. With our choice of tip magnetisation , the \nbright yellow lobes indicate a positive phase  associated  with the stray field at magnetisation  \ntail whilst bright red lobes indicate a negative phase  associated with the stray field  at \nmagnetisation  head. Focusing first upon the L1 nanowires, one can see lobes of strong positive  7 contrast at the upper left of the nanowires and negative  contrast in the lower right of the \nnanowires . Now focusing upon L2, strong  positive  contrast is seen in top right of nanowires, \nwith negative  contrast seen in bottom left. Overall, the vertex config uration is consistent with \na type 2  ice-rule configuration produced by the applied field protocol. We note that near the \nbottom left of the L2 nanowires, faint positiv e contrast is seen (Labelled A). A previous \ninvestigation, which took images in reversed tip configurations identified this as an art ifact \n[24], due to the abrupt upwards change in topography experienced by the tip at this point . Since \nthe signal originating from the artifact  is approximately a factor of two  small er than the signal \nFigure 3: Identification of monopole -excitations. (a) MFM image tak en at remanence \nfollowing a saturating field along the unit vector (1, -1,0) and subsequent 9.5mT field along the \nunit vector ( -1,1,0).  (b) Associated vector map illustrating the magnetic configuration,  \nmonopole -excitations are annotated with bright yello w (Q = -2q) and red (Q = +2q) lobes. Each \nisland represents a bipod, coloured with the local in -plane magnetization, as determined by \nkey. (c) MFM image taken at remanence following a saturating field along unit vector ( -1,-1,0) \nand subsequent 8.0mT field applied along (1,1,0).  (d)  Associated vector map illustrating the \nmagnetic configuration,  and presence of   monopole -excitations  (e-h) Magnified examples of \nthe MFM constrast associated with L1 -L2 junctions where Q = ±2q.  \n \n2.0µm\n2.0µm\n2.0µme\n2.0µm\na b\nc d\ng hf\n 0.25 Deg\n -0.35 Deg\n2.0µm0.25 °\n-0.35 °x\ny\nz 8 originating  from magnetic contrast , its presence does not impede analysis of the magnetic \nconfiguration.  \n \nTo demonstrate that  each sub -lattice can reverse independently, we now take images after \nsaturating fields along different principal  axes. Fig 2c shows the large scale MFM image taken \nat remanence after a saturating field along  unit vector ( -1,1,0) . It is clear that contrast upon L1 \nwires have inverted.  Further inspection of the magnified example  (Fig 2d)  clearly shows the \nlobes of contrast upon L1 have indeed inverted  showing the magnetization here has switched. \nThe contrast upon L2 is found to be unchanged , as expected . The system was then returned to \nthe initial state (Fig 2a) before a saturating field was applied  along the unit vector (1, 1,0). \nExamination of Fig 2 e, now shows contrast upon every L2 nanowire has changed. Close \ninspection of Fig 2f now shows stronger positive  contrast in bottom left and strong negative \ncontrast in top right, suggesting the wires have switched. Overall, these  results  provide \nconfirmation that L1-L2 vertices corresponding to saturated states can be identified. Our \nprevious work  [24] suggests that faint contrast is also expected at the top of L1 coordination \ntwo vertices (black dashed line in Figs 2 b,d,f)  and a t mid points upon L2, close to the L2 -L3 \njunction (bl ue dashed line in Figs 2 b,d,f) . Such contrast is expected even for uniformly \nmagnetise d states, due to a change in sign of M Z at the vertex. Upon L1, the effect of this is to \nsmear out the edge contrast, such that fainter contrast of lower magnitude is seen at the L1 \ncoordination two vertex. At the L2 -L3 vertex, faint contrast is also seen, but we note that this \nis not sufficient  to determine the magnetic state of the L3 layer.  \n \nWith this fundamental understanding we next sought to image  the magnetic configuration of \nvertex states observed during the switching process to determine if monopole -excitations can \nbe identified and tra cked. Figure 3a shows an MFM image following a saturating field along  9 the unit vector (1, -1,0) and subsequent 9.5mT field along the unit vector ( -1,1,0) . Optical \nmagnetrometry  in Fig S2 indicates this is within the field range that switching is expected  upon  \nL1. A vector map of the magnetic configuration (Fig 3b) has been produced through \nobservations of the MFM contrast associated with each L1 -L2 vertex  as well as the surrounding \nwires.  We note that there are multiple independent means to confirm the presence o f a \nmonopole. Firstly, contrast near the L1 -L2 vertex is an excellent indication. If three of the four \nwires have contrast of the same sign, this is a monopole state. This can be further confirmed \nby then checking contrast upon the opposite ends of the wir es. Finally, since the magnetic \nFigure 4 : Direct imaging of magnetic charge upon a 3D artificial spin ice system. (a -\ne) Vector maps illustrating the magnetisation configuration and associated monopole \nexcitations in  five snapshots during a reversal sequence upon the L1 sub -lattice. Here a \nsaturating field was first applied along the unit vector (1, -1,0) after which a field of 8mT \nwas applied along the unit vector ( -1,1,0). Successive images were then ca ptured at \nremanence following 0.25  mT increments. Each island represents a bipod, coloured with \nthe local in -plane magnetization, as determined by key . Each monopole excitation is \nassigned a unique index to track propagation between images. (f-j) Vector ma ps \nillustrating an equivalent reversal of the L2 sub -lattice.  Here the samples was first \nsaturated along the unit vector ( -1,-1,0) after which a field of 6.50mT was applied along \n(1,1,0). Successive images were then captured at 0.25mT increments. Full datasets, \nincluding raw  MFM images can be found in the supplementary information.  \n 8.50 mT\n1\n29.00 mT\n1\n29.75 mT\n310.25 mT\n3\n 4\n10.75 mT\n6.75mT\n1211\n65\n87\n109L1 sub -lattice reversal:\na\nfH\nL2 sub -lattice reversal: H\n8.25mT\n1211 5 7\n109\n815\n16\n1714\n139.00mT\n127\n16\n1714\n13 2119\n20\n189.50mT\n127\n14\n23\n2210.25mT\n23b c d e\ng h i jPositively Charged Excitations\nNegatively Charged Excitations\nx\ny\nz 10 charge upon the wire ends closest to L1 -L2, smears over the vertex area, the absolute magnitude \nof the phase is increased, when compared to an ice -rule state. We have used all three criteria \nsimultaneously  to identify monopo les at the L1 -L2 vertex. Interestingly, so long as a well \ndefined field protocol is used, it is also possible to infer the presence of  monopole s at the L2 -\nL3 vertex. Here, so long as  L3 has been saturated, we expect this sub -lattice to be uniformly \nmagneti sed. However, if the extremities  of two adjacent L2 nanowires both have positive or \nnegative contrast, a monopole is implied at the L2 -L3 vertex.  \n \n \nIn Fig 3a -b every L1 -L2 vertex  in the observed area resembles one of the patterns seen in Fig \n2b and d, with  two exceptions. These are two monopole -excitations, each with a charge of Q = \n-2q, readily identified due to the enhanced MFM signal , which is a factor of 2 greater than the \ncorresponding ice -rule state. Furthermore,  the signal associated with the L1 wire s either side \nof the monopoles  is clearly seen to oppose, whereas the L2 wires are identical and so must be \naligned . Figure 3c -d shows a similar intermediate state following  a saturating field applied \nalong unit vector ( -1,-1,0) and subsequent 8.0mT field applied along (1,1,0) . This allows \nintermediate states to be probed upon the L2 layer.  Here, 9 monopoles are identified through \nobservations of the contrast associated with each L1-L2 junction, as well as the surrounding \nwires. Figure 3e -h shows magnified examples of monopole -excitations with Q = ±2q, in each \ncase one pair of colinear wires exhibits opposing contrast with respect to one another, whilst \nthe other pair of colinear wires show matching patterns of contrast. We note that for both \nintermediate states (Fig 3a ,c), the sub -lattice that extends along the field direction is effectively \ndemagnetise d (M < 0.1M S), so it is intriguing that a vast difference in the density of monopole -\nexcitations is seen between the two images.  \nTracking monopole propagation on the lattice surface and sub -surface   11 To form a more complete  understanding of \nthe m onopole behavior  on the surface and \nsub-surface, we now measure the detailed \nswitching between two saturated states, \ntaking images at 0.25mT intervals. To do \nthis we carry out  direct observations of the \nreversal sequences for the L1 and L2 sub -\nlattices . Figure 4 shows vector maps \nrepresenting snapshots (full MFM data can \nbe found in the supplementary information) \nof the switching process for the upper two \nlayers of the lattice. Here each island \ncorresponds to a bipod on the lattice, as \ndefined in Fig 1d. Ea ch image contains \napproximately 70 wires on L1 and 70 wires \non L2, only counting those where the \nmajority of the wire is within the measured \narea. Analysis herein considers wires within \nthis 8\t×8\t𝜇𝑚\" measured region, this is due \nto a compromise between size of the \nobserved area and  data acquisition time.  \nFigure 4a illustrates the array after \napplication of 8.5 mT along the unit vector \n(-1,1,0). This field magnitude yields the \nfirst evidence of switching along this \n2.3 mT\n2.93 mT\n3.35 mT 3.35 mT2.93 mT2.3 mT\na\nb\ncd\ne\nf\ng\n1.5 2.0 2.5 3.0 3.5 4.00.00.20.40.60.81.0Fraction of States Excited\nµ0H (mT) L1\n L2\nx\ny\nz\nFigure 5 : Simulating the monopole \ndynamics upon a 3D artificial spin -ice. \n(a-c) Arrow maps showing magnetisation  \nconfiguration with field applied along unit \nvector ( -1,1,0). Arrows on L1 and L2 have \nblack borders, arrows on L3 and L4 are \nborderless. (d-f) Arrows maps for field \napplied along (1,1,0) resulting in highly \ncorrelated monopole pairs. (g) Fraction of \nexcit ed states during the L1 (blue) and L2 \n(red) reversal sequences.   12 direction.  Though much of the array remains saturated , six wires (th ree bipods) have switched \nyielding two monopole states, each with charge -2q (Monopoles 1,2). In both cases, the \nmonopoles are found at the intersection between L1 and L2. Further field increments yield \nadditional chains of wires switching (Fig. 4b -d), wit h a further two negative monopoles \n(Monopoles 3, 4) residing at the L1 -L2 junction, after which L1 reaches saturation within the \nsampled area (Fig. 4e).  \n \nFig 4f illustrates the measured region after the array had been saturated along the unit vector ( -\n1,-1,0) and a field of 6.75 mT applied in  (1,1,0) . Eight monopoles can be immediately seen \n(Monopoles 5 -12), all of which seem to have appeared in pairs of ±2q. Here, five monopoles \nreside upon the L1 -L2 junctions, whilst the remaining two reside upon L2 -L3 ju nctions. \nAdditional field increments lead to the creation of further monopoles (Monopoles 12 -18), \nwhilst others move along the L2 nanowires or propagate out of the measured area (Fig 4g -4j).  \n \nThe differences in monopole formation upon the L1 and L2 sub -lattices is striking. Application \nof an external field with component along L1 yields few uncorrelated magnetic charges (Fig \nS10a) within the measured region, which seem to only be observed within a narrow field \nwindow (8 mT - 10.5 mT). We note that whilst this yields a  net charge locally in the measured \narea, charge neutrality is expected across the full lattice. Analysis of the switching also shows \na distinct absence of magnetic  charges  upon surface vertices with coordination two. On the \ncontrary, the L2 switching leads to nucleation of many correlated pairs yielding almost equal \nnumbers of positive and negative magnetic charges (Fig S10b), meaning the net charge within \nthe measured area is close to zero throughout the field range  (Fig S10c). The magnetic charges \nare also formed at a lower field (6.5mT) for the L2 sub -lattice and remain for a wider field \nrange (6.5mT – 10.75mT).   13  \nModelling the 3DASI system  \nCalculating the total energy density of every possible vertex state , within  a micromagnetic \nframework  (Fig S11) is an insightful exercise and provides some initial understanding of the \nsystem. Here it can be seen that the energy density  to create a magnetic charge upon a \ncoordination two, surface vertex  is 3.2 times higher than t hat of a monopole at a coordination  \nfour vertex  suggesting surface charges will be very unfavourable. To understand the \nsignificance of this within the context of switching the entire array , we carry out Monte -Carlo \n(MC) simulations based upon a compass ne edle model (see methods). This is carried out for \nvarying surface energetic factors ( a) and quenched disorder arising from  fabrication ( 𝑑!, see \nmethods). A disorder of 𝑑!=30% showed good agreement with switching field distributions in \nexperimental data. The surface energetics factor  (a)  essentially scales the energy required to \nproduce a monopole upon the coordination two vertex, when compared to a coordination four \nvertex. A series of simulations with varying  a are shown in Fig S1 4. Simulations which \nconsidered degenerate monopole surface energetics ( a=1, Fig S1 4) with a field applied along \nprojection of L1 , (-1,1,0)  showed the presence of magnetic charges upon surface coordination \ntwo vertices  and also short Dirac strings, in contrast to experimental data. Increasing the surface \nenergetics factor to the value calculated in finite element simulations ( a=3.2), now reduces the \nnumber of magnetic charges seen upon surface coordination two vertices  but Dirac string \nlengths are still shorter than seen in experimental results.  \n \nFig 5a -c shows the results of MC simulations performed with enhanced surface energetics \n(a=6.4) for field applied along the unit vector  (-1,1,0). Upon the threshold of switching (Fig \n5b), chains of islands switch upon the L1 sub -lattice producing uncorrelated monopoles and \nlong Dirac strings as seen in the experimental data, before the majority of the array becomes  14 saturated (Fig 5c). Critically, ch arges upon surface coordination two vertices  are now very rare, \nwhich is in agreement with experiment. Fig 5d -f shows MC simulations for the field aligned \nalong unit vector (1,1,0) . Here, a low field immediately produces large numbers of correlated \nmonopol e-antimonopole pairs (Fig 5e), separated by a single lattice spacing, closely aligned \nwith the experimental data. Fig 5g summarises the simulation results by showing the fraction \nof excited states obtained upon L1 and L2, showing excellent qualitative agre ement with the \nexperimental data, presented in Fig S10.  \n \nDiscussion  \nAs in all ferromagnetic materials, the 3DASI studied here passes through a field -driven state \nwhereby the component along the field is effectively de magnetise d. It is interesting to ident ify \ntwo main ways that this can be achieved in this novel 3D nanostructured system. The first \npossibility is that of local demagnetization upon each vertex, whereby the production of \nmonopole/anti -monopole pairs locally yield a net magnetization of zero  upon the relevant sub -\nlattice . A second possibility is the production of stripes of alternating magnetization direction, \nyielding complete demagnetization upon a given sub -lattice. Here magnetic charges can only \nbe found at the stripe ends. A key quantity wh ich will be important in determining the means \nof demagnetization is that of the monopole effective chemical potential, which quantifies the \nextent to which monopoles remain closely correlated. This is defined as µ∗=µ/u, where  µ\tis \nthe chemical potential o f a monopole and  u=µ$Q\"/4πa. We note that when this value \napproaches half the Madelung constant (For diamond lattice, M/2=0.819) [ 26], a highly \ncorrelated monopole crystal is energetically favorable and hence is a possible state during the \nfield driven dyn amics. Within a simple dipolar model, we calculate (See methods) 𝜇∗≈1.179 \nfor four -way junctions upon L2. Surface energetics restrict magnetic charges upon coordination \ntwo vertices , so we must consider both the high -energy, coordination two  intermediate state,  15 modulated by the factor 𝛼, and the final state in which some energy has been spent separating \nthe monopoles from this intermediate state (Fig. S12). The intermediate state, i f stable, would \nimply an effective chemical potential of 𝜇!%&∗=4.331. Though the system must clear this \nenergy barrier to transition to a more favorable state, it is more conventional to only consider \nthe chemical potential with respect to the final st ate. Due to the less favorable Coulomb \ninteraction of the monopoles, the effective chemical potential to produce a monopole across  an \nL1 coordination two vertex  is 𝜇∗≈1.5661 , overall yielding a large r fraction of uncorrelated \ncharges. Though this latter v alue alone may not completely justify the striking differences in \nmonopole dynamics upon the lattice, the high energy intermediate state  is far more compelling \nand we note that this local energy barrier (See Fig S12 ,S13) must be cleared to produce \nmonopole s upon the L1 sub -lattice.  \n \nA key question that remains is the magnitude of surface energetic factor ( a) and why such large \nvalues are required in MC simulations ( a=6.4) when compared to the magnitude implied by \nmicro -magnetics. The surface energetics in these systems arise due to a difference in how the \nmagnetic charge is distributed for two -way and four -way junctions  [24]. In both cases this will \nbe dictated by a balance between exchange and dipolar energies. For coordination two vertices , \nthe reduced effective dimensionality and resulting confinement produces an unfavourably large \nenergy for monopoles upon the vertex. In contrast, the coordination four system  allows the \nmagnetic charge to spread across the vertex area , overall reducing the energy and yielding a \nstable monopole configuration.  It is important to note that even when a=3.2 (value indicated \nby MM simulations)  the resulting MC simulations still bear a far closer resemblance to \nexperiments than when enhanced surface energetics are not considered ( a=1), in terms of string \nlength, monopole density, and density of charges upon surface coordination two vertices . \nHow ever, increasing a beyond the value predicted by MM simulations yields an even closer  16 resemblance to experiments, due to fundamental differences in the two methods. In particular, \nthe MC simulations use a compass needle model, where the magnetic charge ass ociated with \neach wire is distributed evenly across each needle, effectively reducing the energy barrier for \nsurface charges to form. Therefore, a greater value of a is required to suppress surface charges \nand hence approximate the experimental observation s. \n \nIn conclusion, we have demonstrated the fabrication of a 3DASI system, where the magnetic \nconfiguration upon the upper two nanowire layers can be determined. We find a striking \ndifference in the field -driven magnetic monopole  transport  along two principle axes.  With a \nfield applied along  the projection of surface termination,  magnetic imaging shows a low \nnumber of uncorrelated monopoles during the switching, which are always found at \ncoordination four vertices. Applying a field along  the projection of  L2 yields large numbers of \ncorrelated monopoles. Micromagnetic and Monte Carlo simulations, supported by simple \ncalculations within a dipolar framework, suggest it is the difference in effective chemical \npotential , as well as the energy landscape experienced during surface monopole dynamics, \nwhich accounts for the measured differences. We anticipate that our study will inspire a new \ngeneration in artificial spin -ice study whereby the ground state in these 3DASI sys tems are \nexplored as a function of key parameters such as magnetic moment and lattice spacing. \nUltimately, this may also yield the realisation of monopole crystals as predicted in bulk spin -\nice [26] or bespoke  spin-ice ground states only possible in artifi cial systems  of novel  3D \ngeometry. By utilizing a full suite of magnetic imaging techniques including MFM, nanoscale \nballistic sens ing [27] and novel synchrotron -based  methods [28] , it is hoped that full 3D \ncharacterization of the bulk and surface will soon  be possible.  \n \nMethods   17 Fabrication  \nDiamond -bond lattice structures were fabricated upon glass coverslips via two -photon \nlithography (TPL). Substrates were  first cleaned in aceton e, followed by isopropyl alcohol \n(IPA), and dried with a compressed air. Next, droplets of Immersol 518 F immersion oil and \nIPL-780 photoresist are applied to the lower and upper substrate surfaces respectively. Using \na Nanoscribe Photonic Professional GT  system, a polymer scaffold in the diamond -bond lattice \ngeometry was defined within the negative tone photoresist, of dimensions 50 µm × 50 µm × \n10 µm. Samples were developed in propylene glycol monomethyl ether acetate for 20 minutes, \nthen 2 minutes in IPA , to remove any unexposed photoresist. Once again, the samples were  \ndried with a compressed air gun.  \n \nUsing a thermal evaporator, a uniform 50 nm film of Permalloy (Ni 81Fe19) was deposited on \nthe samples from above, yielding a magnetic nanowire lattice up on the polymer scaffold. This \ndeposition requires a 0.06 g ribbon of Ni 81Fe19, washed in IPA, and evaporated in an alumina \ncoated molybdenum boat. A base pressure of 10-6 mBar  is achieved prior to evaporation, the \ndeposition rate is 0.2 nm/s, as measured by a crystal quartz monitor.  \n \nScanning electron microscopy  \nImaging was performed using a Hitachi SU8230 SEM with an accelerating voltage of 10kV. \nImages were taken from top vie w as well as at a 45° tilt with respect to the substrate plane.  \n \nMagnetic force microscopy  \nMFM measurements were performed in tapping mode using a Bruker Dimension 3100 Atomic \nForce Microscope. Commercial low moment MFM tips were magnetised along the tip a xis \nwith a 0.5 T permanent magnet. Once mounted, uniform magnetic fields could be applied  18 parallel to each sub -lattice using a bespoke quadrupole electromagnet, which was fixed upon \nthe surface of the AFM stage. During the application of a field, the MFM t ip was positioned \nseveral mm above the scanning height, such that the tip magnetisation was not influenced. \nMFM data was taken at a lift height of 100 nm. Prior to capturing MFM images, feedback \nsettings were carefully optimised to ensure sample topography  was being accurately measured \non the three uppermost lattice layers (L1, L2, L3).  \n \nIn order to probe the transport of magnetic charge upon the 3DASI surface, the system was \nplaced into a well -defined state by saturating the array along a principal directi on (Hsat \nobtained via optical magnetometry, see Fig. S2). MFM images were then obtained after \nsuccessive field increments in the reverse direction. MFM measures the stray field gradient \nd2Hz/dz2 due to magnetic charges and hence is an ideal methodology to visualise such transport \nacross the surface [2 5].   \n \nFinite element simulations  \nMicro -magnetic simulations of bipod and tetrapod structures were carried out with NMAG28, \nusing finite element method discretisation. Geometries possessing wires with a crescen t-shaped \ncross -section were designed such that the arcs subtend a 160° angle. The inner arc is defined \nfrom a circle with 80 nm radius corresponding to the 160 nm lateral feature size of the TPL \nsystem. The outer arc is based on an ellipse with an 80 nm mi nor radius and 130 nm major \nradius, yielding a thickness gradient with a peak of 50 nm. The length of all wires is set to 780 \nnm, due to computational restraints. All geometries were meshed using adaptive mesh spacing \nwith a lower limit of 3 nm and upper l imit of 5 nm. Simulations numerically integrated the \nLandau -Liftshitz equation upon a finite element mesh. Typical Ni 81Fe19 parameters were used, \ni.e. M S = 0.86×106 Am-1, A = 13×10-12 Jm-1 with zero magnetocrystalline anisotropy.   19  \nMonte Carlo simulations  \nEach nanomagnet is modelled as an infinitesimally thin compass needle with a uniform, linear \nmagnetic moment density mL, with exceptions to this being made at coordination number two \nvertices. The moment orients along the long axis of the island. The inter action between \ncompass needles are equivalent to two equal and opposite magnetic charges with charges m/L \nplaced at their ends with exceptions for the coordination two vertex  energy. The coordination \nfour energy calculated by the micromagnetic simulations corresponds to compass needles with \nL = 0.92a while the micromagnetic energies for coordination two imply an interaction strength \n3.2 times stronger than those between other charges. The compass needle energy obeys \nCoulomb's law with corrections for experimental considerations:  \n \n𝐸!'=𝛼!'𝑑!𝜇$m\"\n4𝜋𝐿\"B1\nC𝒓(!−𝒓('C−1\nC𝒓(!−𝒓)'C−1\nC𝒓)!−𝒓('C+1\nC𝒓)!−𝒓)'CG−𝑑!𝒎!⋅𝑩\t \n \nWhere 𝒓(! and 𝒓)! are the locations of the positive and negative magnetic charge on the ith \nnanomagnet, 𝑩 is the exte rnal field applied either in the L1 or L2 direction with simulated \nmagnitudes ranging between 1.89 and 3.77 mT, 𝒎! is each nanomagnet’s magnetic moment \nwith amplitude m=MV (with 𝑀 being the saturation magnetization, and V the nanomagnet \nvolume), 𝜇$ is the magnetic permeability, and L = 1000 nm is the island length. The \nmagnetization was chosen to be 𝑀=850 kA m-1 in agreement with previous studies on \nnanoislands fabricated in the  same manner. 𝛼!' is a factor which increases the interaction \nstrength between 𝑖𝑗 pairs at coordination number 2 vertices on the L1 sub -lattice with respect \nto 𝑖𝑗 pairs at all other vertices, this captures the enhanced surface energetics indicated b y micro \nmagnetic simulations. Reversals were simulated with 𝛼!' = 1, 3.23, and 6.45 at coordination  20 number 2 vertices on L1, 𝛼!' = 6.45 was found to yield the closest agreement to experimental \nobservations, all other vertices were consistently define d with 𝛼!' = 1. Site disorder 𝑑! is drawn \nfrom the distribution 𝑃(𝑑)=*\n+√\"-𝑒.(\"#$)&\n&'& (𝜎=30% is found to yield good agreement with \nexperimental data in this case). This disorder arises due to subtle variations in 3D nanowire \ngeometry. Systems of the same dimensions as the experiment, one unit cell deep and 15 by 15 \nunit cells wide and long, were simulated with 20 replicas apiece.  \nThe simulations began in the saturated state seen in our experiments. A random spin was \nselected and the energy to f lip that spin was calculated. The corresponding spin flip was carried \nout if the energy lost exceeded a threshold energy corresponding to the magnetic coercivity, an \nalgorithm equivalent to the Metropolis method with zero temperature and used in prior spin  ice \nstudies17,29. Flips were attempted 10 times the number of total spins, sufficient for equilibration. \nThe resulting arrow maps were plotted in Fig. 5a -f and Fig. S13. Additionally, the number of \nexcited states was recorded from these final arrow maps a nd plotted as a function of field in \nFig. 5g.  \n \nDipolar approximation calculations  \nOne can define an effective chemical potential 𝜇∗=𝜇/𝑢, where 𝑢=𝜇$𝑄\"/4𝜋𝑎. Here 𝜇 is \ncalculated within the dipolar approximation. Magnetic moments are located upon a diamond -\nbond lattice. The energy of interaction between moments can then be approximated as:  \n𝐸*\"=𝑢|𝑚X*∙\t𝑚X\"−3(𝑚X*∙\t𝑟̂)(𝑚X\"∙\t𝑟̂)|\n\\𝑟\n𝑎\\/\n\t\t\t\t \nThis choice of deunitization places t he physical value of the lattice parameter a in u. All \ngeometric factors are then derived from a lattice where the lattice constant is one. Considering \nany two spins within a coordination four  geometry yields 𝐸*\"=0\n1√\"𝑢.\t For an ice -rule state  21 there are four low energy pairs and two high energy pairs yielding 𝐸!23=−0\n/√\"𝑢. A doubly \ncharged monopole can be created by flipping a single spin. Each monopole state has three low \nenergy pairs and three high energy pairs, makin g 𝐸45%56573 =0. The energy landscape for \ncreation of an L2 monopole is shown in Fig S12a. The chemical potential can thus be obtained \nvia  𝜇=(𝐸45%56573 −𝐸!23)=0\n/√\"𝑢\t. Therefore, the effective chemical potential, 𝜇∗, for a \ncoordination  four geometry is 0\n/√\" or about 1.179. In units of Kelvin this yields ~700K, as \nexpected for a system in the frozen regime.  \nNow turning to monopole nucleation  upon the L1 sub -lattice , the overall energy landscape is \nshown in Fig S12  (Top) , with a more deta iled depiction shown  in Fig S13. Surface energetics \nrestrict the presence of magnetic charges on the coordination two vertices . Imposing this \nconstraint now requires the resulting monopole -antimonopole pair to separate now \"√\"\n√/ lattice \nspacings apart. The e nergy of monopole creation is the same because two “monopole” \ncoordination four  vertices are converted to a higher energy state, but the long -range \n\tinteractions increase the energy. Approximating this via the dumbbell model’s long -range \ninteractions, 𝐸78=9\n8()*+,-/( where 𝑟2;(8<3 is the distance between the monopoles, the energy \nrequired to produce a surface monopole rises by the difference between the interaction of \nmonopoles a lattice spacing apart and monopoles separated further. This ra ises the cost by \n𝑢\"√\".√/\n\"√\", yielding an effective chemical potential,  𝜇∗, of 0\n/√\"+\"√\".√/\n\"√\" , or approximately \n1.566. From a dynamic perspective, this state must be created by either monopoles traversing \nthe bulk with multiple spin flips or generating a monop ole on the system’s surface. The former \nis prohibited dynamically because it would only be possible with a mix of L1 and L2 direction \nspin flips. This is forbidden because the external field responsible for the flips is only applied \nin one direction at a t ime. The latter option raises the energy further as the coordination two   22 monopole is created with both different geometry and an increased surface energetic factor \nwhen compared to the coordination four  monopole. The emergence of a coordination two  \nmonopol e is from a single favorable interaction, -0\n1√\"𝛼𝑢, to a single unfavorable interaction, \n0\n1√\"𝛼𝑢, each of which are modified by the α factor described in the Monte Carlo simulations. \nThis requires an energy change of 0\n/√\"𝛼𝑢. Averaging this with the coordi nation four  monopole \ngives the average, reduced cost per monopole in this intermediate state, 𝜇!%&∗=0\n/√\"=>*\n\". The \nsystem effectively leaps from a reduced, average energy of -0\n1√\"=>\"\n\" to an average of 0\n*\"√\"𝛼. \nFor our experimentally confirmed 𝛼=6.4, this implies a transition state costing 𝜇!%&∗=4.331. \n \nMagneto -optical Kerr Effect Magnetometry  \nA 0.5 mW, 637 nm wavelength laser was expanded to a diameter of 1 cm, passed through a \nGlan –Taylor polarizer to obtain an s -polarized beam, then focu sed onto the sample using an \nachromatic doublet (f = 10 cm), to obtain a spot size of approximately 10 μm2. During the \nsource -to-sample path the laser is attenuated, approximately reducing the power by a factor of \n4. The reflected beam was also collected u sing an achromatic doublet (f = 10 cm) and passed \nthrough a second Glan –Taylor polarizer, from which the transmitted signal was directed onto \nan amplified Si photodetector, yielding the Kerr signal. After magneto -optical Kerr effect data \nwas captured from the nanowire lattice, a second dataset was obtained from the substrate film. \nThe film data is scaled to the lattice data and subtracted off. This corrected for any Kerr signal \ncontributions originating from the film, during the lattice measurements.  \n \n \n  23 References and Notes:  \n \n1 Castelnovo,  C., Moessner,R. & Sondhi,S. Magnetic monopoles in spin ice. Nature  \n451, 42-45 (2008).  \n2 Jaubert,  L. D. C. & Holdsworth,P. C. W. Signature of magnetic monopole and Dirac \nstring dynamics in spin ice. Nat Phys  5, 258-261 (200 9). \n3 Morris,  D. et al.  Dirac Strings and Magnetic Monopoles in the Spin Ice Dy2Ti2O7. \nScience  326, 411-414 (2009).  \n4  Giblin,  S. R., Bramwell,  S. T., Holdsworth,  P.C.W., Prabhakaran,  D. & Terry,  I. \nCreation and measurement of long -lived magnetic monopole currents in spin ice. Nat \nPhys  7, 252-258 (2011).  \n5  Bramwell,  S. T. & Gingras,  M. J. P. Spin ice state in frustrated magnetic pyrochlore \nmaterials. Science  294, 1495 -1501 (2001).  \n \n6  Zhou,  H. D.  et al.  High pressure route to generate magnetic monopole dimers in spin \nice. Nat Commun  2, 478 (2011).  \n \n7 Jaubert,  L. D. C., Lin,  T., Opel,  T. S., Holdsworth,  P. C. W. & Gingras,  M. J. P. Spin \nice Thin Film: Surface Ordering, Emergent Square ice, and Strain Effec ts. Phys Rev \nLett 118, 207206  (2017).  \n \n8 Bovo,  L., Rouleau,  C. M., Prabhakaran,  D. & Bramwell,  S. T. Phase transitions in \nfew-monolayer spin ice films. Nat Commun  10, 1219  (2019).  \n \n9 Wang,  R. F.  et al.  Artificial 'spin ice' in a geometrically frustrated lattice of nanoscale \nferromagnetic islands Nature  446,102-102 (2007).  \n \n10 Nisoli,  C., Moessner,  R. & Schiffer,  P. Colloquium: Artificial spin ice: Designing and \nimaging magnetic frustration. Rev Mod Phys  85,1473 -1490 (2013).  \n \n11  Zhang,  S. et al.  Crystallites of magnetic charges in artificial spin ice. Nature  500, 553-\n557 (2013).  \n \n12 Branford,  W. R., Ladak, S., Rea d, D. E., Zeissler,  K. & Cohen,  L. F. Emerging \nChirality in Artificial Spin Ice. Science  335,1597 -1600 (2012).  \n \n13 Morgan,  J., Stein,  A., Langridge,  S. & Marrows,  C. Thermal ground -state ordering \nand elementary excitations in artificial magnetic square ice . Nat Phys  7,75-79 (2011).  \n  24 14 Ladak,  S., Read,  D., Tyliszczak,  T., Branford,  W. R. & Cohen,  L. F. Monopole \ndefects and magnetic Coulomb blockade. New J Phys  13,023023 (2011).  \n \n15 Ladak,  S. et al.  Disorder -independent control of magnetic monopole defect population \nin artificial spin -ice honeycombs. New J Phys  14, 045010  (2012).  \n \n16 Perrin,  Y., Canals,  B. and Rougemaille,  N. Extensive degeneracy, Coulomb phase and \nmagnetic monopoles in artificial sq uare ice. Nature  540, 410 (2016).  \n \n17 Farhan,  A. et al.  Emergent magnetic monopole dynamics in macroscopically \ndegenerate artificial spin ice. Sci Adv  5, eaav6380  (2019).  \n \n18  Wang,  Y. et al.  Rewritable artificial magnetic charge ice. Science  352,962-966 \n(2016).  \n \n19 Parkin,  S. S. P., Hayashi, M. & Thomas, L. Magnetic domain -wall racetrack memory. \nScience  320,190-194 (2008).  \n \n20  Parkin,S. & Yang,S. H. Memory on the racetrack. Nat Nanotechnol  10, 195-198 \n(2015).  \n \n21 Hunt,  M., Taverne,  M., Askey , J., May, A. , Van den Berg , A., Ho,  Y.D. ,  Rarity , J., \nand Ladak,  S., Harnessing Multi -Photon Absorption to Produce Three -Dimensional \nMagnetic Structures at the Nanoscale , Materials 13, 761 (2020)  \n \n22 Sahoo,  S., Mondal , S., Williams , G., May, A., Ladak , S. and Barman,  A. Ultrafast \nmagnetization dynamics in a nanoscale three -dimensional cobalt tetrapod structure , \nNanoscale 10, 9981 (2018)  \n \n23 Williams, G., Hunt ,M., Boehm ,B., May ,A., Taverne, Ho ,A., Giblin ,S., Read ,D.E. ,  \nRarity , J., Allenspach , R. and Ladak,  S. Two-photon lithography for 3D magnetic \nnanostructure fabrication , Nano Research 11, 845 (2018)  \n \n24 May,  A., Hunt,  M., van den Berg,  A., Hejazi,  A. & Ladak,  S. Realisation of a 3D \nfrustrated magnetic nanowire lattice. Communications Physics  2,13 (2019).  \n \n25 Ladak,  S., Read,  D. E., Perkins,  G. K., Cohen,  L. F. & Branford,  W. R. Direct \nobservation of magnetic monopole defects in an artificial spin -ice system. Nat Phys  \n6,359-363 (2010).  \n  25 26 Brooks -Bartlett, M.,  Banks, S.,  Jaubert,  L., Harman -Clarke, A. & Ho ldsworth, P. \nMagnetic -Moment Fragmentation and Monopole Crystallization. Phys Rev X  4 \n(2014).  \n \n27 Gilbertson , A.M. , Benstock , D. , Fearn , M. , Kormanyos , A. , Ladak , S., Emeny , M. \nT., Lambert , C. J. , Ashley , T., Solin , S. A.  and Cohen , L.F. , Appl. Phys. Lett. 98, \n062106 (2011)  \n \n28  Donnelly , C.,  Guizar -Sicairos  M., Scagnoli , V., Gliga  S., Holler , M., Raabe , J., and \nHeyderman , L., Nature 547, 328 (2017)  \n \n29 Moller, G. and Moessner, R. “Magnetic multipole analysis of kagome and artificial \nspin-ice dipolar arrays ”, Phys. Rev. B 80, 140409 (2009).  \n \n \nFunding : SL acknowledges funding from the Engineering and Physical Sciences Research \nCouncil (Grant number: EP/R009147/1 /) \nAuthor  contributions : SL conceived of the study and wrote the first draft of the man uscript. \nSample fabrication and Magnetic force microscopy was carried out by AM. Monte Carlo \nsimulations and chemical potential calculations were carried out by MS. Finite element \nsimulations were performed by AM and AV. Optical magnetometry was carried out by JA \nand MH. All authors contributed to writing the final manuscript.  \nCompeting interests : The authors declare no competing interests.  \nData and materials availability : Information on the data presented here, including how to \naccess them, can b e found in the Cardiff University data catalogue.  \n \n \n \n \n "    },    {        "title": "1807.03424v1.Hybrid_State_Free_Precession_in_Nuclear_Magnetic_Resonance.pdf",        "content": "1\nHybrid-State Free Precession in Nuclear Magnetic Resonance\nJakob Assländer1,2,\u0003, Dmitry S. Novikov1,2, Riccardo Lattanzi1,2,3, Daniel K. Sodickson1,2,3, Martijn A. Cloos1,2\n1Center for Biomedical Imaging, Dept. of Radiology, New Y ork University School of Medicine, New Y ork, NY , USA\n2Center for Advanced Imaging Innovation and Research, New Y ork University School of Medicine, New Y ork, NY , USA\n3The Sackler Institute of Graduate Biomedical Sciences, New Y ork University School of Medicine, New Y ork, NY , USA\nThe dynamics of large spin-1/2 ensembles in the pres-\nence of a varying magnetic field are commonly described\nby the Bloch equation. Most magnetic field variations result\nin unintuitive spin dynamics, which are sensitive to small\ndeviations in the driving field. Although simplistic field vari-\nations can produce robust dynamics, the captured informa-\ntion content is impoverished. Here, we identify adiabaticity\nconditions that span a rich experiment design space with\ntractable dynamics. These adiabaticity conditions trap the\nspin dynamics in a one-dimensional subspace. Namely,\nthe dynamics is captured by the absolute value of the\nmagnetization, which is in a transient state, while its\ndirection adiabatically follows the steady state. We define\nthehybrid state as the co-existence of these two states and\nidentify the polar angle as the effective driving force of the\nspin dynamics. As an example, we optimize this drive for\nrobust and efficient quantification of spin relaxation times\nand utilize it for magnetic resonance imaging of the human\nbrain.\nFor many nuclei, the spin gives rise to a magnetic\nmoment, whose dynamics can be used for quantum\ncomputing1and provides a window to study, e.g., the\nchemical structure of molecules, as done in nuclear\nmagnetic resonance2(NMR) spectroscopy, or the com-\nposition of biological tissue, as used for clinical diagnosis\nin magnetic resonance imaging3(MRI). Modeling spin-\nlattice and spin-spin interactions as random magnetic\nfield fluctuations4allows for capturing their macroscopic\neffect by the relaxation times T1andT2, respectively. This\nfacilitates the description of large spin-1/2 ensembles with\nthe classical Bloch equation5, formally akin to the time-\ndependent SchrÃ˝ udinger Equation in a 4D-space:\n¶t0\nBB@x\ny\nz\n11\nCCA=0\nBBB@\u00001\nT2\u0000wzwy 0\nwz\u00001\nT2\u0000wx0\n\u0000wywx\u00001\nT11\nT1\n0 0 0 01\nCCCA0\nBB@x\ny\nz\n11\nCCA. (1)\nHere, ¶tdenotes the partial derivative with respect to\ntime, x,y,zare the spatial components of the magnetiza-\ntion, and 1 is the normalized z-magnetization at thermal\nequilibrium. The Rabi frequencies2wxandwy(induced\nby radio frequency (RF) pulses), together with the Larmor\nfrequency wz, are the external drive of the spin dynamics.\nWhile the Bloch equation is very general, it provides\nlittle intuition to help design robust and efficient exper-\niments. This lack of intuition has biased experimental\ndesign towards elementary drives for which analytic so-\nlutions make the effect of spin relaxation and experimen-\ntal imperfections evident. For example, the workhorses\nof clinical MRI weight the signal intensity either by T1orT2effects by exploiting the simplest spin dynam-\nics, most notably exponential relaxation6–8and steady\nstates9–11. These basic drives span small subspaces\nlike the steady-state ellipse9,12–14, which harbor impov-\nerished spin dynamics compared to the richness found\noutside. More recent approaches strive to break away\nfrom such traditional experimental design in search for\nan improved signal-to-noise efficiency15. However, the\nnon-intuitive nature of the Bloch equation has limited\nthe exploration of this vast experiment design space to\nheuristic guesses15–20.\nThe rationale for this improved encoding efficiency is\nsketched in Fig. 1: Variations of the driving fields result\nin a transient state, which enables one to exploit the\nentire Bloch sphere in search for the optimal encoding of\ncharacteristic parameters such as spin relaxation times.\nThe same plot also points out a risk associated with\nthe transient state: Small magnetic field deviations can\nproduce substantially differing spin trajectories, which\ncan bias the estimation of characteristic parameters.\nThis is particularly problematic in biological tissue, where\ninhomogeneous broadening is inevitable and difficult to\nmodel19,21.\nHere, we formulated conditions under which the sen-\nsitivity to magnetic field deviations and inhomogeneous\nbroadening is greatly mitigated and reveal a large sub-\nspace of drives in which the Bloch equation is tractable.\nOur analysis shows that, under these conditions, the\ndirection of the magnetization adiabatically follows the\none of steady states, while the absolute value of the mag-\nnetization can be in a transient state. In this hybrid state ,\nthe spin dynamics live, therefore, in a one-dimensional\nsubspace and can be described by a 2x2 Hamiltonian:\n¶t\u0012\nr\n1\u0013\n= \n\u0000cos2J\nT1\u0000sin2J\nT2cosJ\nT1\n0 0!\u0012\nr\n1\u0013\n, (2)\nwhere ris the magnetization along the radial direction,\ni.e. its magnitude (cf. Section VII-B for the derivation).\nThis notation identifies the polar angle J(t), which is the\nangle between the z-axis and the magnetization, as the\nrelevant degree of freedom, which describes the joint\neffect of the drives wx(t),wy(t), and wz(t)on the spin\ndynamics. As an example, we show that this hybrid-state\nequation and its solution provide intuition for the encoding\nprocesses of spin relaxation times and are an excellent\nbasis for numerical optimizations of a T1andT2mapping\nexperiment that combines the robustness of the steady\nstate with the encoding efficiency of the transient state.arXiv:1807.03424v1  [physics.med-ph]  9 Jul 20182\nxyztransient state\nxyzhybrid state\nxyzsteady state\n0.20.40.60.81\nφ/π\nxyz\nxyz\nxyz\n0.80.911.11.2\nB1/Bnominal\n1\nFigure 1: In the fully-transient state (here visualized on the left for the example of a random RF-pattern), the spin trajectories on the Bloch\nsphere are, in general, very sensitive to magnetic field inhomogeneities. Deviations of the Larmor frequency are depicted in the top row, where\nfdenotes the phase accumulated over one repetition time T R, and we define f=pas the on-resonance condition. The bottom row sketches\nspin trajectories for deviations in the RF field B1, which alter the Rabi-frequencies. The signal of an NMR sample or a volume element in MRI\n(visualized by the cube) is generated by spins at different Larmor frequencies, which additionally introduces a strong sensitivity of the signal to\nthe particular distribution of Larmor frequencies22. The hybrid state, shown at center, is explicitly designed to mitigate these sensitivities, while\nstill allowing the magnetization to visit the entire Bloch sphere. Fully adiabatic transitions between steady states, shown at right, have the same\nrobustness to magnetic field deviations, however, they trap the magnetization on the steady-state ellipse9,12–14, which diminishes the capabilities\nto encode tissue properties such as relaxation times. The steady-state ellipse is described by setting the left hand side of Eq. (2) to zero.\nI. H YBRID STATE BOUNDARY CONDITIONS\nAs the magnetization described by Eq. (1) is real-\nvalued, we can conclude that the eigenvalues of the\nHamiltonian must either be real-valued or occur in\ncomplex conjugate pairs. One eigenvalue is zero and\ndescribes the steady-state magnetization. Therefore,\nanother eigenvalue must be real-valued. As such, it\ndescribes an exponential decay of the corresponding\ntransient-state component, while the remaining complex\neigenvalues describe oscillatory decays. Ganter pointed\nout that the complex phase makes the latter components\nvery sensitive to deviations in the magnetic field and\nin particular to inhomogeneous broadening22. Fig. 1\nprovides some intuition for this sensitivity: As the com-\nplex phase accumulates during the experiment, the spin\ntrajectory becomes very sensitive to deviations in the\nmagnetic fields. Considering that the measured signal is\ninvariably given by the integral over some distribution of\nLarmor frequencies, which is difficult to model in biologi-\ncal tissue21, contributions of the complex eigenvalues will\nlead to a bias in the estimated relaxation parameters19.\nConversely, if we design our MR experiment such that\nthe cumbersome complex eigenstates are not populated,\nwe achieve robustness to magnetic field deviations and\ninhomogeneous broadening. If we simultaneously pop-\nulate the real-valued transient eigenstate, we liberate\nthe magnetization from the steady-state ellipse and gain\naccess to the entire Bloch sphere (Fig. 1).In general, variations of the driving fields rotate the\neigenvectors and populate all transient eigenstates. A\nTaylor expansion of this eigenbasis rotation (cf. Section\nVII-A) reveals that this population is dominated by the\ngaps between the eigenvalue and the rest of the Hamil-\ntonian’s spectrum, similar to the quantum mechanical\nadiabatic theorem23. The real-valued eigenvalue is close\nto the steady-state eigenvalue, resulting in a very restric-\ntive boundary condition. On the contrary, the complex\neigenvalues are well separated from the rest of the\nspectrum due their complex phase, resulting in a less\nrestrictive boundary condition.\nFor pulsed experiments6, which dominate modern MR,\nwe find the condition\nmaxfj\u0001aj,j\u0001fjg\u001c sin2a\n2+sin2f\n2\u00005\n2(1\u0000E2)(3)\nunder which the complex eigenstates are not populated,\nand\nmaxfj\u0001aj,j\u0001fjg\u001c (1\u0000E1)2(4)\nunder which the real-valued eigenstate is not populated.\nHere, the driving fields are parameterized by the flip angle\naand the accumulated phase f=wzTR, where the\nrepetition time T Rdenotes the time between consecu-\ntive RF pulses, and \u0001aand\u0001fdenote the change of\nthese parameters in consecutive repetitions. Relaxation\nis described by E1,2=exp(\u0000TR/T1,2).\nExperiments in which Eq. (3) holds, but Eq. (4) does\nnot, result in non-trivial, yet tractable spin dynamics that3\nare rich in information content. Since the latter adia-\nbaticity condition is substantially more restrictive, the\nhybrid state theory governs a vast experiment design\nspace. In order to provide some intuition, we can assume\nTR=4.5 ms, and relaxation times of human brain white\nmatter ( T1=781 ms and T2=65 ms)16. In such a case,\nmaxfj\u0001aj,j\u0001fjg\u001c 1 suffices to avoid a population of\nthe complex eigenstates when, e.g., assuming f=p. In\ncontrast, maxfj\u0001aj,j\u0001fjg\u001c 10\u00005would be required to\navoid a population of the real-valued transient eigenstate.\nII. A DIABATICITY AND THE SOLUTION OF THE BLOCH\nEQUATION\nHargreaves et al. showed that the eigenvector cor-\nresponding to the complex eigenvalue is approximately\nperpendicular to the steady-state magnetization24, while\nthe real-valued eigenvalue describes the transient-state\ncomponent parallel to the steady-state magnetization. By\nenforcing Eq. (3), we, thus, effectively force the direction\nof the magnetization to adiabatically follow that of the\nsteady states. If we then simultaneously pick our driving\nfields to violate Eq. (4), the magnitude of the magnetiza-\ntion is in a transient state, and a hybrid of two co-existing\nstates emerges, which we dub hybrid state .\nThe adiabaticity of the magnetization’s direction effec-\ntively decouples the components of the Bloch equation,\nwhich allows us to formulate an analytic solution. For this\npurpose, we transform the Bloch equation into spherical\ncoordinates and provide the solutions for the polar angle\nJ, the phase j, and the radius r, which we here define as\nthe magnitude combined with a sign (cf. Section VII-B for\nthe derivation). Except in the vicinity of the stop bands,\nwhich are defined by jsinfj\u001c1 (cf. supporting Fig. S4),\nthe polar angle can be approximated by\nsin2J=sin2a\n2\nsin2f\n2\u0001cos2a\n2+sin2a\n2. (5)\nThis equation reduces to J=a/2 for f=p, which we\ndefine as the on-resonance condition. In practice, f=p\nis assigned to the on-resonant spin isochromat by the\ncommon phase increment of pin consecutive RF pulses.\nThe phase of the magnetization is approximated by\nj=tan\u00001\u0012cosf\u0000E2\nsinf\u0013\n\u0000Hf sinfg\u0001p+fTE, (6)\nwhere the Heaviside function Hdisambiguates the four-\nquadrants and fTEdescribes the phase of the magneti-\nzation accumulated between the RF pulse and the time\nthe signal is observed, i.e., the echo time TE.\nThe radial component rcaptures the entire spin dy-\nnamics, which is described by a single first order differ-\nential equation (Eq. (2)). This equation is solved by\nr(t) =a(t)\u0001\u0012\nr(0) +1\nT1Zt\n0cosJ(t)\na(t)dt\u0013\n(7)\nwith\na(t) =exp \n\u0000Zt\n0sin2J(x)\nT2+cos2J(x)\nT1dx!\n.Here, tdenotes time and r(0)the initial magnetization.\nAlternatively, we can define the initial magnetization as a\nfunction of the final magnetization, i.e. r(0) =b\u0001r(TC),\nwhere TCdenotes the duration of a single cycle of the\nexperiment. With this boundary condition, the radial Bloch\nequation is solved by Eq. (7) with\nr(0) =b\nT1a(TC)\n1\u0000ba(TC)ZTC\n0cosJ(t)\na(t)dt.\nWhen we set b=1, a periodic boundary condition is ob-\ntained, which requires the magnetization at the beginning\nand the end of each cycle to be equal. Similarly, b=\u00001\nleads to an anti-periodic boundary condition, which im-\nplies an inversion of the magnetization between cycles.\nSuch boundary conditions enable the concatenation of\nmultiple cycles without delays, thus, allowing for efficient\nsignal averaging and a flexible implementation, e.g., of\ntime-consuming 3D imaging experiments.\nIntuitively, Eq. (7) describes a predominant T1en-\ncoding at small J-values (close to the z-axis), and a\npredominant T2encoding as Japproaches p/2, which\ncorresponds to the x-y-plane. When Jis constant, Eq. (7)\nreduces to the exponential transition into steady state\ndescribed by Schmitt et al.25(cf. supporting material).\nSupporting Fig. S4 validates the hybrid-state model\nby comparing Eqs. (5)-(7) to Bloch simulations for the\nexample of anti-periodic boundary conditions.\n0.70.80.91T1(s)\nCRB →0 20 40 60 800.040.050.060.070.08\nσω(rad/s)T2(s)steady state\nhybrid state\ntransient state\nFigure 2: Both, the steady-state and the hybrid-state experiments\nare robust with respect to inhomogeneous broadening (here modeled\nby a Gaussian distribution of Larmor frequencies with the standard\ndeviation sw), while the transient state exhibits a substantial bias with\nincreasing broadening. The observed noise (indicated by the error bars)\nis considerably less in the hybrid state compared to the steady state,\nand for all experiments the observed noise approximates the limit set\nby the Cramér-Rao bound (CRB) well (far left). The relaxation times\nwere estimated from signal simulated with the steady-state pattern\nshown in supporting Fig. S3n, an anti-periodic hybrid-state pattern\n(Fig. 4e), and the transient state is illustrated using the example of the\noriginal magnetic resonance fingerprinting (MRF)15experiment. Note\nthat the steady-state and the hybrid-state experiment have a duration\nofTC=3.8 s, while the MRF experiments lasts for 12.3 s.4\nIII. E FFICIENCY OF THE HYBRID STATE\nThe superior signal-to-noise ratio (SNR) efficiency of\nthe hybrid state in comparison to the steady state be-\ncomes evident when comparing numerically optimized\nexperiments. For this purpose, we simulated the aver-\nage signal obtained from a collection of isochromats\nwith a Gaussian distribution of Larmor frequencies and\nadded white noise to reflect thermal noise. Because the\ninternal frequency distribution in a sample is generally\nunknown, the obtained signals were fitted with their re-\nspective models assuming a single isochromat. Fig. 2\nshows that the transient state leads to increasingly biased\nestimates of the relaxation times as the distribution of\nLarmor frequencies widens ( swincreases). Conversely,\nboth the steady and hybrid state demonstrate a similar\nrobustness with respect to inhomogeneous broadening.\nAs anticipated, the estimates retrieved from the hybrid-\nstate experiment exhibit substantially less noise. The\nhybrid state, thus, unites superior encoding capabilities\nsimilar to the transient state, and robustness deviations\nof the magnetic fields and to inhomogeneous broadening,\nsimilar to the steady state.\nFor a more comprehensive analysis of the noise prop-\nerties of different experiment design spaces, we examine\nthe sum of the relative Cramér-Rao bound ( rCRB ) for\nT1- and T2-encoding. The rCRB provides a lower limit\nfor the noise in the estimated parameters, normalized by\nthe input noise variance, by the square of the respective\nrelaxation time and by TC/TR(Eqs. (35) and (36)). It\ncan be understood as a lower bound for the squared\ninverse SNR efficiency per unit time, and Fig. 2 shows\nthat the simulated noise comes close to this theoretical\nlimit. We numerically searched the parameter space of\npossible drive functions for the lowest combined rCRB .\nDue to the nature of the steady state, its rCRB does not\ndepend on TC, so that the experiment’s duration can be\nchosen freely to meet the experimental needs. Hybrid-\nstate experiments with anti-periodic boundary conditions\nprovide a similar flexibility, since multiple cycles can\nbe concatenated without gaps. Comparing these two\nexperiments, one finds that the hybrid state allows for a\nsubstantially more efficient measurement than the steady\nstate (Fig. 3).\nThe performance of exponential relaxation curves is\nhere demonstrated using the example of the inversion-\nrecovery balanced steady-state free precession (IR-\nbSSFP) experiment†, which is known to have a high SNR\nefficiency25,27. In contrast to the previously discussed ex-\nperiments, the magnetization departs here from thermal\nequilibrium. This requires a long waiting time ( \u0001t\u001dT1)\nbefore the measurement can be repeated. For TC.25s,\nexponential experiments have a lower rCRB compared\nto steady-state experiments, and for TC.5s it is even\nlower compared to anti-periodic hybrid-state experiments\n†Despite the name, this is actually not a steady-state experi-\nment. Instead, one measures the magnetization as it exponentially\napproaches the steady state.0 5 10 15 20 25 30 35 40103104\nTC(s)rCRB (T1) +rCRB (T2)\nsteady-state\nanti-per. bHSFP\nexponential\nIR-bHSFP\norg. DESPOT26\norg. MRF15\norg. pSSFP19\nFigure 3: The depicted relative Cramér-Rao bounds ( rCRB ) are defined\nby Eqs. (35) and (36), and can be understood as a lower bound of\nthe squared inverse SNR efficiency per unit time. One can observe\nthat, for most cycle times (TC), exponential decays as well as steady-\nstate experiments are substantially less efficient than variants that\nexploit the entire experiment design space spanned by the hybrid state,\nnamely the inversion recovery balanced hybrid-state free precession\n(IR-bHSFP) and the anti-periodic bHSFP experiment. For reference,\nsome experiments from literature are shown as well, namely the original\nDESPOT26, MRF15, and pSSFP19experiment. All Cramér-Rao bounds\nwere calculated for the relaxation times T1=781 ms and T2=65 ms.\n(Fig. 3). An optimization of exponential experiments is es-\nsentially the search for the optimal line from the southern\nhalf of the Bloch sphere to the steady-state ellipse (sup-\nporting Fig. S3g). If we take the IR-bSSFP experiment\nand allow J(t)to vary over time, we can exploit the full\nexperiment design space spanned by the hybrid state,\nand we find an improved SNR-efficiency at all TCvalues,\nwith the most dramatic improvement in the case of long\nexperiments. In analogy to the acronym IR-bSSFP , we\nuse the term inversion-recovery balanced hybrid-state\nfree precession (IR-bHSFP) for hybrid-state experiments\nthat start from thermal equilibrium by the application of\nan inversion pulse‡.\nIn this section, we analyzed the noise properties at\na single T1and T2value. Supporting Figs. S1 and S2\ndemonstrate that the conclusions drawn here remain\nvalid throughout large areas in T1-T2-space, and also\nin the presence of deviations of the Larmor and Rabi\nfrequencies.\nIV. S PINDYNAMICS IN THE HYBRID STATE\nOptimizing the driving functions J(t)results in spin\ntrajectories with reproducible features. For example, all\noptimizations resulted in comparatively smooth functions\nJ(t). Note that the optimizations assume a hybrid state,\nbut otherwise do not enforce smoothness, which indi-\ncates that the adiabaticity condition (Eq. (3)) does not\nimpair the T1,2-encoding efficiency. In some segments,\nthe optimization exploits the design limits 0 \u0014J\u0014p/4,\nwhich are imposed for practical reasons. These extreme\nvalues help to achieve a large dr/dT1while minimizing\n‡We focus this analysis on experiments with balanced gradient\nmoments because of their superior SNR properties.5\nIR-bHSFPa\nxzIR-bHSFPa\nxz\n00.10.2 bϑ/π\n−0.200.20.40.6 canti-periodic bHSFPd\nxzanti-periodic bHSFPd\nxz\n00.10.2eϑ/π\n0 1 2 3−1−0.500.5\nTCf\nt(s)r−T1dr\ndT1T2dr\ndT2\nFigure 4: The spin dynamics in hybrid state experiments are depicted\non Bloch spheres ( a,d). The optimized polar angle functions are shown\nin (b,e), with the color scale providing a reference for the trajectories\non the Bloch spheres. The radial component magnetization and its\nnormalized derivatives with respect to the relaxation times are the\nfoundation of computing the relative Cramér-Rao bound and are shown\nin (c,f). Both spin trajectories were jointly optimized for T1andT2and\nthe polar angle was limited to 0 \u0014J\u0014p/4.\ndr/dT2and vice versa. However, in other segments, e.g.,\ndirectly after crossing the origin (turquoise segment), the\nderivative dr/dT2is already close to zero and the mag-\nnetization follows a trajectory with J>0. Similarly, after a\nsegment of J\u00190 (yellow segment), dr/dT2approaches\nzero and the optimized driving function transitions to a\nJ>0, resulting in non-zero signal and disentangled\nencoding of rand dr/dT1. Further, the optimized tra-\njectories do not spend a significant amount of time on\nthe steady-state ellipse. On the contrary, crossing the\nellipse triggers a fast change of J, as highlighted by the\nmagnifications in Fig. 4.\nDescribed hybrid-state spin trajectories result from\nnon-convex optimizations and we can only speculate\nabout their optimality. However, the simple and repro-\nducible structures, together with the simple form of the\ngoverning Eq. (2) provide an excellent basis for a more\ndetailed analysis.\nV. I NVIVOEXPERIMENT\nFig. 5 shows an example application of the hybrid state.\nTheT1- and T2-maps in a sagittal slice through a human\nbrain were acquired with an anti-periodic bHSFP exper-\niment and also serve as a validation of the hybrid-state\nmodel: Fitting the data with the full Bloch model and the\nhybrid-state model resulted in virtually the same T1- and\nT2-maps, which is also confirmed by the values within a\nregion of interest (Bloch model: T1=965\u000623ms, T2=\n48.2\u00063.0ms; hybrid-state model: T1=988\u000623ms,\nT2=49.7\u00062.9ms).\nBloch model\n hybrid-state model\n12345\nT1(s)\n0.050.10.20.3\nT2(s)Figure 5: A single sagittal slice of an in vivo 3D human brain MRI\nscan is depicted. The data were acquired with an anti-periodic bHSFP\nexperiment and were fitted once with the Bloch model (Eq. (1)), and\nonce with the hybrid-state model (Eq. (5)-(7)). The parameter maps\nhave a resolution of 1 mm \u00021 mm\u00022 mm and spatial encoding was\nperformed with a 3D stack-of-stars k-space trajectory28. The red box\nindicates a region of interest used for extracting T1andT2values. Note\nthe logarithmic scale of the color coding. The entire 3D data set can\nbe found in supporting Fig. S5.\nVI. S COPE OF THE HYBRID -STATE MODEL\nAdiabatic passages are frequently used in NMR, MRI,\nas well as quantum computing for robust spin excitation,\ninversion, and refocusing in the presence of magnetic\nfield inhomogeneities29,30. These passages are achieved\nby continuous, slowly varying driving fields, and are\ncommonly assumed to be much faster than spin relax-\nation, such that one enforces adiabatic transitions of the\nmagnetization’s direction, while its magnitude is assumed\nto be constant. Neglecting relaxation, the Hamiltonian\nin Eq. (1) reduces to a generator of a rotation and\nwe can derive the well established adiabaticity condi-\ntionjdwx,y,z/dtj \u001c w2\nx+w2\ny+w2\nzwith the described\nformalism. Here, we generalized adiabatic passages to\npulsed experiments, which allows for exploiting their ro-\nbustness throughout the entire experiment. The hybrid-\nstate adiabaticity condition (Eq. (3)) has a very similar\nstructure to the established adiabaticity condition, apart\nfrom an additional relaxation term, which is required at\ntypical experiment durations at the order of seconds\nto minutes. Gaining a flexible and efficient access to\nrelaxation mechanisms while exploiting the robustness\nof adiabatic passages constitutes the core of the hybrid-\nstate framework.6\nThe robustness of the measured signal to magnetic\nfield deviations, including inhomogeneous broadening,\nis reflected by the hybrid-state equations of motion\n(Eqs. (5)-(7)) being smooth functions of the Larmor and\nRabi frequencies, which are here parameterized by f\nand a, respectively. This property is a direct conse-\nquence of constraining the population of the complex\neigenstates and is particularly important when the line\nshape is unknown, e.g., when measuring biological tissue\nwith balanced-HSFP experiments21. The estimation of\nthe distribution is less problematic in unbalanced ex-\nperiments, such as the fast imaging with steady-state\nprecession11(FISP) experiment, or the reversed PSIF\nexperiment. In these experiments, one places spoiler\ngradient pulses directly before or after the RF pulses,\nwhich desensitize the signal to inhomogeneous broad-\nening at the cost of SNR. The hybrid-state model holds\ntrue for these experiments, and the spoiler gradients can\nbe incorporated by setting fTE=0 orfTE=fin Eq. (6)\nfor FISP and PSIF , respectively.\nFor complex molecules, as well as for complex biologi-\ncal tissues, the Bloch equation is an oversimplified model.\nThis can be observed in Fig. 5, where the measured\nrelaxation times are subject to systematic deviations,\nwhich are most likely caused by magnetization trans-\nfer31–33. Magnetization transfer, as well as diffusion34\nand chemical exchange35, are captured neither by the\nBloch equation, nor by the hybrid-state model in their\nbasic form. However, these effects can be modeled by\nextensions to the hybrid-state model similarly to the\nestablished extensions of the Bloch equation34,35. Such\nextended hybrid-state models can provide a more intu-\nitive understanding of these effects, and pave the road\ntowards more efficient experiment designs to measure\nthem.VII. M ETHODS\nA. Adiabaticity Conditions of the Hybrid State\n1) The Evolution Matrix\nIn order to describe pulsed MR experiments, we analyze the\nspin evolution matrix U2R4\u00024, which is generated by the\nHamiltonian. The matrix Ucan, e.g., be derived by taking the\nmatrix exponential of the Hamiltonian and is not unitary due\nto the relaxation terms (Eq. (1)). Note that an analysis of the\nevolution matrix is largely equivalent to an analysis based on the\nHamiltonian itself. For pulsed experiments, where we assume\none hard, i.e. infinitesimally short, RF pulse, surrounded by\nLarmor precession and relaxation, the evolution matrix is given\nby\nU=E\u0001Rz\u0001Ry\u0001Rz\u0001E, (8)\nwhere\nE=0\nBB@pE2 0 0 0\n0pE2 0 0\n0 0pE11\u0000pE1\n0 0 0 11\nCCA\ndescribes the relaxation of the magnetization with E1,2=\nexp(\u0000TR/T1,2). The rotation matrices\nRy=0\nB@cosa0\u0000sina0\n0 1 0 0\nsina0 cos a0\n0 0 0 11\nCA\nand\nRz=0\nBB@cosf\n2\u0000sinf\n20 0\nsinf\n2cosf\n20 0\n0 0 1 0\n0 0 0 11\nCCA\ndescribe the rotations caused by the RF pulse and free preces-\nsion, respectively.¶\nFor future reference, we also define the derivative of Uwith\nrespect to a, which is given by U0=ERzR0yRzEwith\nR0\ny=0\nB@\u0000sina0\u0000cosa0\n0 0 0 0\ncosa0\u0000sina0\n0 0 0 01\nCA, (9)\nand the derivative of Uwith respect to f, which is given by\nU0=ER0zRyRzE+ERzRyR0zEwith\nR0\nz=1\n20\nBB@\u0000sinf\n2\u0000cosf\n20 0\ncosf\n2\u0000sinf\n20 0\n0 0 0 0\n0 0 0 01\nCCA. (10)\n¶Eq. (8) assumes a symmetric experiment, as it is used e.g. in\nbalanced-SSFP experiments, where one usually measures the mag-\nnetization in the middle between two RF pulses ( TE=TR/2)36. In\nthe case of unbalanced-SSFP experiments, one would usually acquire\nthe magnetization right after each RF pulse and would place a so-\ncalled spoiler gradient after the signal acquisition in order to create\na net gradient moment. In such a FISP11experiment, the evolution\nmatrix would, thus, be given by UFISP=Ry\u0001Rz\u0001E2with the appropriate\nchoice of f, and the reversed PSIF experiment with the spoiler gradient\nprior to the readout would be described by UPSIF=E2\u0001Rz\u0001Ry. Note\nthat derivations for FISP and PSIF lead to the same result as the one\npresented here.7\n2) Eigendecomposition of the Evolution Matrix\nThe eigendecomposition of the evolution matrix is given by\nU=V\u0003V\u00001, (11)\nwhere V2C4\u00024is composed of the right-eigenvectors vd2\nC4\u00021defined by Uvd=ldvd, and \u00032C4\u00024is a diagonal\nmatrix with the eigenvalues ld2Con the diagonal. The\nmagnetization in MR experiments never grows arbitrarily, so\nthatjldj\u0014 1 must be fulfilled for all eigenvalues. Further, if\nthe experiment described by Uhas a non-zero steady-state\nmagnetization, at least one eigenvalue must fulfill jldj=1.\nFor the explicit definition of the evolution matrix in Eq. (8),\nwhich describes one RF pulse surrounded by free precession\nand relaxation, one eigenvalue is given by\nlS=1 (12)\nand the corresponding eigenvector describes the steady-state\nmagnetization. As shown by Ganter22, the remaining eigenval-\nues are approximated by\nlk=1\nh2\u0012\ncos2a\n2sin2f\n2E1+sin2a\n2E2\u0013\n(13)\nl(\u0003)\n?=e\u0006i\n2h2\u0012\nsin2a\n2E1+\u0012\nh2+cos2a\n2sin2f\n2\u0013\nE2\u0013\n(14)\nwith\nh=r\ncos2a\n2sin2f\n2+sin2a\n2(15)\ne\u0006i\n=1\u00002h2\u00062hicosa\n2cosf\n2. (16)\nThese eigenvalues are a first order approximation of the pa-\nrameter\nd=E1\u0000E2\nE1+E2, (17)\nwhich is small for TR\u001cfT1,T2gin most biological tissues22,\nhave an absolute value smaller than one, and describe the\ntransient state. The eigenvalue lkis real-valued and the corre-\nsponding eigenvector is approximately parallel to the steady-\nstate magnetization in the three spatial dimensions22. The\nother two eigenvalues l(\u0003)\n?are in general complex and complex\nconjugate of each other, as indicated by the star. This results\nin the well known oscillatory behavior of the transient state of\nbSSFP experiments24. As shown by Ganter22, the correspond-\ning eigenvectors are approximately perpendicular to the steady-\nstate eigenvector.\n3) The Perturbation Matrix\nA sequence of Nidentical and equidistant RF pulses is simply\ndescribed by UN=V\u0003NV\u00001and describes the transition into\nthe steady state22,24. The description of an experiment with\nvarying driving fields, as required to avoid the steady state,\nis slightly more complicated. To approach this problem, we\ndenote the evolution matrix of the nthrepetition by Unand\nthe spin dynamics in two consecutive repetitions is described\nbyUnUn\u00001=Vn\u0003nV\u00001\nnVn\u00001\u0003n\u00001V\u00001\nn\u00001=Vn\u0003nPn\u0003n\u00001V\u00001\nn\u00001.\nHere, the perturbation matrix\nPn=V\u00001\nnVn\u00001 (18)\ndescribes the transformation from the eigenspace of Un\u00001to\nthe eigenspace of Un.4) Expanding the Perturbation Matrix\nSince an explicit notation of the perturbation matrix is not\nvery enlightening, we approximate its elements by a Taylor\nexpansion. As demonstrated in the supporting material, any\nchanges \u0001kof the parameters k2fa,fghas to be small in\norder to avoid a population of the transient eigenstates. This\nallows us to employ the Taylor expansion Un\u00001=U(kn\u00001) =\nU(kn)\u0000\u0001knU0(kn) +O(\u0001k2n), where U0(kn) = dU/dkjk=kn\ndenotes the derivative evaluated at kn. Assuming that U(kn)is\nnot degenerate, i.e. all eigenvalues are distinct, we can utilize\nthe Taylor series described by Eq. (10.2) in Chapter 2 of Ref.37\nto expand the perturbation matrix (Eq. (18)). The diagonal\nelements are then given by Pd!d=1 and the off-diagonal\nelements by\nPd!f6=d(kn,\u0001kn)\u0019\u0001knuH\nf(kn)U0(kn)vd(kn)\n(ld(kn)\u0000lf(kn))uH\nf(kn)vf(kn), (19)\nwhere the left-eigenvectors are defined by uH\nf(kn)U(kn) =\nlf(kn)uH\nf(kn)and the right-eigenvectors by U(kn)vf(kn) =\nlf(kn)vf(kn). The superscript Hindicates the complex conju-\ngate transpose. Eq. (19) has some similarities to the quantum\nmechanical adiabatic theorem23. In both cases, the matrix\nelements strongly depend on the gap between the eigenvalues.\nLike in quantum mechanical case, lS\u0000lkis purely determined\nby the absolute value of the eigenvalues, since they both are\nreal-valued and positive. This is fundamentally different in the\ncase of lS\u0000l(\u0003)\n?, where the gap is dominated by the complex\nphase of l(\u0003)\n?. In the following, we will show that this key\ndifference opens the door for the hybrid state to emerge.\n5) The Population of the Transient Eigenstates\nIn order to analyze the cumulative population transfer during\nNrepetitions, we describe the corresponding spin dynamics by\nN\nÕ\nn=1UN\u0000n=VN\u00001 \nN\u00001\nÕ\nn=1\u0003N\u0000nPN\u0000n!\n\u00030V\u00001\n0. (20)\nThe goal of this section is to extract the essential elements\nof this matrix product and to derive boundary conditions for\navoiding a population of the individual eigenstates that describe\nthe transient state magnetization. For this purpose, we will first\nshow that only the population transfer from the steady state is\nof relevance.\nThe steady-state left-eigenvector uH\nS= (0, 0, 0, 1 )becomes\nevident by multiplying it from the left to U(Eq. (8)). For either\nparameter variation, we obtain uH\nSU0= (0, 0, 0, 0 )since the last\nrows of R0yand R0zcontain only zeros (Eqs. (9), (10)). With\nEq. (19), it follows that Pd!S=08d6=S, resulting in the\nfollowing structure of the perturbation matrix:\nPn\u0019\n0\nBB@1 0 0 0\nPS!k(kn,\u0001kn) +O(\u0001k2n) 1O(\u0001kn)O(\u0001kn)\nPS!?(kn,\u0001kn) +O(\u0001k2n)O(\u0001kn) 1O(\u0001kn)\nP\u0003\nS!?(kn,\u0001kn) +O(\u0001k2n)O(\u0001kn)O(\u0001kn) 11\nCCA\nHere, only the essential elements are denoted\nexplicitly. The central part of Eq. (20) describes\nthe combined effect of NRF pulses with varying\nparameters onto the eigenvectors and is given by\nN\u00001\nÕ\nn=0\u0003N\u0000nPN\u0000n\u00190\nBB@1 0 0 0\nåN\nn=1PS!k(kn,\u0001kn)ÕN\nk=nlk(kk) +O(\u0001k2n)O(lN)O(\u0001k\u0001lN)O(\u0001k\u0001lN)\nåN\nn=1PS!?(kn,\u0001kn)ÕN\nk=nl?(kk) +O(\u0001k2n)O(\u0001k\u0001lN)O(lN)O(\u0001k\u0001lN)\nåN\nn=1P\u0003\nS!?(kn,\u0001kn)ÕN\nk=nl\u0003\n?(kk) +O(\u0001k2n)O(\u0001k\u0001lN)O(\u0001k\u0001lN)O(lN)1\nCCA. (21)8\nFor the leading order error term, the differences between the\nthree different l(\u0003)\nk,?and the dependency on the experimental\nparameters are neglected, and the product of any combination\nof eigenvalues is denoted by lN. Eq. (21) shows that all matrix\nelements except the first column approach zero for large N\nsincejl(\u0003)\nk,?j<1. This reveals that the population transfer be-\ntween the individual transient eigenstates are negligible, and we\nare left with the population transfer from the steady eigenstate\nto the transient eigenstates, as described by the first column.\nIts entries describe the counteraction of populating the transient\neigenstates, denoted by PS!f(kn,\u0001kn)with f2 fk ,?,?\u0003g,\nand the relaxation of the transient eigenstates in the time span\nbetween their population and the time of observation after N\nrepetitions, denoted by ÕN\nk=nlf(kk).\nThe entries in the first column of Eq. (21) can be bound by\n\f\f\f\f\fN\nå\nn=1PS!f(kn,\u0001kn)N\nÕ\nk=nlf(kk)\f\f\f\f\f\n\u0014max\nk\f\f\f\f\fPS!f(kk,\u0001kk)N\u00001\nå\nn=0ln\nf(kk)\f\f\f\f\f\n\u0019max\nkjPS!f(kk,\u0001kk)j\nj1\u0000lf(kk)j.(22)\nHere, we used the geometric series\nN\u00001\nå\nn=0ln\nf=1\u0000lN\nf\n1\u0000lf\u00191\n1\u0000lf,\nwhere a large Nwas assumed for the second step.\nIn order to derive a limit under which we can neglect the\nindividual transient eigenstates, we compare the correspond-\ning elements of the first column in Eq. (21) to the element\ncorresponding to the steady-state eigenstate, which is one.\nIncorporating Eq. (22), this corresponds to the condition\nmax\nkjPS!f(kk,\u0001kk)j\nj1\u0000lf(kk)j\u001c1, (23)\nwhich ensures that the corresponding eigenstate is not popu-\nlated.\n6) The Hybrid State Adiabaticity Condition\nIn this section, we will use the Taylor expansion in Eq. (19) to\nsolve Eq. (23) for the cases of the perpendicular eigenstates,\ni.e. for f=?(\u0003). Note that PS!?and P\u0003\nS!?, as defined by\nEq. (19), are complex conjugate of each other.\nAssuming that the eigenvectors are normalized to have a unit\n`2-norm, we can bound the numerator of Eq. (19) by\njuH\nf(kn)U0(kn)vd(kn)j\u0014jj U0jj2\u00141. (24)\nThe here employed subordinate matrix norm is given by the\nsquare root of the largest eigenvalue of (U0)HU0and is smaller\nthan one since the U0consists only of rotations and relaxation\nterms (cf. Eq. (53.5), Chapter 1 and Eq. (8.4), Chapter 2 of\nRef.37).\nThe first term of the denominator in Eq. (19), 1 \u0000l(\u0003)\n?, de-\nscribes the gap of the eigenvalues. We can assume jl(\u0003)\n?j=1\nas a worst case scenario and bound this gap by the complex\nphase \n. This gap can only be small when \napproaches\nzero (Eqs. (14)-(16)), so that we can use a Taylor expansion\nof Eq. (16)\nImf˜l(\u0003)\n?g2\u0019sin2a\n2+sin2f\n2(25)to derive the limit\nj1\u0000l(\u0003)\n?j\u0015r\nsin2a\n2+sin2f\n2. (26)\nThe last term in Eq. (19) that requires our attention is uH\n?v?.\nIn order to assess the scenarios under which this product is\nsmall, we can approximate the evolution matrix by U=R+\neD+O(e2), which views it as a small perturbation of the unitary\nrotation matrix R=RzRyRz. The perturbation is of the order\ne=1\u0000pE2, and D=fR,Cgis the anti-commuter of the\nrotation matrix and\nC=0\nB@\u00001 0 0 0\n0\u00001 0 0\n0 0\u00001 1\n0 0 0 01\nCA, (27)\nwhich approximates the relaxation matrix by E\u00191+eCwhen\nassuming d\u001c1. In this perturbation picture, the product\nof left- and right-eigenvectors uH\nfvfof the evolution matrix is\napproximated by\nuH\nfvf\u00191+e2å\nd6=f(˜vH\ndD˜vf)(˜vH\nfD˜vd)\n(˜lf\u0000˜ld)2, (28)\nwhere the tilde indicates the eigenvalues and vectors of R\n(cf. Eq. (19) or Eq. (10.2) in Chapter 2 of Ref.37). The first\nterm results from the property ˜uH\nf˜vf=1 of the eigenvectors\nofR. Due to the orthornormality of the eigenspace of R, we\nfurther eliminated the terms that are linear in e. With the bound\njjDjj2\u00141 and the normalization of the eigenvectors, we follow\nj˜vH\ndD˜vfj\u00141. Further, we can derive the eigenvalues of Rfrom\nEqs. (13) and (14) by setting E1=E2=1 and find ˜lS=˜lk=1\nand˜l(\u0003)\n?=e\u0006i\n. We adopt the bound in Eq. (26) for d2fS,kg\nand for d=?\u0003we findj˜l?\u0000˜l\u0003\n?j2\u00152(sin2a\n2+sin2f\n2)§. By\nsumming over all three terms, we arrive at\njuH\n?v?j\u00151\u00005\n2e2\nsin2a\n2+sin2f\n2. (29)\nInserting the bounds of the individual terms of the perturba-\ntion matrix (Eqs. (24), (26), and (29)) into Eq. (19), and using\n1\u0000E2\u0015e2, we find\n\f\f\fP(\u0003)\nS!?(an,fn,\u0001kn)\f\f\f\u0014\u0001knq\nsin2a\n2+sin2f\n2\nsin2an\n2+sin2fn\n2\u00005/2(1\u0000E2).\n(30)\nThis bound describes how much magnetization is at most\ntransfered from the steady state to the orthogonal eigenstates\nby varying aorfbetween two consecutive repetitions.\nFurther, inserting into Eq. (23) in order to account for the\ncumulative population, and utilizing Eq. (26), we arrive at the\nlimit\nmaxnj\u0001knj\u001csin2an\n2+sin2fn\n2\u00005\n2(1\u0000E2). (3’)\nWhen this adiabaticity condition is fulfilled, we can neglect the\nperpendicular transient eigenstates.\n§This bound neglects the scenario in which l(\u0003)\n?both approach\nnegative one, which is the case when jcosa\n2j\u001c1 orjcosf\n2j\u001c1. Note\nthat this leads to a breakdown of the approximations made for deriving\nEq. (14). Since both eigenvalues have the same complex phase, we\ncan tread those two components jointly and without proof we state\nthat both scenarios result in jjP(1)\nS!?v(1)\n?+P(2)\nS!?v(2)\n?jj2\u001c1 where the\nsuperscript indicates the two formally complex conjugate components.\nIn other words, when the eigenvalues l(\u0003)\n?approach negative one, the\nperpendicular eigenstates are not populated.9\n7) The Steady State Adiabaticity Condition\nIn order to do the same analysis for the parallel transient\neigenstate, we have to rely on the absolute value of lk, since\nit is real-valued and positive. Note that uH\nkvkcannot be bound\nin the same way as done in Eq. (29) since the eigenvalues\n˜lS=˜lkare degenerate. Since the adiabaticity condition of\nthe parallel eigenstate is not essential for this work, we skip the\ndegenerate perturbation theory and assume uH\nkvk\u00191. With the\nbound lk\u0014E1, which result from Eq. (13), and with Eqs. (23)-\n(24), we arrive at the adiabaticity condition\nj\u0001knj\u001c(1\u0000E1)2, (4’)\nwhich ensures that the parallel transient state is negligible.\nB. The Bloch Equation in Spherical Coordinates\nUnder the derived adiabaticity condition, the hybrid state\nemerges, and we observe transient-state behavior only along\nthe direction of the steady-state magnetization. Transforming\nthe Bloch equation into spherical coordinates isolates the\ntransient-state behavior in a single dimension, and the compo-\nnents of the Bloch equation uncouple into first order differential\nequations that can be solved.\nSpherical coordinates are here defined by x=rsinJcosj,\ny=rsinJsinjand z=rcosJ, where ris the radius, Jthe\npolar angle or the angle between the magnetization and the z-\naxis and jis the azimuth or the angle between the x-axis and\nthe projection of the magnetization onto the x-y-plane. In order\nto better highlight effect of inversion pulses, we use the limits\n\u00001\u0014r\u00141, 0\u0014J\u0014p/2, and 0\u0014j<2pto uniquely identify\nthe polar coordinates. Thermal equilibrium is given by r0=1,\nJ0=0 and j0=0, where the latter can be chosen freely.\nSince the azimuth, or phase, adiabatically transitions be-\ntween steady states, we can transform the known Cartesian\nsteady-state solutions (Eqs. (6,7) in Ref.12) to spherical coordi-\nnates, which results in Eq. (6). The polar angle can be derived\nfrom Eqs. (9-11) in Ref.12and is given by\ntanJ=pE2sinaq\n1\u00002E2cosf+E2\n2\nG+pE1(E2(E2\u0000cosf) + (1\u0000E2cosf)cosa)\n(31)\nwith\nG=(1\u0000E1cosa)(1\u0000E2cosf)\n1+pE1\n\u0000(E1\u0000cosa)(E2\u0000cosf)E2\n1+pE1.\nWith a Taylor expansion at E2=1, the polar angle is described\nby\nsin2J=sin2a\n2\nsin2f\n2\u0001cos2a\n2+sin2a\n2\n+ (1\u0000E2)\u0001x+O((1\u0000E2)2)(32)\nwith\nx=4(cosa\u00001)2(pE1\u00001)\n(pE1+1)(cosa+cosf+cosacosf\u00003)2.\nThe factor xis only large, if cos f\u0019(3\u0000cosa)/(cosa+1),\nwhich is only the case, if j1\u0000cosaj\u001c1 andj1\u0000cosfj\u001c1\nare simultaneously fulfilled, i.e. for small flip angle and in the\nvicinity of the stop-band. Consequently, for standard imaging\nscenarios with TR\u001cT2the polar can be approximated by\nEq. (5) apart from the vicinity of the stop band.\nThe spherical coordinate rcaptures the transient-state spin\ndynamics, and we can derive Eq. (2) simply by transforming the\nBloch equation into spherical coordinates14,38.C. B 1-inhomogeneities\nOne can describe the effect of B1-inhomogeneities on the\nspins by a=B1/Bnom.\n1anom., where Bnom.\n1andanom.describe\nthe nominal B1-field and flip angle, respectively. The effect on\nthe polar angle is described by inserting this relation into Eq.\n(5) and successively into Eq. (7).\nIn order to implement anti-periodic boundary conditions, the\nmagnetization must be inverted between successive cycles\n(r(0) =\u0000r(TC)), while changes of Jand jare required\nto remain within limits in order not to violate the adiabaticity\ncondition posed in Eq. (3). Applying a p-pulse with an inhomo-\ngeneous B1-field would lead to severe fluctuations of J, causing\na violation of the adiabaticity condition. In order to mitigate\nthese fluctuations, we surround the inversion pulse by crusher\ngradients. As shown in Refs.39,40, the transversal magnetization\nM?refocuses after inversion pulse with crusher gradients to\nan echo of the size M+\n?=sin2(p/2\u0001B1/Bnom.\n1)M\u0000\n?, where\nthe superscript +and\u0000indicate the magnetization before and\nafter the RF pulse, respectively. The longitudinal magnetization,\non the other hand, is given by M+\nz=cos(pB1/Bnom.\n1)M\u0000\nz. In\nspherical coordinates, this leads to\ntanJ+=sin2(p\n2B1\nBnom.\n1)\ncos(pB1\nBnom.\n1)tanJ\u0000. (33)\nIn the human brain at 3T, one usually observes variations in\nthe range of B1/Bnom.\n12[0.8, 1.2 ]41. Within this range, the\nresulting effect is bound by jJ+/J\u0000\u00001j<0.12 and will be\nneglected in the following.\nIn return, the crusher gradients manipulate r, which is ac-\ncounted for by setting\nb=\u0000s\nsin2J\u0000\u0001sin4pB1\n2Bnom.\n1+cos2J\u0000\u0001cos2pB1\nBnom.\n1(34)\nin Eq. (7). Repeating the inversion pulses with the same spoiling\ngradients can potentially result in higher order spin echoes and\nstimulated echoes, impairing the derived description of the spin\nphysics. However, when using TC\u001dT2, we can assume that\nthose contributions are negligible.\nD. Numerical Optimizations\n1) Cramér Rao Bound\nThe Cramér-Rao bound42,43provides a universal limit for\nthe noise variance of a measured parameter, given that the\nreconstruction algorithm is an unbiased estimator. This very\ngeneral and established metric has been utilized for optimizing\nMR parameter mapping experiments in Refs.44–46amongst oth-\ners, and to MRF in particular in Ref.47. In discretized notation,\nthe Cramér-Rao bound is defined by the inverse of the Fisher\ninformation matrix Fwith the entries Fij=bT\nibj/s2given by\nb1=dx/dPD\nb2=dx/dT1\nb3=dx/dT2.\nHere x2RNtis a vector describing the measured signal or,\nequivalently, the transversal magnetization at Ntdiscrete time\npoints, and s2is the input variance. Each element of the vector\nis given by xn=r(tn)\u0001sinJ(tn). The vectors bidescribe the\nderivatives of the signal evolution with respect to all considered\nparameters. Note that the proton density is here normalized to\nPD=1, so that b1=x.\nIn this work, we focused on quantifying relaxation times, since\nPD, as defined in this work, is modulated by the receive coil10\nsensitivity and provides only a relative measure. We can define\nthe dimensionless relative Cramér-Rao bounds to be\nrCRB (T1) =1\ns2T2\n1TC\nTR(F\u00001)2,2 (35)\nrCRB (T2) =1\ns2T2\n2TC\nTR(F\u00001)3,3. (36)\nThe normalization by the variances cancels out the variance in\nthe definition of the Fisher information matrix, and the normal-\nization by the relaxation time is done to best reflect the T1,2-\nto-noise ratio (defined as T1,2/sT1,2). Further, the multiplication\nwithTC/TRnormalizes the rCRB by duration of the experiment\nsuch that it can be understood as the squared inverse SNR\nefficiency per unit time, given a fixed TR.\n2) Optimal Control\nThe polar angle Jis here treated as the control parameter for\nspin dynamics along the radial direction as by Eq. (2). Thus, we\ncan employ the rich optimal control literature48,49for numerical\noptimization of J(t). We used a Broyden-Fletcher-Goldfarb-\nShanno (BFGS) algorithm50with rCRB (T1) +rCRB (T2)as an\nobjective function. To further improve convergence, the BFGS\nalgorithm is embedded in a scatter search algorithm which tried\n1000 starting points51. The numerical optimization was based\nonJ(\u0001t\u0001n)with a discrete step size of \u0001t=4.5 ms and the\nevaluation points n2 f1, 2, ... , TC/TRg. The gradient of the\nobjective function with respect to T1,T2, and each J(\u0001t\u0001n)\nwas explicitly calculated.\nSince the rCRB intrinsically compares a signal evolution to\nits surrounding in the parameter space, only a single set of\nrelaxation times is necessary for the optimization. Here, we\nused the relaxation times T1=781 ms and T2=65 ms,\ncorresponding to the values measured for white matter as\nreported in Ref.16. All optimizations were initialized with the\npattern provided in the pSSFP paper19and the optimizations\nwere performed with the constraint 0 \u0014J\u0014p/4, which limits\nthe flip angle to a\u0014p/2, ensuring consistent slice profiles by\nvirtue of the linearity in the small tip-angle approximation52, and\naiding compliance with safety considerations by avoiding high\npower large flip-angle pulses.\nE. In Vivo Experiments\nAn asymptomatic volunteer’s brain was imaged following\nwritten informed consent and according to a protocol approved\nby our institutional review board. A measurement was per-\nformed with the anti-periodic bHSFP experiment on a 3T Prisma\nscanner (Siemens, Erlangen, Germany). The 16 head elements\nof the manufacturer’s 20 channel head/neck coil were used for\nsignal reception.\nSpatial encoding was performed with a sagittally oriented\n3D stack-of-stars trajectory, which starts at the outer k-space\nand acquires for one TCdata while incrementing the angle\nof the k-space spoke by twice the golden angle increment53.\nThese large gaps are filled by repeating this procedure one time\nwith the entire k-space trajectory rotated by the golden angle.\nThereafter, the next 3D phase encoding step is performed in\nthe exact same way, while adhering to the Nyquist-Shannon\ntheorem along the slice direction. The acquired resolution of the\nmaps is 1 mm\u00021 mm\u00022 mm at a FOV of 256 mm \u0002256 mm\u0002\n192 mm. The readout dwell time was set to 2.1 \u0016s and an\noversampling factor of 2 was applied. We used a TR=4.5ms\nand the readout was skipped in segments with a polar angle\nclose to zero (gray areas in supporting Fig. S4), so that 601\nspokes were acquired during one TC. The total scan time was\napproximately 12.24 min.\nAlong the fully sampled phase encoding direction, a Fourier\ntransformation was performed and, thereafter, each slice wastreated separately. The raw data were compressed to 8 vir-\ntual receive coils via SVD compression54, followed by image\nreconstruction with the low rank alternating direction method\nof multipliers (ADMM) approach proposed in Ref.55, which\nincludes parallel imaging56–58. The data consistency step of\nthe ADMM algorithm was performed with 20 conjugate gradient\nsteps. In order to prevent non-linear effects from impairing the\nnoise assessment, only a single ADMM iteration was performed\nand no spatial regularization was applied.\nThe employed dictionaries include the parameter values\nT1(s) = 0.1\u00011.01j8j2 f0, 1, ... , 413g, thus covering the\nrange between 100 ms and 6 s in steps of 1%. The dictionaries\ncovered the range of T2values between 10 ms and 3 s in\nsteps of 1%, i.e. T2(s) = 0.01\u00011.01j8j2 f0, 1, ... , 575g.\nThe dictionaries further discretized f2[0,p]into 15 bins\nand B1/Bnom.\n12[0.8, 1.2 ]into 40 bins. The dictionary was\ncompressed to include the singular vectors corresponding to\nthe 12 largest singular values resulting from a singular value\ndecomposition of the dictionary matrix59.\nIn the matching step of each voxel, only fingerprints were\nconsidered that matched the fand B1/Bnom.\n1from separate\nscans. The fmap was acquired with a double-echo SPGR\nexperiment and the B1map with a turboFLASH experiment,\nas described in Ref.41.\nVIII. C ODE AVAILABILITY\nThe source code used for the current study is available from\nthe corresponding author on reasonable request.\nIX. D ATA AVAILABILITY\nThe datasets generated and analyzed during the current\nstudy are available from the corresponding author on reason-\nable request.\nX. A UTHOR CONTRIBUTIONS\nJA and DSN derived the theory. JA performed the numerical\noptimizations, simulations and the experiment. JA, RL and MAC\nanalyzed and interpreted the data. DKS provided consultancy.\nJA wrote the paper with the help of all authors. All authors have\ncritically reviewed the manuscript.\nXI. A CKNOWLEDGEMENTS\nThe authors would like to thank Steffen Glaser, Quentin\nAnsel and Dominique Sugny for fruitful discussions, and for\ngiving insights into their optimal control implementation. The\nauthors would also like to acknowledge Jeffrey Fessler and\nGopal Nataraj for discussions regarding the solution of the\nsimplified Bloch equation.\nThis work was supported by the research grants NIH/NIBIB\nR21 EB020096 and NIH/NIAMS R01 AR070297, and was\nperformed under the rubric of the Center for Advanced Imag-\ning Innovation and Research (CAI2R, www.cai2r.net), a NIBIB\nBiomedical Technology Resource Center (NIH P41 EB017183).\nREFERENCES\n[1] Gershenfeld, N. A. & Chuang, I. L. Bulk Spin-Resonance\nQuantum Computation. Science (80-. ). 275, 350–356\n(1997). URL http://www.sciencemag.org/cgi/doi/10.1126/\nscience.275.5298.350.\n[2] Rabi, I., Zacharias, J., Millman, S. & Kusch, P . A New\nMethod of Measuring Nuclear magnetic Moment. Phys.\nRev. 53, 318 (1938).\n[3] Lauterbur, P . C. Image formation by induced local interac-\ntions. Examples employing nuclear magnetic resonance.\nNature 242, 190–191 (1973).11\n[4] Bloembergen, N., Purcell, E. M. & Pound, R. V. Relaxation\nEffects in Nuclear Magnetic Resonance Absorption. Phys.\nRev. 73, 679–712 (1948). URL https://link.aps.org/doi/10.\n1103/PhysRev.73.679.\n[5] Bloch, F . Nuclear induction. Phys. Rev. 70, 460–474\n(1946).\n[6] Hahn, E. L. Spin Echoes. Phys. Rev. 80, 580–594 (1950).\nURL https://link.aps.org/doi/10.1103/PhysRev.80.580.\n[7] Hennig, J., Nauerth, A. & Friedburg, H. RARE Imaging : A\nFast Imaging Method for Clinical MR. Magn. Reson. Med.\n3, 823–833 (1986).\n[8] Mugler, J. P . & Brookeman, J. R. Threeâ ˘AˇRdimensional\nmagnetizationâ ˘AˇRprepared rapid gradientâ ˘AˇRecho imag-\ning (3D MP RAGE). Magn. Reson. Med. 15, 152–157\n(1990).\n[9] Carr, H. Steady-State Free Precession in Nuclear Mag-\nnetic Resonance. Phys. Rev. 112, 1693–1701 (1958). URL\nhttp://link.aps.org/doi/10.1103/PhysRev.112.1693.\n[10] Haase, A., Frahm, J., Matthaei, D., Hanicke, W. & Merboldt,\nK. D. FLASH imaging. Rapid NMR imaging using low flip-\nangle pulses. J. Magn. Reson. 67, 258–266 (1986).\n[11] Oppelt, A. et al. FISP â ˘AˇT a new fast MRI sequence.\nElectromedica 54, 15–18 (1986).\n[12] Freeman, R. & Hill, H. D. W. Phase and intensity anomalies\nin fourier transform NMR. J. Magn. Reson. 4, 366–383\n(1971).\n[13] Hennig, J., Speck, O. & Scheffler, K. Optimization of signal\nbehavior in the transition to driven equilibrium in steady-\nstate free precession sequences. Magn. Reson. Med. 48,\n801–809 (2002).\n[14] Lapert, M., Assemat, E., Glaser, S. J. & Sugny, D. Under-\nstanding the global structure of two-level quantum systems\nwith relaxation: Vector fields organized through the magic\nplane and the steady-state ellipsoid. Phys. Rev. A - At.\nMol. Opt. Phys. 88, 1–8 (2013).\n[15] Ma, D. et al. Magnetic resonance fingerprinting. Nature\n495, 187–192 (2013). URL http://www.nature.com/nature/\njournal/v495/n7440/full/nature11971.html.\n[16] Jiang, Y ., Ma, D., Seiberlich, N., Gulani, V. & Griswold,\nM. A. MR fingerprinting using fast imaging with steady\nstate precession (FISP) with spiral readout. Magn. Reson.\nMed. 74, 1621–1631 (2015).\n[17] Cloos, M. A. et al. Multiparamatric imaging with het-\nerogenous radiofrequency fields. Nat. Commun. 7,\ndoi:10.1038/ncomms12445 (2016). URL http://www.\nnature.com/doifinder/10.1038/ncomms12445.\n[18] Ma, D. et al. Music-based magnetic resonance fingerprint-\ning to improve patient comfort during MRI examinations.\nMagn. Reson. Med. 75, 2303–2314 (2016).\n[19] Assländer, J., Glaser, S. J. & Hennig, J. Pseudo Steady-\nState Free Precession for MR-Fingerprinting. Magn. Re-\nson. Med. 77, 1151–1161 (2017). URL http://doi.wiley.com/\n10.1002/mrm.26202.\n[20] Jiang, Y . et al. MR fingerprinting using the quick echo\nsplitting NMR imaging technique. Magn. Reson. Med. 77,\n979–988 (2017).\n[21] Ganter, C. Static susceptibility effects in balanced SSFP\nsequences. Magn. Reson. Med. 56, 687–691 (2006).\n[22] Ganter, C. Off-resonance effects in the transient response\nof SSFP sequences. Magn. Reson. Med. 52, 368–375\n(2004).\n[23] Born, M. & Fock, V. Beweis des Adiabatensatzes.\nZeitschrift für Phys. 51, 165–180 (1928).\n[24] Hargreaves, B. A., Vasanawala, S. S., Pauly, J. M. &\nNishimura, D. G. Characterization and reduction of the\ntransient response in steady-state MR imaging. Magn.\nReson. Med. 46, 149–158 (2001).\n[25] Schmitt, P . et al. Inversion recovery TrueFISP: quantifi-\ncation of T1, T2, and spin density. Magn. Reson. Med.51, 661–667 (2004). URL http://www.ncbi.nlm.nih.gov/\npubmed/15065237.\n[26] Deoni, S. C. L., Rutt, B. K. & Peters, T. M. Rapid combined\nT1 and T2 mapping using gradient recalled acquisition in\nthe steady state. Magn. Reson. Med. 49, 515–526 (2003).\n[27] Ehses, P . et al. IR TrueFISP with a golden-ratio-based\nradial readout: Fast quantification of T1, T2, and proton\ndensity. Magn. Reson. Med. 69, 71–81 (2013).\n[28] Chandarana, H. et al. Free-Breathing Radial 3D Fat-\nSuppressed T1-Weighted Gradient Echo Sequence. In-\nvest. Radiol. 46, 648–653 (2011).\n[29] Silver, M. S., Joseph, R. I., Chen, C. N., Sank, V. J. &\nHoult, D. I. Selective population inversion in NMR. Nature\n310, 681–683 (1984).\n[30] Jones, J. A., Vedral, V., Ekert, A. & Castagnoli, G. Geo-\nmetric quantum computation using nuclear magnetic reso-\nnance. Nature 403, 869–71 (2000). URL http://dx.doi.org/\n10.1038/35002528.\n[31] Wolff, S. D. & Balaban, R. S. Magnetization transfer\ncontrast (MTC) and tissue water proton relaxation in vivo.\nMagn. Reson. Med. 10, 135–144 (1989).\n[32] Bieri, O. & Scheffler, K. On the origin of apparent low\ntissue signals in balanced SSFP. Magn. Reson. Med. 56,\n1067–1074 (2006).\n[33] Hilbert, T. et al. Magnetic Resonance Fingerprinting:\nMitigating the Bias in the Quantification of T1 and T2\nCaused by Macromolecules. In Radiol. Soc. North Am.\n2016 Sci. Assem. Annu. Meet. (2016). URL http://archive.\nrsna.org/2016/16010335.html.\n[34] Torrey, H. C. Bloch equations with diffusion terms. Phys.\nRev. 104, 563–565 (1956).\n[35] McConnell, H. M. Reaction Rates by Nuclear Magnetic\nResonance. J. Chem. Phys. 28, 430–431 (1958). URL\nhttp://aip.scitation.org/doi/10.1063/1.1744152.\n[36] Scheffler, K. & Hennig, J. Is TrueFISP a gradient-echo or\na spin-echo sequence? Magn. Reson. Med. 49, 395–397\n(2003). URL http://doi.wiley.com/10.1002/mrm.10351.\n[37] Wilkinson, J. H. The Algebraic Eigenvalue Problem\n(Clarendon Press Oxford, London, 1965).\n[38] Tahayori, B., Johnston, L. A., Mareels, I. M. Y . & Farrell,\nP . M. Novel Insight into Magnetic Resonance through a\nSpherical Coordinate Framework for the Bloch Equation.\nSPIE Conf. Med. Imaging 7258 , 2–4 (2009).\n[39] Hennig, J. Echoes - How to Generate, Recognize, Use\nor Avoid Them in MR- Imaging Sequences; Part 1: Fun-\ndamental and Not So Fundamental Properties of Spin\nEchoes. Concepts Magn. Reson. 3, 125–143 (1991).\n[40] Weigel, M. Extended phase graphs: Dephasing, RF\npulses, and echoes - pure and simple. J. Magn. Reson.\nImaging 41, 266–295 (2015).\n[41] Chung, S., Kim, D., Breton, E. & Axel, L. Rapid B1+ map-\nping using a preconditioning RF pulse with turboFLASH\nreadout. Magn. Reson. Med. 64, 439–446 (2010).\n[42] Rao, C. R. Information and the Accuracy Attainable in the\nEstimation of Statistical Parameters. Bull. Calcutta Math.\nSoc. 37, 81–91 (1945).\n[43] Cramér, H. Methods of mathematical statistics (Princeton\nUniversity Press, Princeton, NJ, 1946).\n[44] Jones, J., Hodgkinson, P ., Barker, A. & Hore, P . Optimal\nSampling Strategies for the Measurement of Spin-Spin\nRelaxation Times. J. Magn. Reson. Ser. B 113, 25–34\n(1996). URL http://www.sciencedirect.com/science/article/\npii/S106418669690151X.\n[45] Jones, J. A. Optimal sampling strategies for the measure-\nment of relaxation times in proteins. J. Magn. Reson. 126,\n283–286 (1997).\n[46] Teixeira, R. P . A. G., Malik, S. J. & Hajnal, J. V.\nJoint System Relaxometry (JSR) and Crámer-Rao Lower\nBound optimisation of sequence parameters: a frame-12\nwork for enhanced precision of DESPOT T1 and T2 Es-\ntimation. Magn. Reson. Med. DOI:10.1002/mrm.26670\n(2017). URL http://onlinelibrary.wiley.com/wol1/doi/10.\n1002/mrm.26670/abstract.\n[47] Zhao, B., Haldar, J. P ., Setsompop, K. & Wald, L. L.\nOptimal Experiment Design for Magnetic Resonance Fin-\ngerprinting. In Eng. Med. Biol. Soc. (EMBC), IEEE 38th\nAnnu. Int. Conf. , 1, 453–456 (2016).\n[48] Conolly, S., Nishimura, D. & Macovski, A. Optimal control\nsolutions to the magnetic resonance selective excitation\nproblem. IEEE Trans. Med. Imaging 5, 106–115 (1986).\n[49] Skinner, T. E., Reiss, T. O., Luy, B., Khaneja, N. & Glaser,\nS. J. Application of optimal control theory to the design of\nbroadband excitation pulses for high-resolution NMR. J.\nMagn. Reson. 163, 8–15 (2003).\n[50] De Fouquieres, P ., Schirmer, S. G., Glaser, S. J. & Kuprov,\nI. Second order gradient ascent pulse engineering. J.\nMagn. Reson. 212, 412–417 (2011). URL http://dx.doi.\norg/10.1016/j.jmr.2011.07.023.\n[51] Ugray, Z. et al. Scatter search and local NLP solvers: A\nmultistart framework for global optimization. INFORMS J.\nComput. 19, 328–340 (2007).\n[52] Hoult, D. I. The solution of the bloch equations in the\npresence of a varying B1 field-An approach to selective\npulse analysis. J. Magn. Reson. 35, 69–86 (1979).\n[53] Winkelmann, S., Schaeffter, T., Koehler, T., Eggers, H. &Doessel, O. An Optimal Radial Profile Order Based on the\nGolden Ratio for Time-Resolved MRI. IEEE Trans. Med.\nImaging 26, 68–76 (2007). URL http://ieeexplore.ieee.org/\nlpdocs/epic03/wrapper.htm?arnumber=4039540.\n[54] Huang, F ., Vijayakumar, S., Li, Y ., Hertel, S. & Duensing,\nG. R. A software channel compression technique for faster\nreconstruction with many channels. Magn. Reson. Imaging\n26, 133–141 (2008).\n[55] Assländer, J. et al. Low rank alternating direction method\nof multipliers reconstruction for MR fingerprinting. Magn.\nReson. Med. 79, 83–96 (2018). URL http://doi.wiley.com/\n10.1002/mrm.26639. 1608.06974.\n[56] Sodickson, D. K. & Manning, W. J. Simultaneous acqui-\nsition of spatial harmonics (SMASH): Fast imaging with\nradiofrequency coil arrays. Magn. Reson. Med. 38, 591–\n603 (1997).\n[57] Pruessmann, K. P ., Weiger, M., Börnert, P . & Boesiger,\nP . Advances in sensitivity encoding with arbitrary k-space\ntrajectories. Magn. Reson. Med. 46, 638–651 (2001).\n[58] Uecker, M. et al. ESPIRiT - An eigenvalue approach to au-\ntocalibrating parallel MRI: Where SENSE meets GRAPPA.\nMagn. Reson. Med. 71, 990–1001 (2014).\n[59] McGivney, D., Ma, D., Saybasili, H., Jiang, Y . & Griswold,\nM. Singular Value Decomposition for Magnetic Resonance\nFingerprinting in the Time Domain. IEEE Trans. Med.\nImaging 33, 2311–2322 (2014)."    },    {        "title": "2205.06399v1.Precession_dynamics_of_a_small_magnet_with_non_Markovian_damping__Theoretical_proposal_for_an_experiment_to_determine_the_correlation_time.pdf",        "content": "arXiv:2205.06399v1  [cond-mat.mes-hall]  13 May 2022Precession dynamics of a small magnet with non-Markovian da mping: Theoretical\nproposal for an experiment to determine the correlation tim e,✩✩\nHiroshi Imamura, Hiroko Arai, Rie Matsumoto, Toshiki Yamaj i, Hiroshi Tsukahara✩\nNational Institute of Advanced Industrial Science and Tech nology (AIST), Tsukuba, Ibaraki 305-8568, Japan\nAbstract\nRecent advances in experimental techniques have made it pos sible to manipulate and measure the magnetization dynamics on\nthe femtosecond time scale which is the same order as the corr elation time of the bath degrees of freedom. In the equations of\nmotion of magnetization, the correlation of the bath is repr esented by the non-Markovian damping. For development of th e science\nand technologies based on the ultrafast magnetization dyna mics it is important to understand how the magnetization dyn amics\ndepend on the correlation time. It is also important to deter mine the correlation time experimentally. Here we study the precession\ndynamics of a small magnet with the non-Markovian damping. E xtending the theoretical analysis of Miyazaki and Seki [J. C hem.\nPhys. 108, 7052 (1998)] we obtain analytical expressions of the prece ssion angular velocity and the e ffective damping constant for\nany values of the correlation time under assumption of small Gilbert damping constant. We also propose a possible experi ment for\ndetermination of the correlation time.\nKeywords: non-Markovian damping, generalized Langevin equation, LL G equation, ultrafast spin dynamics, correlation time\n1. Introduction\nDynamics of magnetization is the combination of precession\nand damping. The precession is caused by the torque due to\nthe internal and external magnetic fields. Typical time scal e\nof the precession around the external field and the anisotrop y\nfield is nanosecond. The damping is caused by the coupling\nwith the bath degrees of freedom such as conduction electron s\nand phonons. The typical time scale of the relaxation of con-\nduction electrons and phonons is picosecond or sub-picosec ond\nwhich is much faster than precession. In typical experiment al\nsituations such as ferromagnetic resonance and magnetizat ion\nprocess, the time correlation of the bath degrees of freedom\ncan be neglected and the magnetization dynamics is well repr e-\nsented by the Landau-Lifshitz-Gilbert (LLG) equation with the\nMarkovian damping term[1–3].\nRecent advances in experimental techniques such as fem-\ntosecond laser pulse and time-resolved magneto-optical Ke rr\neffect measurement have made it possible to manipulate and\nmeasure magnetization dynamics on the femtosecond time\nscale[4–11]. In 1996, Beaurepaire et al. observed the femto sec-\nond laser pulse induced sub-picosecond demagnetization of a\nNi thin film[4], which opens the field of ultrafast magnetiza-\ntion dynamics. The all-optical switching of magnetization in a\n✩Permanent address: High Energy Accelerator Research Organ ization\n(KEK), Institute of Materials Structure Science (IMSS), Ts ukuba, Ibaraki 305-\n0801, Japan\n✩✩This work is partly supported by JSPS KAKENHI Grant Numbers\nJP19H01108 and JP18H03787.\nEmail addresses: h-imamura@aist.go.jp (Hiroshi Imamura),\narai-h@aist.go.jp (Hiroko Arai)ferrimagnetic GdFeCo alloy was demonstrated by Stanciu et a l.\nusing a 40 femtosecond circularly polarized laser pulse[5] . The\nhelicity-dependent laser-induced domain wall motion in Co /Pt\nmultilayer thin films was reported by Quessab et al.[11].\nTo understand the physics behind the ultrafast magnetizati on\ndynamics it is necessary to take into account the time correl a-\ntion of bath in the equations of motion of magnetization. The\nfirst attempt was done by Kawabata in 1972[12]. He derived the\nBloch equation and the Fokker-Planck equation for a classic al\nspin interacting with the surroundings based on the Nakajim a-\nZwanzig-Mori formalism[13–15]. In 1998, Miyazaki and Seki\nconstructed a theory for the Brownian motion of a classical\nspin and derived the integro-di fferential form of the generalized\nLangevin equation with non-Markovian damping[16]. They\nalso showed that the generalized Langevin equation reduces to\nthe LLG equation with modified parameters in a certain limit.\nAtxitia et al. applied the theory of Miyazaki and Seki to the\natomistic model simulations and showed that materials with\nsmaller correlation time demagnetized faster[17].\nDespite the experimental and theoretical progresses to dat e\nlittle attention has been paid to how to determine the correl a-\ntion time experimentally. For development of the science an d\ntechnologies based on the ultrafast magnetization dynamic s it\nis important to determine the correlation time experimenta lly\nas well as to understand how the magnetization dynamics de-\npend on the correlation time.\nIn this paper the precession dynamics of a small magnet with\nnon-Markovian damping is theoretically studied based on th e\nmacrospin model. The magnet is assumed to have a uniaxial\nanisotropy and to be subjected to an external magnetic field\nparallel to the magnetization easy axis. The non-Markovian ity\nPreprint submitted to Journal of Magnetism and Magnetic Mat erials May 16, 2022enhances the precession angular velocity and reduces the da mp-\ning. Assuming that the Gilbert damping constant is much\nsmaller than unity, the analytical expressions of the prece ssion\nangular velocity and the e ffective damping constant are derived\nfor any values of the correlation time by extending the analy sis\nof Miyazaki and Seki[16]. We also propose a possible exper-\niment for determination of the correlation time. The correl a-\ntion time can be determined by analyzing the external field at\nwhich the enhancement of the precession angular velocity is\nmaximized.\nThe paper is organized as follows. Section 2 explains the\ntheoretical model and the equations of motion. Section 3 giv es\nthe numerical and theoretical analysis of the precession dy nam-\nics in the absence of an anisotropy field. The e ffect of the\nanisotropy field is discussed in Sec. 4. A possible experimen t\nfor determination of the correlation time is proposed in Sec . 5.\nThe results are summarized in Sec. 6.\n2. Theoretical model\nWe calculate the magnetization dynamics in a small mag-\nnet with a uniaxial anisotropy under an external magnetic fie ld\nbased on the macrospin model. The magnetization easy axis\nis taken to be z-axis and the magnetic field is applied in the\npositive z-direction. In terms of the magnetization unit vector,\nm=(mx,my,mz), the energy density is given by\nE=K(1−m2\nz)−µ0MsH m z, (1)\nwhere Kis the effective anisotropy constant including the crys-\ntalline, interfacial, and shape anisotropies. µ0is the vacuum\npermeability, Msis the saturation magnetization, His the exter-\nnal magnetic field. The e ffective field is obtained as\nHeff=(Hkmz+H)ez, (2)\nwhere ezis the unit vector in the positive zdirection and Hk=\n2K/(µ0Ms) is the effective anisotropy field.\nThe magnetization precesses around the e ffective field with\ndamping. The energy and angular momentum are absorbed by\nthe bath degrees of freedom such as conduction electrons and\nphonons until the magnetization becomes parallel to the e ffec-\ntive field. The equations of motion of mcoupled with the bath\nis given by the Langevin equation with the stochastic field re p-\nresenting the bath degrees of freedom. If the time scale of th e\nbath is much smaller than the precession frequency the stoch as-\ntic field can be treated as the Wiener process[18] as shown by\nBrown[3].\nSince we are interested in the ultrafast magnetization dyna m-\nics of which time scale is the same order as the correlation ti me\nof the bath degrees of freedom, the stochastic field should be\ntreated as the Ornstein-Uhlenbeck process[18, 19]. As show n\nby Miyazaki and Seki [16] the equations of motion of mtakes\nthe following integro-di fferential form:\n˙m=−γm×(Heff+r)+αm×/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′,(3)whereγis the gyromagnetic ratio, αis the Gilbert damping con-\nstant, and ris the stochastic field. The first term represents the\nprecession around the sum of the e ffective field and the stochas-\ntic field, and the second term represents the non-Markovian\ndamping. The memory function in the non-Markovian damping\nterm is defined as\nν(t−t′)=1\nτcexp/parenleftigg\n−|t−t′|\nτc/parenrightigg\n, (4)\nwhereτcis the correlation time of the bath degrees of freedom.\nThe stochastic field, r, satisfies/angbracketleftri(t)/angbracketright=0 and\n/angbracketleftrj(t)rk(t′)/angbracketright=µ\n2δj,kν(t−t′), (5)\nwhere/angbracketleft/angbracketrightrepresents the statistical mean, and\nµ=2αkBT\nγMsV. (6)\nThe subscripts jandkstand for x,y, orz,kBis the Boltzmann\nconstant, Tis the temperature, Vis the volume of the mag-\nnet, andδj,kis Kronecker’s delta. The LLG equation with the\nMarkovian damping derived by Brown [3] is reproduced in the\nlimit ofτc→0 because lim τc→0ν(t−t′)=2δ(t−t′), where\nδ(t−t′) is Dirac’s delta function. Equation (3) is equivalent to\nthe following set of the first order di fferential equations,\n˙m=−γm×[Heff+δH] (7)\n˙δH=−1\nτcδH−α\nτ2cm−γ\nτcR, (8)\nwhere Rrepresents the stochastic field due to thermal agita-\ntion. Equations (7), (8) are used for numerical simulations . The\nstochastic field, R, satisfies/angbracketleftRj(t)/angbracketright=0 and\n/angbracketleftRj(t)Rk(t′)/angbracketright=µδj,kδ(t−t′). (9)\n3. Precession dynamics in the absence of an anisotropy field\nIn this section the precession dynamics in the absence of an\nanisotropy field, i.e. Hk=0, is considered. The initial di-\nrection of magnetization is assumed to be m=(1,0,0). The\nnumerical simulation shows that the non-Markovian damping\nenhances the precession angular velocity and reduces the da mp-\ning. The numerical results are theoretically analyzed assu ming\nthatα≪1. The analytical expressions of the precession an-\ngular velocity and the e ffective damping constant are obtained.\nThe case with Hk/nequal0 will be discussed in Sec. 4.\n3.1. numerical simulation results\nWe numerically solve Eqs. (7), (8) for H=10 T,α=0.05,\nandτc=1 ps. The temperature is assumed to be low enough\nto set R=0 in Eq. (8). Figure 1(a) shows the trajectory of m\non a unit sphere. The initial direction is indicated by the fil led\ncircle. The plot of the temporal evolutions of mx,my, and mzare\nshown in Fig. 1(b). The magnetization relaxes to the positiv ez\ndirection with precessing around the external field. The res ults\n2t [ps] 0.00 0.01 0.03 \n0.02 0.04 \n100 200 0\nt [ps] \na) b) \nc) d) z\nx yHφ [rad / ps] \n1.76 1.77 1.79 \n1.78 1.80 100 200 0\n100 200 0-1 1\n0mx, m y, m z\nt [ps] \nφ00.05 \nαmzmymx\nyαeff \nFigure 1: (a) Trajectory of mon a unit sphere. The external field of H=\n10 T is applied in the positive zdirection. The initial direction is assumed to\nbem=(1,0,0) as indicated by the filled circle. The other parameters are\nτc=1 ps, andα=0.05. (b) Temporal evolution of mx,my,mz. (c) Temporal\nevolution of the precession angular velocity, ˙φ. The solid red curve shows the\nsimulation result. The dotted black line indicates the resu lt of the Markovian\nLLG equation, i.e. ˙φ0=γH/(1+α2). (d) Temporal evolution of the e ffective\ndamping constant, αeff. The solid red curve shows the simulation result. The\ndotted black line indicates α=0.05.\nare quite similar to that of the Markovian LLG equation, whic h\nimplies that the non-Markovianity in damping causes renorm al-\nization of the gyromagnetic ratio and the Gilbert damping co n-\nstant in the Markovian LLG equation.\nThe renormalized value of the gyromagnetic ratio can be\nobserved as a variation of the precession angular velocity, ˙φ,\nwhere the polar and azimuthal angles are defined as m=\n(sinθcosφ,sinθsinφ,cosθ). Figure 1(c) shows that temporal\nevolution of ˙φ(solid red) together with the precession angular\nvelocity without non-Makovianity, ˙φ0=γH/(1+α2), (dotted\nblack). The precession angular velocity increases with inc rease\nof time and saturates to a certain value around 1.798. The sha pe\nof the time dependence of ˙φis quite similar to that of mzshown\nin Fig. 1(b), which suggests that the non-Markovian damping\nacts as an effective anisotropy field in the precession dynamics.\nThe renormalization of the Gilbert damping constant can be\nobserved as a variation of the temporal evolution of the pola r\nangle, ˙θ. Rearranging the LLG equation for ˙θ, the effective\ndamping constant can be defined as\nαeff=−˙θ/(γHsinθ). (10)\nIn Fig. 1(d)αeffis shown by the red solid curve as a function of\ntime. The effective damping constant is reduced to about one-\nfifth of the original value of α=0.05 (dotted black). Contrary\nto˙φ,αeffdoes not show clear correlation with the dynamics of\nm. During the precession, αeffis kept almost constant.\nThe enhancement of the precession angular velocity and the\nreduction of the Gilbert damping constant due to the non-\nMarkovian damping will be explained by deriving the e ffectiveLLG equation that is valid up to the first order of αin the next\nsubsection.\n3.2. Theoretical analysis\nSince the Gilbert damping constant, α, of a conventional\nmagnet is of the order of 0 .01∼0.1, it is natural to take the\nfirst order ofαto derive the effective equations of motion for\nm. The other parameters related to the motion of mareγ,H,\nandτc. Multiplying these parameters we can obtain the dimen-\nsionless parameter, ξ=γHτc, which represents the increment\nof the precession angle during the correlation time.\nIn the case ofξ<1 Miyazaki and Seki dereived the e ffective\nLLG equation using time derivative series expansion[16]. W e\nfirst briefly review their analysis. Then we derive the e ffective\nLLG equation forξ>1 using the time-integral series expansion\nand show that the e ffective LLG equation has the same form for\nbothξ< 1 andξ> 1. Therefore, it is natural to assume that\nthe derived effective LLG equation is valid for any values of ξ\nincludingξ=1.\n3.2.1. Brief review of Miyazaki and Seki’s derivation of the ef-\nfective LLG equation for ξ<1\nIn Ref. 16, Miyazaki and Seki derived the e ffective LLG\nequation with renormalized parameters using the time deriv a-\ntive series expansion. Similar analysis of the LLG equation\nwas also done by Shul in the study of the damping due to\nstrain[20, 21]. The following is the brief summary of the der iva-\ntion.\nSuccessive application of the integration by parts using ν(t−\nt′)=τc[dν(t−t′)/dt′] gives the following time derivative se-\nries expansion:\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=∞/summationdisplay\nn=1(−τc)n−1dnm\ndtn. (11)\nThen the non-Markovian damping term in Eq. (3) is expressed\nas\nα∞/summationdisplay\nn=1(−τc)n−1/parenleftigg\nm×dnm\ndtn/parenrightigg\n. (12)\nThe first derivative, n=1, is given by\n˙m=−γHm×ez+O(α), (13)\nwhere Ois the Bachmann–Landau symbol. For n=2, substi-\ntution of Eq. (13) into the time derivative of Eq. (13) gives\n¨m=(−γH)2(m×ez)×ez+O(α). (14)\nThe n-th order time derivative is obtained by using the same\nalgebra as\ndn\ndtnm=(−γH)n/bracketleftbig(m×ez)×ez.../bracketrightbig+O(α), (15)\nwhere ezappears ntimes. Expanding the vector products we\nobtain for even order time derivatives\nd2nm\ndt2n=(−1)n(γH)2n/bracketleftbigm−mzez/bracketrightbig+O(α), (16)\n3and for odd order time derivatives\nd2n+1m\ndt2n+1=(−1)n(γH)2n˙m+O(α). (17)\nSubstituting Eqs. (16) and (17) into Eq. (12) the non-\nMarkovian damping term is expressed as\n−∞/summationdisplay\nn=1γ2nm×ez+∞/summationdisplay\nn=0α2n+1m×˙m, (18)\nwhere\nγ2n=αγH m z(−1)n−1ξ2n−1(19)\nα2n+1=α(−1)nξ2n. (20)\nThe sums in Eq. (18) converge for ξ<1. Introducing\n˜γ=γ/parenleftigg\n1+αmzξ\n1+ξ2/parenrightigg\n(21)\n˜α=α\n1+ξ2, (22)\nEq. (3) can be expressed as the following e ffective LLG equa-\ntion with renormalized gyromagnetic ratio, ˜ γ, and damping\nconstant, ˜α:\n˙m=−˜γm×(H+r)+˜αm×˙m+O(α2). (23)\n3.2.2. Derivation of the e ffective LLG equation for ξ>1\nForξ>1 we expand Eq. (3) in power series of 1 /ξusing the\ntime integral series expansion approach. Using the integra tion\nby parts with dν(t−t′)/dt′=ν(t−t′)/τcthe integral part of the\nnon-Markovian damping can be written as\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=1\nτc/integraldisplayt\n−∞˙m(t′)dt′\n−1\nτc/integraldisplayt\n−∞ν(t−t′)/bracketleftigg/integraldisplayt′\n−∞˙m(t′′)dt′′/bracketrightigg\ndt′. (24)\nSuccessive application of the integration by parts gives\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=−∞/summationdisplay\nn=1/parenleftigg\n−1\nτc/parenrightiggn\nJn, (25)\nwhere Jnis the nth order multiple integral defined as\nJn=/integraldisplayt\n−∞/integraldisplayt1\n−∞···/integraldisplaytn−1\n−∞˙m(tn)dtn···dt2dt1. (26)\nFrom Eq. (17), on the other hand, ˙ mis expressed as\n˙m=1\n(−1)n(γH)2nd2n\ndt2n˙m+O(α). (27)\nSubstituting Eq. (27) into Eq. (26) the multiple integrals a re\ncalculated as\nJ2n=1\n(−1)n(γH)2n˙m (28)\nJ2n−1=1\n(−1)n(γH)2n¨m. (29)Then Eq. (25) becomes\n/integraldisplayt\n−∞ν(t−t′) ˙m(t′)dt′=∞/summationdisplay\nn=11\n(−1)n−1ξ2n˙m\n+∞/summationdisplay\nn=1τc\n(−1)nξ2n¨m. (30)\nSubstituting Eq. (30) into the second term of Eq. (3) the non-\nMarkovian damping term is expressed as\nα∞/summationdisplay\nn=11\n(−1)n−1ξ2nm×[ ˙m+τc¨m]+O(α2). (31)\nFrom Eq. (16) ¨ mis expressed as\n¨m=(−1)(γH)2/bracketleftbigm−mzez/bracketrightbig. (32)\nSubstituting Eqs. (31) and (32) into Eq. (3) we obtain\n˙m=−γH∞/summationdisplay\nn=1/bracketleftigg\n1+αmz\n(−1)n−1ξ2n−1/bracketrightigg\nm×ez−γm×r\n+α∞/summationdisplay\nn=11\n(−1)n−1ξ2nm×˙m+O(α2). (33)\nThe sums converge for ξ>1, and the effective LLG equation\nforξ>1 has the same form as ξ<1, i.e. Eq. (23). Since the\neffective LLG equation has the same form for both ξ< 1 and\nξ>1, it is natural to Eq. (23) is valid for any values of ξ.\nAs pointed out by Miyazaki and Seki, and independently by\nSuhl the effect of the non-Markovian damping on the precession\ncan be regarded as the renormalization of the e ffective field [16,\n20, 21]. Equation (23) can be expressed as\n˙m=−γm×/parenleftigg\nH+αHξ\n1+ξ2mz/parenrightigg\nez−γm×r\n+˜αm×˙m+O(α2). (34)\nThe second term in the bracket represents the fictitious unia xial\nanisotropy field originated from the non-Markovian damping .\nThe fictitious anisotropy field increases with increase of ξfor\nξ<1 and takes the maximum value of αH m z/2 atξ=1, i.e.\nγHτc=1. Forξ>1 the fictitious anisotropy field decreases\nwith increase ofξand vanishes in the limit of ξ→∞ because\nthe non-Markovian damping term vanishes in the limit of τc→\n∞. The precession angular velocity, ˙φ, is expected to have the\nsameξdependence as the fictitious anisotropy field and to have\nthe same temporal evolution as mzas shown in Figs. 1(b) and\n1(c).\n3.2.3. The Correlation time dependence of the precession an -\ngular velocity, and e ffective damping constant\nEquation (21) tells us that up to the first order of αthe pre-\ncession angular velocity can be approximated as\n˙φ≃˜γH=γH/bracketleftigg\n1+αmzγHτc\n1+(γHτc)2/bracketrightigg\n, (35)\n4τc’ =1/( γH) \n0.1 1 10 0.01 \nτc [ps] τc [ps] a) b) \nφ [rad / ps] \n1.76 1.77 1.79 \n1.78 1.80 \n0.1 1 10 0.01 \nα, αeff ~\nαeffα0.04 \n0.00 0.02 0.05 \n0.01 0.03 \nααα\neffeffeffαeffαeff~sim. approx. \nFigure 2: (a) The correlation time, τc, dependence of the precession angular\nvelocity,δ˙φ, atθ=5◦forH=10 T. The solid yellow curve shows the ap-\nproximation result, ˜ γH. The dotted black curve shows the simulation results\nobtained by numerically solving Eqs. (7) and (8). The thin ve rtical dotted line\nindicates the critical value of the correlation time, τ′\nc=1/(γH). (b)τcdepen-\ndence of ˜α(solid yellow) andαeff(dotted black). The parameters and the other\nsymbols are the same as panel (a).\nwhere the second term in the square bracket represents the en -\nhancement due to the fictitious anisotropy field.\nIn Fig. 2(a) the approximation result of Eq. (35) at θ=5◦\nwhere ˙φis almost saturated is plotted as a function of τcby the\nsolid yellow curve. The external field and the Gilbert damp-\ning constant are assumed to be H=10 T andα=0.05, re-\nspectively. The corresponding simulation results obtaine d by\nnumerically solving Eqs. (7) and (8) are shown by the dotted\nblack curve. Both curves agree well with each other because\nαis as small as 0.05. The precession angular velocity is maxi-\nmized at the critical value of the correlation time τ′\nc=1/(γH).\nFigure 2(b) shows the τcdependence of ˜α(solid yellow) and\nαeff(dotted black) for the same parameters as panel (a). Both\ncurves agree well with each other and are monotonic decreasi ng\nfunctions ofτc. They vanish in the limit of τc→∞ similar to\nthe non-Markovian damping term.\n4. Effect of an anisotropy field on precession dynamics\nThe theoretical analysis given in the previous section can\nbe applied to the case with Hk/nequal0 by replacingξwithξk=\nγ(H+Hkmz)τc. Following the same procedure as for Hk=0\nEq. (3) can be expressed as\n˙m=−γm×/parenleftig\nH+αHξk\n1+ξ2\nkmz+αHkξk\n1+ξ2\nkm2\nz/parenrightig\nez\n−γm×r+α\n1+ξ2\nkm×˙m+O(α2). (36)\nThe second and the third terms in the bracket can be regarded\nas the fictitious uniaxial and unidirectional anisotropy fie lds\ncaused by the non-Markovian damping. Similar to the re-\nsults for Hk=0 the precession angular velocity is maximized\natξk=1. The renormalized damping constant is given by\nα/(1+ξ2\nk) which is a monotonic decreasing function of ξkand\nvanishes in the limit of ξk→∞ .c) d) a) b) \n24 6 14 12 10 8 0\nH [T] δφ /φ0 [%] 2\n013\nδφ /φ0 [%] 2\n01\n24 6 14 12 10 8 0\nH [T] Hk = 0 H’=1/( γτ c) \nτc = 1 ps \nθ = 5 oθ = 5 oHk = 1 T \nτc = 1 ps H’=1/( γτ c) − HkmzH = 6, 7, 8, 9 T \nH = 2, 3, 4, 5 T \n = 1 ps \n = 5  = 0 \n = 1 ps τ\nθH\nτ = 1 ps  = 1 ps  = 1 ps τc = 1 ps τ = 1 ps τ = 1 ps \nθτH\nτ\n = 5  = 1 ps  = 1 T \n = 1 ps  = 1 ps  = 1 T \n = 1 ps  = 1 ps τc = 1 ps ττ = 1 ps  = 1 ps 3δφ /φ0 [%] 2\n013\nδφ /φ0 [%] 2\n013\nt [ps] 100 200 0\nt [ps] 100 200 0Hk = 0 \nτc = 1 ps Hk = 0 \nτc = 1 ps \nFigure 3: (a)τcdependence ofδ˙φ/˙φ0atθ=5◦. From top to bottom the\nexternal field is H=2,3,4,5 T. The parameters are Hk=0, andτc=1 ps. (b)\nThe same plot as panel (a) for H≥5 T. From top to bottom the external field is\nH=6,7,8,9 T. (c) Hdependence ofδ˙φ/˙φ0atθ=5◦obtained by solving Eqs.\n(7) and (8). The parameters are Hk=0, andτc=1 ps. The critical value of\nthe external field, H′=1/(γτc), is indicated by the thin vertical dotted line. (d)\nThe same plot as panel (c) for Hk=1 T. The thin vertical dotted line indicates\nthe critical value of the external field, H′=1/(γτc)−Hkmz.\n5. A possible experiment to determine the correlation time\nBased on the results shown in Secs. 3 and 4 we propose a\npossible experiment to determine the correlation time, τc. Sim-\nilar to the previous sections we first discuss the case withou t\nanisotropy field, i.e. Hk=0, and then extend the discussion to\nthe case with Hk/nequal0.\nIn Figs. 3(a) and 3(b) we show the temporal evolution of the\nenhancement of angular velocity, δ˙φ/˙φ0, obtained by the solv-\ning Eqs. (7) and (8) for various values of H. The increment\nof the precession angular velocity is defined as δ˙φ=˙φ−˙φ0.\nThe initial state and the correlation time are assumed to be\nm=(1,0,0) andτc=1 ps, respectively. As shown in Fig.\n3(a),δ˙φ/˙φ0increases with increase of HforH≤5T. Once\nthe external field exceeds the critical value of 1 /(γτc)=5.7 T,\nδ˙φ/˙φ0decreases with increase of Has shown in Fig. 3 (b). The\nresults suggest that correlation time can be determined by a na-\nlyzing the external field that maximizes the enhancement of t he\nprecession angular velocity.\nFigure 3(c) shows the Hdependence ofδ˙φ/˙φ0atθ=5◦\nwhereδ˙φ/˙φ0is almost saturated. The enhancement is maxi-\nmized at the critical value of the external field, H′=5.7 T. The\ncorrelation time is calculated as τc=1/(γH′)=1 ps.\nIf the system has a uniaxial anisotropy field, Hk, the en-\nhancement of the precession angular velocity is maximized a t\nH′=1/(γτc)−Hkmzas shown in Fig. 3(d). The correlation\ntime is obtained as τc=1/γ(H′+Hkmz).\nThe above analysis is expected to be performed experimen-\n5tally using the time resolved magneto optical Kerr e ffect mea-\nsurement technique. In the practical experiments the analy sis\ncan be simplified as follows. The polar angle of the initial st ate\nis not necessarily large. It can be small as far as the preces-\nsion angular velocity can be measured. Instead of analyzing\nδ˙φ/˙φ0, one can analyze ˙φ/Hor˙φ/(H+Hkmz) because they are\nmaximized at the same value of Hasδ˙φ/˙φ0. Since the required\nmagnetic field is as high as 10 T, a superconducting magnet [22 ]\nis required.\n6. Summary\nIn summary we theoretically analyze the ultrafast precessi on\ndynamics of a small magnet with non-Markovian damping. As-\nsumingα≪1, we derive the effective LLG equation valid for\nany values ofτc, which is a direct extension of Miyazaki and\nSeki’s work[16]. The derived e ffective LLG equation reveals\nthe condition for maximizing ˙φin terms of Handτc. Based on\nthe results we propose a possible experiment for determinat ion\nofτc, whereτccan be determined from the external field that\nmaximizesδ˙φ/˙φ0.\nReferences\n[1] L. Landau, E. Lifshits, ON THE THEORY OF THE DISPER-\nSION OF MAGNETIC PERMEABILITY IN FERROMAGNETIC\nBODIES, Physikalische Zeitschrift der Sowjetunion 8 (1935 ) 153.\ndoi:10.1016/B978-0-08-010586-4.50023-7 .\n[2] T. Gilbert, Classics in Magnetics A Phenomenological Th eory of Damp-\ning in Ferromagnetic Materials, IEEE Transactions on Magne tics 40 (6)\n(2004) 3443–3449. doi:10.1109/TMAG.2004.836740 .\n[3] W. F. Brown, Thermal Fluctuations of a Single-Domain Par ticle, Physical\nReview 130 (5) (1963) 1677–1686. doi:10.1103/PhysRev.130.1677 .\n[4] E. Beaurepaire, J.-C. Merle, A. Daunois, J.-Y . Bigot, Ul trafast Spin Dy-\nnamics in Ferromagnetic Nickel, Physical Review Letters 76 (22) (1996)\n4250–4253. doi:10.1103/PhysRevLett.76.4250 .\n[5] C. D. Stanciu, F. Hansteen, A. V . Kimel, A. Kirilyuk, A. Ts ukamoto,\nA. Itoh, T. Rasing, All-Optical Magnetic Recording with Cir cu-\nlarly Polarized Light, Physical Review Letters 99 (4) (2007 ) 047601.\ndoi:10.1103/PhysRevLett.99.047601 .\n[6] G. P. Zhang, W. H¨ ubner, G. Lefkidis, Y . Bai, T. F. George, Paradigm of the\ntime-resolved magneto-optical Kerr e ffect for femtosecond magnetism,\nNature Physics 5 (7) (2009) 499–502. doi:10.1038/nphys1315 .\n[7] J.-Y . Bigot, M. V omir, E. Beaurepaire, Coherent ultrafa st magnetism in-\nduced by femtosecond laser pulses, Nature Physics 5 (7) (200 9) 515–520.\ndoi:10.1038/nphys1285 .\n[8] A. Kirilyuk, A. V . Kimel, T. Rasing, Ultrafast optical ma nipulation of\nmagnetic order, Reviews of Modern Physics 82 (3) (2010) 2731 –2784.\ndoi:10.1103/RevModPhys.82.2731 .\n[9] J.-Y . Bigot, M. V omir, Ultrafast magnetization dynamic s of nanostruc-\ntures: Ultrafast magnetization dynamics of nanostructure s, Annalen der\nPhysik 525 (1-2) (2013) 2–30. doi:10.1002/andp.201200199 .\n[10] J. Walowski, M. M¨ unzenberg, Perspective: Ultrafast m agnetism and\nTHz spintronics, Journal of Applied Physics 120 (14) (2016) 140901.\ndoi:10.1063/1.4958846 .\n[11] Y . Quessab, R. Medapalli, M. S. El Hadri, M. Hehn, G. Mali nowski, E. E.\nFullerton, S. Mangin, Helicity-dependent all-optical dom ain wall motion\nin ferromagnetic thin films, Physical Review B 97 (5) (2018) 0 54419.\ndoi:10.1103/PhysRevB.97.054419 .\n[12] A. Kawabata, Brownian Motion of a Classical Spin, Progr ess of Theoret-\nical Physics 48 (6) (1972) 2237–2251. doi:10.1143/PTP.48.2237 .\n[13] S. Nakajima, On Quantum Theory of Transport Phenomena: Steady\nDiffusion, Progress of Theoretical Physics 20 (6) (1958) 948–95 9.\ndoi:10.1143/PTP.20.948 .[14] R. Zwanzig, Ensemble Method in the Theory of Irreversib il-\nity, The Journal of Chemical Physics 33 (5) (1960) 1338–1341 .\ndoi:10.1063/1.1731409 .\n[15] H. Mori, Transport, Collective Motion, and Brownian Mo -\ntion, Progress of Theoretical Physics 33 (3) (1965) 423–455 .\ndoi:10.1143/PTP.33.423 .\n[16] K. Miyazaki, K. Seki, Brownian motion of spins revisite d,\nThe Journal of Chemical Physics 108 (17) (1998) 7052–7059.\ndoi:10.1063/1.476123 .\n[17] U. Atxitia, O. Chubykalo-Fesenko, R. W. Chantrell, U. N owak,\nA. Rebei, Ultrafast Spin Dynamics: The E ffect of Col-\nored Noise, Physical Review Letters 102 (5) (2009) 057203.\ndoi:10.1103/PhysRevLett.102.057203 .\n[18] C. W. Gardiner, Stochastic Methods: A Handbook for the N atural and\nSocial Sciences, 4th Edition, no. 13 in Springer Series in Sy nergetics,\nSpringer, Berlin Heidelberg, 2009.\n[19] G. E. Uhlenbeck, L. S. Ornstein, On the Theory of the\nBrownian Motion, Physical Review 36 (5) (1930) 823–841.\ndoi:10.1103/PhysRev.36.823 .\n[20] H. Suhl, Theory of the magnetic damping constant, IEEE T ransactions on\nMagnetics 34 (4) (1998) 1834–1838. doi:10.1109/20.706720 .\n[21] H. Suhl, Relaxation Processes in Micromagnetics, Oxfo rd University\nPress, 2007. doi:10.1093/acprof:oso/9780198528029.001.0001 .\n[22] H. W. Weijers, U. P. Trociewitz, W. D. Markiewicz, J. Jia ng, D. My-\ners, E. E. Hellstrom, A. Xu, J. Jaroszynski, P. Noyes, Y . Viou chkov,\nD. C. Larbalestier, High field magnets with HTS conductors, I EEE\nTransactions on Applied Superconductivity 20 (3) (2010) 57 6–582.\ndoi:10.1109/TASC.2010.2043080 .\n6"    },    {        "title": "0805.2706v1.Effect_of_spin_diffusion_on_spin_torque_in_magnetic_nanopillars.pdf",        "content": "arXiv:0805.2706v1  [cond-mat.mtrl-sci]  18 May 2008Effect of spin diffusion on spin torque in magnetic nanopillar s\nSergei Urazhdin and Scott Button\nDepartment of Physics, West Virginia University, Morganto wn, WV 26506\nWe present systematic magnetoelectronic measurements of m agnetic nanopillars with different\nstructures of polarizing magnetic layers. The magnetic rev ersal at small magnetic field, the onset\nof magnetic dynamics at larger field, and the magnetoresista nce exhibit a significant dependence\non the type of the polarizing layer. We performed detailed qu antitative modeling showing that the\ndifferences can be explained by the effects of spin-dependent electron diffusion.\nPACS numbers: 72.25.Ba, 72.25.RB, 75.47.De\nAccording to the spin torque (ST) model1, current-\ninduced magnetic switching (CIMS) in magnetic multi-\nlayers is caused by angular momentum transfer from the\nconduction electrons to the magnetic layers. ST is be-\nlieved to occur within atomic distances from the mag-\nnetic interfaces. Nevertheless, theories have shown that\nelectron diffusion in the layers has an important effect\non ST.2,3,4As a simple example, an electron scattering\nbetween two ferromagnets transfers angular momentum\nupon each reflection. However, this transfer is not neces-\nsarily associated with a net charge current I. Therefore,\nefficient utilization of electron scattering can result in re-\nducedIrequired to manipulate magnetic devices with\nST. In a more subtle manifestation, spin-dependent elec-\ntron diffusion causes an asymmetry between the ST in\nantiparallel (AP) and parallel (P) configurations of the\nmagnetic layers.5In an extreme case of such asymmetry,\nST can change direction, resulting in anomalous current-\ninduced behaviors.6\nDespite extensive theoretical work, few experiments\naddressed the effects of diffusion on ST.6,7,8,9,10The\nmain difficulty stems from the limited knowledge about\nthe transport properties of individual layers in magnetic\nnanostructures. Different deposition and measurement\ntechniques yielded significantly different values.11On the\nother hand, both the Magnetoresistance(MR) and CIMS\ndepend onthe samespin-dependent transportproperties.\nTherefore,simultaneousmeasurementsofMRandCIMS,\nand their analysis within the same theoretical framework\ncan lead to better understanding of the electron diffusion\nand its effect on ST.\nWe report systematic measurements of MR and CIMS\nin nanopillar spin valves F 1/N/F2with identical free lay-\ners F2=Py(5), Py=Ni 80Fe20, and different polarizers F 1\nincorporating Co. Thicknesses are given in nm. Large\nspin diffusion length lsf,Comakes Co ideal for studying\nthe effects of diffusion. We used F 1=Co(20) in samples\nlabeledCo20. To separatethe contributions ofthe Coin-\nterfaces and its bulk, we tested samples with F 1=Co(3),\nlabeledCo3, in which the scattering in the bulk of Co(3)\nwas negligible. To eliminate the spin diffusion in the\nsample contacts, we inserted a strongly spin flipping bi-\nlayer Fe 50Mn50(1)/Cu(1) between Co(3) and the bottom\ncontact in samples labeled FeMnCo 3.\nThe multilayers Cu(50)/F 1/Cu(10)/F 2/Cu(200) were/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s50/s52/s54/s56\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s52/s56/s32/s73\n/s67/s32/s40/s109/s65/s41\n/s84/s32/s40/s75/s41/s40/s100/s41 /s67/s111/s50/s48\n/s70/s101/s77/s110/s67/s111/s51/s67/s111/s51/s45/s50 /s48 /s50 /s52/s50/s46/s50/s50/s46/s51/s50/s46/s55\n/s49/s32/s107/s79/s101/s51/s48/s48/s32/s79/s101/s100/s86/s47/s100/s73/s32/s40Ω /s41\n/s73/s32/s40/s109/s65/s41/s53/s48/s32/s79/s101/s40/s97/s41/s84/s61/s50/s57/s53/s32/s75\n/s48 /s53 /s49/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s50\n/s49/s32/s107/s79/s101/s51/s54/s48/s32/s79/s101/s53/s48/s32/s79/s101/s100/s86/s47/s100/s73/s32/s40Ω /s41\n/s73/s32/s40/s109/s65/s41/s40/s98/s41/s84/s61/s53/s32/s75\n/s32/s73/s43\n/s44/s32/s73/s45\n/s32/s40/s109/s65/s41\n/s84/s32/s40/s75/s41/s40/s99/s41\n/s73/s43\n/s73/s45\nFIG. 1: (a) dV/dIvsIat labeled HandT= 295 K. Curves\nare offset for clarity. (b) same as (a), at T= 5 K. (c) I+,I−\nvsTfor aCo20 sample. (d) ICvsTmeasured at H= 500 Oe\nfor the three types of samples as labeled.\ndepositedatroomtemperature295K(RT)bymagnetron\nsputtering at base pressure of 5 ×10−9Torr, in 5 mTorr\nof purified Ar. F 2and part of the Cu(10) spacer were\npatterned into an elliptical nanopillar with approximate\ndimensions 130 ×60 nm. We measured dV/dIwith four-\nprobes and lock-in detection. Positive Iflowed from F1\ntoF2. Magnetic field Hwas in the film plane and along\nthe nanopillar easy axis. At least three nanopillars of\neach type were tested with similar results.\nFigs. 1(a),(b) show dV/dIvsIfor aCo20 sample, ac-\nquired at RT and 5 K, respectively. The data at small\nH= 50 Oe are characterized by hysteretic jumps to\nthe P state with low resistance RPatI−<0, and to\nthe AP state with high resistance RAPatI+>0. At\nH= 300/360 Oe in Figs. 1(a)/(b), the jumps are re-\nplaced by large peaks caused by the reversible transition\nbetween the P and AP states.12The onset of the mag-\nnetic dynamics starting at I=ICappears as a sharp\nincrease of dV/dInearly independent of H(1 kOe data\nin Figs. 1(a),(b)). The approximate equality IC≈I+\nshown with a dashed line indicates that the reversal oc-2\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s82\n/s65/s80/s45/s82\n/s80/s32/s40Ω /s41\n/s84/s32/s40/s75/s41/s67/s111/s50/s48\n/s67/s111/s51\n/s70/s101/s77/s110/s67/s111/s51/s40/s98/s41\n/s32/s32/s32/s32/s32/s32/s32 /s32/s67/s111/s50/s48\n/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s67/s111/s51\n/s32/s70/s101/s77/s110/s67/s111/s51\n/s32/s82\n/s112/s40/s84/s41/s45/s82\n/s80/s40/s53/s75/s41/s32/s40Ω /s41\n/s84/s32/s40/s75/s41/s40/s97/s41\nFIG. 2: (a) P-state resistances RPoffset by values at 5 K,\nand (b) MR vsTfor the three types of samples as labeled.\ncurs when large-amplitude dynamics is excited by ST.\nThe 5 K data exhibit significantly increased reversal cur-\nrents and IC. Fig. 1(c) summarizes the temperature de-\npendence of I+andI−. Both are nearly constant above\n130 K, below which they dramatically increase. Similar\nbehaviors of Co/Cu/Co nanopillars indicate their intrin-\nsic origin from the spin-dependent transport in Co.8\nOnemayattributesomeofthe dependenceon Tshown\nin Fig. 1(c) to the effects of thermal activation. Indeed,\nI+≤ICat RT because thermal fluctuations result in\nreversal slightly before the onset of large-amplitude dy-\nnamics. In contrast, I+≥ICat 5 K because current-\ninduced dynamics can occur before the reversal occurs.\nThe fundamental quantity predicted by the models of ST\nisIC. It is insensitive to thermal fluctuations and sam-\nple shape imperfections, and can be directly determined\nfrom the sharp increase of dV/dIatHlarge enough to\nsuppress hysteretic reversal. Fig. 1(d) summarizes ICvs\nTfor all three different sample structures. FeMnCo 3\ndata are approximately independent of T, whileICfor\nCo3 andCo20 increase when Tis decreased. Compar-\ning panels (c) and (d) reveals that ICclosely follows I+.\nIt is not possible to measure a similar excitation onset\ncurrentI−\nCin the AP state, because transition to the P\nstate is not suppressed at any H. Below, we use I−as\nan approximation for I−\nC.\nSince F 2is identical in all samples, the different be-\nhaviors of ICmust be attributed to the different spin-\ndependent transport properties of F 1. The difference be-\ntweenCo3 andFeMnCo 3 is due to the spin flipping\nin FeMn, which eliminates spin diffusion in the bottom\nCu(50) contact. The difference between the Co20 and\nCo3 data indicates that the effects of spin diffusion in\nCo are stronger than those in Cu. Despite a significant\nincrease of ICinCo20, it does not diverge as would be\nexpected if the sign of ST was reversed.6\nFigs. 2(a),(b) show temperature dependence of RP−\nRP(0) and MR= RAP−RP.RPincreased with Tdue\nto magnon and phonon scattering, and were surprisingly\nconsistent among the samples. Interestingly, there is a\nclear correlation between the variations of MR and ICin\nall samples. As Tdecreasesfrom RT, all MRs increase at\na similar rate, while ICslightly increase. At lower T, the\ntrends for Co3 andFeMnCo 3 remain the same, while/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s49/s48/s50/s48\n/s48/s46/s48/s48 /s48/s46/s48/s52 /s48/s46/s48/s56/s49/s48/s50/s48\n/s32/s84/s32/s40/s75/s41ρ /s32/s40µΩ /s32/s99/s109/s41/s80/s121\n/s67/s111\n/s67/s117/s40/s97/s41\n/s80/s121\n/s67/s111ρ /s32/s40/s84/s61/s53/s75/s41/s32/s40 µΩ /s32/s99/s109/s41\n/s49/s47/s100/s32/s40/s110/s109/s45/s49\n/s41/s67/s117/s40/s98/s41\nFIG. 3: (a) Resistivities of 40 nm thick Py, Co, and Cu films\nmeasuredin VanderPauwgeometry. The CoandCudataare\nfitted with the Bloch-Gruneisen approximation, with Debye\ntemperatures θCo= 373 K and θCu= 265 K. The Py data\nare fitted with a quadratic dependence. (b) Dependencies of\nresidual resistivities on inverse film thickness (symbols) , with\nlinear fits shown.\na decrease of MR in Co20 atT <130 coincides with a\nsharp increase of IC.\nTo understand the dependencies of MR, CIMS, and\nICon the sample structure, we performed simultaneous\ncalculations of spin-dependent transport and ST. Our\nmodel combines a diffusive approximation for the ferro-\nmagnets and outer sample contacts with a ballistic ap-\nproximationfor the Cu(10) spacer between the ferromag-\nnets.5This approximation is consistent with calculations\nbasedontheBoltzmannequation.14Wecombinethecon-\ntinuity conditions for spin currents and spin accumula-\ntion in the spacer between F 1and F 2derived by Slon-\nczewski5(Eqs. (13),(14)) with a small-angle expansion of\nEq. (28) for ST. The resulting expression for ST in terms\nof the spin current Is=I↑−I↓and spin accumulation\n∆µ=µ↑−µ↓in the Cu(10) spacer near the collinear\nmagnetic configuration is\nτ=¯hsin(θ)\n4e(AG∆µ−IS) (1)\nwhereeis the electron charge, ¯ his the Planck’s constant,\nGis twice the mixing conductance introduced in the cir-\ncuit theory,2Ais the area of the nanopillar, and θis\nthe angle between the magnetic moments. At I=IC,τ\ncompensates the damping torque, yielding\nIC=αeγS22πM2\nτ, (2)\nwhereα≈0.03 is the Gilbert damping parameter,15\nγis the gyromagnetic ratio, τis ST determined from\nEquation (1) at I= 1 in appropriate units, and S2=\nM2V/2µBis the total spin of the Py(5) nanopillar. Here,\nVis the volume of F 2, andµBis the Bohr magneton.\nThe magnetization M2of Py varied from 730 emu/cm3\nat 20K to 675emu/cm3at 300K, asdetermined by mag-\nnetometry of Py(5) films prepared under the same con-\nditions as the nanopillars. These values are lower than\nexpected for bulk Py, but consistent with the published\nresults for Py films.163\n/s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s45/s50/s48/s50\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s53/s49/s48/s49/s53/s50/s48\n/s48 /s53/s48 /s49/s48/s48/s45/s50/s48/s50/s52/s54\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s51/s52/s53/s54/s55/s56/s73\n/s67/s44/s32/s73/s45\n/s67/s40/s109/s65/s41\n/s71/s32/s40/s102Ω/s45/s49\n/s109/s45/s50\n/s41/s40/s97/s41\n/s71/s61/s48/s46/s56/s55/s32/s102 Ω/s45/s49\n/s109/s45/s50\n/s73\n/s67/s32/s44/s32/s73/s45\n/s67/s32/s40/s109/s65/s41\n/s116\n/s67/s117/s32/s40/s110/s109/s41/s40/s98/s41\n/s116\n/s67/s117/s61/s53/s53/s32/s110/s109\n/s32/s73\n/s67/s32/s44/s32/s73/s45\n/s67/s32/s40/s109/s65/s41\n/s108\n/s115/s102/s44/s67/s111/s32/s40/s110/s109/s41/s40/s99/s41\n/s108\n/s115/s102/s44/s67/s111/s61/s52/s50/s32/s110/s109\n/s84/s32/s40/s75/s41/s32/s73\n/s67/s32/s40/s109/s65/s41/s40/s100/s41\n/s50/s48/s110/s109/s52/s48/s32/s110/s109/s54/s48/s32/s110/s109\nFIG. 4: (a) Calculated IC,I−\nCvsGforFeMnCo 3. (b) Same\nvstCuforCo3. (c) Same vslsf,CoforCo20, (d) same vsT\nforCo20 samples, for the residual values of lsf,Coas labeled.\nEquations (1) and (2) express ICin terms of ∆ µand\nIS, the same quantities that determine MR in magnetic\nmultilayers. We calculated ∆ µandISself-consistently\nusing a one-dimensional diffusive approximation employ-\ning the standard MR parameters: spin asymmetries β,\nrenormalized resistivities ρ∗=ρ/(1−β2), spin diffusion\nlengthslsfin the layers, and similarly defined parame-\ntersAR∗,γ, andδfor the interfaces.17We estimate these\nparameters from a combination of the published values11\nand our own measurements, as described below.\nThe resistivityofeach layerin oursamples provideses-\nsential information about electron diffusion. Because of\nvariations among published resistivities, we instead de-\ntermined their values from measurements of thin films\nprepared under the same conditions as the nanopillars,\nwith thicknessesverifiedby x-rayreflectometry. Fig. 3(a)\nshowsρ(T)for40nmthickPy,Co,andCufilms,together\nwith fittings for Co and Cu with the Bloch-Gruneisen\napproximation. We obtained better fitting for Py data\nwith a quadratic dependence, indicating that electron-\nmagnon scattering may dominate electron-phonon scat-\ntering.18The dependence of the residual resistivity on\nfilm thickness wasconsistentwith the Fuchs-Sommerfield\napproximation(Fig.3(b)), allowingustoextractthebulk\nresidual values ρPy(0) = 11.3µΩcm,ρCo(0) = 4.4µΩcm,\nandρCu(0) = 1.1µΩcm. We used the extracted bulk\nρ(T) to model all the extended layers in the nanopillars.\nThe effect oflateralconfinement in Py(5) nanopillarswas\napproximated by using the resistivity of a Py(40) film.\nTo estimate lsf(T), we used its empirical inverse re-\nlationship with ρ, along with the bulk residual values\nlsf,Py(0) = 6 nm, and lsf,Cu(0) = 300 nm based on pub-\nlished measurements,11scaled by the somewhat different\nresidual resistivities of our films. If scattering by thermal\nexcitations does not flip electron spins, a weaker depen-/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s50/s52/s54/s56/s49/s48\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s46/s49/s50/s48/s46/s49/s52/s48/s46/s49/s54/s48/s46/s49/s56/s73\n/s67/s32/s40Ω /s41\n/s84/s32/s40/s75/s41/s40/s97/s41\n/s67/s111/s50/s48\n/s67/s111/s51\n/s70/s101/s77/s110/s67/s111/s51\n/s82\n/s65/s80/s45/s82\n/s80/s32/s40Ω /s41\n/s84/s32/s40/s75/s41/s40/s98/s41\n/s67/s111/s50/s48/s70/s101/s77/s110/s67/s111/s51\n/s67/s111/s51\nFIG. 5: (a) Calculated ICvsT, and (b) calculated MR vsT\nfor three sample types as labeled.\ndencelsf(T)∝/radicalbig\n1/ρ(T) is possible. However, we show\nbelow that a dependence even stronger than 1 /ρis more\nlikely. We use βPy=γPy/Cu= 0.7,γCo/Cu= 0.8,βCo=\n0.36 for spin asymmetries, AR∗\nCo/Cu= 0.55fΩm2,\nAR∗\nPy/Cu= 0.5fΩm2for renormalized interface resis-\ntances, and δCo/Cu= 0.2,δPy/Cu= 0.25 for spin flip-\nping coefficients.8,11Their dependence on Tis neglected\ndue to the dominance of the band structure and impurity\nscattering far from the Curie temperature. For FeMn, we\nusedlsf,FeMn ≈0.5 nm, and ρFeMn= 87µΩcm. Scat-\ntering at its interfaces was modeled by adding 0 .5 nm\nto the nominal thickness of FeMn. To account for the\nCu contacts, the calculation included outer Cu layers of\nthickness tCu, determined as described below. These lay-\ners were terminated with fictitious spin sinks.\nTo demonstrate that CIMS is extremely sensitive to\nthe effects ofdiffusion, we now describe how our5 K data\ncan be fitted by appropriate choice of three parameters\nwhose values have the largest uncertainty: conductance\nGin Equation (1), effective MR-active thickness tCuof\nthe Cu contacts, and spin diffusion length lsf,Co. Calcu-\nlations for FeMnCo 3 were significantly affected only by\nG, which controls the asymmetry of CIMS. The values\nofIC/|I−|in mA measured at 5 K for three FeMnCo 3\nsamples were 2 .3/0.8, 1.6/0.6, and 3.1/1.5, giving an av-\nerage ratio IC/|I−|= 2.6. The calculated value increases\nfrom1.46atG= 0.5fΩ−1m−2to6.1atG= 2fΩ−1m−2\n(Fig. 4(a)). The best values IC/|I−\nC|= 3.34/1.27 are ob-\ntained at G= 0.87fΩ−1m−2, in reasonable agreement\nwith band structure calculations.5,19\nSpin diffusion in the bottom Cu layer has little effect\nonCo20 andFeMnCo due to the spin relaxation in Co\nand FeMn, respectively. To determine tCu, we use the ra-\ntiosIC/|I−|of the three Co3 samples, 3 .55/1.0, 4.6/1.5,\nand 4.2/1.2, giving an average ratio IC/|I−|= 3.4. The\ncalculated IC/|I−\nC|increases from 1 .9 fortCu= 0 to 14\nfortCu= 140 nm (Fig. 4(b)), and eventually diverges at\ntCu= 200nm. The best agreementwith data is obtained\nfortCu= 55 nm, resulting in IC/|I−\nC|= 4.4/1.3.\nLastly, diffusion in Co significantly affects CIMS in\nsamplesCo20, but not in Co3 andFeMnCo 3. We deter-\nminelsf,Cofrom the ratio IC/|I−|of fiveCo20 samples,\n8.9/2.1, 7.3/1.6, 9.0/2.0, 8.5/2.0, 8.0/1.7, giving an av-\nerage ratio IC/|I−|= 4.4. Fig. 4(c) illustrates that the4\ncalculated ratio IC/|I−\nC|increases from 1 .0 forlsf,Co= 0\nto 5.2 forlsf,Co= 100 nm. The best agreement with the\ndata is obtained for lsf,Co= 42 nm consistent with the\npublished values.11\nDespite the ability to model the 5 K data, the cal-\nculations did not reproduce the dramatic dependence of\nIConTin Fig. 2 (see below). Therefore, one can at-\ntempt to determine lsf,Cofrom the dependence of ICon\nT. Fig. 4(d) shows calculations for the residual values\nlsf,Co= 20 nm, 40 nm, and 60 nm. Large lsf,Coresults\ninICdecreasing with T, which is inconsistent with the\ndata. Small lsf,Cogives decrease of ICwithTin better\nqualitative agreement with data, but gives unreasonably\nsmallICat 5 K. Consequently, we return to the value\ndetermined from Fig. 4(c).\nFig. 5(a) shows the calculated ICvsTfor the three\nsample types. To interpret these results, we note that\nFigs. 4(b)-(d) exhibited an increase of ICwhen the ef-\nfective MR-active resistance of F 1determined by ρlsf\nwas increased. This relationship was also established an-\nalytically.5,20The experimental correlation between the\ndecreases of MR and increases of ICin Figs.1, 2 is of\nthe same origin. The lack of temperature dependence for\nFeMnCo 3isthereforeconsistentwith negligiblespin dif-\nfusion effects in F 1. In calculations for Co3, the increase\nofICwithTis caused by the increased contribution\ntCulsf,Cuto the effective resistance of F 1. Calculations\nforCo20 show a competition between the contribution of\nthe bulk Co resistivity, which increases with T, and the\ncontributionsfromthe Cu(50) layerandthe outerCo/Cu\ninterface, whichdecreasewith Tduetotheincreasedspin\nflipping in Co. However,both Co3 andCo20 calculations\ndo not reproduce the data, suggesting that the effects of\nthermal scattering should be re-examined.\nThe calculated dependence of MR on Twas in over-all agreement with data for Co3 andFeMnCo 3, but did\nnot reproduce the decrease at T <130 K seen in Co20\ndata (Fig. 5(b)). The calculations overestimated the val-\nues, suggestingthat oursamples maybe largerthan their\nnominal size. However, this seems to contradict the cal-\nculated temperature dependence of RPconsistent with\nthe data(not shown), and the values of ICthat arelarger\nthan the measured 5 K values. This discrepancy can be\nreduced e.g. by decreasing lsf,Py, which results in a de-\ncreased MR without significantly affecting CIMS.\nThe failure of Co20 calculations to capture the de-\ncrease of MR and the increase of ICatT <130 K in-\ndicates that lsf,Codecreases with Tmore rapidly than\nthe accepted lsf∝1/ρ, resulting in the reduction of\nthe effective MR-active resistance ρColsf,Co. One pos-\nsible mechanism for such a strong dependence may be\nelectron-magnon scattering which can result in electron\nspin flipping without significant momentum scattering.\nWe leave more detailed and perhaps alternative explana-\ntions to future studies.\nTo summarize, we performed magnetoelectronic mea-\nsurements of nanopillars with three different polarizing\nmagneticlayers. The samplesexhibited differentcurrent-\ninduced behaviors, attributed to the spin diffusion in the\npolarizing layer. The calculations reproduced the lower\ntemperature behaviors with reasonable values of trans-\nport parameters. However, temperature dependencies\nofmagnetoresistanceandcurrent-induced switchingindi-\ncate that the effects of thermal scattering on spin trans-\nport are more significant than presently believed.\nWe thank Mark Stiles, Jack Bass and Norman Birge\nfor helpful discussions. This work was supported by NSF\nGrant DMR-0747609 and a Research Corporation Cot-\ntrell Scholar Award. SB acknowledges support from the\nNASA Space Grant Consortium.\n1J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n2A.A. Kovalev, A. Brataas, and G.E.W. Bauer, Phys. Rev.\nB 66, 224424 (2002).\n3S. Zhang, P.M. Levy, and A. Fert, Phys. Rev. Lett. 88,\n236601 (2002).\n4A. Shpiro, P.M. Levy, and S. Zhang, Phys. Rev. B 67,\n104430 (2003).\n5J. Slonczewski, J. Magn. Magn. Mater. 247, 324 (2002).\n6O. Boulle, V. Cros, J. Grollier, L.G. Pereira, C. Deranlot,\nF. Petroff, G. Faini, J. Barnas, andA. Fert, Nature Physics\n3, 492 (2007).\n7S.Urazhdin, N.O.Birge, W.P. PrattJr., andJ. Bass, Appl.\nPhys. Lett. 84, 1516 (2004).\n8T. Yang, A. Hirohata, M. Hara, T. Kimura, and Y. Otani,\nAppl. Phys. Lett. 89, 252505 (2006).\n9M. AlHajDarwish, H. Kurt, S. Urazhdin, A. Fert, R.\nLoloee, W. P. Pratt, Jr., and J. Bass, Phys. Rev. Lett.\n93, 157203 (2004).\n10N. Theodoropoulou, A. Sharma, W.P. Pratt Jr., and J.\nBass, Phys. Rev. B 76, 220408(R) (2007).\n11J. Bass and W.P. Pratt Jr., JMMM 200, 274 (1999) andJ. Phys.: Condens. Matter 19, 183201 (2007).\n12S.Urazhdin, N.O.Birge, W.P. PrattJr., andJ. Bass, Phys.\nRev. Lett. 91, 146803 (2003).\n13S.D. Steenwyk, S.Y. Hsu, R. Loloee, J. Bass, and W.P.\nPratt Jr., J. Magn. Magn. Mater. 170, L1 (1997).\n14J. Xiao, A. Zangwill, and M.D. Stiles, Phys. Rev B 70,\n172405 (2004).\n15I.N. Krivorotov, N.C. Emley, J.C. Sankey, S.I. Kiselev,\nD.C. Ralph, and R.A. Buhrman, Science 307, 228 (2005).\n16I.N. Krivorotov, N.C. Emley, A.G.F. Garcia, J.C. Sankey,\nS.I. Kiselev, D.C. Ralph, and R.A. Buhrman, Phys. Rev.\nLett.93, 166603 (2004).\n17T. Valet and A. Fert Phys. Rev. B 48, 7099 (1993).\n18D.A. Goodings, Phys. Rev. 132, 542 (1963).\n19K. Xia, P.J. Kelly, G.E.W. Bauer, A. Brataas, and I.\nTurek, Phys. Rev. B 65, 220401R (2002).\n20S. Urazhdin, R. Loloee, and W. P. Pratt Jr., Phys. Rev. B\n71, 100401 (2005).\n21J. Bass and W.P. Pratt Jr., private communications."    },    {        "title": "2210.04262v1.Method_of_dynamic_resonance_tuning_in_spintronics_of_nanosystems.pdf",        "content": "arXiv:2210.04262v1  [physics.app-ph]  9 Oct 2022Method of dynamic resonance tuning in spintronics of\nnanosystems\nV.I. Yukalov1,2and E.P. Yukalova3\n1Bogolubov Laboratory of Theoretical Physics,\nJoint Institute for Nuclear Research, Dubna 141980, Russia\n2Instituto de Fisica de S˜ ao Carlos, Universidade de S˜ ao Pau lo,\nCP 369, S˜ ao Carlos 13560-970, S˜ ao Paulo, Brazil\n3Laboratory of Information Technologies,\nJoint Institute for Nuclear Research, Dubna 141980, Russia\nE-mails:yukalov@theor.jinr.ru ,yukalova@theor.jinr.ru\nAbstract\nA method is advanced allowing for fast regulation of magneti zation direction in mag-\nnetic nanosystems. The examples of such systems are polariz ed nanostructures, magnetic\nnanomolecules, magnetic nanoclusters, magnetic graphene , dipolar and spinor trapped\natoms, and quantum dots. The emphasis in the paper is on magne tic nanomolecules and\nnanoclusters. The method is based on two principal contriva nces: First, the magnetic\nsample is placed inside a coil of a resonant electric circuit creating a feedback field, and\nsecond, there is an external magnetic field that can be varied so that to dynamically\nsupport the resonance between the Zeeman frequency of the sa mple and the natural fre-\nquency of the circuit during the motion of the sample magneti zation. This method can\nfind applications in the production of memory devices and oth er spintronic appliances.\n11 Introduction\nIn the problem of regulating spin dynamics, there are two principal c hallenges contradicting\neach other. First, it is necessary to possess the ability of keeping fi xed the device magnetization\nfor sufficiently long time. And second, one has to have the capability o f quickly varying the\nmagnetization direction at any required time. The possibility of keepin g fixed the magnetiza-\ntion direction, as such, is not a difficult task that can be easily realized by using the materials\nenjoying strong magnetic anisotropy. The well known examples of m aterials with strong mag-\nnetic anisotropy are magnetic nanomolecules [1–11] and magnetic n anoclusters [12–16]. These\nmaterials can possess large spins and strong magnetic anisotropy k eeping, at low temperature,\nbelowthe blocking temperature, thesample magnetization frozen. At the same time, thestrong\nmagnetic anisotropy prevents the realization of fast magnetizatio n reversal because of two rea-\nsons. First, as will be shown below, magnetic anisotropy induces the dynamic variation of the\neffective Zeeman frequency which cannot be compensated by a con stant magnetic field, and\nsecond, even inverting the external magnetic field, the sample spin cannot be quickly reversed\nbecause of the strong magnetic anisotropy. Thus the dilemma arise s: For fixing during the\nrequired long time the magnetization direction, one needs a rather s trong magnetic anisotropy;\nhowever the latter does not allow for fast magnetization reversal.\nInthepresent paper, weadvanceanoriginalwayoutoftheabove dilemma. Wekeep inmind\nthematerialsenjoyingmagneticanisotropysufficient forfixingthes amplemagnetization. These\ncan be magnetic nanomolecules or magnetic nanoclusters. To some e xtent, the consideration\nis applicable to dipolar and spinor trapped atoms [17–23] and to magne tic graphene (graphene\nwithmagneticdefects) [24,25]. Quantumdotsinmanyaspectsares imilar tonanomolecules [26]\nand also can possess magnetization [27–30] that could be governed .\nThe method we suggest is based on two principal points. First, the c onsidered sample is\nplacedinsideamagneticcoilofanelectriccircuit. Thenthemovingmag netizationofthesample\nproduces electric current in the circuit, which, in turn, creates a m agnetic feedback field acting\non the sample. An effective coupling between the circuit and the samp le appears only when the\ncircuit natural frequency is in resonance with the Zeeman frequen cy of the sample. However,\nthe magnetic anisotropy, as we show, leads to the detuning of the e ffective Zeeman frequency\nfrom the resonance. Moreover, this detuning is dynamic, varying in time. The second pillar\nof the suggested method is the use of a varying external magnetic field realizing the dynamic\ntuning of the effective Zeeman frequency to the resonance with th e circuit natural frequency.\nSince the tuning procedure is dynamic, the method is called the dynamic resonance tuning .\n2 Magnetic nanomolecules and nanoclustres\nSingle-domain nanomolecules and nanoclusters are of special intere st for spintronics. Due to\nstrong exchange interactions, the spins of particles forming the n anomagnet point in the same\ndirection thus creating a common spin and hence the common single ma gnetization vector.\nUnder the action of external fields, the magnetization vector mov es as a whole, which is called\ncoherent motion. At the same time, such nanomagnets usually poss ess a rather strong magnetic\nanisotropy. The standard Hamiltonian of a nanomagnet with a spin Shas the form\nˆH=−µSB·S+ˆHA, (1)\n2wherethefirst istheZeemanterm, µS=−gSµB,gSisaLand´ efactor, µBistheBohrmagneton,\nandSis a spin operator. The second term describes the nanomagnet mag netic anisotropy,\nˆHA=−DS2\nz+E(S2\nx−S2\ny), (2)\nwith the anisotropy parameters\nD=1\n2(Dxx+Dyy)−Dzz, E=1\n2(Dxx−Dyy),\nexpressed through the dipolar tensor DαβandSαbeing spin-operator components.\nThe sample is inserted into a magnetic coil of an electric circuit produc ing a feedback field\nH. The coil axis is in the xdirection. The total magnetic field, acting on the sample,\nB=Hex+B1ey+(B0+∆B)ez, (3)\nconsists of the feedback field H, a small anisotropy field B1, and an external magnetic field\nB0+∆B, withB0being a constant field and∆ Bbeing a regulatedpart of thefield. The electric\ncircuit plays the role of a resonator. The motion of the sample magne tization creates in the\ncircuit electric current satisfying the Kirhhoff equation. In its turn , the electric current forms\nthe feedback magnetic field defined by the equation [31] following fro m the Kirhhoff equation,\ndH\ndt+2γH+ω2/integraldisplayt\n0H(t′)dt′=−4πηresdmx\ndt, (4)\nwhereγis the circuit attenuation, ω, the resonator natural frequency, and ηres≈V/Vresis the\nresonator coil filling factor, Vbeing the sample volume, and Vres, the resonator coil volume.\nThe right-hand side of the equation characterizes the electromot ive force due to the moving\naverage magnetization\nmx=µS\nV/an}bracketle{tSx/an}bracketri}ht. (5)\nWriting down the Heisenberg equations of motion, we average them, looking for the dynam-\nics of the average spin components\nx=/an}bracketle{tSx/an}bracketri}ht\nS, y=/an}bracketle{tSy/an}bracketri}ht\nS, z=/an}bracketle{tSz/an}bracketri}ht\nS, (6)\nwithSbeing the sample spin. In the process of the averaging, we meet the combination of\nspinsSαSβ+SβSα. It would be incorrect to decouple this combination in the simple mean- field\napproximation, since for S= 1/2 this combination has to be zero. The correct decoupling [32],\nthat is exactly valid for S= 1/2 as well as asymptotically exact for large spins, is done in the\ncorrected mean-field approximation\n/an}bracketle{tSαSβ+SβSα/an}bracketri}ht=/parenleftbigg\n2−1\nS/parenrightbigg\n/an}bracketle{tSα/an}bracketri}ht/an}bracketle{tSβ/an}bracketri}ht. (7)\nFor what follows, we need to introduce several notations. We defin e the Zeeman frequency\nω0≡ −µS\n/planckover2pi1B0, (8)\n3and the anisotropy frequencies\nω1≡ −µS\n/planckover2pi1B1, ω D≡(2S−1)D\n/planckover2pi1, ω E≡(2S−1)E\n/planckover2pi1. (9)\nThe dimensionless anisotropy parameter is defined as\nA≡ωD+ωE\nω0. (10)\nThe dimensionless regulated field is given by the expression\nb≡ −µS∆B\n/planckover2pi1ω0. (11)\nFrom the evolution equations, it is seen that the coupling between th e resonant circuit and\nthe sample is characterized by the coupling rate\nγ0≡πηresµ2\nSS\n/planckover2pi1V=πµ2\nSS\n/planckover2pi1Vres. (12)\nFinally, the dimensionless feedback field is denoted by\nh≡ −µSH\n/planckover2pi1γ0. (13)\nThus we come to the system of equations\ndx\ndt=−ωSy+ω1z ,dy\ndt=ωSx−γ0hz ,\ndz\ndt= 2ωExy−ω1x+γ0hy , (14)\nin which the effective Zeeman frequency is\nωS≡ω0(1+b−Az). (15)\nThe feedback-field equation (4) becomes\ndh\ndt+2γh+ω2/integraldisplayt\n0h(t′)dt′= 4dx\ndt. (16)\nOf course, the evolution equations are to be complemented by the in itial conditions x(0) =\nx0for the vector x={x,y,z}and for the feedback field h(0) =h0. In what follows, we set the\ninitial conditions as x0=y0= 0,h0= 0, and z0= 1. These initial conditions correspond to\nthe setup, when there are no alternating fields pushing the spin mot ion and all the following\ndynamics is self-organized.\n43 Dynamic resonance tuning for single spins\nThe initial setup s0= 1 describes the sample with the spin polarization, formed by electro ns,\ndirected up, hence the magnetization directed down, while the exte rnal magnetic field B0is\ndirected up. This implies that the sample is in a metastable state. The s table state corresponds\nto the magnetization up, hence to the spin polarization down.\nBecause ofthestrong magneticanisotropy, themagnetizationisf rozenand, belowtheblock-\ning temperature, it can exist in this metastable state for days and m onths. This is convenient\nfor keeping the information in memory devices. However, if on needs to rewrite or erase the\ninformation, which also is an action typical of memory devices, then t he frozen magnetization\nhinders this. Thus we confront the problem: how could we overcome the anisotropy in order\nto start moving the sample spin and to move it sufficiently fast?\nIf there would be resonance between the resonator natural fre quencyωand the effective\nZeeman frequency ωS, which is possible in the absence of the magnetic anisotropy, when A= 0,\nthen there would appear strong coupling between the resonator f eedback field and the sample\nmagnetization, as a result of which the magnetization could be quickly reversed at the initial\ntime [31–35]. However, the effective Zeeman frequency (15) canno t be tuned to a constant\nresonator natural frequency ω, since the effective Zeeman frequency is not a constant but a\nfunction of the polarization z. For instance, if ωis tuned to ω0, then the relative detuning\nωS−ω0\nω0=b−Az\nvaries in time and, generally, can be very large for large anisotropy p arameters A.\nSuppose, we have been keeping the magnetization fixed for a requir ed timeτ, after which\nwe need to quickly reverse it. To make the detuning small, and moreov er for keeping it small\nduring the whole process of spin reversal, we suggest to set ω0=ωand to vary the regulated\nfieldb=b(t) according to the law of dynamic resonance tuning , so that\nb(t) =/braceleftbigg0, t < τ\nAzreg, t≥τ ,(17)\nwithzregsatisfying the equations\ndxreg\ndt=−ω0yreg+ω1zreg,dyreg\ndt=ω0xreg−γ0hregzresg,\ndzreg\ndt= 2ωExregyreg−ω1xreg+γ0hregyreg,\ndhreg\ndt+2γhreg+ω2/integraldisplayt\nτhreg(t′)dt′= 4dxreg\ndt. (18)\nThe initial conditions for the latter system of equations can be take n either as xreg(τ) =x(τ)\nandhreg(τ) =h(τ) or asxreg(τ) =x0andhreg(τ) =h0.\nIn Fig. 1, we show the process where the spin polarization is frozen, by a strong magnetic\nanisotropy, during the time τ, after which the mechanism of the dynamic resonance tuning\nis switched on. Different initial conditions are compared, as explained in the Figure, demon-\nstrating that they lead to a slight shift of the reversal process. I t is also shown that if the\nresonance condition is not dynamic, but only the initial triggering res onance [36], when the\n50 5 10 15 20t-1-0.500.51z(t)\n(a)\n = 1\n = 10\nno tuningzreg() = z()zreg() = 1\n0 5 10 15 20t-1-0.500.51z(t)\n(b)\n = 10\n = 100\nzreg() = z()zreg() = 1\nno tuning\nFigure 1: Spin polarization z(t) of a nanocluster or nanomolecule as a function of time under dynamic\nresonance tuning, starting at the delay time τ= 10, for different initial conditions, as explained in the\nFigure. The anisotropy parameters are A= 1 and ωE=ω1= 0.01. The line with a long tail describes\nthe process, where the dynamic resonance tuning is not used, but instead the condition of triggering\nresonance at the same initial time is employed. (a) ω=ω0= 10 and γ= 1; (b) ω=ω0= 100 and\nγ= 10. Time is measured in units of γ−1\n0and frequencies, in units of γ0.\ncondition b(τ) =Az(τ) is imposed, then the reversal is not fast but possesses a long tail. Time\nis measured in units of γ−1\n0and frequencies, in units of γ0.\nIn Fig. 2, the spin polarization of a nanocluster or nanomolecule is sho wn for dynamic\nresonance tuning, compared with the polarization without dynamic t uning but under the trig-\ngering resonance at the same delay time. The advantage of dynamic resonance tuning is in an\nultrafast spin reversal, while the triggering resonance leads to long tails.\n0 20 40 60 80 100t-1-0.500.51z(t)(a)\nno tuning = 1\n = 100\n0 10 20 30 40 t-1-0.500.51z(t)(b)  = 5\n = 100\nno tuning\nFigure 2: Spin polarization z(t) of a nanocluster or nanomolecule as a function of time under dynamic\nresonance tuning, starting at the delay time τ= 10 (solid line), compared with the polarization\nwithout dynamic tuning but under the triggering resonance a t the same delay time (dashed line). The\nanisotropy parameters are A= 1 and ωE=ω1= 0.01. (a)ω=ω0= 100 and γ= 1; (b) ω=ω0= 100\nandγ= 5. The absence of dynamic tuning leads to long tails.\nFigure 3 illustrates the reversal of the spin polarization of a nanoclu ster or nanomolecule\nunder dynamic resonance tuning, starting at different delay times. This shows thatit is possible\nto quickly reverse the magnetization at any required time.\n60 5 10 15 20 25 30t-1-0.500.51z(t)\n = 10\n = 100\n = 5 = 1\n = 20 = 10\nFigure 3: Spin polarization z(t) of a nanocluster or nanomolecule as a function of time under dynamic\nresonance tuning, starting at different delay times: τ= 1 (solid line), τ= 5 (dashed line), τ= 10\n(dash-dotted line), and τ= 20 (dotted line). Other parameters are fixed as A= 1,ωE=ω1= 0.01,\nω=ω0= 100 and γ= 10.\n4 Assemblies of nanomolecules or nanoclusters\nIn the previous section, we have considered the realization of dyna mic resonance tuning for sin-\ngle nanomolecules or nanoclusters, which allows for fast magnetizat ion reversal at any required\ntime. The natural question is whether it would be feasible to apply this effect for regulating\nspin dynamics of the assemblies of nanomolecules or nanoclusters.\nThe Hamiltonian of a system of nanomolecules or nanoclusters reads as\nˆH=−µS/summationdisplay\njB·Sj+ˆHA+ˆHD, (19)\nwherej= 1,2,...,Nenumerates the clusters, the first is the Zeeman term and the sec ond is\nthe anisotropy term\nˆHA=−/summationdisplay\njD(Sz\nj)2. (20)\nThe anisotropy parameter Eis usually much smaller than D, so it can be safely omitted.\nIn addition, there is the term responsible for the dipolar interaction s of the constituents\nˆHD=1\n2/summationdisplay\ni/negationslash=j/summationdisplay\nαβDαβ\nijSα\niSβ\nj, (21)\nwith the dipolar tensor\nDαβ\nij=µ2\nS\nr3\nij/parenleftBig\nδαβ−3nα\nijnβ\nij/parenrightBig\n, (22)\nin which\nrij≡ |rij|,nij≡rij\nrij,rij≡ri−rj.\nThe sample is again inserted into a magnetic coil of an electric circuit. T he total magnetic\nfield is the sum\nB=Hex+(B0+∆B)ez (23)\n7of the feedback field H, a constant external magnetic field B0, and a regulated field ∆ B.\nThe feedback field satisfies equation (4), but with the right-hand s ide, describing the elec-\ntromotive force, containing the average magnetization\nmx=µS\nV/summationdisplay\nj/an}bracketle{tSx\nj/an}bracketri}ht. (24)\nThe coil axis is again aligned with the xaxis.\nIn what follows, it is convenient to employ the ladder spin operators S±\nj=Sx\nj±iSy\nj. We\nintroduce the average transverse spin component\nu=1\nSN/summationdisplay\nj/an}bracketle{tS−\nj/an}bracketri}ht, (25)\nthe coherence intensity\nw=1\nSN(N−1)/summationdisplay\ni/negationslash=j/an}bracketle{tS+\niS−\nj/an}bracketri}ht, (26)\nand the longitudinal spin polarization\ns=1\nSN/summationdisplay\nj/an}bracketle{tSz\nj/an}bracketri}ht. (27)\nLocal spin fluctuations, caused by dipolar interactions, are chara cterized by the expressions\nξS=1\n/planckover2pi1/summationdisplay\nj/an}bracketle{taijSz\nj+cijS+\nj+c∗\nijS−\nj/an}bracketri}ht,\nϕS=1\n/planckover2pi1/summationdisplay\nj/angbracketleftBigaij\n2S−\nj−2bijS+\nj−2cijSz\nj/angbracketrightBig\n, (28)\nin which\naij≡Dzz\nij, b ij≡1\n4/parenleftbig\nDxx\nij−Dyy\nij−2iDxy\nij/parenrightbig\n, c ij≡1\n2/parenleftbig\nDxz\nij−iDyz\nij/parenrightbig\n.\nThese expressions are responsible for dipolar spin waves initiating sp in motion at the beginning\nof the process [32].\nThe decoupling of pair spin correlators is accomplished in the correct ed mean-field approx-\nimation\n/an}bracketle{tSα\niSβ\nj/an}bracketri}ht=/an}bracketle{tSα\ni/an}bracketri}ht/an}bracketle{tSα\nj/an}bracketri}ht(i/ne}ationslash=j),\n/an}bracketle{tSα\njSβ\nj+Sβ\njSα\nj/an}bracketri}ht=/parenleftbigg\n2−1\nS/parenrightbigg\n/an}bracketle{tSα\nj/an}bracketri}ht/an}bracketle{tSβ\nj/an}bracketri}ht (29)\naccurately taking into account the terms describing magnetic aniso tropy [31]. Writing down\nthe spin equations of motion and averaging them, we come to the sys tem of equations\ndu\ndt=−i(ωS+ξS−iγ2)u+fs ,dw\ndt=−2γ2w+(u∗f+f∗u)s ,\n8ds\ndt=−1\n2(u∗f+f∗u), (30)\nwith the transverse attenuation\nγ2=1\n/planckover2pi1ρ µ2\nSS , (31)\nwhereρ=N/Vis spin density, and the effective force\nf=−i/parenleftbiggµSH\n/planckover2pi1+ϕS/parenrightbigg\n. (32)\nThe effective Zeeman frequency here is\nωS=ω0(1+b−As), (33)\nwhere the dimensionless regulated field bis defined in Eq. (11).\nThe feedback-field equation (4) can be represented in the integra l form\nH=−4πηres/integraldisplayt\n0G(t−t′) ˙mx(t′)dt′, (34)\nwith the transfer function\nG(t) =/bracketleftBig\ncos(ω′t)−γ\nω′sin(ω′t)/bracketrightBig\ne−γt,\nwhere the frequency is\nω′≡/radicalbig\nω2+γ2.\nThe electromotive force is expressed through\n˙mx=1\n2ρ µSS(u∗+u). (35)\nThe coupling rate is\nγ0=πηresγ2. (36)\nAs usual, the attenuations are small as compared to the related fr equencies:\nγ0\nω≪1,γ\nω≪1,γ2\nω0≪1. (37)\nThe solution to the feedback field, following from Eq. (34), to the fir st order in γ0reads as\nµSH=i/planckover2pi1(uX−X∗u∗), (38)\nwith the coupling function\nX=γ0ωS1−exp(−i∆St−γt)\nγ+i∆SsignωS(39)\nand the dynamic detuning\n∆S=ω−|ωS|. (40)\n9Thepresenceofthesmallparameters(37)makesitstraightforw ardtoresorttotheaveraging\ntechniques [37,38], with considering the dipolar spin fluctuations, ch aracterized by expressions\n(28), assmall randomvariables [39,40]. Dueto thesmall parameter s (37), thefunction u=u(t)\nin Eqs. (30) has to be treated as a fast variable, while the functions w=w(t) ands=s(t),\nas slow variables. With the slow variables playing the role of adiabatic inv ariants, we solve the\nequation for the fast variable, yielding\nu=u0exp/braceleftbigg\n−iΩt−i/integraldisplayt\n0ξS(t′)dt′/bracerightbigg\n−\n−is/integraldisplayt\n0ϕS(t′) exp/braceleftbigg\n−iΩ(t−t′)−i/integraldisplayt\nt′ξS(t′′)dt′′/bracerightbigg\ndt′, (41)\nwhere\nΩ =ωS−i(γ2−Xs).\nThen, substituting the solutions for the feedback field (39) and fo r the fast variable (41) into\nthe equations for the slow variables wands, we average the latter over time and random spin\nfluctuations [41]. As a result, we obtain the guiding-center equation s\ndw\ndt= 2γ2w(αs−1)+2γ3s2,ds\ndt=−γ2αw−γ3s , (42)\nin which we define the coupling function\nα≡ReX\nγ2=γ0γωS\nγ2(γ2+∆2\nS)/braceleftbig\n1−[ cos(∆ St)−δSsin(∆St) ]e−γt/bracerightbig\n, (43)\nthe relative detuning\nδS≡∆S\nγsignωS, (44)\nand the spin-wave attenuation\nγ3=γ2\n2/radicalbig\nω2\nS+γ2\n2. (45)\nAs initial conditions, we take w(0) = 0, which implies the absence of triggering fields, so that\nthe process is self-organized, and we assume the initial average sp in polarization up, so that\ns(0) = 1.\n5 Dynamic resonance tuning for spin assemblies\nSuppose we wish to keep the average spin polarization up until the tim eτand then we need\nto quickly reverse the average spin. For this purpose, we set ω0=ωand arrange the regulated\nmagnetic field according to the law\nb(t) =/braceleftbigg\n0, t < τ\nAsreg, t≥τ ,(46)\nwith the parameter\nA≡ωD\nω0(47)\n10and with sregsatisfying the equations\ndsreg\ndt=−γ2αregwreg−γregsreg,\ndwreg\ndt= 2γ2wreg(αregsreg−1)+2γregs2\nreg. (48)\nIn the latter, the effective coupling function is\nαreg=g/parenleftbig\n1−e−γt/parenrightbig\n, (49)\nwith the coupling parameter\ng≡γ0ω0\nγγ2(50)\nand the attenuation\nγreg≡γ2\n2/radicalbig\nω2+γ2\n2. (51)\nThe initial conditions at time τarewreg(τ) =w(τ) andsreg(τ) =s(τ). At time τ, there is the\nresonance ω0=ω, and in the following times the regulated field (46) varies so that the s ystem\nis dynamically captured into resonance [42].\nIf there is need for repeating the overall process again, this can b e done by either inversing\nthe direction of the external field B0or by rotating the sample as has been described for\norganizing a Morse-code alphabet functioning of spin pulses for sam ples having no magnetic\nanisotropy [43].\nFigures 4 and 5 demonstrate the effect of dynamic resonance tunin g for the assemblies of\nmagnetic nanomolecules or nanoclusters. The average spin of the s ystem can be kept for a\nlong time by a strong magnetic anisotropy. The spin is better frozen for larger anisotropies and\nlarger Zeeman frequencies. Employing dynamic resonance tuning ma kes it possible to realize\nan ultrafast spin reversal at any required time.\n0 5 10 15 t-1-0.500.51s(t)\n = 1 = 5\n = 10(a)\n0 5 10 15 t-1-0.500.51s(t)\n(b)\n = 1 = 5\n = 10\nFigure4: Average spin polarization s(t) of an assembly of magnetic nanomolecules or nanoclusters f or\nω=ω0= 100 and γ= 10 as a function of time under dynamic resonance tuning star ting at different\ndelay times: τ= 1 (solid line), τ= 5 (dashed line), and τ= 10 (dash-dotted line). The anisotropy\nparameters are: (a) A= 5; (b) A= 1.\n110 5 10 15 t-1-0.500.51s(t)(a)\n = 10 = 1\n = 5\n0 5 10 15 t-1-0.500.51s(t)\n(b)\n = 1 = 5\n = 10\nFigure5: Average spin polarization s(t) of an assembly of magnetic nanomolecules or nanoclusters f or\nthe anisotropy parameter A= 5 and γ= 10. The dynamic resonance tuning starts at different delay\ntimes:τ= 1 (solid line), τ= 5 (dashed line), and τ= 10 (dash-dotted line). Resonance frequencies\nare: (a)ω=ω0= 1000; (b) ω=ω0= 100.\n6 Discussion and conclusion\nWe have suggested a method allowing, by using magnetic nanosystem s, such as magnetic\nnanomolecules and nanoclusters, to combine two features that ar e crucially important for the\nfunctioning of memory devices, the possibility of keeping for long time s a frozen magnetiza-\ntion that can be reversed at a required time. The method is based on the following technical\nstratagems. First, it is straightforward to use the property of m agnetic nanomolecules and\nnanoclusters to keep, below the blocking temperature, the direct ion of magnetization frozen in\na metastable state. Second, the sample is inserted into a magnetic c oil of an electric circuit cre-\nating a magnetic feedback field. Third, at a required time, a varying m agnetic field is imposed,\nvarying in such a way that to dynamically support a resonance betwe en the electric circuit\nand the varying Zeeman frequency of the sample. Recall that the e ffective Zeeman frequency\nchanges in time because of the interaction between the moving samp le spin and the magnetic\nanisotropy field. Due to this dynamic resonance, there develops a s trong coupling between the\nsample and the electric circuit, that is between the sample spin and th e resonator feedback\nfield, which realizes an ultrafast spin reversal. This procedure can b e implemented for single\nnanomolecules or nanoclusters as well as for their assemblies. The p rocess is illustrated by\nnumerically solving the spin equations of motion.\nIn order to grasp the typical values of the empirical parameters, let us adduce several\nexamples. Thus the typical parameters of Co, Fe, and Ni nanoclus ters are as follows. A single\ncluster, of volume around V∼10−20cm3, can contain about N∼103−104atoms, so that\nthe total cluster spin can be S∼103−104. The blocking temperature is TB∼(10−100) K.\nWith the magnetic field B0∼1 T, the Zeeman frequency is ω0∼1011s−1. The feedback rate\nis of order γ0∼(1010−1011) s−1. The typical anisotropy parameters are D//planckover2pi1∼107s−1and\nE//planckover2pi1∼106s−1orωD∼1010−1011s−1andωE∼109−1010s−1. Hence the dimensionless\nanisotropy parameter can be A∼0.1−1.\nThe magnetic nanomolecule, named Fe 8, possesses the spin S= 10, blocking temperature\nTB≈1 K, the molecule volume V∼10−20cm3, the anisotropy parameters D/kB= 0.27.5\nK andE/kB= 0.046 K, or D//planckover2pi1∼4×1010s−1andE//planckover2pi1∼1010s−1. Thus, the anisotropy\nfrequencies are ωD∼4×1011s−1andωE∼1011s−1. Then the dimensionless anisotropy\n12parameter is A∼1−4.\nThe magnetic nanomolecule Mn 12also has the spin S= 10 and the blocking temperature\n3.3 K. The spin polarization can be kept frozen for very long times depe nding on temperature\nand defined by the Arrhenius law. For example, at T= 3 K, the spin is frozen for one hour and\nat 2 K, for 2 months. The molecule radius is around 10−7cm and the volume, V∼10−20cm3.\nThe Zeeman frequency, for B0= 1 T, is ω0∼1011s−1. The feedback rate is γ0∼108s−1. The\nanisotropy parameter D/kB≈0.6 K and D//planckover2pi1∼1011s−1, while the value of Eis negligible.\nThe anisotropy frequency is ωD∼1012s−1. Therefore the dimensionless anisotropy parameter\nisA∼10.\nThese values of the parameters have been kept in mind in our numeric al calculations. The\nreversal time is trev≈γ/(γ0ω0s0). Forγ≈γ0andω∼1011s−1, the reversal time is of order\ntrev∼10−11s.\n13References\n[1] Barbara B, Thomas L, Lionti F, Chiorescu I and Sulpice A 1999 J. Magn. Magn. Mater.\n200167\n[2] Caneschi A, Gatteschi D, Sangregorio C, Sessoli R, Sorace L, Cornia A, Novak M A,\nPaulsen C W and Wernsdorfer W 1999 J. Magn. Magn. Mater. 200, 182\n[3] Yukalov V I 2002 Laser Phys. 12, 1089\n[4] Yukalov V I and Yukalova E P 2004 Phys. Part. Nucl. 35348\n[5] Yukalov V I and Yukalova E P 2005 Eur. Phys. Lett. 70306\n[6] Yukalov V I and Yukalova E P 2005 Laser Phys. Lett. 2302\n[7] Friedman J R and Sarachik M P 2010 Annu. Rev. Condens. Matter Phys. 1109\n[8] Miller J S 2014 Mater. Today 17224\n[9] Craig G A and Murrie M 2015 Chem. Soc. Rev. 442135\n[10] Liddle S T and van Slageren J 2015 Chem. Soc. Rev. 446655\n[11] Rana B, Mondal A K, Bandyopadhyay A and Barman A 2021 Nanotechnology 33062007\n[12] Kodama R H 1999 J. Magn. Magn. Mater. 200359\n[13] Hadjipanays G C 1999 J. Magn. Magn. Mater. 200373\n[14] Wernsdorfer W 2001 Adv. Chem. Phys. 11899\n[15] Yukalov V I and Yukalova E P 2011 Laser Phys. Lett. 8804\n[16] Kudr J, Haddad Y, Richtera L, Heger Z, Cernak M, Adam V and Zit ka O 2017 Nanoma-\nterials7243\n[17] Griesmaier A 2007 J. Phys. B 40R91\n[18] Baranov M A 2008 Phys. Rep. 46471\n[19] Baranov M A, Dalmonte M, Pupillo G and Zoller P 2012 Chem. Rev. 1125012\n[20] Stamper-Kurn D M and Ueda M 2013 Rev. Mod. Phys. 851191\n[21] Gadway B and Yan B 2016 J. Phys. B 49152002\n[22] Yukalov V I and Yukalova E P 2018 Eur. Phys. J. D 72190\n[23] Yukalov V I 2018 Laser Phys. 28053001\n[24] Yukalov V I, Henner V K and Belozerova T S 2016 Laser Phys. Lett. 13016001\n14[25] Yukalov V I, Henner, V K, Belozerova T S and Yukalova E P 2016 J. Supercond. Nov.\nMagn.29721\n[26] Birman J L, Nazmitdinov R G and Yukalov V I 2013 Phys. Rep. 5261\n[27] Schwartz D A, Norberg N S, Nguyen Q P, Parker J M and Gamelin D R 2003J. Am.\nChem. Soc. 12513205\n[28] Koole R, Mulder W J M, van Schooneveld M M, Strijkers G J, Meijerin k A and Nicolay\nK 2009Wiley Interdiscip. Rev. Nanomed. Nanobiotechnol. 1475\n[29] Mahajan K D, Fan Q, Dorcena J, Ruan G and Winter J O 2013 Biotechnol. J. 81424\n[30] Tufani A, Qureshi A and Niazi J H 2021 Mater. Sci. Eng. C 118111545\n[31] Yukalov V I 1996 Phys. Rev. B 539232\n[32] Yukalov V I 2005 Phys. Rev. B 71184432\n[33] Yukalov V I, Henner V K and Kharebov P V 2008 Phys. Rev. B 77134427\n[34] Yukalov V I and Yukalova E P 2012 J. Appl. Phys. 111023911\n[35] Kharebov P V, Henner V K and Yukalov V I 2013 J. Appl. Phys. 113043902\n[36] Yukalov V I and Yukalova E P 2022 Laser Phys. Lett. 19046001\n[37] Bogolubov N N and Mitropolsky Y A 1961 Asymptotic Methods in the Theory of Nonlnear\nEquations (New York: Gordon and Breach)\n[38] Freidlin M I and Wentzell A D 1984 Random Perturbations of Dynamical Systems (New\nYork: Springer)\n[39] Gardiner C W 1983 Handbook of Stochastic Methods (Berlin: Springer)\n[40] Yukalov V I and Yukalova E P 2000 Phys. Part. Nucl. 31561\n[41] Yukalov V I and Yukalova E P 2018 Phys. Rev. B 98144438\n[42] Lochak P and Meunier C 1988 Multiphase Averaging for Classical Systems (New York:\nSpringer)\n[43] Yukalov V I and Yukalova E P 2002 Phys. Rev. Lett. 88257601\n15"    },    {        "title": "1804.07080v2.Damping_of_magnetization_dynamics_by_phonon_pumping.pdf",        "content": "Damping of magnetization dynamics by phonon pumping\nSimon Streib,1Hedyeh Keshtgar,2and Gerrit E. W. Bauer1, 3\n1Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n2Institute for Advanced Studies in Basic Science, 45195 Zanjan, Iran\n3Institute for Materials Research & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan\n(Dated: July 11, 2018)\nWe theoretically investigate pumping of phonons by the dynamics of a magnetic film into a non-\nmagnetic contact. The enhanced damping due to the loss of energy and angular momentum shows\ninterferencepatternsasafunctionofresonancefrequencyandmagneticfilmthicknessthatcannotbe\ndescribed by viscous (“Gilbert”) damping. The phonon pumping depends on magnetization direction\nas well as geometrical and material parameters and is observable, e.g., in thin films of yttrium iron\ngarnet on a thick dielectric substrate.\nThe dynamics of ferromagnetic heterostructures is at\nthe root of devices for information and communication\ntechnologies [1–5]. When a normal metal contact is at-\ntached to a ferromagnet, the magnetization dynamics\ndrives a spin current through the interface. This effect\nis known as spin pumping and can strongly enhance the\n(Gilbert) viscous damping in ultra-thin magnetic films\n[6–8]. Spin pumping and its (Onsager) reciprocal, the\nspin transfer torque [9, 10], are crucial in spintronics, as\nthey allow electric control and detection of magnetiza-\ntion dynamics. When a magnet is connected to a non-\nmagnetic insulator instead of a metal, angular momen-\ntum cannot leave the magnet in the form of electronic or\nmagnonic spin currents, but they can do so in the form\nof phonons. Half a century ago it was reported [11, 12]\nand explained [13–16] that magnetization dynamics can\ngenerate phonons by magnetostriction. More recently,\nthe inverse effect of magnetization dynamics excited by\nsurface acoustic waves (SAWs) has been studied [17–20]\nand found to generate spin currents in proximity normal\nmetals [21, 22]. The emission and detection of SAWs was\ncombined in one and the same device [23, 24], and adia-\nbatic transformation between magnons and phonons was\nobserved in inhomogeneous magnetic fields [25]. The an-\ngular momentum of phonons [26, 27] has recently come\ninto focus again in the context of the Einstein-de Haas\neffect [28] and spin-phonon interactions in general [29].\nThe interpretation of the phonon angular momentum in\ntermsoforbitalandspincontributions[29]hasbeenchal-\nlenged [30], a discussion that bears similarities with the\ninterpretation of the photon angular momentum [31]. In\nour opinion this distinction is rather semantic since not\nrequired to arrive at concrete results. A recent quantum\ntheory of the dynamics of a magnetic impurity [32] pre-\ndicts a broadening of the electron spin resonance and a\nrenormalized g-factor by coupling to an elastic contin-\nuum via the spin-orbit interaction, which appears to be\nrelated to the enhanced damping and effective gyromag-\nnetic ratio discussed here.\nA phonon current generated by magnetization dynam-\nics generates damping by carrying away angular momen-\ntum and energy from the ferromagnet. While the phonon\nphonon sinkzmagnet\nnon-magnet0\nphononsmHFigure 1. Magnetic film (shaded) with magnetization mat-\ntached to a semi-infinite elastic material, which serves as an\nideal phonon sink.\ncontribution to the bulk Gilbert damping has been stud-\nied theoretically [33–38], the damping enhancement by\ninterfaces to non-magnetic substrates or overlayers has\nto our knowledge not been addressed before. Here we\npresent a theory of the coupled lattice and magnetiza-\ntion dynamics of a ferromagnetic film attached to a half-\ninfinite non-magnet, which serves as an ideal phonon\nsink. We predict, for instance, significantly enhanced\ndamping when an yttrium iron garnet (YIG) film is\ngrown on a thick gadolinium gallium garnet (GGG) sub-\nstrate.\nWe consider an easy-axis magnetic film with static ex-\nternal magnetic field and equilibrium magnetization ei-\nther normal (see Fig. 1) or parallel to the plane. The\nmagnet is connected to a semi-infinite elastic material.\nMagnetization and lattice are coupled by the magne-\ntocrystalline anisotropy and the magnetoelastic interac-\ntion, giving rise to coupled field equations of motion in\nthe magnet [39–42]. By matching these with the lattice\ndynamics in the non-magnet by proper boundary con-\nditions, we predict the dynamics of the heterostructure\nas a function of geometrical and constitutive parameters.\nWe find that magnetization dynamics induced, e.g., by\nferromagnetic resonance (FMR) excites the lattice in the\nattachednon-magnet. Inanalogywiththeelectroniccase\nwecallthiseffect“phononpumping” thataffectsthemag-\nnetization dynamics. We consider only equilibrium mag-\nnetizations that are normal or parallel to the interface,\nin which the pumped phonons are pure shear waves that\ncarry angular momentum. We note that for general mag-arXiv:1804.07080v2  [cond-mat.mes-hall]  16 Jul 20182\nnetization directions both shear and pressure waves are\nemitted, however.\nWe consider a magnetic film (metallic or insulating)\nthat extends from z=\u0000dtoz= 0. It is subject to suffi-\nciently high magnetic fields H0such that magnetization\nis uniform, i.e. M(r) =M:For in-plane magnetizations,\nH0> Ms, where the magnetization Msgoverns the de-\nmagnetizing field [43]. The energy of the magnet|non-\nmagnet bilayer can be written\nE=ET+Eel+EZ+ED+E0\nK+Eme;(1)\nwhich are integrals over the energy densities \"X(r). The\ndifferent contributions are explained in the following.\nThe kinetic energy density of the elastic motion reads\n\"T(r) =(\n1\n2\u001a_u2(r); z> 0\n1\n2~\u001a_u2(r);\u0000d<z< 0; (2)\nand the elastic energy density [44]\n\"el=(\n1\n2\u0015(P\n\u000bX\u000b\u000b(r))2+\u0016P\n\u000b\fX2\n\u000b\f(r); z> 0\n1\n2~\u0015(P\n\u000bX\u000b\u000b(r))2+ ~\u0016P\n\u000b\fX2\n\u000b\f(r);\u0000d<z< 0;\n(3)\nwhere\u000b;\f2fx;y;zg,\u0015and\u0016are the Lamé parameters\nand\u001athe mass density of the non-magnet. The tilded\nparameters are those of the magnet. The strain tensor\nX\u000b\fis defined in terms of the displacement fields u\u000b(r),\nX\u000b\f(r) =1\n2\u0012@u\u000b(r)\n@r\f+@u\f(r)\n@r\u000b\u0013\n: (4)\nEZ=\u0000\u00160VM\u0001Hextis the Zeeman energy for Hext=\nH0+h(t), where h(t)is time-dependent. ED=\n1\n2\u00160VMTDMis the magnetostatic energy with shape-\ndependent demagnetization tensor DandVthe volume\nof the magnet. For a thin film with zaxis along the sur-\nface normal n0,Dzz= 1while the other components van-\nish.E0\nK=K1V(m\u0002n0)2is the uniaxial magnetocrys-\ntalline anisotropy in the absence of lattice deformations,\nwhere m=M=MsandK1is the anisotropy constant.\nThe magnetoelastic energy Emecouples the magnetiza-\ntion to the lattice, as discussed in the following.\nThe magnetoelastic energy density can be expanded as\n\"me(r) =1\nM2sX\n\u000b;\fM\u000b(r)M\f(r)\n\u0002[B\u000b\fX\u000b\f(r) +C\u000b\f\n\u000b\f(r)]:(5)\nFor an isotropic medium the magnetoelastic constants\nB\u000b\fread [45]\nB\u000b\f=\u000e\u000b\fBk+ (1\u0000\u000e\u000b\f)B?: (6)\nRotational deformations as expressed by the tensor\n\n\u000b\f(r) =1\n2\u0012@u\u000b(r)\n@r\f\u0000@u\f(r)\n@r\u000b\u0013\n(7)are often disregarded [39–42, 46], but lead to a position\ndependence of the easy axis n(r)from the equilibrium\nvalue n0=ezand an anisotropy energy density [29, 47,\n48]\n\"K(r) =K1\nM2s[M\u0002n(r)]2: (8)\nTo first order in the small deformation\n\u000en(r) =n(r)\u0000n0=0\n@\nxz(r)\n\nyz(r)\n01\nA; (9)\n\"K(r) =\"0\nK+ 2K1(n0\u0000mzm)\u0001\u000en(r):(10)\nFrom \n\u000b\f=\u0000\n\f\u000bit follows that (for non-chiral crystal\nstructures) C\u000b\f=\u0000C\f\u000b. For the uniaxial anisotropy\nconsidered here Cxz=Cyz=\u0000K1. The magnetoelastic\ncoupling due to the magnetocrystalline anisotropy thus\ncontributes [47]\n\"K\nme(r) =\u00002K1\nM2sMz(r) [Mx(r)\nxz(r) +My(r)\nyz(r)]:\n(11)\nPureYIGismagneticallyverysoft, sothemagnetoelastic\nconstants are much larger than the anisotropy constant\n[49, 50]\nBk= 3:48\u0002105Jm\u00003; B?= 6:96\u0002105Jm\u00003;\nK1=\u00006:10\u0002102J m\u00003; (12)\nbut this ratio can be very different for other magnets.\nWe find below that for the Kittel mode dynamics both\ncoupling processes cannot be distinguished, even though\nthey can characteristically affect the magnon-phonon\ncoupling for finite wave numbers.\nThe magnetization dynamics within the magnetic film\nis described by the Landau-Lifshitz-Gilbert (LLG) equa-\ntion [51, 52]\n_m=\u0000\r\u00160m\u0002He\u000b+\u001c(\u000b)\nm; (13)\nwhere\u0000\ris the gyromagnetic ratio, the effective mag-\nnetic field which includes the magnetoelastic coupling\nHe\u000b=\u0000rmE=(\u00160VMs); (14)\nand the Gilbert damping torque [52]\n\u001c(\u000b)\nm=\u000bm\u0002_m: (15)\nThe equation of motion of the elastic continuum reads\n[44]\nu(r;t) =c2\nt4u(r;t) + (c2\nl\u0000c2\nt)r[r\u0001u(r;t)];(16)\nwith longitudinal and transverse sound velocities\ncl=s\n\u0015+ 2\u0016\n\u001a; ct=r\u0016\n\u001a; (17)3\nwhere elastic constants and mass density of non-magnet\nand magnet can differ.\nA uniform precession of the magnetization interacts\nwith the lattice deformation at the surfaces of the mag-\nnetic film [13, 14] and at defects in the bulk. The present\ntheorythenholdswhenthethicknessofthemagneticfilm\nd\u001cp\nA, whereAis the cross section area. The Kittel\nmode induces lattice distortions that are uniform in the\nfilm planeu\u000b(r) =u\u000b(z)[14]. The elastic energy density\nis then affected by shear waves only:\n\"el(z) =(\n\u0016\n2\u0000\nu02\nx(z) +u02\ny(z)\u0001\n; z> 0\n~\u0016\n2\u0000\nu02\nx(z) +u02\ny(z)\u0001\n;\u0000d<z< 0;(18)\nwhereu0\n\u000b(z) =@u\u000b(z)=@z. The magnetic field\nHext=\u0000hx(t); hy(t); H 0\u0001Twith monochromatic drive\nhx;y(t) = Re\u0000\nhx;ye\u0000i!t\u0001\nand static component H0along\nthezaxis. At the FMR frequency !?=!H+!Awith\n!H=\r\u00160H0and!A=\r(2K1=Ms\u0000\r\u0016Ms). The equi-\nlibrium magnetization is perpendicular for !?>0. The\nmagnetoelastic energy derived above then simplifies to\nEz\nme=(B?\u0000K1)A\nMsX\n\u000b=x;yM\u000b[u\u000b(0)\u0000u\u000b(\u0000d)];(19)\nwhichresultsinsurfaceshearforces F\u0006(0) =\u0000F\u0006(\u0000d) =\n\u0000(B?\u0000K1)Am\u0006, withF\u0006=Fx\u0006iFy. These forces\ngenerate a stress or transverse momentum current in the\nzdirection (see Supplemental Material)\nj\u0006(z) =\u0000\u0016(z)u0\n\u0006(z); (20)\nwith\u0016(z) =\u0016forz >0and\u0016(z) = ~\u0016for\u0000d < z < 0,\nandu\u0006=ux\u0006iuy, which is related to the transverse mo-\nmentump\u0006(z) =\u001a( _ux(z)\u0006i_uy(z))by Newton’s equa-\ntion:\n_p\u0006(z) =\u0000@\n@zj\u0006(z): (21)\nThe boundary conditions require momentum conserva-\ntion and elastic continuity at the interfaces,\nj\u0006(\u0000d) = (B?\u0000K1)m\u0006;(22)\nj\u0006(0+)\u0000j\u0006(0\u0000) =\u0000(B?\u0000K1)m\u0006;(23)\nu\u0006(0+) =u\u0006(0\u0000): (24)\nWe treat the magnetoelastic coupling as a small pertur-\nbation and therefore we approximate the magnetization\nm\u0006entering the above boundary conditions as indepen-\ndent of the lattice displacement u\u0006. The loss of angular\nmomentum (see Supplemental Material) affects the mag-\nnetization dynamics in the LLG equation in the form of a\ntorque, which we derive from the magnetoelastic energy\n(19),\n_m\u0006jme=\u0006i!c\nd[u\u0006(0)\u0000u\u0006(\u0000d)]\n=\u0006i!cRe(v)m\u0006\u0007!cIm(v)m\u0006;(25)where!c=\r(B?\u0000K1)=Ms(for YIG:!c= 8:76\u0002\n1011s\u00001) andv= [u\u0006(0)\u0000u\u0006(\u0000d)]=(dm\u0006). We can\ndistinguish an effective field\nHme=!c\n\r\u00160Re(v)ez; (26)\nand a damping coefficient\n\u000b(?)\nme=\u0000!c\n!Imv: (27)\nThe latter can be compared with the Gilbert damping\nconstant\u000bthat enters the linearized equation of motion\nas\n_m\u0006j\u000b=\u0006i\u000b_m\u0006=\u0006\u000b!m\u0006: (28)\nWith the ansatz\nu\u0006(z;t) =(\nC\u0006eikz\u0000i!t; z> 0\nD\u0006ei~kz\u0000i!t+E\u0006e\u0000i~kz\u0000i!t;\u0000d<z< 0;\n(29)\nwe obtain\nv=Ms!c\n!\rd~\u001a~ct2h\ncos(~kd)\u00001i\n\u0000i\u001act\n~\u001a~ctsin(~kd)\nsin(~kd) +i\u001act\n~\u001a~ctcos(~kd);(30)\nand the damping coefficient for perpendicular magneti-\nzation\n\u000b(?)\nme=\u0010!c\n!\u00112Ms\n\rd~\u001a~ct\u001act\n~\u001a~ct4 sin4\u0010~kd\n2\u0011\nsin2(~kd) +\u0010\n\u001act\n~\u001a~ct\u00112\ncos2(~kd);\n(31)\nwhere!=ctk= ~ct~k. The oscillatory behavior of the\ndamping\u000b(?)\nmecomes from the interference of the elastic\nwaves that are generated at the top and bottom surfaces\nof the magnetic film. When they constructively (destruc-\ntively) interfere at the FMR frequency, the damping is\nenhanced(suppressed), becausethemagnon-phononcou-\npling and phonon emission are large (small).\nWhen\u001act\u001c~\u001a~ct(soft substrate) or when acoustic\nimpedances are matched ( \u001act= ~\u001a~ct), damping at the\nresonance ~kd= (2n+ 1)\u0019withn2N0[14] simplifies to\n\u000b(?)\nme!\u0010!c\n!\u001124Ms\n\rd\u001act: (32)\nWhen\u001act\u001d~\u001a~ct(hard substrate), the magnet is acousti-\ncally pinned at the interface and the acoustic resonances\nare at ~kd= (2n+ 1)\u0019=2[14] with\n\u000b(?)\nme!\u0010!c\n!\u00112Ms\n\rd~\u001a~ct\u001act\n~\u001a~ct: (33)\nIn contrast to Gilbert damping, \u000b(?)\nmedepends on the\nfrequency and vanishes in the limits !!0and!!1.\nTherefore, it does not obey the LLG phenomenology4\nand in the non-linear regime does not simply enhance\n\u000bin Eq. (15). The magnetization damping \u000b0in bulk\nmagnetic insulators, on the other hand, is usually of\nthe Gilbert type. It is caused by phonons as well, but\nnot necessarily the magnetoelastic coupling. A theory\nof Gilbert damping [38] assumes a bottleneck process\nby sound wave attenuation, which appears realistic for\nmagnets with high acoustic quality such as YIG. In the\npresent phonon pumping model, energy and angular mo-\nmentum is lost by the emission of sound waves into an\nattached perfect phononwaveguide, so thepumpingpro-\ncess dominates. Such a scenario could also dominate the\ndamping in magnets in which the magnetic quality is rel-\natively higher than the acoustic one.\nWhen the field is rotated to Hext =\u0000\nhx(t); H 0; hz(t)\u0001T, the equilibrium magnetization\nis in the in-plane ydirection and the magnetoelastic\nenergy couples only to the strain uy,\nEy\nme=(B?\u0000K1)A\nMsMz[uy(0)\u0000uy(\u0000d)]:(34)\nThe FMR frequency for in-plane magnetization !k=\n!Hp\n1\u0000!A=!Hwith!A< !H. The magnetoelastic\ncoupling generates again only transverse sound waves.\nThe linearized LLG equation including the phononic\ntorques reads now\n_mx= (!H+!me)mz\u0000\r\u00160hz\u0000!Amz\n+ (\u000b+\u000bme) _mz; (35)\n_mz=\u0000!Hmx+\r\u00160hx\u0000\u000b_mx; (36)\nwhere\u000bmeis given by Eq. (27) and !me=\r\u00160Hmewith\neffective field Hme=Hme\u0001ezgiven by Eq. (26). Both\nHmeand\u000bmecontribute only to _mx. The phonon pump-\ning is always less efficient for the in-plane configuration:\n\u000b(k)\nme=1\n1 + (!k=!H)2\u000b(?)\nme: (37)\nAs an example, we insert parameters for a thin YIG\nfilm on a semi-infinite gadolinium gallium garnet (GGG)\nsubstrate at room temperature. We have chosen YIG\nbecause of its low intrinsic damping and high quality in-\nterface to the GGG substrate. Substantially larger mag-\nnetoelastic coupling in other materials should be offset\nagainst generally larger bulk damping. For GGG, \u001a=\n7:07\u0002103kg m\u00003,cl= 6411 m s\u00001, andct= 3568 m s\u00001\n[53]. For YIG, Ms= 1:4\u0002105A m\u00001,\r= 1:76\u0002\n1011s\u00001T\u00001,~\u001a= 5170 kg m\u00003,~cl= 7209 m s\u00001,~ct=\n3843 m s\u00001, and!c= 8:76\u00021011s\u00001[49, 50]. The ratio\nof the acoustic impedances ~\u001a~ct=\u001act= 0:79. The damp-\ning enhancement \u000b(?)\nmeis shown in Fig. 2 over a range\nof FMR frequencies and film thicknesses. The FMR fre-\nquencies!?=!H+!Aand!k=!Hp\n1\u0000!A=!Hfor\nthe normal and in-plane configurations are tunable by\nthe static magnetic field component H0via!H=\r\u00160H0.\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n<10−510−410−3\nα(⊥)\nmeFigure 2. Damping enhancement \u000b(?)\nmeby phonon pumping\nin a YIG film on a semi-infinite GGG substrate, as given by\nEq. (31).\nThe damping enhancement peaks at acoustic resonance\nfrequencies \u0017\u0019n~ct=(2d). The counter-intuitive result\nthat the damping increases for thicker films can be un-\nderstood by the competition between the magnetoelastic\neffect that increases with thickness at the resonances and\nwins against the increase in total magnetization. How-\never, with increasing thickness the resonance frequencies\ndecrease below a minimum value at which FMR can be\nexcited. For a fixed FMR frequency \u000bme!0ford!1:\nFor comparison, the Gilbert damping in nanometer thin\nYIG films is of the order \u000b\u001810\u00004[54] which is larger\nthan corresponding values for single crystals. We con-\ncludethattheenhanceddampingisatleastpartlycaused\nby interaction with the substrate and not by a reduced\ncrystal quality.\nThe resonances in the figures are very broad because\nthe\u001act\u0019~\u001a~ctimplies very strong coupling of the discrete\nphonons in the thin magnetic layer with the phonon con-\ntinuum in the substrate. When an acoustic mismatch\nis introduced, the broad peaks increasingly sharpen, re-\nflecting the increased lifetime of the magnon polaron res-\nonances in the magnet.\nThe frequency dependent effective magnetic field H(?)\nme\nis shown in Fig. 3. The frequency dependence of H(?)\nme\nimplies a weak frequency dependence of the effective gy-\nromagnetic ratio\n\r(?)\ne\u000b=\r \n1 +\r\u00160H(?)\nme\n!!\n: (38)\nIn the limit of vanishing film thickness, \u00160H(?)\nme!\n\u0000(B?\u0000K1)2=(Ms~\u0016).\nWe assumed that the non-magnet is an ideal phonon\nsink, which means that injected sound waves do not\nreturn. In the opposite limit in which the phonons\ncannot escape, i.e. when the substrate is a thin film\nwith high acoustic quality, the additional damping van-\nishes. This can be interpreted in terms of a phonon5\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n−60−45−30−15015304560\nµ0H(⊥)\nme[µT]\nFigure3. Effectivefield H(?)\nmegeneratedbythemagnetoelastic\ngeneration of phonons in a YIG film on a semi-infinite GGG\nsubstrate, as given by Eq. (26).\naccumulation that, when allowed to thermalize, gener-\nates a phonon chemical potential and/or non-equilibrium\ntemperature. The non-equilibrium thermodynamics of\nphonons in magnetic nanostructures is subject of our on-\ngoing research.\nThe damping enhancement by phonons may be com-\npared with that from electronic spin pumping [6–8],\n\u000bsp=\r~\n4\u0019dMsh\ne2g; (39)\nwhich is inversely proportional to the thickness dof the\nmagnetic film and does not depend on the FMR fre-\nquency, i.e. obeys the LLG phenomenology. Here, g\nis the spin mixing conductance per unit area at the in-\nterface. While phonons can be pumped into any elastic\nmaterial, spin pumping requires an electrically conduct-\ning contact. With a typical value of hg=e2\u00181018m\u00002\nthe damping enhancement of YIG on platinum is \u000bsp\u0018\n10\u00002nm=d:The physics is quite different, however, since\n\u000bsp;in contrast to \u000bme, does not require coherence over\nthe interface.\nIn conclusion, the pumping of phonons by magnetic\nanisotropy and magnetostriction causes a frequency-\ndependent contributions to the damping and effective\nfield of the magnetization dynamics. The generation\nof phonons by magnetic precession can cause significant\ndamping in a magnetic film when grown on an insulating,\nnon-magnetic substrate and partly explains the increased\ndampinginvariablyobservedforthinnerfilms. Theimpli-\ncations of further reaching ramifications, such as phonon-\ninduced dynamic exchange interactions, phonon accumu-\nlations and phonon spin Seebeck effect require additional\nresearch.\nThis work is financially supported by the Nederlandse\nOrganisatievoorWetenschappelijkOnderzoek(NWO)as\nwell as by the Grant-in-Aid for Scientific Research on In-\nnovative Area, ”Nano Spin Conversion Science” (Grant\nNo. 26103006). H. K. acknowledges support from theIran Science Elites Federation. We acknowledge use-\nful discussions with Yaroslav Blanter, Rembert Duine,\nAkashdeep Kamra, Eiji Saitoh, and Sanchar Sharma.\n[1] S. Bader and S. Parkin, Annu. Rev. Condens. Matter\nPhys. 1, 71 (2010).\n[2] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J.\nPhys. D: Appl. Phys. 43, 264001 (2010).\n[3] F. Pulizzi, Nat. Mater. 11, 367 (2012).\n[4] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. Hillebrands, Nat. Phys. 11, 453 (2015).\n[5] S. A. Nikitov, D. V. Kalyabin, I. V. Lisenkov, A. Slavin,\nY. N. Barabanenkov, S. A. Osokin, A. V. Sadovnikov,\nE. N. Beginin, M. A. Morozova, Y. A. Filimonov, Y. V.\nKhivintsev, S. L. Vysotsky, V. K. Sakharov, and E. S.\nPavlov, Phys. Usp. 58, 1002 (2015).\n[6] Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[7] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n[8] A. Kapelrud and A. Brataas, Phys. Rev. Lett. 111,\n097602 (2013).\n[9] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[10] J. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n[11] H. Bömmel and K. Dransfeld, Phys. Rev. Lett. 3, 83\n(1959).\n[12] M. Pomerantz, Phys. Rev. Lett. 7, 312 (1961).\n[13] R. L. Comstock and R. C. LeCraw, J. Appl. Phys. 34,\n3022 (1963).\n[14] M. H. Seavey, Proc. IEEE 53, 1387 (1965).\n[15] T. Kobayashi, R. Barker, and A. Yelon, IEEE Trans.\nMagn. 7, 755 (1971).\n[16] T. Kobayashi, R. Barker, and A. Yelon, IEEE Trans.\nMagn. 8, 382 (1972).\n[17] M. Weiler, L. Dreher, C. Heeg, H. Huebl, R. Gross, M. S.\nBrandt, and S. T. B. Goennenwein, Phys. Rev. Lett.\n106, 117601 (2011).\n[18] L. Dreher, M. Weiler, M. Pernpeintner, H. Huebl,\nR. Gross, M. S. Brandt, and S. T. B. Goennenwein,\nPhys. Rev. B 86, 134415 (2012).\n[19] P. G. Gowtham, T. Moriyama, D. C. Ralph, and R. A.\nBuhrman, J. Appl. Phys. 118, 233910 (2015).\n[20] X. Li, D. Labanowski, S. Salahuddin, and C. S. Lynch,\nJ. Appl. Phys. 122, 043904 (2017).\n[21] K. Uchida, T. An, Y. Kajiwara, M. Toda, and E. Saitoh,\nAppl. Phys. Lett. 99, 212501 (2011).\n[22] M. Weiler, H. Huebl, F. S. Goerg, F. D. Czeschka,\nR. Gross, and S. T. B. Goennenwein, Phys. Rev. Lett.\n108, 176601 (2012).\n[23] S. Bhuktare, A. Bose, H. Singh, and A. A. Tulapurkar,\nSci. Rep. 7, 840 (2017).\n[24] S. Bhuktare, H. Singh, A. Bose, and A. A. Tulapurkar,\n(2017), arXiv:1712.02711.\n[25] J.Holanda, D.S.Maior, A.Azevedo, andS.M.Rezende,\nNat. Phys. 14, 500 (2018).\n[26] A. T. Levine, Nuovo Cimento 26, 190 (1962).\n[27] A. G. McLellan, J. Phys. C: Solid State Phys. 21, 1177\n(1988).\n[28] L. Zhang and Q. Niu, Phys. Rev. Lett. 112, 085503\n(2014).6\n[29] D. A. Garanin and E. M. Chudnovsky, Phys. Rev. B 92,\n024421 (2015).\n[30] S. C. Tiwari, (2017), arXiv:1708.07407.\n[31] E. Leader, Phys. Lett. B 756, 303 (2016).\n[32] J. H. Mentink, M. I. Katsnelson, and M. Lemeshko,\n(2018), arXiv:1802.01638.\n[33] T. Kobayashi, R. C. Barker, and A. Yelon, J. Phys.\nColloques 32, C1 (1971).\n[34] T. Kobayashi, R. C. Barker, J. L. Bleustein, and\nA. Yelon, Phys. Rev. B 7, 3273 (1973).\n[35] T. Kobayashi, R. C. Barker, and A. Yelon, Phys. Rev.\nB7, 3286 (1973).\n[36] E. Rossi, O. G. Heinonen, and A. H. MacDonald, Phys.\nRev. B 72, 174412 (2005).\n[37] A. Widom, C. Vittoria, and S. D. Yoon, J. Appl. Phys.\n108, 073924 (2010).\n[38] C. Vittoria, S. D. Yoon, and A. Widom, Phys. Rev. B\n81, 014412 (2010).\n[39] E. Abrahams and C. Kittel, Phys. Rev. 88, 1200 (1952).\n[40] C. Kittel and E. Abrahams, Rev. Mod. Phys. 25, 233\n(1953).\n[41] C. Kittel, Phys. Rev. 110, 836 (1958).\n[42] M. I. Kaganov and V. M. Tsukernik, Sov. Phys. JETP 9,\n151 (1959).[43] H. Knüpfer and C. B. Muratov, J. Nonlinear Sci. 21, 921\n(2011).\n[44] L. D. Landau and E. M. Lifshitz, Theory of Elasticity ,\n2nd ed., Course in Theoretical Physics Vol. 7 (Pergamon\nPress, Oxford, 1970).\n[45] A. Rückriegel, P. Kopietz, D. A. Bozhko, A. A. Serga,\nand B. Hillebrands, Phys. Rev. B 89, 184413 (2014).\n[46] C. Kittel, Rev. Mod. Phys. 21, 541 (1949).\n[47] D. A. Garanin and E. M. Chudnovsky, Phys. Rev. B 56,\n11102 (1997).\n[48] R. Jaafar, E. M. Chudnovsky, and D. A. Garanin, Phys.\nRev. B 79, 104410 (2009).\n[49] A. G. Gurevich and G. A. Melkov, Magnetization Oscil-\nlations and Waves (CRC, Boca Raton, FL, 1996).\n[50] P. Hansen, J. Appl. Phys. 45, 3638 (1974).\n[51] L. D. Landau and E. Lifshitz, Phys. Z. Sowjet. 8, 153\n(1935).\n[52] T. L. Gilbert, IEEE Trans. Magn. 40, 3443 (2004).\n[53] Z. Kleszczewski and J. Bodzenta, Phys. Status Solidi B\n146, 467 (1988).\n[54] H. Chang, P. Li, W. Zhang, T. Liu, A. Hoffmann,\nL. Deng, and M. Wu, IEEE Magn. Lett. 5, 1 (2014).\n[55] E. B. Magrab, Vibrations of Elastic Systems (Springer,\nDordrecht, 2012).1\nSupplemental Material\nInSecs.IandIIofthisSupplementwederivetheangu-\nlar and transverse momentum of transverse elastic waves\nand the corresponding momentum currents. In Sec. III\nwe give results for a magnet sandwiched between two\nnon-magnets and in Sec. IV we present a theory for the\nmagnetizationdampingenhancementfrompumpingflex-\nural waves into a thin beam.\nI. ANGULAR MOMENTUM\nThe magnetization M=Msmof an uniformly mag-\nnetized magnet with saturation magnetization Msand\nvolumeVis associated with the angular momentum\nS=\u0000MsV\n\rm; (S1)\nwhere\u0000\ris the gyromagnetic ratio. The angular mo-\nmentum density relative to the origin of an elastic body\nwith displacement field u(r;t)and constant mass density\n\u001areads\nl(t) =\u001a(r+u(r;t))\u0002_u(r;t): (S2)\nWith uniaxial anisotropy axis along z, FMR generates\nthe transverse elastic wave\nu(z;t) = Re2\n40\n@ux\nuy\n01\nAeikz\u0000i!t3\n5; (S3)\nwith dispersion relation !=ctk. Defining the time av-\nerage\nhf(t)i= lim\nT!11\nTZT\n0dtf(t); (S4)\nhli= (0;0;hlzi)can be expressed as\nhlzi=\u001ahux_uy\u0000_uxuyi\n=\u0000\u001a!\n4\u0010\nju+j2\u0000ju\u0000j2\u0011\n; (S5)\nwhereu\u0006=ux\u0006iuyand where we used\n\nRe(ae\u0000i!t)Re(be\u0000i!t)\u000b\n=1\n2Re (a\u0003b):(S6)\nThe non-magnet harbors a constant phonon angular mo-\nmentum density in the z-direction, which implies pres-\nence of a phonon angular momentum current Ahjz\nliat\nthe interface to the magnet with area Athat is ab-\nsorbed at the phonon sink. In our model the angular\nmomentum loss rate of the magnet by phonon pumping\nh_Szjmei=\u0000Ahjz\nliand\nD\n_Sz\f\f\f\nmeE\n=hfSz;Ez\nmegi=MsV\u000b(?)\nme!\n4\r\u0010\njm+j2\u0000jm\u0000j2\u0011\n(S7)wheref;gis the Poisson bracket and Ez\nmethe magnetoe-\nlastic energy Eq. (19) in the main text. From Eq. (31)\nand\nu\u0006=Cm\u0006; (S8)\nwith\nC=(B?\u0000K1)h\ncos(~kd)\u00001i\nik\u0016cos(~kd) +~k~\u0016sin(~kd); (S9)\nwe obtain the relation\nhjz\nli=cthlzi; (S10)\nwhichagreeswiththesimplephysicalpictureofanelastic\nwave carrying away its angular momentum density hlzi\nwith transverse sound velocity ct.\nII. TRANSVERSE MOMENTUM\nFor the transverse elastic wave (S3) in a magnet ex-\ntending from z=z0toz=z1(withz1> z0), the time\nderivative of the transverse momentum P\u0006=Px\u0006iPy\nreads\n_P\u0006=\u001aZ\nVd3ru\u0006(z;t)\n=\u0016A\u0002\nu0\n\u0006(z1;t)\u0000u0\n\u0006(z0;t)\u0003\n:(S11)\nThe change of momentum can be interpreted as a trans-\nverse momentum current density j\u0006(z0) =\u0000\u0016u0\n\u0006(z0)\nflowing into the magnet at z0and a current j\u0006(z1) =\n\u0000\u0016u0\n\u0006(z1)flowing out at z1. The momentum current\nis related to the transverse momentum density p\u0006(z) =\n\u001a_u\u0006(z)by\n_p\u0006(z) =\u0000@\n@zj\u0006(z); (S12)\nwhich confirms that\nj\u0006(z;t) =\u0000\u0016u0\n\u0006(z;t): (S13)\nThe instantaneous conservation of transverse momentum\nisaboundaryconditionsattheinterface. Itstimeaverage\nhj\u0006i= 0, but the associated angular momentum along z\nis finite, as shown above.\nIII. SANDWICHED MAGNET\nWhen a non-magnetic material is attached at both\nsides of the magnet and elastic waves leave the magnet\natz= 0andz=\u0000d, the boundary condition are\nj\u0006(\u0000d\u0000)\u0000j\u0006(\u0000d+) = (B?\u0000K1)m\u0006;(S14)\nj\u0006(0+)\u0000j\u0006(0\u0000) =\u0000(B?\u0000K1)m\u0006;(S15)\nu\u0006(0+) =u\u0006(0\u0000); (S16)\nu\u0006(\u0000d+) =u\u0006(\u0000d\u0000); (S17)2\nwithd\u0006=d\u00060+. Since the Hamiltonian is piece-wise\nconstant\nu\u0006(z;t) =8\n><\n>:C\u0006eikz\u0000i!t; z> 0\nD\u0006ei~kz\u0000i!t+E\u0006e\u0000i~kz\u0000i!t;\u0000d<z< 0\nF\u0006e\u0000ikz\u0000i!t: z< \u0000d;\n(S18)\nUsing the boundary conditions\nv=u\u0006(0)\u0000u\u0006(\u0000d)\ndm\u0006=Ms!c\n!\rd~\u001a~ct2\ni\u001act\n~\u001a~ct\u0000cot(~kd\n2);(S19)\nleading to the damping coefficient\n\u000b(?)\nme=\u0010!c\n!\u00112Ms\n\rd~\u001a~ct2\n\u001act\n~\u001a~ct+~\u001a~ct\n\u001actcot2\u0010~kd\n2\u0011:(S20)\nWhen ~\u001a~ct=\u001act,\n\u000b(?)\nme=\u0010!c\n!\u001122Ms\n\rd\u001actsin2 ~kd\n2!\n;(S21)\nwhich differs from the sin4\u0010\n~kd=2\u0011\ndependence obtained\nfor the magnet|non-magnet bilayer. This result can be\nexplained by the phonon angular momentum leaking at\ntwo interfaces that should enhance the damping for thin\nmagneticfilms. However, thephononpumpingisacoher-\nent process that couples both interfaces, so the damping\nis not increased simply by a factor of 2 as in case of in-\ncoherent spin pumping of a magnetic film sandwiched by\nmetals. The position of the resonances, ~kd= (2n+1)\u0019=2\nwithn2N0, are independent of the ratio \u001act=~\u001a~ctwith\n\u000bme=\u0010!c\n!\u001122Ms\n\rd\u001act: (S22)\n\u000b(?)\nmeand the effective magnetic field for YIG sandwiched\nbetween two infinitely thick GGG layers are shown in\nFigs. S1 and S2.\nIV. FLEXURAL (BENDING) WAVES IN THIN\nBEAMS\nIn the main text we focus on the generation of trans-\nverseorlongitudinalsoundwaves. Infree-standingstruc-\ntured samples such as cantilevers, additional modes be-\ncome important that can be excited by magnetization\ndynamics as well. This can be illustrated by a thin\ncylindrical elastic beam (see Fig. S3) with cross section\nareaA=\u0019r2, in which flexural waves are generated by\nthe magnet of volume V=Adattached to the top of\nthe beam. The elastic energy according to the Euler-\nBernoulli beam theory [44]\nEel=ZL\n0dz\u00141\n2\u001aA_u2(z;t) +1\n2EYI?u002(z;t)\u0015\n;(S23)\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n<10−510−410−3\nα(⊥)\nmeFigure S1. Phonon pumping-enhanced \u000b(?)\nmein a YIG film\nsandwiched between two infinitely thick GGG layers.\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n−40−30−20−10010203040\nµ0H(⊥)\nme[µT]\nFigure S2. Phonon pumping effective field H(?)\nmein a YIG film\nsandwiched between two infinitely thick GGG layers.\nleads to the equation of motion for the flexural waves [44]\n\u001aAu\u0006(z;t) +EYI?u(4)\n\u0006(z;t) = 0;(S24)\nwhereI?=R\ndAx2=\u0019r4=4and elastic modulus EY=\n\u0016(3\u0015+ 2\u0016)=(\u0015+\u0016). The dispersion relation of the flex-\nural waves is quadratic,\n!=s\nEYI?\n\u001aAk2: (S25)\nWhen the dimensions of the magnet are much smaller\nthan the wavelength of the elastic waves, the magne-\ntoelastic coupling is suppressed and magnetization and\nlattice are coupled exclusively by the magnetocrystalline\nand, in contrast to the bulk magnet, also the shape\nanisotropies,\nEA=EK+ED; (S26)\nwith\nEK=K1V(m\u0002n)2; (S27)\nED=1\n2\u00160VMTDM: (S28)3\nm\nH\nz 0magnet non-magnet\nflexural wave\nFigure S3. Thin film magnet (shaded) with magnetization m\nattached to a thin semi-infinite elastic beam.\nFor a thin magnetic film Dzz=D3= 1. When the\nmagnet become very thick ( d\u001dr), i.e. a needle with\nits point forming the contact, and a coordinate system\nwithzaxis along the surface normal n,Dxx=Dyy=\nD?=\u00001=2. All otherD\u000b\fvanish. In contrast to the\nextended bilayer treated in the main text nis now a\ndynamic variable with n\u0006=\u0000u0\n\u0006(0;t). The mechanical\ntorque exerted by the magnet on the elastic beam reads\n\u001c=_L=VMs\n\r_m+_J; (S29)\nwhere _J=\u00160VMsm\u0002Hextis the torque exerted by the\nexternal magnetic field on the total angular momentum.\nFor a magnet with equilibrium magnetization mkn0\n\u001c\u0006=\u0006if(m\u0006\u0000n\u0006); (S30)\nwheref=VMs!A=\rand\n!A=(\n2\rK1=Ms\u0000\r\u00160Ms;thin film\n2\rK1=Ms+1\n2\r\u00160Ms;needle:(S31)\nIn order to compute the angular momentum current\npumped into the attached non-magnet, _L\u0006=\u001c\u0006, we\nhave to specify four boundary conditions. Two are pro-\nvided by the assumption that the beam is infinitely long\nso that the are no reflections. The absence of shear forces\nat the boundary is expressed by u000(0;t) = 0, while the\nbending by the torque follows from the principle of least\naction [55]\nu00\n\u0006(0;t) =\u0006i\u001c\u0006\nEYI?: (S32)The general solution for the differential equation can be\nwritten\nu\u0006(z;t) =e\u0000i!t\u0000\nA\u0006eikz+B\u0006e\u0000kz\u0001\n;(S33)\nbecause there are no back-reflections in the semi-infinite\nbeam. We find\nn\u0006=wm\u0006\nwith\nw=\u00002f\nEYI?k\u0012\n1 +i\u00002f\nEYI?k\u0013\u00001\n;(S34)\nand the following source term in the LLG equation,\n_m\u0006jan=\u0007i!An\u0006\n=\u0006i!ARe(w)m\u0006\u0007!AIm(w)m\u0006;(S35)\nThe first term on the right-hand-side is a field-like torque\nequivalent to the effective field\n\u00160Han=!A\n\rRe(w)ez; (S36)\nand the second one a damping-like torque with damping\ncoefficient\n\u000ban=\u0000!A\n!Imw: (S37)\nSince for weak magnetoelastic coupling we expect \u000ban\u001c\n1and thereforejwj \u001c 1, which is equivalent to\n2f=(EYI?k)\u001c1, we may approximate\nw\u0019f(1\u0000i)\nEYI?k; (S38)\n\u000ban\u0019VMs!2\nA\n!k\rI?EY; (S39)\n\u00160Han\u0019\u0000VMs!2\nA\n\r2EYI?kez: (S40)\nThe damping enhancement scales as\n\u000ban/V\nA2!3\n2: (S41)\nFor a needle-shaped YIG magnet attached to GGG with\nEY= 2:5\u00021011Paand!A= 1:4\u00021010s\u00001\n\u000ban\u00198:6\u000210\u00006d=nm\n(\u0017=GHz)3\n2(r=nm)2;(S42)\n\u00160jHanj\u00193:1\u000210\u00007d=nm\n(\u0017=GHz)1\n2(r=nm)2T;(S43)\nwhich are very small numbers even at nanoscale dimen-\nsions."    },    {        "title": "1109.1414v1.A_theoretical_analysis_of_inertia_like_switching_in_magnets__applications_to_a_synthetic_antiferromagnet.pdf",        "content": "arXiv:1109.1414v1  [cond-mat.mtrl-sci]  7 Sep 2011A theoretical analysis of inertia-like switching in magnet s: applications to a synthetic\nantiferromagnet\nSatadeep Bhattacharjee, Anders Bergman, Andrea Taroni, Joh an Hellsvik, Biplab Sanyal and Olle Eriksson\nDepartment of Physics and Astronomy, Box 516, 75120, Uppsal a University, Uppsala, Sweden\n(Dated: November 1, 2018)\nThemagnetization dynamicsofasyntheticantiferromagnet subjecttoashortmagneticfieldpulse,\nhas been studied by using a combination of first-principles a nd atomistic spin dynamics simulations.\nWeobserveswitchingphenomenaonthetimescale oftensofpi coseconds, andinertia-likebehavior in\nthe magnetization dynamics. We explain the latter in terms o f adynamic redistribution of magnetic\nenergy from the applied field pulse to other possible energy t erms, such as the exchange interaction\nand the magnetic anisotropy, without invoking concepts suc h as inertia of an antiferromagnetic\nvector. We also demonstrate that such dynamics can also be ob served in a ferromagnetic material\nwhere the incident field pulse pumps energy to the magnetic an isotropy.\nPACS numbers: 75.78.-n, 75.50.Ee, 75.50.Ss, 75.75.-c\nI. INTRODUCTION\nStudies of magnetization reversal by an external mag-\nnetic field is a central paradigm which relates to mag-\nnetic data storage.1The magnetic random access mem-\nory(MRAM) device2basedonmagneto-resistance(MR),\nwhere magnetic bits can be described in terms high (1)\nor low (0) resistance of MR units, is found to be one\nof the most promising memory devices for the near fu-\nture. The magneto-resistance unit consists of a non-\nmagnetic layer sandwiched between two magnetic metal-\nlic layers. The magneto-resistance could be either gi-\nant magneto-resistance (GMR)3,4or tunnel magneto-\nresistance (TMR)5,6depending on the sandwiched layer,\nwhich could be either metallic or insulating. Such mem-\nory devices, with a very dense architecture, capable of\nperforming with low power consumption is proposed to\nbe aUniversal memory device with advantages of other\nmemory devices such as static RAM (SRAM), dynamic\nRAM (DRAM) and flash memory7. An MRAM, which is\ncapable of performing at the speed of SRAM with dense\narchitecture comparable to DRAM and non-volatility,\n(i.e. there is no loss of information when the power is\nswitched off) is to some extent limited by it’s writing\nspeed. This speed depends on the magnetization rever-\nsal time, where the magnetization of a given magnetic\nunit is reversed.\nThe potential for applications of magnetization dy-\nnamics in technology, combined with the new knowledge\nof fundamental magnetic interactions in nano-sized ma-\nterials, has caused a large increase in the interest in this\nfield. Several mechanisms for shorter switching speeds\nhave been proposed, e.g.toggle switching7, all optical\nswitching8etc. One recent suggestion is the so called in-\nertiadrivenswitching.9Herea shortmagneticfield pulse,\noptically generated, of 100 fs, was found to cause a dy-\nnamic behavior of the magnetism of HoFeO 3, which went\non for tens of picoseconds. In the paper of Ref. 9 it was\npointed out that this behavior is expected only for mate-\nrials with antiferromagneticexchangeinteractions, either\nfor bulk compounds with intrinsic antiferromagnetic ex-change (e.g. like HoFeO 3) or in artificial magnetic tri- or\nmultilayers, with antiferromagnetic interlayer exchange.\nIt was argued that for materials with antiferromagnetic\nexchange interactions, the switching can be described by\na second order differential equation of the so called anti-\nferromagnetic unit vector, and hence have features asso-\nciated to an inertia. It was in particular argued that the\ninertia accumulated during the short pulse, wasresponsi-\nble for the magnetization dynamics that lasted even after\nthe pulse was switched off.\nIn this paper we address the microscopic mecha-\nnisms behind this kind of switching for a magnetic tri-\nlayer,using atomisticspin-dynamicssimulations.10,11We\nhave chosen for our studies a Fe/Cr/Fe model system,\nwith antiferromagnetic interlayer exchange interaction\n(IEC). A schematic figure of the system considered is\nshown in Fig.1. The exchange constants were calcu-\nlated from first principles by mapping the ground state\nelectronic structure to a Heisenberg Hamiltonian, us-\ning the Liechtenstein-Katnelson-Gubanov method.12We\nhave calculated the time-evolution of the magnetizations\nof a Fe/Cr/Fe trilayer using spin dynamics simulation\nbased on Landau-Lifshitz-Gilbert method, using the Up-\npASDpackage.11Inoursimulationswehaveanalyzedthe\ndifferent terms that play a role in the switching behavior\nof materials with antiferromagnetic exchange, which are\nexposed to ultra short pulses. Although our simulations\nreproduce much of the observations reported in Ref.9,\nour interpretation of the microscopic mechanisms is dif-\nferent, and we argue that a more appropriate description\nof the mechanism behind this switching is described as\ndue to a redistribution of the energy from the ultra fast\nmagnetic pulse to the antiferromagnetic exchange inter-\naction and magnetic anisotropy of the material. After\nthe pulse is switched off the dynamics is governed by\nan effective field, now composed only of the exchange\nand anisotropy fields, that temporally evolves for much\nlonger time scales than the initial pulse. The dynami-\ncally evolving magnetization carries the effect of the field\npulse applied at the initial phase of the simulation and\nthe effect persists through the course of simulation.2\nII. METHOD\nThe temporal evolution of sub-lattice magnetizations\nare calculated using the dynamical equation of Landau-\nLifshitz-Gilbert form expressed in terms of atomic\nmoments13\ndmi(t)\ndt=−γ\n1+α2mi(t)×[Hi\neff+α\nMs(mi(t)×Hi\neff)]\n(1)\nwheremi(t) is the atomic moment on the ithsite at time\nt.γisthegyro-magneticratioand αistheGilbert damp-\ning factor. The effective field Hi\neffon theithatom is cal-\nculated from the effective magnetic Hamiltonian given\nby\nHMag=Hex+HMA+Ha, (2)\nthrough\nHi\neff=−∂HMag\n∂mi(t).\nThe sub-lattice magnetization for a given Fe or Cr sub-\nlattice is simply\nM(t) =/summationdisplay\nimi(t).\nThe first term in the Hamiltonian describes the inter-\natomic exchange interaction between between all atoms\nof the simulation cell. It can be separated into two parts;\n(i) pair-exchange between the atomic moments in the\nsame layer (intra-layer exchange coupling) and (ii) the\nexchange coupling between the moments residing in dif-\nferent layers (inter-layer exchange coupling). The second\nterm of Eqn.2 represents the magnetic anisotropy. The\nform of magnetic anisotropy Hamiltonian we have con-\nsidered is HMA=Ksin2θ, whereθis the angle which\nthe sublattice magnetization makes with anisotropy axis.\nK is the anisotropy energy density and can be written\nas the sum of two contributions K=Kmca+Kshape.\nFurthermore, Kmcais the energy density which corre-\nspondstomagneto-crystallineanisotropyand Kshapecor-\nresponds to the energy density for the shape anisotropy\narising from dipole-dipole interaction. We choose K such\na way that the hard axis is considered to be along the\nz-direction and the xy-plane is considered as easy plane,\ndue to the shape anisotropy of thin magnetic films and\ndue to the possible intrinsic uniaxial anisotropy. The last\nterm of Eqn.2 is interaction due to the applied magnetic\nfield. For the present case it describes the pulsed field we\napply during a short time of the simulations. Equation\n1 was solved numerically using the procedure of the Ref.\n14.\nFor T= 0 K we consider the macrospin approximation,\ni.e. all atomistic spins within one layer are assumed to\nco-rotate. For finite temperature simulations, one has to\ndeal with this problem in an atomistic way, consideringa large number of atoms in the simulations.11As men-\ntioned, the exchange constants were calculated from first\nprinciples by mapping the ground state electronic struc-\nturetoaHeisenbergHamiltonian. Thegroundstateelec-\ntronicstructurewascalculatedusingthe Korringa-Kohn-\nRostocker Green’s function method within the atomic\nsphere approximation.15,16In reality the interlayer ex-\nchange between the Fe layers through the Cr spacer layer\nare influenced by the interface roughness of the Fe-Cr in-\nterface, and to simulate this we rescaled our interlayer\nexchange interactions with a factor of 10, which is ac-\ncording to previous analysis.17,18\nWe considereachFe sublattice hasmagnetization M=\nMsˆ n(t), where ˆ n(t) is the unit vector. The magnitude of\nthe effective anisotropy field is given by FMA=2K\nMscosθ.\nMsis the saturation moment for the Fe sublattice. The\ninitial value of a Fe sublattice magnetization is along\nthe positive or negative x-direction, as shown in Fig.1,\ni.e.M=±Msˆ ex. For all simulations we used a square\nshaped pulse of the external field. At time t=0, we apply\na field of constant value in the x-z plane with constant\nvaluedescribedby Ha= [Hx0Hz]. We studied thecases\nwhere the magnetization was in-plane, due to the shape\nanisotropy, without any further in-plane anisotropy.\nFIG. 1: (Color online) (Color online) The considered geom-\netry of the atomistic simulations. The red slabs represent\nFe and the blue slab Cr. The arrows in the slabs represent\ndirections of magnetization.\nIII. RESULTS\nWe start by showingin Fig.2an example of the magne-\ntization dynamics of the tri-layer in Fig.1. Note that the\nmagnetization shown in the figure is only for the top Fe-\nslab in Fig.1 (normalized to unity). The results shown\nare for a field pulse with H x= 3 T and H z=5 T, for a\nduration of 5 picoseconds (ps). The Gilbert damping\nconstant used was 0.02 and the strength of the uniax-\nial anisotropy constant was 0.06 mRy/atom, providing3\nFIG. 2: (Color online) The time-evolution of the x-, y- and\nz-component of the top-most Fe sublattice magnetization of\nthe tri-layer shown in Fig.1. We have used H x=3T, H z=5T,\n|K|=0.06 mRy for α= 0.02\nan easy plane magnetization. At time t=0, we applied\na field pulse. From the figure, we see that as soon as\nthe pulse is applied, the magnetization is brought out\nof plane, and all the components of magnetizations are\nnon-zero. The magnetization then precess under the in-\nfluence of the external field, as well as the internal fields,\nprovided by the anisotropy field and the exchange field\ndue to the IEC. From the figure it is clear that even af-\nter the external magnetic field is switched of (i.e. after\n5 ps) the system continues to evolve in time, and the\nmagnetization dynamics continues over the entire time-\ninterval shown in the figure. At sufficiently long simula-\ntion time it seems however that the x-component of the\nmagnetism approaches the negative x-direction. Hence\nfor this choice of external pulse a switching of the mag-\nnetism occurs, and the direction ofall magnetic moments\nillustrated in Fig.1 have become reversed. The data in\nFig. 2 are similar to those demonstrated experimentally\nin Ref. 9, in that the magnetization dynamics continue\nlong after the external pulse has been first applied and\nthen removed from the system.\nBysweepingthestrengthofthex-componentoftheap-\nplied field we can realize different situations, in which the\nsublattice magnetization of one of the Fe slabs in Fig.1\nrotates half a turn (180 degrees), a whole turn (360 de-\ngrees) or even more than one turn. In Fig.3 we illustrate\nthisresult, inaphasediagramoftheswitchingbehaviors.\nIn the simulations shown in Fig.3 we have used the same\nz-component ofthe applied field, the same Gilbert damp-\ningfactor,pulsingtime, andthesameuniaxialanisotropy\nconstant, asused in Fig.2. The figureonly containsinfor-\nmation about the Fe magnetization of the top Fe slab of\nFig.1. In the phase diagram of Fig.3 the time of the sim-\nulations is shown on the y-axis and the magnitude of the\nx-component of the applied field is shown on the x-axis.\nThe switched and unswitched regions are represented interms of a color coding. The red color specifies that the\nmagnetization has turned n·360 degrees (n=0,1,2,...),\nwhereas blue color specifies that the magnetization has\nturnedn·360+180degrees(n=0,1,2,...). Takingasanex-\nample H x= 5T, Fig.3a shows that when the pulse arrives\nthe magnetism of the top Fe slab is along the positive x-\ndirection (red color), but after 5-6 ps the magnetization\nhas reversed to the negative x-direction, and after 10 ps\na rotation of 360 degrees to the positive x-direction has\noccurred. In between, the magnetism of course points\nalong the positive or negative y-direction (this compo-\nnent is not shown), and hence exhibits a rotational be-\nhavior. After some 30 ps the magnetism finally settles\ndown along the negative x-direction. Overall, the data\nin Fig.3 shows that the magnetization dynamics contin-\nues long after the external pulse has been turned off, in\na similar fashion as observed experimentally.9\nAnotherinterestingfeatureofthedatain Fig.3aisthat\nthe results are not symmetric with respect to positive or\nnegative x-component to the applied field. This might\nat first sight seem puzzling, since the geometry shown\nin Fig.1 is very symmetric with respect to positive and\nnegative x-direction. However, the data in Fig.3a depict\nonly the magnetization of the topmost Fe layer, and for\nthis layer alone there is no obvious symmetry for cases\nwhen the applied pulse is along the positive or negative\nx-direction, and the patterns in Fig.3a for positive and\nnegative values of H xare different. However, for all val-\nues of H xwe find that after sufficiently long time the\nmagnetism has settled down into a given orientation.\nThere are two torques that in general are relevant for\nthe magnetization dynamics, a precessional torque and\na damping torque, and the influence of the latter is pro-\nvided in the simulations by the strength of the Gilbert\ndamping parameter. Hence, one may expect that the\nphase diagram should depend critically on this parame-\nter and to investigate this possibility, we show in Fig. 3b\na simulation with all parameters kept exactly the same\nas for the simulation shown in Fig.3a, with the one dif-\nference that the Gilbert damping parameter was set to\n0.002, i.e. ten times lower than that used for Fig.3a.\nIn this case the rotation of the magnetism continues\nthroughout the entire period of the simulations, at least\nforpositivevaluesofH x, whichshowsthatit isthe damp-\ning parameter that primarily determines when the mag-\nnetization stops rotating.\nAs the pulse hits the sample (simulation box) the an-\ngle between the two Fe slabs start to deviate from the\n180 degree antiferromagnetic coupling. To illustrate this\nfact we show in Fig.4 a contour plot for the angle be-\ntween two Fe sublattice magnetizations, in which red col-\noration signifies 180 degree antiferromagnetic coupling.\nThis plot corresponds to the simulations shown in Fig.3a\nand Fig.3b. As is clear from Fig.4a,b, the magnetiza-\ntion of the two Fe sublattices are initially at 180 degree\nwith each other, but after the external field is switched\non and time evolves, the angle between the sublattices\nis reduced from 180 degrees and oscillates with time, es-4\nFIG. 3: (Color online) The switching phase diagram of an\nFe/Cr/Fe-trilayer at T=0 K. The duration of the magnetic\npulse is 5 ps, and the x-component of the magnetic field is\nshown on the x-axis, whereas the z-component is 5 T. A uni-\naxial anisotropy of 0.06 mRy is considered in the calculatio ns,\nproviding an easy plane magnetization. The x-component of\nthe magnetization of the top Fe slab of Fig.1 is shown in a\ncolor scheme. The blue/red areas indicate the regions where\nswitching has occurred/not occurred (see also text for de-\nscription). In a) we show simulations for a damping constant\nof 0.02 and in b) the damping constant is 0.002. H z= 5T,\n|K|=0.06 mRy\npecially when H x/negationslash= 0. Fig.4a,b shows data for two dif-\nferent damping parameters, and we note that these two\ndifferent damping parameters result in slightly different\nmagnetization dynamics, in that it takes longer time for\nthe angle between the two sublattices to approach 180\ndegrees for the weakly damped system. However, for\nboth simulations a 180 degree coupling is obtained when\nt→ ∞. The minimum angle is seen when the magnitude\nof the x-component of the applied field is largest, after 5\nps (i.e. the duration of the pulse), where the two sublat-\ntice magnetizations are approximately perpendicular to\neach other. This means that, with the applied field as a\ndriving force, the angle between the two sublattices is in-\ncreasinglydeviating, albeit in an oscillatingfashion, from\nantiferromagneticcoupling. When the anglebetweentwo\nsublattices is less than 180 degrees an internal exchange\nfield is build up in each Fe slab, which causes a dynam-\nical response. In addition, a non-zero z-component to\nthe magnetization (e.g. as shown in Fig.2) builds up an\nanisotropy field. These two contributions to the effective\nfield, which drives the magnetization dynamics, are non-\nzero long after the external field has been switched of. In\nfact, they are zero only after the magnetization has set-\ntled down to a constant value. Hence, the effective field\nis in this case dynamic, and the torques due to this fieldhas as usual two components; the damping part and the\nprecessional term.\nThe data in Fig.4 demonstrate an important result,\nnamely that during the short time that the external field\nis applied, a significant energy is established both in\nthe exchange interaction term, Hex, as well as in the\nanisotropy energy, HMAE, of Eqn.2. In a way one can\nview the whole process as being due to energy provided\nby theHa-term of Eqn.2, that during the pulse duration\npumps energy into the other two terms, HexandHMAE.\nEven ifHais non-zero only for a short time, the other\ntwo terms will be non-zero for a much longer period, and\nthe effective field composed of contributions due to these\ntwo terms depending on mi(t), carries some sort of in-\nformation of the field pulse applied at an earlier stage.\nTo demonstrate this in a more formal way, we plot in\nFig.5 the temporal evolution of the different components\nof magnetic energy during one of the simulations. From\nFIG. 4: (Color online) Evolution of angle between the sublat -\ntice magnetizations (a) for damping factor of 0.02 (b) with a\ndamping factor of 0.002. H z= 5 T,|K|=0.06 mRy, α=0.02\nFig.5 one can see that the Zeeman energy is negative\nduring the duration of the pulse whereas the exchange\nand anisotropy energy are positive. The sum of the three\ncomponents, i.e. the total energy is negative during the\nduration of the pulse, which means that during the time\nthat the pulse is applied, the energy is lowered and the\nsystem tries to relax into a new ground state configu-\nration. When the external pulse is switched off, Fig.5\nshows that there is still positive energy in the exchange\nand anisotropy contribution to the energy and the total\nenergyisnowpositive. Inthisconfiguration,thereisboth\na z-component of the magnetization as well non-collinear\ncoupling between the moments in the two Fe-slabs. The\nsystem then relaxes back to the original configuration of\ncollinear antiferromagnetic coupling of the Fe-slabs with-\nout a z-component to the magnetism and it does this in\nthe anisotropy and exchange field that has been build up5\nduring the duration of the external field . The relaxation\nback to collinear configuration takes considerably longer\ntime than the duration of the field as Fig.5 shows. From\nFig.5 it can be seen that the energy due to applied field\n(Zeeman energy) acts as a reservoir for the other two\nterms (exchange and anisotropy) of the magnetic Hamil-\ntonian. Hence, in order to analyze the magnetization\ndynamics provided in Figs.2 and Fig. 3, one does not\nneed to invoke concepts like inertia of antiferromagnetic\nunit vectors9. We suggest a simpler interpretation based\non a redistribution of energy among the three terms in\nEqn.2, in particulararedistribution ofenergyfrom Hato\nHexandHMAE, duringthe time period that the external\nfield is applied.\nFIG. 5: (Color online) Time evolution ofdifferentcomponent s\nof magnetic energy (Exchange, Anisotropy, Applied), plott ed\nalongside with total magnetic energy. H z= 5 T,α=0.02\nThe analysis in the preceding paragraphindicates that\nalso the strength of the anisotropy influenced the mag-\nnetization dynamics and to investigate this further, we\nplot in Fig.6, simulated data as a function of uniaxial\nanisotropy constant, K. All other parameters were the\nsame as for the simulation of Fig.3a. First we note that\nwhen K=0, the tendency of switching is reduced, since in\nthis case the dynamics is provided by the exchange field\nalone, via the IEC. However, when K increases to 0.01-\n0.04 mRy, the tendency for switching increases and the\nmagnetizationisseentoprecesstwotimesbeforestabiliz-\ning along the positive x-direction. For even larger values\nof anisotropyenergy, K largerthan ≈0.05 mRy, only one\nrevolution is found in the simulations, before the magne-\ntization is stabilized. Hence there is a critical value of K\nwhich provides an optimally large amount of revolutions\nin the magnetization.\nWe have also investigated the case when the inter-\nlayer exchange coupling is zero, in Eqn.2, by perform-\ning simulations to only the top-most Fe slab, of Fig. 1,\ni.e. a purely ferromagnetic system having an easy-plane\nanisotropy. The simulated results are shown in Fig.7 andwe note that also in this case can the magnetization be-\nhaveasifit had’inertia’, i.e. it proceedsafteranexternal\nfield is switched off. An analysis relying on antiferromag-\nnetic unit vectors becomes cumbersome in this case, but\nthe pumping of energy from Hato the two other compo-\nnents given in Eqn.2, in this case the anisotropy energy,\nprovides an easy interpretation. The data in Fig.7, are\nactually consistent with the experiments performed in\nRef. 19, where a short pulse was applied to a ferromag-\nnet thin film.\nFIG. 6: (Color online) Switching times as function of out-of -\nplane anisotropy. The duration of the magnetic pulse is 5 ps,\nthe z-component of the applied magnetic field is 5T and the\nx-component is 3T. (All other parameters are same as Fig.2)\nFIG. 7: (Color online) Switching phase diagram for Fe 3lay-\ners with the same set of parameters as indicated in Fig.3 for\nFe3/Cr4/Fe3trilayer, except the IEC. H z= 5 T, |K|=0.06\nmRy,α=0.026\nIV. CONCLUSIONS\nIn conclusion we have performed atomistic spin dy-\nnamics simulations for a synthetic antiferromagnet sub-\njected to a short magnetic pulse. We have demonstrated\nthat the magnetization dynamics continueslong after the\npulse has been applied and removed, in a fashion that is\nconsistent with the experimental results of Ref. 9.\nSeveral materials parameters, like the damping and\nstrength of the magnetic anisotropy, are found to in-\nfluence the details of the magnetization dynamics. We\nsuggest that such magnetization dynamics could be un-\nderstood in terms of a redistribution of energy, during\nthe time that the pulse is applied, so that the magnetic\nanisotropyenergyandinterlayerexchangeenergybecome\nin an out-of-equilibrium state. The magnetic energy due\nto the incident pulse hence becomes distributed dynam-ically to the effective interlayer exchange coupling be-\ntween the Fe layers and it also modifies the magnetic\nanisotropy energy. The latter term ensures that the ob-\nserved behavior of a magnetization dynamics that pro-\ngresses long after an external field has been applied and\nswitched off, is possible also in ferromagnetic materials.\nV. ACKNOWLEDGMENT\nWe are grateful to the Swedish Research Council for\nsupport. O.E. acknowledges also support from the KAW\nfoundation and the ERC (grant 247062 - ASD). Simu-\nlations performed on the Swedish Supercomputer Cen-\ntral NSC, via a grant from SNAC. Valuable discussions\nwith the late Prof. H.-C. Siegmann, Prof. J. St¨ ohr and\nA.Kirilyuk are acknowledged.\n1E. Beaurepaire, J.-C.Merle, A. Daunois, and J.-Y. Bigot,\nPhys. Rev. Lett. 76, 4250 (1996).\n2Y Zheng, Yuankai Wu, Yihong Li, Kebin Qiu, Jinjun Han,\nGuchang Guo, Zaibing Luo, Ping An, Lihua Liu, Zhiy-\nong Wang, Li Tan, Seng Ghee Zong, Baoyu Liu, Bo, J. of\nNanosci. and Nanotechno., 7, 117 (2007).\n3M. N. Baibich, J. M. Broto, A. Fert, F. Nguyen Van Dau,\nand F. Petroff,P. Etienne, G. Creuzet, A. Friederich, and\nJ. Chazelas, Phys. Rev. Lett 612472 (1988).\n4G. Binasch, P. Grunberg, F. Saurenbach, and W. Zinn,\nPhys. Rev. B 39, 4828 (1989).\n5W. H. Butler, X.-G. Zhang, T. C. Schulthess, and J. M.\nMacLaren, Phys. Rev. B 63, 054416 (2001).\n6J. S. Moodera, Lisa R. Kinder, Terrilyn M. Wong, and R.\nMeservey, Phys. Rev. Lett. bf 74, 3273 (1995).\n7J Akerman, Science, 308, 508 (2005)\n8C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A.\nTsukamoto, A. Itoh, and Th. Rasing, Phys. Rev. Lett. 99,\n047601 (2007).\n9A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev,\nA. Kirilyuk, and Th. Rasing, Nature Phys. 5, 727 (2009).\n10V. P. Antropov, M. I. Katsnelson, B. N. Harmon ,M.\nvan Schilfgaarde and D.Kusnezov, Phys. Rev. B 54, 1019\n(1996).11B.Skubic, J.Hellsvik, L.Nordstr¨ om, O.Eriksson, Jour-\nnal of Physics Condensed Matter, 20315203 (2008).\nhttp://www.fysik.uu.se/cmt/asd/ .\n12A I Liechtenstein, M I Katnelson and V A Gubanov,\nJ.Phys. F:Met.Phys. 14, L125 (1984); A I Liechtenstein,\nM I Ketnelson V.P. Antropov and V A Gubanov, J.\nMag.Mater. 67, 65 (1987).\n13Jacques Miltat, Goncalo Albuquerque and Andre Thiavill,\nSpin dynamics in confined magnetic structures , (Springer\nVerlag, more info here e.g. year of publication etc.)\n14J H Mentink, M V Tretyakov, A Fasolino, M I Katsnelson\nand Th. Rasing, J. Phys. Cond. Matter. 22176001 (2010).\n15J Korringa, Physica, 13, 392 (1947).\n16W Kohn W and N Rostoker, Phys. Rev. 94, 1111 (1954).\n17J. Kudrnovsky, V. Drchal, I. Turek, M. Sob and P. Wein-\nberger, Phys. Rev. B 53, 5125 (1996).\n18E. Holmstr¨ om, L.Bergqvist, B. Skubic, B.Hj¨ orvarsson,\nL.Nordstr¨ om, I.Abrikosov, P.Svedlindh and O. Eriksson,\nPNAS101, 4742 (2004).\n19C. Stamm, I. Tudosa, H. C. Siegmann, and J. St¨ ohr, A.\nYu. Dobin, G. Woltersdorfand B. Heinrich, A. Vaterlaus,\nPhys. Rev. Lett. 94, 197603 (2005)."    },    {        "title": "2209.14179v1.Unidirectional_magnetic_coupling.pdf",        "content": "Unidirectional magnetic coupling\nH. Y. Yuan,1R. Lavrijsen,2and R. A. Duine1, 2\n1Institute for Theoretical Physics, Utrecht University,\nPrincetonplein 5, 3584 CC Utrecht, The Netherlands\n2Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\n(Dated: September 29, 2022)\nWe show that interlayer Dzyaloshinskii-Moriya interaction in combination with non-local Gilbert\ndamping gives rise to unidirectional magnetic coupling. That is, the coupling between two magnetic\nlayers | say the left and right layer | is such that dynamics of the left layer leads to dynamics of\nthe right layer, but not vice versa. We discuss the implications of this result for the magnetic sus-\nceptibility of a magnetic bilayer, electrically-actuated spin-current transmission, and unidirectional\nspin-wave packet generation and propagation. Our results may enable a route towards spin-current\nand spin-wave diodes and further pave the way to design spintronic devices via reservoir engineering.\nIntroduction. | Non-reciprocal transmission of elec-\ntrical signals lies at the heart of modern communication\ntechnologies. While semi-conductor diodes, as an exam-\nple of an electronic component that underpins such non-\nreciprocity, have been a mature technology for several\ndecades, new solutions are being actively pursued [1, 2].\nSuch research is spurred on by the emergence of quan-\ntum technologies that need to be read out electrically\nbut should not receive unwanted back-action from their\nelectronic environment.\nComplementary to these developments, spintronics has\nsought to control electronic spin currents and, more\nrecently, spin currents carried by spin waves | i.e.,\nmagnons | in magnetic insulators [3]. Devices that im-\nplement non-reciprocal spin-wave spin currents have been\nproposed [4{7]. Most of these proposals rely on dipolar\ninteractions [8{11] or Dzyaloshinskii-Moriya interactions\n(DMI) [12{16]. Other proposals involve the coupling of\nthe spin waves to additional excitations such that the spin\nwaves are endowed with non-reciprocity. Examples are\nthe coupling of the spin waves to magnetoelastic, optical,\nand microwave excitations [17{22].\nMost of these proposals have in common that they con-\nsider spin-wave dispersions that are asymmetric in wave\nvector. For example, due to the DMI spin waves at one\nparticular frequency have di\u000berent wave numbers and ve-\nlocities for the two di\u000berent directions. There are there-\nfore spin waves travelling in both directions. This may\nbe detrimental for some applications. For example, one\nwould like to shield quantum-magnonic technologies from\nspin-current noise [23], and completely quench the spin-\ncurrent transmission in one of the two directions along a\nwire.\nHere, we propose a set-up that realizes unidirectional\nmagnetic coupling between two magnetic layers or be-\ntween two magnetic moments. The ingredients are DMI\nand dissipative coupling between the two layers or mo-\nments. The dissipative coupling takes the form of a non-\nlocal Gilbert damping and may arise, for example, from\nthe combined action of spin pumping and spin transfer.Then, one magnet emits spin current when it precesses,\nwhich is absorbed by the other. The resulting dissipa-\ntive coupling turns out to, for certain parameters, pre-\ncisely cancel the DMI in one direction. As a result, an\nexcitation of one of the magnets leads to magnetization\ndynamics of the other, but not vice versa. This yields\nspin-wave propagation that is truly uni-directional: for\nspeci\fc direction and magnitude of the external \feld, all\nspin waves travel in one direction only.\nMinimal model. | Let us start with the minimal set-\nup that demonstrates the unidirectional coupling. We\n\frst consider two identical homogeneous magnetic lay-\ners that are coupled only by an interlayer DMI with\nDzyaloshinskii vector Dand by interlayer spin pumping\n(see Fig. 1). The magnetization direction in the layers\nis denoted by mi, wherei2f1;2glabels the two lay-\ners. We also include an external \feld H. The magnetic\nenergy is given by\nE[m1;m2] =D\u0001(m1\u0002m2)\u0000\u00160MsH\u0001(m1+m2);(1)\nwhereMsis the saturation magnetization of both layers\nand\u00160is the vacuum susceptibility. The magnetization\ndynamics of layer 1 is determined by the Landau-Lifshitz-\nGilbert (LLG) equation\n@m1\n@t=\r\nMsm1\u0002\u000eE\n\u000em1+\u000bnlm1\u0002@m2\n@t; (2)\nwhere\ris the gyromagnetic ratio and \u000bnlcharacterizes\nthe strength of the non-local damping that in this set-up\nresults from the combination of spin pumping and spin\ntransfer torques, as described in the introduction. The\nequation of motion for the magnetization dynamics of the\nsecond layer is found by interchanging the labels 1 and\n2 in the above equation. Working out the e\u000bective \feldsarXiv:2209.14179v1  [cond-mat.mes-hall]  28 Sep 20222\nFM\nFM\nm2\nm1xy\nz, H, D\nFIG. 1. Schematic of two magnetic moments coupled by an\ninterlayer DMI and by interlayer spin pumping. The dynam-\nics of m1induces the motion of m2, but not vice versa for\nappropriate parameters.\n\u000eE=\u000emiyields\n@m1\n@t=\r\nMsm1\u0002(m2\u0002D\u0000\u00160MsH) +\u000bnlm1\u0002@m2\n@t;\n(3a)\n@m2\n@t=\r\nMsm2\u0002(D\u0002m1\u0000\u00160MsH) +\u000bnlm2\u0002@m1\n@t;\n(3b)\nwhere the sign di\u000berence in e\u000bective-\feld contribution\nfrom the DMI stems from the asymmetric nature of the\nDMI. We show now that depending on the magnitude and\ndirection of the e\u000bective \feld, this sign di\u000berence leads\nfor one of the layers to cancellation of the torques due\nto interlayer DMI and non-local damping. As the can-\ncellation does not occur for the other layer, and because\nthe DMI and non-local damping are the mechanisms that\ncouple the layers in the model under consideration, this\nleads to uni-directional magnetic coupling.\nTaking the external \feld to be much larger than the\ninterlayer DMI, i.e., \u00160jHj\u001djDj=Ms, and taking \u000bnl\u001c\n1, we may replace @mi=@tby\u0000\r\u00160mi\u0002Hon the right-\nhand side of Eqs. (3) because the external \feld then is\nthe dominant contribution to the precession frequency.\nFor the \feld H=D=\u000bnl\u00160Ms, one then \fnds that\n@m1\n@t=\u0000\r\n\u000bnlMsm1\u0002D; (4a)\n@m2\n@t=2\r\nMsm2\u0002(D\u0002m1)\u0000\r\n\u000bnlMsm2\u0002D:(4b)\nHence, the coupling between the two magnetic layers is\nunidirectional at the \feld H=D=\u000bnl\u00160Ms: the magne-\ntization dynamics of layer 1 leads to dynamics of layer\n2 as evidenced by Eq. (4b), but not vice versa as im-\nplied by Eq. (4a). This one-way coupling is reversed by\nchanging the direction of the \feld to \u0000Hor the sign of\nthe non-local coupling \u000bnl.\nMagnetic susceptibility. | Let us now take into ac-\ncount the Gilbert damping within the layers, exchange,\nand anisotropies and discuss the in\ruence of the unidi-\nrectional coupling on the magnetic susceptibility. Theenergy now reads\nE[m1;m2] =\u0000Jm1\u0001m2+D\u0001(m1\u0002m2)\n\u0000\u00160MsH\u0001(m1+m2)\u0000K\n2\u0000\nm2\n1;z+m2\n2;z\u0001\n;(5)\nwith the constant Kcharacterizing the strength of the\nanisotropy and Jthe exchange. We shall focus on the\nferromagnetic coupling ( J >0) without loss of generality.\nThe LLG equation now becomes\n@m1\n@t=\r\nMsm1\u0002@E\n@m1+\u000bm1\u0002@m1\n@t\n+\u000bnlm1\u0002@m2\n@t; (6)\nwith\u000bthe Gilbert damping constant of each layer, and\nwhere the equation for the second layer is obtained from\nthe above by interchanging the labels 1 and 2. We\ntake the external \feld in the same direction as the\nDzyaloshinskii vector and D=D^z,H=H^z, while\n\u00160MsH;K\u001dD, so that the magnetic layers are aligned\nin the ^z-direction. Linearizing the LLG equation around\nthis direction we write mi= (mi;x;mi;y;1)Tand keep\nterms linear in mi;xandmi;y. Writing\u001ei=mi;x\u0000imi;y,\nwe \fnd, after Fourier transforming to frequency space,\nthat\n\u001f\u00001(!)\u0012\u001e1(!)\n\u001e2(!)\u0013\n= 0: (7)\nTo avoid lengthy formulas, we give explicit results below\nfor the case that J= 0, while plotting the results for\nJ6= 0 in Fig. 2. The susceptibility tensor \u001fij, or magnon\nGreen's function, is given by\n\u001f(!) =1\n((1 +i\u000b)!\u0000!0)2\u0000(\rD=Ms)2\u0000\u000b2\nnl!2)\n\u0002\u0012(1 +i\u000b)!\u0000!0i(\rD=Ms\u0000\u000bnl!)\n\u0000i(\rD=Ms+\u000bnl!) (1 +i\u000b)!\u0000!0\u0013\n;(8)\nwith!0=\r(\u00160H+K=Ms) the ferromagnetic-resonance\n(FMR) frequency of an individual layer. The poles of\nthe susceptibility determine the FMR frequencies of the\ncoupled layers and are, for the typical case that \u000b;\u000b nl\u001c\n1, given by\n!\u0006=!r;\u0006\u0000i\u000b!r;\u0006; (9)\nwith resonance frequency\n!r;\u0006=\r(\u00160H+K=Ms\u0006D=Ms): (10)\nWhen\r\u00160H= (1\u0007\u000bnl)D=(\u000bnlMs)\u0000K=Ms\u0019\nD=(\u000bnlMs)\u0000K=Mswe have for J= 0 that\u001f12(!r;\u0006) = 0\nwhile\u001f21(!r;\u0006)6= 0, signalling the non-reciprocal cou-\npling. That is, the excitation of layer 1 by FMR leads\nto response of magnetic layer 2, while layer 1 does not\nrespond to the excitation of layer 2. For opposite direc-\ntion of \feld the coupling reverses: the excitation of layer3\n|χ21(J=0)|\n|χ21(J=0.5D)|\n|χ21(J=15D)|\n|χ12(J=0)|\n|χ12(J=0.5D)|\n|χ12(J=15D)|\n0.96 0.98 1.00 1.02 1.040100200300400\nω/ωH\nFIG. 2. Magnetic susceptibilities of two magnetic layers as a\nfunction of frequency at di\u000berent exchange couplings. !H\u0011\n\r(\u00160H+K=M s). The resonance frequencies are located at\nthe peak positions. The parameters are D=! H= 0:001;\u000bnl=\n0:001;\u000b= 0:002.\n2 by FMR leads in that case to response of magnetic\nlayer 1, while layer 2 does not respond to the excitation\nof layer 1. As is observed from Fig. 2, for \fnite but\nsmallJ\u001cD, the coupling is not purely unidirectional\nanymore but there is still a large non-reciprocity. For\nJ\u001dD, this non-reciprocity is washed out.\nElectrically-actuated spin-current transmission. | In\npractice, it may be challenging to excite the individual\nlayers independently with magnetic \felds, which would\nbe required to probe the susceptibility that is determined\nabove. The two layers may be more easily probed inde-\npendently by spin-current injection/extraction from ad-\njacent contacts. Therefore, we consider the situation that\nthe two coupled magnetic layers are sandwiched between\nheavy-metal contacts (see Fig. 3(a)). In this set-up, spin\ncurrent may be transmitted between the two contacts\nthrough the magnetic layers.\nFollowing the Green's function formalism developed by\nZheng et al. [24], the spin-current from the left (right)\nlead to its adjacent magnetic layer is determined by the\ntransmission function of the hybrid system T12(T21)\ngiven by\nTij(!) = Trh\n\u0000i(!)G(+)(!)\u0000j(!)G(\u0000)(!)i\n: (11)\nHere,G(+)(!) is the retarded Green's function for\nmagnons in contact with the metallic leads that is de-\ntermined by Dyson's equation\u0002\nG(+)\u0003\u00001(!) =\u001f\u00001(!)\u0000\n\u0006(+)\n1(!)\u0000\u0006(+)\n2(!), where the retarded self energy\n~\u0006(+)\ni(!) accounts for the contact with the metallic lead\ni. These self energies are given by\n~\u0006(+)\n1(!) =\u0000i~\u000b0\n1\u0012\n!0\n0 0\u0013\n; (12)and\n~\u0006(+)\n2(!) =\u0000i~\u000b0\n2\u00120 0\n0!\u0013\n: (13)\nThe rates for spin-current transmission from the heavy\nmetal adjacent to the magnet iinto it, are given by\n\u0000i(!) =\u00002Imh\n\u0006(+)\ni(!)i\n=~. The couplings \u000b0\ni=\n\rRe[g\"#\ni]=4\u0019Msdiare proportional to the real part of the\nspin-mixing conductance per area g\"#\nibetween the heavy\nmetal and the magnetic layer i, and further depend on\nthe thickness diof the magnetic layers. Finally, the ad-\nvanced Green's function is G(\u0000)(!) =\u0002\nG(+)\u0003y.\nIn the analytical results below, we again restrict our-\nselves to the case that J= 0 for brevity, leaving the\ncaseJ6= 0 to plots. Using the above ingredients,\nEq. (11) is evaluated. Taking identical contacts so that\n\u000b0\n1=\u000b0\n2\u0011\u000b0, we \fnd that\nT12=4(\u000b0)2!2(\rD=Ms+\u000bnl!)2\njC(!)j2; (14)\nwhile\nT21=4(\u000b0)2!2(\rD=Ms\u0000\u000bnl!)2\njC(!)j2; (15)\nwith\nC(!) = [!H\u0000(1 +i(\u000b\u0000\u000bnl+\u000b0))!]\u0001\n[!H\u0000(1 +i(\u000b+\u000bnl+\u000b0))!]\u0000(\rD=Ms)2:(16)\nFrom the expression for C(!) it is clear that, since\n\u000b;\u000b nl;\u000b0\u001c1, the transmission predominantly occurs\nfor frequencies equal to the resonance frequencies !r;\u0006\nfrom Eq. (9). Similar to the discussion of the suscepti-\nbilities, we have for \felds \r\u00160H=D=\u000b nl\u0000K=Msthat\nthe transmission T12(!=D=\u000b nl)6= 0, while T21(!=\nD=\u000b nl) = 0. As a result, the spin-current transmis-\nsion is unidirectional at these \felds. For the linear spin-\nconductances Gij, given byGij=R\n~!(\u0000N0(~!))Tij(!),\nwe also have that G126= 0, while G21= 0. Here,\nN(~!) = [e~!=kBT\u00001]\u00001is the Bose-Einstein distri-\nbution function at thermal energy kBT. For the oppo-\nsite direction of external \feld we have G12= 0, while\nG216= 0. Like in the case of the susceptibility discussed\nin the previous section, a \fnite but small exchange cou-\npling makes the spin current transport no longer purely\nunidirectional, while maintaining a large non-reciprocity\n(see Fig. 3(b)).\nSpin-wave propagation. | Besides the unidirectional\ncoupling of two magnetic layers, the above results may\nbe generalized to a magnetic multilayer, or, equivalently,\nan array of coupled magnetic moments that are labeled\nby the index isuch that the magnetization direction of\nthei-th layer is mi. This extension allows us to engi-\nneer unidirectional spin-wave propagation as we shall see4\nm1\nm2\nLead Lead FM FM(a)\n(b)\nT21(J=0)\nT21(J=0.5D)\nT21(J=15D)\nT12(J=0)\nT12(J=0.5D)\nT12(J=15D)\n0.96 0.98 1.00 1.02 1.040.0000.0020.0040.0060.008\nω/ωH\nFIG. 3. (a) Schematic of the system that the two coupled\nmagnetic layers are sandwiched between two heavy-metal con-\ntacts. (b) Transmission of the hybrid system as a function of\nfrequency.\nbelow. We consider the magnetic energy\nE[m] =X\nk[D\u0001(mk\u0002mk+1)\u0000\u00160MsH\u0001mk];(17)\nand \fnd | within the same approximations as for our\ntoy model above | for the magnetization dynamics that\n@mk\n@t=2\r\nMsmk\u0002(D\u0002mk\u00001)\u0000\r\n\u000bnlMsmk\u0002D;(18)\nfor the \feld H=D=\u000bnl\u00160Ms. This shows that for these\n\felds the magnetic excitations travel to the right | cor-\nresponding to increasing index k| only. The direction\nof this one-way propagation is reversed by changing the\nmagnetic \feld to \u0000Hor by changing the sign of the non-\nlocal damping.\nTo study how spin waves propagate in an array of cou-\npled magnetic moments described by the Hamiltonian in\nEq. (17). We start from the ground state mk= (0;0;1)T\nand perturb the left-most spin ( k= 0) to excite spin\nwaves. Since the dynamics of this spin is not in\ruenced\nby the other spins for the \feld H=D=\u000b nl\u00160Ms, its\nsmall-amplitude oscillation can be immediately solved\nas\u001e0(t) =\u001e0(t= 0) exp(\u0000i!0t\u0000\u000b!0t) with\u001ek=\nmk;x\u0000imk;yas used previously. The dynamics of the\nspins to the right of this left-most spin is derived by solv-\ning the LLG equation (18) iteratively, which yields\n\u001ek(t) =\u001e0(t= 0)e\u0000i!0te\u0000\u000b!0t\nk!(\u00002\u000bnl!0t)k;(19)wherek= 0;1;2;:::N\u00001.\nTo guarantee the stability of the magnetization dynam-\nics, the dissipation matrix of the N-spin system should\nbe negative-de\fnite, which imposes a constraint on the\nrelative strength of Gilbert damping and non-local damp-\ning, i.e.,\u000b > 2\u000bnlcos\u0019\nN+1. For an in\fnitely-long chain\nN!1 , we have\u000b>2\u000bnl. Physically, this means that\nthe local dissipation of a spin has to be strong enough to\ndissipate the spin current pumped by its two neighbors.\nFor a spin chain with \fnite number of spins, \u000b= 2j\u000bnljis\nalways su\u000ecient to guarantee the stability of the system.\nTaking this strength of dissipation simpli\fes Eq. (19) to\n\u001ek(t) =\u001e0(t= 0)e\u0000it=(\u000b\u001c)e\u0000t=\u001c\nk!(\u0000t=\u001c)k; (20)\nwhere\u001c\u00001=\u000b!0is the inverse lifetime of the FMR\nmode. This spatial-temporal pro\fle of spins is the same\nas a Poisson distribution with both mean and variance\nequal to\u001b=t=\u001cexcept for a phase modulation, and it\ncan be further approximated as a Gaussian wavepacket\non the time scale t\u001d\u001c, i.e.\n\u001e(x) =\u001e0(t= 0)e\u0000it=(\u000b\u001c)\np\n2\u0019\u001be\u0000(x\u0000\u001b)2\n2\u001b: (21)\nSuch similarity suggests that any local excitation of the\nleft-most spin will generate a Gaussian wavepacket prop-\nagating along the spin chain. The group velocity of the\nmoving wavepacket is v=a=\u001c, whereais the distance\nbetween the two neighboring magnetic moments. The\nwidth of the wavepacket spreads with time as ap\nt=\u001c,\nwhich resembles the behavior of a di\u000busive particle. Af-\nter su\u000eciently long time, the wavepacket will collapse.\nOn the other hand, the excitation is localized and can-\nnot propagate when the right-most spin ( k=N\u00001) is\nexcited, because its left neighbor, being in the ground\nstate, has zero in\ruence on its evolution. These results\ndemonstrate the unidirectional properties of spin-wave\ntransport in our magnetic array.\nDiscussion, conclusion, and outlook. | We have\nshown that the ingredients for unidirectional coupling be-\ntween magnetic layers or moments are that they are cou-\npled only by DMI and non-local Gilbert damping. While\nin practice it may be hard to eliminate other couplings,\nthe DMI and non-local coupling need to be su\u000eciently\nlarger than the other couplings to observe unidirectional\ncoupling.\nThere are several systems that may realize the unidi-\nrectional coupling we propose. A \frst example is that of\ntwo magnetic layers that are coupled by a metallic spacer.\nSuch a spacer would accommodate non-local coupling via\nspin pumping and spin transfer. For a spacer that is\nmuch thinner than the spin relaxation length, we \fnd,\nfollowing Refs. [25{27], that \u000bnl=\r~Re[~g\"#]=4\u0019dMs,\nwith ~g\"#the spin-mixing conductance of the interface\nbetween the magnetic layers and the spacer, dthe thick-\nness of the magnetic layers. For simplicity, we took the5\nmagnetic layers to have equal properties. The two mag-\nnetic layers may be coupled by the recently-discovered\ninterlayer DMI [28, 29], tuning to a point (as a function\nof thickness of the spacer) where the ordinary RKKY ex-\nchange coupling is small. We estimate \u000bnl= 4:5\u000210\u00003\nford= 20 nm, Re[~g\"#] = 4:56\u00021014\n\u00001m\u00002and\nMs= 1:92\u0002105A=m (YIGjPt). The required mag-\nnetic \feld for unidirectional magnetic coupling is then\naround 4.5 T for D= 1 mT. Another possible platform\nfor realizing the unidirectional coupling is the system of\nFe atoms on top of a Pt substrate that was demonstrated\nrecently [30]. Here, the relative strength of the DMI and\nexchange is tuned by the interatomic distance between\nthe Fe atoms. Though not demonstrated in this experi-\nment, the Pt will mediate non-local coupling between the\natoms as well. Hence, this system may demonstrate the\nunidirectional coupling that we proposed.\nThe non-local damping is expected to be generically\npresent in any magnetic material and does not require\nspecial tuning, though it may be hard to determine its\nstrength experimentally. Hence, an attractive implemen-\ntation of the unidirectional coupling would be a magnetic\nmaterial with spins that are coupled only via DMI, with-\nout exchange interactions. While such a material has\nto the best of our knowledge not been discovered yet,\nit is realized transiently in experiments with ultrafast\nlaser pulses [31]. Moreover, it has been predicted that\nhigh-frequency laser \felds may be used to manipulate\nDMI and exchange, even to the point that the former is\nnonzero while the latter is zero [32, 33].\nPossible applications of our results are spin-wave and\nspin-current diodes and magnetic sensors, where a weak\n\feld signal can be ampli\fed and transported through\nthe unidirectional coupling to the remote site to be read\nout without unwanted back-action. Finally, we remark\nthat the unidirectional magnetic coupling that we pro-\npose here may be thought of as reservoir engineering, cf.\nRef. [34]. In our proposal, the reservoir is made up by the\ndegrees of freedom that give rise to the non-local damp-\ning, usually the electrons. We hope that this perspective\nmay pave the way for further reservoir-engineered mag-\nnetic systems\nAcknowledgements. | It is a great pleasure to\nthank Mathias Kl aui and Thomas Kools for discus-\nsions. H.Y.Y acknowledges the European Union's Hori-\nzon 2020 research and innovation programme under\nMarie Sk lodowska-Curie Grant Agreement SPINCAT\nNo. 101018193. R.A.D. is member of the D-ITP consor-\ntium that is funded by the Dutch Ministry of Education,\nCulture and Science (OCW). R.A.D. has received fund-\ning from the European Research Council (ERC) under\nthe European Union's Horizon 2020 research and inno-\nvation programme (Grant No. 725509). This work is in\npart funded by the project \\Black holes on a chip\" with\nproject number OCENW.KLEIN.502 which is \fnanced\nby the Dutch Research Council (NWO).[1] Directions for non-reciprocal electronics, Nat. Electron.\n3, 233 (2020).\n[2] S.-W. Cheong, D. Talbayev, V. Kiryukhin, and A. Sax-\nena, Broken symmetries, non-receprocity, and multifer-\nroicity, npj Quant. Mater. 3, 19 (2018).\n[3] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B.\nHillebrands, Magnon spintronics, Nat. Phys. 11, 453\n(2015).\n[4] M. Jamali, J. H. Kwon, S.-M. Seo, K.-J. Lee and H. Yang,\nSpin wave nonreciprocity for logic device applications,\nSci. Rep. 3, 3160 (2013).\n[5] J. Lan, W. Yu, R. Wu, and J. Xiao, Spin-Wave Diode,\nPhys. Rev. X 5, 041049 (2015).\n[6] M. Grassi, M. Geilen, D. Louis, M. Mohseni, T. Br acher,\nM. Hehn, D. Stoe\u000fer, M. Bailleul, P. Pirro, and Y.\nHenry, Slow-Wave-Based Nanomagnonic Diode, Phys.\nRev. Applied 14, 024047 (2020).\n[7] K. Szulc, P. Graczyk, M. Mruczkiewicz, G. Gubbiotti,\nand M. Krawczyk, Spin-Wave Diode and Circulator\nBased on Unidirectional Coupling, Phys. Rev. Applied\n14, 034063 (2020).\n[8] R. W. Damon and J. R. Eshbach, Magnetostatic modes\nof a ferromagnet slab, J. Phys. Chem. Solids 19, 308\n(1961).\n[9] R. E. Camley, Nonreciprocal surface waves, Surf. Sci.\nRep.7, 103 (1987).\n[10] T. An, V. I. Vasyuchka, K. Uchida, A. V. Chumak,\nK. Yamaguchi, K. Harii, J. Ohe, M. B. Jung\reisch, Y.\nKajiwara, H. Adachi, B. Hillebrands, S. Maekawa, and\nE. Saitoh, Unidirectional spin-wave heat conveyer, Nat.\nMater. 12, 549 (2013).\n[11] J. H. Kwon, J. Yoon, P. Deorani, J. M. Lee, J. Sinha,\nK. J. Lee, M. Hayashi, and H. Yang, Giant nonreciprocal\nemission of spin waves in Ta/Py bilayers, Sci. Adv. 2,\ne1501892 (2016).\n[12] R. L. Melcher, Linear contribution to spatial dispersion\nin the spin-wave spectrum of ferromagnets, Phys. Rev.\nLett. 30, 125 (1972).\n[13] L. Udvardi and L. Szunyogh, Chiral Asymmetry of the\nSpin-Wave Spectra in Ultrathin Magnetic Films, Phys.\nRev. Lett. 102, 207204 (2009).\n[14] J.-H. Moon, S.-M. Seo, K.-J. Lee, K.-W. Kim, J. Ryu,\nH.-W. Lee, R. D. McMichael, and M. D. Stiles, Spin-wave\npropagation in the presence of interfacial Dzyaloshinskii-\nMoriya interaction. Phys. Rev. B 88, 184404 (2013).\n[15] F. Garcia-Sanchez, P. Borys, R. Soucaille, J.-P. Adam, R.\nL. Stamps, and J.-V. Kim, Narrow magnonic waveguides\nbased on domain walls, Phys. Rev. Lett. 114, 247206\n(2015).\n[16] H. Wang, J. Chen, T. Liu, J. Zhang, K. Baumgaerl,\nC. Guo, Y. Li, C. Liu, P. Che, S. Tu, S. Liu, P. Gao,\nX. Han, D. Yu, M. Wu, D. Grundler, and H. Yu, Chi-\nral spin-wave velocities induced by all-garnet interfacial\nDzyaloshinskii-Moriya interaction in ultrathin yttrium\niron garnet \flms, Phys. Rev. Lett. 124, 027203 (2020).\n[17] Y.-P. Wang, J. W. Rao, Y. Yang, P.-C. Xu, Y. S. Gui, B.\nM. Yao, J. Q. You, and C.-M. Hu, Nonreciprocity and\nUnidirectional Invisibility in Cavity Magnonics, Phys.\nRev. Lett. 123, 127202 (2019).\n[18] T. Yu, Y. M. Blanter, and G. E. W. Bauer, Chiral Pump-\ning of Spin Waves, Phys. Rev. Lett. 123, 247202 (2019).6\n[19] S. Tateno and Y. Nozaki, Highly Nonreciprocal Spin\nWaves Excited by Magnetoelastic Coupling in a Ni/Si\nBilayer, Phys. Rev. Applied 13, 034074 (2020).\n[20] P. J. Shah, D. A. Bas, I. Lisenkov, A. Matyushov, N.\nX. Sun, and M. R. Page, Giant nonreciprocity of surface\nacoustic waves enabled by the magnetoelastic interaction,\nSci. Adv. 6, eabc5648 (2020).\n[21] X. Zhang, A. Galda, X. Han, D. Jin, and V. M. Vinokur,\nBroadband Nonreciprocity Enabled by Strong Coupling\nof Magnons and Microwave Photons, Phys. Rev. Applied\n13, 044039 (2020).\n[22] Zhizhi Zhang, Zhenyu Wang, Huanhuan Yang, Z.-X.\nLi, Yunshan Cao, Peng Yan, Nonreciprocal Spin Waves\nDriven by Left-Hand Microwaves, arXiv:2202.00780\n[cond-mat.mes-hall].\n[23] H. Y. Yuan and Yunshan Cao and Akashdeep Kamra,\nand Rembert A. Duine and Peng Yan, Quantum magnon-\nics: When magnon spintronics meets quantum informa-\ntion science, Phys. Rep. 965, 1 (2022).\n[24] J. Zheng, S. Bender, J. Armaitis, R. E. Troncoso, and R.\nA. Duine, Green's function formalism for spin transport\nin metal-insulator-metal heterostructures, Phys. Rev. B\n96, 174422 (2017).\n[25] Y. Tserkovnyak, A, Brataas, G. E. W. Bauer, and\nBertrand I. Halperin, Nonlocal magnetization dynamics\nin ferromagnetic heterostructures Rev. Mod. Phys. 77,\n1375 (2005).\n[26] B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas,\nR. Urban, and G. E. W. Bauer, Dynamic exchange cou-\npling in magnetic bilayers, Phys. Rev. Lett. 90, 187601\n(2003).[27] G. Woltersdorf, O. Mosendz, B. Heinrich, and C. H.\nBack, Magnetization dynamics due to pure spin currents\nin magnetic double layers, Phys. Rev. Lett. 99, 246603\n(2015).\n[28] A. Fernandez-Pacheco, E. Vedmedenko, F. Ummelen,\nR. Mansell, D. Petit, and R. P. Cowburn, Symmetry-\nbreaking interlayer Dzyaloshinskii-Moriya interactions in\nsynthetic antiferromagnets, Nat. Mater. 18, 679 (2019).\n[29] D. S. Han, K. Lee, J.-P. Hanke, Y. Mokrousov, K.-W.\nKim, W. Yoo, Y. L. W. van Hees, T.-W. Kim, R. Lavri-\njsen, C.-Y. You, H. J. M. Swagten, M.-H. Jung, and\nM. Kl aui, Long-range chiral exchange interaction in syn-\nthetic antiferromagnets, Nat. Mater. 18, 703 (2019).\n[30] M. Steinbrecher, R. Rausch, K. T. That, J. Hermenau, A.\nA. Khajetoorians, M. Pottho\u000b, R. Wiesendanger, and J.\nWiebe, Non-collinear spin states in bottom-up fabricated\natomic chains, Nat. Commun. 9, 2853 (2018).\n[31] N. Kerber, D. Ksenzov, F. Freimuth, F. Capotondi, E.\nPedersoli, I. Lopez-Quintas, B. Seng, J. Cramer, K. Litz-\nius, D. Lacour, H. Zabel, Y. Mokrousov, M. Kl aui, and\nChristian Gutt, Nat. Commun. 11, 6304 (2020).\n[32] A. Stepanov, C. Dutreix, and M. I. Katsnelson, Dynami-\ncal and Reversible Control of Topological Spin Textures,\nPhys. Rev. Lett. 118, 157201 (2017).\n[33] J. M. Losada, A. Brataas, and A. Qaiumzadeh, Ultrafast\ncontrol of spin interactions in honeycomb antiferromag-\nnetic insulators, Phys. Rev. B 100, 060410 (2019).\n[34] A. Metelmann and A. A. Clerk, Nonreciprocal photon\ntransmission and ampli\fcation via reservoir engineering,\nPhys. Rev. X 5,021025 (2015)."    },    {        "title": "1612.06951v1.Critical_current_density_of_a_spin_torque_oscillator_with_an_in_plane_magnetized_free_layer_and_an_out_of_plane_magnetized_polarizer.pdf",        "content": "arXiv:1612.06951v1  [cond-mat.mtrl-sci]  21 Dec 2016Critical current density of a spin-torque oscillator with a n in-plane magnetized free\nlayer and an out-of-plane magnetized polarizer\nR. Matsumoto1and H. Imamura1,a)\n1National Institute of Advanced Industrial Science and Tech nology (AIST),\nSpintronics Research Center, Tsukuba, Ibaraki 305-8568,\nJapan\n(Dated: 21 June 2021)\nSpin-torque induced magnetization dynamics in a spin-torque oscillat or with an in-\nplane(IP)magnetizedfreelayerandanout-of-plane(OP)magnet izedpolarizerunder\nIP shape-anisotropy field ( Hk) and applied IP magnetic field ( Ha) was theoretically\nstudied based on the macrospin model. The rigorous analytical expr ession of the\ncritical current density ( Jc1) for the OP precession was obtained. The obtained\nexpression successfully reproduces the experimentally obtained Ha-dependence of Jc1\nreported in [D. Houssameddine et al., Nat. Mater. 6, 447 (2007)].\nPACS numbers: 75.78.-n, 85.75.-d, 85.70.Kh, 72.25.-b\nKeywords: Spin-torque oscillator\na)Electronic mail: h-imamura@aist.go.jp\n1pmFree layer\nPolarizer z\nxy\n-e(a) (b) Ha\n(J < 0)\nFIG. 1. (a) Spin-torque oscillator consisting of in-plane ( IP) magnetized free layer and out-of-plane\n(OP) magnetized polarizer layer. IP magnetic field ( Ha) is applied parallel to easy axis of the free\nlayer. Negative current density ( J <0) is defined as electrons flowing from the polarizer layer to\nthe free layer. The unit vector mrepresents the direction of magnetization in the free layer . (b)\nDefinitions of Cartesian coordinates ( x,y,z), polar angle ( θ) and azimuthal angle ( φ).\nA spin-torque oscillator (STO)1–6with an in-plane (IP) magnetized free layer and an\nout-of-plane (OP) magnetized polarizer7–15has been attracting a great deal of attention as\nmicrowave field generators12,16–20and high-speed field sensors21–26. The schematic of the\nSTO is illustrated in Fig. 1(a). When the current density ( J) of the applied dc current\nexceeds the critical value ( Jc1), the 360◦in-plane precession of the free layer magnetization,\nso-called OP precession, is induced by the spin torque. Thanks to th e OP precession, a\nlarge-amplitude microwave field can be generated,12,14,15and a high microwave power can\nbe obtained through the additional analyzer.8\nThecritical current density, Jc1, fortheOPprecession ofthistypeofSTO hasbeenexten-\nsively studied bothexperimentally8,15andtheoretically.7,9–11,13,27In 2007, D. Houssameddine\net al. experimentally found that Jc1was approximately expressed as Jc1∝Hk+2Hawhere\nHkis IP shape-anisotropy field and Hais the applied IP magnetic field. In theoretical stud-\nies, the effect of HkandHaonJc1has been studied analytically and numerically. U. Ebels\net al. proposed an apporximate expression of Jc1, however, as we shall show later, it gives\nexact solution only in the limit of Ha= 0 and Hk→0. Lacoste et al. obtained the lower\ncurrent boundary for the existence of OP precession13which gives some insights into Jc1,\nhowever, it could be lower than Jc1. To our best knowledge, Jc1of this type of STO is\nstill controversial and a systematic understanding of Jc1in the presence of HkandHais\nnecessary.\nIn this letter, we theoretically analyzed spin-torque induced magne tization dynamics in\n2the STO with anIP magnetized freelayer and anOP magnetized polariz er inthe presence of\nHkandHabased on the macrospin model. We obtained the rigorous analytical e xpression of\nJc1and showed that it successfully reproduces the experimentally obt ainedHa-dependence\nof the critical current reported by D. Houssameddine et al.8\nThe system we consider is schematically illustrated in Figs. 1(a) and (b ). The shape of\nthe free layer is either a circular cylinder or an elliptic cylinder. The late ral size of the nano-\npillar is assumed to be so small that the magnetization dynamics can be described by the\nmacrospin model. Directions of the magnetization in the free layer an d in the polarizer are\nrepresented by the unit vectors mandp, respectively. The vector pis fixed to the positive\nz-direction. The negative current is defined as electrons flowing fro m the polarizer to the\nfree layer. The applied IP magnetic field, Ha, is assumed to be parallel to the magnetization\neasy axis of the free layer. The easy axis is parallel to x-axis.\nThe energy density of the free layer is given by28\nE=1\n2µ0M2\ns(Nxm2\nx+Nym2\ny+Nzm2\nz)\n+Ku1sin2θ−µ0MsHasinθcosφ. (1)\nHere (mx,my,mz) = (sinθcosφ, sinθsinφ, cosθ), andθandφare the polar and azimuthal\nangles of mas shown in Fig. 1(b). The demagnetization coefficients, Nx,Ny, andNzare\nassumed to satisfy Nz≫Ny≥Nx.Ku1is the first-order crystalline anisotropy constant, µ0\nis the vacuum permeability, Msis the saturation magnetization of the free layer, and Hais\napplied IP magnetic field.\nHereafter we conduct the analysis with dimensionless expressions. The dimensionless\nenergy density of the free layer is given by\nǫ=1\n2(Nxm2\nx+Nym2\ny+Nzm2\nz)\n+ku1sin2θ−hasinθcosφ. (2)\nHere,ku1andhaare defined as ku1=Ku1/(µ0M2\ns) andha=Ha/Ms. We discuss on the\nspin-torque induced magnetization dynamics at ha≥0 in this letter, however, the dynamics\natha<0 can be calculated in the similar way.\nThe spin-torque induced dynamics of min the presence of applied current is described\n3by the following Landau-Lifshitz-Gilbert equation,28\n(1+α2)dθ\ndτ=hφ+χsinθ+αhθ, (3)\n(1+α2)sinθdφ\ndτ=−hθ+α(hφ+χsinθ), (4)\nwhereτ,χ,hθ, andhφare the dimensionless quantities representing time, spin torque, an d\nθ,φcomponents of effective magnetic field, heff, respectively. heffis given by heff=−∇ǫ.\nαis the Gilbert damping constant. The dimensionless time is defined as τ=γ0Mst, where\nγ0= 2.21×105m/(A·s) is the gyromagnetic ratio and tis the time. hθandhφare given by\nhθ=cosθ/bracketleftbigg\n2sinθ/parenleftbigghk\n2cos2φ−keff\nu1/parenrightbigg\n+hacosφ/bracketrightbigg\n, (5)\nhφ=−hk\n2sinθsin2φ−hasinφ. (6)\nHerehkisdimensionless IPshape-anisotropyfieldbeingexpressedas hk=Ny−Nx=Hk/Ms.\nkeff\nu1is defined as keff\nu1=Keff\nu1/(µ0M2\ns) =ku1−(Nz−Ny)/2.Keff\nu1is the effective first-order\nanisotropy constant where the demagnetization energy is subtra cted. Since we areinterested\nin the spin-torque induced magnetization dynamics of the IP magnet ized free layer, we\nconcentrate on keff\nu1<0. The prefactor of the spin-torque term, χ, is expressed as\nχ=η(θ)/planckover2pi1\n2eJ\nµ0M2sd, (7)\nwhereη(θ) =P/(1+P2cosθ) is spin-torque efficiency, Pis the spin polarization, Jis the\napplied current density, dis the thickness of the free layer, e(>0) is the elementary charge\nand/planckover2pi1is the Dirac constant. For convenience of discussion, the sign of Eq . (7) is taken to\nbe opposite to that in Ref. 28.\nIn the absence of the current, i.e., J= 0, the angles of the equilibrium direction of mare\nobtained as θeq=π/2 andφeq= 0 by minimizing ǫwith respect to θandφ. Application of J\nchangesθandφfromits equilibrium values. If themagnitude of Jis smaller thanthe critical\nvalue, the magnetization converges to a certain fixed point.29The equations determining the\npolar and azimuthal angles of the fixed point ( θ0,φ0) are obtained by setting dθ/dτ= 0 and\ndφ/dτ= 0 as\nh0\nθ= 0, (8)\nh0\nφ=−χsinθ0. (9)\n4FIG. 2. (a) Function, sin2 φ+ξsinφ, is plotted as against φ. Value of ξis varied from 0.0 to\n4.0. (b) Spin-torque magnitude ( χ) dependence of φat fixed point ( φ0) in the presence of IP\nshape anisotropy field. χis defined in Eq. (7), and it is proportional to J. Curves represent the\nanalytical results obtained by Eq. (10). Open or solid circl es, squares, and triangles represent\nnumerical calculation results.\nThe fixed point around the equilibrium direction ( θeq=π/2,φeq=0) are obtained as follows.\nAssuming |φ0| ≤π/2, i.e., cos φ0≥0 and noting keff\nu1<0, one can see that the quantity\nin the square bracket of Eq. (5) is positive and θ0=π/2 to satisfy h0\nθ= 0. Substituting\nθ0=π/2 toh0\nφ=−χsinθ0, the equation determining φ0is obtained as\nsin2φ0+ξsinφ0= 2χ/hk, (10)\nwhereξ= 2ha/hk. Since Eq. (10) does not contain the Gilbert damping constant, α,φ0\nis independent of α. In Fig. 2(a), the function, sin2 φ+ξsinφ, is plotted against φfor\nvarious values of 0 ≤ξ≤4. One can clearly see that the azimuthal angle of the maximum\n(minimum) increases (decreases) towards π/2 (−π/2) with increase of ξ. The azimuthal\nangle of the fixed point is given by the intersection of this sinusoidal c urve and a horizontal\nline at 2χ/hk, andit increases withincrease of 2 χ/hkas shown in Fig. 2(b). In Fig. 2(b), the\ncurves represent the analytical results obtained by Eq. (10) and the symbols represent the\nnumerical results obtained by directly solving the Eqs. (3) and (4) w ithα= 0.02,hk= 0.01,\nandkeff\nu=−0.4. The analytical and simulation results agree very well with each oth er. We\n5also performed numerical simulations for wide range of αand confirmed that the numerical\nresults of φ0are independent of αas predicted by the analytical results. In the numerical\nsimulations, the current density was gradually increased from zero . At each current density,\nthe simulation was run long enough for the polar and azimuthal angles to be converged to\nθ0andφ0.\nNumerical simulations showed thatthereexists acritical current d ensity,Jc1, abovewhich\nthe OP precession is induced. For J >0,Jc1is obtained by calculating the maximum value\n(Λ) of the left hand side (LHS) of Eq. (10). If 2 χ/hkis larger than Λ, there is no fixed point\nand the limit cycle corresponding to the OP precession is induced. Her eafter we consider\nthe case of J >0, however, the critical current density for J <0 can be obtained in the\nsimilar way by calculating the minimum value.\nAt the maximum, the derivative of the LHS of Eq. (10) with respect t oφ0is zero, that\nis,\n2cos2φ0+ξcosφ0= 0. (11)\nExpressing cosine functions by tan φ0, one can easily obtain the solution of Eq. (11) as\nφc1= arctan/bracketleftbigg1\n2√\n2/radicalBig\nξ2+8+ξ/radicalbig\nξ2+32/bracketrightbigg\n, (12)\nwhere the subscript “c1” stands for the critical value correspon ding toJc1. Fig. 3(a) shows\nξdependence of φc1given by Eq. (12). φc1=π/4 forξ= 0, i.e., ha= 0. It monotonically\nincreases with increase of ξand reaches π/2 in the limit of ξ→ ∞, i.e.,hk→0.\nThe maximum value, Λ, can be obtained by substituting φ=φc1into the LHS of Eq.\n(10) as\nΛ =√\nX+8/parenleftbig\nξ√\nX+16+4√\n2/parenrightbig\nX+16, (13)\nwhereX=ξ(ξ+/radicalbig\nξ2+32). Equating this maximum value with 2 χ/hkand using Eq. (7),\nthe critical current density is obtained as\nJc1=eµ0MsdHk\n/planckover2pi1P√\nX+8/parenleftbig\nξ√\nX+16+4√\n2/parenrightbig\nX+16. (14)\nThis is the main result of this letter. It should be noted Jc1is also independent of α. In the\nabsence of the applied IP magnetic field, i.e., Ha= 0, Eq. (14) becomes\nJc1/vextendsingle/vextendsingle\nHa=0=eµ0MsdHk\n/planckover2pi1P. (15)\n6(a)\nHa (kA/m) Ic (mA) (b)\nξ φc1  (deg.) \nFIG. 3. (a) Analytically-calculated ξdependence of critical φ(φc1).ξis ratio between Haand IP\nshape anisotropy field ( Ha), beingξ= 2Ha/Ha. (b)Hadependence of critical current ( Ic) for OP\nprecession. Solid blue curves represent plots of analytica l expression (Eq. (14)). Hkof 4 kA/m\nis assumed. Open blue circles represent critical current ab ove which the OP precession can not\nbe maintained. Red dots represent past experimental result s (redrawn from Ref. 8). Dotted gray\nlines represent the empirically approximated value propos ed in Ref. 8.\nIn the limit of Hk→0, it reduces to\nlim\nHk→0Jc1=2eµ0MsdHa\n/planckover2pi1P. (16)\nFor small magnetic field such that Ha≪Hk, i.e.,ξ≪1, it can be approximated as\nJc1≃eµ0Msd\n/planckover2pi1P/parenleftBig\nHk+√\n2Ha/parenrightBig\n, (17)\nby noting that the Taylor expansion of Λ around ξ= 0 is given by Λ = 1+ ξ/√\n2+ξ2/16+\nO(ξ3).\nOnce thecurrent density, J, exceeds Jc1, theOP precession is excited andfurther increase\nofJmoves the trajectory towards the south pole ( θ=π). Around θ= 0 and π, there exist\nthe fixed points other than θ0=π/2, which are determined by\n2sinθ0(hk\n2cos2φ0−keff\nu1)+hacosφ0= 0, (18)\nhk\n2sinθ0sin2φ0+hasinφ0=χsinθ0. (19)\n7After some algebra, the fixed point is obtained as\nθ0= arcsin\nha/radicalBig/parenleftbig\nkeff\nu1/parenrightbig2+χ2\n2/bracketleftBig/parenleftbig\nkeff\nu1/parenrightbig2+χ2/bracketrightBig\n−hkkeff\nu1\n, (20)\nφ0=−arctanχ\nkeff\nu1, (21)\nwhereπ/2<|φ0| ≤π. In the absence of the applied IP magnetic field, i.e., ha= 0, the polar\nangle of the fixed point is θ0= 0 orπ. It is difficult to obtain the exact analytical expression\nfor the critical current density, Jc2, above which the OP precession can not be maintained,\nandmstays at the fixed point given by Eqs. (20) and (21). Since the aver age polar angle of\nthe trajectory of the OP precession is determined by the competit ion between the damping\ntorque and spin torque, this critical current density should depen d onα. The approximate\nexpression was obtained by Ebels et al.11as\nJc2≃ −4αedKeff\nu1\n/planckover2pi1P, (22)\nwhich agrees well with the macrospin simulation results.\nLet us compare our results with the experimental results reporte d by D. Houssameddine\net al.8Figure 3(b) shows the applied IP magnetic field, Ha, dependence of critical current\n(Ic) for the OP precession. The analytical results of Eq. (14) are plot ted by the solid\n(blue) line and the experimental results are plotted by the (red) do ts. The critical current\ncorresponding to Jc2are also shown by open (blue) circles. In the analytical calculation,\nthe following parameters indicated in Ref. 8 are assumed: α= 0.02,Ms= 866 kA/m, the\njunction area is 30 ×35×πnm2,d= 3.5 nm,P= 0.3,Hk= 4 kA/m. The dotted (gray)\nlines represent the approximated values proposed in Ref. 8, Ic∝Hk+2Ha. One can clearly\nsee that the analytical results of Eq. (14) reproduces the exper imental results very well.\nThe agreement is much better than the approximated values of Ref . 8. As shown in Eq.\n(17), the critical current for small magnetic field can be approxima ted asIc∝Hk+√\n2Ha\nrather than Ic∝Hk+2Ha.\nIn summary, we theoretically studied spin-torque induced magnetiz ation dynamics in an\nSTO with an IP magnetized free layer and an OP magnetized polarizer. We obtained the\nrigorous analytical expressions of Jc1for the OP precession in the presence of IP shape-\nanisotropy field ( Hk) and applied IP magnetic field ( Ha). The expression reproduces the\n8experimental results very well and revealed that the critical curr ent is proportional to Hk+\n√\n2HaforHa≪Hk.\nACKNOWLEDGMENTS\nThis work was supported by JSPS KAKENHI Grant Number 16K17509 .\nREFERENCES\n1J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n2L. Berger, Phys. Rev. B 54, 9353 (1996).\n3M. Tsoi, A. G. M. Jansen, J. Bass, W.-C. Chiang, M. Seck, V. Tsoi, an d P. Wyder,\nPhys. Rev. Lett. 80, 4281 (1998).\n4S.I. Kiselev, J.C. Sankey, I. N.Krivorotov, N. C.Emley, R.J. Sch oelkopf, R.A. Buhrman,\nand D. C. Ralph, Nature 425, 380 (2003).\n5A. M. Deac, A. Fukushima, H. Kubota, H. Maehara, Y. Suzuki, S. Yu asa, Y. Nagamine,\nK. Tsunekawa, D. D. Djayaprawira, and N. Watanabe, Nat. Phys. 4, 803 (2008).\n6A. Slavin and V. Tiberkevich, IEEE Transactions on Magnetics 45, 1875 (2009).\n7K. J. Lee, O. Redon, and B. Dieny, Appl. Phys. Lett. 86, 022505 (2005).\n8D. Houssameddine, U. Ebels, B. Delaet, B. Rodmacq, I. Firastrau, F. Ponthenier,\nM. Brunet, C. Thirion, J.-P. Michel, L. Prejbeanu-Buda, M.-C. Cyrille , O. Redon, and\nB. Dieny, Nat. Mater. 6, 447 (2007).\n9I. Firastrau, U. Ebels, L. Buda-Prejbeanu, J. C. Toussaint, C. T hirion, and B. Dieny,\nJ. Magn. Magn. Mater. 310, 2029 (2007).\n10T. J. Silva and M. W. Keller, IEEE Trans. Magn. 46, 3555 (2010).\n11U. Ebels, D. Houssameddine, I. Firastrau, D. Gusakova, C. Thirion , B. Dieny, and L. D.\nBuda-Prejbeanu, Phys. Rev. B 78, 024436 (2008).\n12H. Suto, T. Yang, T. Nagasawa, K. Kudo, K. Mizushima, and R. Sato ,\nJ. Appl. Phys. 112, 083907 (2012).\n13B. Lacoste, L. D. Buda-Prejbeanu, U. Ebels, and B. Dieny,\nPhys. Rev. B 88, 054425 (2013).\n914S. Bosu, H. Sepehri-Amin, Y. Sakuraba, M. Hayashi, C. Abert, D. S uess, T. Schrefl, and\nK. Hono, Appl. Phys. Lett. 108, 072403 (2016).\n15R. Hiramatsu, H. Kubota, S. Tsunegi, S. Tamaru, K. Yakushiji, A. F ukushima, R. Mat-\nsumoto, H. Imamura, and S. Yuasa, Appl. Phys. Express 9, 053006 (2016).\n16X. Zhu and J. G. Zhu, IEEE Trans. Magn. 42, 2670 (2006).\n17J. G. Zhu, X. Zhu, and Y. Tang, IEEE Trans. Magn. 44, 125 (2008).\n18Y. Wang, Y. Tang, and J.-G. Zhu, J. Appl. Phys. 105, 07B902 (2009).\n19M. Igarashi, Y. Suzuki, H. Miyamoto, Y. Maruyama, and Y. Shiroishi,\nIEEE Trans. Magn. 45, 3711 (2009).\n20K. Kudo, H. Suto, T. Nagasawa, K. Mizushima, and R. Sato,\nAppl. Phys. Express 8, 103001 (2015).\n21K. Kudo, T. Nagasawa, K. Mizushima, H. Suto, and R. Sato,\nAppl. Phys. Express 3, 043002 (2010).\n22P. M. Braganca, B. A. Gurney, B. A. Wilson, J. A. Katine, S. Maat, a nd J. R. Childress,\nNanotechnology 21, 235202 (2010).\n23H. Suto, T. Nagasawa, K. Kudo, K. Mizushima, and R. Sato,\nAppl. Phys. Express 4, 013003 (2011).\n24W. H. Rippard, A. M. Deac, M. R. Pufall, J. M. Shaw, M. W. Keller, S. E. Russek,\nG. E. W. Bauer, and C. Serpico, Phys. Rev. B 81, 014426 (2010).\n25H. Kubota, K. Yakushiji, A. Fukushima, S. Tamaru, M. Konoto, T. N ozaki,\nS. Ishibashi, T. Saruya, S. Yuasa, T. Taniguchi, H. Arai, and H. Ima mura,\nAppl. Phys. Express 6, 103003 (2013).\n26H. Suto, T. Nagasawa, K. Kudo, K. Mizushima, and R. Sato,\nNanotechnology 25, 245501 (2014).\n27H. Morise and S. Nakamura, Phys. Rev. B 71, 014439 (2005).\n28M. D. Stiles and J. Miltat, “Spin-transfer torque and dynamics,” in\nSpin Dynamics in Confined Magnetic Structures III , Topics in Applied Physics, Vol.\n101, edited by B. Hillebrands and A. Thiaville (Springer Berlin Heidelberg , 2006) pp.\n225–308.\n29G. Bertotti, I. D. Mayergoyz, and C. Serpico, in\nNonlinear Magnetization Dynamics in Nanosystems , Elsevier Series in Electromag-\nnetism (Elsevier, 2009) pp. 1–466.\n10"    },    {        "title": "1210.3933v1.Dynamic_Control_of_Magnetically_Trapped_Indirect_Excitons_by_Using_External_Magnetic_Bias.pdf",        "content": "Dynamic Control of Magnetically Trapped Indirect Ex citons by Using External Magnetic Bias \nAhmed M. Abdelrahman and Byoung S. Ham * \n Photon Information Processing Center, School of Ele ctrical Engineering, \n Inha University, Incheon 402-751, South Korea \n* bham@inha.ac.kr   \n \nAbstract \nWe demonstrate an on demand spatial control of exci tonic magnetic lattices for the potential \napplications of excitonic-based quantum optical dev ices.  A two dimensional magnetic lattice of \nindirect excitons can form a transition to one dime nsional lattice configuration under the \ninfluence of external magnetic bias fields. The tra nsition is identified by measuring the spatial \ndistribution of two dimensional photoluminance for several values of the external magnetic bias \nfields. The number of the trapped excitons is found  to increase between sites along a \nperpendicular direction exhibiting two to one dimen sional lattice transition. This work may apply \nfor various controllable quantum simulations, such as superfluid-Mott-insulators, in quantum \noptical devices.  \nPACS number (s): 71 .35 .Ji, 52 .55 .Jd, 52 .55 .Lf \n \nIntroduction \nTrapping of excitonic quasi-particles in solid stat e medium using electric or magnetic \nfields has recently become an active field of inter est in the area of semiconductor physics. \nThe two trapping mechanisms are rapidly gaining cre dits for their promises to underlie basic \nconcepts of condensed matter systems. For example, it is now possible to create one and two \ndimensional lattice-like configurations of electric ally trapped indirect excitons (bound state of \nelectron-holes) in a system of coupled quantum well s (CQWs) 1-3. Electrostatic traps were \nobserved in cold excitonic gases, whose advantages are extending the lifetime of the trapped \nexcitonic particles as well as allowing to observe the direct emission of their spontaneous \ncoherence of the trapped indirect excitons 4. In such approach the gate voltage controls the \nexciton’s energy, where the applied electric field perpendicular to the growth direction of the \nCQWs shifts the energy of the created indirect exci tons 5. Moreover, the spatial distribution of \nthe electric fields can configure a profile of conf ining potential for the excitonoic particles 5-7. \nAccordingly, the electrical variable trapping field  and its configurability made it possible to \ncreate in situ  gate control for manipulating the trapped excitoni c particles on a time scale that \nis shorter than their lifetimes8. \nOn the other hand, magnetic field confinement has e merged recently as an alternative \nstable trapping mechanism that can be used to confi ne excitonic particles in solid mediums 9-13. \nAfter its successful implementation in an atomic me dium, magnetic traps have shown \nextremely stable trapping and a long coherence life time of the trapped particles 14. These \nmagnetic traps can also be configured to shape one and two dimensional magnetic lattices of \nthe trapped particles 13,14,16 . The impeded magnetic trapping fields in semicondu ctors can be \nrealized by integrating a fabricated permanent magn etic material, such as a form of magnetic \nthin film, with a system of CQWs 12-13. The magnetic thin film can then be patterned \naccording to a desirable spatial distribution of th e magnetic trapping potentials. Similar to \nelectrical traps, the shape and depth of the magnet ic traps can be controlled by using external \nmagnetic bias fields.  \nIn this approach, excitonic particles are created i n CQWs and trapped by magnetic \nfields representing two dimensional magnetic lattic es of indirect excitons. This approach can \nbe adopted to simulate condensed matter systems whe re strongly correlated systems such as \nthe transition of superfluid-Mott-insulator (SF-MI)  can be achieved 17. In this article, we \ndemonstrate a dimensional active control of the mag netic lattice configuration of indirect \nexcitons by applying external magnetic bias fields.   \n \nTwo-Dimensional Magnetic Lattices \nTo create a magnetic lattice of indirect excitons, periodically distributed magnetic field local \nminima Bmin  and maxima  Bmax  are projected onto the indirect-excitons formation  plane of the CQWs 13. The inhomogeneous \noriginated from a specific pattern \nexample of two-dimensio nal arrays of \nmagnetic field local minima  \nonto the plane of the CQWs, \ntrapping of ultracold atoms 14-1\nAn analytical model to \nzmin  from the surface of the thin film \n \n/g1828/g4666/g1876/uni002C/g1877/uni002C/g1878/g4667/g3404/g4674/g1828/g3051/g2879/g3029/g3036/g3028/g3046 /g2870/g3397/uni0009/g1828/g3052\n/g3397/uni0032/g1828/g3042/g2870/g1857/g2879/g3081/g4666/g3053/g2879/g3099/g4667/g3427/g1828/g3051/g2879/g3029/g3036/g3028/g3046 /uni0009/g1871/g1861/g1866/g4666\n    \nExternal magnet ic bias field \npermanently magnetized thin film is characterized b y the parameter \nto 2 µm in both the simulation and \ndefined by /uni0009/g1828/g3042/g3404/g1828/g3560/g4666/uni0031/g3398/g1857/g2879/g3081/g3099 \neach fabricated pattern and their separating distances \nmagnetic induction, /g1828/g3560/g3404/g3091/g3290/g3014\n/g3095\ndistribution of the magnetic lattice sites (single traps \nposition of the local field minima \n/g1866/g3052/g3404/uni0030/uni002C/g3399/uni0031/uni002C/g3399/uni0032/uni002C/uni2026 , and /g1878/g3040/g3036/g3041 \n(a) \n(c) \nFig. 1.  (a) S imulation model of \nto generate the two- dimensional magnetic lattice ( \nlattice is created with periodicity of \nthe distributed single traps a cross \nand (c) numerically calculated trapping fields acro ss \nasymmetrical distribution of the sites along the z \n(dashed line) calculations of the inhomogeneous mag netic field at \n \ninhomogeneous magnetic fields that used to trap the excitonic \nspecific pattern  permanently magnetized thin film 16. Fig \nnal arrays of the permanent magnetic s quare holes \n and maxima introduce periodic confining space \nCQWs, where the confinement mechanism acts a role \n16.  \nto  describe the confining magnetic fields at a working distance \nfrom the surface of the thin film can be written as follow 16: \n/g3052/g2879/g3029/g3036/g3028/g3046 /g2870/g3397/g1828/g3053/g2879/g3029/g3036/g3028/g3046 /g2870/g3397/uni0032/g1828/g3042/g2870/g1857/g2879/g2870/g3081/g4666/g3053/g2879/g3099/g4667/g4670/uni0031/g3397/uni0063/uni006F/uni0073 \n/g4666/g2010/g1876 /g4667/uni0009/uni0009/g3397/g1828/g3052/g2879/g3029/g3036/g3028/g3046 /uni0009/g1871/g1861/g1866/g4666/g2010/g1877 /g4667/g3397/g4666/uni0063/uni006F/uni0073 /g4666/g2010/g1876 /g4667/g3397/uni0063/uni006F/uni0073 /uni0009\n     \n \nic bias field s are denoted by /g1828/g3051/uni002C/g3052/uni002C/g3053/g2879/g3029/g3036/g3028/g3046 , and the thickness of the \npermanently magnetized thin film is characterized b y the parameter τ, which is set to \nm in both the simulation and the experiment. The Bo is the reference magnetic \n/g3081/g3099 /g4667/uni0009 with /g2010/g3404/g2024/g2009/uni2044, where α represents both the length \nand their separating distances αs, as shown in Fig. 1(a \n/g3014/g3301\n/g3095, where Mz is the magnetization of the thin \ndistribution of the magnetic lattice sites (single traps ) is periodic according to Eq. \nlocal field minima Bmin  are /uni0009/g1876/g3040/g3036/g3041 /g3404/g1866/g3051/g2009/uni002C/uni0009/g1866/g3051/g3404/uni0030/uni002C/g3399/uni0031/uni002C/g3399\n/g3406/g3080\n/g3095/uni006C/uni006E /g4670/g1828/g3042/g4671. \n (b) \n (d) \n \nimulation model of a permanent magnetic material thin film with thickn ess of \ndimensional magnetic lattice ( Inset (1) shows a block of 9×9 square holes). The \ncreated with periodicity of /g2009/g3046/g3404/g2009/g3035/g3404/g2009/g3404/uni0032/g2020/g1865 . (b) Density plot of simulated lattice show \ncross  the x/y- plane at the local minima using analytical expressi on (Eq.1) \nand (c) numerically calculated trapping fields acro ss  the x/z-plane where the dashed line indicates the \nasymmetrical distribution of the sites along the z -axis. (d) N umerical (solid line) and the analytical \n(dashed line) calculations of the inhomogeneous mag netic field at a trapping level ( i.e. \nexcitonic particles are \nFig ure 1(a) shows an \nquare holes . The projected \nspace  (single traps) \na role  of magnetic \nat a working distance \n/uni0063/uni006F/uni0073 /uni0009/g4666/g2010/g1876/g4667/uni0063/uni006F/uni0073/uni0009/g4666/g2010/g1877/g4667 /g4671/g3397\n/uni0009/g4666/g2010/g1877/g4667/g4667/g1828/g3053/g2879/g3029/g3036/g3028/g3046 /g3431/g4675/g2869/g2870/g3415\n/uni0009\n  (1) \nand the thickness of the \nwhich is set to be equal \nis the reference magnetic field \nrepresents both the length  αh of \n, as shown in Fig. 1(a ). The /g1828/g3560/uni0009 is \nthe magnetization of the thin film. The \n) is periodic according to Eq.  (1), and the \n/g3399/uni0032/uni002C/uni2026, /g1877/g3040/g3036/g3041 /g3404/g1866/g3052/g2009/uni002C\n \n \na permanent magnetic material thin film with thickn ess of τ=2 µm used \nnset (1) shows a block of 9×9 square holes). The \n. (b) Density plot of simulated lattice show s \nplane at the local minima using analytical expressi on (Eq.1) , \nwhere the dashed line indicates the \numerical (solid line) and the analytical \ni.e. z/g3404/g1878/g3040/g3036/g3041 /g4666/g2020/g1865/g4667 ). \nFigure 1(a) shows the model used to simulate the \nmagnetic lattice created at the plane of CQWs simulated \n(1). In the experimental s etup \nwhere a set of n×n square holes is regarded as \nThe 3×3 blocks are fabricated \nin the experimental details, the coupled \nHowever, due to the asymmetrical effect \nlattice sites) across the x,y/z \nindividual lattice site) experience \nis indicated by the dashed line in Fig. 1(c) \nhave smaller values of the magnetic local minima \nsites whose values are higher  \nwells (i.e., /g1828/g3040/g3036/g3041 /g3030/g3032/g3041/g3047/g3032/g3045 /g4666/g1833/g4667/g3407\ncreates periodically distributed \nafter magnetization 16. The potential \nbetween the lattice sites , where the tunneling is the key mechanism of \ncontrol of the excitonic lattice transition \n \n(a) \n   (d) \nFig. 2.  Numerical simulations \nThe magnetic lattice is initially set at zero \n/g2009/g3046/g3404/g2009/g3035/g3404/uni0032/g2020/g1865  /g1839/g3053/g3404/uni0032/uni002E/uni0038\nIncreasing z-bias field ( /g1828/g3051/g2879\nto depart away from the thin film surface deeper in to \nplane. (f) A collapse of the local minima occurs at high v alues of \nmagnetic lattice is reshaped as shown in Fig. 3(f).\n \nThe gradient (or the curvature) \nparticular interest, where trapping field with steeper gradient \nclose to zero) often develops  \nthis destructive process has not \nexcitons. It is important at this \navoided in the case of magnetic trapping in semicon ductor \nminima can precisely be allocated at the confin \nThe trapping frequencies \nmodel used to simulate the  magnetic lattice, and \nthe plane of CQWs simulated by the analytical \netup  each fabricated single pattern takes the shape of \nsquare holes is regarded as one block as shown in the inset of Fig. 1(a) \n3×3 blocks are fabricated , where each block consists of 9×9 square holes \nin the experimental details, the coupled quantum wells are allocated at \nHowever, due to the asymmetrical effect  (that is the inhomogeneous spatial \nacross the x,y/z -plane, each confining area at the CQWs plane (i.e. at each \nexperience s different minimum value of trapping potential \ndashed line in Fig. 1(c) . In other words, at zero bias fields the center sites \nhave smaller values of the magnetic local minima /g1828/g3040/g3036/g3041 /g4666/g1833/g4667/uni0009 in comparison with \n when measured at the confining plane of the coupled quantum \n/g3407/uni0009/g1828/g3040/g3036/g3041 /g3032/g3031/g3034/g3032 /g4666/g1833/g4667). The asymmetrical distribution of the sites \ncreates periodically distributed potential tilts across the x/y-plane which is \npotential tilt is essential to allow the tunneling of trapped particles \n, where the tunneling is the key mechanism of \nlattice transition . \n (b)  (c) \n (e)  (f) \n \nNumerical simulations  for the effect of external magnetic bias field along th e z \nThe magnetic lattice is initially set at zero -bias: /g1828/g3053/g2879/g3029/g3036/g3028/g3046 /g3404/g1828/g3051/g2879/g3029/g3036/g3028/g3046 /g3404\n/uni0038/g1863/g1833 ; /g2028/g3404/uni0032/g2020/g1865 ; n = 9 with local minima at /g1878/g3040/g3036/g3041 \n/g2879/g3029/g3036/g3028/g3046 /g3404/g1828/g3052/g2879/g3029/g3036/g3028/g3046 /g3404/uni0030/uni002C/g1828/g3053/g2879/g3029/g3036/g3028/g3046 /g3405/uni0030) causes the magnetic \nto depart away from the thin film surface deeper in to  and beyond the coupled quantum well \n(f) A collapse of the local minima occurs at high v alues of /g1828/g3053/g2879/g3029/g3036/g3028/g3046 \nmagnetic lattice is reshaped as shown in Fig. 3(f).  \nthe curvature)  of the magnetic field at the trapping po \ntrapping field with steeper gradient  and zero local minima (or very \n the so-called Majorana spin-flip in magnetic tr \nthis destructive process has not yet been observed in such magnetic  \nat this stage to emphasize that the zero local minima can easily be \navoided in the case of magnetic trapping in semicon ductor s via fabrication; \nminima can precisely be allocated at the confin ing plane of the quantum wells. \nfrequencies /uni0009/g2033/g3051/uni002C/g3052, along the x and y confining directions \nand Fig. 1(b) shows a \nanalytical expression in Eq. \nthe shape of a square hole, \nblock as shown in the inset of Fig. 1(a) . \neach block consists of 9×9 square holes . As explained \nquantum wells are allocated at /g1878/g3404/g1878/g3040/g3036/g3041 /g4666/g2020/g1865/g4667 . \ninhomogeneous spatial  distribution of the \nconfining area at the CQWs plane (i.e. at each \npotential : This effect \nzero bias fields the center sites \nin comparison with  the edge \nthe coupled quantum \nThe asymmetrical distribution of the sites \nwhich is initially existed \nof trapped particles \n, where the tunneling is the key mechanism of the present active \n  \n \nthe effect of external magnetic bias field along th e z -axis. (a) \n/g1828/g3052/g2879/g3029/g3036/g3028/g3046 /g3404/uni0030 ); \n/g3406/uni0034/g2020/g1865 . (b-e) \n) causes the magnetic local minima \nbeyond the coupled quantum well  \n in which the \nof the magnetic field at the trapping po sition is of a \nand zero local minima (or very \nin magnetic tr aps 18. However, \n lattices of indirect \nthe zero local minima can easily be \nfabrication;  the non-zero local \nquantum wells.  \nconfining directions  (that is across \nthe x/y- plane of confinement) \nmagnetic fields at each site,  \ntrapping field along x/y-axis as follows \n \n/g3105/g3118/g3003\n/g3105/g3051/g3118/g3404/g3398/g2010/g2870/g1828/g3042/g1857/g3081/g4666/g3053\n/g3105/g3118/g3003\n/g3105/g3052/g3118/g3404/g3398/g2010/g2870/g1828/g3042/g1857/g3081/g4666/g3053\n \nThe trapping frequencies along x and y ax \n \n/g2033/g2200/g3404/uni0009/g3081\n/g2870/g3095/g3495\nThe trapping frequency along the z \nand /g1859/g3007 represent magnetic quantum number, the Bohr magneto n \nrespectively. Each localized  \nspace of the confinement and the depth of the traps , \ndefined by /uni0009/uni2206/g1828/g4666/g2200/g4667/g3404/uni007C/g1828/g3040/g3028/g3051 /g4666\n/uni0393/g4666/g2200/g4667/g3404/uni0009/g3091/g3251/g3034/g3255/g3040/g3255\n/g3038/g3251/uni2206/g1828/g4666/g2200/g4667, wh ere \nFigures 2-5 show the numerical simulat \ncreated with the following parameters \n/g1866/g3404/uni0039/uni0009  holes. The working distance is found to be located at \nthe magnetic lattice is initially set at \n \n(a) \n(d)\nFig. 3.  Numerical calculations for \nat /g1878/g3040/g3036/g3041  with (a) no external magnetic bias field \nlattice with the sa me initial parameter as in Fig. \nand deformation in their confining spaces wi \n/g1828/g3052/g2879/g3029/g3036/g3028/g3046 /g3404/uni0030/uni002C/g1828/g3053/g2879/g3029/g3036/g3028/g3046 /g3405/uni0030. \n/g1878/g3040/g3036/g3041 /g3406/uni0034/g2020/g1865 . \nplane of confinement)  depend strongly on the gradie nt of the \n where in our calculations we determine the curvatur es of the \nas follows : \n/g3053/g2879/g3099/g4667/g3428/g2913/g2925/g2929 /g4666/g3081/g3051 /g4667/g2913/g2925/g2929/g4666/g3081/g3052/g4667\n/g3493/g2870/g2878/g2870/g2913/g2925/g2929 /g4666/g3081/g3051 /g4667/g2913/g2925/g2929/g4666/g3081/g3052/g4667/g3397/g2913/g2925/g2929 /g3118/g4666/g3081/g3051 /g4667/g2929/g2919/g2924 /g3118/g4666/g3081/g3052/g4667\n/g4666/g2870/g2878/g2870/g2913/g2925/g2929 /g4666/g3081/g3051 /g4667/g2913/g2925/g2929/g4666/g3081/g3052/g4667/g4667/g3119\n \n/g3053/g2879/g3099/g4667/g3428/g2913/g2925/g2929 /g4666/g3081/g3051 /g4667/g2913/g2925/g2929/g4666/g3081/g3052/g4667\n/g3493/g2870/g2878/g2870/g2913/g2925/g2929 /g4666/g3081/g3051 /g4667/g2913/g2925/g2929/g4666/g3081/g3052/g4667/g3397/g2913/g2925/g2929 /g3118/g4666/g3081/g3052 /g4667/g2929/g2919/g2924 /g3118/g4666/g3081/g3051/g4667\n/g4666/g2870/g2878/g2870/g2913/g2925/g2929 /g4666/g3081/g3051 /g4667/g2913/g2925/g2929/g4666/g3081/g3052/g4667/g4667/g3119\nalong x and y ax es are defined as:  \n/g3495/g2020/g3003/g1859/g3007/g1865/g3007/g3105/g3118/g3003\n/g3105/g2200/g3118 with /g2200/uni2208/g4668/g1876/uni002C/g1877/g4669.   (4)  \n \nalong the z -axis is defined by /g2033/g3053/g3404/g2033/g3051/g3397/g2033/g3052. In Eq. (4 \nrepresent magnetic quantum number, the Bohr magneto n , and the Lande g \n minimum is surrounded by magnetic barriers that define the \nspace of the confinement and the depth of the traps , and th e magnitude of the barrier is \n/g4666/g2200/g4667/uni007C/g3398/uni007C/g1828/g3040/g3036/g3041 /g4666/g2200/g4667/uni007C, while the depth of each trap is defined by \nere  /g1837/g3003 is the Boltzmann constant. \nshow the numerical simulat ion results of a two- dimensiona \nthe following parameters : /g2009/g3046/g3404/g2009/g3035/g3404/uni0032/g2020/g1865 , /g2028/g3404/uni0032/g2020/g1865 , \nholes. The working distance is found to be located at /g1878/g3040/g3036/g3041 /g3404/g4666/uni0032/g2009\nis initially set at /g1828/g3051/g2879/g3029/g3036/g3028/g3046 /g3404/g1828/g3052/g2879/g3029/g3036/g3028/g3046 /g3404/g1828/g3053/g2879/g3029/g3036/g3028/g3046 /g3404/uni0030 as shown in Fig. \n (b)  (c) \n (e)  (f) \n \nNumerical calculations for magnetic field local minima distributed across the \nno external magnetic bias field  (/g1828/g3053/g2879/g3029/g3036/g3028/g3046 /g3404/g1828/g3051/g2879/g3029/g3036/g3028/g3046 /g3404/g1828/g3052/g2879/g3029/g3036/g3028/g3046 \nme initial parameter as in Fig.  2. (b-f) The shift of the lattice site location \nand deformation in their confining spaces wi th respect to the external z- bias field \n All x/y- planes are simulated at the local minima along the z \nnt of the inhomogeneous \nwhere in our calculations we determine the curvatur es of the \n/g4667/g3119/g3118/g3415/g3432, (2) \n/g4667/g3119/g3118/g3415/g3432. (3) \n \nIn Eq. (4 ), /g1865/g3007, /g2020/g3003, \nand the Lande g -factor, \nis surrounded by magnetic barriers that define the \ne magnitude of the barrier is \nwhile the depth of each trap is defined by \ndimensiona l magnetic lattice \n, /g1839/g3053/g3404/uni0032/uni002E/uni0038/g1837/g1833 , and \n/g2009/g3399/uni2206/g1878/g4667/g2020/g1865 , where \nas shown in Fig. 2(a).  \n \n \nmagnetic field local minima distributed across the x/y-plane \n/g3029/g3036/g3028/g3046 /g3404/uni0030) for a \nthe lattice site location s \nbias field : /g1828/g3051/g2879/g3029/g3036/g3028/g3046 /g3404\nplanes are simulated at the local minima along the z -axis: \n \n \nTo induce the tunneling of the magnetically trapped  particles or to dis \nlocations across the x/y- plane and/or along the z \napplied along the x/y- axis and/or the z \n/g1828/g3053/g2879/g3029/g3036/g3028/g3046  field, results in reducing the tunn \nthe tunnelin g rate of the trapped exciton \nand 3 are for the effect of the \nmagnetic z-bias fi eld increases \npermanently magnetized thin film and the \nthe actual location of the lattice site \nbe allocate d at different distances \nthe magnetic field at the trapping point is increas ed \n(increased) number of trapped excitons. The z \nspace and to redistribute the  \ntransition between different interesting configurat ions \n \n(a) \n(d) \nFig. 4.  (a) Using the same initial parameters of Fig. \nmagnetic bias field along the x \n(b-f) The effects of the appl \n/uni0030/uni002C/g1828/g3051/g2879/g3029/g3036/g3028/g3046 /g3405/uni0030. Oscillations between \nline in (b)) magnetic lattice \n  \nOn the other hand, external magnetic \nreshape the magnetic lattices from two \nin Figs. 5(b)-(f), external magnetic \nevolve from a two dim ensional \nand x/y- axis external magnetic bias fields may allow simula ting condensed matter systems \nthrough reshaping the sites. For instance, \nBrillouin zones by redistributing th \nlattice according to accurately adjusted external magnetic bias fields (this description also \nincludes op tically trapped atomic species \n \nTo induce the tunneling of the magnetically trapped  particles or to dis \nplane and/or along the z -axis, the external magnetic bias fields \naxis and/or the z -axis. The application of the bias fields, specifically \nin reducing the tunn eling barrier between the sites and hence increasin g \ng rate of the trapped exciton s. The numerical simulation results \nthe applied external bias field along the z -\neld increases  (or decreases) the distance /g1878/g3040/g3036/g3041  between the surface of the \npermanently magnetized thin film and the initial position of the local field minima \nlattice site  along the z-axis . As shown in Fig. 2 \nd at different distances  from the plane of the coupled quantum wells in whic h case \nthe magnetic field at the trapping point is increas ed  (decreased) resulting in reduced \nnumber of trapped excitons. The z -bias field also causes to deform the con \n lattice sites. As shown in Figs. 3(b)-(f), such \ntransition between different interesting configurat ions  of the magnetic lattice \n (b)  (c) \n (e)  (f) \n \nsame initial parameters of Fig.  2 a magnetic lattice subject to external \nmagnetic bias field along the x -axis /g1828/g3051/g2879/g3029/g3036/g3028/g3046  is simulated: /g1828/g3051/g2879/g3029/g3036/g3028/g3046 /g3404/g1828/g3052/g2879/g3029/g3036/g3028/g3046 /g3404\nappl ied external bias field along the x-axis: /g1828/g3052/g2879/g3029/g3036/g3028/g3046 \n. Oscillations between asymmetric (dashed line in (a)) and symmetric \nmagnetic lattice s can be controlled externally using the magnetic bias fields. \nexternal magnetic  bias fields along the x- axis and/or \nfrom two -dimension to one-dimension configuration \nmagnetic bias field along the x- axis can make the \nensional to a one dimensional distribution . A combination of z \naxis external magnetic bias fields may allow simula ting condensed matter systems \nthe sites. For instance, it is possible to simulate the transition from/ \nby redistributing th e magnetically trapped exciton ic particles \naccording to accurately adjusted external magnetic bias fields (this description also \ntically trapped atomic species 19). \nTo induce the tunneling of the magnetically trapped  particles or to dis place the site \nmagnetic bias fields are \nof the bias fields, specifically  \neling barrier between the sites and hence increasin g \nnumerical simulation results  shown in Figs. 2 \n-axis. The external \nbetween the surface of the \nlocal field minima /g1828/g3040/g3036/g3041 , i.e., \n. As shown in Fig. 2 (e), /g1828/g3040/g3036/g3041/uni002F/g3040/g3028/g3051  can \nfrom the plane of the coupled quantum wells in whic h case \nresulting in reduced \nto deform the con fining \nsuch  effect allows the \nof the magnetic lattice . \n \n \na magnetic lattice subject to external \n/g3404/uni0009/g1828/g3053/g2879/g3029/g3036/g3028/g3046 /g3404/uni0030.  \n/g3029/g3036/g3028/g3046 /g3404/g1828/g3053/g2879/g3029/g3036/g3028/g3046 /g3404\nsymmetric  (dashed \nmagnetic bias fields.  \naxis and/or y-axis are used to \nconfiguration . As shown \naxis can make the lattice sites to \n. A combination of z -axis \naxis external magnetic bias fields may allow simula ting condensed matter systems \nit is possible to simulate the transition from/ to \nic particles  across the \naccording to accurately adjusted external magnetic bias fields (this description also \nExperimental Realizatio n of \nIn our recent experimental results \ndimensional magnetic lattice \nexperiments, the CQWs are grown by \nGaAs quantum wells separated by a 4 nm Al \n200 nm Al 0.33 Ga 0.67 As co nducting layer \nand at the bottom of the sample to monitor the elec tric field \ngenerating indirect-excitons. A\n(GGG) of thickness ≈3 /uni03BCm is deposited \nsystem. T he thickness of the \ndistance z min  for allocating the magnetic field \nquantum well layers. The permanent magnetic material \nof ≈ 2 /uni03BCm is deposited on top of the s \ntechnique.  \n \n(a) \n(d) \n \nFig. 5.  External magnetic x \ndimensional magnetic lattice \nbias fields such as the transition to a one dimensional configuration (\nthe same as those in Fig. 2 \n/uni0030/uni002C/g1828/g3051/g2879/g3029/g3036/g3028/g3046 /g3405/uni0030at /g1878/g3040/g3036/g3041 /g3406/uni0034/g2020/g1865 \n \nThe integrated magnetic- CQWs sample is placed in an optical cryostat with a  temperature \nfixed at 13K, and the emitted PL images are collecte \nmore details of the experiment \nself trapped according to the two dimensional distribution of th e confining magnetic field, as \nshown in Fig. 6(a). As simulated in Fig \nused to induce the bidirectional tunneling process between sites along the y \nthe observed 2D to 1D lattice \ngradually increasing the external magnetic \nreconfigure the shape of the magnetic lattice. \nchanges according to the value of the externally ap plied magnetic field \nredistribution in Fig. 6 strongly \nn of On-demand 2D to 1D Lattice Transition \nMagnetic Lattice  \nIn our recent experimental results 13, we observed the formation of \ndimensional magnetic lattice at a plane of coupled quantum wells. \ngrown by molecular beam epitaxy, where the \nGaAs quantum wells separated by a 4 nm Al 0.33 Ga 0.67 As barrier . This CQWs is \nnducting layer  on both sides. Metal contacts are deposited at the top \nand at the bottom of the sample to monitor the elec tric field  along the \nA nonmagnetic material gadolinium gallium garnet Gd \nm is deposited , using an rf-sputtering technique,  \nhe thickness of the GGG nonmagnetic spacer is used to determin \nthe magnetic field  trapping local minima and maxima wit \nlayers. The permanent magnetic material  (Bi 2Dy 1Fe 4Ga 1O12 \non top of the s ample of the (GGG + CQWs ) using rf \n (b)  (c) \n (e)  (f) \n \nxternal magnetic x -bias field-induced lattice sites across the x/y- plane. \ndimensional magnetic lattice can align themselves in specific patterns according  to the external \nthe transition to a one dimensional configuration ( a-f). Initial parameters are \nin Fig. 2 . (a) /g1828/g3051/g2879/g3029/g3036/g3028/g3046 /g3404/g1828/g3052/g2879/g3029/g3036/g3028/g3046 /g3404/uni0009/g1828/g3053/g2879/g3029/g3036/g3028/g3046 /g3404/uni0030. (b-f) /g1828/g3052/g2879/g3029/g3036/g3028/g3046 \n/g2020/g1865 . Black arrows indicate the direction of the applied field. \nCQWs sample is placed in an optical cryostat with a  temperature \nand the emitted PL images are collecte d using an objective microscope \nthe experiment al methods see ref. 13. The indirect ex citons are \naccording to the two dimensional distribution of th e confining magnetic field, as \nsimulated in Fig . 5, external magnetic bias field \nused to induce the bidirectional tunneling process between sites along the y \nlattice transition as shown in Fig. 6.  As shown  in Fig. 6 \nexternal magnetic  bias field on the trapped indirect excitons \nthe shape of the magnetic lattice. Because the number of trapped particles per site \nchanges according to the value of the externally ap plied magnetic field \nstrongly  supports the theoretical calculations in Fig. 5 \nTransition  in an Excitonic \n, we observed the formation of an excitonic two \nplane of coupled quantum wells.  For the present \nwhere the re are two 8 nm \n. This CQWs is  deposited by a \nMetal contacts are deposited at the top \nalong the z-direction for \nnonmagnetic material gadolinium gallium garnet Gd 3Ga 5O12  \n on top of the CQW \ndetermin e the effective \nlocal minima and maxima wit hin the \n12 ) with a thickness \n) using rf -sputtering \n \n \nplane. Sites in a two \ncan align themselves in specific patterns according  to the external \n. Initial parameters are \n/g3029/g3036/g3028/g3046 /g3404/g1828/g3053/g2879/g3029/g3036/g3028/g3046 /g3404\nBlack arrows indicate the direction of the applied field.  \nCQWs sample is placed in an optical cryostat with a  temperature \nd using an objective microscope : For \ncitons are  found to be \naccording to the two dimensional distribution of th e confining magnetic field, as \nexternal magnetic bias field along the x-axis is \nused to induce the bidirectional tunneling process between sites along the y -axis resulting in \nin Fig. 6  the effect of \non the trapped indirect excitons is to \nBecause the number of trapped particles per site \nchanges according to the value of the externally ap plied magnetic field , the exciton \nin Fig. 5 .  \nFig. 6.  Experimental results of the \nlattice configuration. The magnetic lattice is created with \n/g2028/g3404/uni0032/g2020/g1865 , and n = 9 . The \n/g1878/g3040/g3036/g3041 /g3406/uni0032/uni002E/uni0035/uni0009/g2020/g1865 . External magnetic bias \nincreased from 0 to 75 G, \n0.2V. (i) Gate voltage increases to \n \nAlthough the trapped excitonic gases \n6 is best interpreted in terms of a possibly reversed  cycle of self \nthe intra-sit es superfluid state \nexternal magnetic bias fields (\nA scan ning property of \nincreased number of trapped particles \nshows the shift of trap position \nwhere,  as a result, the external bias field causes \nfield direction, i.e., x- direction. \nsites along the y- axis for different values of the external \naxis. The number of trapped particles between sites \nthe value of the external bias field increases (alo ng x \nThe results in Figs. 6- 8 are in agreement with the simulation r \nmodification in the site’s position and \nobserved when applying external bias fields \nalso occurs along the y-axis at the field bias constr \n \n \n \nxperimental results of the  magnetically trapped indirect excitons showing \nThe magnetic lattice is created with /g2009/g3046/g3404/g2009/g3035/g3404/uni0032/g2020/g1865 \n. The local filed minima projected at the coupled quantum  wells level \nExternal magnetic bias  field is applied along the x- axis and gradually \nincreased from 0 to 75 G, /uni0009/g1828/g3052/g2879/g3029/g3036/g3028/g3046 /g3404/g1828/g3053/g2879/g3029/g3036/g3028/g3046 /g3404/uni0030/uni002C /g1828/g3051/g2879/g3029/g3036/g3028/g3046 /g3404/uni0030/uni2192/uni0037/uni0035/g1833 . (a-h) Gate volt \nincreases to  0.3V to emphasize the excitonic concentration in \nAlthough the trapped excitonic gases are not sufficiently cooled, the transition \nis best interpreted in terms of a possibly reversed  cycle of self -trapped excitonic \nes superfluid state 20-23. The reverse process can be achieved by decreasing th \n(to be discussed elsewhere).  \nning property of  selected lattice sites along the y-axis in Fig. \nincreased number of trapped particles with the external magnetic x- bias field \nposition s along the x- axis with increased number of trapped \nexternal bias field causes a slight shift of the sites along the applied \ndirection.  In Fig. 8 , trapped excitons are compared \naxis for different values of the external magnetic bias field applied along x \ntrapped particles between sites  (in the valley along y \nthe value of the external bias field increases (alo ng x -axis).  \n8 are in agreement with the simulation r esults \nposition and the shape along the x-axis of the lattice \nexternal bias fields . The two to one dimensional lattice transition \nat the field bias constr ains as simulated.  \n \nshowing 2D to 1D in \n /g1839/g3053/g3404/uni0032/uni002E/uni0038/g1863/g1833 , \nlocal filed minima projected at the coupled quantum  wells level  is \naxis and gradually \nGate volt age is \nconcentration in  1D space. \ntransition effect in Fig. \ntrapped excitonic clouds to \nreverse process can be achieved by decreasing th e \nin Fig. 7(a) shows \nbias field . Figure 7(b) \naxis with increased number of trapped  excitons \nslight shift of the sites along the applied \n, trapped excitons are compared  for two adjacent \nbias field applied along x -\nalong y -axis) increases as \nesults  of Fig. 5, a slight \nof the lattice can be \nto one dimensional lattice transition \n(a) \n(b) \nFig. 7.  (a) Measured PL intensity of the trapped indirect excitons \nlattice sites along the y- axis \nfield increases the number of trapped particles between \n1D magnetic lattice transformation \nbefore and after the external \noriginal positions with increased number of trapped particles. \nsame as in Fig. 6. \n \nFig. 8.  PL measurements \nmagnetic bias filed applied in x \nin the insets (1)- (4) more indirect \ndecreased magnetic field values between sites along  y \nsame as in Fig. 6. \n   \n \n \nPL intensity of the trapped indirect excitons across a selected numbe \naxis , before and after the external magnetic bias field . External x \nnumber of trapped particles between lattice sites (along the y -\ntransformation . (b) Measured PL intensity of the lattice sites \nexternal x-bias field. The sites are also noticed to be displaced from the ir \nwith increased number of trapped particles. Experimental conditions \nmeasurements of two adjacent lattice sites along the y-axis for different \napplied in x -axis. According to analytical calculations of the magneti c field \n(4) more indirect  excitons are trapped between the sites because of t he \ndecreased magnetic field values between sites along  y -axis. Experimental conditions \n \n \nacross a selected numbe r of \n. External x -bias \n-axis) for 2D to \nsites along x-axis \nThe sites are also noticed to be displaced from the ir \nExperimental conditions are the \n \ndifferent  external \nAccording to analytical calculations of the magneti c field \nexcitons are trapped between the sites because of t he \nExperimental conditions  are the \nEmphasizing the fact that the current approach is s uitable to simulate condensed \nmatter systems, one can consider the magnetic confi ning field to act as the potential confining \nfields between ions, and the magnetically trapped e xcitonic particles play the role of electrons. \nIn such a model the parameters are tunable without crystal-like disorders. The magnetic \nconfinement approach can be applied to accommodate a strongly correlated system such as \nthe transition of superfluid-Mott-insulator (MI) pr oposed by Jaksch et al. 17 and achieved in \natomic medium by Greiner et al.  24-26 . Moreover, the preparation of the lattice in a MI state is \na crucial step towards an efficient quantum registe r, where entanglement between the \nmagnetically trapped excitons can be achieved by co ntrolled collision using external \nmagnetic bias fields 27 . \n \nConclusion  \nA dimensional active control of a magnetic lattice of excitonic particles was \ndemonstrated. Magnetically trapped indirect exciton s in a two dimensional lattice \nconfiguration can be controlled to transit into a o ne dimensional magnetic lattice of excitons \nwith applied external magnetic bias fields. The dim entional transition is identified by \nmeasuring the spatial distribution of the 2D PL int ensity for several values of the external \nbias fields where the number of the trapped exciton s increases in the valley between sites, at a \ncertain direction, exhibiting a 1D lattice configur ation. In the current experiment, although \nthe trapped excitonic gases were not sufficiently c ooled to excitonic condensates (namely \nBose-Einstein condensation of excitons), the observ ed transition can be interpreted in terms \nof self-trapped excitonic clouds to intra-sites  superfluidity transition. \n \nAcknowledgment \nThis work was supported by the Creative Research In itiative Program (No. 2012-0000228) of the \nKorean Ministry of Education, Science and Technolog y via the National Research Foundation. BSH \nacknowledge that this work was also supported by th e Korea Communications Commission , S. \nKorea, under the R&D program supervised by the Kore a Communications Agency (KCA-2012-12-\n911-04-003) . \n \nReferences \n[1] M. Remeika, J. C. Graves, A. T. Hammack, A. D. Meyertholen, M. M. Fogler, L. V. Butov, M. Hanson, and \nA.  C. Gossard, Phys. Rev. Lett. 102 , 186803 (2009). \n[2] M. Remeika, M. Fogler, L. Butov, M. Hanson, and  A. Gossard, Appl. Phys. Lett. 100 , 061103 (2012). \n[3] Y. Kuznetsova, A. High, and L. Butov, Appl. Phy s. Lett. 97 , 201106 (2010). \n[4] High, A. A.; Leonard, J. R.; Remeika, M.; Butov , L. V.; Hanson, M.; Gossard, A. C. Nano Lett. 12 , 2605. \n(2012). \n[5] M. Hagn, A. Zrenner, G. Böhm, G. Weimann, Appl.  Phys. Lett. 67 , 232−235 ( 1995). \n[6] A. Gärtner, A. Holleitner, J. Kotthaus, D. Schu h, Appl. Phys. Lett. 89 , 052108 (2006). \n[7] S. Zimmermann, G. Schedelbeck, A. Govorov, A. W ixforth, J. Kotthaus, M. Bichler, W. Wegscheider, G . \nAbstreiter, Appl. Phys. Lett. 73 , 154−156 (1998). \n[8] Chen, G.; Rapaport, R.; Pffeifer, L. N. ; West, K.; Platzman, P. M. ; Simon, S.; Vörös, Z.; Snoke, D. Phys. \nRev. B, 74 , 045309 (2006). \n[9] J. A. K. Freire, A. Matulis, F. M. Peeters, V. N. Freire and G. A. Farias, Phys. Rev. B 61 , 2895 (2000). \n[10] J. A. K. Freire, F. M. Peeters, A. Matulis, V.  N. Freire, and G. A. Farias, Phys. Rev. B  62 , 7316 (2000). \n[11] Z. G. Koinov, G. Collins, and M. Mirassou, Phy s. Stat. Sol. (b) 243 , 4046 (2006). \n[12] A. Abdelrahman, H. Kang, S. Y. Yim, M. Vasilie v, K. Alameh, and Y. T. Lee, J. Appl. Phys. 110 , 013710 \n(2010). \n[13] A. Abdelrahman and Byoung S. Ham Phys. Rev. B 86 , 085445 (2012). \n[14] S. Whitlock, R. Gerritsma, T. Fernholz, and R.  Spreeuw, New J. Phys. 11, 023021 (2009). \n[15] M. Singh, R.l McLean, A. Sidorov, and P. Hanna ford, Phys. Rev. A 79 , 053407 (2009). \n[16] A. Abdelrahman, M. Vasiliev, K. Alameh, and P.  Hannaford, Phys. Rev. A 82, 012320 (2010). \n[17] D. Jaksch , C. Bruder, J. I. Cirac, C. W. Gard iner, and P. Zoller, Phys. Rev. Lett. 81 , 3108 (1998)  \n[18] D. M. Brink and C. V . Sukumar, Phys. Rev. A 74 , 035401 (2006) \n[19] D. Clément, N. Fabbri, L. Fallani, C. Fort and  M. Inguscio New. J. Phys. 11 , 103030 (2009). \n[20] S. Backhaus et al ., Science 278 , 1435 (1998). [21] D. S. Hall et al., Phys. Rev. Lett. 81 , 1543 (1998). \n[22] B. P. Anderson and M. A. Kasevich, Science 282 , 1686 (1998). \n[23] A. Smerzi, S. Fantoni, S. Giovanazzi, S.R. She noy, Phys. Rev. Lett. 79 , 4950 (1997). \n[24] M. Greiner, O. Mandel, T. Esslinger, T. W. Hän sch  and I.l Bloch , Nature 415 , 39 (2002). \n[25] W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simo n, J. I. Gillen, S. Fölling , L. Pollet and M. Grei ner, Science . \n329,  547 (2010). \n[26] M. Greiner, O. Mandel, T. Hänsch and I. Bloch,  Nature 419 , 51 (2002). \n[27] D. Jaksch, H.-J. Briegel, J. I. Cirac, C. W. G ardiner and P. Zoller, Phys. Rev. Lett. 82 , 1975 (1999). \n "    },    {        "title": "1506.01805v3.On_the_magnetization_process_of_ferromagnetic_materials.pdf",        "content": "On the magnetization process in ferromagnetic materials\nRuben Khachaturyan\u0003and Vahram Mekhitariany\nInstitute for Physical Research, NAS of Armenia, Ashtarak, Armenia\n(Dated: June 18, 2021)\nThe present article concludes that a ferromagnetic sample could be considered as a paramagnetic\nsystem where roles of magnetic moments play magnetic domains. Based on this conclusion and\ntaking into account presence of an anisotropy \feld the formula which describes magnetization de-\npendence on the external magnetic \feld is derived. Expressions for a remanent magnetization and a\ncoercive force are presented. The new parameter to characterize a magnetic sti\u000bness of a material is\nintroduced. A physical expression for a dynamic magnetic susceptibility as a function of materials\ncharacteristics, external magnetic \feld, and temperature is given.\nI. INTRODUCTION\nA physical theory permits correctly involve all inter-\nactions in a magnetization process and to reveal rela-\ntionship between structure and physical properties of a\nmagnetic material.\nApplicable mathematical model could be derived from\nsuch a theory. This model will give a possibility to inves-\ntigate real physical and structural properties of magnetic\nmaterials from experimental data. It is essential for syn-\nthesis of new materials with desired properties.\nNowadays there are several models for describing mag-\nnetization of ferromagnetic materials. More detailed de-\nscription and analyzes of advantages and disadvantages\nof these models one can \fnd in works [1]-[4].\nIn the present paper, there suggests a new theory of\nmagnetization and an attempt to derive an applicable\ngeneral mathematical model to describe magnetization\ncurve for soft and sti\u000b magnetic materials. Such generic\nspecial points in magnetization curve are also elicited\nfrom the model.\nA formula for dynamic magnetic susceptibility is de-\nrived from a magnetization \feld dependence.\nA new parameter which can numerically characterizes\nthe magnetic sti\u000bness of a material is introduced and its\nphysical interpretation is given.\nII. MAIN IDEA\nTwo competing interactions could be distinguished in\nferromagnetic materials: an exchange interaction ( exch)\nwhich tends to orient magnetic moments in the same di-\nrection and by this magnetizes the system and dipole-\ndipole interaction ( dip-dip ) which tends to orient mag-\nnetic moments antiparallel to each other and by this\ndemagnetize the system. A relevant di\u000berence between\nthese interactions is that exch acts between nearest atoms\nand its energy is independent of a magnetic moment of\n\u0003rubenftf@gmail.com\nyvahram.mekhitarian@gmail.coma system. On the contrary, dip-dip energy rises as mag-\nnetic moment increases. Increasing a material size a dip-\ndipenergy can overcome exch energy. In this case, two\nand more domains structures become more favorable. It\nis schematically shown in \fgure 1.\nAsexch energy is much bigger than dip-dip energy be-\ntween nearest atoms magnetic moments inside a domain\nare \frmly connected in the same direction. The magnetic\nshell where dip-dip energy becomes equal to exch energy\ncould be accepted as a border of the domain and by this\nde\fning a size of a domain [5]. So, after a domain was\ncompleted the next shell of magnetic moments will recline\nin the opposite direction, \fgure 1. It should be mentioned\nthat magnetic moments inside a domain are not strictly\ndirected in the same direction. They recline under an\nangle to each other from shell to shell until the domain\nwould not be \fnished. The transition layers behaviour\nis detailed described in the work of Landau-Lifshitz [6].\nBecause of it, a magnetic moment of a domain is less\nthan a sum of magnetic moments of atoms inside it.\n() dipdip E V−exchE const=E\nsize\nFIG. 1. Diagram representation how exch energy and dip-dip\nenergy behave with increasing of size.\nOn the assumption of foregoing we could consider do-\nmains like solitary magnetic particles. The separated\nquasiparticles we would call supermagneton ( sm) analog-\nically to R.Harrison [7]-[9] where domains are also sup-\nposed to be a unit magnetic moments in the magnetiza-\ntion processes.\nWe know that exch is compensated by dip-dip between\nsms. It means that sms magnetic moments are exempt\nfrom exch. So, the problem of ferromagnetic materials isarXiv:1506.01805v3  [cond-mat.mtrl-sci]  15 Jul 20162\nbrought to a problem of paramagnetic materials where\nsms play the role of magnetic moments.\nSeeing that there are axes of easiest magnetization in\nferromagnetic materials [10] sms are distributed in a \feld\nof anisotropy according to the Boltzmann distribution.\nFor simplicity, we would consider a case of uniaxial\nanisotropy. Sms with a positive projection on any se-\nlected direction along the anisotropy axis separated from\nsms with negative projection by anisotropy barrier, \fg-\nure 2.\nθaEθ\nθπ+aEE\nπ\n20π\nFIG. 2. Sms separated by anisotropy barrier.\nIII. III. MAGNETIZATION PROCESS\nBy applying magnetic \feld Hto a ferromagnetic ma-\nterial a magnetic \feld Bis induced inside. Sms with\npositive projection on the magnetic \feld direction obtain\nenergy - m+B, and sms with negative projection obtain\nenergy m\u0000B, where m+andm\u0000are magnetic moments\nofsms with positive and negative projection respectively.\nTaking into account that m++m\u0000= 2m,mis mag-\nnetic moment of sms in zero \feld, we can conclude that\nm\u0000B+m+B= 2mB. It means that domains could be\ne\u000bectively replaced by sms magnetic moment of which\nremains unchanged and equal to a magnetic moment of\ndomain in nonmagnetized state.\nSo, potential wells shift on a value 2 mB, as shown on\n\fgure 3.\nFIG. 3. Shift of potential wells in the induced \feld B.As a result of the energetic shift magnetization of the\nsample is appeared.\nTo estimate the magnetization one needs to calculate a\ndi\u000berence between magnetic moments with positive and\nnegative projections. Here and in further positive and\nnegative directions would be considered relative to the\ndirection in which external magnetic \feld is applied.\nTaking into account that at any \feld value distribution\nofsms is obey to the Boltzmann statistics we can calcu-\nlate a di\u000berence between amount of smswith negatie and\npositive projections:\nN1\u0000N2=\nZ\ng(E+Ea)e\u0000E\nkBTdE\u0000\n\u0000Z\ng(E\u0000Ea\u00002mB)e\u0000E\nkBTdE=\n=e\u0000Ea\nkBT\u0000e\u00002mB\u0000Ea\nkBT(1)\nN1+ N 2=e\u0000Ea\nkBT\u0010\n1 +e\u00002mB\u00002Ea\nkBT\u0011\n(2)\nBy the same way we get:\nN1+ N 2=e\u0000Ea\nkBT\u0010\n1 +e\u00002mB\u00002Ea\nkBT\u0011\n(3)\nDividing 2 on 3 we get:\nN1-N2=1\u0000e\u00002(mB+Ea)\nkBT\n1 +e\u00002(mB+Ea)\nkBT=\n= (N 1+N2) tanh\u00142 (mB\u0000Ea)\nkBT\u0015(4)\nMultiplying 4 on mand taking into account that by de\u000b-\ninition:\n\u001a\nMS=m(N1+ N 2)\nM=m(N1- N2)(5)\n\fnally we obtain:\nM=MStanh\u00142 (mB\u0000Ea)\nkBT\u0015\n(6)\nwhere MSis saturation magnetization and Mis magne-\ntization in \feld B.\nTo explore magnetization process more thoroughly it is\nnecessary to understand \feld distribution inside a mate-\nrial during magnetization. As direction of external \feld\nis given we will concentrate on distribution of dipolar\n\feld.\nThe magnetic \feld which is induced by magnetic mo-\nment in any point could be calculated in \frst approxima-\ntion as [12]:\n~H=3^n(~ m\u0001^n)\u0000~ m\nr3(7)3\nFIG. 4. Dipolar \feld distribution around the magnetic mo-\nment. aand bare lines where magnetic \feld change their\nprojection sing on magnetization direction.  is an angle\nbetween magnetization direction and lines aandb. 1, 3 are\nspace regions where dipolar \feld has positive projection on\nmagnetic moment direction and 2, 4 are space regions where\ndipolar \feld has negative projection on magnetic moment di-\nrection.\nwhere~ nis a unit vector in the direction to the point, and\nris distance between magnetic moment and the point\nwhere the \feld is calculated.\nIt is seen from 7 that Hhas negative or positive pro-\njection on the direction of magnetic moment in di\u000berent\npoints. It is not di\u000ecult to found points where Hchanges\nits projection sign on direction of the magnetization from\n7.\n3_n\u0010\n~M\u0001_n\u0011\n\u0000~M= 0 (8)\n8\n<\n:3\u0010\n~M_n\u00112\n\u0000~M_n= 0\n~M_n=Mcos ( )(9)\ncos ( ) =p\n3\n3(10)\n \u001955\u000e(11)\nThese points belong to the lines aandbwhich decline\nunder angles  to magnetization direction as shown on\n\fgure 4. Lines aandbdemarcate space around magnetic\nmoment on four regions: 1, 2, 3, 4.\nSo,Hhas positive projection in any point which be-\nlongs to regions 1 and 3 with biggest value when or and\nhas negative projection in any pint of region 2 and 4 with\nbiggest value ~H=2~ m\nr3when = 0\u000eor 180\u000e.\nBy this, dipolar \feld plays both magnetize (positive)\nand demagnetize (negative) roles. Due to positive in-\n\ruence of depolar \feld it is possible to magnetize bulk\nsamples.\nThus magnetic \feld inside a sample could be repre-\nsented as:\nB=H\u0000\u0011M\na3(12)where\u0011M\na3is a dipolar term, ais a distance between near-\nest domains domains (linear size of a sm),\u0011is a coe\u000ecient\nwhich represents di\u000berence between positive and negative\ndipolar in\ruences.\nAs an example, we will consider two-domain rod as\nshown on \fgure 5. Lets compare processes of magneti-\nzation of such a rod when the external magnetic \feld is\ndirected along longitude (x axis) and when the external\nmagnetic \feld is directed along width (y or z axes).\nz\nyx\nFIG. 5. Two domain rod.\nWhen the rod is magnetized along x axis the dipolar\n\feld of one domain will magnetize the second domain be-\ncause it belongs to the region of the space where dipolar\n\feld has positive projection on magnetization direction.\nThe diagram of the process is shown in \fgure 6. By this\ndipolar \feld will help to magnetize the system. In this\ncase s-shaped hysteresis loop would be observed [7]-[8].\nWhen the rod is magnetized along z axis one domain will\n \nx\nH\nFIG. 6. The rod is magnetized along longitude.\ndirect the second domain in the opposite direction be-\ncause it belongs to the region of the space where dipolar\n\feld has negative projection on magnetization direction\nas shown in \fgure 8. In this case \u0011= 1. Because of\nall written above, the magnetic energy term in 6 should\nbe replaced by \u0011mMa, where\u0011is a parameter which\ndepend on di\u000berence between demagnetizing and magne-\ntizing parts of dip-dip in\ruences, by this \u0011depends on\nshould depend on surface and surface volume ratio. At\neach certain values of external magnetic \feld the certain\ndistribution of \feld inside a material exists. This \feld\ndistribution changes with external \feld, and by this \u0011\nchange with external \feld as well.\nM=MStanh\u0014mH\u0000\u0011m\na3M+Ea(\r)\nkBT\u0015\n(13)4\n \nz\nH\nFIG. 7. The rod is magnetized along width.\nAs equation 13 is trancendent it could be rewritten in\nmore convenient form as done in [7]-[9]:\nH=\u0011\na3M+kBT\n2mln\u0012MS\u0000M\nMS+M\u0013\n+Ea(\r)\nm(14)\nIt should be noticed that the last term in 14 is indepen-\ndent on the value of magnetic moment as both magnetic\nmoment of the domains and anisotropy energy term are\nboth volume dependant.\nIV. REMANENT MAGNETIZATION AND\nCOERCIVE FORCE\nAfter the external magnetic \feld was abolished a dip-\ndiptends to demagnetize the sample. Because of this\nsms would turn from positive projection to negative\nthrough anisotropy barrier. But not all sms can over-\ncome anisotropy barrier and the part of them would re-\nmain with positive projection. So, after the external\nmagnetic \feld was abolished ferromagnetic sample would\nsteel remain in magnetize condition. This magnetization\nis called remanent magnetization. It is represented in\n\fgure 9. In order to get expression for remanent magne-\ntization it is necessary to put H=0 into 13:\nMR=MStanh\u0014\u0000\u0011m\na3MR+Ea(\r)\nkBT\u0015\n(15)\nwhere MRis remanent magnetization.\nAs is known a coercive force is a magnetic \feld which\nshould be applied to the sample to demagnetize it. So to\nget an expression of coercive force one need to put M=0\ninto 14:\nHC(T) =\u0000Ea(\r)\nm(16)\nIt is seen that coercive force depends on the \feld di-\nrection (angle between \feld and anisotropy axis).\nV. MAGNETIC STIFFNESS\nThe next parameter can characterize magnetic sti\u000bness\nof ferromagnetic materials:\nk=e\u00002\u0011mMRa\u0000Ea(\r)\nkBT (17)\n \nSRMM\naE\nEFIG. 8. sms overcome anisotropy barrier when sample change\nits state from saturated magnetization to remanent magneti-\nzation.\nDividing 2 on 3 it is possible to show that k=N2\nN1and\nrepresents the ratio of the amount of sms which overcame\nanisotropy barrier to the amount of sms which remain\nwith positive projection after the external magnetic \feld\nwas abolished, \fgure ??.\nValues of kcould change in the range from 0 to 1. The\nbigger the value of kthe softer magnetic material is and\nvice verse. For example, in case of strong anisotropy no\nofsms are able to overcome anisotropy barrier, it means\nthat there are no sms with negative projection ( k= 0)\nandMR=MSrectangular-like hysteresis loop, \fgure\n12 a). In the case when all sms were able to overcome\nanisotropy barrier, the amount of sms with negative pro-\njection is equal to the amount of positive projection ( k\n= 1 ) and consequently MR= 0, \fgure 12 c).\n \n)a\n)b\nFIG. 9. Three possible case of sms distribution in a state of\nremanent magnetization: a) k= 0; b)k= 1;; Cases a) and\nb) corresponds to a magneto sti\u000b material and magneto soft\nmaterials correspondingly.\nVI. DYNAMIC MAGNETIC SUSCEPTIBILITY\nA low which describes magnetization dependence on\nexternal magnetic \feld gives possibility to \fnd out a law\nfor magnetic susceptibility:\n\u001f=dM\ndH=MSd\ndHtanh\u0014m(H\u0000\u0011Ma) +Ea(\r)\nkBT\u0015\n=\n=MSm\u0001\u0010\n1\u0000d\u0011\ndHMa\u00004\u0019\u0011\na3\u001f\u0011\n+MSdEa(\r)/dH\nkBT\u0001ch2h\nm(H\u0000\u0011Ma)+Ea(\r)\nkBTi\n(18)5\n\u001f=MSh\nm\u0010\n1\u0000d\u0011\ndHMa\u0011\n+dEa(\r)/dHi\nkBTch2h\nm(H\u0000\u0011Ma)+Ea(\r)\nkBTi\n+\u0011\na3mMS(19)\nIt should be noted that \u0011decreases with increasing of\nHthusd\u0011/dHM < 0.\nThis term is one of factors which are responsible for\nhigh value of magnetic susceptibility of ferromagnetic\nmaterials.\nIt is seen that a magnetic susceptibility depends on\nthe external magnetic \feld, the dipolar interaction, the\ntemperature of the sample and the anisotropy energy, as\nit was expected.\nIn case of high temperature or small external magnetic\n\feld (tanh ( x)!x;\u0011!1) from 19 one can get:\n\u001f=MS\u0010\nm+dEa(\r)/dH\u0011\nkBT+mMS\u000e\na3(20)\nVII. CONCLUSION\nThere was shown that a magnetic sample could be con-\nsidered as a set of magnetic particles, called superpara-\nmagnetons ( sms). Magnetic moment of smis equal to\na magnetic moment of domain plus magnetic moment\nof the domain wall, before external magnetic \feld was\napplied. The stark di\u000berence of the smis that it's mag-\nnetic moment doesn't change during magnetisation pre-cess and sms are free from exchange interaction what\ngives a possibility to apply the Boltzmann statistics for\nthem as it done for paramagnetic samples.\nBased on this assumption there was derived the analyt-\nical excretion to describe a dependence of magnetization\nof a ferromagnetic material on an internal 6.\nConsidering dipolar \feld distribution inside a material\nan analytical expression of magnetization dependence on\nan external magnetic \feld 13 is also deduced.\nIt is important that all energies that take place in\nthe process of magnetization are included additively. It\nmeans that additional energies like energies on domain\nwalls pinning could be easily added in the formula in 13.\nThere were derived expressions for a remanent mag-\nnetization 15 and a coercive force 16 as special points\nonM(H) curve. It is seen that temperature dependence\nof remanent magnetization bears exponential character\nand depend on dip-dip in a sample and on an anisotropy\nbarrier. It is also seen how coercive force depend on\nanisotropy barrier and a magnetic moment of a domain.\nA new parameter which characterizes a magnetic sti\u000b-\nness of a material and its temperature dependence is in-\ntroduced 17.\nFrom M(H) function there was derived an expression\nfor a magnetic susceptibility 19. It is shown that in ex-\ntremal cases, like high temperature or low \feld, magne-\ntization depends on \feld linearly, and magnetic suscepti-\nbility is \feld independent like in the paramagnetic case.\nIt is essential to note that expressions 19 20 are appli-\ncable at all temperature regions.\n[1] D.C Jiles, X. Fang, W. Zhang, Handbook of Advanced\nMagnetic Materials , (2006).\n[2] F.Liorzou, B. Phelps, and D.L. Atherton, Macroscopic\nmodels of magnetization ,IEEE transactions on magnet-\nics, vol.36, No.2., (2000).\n[3] Sergey E. Zirka, Yuriy L. Moroz, Robert G. Harrison, and\nKrzysztof Chwastek, On the physical aspects of the Jiles-\nAtherton hysteresis models , J. Appl. Phys. 112, (2012).\n[4] D.C. Jiles and D.L. Atherton, \"Theory of ferromagnetic\nhysteresis\", J. Magn. Magn. Mater. 61, 48, (1986).\n[5] J. Frankel and J. Dorfman; \"Spontaneous and Induced\nMagnetisation in Ferromagnetic Bodies\", (1930).\n[6] L. Landau, E. Lifshits, \"On the theory of the dispersion of\nmagnetic permeabillity in ferromagnetic bodies\", Phys.\nZeitsch. der Sow. 8, pp. 153-169, (1935).\n[7] R.G. Harrison, \"Physical model of spin ferromagnetism\",\n(2003).[8] R.G. Harrison, \"Variable-Domain-Size theory of spin fer-\nromagnetism\", (2004).\n[9] R.G. Harrison, \"Physical theory of ferromagnetic \frst-\norder return curves\", IEEE TRANSITION ON MAG-\nNETICS, Vol. 45 NO, 4, (2009).\n[10] L. D. Landau and E. M. Lifshitz, Course of Theoretical\nPhysics, Vol. 8, (2005).\n[11] C. Kittel, \"Theory of the structure of ferromagnetic do-\nmains in \flms and small particles\", Phys.Rev. vol.70,\nNO. 11 and 12, (1946).\n[12] L. D. Landau and E. M. Lifshitz, Course of Theoretical\nPhysics, Vol. 2: The Classical Theory of Fields (Nauka,\nMoscow, 1988; Pergamon, Oxford, 1975).\n[13] Alberto P. Gimaraes, Principles of Nanomagnetism,\nSpringer, 2009."    },    {        "title": "1911.06031v1.Suspensions_of_magnetic_nanogels_at_zero_field__equilibrium_structural_properties.pdf",        "content": "Suspensions of magnetic nanogels at zero field: equilibrium structural\nproperties\nIvan S. Novikaua,<, Elena S. Mininaa, Pedro A. Sánchezb,cand Sofia S. Kantorovicha,b\naComputational Physics, University of Vienna, Sensengasse 8, Vienna, Austria.\nbUral Federal University, 51 Lenin av., Ekaterinburg, 620000, Russian Federation.\ncInstitute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf e.V., D-01314 Dresden, Germany.\nARTICLE INFO\nKeywords :\nmagnetic nanogels\nmagnetic self-assembly\nLangevin Dynamics\nstructure factorABSTRACT\nMagnetic nanogels represent a cutting edge of magnetic soft matter research due to their numerous\npotentialapplications. Here,usingLangevindynamicssimulations,weanalysetheinfluenceofmag-\nneticnanogelconcentrationandembeddedmagneticparticleinteractionsontheself-assemblyofmag-\nnetic nanogels at zero field. For this, we calculated radial distribution functions and structure factors\nfor nanogels and magnetic particles within them. We found that, in comparison to suspensions of\nfree magnetic nanoparticles, where the self-assembly is already observed if the interparticle inter-\naction strength exceeds the thermal fluctuations by approximately a factor of three, self-assembly\nof magnetic nanogels only takes place by increasing such ratio above six. This magnetic nanogel\nself-assemblyisrealisedbymeansoffavourableclosecontactsbetweenmagneticnanoparticlesfrom\ndifferent nanogels. It turns out that for high values of interparticle interactions, corresponding to the\nformationofinternalringsinisolatednanogels,intheirsuspensionslargermagneticparticleclusters\nwithlowerelasticpenaltycanbeformedbyinvolvingdifferentnanogels. Finally,weshowthatwhen\ntheself-assemblyofthesenanogelstakesplace,ithasadrasticeffectonthestructuralpropertieseven\nif the volume fraction of magnetic nanoparticles is low.\n1. Introduction\nThe concept of microgel is almost 70 years old [1]. The\ntermreferstoacolloidalsoftparticlemadeofapermanently\ncrosslinkednetworkofpolymers,whosesizecanrangefrom\ntensofnanometerstoseveralmicrometers[2,3]. Microgels\ncan be made responsive to different stimuli, as temperature,\npH or external fields [4, 5, 6, 7]. As a result, they have a\ngreat potential for many technological and bio-medical ap-\nplications[8,9]. Microgelsunder í100nmindiameter,also\nknownasnanogels,areespeciallypromisingfordrugdeliv-\nery [10]. Nowadays, the amount of studies devoted to these\nsystems is growing very fast [11, 12].\nAmong the different responsive behaviours obtained for\nmicro- and nanogel particles, the response to external mag-\nnetic fields is particularly appealing. This is achieved ex-\nperimentallybyembeddingmagneticnanoparticles(MNPs)\nintothepolymernetwork[13,14]. Despitesuchinterest,the\ntheoretical understanding of magnetic micro- and nanogels\nis still rather limited due to the challenges involved in their\nmodeling.\nRecently,weintroducedacoarse-grainedcomputersim-\nulationmodelofmagneticnanogelsthatallowedustostudy\nthe influence of the magnetic filler concentration on the\nstructure of single nanogel particles [15]. Regarding the\npolymer network, the model represents qualitatively the in-\nternal structure of nanogels obtained by electrochemically\nor photonically induced crosslinking of polymer precursors\n<Corresponding author\nivan.novikau@univie.ac.at (I.S. Novikau); r.p.sanchez@urfu.ru\n(P.A. Sánchez); sofia.kantorovich@univie.ac.at (S.S. Kantorovich)\nORCID(s): 0000-0003-0841-6820 (P.A. Sánchez); 0000-0001-5700-7009\n(S.S. Kantorovich)confined in emulsion nanodroplets [16, 17, 18].\nIn this study we employ the aforementioned coarse-\ngrained nanogel model to investigate the equilibrium struc-\nturalpropertiesofmagneticnanogelsuspensionsinabsence\nof an applied external field. To the best of our knowledge,\nno theoretical study on magnetic nanogel suspensions is yet\navailable in literature.\nThestructureofthepaperisthefollowing. First,inSec-\ntion2,webrieflydescribethemodelandsimulationmethod.\nIn Section 3, we discuss the influence of nanogel concen-\ntration and the impact of magnetic interaction strength on\nthe structural properties of nanogels suspensions. Finally, a\nbrief summary of the work can be found in Section 4.\n2. Simulation Approach\nOur nanogel model is based on a bead-spring represen-\ntation of the polymer chains and the embedded magnetic\nparticles [18]. Briefly, the setup of each nanogel particle is\nperformed in the following way. First, polymer precursors\nare modelled as chains of spherical beads with unit dimen-\nsionless mass and diameter. They have a steric repulsion\ngivenbyatruncatedandshiftedLennard-Jonespotential,or\nWeeks-Chandler-Andersen (WCA) potential [19]:\nUWCA.r/ =<\n4\u0004r*12*r*6\u0005+ 1; rÍ21_6\n0; r> 21_6;(1)\nwhereristhecentre-to-centredistancebetweentheinteract-\ning particles. Here, we set the depth of the Lennard-Jones\npotential well to unity, thus introducing a scaling for all en-\nergiesinthesimulationprotocol. Thesebeadsformthepoly-\nmer chain backbones by means of FENE springs connected\nI. S. Novikau et al.: Submitted preprint Page 1 of 5arXiv:1911.06031v1  [cond-mat.soft]  14 Nov 2019Suspensions of magnetic nanogels at zero field\nFigure 1: Bead-spring representation of the internal structure\nof a model magnetic nanogel. Arrows inside particles indicate\nthe presence of a point magnetic dipole.\nto their centres:\nUFENE .r/ = *1\n2\u000ffr2\nflnL\n1 *0\nr\nrf12M\n; (2)\nwhere\u000ff= 22:5is the dimensionless interaction strength\nandrf= 1:5isthemaximumbondextension. Nanogelpar-\nticles are obtained by equilibrating Np= 6polymer chains\nwithL= 100beads each inside a spherical confinement\nwallwithvolumefractionofapproximately \u001epù 0:1. After\nequilibration,interchaincrosslinksarerandomlyintroduced\naccordingtoaminimuminterparticledistancecriterium,up\nto reach a fraction of crosslinks of \u001elinks = 0:17. Each\ncrosslink consists in a elastic spring connecting the centres\nof the newly bonded pair of particles. In order to speed up\nthe crosslinking process, harmonic springs are used for this\npurpose:\nUh.r/ = *1\n2Kr2: (3)\nBy usingK= 10, we ensured that the mechanical effect of\nthesespringsisequivalenttotheFENEbondsconnectingthe\nprecursor backbones. For further details on the crosslink-\ning protocol see Reference [18]. Regarding the magnetic\nparticles, for simplicity they are introduced by assigning a\npermanentmagneticdipole,  \u0016,atthecentreofrandomlyse-\nlected beads, up to a fraction of \u001em= 0:1. Therefore, these\nmagnetic beads interact by means of the dipole-dipole pair\npotential:\nUdd\u0000 rij\u0001=\u0000 \u0016i\f \u0016j\u0001\nr3*3\u0000 \u0016i\f rij\u0001\u0000 \u0016j\f rij\u0001\nr5;(4)\nwhere \u0016i, \u0016jaretherespectivedipolemomentsoftheinter-\nacting particles and  rijis the displacement vector connect-\ning their centres. These long range magnetic interactions\nwere calculated using the dipolar P3M algorithm [20]. Fig-\nure 1 shows a sketch of this bead-spring representation of\nour model magnetic nanogel particles.\nFinally, suspensions were simulated by placing 100\nnanogel particles obtained from the procedure described\nabove into a periodic cubic box with a volume fraction of\nbeads fixed either to 0.1 or 0.2. It is worth noting that each\nindividual nanogel was previously equilibrated. In total,10 different equilibrium configurations with different cross-\nlinkerandmagneticparticleintrinsicdistributionswereused\nto form the suspension. This allowed us to avoid the de-\npendence of self-assembly on individual nanogel topology.\nMolecular dynamics simulations of such systems were per-\nformed with the simulation package ESPREsSo [21]. A\nLangevin thermostat with fixed dimensionless temperature\nT= 1was used to mimic the thermal fluctuations of the\nbackgroundfluid. Thesystemwasfirstequilibratedbymak-\ning2\f107integrationsteps,usingafixedtimestep \u000et= 0:01.\nsubsequent measurements were obtained for 8  107inte-\ngration steps. Each set of parameters was sampled with 5\nindependent runs using different initial configurations.\nNote that, in the system of dimensionless units defined\nabove, the conventional dipolar coupling parameter, that\nmeasures the ratio between the dipole-dipole interactions\nand the thermal fluctuations, can be simply defined as \u0015=\n\u00162.\n3. Results and Discussions\nWe start the analysis with the visual inspection of the\nequlibrated suspensions. As long as the system is rather\ncrowded, in Fig. 2(a), we show only the magnetic particles\nexplicitly,whereasthewholestructuresofeachnanogelpar-\nticle is represented by its convex hulls. This snapshot was\nobtainedfor \u0015= 6andoverallvolumefractionof0.1. Inthe\nlowestpartofthesnapshot(closetothecentre),onecanfind\ntwo nanogels that seem to form a cluster. From our previ-\nous work [15], it is known that inside an isolated magnetic\nnanogelwith \u0015= 6,magneticparticlestendtoself-assemble\nandformlongchainsclosetotheperipheryofthenanogelto\nminimise the curvature. However, the translational motion\nrequired for self-assembly is often penalised by the elastic\nnetwork. In case of suspension, dipolar energy can be ad-\nditionally minimised if magnetic particles close to the sur-\nfaceofonenanogelformfavourablecontactswithmagnetic\nnanoparticlesofaneighbouringnanogel,thus,leadingtothe\nnanogel self-assembly, as shown in Fig. 2(b).\nInordertounderstandhowintensivelythenanogelsself-\nassemble, in Figs. 3(a) and 3(b), we plot radial distribution\nfunctionscalculatedfornanogelscentresofmass. Foravol-\nume fraction of 10% (light blue curve, Fig. 3(a)), one can\nhardly see any signature of self-assembly. The first peak of\nthe RDF is rather small and the first minimum is not very\npronounced. The situation changes for higher nanogel con-\ncentrations. Doubling the volme fraction results in a very\nclearlypronouncedfirstpeakat r= 12. Itisworthmention-\ningherethattheaveragenanogelradiusofgyrationis Rg= 6\nandremainedunchangedduringthesimulation( ,5~ofthe\ninitialvalue). Thus, r= 12correspondstotheclosecontact\nof two nanogels similar to the one shown in Fig. 2(b). To\nexclude the possibility of having simply density fluctuation\neffects captured by the RDF, we also calculate the RDFs of\nWCA spheres suspensions with radii equal to Rgand plot\nthem with thin red lines in Fig. 3(a). The comparison be-\ntween WCA spheres and actual magnetic nanogels clearly\nI. S. Novikau et al.: Submitted preprint Page 2 of 5Suspensions of magnetic nanogels at zero field\n(a)\n (b)\nFigure 2: (a) Simulation snapshot of a nanogel suspension.\nVolume fraction is 10 %. \u0015= 6. Only magnetic particles are\nshownexplicitly, nanogelsarerepresentedbytheirconvexhulls.\n(b) Zoomed in three nanogels with all beads shown explicitly.\nMagnetic particles are shown in red, nonnmagnetic beads of\neach nanogel have the same colour (light blue, blue and dark\nblue).\nreveals the signature of self-assembly in the latter for the\ncase 20% volume fraction: the existence of a second max-\nimum and a deep first minimum exhibited by the RDF of\nnanogel suspension. Apart from concentration impact, self-\nassemblyof magneticnanogels canbealso tunedbychang-\ning the interaction strength between MNPs. This is illus-\ntrated in Fig. 3(b), where we plot the RDFs calculated for\nthe centres of mass of nanogels with different interaction\nstrengths for their magnetic beads. In case of \u0015= 4, the\nRDF has a perfect shape of a noninteracting WCA-sphere\nliquid. In contrast, for \u0015= 8the self-assembly is clearly\ntaking place.\nFurthermicroscopicinformationcanberevealedbycal-\nculating the RDFs corresponding only to the magnetic par-\nticles. This result is shown in Figs. 3(c) and 3(d). As ex-\npected,magneticparticlesinnanogelsandacrossthemself-\nassembleandthetendencytoformlargerclustersgrowswith\nincreasingnanogelconcentrationaswellaswithvalueof \u0015.\nThus, for example, for \u0015= 8(dark blue in Fig. 3(d)), one\nfindswell-definedpeaksuptothefourthcoordinationshell,\nmeaning that the chain-like magnetic structures formed in\nthe system are rather frequent and have a significant length.\nTo answer the question whether the self-assembly of\nmagnetic nanogels can be detected experimentally, we cal-\nculated their structure factors (SFs). Analogously to RDFs,\nwecomputedSFsforoverallnanogelsandformagneticpar-\nticles only. Note these SFs, presented in Fig. 4, are cal-\nculated directly from the simulation data using the proce-\ndure described in Reference [22] and not as Fourier trans-\nform of the RDFs from Fig. 3. The first peaks, qí 0:5, in\nFigs.4(a)and4(b)correspondtoapproximately 2Rgí 12in\nrealspace. Theyreflectthenumberofnanogelpairsinclose\ncontact. Theirheightgrowsandtheirpositionshiftsslightly\nto the right with both, increasing concentration (4(a)) and\ngrowing value of \u0015(4(b)). The shift of the peaks can be ex-\nplained by two effects: first, the radius of gyration of mag-\n(a)\n (b)\n(c)\n (d)\nFigure 3: Radial distribution functions (RDFs.) (a) and (b) –\nRDFs calculated for the nanogels centres of mass. Nanogels\nradius of gyration is on average equal to six. (c) and (d) –\nRDFs computed using exclusively magnetic particles. (a) and\n(c) show the differences in RDFs caused by changes in nanogel\nvolume fraction; (b) and (d) show the impact of \u0015on the\nRDFs. RDFs for equivalent suspensions of WCA spheres (SS)\nare also included in (a).\nnetic nanogels decreases with \u0015[15]; second, when form-\ningacontactthroughmagneticnanoparticles,thesenanogels\ngetdeformedandcaninterpenetratetoacertainextent. The\nscale of these effects can be estimated from the shift of the\nSF first peak to be around 10% for \u0015= 6in case of grow-\ning volume fraction, or to be around 20 % if the value of \u0015\nchangesfrom6to8andthevolumefractionisfixedto10%.\nFig.4(c)showstheSFofmagneticnanoparticlescalculated\nfor different volume fractions and fixed \u0015= 6. It is clearly\nseen that the overall volume fraction affects only the region\nofsmallq. Infact,doublingthevolumefractionofmagnetic\nnanogels seems to not affect qualitatively the self-assembly\nof magnetic nanoparticles, as it was also seen in Fig. 3(c).\nFor\u0015= 4, the shape of SF (light blue curve, Fig. 4(b)) sug-\ngests that the self-assembly is insignificant. This observa-\ntion is confirmed by Fig. 4(d), where the peak at qí 7,\ncorresponding to the close contact of two magnetic beads,\nis negligible. In contrast, for \u0015= 8we can clearly observe\nthe latter peak. In Figs. 4(c) and 4(d) SFs have 2 peaks for\nq<2. Inrealspacethiscorrespondstodistanceslargerthan\ní 3. Whereas the first peak at qí 0:5is clearly related\nto the close contact of two nanogels at distance 2Rgí 12,\nthe second peak is more difficult to interpret. Correspond-\ningtorealdistancesontheorderof Rg,mostprobablyshows\nthecorrelationlengthofmagneticparticlesinsideindividual\nnanogels.\nFinally, having analysed all evidences of magnetic\nnanogel self-assembly and described the qualitative trends,\nI. S. Novikau et al.: Submitted preprint Page 3 of 5Suspensions of magnetic nanogels at zero field\n(a)\n (b)\n(c)\n (d)\nFigure 4: Structure factors (SFs.) (a) and (b) – SFs calculated\nfor whole nanogels structures. (c) and (d) – SFs computed\nusing exclusively magnetic particles. (a) and (c) show the dif-\nferences in SFs caused by changes in nanogel volume fraction;\n(b) and (d) show the impact of \u0015on the SFs.\n\u0015= 4\u0015= 6, 10 %\u0015= 8\u0015= 6, 20 %\nMNPs 3.7 7.6 20.0 7.5\nnanogels 0.2 2.1 5.2 2.6\nTable 1\nCluster sizes measured for MNPs only (upper row) and for\nwhole nanogels (lower row). For cluster definition, see the\nmain text. The errorbars for these values are below 5 per cent.\ninTable1wecollectmeanclusterssizesmeasuredfor,both\nmagneticnanogelsandindividualmagneticparticles,forall\nsystems investigated here. For magnetic particles we used\nthe distance-energy criteria of Reference [22], whereas for\nwhole nanogels pairs we considered them to be connected\nonlyiftheyhadasharedclusterofmorethanthreemagnetic\nparticles, with at least two of them belonging to each of the\nnanogels. One can see that all the implicit evidences de-\nscribed above are fully confirmed by the clusters sizes: we\nsee no aggregation for magnetic nanogels for \u0015= 4, even\nthough, magnetic nanoparticles do agregate; for \u0015= 6we\nsee a moderate level of nanogel self-assembly enhanced by\nthe increase of the volume fraction; for \u0015= 8the nanogel\nself-assembly is very pronounced. It is worth noting here\nthat a cluster of 5 nanogels is a relatively large object that\ncannot but affect rheological and mechanical properties of\nthesesuspensions. Thestudyofthesepropertiesiscurrently\nin progress.\n4. Conclusions\nIn this paper we thoroughly analysed the influence of\nmagneticnanogelconcentrationandmagneticparticleinter-actions on the self-assembly of magnetic nanogels in zero\nfield. Our results show that, whereas \u0015g4can be con-\nsidered as strong self-assembly conditions for free mag-\nneticnanoparticles,magneticnanogelscontainingthesepar-\nticles only start forming clusters at \u0015g6. It is worth\nreminding here that for individual magnetic nanogels, if\nthe value of \u0015exceeds 6-7, the initial susceptibility de-\ncreases due to the fact that polymer matrix cannot hinder\nanymore the formation of rings of magnetic nanoparticles\ninside the nanogel [15]. It was found here that this effect\nvanished in suspensions due to the possibility of forming\nlarger magnetic clusters with lower elastic penalty involv-\ningparticlesfromdifferentnanogelsinclosecontact. These\nconnections were shown to be responsible for nanogel self-\nassembly. The valuesof \u0015for whichreal nanogelswill start\nforming bridges, containing magnetic particles, and self-\nassemble will, however, strongly depend on the crosslink-\ning degree of the nanonogels, as the latter parameter was\nshown to have a nontrivial impact on the intrinsic mag-\nneticparticleself-assemblywithinindividualnanogels[15].\nThe formation of bridges, albeit only qualitatively, was re-\nported recently in Ref. [23] (see, Fig. 4a). The degree\nofclusterisationofmagneticnanoparticlesinsideindividual\nnanogels, as well as between them, can be verified by scat-\ntering techniques [24, 25] as we show by calculating mag-\nnetic nanogel structure factors. Our findings revealed three\nlength-scales inherent to magnetic nanoparticles in these\nsystems. The largest scale (region of small wave vectors q)\ncorrespondstotwicethenanogelradiusofgyrationandcan\nbeattributedtocorrelationsofmagneticnanoparticlesfrom\ndifferent nanogels forming clusters; the second length-scale\ncorresponds to the nanogel radius of gyration and may re-\nflectthecorrelationlengthofthemagneticnanoparticlesin-\nside one nanogel; finally the smallest scale, corresponding\nto high values of wave vectors q, shows close contacts of\nnanoparticles. Heights and positions of all three SF peaks\nwere shown to be sensitive to the value of \u0015. Interest-\ningly,onlytheregionofsmall qwasfoundtodependonthe\nnanogel volume fraction. The latter, however, plays a sig-\nnificant part in the amplitude of SFs if calculated for whole\nnanogel structures. Summarising our findings, one can ex-\npect cluster formation in suspension of magnetic nanogels,\nin absence of external magnetic fields, only if the interac-\ntion between embedded ferromagnetic nanoparticles is sig-\nnificantly higher than the values of thermal energy. How-\never,iftheself-assemblytakesplace,itmighthaveadrastic\neffectonthestructuralpropertiesevenforrelativelylowvol-\nume fraction of the magnetic filler.\n5. Acknowledgements\nThisresearchhasbeensupportedbytheRussianScience\nFoundation Grant No.19-12-00209. Authors acknowledge\nsupport from the Austrian Research Fund (FWF), START-\nProjekt Y 627-N27. Computer simulations were performed\nat the Vienna Scientific Cluster (VSC-3).\nI. S. Novikau et al.: Submitted preprint Page 4 of 5Suspensions of magnetic nanogels at zero field\nReferences\n[1] W.O.Baker,Microgel,anewmacromolecule,Ind.Eng.Chem.41(3)\n(1949) 511–520 (1949). doi:10.1021/ie50471a016 .\n[2] A. Fernandez-Nieves, H. M. Wyss, J. Mattsson, D. A. Weitz (Eds.),\nMicrogel Suspensions: Fundamentals and Applications, WILEY-\nVCH Verlag & Co. KGaA, Weinheim, Germany, 2011 (2011). doi:\n10.1002/9783527632992 .\n[3] Y. B. Hamzah, S. Hashim, W. A. W. A. Rahman, Synthesis of poly-\nmeric nano/microgels: a review, J. Polym. Res. 24 (9) (2017) 134\n(2017). doi:10.1007/s10965-017-1281-9 .\n[4] I. Gorelikov, L. M. Field, E. Kumacheva, Hybrid microgels pho-\ntoresponsive in the near-infrared spectral range, J. Am. Chem. Soc.\n126 (49) (2004) 15938–15939 (2004). doi:10.1021/ja0448869 .\n[5] P. S. Mohanty, P. Bagheri, S. Nöjd, A. Yethiraj, P. Schurtenberger,\nMultiple path-dependent routes for phase-transition kinetics in ther-\nmoresponsive and field-responsive ultrasoft colloids, Phys. Rev. X 5\n(2015) 011030 (2015). doi:10.1103/PhysRevX.5.011030 .\n[6] H.Kobayashi,R.G.Winkler,Universalconformationalpropertiesof\npolymers in ionic nanogels, Sci. Rep. 6 (2016) 19836 (2016). doi:\n10.1038/srep19836 .\n[7] S. Backes, P. Krause, W. Tabaka, M. U. Witt, D. Mukherji, K. Kre-\nmer, R. von Klitzing, Poly(n-isopropylacrylamide) microgels under\nalcoholic intoxication: When a LCST polymer shows swelling with\nincreasing temperature, ACS Macro Lett. 6 (10) (2017) 1042–1046\n(2017). doi:10.1021/acsmacrolett.7b00557 .\n[8] H. A. Son, S. K. Choi, E. S. Jeong, B. Kim, H. T. Kim, W. M. Sung,\nJ. W. Kim, Microbial activation of bacillus subtilis-immobilized mi-\ncrogel particles for enhanced oil recovery, Langmuir 32 (35) (2016)\n8909–8915 (2016). doi:10.1021/acs.langmuir.6b02010 .\n[9] J.Pu,J.Geng,P.Han,B.Bai,Preparationandsalt-insensitivebehav-\nior study of swellable, Cr3+-embedded microgels for water manage-\nment,J.Mol.Liq.273(2019)551–558(2019). doi:10.1016/j.molliq.\n2018.10.070 .\n[10] S. V. Vinogradov, Nanogels in the race for drug delivery,\nNanomedicine5(2)(2010)165–168(2010). doi:10.2217/nnm.09.103 .\n[11] L. Rovigatti, N. Gnan, L. Tavagnacco, A. J. Moreno, E. Zaccarelli,\nNumericalmodellingofnon-ionicmicrogels: anoverview,SoftMat-\nter 15 (2019) 1108–1119 (2019). doi:10.1039/C8SM02089B .\n[12] A. Martín-Molina, M. Quesada-Pérez, A review of coarse-grained\nsimulationsofnanogelandmicrogelparticles,J.Mol.Liq.280(2019)\n374–381 (2019). doi:10.1016/j.molliq.2019.02.030 .\n[13] L. Jiang, Q. Zhou, K. Mu, H. Xie, Y. Zhu, W. Zhu, Y. Zhao, H. Xu,\nX.Yang,ph/temperaturesensitivemagneticnanogelsconjugatedwith\ncy5.5-labledlactoferrinformrandfluorescenceimagingofgliomain\nrats, Biomaterials 34 (30) (2013) 7418 – 7428 (2013). doi:10.1016/\nj.biomaterials.2013.05.078 .\n[14] S.Backes,M.U.Witt,E.Roeben,L.Kuhrts,S.Aleed,A.M.Schmidt,\nR. von Klitzing, Loading of pnipam based microgels with cofe2o4\nnanoparticles and their magnetic response in bulk and at surfaces, J.\nPhys. Chem. B 119 (36) (2015) 12129–12137 (2015). doi:10.1021/\nacs.jpcb.5b03778 .\n[15] E. S. Minina, P. A. Sánchez, C. N. Likos, S. S. Kantorovich, The in-\nfluenceofthemagneticfillerconcentrationonthepropertiesofami-\ncrogel particle: Zero-field case, J. Magn. Magn. Mater. 459 (2018)\n226–230 (2018). doi:10.1016/j.jmmm.2017.10.107 .\n[16] A. Galia, S. Lanzalaco, M. A. Sabatino, C. Dispenza, O. Scialdone,\nI. Sirés, Crosslinking of poly(vinylpyrrolidone) activated by electro-\ngenerated hydroxyl radicals: A first step towards a simple and cheap\nsyntheticrouteofnanogelvectors,Electrochem.Commun.62(2016)\n64–68 (2016). doi:10.1016/j.elecom.2015.12.005 .\n[17] S. Mavila, O. Eivgi, I. Berkovich, N. G. Lemcoff, Intramolecular\ncross-linking methodologies for the synthesis of polymer nanoparti-\ncles, Chem. Rev. 116 (3) (2016) 878–961 (2016). doi:10.1021/acs.\nchemrev.5b00290 .\n[18] E. S. Minina, P. A. Sánchez, C. N. Likos, S. S. Kantorovich, Study-\ningsynthesisconfinementeffectsontheinternalstructureofnanogels\nin computer simulations, Journal of Molecular Liquids 289 (2019)\n111066 (2019). doi:10.1016/j.molliq.2019.111066 .[19] J. D. Weeks, D. Chandler, H. C. Andersen, Role of repulsive forces\nin determining the equilibrium structure of simple liquids, J. Chem.\nPhys. 54 (1971) 5237–5247 (1971). doi:10.1063/1.1674820 .\n[20] J. J. Cerdà, V. Ballenegger, O. Lenz, C. Holm, P3m algorithm for\ndipolar interactions, J Chem Phys 129 (2008) 234104 (2008). doi:\n10.1063/1.3000389 .\n[21] F. Weik, R. Weeber, K. Szuttor, K. Breitsprecher, J. de Graaf,\nM. Kuron, J. Landsgesell, H. Menke, D. Sean, C. Holm, ESPResSo\n4.0 – an extensible software package for simulating soft matter sys-\ntems, Eur. Phys. J. ST 227 (14) (2019) 1789–1816 (2019). doi:\n10.1140/epjst/e2019-800186-9 .\n[22] E. Pyanzina, S. Kantorovich, J. J. Cerdà, A. Ivanov, C. Holm, How\nto analyse the structure factor in ferrofluids with strong magnetic in-\nteractions: a combined analytic and simulation approach, Mol. Phys.\n107 (4–6) (2009) 571–590 (2009). doi:10.1080/00268970902893149 .\n[23] M. U. Witt, S. Hinrichs, N. Möller, S. Backes, B. Fischer, R. Von\nKlitzing, Distribution of CoFe 2 O 4 Nanoparticles Inside PNIPAM-\nBased Microgels of Different Cross-linker Distributions, Journal of\nPhysical Chemistry B 123 (10) (2019) 2405–2413 (2019). doi:10.\n1021/acs.jpcb.8b09236 .\n[24] T. Eckert, W. Richtering, Thermodynamic and hydrodynamic inter-\naction in concentrated microgel suspensions: Hard or soft sphere be-\nhavior?, Journal of Chemical Physics 129 (12) (2008). doi:10.1063/\n1.2978383 .\n[25] J. A. Galicia, F. Cousin, E. Dubois, O. Sandre, V. Cabuil, R. Perzyn-\nski, Local structure of polymeric ferrogels, Journal of Magnetism\nand Magnetic Materials 323 (10) (2011) 1211–1215 (2011). doi:\n10.1016/j.jmmm.2010.11.008 .\nI. S. Novikau et al.: Submitted preprint Page 5 of 5"    },    {        "title": "0811.1838v1.Magnetic_Response_of_NiFe2O4_nanoparticles_in_polymer_matrix.pdf",        "content": "(a) NiFe 2O4bulk \n(b) NFNP (c) Nanocomposite sample \nFig. 1 Intensity (arb. unit) \n10 20 30 40 50 60 70 80 90 100   \nNC100  \n \n \nNC200   \n \n2θ(0)NC50    \nNC500                \n  \n               \n  \n \n         \n                 Fig. 2  rij ~80  nm rij ~10 nm     \n(a)  NFNP \n(b)  NC100 10 nm     0 100 200 300 246810 12 14 \n0 100 200 300 020 40 60 80 0 100 200 300 010 20 30 40 \n0 100 200 300 01234\n0 100 200 300 050 100 150 200 250 \nTm2 Tm1 NC100 \n  (b) \nχ//  (10 -5  emu/g/Oe)   χ/ (10 -4  emu/g/Oe) \nT (K) 0510 \n \nχ//  (10 -5  emu/g/Oe)  \nNC500 \n (d) χ/ (10 -4  emu/g/Oe) \nT (K) 020 40 60 80 \n NC200 \n  (c)  \nT (K) 010 20 30 40   NC50 \n  (a) \nχ// (10 -5  emu/g/Oe)  \nT (K) 01234\n \nNFNP \n (e)   \nT (K)) 050 100 150 200 \nFig. 3 \n 0 50 100 150 200 250 300 350 0.02 0.03 0.04 0.05 0.06 \n0 50 100 150 200 250 300 350 0.08 0.10 0.12 0.14 NC50 NC200 \n  M (emu/g) \nT (K) 0.1 0.2 0.3 0.4 \nTirr (240 K) TB (125 K) TB (240 K) Tirr (295 K) (a)  100 Oe \n \nTB (90 K) \nFig. 4 (b) 1 kOe \nNC100 \nNC50  M (emu/g) \nT (K) 0.30 0.35 0.40 0.45 0.50 \nTirr  = 130 K Tirr = 170 K \n \n 0 20000 40000 60000 0510 15 20 25 0 100 200 300 0.5 1.0 1.5 2.0 2.5 3.0 \n0 200 400 0510 15 20 \nNC100 NC200 NC500 NFNP (b) \n  M (emu/g) \nH (Oe) H = 1 kOe \n(75 K) (a) \nTB (140 K) \nNC500 \nNC200 \nNC100 \nNC50 NFNP \nFig. 5.  \n MZFC(T)/M(5 K) \nT (K) \nMR (emu/g)  HC (Oe) \nNFNP quantity in matrix (mg) 0.00 0.02 0.04 0.06 0.08 0.10 \nNFNP \n Magnetic Response of NiFe 2O4 nanoparticles in polymer \nmatrix \nA. Poddar a, R.N. Bhowmik b, Amitabha De a and Pintu Sen c  \n \naSaha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata-700064, India  \n                b Department of Physics, Pondicherry University, R. V.Nagar, Kalapet,  Pondicherry-605014, India \n                                             c Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Kolkata-700064, India \n                          E-mail address of the corresponding author(RNB): rnbhowmik.phy@pondiuni.edu.in \n \nWe report the magnetic properties of magnetic nano-compo site, consisting of different \nquantity of NiFe 2O4 nanoparticles in polymer matrix. The nanoparticles exhibit ed a typical \nmagnetization blocking, which is sensitive on the variati on of magnetic field, mode of zero \nfield cooled/field cooled experiments and particle quantity i n the matrix. The samples with \nlower particle quantity showed an upturn of magnetization  down to 5 K, whereas the \nblocking of magnetization dominates at lower temperatures as the particle quantity increases \nin the polymer. We examine such magnetic behaviour in term s of the competitive magnetic \nordering between core and surface spins of nanoparticles,  taking into account the effect of \ninter-particle (dipole-dipole) interactions on nanopart icle magnetic dynamics.  \n \nKey words: Magnetic Nanocomposite, Core-Shell model, Mag netic dipole interactions, \nSuperparamagnetic Blocking, Ferrimagnetic Nanoparticle. \n I. INTRODUCTION \nThe research activities on magnetic nanoparticles (MNPs ) have seen many unusual \nphenomena [1, 2] over the last few decades. Some of the r ecently focused phenomena are: \nsuperparamagnetism, surface magnetism, spin glass, exchan ge bias effect, variation of particle \nmagnetic moment and magnetic ordering temperature. The understanding of such magnetic \nphenomena has a direct or indirect importance on the theoretical modeling or applying the magnetic \nnanomaterials in technology. So many concurrent processes  (inter-particle interactions, surface spin \ncanting, interfacial/grain boundary effects, site excha nge of cations among A and B sublattices, \netc.) are involved in the magnetic properties of ferrite nanoparticles [3, 4, 5]. Most of the magnetic \nnanomaterials have shown superparamagnetic blocking of nano particles below a certain finite \ntemperature. This behaviour of magnetic nanoparticles is believed to be a time domain problem, \nrelated to the inter-particle interactions and surface spin morphology of the particles. The \nsuperparamagnetic properties have been examined using techniques,  like: magnetometry (dc \nmagnetization, ac susceptibility) and Mössbauer spectrosco py [5, 6]. Kodama et al. [4] had \nproposed a core-shell model to explain the (ferri)magne tic properties of NiFe 2O4 nanoparticles. It \nhas been realized that the concept of core-shell spin s tructure may be the better description for \nnanoparticle magnetism [5, 7, 8]. Despite extensive work, the effect of core-shell spin structure and \ninter-particle interactions on the properties of magnet ic nanoparticle is not clear. Recently, attempts \nare being made to study nano-composite materials, where ma gnetic nanoparticles are dispersed in a \nsuitable (magnetic/non-magnetic) matrix [9, 10, 11]. The m agnetic nano-composites are interesting \nnot only to formulate the mechanism of inter-particle interactions and surface magnetism, but the \nmaterials are also emerging as the potential candidate t o replace many conventional materials in \nscience and technology [12, 13 , 14].  In this paper, we study the magnetic nano-composite, invo lving NiFe 2O4 nanoparticles in \nthe polymer (3, 4-ethylenedioxythiophene) ( PEDOT ) matrix. The interesting fact is that NiFe 2O4 is \na typical ferrimagnet with relatively large magnetic moment (~55 emu/g) and ordering temperature \n(~850K) [15]. These magnetic parameters are important to sy nthesize nano-composite materials \nwith reasonably large magnetic moment for room temperature  applications. We have dispersed \ndifferent quantity of NiFe 2O4 nanoparticles (from same batch) in a fixed volume of PEDOT matrix. \nIn this process, the complication related to the site e xchange of cations, surface to volume ratio and \nsize distribution as a function of varying particle size  can be avoided.  \n \nII. EXPERIMENTAL  \nNiFe 2O4 nanoparticles are prepared by sol-gel procedure. The stoi chiometric amount of \nFe(NO 3)3, 9H 2O and Ni(NO 3)2,6H 2O are mixed (mass ratio 2:1) and dissolved in ethylene glyco l \nat~ 40 0C. The sol of metal salts is heated at ~ 60 0C to obtain gel. The gel product is dried at ~100 0C \nand fired at ~ 400 0C for 24 hours. Finally, the sample is cooled to 300 K. The  PEDOT nano-\ncomposite is prepared by polymerization of 3, 4-ethylenedioxyt hiophene (EDOT) in the colloidal \ndispersion containing specified quantity of NiFe 2O4 nanoparticles (NFNP). Polymerization is \nallowed for 20 hours under vigorous stirring, resulting in a dark blue coloured nano-composite in \nthe dispersed phase. Ethanol is added as a non-solvent to obtain the precipitate, which is washed \nand dried in vacuum. The nano-composites are denoted as NCX, where X indicates the quantity (50 \nmg, 100 mg, 200 mg, 500 mg) of NiFe 2O4 nanoparticles dispersed in 3 ml volume of polymer. The \npellet form of sample is used for characterization and measurement. The crystallographic phase of \nthe samples is characterized by X-ray Diffraction spec trum at room temperature (300 K) using Cu \nKα radiation from Philips PW1710 diffractometer. Particle s ize of the NFNP sample is determined from transmission electron micrographs (TEM). The ac s usceptibility and dc magnetization of the \nsamples are measured in the temperature range 5 K to 340 K, using SQUID magnetometer \n(Quantum Design, USA). The dc magnetization is recorde d under zero field cooled (ZFC) and field \ncooled (FC) modes. \n \nIII. RESULTS \nFig. 1 (a-b) shows that the XRD spectrum of nanoparticl e NiFe 2O4 (NFNP) sample is \nconsistent with the standard pattern of bulk NiFe 2O4 sample. XRD patterns of both the samples are \nmatched to cubic spinel structure with space group Fd3m. The crystal structure of the samples is \ndetermined by standard full profile fitting method using FULLP ROF Program. The lattice \nparameter ( a) is ∼ 8.343 (2) Å and 8.351(3) Å for bulk sample and NFNP sample, res pectively. The \nsmall variation of lattice parameter in NFNP sample wi th respect to the bulk sample may be \nattributed to the nanocrystalline nature of the material , characterized with broad peak lines. The \nparticle size of NFNP sample (~ 7 nm) is estimated using Debye-Scherrer formula to (311) and \n(440) XRD peaks of the spectrum. The XRD pattern of NFNP sa mple is modified in the presence \nof polymer matrix (Fig. 1c). The nano-composite samples with lower particle quantity exhibit a \nbroad background in XRD spectrum. The background is contribute d from the amorphous polymer \nmatrix [16]. The increase of particle quantity increases th e crystalline fraction in the nano-\ncomposite, as seen from the appearance of crystalline pe aks in NC500 sample. The TEM picture \n(Fig. 2a) suggests that NFNP particles are in nanocrystall ine state with clear lattice fringe and size \nis ~ 10 nm. This value is close to the particle size (~ 7 nm ) from XRD peaks. The TEM picture \nNC100 sample (Fig. 2b) shows that NiFe 2O4 nanoparticles are coated by the amorphous matrix of \npolymer. The TEM picture suggested that particles are in con tact for NFNP sample, while the particles are well separated in the nano-composite samp le. The estimated inter-particle distance ( rij ) \n(~ 10 nm and 80 nm for NFNP and NC100 samples, respectively) is increasing with the decrease of \nparticle quantity. \nWe, now, investigate the magnetic dynamics of nano-compo sites by measuring the ac \nsusceptibility and dc magnetization. The real ( χ/) and imaginary ( χ// ) parts of ac susceptibility data, \nmeasured at ac field (h rms ) ~ 1 Oe and frequency ( ν) = 10 Hz and 997 Hz, are shown in Fig. 3. The \nnotable feature is that magnitude of susceptibility drastica lly decreases as the particle quantity \ndecreases in the matrix. The samples showed a typical magnetic freezing/blocking behaviour below \ncertain temperature. The freezing of χ/ is appearing at higher temperature (T m1 ) than the freezing of \nχ//  at T m2  (T m2 < T m1 ), as shown by arrow in Fig. 3b for NC100 sample.  The ty pical values of peak \ntemperature (T m1 ) may be noted as 180 K, 274 K at 10 Hz and 220 K, 300 K at 997 Hz fo r NC50 \nand NC100 samples, respectively. However, frequency depende nt shift of χ/ peak is not seen for \nNC200 and NC500 samples, except the onset of gradual decrease in χ/ below 270 K and 280 K for \nNC200 and NC500 samples, respectively. The χ/ data do not show any peak up to the measurement \ntemperature 340 K for NFNP sample. It seems that T m1  may be ~350 K for NFNP sample. On the \nother hand, the freezing temperature of χ//  at T m2 is clearly observed below 340 K for all samples \nand T m2  strongly depends on the frequency of applied ac magnetic field. For example, T m2 is ~ 40 \nK, 190 K, 218 K, 225 K, 300 K at 10 Hz and ~ 80 K, 216 K, 244 K, 248 K, 320 K f or the samples \nNC50, NC100, NC200, NC500 and NFNP, respectively. The result cl early suggests that \nblocking/freezing temperature of the samples increases wit h the increase of particle quantity in the \nmatrix. The estimated frequency shift per decade of frequenc y [ Δ = ΔTm2 /{T m2 (10 Hz)ln ν}] is \n0.220, 0.030, 0.024, 0.022 and 0.014 for the samples NC50, NC100, NC200, NC500 and  NFNP, \nrespectively. The decrease of Δ with increasing particle quantity in the present nano-compo site system can be attributed to the increase of inter-part icle interactions. However, the freezing \nbehaviour can not be classified either an ideal spin gl ass or superparamagnetic blocking in the \nsamples, because the typical value of Δ is ~ 1 for ideal superparamagnetic system and ~ 0.001 for \nspin glass system [17]. \nThe dc field effect on the magnetic dynamics is underst ood from the measurement of zero \nfield magnetization (MZFC) and field cooled magnetizati on (MFC) of the samples. The MZFC and \nMFC at 100 Oe for NC50 and NC200 samples are shown in Fig. 4a. The features suggest the \nblocking of magnetic nanoparticles, associated with the decrease of MZFC below the blocking \ntemperature T B and separation between MFC and MZFC below the irrever sibility temperature T irr \n(\u0001TB). Below T irr , MFC of both the samples increases down to 5 K, except  the increase is more \nrapid in NC50 than NC200 sample. It may be noted that T B ~125 K and 240 K at 100 Oe for NC50 \nand NC200 samples is less in comparison with T m1  ~180 K and 270 K, as mentioned earlier from \nthe χ/ data. This is due to the effect of increasing dc field on the blocking temperature of particles \n[18]. The field effect is much more pronounced by increasing t he field at 1 kOe (Fig. 4b). The \nnovel feature is that MZFC of NC100 sample at 1 kOe show s blocking behaviour below T B ~80 K \nand an upturn in magnetization at lower temperature. The c ompetition between low temperature \nupturn and superparamagnetic blocking about T B results in a minimum at ~ 20 K for the NC100 \nsample. Interestingly, a typical blocking behaviour in MZ FC, as observed at 100 Oe (Fig. 4a), is \nnot visible at 1 kOe for NC50 sample down to 5 K, except the  observation of a low temperature \nupturn. The MFC (T) in both (NC50 and NC100) samples increase s with similar manner below T irr . \nHowever, the irreversible effect between MFC(T) and MZ FC(T) curves at 1 kOe is significantly \ndifferent in terms of splitting (between MFC and MZFC ) and irreversibility temperature T irr  (e.g.,~ \n150 K and 200 K for NC50 and NC100 respectively). A systematic ev olution of magnetization at 1 kOe with increasing particle quantity is seen from the MZF C(T) curves, normalized by 5 K data \n(Fig. 5a).  The magnitude of MZFC at 5 K (~0.154, 0.384, 0.734, 1.431 a nd 3.727 in emu/g unit for \nNC50, NC100, NC200, NC500 and NFNP samples, respectively) increa ses with particle quantity. \nThe interesting feature is that blocking temperature (T B) decreases not only by the increment of \nfield, but also by the decrease of particle quantity in the  matrix. This is clear from the fact that T B \n~140 K (at 1 kOe)  of NFNP sample decreases to ~110 K (at 1 kO e) for NC500 sample,  ~100 K (at \n1 kOe) for NC200 sample, ~75 K (at 1 kOe) for NC100 sample and n o typical blocking of \nmagnetization at 1 kOe down to 5 K for NC50 sample. It is also noted that not MZFC alone, the \nMFC of NC50 and NC100 samples are also showing rapid increase  at lower temperature. The low \ntemperature upturn in MZFC is not seen for samples (NC200 and NC500) with sufficiently large \nquantity of magnetic nanoparticles. The magnetic momen t of the nano-composite samples can be \ncompared directly from the field dependence of magnetizat ion (isotherms) data (Fig. 5b). The \nisotherms suggest that the typical ferrimagnetic chara cter of NiFe 2O4 nanoparticles are also retained \nin the nano-composite samples. The spontaneous magnetiz ation can be estimated by the \nextrapolation of high field (> 40 kOe) magnetization data  to zero field value. The obtained value of \nspontaneous magnetization is gradually increasing (~ 0.54 emu/ g at 300 K for NC100, ~ 1.94 \nemu/g at 300 K for NC200, ~ 4.49 emu/g at 300 K for NC500 and ~ 21 emu/g a t 300 K for NFNP) \nwith the increase of particle quantity in the polymer matr ix. This observation is also consistent with \nthe variation of peak susceptibility at T m1  (ac susceptibility data) and peak magnetization at T B (dc \nmagnetization data) with the increase of particle quanti ty. It may be mentioned that NC50 sample \nexhibited a typical non-linear increase of M(H) data at 300 K, without any spontaneous \nmagnetization. The M(H) variation of NC50 sample is n ot shown in Fig. 5 (b), because the \nmagnitude is very low in comparison with other nano-composi te samples. The interesting feature is that both coercivity (H C) and remanent magnetization (M R), obtained from M(H) data, are showing \n(inset of Fig. 5b) a maximum at the intermediate partic le quantity, followed by zero value for both  \nNC50 sample and NFNP sample (indicated by dotted lines). S imilar variation of H C and M R with \nparticle quantity in the matrix has been found in other nan o-composite [12], where such magnetic \nproperties have been attributed to the effect of dipolar  inter-particle interactions. \n \nIV. DISCUSSION \nWe, now, try to understand some unique magnetic features of  the present nano-composite \nsamples. These features are reflected by a systematic v ariation in the shape of low temperature \nmagnetization curves, where a typical blocking behaviour of magnetic particles is transformed into \na magnetization upturn with the decrease of particle quant ity in the matrix. The average blocking \ntemperature (T B) of the particles is defined at the peak/maximum of MZFC  curves. The shift of \nmagnetization maximum with increasing magnetic field is understood as an effect of magnetic field \non the blocking of particle magnetization. In conventiona l superparamagnetic blocking, MZFC \ndecreases with temperature below T B, and one could expect either continuous increase or flat ness \nof MFC below T B [19]. In our case, the shift of MZFC maximum is not due to increase of magnetic \nfield alone, also due to the change of particle quantity i n the matrix. The fact is that low \ntemperature magnetic upturn is seen in both MZFC and MFC i n our samples with lower particle \nquantity. The low temperature magnetization upturn has been f ound in a few composite systems [3, \n20, 21]. Although various aspects, e.g., random anisotropy eff ect [18], surface spin contributions \n[3], spin reorientation [20] and precipitation of superpara magnetic type small particles [21], have \nbeen suggested, but the mechanism is not very clear till d ate. It is highly unexpected that the low \ntemperature magnetic upturn is due to some precipitated smal l (superparamagnetic) particles in the samples, because the same NiFe 2O4 nanoparticles with different quantity are mixed in the matrix. \nOne could expect more precipitation in the samples with hi gher particle quantity. Consequently, \nlow temperature upturn is expected to be more prominent for  samples with higher particle quantity. \nHowever, this picture is not consistent with our exper imental observations. We offer alternate \nexplanation from the fact that similar low temperatur e magnetic behavior has been found in some \nantiferromagnetic nanoparticles [22, 23]. We demonstrate the origin of such low temperature \nmagnetism in NiFe 2O4 nano-composites in a simple and realistic manner, by inco rporating the core-\nshell model proposed for antiferromagnetic nanoparticles ( AFMNPs) [24] and considering the \nmagnetic dipole-dipole interactions E ij  = ( \u00020/4 \u0003)[ \u0000i. \u0004j/r ij 3-3( \u0005i.rij )( \u0006j.rij )/r ij 5] [21]. The dipole-dipole \ninteractions vary as a function of the magnitude of spi n moments ( \u0007i and \bj), direction ( θij ) of the \nmoments with respect to the line of joining ( rij ) and the distance of separation (r ij ).  The core spin \nmoments in a typical AFMNP are compensated. The intera ctions between two spin moments \n(dipoles) may be from intra-particle or inter-particle s. The inter-particles interactions, and also the \ncore-shell interactions, are neglected in antiferroma gnet due to small magnetic moment of particles. \nA significant number of uncompensated shell spins in the a bsence of strong exchange interactions \ncontributes to the paramagnetic or superparamagnetic like magnetic upturn at lower temperatures, \nfollowing the law [24]: MZFC ∝ T -γ ( γ ≤ 1). The core-shell model for antiferromagnetic \nnanoparticle [24] has also found its application in ferri magnetic nanoparticles [25]. The exchange \ninteractions between core-shell spins are not neglecte d in ferrimagnetic (as in: NiFe 2O4) \nnanoparticles due to large ferrimagnetic moment of core spins. The magnetic ordering of core spins \ndominates over the ordering of shell (surface) spins. Suc h magnetic competition strongly depends \non the surface spin configuration and contributions from inter-particle interactions [9, 10, 11, 26]. J. \nNogues et al [8] had described the essential role of shells in stabiliz ing the magnetism of core-shell nanoparticles, where the shell mediated interactions may  introduce an induced magnetic state in the magnetic \nnanoparticle. Our  experimental results suggested that the ferrimagnetic character of NiFe 2O4 \nnanoparticles is maintained in the nano-composites. We as sume that the change of surface \nmorphology for each particle, due to pinning of spins to th e polymer bonds [27], remains identical \nas long as NiFe 2O4 nanoparticles are concerned in the PEDOT matrix. We, further, assume that the \nsurface spin ordering (surface spin morphology) depends on the effective field, created by the \nexchange interactions between shell-core spins (intra- particle interactions) and the interactions \nbetween neighbouring particles (inter-particle interacti ons). It may be mentioned that change of \nexchange interactions of the particles are not playing sig nificant role with the change of particle \nquantity in the matrix, because distortion in exchange interactions occur only between core and \nshell spins. The interactions between two particles are  included in the form of dipole-dipole \ninteractions and this effect is significant with the i ncreasing particle quantity in the matrix. The \nincrease of inter-particle interactions with particle qua ntity is realized from a systematic decrease of \nthe shift of ac susceptibility maximum per decade of frequenc y ( Δ).  \nNow, we estimate the relative strengths of dipole-dipo le interactions from the variation of \ninter-particle separation ( rij ) ~ 10 nm and ~ 80 nm (from TEM picture) and magnetic moment ( \t) at \n300 K ~ 0.54 emu/g and ~ 21 emu/g for NFNP and NC100 samples (from M- H data), respectively.  \nWe simplify the dipole-dipole interaction term as E ij  ~ -(2 \n0/4 \u000b)\fi\n\rj cos θij /r ij 3 ~ -(2 \u000e0/4 \u000f)\u00102\n cos θij \n/r ij 3. The ratio of dipole-dipole interactions for NFNP an d NC100 samples is (E ij )NFNP /(E ij )NC100  ~ \n(\u0011NFNP /\u0012NC100 )2 (r NC100 /r NFNP )3 (cos θNFNP /cos θNC100 )2. Replacing (E ij )NFNP /(E ij )NC100  by \n(T m1 )NFNP /(T m1 )NC100 (from χ/ peak temperature at 10 Hz) ~ 350 K/274 K and substituting all th e \nparameters values, we get cos θNFNP /cos θNC100  ~0.00128. This suggests that the angle ( θij ) between \ntwo dipole moments of NC100 sample is less than that for the NFNP sample. The θij  is restricted to 00 < θij  <90 0.  The lower value of θij  for the samples with lower particle quantity is expl ained from \nthe fact that dipole-dipole interactions are not large e nough, due to large inter-particle separation \n(r ij ), to significantly modify the surface spin ordering. Th e individual particle moments, largely \ncontrolled by the superparamagnetic type contributions fr om shell spins, are relatively free to \nrespond to the external magnetic field. The superparamagn etic nature (less inter-particle \ninteractions) in NC50 samples is suggested from a typical M(H) curve at 300 K with zero \ncoercivity. As soon as the small inter-particle inter actions are minimized, the superparamagnetic \nresponse of the shell spins dominates in controlling the low temperature magnetic behaviour of the \nmaterial. The increase of particle concentration in th e matrix contributes to a significant amount of \ndipole-dipole interactions [3, 9, 11, 28], which effectively  increase the anisotropy energy (E eff  ∝ \nTB) of the nanoparticles. The particle moments are rando mly blocked along the local anisotropy \naxes, created by the inter-particle (dipole-dipole) inte ractions from the neighbouring particles. So, \nthe magnetic behaviour is no more a surface dominant effe ct, rather the particles moments are \ncollectively blocked below the average blocking temperature ( T B) of the sample.  \n \nV. CONCLUSIONS \nThe experimental data indicated that the shape of low temperature magnetization curves and \nmagnetization blocking of NiFe 2O4 nanoparticles in PEDOT polymer matrix depends not only on \nthe factors like: particle size and core-shell morpholog y alone, but also on the factors like:  \nvariation of magnetic field, variation of particle qua ntity in the matrix and mode of magnetization \nmeasurement. The correlated effect of core-shell spin  structure and inter-particle interactions are \nused to understand the shape of magnetization curves. The  paramagnetic contributions of shell \nspins are significant for the samples with lower parti cle quantity in the matrix. The inter-particle \ninteractions, contributed by dipole-dipole interactions between particle moments, are dominating in samples with higher particle quantity. The magnetizatio n upturn at lower temperatures or the \nmagnetization maximum at finite temperature is an effec t of modified surface spins dynamics, \ndepending on the quantity of magnetic particles in the polym er matrix.  \n \nAcknowledgment:  The authors thank Pulak Ray for providing TEM data. \n \nReferences: \n[1] J L Dormann, L Bessais and D Fiorani, J. Phys. C: So lid State Phys. 21 , 2015 (1988)  \n[2] J. Nogués et al., Physics Reports 422 , 65 (2005) \n[3] E. Tronc et al., J. Magn. Magn. Mater. 221 , 63 (2000) \n[4] R.H. Kodama, A.E. Berkowitz, E.J. McNiff Jr., S. Foner , Phys. Rev. Lett. 77,  394 (1996). \n[5] V. Sepelak, I. Bergmann, A. Feldhoff, P. Heitjans, F . Krumeich, D. Menzel, F.J. Litterst, S.J.  \n \n      Campbell and K. D. Becker, Chem. Mater . 111 , 5026   (2007).  \n \n[6] T. Jonsson, P. Nordblad, and P. Svedlindh, Phys. Rev. B  57, 497 (1998) \n[7] Òscar Iglesias, Xavier Batlle, and Amílcar Labarta, Phys. Rev. B 72 , 212401 (2005) \n[8] J. Nogues, V. Skumryev, J. Sort, S. Stoyanov, and D. G ivord, Phys. Rev. Lett. 97 , 1572003 \n(2006) \n[9] J.L. Dormann et al., Phys. Rev. B 53 , 14291 (1996) \n[10] J. Zhang, C. Boyd, and W. Luo, Phys. Rev. Lett. 77 , 390 (1996)  \n[11] C. Djurberg , P. Svedlindh and P. Nordblad, Phys. Rev. Lett. 79 , 5154 (1997)    \n[12]  A. Ceylan, C.C. Baker, S.K. Hasanain,, S.I. Shah ,  Phys. Rev. B, 72 , 134411 (2005)  \n[13] R.F. Ziolo  et al., Science 257 , 219 (1992)  [14] J. L. Wilson, P. Poddar, N. A. Frey, H. Srikanth, K.  Mohomed, J. P. Harmon, S. Kotha and J.   \nWachsmuth, J. Appl. Phys. 95 , 1439 (2004) \n[15] U. Lüders, M. Bibes, Jean-François Bobo, M. Cantoni,  R. Bertacco, and J. Fontcuberta, Phys. \nRev. B 71 , 134419 (2005) \n[16] P. Dallas, V. Georgakilas, D. Niarchos, P. Komninou, T. Kehagias and D. Petridis, \nNanotechnology 17, 2046 (2006) \n \n[17] J.A. Mydosh, Spin Glasses: an Experimental Introducti on, Taylor & Francis, London, 1993 \n[18] W. Luo, S.R. Nagel, T. F. Rosenbaum, and R. E. Rose nsweig, Phys. Rev. Lett. 67 , 2721 \n(1991)  \n[19] Qi Chen and Z.J. Zhang, Appl. Phys. Lett. 73 , 3156 (1998) \n[20] R.V. Upadhyay, K. Parekh, L. Belova, K.V. Rao, J. Ma gn.Magn. Mater. 311 , 106 (2007) \n[21] R.D. Zysler, C.A. Ramos, E. De Biasi, H. Romero, A. Ortega, D. Fiorani, J.Magn. Magn. \nMater. 221 , 37 (2000) \n[22] L. He and C. Chen, N. Wang, W. Zhou, and L. Guo, J. Appl. Phys. 102 , 103911 (2007) \n[23] A. Punnoose, H. Magnone, M.S. Seehra, J. Bonevich, Phys. Rev. B 64, 174420 (2001). \n[24] R.N. Bhowmik, R. Nagarajan, and R. Ranganathan, Phys. Rev. B 69 , 054430 (2004). \n[25 ] T. Zhang, T. F. Zhou, T. Qian, and X. G. Li, Phys. Rev.  B 76 , 174415 (2007) \n \n[26] P. Poddar, J.L. Wilson, H Srikanth, S.A. Morrison a nd E.E. Carpenter, Nanotechnology 15, \nS570 (2004) \n[27] S.R. Ahmed, S.B. Ogale, G.C. Papaefthymiou, R. Ramesh , P. Kofinasa, Appl. Phys. Letter. \n80 , 1616 (2002) \n[28] M. Blanco-Manteco´n, K. O’Grady, J. Magn. Magn. Mater. 296 , 124 (2006) \n \n Figure Captions: \nFig. 1 (a) XRD spectrum for bulk (a) and nanoparticle (b) NiFe2O4 samples, alongwith  \nFull profile fit data. The  XRD spectrum of nanocomposite  samples are shown in  (c) .  \n \nFig. 2. The TEM pictures are shown for NFNP sample (a) and NC100 sample (b). The arrow \nindicates the average inter-particle separation (r ij ) of the particles. \n \nFig. 3. Real χ/ (left scale) and imaginary χ//  (right scale) parts of ac susceptibility data for diffe rent \nnano-composite samples, measured at h rms  = 1 Oe and ν = 10 Hz (circle) and 997 Hz (square). Solid \nand Open symbols represent χ/ and χ// , respectively. The arrows (Fig. 2b) indicate the shift o f χ/ and \nχ// maximum with increasing frequency. \n \nFig. 4. The temperature dependence of MZFC and MFC data at 100 Oe for NC50 and NC200 \nsamples in (a) and at 1 kOe for NC50 and NC100 samples in (b ). T B and T irr  of the samples are \nindicated by (up-down) arrows. The side arrows represent t he magnetization axis for the \ncorresponding samples. \n \nFig. 5. (a) MZFC (normalized by 5 K data) at 1 kOe for diff erent nano-composite samples. The \narrows indicate the position of blocking temperature (T B) of the samples. (b) Field dependence of \nmagnetization at 300 K for the selected samples. Inset of Fig. 5(b) shows the variation of coercive \nfield (H C) and remanent magnetization (M R) with particle quantity in the matrix. The dotted lines \njoin the H C and M R of NC500 sample with NFNP sample. \n "    },    {        "title": "2307.06403v1.Quasi_static_magnetization_dynamics_in_a_compensated_ferrimagnetic_half_metal____Mn__2_Ru__x_Ga.pdf",        "content": "Quasi-static magnetization dynamics in a compensated ferrimagnetic half-metal -\nMn 2RuxGa\nAjay Jha,1,∗Simon Lenne,1Gwena¨ el Atcheson,1Karsten Rode,1J.M.D. Coey,1and Plamen Stamenov1\n1CRANN, AMBER, and School of Physics, Trinity College Dublin, Dublin 2, Ireland\nExploring anisotropy and diverse magnetization dynamics in specimens with vanishing magnetic mo-\nments presents a significant challenge using traditional magnetometry, as the low resolution of exist-\ning techniques hinders the ability to obtain accurate results. In this study, we delve deeper into the\nexamination of magnetic anisotropy and quasi-static magnetization dynamics in Mn 2RuxGa (MRG)\nthin films, as an example of a compensated ferrimagnetic half-metal, by employing anomalous Hall\neffect measurements within a tetragonal crystal lattice system. Our research proposes an innovative\napproach to accurately determine the complete set of anisotropy constants of these MRG thin films.\nTo achieve this, we perform anomalous Hall voltage curve fitting, using torque models under the\nmacrospin approximation, which allow us to obtain out-of-plane anisotropy constants K1= 4.0×104\nJ m−3(K1/M= 0.655 T) and K2= 2.54×104J m−3(K2/M= 0.416 T), along with a weaker\nin-plane anisotropy constant K3= 3.48×103J m−3(K3/M= 0.057 T). By additionally employing\nfirst-order reversal curves (FORC) and classical Preisach hysteresis (hysterons) models, we are able\nto validate the efficacy of the macrospin model in capturing the magnetic behavior of MRG thin\nfilms. Furthermore, our investigation substantiates that the complex quasi-static magnetization dy-\nnamics of MRG thin films can be effectively modelled using a combination of hysteronic and torque\nmodels. This approach facilitates the exploration of both linear and non-linear quasi-static magne-\ntization dynamics, in the presence of external magnetic field and/or current-induced effective fields,\ngenerated by the spin-orbit torque and spin transfer torque mechanisms. The detailed understanding\nof the quasi-static magnetization dynamics is a key prerequisite for the exploitation of in-phase and\nout-of-phase resonance modes in this material class, for high-bandwidth modulators/de-modulators,\nfilters and oscillators for the high-GHz and low-THz frequency bands.\nI. INTRODUCTION\nSpintronics-based devices have emerged as highly-\npromising candidates for next-generation telecommuni-\ncation applications due to their potential for efficient con-\ntrol of magnetic moments via electrical methods, ultra-\nfast operating speeds, and ultralow power dissipation [1].\nIn the pursuit of these capabilities, antiferromagnetic\n(AFM) materials have demonstrated exceptional advan-\ntages over their ferromagnetic (FM) counterparts when\nincorporated into spintronic devices [2, 3]. FM materi-\nals are limited by their large stray field interactions and\nslow switching speeds (on the order of nanoseconds) [4, 5],\nwhich hampers their utility in memory and switching de-\nvices. In contrast, AFM materials project no stray fields\nand possess ultrafast spin dynamics (on the order of a\nfew picoseconds to hundreds of picoseconds) [6], mak-\ning them attractive for spintronics applications. How-\never, the control and detection of magnetization in AFM\nmaterials remain challenging due to their zero magnetic\nmoment and zero Fermi level spin polarization.\nTo address this technological gap, compensated fer-\nrimagnetic half-metal (CFHM) [7, 8] materials have\nemerged as an excellent alternative to AFM materials.\nSimilar to AFM materials, CFHMs also consist of two\nantiferromagnetically coupled spin sublattices, and their\nspin contribution can be conveniently tuned by adjust-\ning the composition and/or temperature. Moreover, due\n∗ajha@tcd.ieto the presence of inequivalent spin sites, these sublat-\ntices contribute unequally at the Fermi level, resulting\nin a semiconducting band-gap in one of the spin chan-\nnels and a zero band gap in the other [9]. This disparity\ngives rise to a high spin polarization for the conduction\nelectrons.\nAt the magnetic compensation point, CFHMs exhibit\nbehaviour akin to AFM materials, demonstrating ultra-\nfast magnetization dynamics. However, unlike AFM ma-\nterials, the detection and manipulation of CFHM magne-\ntization remain feasible due to the distinct responses of\nthe two spin sublattices to electrical and optical excita-\ntion [10–12]. These unique characteristics of CFHM pave\nthe way for the development of novel spintronics devices\nwith improved performance and functionality.\nFollowing theoretical predictions [13], the first exper-\nimental observation of a CFHM thin film was achieved\nwith the Mn 2RuxGa (MRG) class of materials [9]. MRG\ncrystallises in the inverse-Heusler XA structure, space\ngroup F¯43m, as depicted in figure 1. Within this struc-\nture, Mn occupies two distinct and non-equivalent sub-\nlattice sites: 4 a(Mn4a) and 4 c(Mn4c). The Mag-\nnetic moments of 4 aand 4 csublattices exhibit antifer-\nromagnetic coupling, whereas moments located on iden-\ntical sites display ferromagnetic coupling. Owing to the\nnon-equivalent crystallographic surroundings, the mag-\nnetic moments of both sublattices display quite distinct\ntemperature-dependent characteristics. The site-specific\nmagnetic moment attributable to Mn4apossesses weaker\ntemperature dependence in contrast to Mn4csublattice\nmoment [14]. As such, it is possible to attain an idealarXiv:2307.06403v1  [cond-mat.mtrl-sci]  12 Jul 20232\nMn4a\nGa4b\nMn4c\nRu4d\nFIG. 1. Schematic representation of the crystal structure of\nthe MRG compound (Mn 2RuGa). The structure features four\ninterpenetrating face-centered cubic lattices, with Mn4a(red)\natoms occupying the 4 aWyckoff positions and Mn4c(green)\natoms at the 4 cpositions. Ru (black) and Ga (blue) atoms\nare situated at the 4d and 4b Wyckoff positions, respectively.\nThe magnetic moments at the Mn4aand Mn4csites exhibit\nantiferromagnetic coupling. In this depiction, both the 4 c\nand 4 dpositions are fully occupied by Mn and Ru atoms,\nrespectively.\nmagnetic compensation for MRG by modulating its com-\nposition and/or inducing crystal lattice distortion.\nMRG displays pronounced c-axis magnetic anisotropy\ndue to the evanescence of magnetic moments, with an\nanisotropy field surpassing 14 T in proximity to the\ncompensation point. Furthermore, the Fermi level of\nMRG is predominantly influenced by the electronic states\noriginating from Mn situated at the 4 cposition, sub-\nsequently dictating the transport phenomena via Mn4c\nelectrons [15]. Additionally, MRG displays half-metallic\nproperties, as corroborated by Density Functional The-\nory (DFT) calculations [16] and Point Contact An-\ndreev Reflection (PCAR) spectroscopy (spin polariza-\ntion obtained as high as P > 60 %) [9, 10]. The\nhighly spin polarized carriers lead to a large anoma-\nlous Hall effect (AHE) [15] and magneto-optic Kerr effect\n(MOKE) [11, 17] even at the perfect magnetic compensa-\ntion. Therefore, the distinctive amalgamation of a van-\nishing net magnetic moment, high spin polarization at\nthe Fermi level, and high magnetic anisotropy designates\nMRG as a promising contender material for active layers\nof next-generation spintronics devices.\nBy employing MRG as an active layer in the spin-\noscillator, sub-THz chip-to-chip communication could be\nachieved, as its spin excitations were found to reside in\nthe necessary terahertz gap [18]. The sub-THz excita-\ntions of MRG were ascribed to its low magnetic moment,\nhigh uniaxial anisotropy field, and low Gilbert damp-\ning [18, 19]. In addition, the tunability of the anisotropy\nconstant and the moment in MRG afford the flexibil-\nity to modify the resonance frequencies of oscillators\nconstructed with MRG. Consequently, determining the\nanisotropy constants of MRG thin-films is a crucial pre-\nliminary step in examining their magnetization dynamics\nunder the influence of external stimuli.\nInvestigating anisotropy and other magnetization dy-namics in a sample with a negligible magnetic moment\nis unattainable using conventional magnetometry tech-\nniques (VSM, SQUID, etc.) due to insufficient resolu-\ntion and sensitivity. Furthermore, for a sample with\nan extremely small magnetic moment ( M), both the\nanisotropy field ( Ha= 2K/M ) and coercive field ( Hc)\ntypically diverge, rendering the measurement of mag-\nnetic anisotropy unfeasible with exceedingly large mag-\nnetic fields ( µ0H > 14 T) [15]. Generally, anisotropy\nis assessed by applying an external magnetic field at a\nspecific angle ( θH) to the magnetic easy axis and mon-\nitoring the corresponding changes in physical proper-\nties such as magnetization [20–22], anomalous Hall effect\n(AHE) [23, 24], and magneto-optical properties [25–27].\nThe acquired data are then conventionally fitted using\nthe torque balance method, which ultimately yields the\nanisotropy constants of the specimen.\nIn this study, an analysis of magnetic anisotropy\nand quasi-static magnetization dynamics in MRG thin-\nfilms, featuring a tetragonal crystal structure, is con-\nducted through electrical transport measurement tech-\nniques (AHE). MRG demonstrates a pronounced uniaxial\nout-of-plane anisotropy and a small yet significant four-\nfold in-plane anisotropy, which originates from substrate-\ninduced compressive strain. MRG exhibits a substan-\ntial anomalous Hall effect, alongside a high magnetic\nanisotropy field and high Fermi-level spin polarization, a\ncombination that enables direct probing of the anisotropy\nin MRG thin-films via electrical means. The manipula-\ntion of the magnetization vector ( M) of MRG within a\n3D space, under the influence of a magnetic field, en-\nables the examination of various anisotropy constants\nof the film. To characterize the equilibrium or dynamic\nresponse of the magnetization vector within an applied\nor induced effective field, accounting for the magnetic\nanisotropy of the sample is crucial. Generally, the equa-\ntion of motion for magnetization is spatially non-uniform\n(described by a micromagnetic model) or, in a much sim-\npler case, spatially uniform (explained by a macrospin\nmodel). This study employs the anomalous Hall effect\nto examine magnetic anisotropy in MRG, within the\nmacrospin model framework, in combination with a dis-\ntribution of hysterons, with finite magnetic viscosity and\nnegligible interaction field.\nThis paper commences with a discussion of the sample\npreparation and characterization techniques employed in\nthis study (section II). Subsequently, section III A intro-\nduces the modeling of hysteresis in MRG under a clas-\nsical Preisach model and the first-order reversal curves\n(FORC) method, wherein the validity of the macrospin\nmodel for MRG is established within the FORC and\nPreisach frameworks. A comprehensive torque model for\nevaluating the anisotropy constants of MRG using AHE\nis explored in section III B. Moreover, section III C ex-\namines various intricate static and quasi-static magneti-\nzation dynamics of MRG through a ’combined’ Preisach\nand torque model. Lastly, conclusions are drawn in sec-\ntion IV.3\nII. EXPERIMENTAL DETAILS\nEpitaxial thin films of Mn 2RuxGa were fabricated us-\ning a DC magnetron sputtering system on a 10 ×10 mm2\nMgO (001) substrate. The films were co-sputtered in an\ninert environment (argon gas) from Mn 2Ga and Ru tar-\ngets onto the substrate, which was maintained at 320◦C.\nAdditional details regarding film growth and characteri-\nzation can be found in a separate publication [10]. This\nstudy focuses on the x= 0.9 stoichiometry and a film\nthickness of approximately 40 nm. The compensation\ntemperature ( Tcomp) of this sample is considerably higher\nthan room temperature, at Tcomp∼350 K, as deter-\nmined by SQUID®magnetometry measurements. To\nprevent oxidation, the films were in-situ capped with\napproximately ∼3 nm of amorphous AlO x, deposited\nat room temperature. The substrate-induced compres-\nsive strain ( c/a≈1.02) facilitated the out-of-plane\nmagneto-crystalline anisotropy in the film. To investi-\ngate the transport properties, the films were patterned\ninto micron-sized (60 ×20µm2) Hall bars, using UV pho-\ntolithography and Ar-ion milling. A subsequent round of\nlithography and metal deposition was performed to es-\ntablish the contact pads and to minimize the series re-\nsistance contribution of the corresponding contacts, con-\nsisting of Ti (5 nm)/Au (50 nm).\nThe electronic transport properties were measured us-\ning the Quantum Design Physical Property Measurement\nSystem ( PPMS®) in the temperature range of 2 K ≤T\n≤300 K and magnetic field strengths of |µ0H| ≤14 T.\nThe longitudinal and transverse voltages were measured\nby applying a lock-in demodulation technique at the\nfirst harmonic with low excitation frequency, typically\nfAC∼517 Hz, which was significantly smaller than the\nresonance frequencies of MRG. To determine the angular\ndependence of the resistivity, measurements were taken\non a rotating platform within the PPMS, with an angular\nresolution of 0.01 deg. Additionally, First Order Reversal\nCurves (FORC) were measured using a field resolution of\n5 mT at room temperature in a 1 T GMW®electromag-\nnet.\nIII. RESULTS AND DISCUSSION\nA. Hysteresis model\nThis section discussess the approach taken to model\nthe switching of the magnetization (magnetic hystere-\nsis) using the classical Preisach (hysterons) model. Hys-\nteresis modeling has been an active area of research for\ndecades, owing to both physical and mathematical in-\nterest. The magnetic hysteresis of ferromagnetic ma-\nterials is the most famous example of hysteresis. It is\nwidely accepted that the multiplicity of metastable states\nis the origin of hysteresis. Consequently, a micromag-\nnetic model must be considered for hysteresis modeling.\nIn 1935, Preisach [28] proposed a classical micromagneticmathematical approach to describe the hysteretic effect.\nThe Preisach model (PM) employs a large number of\ninteracting magnetic entities (referred to as hysterons),\neach of which has a rectangular hysteresis loop (fig-\nure 2a). These hysterons are characterized by the opera-\ntorRh,k(x), where xis an arbitrary input variable, such\nas an applied magnetic field. Hysteresis arises from the\ncollective behavior of numerous hysterons, which switch\nfully at a discrete applied field. The value of Rh,k(x)\nrelies on the applied field history. For instance, if the ap-\nplied field ( x) starts from the saturation state ( x=∞),\nRh,k(x) initiates at Rh,k(∞) = 1. The value of Rh,k(x)\ntransitions to −1 when the applied field falls below the\nvalue h, and Rh,k(x) returns to +1 when the field value\nexceeds k. Typically, the switching fields handkare not\nidentical.\nThe interaction field experienced by a hysteron is de-\nfined by Hu= (h+k)/2, resulting in an asymmetric\nelementary hysteron. In contrast, a hysteron with no in-\nteraction is symmetric. The coercive field of a hysteron\nis defined as Hc= (h−k)/2. In a realistic sample, the\nhysteresis property is a weighted sum of a large number\nof hysterons, as described in equation 1:\ny(x) =NX\ni=1ϕ(hi, ki)Rhi,ki(x). (1)\nHere, the weighting factor ϕ(h, k) represents the distribu-\ntion of the switching fields handkand is commonly re-\nferred to as the switching field distribution (SFD) or hys-\nteron distribution (HD). Figure 2b illustrates a schematic\nrepresentation of the Preisach model.\nxRh,k\nHc\nHuh k+1\n−1(a)\nRh1,k1ϕ(h1, k1)\nRh2,k2ϕ(h2, k2)\nRhN,kN\nϕ(hN, kN)xΣy(b)\nFIG. 2. Illustration of the Preisach model. (a) A depiction\nof an elementary hysteron, a key component of the Preisach\nmodel, which exhibits unequal reversal fields with values h\nandk. The hysteron’s state is determined by the input vari-\nablex, as well as its history and strength. Hysteresis results\nfrom the collective interaction of numerous hysterons. (b) The\ndiscrete Preisach model of hysteresis, where a large number of\nhysterons are assumed to be connected in parallel, each with\na corresponding weighting factor ϕ(h, k). In this model, x\nrepresents an arbitrary excitation variable (e.g., applied mag-\nnetic field), and ysignifies the resulting hysteretic physical\nproperty (e.g., magnetic moment).\nIn the continuum limit, the discrete model is trans-4\nformed into the following expression:\ny(x) =ZZ\nk≥hϕ(h, k)Rh,k(x) dhdk , (2)\nwhere, xis an arbitrary variable (e.g., applied mag-\nnetic field) and yis the resultant hysteresis output (e.g.,\nmagnetic moment, anomalous Hall voltage, etc.). The\nmost challenging aspect of the Preisach model involves\nuniquely defining the distribution function ϕ(h, k). Nev-\nertheless, for an assembly of weakly interacting hys-\nterons, it is possible to assume that the distribution\nfunction ϕ(h, k) follows a specific statistical distribution.\nCommon choices include the Gaussian function [29, 30],\nGauss-Lorentzian function [31], and Lognormal-Gaussian\ndistribution function [32], among others. However, this\napproach faces the issue of lacking justification for select-\ning one particular distribution over others [32]. An alter-\nnative method entails using a linear combination of a set\nof functions as a basis. The drawback of this approach is\nthe requirement of a large set of basis functions and their\ncoefficients to obtain a Preisach distribution with a rel-\natively continuous output (y) [33], which rapidly strains\ncomputational capabilities, even for modern computers.\nThe first-order reversal curves (FORC) method of-\nfers an experimental technique for obtaining a unique\nPreisach distribution, as long as the sample of inter-\nest meets the necessary and sufficient Mayergoyz con-\nditions [34]. The FORC method is both easily achievable\nexperimentally and highly reproducible, given that it be-\ngins by saturating the sample each time. It has been\nemployed to examine various magnetic systems, such as\npermanent magnets [35, 36], geological samples [37, 38],\nnanowires [39, 40], and more. Moreover, FORC can dif-\nferentiate between interacting and noninteracting single\ndomain (SD), pseudo single-domain (PSD), and multi-\ndomain (MD) systems [37, 41]. In fact, the FORC\nmethod can be extended to any system exhibiting hys-\nteresis behavior, including ferroelectric samples [42, 43].\nAdditionally, FORC studies on certain magnetic systems\ncan be complemented by AHE measurements, where elec-\ntrical probing presents a decisive advantage over standard\nmagnetic moment measurements [44].\nThe FORC measurement using AHE commence by sat-\nurating the sample in a sufficiently high positive mag-\nnetic field. Subsequently, the field is decreased to a lower\nfield value on the main hysteresis loop (MHL), referred\nto as the reversal field ( HR), and the Hall resistance\nRxy(H, H R) is measured by sweeping the applied field\nHback to the saturation field. The resulting AHE resis-\ntance, Rxy(H, H R), constitutes a minor curve within the\nMHL (figure 3a). This procedure is repeated for numer-\nous uniformly spaced values of HRandH. The FORC\ndistribution is acquired through the second-order mixed\nderivative, as defined by equation 3 :\nϕ(H, H R) =−1\n2∂2\n∂H∂H R[Rxy(H, H R)]. (3)The FORC distribution was assessed through the ap-\nplication of a locally fitted second-order polynomial sur-\nface. A gradient smoothing factor was incorporated\ninto the algorithm to suppress numerical artifacts. Con-\nventionally, FORC diagrams are depicted in terms of\nthe coercivity field ( Hc) and the interaction field ( Hu),\nwhich can be derived using Hc= (HR−H)/2 and\nHc= (HR+H)/2. The resulting FORC diagram is dis-\nplayed in figure 3b, where a central ridge is observed\naround µ0Hu=−0.01 T and µ0Hc= 0.39 T. Figure 3c\nillustrates the local interaction field distribution of the\nMRG. A narrow distribution of Hu, with a central point\natµ0Hu=−0.01 T, emphasizes the lack of any signifi-\ncant interactions (dipolar, etc.) between the elementary\nunits (hysterons) that comprise the MRG. Consequently,\nin the absence of inter-particle interactions, the overall\nsystem can be reasonably approximated using the Stoner-\nWohlfarth (SW) model [45].\nThe coercive field distribution of the FORC diagram\nis depicted in figure 3d, with the peak of the distribu-\ntion centered at µ0Hc= 0.39 T. In the absence of in-\nteractions, coercive field distribution also represents the\nswitching field distribution (SFD) of hysterons. A sta-\ntistical analysis of SFD was conducted within the frame-\nwork of the Preisach model. For this analysis, a pseudo-\nVoigt distribution is employed, defined as:\nV(Hc, Hc0,Γ) = ηG(Hc, Hc0,Γ) + (1 −η)L(Hc, Hc0,Γ),\n(4)\nwhere, G(Hc, Hc0,Γ) and L(Hc, Hc0,Γ) are normalized\nGaussian and Lorentzian function. Γ is the common\nFWHM and Hc0is peak center. η(0≤η≤1) serves\nas a weighting factor that transitions the overall profile\nbetween pure Gaussian and pure Lorentzian distributions\nby adjusting the factor from 1 to 0, respectively.\nThe coercive field distribution within the FORC dia-\ngram can be suitably fitted using equation 4. The elon-\ngated tail of the coercivity distribution is attributable to\nthe magnetic viscosity resulting from the thermal fluctu-\nations of metastable states. In MRG, magnetic viscosity\npredominantly stems from the rotation of the magneti-\nzation vector, as contributions from domain wall motion\nare substantially hindered by defects and disorder present\nwithin the film [12]. Therefore, viscosity can be expressed\nas the sum of exponentially decaying metastable states.\nConvolution of these states with the pseudo-Voigt func-\ntion results in the Preisach distribution or switching field\ndistribution (SFD), as demonstrated in equation 5:\nD(Hc, Hc0,Γ, τ) =∞Z\n−∞V[(Hc−ξ), Hc0,Γ]1\nτ\u0014\nexp\u0012−ξ\nτ\u0013\u0015\ndξ ,\n(5)\nhere, τrepresents the magnetic viscosity parameter, mea-\nsured in units of magnetic field. Figure 3d provides clear\nevidence of a strong agreement between the experimen-\ntally obtained coercive field distribution under FORC\nmethod and the theoretical prediction provided by the5\n(a)\n−1.0−0.5 0.0 0.5 1.0\nµ0H(T)−1.0−0.50.00.51.0Normalized Rxy\nHR Rxy(H,HR)FORC\nMHL (b)\n0.0 0.2 0.4 0.6 0.8\nµ0Hc(T)−0.10−0.050.000.050.10µ0Hu(T)\n0.1\n0.20.5\n0.8\n0.9\n00.51φ\n(c)\n−0.2−0.1 0.0 0.1 0.2\nµ0Hu(T)−0.20.00.20.40.60.81.0Distribution (a.u.)FORC(d)\n0.0 0.2 0.4 0.6 0.8 1.0\nµ0Hc(T)0.00.20.40.60.81.0Distribution (a.u.)FORC\nPreisach Model\nFIG. 3. Investigation of FORC and Preisach model for MRG. (a) FORC measurements of MRG using the AHE. The main\nhysteresis loop (MHL) is depicted as a blue solid curve, acquired when the magnetic field is swept from ±1 T. A minor AHE\ncurve is subsequently obtained by starting from the saturation point and returning the magnetic field value to a lower field on\nthe MHL, referred to as the reversal field ( HR). The field is then brought back to the saturation point, forming a single FORC\ncurve (dashed red curve). This process is repeated for numerous uniformly spaced values of HRand applied magnetic field\n(H), resulting in the FORC diagram covering the area within the MHL (black data points). (b) FORC distribution for MRG,\nderived using equation 3. The distribution is attained by fitting the FORC grid utilizing a local second-order polynomial. The\ndistribution is presented in transformed interaction field ( Hu) and coercivity field ( Hc) axes for convenience. (c) Interaction\nfield ( Hu) distribution of MRG derived from the FORC distribution. A narrow Hudistribution, centered at µ0Hu=−0.01 T,\nhighlights the absence of long range interactions (dipolar) between the hysterons comprising the MRG. (d) Resultant coercive\nfield distribution Hc, centered at µ0Hc= 0.39 T, from the FORC distribution (black line) and the estimated coercive field\ndistribution as per the Preisach model (red line). The congruence between the two curves validates the proposed Preisach\nmodel for the MRG system.\nPreisach distribution (equation 5). This finding supports\nthe conclusion that the distribution described by equa-\ntion 5 can be safely considered as a unique Preisach dis-\ntribution of the MRG samples. It is worth noting that\nappropriate normalization methods (amplitude or arial)\nmust be implemented in order to accurately signify the\ndeterministic switching of hysterons. These findings not\nonly contribute to a better understanding of the switch-\ning behavior of MRG samples, but also have important\nimplications for the development of more robust models\nfor other similar systems.\nUpon obtaining the requisite hysteron distribution,\na hysteresis curve for the MRG can be seamlessly de-\nrived by integrating this distribution into the Preisachmodel (equation 2.). Figure 4a demonstrates a remark-\nable congruence between the experimental AHE hystere-\nsis data obtained at 300 K and the fit generated through\nthe Preisach model, with the corresponding resultant\nPreisach distribution presented in figure 4b. Further-\nmore, this model has been expanded to encompass out-of-\nplane hysteresis measurements of MRG at various other\ntemperatures.\nFigure 5 illustrates the AHE hysteresis loops and cor-\nresponding Preisach distributions at select temperature\nvalues, such as 200 K, 100 K, and 5 K. As evidenced by\nthese results, the model captures the experimental intri-\ncacies with remarkable precision, thereby underscoring\nits ability to accurately represent the extensive range of6\n(a)\n−1.0−0.5 0.0 0.5 1.0\nµ0H(T)−1.0−0.50.00.51.0Normalized Rxy\n300 KData\nFit\n(b)\n0.0 0.2 0.4 0.6 0.8\nµ0Hc(T)0.00.20.40.60.81.0Distribution (a.u.)300 KHysteron\nDistribution\nFIG. 4. Hysteresis of MRG characterized using the Preisach\nmodel. (a) The experimentally acquired AHE hysteresis loop\n(black circles) recorded at 300 K, obtained when the applied\nmagnetic field was swept perpendicular to the film plane.\nThe estimated hysteresis (red line), derived from the Preisach\nmodel (equations 2 and 5), demonstrates a high degree of\nagreement with the experimental data, effectively capturing\nthe key features of the observed hysteresis loop. (b) The\nPreisach distribution (PD) for this MRG sample, which de-\npicts the distribution of elementary hysterons, is centered at\n0.40 T exhibits a notably narrow distribution.\nhysteresis observed in MRG using with only a limited\nnumber of parameters ( Hc0,Γ and τ). Additional insights\ninto the magnetic properties can also be gleaned from this\nmodel. Figure 6 depicts the variations in the center point\n(Hc0) and magnetic viscosity ( τ) as functions of temper-\nature for hysteresis curves measured at diverse tempera-\nture range. A notably weak dependency of the magnetic\nviscosity parameter on temperature is observed, which\ncan be ascribed to the dominant influence of anisotropy\non the overall energy landscape of MRG. Consequently,\nany weaker thermodynamic fluctuations exert negligible\nimpact on the static dynamics of the magnetization, caus-\ning the magnetic domains of MRG to remain frozen over\na wide temperature range.\nThe relationship between the center point ( Hc0), which\n−101\n300 K(a)\nData\nFit(b)\n−101\n200 K(c)\nData\nFit(d)\n−1 0 1\nµ0H(T)−101\n5 K(e)\nData\nFit\n0.0 0.5 1.0\nµ0Hc(T)(f)01\n01\n01Normalized Rxy\nDistribution (a.u.)FIG. 5. Temperature-dependent hysteresis of MRG char-\nacterized using the Preisach model. The experimentally\nrecorded AHE hysteresis loops (black circles) measured at\n200 K, 100 K, and 5 K are presented in panels (a), (c), and\n(e) respectively, with their corresponding Preisach model es-\ntimations (red lines) capturing the experimental data in great\ndetail. The estimated Preisach distribution of each model are\ndepicted in panels (b), (d), and (f), providing insights into\nthe hysteron distributions and their temperature-dependent\nbehavior.\nis also known as the sample’s coercivity, and temperature\nis characterized by two distinct regimes. At elevated tem-\nperatures, the coercive field experiences an increase due\nto the diminishing net moment as it approaches the com-\npensation point ( Tcomp = 375 K); conversely, at lower\ntemperatures, the rise in effective anisotropy prevails.\nB. Torque model\nIn the investigation of magnetization dynamics, em-\nploying the macrospin approximation serves as a highly\neffective approach for analysis. In this approximation,\nthe spatial variation of the magnetization remains con-\nstant throughout the equation of motion. The static and\nquasi-static magnetization dynamics of MRG can be ac-\ncurately represented under the macrospin approximation,\nas it accounts for the absence of hysteron interaction,\nwhich is clearly illustrated in figure 3c. The torque model\nis constructed based on the macrospin approximation,\nwhere the equilibrium direction of the magnetization is\ndetermined by counterbalancing the torque that arises\nfrom anisotropy fields with the Zeeman torque. For the\ntetragonal MRG system, the torque balance equation can\nbe efficiently derived from the magnetic anisotropy free\nenergy expression, in which θMandφMrepresent the\npolar and azimuthal angles of the magnetization vector\nM:\nE=K1sin2(θM) +K2sin4(θM) + K3sin4(θM) cos(4 φM)\n−µ0H·M. (6)7\n(a)\n0.200.250.300.350.40Hc0(T)\n(b)\n0 100 200 300\nT(K)0.000.010.020.03τ(T)\nslope =−5.50×10−6T\nKData\nLinear fit\nFIG. 6. Temperature-dependent analysis of coercivity and\nmagnetic viscosity in MRG. (a) The center-point ( Hc0) of\nthe Preisach distribution, which also represents the sample’s\ncoercivity, as a function of temperature. The value of Hc0\nincreases for both high and low temperature ranges. The in-\ncrease at higher temperatures is attributed to the approach to-\nwards the compensation temperature ( Tcomp = 375 K), while\nat lower temperatures, the rise in the anisotropy constant\nleads to an increase in the central-point value. The solid line\nserves as a guide for the data points. (b) Viscosity parameter\n(τ) as a function of temperature. This relationship suggests\nthat magnetic viscosity remains approximately independent\nwithin the measured temperature range, indicating that the\nmagnetic domains for MRG are essentially frozen over a wide\ntemperature range.\nHere, the first and second-order uniaxial out-of-plane\nanisotropy constants are denoted by K1andK2, respec-\ntively, while K3signifies the four-fold in-plane anisotropy\nconstant. By evaluating the extrema of equation 6 with\nrespect to θMandφM, the equilibrium magnetization\ndirection can be determined. For instance, the polar\nequilibrium position can be ascertained by solving the\nsubsequent equation:\n∂E\n∂θM= 2K1+ [4K2+ 4K3cos(4 φM)] sin2(θM)\n−µ0HM sin(θH−θM)\nsin(θM) cos( θH)= 0. (7)In the aforementioned equation, it is assumed that the\nin-plane anisotropy ( K3) is relatively weak; therefore, M\nadheres to the applied magnetic field ( H) along the az-\nimuthal direction with a slight delay, i.e., φM≈φH.\nHere, the polar angle and the azimuthal angle of the ap-\nplied magnetic field are represented by θHandφH, re-\nspectively. In this work, we utilize the anomalous Hall\neffect (AHE) to examine the anisotropy constants, which\nis particularly sensitive to the out-of-plane component of\nthe Mn4cmoment, therefore:\nVxy∝Mcos(θM),\n=⇒cos(θM) =Vxy\nVNxy=vz,(8)\nwhere, VN\nxydenotes the AHE voltage when the magne-\ntization ( M) is aligned with the normal to the sample\n(θM= 0), and vzrepresents the normalized AHE voltage.\nConsequently, the equilibrium condition (equation 7) is\nreduced to:\n2K1\nM+\u00124K2\nM+4K3\nMcos(4 φM)\u0013\u0000\n1−v2\nz\u0001\n= \nµ0Hsin(θH−θM)\nvzp\n1−v2z!\n.(9)\nTo determine the values of K1,K2, and K3, rotational\nscans were conducted in various geometric configurations.\nIt is important to recognize that the recorded trans-\nverse resistance consists of five distinct contributions,\nwhich include: the ordinary Hall effect (OHE), the\nanomalous Hall effect (AHE), the planar Hall effect\n(PHE), the ordinary Nernst effect (ONE), and the\nanomalous Nernst effect (ANE). These contributions are\nrepresented in equation 10 [46, 47]:\nRxy=ROHE\nxy +RAHE\nxy +RPHE\nxy +RONE\nxy +RANE\nxy.(10)\nIn the context of MRG, the ordinary Nernst effect (\nRONE\nxy ) and the anomalous Nernst effect ( RANE\nxy ) were\neffectively minimized by employing an exceedingly small\ninput bias current signal ( IRMS≈50µA). This approach\nensured the absence of any significant thermal gradient\nwithin the observed sample. To further mitigate the tem-\nperature gradient across the Hall bar, temperatures were\nstabilized using a helium partial pressure ( P∼100 Torr)\nwithin a PPMS tool. The sample was carefully rotated\nat a very slow rate to minimize temperature destabiliza-\ntion that could arise from friction within the sample’s\nrotator gears. The ordinary Hall effect ( RONE\nxy ) was\ndetermined by measuring the slope of the AHE at high\nmagnetic fields ( |µ0H|>8 T), as illustrated in figure 7a.\nThe Hall coefficient calculated for MRG yielded a value\nofRH=−4.41×10−10m3C−1, which corresponds to\na carrier concentration ne= 1.42×1022cm−3. In MRG,\nthe Hall effect is predominantly governed by the minor-\nity carrier at the Fermi level due to the material’s high\nspin-polarization [9, 15].8\nThe planar Hall effect ( RPHE\nxy) was evaluated by ro-\ntating the magnetic field within the plane of the sample.\nFigure 7b displays the anisotropic magneto-resistance\n(AMR) and the planar Hall effect (PHE) measured at\nroom temperature in the presence of a magnetic field\nwith a value of 1 .9 T. The observed PHE is three orders of\nmagnitude smaller than the recorded AHE, thus allowing\nit to be safely disregarded from equation 10. As a result,\nthe primary dominant contribution to the transverse Hall\nresistance is due to the AHE. Nevertheless, OHE has also\nbeen considered in the model to acknowledge its signifi-\ncant contribution, particularly at high applied magnetic\nfields.\n(a)\n−15−10−5 0 5 10 15\nµ0H(T)−1.0−0.50.00.51.0Rxy(Ω)300 K\nslope =−1.10×10−2ΩT−1AHE\nSlope\nSlope\n(b)\n0 90 180 270 360\nϕH(◦)177.50177.51177.52177.53177.54RAMR\nxy (Ω)AMRAMR\n−1.5−1.0−0.50.00.51.01.5\nRPHE\nxy (mΩ)PHEPHE\nFIG. 7. AHE, AMR, and PHE measurements of MRG at\n300 K using a micron-sized (60 ×20µm2) Hall bar. (a) AHE\nis recorded with the magnetic field applied perpendicular to\nthe sample within a range of ±14 T. The AHE hysteresis\nloop exhibits a coercivity value of 0 .40 T. The ordinary Hall\neffect (OHE) contribution of MRG is evaluated by calculating\nthe slope of the curve at high magnetic fields ( |µ0H|>8 T).\n(b) AMR and PHE of MRG are measured at 300 K when a\n1.9 T magnetic field is rotated within the sample plane. The\nPHE (of the order of a few milliOhm) has a contribution three\norders of magnitude smaller than that of AHE. Consequently,\nthe PHE contribution from the transverse Hall effect can be\nsafely disregarded.\nTo investigate the out-of-plane anisotropy constants\n(K1andK2), the AHE was conducted in the measure-ment geometry depicted in figure 8a. In this configu-\nration, the sample was rotated in such a way that the\napplied magnetic field effectively rotated within the yz-\nplane. Figure 8b presents the three rotational AHE loops\nmeasured at T = 300 K, under constant applied mag-\nnetic fields of 1 T, 2 T, and 14 T. The acquired data\nexhibit two distinct regimes: the first regime showcases a\ncontinuous change in the resistance (non-hysteretic seg-\nments, for θH<90◦,θH>270◦and in the vicinity of\nθH= 180◦), attributable to the smooth coherent rotation\nof the magnetization against the anisotropy field, while\nthe second regime displays an abrupt change in the resis-\ntance (hysteretic segments, for 90◦< θH<270◦) due to\nthe switching of the magnetic moment from out-of-plane\nto in-plane or vice versa. The non-hysteretic portions\nof the data were fitted using the torque model. Since\nthe anisotropy constants are independent of the applied\nexternal field (at least up to the first order), the non-\nhysteretic segments of the data should be modelled us-\ning common fitting parameters. Figure 8c illustrates the\nrecorded data alongside the corresponding best fits uti-\nlizing common anisotropy parameters. The data align\nwell with the model, yielding anisotropy constants of\nK1\nM= 0.655 T andK2\nM= 0.416 T, where Mdenotes the\nmagnitude of saturation magnetization. The sample’s\nsaturation magnetization, obtained from SQUID mea-\nsurement, is 61 kA m−1. Consequently, the first and sec-\nond order out-of-plane anisotropy constants of MRG are\nK1= 4.0×104J m−3andK2= 2.54×104J m−3, re-\nspectively. It is crucial to note that in this measurement\ngeometry, the AHE is insensitive to in-plane anisotropy\ndue to the absence of azimuthal rotation of magnetiza-\ntion. The in-plane anisotropy constant ( K3) can be ex-\namined in a measurement geometry where the azimuthal\ndirection of magnetization is varied.\nProgressing with the study, the in-plane anisotropy\nwas examined using the AHE in the measurement geom-\netry depicted in figure 8d. In this configuration, the sam-\nple was rotated in such a way that the applied magnetic\nfield effectively rotated within the plane of the sample\n(xz-plane). The presence of in-plane anisotropy causes\nthe AHE signal to oscillate as a function of the azimuthal\nangle ( φM) of the magnetization, revealing the four-fold\nanisotropy of MRG. Figure 8e and 8f display the scans\nobtained at 300 K when constant magnetic fields of 1 T\nand 1 .5 T were applied in the plane of the sample, respec-\ntively. The unequal amplitude of oscillation arises from\na small offset ( ∼6 deg) of the sample from the xz-plane,\nwhich subsequently causes sample wobbling during ro-\ntation and introduces an additional term – a non-zero\nnormal component – affecting the magnetization vector\nposition θM. The equilibrium position of the magnetiza-\ntion vector, under the influence of the external magnetic\nfield, is numerically obtained using the torque model with\na correction for wobbling taken into consideration (the\nimplementation involved employing Rodrigues’ rotation\nformula, which utilizes the appropriate axis of rotation).9\n(a)\nx\nyz\nM\nHˆn\nθHθM\nφM≈φHω(b)\n0 90 180 270 360\nθH(◦)−1.0−0.50.00.51.0Normalized Rxy\nAHE 1 T\nAHE 2 T\nAHE 14 T (c)\n0 30 60 90\nθH(◦)−1.0−0.50.00.51.0Normalized RxyData 1 T\nFit 1 T\nData 2 T\nFit 2 T\nData 14 T\nFit 14 T\n(d)\nxyz\nMH\nˆnφHω\nθH\nθM(e)\n0 90 180 270 360\nϕH(◦)0.940.960.981.00Normalized RxyData 1 T\nFit 1 T (f)\n0 90 180 270 360\nϕH(◦)0.670.720.770.82Normalized RxyData 1.5 T\nFit 1.5 T\nFIG. 8. AHE measurement at 300 K in two different geometric arrangements to evaluate the out-of-plane anisotropy constants\n(K1,K2) and in-plane anisotropy constant ( K3) of MRG. (a) Out-of-plane rotational measurement geometry for investigating\nK1andK2. In this configuration, a constant applied magnetic field ( H) is effectively rotated in the yz-plane, varying the field\nangle θHand consequently changing the magnetization angle θM, which is recorded using AHE. (b) Rotational AHE data as a\nfunction of field angle ( θH) for constant applied magnetic fields of 1 T, 2 T, and 14 T. Each loop contains two distinct regimes:\na hysteretic part attributed to the abrupt switching of magnetization, and a non-hysteretic part due to the coherent rotation of\nmagnetization experiencing net torque. The detailed fits to this data are presented on figure 10. (c) The recorded non-hysteretic\nsegments of curves (scatter plots) for θH<90◦are fitted with the balanced torque model (solid lines), given by equation 9.\nAnisotropy constants K1/M= 0.655 T, K2/M= 0.416 T are obtained by fitting these AHE curves. (d) In-plane measurement\ngeometry for investigating K3, where the field is effectively rotated within the sample plane (i.e., θH≈90◦andφHis varied),\nresulting in the modulation of θMas a function of φH. (e) In-plane AHE data (black scattered circles) as a function of field\nrotation angle ( φH) for a field value of 1 T. The four troughs and valleys in the curve are associated with the four-fold in-plane\nanisotropy of MRG. The unequal amplitude of oscillation is attributed to sample offset (sample plane slightly tilted away from\nthexz-plane). Incorporating sample offset in the torque model provides an excellent estimation of in-plane rotational AHE\n(red solid line), with the estimated K3/M= 0.057 T. (f) In-plane AHE data (black scattered circles) as a function of φH\nfor an applied field of 1 .5 T. The estimated curve (red solid line) is obtained for K1/M= 0.655 T, K2/M= 0.416 T, and\nK3/M= 0.057 T.\nThe extracted value of the in-plane anisotropy constant\nisK3\nM= 0.057 T, or K3= 3.48×103J m−3, which\nis an order of magnitude smaller than the out-of-plane\nanisotropy constants K1andK2. It is noteworthy that\nan increase in field strength results in a more pronounced\nhysteresis of AHE in the in-plane configuration (compar-\ning figure 8e and 8f). This phenomenon occurs because,\nat sufficiently high magnetic fields, the sample wobbling\nleads to partial switching of magnetic moments. Model-\ning such complex data, where both coherent rotation and\nmagnetization switching take place, can be accomplished\nby combining both the Preisach and torque models.C. Combined Preisach and torque (CPT) model\n1. In-plane field hysteresis loop\nThe intricate quasi-static magnetization dynamics can\nbe elucidated by combining the torque and Preisach mod-\nels, wherein the magnetic moment exhibits both coherent\nrotation and abrupt switching events under the influ-\nence of suitable stimuli. One such mixed behavior can\nalso be observed when a high magnetic field is swept\nwithin the plane of the sample. To examine such dy-\nnamics, AHE was measured in the measurement geom-\netry depicted in figure 9a. In this setup, the magnetic\nfield was swept within the plane of the sample (along the\nz-axis) from ±14 T. The recorded AHE signal exhibits10\na combination of hysteretic and rotational behavior, as\nshown in figure 9b. Due to an unavoidable minor off-\nset (δ) of the sample while mounting it on the rotary\nstage of the PPMS, the applied magnetic field does not\nlie exactly within the plane of the sample (see the inset\nof figure 9b). An offset of approximately θ∼6 deg was\npresent, the exact offset value can be determined by fit-\nting the AHE data. As a result, the AHE signal has two\ndistinct regimes: (i) when the magnetic field (H) forms an\nacute angle with the normal to the sample ( θH<90 deg),\na coherent rotation of the magnetization vector is ob-\nserved; and (ii) when the magnetic field makes an obtuse\nangle with the normal ( θH>90 deg), the switching of\nthe magnetic moment occurs when the field projection\nsurpasses the coercive field. It should be noted that the\ncoercive field in this scenario is scaled according to equa-\ntion 11:\nHc=Hcn\ncos(θH), (11)\nhere, Hcnrepresents the coercivity of the MRG sample\nwhen the magnetic field is applied along the direction\nnormal to the sample plane ( θH= 0).\nA comprehensive trajectory of magnetization under\nthe influence of the applied magnetic field is meticulously\ndemonstrated in figure 9b. This representation encom-\npasses the combined behaviour of the rotation of mag-\nnetic moments and their corresponding switching events,\nwhich can be effectively described through equation 12;\nRxy=RTM\nxy·RPM\nxy (12)\nwhere, RTM\nxyandRPM\nxydenote the respective contribu-\ntions arising from the torque model (as explicated in\nequation 9 ) and the Preisach model (as detailed in\nequation 5). To determine the solution for equation 12,\nthe Levenberg-Marquardt algorithm was employed, while\nmaintaining the anisotropy constants K1/Mat a fixed\nvalue of 0 .655 T and K2/Mat 0.416 T. The resultant\nestimated curve, in conjunction with the data, is dis-\nplayed in figure 9b, which reveals an exceptional con-\ngruence with the gathered data. Furthermore, figure 9c\ndepicts the estimated hysteron distribution for the cor-\nresponding AHE curve, with the central point of distri-\nbution ( Hc0) determined to be 3 .88 T. Notably, this dis-\ntribution curve also exhibits a strong resemblance to the\nhysteron distribution curve obtained when the magnetic\nfield is applied perpendicular to the sample (as shown in\nfigure 4b), with the field axis scaled according to equa-\ntion 11. This consistency highlights the robustness of\nthe analysis and further validates the effectiveness of the\nmodel.\nThough the combined Preisach and torque (CPT)\nmodel, as delineated by equation 12, effectively predicts\nthe intricate magnetization dynamics, implementing this\nequation to depict complex quasi-static magnetic dynam-\nics presents a considerable computational challenge due\nto the multiparameter nature of the equation. Never-\ntheless, the awareness that the Preisach distribution fora specific temperature can be independently determined\nthrough a pure switching event (out-of-plane hysteresis\ncurve) allows for further simplification of the combined\nmodel. This is achieved by further considering an effec-\ntive out-of-plane anisotropy field ( Heff) to resolve the\npure torque model component. The rationale for utiliz-\ning an effective anisotropy field stems from the fact that\nMRG exhibits substantial and dominating out-of-plane\nanisotropy, which is also evident from the steep square\nhysteresis loop observed when the field is swept perpen-\ndicular to the sample (figure 7a). Under this approxima-\ntion, the equilibrium position of magnetization ( M) can\nbe attained by counterbalancing the torques acting upon\nit, (equation 13 )\nM×µ0Heff=M×µ0H, (13)\nwhere, HeffandHrepresent the effective out-of-plane\nanisotropy field and the applied external magnetic field,\nrespectively. In figure 9d, the data and corresponding fit\nare presented, which utilize the Heffmodel with a single\nfree parameter ( Heff) and the Preisach model that has\nbeen determined previously. The calculated Heffvalue\nfrom the fitting is 1 .46 T. The model captures all details\nof the AHE, thereby validating the proposed approxima-\ntion. It is important to note that by comparing figures 9b\nand 9d, the distinction between the two models can be\ndiscerned. At high magnetic field values ( |µ0H| ≥5 T),\nthe effective anisotropy field model slightly deviates from\nthe data and does not accurately capture the curvature of\nthe data as effectively as the complete model (figure 9b).\nThis is due to the model’s assumption of a unique fixed\nHeffvalue for all Morientations. In contrast, the mag-\nnitude of Hefffor a tetragonal crystal system relies on\nthe magnetization direction, and its magnitude typically\ndecreases as Mdeviates from the out-of-plane direction\n(easy-axis). Consequently, a Heff(θM) is necessary to\ncapture the data in greater detail for all possible mag-\nnetic field values. Nonetheless, it is adequate to assume\nthat a single fixed Heffperforms remarkably well, at\nleast up to the magnetic field strength employed in this\nstudy (14 T ≤ |µ0H|). This approximation offers a sig-\nnificant advantage in describing complex magnetization\ndynamics by substantially reducing the number of free\nparameters in the model.\n2. Out-of-plane rotational hysteresis loop\nThe efficacy of the CPT model is further substantiated\nby applying it to magnetization dynamics derived from\nout-of-plane rotational hysteresis curves, as illustrated in\nfigure 10. In this experiment, AHE curves were acquired\nutilizing the measurement geometry shown in figure 10a.\nThis setup involves rotating a constant applied magnetic\nfield within the yz-plane, causing the equilibrium posi-\ntion of the magnetic moment ( θM) to reside within the11\n(a)\nxyz\nMH\nˆnθH\nθM\n−14 T0 T+14 T\n+H\n−H (b)\n−15−10−5 0 5 10 15\nµ0H(T)−1.0−0.50.00.51.0Normalized Rxy\nAB\nC\nD1○\n2○\n3○4○AHE Data\nCPT Model\n−z\nxyz\nMH\nˆn θHθM\nδδθH<90◦\n−z\nxyz\nMHˆn\nθH\nθMδδθH>90◦\n(c)\n0.0 2.5 5.0 7.5 10.0 12.5 15.0\nµ0Hc(T)0.00.20.40.60.81.01.2Distribution (a.u.)CPT Model\nHysteron Distribution (d)\n−15−10−5 0 5 10 15\nµ0H(T)−1.0−0.50.00.51.0Normalized RxyAHE Data\nHeff−Model\nFIG. 9. In-plane field loop study for the combined hysteresis and torque model at 300 K. (a) Measurement geometry of\nin-plane field loop study, where the magnetic field is swept between ±14 T in the sample plane ( θH≈90◦), along the z-axis.\n(b) Resultant experimentally obtained AHE curve (black open circles). An unavoidable sample offset ( δ) with respect to the\nfield axis ( z) leads to this intricate magnetization dynamics, where the magnetization direction switches sign when the normal\ncomponent of the field exceeds the sample’s coercivity. The insets illustrate the geometrical alignment of magnetization and\napplied magnetic field when the field angle ( θH) forms an acute angle with the sample’s normal direction ˆ n(right inset) and\nan obtuse angle with ˆ n(left inset). Paths 1○and 3○represent portions of the curve where magnetization experiences coherent\nrotation, while paths 2○and 4○display the behavior when magnetization dynamics is dominated by switching events. The\nestimated AHE (red line) within the combined Preisach and torque (CPT) model shows excellent agreement with the data.\n(c) Coercivity distribution (hysteron distribution) derived from the CPT model. This distribution curve also exhibits strong\nresemblance to the hysteron distribution curve obtained when the magnetic field is applied perpendicular to the sample (as\nshown in figure 3d), with the field axis scaled according to equation 11. (d) AHE hysteresis data (black circles) modeled using\nan effective out-of-plane anisotropy approximation (red line). Under this approximation, the torque model is simplified with\na single out-of-plane anisotropy field ( Heff). By directly incorporating a predetermined hysteron distribution into the model,\nthe complexity of the model is greatly reduced, with the added benefit of having a minimal number of free parameters.\nsame plane. A collection of AHE data, recorded at ap-\nplied magnetic field values of 1 T, 2 T and 14 T, and their\ncorresponding CPT fits, are displayed in figures 10b, 10c,\nand 10d, respectively. The CPT model well-describes the\ndataset across all applied magnetic fields. It is important\nto note that as the magnitude of the applied magnetic\nfield escalates, the hysteretic contribution to the AHE\nstarts to decrease relative to the non-hysteretic contri-\nbution, resulting in a reduced hysteretic width. In cases\nwhere the field strength reaches exceptionally high lev-\nels, the Zeeman term prevails over the anisotropy term,thereby causing the magnetization to effectively align\nwith the magnetic field direction. Consequently, at a\n14 T field, the hysteresis width has virtually disappeared\n(figure 10d). In the context of the current CPT fitting\napproach, all parameters, including the effective out-of-\nplane anisotropy field ( Heff) and the coefficients of hys-\nteron distribution ( Hc0,Γ and τ), are maintained as con-\nstant values. These parameters have been previously de-\ntermined through the fitting of other AHE curves, as\nelaborated upon in the preceding sections. As a result,\nthe derived fitting curve successfully captures both the12\n(a)\nx\nyz\nM\nHˆn\nθHθM\nφM≈φHω (b)\n0 90 180 270 360\nθH(◦)−1.0−0.50.00.51.0Normalized Rxy\nData 1 T\nCPT Model\n(c)\n0 90 180 270 360\nθH(◦)−1.0−0.50.00.51.0Normalized Rxy\nData 2 T\nCPT Model (d)\n0 90 180 270 360\nθH(◦)−1.0−0.50.00.51.0Normalized Rxy\nData 14 T\nCPT Model\nFIG. 10. Investigation of out-of-plane rotational hysteresis loop within the CPT model. (a) Measurement geometry where a\nconstant magnetic field is effectively rotated in the yz-plane. As a result, the magnetization angle ( θM) changes within the yz-\nplane as the field angle ( θH) varies. Note that this set of data have already been presented in figure 8b. AHE loops as a function\nofθHfor the applied field values of 1 T, 2 T and 14 T are shown in (b), (c), and (d) respectively. The CPT model under the\neffective anisotropy ( Heff) approximation accurately captures the behavior of the data at each field (red line). All parameters,\nsuch as the effective out-of-plane anisotropy field ( Heff) and the coefficients of the hysteron distribution ( Hc0,Γ and τ), remain\nconstant in the current CPT fitting approach. These parameters were determined beforehand by fitting other AHE curves, as\nexplained in the preceding sections.\nhysteretic and non-hysteretic aspects of the AHE curve\nwith remarkable precision, for both low ( µ0H= 1 T) and\nhigh applied magnetic fields ( µ0H= 14 T), while virtu-\nally eliminating the need for free parameters.\nIV. CONCLUSION\nIn this work, we have developed a comprehen-\nsive methodology for determining the various magnetic\nanisotropy constants of low-moment MRG thin films. To\nachieve this, we initially investigated hysteretic phenom-\nena using the Preisach model, also known as the hys-\nteron model. The applicability of the Preisach model\nwas subsequently experimentally verified through the im-\nplementation of the first-order reversal curves (FORC)\nmethod, which enabled us to identify the unique hys-\nteron distribution of the sample under investigation.\nThe FORC method provided crucial insights, specificallyhighlighting the absence of long-range magnetic interac-\ntions within the hysterons, which allowed for the utiliza-\ntion of the macrospin model (Stoner-Wohlfarth model)\nto describe the quasi-static magnetization dynamics of\nMRG. Furthermore, the Preisach model confirmed that\nMRG samples exhibit relatively weak variations in mag-\nnetic viscosity with temperature, signifying the pres-\nence of a frozen domain structure. To determine the\nanisotropy constants of the MRG samples, we employed\na detailed torque model within the macrospin approxi-\nmation framework. Anomalous Hall effect (AHE) mea-\nsurements were carried out in various suitable geometries,\nwhich facilitated the deduction of out-of-plane anisotropy\nconstants K1= 4.0×104J m−3(K1/M= 0.655 T)\nandK2= 2.54×104J m−3(K2/M= 0.416 T), and\nan in-plane anisotropy constant K3= 3.48×103J m−3\n(K3/M= 0.057 T) through data fitting with the torque\nmodel. Additionally, we successfully investigated more\ncomplex quasi-static magnetization dynamics, character-13\nized by the combination of hysteretic and non-hysteretic\ncomponents in AHE, using a combined Preisach and\ntorque (CPT) model with virtually no free parameters.\nOur study demonstrates the efficacy of this methodology\nnot only in determining the magnetic anisotropy of low\nmoment magnetic samples (MRG), but also in explaining\nother complex magnetization dynamics within a unified\nmodel. The proposed method can be readily extended\nto other magnetic systems that lack hysteronic interac-\ntions, exhibit narrow hysteron distributions, and display\nfrozen-domain behaviour. This comprehensive approach\nwill undoubtedly prove valuable in studying both lin-ear and non-linear quasi-static magnetization dynamics\nof MRG in external fields and/or current-induced effec-\ntive fields resulting from spin-orbit torque/spin-transfer\ntorque.\nACKNOWLEDGEMENTS\nA.J., S.L., G.P., K.R., J.M.D.C and P.S. acknowledge\nfunding from TRANSPIRE FET Open H2020 and SFI,\nAMBER and MANIAC programmes.\n[1] C. Chappert, A. Fert, and F. N. Van Dau, The emergence\nof spin electronics in data storage, Nature materials 6,\n813 (2007).\n[2] T. Jungwirth, X. Marti, P. Wadley, and J. Wunder-\nlich, Antiferromagnetic spintronics, Nature nanotechnol-\nogy11, 231 (2016).\n[3] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono,\nand Y. Tserkovnyak, Antiferromagnetic spintronics, Rev.\nMod. Phys. 90, 015005 (2018).\n[4] A. Khvalkovskiy, D. Apalkov, S. Watts, R. Chepulskii,\nR. Beach, A. Ong, X. Tang, A. Driskill-Smith, W. But-\nler, P. Visscher, et al. , Basic principles of STT-MRAM\ncell operation in memory arrays, Journal of Physics D:\nApplied Physics 46, 074001 (2013).\n[5] V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich,\nA. Slavin, D. Baither, G. Schmitz, and S. O. Demokri-\ntov, Magnetic nano-oscillator driven by pure spin current,\nNature materials 11, 1028 (2012).\n[6] P. Wadley, B. Howells, J. ˇZelezn` y, C. Andrews, V. Hills,\nR. P. Campion, V. Nov´ ak, K. Olejn´ ık, F. Maccherozzi,\nS. Dhesi, et al. , Electrical switching of an antiferromag-\nnet, Science 351, 587 (2016).\n[7] W. E. Pickett, Spin-density-functional-based search for\nhalf-metallic antiferromagnets, Phys. Rev. B 57, 10613\n(1998).\n[8] I. Galanakis, K. ¨Ozdo˘ gan, E. S ¸a¸ sıo˘ glu, and B. Akta¸ s, Ab\ninitio design of half-metallic fully compensated ferrimag-\nnets: The case of Cr 2MnZ(Z= P, As, Sb, and Bi),\nPhys. Rev. B 75, 172405 (2007).\n[9] H. Kurt, K. Rode, P. Stamenov, M. Venkatesan, Y. C.\nLau, E. Fonda, and J. M. Coey, Cubic Mn 2Ga Thin films:\nCrossing the spin gap with ruthenium, Physical Review\nLetters 112, 2 (2014).\n[10] K. E. Siewierska, G. Atcheson, A. Jha, K. Esien,\nR. Smith, S. Lenne, N. Teichert, J. O’Brien, J. M. D.\nCoey, P. Stamenov, and K. Rode, Magnetic order\nand magnetotransport in half-metallic ferrimagnetic\nMnyRuxGa thin films, Phys. Rev. B 104, 064414 (2021).\n[11] C. Banerjee, N. Teichert, K. Siewierska, Z. Gercsi,\nG. Atcheson, P. Stamenov, K. Rode, J. Coey, and\nJ. Besbas, Single pulse all-optical toggle switching of\nmagnetization without gadolinium in the ferrimagnet\nMn2RuxGa, Nature communications 11, 1 (2020).\n[12] N. Teichert, G. Atcheson, K. Siewierska, M. N. Sanz-\nOrtiz, M. Venkatesan, K. Rode, S. Felton, P. Stamenov,\nand J. Coey, Magnetic reversal and pinning in a perpen-dicular zero-moment half-metal, Physical Review Mate-\nrials5, 034408 (2021).\n[13] H. Van Leuken and R. De Groot, Half-metallic antiferro-\nmagnets, Physical review letters 74, 1171 (1995).\n[14] D. Betto, N. Thiyagarajah, Y.-C. Lau, C. Piamonteze,\nM.-A. Arrio, P. Stamenov, J. Coey, and K. Rode, Site-\nspecific magnetism of half-metallic Mn 2RuxGa thin films\ndetermined by X-ray absorption spectroscopy, Physical\nReview B 91, 094410 (2015).\n[15] C. Fowley, K. Rode, Y.-C. Lau, N. Thiyagarajah,\nD. Betto, K. Borisov, G. Atcheson, E. Kampert, Z. Wang,\nY. Yuan, S. Zhou, J. Lindner, P. Stamenov, J. M. D.\nCoey, and A. M. Deac, Magnetocrystalline anisotropy\nand exchange probed by high-field anomalous Hall ef-\nfect in fully compensated half-metallic Mn 2RuxGa thin\nfilms, Physical Review B 98, 220406 (2018).\n[16] M. ˇZic, K. Rode, N. Thiyagarajah, Y. C. Lau, D. Betto,\nJ. M. Coey, S. Sanvito, K. J. O’Shea, C. A. Ferguson,\nD. A. Maclaren, and T. Archer, Designing a fully com-\npensated half-metallic ferrimagnet, Physical Review B\n93, 1 (2016), arXiv:1511.07923.\n[17] K. Siewierska, N. Teichert, R. Sch¨ afer, and J. Coey, Imag-\ning domains in a zero-moment half metal, IEEE Trans-\nactions on Magnetics 55, 1 (2018).\n[18] R. E. Troncoso, K. Rode, P. Stamenov, J. M. D. Coey,\nand A. Brataas, Antiferromagnetic single-layer spin-orbit\ntorque oscillators, Phys. Rev. B 99, 054433 (2019).\n[19] N. Awari, S. Kovalev, C. Fowley, K. Rode, R. A. Gal-\nlardo, Y. C. Lau, D. Betto, N. Thiyagarajah, B. Green,\nO. Yildirim, J. Lindner, J. Fassbender, J. M. Coey, A. M.\nDeac, and M. Gensch, Narrow-band tunable terahertz\nemission from ferrimagnetic Mn 3−xGa thin films, Ap-\nplied Physics Letters 109, 10.1063/1.4958855 (2016).\n[20] X. Fan, D. Xue, C. Jiang, Y. Gong, and J. Li, An ap-\nproach for researching uniaxial anisotropy magnet: Ro-\ntational magnetization, Journal of Applied Physics 102,\n123901 (2007).\n[21] Y. Endo, O. Kitakami, S. Okamoto, and Y. Shimada,\nDetermination of first and second magnetic anisotropy\nconstants of magnetic recording media, Applied Physics\nLetters 77, 1689 (2000).\n[22] E. A. Jagla, Hysteresis loops of magnetic thin films with\nperpendicular anisotropy, Physical Review B 72, 094406\n(2005).\n[23] K. Okamoto, A new method for analysis of magnetic\nanisotropy in films using the spontaneous hall effect,14\nJournal of magnetism and magnetic materials 35, 353\n(1983).\n[24] H. Sato, M. Pathak, D. Mazumdar, X. Zhang,\nG. Mankey, P. LeClair, and A. Gupta, Anomalous Hall\neffect behavior in (100) and (110) CrO 2thin films, Jour-\nnal of Applied Physics 109, 103907 (2011).\n[25] G. Suran, M. Naili, H. Niedoba, F. Machizaud, O. Acher,\nand D. Pain, Magnetic and structural properties of Co-\nrich CoFeZr amorphous thin films, Journal of magnetism\nand magnetic materials 192, 443 (1999).\n[26] D. Berling, S. Zabrocki, R. Stephan, G. Garreau,\nJ. Bubendorff, A. Mehdaoui, D. Bolmont, P. Wetzel,\nC. Pirri, and G. Gewinner, Accurate measurement of the\nin-plane magnetic anisotropy energy function E a(θ) in\nultrathin films by magneto-optics, Journal of magnetism\nand magnetic materials 297, 118 (2006).\n[27] R. Cowburn, A. Ercole, S. Gray, and J. Bland, A new\ntechnique for measuring magnetic anisotropies in thin\nand ultrathin films by magneto-optics, Journal of applied\nphysics 81, 6879 (1997).\n[28] F. Preisach, ¨Uber die magnetische nachwirkung,\nZeitschrift f¨ ur physik 94, 277 (1935).\n[29] E. Della Torre, Magnetization calculation of fine parti-\ncles, IEEE Transactions on Magnetics 22, 484 (1986).\n[30] G. K´ ad´ ar, E. Kisdi-Koszo, L. Kiss, L. Potocky, M. Za-\ntroch, and E. Della Torre, Bilinear product preisach mod-\neling of magnetic hysteresis curves, IEEE Transactions on\nMagnetics 25, 3931 (1989).\n[31] J. Fuzi, Analytical approximation of preisach distribu-\ntion functions, IEEE transactions on magnetics 39, 1357\n(2003).\n[32] O. Henze and W. M. Rucker, Identification procedures\nof preisach model, IEEE Transactions on magnetics 38,\n833 (2002).\n[33] W. S. Galinaitis, D. S. Joseph, and R. C. Rogers, Param-\neter identification for preisach models of hysteresis, in\nInternational Design Engineering Technical Conferences\nand Computers and Information in Engineering Confer-\nence, Vol. 80289 (American Society of Mechanical Engi-\nneers, 2001) pp. 1409–1417.\n[34] I. Mayergoyz, Mathematical models of hysteresis, IEEE\nTransactions on magnetics 22, 603 (1986).\n[35] H. Chiriac, N. Lupu, L. Stoleriu, P. Postolache, and\nA. Stancu, Experimental and micromagnetic first-order\nreversal curves analysis in NdFeB-based bulk “exchange\nspring”-type permanent magnets, Journal of magnetism\nand magnetic materials 316, 177 (2007).\n[36] P.-A. Chen, C.-Y. Yang, S.-J. Chang, M.-H. Lee, N.-K.\nTang, S.-C. Yen, and Y.-C. Tseng, Soft and hard natures\nof Nd 2Fe14B permanent magnet explored by first-order-\nreversal-curves, Journal of magnetism and magnetic ma-terials 370, 45 (2014).\n[37] A. P. Roberts, C. R. Pike, and K. L. Verosub, First-order\nreversal curve diagrams: A new tool for characterizing\nthe magnetic properties of natural samples, Journal of\nGeophysical Research: Solid Earth 105, 28461 (2000).\n[38] A. R. Muxworthy and A. P. Roberts, First-order rever-\nsal curve (forc) diagrams, in Encyclopedia of Geomag-\nnetism and Paleomagnetism , edited by D. Gubbins and\nE. Herrero-Bervera (Springer Netherlands, Dordrecht,\n2007) pp. 266–272.\n[39] F. B´ eron, L. Clime, M. Ciureanu, D. M´ enard, R. W.\nCochrane, and A. Yelon, First-order reversal curves dia-\ngrams of ferromagnetic soft nanowire arrays, IEEE trans-\nactions on magnetics 42, 3060 (2006).\n[40] F. B´ eron, L.-P. Carignan, D. M´ enard, and A. Yelon, Mag-\nnetic behavior of Ni/Cu multilayer nanowire arrays stud-\nied by first-order reversal curve diagrams, IEEE Trans-\nactions on Magnetics 44, 2745 (2008).\n[41] C. R. Pike, A. P. Roberts, and K. L. Verosub, Character-\nizing interactions in fine magnetic particle systems using\nfirst order reversal curves, Journal of Applied Physics 85,\n6660 (1999).\n[42] A. Stancu, D. Ricinschi, L. Mitoseriu, P. Postolache, and\nM. Okuyama, First-order reversal curves diagrams for\nthe characterization of ferroelectric switching, Applied\nPhysics Letters 83, 3767 (2003).\n[43] J.-G. Ram´ ırez, A. Sharoni, Y. Dubi, M. G´ omez, and I. K.\nSchuller, First-order reversal curve measurements of the\nmetal-insulator transition in VO 2: Signatures of persis-\ntent metallic domains, Physical Review B 79, 235110\n(2009).\n[44] Z. Diao, N. Decorde, P. Stamenov, K. Rode, G. Feng,\nand J. Coey, Magnetization processes in micron-scale\n(CoFe/Pt) nmultilayers with perpendicular anisotropy:\nFirst-order reversal curves measured by extraordinary\nHall effect, Journal of Applied Physics 111, 07B538\n(2012).\n[45] E. C. Stoner and E. Wohlfarth, A mechanism of magnetic\nhysteresis in heterogeneous alloys, Philosophical Transac-\ntions of the Royal Society of London. Series A, Mathe-\nmatical and Physical Sciences 240, 599 (1948).\n[46] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and\nN. P. Ong, Anomalous hall effect, Rev. Mod. Phys. 82,\n1539 (2010).\n[47] T. Miyasato, N. Abe, T. Fujii, A. Asamitsu, S. Onoda,\nY. Onose, N. Nagaosa, and Y. Tokura, Crossover behav-\nior of the anomalous hall effect and anomalous nernst\neffect in itinerant ferromagnets, Phys. Rev. Lett. 99,\n086602 (2007)."    },    {        "title": "1111.6632v1.Dynamical_friction_in_a_gaseous_medium_with_a_large_scale_magnetic_field.pdf",        "content": "arXiv:1111.6632v1  [astro-ph.CO]  28 Nov 2011Draft version November 8, 2018\nPreprint typeset using L ATEX style emulateapj v. 11/10/09\nDYNAMICAL FRICTION IN A GASEOUS MEDIUM WITH A LARGE-SCALE MA GNETIC FIELD\nF. J. S´anchez-Salcedo1\nDraft version November 8, 2018\nABSTRACT\nThe dynamical friction force experienced by a massive gravitating b ody moving through a gaseous\nmedium is modified by sufficiently strong large-scalemagnetic fields. Us ing linear perturbation theory,\nwe calculate the structure of the wake generated by, and the gra vitational drag force on, a body\ntraveling in a straight-line trajectory in a uniformly magnetized mediu m. The functional form of the\ndrag force as a function of the Mach number ( ≡V0/cs, whereV0is the velocity of the body and csthe\nsound speed) depends on the strength of the magnetic field and on the angle between the velocity of\nthe perturber and the direction of the magnetic field. In particular , the peak value of the drag force is\nnot near Mach number ∼1 for a perturber moving in a sufficiently magnetized medium. As a rule o f\nthumb, we may state that for supersonic motion, magnetic fields ac t to suppress dynamical friction;\nfor subsonic motion, magnetic fields tend to enhance dynamical fric tion. For perturbers moving along\nthe magnetic field lines, the drag force at some subsonic Mach numbe rs may be stronger than it is\nat supersonic velocities. We also mention the relevance of our finding s to black hole coalescence in\ngalactic nuclei.\nSubject headings: black hole physics — hydrodynamics — ISM: general — waves\n1.INTRODUCTION\nAn object moving in a background medium induces\na gravitational wake. The asymmetry of the mass den-\nsitydistribution upstreamanddownstreamfrom the per-\nturber produces a drag on the body, which is often re-\nferredto asgravitationaldragordynamicalfriction (DF)\nforce. A body in orbitalmotion mayundergo a radialde-\ncay of its orbit due to the loss of angular momentum by\nthe negative torque caused by DF drag. Chandrasekhar\n(1943) derived the dynamical friction on a massive par-\nticle passing through a homogeneous and isotropic back-\nground of light stars. His formula is applied to estimate\nthe merger timescale of satellite systems or to study the\naccretion history of galaxies. Bondi & Hoyle (1944) con-\nsideredtheproblemofthemassaccretionbyapointmass\nMtravelling at velocity V0in a collisional homogeneous\nmedium of sound speed csin the limit where the per-\nturber moves at supersonic velocities relative to the am-\nbient gas(i.e. high Machnumbers). Ifthe perturberisan\naccretor, streamlines with small impact parameter may\nbecome bound because of energy dissipation in shocks,\nandcanbeaccretedtotheperturber. Hence, theforceon\ntheperturberconsistsoftwoparts; thegravitationaldrag\nand the momentum accretion force. The latter contribu-\ntion may be decelerating or accelerating (Ruffert 1996).\nIf the size ofthe perturber is largerthan the Bondi-Hoyle\naccretion radius defined as RBH≡GM/[c2\ns(1 +M2)]\nwithM=V0/cs, the density and velocity structure of\nthe wake, at any Mach number, can be inferred analyti-\ncally in linear theory because the body produces a small\nperturbation in the ambient gaseous medium at any lo-\ncation. The gravitational drag is inferred as the grav-\nitational attraction between the perturber and its own\nwake (e.g., Dokuchaev 1964; Rephaeli & Salpeter 1980;\n1Instituto de Astronom´ ıa, Universidad Nacional Aut´ onoma\nde M´ exico, Ciudad Universitaria, 04510 Mexico City, Mexic o;\njsanchez@astroscu.unam.mxJust & Kegel 1990; Ostriker 1999; Kim & Kim 2007;\nS´ anchez-Salcedo 2009; Namouni 2010).\nThe studies of the gravitational drag in gaseous media\nhave enjoyed widespread theoretical application, ranging\nfrom protoplanets to galaxy clusters. It seems to play\na significant role in the growth of planetesimals (Hor-\nnung, Pellat & Barge 1985; Stewart & Wetherill 1988),\nthe eccentricity excitation of planetary embryo orbits\n(Ida 1990; Namouni et al. 1996), the orbital decay of\ncommon-envelope binary stars (e.g., Taam & Sandquist\n2000; Nordhaus & Blackman 2006; Ricker & Taam 2008;\nMaxted et al. 2009; Stahler 2010), the evolution of the\norbits of planets around the more massive stars (Villaver\n& Livio 2009), the evolution of low-mass condensations\ninthecoresofmolecularclouds(Nejad-Asghar2010),the\nmass segregation of massive stars in young clusters em-\nbedded in dense molecular cores (Chavarr´ ıa et al. 2010),\nthe orbital decay of kpc-sized giant clumps in galaxies at\nhigh redshift (Immeli et al. 2004; Bournaud et al. 2007),\nor the heating of intracluster gas by supersonic galaxies\n(El-Zant et al. 2004; Kim et al. 2005; Kim 2007; Conroy\n& Ostriker 2008). Special work has been devoted to un-\nderstand the role of gaseous DF in the orbital decay of\nstars and supermassive black holes as a result of hydro-\ndynamic interactions with an accretion flow in galactic\nnuclei (Narayan 2000). Mergers of supermassive black\nhole binaries may be accelerated on sub-parsec scales by\nangular momentum loss to surrounding gas (Armitage\n& Natarajan 2005). In particular, gaseous DF expedites\nthegrowthofSMBHbymergersin collidinggalaxies(Es-\ncalaetal.2004,2005; Dotti etal.2006;Mayeretal.2007;\nTanaka & Haiman 2009; Colpi & Dotti 2011).\nLess developed is the corresponding theory of DF in\na magnetized gaseous medium. As far as we know, the\nanalytic estimate of the gravitational drag for a body\nmoving on a rectilinear trajectory parallelto the uniform\nunperturbed magnetic field lines by Dokuchaev (1964) is\nthe only work in this area. He concluded that the DF2 S´ANCHEZ-SALCEDO\nforce on a supersonic body is reduced by a factor that\ndepends on the ratio between the Alfv´ en speed and the\nsound speed. Since large-scale magnetic fields are ubiq-\nuitous in many astronomical systems such as molecular\nclouds (Tamura & Sato 1989; Goodman & Heiles 1994;\nMatthews & Wilson 2002; Heiles & Crutcher 2005) or\ngalactic nuclei, it is important to understand how the\nDF force is affected by the presence of ordered large-\nscale magnetic fields. In fact, young stellar systems and\nlow-mass condensations orbiting in the potential of their\nbirth clusters can interact with the surrounding dense\nand magnetized molecular interstellar medium during\nthe dispersal of the cluster’s gas. In the Galactic cen-\nter, structures associated with ordered magnetic fields,\ncalled arches and threads, are detected in radio contin-\nuum maps (Yusef-Zadeh et al. 1984). The magnetic field\nconfiguration of the Galactic center has been viewed as\npoloidal in the diffuse, interstellar (intercloud) medium\nand approximately parallel to the Galactic plane only in\nthe dense molecular clouds (Nishiyama et al. 2010). On\nthe scale of 400 pc, fields of 100 µG have been reported\n(Chuss et al. 2003; Crocker et al. 2010).\nThe importance of gaseous DF in the evolution and\ncoalescence of a massive black hole binary is motivated\nby both observational and theoretical work that indicate\nthe presence of large amounts of gas in the central re-\ngion of merging galaxies. During the merger of galaxies,\nthe inflow of gas material towards the galactic center\ndriven by tidal torques associated with bar instabilities\nand shocks will sweep up and amplify the magnetic field\nin the central region (Callegari et al. 2009; Guedes et\nal. 2011). Observations of gas-rich interacting galaxies\nsuch as the ultraluminous infrared galaxies (ULIRGs)\nshowthattheircentralregionscontainmassiveanddense\nclouds of molecular and atomic gas (Sanders & Mirabel\n1996). ULIRGs are natural locations to expect very\nstrong magnetic fields (Thompson et al. 2006; Robishaw\net al. 2008; Thompson et al. 2009).\nInthispaperwewillstudytheDFinagaseousmedium\non a body moving on rectilinear orbit in a homogeneous,\nuniform magnetized cloud. This is the simplest idealized\nextensionoftheunmagnetizedcaseandisthe firststepin\nunderstanding the role of ordered magnetic fields. Previ-\nous works have shown that, although the formulae of the\ngaseous drag force in a unmagnetized gas medium, were\nderived for rectilinear orbits in homogeneous and infi-\nnite media (Dokuchaev 1964; Rephaeli & Salpeter 1980;\nJust & Kegel 1990; Ostriker 1999; S´ anchez-Salcedo &\nBrandenburg 1999; Kim & Kim 2009; Namouni 2010;\nLee & Stahler 2011; Cant´ o et al. 2011), simple ‘local’\nextensions have been proven very successful in more re-\nalistic situations, e.g. smoothly decaying density back-\ngrounds or when the perturber is moving on a circular\norbit(S´ anchez-Salcedo&Brandenburg2001;Kim&Kim\n2007; Kim et al. 2008; Kim 2010). As a useful starting\npoint for understanding the role ofa large-scalemagnetic\nfield, we also consider that the unperturbed medium is\nhomogeneous and uniformly magnetized. A discussion\non the DF in other initial force-free configurations will\nbe given in a separate paper. Even in the simple case\nof a uniformly magnetized medium, the magnetic field\nproduces qualitatively new phenomena.\nThe paper is organizedas follows. In §2, we discuss the\nbasic concepts on the ideal problem of a particle travel-ing at constant speed through a uniform gas, both in\nthe purely hydrodynamic case and when the plasma is\nmagnetized. In §3, we outline the linear derivation for\ncalculating the steady-state density wake generated by\nan extended body moving along the magnetic fields, give\nananalyticalsolutionoftheproblemandcompareitwith\nprevious work. The time-dependent linear perturbation\ntheory is presented in §4.§5 describes the structure of\nthe resulting wake and evaluate the DF force as a func-\ntion of Mach number, for different angles between the\ndirection of the perturber’s velocity and the direction of\nthe magnetic field. In §6, we summarize our results and\ndiscuss their implications.\n2.DYNAMICAL FRICTION IN GASEOUS MEDIA: BASIC\nFORMULAE\n2.1.Unmagnetized medium\nUnderassumptionofasteadystate,Dokuchaev(1964),\nRuderman & Spiegel (1971) and Rephaeli & Salpeter\n(1980) derived the drag force on a point mass Mmov-\ning at velocity V0on a straight-line trajectory through a\nuniform medium with unperturbed density ρ0and sound\nspeedcs. For subsonic perturbers ( M ≡V0/cs<1,\nwhereMis the Mach number), these authors found that\nthe drag force is zero because of the front-back symme-\ntry of the density distribution about the perturber. For\nthe steady-state supersonic case, the drag force takes the\nform\nFDF=4πρ0(GM)2\nV2\n0lnΛ, (1)\nwhere Λ = rmax/rmin, beingrmaxandrminthe maximum\nand minimum radii of the effective gravitational inter-\naction of a perturber with the gas. For extended per-\nturbers,rminis its characteristic size, whereas for point-\nlike perturbers rminis of the order of the Bondi-Hoyle\nradiusRBH(Cant´ o et al. 2011), as defined in §1.\nUsing a time-dependent analysis in the unmagnetized\ncase, Ostriker (1999) found that (1) the force is not zero\nat the subsonic regime because, although a subsonic per-\nturber generates a density distribution with contours of\nconstant density corresponding to ellipsoids, there are\nalways cut-off ones within the sonic sphere that exert a\ngravitational drag, and (2) rmaxincreases with time in\nthe supersonic case. Morespecifically, she found that the\nCoulomb logarithm is given by:\nlnΛ =1\n2ln/parenleftbigg1+M\n1−M/parenrightbigg\n−M, (2)\nforM<1 andt > rmin/(cs−V0), and\nlnΛ =1\n2ln/parenleftbig\n1−M−2/parenrightbig\n+ln/parenleftbiggV0t\nrmin/parenrightbigg\n,(3)\nforM>1 andt > rmin/(V0−cs). The perturber is\nassumed to be formed at t= 0. The transition be-\ntween the subsonic to the supersonic regime is smooth\nwithout any divergence at a Mach number of unity (see\nFig. 3 in Ostriker 1999). S´ anchez-Salcedo & Branden-\nburg (1999) tested numerically that Ostriker’s formula\nis very accurate for non-accreting extended perturbers.\nIn many astrophysical situations, one needs to assign\na softening radius to the gravitational potential which\nin turn determines rminwithout any ambiguity. For aDYNAMICAL FRICTION 3\nbody described with a Plummer model with core radius\nRsoft≫RBH, S´ anchez-Salcedo & Brandenburg (1999)\nfound that rmin≃2.25Rsoft.\nFor point-mass accretors, the friction force has been\nderived by Lee & Stahler (2011) in the subsonic regime\nand by Cant´ o et al. (2011) in the hypersonic limit.\n2.2.Magnetized medium\nThe presence of a small-scale magnetic field tangled at\nscales below rminwill change the speed of sound. For\nisotropic compression of a random magnetic field, the\neffective sound speed is ( c2\ns+2\n3c2\na)1/2(e.g., Zweibel 2002),\nwherecais the Alfv´ en speed of the random small-scale\ncomponent of the magnetic field. Therefore, in order to\ninclude the effect of a small-scale magnetic field, one has\nto replace the sound speed by the effective sound speed\nin the definition of Min Eqs. (2) and (3).\nTheextensionofthedragforceformulaeisbynomeans\nstraightforward if the gaseous medium is permeated by\na regular magnetic field. Dokuchaev (1964) derived the\ngravitationaldragforcein the steady statefor perturbers\nmovingalongthe linesofthe unperturbed magneticfield.\nHe found that the DF drag is\nFDF=/parenleftbigg\n1−c2\nA\nV2\n0/parenrightbigg4πρ0(GM)2\nV2\n0lnΛ,(4)\natV0>(c2\ns+c2\nA)1/2, wherecAis the Alfv´ en speed of the\nregularmagneticfield, andit iszerofor V0<(c2\ns+c2\nA)1/2.\nBy comparing Eqs. (1) and (4), we see that the drag in a\nuniform magnetized background is never larger than in\nthe unmagnetized case. According to Dokuchaev (1964),\nthe gravitational drag on a body with velocity V0in a\nuniformly magnetized medium is equal to the drag on a\nbody with velocity V0/(1−c2\nA/V2\n0)1/2in a unmagnetized\nmedium. Therefore, if one naively uses the nonmagnetic\nformulae (1)-(3) by replacing the sound speed csby the\nmagnetosonic speed ( c2\ns+c2\nA)1/2would yield to wrong\nresults. In the next Section, we will show, however, that\nthe paper of Dokuchaev (1964) contains an error and it\nis not true that the drag force in the magnetized medium\ncase is always smaller than in the unmagnetized case.\nAs Ostriker (1999) demonstrated in the field-free case,\nthe steady state result found by Dokuchaev that the\nnet force is zero at V0<(c2\ns+c2\nA)1/2, because of the\nfront-back symmetry of the density perturbation in the\nmedium, may be misleading. It is also unclear how lnΛ\nvaries in time for the magnetized supersonic case. More-\nover, it left unexplored how the drag force depends on\nthe angle between the velocity of the perturber and the\ndirection of the magnetic field. Before addressing these\nquestions, however, it is still worthwhile finding analyt-\nical solutions for the perturbed steady density and the\nresulting drag force in the simplest scenario in which the\nvelocity of the perturber and the magnetic field are par-\nallel. Such a exact treatment will allow us to gain insight\ninto more complicated situations. This will be done in\nthe next Section.\n3.AXISYMMETRIC CASE: VELOCITY OF THE\nPERTURBER PARALLEL TO THE DIRECTION OF THE\nMAGNETIC FIELD\nWe consider a gravitational perturber moving on a\nstraight-line at constant velocity in a medium with un-perturbed density ρ0and thermal sound speed cs. In\nthe absence of magnetic fields, the linearized equations\nof motion can be reduced to a nonhomogeneous wave\nequation for the relative perturbation ( ρ−ρ0)/ρ0(e.g.,\nRuderman & Spiegel 1971; Ostriker 1999). Once a uni-\nform magnetic field, B0, parallel to the direction of per-\nturber’s velocity is included, Dokuchaev (1964) showed\nthattherelativeperturbationobeysanequationoffourth\norder in tand solved it using a double Fourier-Hankel\ntransformation. As it will become clear later, we prefer\nto describe the evolution of the system through wave-\nequations because it facilitates the physical interpreta-\ntion of the problem and because the contact with the\nanalysis of Ostriker (1999) is easier. In addition, the\nextension of the equations for a case where the angle be-\ntweenB0and the velocity of the perturber is arbitrary,\nbecomes straightforward in our approach.\n3.1.Perturbed density distribution\nWe study first the completely steady flow created by a\nmass on a constant-speed trajectory parallel to the lines\nofthe unperturbed magnetic field B0=B0ˆz. To do this,\nconsideraparticle at the originofourcoordinatesystem,\nsurrounded by a gas whose velocity far from the particle\nisV0=−V0ˆz, withV0>0. We will further assume\nthat the gas evolves under flux-freezing conditions. Our\nanalysis begins with the linearized MHD equations to\ndescribe the medium’s response to the perturber’s pres-\nenceρ=ρ0+ρ′,V=V0+v′andB=B0+B′, in a\nstationary sate ( ∂/∂t= 0)\nρ0∇·v′+V0·∇ρ′= 0, (5)\n(V0·∇)v′=−c2\ns∇ρ′\nρ0−∇Φ+1\n4πρ0(∇×B′)×B0,(6)\n∇×(V0×B′)+∇×(v′×B0) = 0,(7)\n∇·B′= 0, (8)\nwhere Φ is the gravitationalpotential created by the per-\nturber. ThePoissonequationlinksthepotentialwith the\ndensity profile of the perturber ρp:\n∇2Φ = 4πGρp. (9)\nThe Lorentz force, which provides the magnetic back-\nreaction on the flow pattern, is given by\n(∇×B′)×B0=B0/parenleftbigg/bracketleftbigg∂B′\nx\n∂z−∂B′\nz\n∂x/bracketrightbigg\nˆx−/bracketleftbigg∂B′\nz\n∂y−∂B′\ny\n∂z/bracketrightbigg\nˆy/parenrightbigg\n.\n(10)\nHence, the divergence of the Lorentz force is\n∇·[(∇×B′)×B0] =\n=B0/parenleftBigg\n∂2B′\nx\n∂x∂z−∂2B′\nz\n∂x2−∂2B′\nz\n∂y2+∂2B′\ny\n∂y∂z/parenrightBigg\n=−B0∇2B′\nz. (11)\nIn the last equality we have used that ∇ ·B′= 0. By\nsubstituting equations (5) and (11) in the divergence of4 S´ANCHEZ-SALCEDO\nthe equation of motion2, we have\n−V2\n0\nρ0∂2ρ′\n∂z2=−c2\ns\nρ0∇2ρ′−∇2Φ−B0\n4πρ0∇2B′\nz.(12)\nBy comparing the second and third terms in the right-\nhand side of the equation above, we see that, formally,\nthemagneticback-reactionterm ∇2B′\nzismathematically\nequivalent to having an external potential term. How-\never, whilst Φ is known (Eq. 9), B′\nzis coupled to the\nfluid motions through the flux-freezing equation (7).\nNext, we need an independent equation for B′\nzto close\nthe system. This can be accomplished using the third\ncomponent of the induction equation (7), which has the\nform\nB0/parenleftbigg\n∇·v′−∂v′\nz\n∂z/parenrightbigg\n=V0∂B′\nz\n∂z. (13)\nOur strategy is to eliminate v′in Equation (13). The\nthird component of the equation of motion (6) can be\nwritten as\n−V0∂v′\nz\n∂z=−c2\ns\nρ0∂ρ′\n∂z−∂Φ\n∂z. (14)\nThis equation does not depend explicitly on the frozen-in\nmagnetic field because the z-component of the Lorentz\nforce vanishes in the linear approximation (see Eq. 10).\nSubstituting Eqs. (5) and (14) in Eq. (13), we obtain\nthe desired equation\nB0/parenleftbigg1\nρ0∂ρ′\n∂z−1\nρ0M2∂ρ′\n∂z−1\nV2\n0∂Φ\n∂z/parenrightbigg\n=∂B′\nz\n∂z,(15)\nwherewerecallthat M ≡V0/csisthe(sonic)Machnum-\nber. Once again, the magnetic term ∂B′\nz/∂zis formally\nidentical to ∂Φ/∂z, but some caution should be used\nwhen interpreting it; the z-component of the Lorentz\nforce is not ∂B′\nz/∂zbut zero.\nEquations (12) and (15) constitute a system of two\ncoupled linear differential equations for ρ′andB′\nzwhich\nmay be solved once we have chosen suitable bound-\nary conditions. Defining the dimensionless perturbations\nα≡ρ′/ρ0andβ≡B′/B0, the equations to solve are:\nM2∂2α\n∂z2=∇2α+1\nc2s∇2Φ+Υ2∇2βz,(16)\n(M2−1)∂α\n∂z−1\nc2s∂Φ\n∂z=M2∂βz\n∂z,(17)\nwhere Υ ≡cA/csandcAthe Alfv´ en speed in the un-\nperturbed medium. In the limit of vanishing magnetic\nfield,βz= Υ = 0 and Equation (16) reduces to that of\nthe wake of a body in a unnmagnetized medium (e.g.,\nS´ anchez-Salcedo 2009).\nFor a point-like perturber of mass M, an analytical so-\nlution can be derivedfor the density enhancement, veloc-\nity and magnetic fields in the wake. In order to calculate\n2The curl of the equation of motion provides a relationship\nbetween the vorticity ωand the current density J′:\n−V0∂ω\n∂z=B0\ncρ0∂J′\n∂z.\nIn linear theory, the baroclinic term vanishes and the Loren tz term\nis the only able to generate vorticity, even if the gravitati onal force\nis irrotational.the dynamical friction force exerted on the body, we only\nneed the gas density enhancement in the wake, which is\nderived in Appendix A and is given by\nα(R,z) =λ(1−η)GM\nξc2s1/radicalbig\nz2+R2γ2,(18)\nwhereR=/radicalbig\nx2+y2is the cylindrical radius and\nη= (cA/V0)2= (Υ/M)2, (19)\nξ= 1+(1 −M−2)Υ2= 1−η+Υ2,(20)\nγ2= 1−M2\nξ, (21)\nand\nλ=\n\n1 ifM<Mcrit;\n2 ifMcrit<M<min(1,Υ) andz/R >|γ|;\n1 if min(1 ,Υ)<M<max(1,Υ);\n2 ifM>max(1,Υ) andz/R <−|γ|;\n0 otherwise.\n(22)\nHere, the critical Mach number is defined as\nMcrit≡/parenleftbig\n1+Υ−2/parenrightbig−1/2. (23)\nBecause of the linear-theory assumption, Equation (18)\nis properly valid only for ( z2+γ2R2)1/2≫(1−\nη)GM/(ξc2\ns). The nonmagnetic steady-state solution for\ndensity in the wake past a gravitating body is recovered\nwhen Υ = 0.\nFor clarity, it is convenient to distinguish four intervals\ndepending on the value of the Mach number of the body:\nM<Mcrit(case or interval I), Mcrit<M<min(1,Υ)\n(case or interval II), min(1 ,Υ)<M<max(1,Υ) (case\nIII) and M>max(1,Υ) (interval IV). γ2is a posi-\ntive number in cases I and III, whereas it is negative\nin cases II and IV. In the latter cases where γ2<0,\nthe density perturbation αvanishes at some spatial lo-\ncations. In case IV, for instance, αoutside the cone\ndefined by the surface z=−|γ|Ris actually zero. Turn-\ning to Eq. (15), we see that the magnetic perturbation\nB′\nzin these regions does not vanish but obeys the fol-\nlowing relation, ∂B′\nz/∂z=−(B0/V2\n0)∂Φ/∂z. Now, from\nEq. (14), the axial component of the velocity satisfies\na similar equation, ∂v′\nz/∂z= (1/V0)∂Φ/∂z. Using the\nfact that ∇·v′= 0 in regions of constant density (see\nEq. 5), it is simple to show that B′is parallel to v′in re-\ngions where α= 0, and thus the magnetic configuration\nis force-free in these zones.\nOnceρ/ρ0isknown,thegravitationaldragcanbecom-\nputed; this will be done in Section 3.4. Nevertheless, in\nordertogainmoreinsightintothephysicsofthewake,we\nwilldescribe the morphologyand structureofthe steady-\nstate wake in the next Section.\n3.2.Physical interpretation\nConsider first subsonic perturbers. In the limit M →\n0, we have γ→1,ξ→ −Υ2/M2, (1−η)→ −Υ2/M2\nandλ= 1. Therefore, the density enhancement is\nGM/(c2\nsr), which is Υ-independent, and corresponds\nto the linearized solution of the hydrostatic envelope,\nρ/ρ0= exp/bracketleftbig\nGM/(c2\nsr)/bracketrightbig\n, around a stationary perturberDYNAMICAL FRICTION 5\n(e.g., Ostriker1999). Inthis case, the magneticfieldlines\nremain straight and the whole magnetic configuration is\nforce-free.\nThe surfaces of constant density for subsonic per-\nturbers may be either ellipsoids or hyperbolae, depend-\ning on the Mach number. At M<Mcritit holds that\nγ2>1 and the isodensity surfaces are ellipsoids elon-\ngatedalong the trajectory of the perturber with eccen-\ntricitye=M//radicalbig\n|ξ|. This is in sharp contrast to what\nhappenswithoutanymagneticfieldwheretheellipsesare\nelongated along Rfor perturbers at any subsonic Mach\nnumber(γ2<1). Therefore, the existence of ellipsoidal\ndensity isocontours elongated along zis a clear signature\nof curved magnetic fields.\nIn thenonmagnetic subsonic case ,v′\nR<0 atz >0 and\nv′\nz>0atanyz, signifyingthattheincomingfluid isveer-\ning towards the perturber, but then turning away again\nonce it passesthe body. The result of v′\nz>0 implies that\nthe gas is being dragged by the gravitational pushing of\nthe body. In order to understand the morphology of the\nwake in a magnetized medium, in the following we cal-\nculatev′\nRandv′\nzfor a perturber with Mach number in\nthe interval I.\nFrom Eq. (14) and using the result for αin Eq. (18),\nwe find that\nV0∂v′\nz\n∂z=−λ(1−η)\nξGMz\n(z2+R2γ2)3/2+GMz\n(z2+R2)3/2,\n(24)\nwhich leads to\nv′\nz=−GM\nV0/parenleftbigg1\n(z2+R2)1/2−λ(1−η)\nξ(z2+R2γ2)1/2/parenrightbigg\n.(25)\nIt is simple to show that v′\nz>0 in case I, regardless the\nvalue of Υ.\nThe radial component of the velocity can be found us-\ning Eq. (5),\n1\nR∂Rv′\nR\n∂R=−∂v′\nz\n∂z+V0∂α\n∂z=−V0\nM2/parenleftbigg\n(1−M2)∂α\n∂z+1\nc2s∂Φ\n∂z/parenrightbigg\n.\n(26)\nSince we already know αand Φ, this equation can be\nsolved to obtain the radial velocity:\nv′\nR=GMz\nV0R/parenleftbigg1\n(z2+R2)1/2−λ(1−M2)(1−η)\nγ2ξ(z2+R2γ2)1/2/parenrightbigg\n.\n(27)\nFromtheequationabove,itfollowsthat, incaseI, v′\nR>0\nattheheadoftheperturber,regardlessthemagneticfield\nstrength. Thus, a parcel of fluid in the upstream region\ncirculates around the perturber, reaching its maximum\nR-value at z= 0 and then turning back again. Since\nfrozen-in magnetic field lines are dragged by the gas, the\nupstream magnetic field lines are decompressed radially,\nresulting in arched magnetic field lines with negative B′\nz-\nvalues. In fact, integration of Eq. (15), gives\nB′\nz=GMB 0\nV2\n0/parenleftbigg1\n(z2+R2)1/2−λ(1−M2)(1−η)\nξ(z2+R2γ2)1/2/parenrightbigg\n.\n(28)\nThis clearly states that B′\nz≤0 (here we assume B0>0).\nThus, an anticorrelation between density and B′\nzarises.\nAt Mach numbersclose to Mcrit, that is, M=Mcrit−\nǫwithǫa very small positive number, the density profile,at not extremely large zdistances, is\nα(R,z)≃Υ3/2GM/c2\ns√\n2ǫ(1+Υ2)1/4R. (29)\nHence, the density enhancement is large and its z-\ngradient very small.\nAtsubsonic Mach numbers in the interval Mcrit<\nM<min(1,Υ), the surfaces of constant density exhibit\nno front-back symmetry. The isodensity contours corre-\nspond to hyperbolae in the z−Rplane, as occurs for\nsupersonic perturbers in the absence of magnetic fields,\nbut nowthe density perturbation is null in the rearMach\ncone and is non-vanishing in a modified Mach cone lead-\ning the perturber. The physical reason is as follows. The\ndensity perturbation αis non-positive at any location\nbecause the streamlines diverge at the front cone ( η >1\nandξ >0).αat the edges of the cone is minus infinity\nfor a point mass. In the front cone, ∂v′\nz/∂z >0, mean-\ning that the flow in that region is being accelerated by\nthe inward net pressure force. Across the edge of the\nmodified Mach cone, there is a rapid rise in pressure and\ndensity, and the gas velocity quickly slows. In fact, the\ncausality criterion used in Appendix A is tantamount to\nselecting the solution in which a rapid flow is slowed in a\nshort distance, as occurs in shock waves. Indeed, we will\nfind numerically further below that the system adopts\nthis solution ( §5.1).\nIn case III, the body moves at intermediate velocities,\ni.e. either in the range cs< V0< cA(if Υ>1) or in the\nrangecA< V0< cs(if Υ<1). It is only in this case that\nγlies in the range0 < γ <1 and, therefore, the ellipsoids\nare flattened along z. Remind that in the unmagnetized\nbackground, a subsonic perturber also produces ellipti-\ncal density distributions flattened along z(e.g., Ostriker\n1999); the latter is a particular case of cA< V0< cs.\nForcs< V0< cA(which requires that Υ >1), however,\nthe density perturbation is negative ( α≤0). In this\ncase, the thermal pressure decreases towards the body.\nTherefore, ahead of the perturber the gas is accelerated\nalong the z-direction by the gravitational force plus the\npressure gradient (it holds that ∂v′\nz/∂z >0 atz >0). A\nnegative αis a consequence of the action of the pressure\ngradient plus the reduction of the radial convergence of\nthe flow due to the presence of the orderedmagnetic field\nthat preferentially allows motions along z, which lead to\nan accelerated flow falling towards the perturber. In the\nradial direction, the magnetic field lines are compressed\natz >0 because vR<0 upstream of the perturber.\nFrom Eqs. (18)-(22) we see that if the motion of the\nperturber is both supersonic and super-Alfv´ enic, which\ncorresponds to case IV, the density disturbance is con-\nfined to a rear cone defined by the condition z <−|γ|R.\nInthisregime, the surfacesofconstantdensitywithin the\nwakecorrespondto similarhyperbolaein the z−Rplane,\nat the rear of the body, with eccentricity e=M/√ξ.\nFromEq.(20), itissimpletoshowthat ξ >1inthiscase.\nTherefore, given a sonic Mach number M, the angular\naperture of the cone is larger in the presence of mag-\nnetic fields. In analogy to the unmagnetized medium,\none could define the effective speed of propagation of\nthe disturbance as vp=√ξcsin case IV. At large Mach\nnumbers (say M ≫√ξ), the cone is verynarrow( e≫1)\nand, as expected, the propagation speed coincides with6 S´ANCHEZ-SALCEDO\nthe magnetoacoustic velocity, vp≃/radicalbig\nc2s+c2\nA. Hence,\nat these large Mvalues, the stationary flow is similar\nto that in the nonmagnetic case but replacing the sound\nspeed by the magnetosonic speed.\nNow consider case IV but when the perturber moves\nat the same velocity as the effective speed of the distur-\nbance, so that V0=vpor, equivalently, M=√ξ. In the\nnon-magnetic case, this condition corresponds to ξ= 1\nand, therefore, the velocity of the body is in resonance\nwith the sound speed in the medium. One could naively\nthink that, at M=√ξ, the response of the medium\nis maximum because of the resonance between V0and\nvp. This is not true for Υ >1 because M=√ξimplies\nM= Υ (using Eq. 20), η= 1 (from Eq. 19) and, thereby,\nα= 0. We learn that a mass moving at the Alfv´ en speed\nin a medium with Υ >1, does not generate any density\ndisturbance in the ambient gas because the velocity field\nof the stationary flow is divergence-free ( ∇·v′= 0).\n3.3.A comparison with Dokuchaev (1964)\nAs already mentioned, Dokuchaev (1964) calculated,\nfor the first time, the properties of the wake created by\na star moving along the field lines, by treating it as a\nlinear perturbation. His analysis started from the time-\ndependent linearized equations of magnetohydrodynam-\nics, including a source term Qin the continuity equa-\ntion, representing the gas replenishment by the star. Al-\nthough he used the time-dependent equations, he tac-\nitly assumed that the object’s gravitationalfield is active\nsincet=−∞, so that the wake is in a steady state. For\nthe case without mass injected by the star ( Q= 0), the\nphysical stand points used by Dokuchaev (1964) are ex-\nactly the same as those adopted in §3.1, except that he\nchose a reference frame in which the unperturbed back-\nground gas is at rest.\nDokuchaev (1964) found closed fourth-order differen-\ntialequationsfor ρandtheradialcomponent vroftheve-\nlocity. Through Fourier-Hankel transformations, he was\nable to solve the differential equation for ρ. He found a\nsimilar expression for ρas that given in Eq. (18) but he\nfailed to separate correctly the different intervals for λ\n(Eq. 22) and the intervals at which the isocontours are\nellipsoids or hyperboloids. In particular, he claimed that\nthe isocontours are ellipsoids at any Mach number below\n(c2\ns+c2\nA)1/2/cs= (1+Υ2)1/2, which is misleading.\n3.4.Gravitational drag force in the axisymmetric case\nOnce we have the gas density enhancement α(r) in the\nambientmedium, wecancalculatethe gravitationalforce\nexerted on the body by its own wake:\nFDF= 2πGMρ 0/integraldisplay /integraldisplay\ndzdR Rα (r)z\n(z2+R2)3/2ˆz.\n(30)\nThe net drag is zero when the isodensity contours are\nellipsoids, i.e. when γ2>0, because the wake exhibits\nfront-back symmetry. At values γ2<0, however, the\nregion of perturbed density is confined to a cone and the\ndrag force is nonvanishing. Evaluating the integrals in\nspherical coordinates ( R=rsinθandz=rcosθ), and\nusing the variable µdefined as µ= cosθ, the drag force\ncan be expressed as:\nFDF= (1−η)4πG2M2ρ0\nV2\n0I/integraldisplayrmax\nrmindr\nrˆz,(31)where\nI=1\n2/integraldisplay1\n−1dµµλM2/ξ/radicalbig\n1−ξ−1M2+µ2ξ−1M2.(32)\nAs already said, the drag force is nonzero in cases II and\nIV, where γ2<0 and thus ξ >0. In case II, λ= 2\nfor allµbetween µlower= (M2−ξ)1/2/Mand 1, so\nthatI= 1. In case IV, λ= 2 for all µbetween −1\nandµupper=−(M2−ξ)1/2/Mand thus I=−1. Since\n1−η <0 in case II, we can write (1 −η)I=|1−η|and\nthe resultant expression for the force is:\nFDF=/braceleftBigg−|1−η|FlnΛˆzifM>max(1,Υ)\norMcrit<M<min(1,Υ);\n0 otherwise\n(33)\nwhere\nF=4πG2M2ρ0\nV2\n0, (34)\nand lnΛ is the Coulomb logarithm. Note that the den-\nsity divergesin the wake at Mach numbers close to Mcrit\nbecauseξ→0. However, the drag force is finite because\nthe openingangleofthe cone becomesverynarrow. Still,\nthe drag force peaks at Mach numbers close to Mcritbe-\ncause the factor |1−η|/V2\n0in the formula for αincreases\nwhenMdecreases.\nFigure 1 shows the DF force felt by the body at\nt= 100rmin/cs, as a function of the Mach number and\nfor different values of Υ. In analogy to the unmagnetized\ncase, we take Λ = V0t/rmin= 100V0/cs. Dokuchaev\n(1964) claimed that the drag force is nonzero only at\nM>(1 + Υ2)1/2(see§2.2). This is incorrect. For in-\nstance, for Υ = 1, the drag force is different from zero at\nM>Mcrit= 0.70. If the ratio between the Alfv´ en and\nsound speeds is of Υ2= 2, the DF force is nonvanishing\nin the intervals 0 .816<M<1, andM>1.41. In fact,\nthereexists alwaysasubsonicvelocityrangeat which the\ndrag force is nonzero.\nAs long as Υ /negationslash= 0, the DF force has two local max-\nima; one located at Mcritand the other one at M=\nmax(1,√\n2Υ). The dragforce strength at Mcritincreases\nwith Υ, whereas the drag force at the second local maxi-\nmum decreaseswith Υ. As Figure 1 clearly shows, at low\nΥ-values, the width of the interval with FDF/negationslash= 0 around\nMcritbecomes very narrow. For instance, the width of\nthat interval is only of 4 ×10−3for Υ = 0 .2. Hence,\nthe drag force at subsonic values is irrelevant for astro-\nphysical purposes when the Alfv´ en speed is sufficiently\nsmall as compared to the sound speed. For Υ >0.4,\nthe drag force at the local maximum Mcritis always\nlarger than the drag force at the other local maximum\nM= max(1 ,√\n2Υ). In the particular case of Υ = 1 .41,\nthe drag force is a factor >6 stronger in the interval\n0.816<M<1 than it is at M>1.\n4.TIME-DEPENDENT EQUATIONS\nThesteadystateanalysisin the axisymmetriccasepre-\ndicts zero drag force at certain Mach numbers because\nthe perturber is surrounded by complete ellipsoids that\nexert no net force. As Ostriker (1999) demonstrated in\nthe field-free case, the time-dependent analysis in which\nthe body is dropped suddently at t= 0 allows to captureDYNAMICAL FRICTION 7\nFig. 1.— Gravitational drag force as a function of Mach number, at t= 100rmin/cs, as predicted by the steady-state linear-theory in the\naxisymmetric case, for different values of Υ.\nFig. 2.— Color map of the density ρ/ρ0(left panel) and magnetic\nfieldB′\nz/B0(right panel), in the ( R,z)-plane for a case with V0= 0\nand Υ = 1 .41, in a natural logarithmic scale.\nthe asymmetric density shells in the far field which exert\na gravitational drag on the body. Other advantage of\nthe time-dependent approach is that, contrary to what\nhappens when assuming steady-state, the ambiguity in\nthe definition of the maximum cut-off distance rmaxis\nfixed.\nWithout loss of generality, it is convenient to use the\ngasframeofreferenceinwhichtheambientgasisinitially\nat rest, the initial magnetic field is along the z-axis and\nthe body moves with velocity V0,yˆy+V0,zˆz. The first\norder continuity equation is\n∂ρ′\n∂t+ρ0∇·v′= 0, (35)the MHD Euler equation\n∂v′\n∂t=−c2\ns∇ρ′\nρ0−∇Φ+1\n4πρ0(∇×B′)×B0,(36)\nand the induction equation:\n∂B′\n∂t=∇×(v′×B0). (37)\nThe medium initially uniform will respond to the grav-\nitational pull of the body through the emission of fast\nand slow Alfv´ en waves and sound waves. In the follow-\ning we will manipulate the above equations to obtain a\nclosed system of two differential equations for ρ′andB′\nz\nin analogy to the steady-state case.\nUsing Eq. (11) in the divergence of Equation (36)\n∂(∇·v′)\n∂t=−c2\ns\nρ0∇2ρ′−∇2Φ−B0\n4πρ0∇2B′\nz.(38)\nBy substituting Eq. (35) into Eq. (38), we obtain\n1\nρ0∂2ρ′\n∂t2=c2\ns\nρ0∇2ρ′+∇2Φ+B0\n4πρ0∇2B′\nz.(39)\nIn terms of αandβz, it yields\n∂2α\n∂t2=c2\ns∇2α+∇2Φ+c2\nA∇2βz.(40)\nHere, the magnetic effect on the density perturbation\nappears as a inhomogeneous term. We may recover the\nclassical non-magnetic equation for αby taking cA= 0.\nOn the other hand, the third component of the induc-\ntion equation (Eq. 37) implies:\n∂B′\nz\n∂t=−B0/parenleftbigg∂v′\nx\n∂x+∂v′\ny\n∂y/parenrightbigg\n. (41)\nEquations (35) and (41) give\n∂B′\nz\n∂t=B0/parenleftbigg1\nρ0∂ρ′\n∂t+∂v′\nz\n∂z/parenrightbigg\n. (42)\nFrom the third component of the equation of motion\n(Eq. 36)\n∂v′\nz\n∂t=−c2\ns\nρ0∂ρ′\n∂z−∂Φ\n∂z, (43)8 S´ANCHEZ-SALCEDO\nFig. 3.— Color map of the density ρ/ρ0, in the ( R,z)-plane, for the cylindrical case with M= 0.75 (upper panels) and for M= 0.9\n(lower panels), in a natural logarithmic scale. Both models have Υ = 1 .41, implying Mcrit= 0.816. Therefore, M= 0.75 falls into the\ninterval I, while M= 0.9 lies in the interval II (see §3.1). To easy comparison, the density map in the steady-stat e for a perturber seated\natR=z= 0 is shown in the left panels. The central and right panels di splay the density in the wake at two snapshots, when the pertu rber\nis dropped suddenly at t= 0 at the origin of the coordinate system. The time of the snap shots is given in the lower right hand corner of\neach panel in units of tcross.\nwe know that\n∂\n∂t/parenleftbigg∂v′\nz\n∂z/parenrightbigg\n=−c2\ns\nρ0∂2ρ′\n∂z2−∂2Φ\n∂z2. (44)\nInserting Eq. (44) into the temporal derivative of\nEq. (42), one finds\n1\nB0∂2B′\nz\n∂t2=1\nρ0∂2ρ′\n∂t2−c2\ns\nρ0∂2ρ′\n∂z2−∂2Φ\n∂z2.(45)\nIn dimensionless form:\n∂2βz\n∂t2=∂2α\n∂t2−c2\ns∂2α\n∂z2−∂2Φ\n∂z2. (46)Putting together, the equations (40) and (46) to solve\ncan be written as\n✷sα=∇2˜Φ+Υ2∇2βz, (47)\n✷Aβz= Υ−2/parenleftbigg∂2\n∂x2+∂2\n∂y2/parenrightbigg\n(α+˜Φ),(48)\nwhere˜Φ is the gravitational potential in units of c2\ns\n(i.e.˜Φ = Φ/c2\ns) and we have used the Lorentz invariant\nD’Alembertian ✷, defined as:\n✷lφ=/parenleftbigg1\nc2\nl∂2\n∂t2−∇2/parenrightbigg\nφ. (49)DYNAMICAL FRICTION 9\nFig. 4.— Same as Fig. 3 but for M= 1.2 (upper panels), which falls into the interval III, and M= 1.4 (lower panels). Again Υ = 1 .41.\nThe first equation (Eq. 47) governs the evolution of the\ndensity in the presence of a gravitational potential and\nmagnetic fields. The second equation (Eq 48) describes\nthe evolution of a frozen-in magnetic field when the gas\nis subject to pressure gradients and to an external gravi-\ntational potential. The inhomogeneous term in Eq. (48)\ndoes not have z-derivatives because gas motions in that\ndirection does not compress, stir or stretch the back-\nground magnetic field. The resulting equations (47) and\n(48) conform to a set of two coupled non-homogeneous\nwave equations. For a point-mass perturber, it is sim-\nple to find αin the Fourier-Laplace space, ˆ α(k,ω), but\nthe inverse Fourier-Laplace integral cannot be given in a\nclosed analytic form.\nIn order to gain physical insight, consider first a two-\ndimensional example. If Φ = Φ( x,y), that is, if the per-turber is an infinite cylinder with a certain radial den-\nsity profile ρp=ρp(R), thenβz=αbecause of the flux-\nfreezingcondition, and αsatisfiesa simple waveequation\nwith magnetoacoustic speed:\n/parenleftbigg1\nc2s+c2\nA∂2\n∂t2−∇2/parenrightbigg\nα=1\n1+Υ2∇2˜Φ.(50)\nThe physical reason is that motions are always perpen-\ndicular to the frozen-in magnetic field lines. Magneto-\nhydrodynamical equilibrium is reached within the mag-\nnetosonic cylinder. At a later stage, Parker instabilities\ncan develop (S´ anchez-Salcedo & Santill´ an 2011).\nIn the purely hydrodynamical problem, the equation\ngoverning the evolution of αis✷sα=∇2˜Φ. If the per-\nturberisapointsource, wehave ✷sα= (4πGM/c2\ns)δ(x−10 S´ANCHEZ-SALCEDO\nFig. 5.— Same as Fig. 3 but for M= 1.7. Again Υ = 1 .41. This Mach number lies in the interval IV.\nFig. 6.— Distributions of the perturbed density α(solid lines) and the z-component of the perturbed magnetic field (dashed lines) al ong\na cut at R= 0 at two different times, for the same model as that shown in Fi g. 5 (M= 1.7 and Υ = 1 .41).DYNAMICAL FRICTION 11\nFig. 7.— Temporal evolution of the gravitational drag force in the ax isymmetric model, at six different Mach numbers between 0 .55 and\n2, for Υ = 1 (left panel), and for Υ = 1 .41 (right panel). The number on each curve is the Mach number M.12 S´ANCHEZ-SALCEDO\nV0,xt)δ(y−V0,yt)δ(z−V0,zt)H(t). Hence, the density re-\nmains unperturbed outside the causal region for sound\nwaves (see Ostriker 1999). In a magnetized medium,\nhowever, the situation is different because Equation (48)\nfor the perturbed magnetic field βzhas a source term\n(∂2Φ/∂x2+∂2Φ/∂y2) which does not vanish even out-\nside the causal region for magnetosonic waves.\nIn the next Section we will solve the coupled wave-\nequations numerically. To do so, the perturber gravita-\ntional potential will be modeled by a smooth core Plum-\nmer potential:\nΦ(r,t) =−GMH(t)/radicalbig\nx2+(y−V0,yt)2+(z−V0,zt)2+R2\nsoft,\n(51)\nwhereRsoftis the softening radius and His a Heaviside\nstep function. Stellar and globular clusters can be ac-\ncurately described by Plummer potentials. These type\nof models were also used in S´ anchez-Salcedo & Branden-\nburg (1999, 2001), Kim & Kim (2009), and Kim (2010)\nto study DF.\n5.RESULTS\nThe coupled inhomogeneous wave equations (47) and\n(48) were solved using a finite difference scheme in a uni-\nform grid. The scheme is second order in space and third\norder in time. The temporal algorithm was described\nin S´ anchez-Salcedo & Brandenburg (2001). Calculations\nstart with a uniform background density and magnetic\nfield, and the body is initially placed at the origin of the\ncoordinate system with velocity V0,yˆy+V0,zˆz. For the\naxisymmetric case, which occurs when V0,y= 0, the cal-\nculations were carried out on a two-dimensional ( R,z)-\nplane in cylindrical symmetry. In the general three-\ndimensional case ( V0,y/negationslash= 0), the variables αandβzare\nsymmetric about the plane x= 0. Hence, we considered\na finite domain with x∈[0,Lx,max] and used symmet-\nric boundary conditions at x= 0, and outflow boundary\nconditions in the other five capsof the computational do-\nmain. However, the size of the domain was taken large\nenough to ensure that the perturbed density and mag-\nnetic field do not reach the boundaries.\nAs a test of the algorithm, we studied the convergence\nof homogeneous wave modes by perturbing a uniform\nbackground medium. We further tested convergence of\nour models for several resolutions and found that four\nzones per Rsoftsuffice to have converged results.\nWe take Rsoft,cs, andtcross=Rsoft/csas the units of\nlength, velocity, and time, respectively. A model can\nthus be specified with four dimensionless parameters:\nGM/(c2\nsRsoft),M, Υ and Θ ≡atan(V0,y/V0,z). Θ is the\nangle between V0andB0. FixedM, Θ and Υ, the vari-\nablesαandβzdepend linearly on GM/(c2\nsRsoft). Hence,\nin our calculations we always take GM/(c2\nsRsoft) =\n0.01/3 and explore how the density and the magnetic\nfield in the wake depend on the other three parameters.\n5.1.Axisymmetric case\nWefirstrunmodelswiththemagneticfieldtermsswich\noff and compare the density enhancement and the gravi-\ntationaldragwithpreviouslinearcalculationsinOstriker\n(1999) and S´ anchez-Salcedo & Brandenburg (1999). We\nfound excellent agreement, backing up our numericalmodel. Inthefollowing,wewillpresentresultsforabody\nmoving along the field lines of the unperturbed magnetic\nfield, which corresponds to Θ = 0.\nThe simplest scenario occurs when the gravitational\nperturber is dropped at t= 0 at rest ( V0= 0). As dis-\ncussed at the begining of §3.2, the steady-state density\ndistribution is identical as that without any magnetic\nfield. However, the inital relaxation stage and the far\ndensity distribution are sensitive to magnetic effects. In\nfact, while the problem has spherical symmetry at any\ntime in the purely hydrodynamical case, this symmetry\nis broken in a magnetized medium because the magnetic\nfield dictates a preferential direction. Figure 2 shows\nmaps of density and B′\nzfor a case with Υ = 1 .41. We\nsee that the density distribution in the vicinity of the\nbody is indeed spherically symmetric and the magnetic\nfield takes essentially its initial value, implying that this\nparthasreachedhydrostaticequilibrium. However,there\naretwosymmetric underdense regionsalongthe z-axisin\nthe outer parts. Physically, the origin of them is that the\nmagnetic field reduces the flow convergence toward the\nsymmetry axis because magnetic forces mainly affect the\nradial component of the velocity vR. This loss of radial\nconvergence produces a wave with negative density en-\nhancement ( α <0) but a positive magnetic enhancement\n(βz>0) because ofthe compressionofthe magnetic field\nlines in the radial direction.\nFigure 3 shows snapshots of the density at the ( R,z)\nplaneforΥ = 1 .41andtwosubsonicvelocities( M= 0.75\nandM= 0.9). In the time-dependent analysis, both\ncases present a region of negative density enhancement\n(i.e.α <0) at the head of the perturber. As Mincreases\nfrom 0 to 0 .9, the underdense region at the head of the\nbody remains and gets deeper, while the underdensity in\nthe downstream region becomes less pronounced. This\nis simply consequence of the Doppler effect; gradients\nbecome steeper upstream (remind that this also occurs\nin the purely hydrodynamic case). By comparing the\ndensity at two times (the central and the right panels of\nFigure 3), we see that the evolution of the density looks\nself-similar.\nAtM= 0.75, the steady-state analysis predicts a null\ndrag force because of the front-back symmetry in the\nwake (see upper left panel of Fig. 3). In the finite-time\ncase, complete ellipsoids are visible only in the vicinity of\nthe body. For instance, at t= 173.6tcross(with Υ = 1 .41\nandM= 0.75), the isodensity contours are not longer\nellipsoids at distances beyond ≃8Rsoftfrom the body’s\ncenter. For comparison, in the absence of magnetic fields\n(Υ = 0) and at M= 0.75, ellipsoids within a radius of\n43Rsoftare complete at t= 173.6tcross. This difference\nis a consequence of the coupling between αandβz. At\nz= 136RsoftandR= 0 (i.e. in the symmetry axis),\na steep front separates the fluid into two regions; one\nwithα >0 andβz<0, from another with α <0 and\nβz>0Thisfrontleadstoadecelerationofthe gas,which\nexpands radially.\nAtM= 0.9, a cone of negative density enhancement\nis located at the head of the perturber (see Fig. 3), as\nthat predicted in §3.1, and a region of positive density\nenhancement at the rear. Because of the back-reaction\nof the magnetic field, the isodensity contours of the tail\nare not incomplete ellipsoids at all. Note that the body\nis at the apex of the cone, whereas the overdensity isDYNAMICAL FRICTION 13\nFig. 8.— Gravitational drag force for the time-dependent axisymmet ric models against the Mach number at t= 40tcross(left panel) and\natt= 200tcross(right panel). Symbols correspond to the numerical models. The solid curves plot Ostriker’s formula, while the remaind er\ncurves draw the drag force as predicted by Equation (33), ado ptingrmin= 2.25Rsoftandrmax=V0t.\ndetached from the perturber. It is important to remark\nthat, according to the analysis in §3.4 and Figure 1, the\nmaximum drag force for Υ = 1 .41 occurs at a Mach\nnumber close to 0 .9.\nFigure 4 displays density maps for Υ = 1 .41 and two\nsupersonic Mach numbers: M= 1.2 (sub-Alfv´ enic per-\nturber) and M= 1.4 (trans-Alfv´ enic perturber). In the\nfirst case, the steady-state analysis predicts ellipsoidal\nisocontours. In the time-dependent case, however, ellip-soids are incomplete far enough away upstream from the\nperturber and, again, an overdensity wave at the rear\nof the perturber moves away from the body. When the\npertuber moves at M= 1.4, a teneous ellipsoidal under-\ndenseenvelop is still visible.\nThe overdensity behind the body has the shape of a\ntulip. This tulip-shaped overdensity also appears at the\nrear of the body at M= 1.4 and at M= 1.7 (see Figs. 4\nand 5). A feature of the tulip-shaped overdensity wave14 S´ANCHEZ-SALCEDO\nFig. 9.— Snapshots of the density map along a cut-off through the ( y,z)-plane at x= 0, in natural logarithmic scale, for Θ = 45◦(upper\npanels) and for Θ = 90◦(lower panels), at three different Mach numbers. The Mach num berMis indicated at the right corner in each\npanel. The perturber was dropped at t= 0 and moves on a rectilinear orbit in the ( y,z)-plane. In all panels, Υ = 1 .41.\nFig. 10.— Gravitational DF force as a function of Mach number\nfor different Υ-values at t= 40tcross. The perturber moves per-\npendicular to the magnetic field lines (i.e. Θ = 90◦). The solid\ncurve plots Ostriker’s formula, which was derived for unmag ne-\ntized media, with rmin= 2.25Rsoftandrmax=V0t. To make the\nplot readable, the symbols at M= 0.3,0.55,0.75 and 2 have been\nslightly shifted in the horizontal direction.Fig. 11.— Temporal evolution of the gravitational drag force for\nΥ = 1. The perturber moves in a direction perpendicular to the\nmagnetic field lines. The numbers given at each curve represe nts\nthe Mach number.\nis that it has a negative magnetic enhancement. The\ntulip-shaped overdensity is a consequence of our initial\nconditionsand, asexpected, itisdetachedfromthebody.\nIn order to illustrate the birth of the tulip-shaped wave,\nFigure 6 shows the density and magnetic perturbations\nalongthe symmetry axisfor M= 1.7, at twoearlytimes.\nInitially, βzincreases due to the compression of mag-\nnetic field lines (see the panel at t= 2.83tcross). At the\nfar edge of the tail, z≃ −3Rsoft, an underdense regionDYNAMICAL FRICTION 15\nwith positive βzappears. Later on (see the profiles at\nt= 6.47tcross), the overdensity loses gravitational sup-\nport and expands behind the body, decreasing the mag-\nnetic field strength, until the magnetic pressure plus the\nmagnetic tension provides sufficient radial confinement\nto the tulip-shaped structure, allowing it to remain over\nlong times.\nIn Figure 5 it is simple to identify the modified Mach\ncone dragged by its point by the perturber. At t=\n64.7tcross, the Mach cone at the rear of the body is well-\ndefined. Clearly, thetimescaleforthe developmentofthe\nMach cone at the rear for M= 1.7 is shorter than the\ntimescale to form the upstream Mach front at M= 0.9.\nIn fact, Figure 3 shows that the cone for M= 0.9 is not\nwell developed at t= 57.7tcross.\nFigure 7 shows the gravitational DF drag as a func-\ntion of time for M= 0.55,0.75,0.9,1.1,1.4 and 2. For\nΥ = 1 and M= 0.55, the drag force clearly saturates\nin∼30tcross. In the remainder cases the drag force in-\ncreases with time. However, the drag force on a per-\nturber moving in a medium with Υ = 1 .41, saturates for\nM= 0.55,0.75,1.1, and 1.4. This means that the DF\nforce saturates to a constant value at those Mach num-\nbers that the steady-state analysis predicts a null drag\nforce.\nIn Fig. 8 we plot the drag force at t= 40tcrossand\nt= 200tcrossfor different values of Υ, together with the\npredicted force with rmin= 2.25Rsoft. We see that for\nΥ = 0, there is a perfect agreement between the Os-\ntriker formula and the inferred values, confirming the\nresult that rmin= 2.25Rsoftreported in S´ anchez-Salcedo\n& Brandenburg (1999). The drag force formula given in\nEq. (34), with rmin= 2.25Rsoft, overestimates the drag\nforce in a neighbourhood of Mcrit, where the drag force\nasafunctionof M, becomesverycuspy. Asexpected, the\nsteady-state formula is more accurate at long timescales,\nexcept when it predicts a zero net force. Roughly speak-\ning,wemaysaythat, fortheaxisymmetriccase,thegrav-\nitational drag in a magnetized medium is always smaller\nor equal as the drag force in the unmagnetized case for\nsupersonic perturbers ( M>1), whereas the drag is al-\nwayslargeror equal in a magnetized medium at subsonic\nperturbers ( M<1).\n5.2.Magnetic field perpendicular to the direction of\nmotion of the perturber\nIn§5.1, we have focused on the case where the angle\nformed between the velocity of the perturber and the\nmagnetic field, Θ, was equal to 0. In such a situation,\nthe problem has axial symmetry and it is possible to find\nthe analytical solution in the steady state. However, it\nis by no means clear how the gravitational drag force\ndepends on Θ. A visual comparison of the density wake\nstructures for Θ = 0, Θ = π/4 and Θ = π/2 in Figures\n3 and 9 would lead us to think that the resulting wake\nat Θ =π/4 is more similar to the case with Θ = π/2\nthan to Θ = 0. In particular, we would like to stress\nthe remarkably different structure of the wake for Θ = 0\n(Fig. 3) and Θ = π/2 (Fig. 9) at M= 0.9.\nIn this section we will discuss in detail the extreme\ncase where the perturber moves perpendicular to the\nfield lines. In Appendix B, the perturbed density is\ngiven in Fourier space. At subsonic Mach numbers and\nΘ =π/2, the perturber is surrounded by a ellipsoid-likeenvelope but also presents a tail with positive and neg-\nativeα-values separated by a sharp front (see Fig. 9 for\nΥ = 1.41). Approaching to the upstream axis of motion\nof the body, the y-axis, the plowing up of field lines in-\ncreases the total pressure. At low Mach numbers, say\nM= 0.3, underdense regions are now formed in the di-\nrectionoftheambientfieldlines, whicharelaggedbehind\nthe body (remind that regions with negative αappear\nalong the field lines; see Fig. 2). At M= 0.9, even if\nthe motion is subsonic and sub-Alfv´ enic, a magnetic bow\nwave with sharp edges and opening angle arctan[ cA/V0]\nis apparent in Fig. 9. Note that when we say that it is\nsub-Alfv´ enic, we only mean cA/V0<1. However, some\ncaution should be used when interpreting this ratio be-\ncause the velocities are oriented in different directions.\nSince the velocity of the perturber is always orthogonal\nto the ambient direction of propagation of Alfv´ en waves,\nthe Alv´ en speed in the direction of motion of the per-\nturber is zero and, thus, the body is always infinitely\nsuper-Alfv´ enic in the direction of motion. The morphol-\nogy of the wake is the result of a competition between\nthe gravitational focusing of gas by the perturber and\nthe drainage of gas along magnetic field lines. We should\nwarn here that the wake is not axisymmetric and thus\nthe density map in the ( x,y)-plane is different than the\nmap in the ( y,z)-plane.\nWhen the perturber travels faster than the magne-\ntosonicvelocity cs(1+Υ2)1/2, a magnetosonicMachcone\nisformedattherearofthepertuber; theentireperturbed\ndensity distribution lags the perturber. In the ( y,z)-\nplane, the perturber creates two magnetic bow waves;\nthe Alfv´ enic wave with opening angle arctan[ cA/V0] and\nthe magnetosonic wave with opening angle arctan[( c2\nA+\nc2\ns)1/2/V0].\nThe gravitational drag force is the result of the con-\ntribution of all the parcels in the domain and it is not\npossible to estimate its value just by comparing the den-\nsity structure by eye. Figure 10 shows the gravitational\ndrag as a function of perturber’s Mach number for dif-\nferent values of Υ, together with the gravitational drag\nin the unmagnetized case using Ostriker’s formula. All\nthe points at Mach number larger than 0 .7 lie on or be-\nlow Ostriker’s curve, implying that at M>0.7 the drag\nforce is equal or smaller than it is in the unmagnetized\nmedium. AtΥ ≤0.5, the effectofincludingthemagnetic\nfield on the drag force is rather small. Interestingly, at\nΥ≥0.5, the strength of the drag for Mach numbers\n>(1.7+Υ2)1/2is identical as it is in the unmagnetized\ncase.\nFor Υ = 1 .41, the drag force shows a plateau between\nM= 0.6 and 1.4 (see Figure 10). In general, the drag\nforce is remarkably supressed at Mach numbers around\n∼1 in the magnetized case as compared to the unmag-\nnetized case, as long as Υ >0.5 (see, for instance the\ndrag at 0 .9≤ M ≤ 1.4 for Υ = 1). In fact, the tem-\nporal evolution of the drag force is given in Figure 11\nfor Υ = 1. The DF force on perturbers moving at Mach\nnumbers 0 .75≤ M ≤ 1.2 saturates asymptotically to a\nconstant value. Hence, at 0 .75≤ M<1, the drag forces\nreach a steady-state value either the medium is magne-\ntized or not. However, we know that the unmagnetized\ndragforceincreaseslogarithmicallyintimeforsupersonic\nperturbers. This implies that the drag force may be sup-16 S´ANCHEZ-SALCEDO\nFig. 12.— Components of the gravitational DF force parallel (left pan el) and perpendicular (right panel) to the direction V0(at\nt= 40tcross) versus M, for three different values of Υ (Υ = 0 .71, asterisks; Υ = 1, crosses; Υ = 1 .41, diamonds). The angle between the\nperturber velocity and the magnetic field is 45◦. The key to symbols is the same as in Figure 10.\nFig. 13.— Component of the DF force perpendicular to the di-\nrection of motion as a function of time for Υ = 1 .41 and Θ = 45◦.\nThe number on each curve is the Mach number.\npressedby one orderofmagnitude at 1 <M<1.2 in the\nmagnetized as compared to the unmagnetized medium\nbecause the magnetized drag force saturates to a con-\nstant value. On the other hand, at M<0.75 (again\nΥ = 1) there is no indication that the drag force satu-\nrates, at least up to t= 40tcross, implying that the DF\nforce in a magnetized medium may be larger by a factor\nof a few than the drag in the unmagnetized case.\nIn summary, when the magnetic fields are relevant,\nthat is for Υ >0.5, we distinguish three ranges. At\nhigh Mach numbers [i.e. M>(1.7+Υ2)1/2], the drag is\nthe same as in the unmagnetized case. At intermediate\nMach numbers (1 <M ≤ M−1\ncrit), the drag is highly sup-\npressed. Finally, at low Mach numbers, the drag force is\nstronger in the MHD case than in a purely hydrodynam-\nical medium.\n5.3.Intermediate angle between perturber’s velocity and\nmagnetic field. Θ = 45◦\nThe upper panels of Figure 9 exhibit the complex mor-\nphologyofthe wakewhen Θ = π/4. As it is obviousfromthese panels, the gravitationaldrag will have two compo-\nnents: one parallel to V0, which produces the drag and\nloss of kinetic energy by the perturber, and one compo-\nnent perpendicular to V0, which would change the di-\nrection of V0. Given the symmetry of the problem, both\ncomponents lie in the ( y,z)-plane. We will start our dis-\ncussion by considering the gravitational drag force.\nFor those simulations presented in Figure 12 with Θ =\nπ/4, the drag force saturates within t= 40tcrossonly for\nthe model with M= 0.9 and Υ = 0 .71. The maximum\nof the drag force occurs at Mach numbers near ≃ M−1\ncrit.\nAt intermediate Mach numbers, say at 1 .2–1.6 for Υ =\n1.41, the drag force may decrease by a factor of 2–3 as\ncompared to the force without magnetic fields. At high\nMach numbers, M>M−1\ncrit, the drag force is slightly\nsuppressed as compared to the unmagnetized case, but\nthis reduction is more modest than for Θ = 0◦. Once\nagain, at low Mach numbers ( M<0.75), the drag force\nis stronger than in the unmagnetized case.\nAs already said, the component of the force perpendic-\nular to the velocity of the perturber, Fperp, will tend to\ninduce achangein the directionofthe velocity(note that\nwe force the body to move along a straight line). We will\nuse the following sign convention for Fperp. For an angle\nΘ in the interval 0 ≤Θ≤π/2,Fperp>0 will mean that\n˙Θ>0, in our convention. In Figure 12, Fperpis shown\nas a function of Mach number. The magnitude of Fperp\nmay be comparable to the drag force. For instance, at\nM= 1.2, the perpendicular force is only a factor of 2\nsmaller for Υ = 1 .41 and a factor of 3 for Υ = 1. Given a\ncertain supersonic velocity, Fperpincreases with Υ, while\nFDFshows the opposite behaviour. For supersonic mo-\ntions with angle Θ = π/4,Fperpis always positive and\nincreases monotonically in time (see Fig. 13). This im-\nplies that Fperpwill tend to redirect perturber’s velocity\nto a higher Θ. For the cases shown in Figure 12, the\nperpendicular force saturates in the run of the calcula-DYNAMICAL FRICTION 17\ntion (t= 40tcross) only in two cases; for M= 0.5 and\nΥ = 0.71, and for M= 0.75 and Υ = 1 .41 (this latter\ncase is shown in Fig. 13). In some cases with subsonic\nMachnumbers, theperpendicularcomponentoftheforce\nis initially positive, achieves a maximum and then stars\na linear decline up to negative values (see Fig. 13).\n5.4.Dependence of the drag force on Θ\nIn Figure 14 we plot the drag force as a function of Θ,\nfor Υ = 1 and 1 .41. The dependence of FDFon Θ is\nnot always monotonic. The strongest variation of FDF\nwith Θ occurs for M= 0.9. For this Mach number, the\ndrag force may decrease by a factor of 2–3 from Θ = 0\nto Θ = 30◦. ForM= 1.2, the drag force may change up\nto a factor of 2 depending on the Θ-value. For M= 0.3\nand 1.4, the drag force depends gently on the angle.\nIn many astrophysical scenarios, the perturber will be\nsubject to an external gravitational potential and will\ndescribe a nonrectilinearorbit. S´ anchez-Salcedo& Bran-\ndenburg (2001) numerically treated the orbital decay\nof a perturber in orbit around a unmagnetized gaseous\nsphere. They found that the “local approximation”, that\nis estimating the drag force at the present location of the\nperturberasifthemedium werehomogeneousbut taking\nappropriately the Coulomb logarithm, is very successful.\nConsider now a perturber on a circular orbit in a magne-\ntizedmedium. Iftheorbitliesinaplaneperpendicularto\nthe magnetic field, the attack angle Θ is always π/2. In\nthe local approximation, the maximum drag for Θ = π/2\nand Υ = 0 occurs at M ≈1 and at M ≈1.7 for Υ = 1.\nHowever, if the plane of the circular orbit is parallel to\nthe direction of the initial magnetic field, Θ will change\nperiodically in time as Θ = Θ 0+Ω0t. Therefore, if the\nlocal approximation is valid, one can estimate the mean\ndrag force over a rotation period, which is approximately\nequivalent to take the mean value of FDFover Θ. In par-\nticular, for Υ = 1, the Θ-average drag force is maximum\natM ≈1.4. This example illustrates how FDFmay de-\npend on Υ and on the inclination of the orbit respect to\nthe magnetic field lines. A more detailed analysis of the\ndrag force on a body on a circular orbit will be given\nsomewhere else.\n6.SUMMARY AND DISCUSSION\nUnderstanding the nature of the DF force experienced\nby a gravitational object that moves against a mass den-\nsity background is of great importance to describe the\nevolution of gravitational systems. In this work, we in-\nvestigated the DF on a body moving in rectilinear tra-\njectory through a gaseous medium with a magnetic field\nuniform on the scales considered. In linear theory, the\nproblem is largely characterized by three dimensionless\nparameters, M, which is defined as the ratio of the par-\nticle velocity to the sound speed of the uniform gas, Υ,\ndefinedastheratiobetweentheAlfv´ enandsoundspeeds,\nandΘ,theanglebetweenthemagneticfielddirectionand\nthe particle velocity. We find that magnetic effects may\nalter the drag force, especially for Υ >0.5, because the\nmagneticfieldaffectstheflowvelocityfieldintheperpen-\ndicular direction of the ambient field lines, and thereby\nthe morphology of the wake. Note that the plasma beta,\ndefined as the ratio of gas to magnetic pressure, is 2 /Υ2\nfor an isothermal system.Fig. 14.— Dependence of the drag force on the angle Θ for Υ = 1\n(upper panel) and Υ = 1 .41 (lower panel). The numbers given at\neach curve represent the Mach number M. The drag force was\ncomputed at t= 40tcross.\nThere are two major differences between the magne-\ntized and unmagnetized case. One conceptual difference\nis that, while gravitational focusing in a unmagnetized\nmedium always generates a positive density enhance-\nment, this is not the case in a magnetized medium (see,\ne.g., Figs. 2, 3 and 9). A second result is that the peak\nvalueofthe dragforceis notnear M= 1fora massmov-\ning in a magnetized medium. In fact, the sharp peak of\nFDFatM= 1 found in the Υ = 0 case is no longer\npresent in a magnetized medium with Υ >0.5. For in-\nstance, for a perturber in perpendicular motion to the\nfield lines (Θ = π/2) in a medium with Υ = 1 .41, the\ndrag force is essentially constant from M= 0.5 to 1.4\nand its maximum is located around M= 2 (see Fig. 10).\nThe flat plateau in the drag force between M= 0.5 and\n1.4 is partly because of the extra rigidity of the magnetic\nfield in the xandydirections.\nFor a body traveling along the field lines, i.e. Θ = 0,\nthe steady-state problem can be treated analytically. We\nfocusfirstonthiscase. ForΥ /negationslash= 0, thedragforcepresents\ntwo local maxima (see Fig. 1); one is located in the sub-\nsonic branch (at Mcrit) and the other peak value is at\nthe supersonic branch [at M= max(1 ,√\n2Υ)]. When\nthe velocity of the perturber is supersonic and super-\nAlfv´ enic (and Θ = 0), the DF force in a magnetized\nmedium is weaker than it is in the unmagnetized case\nby a factor of (1 −η), withη= (cA/V0)2. The physi-\ncal reason is that the medium becomes more rigid in the\nradial direction and, hence, the opening aperture of the\nmodified Mach cone is the same as that in a unmagne-\ntized medium with effective sound speed ( c2\ns+c2\nA)1/2,\nbut the density enhancement is smaller by a factor of\n(1−η). By contrast, the drag force for subsonic veloc-\nities is stronger if the medium is uniformly magnetized.\nFor Θ = 0, an underdense region is formed upstream\nbecause of the gas channeling along the direction of the\nmagnetic field, following the path of less resistance. The\nsteady-state theory predicts that the gravitational drag\non a body with Θ = 0 vanishes at Mach numbers in18 S´ANCHEZ-SALCEDO\nthe following two ranges: (1) at M<Mcritand (2)\nat min(1 ,Υ)<M<max(1,Υ). However, using time-\ndependent analysis we find that the DF force asymptoti-\ncally approaches to a nonzero steady-state value at these\nMach numbers. For Υ >0.4 (still Θ = 0), the drag force\nis maximized for perturbers moving at a Mach number\nclose toMcrit(Fig. 8). At Mach numbers around Mcrit,\nthedensityenhancementislargebutnegativeinaconein\nfront of the body. At those Mach numbers, the DF may\nbe even more efficient than in the stellar case. For exam-\nple, for a medium with Υ = 1 .41, the drag force peaks\nbetween M= 0.6 andM= 1.1. As a consequence of the\nstronger DF force, subsonic massive objects in a orbit\nelongated along the magnetic field lines in a constant-\ndensity core of a nonsingular gaseous sphere will suffer\na orbital decay faster if the medium is pervased by a\nlarge-scale magnetic field.\nWe have also explored the Θ-dependence of the DF\ndrag. For Mach numbers around Mcrit, the drag force\nexhibitsthestrongestvariationswithΘ(seeFig.14). For\nmagnetized media with Υ ≥1 and regardless the exact\nvalue of Θ, we find that (1) the drag force for subsonic\nperturbers is higher by a factor of 1 .5–2 than it is for the\nunmagnetizedDFdrag,and(2)forsupersonicperturbers\n(M>1), the magnetized drag force is always weaker\nthantheunmagnetizeddragforce. AtintermediateMach\nnumbers, 1 .1≤ M ≤1.4, the drag force is a factor of 2–3\nweaker than it is in the absence of magnetic fields3. At\nhigh Mach numbers, M>(1.7+Υ2)1/2, the suppresion\nof the drag force is more important at small values of Θ\n(Fig. 14). At these high Mach numbers and for an angle\nof Θ =π/2, the drag forces are similar with and without\nmagnetic fields (Fig. 10).\nAs a consequence, supersonic massive objects maymake their way more slowly to the center of the sys-\ntem if the medium is pervased by a large-scale mag-\nnetic field. As a model problem, consider a singular\nisothermal spherical cloud threaded by a uniform mag-\nnetic field and a small-scale random magnetic field with\nAlfv´ enspeed caeverywhereconstant. The density profile\nof the cloud is given by ρ(r) = (c2\ns+c2\na/2)/2πGr2, where\ncsis the isothermal sound speed. The circular speed\nisV0=/radicalbig\n2c2s+c2a. Since the effective sound speed is/radicalbig\nγc2s+2c2a/3, the Mach number of a body on a quasi-\ncircular orbit is\nM=/parenleftbigg2c2\ns+c2\na\nγc2s+2\n3c2a/parenrightbigg1/2\n, (52)\nwhich varies from 1 .1 to 1.4 depending on the value of\ncaand whether the perturbations are isothermal or adi-\nabatic4. If the uniform magnetic field component has a\nΥ-value between 1 and 1 .41, the time for the perturber’s\norbit to decay will be a factor of 2–5 larger than the\ncorresponding decay time for Υ = 0. Our results demon-\nstrate that, in the presence of ordered magnetic fields\nwith Υ>0.7, the role of the magnetic field on the drag\nforce should be taken into account to have accurate esti-\nmates of the timescales of orbital decay via gravitational\nDF.\nThe author would like to thank Miguel Alcubierre and\nJuan Carlos Degollado for interesting discussions. Con-\nstructive comments by an anomymousreferee are greatly\nappreciated. This work has been partly supported by\nCONACyT project 60526.\nAPPENDIX\nA. FOURIER TRANSFORMATION: AXISYMMETRIC CASE\nThe three-dimensional Fourier transform of a perturbed variable f(r) is given by\nˆf(k) =1\n(2π)3/2/integraldisplay\nR3f(r)e−ik·rd3r. (A1)\nIn the Fourier space, Equations (16) and (17) are transformed in to:\nM2k2\nzˆα=k2ˆα−4πG\nc2sˆρp+Υ2k2ˆβz, (A2)\ni(M2−1)kzˆα=i\nc2skzˆΦ+iM2kzˆβz, (A3)\nwhereρpis the mass density of the perturber, thus ∇2Φ = 4πGρp. In order to have an equation for ˆ α, we will eliminate\nˆβz. From Eq. (A3), we have\nˆβz=1\nM2/bracketleftBigg\n(M2−1)ˆα−ˆΦ\nc2s/bracketrightBigg\n, (A4)\nand substituting into Eq. (A2) we find\n/bracketleftbig\nM2k2\nz−k2/parenleftbig\n1+(1−M−2)Υ2/parenrightbig/bracketrightbig\nˆα=4πG\nc2s/parenleftbiggΥ2\nM2−1/parenrightbigg\nˆρp,. (A5)\n3Thisfactor may be largerat later timesbecause the magnetiz ed\ndrag force saturates, whereas it increases logarithmicall y in time\nin the unmagnetized case. See Figure 8 for an evolved stage.\n4In the nonmagnetic simulations of the orbital decay of a sing leblack hole due to gaseous DF in Escala et al. (2004), the veloc ity of\nthe black hole is initially supersonic ( M= 1.4) and remains barely\nsupersonic through most of the simulation.DYNAMICAL FRICTION 19\nIn the absence of magnetic fields (Υ = 0), the above equation reduc es to\n(M2k2\nz−k2)ˆα=−4πG\nc2sˆρp, (A6)\nand the standard steady-state equation for the wake past a gra vitating body is recovered. At velocities much larger\nthan the Alfv´ en speed, M ≫cA/cs= Υ, Equation (A5) is simplified to\n/bracketleftBig/parenleftbig\n1+Υ2/parenrightbig−1M2k2\nz−k2/bracketrightBig\nˆα=−4πG\n(1+Υ2)c2sˆρp. (A7)\nBy comparing the above equation with Equation (A6), we see that th e response of the medium in this case is indis-\ntinguishable to that of an unmagnetized medium with sound speed ( c2\ns+c2\nA)1/2=cs(1+Υ2)1/2.\nIt is interesting to note that when V0=cA/negationslash= 0, the right-hand-side of Equation (A5) vanishes and thereby th e\nsolution is α= 0, implying that the steady-state configuration satisfies ∇·v′= 0. Obviously, the drag force is exactly\nzero in this configuration.\nThere exist two situations where the differential equation (A5) is no t well-posed: (1) at M= 1 and (2) at the critical\nMach number, Mcrit, satisfying that\n1+(1−M−2\ncrit)Υ2= 0. (A8)\nSo that\nMcrit≡/parenleftbig\n1+Υ−2/parenrightbig−1/2. (A9)\nIt is clear that Mcrit<1. If the dynamics is dominated by the magnetic field, i.e. when Υ ≫1, thenMcrit→1. If\nnot specified, we will consider M /negationslash= 1 and M /negationslash=Mcritthroughout this section.\nWe will now calculate the solution of Eq. (A5) when the perturber is a p oint mass M, so that ˆ ρp=M/(2π)3/2, which\ncorresponds to the Fourier transformation of ρp(r) =Mδ(r), to obtain the Green’s function. Using the convolution\ntheorem, it is possible to evaluate αfor any general distribution ρp. Hence, we solve for\nα(r) =(1−η)GM\n2π2c2s/integraldisplay1\nξk2−M2k2zeik·rd3k, (A10)\nwhere\nη=/parenleftbiggΥ\nM/parenrightbigg2\n=/parenleftbiggcA\nV0/parenrightbigg2\n, (A11)\nand\nξ= 1+(1 −M−2)Υ2. (A12)\nTheintegral(A10)along kzisevaluatedbytransformingtothecomplexplane. Itisconvenientt odefineγ2≡1−M2/ξ.\nEither if Mstands in the range 0 <M<Mcritor in the range min(1 ,Υ)<M<max(1,Υ), then γ2>0 and thus\nnone of the poles lie on the real axis. Hence we may use Cauchy’s resid ue theorem to obtain:\nα(r) =(1−η)GM\nξc2s1/radicalbig\nz2+R2γ2. (A13)\nFor Mach numbers in any of the two ranges: Mcrit<M<min(1,Υ) and M>max(1,Υ), the integrand has\npoles on the real axis. Hence we make ‘indentations’ in the contour a t the position of the poles. We consider first\nMach numbers larger than max(1 ,Υ). Then, for z >0, we close the contour at + i∞, leaving the poles outside the\ncontour to preserve causality, whereas for z <0, we consider a domain containing the lower half-plane, that is where\nIm(kz)<0, and the contour slightly above the real axis, so that the two pole s lie inside the contour. More specifically,\nforz <0, the integration over kzcan be evaluated as\n/integraldisplay∞\n−∞eikzz\nk2−M2ξ−1k2zdkz=/integraldisplay∞\n−∞eikzz\nk2x+k2y−γ2\n1k2zdkz=−2π\nγ1kRsin/parenleftbiggkRz\nγ1/parenrightbigg\n, (A14)\nwhereγ2\n1≡ −γ2=ξ−1M2−1 andk2\nR=k2\nx+k2\ny. The integrationover kxandkycan be carried out in polarcoordinates\n(kx=kRcosφandky=kRsinφ):\n−2π\nγ1/integraldisplay∞\n0/integraldisplay2π\n0sin/parenleftbiggkRz\nγ1/parenrightbigg\neikRcosφdφ dkR\n=−4π2\nγ1/integraldisplay∞\n0sin/parenleftbiggkRz\nγ1/parenrightbigg\nJ0(kRR)dkR\n=/braceleftBigg\n4π2√\nz2−γ2\n1R2ifz <−γ1R;\n0 otherwise.(A15)20 S´ANCHEZ-SALCEDO\nAt Mach numbers in the interval Mcrit<M<min(1,Υ), causality is perserved at z >0 if we consider a domain\ncontaining the upper half-plane and the two poles lie inside the contou r. Putting all together, the solution is\nα(r) =λ(1−η)GM\nξc2s1/radicalbig\nz2+R2γ2, (A16)\nwhere\nλ=\n\n2 ifM>max(1,Υ) andz/R <−|γ|;\n2 ifMcrit<M<min(1,Υ) andz/R >|γ|;\n1 ifM<Mcritor if min(1 ,Υ)<M<max(1,Υ);\n0 otherwise.\nThis result can be compared with that in Dokuchaev (1964) by noting that he used the variable Ato denote the\ncombination (1 −η)/ξ. Dokuchaev (1964) found the same functional form for α, but failed to divide correctly the cases\naccording to the Mach number. In the absence of a background ma gnetic field, which corresponds to ξ= 1 and η= 0,\nwe recover the classical form derived by previous authors.\nB. FOURIER TRANSFORMATION: PERTURBER’S VELOCITY PERPENDICULAR TO THE MAGNETIC FIELD\nWe consider a gravitational perturber moving at constant velocity v0ˆyin an unperturbed medium with density ρ0,\nsound speed csand a magnetic field B0=B0ˆz. Therefore, the velocity of the perturber and the magnetic field a re\nperpendicular. We are interested in the stationary wake formed be hind the body. To do this, we assume that perturber\nis at rest and feels a wind with velocity −V0ˆyat infinity. Following the same approach as in Appendix A, the two\ncoupled governing equations in this geometry are given by\nM2∂2α\n∂y2=∇2α+1\nc2s∇2Φ+Υ2∇2βz, (B1)\nM2∂2α\n∂y2=∂2α\n∂z2+1\nc2s∂2Φ\n∂z2+M2∂2βz\n∂y2. (B2)\nIn the Fourier space, these equations are transformed into\n−M2k2\nyˆα=−k2ˆα+4πG\nc2sˆρp−Υ2k2ˆβz, (B3)\nM2k2\nyˆα=k2\nzˆα+1\nc2sk2\nzˆΦ+M2k2\nyˆβz. (B4)\nIn order to have an equation for ˆ α, we eliminate ˆβzusing\nˆβz=1\nΥ2k2/bracketleftbigg4πG\nc2sˆρp+(M2k2\ny−k2)ˆα/bracketrightbigg\n, (B5)\nand we obtain the solution of the density disturbance in Fourier spac e:\n/parenleftbigg\n(1+Υ2)k2\nyk2−M2k4\ny−Υ2\nM2k2\nzk2/parenrightbigg\nˆα=\n4πG\nc2s/parenleftbigg\nk2\ny−Υ2\nM2k2\nz/parenrightbigg\nˆρp. (B6)\nOf course, the structure of αin this case is different than in the axisymmetric case (Eq. A5). The inv erse Fourier\ntransform cannot be derived analytically but since the above equat ion is well-posed for any M-value if Υ /negationslash= 0, we\nexpect the drag force to be a continuous function of M.\nREFERENCES\nArmitage, P. J., & Natarajan, P. 2005, ApJ, 634, 921\nBondi, H., & Hoyle, F. 1944, MNRAS, 104, 273\nBournaud, F., Elmegreen, B. G., & Elmegreen, D. M. 2007, ApJ,\n670, 237\nCallegari, S., Mayer, L., Kazantzidis, S., Colpi, M., Gover nato,\nF., Quinn, T., & Wadsley, J. 2009, ApJ, 696, L89\nCant´ o, J., Raga, A. C., Esquivel, A., & S´ anchez-Salcedo, F . J.\n2011, MNRAS, arXiv:1108.3032\nChandrasekhar, S. 1943, ApJ, 97, 255\nChavarr´ ıa, L., Mardones, D., Garay, G., Escala, A., Bronfm an, L.,\n& Lizano, S. 2010, ApJ, 710, 583Chuss, D. T., Davidson, J. A., Dotson, J. L., Dowell, C. D.,\nHildebrand, R. H., Novak, G., & Vaillancourt, J. E. 2003, ApJ ,\n599, 1116\nColpi, M., & Dotti, M. 2011, Advanced Science Letters, 4, 181\nConroy, C., & Ostriker, J. P. 2008, ApJ, 681, 151\nCrocker, R. M., Jones, D. I., Melia, F., Ott, J., & Protheroe, R.\nJ. 2010, Nature, 463, 65\nDokuchaev, V. P. 1964, Soviet Astron., 8, 23\nDotti, M., Colpi, M., & Haardt, F. 2006, MNRAS, 367, 103\nEl-Zant, A. A., Kim, W.-T., & Kamionkowski, M. 2004, MNRAS,\n354, 169DYNAMICAL FRICTION 21\nEscala, A., Larson, R. B., Coppi, P. S., Mardones, D. 2004, Ap J,\n607, 765\nEscala, A., Larson, R. B., Coppi, P. S., Mardones, D. 2005, Ap J,\n630, 152\nGoodman, A. A., & Heiles, C. 1994, ApJ, 424, 208\nGuedes, J., Madau, P., Mayer, L., & Callegari, S. 2011, ApJ, 7 29,\n125\nHeiles, C., & Crutcher, R. 2005, Cosmic Magnetic Fields, Not es\nin Physics, Eds. R. Wielebinski, R. Beck, 664, 137\nHornung, P., Pellat, R., & Barge, P. 1985, Icarus, 64, 295\nIda, S. 1990, Icarus, 88, 129\nImmeli, A., Samland, M., Gerhard, O., & Westera, P. 2004, A&A ,\n413, 547\nJust, A., & Kegel, W. H. 1990, A&A, 232, 447\nKim, W.-T. 2007, ApJ, 667, L5\nKim, W.-T. 2010, ApJ, 725, 1069\nKim, W.-T., El-Zant, A. A., & Kamionkowski, M. 2005, ApJ,\n632, 157\nKim, H., & Kim, W.-T. 2007, ApJ, 665, 432\nKim, H., & Kim, W.-T. 2009, ApJ, 703, 1278\nKim, H., Kim, W.-T., & S´ anchez-Salcedo, F. J. 2008, ApJ, 679 ,\nL33\nLee, A. T., & Stahler, S. W. 2011, MNRAS, 416, 3177\nMatthews, B. C., & Wilson, C. D. 2002, ApJ, 574, 822\nMaxted, P. F. L. et al. 2009, MNRAS, 400, 2012\nMayer, L., Kazantzidis, S., Madau, P., Colpi, M., Quinn, T., &\nWadsley, J. 2007, Science, 316, 1874\nNamouni, F. 2010, MNRAS, 401, 319\nNamouni, F., Luciani, J. F., & Pellat, R. 1996, A&A, 307, 972\nNarayan, R. 2000, ApJ, 536, 663Nejad-Asghar, M. 2010, MNRAS, 406, 1253\nNishiyama, S. et al. 2010, ApJ, 722, L23\nNordhaus, J., & Blackman, E. G. 2006, MNRAS, 370, 2004\nOstriker, E. C. 1999, ApJ, 513, 252\nRephaeli, Y., & Salpeter, E. E. 1980, ApJ, 240, 20\nRicker, P. M., & Taam, R. E. 2008, ApJ, 672, L41\nRobishaw, T., Quataert, E., & Heiles, C. 2008, ApJ, 680, 981\nRuderman, M. A., & Spiegel, E. A. 1971, ApJ, 165, 1\nRuffert, M. 1996, A&A, 311, 817\nS´ anchez-Salcedo, F. J. 2009, MNRAS, 392, 1573\nS´ anchez-Salcedo, F. J., & Brandenburg, A. 1999, ApJ, 522, L 35\nS´ anchez-Salcedo, F. J., & Brandenburg, A. 2001, MNRAS, 322 , 67\nS´ anchez-Salcedo, F. J., & Santill´ an, A. 2011, Rev.Mex.A. &A., 47,\n15\nSanders, D. & Mirabel, I. 1996, ARA&A, 34, 749\nStahler, S. W. 2010, MNRAS, 402, 1758\nStewart, G. R., & Wetherill, G. W. 1988, Icarus, 74, 542\nTaam, R. E., & Sandquist, E. L. ARA&A, 38, 113\nTamura, M., & Sato, S. 1989, AJ, 98, 1368\nTanaka, T., & Haiman, Z. 2009, ApJ, 696, 1798\nThompson, T. A., Quataert, E., & Murray, N. 2009, MNRAS,\n397, 1410\nThompson, T. A., Quataert, E., Waxman, E., Murray, N., &\nMartin, C. L. 2006, ApJ, 645, 186\nVillaver, E., & Livio, M. 2009, ApJ, 705, L81\nYusef-Zadeh, F., Morris, M., & Chance, D. 1984, Nature, 310, 557\nZweibel, E. G. 2002, ApJ, 567, 962"    },    {        "title": "1908.08885v1.Massively_parallel_atomistic_simulation_of_ultrafast_thermal_spin_dynamics_of_a_permalloy_vortex.pdf",        "content": "Massively parallel atomistic simulation of ultrafast thermal spin dynamics of a permalloy vortex\nDaniel Meilak, Sarah Jenkins, Rory Pond, and Richard F. L. Evans\u0003\nDepartment of Physics, The University of York, York, YO10 5DD, UK\n(Dated: November 3, 2021)\nUltrafast magnetization dynamics probes the most fundamental properties of magnetic materials, exploring\nquestions about the fundamental interactions responsible for magnetic phenomena. Thermal effects are known\nto be extremely important for laser-induced dynamics in metallic systems, but the dynamics of topological\nmagnetic structures are little understood. Here we apply a massively parallel atomistic spin dynamics simulation\nto study the response of a permalloy vortex to a 50 fs laser pulse. We find that macroscopically the short\ntimescale dynamics are indistinguishable from the bulk, but that strong edge spin waves lead to a complex\ntime evolution of the magnetic structure and long-lived oscillations on the nanosecond timescale. In the near\nfuture such simulations will provide unprecedented insight into the dynamics of magnetic materials and devices\nbeyond the approximations of continuum micromagnetics.\nSince the pioneering experiment of Beaurepaire et al [1]\ndemonstrating sub-picosecond demagnetization of Ni in re-\nsponse to laser excitation, ultrafast magnetism has become\none of the most active areas of research in the field. This\nhas lead to a wide range of phenomena including iner-\ntia driven spin excitations[2], helicity-dependent all-optical\nswitching [3, 4], and thermally induced switching [5, 6]. A\ntheoretical understanding of the physical origins of the diver-\nsity of phenomena is at an early stage, but it is clear that el-\nevated temperatures and transverse spin fluctuations play an\nimportant if not dominating role in laser induced magnetisa-\ntion dynamics. Most models and interpretations of ultrafast\nmagnetisation dynamics have relied on a single domain ap-\nproximation, where the magnetisation is assumed to be ap-\nproximately uniform in the sample. However, this approxi-\nmation breaks down in the case of materials with topologi-\ncal magnetic structures such as magnetic domains, vortices,\nBloch points or Skyrmions. It is an open question how these\ninhomogeneous structures may affect the response of a ma-\nterial to ultrafast laser excitation, though initial experimental\nstudies have suggested a range of complex and statistical dy-\nnamic effects [7, 8].\nPermalloy (Ni 80Fe20) is an ideal material for studying mag-\nnetisation dynamics experimentally due to its intrinsically\nlow magnetocrystalline anisotropy and compatibility with\nnanoscale patterning. This combination of relatively high\nmagnetic moment and low anisotropy also led to theoreti-\ncal understanding of topological magnetic vortex structures in\nnanodots [9, 10]. Despite its relative simplicity, there are open\nquestions regarding the dynamics and temperature dependent\nproperties of magnetic vortices which are not properly ad-\ndressed with continuum micromagnetic models. In particu-\nlar, high temperatures and rapid spatial variations of the mag-\nnetization and ultrafast dynamics are not accessible to con-\nventional micromagnetic models [8, 10–12], though Landau-\nLifshitz-Bloch models may describe the essential thermody-\nnamic behaviour[13–15].\nWith the recent development of massively parallel atomistic\nspin dynamics codes [16, 17] and wider availability of High\nPerformance Computing resources, it is now possible to simu-\nlate a sufficiently large nanodot with atomistic resolution suchthat a magnetic vortex is the natural ground state. Unlike pre-\nvious micromagnetic simulations[8, 10, 11], thermal effects\nare naturally included and high energy spin waves are simu-\nlated directly. To simulate the thermal properties of Permalloy\nunder the influence of temperature and laser pulses we con-\nstruct an atomistic spin model[16] writing a spin Hamiltonian\nwith Heisenberg exchange of the form\nH=\u0000å\ni<jJi jSi\u0001Sj+kc\n2\u0000\nS4\nx+S4\ny+S4\nz\u0001\n\u0000å\nimiSi\u0001Bdip(1)\nwhere SiandSjare unit spin directions at local sites iand\nneighbouring sites jrespectively, Ji j=3:78\u000210\u000021J/link is\nthe exchange interaction between nearest neighbouring spins,\nkc=3:355\u000210\u000026J/atom is the cubic anisotropy constant, mi\nis the atomic spin moment at lattice site iandBdipis the dipole\nfield originating from the sample. The spins are constructed\non a face-centred cubic lattice with local moments of mFe=\n2:9mBandmNi=0:62mBfor Fe and Ni sites respectively[18].\nTo model the equilibrium temperature dependent properties\nof Permalloy we simulate the temperature dependent mag-\nnetization of a (10 nm)3with periodic boundary conditions\nshown in Fig. 1 using a Monte Carlo Metropolis simulation\nwith an adaptive move[19] as implemented in the VAMPIRE\nsoftware package [16, 20]. To account for the quantum na-\nture of the thermal fluctuations we apply spin temperature\nrescaling[21] with an exponent h=1:63 fitted from experi-\nmental data for bulk Permalloy[22], giving quantitative agree-\nment with experimental data for the macroscopic tempera-\nture dependent properties and Curie temperature. At thermal\nequilibrium the Fe and Ni sublattices in the Permalloy are\nstrongly exchange coupled and therefore have the same tem-\nperature dependence for their intrinsic magnetic properties.\nWhile the equilibrium properties can be effectively simulated\nusing Monte Carlo simulations, we use atomistic spin dynam-\nics simulations to model the dynamic properties of Permalloy.\nThe time evolution of atomistic spins is treated by numeri-\ncal solution of the stochastic Landau-Lifshitz-Gilbert equation\nwith Langevin dynamics applied at the atomistic level [23]\n¶Si\n¶t=\u0000g\n1+l2[Si\u0002Beff+lSi\u0002(Si\u0002Beff)]; (2)arXiv:1908.08885v1  [cond-mat.mtrl-sci]  23 Aug 20192\n0.00.20.40.60.81.0\n 0  200  400  600  800  1000Magnetisation ( m/ms)\nTemperature (K)Fe\nNi\nFIG. 1. (Color online) Simulated temperature dependent magnetiza-\ntion for Ni 80Fe20Permalloy using Monte Carlo simulations. The Fe\nand Ni sublatttices show full ferromagnetic alignment at equilibrium\ndue to the strong ferromagnetic exchange.\nwhere l=0:0064 is the microscopic Gilbert damping con-\nstant, g=1:76\u00021011T\u00001s\u00001is the absolute value of the gy-\nromagnetic ratio. The effective field Beffis calculated from the\nderivative of the spin Hamiltonian given by Eq. 1 with respect\nto the local spin moment Siplus a random thermal Langevin\nfield\nBeff=\u00001\nmi¶H\n¶Si+G(t)s\n2lkBT0e\ngmiDt(3)\nwhere Gis a Gaussian distributed 3D random number and\nT0\neis the rescaled electron temperature. As with the equilib-\nrium properties, the classical Heisenberg model fails to repro-\nduce the correct dynamics of real magnetic materials due to\nthe overestimation of the thermal spin fluctuations [21]. We\ntherefore apply a rescaled temperature to the Langevin ther-\nmal field, replicating the quantum statistical effects of the ther-\nmal bath on the spin system, where T0\neis given by\nT0\ne=Tc\u0012Te\nTc\u0013h\n(4)\nwhere Tcis the Curie temperature, Teis the temperature of\nthe electron bath and h=1:63 is the temperature rescaling\nexponent. Above Tcno rescaling is applied and the spin sys-\ntem is assumed to follow classical statistics. The temperature\nrescaling approach corrects for the significant differences in\nultrafast dynamics between the classical Heisenberg approach\nand experimental data for bulk Ni [21], and we now utilise the\nsame approach to study ultrafast demagnetization in Permal-\nloy. The time evolution of the electron temperature is sim-\nulated using the two-temperature model [5, 21, 24] and the\ndipole fields are computed using a tensor macrocell approx-\nimation [25] with a 1 nm cell size. The sLLG equation is\nintegrated using a second-order predictor-corrector Heun nu-\nmerical scheme [16] due to its low computational cost and\n0.00.20.50.81.0\n 0  0.2  0.4  0.6  0.8  1  1.2Normalized magnetization ( M / Ms )\nTime (ps)Ni\nFeFIG. 2. (Color online) Simulated ultrafast demagnetization dynam-\nics for Permalloy for a (20 nm)3representative of a bulk saturated\nsample. The Fe and Ni sublattices exhibit different dynamics due\nto different values of magnetic moment, corresponding to approxi-\nmately the same dissipation rate of angular momentum.\ninherent suitability for parallelization using a domain decom-\nposition approach [16].\nThe response of a magnetic material to a laser pulse\nis often complex with a range of physical effects such as\ndemagnetization[1, 26, 27] and switching [3–6] but thermal\neffects have been shown to be the dominant mechanism for\ndemagnetization in ferromagnetic metals [13, 21, 28] and\nswitching in ferrimagnetic GdFe alloys [6, 29, 30]. In two-\ncomponent ferromagnetic alloys the ultrafast demagnetization\ndynamics of each magnetic component are different due to\ndifferent magnetic moments and weakly due to the different\nmicroscopic damping [14, 18, 31, 32]. The dependence of the\ndemagnetization time tdemag on the magnetic moment mand\nGilbert damping areflect the time required to dissipate angu-\nlar momentum from the spin system to the electrons and lat-\ntice, characterized by the approximate proportionality [18, 33]\ntdemagµm\na: (5)\nThe question remains whether there is any role of topo-\nlogical magnetic structures on ultrafast demagnetization dy-\nnamics, and so for comparison we first model the response\nof a (10 nm)3block of Permalloy to a laser pulse with 50 fs\nwidth with two-temperature model parameters assumed ap-\nproximately the same as Nickel as used in [13]. The system\nincludes periodic boundary conditions and is representative of\nbulk Ni saturated to a single magnetic domain and equilibrated\natT0\ne=300K. The short timescale dynamics for the Fe and Ni\nsublattice are shown in Figure. 2, showing rapid demagneti-\nsation on the sub-picosecond timescale and distinct dynam-\nics for the Fe and Ni sublattices despite their strong exchange\ncoupling. Our results are in close agreement with Radu et\nal[18] showing an approximate factor 2 in the characteristic\ndemagnetization time for Fe and Ni in Ni80Fe20Permalloy, al-3\n2.5 ps3 ps5 ps\n10 ps20 ps40 ps100 ps4 ps0 ps1 ps1.5 ps2 psfgijklmhbcde20 nm0255075100\n0.010.11101001000aNormalizedmagnetization(%)\nTime(ps)NiFea\nFIG. 3. (Color online) (a) Simulated ultrafast demagnetization dynamics of the z-component of the magnetization for the Permalloy nanodot\nnormalised to the initial room-temperature value including the vortex configuration. As with the bulk-like sample the demagnetization rate is\ndifferent for Fe and Ni sublattices, but away from the non-linear dynamics induced by the laser pulse the Ni and Fe sublattices are collinear.\nAfter the initial pulse oscillations of the z-component of the magnetization are induced and long-lived. (b) Initial thermalised magnetic state of\nthe vortex before the laser excitation is applied. (c-e) Snapshots of the magnetic configuration immediately after the laser excitation showing\nultrafast demagnetization and extremely small magnetic textures which induce high frequency spin waves leading to the observed oscillations\nof the magnetization. (f-l) Time evolution of the magnetic configurations and coalesence of magnetic domains due to magnon transport. (m)\nQuasi-relaxed magnetic state 100 ps after the laser pulse showing recovery of the vortex state. The non ground-state configuration takes over\n1 ns to relax (beyond the timescale of the simulation) to a truly relaxing and thermalise vortex structure.\nthough in the case of our simulations the excitation is stronger\ncausing a larger degree of demagnetization.\nWe now perform the same simulation for a Permalloy nan-\nodot using 960 processors cores with a runtime of approx-\nimately 36 hours. The phase diagram for the vortex state\ndepends on the lateral size (radius) and thickness to achieve\nthe appropriate balance of exchange energy and dipole fields\n[10]. Given the computational cost of atomistic calculations,\nwe consider the smallest structure that can support a vortex\nwith a diameter of 70 nm and 20 nm thickness. To generate\nthe vortex magnetic structure we quench the system from arandom spin state (infinite temperature) to zero allowing the\nspins to evolve dynamically to form a ground state configu-\nration with critical damping (l=1). This naturally forms a\nvortex state within 100 ps which is fully relaxed for a total\nof 1 ns of simulation time. The system is then re-thermalized\nat fixed temperature T0\ne=300K. The system is excited by\nthe thermal laser pulse and the dynamic response of the Fe\nand Ni sublattices is shown in Fig. 3(a) with time shown on\na logarithmic scale. As for the bulk Permalloy, the Fe and\nNi sublattices exhibit different characteristic demagnetization\ntimescales owing to their different magnetic moments. On4\nthe 1-10 ps the sublattices partially re-magnetize due to the\nrapid reduction of the electron temperature due to heat trans-\nfer to the lattice. For times greater than 10 ps the Fe and Ni\nsublattices recover their full collinear character and act as a\nsingle ferromagnetic material, while exhibiting oscillations of\nthe perpendicular component of the magnetization for over a\nnanosecond.\nTo gain further insight into the dynamics we plot the mag-\nnetic configuration of atoms at the centre plane of the nan-\nodot in Fig. 3(b)-(m). The classic vortex pattern is visible\nin Fig. 3(b) before the pulse arrives, superimposed with ther-\nmal noise originating from the local random spin fluctuations.\nWe note that a similar effect is seen for Lorentz microscopy\nimages of Permalloy dots and such thermal spin fluctuations\ncould also be considered as a source of noise in such im-\nages. One ps after the laser pulse arrives the vortex pattern is\ndisrupted with high frequency noise due to the ultrafast spin\nwave modes excited by the pulse, shown in Fig. 3(c). Over the\nnext few ps the high frequency noise is damped forming a rich\ndomain pattern that evolves quickly in time, consisting of an\nunderlying vortex structure with superimposed noise indicat-\ning out-of-plane magnetization components. This evolution is\nreminiscent of magnon coalescence seen in ferrimagnetic al-\nloys with similarly fast dynamics [12]. Additionally we see\nthe formation of propagating ultrafast edge spin wave modes\n(See supplementary movie x1) that rapidly form and continue\nfor hundreds of picoseconds after the laser pulse. The edge\nspin waves are the origin of the long-lived oscillations of the\nperpendicular magnetization component after the laser pulse.\nThe strength of the oscillations decays slowly, likely over sev-\neral nanoseconds due to the low damping of Permalloy. Nev-\nertheless the underlying vortex state is clearly visible after\n100 ps, showing recovery back to the initial vortex magne-\ntization state. Comparison with recent experimental measure-\nments of vortex core demagnetization [34] show comparable\nresults, with a strong demagnetization of the central vortex re-\ngion and fast recovery to a new equilibrium value on the sub\n10 ps timescale. For larger dots thermal excitation is known to\ninduce multiple vortices [8], but structures of such size are not\nyet computationally accessible with atomistic simulations.\nFinally we compare the initial demagnetization for the\nsmall bulk-like sample and vortex sample of Py, shown in\nFig. 4. The magnetization is normalized to the initial value\nof the perpendicular magnetization, which is around 95% for\nthe bulk-like sample and 12.5% for the nanodot including the\nvortex. Surprisingly the data is almost exactly the same for\nboth cases, both in terms of the demagnetization time and\nalso the behaviour of the individual magnetic sublattices. This\nsuggests that the relationship between the demagnetisation\ntime and the magnetic moment in Eq. 5 is applicable only\nat the atomic scale, and that macroscopic magnetic textures\nplay little role in magnetic processes on the sub-picosecond\ntimescale.\nIn conclusion, we have applied large-scale massively paral-\nlel atomistic spin dynamics simulations to study the ultrafast\ndynamics of a magnetic vortex subject to an ultrafast thermal\n  0 25 50 75100\n 0  0.2  0.4  0.6  0.8  1  1.2Normalized magnetization (%)\nTime (ps)Ni (vortex)\nFe (vortex)\nNi (bulk)\nFe (bulk)FIG. 4. (Color online) Comparative simulated demagnetization of Ni\nand Fe sublattices of Permalloy in bulk and nanodot configurations\nnormalized to the perpendicular magnetization before the pulse. The\ndata show almost exact agreement, demonstrating that topological\nmagnetic structures have no perceptible effect on ultrafast demagne-\ntization, and that the characteristic demagnetisation time is charac-\nteristic of atomic scale magnetic properties.\nlaser pulse. We find that the vortex structure is stable for even\nstrong laser excitation, returning to a clear vortex state after\nonly 5 ps. We also find that the characteristic demagnetization\ntime is unaffected by topological magnetic structures which\nhas important consequences for the manipulation of domain\nwalls and Skyrmions by purely thermal means. More gener-\nally, large scale atomistic spin dynamics models provide un-\nprecedented detail for studying the underlying dynamic pro-\ncesses and are approaching lengthscales and timescales ac-\ncessible with complementary experimental techniques. This\npresents an exciting opportunity to study the dynamics of\ncomplex magnetic structures and devices such as Skyrmions\nand domain walls including thermal effects beyond the capa-\nbilities of the continuum micromagnetic approximation. A\nparticularly interesting topic could be the formation and evo-\nlution of Bloch points which are impossible to properly re-\nsolve with a continuum approximation but are currently at the\nlimit of computational power available today. The continuing\nincrease in availability of general purpose computing power\nwill likely make the calculations presented here possible on a\ndesktop computer within the next 10-20 years.\nACKNOWLEDGMENTS\nThis work used the ARCHER UK National Supercomput-\ning Service (http://www.archer.ac.uk) using code enhance-\nments implemented and funded under the ARCHER embed-\nded CSE programme (eCSE0709 and eCSE1307). Data anal-\nysis was performed using the VIKING supercomputing cluster\nprovided by the University of York.5\n\u0003richard.evans@york.ac.uk\n[1] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y . Bigot, Phys.\nRev. Lett. 76, 4250 (1996).\n[2] A. V . Kimel, A. Kirilyuk, P. A. Usachev, R. V . Pisarev, A. M.\nBalbashov, and T. Rasing, Nature 435, 655 (2005).\n[3] C. D. Stanciu, F. Hansteen, A. V . Kimel, A. Kirilyuk,\nA. Tsukamoto, A. Itoh, and T. Rasing, Phys. Rev. Lett. 99,\n047601 (2007).\n[4] S. Mangin, M. Gottwald, C.-H. Lambert, D. Steil, V . Uhl ´ır,\nL. Pang, M. Hehn, S. Alebrand, M. Cinchetti, G. Malinowski,\nY . Fainman, M. Aeschlimann, and E. E. Fullerton, Nature Ma-\nterials 13, 286 EP (2014), article.\n[5] I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A.\nDurr, T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell,\nA. Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing, and A. V .\nKimel, Nature 472, 205 (2011).\n[6] T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell, U. Atxi-\ntia, O. Chubykalo-Fesenko, S. El Moussaoui, L. Le Guyader,\nE. Mengotti, L. J. Heyderman, F. Nolting, A. Tsukamoto,\nA. Itoh, D. Afanasiev, B. A. Ivanov, A. M. Kalashnikova, K. Va-\nhaplar, J. Mentink, A. Kirilyuk, T. Rasing, and A. V . Kimel,\nNat. Commun. 3, 666 (2012).\n[7] L. Le Guyader, S. El Moussaoui, M. Buzzi, R. V . Chopdekar,\nL. J. Heyderman, A. Tsukamoto, A. Itoh, A. Kirilyuk, T. Ras-\ning, A. V . Kimel, and F. Nolting, Applied Physics Letters 101,\n022410 (2012), https://doi.org/10.1063/1.4733965.\n[8] X. Fu, S. D. Pollard, B. Chen, B.-K. Yoo, H. Yang, and\nY . Zhu, Science Advances 4(2018), 10.1126/sciadv.aat3077,\nhttps://advances.sciencemag.org/content/4/7/eaat3077.full.pdf.\n[9] T. Shinjo, T. Okuno, R. Hassdorf, y. K.\nShigeto, and T. Ono, Science 289, 930 (2000),\nhttps://science.sciencemag.org/content/289/5481/930.full.pdf.\n[10] W. Scholz, J. Fidler, T. Schrefl, D. Suess, R. Dittrich, H. Forster,\nand V . Tsiantos, Comp. Mater. Sci. 28, 366 (2003), proceedings\nof the Symposium on Software Development for Process and\nMaterials Design.\n[11] G. M. Wysin and W. Figueiredo, Phys. Rev. B 86, 104421\n(2012).\n[12] E. Iacocca, T.-M. Liu, A. H. Reid, Z. Fu, S. Ruta, P. W. Gran-\nitzka, E. Jal, S. Bonetti, A. X. Gray, C. E. Graves, R. Kukreja,\nZ. Chen, D. J. Higley, T. Chase, L. Le Guyader, K. Hirsch,\nH. Ohldag, W. F. Schlotter, G. L. Dakovski, G. Coslovich, M. C.\nHoffmann, S. Carron, A. Tsukamoto, A. Kirilyuk, A. V . Kimel,\nT. Rasing, J. St ¨ohr, R. F. L. Evans, T. Ostler, R. W. Chantrell,\nM. A. Hoefer, T. J. Silva, and H. A. D ¨urr, Nature Communica-\ntions 10, 1756 (2019).\n[13] U. Atxitia, O. Chubykalo-Fesenko, J. Walowski, A. Mann, and\nM. M ¨unzenberg, Phys. Rev. B 81, 174401 (2010).\n[14] D. Hinzke, U. Atxitia, K. Carva, P. Nieves, O. Chubykalo-\nFesenko, P. M. Oppeneer, and U. Nowak, Phys. Rev. B 92,\n054412 (2015).\n[15] K. M. Lebecki and U. Nowak, Phys. Rev. B 89, 014421 (2014).\n[16] R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler, M. O. A.\nEllis, and R. W. Chantrell, J. Phys.: Condens. Matter 26,\n103202 (2014).\n[17] B. Skubic, J. Hellsvik, L. Nordstr ˜A{m, and O. Eriksson, Jour-\nnal of Physics: Condensed Matter 20, 315203 (2008).\n[18] I. Radu, C. Stamm, A. Eschenlohr, F. Radu, R. Abrudan,\nK. Vahaplar, T. Kachel, N. Pontius, R. Mitzner, K. Holl-\ndack, A. F ˜A{hlisch, T. A. Ostler, J. H. Mentink, R. F. L.\nEvans, R. W. Chantrell, A. Tsukamoto, A. Itoh, A. Kiri-lyuk, A. V . Kimel, and T. Rasing, SPIN 05, 1550004 (2015),\nhttps://doi.org/10.1142/S2010324715500046.\n[19] J. D. Alzate-Cardona, D. Sabogal-Su ´arez, R. F. L. Evans, and\nE. Restrepo-Parra, Journal of Physics: Condensed Matter 31,\n095802 (2019).\n[20] VAMPIRE software package, (2019), version 5.0 Available from\nhttps://vampire.york.ac.uk/.\n[21] R. F. L. Evans, U. Atxitia, and R. W. Chantrell, Phys. Rev. B\n91, 144425 (2015).\n[22] C. Luo, Y . Fu, D. Zhang, S. Yuan, Y . Zhai, S. Dong, and\nH. Zhai, Journal of Magnetism and Magnetic Materials 374,\n711 (2015).\n[23] M. O. A. Ellis, R. F. L. Evans, T. A. Ostler, J. Barker, U. Atxitia,\nO. Chubykalo-Fesenko, and R. W. Chantrell, Low Temperature\nPhysics 41, 705 (2015), https://doi.org/10.1063/1.4930971.\n[24] S. I. Anisimov, B. L. Kapeliovich, and T. L. Perelman, Zhurnal\nEksperimentalnoi i Teoreticheskoi Fiziki 66, 776 (1974).\n[25] G. J. Bowden, G. B. G. Stenning, and G. van der Laan, Journal\nof Physics: Condensed Matter 28, 066001 (2016).\n[26] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf,\nM. Fahnle, T. Roth, M. Cinchetti, and M. Aeschlimann, Nat.\nMater. 9, 259 (2010).\n[27] W. Zhang, W. He, L.-C. Peng, Y . Zhang, J.-W. Cai, R. F. L.\nEvans, X.-Q. Zhang, and Z.-H. Cheng, Nanotechnology 29,\n275703 (2018).\n[28] T. Roth, A. J. Schellekens, S. Alebrand, O. Schmitt, D. Steil,\nB. Koopmans, M. Cinchetti, and M. Aeschlimann, Phys. Rev.\nX2, 021006 (2012).\n[29] U. Atxitia, T. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell,\nand O. Chubykalo-Fesenko, Phys. Rev. B 87, 224417 (2013).\n[30] Y . Yang, R. B. Wilson, J. Gorchon, C.-H.\nLambert, S. Salahuddin, and J. Bokor, Sci-\nence Advances 3 (2017), 10.1126/sciadv.1603117,\nhttps://advances.sciencemag.org/content/3/11/e1603117.full.pdf.\n[31] C. La-O-V orakiat, M. Siemens, M. M. Murnane, H. C. Kapteyn,\nS. Mathias, M. Aeschlimann, P. Grychtol, R. Adam, C. M.\nSchneider, J. M. Shaw, H. Nembach, and T. J. Silva, Phys.\nRev. Lett. 103, 257402 (2009).\n[32] S. Mathias, C. La-O-V orakiat, P. Grychtol, P. Granitzka,\nE. Turgut, J. M. Shaw, R. Adam, H. T. Nembach, M. E.\nSiemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschli-\nmann, M. M. Murnane, and H. C. Kapteyn, Proceedings\nof the National Academy of Sciences 109, 4792 (2012),\nhttps://www.pnas.org/content/109/13/4792.full.pdf.\n[33] R. F. L. Evans, T. A. Ostler, R. W. Chantrell, I. Radu,\nand T. Rasing, Applied Physics Letters 104, 082410 (2014),\nhttps://doi.org/10.1063/1.4867015.\n[34] N. Rubiano da Silva, M. M ¨oller, A. Feist, H. Ulrichs, C. Ropers,\nand S. Sch ¨afer, Phys. Rev. X 8, 031052 (2018)."    },    {        "title": "1501.07159v1.Superconductor_Dynamics.pdf",        "content": "Superconductor D ynamics  \nF. Gömöry1 \nInstitute of Electrical Engineering, Slovak Academy of Sciences, Bratislava, Slovakia \nAbstract  \nSuperconductors used in magnet technology could carry extreme currents \nbecause of their ability to  keep  the magnetic flux  motionless . The dynamics \nof the magnetic flux interaction with superconductor s is controlled by this \nproperty. The cases of electrical t ransport in a round wire and the \nmagnetization of wires of various shapes (circular, elliptical, plate) in an \nexternal magnetic field  are analysed . Resistance to the magnetic field \npenetration means that the field produced by the superconducting magnet is  \nno longer proportional to the supplied current . It also  leads to a dissipation  of \nelectromagnetic energy . In conductors  with unequal transverse  dimensions , \nsuch as flat cables, the orientation with respect to the magnetic field plays  an \nessential role. A reduction  of magnetization currents  can be achieved by \nsplitting the core of a superconducting wire into fine filaments ; however , \nnew kinds  of electrical currents that coupl e the filaments consequently \nappear. Basic formulas allowing qualitative analyses of various flux dynamic \ncases are presented.  \n \nKeywords : type II superconductors, flux pinning, critical state model . \n1 Introduction   \nThe m ain motivation for using type II superconductors  (in these materials the magnetic flux  is divided \ninto flux quanta called flux lines)  in electromagnets originates in their ability to carry large electrical \ncurrents without dissipation. This property, which enabl es the generation of stationary magnetic fields \nbeyond the reach of traditional technology based on metallic conductors, is the consequence of a \nmechanism that hinders the  motion of magnetic flux inside such a superconductor  [1, 2] . Flux lines \ninteract with pinning centres that usually are imperfections in the composition or in the structure.  This \nphenomenon is illustrated schematically in Fig.  1, which show s a superconducting plate with finite \nthickness in  the x-direction and very large dimensions  along the other two axes  exposed to a  magnetic \nfield parallel to the z –y plane. For simplicity , just one kind of pinning centre is shown on the right -\nhand side of Fig. 1 ( see the white objects of irregular shape randomly distributed in the  volume of the \nsuperconductor ). When the applied magnetic field, B ex, is switched on, the magnetic flux enters the \nplate in the form of flux lines , each carr ying the elementary quantum of magnetic flux, Φ0 = 2 × 10–15 \nWb. A rough estimate of the distance b etween  the flux lines can be obtained by assuming the absence \nof any mechanism hindering the flux penetration, i.e. neglecting also a flux pinning. In that case, the \nlines would distribute uniformly and the  flux density inside the plate would be roughly equal to the \nexternal applied field, Bex. Let us assume that the flux lines form the quadratic  lattice shown in Fig.  2. \nIts line spacing could be then calculated as  \n0\nexΦaB= ,     (1) \n1 elekgomo@savba.sk  \n \n                                                        resulting in the applied field Bex = 1 T at distance a = 45  nm. In other words , there are a very large \nnumber of flux lines interacting with a superconducting wire with a typical diameter of 1 mm . This \nconclusion remains valid after detailed calculations , indicating that a hexagonal lattice would be \npreferred.  Now consider the influence of flux pinning. Because the pinning centres  obstruct the flux \npenetration, we will find the highest density of flux lines close to the sample surface , and it will \ndecrease inwards. By applying the  Maxwell equation in quasi -static conditions (i.e. neglecting the  \ndisplacement current) and taking into account all the symmetries , we find for the plate shown in Fig. 1 \nthat \n  )(d)(d\nxjxxB\nzy−= . (2) \n \nFig. 1 : Left: slab from a hard superconductor exposed to a parallel magnetic field B ex. Right: the existence of \nimperfections (white spots in the upper figure) causes a non -uniformity in the concentration of magnetic flux \nlines (lower figure). Magnetic flux can not move freely inside a hard superconductor because t he imperfections \nact as pinning centres. Macroscopic currents resulting from the gradient magnetic flux density are indicated in \nthe left-hand picture.  \n \nFig. 2 : Regular lattice of flux lines created in a type II superconductor when exposed to the magnetic field in the  \nz-direction. The c entral part s (black) denote the normal cores  surrounded by circulating supercurrents.  \n  This expression can be interpreted in the following way: a macroscopic electrical current  is \nequivalent to a gradient in the magnetic flux density . It is valid for both normal conductors and \nsuperconductors . In a metal the currents (called the eddy currents) dissipate because of normal metal \nresistivity and the field gradients disappear with time. Only in AC fields with periods shorter tha n the \ndissipation time will field gradients persist , leading to the so- called ‘skin effect ’. However , in the \nsuperconductor shown in Fig. 1 the gradient of the density of flux lines —thus of the flux density \nitself —persists in the quasi -static conditions of switching on the external field . Thus  Eq. (2) predicts \nthe existence of persistent macroscopic currents flowing in the z -direction. The density of the flux \nlines is lower than th at equivalent to the applied field. In other words , the magnetic field in a \nsuperconducting plate is  ‘screened ’ by these currents,  which produc e a magnetic field with a polarity \nthat oppos es the applied field  Bex. \nThis simple example shows that the  ‘freezing ’ of the flux line  positions  by pinning forces is \nessential to reach the large stationary densities of electrical current  necessary in the construction of \nelectromagnets able to generate static magnetic fields exceeding 10  T. However , the mechanism that \nserves to keep the distribution of the magnetic field and current density u nchanged oppos es any \nchange in the magnetic field . Such superconductors are referred to as ‘ hard’ because in a cyclic \nmagnetization process they exhibit hysteresis resembling the behaviour of a magnetic material with \nlarge coercivity. The mechanism  of flux  pinning , however, causes  hysteresis of the magnetic flux \ndistribution with respect to the current supplied to the magnet winding . As a consequence, the \ndynamics of the magnetic flux movement in an electromagnet made from a hard superconductor is \ncontrolle d by the flux pinning mechanism.  \n2 Critical state model   \nThe i nteraction of flux lines with pinning centres is understood for some simple cases, and there is \nextensive research activity dedicated to this issue. Nevertheless, the complexity of superconducting \nmaterials and the variety of pinning centres encountered ma ke it difficult to establish a  procedure \nallowing t he predict ion of  the macroscopic volume density of the pinning force directly from \nmicroscopic calculations. It is more practical to formulate a phenomenological model describing the \nmacroscopic behaviour of a hard superconductor. Its simplest version was developed by Bean [3] , and \nis commonly known as the ‘ critical state model’ . It is valid on a macroscopic scale, ignoring the \ndetails of a magnetic field distribution around an individual flux line. Working with  an average taken \nover many flux lines, the c ritical state model states that in any (macroscopic) part of a hard \nsuperconductor one can find either no electr ical current or a  current with density  equal to the so- called \n‘critical current density ’, jc. In the original formulation its value is constant and it characterizes fully \nthe properties of  a material that is a hard superconductor, i.e. a type II supercondu ctor with flux \npinning. Several later adjustments  were  incorporated in order to include  the properties of \nsuperconductors in a more realistic way, e.g. the suppression of the critical current density by the \nmagnetic field [4]  or its dependence on the elect rical field [5 , 6]. With these modifications the results \ncan be better compared to experimental data ; however , the results are no longer  analytical. Therefore \nthe original version with jc = const. is still used to obtain a basic understanding of the proble m, and we \nwill use this in the following.  \nThe actual value of the current density in a hard superconductor is controlled by the pre ceding  \nhistory . No current flow s in the regions where the magnetic flux vortices ha d not penetrated before , \nand in the r est of the superconductor the flux line density gradient provides a macroscopic current \ndensity according to Eq. (2) . Because the density of the flux lines controls  the magnetic field in a \nsuperconductor , the critical state  model  states that  \n \nc0 in the places where 0,\nelsewhere.Bjj= =\n (3) \n  Interestingly , the electrical field that is commonly used to calculate electrical currents in normal \nmetals is not explicitly found in this expression. However, because another Maxwell equation states \nthat any change in  magnetic field is  accompanied by the appearance of an electrical field, all the \nregions in a hard superconductor in which a non-zero current density appears will have had to \nexperience a finite electrical field during the transient process of magnetic field penetration. In  fact, \nthis is the way to in clude  the history of the magnetization process in to the analysis of flux dynamics in \nhard superconductors.  \nIn spite of its simplicity , the crit ical state model defined by Eq. (3) has provided many useful \npredictions that can  also be reached in an analytical way.  \n2.1 Magnetic flux dynamics at current transport  \nLet us consider a long wire with circular cross -section (its length, l , significantly exceed s its \nradius, R ), made  from a hard superconductor  characterized by the critical current density j c. The \nmaximum value of DC current that the wire can carry without resistance is  \n 2\nccI jRπ= . (4) \nWhen the current I1 (lower than Ic) is supplied to the wire in the stand -alone conditions (no \nadditional magnetic field appli ed), only a part of its cross- section will carry the electrical current and \nexperience a non -zero magnetic field. The pr ocess of injecting I1 (through the termina tions  that are at \nthe wire ends, far from the portion of wire under investigation) is accompanied by the appearance of  a \ntemporary electrical field that  provokes electrical currents that could not be deleted because of flux \npinning. The only solution compatible with the critical state equation (2) is th e one shown in the \nmiddle part of Fig.  3: a current with density j c fills the outer shell with a cross- section that is just right \nto collect the necessary value of transported current . The e xistence of parallel paths along the wire axis \nwith zero current density (in the centre) and non- zero cu rrent density (in the outer shell) does not \nviolate any law of electromagnetism, because in the stationary state the electrical field is zero in the \ncurrent -free central zone as well as in the outer shell —the latter is a consequence of superconductor \nzero resistivity. Nevertheless, in the transient process of establishing the transport current , there h as \nbeen  an electrical field in the outer shell. Calculating its distribution in space and time together with \nthe time evolution of the current density distribution allows the determin ation of  the cost of \nelectromagnetic energy (furnished by the supply of wire current) necessary to establish the final distribution [7]. A further increase in current  to the value I\n2 > I1 (but still below I c) will produce \ndistribut ions similar to that shown in the centre of Fig.  3, with the subsequent shrinking  of the central \ncurrent -free zone. Eventually , when the transport current reaches Ic, all the wire  section will be filled \nby the critical current density , jc. \n \nFig. 3 : Series of distributions of magnetic field (lines of constant vector potential) and electrical current density \n(grey scale) calculated for a round wire of radius R  = 0.5  mm and a critical current of I c = 100 A during the first \nincrease of transported current.  \n  Probably the most striking consequence of the critical state model is reached when the wire \ncurrent decreases. Because the critical current density once induced in a hard superconductor can not \nbe cancelled, the only way to reduce the total current transported in the wire is by reversing the current \npolarity in a part of it. Similar to the case of current increase , it is the outer shell where this process \nstarts , as can be found through a detailed analysis of the electrical field at the transition of total current \nin the wire from I c  to I3 < Ic. The series of distributions at decreasing current is shown in Fig.  4. Note \nthat the entire section of wire is filled by the critical current density even when the total current is \nreduced to zero. The d ifference between this distribution and the original one shown on the left of \nFig. 3 is due to flux pinning. History dependence (hysteresis) in the current distribution indicates that, \nfrom the thermodynamic point of view, the process is irreversible and that a part of the \nelectromagnetic energy provided by the current supply has been converted to heat. This quantity is \ncalled the AC loss because it is commonly determined in a cyclic process, in this example at the \ntransport of alternating current with sinusoidal waveform.  \n \nFig. 4 : Series of distributions of magnetic field (lines of constant vector potential) and electrical current density \n(grey scale) calculated for a  round wire of radiu s R = 0.5  mm and a critical current of I c = 10  A during the \nreduction of transported current from I c down to zero . \n \nFig. 5 : Series of distributions of magnetic field (lines of constant vector potential) and electrical current density \n(grey scale) calculated for a round wire of radius R = 0.5  mm and a critical current of Ic = 100  A during the \nsweep of transported current from 8 0% of I c, to −80%, and then to zero. The c entral part s (white) denote the \ncurrent - and field -free neutral zone where the electrical field remained zero  throughout . \nUsually a superconducting wire operat es at transport currents that are always lower than the \ncritical current. Then the central part of the wire remains free of current at all time s. Considerations \nwithin the critical state model state that no magnetic field (no flux lines) has penetrated to this so -\ncalled ‘neutral zone ’, and the electrical fi eld remained zero also during the change of field and current \ndistribution. This property can be used to evaluate the AC loss in the wire carrying  an AC current \nIac(t) = Iasinωt with the amplitude Ia and frequency f =ω/2π. In fact, the voltage measured on such a \n  wire can be derived  from the path integral of the electrical  field on the  rectangular loop shown in \nFig. 6. Because the inner leg of the loop is in the neutral zone, the electric field on that part is zero. \nTwo lines perpendicular to the wire axis do not contribute either , because the electrical field in the \naxial direction could not exist in this geometry. All the voltage induced in the loop due to the change \nof the linked magnetic flux with time  is on the outer surface of the wire and can be picked  by two \nvoltage taps:  \n tΦU∂∂−= . (5) \nThis relation is very useful in expressing the loss of electromagnetic energy during one cycle with \nperiod T =  2π/ω, with the result  \n ac ac dd\nTW UI t I Φ = =∫∫, (6) \nwhich is simply  the area of loop enclosed by the Φ(Iac) dependence during one cycle of AC current. \nThe loops for two cycles are shown in Fig. 7. The absence of time as an independent variable means \nthat the shape of this loop does not depend on the frequency f = ω/2π. This feature is common for all \ncases for which the flux pinning controls the flux penetration. The a ccompanying dissipation is \ntherefore called the hysteresis loss.  \n \nFig. 6 : The voltage measured by a pair of taps on the surface of a round wire form ed of  hard superconductor \nduring transport of electrical current. The voltage can be calculated from the change in  magnetic flux enclosed in \nthe rectangular loop consisting of two radial sections perpendicular to the wire axis, with the longitudinal section \nenclosed in the neutral zone and the line connecting two voltage taps. Electrical fields on all p arts except  the \nwire surface are zero.  \n \nFig. 7 : Dependence of magnetic flux enclosed between the wire surface and the neutral zone,  Φ, on the wire \ncurrent, Iac, at an amplitude equal to 80% of I c (squares) and at I c (diamonds). It begins at (0,0) with the initial \npart shown in Fig. 3 ; after the first turn in the current ramping,  repetitive cycles with closed loops follow.  \n  2.2 Magnetic flux dynamics in transient magn etic field  \nThe example of AC transport by a round wire has demonstrated the main features of magnetic flux \ndynamics  in a hard superconductor. Closer to the actual conditions in the winding of an electromagnet \nis the case when such a wire is exposed to an e xternal magnetic field that chang es in time.  In contrast \nto the previous case,  the solutions cannot be found analytically. Fortunately, numerical methods \nincorporating the critical state principles are available [8–10 ]. \n2.2.1 Wire with circular cross- section   \nFigure 8 present s the result s of calculations  for some significant points  in the magnetization cycle of a \nround wire exposed to a transversal magnetic field . Braking of the magnetic flux penetration is \nequivalent to  the appearance of a current loop (two currents of opposite direction) generating a \nmagnetic field that opposes the change in  field inside the superconductor. At the first field increase the \ncentral part remains free of currents. Th is would persist in case s when the maximum field ( the \namplitude of the field in the case of cyclic AC magnetization) does not exceed the so -called \n‘penetration field’ , Bp, defined as the field at which all section s of the wire are filled with critical \ncurrent density.  \nIn order to characterize the process of magnetic flux penetration into the wire exposed to an \nexternal magnetic field , the dependence of its magnetization, M , on the actual value of the external \nfield, Bex, is commonly used. This is illustrated in Fig. 9 , where the loop calculated for the wire shown \nin Fig. 8 is presented . At the penetration field, B p, the sample magnetization reaches the saturation \nmagnetization, Ms. Its value can be obtained from the following consideration, illustrated by Fig.  10, \nwhich show s the current distribution in a wire with its length parallel to the z -coordinate exposed to \nthe magnetic field in  the y-direction [11]. The magnetic moment will have only the y -component that \nis at any instant of the magnetization cycle calculated  by performing  the integration  (see also \nAppendix A)  \n yxyxj m\nSdd),(∫−=  (7) \nover the wire cross- section, S. When the saturation state has been reached —this is the situation \nactually shown in Fig. 10— this expression can b e rewritten as  \n \n          sc c c c\n00 0dd dd 2 dd\nxx xm x j xy x j xy j xxy jS x\n<> >= −+ = =∫∫ ∫, (8) \nwhere x is the average distance of the  x-coordinate from the axis of symmetry (in Fig. 10 this is the y -\naxis) in the wire cross -section , i.e.   \n ∫=\nSSxSx d1. (9) \nThe quantity  x2 may be interpreted as the wire dimension transverse  to the applied field . It \ncan be evaluated in a similar way as for a wire of any shape S, although the integration could be more \ncomplicated when the boundary between two orientations of current density does not coincide with \nany of the principal coordinate axes.  The saturation magnetization is given by  \n  s\nscmM jxS= = . (10) \nFor a  round wire with radius R , one obtains π34Rx=  and sc4\n3RMjπ= , respectively.   \n   \nFig. 8 : Series of distributions of magnetic field (lines of constant vector potential) and electrical current density \n(grey scale) calculated for a  round wire of radius R  = 0.5  mm and critical current of I c = 100 A during the \napplication o f an external magnetic field, Bex, in a direction transvers e to the wire axis. Actual values of B ex (in \nmilli-tesla) are  indicated above the plots.  \n \nFig. 9 : Dependence of magnetization on the applied field calculated for the round wire of  Fig. 8 \n \nFig. 10: Saturation of round wire with screening currents  \n  2.2.2 Wire with elliptical cross -section —effect  of magnetic field orientation  \nThe i mportance of the transvers e dimension can be nicely illustrated by  the following example . \nAssume a wire with an elliptic al cross -section , the relation between the main axes (commonly called \nthe aspect ratio) being  a:b = 10:1. Two typical cases of its orientation with respect to the applied field \nare show n in Fig.  11. When the field is parallel to the minor axis —this  is called  the ‘perpendicular ’ \norientation— the values of the transverse  dimension and the saturation magnetization will be  \n sc44;33aax Mjππ⊥= = , (11) \nrespectively. Rotating the wire by 90 ° will lead to the configuration called ‘parallel ’, with the \nfollowing values of transvers e dimension and saturation magnetization:  \n  || s c44;33bbx Mjππ= = . (12) \nBecause a >b , the values for the perpen dicular field are larger than those for the parallel field , by \nexactly the same ratio as the ellipse axes.  \n \nFig. 11:  Exposure of wire with elliptic al cross -section to a perpendicular or parallel magnetic field  \nLet us now analy se the difference in the magnetic flux dynamics between two orientations of a \nwire with an elliptic al cross -section. For this purpose we compare the magnetization loops calculated \nin two cases (by a numerical procedure) and plotted in Fig. 12. The difference in  the loop ‘thickness ’ \ncontrolled by the saturation magnetization  is spectacular . This is b ecause  the same ratio ( a/b = 10) \nexists  for the values of Ms. Much less prominent is the difference in the penetration field s, Bp, which \ncan be identified as the field occurring when the m agnetization reaches its saturat ion value . The flux \nlines before entering the wire deform in order to adapt  to become  roughly parallel to the surface. This \nbuckling creates more line tension in the perpendicular orientation ; therefore the push to enter the wire \nis bigger , and the penetration process  is quicker than in the parallel case. However , this difference is \nfar lower (a factor of about 2 in this case) than in the saturation magnetization. Because the dissipation \nof the electromagnetic energy in one cycle— commonly called the AC loss— per unit length of wire is \ngiven by a formula indicating that it is proportional to the loop area , \n     l exd Q mB=∫  [J·m −1],      (13) \nthe AC loss in the perpendicular field is much  bigger than in the parallel field.  \n  The m ain consequences for electromagnets manufactured fro m wires containing hard \nsuperconductors, arising from the existence of magnetization currents in these materials,  are as \nfollow s: \n• magnetization currents generate a macroscopic magnetic field that is not proportional to the \nsupplied current , e.g. after switching off the current a remanent field remains in the bore ; \n• ramping  of the magnetic field generates a dissipation that warms up the magnet winding, so a \ncooling system must be designed to remove this AC loss . \nIn a magnet designed to use the transport capacity of a superconducting material  in a reasonable way , \nthe magnetization currents saturate the wires in a substantial  portion of the winding. In other words , \nthe local magnetic field is far larger than the penetrat ion field of the wire, Bp. Then the essential  \nparameter controlling the behaviour is the saturation magnetization. The v olume density of the AC \nloss  i.e. the dissipation Q  released in the superconductor volume V   in the cycle with AC field \namplitude B a >> B p can be estimated  as \n sa4QMBV= . (14) \nTwo cases for the orientation of the magnetic field with respect to a wire of elliptic cross -section with \naspect ratio 10 will then result in the prediction of an AC loss 10 times higher in the perpendicular \ncase because of  the difference in the saturation magn etization.  \n \nFig. 12:  Magnetization loops calculated for a wire of elliptic al cross- section, where a = 1 mm, b = 0.1  mm, \nIc = 100 A, exposed to the magnetic fields of parallel (dashed) and perpendicular ( solid ) orientation with respect \nto the major semi -axis.  \n2.2.3 Slab in parallel field   \nThe distributions shown in Figs. 8 and 11 and the magnetization loops plotted in Figs. 9 and 12 have \nbeen calculated numerically because there is no analytical solution available for the case of a wire \nmagnetized in a perpendicular field. One can avoid setting up a numerical model by using the \n  analytical approximation of these results [ 12]. On the other hand, a nalytical formulas are available for \nthe case of a superconducting slab of thickness w, as shown in Fig. 1. Magnetic flux profiles and \ncurrent distributions obtained at a cyclical change of magnetic field applied parallel to the slab are \nplotted in Fig.  13 for the field amplitude just reaching  the penetration field value  and in Fig. 14 for \nlarger field amplitudes. The value of B p is obtained with the help of Eq. (2) under the assumption that \nthe field just reaches the centre of the slab , which is at a distance w /2 from the surface:  \n   p,slab 0 c2wBjµ= . (15) \nThe saturation magnetization is easily calculated from Eq. (10) by taking into account that, for a slab \nwith thickness w in a parallel applied field , 4 xw= : \n p,slab\ns,slab c\n0 4BwMjµ= = . (16) \nThe volume density of loss in cyclic magnetization in an AC field with amplitude B a was derived for \nthis case [ 3] as  follows:  \n 3\na\nap\np\n02\npa p a p2for ,13\n42 for ,3BBBQ B\nV\nBB B B Bµ< =\n−> (17) \nwith B p = B p,slab given by Eq. (15).  \n \nFig. 13:  Series of profiles of magnetic field (upper part) and electrical current density (lower part) at the first \nincrease (left) and subsequent reduction (right) of the magnetic field applied in a parallel direction to a slab \ninfinite in the y–z plane. A particular case of maximal field equal to the penetration field, B p, is shown.  \n   \nFig. 14:   Similar to Fig. 13 , but the field increases from B p (left) and is then reduced to a negative value (right). \nNote that the  entire  section of the slab is filled with critical current density.  \nThe p lot of this dependence for slabs with different thickness es and critical current densit ies is \nshown in Fig. 15. Among other  things,  it illustrates the fact that two different slabs exhibit the same \nvolume density of loss provided the penetration field, B p, is the same. An important feature of loss \ndependence is the change of slope when the amplitude trespasse s the value of B p: from 3\na QB∝  \nobserved below penetration, it reduces to a QB∝  for large amplitudes B a >> Bp in accordance with  \nthe general formula (14). In spite of fact that the slab geometry is far from describing a wire used to \nwind a superconducting magnet , Eq. (17) was widely used to pr edict the loss in superconducting \nmagnets developed for a generation of magnetic fields varying in time, such as  dipoles for particle \naccelerators.  \n \nFig. 15:  Dependence of the volume density of AC loss calculated with the help of Eq. ( 17) for four slabs \ndiffering in thickness w and critical current density jc. The loss es calculated for two slabs with identical value s of \nthe penetration field calculated from Eq. (15) are the same.  \n  Nowadays , with increased computing power and numerical methods developed for the \ncalculation of magnetic flux dynamics in hard superconductors , such approach serves only for \nqualitative considerations and rough estimations. Interestingly , the results [e.g. 13, 14] found by such \ncomputations always follow the general features shown in Fig. 16: there is a change in the slope at \namplitudes comparable to the penetration field and an inverse order of loss le vel below and above this \n‘knee ’. At low amplitudes , a smaller B p means higher lo sses; at large amplitudes , it results in lower \nlosses. \n \nFig. 16:  Electrical loop created by the currents coupling two superconducting filaments when exposed to the \nmagnetic field Bex changing in time (in this particular case , it is increasing).  \n3 Multicore wires and coupling currents  \nOne i mportant consequence of Eq. (10) is that the only way to reduce  the magnetization currents \nwithout lowering the transport capacity of a superconducting wire —as well as the AC loss at large AC \nfield amplitudes given by Eq. (14) —is to minimi ze the dimension of the wire that is perpendicular to \nthe applied field. In the case of flat conductors, such as Rutherford cables or tapes coated with high-\ntemperature superconductors , this means one must avoid the exposure of the magneti c field \nperpendicular to the flat face. This is not always possible, and therefore a remedy was developed that \ninvolved splitting  the superconducting core in to many  filaments. Imagine dividing a wire of  circular \ncross -section  with radius R0 into N circular filaments. Maintaining the critical current requires that the \ntotal cross- section of the superconducting material be the same, so the filament radius will be\nN R RN 0= . Then, according to Eq. (10), the saturation magnetization will be  reduced by a \nfactor  N. Therefore the wires used in magnets generating a pulse or AC magnetic field are \ncomposites containing fine (5–50 µm in diameter) superconducting filaments in a metallic matrix. The \nmatrix —besides  assisting  thermal and mechanical stabili zation—allows a transfer of electrical current \nin the transverse  direction and the optimal distribution of this current  among the superconducting \nfilaments.  However,  another phenomenon influencing the macroscopic magnetization and the AC loss \nwill appear  at the ramp of the magnetic field : electrical currents connecting the filaments across the \nmetallic matrix in closed loops , commonly called the coupling currents.  \nThe mechanism of magnetically induced coupling is illustrated  schematically  for just two \nfilaments in Fig. 16. The current loop provoked  by the change in applied magnetic field consists of the  \npart running along the superconducting filaments , but it must be closed either across the normal \nconducting matrix or at the e nds of filaments . Let us now compare the flux dynamics in two perfectly \ncoupled parallel round filaments for  the case when the filaments would be completely insulated. \nDistributions of current density and magnetic field calculated for these two cases are shown for some \nrepresentative instants of the magnetization cycle in Fig. 17. When the connection between filaments \nis perfect, all the current running along one filament will return through the second one:  \n  \n1,coupled 2,coupledddjS jS=−∫∫, (18) \n  where the indices 1 and 2 denote t he cross -sections of the two filaments , respectively . In the case of \nnon-existing coupling , \n  \n1,uncoupled 2,uncoupledd d0jS jS= = ∫∫, (19) \ni.e. all the current induced by the field change must return through the same filament. Let us evaluate \nthe transverse  dimension in these two cases. For uncoupled round filaments, each with radius R , the \nvalue of xis the same as for  the single round wire with radius R:   \n 2,uncoupled4\n3Rxπ= . (20) \nIn the case of perfect coupling , the maximum magnetic moment is determined by the distance between \nfilaments, d, and one can estimate that   \n 2,coupled2dx=. (21) \n \nFig. 17 : Distributions of magnetic field (lines of constant vector potential) and electrical current density (gr ey \nscale) calculated for a  pair of round wires during the change in applied magnetic field, Bex in a direction \ntransvers e to the plane connecting the axis of the two wires. The upper part shows the result obtained when \nassuming a perfect galvanic connection at the terminations of the wires ; the lower part shows the distributions \ncalculated for two insulated wires.   \nMagnetization loops calculated by a numerical finite element method [10] are given for these \ntwo cases in Fig. 1 8. The proportion of values for the saturation magneti zation is given by the ratio \nof x, which is  evaluated using Eqs . (20)  and (21). U sing the  values R  = 1 mm and d = 4  mm, one finds \nthat Ms  in the coupled case should be 4.7 times higher than in the uncoupled case, which is in good \nagreement with the numerically  calculated  result .  \nThe existence of coupling currents means that splitting a single superconducting  core into many \nfilaments is not  in itself sufficient to depress the magnetization currents and reduce AC loss. An \nadditional measure is required to uncouple the  filaments.  This was achieved by means of a  \ntransposition  through twisting the whole filamentary zone . The polarity of the electrical f ield induced \nby the change in the external magnetic field alternates in the half -loops betwe en filaments , as shown in \nFig. 19. The n et voltage generated along one  filament is  therefore negligible , and thus there is no \ndriving force to create interfilament currents. However , within one half of the twist pitch, lp, a \npotential difference between parallel filaments remain s, leading to a  current traversing the matrix. This \n  mechanism can be interpreted as a diffusion of magnetic flux opposed by coupling currents  [1] with \ntime constant  \n   2\np 0\nt22lµτρπ=\n, (22) \nwhere ρt is the effective transverse resistivity of the multifilament composite. This quantity is \nestimated from the resistivity of the matrix, ρm, and the volume fraction occupied in the wire by the \nsuperconductor, λ, taking into account that superconducting filaments provide shorts for currents:   \n tm1\n1λρρλ−=+. (23) \nThis is the lower limit of the effective transverse resistivity . For cases when the interfaces between the \nfilaments and matrix create obstacles for the current flow , e.g. because the formation  of an oxide layer , \nthe values of transverse resistivity could be significantly higher.  \n \nFig. 18:  Magnetization loops evaluated from the distributions shown in Fig. 17. Full line : coupled wires ; dashed \nline: uncoupled wires.  \nIn contrast to the magnetization currents generated by the flux pinning in a superconductor , the \ncoupling currents represent a r amp-rate-dependent phenomenon. The screening field created by the \ncoupling currents changes in time , with the characteristic time constant given by Eq. (22). This can be \nmeasured e.g. as the delay of the magnetic field in the closed vicinity of the wire with respect to the \napplied field , as illustrated in Fig. 2 0. Then the magnetization loops will be ramp- rate dependent and \none must always indicate the waveform of the magnet current used in experiment, otherwise the \ninterpretation of results is not possible.  \nBecause the coupling currents are controlled by the normal resistance of the matrix , they exhibit \nproperties similar to  normal eddy currents. In particular, the behaviour at any cyclic change can be \npredicted on knowing the response at various frequencies. The volume density of the AC loss due to \n  coupling currents caused by the external magnetic field ()ex a sin BB t ω=  is predicted by the  \nfollowing  formula  [15]: \n  2\nc a0\nw0 1QB\nVχ πωτ\nµ ωτ=+, (24) \nwhere Vw is the volume of the whole wire (both supercondu ctor and matrix) and χ0 is the magnetic \nsusceptibility for a completely screened wire [16]. Th e latter  quantity could be determined \nexperimentally at low temperature and small B a, or calculated from the shap e of the wire. For the wires \nwith cross -sections that could be approximated by an ellipse with axes a and b, respectively, placed in \na magnetic field oriented in parallel to the minor semi -axis, b, its value approaches χ0 = 1 + a/b [17]. \nAccordingly, for the round wire , χ0 = 2. In weak magnetic fields with amplitudes well below the \npenetration field, the coupling loss should be corrected by this  factor , taking into account a total \nexpulsion of flux from the superconducting filaments ; Eq. (24) i s modif ied to   \n \n ()\nap2\nc a0\nw011BBQB\nVχ πωτλµ ωτ<<= −+. (25)      \nIn reality one often obtains a result that lies between  the predictions of Eqs. (24) and (25) , as shown in \nFig. 2 1. In this plot the AC loss is expressed in terms of the imaginary part of the complex magnetic  \nsusceptibility  [18] , \n 0\n2\nw a''Q\nV Bµχ\nπ= . (26) \nSuch a representation is similar to the ‘loss function’  but allows better quantitative comparison.  \n \nFig. 19 : The polarity of the electrical field induced between two twisted wires alternates every half-pitch  \n \nFig. 20:  The m agnetic field inside or in the vicinity of the wire, B i, compared to the applied field, B ex, increasing \nwith constant ramp rate, is delayed in time. From this delay , the time constant of  the coupling currents, τ, can be \nestimated . \n   \nFig. 21 : Imaginary part of the complex magnetic susceptibility measured on a multifilamentary wire compared \nwith two predictions that differ in the extent of filament penetration by an applied magnetic field. The solid  line \nhas been calculated from Eq.  (24), assuming complete filament penetration ; the  dashed line is from Eq. (25),  \nderived for completely screened filaments.  \n4 Conclusions   \nThe p inning of magnetic flux, which is  necessary to secure an exceptionally high current transport \ncapacity  in type II superconductors , is at the same time an obstacle when  a variation in the magnetic \nfield occurs . This is b ecause a dissipation in the cyclic regime (e.g. when transporting AC current) \nappear s. Also , a superconduct ing wire in a DC magnet must undergo a chang e in magnetic field at the \nramp necessary to reach the operating field.  \nThe d ynamics of magnetic flux penetration in to a hard superconductor (i.e. a type II \nsuperconductor able to pin the magnetic flux  in its volume ) is described by a complex process  that \ndepend s on the properties of the superconduct ing material as well as the  architecture of the \nsuperconducting wire  and the orientation of magnetic field. Many basic features of this process can be \nmodelled on a macroscopic scale with the help of  the critical state model , assuming that , in a hard \nsuperconductor , the local value of the current density could be either zero or jc.  \nInfiltration of a magnetic field provokes magnetization currents, which have  negative \nconsequences on the quality of the magnetic field generated by the superconducting magnet . The se \ncurrents  also lead to  a dissipation of electromagnetic energy. The most practical way of minimizing \nthese currents is through a reduction of the superconductor’s dimension transvers e to the mag netic \nfield. This is why composite wires for pulsed magnets contain fine superconducting filaments , which \nin turn should be twisted in order to reduce the coupling currents induced in the interfilament loops.  \nThe b asic principles erquired to understand the se problems have been explained in this paper.  \nThe c urrent state of numerical methods allows detailed calculations that assum e a more realistic \ndescription of the superconductor properties than is  possible with the original critical state model with \njc = constant. For example,  a magnetic- field dependence of the critical current density and its \nanisotropy, and/or non- uniformity of j c in the volume of the superconductor , can be in cluded  in order \nto interpret the experimental data obtained from industrially pr oduced materials  [19]. There is \ncurrently a great deal  of activity directed towards a more exact prediction of flux dynamics in non-\ntraditional superconductors , and the topic is far from being completely managed . A recent review of \nthe achievements in this field is given in [20].  \n  Appendix A  \nThe quantity used here to define  the magnetic field s is the induction, B , in units  of tesla (T), where  \n1 T = 1 V·s·m−2. This is the quantity measured by magnetic field sensors and an exerting force o n a \nmoving charged particle.  In a material with a magnetic response , one finds  loops of electrical currents \n(see Fig. A1 , left panel ). The a rea of the loop, S\n, and the circulating current, I, determine its magnetic \nmoment  \n m IS= [Am2]. (A.1)  \nThe magnetization of a sample with volume V  is defined as the volume density of magnetic moments  \n VmM∑=\n [A·m−1]. (A.2)  \nQuite often in the literature on superconductors an alternative definition is use d that expresses the \nmagnetization in tesla . This is obtained by multiplying Eq. (A.2) by the permeability of vacuum, \nµ0 = 4π·10−7 H·m−1. In all other aspects they are identical.  \n \nFig. A1 : Magnetic moment of a circular loop (left) and the magnetization of a macroscopic sample (right)  \nAcknowledgement  \nThis work was supported in part by the VEGA grant agency under contract N o. 1/0162/11, by the \nEuropean Commission EURATOM project FU07- CT-2007 -00051 and co-funded by the Slovak \nResearch and Development Agency under the  contract  No. DO7RP -0003 -12. \nReferences  \n \n[1] M.N. Wilson, Superconducting Magnets  (Clarendon , Oxford, 1983) . \n[2] W.J. Carr Jr, AC Loss and M acroscopic Theory of Superconductors , 2nd ed. (Taylor and \nFrancis, London, 2001) . \n[3] C.P. Bean , Phys. Rev. Lett . 8 (1962)  250–253.  \n[4] Y.B. Kim, C.F. Hempstead and A.R. Strnad, Phys. Rev. Lett.  9 (1962) 306–309.  \n[5] E.H. Brandt, Phys. Rev. B  54 (1996) 4246–4264.  \n[6] N. Amemiya, S. Murus awa, N. Banno and K. Miyamoto , Physica C  310 (1998) 16 –29. \n[7] W.T. Norris,  J. Phys. D  3 (1970) 489 –507.  \n[8] A.M. Campbell, Supercond. Sci. Technol. 20  (2007) 292 –295.  \n[9] C. Gu and Z. Han , IEEE Trans. Appl. Supercond. 15 (2005) 2859–2862.  \n[10] F. Gömöry, M. Vojenčiak, E. Pardo, M. Solovyov and J. Šouc  (2010) , Supercond. Sci . \nTechnol. 23 (2010) 034012.  \n  [11] M. Ashkin , J. Appl. Phys. 50 (1979) 7060–7066.  \n[12] F. Gömöry, R. Tebano, A. Sanchez, E. Pardo, C. Navau, I. Hušek, F. Strýček  and P. Kováč , \nSupercond. Sci . Technol . 15 (2002) 1311–1315.  \n[13] E. Pardo, D.X. Chen,  A. Sanchez and C. Navau , Supercond. Sci. Technol. 17 (2004) 83–87. \n[14] T. Yazawa, J.J. Rabbers, B. ten Haken, H.H.J. ten Kate and Y. Yamad a, Physica C  310 (1998) \n36–41.  \n[15] A.M. Campbell, Cryogenics  22 (1982) 3–16.  \n[16] K. Kwasnitza and S. Clerc, Physica C 233 (1994) 423 –435. \n[17] P. Fabbricatore, S. Farinon, S. Innocenti and F. Gömöry, Phys. Rev. B  61 (2000) 6413–6421.  \n[18] P. Fabbricatore, S. Farinon, S. Incardone, U. Gambardella, A. Saggese and G. Volpini , J. Appl. \nPhys.  106 (2009) 083905.  \n[19] F. Gömöry , in High  Temperature Superconductors (HTS) for E nergy Applications , Ed. \nZ. Melhem ( Woodlhead Publ ishing  Ltd. , Oxford, 2012 ), pp. 216–256.  \n[20] F. Grilli, E. Pardo, A. Stenvall, D.N. Nguyen, W. Juan and F. Gömöry , IEEE Trans. Appl. \nSupercond. 24 (2014)  8200433.  \n \n  "    },    {        "title": "1906.06506v1.Bursty_magnetic_friction_between_polycrystalline_thin_films_with_domain_walls.pdf",        "content": "Bursty magnetic friction between polycrystalline thin films with domain walls\nIlari Rissanen1∗and Lasse Laurson2\n1Helsinki Institute of Physics and Department of Applied Physics,\nAalto University, P.O. Box 11100, FI-00076 Aalto, Espoo, Finland and\n2Computational Physics Laboratory, Tampere University, P.O. Box 692, FI-33014 Tampere, Finland\nTwo magnets in relative motion interact through their dipolar fields, making individual magnetic\nmoments dynamically adapt to the changes in the energy landscape and bringing about collective\nmagnetization dynamics. Some of the energy of the system is irrevocably lost through various cou-\npling mechanisms between the spin degrees of freedom and those of the underlying lattice, resulting\nin magnetic friction. In this work, we use micromagnetic simulations to study magnetic friction in\na system of two thin ferromagnetic films containing quenched disorder mimicking a polycrystalline\nstructure. We observe bursts of magnetic activity resulting from repeated domain wall pinning due\nto the disorder and subsequent depinning triggered by the dipolar interaction between the moving\nfilms. These domain wall jumps result in strong energy dissipation peaks. We study how the prop-\nerties of the polycrystalline structure such as grain size and strength of the disorder, along with\nthe driving velocity and the width of the films, affect the magnetization dynamics, average energy\ndissipation as well as the statistical properties of the energy dissipation bursts.\nPACS numbers: 75.78.-n,76.60.Es,75.70.Kw\nI. INTRODUCTION\nCrystalline structures of solids found in nature are\nrarely perfect, but instead contain many kinds of im-\npurities, defects and grains. These irregularities deter-\nmine the mechanical, thermal and electromagnetic prop-\nerties of materials to a great extent, examples of which\ncan be found from the production of electrical steels1to\ndoping semiconductors2. Modern fabrication techniques\nhave made it possible to purposefully engineer materials\nat very small scales to have desirable micro- and macro-\nscopic properties.\nWhen it comes to magnetic properties of materials,\nan interesting consequence of the imperfect lattice struc-\nture is the creation of energetically preferable locations\nin which magnetic substructures, such as domain walls\nand vortices, can become pinned. The pinning con-\ntributes to multiple static and dynamic attributes of\nthe magnet, from affecting properties such as coerciv-\nity and permeability3, to giving rise to dynamics such\nas Barkhausen jumps/avalanches4, in which the domain\nwalls inside a magnet jump from one configuration to\nanother during a magnetization process.\nChanges in magnetization, such as the aforementioned\nBarkhausen avalanches, incur energy losses due to var-\nious coupling mechanisms between the spin degrees of\nfreedom and the lattice5. Along with eddy current losses,\nthese losses due to magnetic dynamics, including hys-\nteretic losses due to domain wall jumps and anoma-\nlous losses, are relevant in applications where there are\nhigh-frequency alternating electromagnetic fields and/or\ncomponents moving in such fields6, such as in magnetic\nbearings7, magnetic gears8and electric motors9. In thin\nfilms and insulators, the hysteretic losses are particularly\nimportantduetoeddycurrentsbeinglargelynegligible10.\nWhen associated with motion, it is natural to call the\nmagnetic losses ”magnetic friction”.Inthisstudy, weinvestigatehowadisorderedpolycrys-\ntalline structure and the related domain wall pinning and\ndepinning influence the magnetic domain wall dynamics\nand the resulting magnetic friction between thin films in\nrelative motion. We focus on two things: the influence of\nthe dimensions of the films and parameters such as grain\nsize and strength of the disorder on the average energy\ndissipation, and the statistics of the fluctuations in the\nenergy dissipation due to bursts of domain wall motion\nin the system.\nThe paper is structured as follows. In Sec. II, we go\nthroughthetheoreticalbackground ofeffectsofgrainsize\non the properties of polycrystalline magnets and domain\nwall motion in a disordered medium. Sec. III explains\nour micromagnetic simulation scenario and the relevant\ndetails regarding the interaction of magnets and energy\ndissipation. Theresultsofthesimulationsareprovidedin\nSec. IV, and the conclusions of this study are elaborated\non in Sec. V.\nII. DOMAIN STRUCTURE AND DYNAMICS\nIN POLYCRYSTALLINE MAGNETS\nBetween perfectly monocrystalline structure and com-\npletely disordered (amorphous) structure are polycrys-\ntalline solids, a common form of structure found in e.g.\nmetals and ice. Polycrystalline solids consist of multi-\nple single-crystal grains (crystallites) with more or less\nrandom sizes and crystallographic orientations, deter-\nmined by conditions in which the solid is formed. In\nmagnetic materials, the individual grains influence the\ntotal domain structure of the magnet, their contribu-\ntion determined by the orientation of grain surface and\ngrain boundaries relative to the orientation of the easy\nanisotropy axis/axes and the interaction between nearby\ngrains11.arXiv:1906.06506v1  [cond-mat.mes-hall]  15 Jun 20192\nThe magnetic properties of a ferromagnet are greatly\naffected by the degree of structural order, or crystallinity,\nof the material. It has been found, for example, that due\nto domain wall pinning at grain boundaries, both the co-\nercivity and remanent magnetization of nanocrystalline\nmagnets can be tuned by altering the grain size12. The\nmaximum of these properties is attained when the grain\nsize equals the typical size of magnetic nanostructures\nsuch as domain walls, leading to strong pinning and thus\ndomains that resist changes of size and shape by an ap-\nplied external field.\nThe grain size dependence is mostly the result of the\ncompetition of magnetocrystalline anisotropy and ex-\nchange interaction defining how strongly domain walls\nbecome pinned at the grain boundaries. When the grains\nare smaller than the typical width of domain walls in the\nmaterial, the exchange interaction prevents the magneti-\nzationfromcompletelyaligningintothepreferredmagne-\ntizationdirectionofeachgrain,averagingthepinningdis-\norder over multiple grains and thus lowering the effective\nanisotropy. Grains approximately the size of a domain\nwall strike a balance between following the anisotropy di-\nrection and having a slowly varying magnetization, thus\nachieving the strongest pinning effect. With further in-\ncreasing grain size the possible pinning volume decreases,\nan extreme example being a single crystal magnet with\nno grain boundaries, in which a domain wall can in prin-\nciple move freely. In larger grains, the exchange energy\nalso plays a decreasing role in the magnetization reversal,\nso that the magnetization in each grain can be switched\nmore easily.13\nA. Domain wall jumps in disordered media\nA consequence of domain wall pinning is that the do-\nmain wall motion during magnetization processes of fer-\nromagnets is not continuous, but consists of periods of\ninactivity followed by short bursts of movement. The\nmagneticdynamicsisthusdominatedbyintermittentdo-\nmain wall jumps, the character of which depends on the\nstrength of the field driving the magnetization.\nClose to the depinning field strength Hdwhere the do-\nmain walls become completely unpinned, one encounters\nthe Barkhausen effect11, in which the domain wall mo-\ntion is dominated by large-scale avalanches across the\nsystem. The size distribution of Barkhausen jumps or\navalanches has been found to contain universal char-\nacteristics similar to many other forms of crackling\nnoise, such as earthquakes14and microfractures15. The\nsize distribution P(S)typically following a power law\nP(S)∝S−τS, with well-defined exponents τSfor a wide\nrange of avalanche sizes, suggesting critical behavior16.\nDue to interest in the statistics of critical phenom-\nena and avalanche dynamics in disordered systems,\nBarkhausen noise has been quite extensively studied,\nboth in 3-dimensional magnets17and thin films18–21. As\nthe domain walls tend to get pinned at impurities andgrain boundaries in the material, the Barkhausen signal\nduring a magnetization process can potentially serve as a\nmeasure of probing e.g. the grain size of a ferromagnetic\nmaterial22.\nAnother class of domain wall motion in disordered\nmedia, taking place at field strengths below the de-\npinning field, is the domain wall creep regime. In\nthis regime, small segments of the domain wall un-\ndergo motion approximately independently due to ther-\nmal activation23. Studied both experimentally24and\nwithsimulations25, thedomainwallcreephasbeenfound\nto include avalanches that obey slightly different scal-\ning than the Barkhausen avalanches at the depinning\nthreshold26,27. Thedomainwallroughnessandavalanche\nstatistics in the creep regime have generally been found\nto follow the theory of an elastic interface in a random\npinning landscape23,28.\nCompared to the aforementioned types of driven do-\nmain wall motion, where driving is accomplished by an\nexternal field and thermal effects, in our system the do-\nmain wall motion in one thin film is instigated due to\nthe interaction with the stray field of the other film. The\nchange in magnetization in one film affects its own stray\nfield, further changing the response of the other magnet.\nAdditionally, in our films the domain walls are confined\nto a much smaller space with a relatively small number\nof grains. Our interest lies in if and how these differences\naffect the size and duration statistics of the domain wall\njumps.\nIII. SIMULATION SETUP\nWe simulate polycrystalline thin films in relative mo-\ntion using micromagnetic simulations. Our simulation\nsetup consists of two thin films, the upper of which\nis driven towards the +x-direction with a constant\nvelocityvwhile the lower film is held in place. The equa-\ntion of motion for the moving film is solved simultane-\nously with magnetization dynamics, which are governed\nby the Landau-Lifshitz-Gilbert equation\n∂m\n∂t=−γ0\n1 +α2/parenleftbig\nHeff×m+αm×(m×Heff)/parenrightbig\n,(1)\nwhereγ0is the product of electron gyromagnetic ratio γ\nand the permeability of vacuum µ0,mis the normalized\nmagnetization vector, Heffis the effective field and αis\nthe phenomenological Gilbert damping constant. The\neffective field takes into account the exchange interac-\ntion, magnetocrystalline anisotropy, the demagnetizing\nfield and the external field, the last of which was ab-\nsent in the simulations of this work. We use micromag-\nnetic solver Mumax329augmented with our smooth mo-\ntion package30to simultaneously solve the motion and\nmagnetization dynamics.\nThe micromagnetic parameters were chosen to rep-\nresent a hard, uniaxial material, with CoCrPt-like31\nparameters Msat= 300kA/m, Aex= 10−12J/m,3\nKu1Ku2c) z\nθ\nFigure 1. a)The two films and the initial magnetization\nconfiguration (periodic images not shown). The color wheel\nshows the orientation of the magnetization in-plane, whereas\nblack and white correspond to −zand +z-directions, respec-\ntively. b)The schematic depiction of the Voronoi tessel-\nlated grains of the two films. c)The realization of the disor-\nder: anisotropy vectors with randomized deviations from the\nz−axis in a pair of grains.\nKu= 200kJ/m3andα= 0.01. The uniaxial anisotropy\neasy axis points along the z−axis (out of film plane).\nThe films were 20 nm thick and 424 nm long (though\nthe periodicity makes them effectively infinite), with\nthe film width being one of the studied variables in\nthis study. The simulation domain was discretized into\n4 nm×4 nm×4 nm cells, and the distance between\nthe films was set to 5 cells (20 nm), so that they can be\nconsidered to interact only via the demagnetizing field.\nWe ignore thermal effects, running the simulations in\n0 K temperature.\nThe initial magnetization is a simple structure of two\ndomains, starting aligned in both films, with one do-\nmain having +z-directional magnetization and the other\ndomain having−z-directional magnetization (Fig. 1 a).\nThe simulation volume is periodic in the driving direc-\ntion, resulting in two Bloch domain walls between the\ntwo domains.\nIn this paper we focus on polycrystallinity, ignoring\nother forms of lattice irregularities that could cause pin-\nning. The films were made polycrystalline by dividing\nthem into grains with Voronoi tessellation32(Fig. 1 b).\nAs the theoretical estimate for the domain wall width us-\ning the aforementioned material parameters is approxi-\nmatelyldw=π/radicalbig\nAex/Ku≈22nm, we simulate the films\nwith three average grain diameters /angbracketleftD/angbracketright, 10 nm, 20 nm\nand 40 nm, taking into account the theoretical consid-\neration of strongest pinning being found with grain size\nroughly equal to the domain wall width.\nThere are multiple ways to realize the disorder in the\ntessellated films, e.g. weakening the exchange interac-\ntionbetweengrainsorchangingmaterialparameterssuch\nasMsatwithin the individual grains. In our system,\nthe magnetocrystalline anisotropy is the dominating en-\nergy term, and thus we chose to simulate the disorder\nby deviating the direction of the anisotropy vector from\nthez−axis by a random amount in each grain, with\n-0.10-0.050.000.050.10F\n0 100 200 300 400 500\nt [ns]\nm[nN]Figure 2. In a completely pinned system, the energy is peri-\nodically stored and released due to the domains aligning and\nmisaligning, resulting in an oscillating magnetic force being\nexerted on the films. Due to the pinning, there’s negligible\ndissipation, and the magnetic force between the films is zero\non average.\nx−andy−components ∆xand∆ydrawn from the nor-\nmal distribution with mean 0 and equal standard devia-\ntionsσx=σy=σ. The length of the anisotropy vector\nis 1 in thez−direction, and thus together the deviations\nform an angle θ= tan−1(/radicalbig\n∆x2+ ∆y2)from thez−axis\n(Fig. 1 c). After the randomized deviation, the vector is\nnormalized to unit length again to keep the magnitude\nof the anisotropy constant.\nA. The interaction of magnets and magnetic losses\nIn our simulation scenario with two films, the do-\nmain dynamics of the films couple via the demagnetizing\nfields of the mutually changing magnetization. Initially,\nthe magnets relax into an equilibrium in which the to-\ntal energy of the system is minimized, usually meaning\naligned domains in films close together. As the motion of\nthe driven upper film moves the domains within it, the\n+zand−zdomains of the stationary and moving film\nbecome misaligned, increasing stray field energy. The re-\nsulting dynamics depends on how strongly the domain\nwalls are pinned in the films.\nIn the case of weak pinning, the stray field of the mag-\nnets exceed the depinning field, and the domain walls\nin the films tend to match positions. Depending on the\nrandomgrainpatternanddisorderstrengthinthegrains,\nthis means that the domain walls either stay still on av-\nerage (stationary film has stronger pinning) or move to-\nwardsthedrivingdirectionatafractionofthedrivingve-\nlocity (moving film has stronger pinning). Contrariwise,\nif the pinning is strong, the stray field is not enough to\ndepin the domain wall(s), in which case the domain walls\nstay misaligned when the moving film is driven forward,\nincreasing stray field energy. In this case, the resulting4\nenergy gradient tries to drive the displaced film back to\nit’s original location. The magnetic force acting on the\nmoving film is determined by\nFm=µ0VcellMsat/summationdisplay\ni∈{u}∇(mi·Hi\nl),(2)\nwhereµ0is the permeability of vacuum, Vcellis the dis-\ncretization cell volume, Msatis the saturation magnetiza-\ntion,{u}denotes the discretization cells belonging to the\nupper (moving) film, miis the magnetic moment vector\nin discretization cell iandHi\nlis the demagnetizing field\nof the lower (stationary) film acting on cell i. In a com-\npletely pinned system the magnetic force oscillates in-\ndefinitely (Fig. 2) due to the periodic misalignment and\nrealignment of the up and down domains.\nThough Fmresists the motion of the moving film, it\nis not dissipative per se, and thus is not contributing\nto the magnetic losses and thus magnetic friction. The\nmagnetic losses mainly originate from the pinning and\ndepinning of domain walls, i.e. hysteresis losses33. In\nthe micromagnetic picture, the energy dissipation comes\nfrom the relaxation of the magnetic moments according\nto the LLG equation after a domain wall jump. The\nequation for the power dissipation can be derived with\nthe help of the LLG equation34,\nP=αµ0γMsatVcell\n1 +α2N/summationdisplay\ni=1/parenleftbig\nmi×Heff,i/parenrightbig2,(3)\nwhere miandHeff,idenote the local magnetization and\neffective field in discretization cell i, respectively, and N\nis the total number of discretization cells.\nThe average friction force can be calculated from the\npower dissipation divided by the velocity of the moving\nfilm/angbracketleftFfric/angbracketright=/angbracketleftP/angbracketright/v. Unless otherwise stated, we use\ndriving velocity v= 1m/s, so that the average dissipa-\ntion power indicates also the magnitude of the average\nfriction force.\nIV. RESULTS AND DISCUSSION\nA. Domain dynamics and energy dissipation\nWe first charted the domain wall dynamics qualita-\ntively using a film of small width ( w= 140nm) and av-\nerage grain size/angbracketleftD/angbracketright= 20nm, varying the standard devi-\nationσto examinehowthe randomness inthe anisotropy\nvector deviations affected the domain wall dynamics. For\nconvenience, we use a single measure for the strength\nof the disorder, θσ, defined as the angle in which both\n∆xand∆yare equal to one standard deviation σ,\nθσ= tan−1(/radicalbig\nσ2+σ2) = tan−1(√\n2σ).\nWefoundthatforlowvaluesofdisorder, θσ<2◦,thepin-\nning is weak, resulting in mostly smooth changes in mag-\n0 4 8 12 16\nθ σ[°]04080120P [pW]D = 10 nm\nD = 20 nm\nD = 40 nmFigure 3. The magnetic losses as a function of θσ, the maxi-\nmumangledeviationfromthe z-axisfortheanisotropyvector,\nfor the three different grain sizes. The results for the largest\nvalues ofθσare more noisy due to the pinning depending\nstrongly on the random grain configuration.\n100 0 20 40 60 80\nt  0300600900120015001800P [pW]θ σ ≈ 1\nθ σ ≈ 8\nθ σ ≈ 16\n[ns]\nFigure 4. The energy dissipation for the first 100 ns for three\ndifferent disorder strengths θσ, showing the largest and most\nfrequent avalanches for a system with medium strength disor-\nder. The system is 600 nm wide with average grain diameter\n/angbracketleftD/angbracketright= 20nm.\nnetization, with the domain walls moving almost conti-\nnously in response to the driving. In the regime of larger\ndeviations ( 2◦< θσ<16◦), the dynamics consists of\navalanche-like bursts of motion of the domain walls, in-\nterspersed with periods of negligible activity due to the\ndomain walls being pinned. The extent of pinning and\nthe sizes of individual avalanches depend on the strength\nof the pinning, with a further increase in θσtypically\nresulting in the domain walls not depinning at all after\nsome initial reconfiguration. In this case the magnetiza-5\ntion is completeley rigid, eliminating magnetic losses and\nthus magnetic friction.\nBased on these qualitative observations, we simulated\nθσvalues from 1◦to16◦for the three different grain sizes\n(Fig. 3). For these simulations, we used 600 nm wide\nfilms so that the films can fit a large number of grains\nalong the domain wall, mitigating the random noise in\nthe results. The results are also averaged over several\nrandom realizations of the grain structure. Examples of\nthe dissipation signal in single simulations with varying\nstrengths of disorder are shown in Fig. 4.\nAt small disorder strengths, there’s very little domain\nwall pinning with all grain sizes, and thus the grain size\nhas a negligible effect on the average energy dissipation,\nthe magnitude of which was roughly /angbracketleftP/angbracketright= 10−30pW,\nwhich is still quite high compared to purely monocrys-\ntalline systems35. When the domain walls begin to pin\nmore strongly, the effect of grain size becomes more\npronounced. As expected from the initial simulations,\nfor the average grain diameter /angbracketleftD/angbracketright= 20nm, the lowest\nand highest θσvalues both show diminishing magnetic\nlosses, due to either having practically no pinning at all\n(θσ≤1◦) or mostly pinned domain walls ( θσ≥16◦).\nAs can be seen from Fig. 4, the few avalanches that oc-\ncur at the strongest disorder are sharp and short, with\nlong downtimes in between. The peak dissipation, where\nthe average dissipation power and thus friction force is\nroughly an order of magnitude stronger, was found to lie\natroughlyinthemiddleofthetwoextremes, θσ= 7◦−9◦.\nIn this regime, the magnetization dynamics is governed\nby individual parts of the domain wall undergoing inter-\nmittent wide avalanches. The average friction force in\nthis regime,/angbracketleftFfric/angbracketright≈0.1nN, is quite high compared to\nforces usually encountered in non-contact friction36.\nFor/angbracketleftD/angbracketright= 10nm, we observe the previously-discussed\naveraging13effect due to having smaller grain size\nthan the domain wall width, and thus lowered effective\nanisotropy. While not exactly half, the measured dissipa-\ntion for small θσis much lower, and the dissipation peak\nis found at almost double the angle deviation compared\nto/angbracketleftD/angbracketright= 20nm, since at this point the larger deviations\nin the anisotropy direction balance out the averaging due\nto grain size. The overall low dissipation with the largest\ngrain size/angbracketleftD/angbracketright= 40nm is likely a combination of the\neasier switching of each grain and the lowered pinning\nvolume due to the films still not being large enough to\naccommodate many grains along the direction of the do-\nmain wall ( y−direction). The dissipation for the largest\ngrain size also display the most noise, since the small\nnumber of grains leads to the results depending more on\nthe grain configuration.\nBeing interested in how width of the film wcompared\nto the average grain size affects the dissipation, we varied\nthe width of the films in the y−direction from 20 nm up\nto 1.2µm. The energy dissipation of the system as a\nfunction of the width is depicted in Fig. 5. In these sim-\nulations, we used θσ= 8◦, and since the angle is not at\nthe peak of dissipation for 10 nm and 40 nm grains, the\n0 400 800 1200\nw [nm]050100150200P [pW]D = 10 nm\nD = 20 nm\nD = 40 nm µW/m\nk = 57 ± 20µW/m\nµW/m\nk = 78 ± 4k = 86 ± 7Figure 5. The average dissipation power as a function of film\nwidthwwithθσ= 8◦and driving velocity v= 1m/s. For\n/angbracketleftD/angbracketright= 10nm and/angbracketleftD/angbracketright= 20nm, the curve becomes linear\nwhen the size of the film exceeds roughly 15 times the average\ngrain size. The result is likely similar for largest grain size,\nbut due to noise the linearity isn’t as clear.\n0 2 4 6 8 10\nv [m/s]020040060080010001200〈P〉 [pW]〈D〉  = 10 nm\n〈D〉  = 20 nm\n〈D〉  = 40 nm\n〈Ffric〉  = 110 pN\n〈Ffric〉  = 76 pN\n〈Ffric〉  = 46 pN\nFigure 6. The average dissipation power as a function of ve-\nlocity with θσ= 8◦andw= 600nm. The relationship is\nlinear for all grain sizes, implying that the average friction\nforce is independent of velocity in this range.\nmagnetic losses with these grain sizes tend to be consis-\ntently below that of the /angbracketleftD/angbracketright=20 nm films for larger\nvalues ofw. When the film can accommodate only a\nfew grains along the domain wall, as is the case for the\nsmallest widths, the dissipation is quite random, likely\ndepending strongly on the random pattern of grains and\nanisotropy vector angles. This also causes the smallest\ngrain size/angbracketleftD/angbracketright= 10nm to have the strongest dissipa-\ntion initially, mostly because having multiple grains in\ny−direction and relatively weaker pinning makes it more\nlikely for the domain wall to actually depin and dissipate\nenergy, whereas smaller number of larger grains have a6\nhigher chance to cause complete pinning of the domain\nwall. For/angbracketleftD/angbracketright= 40nm in particular, the films on the\nsmaller side contain only one or two grains along the do-\nmain wall.\nAfter a certain point, seemingly about 14 - 20 grains\nfitting in the width of the film, the dissipation becomes\nmore regular and starts to grow approximately linearly\nwith size. This makes sense, as the domain wall jumps\noccurring in the medium disorder regime are mainly re-\nsponsible for the dissipation, and the domain wall ex-\ntends across the film in the y−direction and thus scales\ndirectly with the film width. The slopes of the linear por-\ntions for each grain are very close within error margins\n(k≈75−85µW/m), though with the largest grain size\n/angbracketleftD/angbracketright=40 nm, the curve has significantly more noise and\nthe linearity seems to appear quite late. In the linear\nregime, only parts of the domain walls usually depin and\nmove at a time, whereas in low width films, the small\namount of grains tends to increase the likelihood of film-\nwide jumps, in which the domain walls pin and depin\ncompletely at once. This results in the nonlinear (and\nnoisy) initial growth of the dissipation power with size.\nFinally, we investigated the effect of the driving veloc-\nity on the average power dissipation. It turns out that\nfor velocities ranging between 0.2 m/s and 10 m/s, the\nrelationship between the velocity and dissipation power\nis linear for all grain sizes (Fig. 6), meaning that the\naverage friction force /angbracketleftFfric/angbracketright=/angbracketleftP/angbracketright/vis independent of\nvelocity in this range. The result is analogous to hys-\nteresis losses in hysteresis loop experiments, in which the\npower dissipation has been found linearly dependent on\nthe applied field frequency37. The reason for the lin-\near dependence is that the increase in frequency reduces\ndowntime between individual domain wall jumps with-\nout significantly affecting the jumps themselves. In our\nsetup, an increase in driving velocity has a similar effect.\nHigh velocities have the magnetic film in a near constant\nstate of excitation due to new avalanches starting before\nprevious ones have stopped.\nVelocities significantly exceeding 10 m/s can result in\nthe domain pattern breaking down, resulting in single-\ndomain films and negligible dissipation. Thus the ob-\nserved linear relationship can break at high velocity. Go-\ning to much lower velocities is impractical with micro-\nmagnetic simulations as the simulation times grow con-\nsiderably and the results become noisy due to only a few\navalanches occurring, requiring more averaging.\nB. Domain wall roughness and avalanche statistics\nTo acquire sufficient statistics about the domain wall\navalanches, a large number of relatively long simulations\nis required. As such, we study the avalanche statistics\nusing average grain size of /angbracketleftD/angbracketright= 20nm withθσ= 8◦,\nsince these parameters results in most avalanches based\non the earlier simulations. We limited the study to the\neffects of film width and velocity on the avalanche statis-\n0 50 100 150 200\nt [ns]0100200300400500600700P [pW]T\nSFigure 7. An example of a power dissipation signal found\nin this study, with multiple avalanche-like bursts of magnetic\nactivity. The size (S) of an avalanche is defined as the total\namount of energy dissipated and the duration (T) as the to-\ntal time the signal stays over the avalanche threshold (red\nline). This particular case was simulated with film width\nw= 400nm and driving velocity v= 1m/s.\n1 10 100\nL [nm]0110100[u( +L)-u( )]2\n2ζ= 1.36y y\nFigure 8. The correlation function of domain wall displace-\nments in a 800nm wide system with /angbracketleftD/angbracketright= 20nm grains,\nused to find the roughness exponent ζ≈0.68±0.05.\ntics, using three different values for the width, 200 nm,\n400 nm and 800 nm, and three velocities, 1 m/s, 3 m/s\nand 5 m/s.\nTo capture the avalanches from the power dissipa-\ntion signal P(t)(Fig. 7), a threshold needs to be set to\ndifferentiate between an avalanche and a pinned state.\nDepending on the threshold, a single avalanche could\nbe split into multiple sub-avalanches, thus changing the\nshape of the distribution. We found that having a\nthreshold of 0.3/angbracketleftP/angbracketrightresulted in a suitable amount of\navalanches without incurring significant splitting. We\ncut off avalanches smaller than 0.1 aJ, since smaller7\n0.1 1.0 10.0\nS [aJ]10141015101610171018P(S) [arb. units]S= 0.70τ\n0.1 1.0 10.0\nS [aJ]10141015101610171018P(S) [arb. units]v = 1 m/s\nv = 3 m/s\nv = 5 m/s\n1 10 100\nT [ns]P(T) [arb. units]v = 1 m/s\nv = 3 m/s\nv = 5 m/s\n0 10 20 30 40 50 60\nT [ns]P(T) [arb. units]468×107\n10\n2\n0468×107\n10\n2\n0a) b)\nFigure 9. a)The avalanche size distribution for a w= 800nm film, with a power law fitted via the maximum likelihood method.\nThe inset shows how increasing velocity softens the cutoff due to more avalanche overlap resulting in larger avalanches. b)The\navalanche duration seemingly follows the log-normal distribution. Larger velocities result in more broad distribution and the\nshifting of the peak to lower values.\navalanches were usually the result of the noise in the sig-\nnal just momentarily crossing the avalanche threshold.\nA relevant quantity related to elastic interfaces moving\nin a disordered medium, such as the domain walls in our\ncase, is the interface roughness exponent ζ. Assuming\nnon-anomalous scaling38, the roughness exponent can be\nfoundthroughthe displacement-displacement correlation\nfunctionofthedomainwalldisplacement u(y)perpendic-\nular to the wall,/angbracketleft[u(y+L)−u(y)]2/angbracketright∝L2ζ, whereLis the\ndistance between two points of the domain wall. Taking\nsnapshots of a single domain wall in the upper film after\n60 ns of driving, averaged over 5 random realizations of\nthe disorder, we find ζ≈0.68±0.05(Fig. 8) for the three\nfilm widths.\nThe avalanche size and duration distributions from\nsimulations with film width w= 800nm and driving\nvelocityv= 1m/s are depicted in Fig. 9. The size distri-\nbution resembles a power law P(S)∝S−τSover roughly\none decade S=0.1 aJ - 2 aJ, after which there’s a cut-\noff. A likely source for the cutoff is the fact that the stray\nfields of the films attempt to align the domain wall loca-\ntions in the upper and lower films to minimize the stray\nfield energy, meaning that it is difficult for the domain\nwall experiencing an avalanche to jump past its corre-\nsponding domain wall in the other film. The extent of\nthe jump is thus limited by the domain wall width and\nthe grain size in the direction perpendicular to the wall.\nFitting a power law using the maximum likelihood\nmethod39with the NCC toolbox40, we find a size expo-\nnentτS≈0.70. Similar exponents have been predicted\nfor weak-field driven, thermally activated avalanches in\nthe domain wall creep regime24,26. The avalanche size\nexponent for systems with short-range disorder can also\nbe estimated theoretically from the dimensionality of theinterfacedand the roughness,\nτS= 2−2\nd+ζ.\nIn our case, the roughness and dimensionality ( d= 1)\npredictτS≈0.77−0.84forthesizedistributionexponent,\nmatching quite well to the simulation results.\nInterestingly, the avalanche duration distribution in\nour simulations seemingly follows a log-normal distribu-\ntion instead of a power law (Fig. 9 b). The best fit\nto the results gives parameter values σ= 0.6182and\nµ=−18.52with a mean avalanche duration /angbracketleftT/angbracketright=\nexp(µ+σ2/2)≈11ns. A possible reason for this form of\nthe distribution is similar to the cut-off of the size distri-\nbution, in that the time in which the avalanche relaxes is\nroughly independent of the lateral size of the avalanche,\nand thus the duration of the jumps is mainly determined\nof the forward motion of the domain walls which is lim-\nited by the stray fields of the films.\nIn a similar fashion to what was observed for the av-\nerage energy dissipation, increasing the driving veloc-\nity reduces the downtime between individual avalanches,\nthough the avalanches themselves are not strongly af-\nfected. However, distinguishing between individual\navalanches becomes a challenge, as new avalanches tend\nto start before the system has relaxed. Thus the velocity\nhas a visible effect on the avalanche size and duration\ndistributions, illustrated in the insets of Figs. 9 aandb.\nHighvelocitiesdidn’tsignificantlyaffecttheexponent τS,\nbuttheincreasedoverlapintheavalanchesresultedinthe\ncutoff having a slightly softer falloff. The duration distri-\nbution becomes broader when velocity is increased, with\nboth short and long avalanches becoming more common,\nlikely due to the increased noise resulting in momentary\ncrossings of the avalanche threshold and the overlapping8\n1.0 10.0 100.0\nT [ns]2468×107\n10 P(T) [arb. units]w = 200 nm\nw = 400 nm\nw = 800 nm\n0\n0.1 1.0 10.0\nS [aJ]10141015101610171018P(S) [arb. units]S= 0.70τ\n0.1 1.0 10.0 100.0\nS [aJ]101410161018P(S) [arb. units]w = 200 nm\nw = 400 nm\nw = 800 nma) b)\nFigure 10. a)The avalanche size distribution as a function of film width. System-wide avalanches contribute to the sizes\nbeyond the cut-off for w= 200nm.a)The avalanche duration distribution as a function of film width. Unlike in the case of\nvelocity, there’s no visible shift in the peak of the distribution, though larger film widths display a somewhat similar widening\nof the distribution as was observed at higher velocities.\navalanches combining the duration of multiple individ-\nual avalanches. However, the average avalanche duration\nremained approximately the same.\nThe film width had a lesser influence on the avalanche\nsize and duration distributions. The only discernible ef-\nfects were the increase of the number of large ( Saround\nthe cutoff) avalanches with the smallest film width of\n200 nm and a small widening of the duration distribu-\ntion with size, shown in Figs. 10 aand b. The sub-\nstantial increase in avalanche sizes in the smallest sys-\ntem is likely explained by the film-wide avalanches still\noccurring relatively often at this width. It’s possible that\nthe system-wide avalanches are also distributed accord-\ning to a power-law, but the limited amount of avalanche\ncounts make this difficult to ascertain. The size distri-\nbution for 400 nm and 800 nm wide films were almost\nidentical, showing that the avalanche size cutoff does not\nscale indefinitely with the system width, thus enabling\nthe normalization of the size distribution and the ob-\nserved lowτS<1.\nV. CONCLUSION\nWe simulated the interaction of two polycrystalline\nthin films in relative motion, investigating how the av-\nerage energy dissipation is influenced by the disordered\nstructure, determined by the grain diameter and the\nstrength of the disorder, along with external parameters\nsuch as film width and the driving velocity. We also stud-\nied the size and duration distributions of domain wall\njumps in this two-film system, and how the distributions\nare affected by the width of the film and the driving ve-\nlocity.Our results for the average energy dissipation indicate\nthat the magnetic losses and thus magnetic friction are\nat their highest when the domain walls of the system\nare strongly but not completely pinned, such that the\nstrength of the stray field of the films is just enough to\ndepin the domain walls consistently. In this regime, the\naverage losses were roughly an order of magnitude higher\nthan either mostly freely moving or very strongly pinned\ndomain walls. The domain wall motion is characterized\nby frequent jumps, in which parts of the domain wall\nexperience avalanches roughly independently, the peak of\nthe dissipation depending on the combination of average\ngrain size and the strength of the disorder. The energy\ndissipation was found to be linearly proportional to the\nsliding velocity, a result akin to hysteretic losses arising\nfrom domain wall jumps in a magnetization process. The\ndissipation also scales linearly with film width, provided\nthat length scales above a certain grain size dependent\nwidth are considered.\nThe domain wall avalanches were observed to be ini-\ntiated by the misalignment of the up and down domains\nin the films, which increased the stray field energy un-\ntil an avalanche realigned the domain walls. The sizes\nof the domain wall jumps seemingly followed a power-\nlaw+cutoff distribution with a relatively small exponent,\na value similar to what has been found for thermally ac-\ntivated domain wall creep, while the avalanche duration\ndistribution followed a log-normal distribution, presum-\nably due to the limited extent of the avalanches. The\navalanche size exponent predicted from the roughness\nand dimensionality of the wall agreed quite well with the\nexponent obtained from the simulations. Aside from a\nminor increase in the number of largest avalanches, the\nfilm width and the driving velocity did not significantly\nalter the avalanche distributions.9\nOverall, our results reveal intriguing physics arising\nfrom the coupled collective dynamics of interacting do-\nmain walls, resulting in bursty magnetic non-contact fric-\ntion with a relatively large magnitude. The domain wall\ninteraction as a driving force for the avalanches produced\natypical, non-criticalstatistics, meritingfurtherstudyre-\ngarding the domain wall and disorder characteristics and\ntheir relation to the avalanche distributions.ACKNOWLEDGMENTS\nWe acknowledge the support of the Academy of Fin-\nland via an Academy Research Fellowship (LL, projects\nno. 268302 and 303749), and the Centres of Excellence\nProgramme (2012-2017, project no. 251748). We ac-\nknowledge the computational resources provided by the\nAaltoUniversitySchoolofScienceScience-ITprojectand\nthe CSC.\n∗ilari.rissanen@aalto.fi\n1A. Moses, “Advanced soft magnetic materials for power\napplications,” (2007).\n2T. Markvart and L. Castañer, in Solar Cells (Second Edi-\ntion), edited by A. McEvoy, L. Castañer, and T. Markvart\n(Elsevier, 2013) second edition ed., pp. 27 – 54.\n3G. Herzer, IEEE Transactions on Magnetics 26, 1397\n(1990).\n4G. Durin and S. Zapperi, in The Science of Hysteresis ,\nedited by G. Bertotti and I. D. Mayergoyz (Academic\nPress, Oxford, 2006) pp. 181 – 267.\n5M. Fähnle and D. Steiauf, “Dissipative magnetization dy-\nnamicsclosetotheadiabaticregime,”in Handbook of Mag-\nnetism and Advanced Magnetic Materials (American Can-\ncer Society, 2007).\n6J. M. D. Coey, Magnetism and Magnetic Materials (Cam-\nbridge University Press, 2010).\n7D. C. Meeker, A. V. Filatov, and E. H. Maslen, IEEE\nTransactions on Magnetics 40, 3302 (2004).\n8M.Desvaux,H.Bildstein,B.Multon,H.B.Ahmed,S.Sire,\nA. Fasquelle, and D. Laloy, in 2018 Thirteenth Interna-\ntional Conference on Ecological Vehicles and Renewable\nEnergies (EVER) (2018) pp. 1–7.\n9J. Lee, Y. Kim, S. Rhyu, I. Jung, S. Chai, and J. Hong,\nIEEE Transactions on Magnetics 48, 935 (2012).\n10A. Hubert, R. Schäfer, and A. Rudolf Schafer, Mag-\nnetic Domains: The Analysis of Magnetic Microstructures\n(Springer, 1998).\n11B. D. Cullity and C. D. Graham, Introduction to Magnetic\nMaterials , 2nd ed. (Wiley-IEEE Press, 2008).\n12W. Rave and K. Ramst ´’ock, Journal of Magnetism and\nMagnetic Materials 171, 69 (1997).\n13J. Ping Liu, O. Gutfleisch, E. Fullerton, and D. Sellmyer,\nNanoscale magnetic materials and applications (Springer\nUS, 2009).\n14J. Sethna, K. Dahmen, and C. Myers, Nature 410, 242\n(2001).\n15A. Vespignani, H. Stanley, and S. Zapperi, Nature 388,\n658 (1997).\n16G. Durin and S. Zapperi, Physical review letters 84, 4705\n(2000).\n17S. Zapperi, P. Cizeau, G. Durin, and H. E. Stanley, Phys.\nRev. B 58, 6353 (1998).\n18T. Herranen and L. Laurson, Phys. Rev. Lett. 122, 117205\n(2019).\n19J. A. J. Burgess, A. E. Fraser, F. F. Sani, D. Vick, B. D.\nHauer, J. P. Davis, and M. R. Freeman, Science 339, 1051\n(2013).\n20S.Papanikolaou,F.Bohn,R.L.Sommer,G.Durin,S.Zap-peri, and J. P. Sethna, Nature Physics 7, 316 (2011).\n21A.BenassiandS.Zapperi,Phys.Rev.B 84,214441(2011).\n22R. Ranjan, D. C. Jiles, O. Buck, and R. B. Thomp-\nson, Journal of Applied Physics 61, 3199 (1987),\nhttps://doi.org/10.1063/1.338900.\n23S. Lemerle, J. Ferré, C. Chappert, V. Mathet, T. Gia-\nmarchi, and P. Le Doussal, Phys. Rev. Lett. 80, 849\n(1998).\n24M. P. Grassi, A. B. Kolton, V. Jeudy, A. Mougin,\nS. Bustingorry, and J. Curiale, Phys. Rev. B 98, 224201\n(2018).\n25L. D. Geng and Y. M. Jin, EPL (Europhysics Letters) 116,\n36002 (2016).\n26E. E. Ferrero, L. Foini, T. Giamarchi, A. B. Kolton, and\nA. Rosso, Phys. Rev. Lett. 118, 147208 (2017).\n27V. Repain, M. Bauer, J.-P. Jamet, J. Ferré, A. Mougin,\nC. Chappert, and H. Bernas, Europhysics Letters (EPL)\n68, 460 (2004).\n28P. Le Doussal, A. A. Middleton, and K. J. Wiese, Phys.\nRev. E 79, 050101(R) (2009).\n29A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen,\nF. Garcia-Sanchez, and B. Van Waeyenberge, AIP Adv.\n4, 107133 (2014).\n30I. Rissanen and L. Laurson, Phys. Rev. E 97, 053301\n(2018).\n31D. Weller, A. Moser, L. Folks, M. Best, W. Lee, M. Toney,\nM. F. Schwickert, J.-U. Thiele, and M. Doerner, Magnet-\nics, IEEE Transactions on 36, 10 (2000).\n32J. Leliaert, B. Van de Wiele, A. Vansteenkiste, L. Laur-\nson, G. Durin, L. Dupre, and B. V. Waeyen-\nberge, Journal of Applied Physics 115, 17D102 (2014),\nhttps://doi.org/10.1063/1.4854956.\n33G. Bertotti and F. Fiorillo, in Reference Module in Mate-\nrials Science and Materials Engineering (Elsevier, 2016).\n34M. P. Magiera, L. Brendel, D. E. Wolf, and U. Nowak,\nEPL (Europhysics Letters) 87, 26002 (2009).\n35I. Rissanen and L. Laurson, “Magnetic non-contact fric-\ntion from domain wall dynamics actuated by oscillatory\nmechanical motion,” (2018), arXiv:1809.07130.\n36E. Gnecco and E. Meyer, Elements of Friction Theory and\nNanotribology (Cambridge University Press, 2015).\n37G. Bertotti, in Encyclopedia of Materials: Science and\nTechnology , edited by K. J. Buschow, R. W. Cahn, M. C.\nFlemings, B. Ilschner, E. J. Kramer, S. Mahajan, and\nP. Veyssiére (Elsevier, Oxford, 2001) pp. 4798 – 4804.\n38J. J. Ramasco, J. M. López, and M. A. Rodríguez, Phys.\nRev. Lett. 84, 2199 (2000).\n39A. Clauset, C. Shalizi, and M. Newman, SIAM Review\n51, 661 (2009), https://doi.org/10.1137/070710111.10\n40N. Marshall, N. M. Timme, N. Bennett, M. Ripp, E. Lautzenhiser, and J. M. Beggs, Frontiers in Physiol-\nogy7, 250 (2016)."    },    {        "title": "2209.10567v1.Dynamics_of_photo_induced_ferromagnetism_in_oxides_with_orbital_degeneracy.pdf",        "content": "Dynamics of photo-induced ferromagnetism in oxides with orbital degeneracy\nJonathan B. Curtis,1, 2,\u0003Ankit Disa,3, 4Michael Fechner,4Andrea Cavalleri,4, 5and Prineha Narang1,y\n1College of Letters and Science, University of California, Los Angeles, CA 90095, USA\n2Department of Physics, Harvard University, Cambridge, MA 02138, USA\n3Applied and Engineering Physics, Cornell University, Ithaca, NY 14853, USA\n4Max Planck Institute for the Structure and Dynamics of Matter, Hamburg, DE\n5Clarendon Laboratory, Department of Physics, Oxford University, Oxford, UK\n(Dated: September 23, 2022)\nBy using intense coherent electromagnetic radiation, it may be possible to manipulate the prop-\nerties of quantum materials very quickly, or even induce new and potentially useful phases that are\nabsent in equilibrium. For instance, ultrafast control of magnetic dynamics is crucial for a number of\nproposed spintronic devices and can also shed light on the possible dynamics of correlated phases out\nof equilibrium. Inspired by recent experiments on spin-orbital ferromagnet YTiO 3we consider the\nnonequilibrium dynamics of Heisenberg ferromagnetic insulator with low-lying orbital excitations.\nWe model the dynamics of the magnon excitations in this system following an optical pulse which\nresonantly excites infrared-active phonon modes. As the phonons ring down they can dynamically\ncouple the orbitals with the low-lying magnons, leading to a dramatically modi\fed e\u000bective bath\nfor the magnons. We show this transient coupling can lead to a dynamical acceleration of the mag-\nnetization dynamics, which is otherwise bottlenecked by small anisotropy. Exploring the parameter\nspace more we \fnd that the magnon dynamics can also even completely reverse, leading to a nega-\ntive relaxation rate when the pump is blue-detuned with respect to the orbital bath resonance. We\ntherefore show that by using specially targeted optical pulses, one can exert a much greater degree\nof control over the magnetization dynamics, allowing one to optically steer magnetic order in this\nsystem. We conclude by discussing interesting parallels between the magnetization dynamics we \fnd\nhere and recent experiments on photo-induced superconductivity, where it is similarly observed that\ndepending on the initial pump frequency, an apparent metastable superconducting phase emerges.\nI. INTRODUCTION\nThe idea of using strong optical \felds to gain control\nover phases of quantum matter is a tantalizing one. Op-\ntical control of ferroelectric [1, 2], structural [3{8], super-\nconducting [9{15], charge density wave [16{18], and mag-\nnetic orders [19{34] have all been proposed theoretically\nor demonstrated experimentally. Essentially, by inducing\nstrongly nonequilibrium scenarios, one can explore an en-\nlarged nonequilibrium phase diagram which may allow\nfor the access of novel phenomena and functionalities.\nOne way to realize such strongly nonequilibrium sce-\nnarios is by resonantly driving the system, inducing co-\nherent oscillations in the Hamiltonian, thereby breaking\ntime-translation symmetry and driving the system away\nfrom the thermal regime. These coherent oscillations\ncan potentially excite parametric resonances [35{39], in-\nduce novel topology [40{46], produce tunable interac-\ntions [30, 47{49], generate non-thermal correlations [50],\nand even lead to e\u000bective cooling mechanisms [51{53].\nIn particular, the dynamics of correlated Mott insulators\nhosting orbital degrees of freedom [54] is known to be\nquite rich, exhibiting spin-orbital separation [55, 56], tun-\nable exchange [57, 58], and hidden phases [59] Similarly,\nferromagnets with high levels of spin-rotation symmetry\ncan exhibit many exotic phenomena away from equilib-\n\u0003joncurtis@ucla.edu\nyprineha@ucla.edurium [60{62].\nThe interplay between spin and orbital \ructuations\ncan lead to dramatic e\u000bects including magnon soften-\ning [63], entangled spin-orbital phases [64{71], and pro-\nnounced magnetic \ructuations [72, 73] and subsequent\nphase transition [74{76]. Inspired by recent experiments\non ferromagnetic Mott-insulator YTiO 3(YTO) [19], we\nexamine the nonequilibrium dynamics of magnons in a\nmodel quasi-degenerate orbital system driven out of equi-\nlibrium by coherently oscillating optical phonons.\nIn YTO the 3 d1conduction band is formed from the\ntitaniumt2gshell which naively has a three-fold degen-\neracy enforced by a cubic lattice symmetry [66]. How-\never, in YTO and many other compounds, this cubic\nsymmetry is broken at low temperatures by a GdFeO 3-\ntype structural distortion, which then lifts the resulting\norbital degeneracy [77{83]. In this case the system has\nnon-degenerate, but potentially low-lying orbital excita-\ntions [70, 84{88]. In equilibrium settings, the lifted or-\nbital degeneracy leads to a decoupling between spin and\norbital excitations and for most magnetic purposes their\ncross-coupling can be ignored.\nCan these orbital excitations, which are essentially ab-\nsent from equilibrium processes, signi\fcantly modify the\nout-of-equilibrium dynamics? This question is not purely\nacademic; the ability to control magnetic order on ul-\ntrafast timescales [21, 25, 28, 49] is crucial for many\nspintronic technologies, and may also help design better\nferromagnets which can operate at higher temperatures\nmore e\u000eciently. We answer this question in the a\u000erma-\ntive, provided the orbitals are relatively low-lying andarXiv:2209.10567v1  [cond-mat.str-el]  21 Sep 20222\nmay come close in energy to relevant optically driven de-\ngrees of freedom, such as infrared-active phonons, which\ncan reside in the 1-20 terahertz regime. In this case, by\njudiciously choosing the parameters of the optical driv-\ning applied, one can speed-up, slow-down, and even re-\nverse the magnetization dynamics. This then paves the\nway for novel control routes in quasi-degenerate spin-\norbital systems [19, 89] as well as potentially other sys-\ntems [20, 34, 49].\nThe remainder of this paper is structured as follows.\nIn Sec. II we outline the model system considered, and\nmotivate various parameter choices. Then, in Sec. III we\nshow how in equilibrium the orbitals essentially serve to\nprovide a bath for angular momentum for the magnons.\nIn Sec. IV we examine the dynamics of this system in\nequilibrium and estimate the equilibrium relaxation time-\nscale. In Sec. V we then explore the nonequilibrium dy-\nnamics of this system following a simulated impulsive\ndrive of optical phonons, presenting the main results of\nthis work. Finally, we discuss the implications of our re-\nsults in Sec. VI, where we conclude by discussing interest-\ning parallels between this system and recent experiments\non light-induced superconductivity. In Appendix A we\nshow how to map the t2gorbital levels into an e\u000bective\nangular momentum. In Appendix B we present details\non the nonequilibrium Keldysh technique as applied to\nthe Holstein-Primako\u000b spin-wave expansion, and in Ap-\npendix D we show how to reduce these equations to a\nsimple equation of motion for the magnon occupation.\nII. MODEL\nHere we introduce a simple model for ferromagnetic\nspins interacting with a quasi-degenerate orbital bath.\nThough this is inspired by YTiO 3(YTO), we emphasize\nwe are considering a more abstract model, and we expect\nour results to be relevant to other high-symmetry ferro-\nmagnetic insulators with low-lying orbital excitations. In\nparticular, we consider a single electron occupying a low-\nlyingt2gorbital manifold. In the cubic limit there is a\nlarge threefold degeneracy which can lead to pronounced\norbital \ructuations. In reality this cubic degeneracy is\nlifted by the GdFeO 3structural distortion which renders\nthe crystal structure orthorhombic and induces a \fnite\ncrystal \feld splitting \u0001 between the lowest and next-\nlowest orbitals on each Ti site. This leads to a model\nwhere each site has an orbital pseudospin-1/2 ^\u001cjin ad-\ndition to the actual electron spin ^Sjon each site.\nWe consider a three-dimensional isotropic ferromag-\nnetic Heisenberg model along side a local orbital degree\nof freedom with Hamiltonian\n^H=\u0000JX\nj;\u000e^Sj\u0001^Sj+\u000e\u0000Jz\n0X\nj;\u000e^Sz\nj^Sz\nj+\u000e\n+X\nj\u0001\n2^\u001c3\nj+\u0015^Lj\u0001^Sj:(1)\n3d1t2g|0⟩|1⟩|2⟩\n̂L=n̂τ2(a)λΓ|0⟩|1⟩(b)FIG. 1. (a) Splitting of Ti t2gshell into non-degenerate levels\nby the GdFeO 3distortion, which are then occupied by a single\nelectron giving S= 1=2. Interorbital coherences lead to angu-\nlar momentum L, which is largely quenched in equilibrium.\n(b) Focusing on the lowest two-levels j0iandj1iwe \fnd spin-\n\ripT1processes in the orbital ground-state obtained from\nvirtual orbital transitions. Spin-orbit coupling \u0015can lead to a\nsimultaneous orbital excitation along with a spin-\rip. This is\nthen followed by a spin-independent orbital decay with rate\n\u0000, shown in the level diagram. Ultimately, the decay rate is\ngoverned by the spectral overlap of the orbital bath (shown\non the left schematically) with the spin-transition, which is\nsmall leading to a long-lifetime.\nThe \frst two terms are the isotropic Heisenberg ex-\nchange, with J\u00182:75 meV [90] for the case of YTO,\nand an easy-axis exchange which is chosen to counter the\norbital bath-induced Lamb shift, leading to the renor-\nmalized spin-wave gap which for the case of YTO was\nestimated to be 0.02 meV, though the upper bound was\nqutie a bit larger, of order 0.3 meV [90]. We will con-\nsider a modestly-sized renormalized gap of \n 0= 0:1\nmeV in this work. Here jlabels the lattice sites Rjand\n\u000e=ex;ey;ezlabels the nearest-neighbors along the three\nprinciple axes.\nThe third term, involving ^ \u001c3\nj, corresponds to the local\ncrystal-\feld excitation gap. There is considerable uncer-\ntainty about the value of this parameter, with theoreti-\ncal estimates ranging from nearly zero to over 300 meV.\nIn Ref. [91] it was estimated by Raman scattering that\n2\u0001\u001850 meV, while Ref. [92] found energies closer to\n2\u0001\u0018235 meV. Using resonant inelastic x-ray scatter-\ning (RIXS), Ref. [87] found evidence for collective orbital\nexcitations with a gap of order 120 meV. We will con-\nsider \u0001 = 90 meV here, though more experiments with\ngreater resolution and sensitivity are probably needed in\nthe case of YTO. The last term describes the atomic L\u0001S\nspin-orbit coupling, which leads to a torque on the spin\nin the presence of an orbital angular momentum L.\nThe orbital angular momentum can be obtained by\nprojecting the full three-dimensional t2gangular momen-\ntum onto the lowest crystal-\feld levels, leading to the\nexpression\n^Lj=nj^\u001c2\nj: (2)3\nThe unit vector njis orthogonal to the two participat-\ning orbitals, and characterizes the \\soft\" axis for orbital\nangular momentum (see Appendix A). The operator ^ \u001c2\nj\ncharacterizes the instantaneous orbital many-body state\nand in particular ^ \u001c2\njis odd under time-reversal (which\nsquares to +1 for the L= 1 orbitals), satisfying the se-\nlection rules. This is coupled to the spin angular momen-\ntum by the atomic spin orbit interaction \u0015, which in fact\nneed not be small. Reasonable estimates place \u0015\u001815\nmeV for a light 3 dtransition metal such as Ti [82]. Fi-\nnally, we note that njwill in general point in a di\u000berent\ndirection on each of the four Ti sublattices depending on\nthe local crystal-\feld environment. For more details, we\nrefer to Appendix B. Finally, we will assume that due to,\ne.g. phonons or orbital interactions the orbital excitation\nitself obtains a \fnite T1linewidth \u0000. We estimate \u0000 \u0018\n15 meV as well, although is not known with great cer-\ntainty and may appear signi\fcantly more broad in, e.g\na two-orbital spectral function, which may appear in the\nRaman and RIXS measurements.\nWe can imagine that in the magnetically ordered\nphase, each site has a local level scheme as illustrated\nin Fig. 1(a), where we show the crystal-\feld splitting of\nthe Tit2gstates and their corresponding spin and orbital\nangular momentum. The main idea is that magnetization\ndynamics is often intrinsically slow due to the bottleneck\nassociated to transfer of spin angular momentum in to a\nbath, such as the orbital. One such route is illustrated in\nFig. 1(b), which shows how in second order perturbation\ntheory this model can give rise to a \fnite longitudinal\nmagnetization relaxation rate. We argue that the phonon\ndynamics induced by the strong optical pulse can lead to\nan acceleration of this relaxation time out of equilibrium,\nleading to the possibility of pump-enhanced magnetiza-\ntion dynamics. Note that, unlike Ref. [58] which consid-\nered the impact of dynamics on the superexchange inter-\nactions, we are more concerned here with the impact on\nthe spin-orbit coupling.\nFinally, we comment on the coupling to the drive. In\nthe experiment [19], the pump was performed using a\nmid-IR pulse which strongly couples to lattice degrees of\nfreedom, rather than e.g. an optical pulse which traverses\nthe Mott gap. This pump was tuned to be resonant with\nvarious di\u000berent infrared active phonon modes and used\nto strongly drive these vibrations. Based on ab initio cal-\nculations, we argue that one of the dominant e\u000bects of\nthis pump is a strong modulation of the crystal \feld ma-\ntrix and in particular, we \fnd that for relevant \ruences\nthis may lead to a sizeable change in the eigenvector nj.\nThis in turn leads to a dynamical modulation of the or-\nbital angular momentum Lj=nj(t)^\u001c2\nj, which will now\nacquire sidebands at twice the phonon frequency.\nWe model this by writing\nnj(t)\u0018nj+\u000enjQ2\nIR(t): (3)\nHereQIRis the generalized coordinate describing the\ninfrared-active phonon which is directly driven by light\n(in general there may be di\u000berent or multiple modeswhich are excited depending on the frequency and polar-\nization used in the pump and the absoprtion spectrum of\nthe material). The coupling to Q2\nIR(t) is due to the fact\nthat the orbital angular momentum is a Raman active\ntransition whereas the infrared active phonon is polar.\nThis is in fact very important since this will induce os-\ncillations at twice the phonon frequency. For an \n ph=\n9 THz optical phonon mode, this leads to sidebands for\nthe spin-orbit coupling at a frequency of 2\n ph\u001880 meV.\nThis comes close to the orbital resonance in this model\nat 90 meV. We will study in particular how the dynamics\ndepends on the drive frequency \n d. We now proceed to\ndetermine the equilibrium structure of this model before\nproceeding on to compute the nonequilibrium dynamics.\nIn particular, we show that the orbitals can act as a bath\nfor angular momentum for the spins even in equilibrium.\nThough we do not speci\fcally consider YTO, for cer-\ntain rough estimates of parameter values and feasibil-\nity analysis we have used ab initio calculations based on\nYTO. We estimate interactions between phonon and the\ncrystal \feld parameters by performing \frst-principles cal-\nculations in the framework of Density Functional Theory\n(DFT). All technical details are listed in Appendix C.\nOur approach is inspired by Ref. [78, 79], where we \frst\ncompute the full DFT bandstructure within the local\ndensity approximation (LDA). Using this we construct\nlocalizedt2gWannier-functions using appropriate pro-\njectors according to Ref. [93]. To estimate the modula-\ntion of the crystal \feld parameters due to the phonon\ndistortion, we performed frozen phonon computations.\nTherefore, we recalculated the electronic structure and\nWannier-functions for crystal structures which have been\nmodulated according eigenvectors of polar eigenmodes\n(QIR). Tthis allows us to estimate changes in the crystal\nHamiltonian for distinct polar distortions.\nIII. ORBITAL BATH\nWe now analyze the spin-orbit coupling in equilibrium.\nTo later accommodate the nonequilibrium calculations,\nwe will implement at the outset the Schwinger-Keldysh\nformalism for describing this system. We begin by treat-\ning the orbitals in a Gaussian approximation, valid for\nsmall orbital excitation amplitudes. For details we again\nrefer to Appendix B, though for a more complete treat-\nment we refer the reader to Ref. [94].\nIn the Keldysh formalism we have a doubling of the de-\ngrees of freedom, which can be arranged into a \\classical\"\npartSj;clcharacterizing the expectation value, and the\n\\quantum\" part Sj;qwhich characterizes the \ructuations\nabout the expectation value. Applying this formalism to\nthe spin-orbit interaction we \fnd a Keldysh action of\nSsoc=\u0000\u0015X\njZ\ndt[Ljq(t)\u0001Sj;cl(t) +Ljcl(t)\u0001Sj;q(t)]:\n(4)\nThis then appears as a contribution to a path integral4\nZ=R\nD[S;L]eiSwhich can be used to generate nonequi-\nlibrium correlation functions, such as the magnetization\nhSj(t)i.\nWe remark to the reader that in this expression, Sj\u000b(t)\n(\u000b=cl;q) should be understood as a stand-in for an ap-\npropriate representation of the spin operator in terms of\na canonical bosonic or fermionic \feld. In this work we\nwill focus on the dynamics in the ordered phase, wherein\nthe operators Sj(t) can be expanded in terms of the\nHolstein-Primako\u000b bosons perturbatively in 1 =Swhere\nSis the spin length. This is strictly valid only at low-\ntemperatures with T\u001cTCand even then, it su\u000bers from\nthe fact that in YTO S=1\n2is small. It therefore remains\nan important problem for future studies to extend this\ntreatment to include the \ructuation regime near TCby,\ne.g. an expansion in terms of Schwinger bosons instead,\nwhich can better handle the dynamics in the disordered\nphase. Nevertheless, we expect that for low magnon den-\nsities, this ought to be at least qualitatively acceptable.\nWe proceed by integrating out the local orbital angu-\nlar momentum, treating it as a bath under a Gaussian\napproximation. This bath can be characterized by the\ncorrelation functions\n^DR(t;t0) =\u0000ihLj;cl(t)Lj;q(t0)i=nj(t)nj(t0)DR(t;t0)\n(5a)\n^DA(t;t0) =\u0000ihLj;q(t)Lj;cl(t0)i=nj(t)nj(t0)DA(t;t0)\n(5b)\n^DK(t;t0) =\u0000ihLj;cl(t)Lj;cl(t0)i=nj(t)nj(t0)DK(t;t0):\n(5c)\nThis is in turn expressed in terms of a scalar dynamical\nresponse function D(t;t0) and the unit vectors nj(t), as\nemphasized in the second equalities, and elaborated on\nin Appendix B. In equilibrium, these are completely de-\ntermined given knowledge of the orbital spectral functionand the thermal occupation function coth( \f!=2).\nIn the Gaussian approximation, we can model the spec-\ntral function for ^ \u001c2\njas that of a damped harmonic oscil-\nlator. In particular, we assume the orbital angular mo-\nmentum has linear response equations of motion of\nd\u001c2\ndt= \u0001\u001c1\u0000\u0000\u001c2(6a)\nd\u001c1\ndt=\u0000\u0001\u001c2\u0000Fext(t): (6b)\nFext(t) is an external force which acts on the angular mo-\nmentum L/^\u001c2, which is canonically conjugate to ^ \u001c1(the\nxPauli matrix whose expectation value corresponds to\nan interorbital density rather than angular momentum).\nThis leads to a spectral function of\nA(!) =\u00001\n\u0019=\u0001\n!2+i!\u0000\u0000\u00012=1\n\u0019!\u0000\u0001\n(!2\u0000\u00012)2+ \u00002!2:\n(7)\nIn the limit of \u0000 !0, this reduces to the spectrum\nfound from the Hamiltonian (1). A realistic estimate for\n\u0000, based on Raman data [91] is that \u0000 \u001815 meV, though\nit seems there is a great amount of uncertainty about\nthis parameter [95]. From the spectral function, we \fnd\nthe frequency domain Green's functions from Kramers-\nKronig relations as\nDR(!) =\u0001\n!2+i\u0000!\u0000\u00012(8a)\nDA(!) =\u0001\n!2\u0000i\u0000!\u0000\u00012(8b)\nDK(!) =\u00002\u0019icoth\u0012\f!\n2\u0013\nA(!): (8c)\nThis then generates an e\u000bective action for the spin af-\nter integrating out the bath in the Gaussian approxima-\ntion (for the orbitals) of\nSe\u000b=\u0000\u00152\n2Z\ndtZ\ndt0X\nj(Sj;cl(t0)\u0001nj(t0);Sj;q(t0)\u0001nj(t0))\u0012\n0DA(t0;t)\nDR(t0;t)DK(t0;t)\u0013\u0012\nnj(t)\u0001Sj;cl(t)\nnj(t)\u0001Sj;q(t)\u0013\n: (9)\nAs a \fnal step, we simplify by averaging over the four\ntitanium sublattices. For a long-wavelength spin-wave, it\nis reasonable to expect the magnon to only be sensitive to\nthe average of the four titanium sites. It is worth point-\ning out that this possibly fails for a short-wavelength\nmagnon, which is localized to the order of one unit cell.\nIn this case it is possible that the distinct nature of theorbital bath on each site may be important and could be\nan important source of quantum \ructuations, though we\nleave this for future work.\nProceeding on, if we average over the four sites, we\ngenerate an e\u000bective local action describing the orbital\nbath of\nSe\u000b=\u0000\u00152\n2Z\ndtZ\ndt0X\njh\nSj;cl(t0)\u0001^DA(t0;t)\u0001Sj;q(t) +Sj;q(t0)\u0001^DR(t0;t)\u0001Sj;cl(t) +Sj;q(t0)\u0001^DK(t0;t)\u0001Sj;q(t)i\n:\n(10)5\nFIG. 2. Matrix elements of sublattice averaged anisotropy ten-\nsorN=n\nnin equilibrium. We see that the o\u000b-diagonal el-\nements are zero, indicating the eigenbasis is aligned with the\ncrystalline axes. We also see the eigenvalues are nondegener-\nate, due to the orthorhombicity of the crystal. The matrix is\nfurther partitioned into the ab-plane (x;y) components and\nthecaxis (z) components. To a good approximation, the ma-\ntrix projection along the c-axis is zero, while the anisotropy\nin theab-plane is appreciable but not extreme.\nThis involves the sublattice averaged anisotropy tensor\nN(t0;t) =n(t0)\nn(t) through\n\u0014D(t0;t) =N(t0;t)\u0014D(t0;t): (11)\nThis tensor is presented in Fig. 2, which illustrates the\nmatrix elements along the x;y;z axes (note the x;y;z\naxes may not necessarily align with the a;b;c axes of the\ncrystal but rather are de\fned by the orientation of the\ncrystal-\feld levels).\nIV. EQUILIBRIUM SPIN-ORBIT COUPLING\nWe now analyze the spin-orbit coupling in equilib-\nrium. At low temperatures, we can expand around the\nfully-polarizedj\"\"\":::iground state via the Holstein-\nPrimako\u000b expansion. We then describe magnons in terms\nof canonical bosons ^bjvia the formal mapping\n^Sz\nj=S\u0000^by\nj^bj (12a)\n^S+\nj=q\n2S\u0000^by\nj^bj^bj (12b)\n^S\u0000\nj=^by\njq\n2S\u0000^by\nj^bj: (12c)\nWe then expand in the large- Slimit up to order 1 =Sto\n\fnd the linear spin-wave Hamiltonian. The Heisenberg\ninteraction (along with the external \feld along ez) gives\nthe standard form, which is diagonalized in momentum\nspace to give\n^H(2)\n0=X\np\np^by\np^bp; (13)with dispersion relation (for spin S=1\n2)\n\np= 6SJ\u0014\n1\u00001\n3(cospx+ cospy+ cospz)\u0015\n+ 6SJz\n0:\n(14)\nThis has a gap set by the easy-plane anisotropy energy\n6SJz\n0, and has a bandwidth of order 12 SJ\u001818meV.\nAt this point, we still need to include the Lamb shift\ndue to orbital \ructuations. This dispersion relation is de-\npicted in Fig. 3(a), along with the corresponding single-\nparticle density-of-states (DOS) in Fig. 3(b), computed\nusing Monte Carlo sampling.\nWe now include the orbital self-energy which can\nbe written in the frequency domain due to the time-\ntranslational invariance in equilibrium. We expand to\norderS, neglecting the linear term which should van-\nish when expanding around the ground state. We \fnd a\nGaussian action for the magnons of\nS(2)=Z\npbp\u0002\u0014G\u00001\n0(p)\u0000\u0014\u0006(p)\u0003\nbp: (15)\nWe expand the orbital bath term to O(S) in Appendix B\nin order to \fnd the magnon self-energy \u0014\u0006(p).\nOne can calculate the anisotropy due to the orbital\n\ructuations and \fnd that due to the orthorhombic na-\nture of YTO, it has three distinct eigenvectors. Calcu-\nlating the projections, we \fnd essentially no projection\nalong thecaxis, with approximately 75% and 25% along\nthe two in-plane directions. This has the result of induc-\ning an easy and hard axis for \ructuations, which leads to\nquantum \ructuations manifested by the anomalous cor-\nrelation functions of hbpb\u0000pi. In our simplistic treatment,\nwe neglect these, \fnding an approximately U(1) system,\nwith dominant eigenvalue of :5 and an anisotropy of :25\nsplitting the two principal axes.\nFor simplicity, we will neglect the anisotropy so we\nmay obtain a diagonal self-energy, leading to a Green's\nfunction of\nGR(!;p) =\u0002\n!\u0000\np\u0000S\u00152N+\u0000DR(!)\u0003\u00001: (16)\nWe haveN+\u0000\u0018:48, which is the isotropic projection\nof the in-plane components of the anisotropy tensor. We\nplot the magnon spectral function in the ( !;p) plane in\nFig. 4, however it is worth brie\ry examining the e\u000bects\nof the orbital bath perturbatively.\nDue to the large separation of scales between the or-\nbital and spin degrees of freedom, we can analyze the cor-\nrections to the magnon spectrum perturbatively. We \fnd\na Lamb shift due to the coupling to the reservoir which\nshifts the magnon band gap (it is essentially a source of\nsingle-ion anisotropy of the easy-plane type). We then\n\fnd renormalized magnon gap of\n\n0= 6SJz\n0\u0000\u00152SN+\u0000=\u0001: (17)\nThis is used to \fx the counterterm Jz\n0by matching this\nto experiment. It is empirically observed that the gap for6\nΓXRΓq[a-1]369121518Ω[meV]\n(a)(b)(c)\nFIG. 3. (a) Linear spin-wave dispersion relation along \u0000 \u0000X\u0000R\u0000\u0000 cut in Brillouin zone using idealized cubic model with\nJ= 2:75 meV and magnon gap of \n 0= 0:1 meV (after renormalizing away the Lamb shift from the orbital bath). (b) Magnon\ndensity of states (DOS) \u001a(E) =R\nq\u000e(E\u0000\nq) obtained by Monte Carlo sampling dispersion relation in (a). This is used later\nwhen we evaluate the integrals of the kinetic equation. (c) Equilibrium occupation of magnons with dispersion in (a), in terms of\nthe magnetic moment Mz= 2\u0016B(S\u0000neq). We expect the Holstein-Primako\u000b expansion to underestimate the role of \ructuations\nsince 1=Sis not small in reality.\nFIG. 4. Magnon spectral function Amag(!;p) =\n\u00001\n\u0019=GR(!;p) including orbital bath for N+\u0000=:5,\n\u0015= 15 meV, \u0001 = 90 meV, and \u0000 = 10 meV. Plotted along\nsame dispersion contour as Fig. 3. Damping is approximately\nproportional to frequency, so that the linewidth \rp\u0018\np.\nThe Lamb shift due to the orbital bath is renormalized away\nso that the magnon gap is the physically measured gap of\n\n0= 0:1 meV.\nmagnons is quite small [90], which is in and of itself an\ninteresting fact though we won't dwell on this here. We\nalso \fnd a \fnite lifetime is generated for the magnons via\ntheir interaction with the bath. This has a strong energy\ndependence and is found to be\n\r(Ep) =\u0019S\u00152N+\u0000A(Ep) =S\u00152N+\u0000\u0000Ep\n\u00013: (18)\nIn particular, the imaginary part scales with Ep, indicat-\ning it is essentially a form of Ohmic Gilbert damping dueto the orbital bath. Taking estimates for YTO parame-\nters of \u0001\u001890 meV, \u0000\u001815 meV, and \u0015\u001815 meV, we\n\fnd a lifetime in ns of\n\u001cp= 3:6 nsmeV\nEp: (19)\nWe note to the reader that the \u001cp(the lifetime for\nmagnon with momentum p) is completely distinct from\n^\u001c1;^\u001c2which correspond to the orbital operators, and also\nfrom\u001cdwhich corresponds to the lifetime of the phonon\nring-down. These all should occur in separate contexts,\nbut emphasize this distinction here to avoid confusion. At\nT\u001810 K, we have typical magnon energies of Ep\u00181\nmeV and thus we have a typical lifetime for the magneti-\nzation relaxation of order 3.6 ns according to this model,\nthough other channels for spin-\rip processes may reduce\nthis time according to Matthiessen's rule. We now pro-\nceed on to study the nonequilibrium dynamics of this\nsystem.\nV. NONEQUILIBRIUM DYNAMICS\nWe now discuss the e\u000bect of the strong optical pulse.\nFocusing our attention on to the most striking of the\nthree pump frequencies from Ref. [19], which is the pump\nat 9 THz, we start by describing how this pump e\u000bects\nthe orbital state.\nA. Pump Model\nAs per the estimates of the experiment [19], we con-\nsider a terahertz pulse which resonantly drives an IR ac-\ntive phonon mode; in the experiment [19] these were at\nfrequencies of 4 THz, 9 THz, and 17 THz. Even though\nthe pump itself is quite short, the coherent oscillations it\ninitiates in the phonon mode are estimated to live much\nlonger, with a ring-down time of order of 20-30 ps. We7\nλsocΓorb2Ωph\ntτphΩphFastSlow(a)\n(b)(c)\nFIG. 5. (a) Time scales of pump-induced dynamics. Inci-\ndent THz pulse (red) resonantly excites a phonon mode\nwhich then exhibits coherent oscillations (blue) for time scale\n\u001cph\u001d2\u0019=\nph. (b) These dynamics also lead to an accel-\neration of the spin-orbit mediated magnetization dynamics\nwhich leads to faster dynamics during the oscillations due to\nthe appearance of a new channel for spin-\rip decay via the\nphonon-induced sidebands. (c) After the oscillations decay,\nthe dynamics returns to the slower time scale present in equi-\nlibrium.\ntherefore focus on the magnon dynamics which are in-\nduced by these coherent ring-down dynamics rather than\nthe initial pulse which is a quite short duration. We use\na ring-down model of the form\nQIR(t) =Q0e\u0000t=\u001cdsin(\ndt)\u0012(t); (20)\nwith initial (and maximal) excitation amplitude Q0, cen-\ntral frequency \n d, and ring-down time \u001cd.\nFor our purposes, we will assume the pump has two\nmain e\u000bects; \frst, it is assumed to induce a transient\nchange in the spin-exchange Jdue to a standard spin-\nphonon coupling mechanism. The origin of this mecha-\nnism is not the main focus of this work, though it may\nalso be interesting. We simply model this as a coupling\nbetweenQIRandSj\u0001Sj+\u000eof the form\nHsp\u0000ph=X\nj;\u000e\u0000\f^Sj\u0001^Sj+\u000eQ2\nIR(t): (21)\nThis leads to a transient change in Jsuch that we have\ninstantaneous value of J(t) =J+\fQ2\nIR(t). We focus on\nthe recti\fed part of this, which leads to a change in the\nexchange of\n\u0001J(t) =1\n2\fQ2\n0e\u00002t=\u001cd= \u0001J(0)e\u00002t=\u001cd: (22)\n1.00\n 0.75\n 0.50\n 0.25\n 0.00 0.25 0.50 0.75 1.00\nQIR [uÅ]\n84868890929496L2 [%]\nFIG. 6. Change in in-plane projection of angular momentum\nunit vector njfor 9 THzB2upolarized phonon mode in YTO.\nWe expect for realistic \ruence, the peak amplitude is of or-\nder 1 in these units, leading to an appreciable change in the\neigenvalues of the anisotropy tensor Nwhich oscillate at fre-\nquencies\u00062\nd. This motivates the amplitude parameter Ad\nof orderAd\u0018p\n:1 =:3.\nWe consider two cases |pump-induced enhancement of\n\u0001J(0) =:5Jand pump-induced destruction \u0001 J(0) =\n\u0000:5J.\nIn addition to this, we also have argued that the pump\ninduces substantial changes to the excited crystal-\feld\neigenvector, which in turn leads to a dynamical modula-\ntion of the spin-orbit coupling between the magnons and\norbital bath. This is motivated by Fig. 6, which shows\nhow the orbital angular momentum associated to the\n\frst-excited crystal-\feld transition changes with QIRin\nthe case of YTO.\nWe see there is a quadratic coupling between the\nphonon mode Q2\nIRand the crystal \feld eigenvector nj\nsuch that we write\nnj\u0018nj+\u000enQ2\nIR(t): (23)\nThis also has a recti\fed part, which may lead to an in-\nteresting pump-induced renormalization of the magnetic\nanisotropy [29, 30], however here we will focus instead on\nthe dynamic harmonics, which can dramatically change\nthe nature of magnetic relaxation in this system. This is\nmodeled as a change in the anisotropy tensor, which is\nobtained by averaging this over the four Ti sublattices\n(we refer to Appendices B and D).\nWe write this as\nN(t;t0) =Neq\n\u0002\u0002\nA0(t) +A1(t)e\u0000i\ndt+A\u00001(t)ei\ndt\u0003\n\u0002h\nA0(t0) +A\u0003\n1(t0)e+i\ndt0+A\u0003\n\u00001(t0)e\u0000i\ndt0i\n(24)\nwhere roughly, A2\n0+jA1j2+jA\u00001j2= 1 models the ro-\ntation of the excited state unit vector without changing8\nthe net length, such that the projection onto the c-axis\nchanges byjA\u00001j2+jA\u00001j2. We assume the phase is not\nimportant, and take\nA1(t) =A\u00001(t) =1\n2Ade\u00002t=\u001cd; (25)\nin line with the same pump-pro\fle as the one which drives\nthe change in exchange.\nIn order to determine the e\u000bect of this Floquet-driven\ncoupling to the orbital bath we will work in the approx-\nimation that we may still separate the time scales asso-\nciated to (i) the magnon dynamics, (ii) the transience of\nthe drive, (iii) the orbital dynamics. It is very interesting,\nhowever challenging, to relax this hierarchy and allow\nfor a complete breakdown of separation of time scales.\nThis is left open for future works to handle. Addition-\nally, though this model captures the essential physics, it\nis still only qualitatively motivated by YTO calculations,\nand in future work a more detailed calculation of how ex-\nactly the changes in crystal-\feld evolve for each phonon\nmode would be warranted. In particular, it may be the\ncase that di\u000berent phonon modes are more or less e\u000bec-\ntive at modulating various components of this tensor and\nmay allow for a more selective control over the e\u000bects we\ndescribe here.\nB. Quasiparticle Dynamics\nWe now examine the magnon dynamics in the presence\nof a hypothetical Floquet modulation of the spin-orbit\ninteraction. As argued in the previous section, this is a\nreasonable model of the pumped phonon's e\u000bect on the\nspin-orbit coupling. To simplify matters, we assume that\nthe pump doesn't actually change the orbital correlations\nor \ructuations, but rather changes the coupling of the\nmagnons to the orbital bath.\nBy using the Keldysh technique we are able to calculate\nthe real-time dynamical evolution of the magnon correla-\ntion functions, as detailed in Appendix B. The key object\nof interest in this work is the magnon occupation func-\ntion, which is encoded in the Keldysh correlation function\nGK\np(t;t0), here taken to be diagonal in momentum space.\nFrom this, we can then obtain the net magnetization as\na function of time.\nWe further utilize the separation of time-scales be-\ntween the evolution under the pump pro\fle and the inter-\nnal frequency scales by taking the Wigner-transform of\nGK, which encodes the full two-time dependence in terms\nof a \\center-of-mass\" time, which corresponds to the slow\nevolution, and the frequency, which encodes the rapid\noscillations in the relative time di\u000berence. The Wigner\ntransformed Keldysh function is\nGK\np(T;!) =Z\nd\u001cGK\np(T+\u001c\n2;T\u0000\u001c\n2)ei!\u001c: (26)\nFrom this, we can extract the total magnon density as afunction of time as\nn(t) =1\n2Z\np\u0012Zd!\n2\u0019iGK\np(t;!)\u00001\u0013\n; (27)\nand the corresponding magnetization is then found to be\nMz(t) = 2\u0016B(S\u0000n(t)): (28)\nBy systematically expanding in terms of gradients of\nthe slowly-varying pump pro\fle, we derive in Appendix D\nan e\u000bective relaxation-time approximation for this, which\nto the very lowest order reads\n@GK\np(T;!)\n@T\n= 2i=\u0006R(T;!)\u0002\n\u0000iGK\np(T;!) + 2\u0019Amag(T;!;p)Forb(!)\u0003\n:\n(29)\nHereAmag(T;!;p) =\u00001=\u0019=GR\np(T;!) is the instanta-\nneous magnon spectral function, which depends on time\nin the instance where the pump changes, e.g. the spin-\nexchange, as it does in this system. We also see the ap-\npearance of the orbital occupation function, which we\nassume remains in equilibrium at temperature Torb, such\nthatForb(!) = coth!\n2Torb.\nWe now study the dynamics of this system under the\nquasiparticle approximation, such that we can replace\nthe frequency dependence by the instantaneous on-shell\nfrequency. This gives us a simple equation we can solve\nfor the quasiparticle occupation function fp(T) of\n@fp(T)\n@T=\u00001\n\u001cp(T)\u0010\nfp(T)\u0000f(bath)\np (T)\u0011\n; (30)\nwhere 1=\u001cp(T) is the instantaneous relaxation rate\nat timeT, derived from the magnon self-energy, and\nf(bath)\np (T) is the instaneous equilibrium occupation set\nby the orbital bath occupation function projected onto\nthe magnon spectral density. For the details, we refer to\nAppendix D.\nIf we only include the change in J, and therefore only\ninclude the instantaneous change in the spectral function,\nwe see a meager response to the pump. This is shown in\nFig. 7, which shows the change in instantaneous magne-\ntization following a transient increase in Jdue to the co-\nherent phonon rind-down, schematically illustrated above\nthe numerical plot. We plot the change in magnetization\n\u0001Mz(t) as a percent relative to the maximum possible\nchange, which would be 2 \u0016B(S\u0000n(0)) so that if the ini-\ntial moment is :9\u0016Band it increases to :95\u0016Bthis would\nby 50% of the maximum possible increase.\nThough the magnetization does generally follow the\npump-induced change \u0001 J, which here was set to :5J(0),\nit is a relatively mediocre response since the dynamics\nare still quite bottlenecked by the long-relaxation time,\n\u001cpwhich is of order nanoseconds for a thermal magnon,\nwhereas the duration of the pump-induced oscillations\nare at most 50 ps.9\nΔJ(t)τphΔJ=50%T=15K\nFIG. 7. Change in magnetization following a pump-induced\nchange in the ferromagnetic exchange due to the recti\fed spin-\nphonon coupling, modeled here as a transient \u0001 J(t)\u0018Q2\nIR(t)\nwhich follows the impulsive initial pulse, illustrated atop the\nframe. For a ring-down time of order \u001cph\u001830 ps and an\ninitial change in the exchange of \u0001 J(0)=Jeq= 50% we \fnd\nmagnetization dynamics in the plot below, which showns the\nchange in magnetization \u0001 Mz(t) in terms a percent of the\nmaximum possible enhancement \u0001 Mmax(corresponding to a\ncomplete saturation of the magnetization). Ring-down period\nis shaded red. This is not including the resonant enhancement\nof the magnetization dynamics.\nHowever, as we argued before, the nonequilibrium dy-\nnamics induced by the pump can potentially have exhibit\naccelerated timescales, as illustrated in Fig. 5. Due to a\ncombination of high-frequency oscillations at 2\n d\u001880\nmeV and low-lying orbital excitations with \u0001 \u001890 meV\nor so, we can \fnd a transient acceleration of the relax-\nation rate, quanti\fed by 1 =\u001cp=\u00002=\u0019=\u0006R(T;!), mak-\ning the system essentially relax faster than in equilibrium\nduring the driving period. This is con\frmed by calcu-\nlating the e\u000bective magnon lifetime in the presence of\nsteady-state coherent oscillations. In Fig. 8 we plot the\nmagnon lifetime \u001cpas a function of the magnon kinetic\nenergyEpfor di\u000berent pump frequencies !dand ampli-\ntudesAd[96].\nTo see whether the increased relaxation rate has any\ne\u000bect in practice, we carry out the simulations of Eq. (30)\nnow including both the pump-induced change in J(t) as\nwell as the pump-induced change in relaxation rate. This\nis presented in Fig. 9 which shows the equivalent \u0001 Jas in\nFig. 7 but now including the pump-accelerated relaxation\nrate for di\u000berent frequencies \n dat \fxed \ruence Ad=:3.\nWe see that when the pump approaches resonance with\nthe orbital excitation, the dynamics greatly accelerates\nand as a result, the magnetization can grow much more\n(a)(b)FluenceFrequencyFIG. 8. (a) Plot of magnon lifetime \u001cp= 1=\rpin the presence\nof the coherent phonon enhancement as a function of magnon\nkinetic energy Epfor di\u000berent drive frequencies. We \fx \ruence\nparameterAd=:3 and \fx orbital parameters to \u0001 = 90 meV\nand \u0000 = 10 meV, with \u0015= 15 meV. The lifetime is reduced by\nthe magnon appearance of sidebands at \u0001 \u00062\nd. Near \nd=\n40 meV this process nears resonance and the decay rate is\nmaximally enhanced by nearly two orderes of magnitude. (b)\nWe study for varying drive \ruence parameter Adat \fxed \n d=\n35 meV for the same orbital parameters. The dependence in\nthis model is monotonic, though in a more re\fnded model we\nwould expect some saturation as Ad!1.\nover the same\u001830 ps window of growth time.\nCuriously, we see that around \n d= 45 meV, the e\u000bect\nseems to completely dissappear, and the resulting magne-\ntization growth is almost completely stunted. In fact, this\nis a manifestation of the pump actually passing through\nthe orbital resonance and changing from red-detuning to\nblue-detuning. If we continue to increase the drive fre-\nquency further, we \fnd that the relaxation rate actually\nbecomes negative|an e\u000bect which is clearly impossible\nin equilibrium. This negative relaxation rate essentially\nindicates that in the rotating frame the orbital bath is\npopulation-inverted with respect to the magnon system.\nTherefore, the bath actually acts as a gain medium rather\nthan a retarder. The resulting dynamics are shown in\nFig. 10 where we simulate both an initial increase in ex-\nchange, as in Fig. 9, as well a pump-induced reduction in\nJof \u0001J=J(0) =\u000050%. We see that the response is most10\nΩd=0meV\nΩd=30meV\nΩd=35meV\nΩd=40meV\nΩd=45meVΔJ=50%T=15K\nFIG. 9. Fractional change in magnon occupation following\npump-induced change in exchange \u0001 Jwhile also including the\nenhancement of the relaxation rate due to the phonon ring-\ndown. For di\u000berent pump frequencies (here we only model the\npump frequency as changing the spin-\rip time) we see a dra-\nmatic increase in the maximum change in magnetization upon\napproaching the resonance condition around \n d\u001840 meV.\nFor pump frequencies above this, the e\u000bect quickly reverses\nand by \nd\u001845 meV we see the dynamics has actually slowed\nsubstantially.\npronounced when \n dis around\u00065 meV detuned from\nthe \u0001=2 = 45 meV point. We also see that the negative\nrelaxation rate essentially leads to an e\u000bectively reversed\nsign of \u0001J, leading to growth in magnon number when\nit should become less ferromagnetic, and vice versa.\nTherefore, we see that not only can one try to acceler-\nate magnetic dynamics away from equilibrium by mod-\nulating the coupling to the orbital bath, but one may\neven potentially slow the dynamics down (in our example,\nby tuning \n d\u001845 meV) or reverse them altogether by\nchanging from red- to blue-detuning. This is a genuinely\nnonequilibrium process and may potentially explain the\napparent opposite trend between the equilibrium spin-\nphonon coupling and pump-induced response in YTO in\nthe recent experiment [19].\nWe can more systematically map this e\u000bect out by\nplotting the most extreme value of the time-traces as a\nfunction of \n dand temperature, shown in Fig. 11(a) as a\ndensity plot, and in Fig. 11(b) for two line-cuts at \fxed\ntemperature T. We see quite clearly that the dynam-\nics are most dramatically a\u000bected near the resonance of\n2\nd= \u0001, and upon passing through the resonance the\nsign of the e\u000bect changes.\nThus, we see that going beyond the \\quasistatic\" pic-\nture and actually considering how the coupling to the\norbital bath changes in the presence of nonequilibrium\ndynamics can lead to striking, and potentially useful\nchanges to the magnetization dynamics. Crucially, the\ne\u000bect we outline here relies on a relatively low-lying or-\nbital excitation which couples to the spins and also the\npumped phonon modes. If the orbitals are too low-lying\nthen they will exhibit strong \ructuations and cannot betreated as a bath, as we have here. On the other hand,\nif they are too high in excitation energy, they cannot be\ne\u000bectively coupled to by phonon oscillations and there-\nfore are cannot realistically participate in the dynam-\nics. Thus, quasi-degenerate magnetic insulators present\na special opportunity for this type of \\bath-control,\" al-\nthough as we will discuss next, this type of physics may\nbe able to be extended to more general systems such as\nantiferromagnets, superconductors, or potentially other\ncorrelated phases.\nVI. DISCUSSION\nWe now summarize our \fndings. We considered a sim-\nple model for the nonequilibrium dynamics of magnons\nin a Heisenberg ferromagnetic insulator with low-lying\n\\quasi-degenerate\" orbitals, as may be be realized in or-\nthorhombic titanates RTiO 3(R= Y, Sm, Gd), and possi-\nbly other compounds. By using a powerful terahertz pulse\nto resonantly excited optical phonons we argued that rel-\natively long-lasting nonequilibrium dynamics can be in-\nduced by the coherently oscillating phonon modes, which\nmay have lifetimes lasting up to 30 ps. These phonon\noscillations may lead to transient modi\fcations to the\nsuperexchange though, e.g. the recti\fed part of the spin-\nphonon coupling \u0018Q2\nIRSj\u0001Sk; this may then lead to\ndynamic changes in the magnetic free-energy landscape\nwhich can potentially be used to optical drive the mag-\nnetization and control the phase diagram. However, this\ndynamics is often plagued by a bottleneck due to small\nspin-orbit coupling which leads to an approximate con-\nservation law for magnetization, leading to slow di\u000busive\ndynamics on the relevant time scales.\nThis bottleneck can be circumvented in a nonequilib-\nrium setting, as we showed in Sec. V. In particular, in the\npresence of low-lying orbital excitations, the coupling be-\ntween magnons and the orbital angular momentum can\nbecome unquenched in the presence of phonon dynamics,\nwhich may lead to \\stimulated emission\" type processes\ninto the orbital bath. This can in principle lead to a sig-\nni\fcant acceleration in time scale for the magnetization\ndynamics, allowing for more e\u000bective optical control on\nrelevant time scales. Furthermore, we found that in prin-\nciple it is even possible to reverse the nature of the cou-\npling to the bath by changing from red- to blue-detuning\nwith respect to the orbital bath, allowing for an even\ngreater degree of control over the magnetization dynam-\nics.\nMore generally, our results should be able to be applied\nto other systems of interest including antiferromagnetic\ninsulators, spin liquids, and other correlated insulators.\nThe key component is the ability to induce a dynami-\ncal coupling between the degrees of freedom of interest\n(such as spins) and the bath degrees of freedom. In ad-\ndition to controlling the bath decay rates this may also\nallow to control the bath-induced Lamb shift, which in\nthe case we consider here enters as an e\u000bective single-11\n(a)(b)ΔJ=50%ΔJ=−50%\nFIG. 10. Magnetization dynamics for frequencies below and above resonance. (a) For a transient increase in Jof 50% ferro-\nmagnetism should increase in equilibrium, however in the presence of a high-frequency drive this can amplify, diminish, or\neven reverse as the frequency passes through resonance with the bath. For \n d= 40 meV the relaxation rate reaches near\nmaximal enhancement and the magnetization is most responsive to the pump-induced increase in J, while for \n d= 50 meV\nit has already passed to the other side of the resonance. The bath now acts to induce \\gain\" rather than loss and drives the\nmagnetization opposite to the naive result, quite dramatically. (b) If we consider instead a pump-induced reduction in Jof\n=50% the same features qualitatively persist, with opposite directions. In this case, driving above the resonance leads to a\nsubstantial enhancement of magnetization.\n(a)(b)\nFIG. 11. (a) Plot of maximum change in magnetization as a function of initial temperature and pump frequency. (a) Color map\nin (\nd;T) plane. We see that crossing through the resonance at \n d= \u0001=2 there is a dramatic change in the sign of the e\u000bect,\nand that the greatest change occurs in this region. (b) Line cuts at low temperature ( T= 1 K) and high temperature ( T= 10\nK). We see that the e\u000bect is slightly more e\u000ecient at increasing the magnetization when temperature is low, while it is more\ne\u000bective at reducing the magnetization at higher temperatures.\nion anisotropy. Thus, it may also be possible to control\nthe anisotropy dynamically, as proposed in the recent ex-\nperiment [29] and theory [30]. Control over the isotropic\nsuperexchange interaction may also be possible through\nthe mechanism we outline here as well as through similar\nmechanisms [20, 24, 31, 49, 57].\nOur results may also be relevant to recent experi-\nments on nonequilibrium light-induced superconductiv-\nity [97] in fullerides [9, 11, 12], organics salts [13, 98],\nand cuprates [14, 15, 99]. In this case, we argue that\nthere are a number of parallels which make it even moreinteresting to understand this physics. Chief among these\nare the observations of pump-induced signatures of the\nordered phase above the equilibrium transition temper-\nature, long-lived resilience of this long-range order, and\nenhancement of \\coherence\" below the ordering tempera-\nture. In the case of superconductors, the e\u000bects of pump-\ninduced order are seen most clearly in systems which are\nstrongly coupled and don't exhibit a simple mean-\feld\ntransition [100, 101] (e.g. cuprates, fullerides), and this\nis also the case for the magnetic order in the recent ex-\nperiment on YTO in Ref. [19], which appears to exhibit12\na \\magnetic pseudogap.\" It is also possible that a simi-\nlar equilibrium slowing-down of reaction pathways occurs\nin these systems, which reside near to a metal=insulator\ntransition [102, 103].\nAlthough in the current work we don't address the\n\\pseudogap regime,\" it should be possible to extend our\nresults to include strong magnetic \ructuations via, e.g.\nthe Schwinger boson technique or various slave-particle\nmappings, which can be extended to nonequilibrium set-\ntings [61, 104{107]. It may turn out that the equiva-\nlent problem in the superconducting case will actually be\nmore tractable since in this case the theory for a \ructuat-\ning superconductor is more amenable to nonequilibrium\ndiagrammatic approaches [101, 108, 109].\nWe also comment that similar ideas have recently been\ndiscussed in the context of \\pump-induced sideband cool-\ning\" for various solid-state systems by various groups [51{\n53]. In particular, it was proposed that recent experi-\nments on light-induced superconductivity [9] could be\nunderstood by a Floquet sideband cooling utilizing an\nintermediate bath state provided by an internal excitonic\nresonance of the fullerene molecules [51]. This was later\nextended to the case of a quantum spin-system with a\ndynamical coupling induced to a complementary bath\nsystem [52]. In this respect, this is very similar to the\nsystem we are proposing here, where the orbitals serve\nas an analogue to the excitonic bath of Ref. [51]. How-\never, our results should still be present even if the true\nsideband cooling does not materialize. In particular, it\nis likely that both processes will be happening in a true\ndriven system.\nTo conclude, we have examined the nonequilibrium\nspin-orbital dynamics in a ferromagnetic insulator and\nfound that away from equilibrium there is a rich va-\nriety of dynamical processes which can happen even\nin a relatively simple quasiparticle description. Experi-\nmentally, this is possibly relevant to various ferromag-\nnetic insulators realized in ferromagnetic rare-earth ti-\ntanatesRTiO 3, and may be more generally applica-ble to strongly-correlated spin-orbital systems such as\nNiPS 3[29, 30], CuSb 2O6[89], other titanates [24, 25, 64,\n78], manganites [110, 111], vandates [26, 64, 112, 113],\nand a number of other compounds [114]. There may also\nbe connections to charge-density wave physics, which can\nalso be manipulated by light [16, 115].\nWe also argued that our results may be analogous to\nrecent experiments on photoinduced superconductivity.\nIn future works it will be important to consider extend-\ning our results to the strongly \ructuating regime near,\nand above TCas well as to incorporate the truly dy-\nnamical terms which break time-translational symme-\ntry [37, 43, 116]. In addition, considering systems which\ndo have degenerate orbitals (or exhibit genunie sponta-\nneous orbital ordering) would be of great interest, with\nmany exotic phenomena already known to occur [56]. It\nwill also be necessary to develop closer connection to spe-\nci\fc materials in order to make contact with current and\nfuture experiments. Experiments using ultrafast x-ray\nscattering may be able to directly con\frm these nonequi-\nlibrium dynamics [55, 117], though this is likely to be\nquite challenging theoretically.\nACKNOWLEDGMENTS\nThe authors. would like to acknowledge crucial dis-\ncussions with Pavel Dolgirev, Eugene Demler, An-\ndrey Grankin, Andy Millis, David Hsieh, Mohammad\nMaghrebi, Benedetta Flebus, Aaron M uller, and Zhiyuan\nSun. This work is primarily supported by the Quantum\nScience Center (QSC), a National Quantum Information\nScience Research Center of the U.S. Department of En-\nergy (DOE). P.N. acknowledges support as a Moore In-\nventor Fellow through Grant No. GBMF8048 and grate-\nfully acknowledges support from the Gordon and Betty\nMoore Foundation as well as support from a Max Planck\nSabbatical Award that enabled this collaborative project.\n[1] T. F. Nova, A. S. Disa, M. Fechner, and A. Cavalleri,\nMetastable ferroelectricity in optically strained SrTiO 3,\nScience 364, 1075 (2019), 1812.10560.\n[2] X. Li, T. Qiu, J. Zhang, E. Baldini, J. Lu, A. M. Rappe,\nand K. A. Nelson, Terahertz \feld{induced ferroelectric-\nity in quantum paraelectric SrTiO3, Science 364, 1079\n(2019), 1812.10785.\n[3] S. Wall, S. Yang, L. Vidas, M. Chollet, J. M. Glow-\nnia, M. Kozina, T. Katayama, T. Henighan, M. Jiang,\nT. A. Miller, D. A. Reis, L. A. Boatner, O. Delaire, and\nM. Trigo, Ultrafast disordering of vanadium dimers in\nphotoexcited VO 2, Science 362, 572 (2018).\n[4] A. Sood, X. Shen, Y. Shi, S. Kumar, S. J. Park, M. Za-\njac, Y. Sun, L.-Q. Chen, S. Ramanathan, X. Wang,\nW. C. Chueh, and A. M. Lindenberg, Universal\nphase dynamics in VO 2switches revealed by ultra-\nfast operando di\u000braction, Science 373, 352 (2021),2102.06013.\n[5] A. Cavalleri, C. T\u0013 oth, C. W. Siders, J. A. Squier,\nF. R\u0013 aksi, P. Forget, and J. C. Kie\u000ber, Femtosecond\nStructural Dynamics in VO 2during an Ultrafast Solid-\nSolid Phase Transition, Physical Review Letters 87,\n237401 (2001).\n[6] D. M. Juraschek, Q. N. Meier, and P. Narang, Para-\nmetric Excitation of an Optically Silent Goldstone-Like\nPhonon Mode, Physical Review Letters 124, 117401\n(2020), 1912.07349.\n[7] A. S. McLeod, J. Zhang, M. Q. Gu, F. Jin, G. Zhang,\nK. W. Post, X. G. Zhao, A. J. Millis, W. B. Wu,\nJ. M. Rondinelli, R. D. Averitt, and D. N. Basov,\nMulti-messenger nanoprobes of hidden magnetism in a\nstrained manganite, Nature Materials 19, 397 (2020),\n1910.10361.\n[8] R. I. Tobey, D. Prabhakaran, A. T. Boothroyd, and13\nA. Cavalleri, Ultrafast Electronic Phase Transition in\nLa1/2Sr3/2MnO4 by Coherent Vibrational Excitation:\nEvidence for Nonthermal Melting of Orbital Order,\nPhysical Review Letters 101, 197404 (2008).\n[9] M. Mitrano, A. Cantaluppi, D. Nicoletti, S. Kaiser,\nA. Perucchi, S. Lupi, P. D. Pietro, D. Pontiroli,\nM. Ricc\u0012 o, A. Subedi, S. R. Clark, D. Jaksch, and\nA. Cavalleri, Possible light-induced superconductivity\nin K 3C60at high temperature, Nature 530, 461 (2016),\n1505.04529.\n[10] S. Rajasekaran, E. Casandruc, Y. Laplace, D. Nicoletti,\nG. D. Gu, S. R. Clark, D. Jaksch, and A. Cavalleri,\nParametric ampli\fcation of a superconducting plasma\nwave, Nature Physics 12, 1012 (2016), 1511.08378.\n[11] M. Buzzi, G. Jotzu, A. Cavalleri, J. I. Cirac, E. A. Dem-\nler, B. I. Halperin, M. D. Lukin, T. Shi, Y. Wang, and\nD. Podolsky, Higgs-Mediated Optical Ampli\fcation in\na Nonequilibrium Superconductor, Physical Review X\n11, 011055 (2021), 1908.10879.\n[12] M. Budden, T. Gebert, M. Buzzi, G. Jotzu, E. Wang,\nT. Matsuyama, G. Meier, Y. Laplace, D. Pontiroli,\nM. Ricc\u0012 o, F. Schlawin, D. Jaksch, and A. Cavalleri, Ev-\nidence for metastable photo-induced superconductivity\nin K 3C60, Nature Physics 17, 611 (2021), 2002.12835.\n[13] M. Buzzi, D. Nicoletti, S. Fava, G. Jotzu, K. Miya-\ngawa, K. Kanoda, A. Henderson, T. Siegrist, J. A.\nSchlueter, M.-S. Nam, A. Ardavan, and A. Cavalleri,\nPhase Diagram for Light-Induced Superconductivity\nin\u0014-(ET)2-X, Physical Review Letters 127, 197002\n(2021), 2106.14244.\n[14] K. A. Cremin, J. Zhang, C. C. Homes, G. D. Gu, Z. Sun,\nM. M. Fogler, A. J. Millis, D. N. Basov, and R. D.\nAveritt, Photoenhanced metastable c-axis electrody-\nnamics in stripe-ordered cuprate La1.885Ba0.115CuO4,\nProceedings of the National Academy of Sciences 116,\n19875 (2019), 1901.10037.\n[15] B. Liu, M. F orst, M. Fechner, D. Nicoletti, J. Porras,\nT. Loew, B. Keimer, and A. Cavalleri, Pump Frequency\nResonances for Light-Induced Incipient Superconduc-\ntivity in YBa 2Cu3O6:5, Physical Review X 10, 011053\n(2020), 1905.08356.\n[16] A. Kogar, A. Zong, P. E. Dolgirev, X. Shen, J. Straqua-\ndine, Y.-Q. Bie, X. Wang, T. Rohwer, I.-C. Tung,\nY. Yang, R. Li, J. Yang, S. Weathersby, S. Park, M. E.\nKozina, E. J. Sie, H. Wen, P. Jarillo-Herrero, I. R.\nFisher, X. Wang, and N. Gedik, Light-induced charge\ndensity wave in LaTe3, Nat. Phys. 16, 159 (2020),\n1904.07472.\n[17] M.-C. Lee, C. H. Kim, I. Kwak, C. W. Seo, C. Sohn,\nF. Nakamura, C. Sow, Y. Maeno, E.-A. Kim, T. W. Noh,\nand K. W. Kim, Strong spin-phonon coupling unveiled\nby coherent phonon oscillations in Ca2RuO4, Physical\nReview B 99, 144306 (2019), 1712.03028.\n[18] T. Rohwer, S. Hellmann, M. Wiesenmayer, C. Sohrt,\nA. Stange, B. Slomski, A. Carr, Y. Liu, L. M. Avila,\nM. Kall ane, S. Mathias, L. Kipp, K. Rossnagel, and\nM. Bauer, Collapse of long-range charge order tracked\nby time-resolved photoemission at high momenta, Na-\nture471, 490 (2011).\n[19] A. S. Disa, J. Curtis, M. Fechner, A. Liu, A. v. Hoe-\ngen, M. F orst, T. F. Nova, P. Narang, A. Maljuk, A. V.\nBoris, B. Keimer, and A. Cavalleri, Optical Stabiliza-\ntion of Fluctuating High Temperature Ferromagnetism\nin YTiO 3, arXiv (2021), 2111.13622.[20] D. Afanasiev, J. R. Hortensius, B. A. Ivanov, A. Sasani,\nE. Bousquet, Y. M. Blanter, R. V. Mikhaylovskiy, A. V.\nKimel, and A. D. Caviglia, Ultrafast control of magnetic\ninteractions via light-driven phonons, Nature Materials\n20, 607 (2021), 1912.01938.\n[21] A. V. Kimel, B. A. Ivanov, R. V. Pisarev, P. A. Usachev,\nA. Kirilyuk, and T. Rasing, Inertia-driven spin switch-\ning in antiferromagnets, Nature Physics 5, 727 (2009).\n[22] B. Liu, W. Niu, Y. Chen, X. Ruan, Z. Tang, X. Wang,\nW. Liu, L. He, Y. Li, J. Wu, S. Tang, J. Du, R. Zhang,\nand Y. Xu, Ultrafast Orbital-Oriented Control of Mag-\nnetization in Half-Metallic La0.7Sr0.3MnO3 Films, Ad-\nvanced Materials 31, 1806443 (2019).\n[23] U. F. P. Seifert and L. Balents, Optical excitation of\nmagnons in an easy-plane antiferromagnet: Application\nto Sr 2IrO4, Physical Review B 100, 125161 (2019),\n1905.01313.\n[24] M. Gu and J. M. Rondinelli, Nonlinear phononic control\nand emergent magnetism in Mott insulating titanates,\nPhysical Review B 98, 024102 (2018), 1710.00993.\n[25] D. J. Lovinger, E. Zoghlin, P. Kissin, G. Ahn, K. Ahadi,\nP. Kim, M. Poore, S. Stemmer, S. J. Moon, S. D. Wil-\nson, and R. D. Averitt, Magnetoelastic coupling to co-\nherent acoustic phonon modes in the ferrimagnetic insu-\nlator GdTiO3, Physical Review B 102, 085138 (2020),\n2009.10222.\n[26] D. J. Lovinger, M. Brahlek, P. Kissin, D. M. Kennes,\nA. J. Millis, R. Engel-Herbert, and R. D. Averitt, In\ru-\nence of spin and orbital \ructuations on Mott-Hubbard\nexciton dynamics in LaVO3 thin \flms, Physical Review\nB102, 115143 (2020), 2009.10219.\n[27] A. Ron, S. Chaudhary, G. Zhang, H. Ning, E. Zoghlin,\nS. D. Wilson, R. D. Averitt, G. Refael, and D. Hsieh, Ul-\ntrafast Enhancement of Ferromagnetic Spin Exchange\nInduced by Ligand-to-Metal Charge Transfer, Physical\nReview Letters 125, 197203 (2020), 1910.06376.\n[28] A. d. l. Torre, K. L. Seyler, M. Buchhold, Y. Baum,\nG. Zhang, N. J. Laurita, J. W. Harter, L. Zhao, I. Phin-\nney, X. Chen, S. D. Wilson, G. Cao, R. D. Averitt,\nG. Refael, and D. Hsieh, Decoupling of static and dy-\nnamic criticality in a driven Mott insulator, Communi-\ncations Physics 5, 35 (2022), 2112.08397.\n[29] D. Afanasiev, J. R. Hortensius, M. Matthiesen,\nS. Ma~ nas-Valero, M. \u0014Si\u0014 skins, M. Lee, E. Lesne, H. S.\nJ. v. d. Zant, P. G. Steeneken, B. A. Ivanov, E. Coro-\nnado, and A. D. Caviglia, Controlling the anisotropy\nof a van der Waals antiferromagnet with light, Science\nAdvances 7, eabf3096 (2021), 2010.05062.\n[30] U. F. P. Seifert, M. Ye, and L. Balents, Ultrafast optical\nexcitation of magnetic dynamics in van der Waals mag-\nnets: Coherent magnons and BKT dynamics in NiPS 3,\nPhysical Review B 105, 155138 (2022), 2111.03674.\n[31] S. F. Maehrlein, I. Radu, P. Maldonado, A. Paarmann,\nM. Gensch, A. M. Kalashnikova, R. V. Pisarev, M. Wolf,\nP. M. Oppeneer, J. Barker, and T. Kampfrath, Dissect-\ning spin-phonon equilibration in ferrimagnetic insula-\ntors by ultrafast lattice excitation, Science Advances 4,\neaar5164 (2018), 1710.02700.\n[32] A. V. Kimel, R. V. Pisarev, J. Hohlfeld, and T. Rasing,\nUltrafast Quenching of the Antiferromagnetic Order in\nFeBO3: Direct Optical Probing of the Phonon-Magnon\nCoupling, Physical Review Letters 89, 287401 (2002).\n[33] S. V. Titov, W. T. Co\u000bey, Y. P. Kalmykov, M. Zari-\nfakis, and A. S. Titov, Inertial magnetization dynamics14\nof ferromagnetic nanoparticles including thermal agita-\ntion, Physical Review B 103, 144433 (2021).\n[34] A. S. Disa, M. Fechner, T. F. Nova, B. Liu, M. F orst,\nD. Prabhakaran, P. G. Radaelli, and A. Cavalleri, Po-\nlarizing an antiferromagnet by optical engineering of the\ncrystal \feld, Nature Physics 16, 937 (2020), 2001.00540.\n[35] M. H. Michael, A. v. Hoegen, M. Fechner, M. F orst,\nA. Cavalleri, and E. Demler, Parametric resonance of\nJosephson plasma waves: A theory for optically am-\npli\fed interlayer superconductivity in YBa2Cu3O6+x,\nPhysical Review B 102, 174505 (2020), 2004.13049.\n[36] M. Knap, M. Babadi, G. Refael, I. Martin, and\nE. Demler, Dynamical Cooper pairing in nonequilibrium\nelectron-phonon systems, Physical Review B 94, 214504\n(2016), 1511.07874.\n[37] M. Babadi, M. Knap, I. Martin, G. Refael, and\nE. Demler, Theory of parametrically ampli\fed electron-\nphonon superconductivity, Physical Review B 96,\n014512 (2017), 1702.02531.\n[38] P. E. Dolgirev, A. Zong, M. H. Michael, J. B. Curtis,\nD. Podolsky, A. Cavalleri, and E. Demler, Periodic dy-\nnamics in superconductors induced by an impulsive op-\ntical quench, arXiv (2021), 2104.07181.\n[39] M. Bukov, S. Gopalakrishnan, M. Knap, and E. Dem-\nler, Prethermal \roquet steady states and instabilities\nin the periodically driven, weakly interacting bose-\nhubbard model, Phys. Rev. Lett. 115, 10.1103/phys-\nrevlett.115.205301 (2015), 1507.01946.\n[40] T. Oka and H. Aoki, Photovoltaic Hall e\u000bect in\ngraphene, Phys. Rev. B 79, 081406 (2009), 0807.4767.\n[41] T. Kitagawa, E. Berg, M. Rudner, and E. Demler, Topo-\nlogical characterization of periodically driven quantum\nsystems, Phys. Rev. B 82, 235114 (2010), 1010.6126.\n[42] N. H. Lindner, G. Refael, and V. Galitski, Floquet topo-\nlogical insulator in semiconductor quantum wells, Nat.\nPhys. 7, 490 (2011), 1008.1792.\n[43] K. I. Seetharam, C.-E. Bardyn, N. H. Lindner, M. S.\nRudner, and G. Refael, Controlled Population of\nFloquet-Bloch States via Coupling to Bose and Fermi\nBaths, Physical Review X 5, 041050 (2015).\n[44] F. Tonielli, J. C. Budich, A. Altland, and S. Diehl, Topo-\nlogical Field Theory Far from Equilibrium, Physical Re-\nview Letters 124, 240404 (2020), 1911.07834.\n[45] M. Vogl, M. Rodriguez-Vega, B. Flebus, A. H. MacDon-\nald, and G. A. Fiete, Floquet engineering of topological\ntransitions in a twisted transition metal dichalcogenide\nhomobilayer, Physical Review B 103, 014310 (2021),\n2011.10948.\n[46] M. Nuske, L. Broers, B. Schulte, G. Jotzu, S. A. Sato,\nA. Cavalleri, A. Rubio, J. W. McIver, and L. Mathey,\nFloquet dynamics in light-driven solids, Physical Re-\nview Research 2, 043408 (2020), 2005.10824.\n[47] J.-Y. Shan, M. Ye, H. Chu, S. Lee, J.-G. Park, L. Ba-\nlents, and D. Hsieh, Giant modulation of optical nonlin-\nearity by Floquet engineering, Nature 600, 235 (2021).\n[48] S. Chaudhary, A. Ron, D. Hsieh, and G. Refael, Con-\ntrolling ligand-mediated exchange interactions in pe-\nriodically driven magnetic materials, arXiv (2020),\n2009.00813.\n[49] R. V. Mikhaylovskiy, T. J. Huisman, V. A. Gavrichkov,\nS. I. Polukeev, S. G. Ovchinnikov, D. Afanasiev, R. V.\nPisarev, T. Rasing, and A. V. Kimel, Resonant Pumping\nofd-dCrystal Field Electronic Transitions as a Mecha-\nnism of Ultrafast Optical Control of the Exchange In-teractions in Iron Oxides, Physical Review Letters 125,\n157201 (2020).\n[50] P. E. Dolgirev, M. H. Michael, A. Zong, N. Gedik, and\nE. Demler, Self-similar dynamics of order parameter\n\ructuations in pump-probe experiments, Physical Re-\nview B 101, 174306 (2020), 1910.02518.\n[51] A. Nava, C. Giannetti, A. Georges, E. Tosatti, and\nM. Fabrizio, Cooling quasiparticles in A3C60 fullerides\nby excitonic mid-infrared absorption, Nature Physics\n14, 154 (2018), 1704.05613.\n[52] M. Fabrizio, Selective Transient Cooling by Impulse Per-\nturbations in a Simple Toy Model, Physical Review Let-\nters120, 220601 (2018), 1801.06622.\n[53] P. Werner, M. Eckstein, M. M uller, and G. Refael,\nLight-induced evaporative cooling of holes in the Hub-\nbard model, Nature Communications 10, 5556 (2019),\n1904.00822.\n[54] K. Kugel' and D. Khomskii, The Jahn-Teller e\u000bect and\nmagnetism: transition metal compounds, Sov. Phys.\nUsp.25, 231 (1982).\n[55] A. M uller, F. Grandi, and M. Eckstein, Ultrafast con-\ntrol of spin-orbital separation probed with time-resolved\nRIXS, arXiv (2021), 2111.12514.\n[56] K. Wohlfeld, M. Daghofer, S. Nishimoto, G. Khaliullin,\nand J. v. d. Brink, Intrinsic Coupling of Orbital Excita-\ntions to Spin Fluctuations in Mott Insulators, Physical\nReview Letters 107, 147201 (2011).\n[57] M. Barbeau, M. Eckstein, M. Katsnelson, and\nJ. Mentink, Optical control of competing exchange in-\nteractions and coherent spin-charge coupling in two-\norbital Mott insulators, SciPost Physics 6, 027 (2019),\n1803.03796.\n[58] J. Liu, K. Hejazi, and L. Balents, Floquet Engineer-\ning of Multiorbital Mott Insulators: Applications to Or-\nthorhombic Titanates, Physical Review Letters 121,\n107201 (2018), 1801.00401.\n[59] J. Li, H. U. R. Strand, P. Werner, and M. Eckstein,\nTheory of photoinduced ultrafast switching to a spin-\norbital ordered hidden phase, Nature Communications\n9, 4581 (2018), 1806.01567.\n[60] J. F. Rodriguez-Nieva, A. Schuckert, D. Sels, M. Knap,\nand E. Demler, Transverse instability and universal de-\ncay of spin spiral order in the Heisenberg model, Phys-\nical Review B 105, L060302 (2022).\n[61] J. F. Rodriguez-Nieva, D. Podolsky, and E. Demler,\nProbing hydrodynamic sound modes in magnon \ruids\nusing spin magnetometers, Phys. Rev. B 105, 174412\n(2022).\n[62] S. Bhattacharyya, J. F. Rodriguez-Nieva, and E. Dem-\nler, Universal prethermal dynamics in Heisenberg ferro-\nmagnets, Phys. Rev. Lett. 125, 230601 (2020).\n[63] D. K. Singh, B. Kamble, and A. Singh, Orbital \ruc-\ntuations, spin-orbital coupling, and anomalous magnon\nsoftening in an orbitally degenerate ferromagnet, Phys-\nical Review B 81, 064430 (2010), 0910.5321.\n[64] G. Khaliullin and S. Maekawa, Orbital Liquid in Three-\nDimensional Mott Insulator: LaTiO3, Physical Review\nLetters 85, 3950 (2000), cond-mat/0008032.\n[65] G. Khaliullin and S. Okamoto, Quantum Behavior of\nOrbitals in Ferromagnetic Titanates: Novel Orderings\nand Excitations, Physical Review Letters 89, 167201\n(2002), cond-mat/0209434.\n[66] G. Khaliullin and S. Okamoto, Theory of orbital state\nand spin interactions in ferromagnetic titanates, Physi-15\ncal Review B 68, 205109 (2003), cond-mat/0510175.\n[67] A. M. Ole\u0013 s, G. Khaliullin, P. Horsch, and L. F. Feiner,\nFingerprints of spin-orbital physics in cubic Mott in-\nsulators: Magnetic exchange interactions and optical\nspectral weights, Physical Review B 72, 214431 (2005),\ncond-mat/0601381.\n[68] A. M. Ole\u0013 s, P. Horsch, L. F. Feiner, and G. Khali-\nullin, Spin-Orbital Entanglement and Violation of the\nGoodenough-Kanamori Rules, Physical Review Letters\n96, 147205 (2006), cond-mat/0604417.\n[69] W. Brzezicki, A. M. Ole\u0013 s, and M. Cuoco, Spin-Orbital\nOrder Modi\fed by Orbital Dilution in Transition-Metal\nOxides: From Spin Defects to Frustrated Spins Polariz-\ning Host Orbitals, Physical Review X 5, 011037 (2015),\n1408.1838.\n[70] J. v. d. Brink, P. Horsch, F. Mack, and A. M. Ole\u0013 s, Or-\nbital dynamics in ferromagnetic transition-metal oxides,\nPhysical Review B 59, 6795 (1999), cond-mat/9812123.\n[71] G. Khaliullin, Order from disorder: Quantum spin gap\nin magnon spectra of LaTiO3, Physical Review B 64,\n212405 (2001), cond-mat/0108016.\n[72] L. F. Feiner, A. M. Ole\u0013 s, and J. Zaanen, Quantum\nMelting of Magnetic Order due to Orbital Fluctua-\ntions, Physical Review Letters 78, 2799 (1997), cond-\nmat/9701098.\n[73] M. Mochizuki and M. Imada, Magnetic and Orbital\nStates and Their Phase Transition of the Perovskite-\nType Ti Oxides: Strong Coupling Approach, Journal of\nthe Physical Society of Japan 70, 1777 (2001), cond-\nmat/0102223.\n[74] P. Pal, X. Liu, M. Kareev, D. Choudhury, and\nJ. Chakhalian, Emergent antiferromagnetism of YTiO3\nin YTiO3-CaTiO3 superlattices, Physical Review B 98,\n045420 (2018), 1804.10333.\n[75] Z. Wang, D. Gautreau, T. Birol, and R. M. Fernan-\ndes, Strain-tunable metamagnetic critical end-point in\nMott insulating rare-earth titanates, arXiv (2021),\n2105.01559.\n[76] T. Katsufuji, Y. Taguchi, and Y. Tokura, Transport\nand magnetic properties of a Mott-Hubbard system\nwhose bandwidth and band \flling are both control-\nlable: R 1\u0000xCaxTiO 3+y=2, Physical Review B 56, 10145\n(1997).\n[77] E. Pavarini, S. Biermann, A. Poteryaev, A. I. Licht-\nenstein, A. Georges, and O. K. Andersen, Mott Tran-\nsition and Suppression of Orbital Fluctuations in Or-\nthorhombic 3d1 Perovskites, Physical Review Letters\n92, 4 (2004), cond-mat/0309102.\n[78] E. Pavarini, A. Yamasaki, J. Nuss, and O. K. Ander-\nsen, How chemistry controls electron localization in 3d1\nperovskites: a Wannier-function study, New Journal of\nPhysics 7, 188 (2005), cond-mat/0504034.\n[79] X.-J. Zhang, E. Koch, and E. Pavarini, Origin of orbital\nordering in YTiO 3and LaTiO 3, Physical Review B 102,\n035113 (2020).\n[80] M. Mochizuki and M. Imada, Orbital physics in the per-\novskite Ti oxides, New Journal of Physics 6, 154 (2004),\ncond-mat/0410650.\n[81] I. V. Solovyev, Lattice distortion and magnetism of 3d-\nt2gperovskite oxides, Phys. Rev. B 74, 054412 (2006),\ncond-mat/0602035.\n[82] I. Solovyev, Superexchange interactions in orthorhom-\nbically distorted titanates RTiO 3(R= Y, Gd, Sm, La),\nN.J. Phys. 11, 093003 (2009), 0906.2035.[83] T. Mizokawa, D. I. Khomskii, and G. A. Sawatzky, In-\nterplay between orbital ordering and lattice distortions\nin LaMnO3, YVO3, and YTiO3, Physical Review B 60,\n7309 (1999), cond-mat/9906010.\n[84] S. Ishihara, T. Hatakeyama, and S. Maekawa, Magnetic\nordering, orbital ordering, and resonant x-ray scattering\nin perovskite titanates, Physical Review B 65, 064442\n(2002), cond-mat/0111249.\n[85] S. Ishihara, Orbital wave and its observation in orbital-\nordered titanates and vanadates, Physical Review B 69,\n075118 (2004).\n[86] C. Ulrich, G. Ghiringhelli, A. Piazzalunga,\nL. Braicovich, N. B. Brookes, H. Roth, T. Lorenz,\nand B. Keimer, Orbital excitations in YTiO 3and\nLaTiO 3probed by resonant inelastic soft x-ray\nscattering, Phys. Rev. B 77, 113102 (2008).\n[87] C. Ulrich, L. J. P. Ament, G. Ghiringhelli, L. Braicovich,\nM. M. Sala, N. Pezzotta, T. Schmitt, G. Khaliullin,\nJ. v. d. Brink, H. Roth, T. Lorenz, and B. Keimer,\nMomentum Dependence of Orbital Excitations in Mott-\nInsulating Titanates, Physical Review Letters 103,\n107205 (2009), 0908.2497.\n[88] S. Krivenko, Interplay between crystal-\feld splitting\nand superexchange interaction in the t2g orbital Mott\ninsulator, Physical Review B 85, 064406 (2012).\n[89] D. T. Maimone, A. B. Christian, J. J. Neumeier, and\nE. Granado, Coupling of phonons with orbital dynam-\nics and magnetism in CuSb2O6, Physical Review B 97,\n174415 (2018), 1803.00559.\n[90] C. Ulrich, G. Khaliullin, S. Okamoto, M. Reehuis,\nA. Ivanov, H. He, Y. Taguchi, Y. Tokura, and\nB. Keimer, Magnetic Order and Dynamics in an\nOrbitally Degenerate Ferromagnetic Insulator, Phys-\nical Review Letters 89, 167202/1 (2002), cond-\nmat/0209400.\n[91] S. Sugai, A. Kikuchi, and Y. Mori, Raman scattering\nof orbital waves in YTiO3, Physical Review B 73, 22\n(2006).\n[92] C. Ulrich, A. G ossling, M. Gr uninger, M. Guennou,\nH. Roth, M. Cwik, T. Lorenz, G. Khaliullin, and\nB. Keimer, Raman Scattering in the Mott Insulators\nLaTiO3 and YTiO3: Evidence for Orbital Excitations,\nPhysical Review Letters 97, 157401 (2006).\n[93] N. Marzari and D. Vanderbilt, Maximally Localized\nGeneralized Wannier Functions for Composite Energy\nBands, Phys. Rev. B 56, 12847 (1997).\n[94] A. Kamenev, Field Theory of Non-Equilibrium Systems\n(Cambridge University Press, Cambridge, 2011).\n[95] It also seems like there is a good amount of uncertainty\nabout the resonance frequency as well, and whether fea-\ntures are one-orbital, two-orbital, or of other origin all\ntogether.\n[96] In fact, Ad\u0018Q2\nIRis itself scaling linearly with \ruence,\nso that scaling quadratically with A2\ndimplies quadratic\n\ruence dependence.\n[97] A. Cavalleri, Photo-induced superconductivity, Con-\ntemp. Phys. 59, 31 (2018).\n[98] M. Buzzi, D. Nicoletti, M. Fechner, N. Tancogne-\nDejean, M. A. Sentef, A. Georges, T. Biesner, E. Uykur,\nM. Dressel, A. Henderson, T. Siegrist, J. A. Schlueter,\nK. Miyagawa, K. Kanoda, M.-S. Nam, A. Ardavan,\nJ. Coulthard, J. Tindall, F. Schlawin, D. Jaksch, and\nA. Cavalleri, Photomolecular High-Temperature Super-\nconductivity, Physical Review X 10, 031028 (2020),16\n2001.05389.\n[99] A. v. Hoegen, M. Fechner, M. F orst, N. Taherian,\nE. Rowe, A. Ribak, J. Porras, B. Keimer, M. Michael,\nE. Demler, and A. Cavalleri, Ampli\fcation of Supercon-\nducting Fluctuations in Driven YBa 2Cu3O6+x, Physical\nReview X 12, 031008 (2022).\n[100] G. Jotzu, G. Meier, A. Cantaluppi, A. Cavalleri, D. Pon-\ntiroli, M. Ricc\u0012 o, A. Ardavan, and M.-S. Nam, Su-\nperconducting \ructuations observed far above T cin\nthe isotropic superconductor K 3C60, arXiv (2021),\n2109.08679.\n[101] A. Larkin and A. Varlamov, Theory of Fluctuations in\nSuperconductors (Oxford University Press, 2005).\n[102] M. Imada, A. Fujimori, and Y. Tokura, Metal-insulator\ntransitions, Reviews of Modern Physics 70, 1039 (1998).\n[103] S. Sayyad and M. Eckstein, Slowdown of the Electronic\nRelaxation Close to the Mott Transition, Physical Re-\nview Letters 117, 096403 (2016), 1601.02994.\n[104] A. Schuckert, A. P. Orioli, and J. Berges, Nonequilib-\nrium quantum spin dynamics from two-particle irre-\nducible functional integral techniques in the Schwinger\nboson representation, Physical Review B 98, 224304\n(2018), 1806.02347.\n[105] M. Babadi, E. Demler, and M. Knap, Far-from-\nEquilibrium Field Theory of Many-Body Quantum Spin\nSystems: Prethermalization and Relaxation of Spin Spi-\nral States in Three Dimensions, Phys. Rev. X 5, 041005\n(2015), 1504.05956.\n[106] M. N. Kiselev and R. Oppermann, Schwinger-Keldysh\nSemionic Approach for Quantum Spin Systems, Phys.\nRev. Lett. 85, 5631 (2000), cond-mat/0008060.\n[107] A. Shnirman and Y. Makhlin, Spin-Spin Correlators in\nthe Majorana Representation, Physical Review Letters\n91, 207204 (2003), cond-mat/0305064.\n[108] Y. Lemonik and A. Mitra, Transport and spectral signa-\ntures of transient \ructuating super\ruids in the absence\nof long-range order, Phys. Rev. B 100, 10.1103/phys-\nrevb.100.094503 (2019), 1905.08906.\n[109] Y. Lemonik and A. Mitra, Quench dynamics of super-\nconducting \ructuations and optical conductivity in a\ndisordered system, Phys. Rev. B 98, 214514 (2018),\n1804.09280.\n[110] J. v. d. Brink, G. Khaliullin, and D. Khomskii, Orbital\ne\u000bects in manganites (Springer, Dordrecht, 2002) pp.\n263{301.\n[111] L. F. Feiner and A. M. Ole\u0013 s, Electronic origin of mag-\nnetic and orbital ordering in insulating LaMnO3, Phys-\nical Review B 59, 3295 (1999), cond-mat/9805011.\n[112] J. Fujioka, T. Yasue, S. Miyasaka, Y. Yamasaki,\nT. Arima, H. Sagayama, T. Inami, K. Ishii, and\nY. Tokura, Critical competition between two distinct\norbital-spin ordered states in perovskite vanadates,\nPhysical Review B 82, 144425 (2010).\n[113] G. Khaliullin, P. Horsch, and A. M. Ole\u0013 s, Spin Order\ndue to Orbital Fluctuations: Cubic Vanadates, Physical\nReview Letters 86, 3879 (2000), cond-mat/0103300.\n[114] R. R uckamp, E. Benckiser, M. W. Haverkort, H. Roth,\nT. Lorenz, A. Freimuth, L. Jongen, A. M oller, G. Meyer,\nP. Reutler, B. B uchner, A. Revcolevschi, S.-W. Cheong,\nC. Sekar, G. Krabbes, and M. Gr uninger, Optical study\nof orbital excitations in transition-metal oxides, New\nJournal of Physics 7, 144 (2005), cond-mat/0503405.\n[115] X. Shi, W. You, Y. Zhang, Z. Tao, P. M. Oppe-\nneer, X. Wu, R. Thomale, K. Rossnagel, M. Bauer,H. Kapteyn, and M. Murnane, Ultrafast electron\ncalorimetry uncovers a new long-lived metastable state\nin 1T-TaSe2 mediated by mode-selective electron-\nphonon coupling, Science Advances 5, eaav4449 (2019),\n1901.08214.\n[116] M. Genske and A. Rosch, Floquet-Boltzmann equation\nfor periodically driven Fermi systems, Physical Review\nA92, 062108 (2015), 1508.04551.\n[117] M. Mitrano and Y. Wang, Probing light-driven quantum\nmaterials with ultrafast resonant inelastic X-ray scatter-\ning, Communications Physics 3, 184 (2020), 2009.11315.\n[118] A. Georges, L. d. Medici, and J. Mravlje, Strong Corre-\nlations from Hund's Coupling, Annual Review of Con-\ndensed Matter Physics 4, 137 (2013), 1207.3033.\n[119] G. Kresse and J. Furthm uller, E\u000ecient iterative schemes\nforab initio total-energy calculations using a plane-wave\nbasis set, Phys. Rev. B 54, 11169 (1996).\n[120] T. A. and I. Tanaka, First principles phonon calcula-\ntions in materials science, Scripta Mat. 108, 1 (2015).\n[121] G. Kresse and D. Joubert, First principles phonon cal-\nculations in materials science, Phys. Rev. B 59, 1758\n(1999).\n[122] H. Monkhorst and J. Pack, Special points for Brillouin-\nzone integrations, Phys. Rev. B 13, 5188 (1976).\nAppendix A: Orbital Angular Momentum\nHere we present details of the orbital angular momen-\ntum projected onto the t2gstates. The orbital angular\nmomentum of the full dshell is in general characterized\nby a totalL= 2 operator. In the presence of a cubic crys-\ntal \feld splitting, this is further split into two eglevels\nwith quenched angular momentum, and three t2glevels\nwhich in general may have an unquenched e\u000bective angu-\nlar momentum of Le\u000b= 1 roughly corresponding to the\nvector representation of the three orbitals [118].\nExplicitly, we have the representation of the e\u000bective\nangular momentum operators given in terms of the t2g\nstatesjai=jyzi;jbi=jzxi;jci=jxyion sitejas\n^Ll\nj=\u0000i\u000flmn(jn;jihm;jj\u0000jm;jihn;jj): (A1)\nNote that this is odd under time-reversal symmetry and\nhas purely o\u000b-diagonal matrix elements written in terms\nof the Cartesian orbitals.\nIn the presence of further splitting induced by the\nGdFeO 3lattice distortion, the orbital angular momen-\ntum becomes quenched. We can then project it onto the\nlowest two states of the intra- t2gdistortion, which we call\nj0iandj1i. If we express the angular orbital momentum\nin the basis of the crystal-\feld matrix we \fnd the relevant\noperator is\n^Lj= (\u0000ij1;jih0;jj\u0000j0;jih1;jj)e2=e2^\u001c2\nj; (A2)\nwhere the second equality expresses this in a vector rep-\nresentation in terms of the unit vector which points along\nthe direction ezfor thexyorbital, and so on in a cyclic\nway. The Pauli matrix ^ \u001c2\njis the relevant second-quantized\noperator resulting from the projection of the full orbital17\nstate operator onto the two lowest orbitals. More gener-\nally, we may have the crystal-\feld matrix evolving with a\nparameter (such as a phonon coordinate), in which case\nwe express this in terms of the relevant crystal-\feld wave-\nfunctions as\n^Lj=n^\u001c2\nj; (A3)\nwhere the matrix elements of the orbital angular momen-\ntum vector are obtained via the Levi-Civita symbol as\nnc=ea\u0002eb (A4)\nwhere eais theath(real, by time-reversal symmetry)\neigenvector of the crystal-\feld.\nAppendix B: Keldysh Holstein-Primako\u000b\nWe introduce the spin \felds on the forward and back-\nward contours, expanded up to O(S0) as\nSj;\u0006=Se3+p\nS\u0002\ne\u0000bj\u0006+e+bj\u0006\u0003\n\u0000e3bj\u0006bj\u0006(B1)with\ne\u0006=1p\n2(e1\u0006ie2): (B2)\nWe therefore \fnd the expansion for the classi-\ncal/quantum \felds of\nSj;cl=S1p\n2e3+p\nS\u0002\ne\u0000bjcl+e+bjcl\u0003\n\u0000e31p\n2(bjcclbjcl+bjqbjq)\n(B3a)\nSj;q=p\nS\u0002\ne\u0000bjq+e+bjq\u0003\n\u0000e31p\n2(bjclbjcl+bjqbjcl):\n(B3b)\nWe require the products S\u000b\njcl(t)S\f\njq(t0) and S\u000b\njq(t)S\f\njq(t0)\nup to order O(S).\nWe \fnd at quadratic order the contributions\nSjq(t)Sjcl(t0) =S\u0002\nT++bjq(t)bjcl(t0) +T+\u0000bjq(t)bjcl(t0) +T\u0000+bjq(t)bjcl(t0) +T\u0000\u0000bjq(t)bjcl(t0)\u0003\n\u0000S\n2T33\u0000\nbjcl(t)bjq(t) +bjq(t)bjcl(t)\u0001\n;(B4)\nand\nSjq(t)Sjq(t0) =S\u0002\nT++bjq(t)bjq(t0) +T+\u0000bjq(t)bjq(t0) +T\u0000+bjq(t)bjq(t0) +T\u0000\u0000bjq(t)bjq(t0)\u0003\n: (B5)\nHere we have introduced the tensors\nT++=e+\ne+ (B6a)\nT+\u0000=e+\ne\u0000 (B6b)\nT\u0000+=e\u0000\ne+ (B6c)\nT\u0000\u0000=e\u0000\ne\u0000 (B6d)\nT33=e3\ne3: (B6e)\nWe have an e\u000bective action due to the orbital bath of\nSe\u000b=\u0000S\u00152\n2X\njZ\nt;t0tr\u001a\nnj(t0)\nnj(t)\u0001\u0014\nDK(t;t0)\u0000\nT++bjq(t)bjq(t0) +T+\u0000bjq(t)bjq(t0) +T\u0000+bjq(t)bjq(t0) +T\u0000\u0000bjq(t)bjq(t0)\u0001\n+DR(t;t0)\u0012\nT++bjq(t)bjcl(t0) +T+\u0000bjq(t)bjcl(t0) +T\u0000+bjq(t)bjcl(t0) +T\u0000\u0000bjq(t)bjcl(t0)\u00001\n2T33\u0000\nbjcl(t)bjq(t) +bjq(t)bjcl(t)\u0001\u0013\n+DA(t;t0)\u0012\nT++bjcl(t)bjq(t0) +T+\u0000bjcl(t)bjq(t0) +T\u0000+bjcl(t)bjq(t0) +T\u0000\u0000bjcl(t)bjq(t0)\u00001\n2T33\u0000\nbjcl(t0)bjq(t0) +bjq(t0)bjcl(t0)\u0001\u0013\u0015\u001b\n:\n(B7)\nHere the trace is taken over the spin tensor indices.\nWe will herein replace the sublattice speci\fc angular mo-\nmentum vectors nwith the sublattice averaged matrix\nN(t0;t) =nj(t0)\nnj(t): (B8)We also will for the most part throw away the anomalous18\ncorrelations, assuming they are small, though this may be an interesting direction for the future. We then \fnd an\ne\u000bectiveU(1) symmetry for the magnons, getting\nSe\u000b=\u0000S\u00152\n2X\njZ\nt;t0tr\u001a\nN(t0;t)\u0001\u0014\nDK(t;t0)\u0000\nT+\u0000bjq(t)bjq(t0) +T\u0000+bjq(t)bjq(t0)\u0001\n+DR(t;t0)\u0012\nT+\u0000bjq(t)bjcl(t0) +T\u0000+bjq(t)bjcl(t0)\u00001\n2T33\u0000\nbjcl(t)bjq(t) +bjq(t)bjcl(t)\u0001\u0013\n+DA(t;t0)\u0012\nT+\u0000bjcl(t)bjq(t0) +T\u0000+bjcl(t)bjq(t0)\u00001\n2T33\u0000\nbjcl(t0)bjq(t0) +bjq(t0)bjcl(t0)\u0001\u0013\u0015\u001b\n:(B9)\nAt this point we can read out the retarded self-energy\n\u0006R(t;t0) =S\u00152\n2\u0002\nN+\u0000(t0;t)DR(t;t0) +N\u0000+(t;t0)DA(t0;t)\u0003\n\u0000S\u00152\n4\u0012Z\ndt00N33(t00;t)DR(t;t00)\u000e(t\u0000t0)\u00001\n2Z\ndt00N33(t0;t00)DA(t00;t0)\u000e(t\u0000t0)\u0013\n:(B10)\nand the Keldysh self energy as\n\u0006K(t;t0) =cS\u00152\n2\u0002\nN+\u0000(t0;t)DK(t;t0) +N\u0000+(t;t0)DK(t0;t)\u0003\n: (B11)\nIn the retarded self-energy, the last two terms describe\nthe drive-induced dephasing ( T2process), which only en-\nters when the e\u000bective magnon gap due to the orbital\n\ructuations is time-dependent. We will leave this study\nto future works, and ignore it in this case as we assume\nthe projection of the angular momentum matrix elements\nis small along the caxis.\nTo summarize, once we discard the anomalous terms\nand the pump-induced dephasing we are left with the\nmagnon self-energies of\n\u0006R(t;t0) =S\u00152\n2\u0002\nN+\u0000(t0;t)DR(t;t0) +N\u0000+(t;t0)DA(t0;t)\u0003\n(B12a)\n\u0006A(t;t0) =S\u00152\n2\u0002\nN+\u0000(t0;t)DA(t;t0) +N\u0000+(t;t0)DR(t0;t)\u0003\n(B12b)\n\u0006K(t;t0) =S\u00152\n2\u0002\nN+\u0000(t0;t)DK(t;t0) +N\u0000+(t;t0)DK(t0;t)\u0003\n:\n(B12c)\nWe are now tasked with using these to solve the equations\nof motion in the driven case. Note that\nN+\u0000(t0;t) =e+\u0001n(t0)n(t)\u0001e\u0000=N\u0000+(t;t0):(B13)Appendix C: Density Functional Theory\nCalculations\nWe performed our computations with the Vienna ab-\ninitio simulation package VASP.6.2 [119]. For the phonon\ncalculations we used the Phonopy software package [120]\nand the Wannier90 package for Wannierization [93]. Our\ncomputations further utilized pseudopotentials gener-\nated within the Projected Augmented Wave (PAW) [121]\nmethod. Speci\fcally, we take the following con\fgurations\nfor default potentials: Ti 3 p64s13d3, Y 4s24p65s24d1, and\nO 2s22p4. We applied the Local Spin Density Approxima-\ntion (LsDA) approximation for the exchange-correlation\npotential, which we augment with the Hubbard U\u0000Jpa-\nrameter to account for the localized nature of the d-states\nof Ti. We use U= 4 eV and J= 0:0 eV. As a numer-\nical setting, we used a 9 \u00029\u00027 Monkhorst [122] gen-\neratedk-point-mesh sampling of the Brillouin zone and\na plane-wave energy cuto\u000b of 600 eV. We iterate self-\nconsistent calculations until the change in total energy\nhas converged up to 10\u00008eV.\nAppendix D: Two-Time Equations\nHere we elaborate on the details associated to comput-\ning the various non-equilibrium Green's function which\nenter into the magnon kinetic equation. We focus on the\ntime-frequency domain transforms, assuming the space\nand momentum dependencies are trivial.19\nTo begin with, we invoke the formula for the relation\nof the Wigner transform of two products. We consider\ntwo correlation functions with known Wigner transforms\nA(T1;!1) andB(T2;!2). We want the Wigner transform\nof their convolution, expressed in terms of the two-time\nfunctionsA;B as\nC(T; \n) =Z\nd\u001cei\n\u001cZ\ndtA(T+\u001c=2;t)B(t;T\u0000\u001c=2):\n(D1)\nThis expression can be found in [94] and is formally given\nas an exponential derivative operation as\nC(T; \n) =A(T; \n) exp\u0012\n\u0000i\n2h \u0000@T\u0000 !@\n\u0000 \u0000@\n\u0000 !@Ti\u0013\nB(T; \n):\n(D2)\nThis is only useful if one can expand the relevant func-\ntions in terms of slowly-varying in both time and fre-\nquency, which in turn relies on a separation of scales be-\ntween the dynamics and frequencies.\nWe also use the related formula, relevant for theWigner transform of the point-wise product,\nD(T;!) =Z\nd\u001cei\n\u001cA(T+\u001c\n2;T\u0000\u001c\n2)B(T+\u001c\n2;T\u0000\u001c\n2);\n(D3)\nwhich yields\nD(T; \n) =Zd!\n2\u0019A(T; \n\u0000!)B(T;!): (D4)\nWe now apply this to the Green's functions. First, we\nconsider the magnon retarded Green's function, which\nobeys the integral equation\n(i@t\u0000\np)GR(t;t0)\u0000Z\ndt00\u0006R(t;t00)GR(t00;t0) =\u000e(t\u0000t0):\n(D5)\nIn the absence of the drive, this is solved in the frequency\ndomain, and we obtain the standard result which in par-\nticular amounts to a form of Gilbert damping at low fre-\nquencies.\nIn this work we will still retain the separation between\nthe evolution times, which are of order of 20-2000 ps, and\nthe time-scales of the internal degrees-of-freedom which\nare from 20-800 fs or so. This allows us to e\u000eciently em-\nploy the equations of motion using the Wigner transfor-\nmations and the Moyal expansions.\nThe lowest order in the Moyal expansion is simply the\nproduct. We retain expansion up to \frst order, giving\nequation of motion for the retarded Green's function\ni\n2\u0002\n1\u0000@!\u0006R(T;!)\u0003\n@TGR\np(T;!) +\u0014\n!\u0000\np\u0000\u0006R(T:!) +i\n2@T\u0006R(T;!)@!\u0015\nGR\np(T;!) =1: (D6)\nThis yields a \frst order di\u000berential equation for the\nGreen's function, though it remains non-local in fre-\nquency space due to the the changing self-energy. When\nsolving, we also supplement with the initial condition\nthat\nGR\np(\u00001:!) =1\n!\u0000\np\u0000\u0006R(!): (D7)\nTo complete this, we need to express the self-energy\nas a Wigner transform as well. We use the product for-\nmula to \fnd Wigner-transform (applied to R;A;K self-energies)\n\u0014\u0006(T; \n) =1\nNeq\n+\u0000Zd!\n2\u0019N+\u0000(T;!)\u0014\u0006eq(\n\u0000!):(D8)\nHere \u0006 eq(!) is the equilibrium self-energy and depends\nonly on frequency. Neq\n+\u0000is the equilibrium angular mo-\nmentum projection, while N+\u0000(T;!) is the Wigner\ntransform of the modulated angular momentum tensor.\nWe model the modulation via\nN+\u0000(t;t0) =Neq\n+\u0000(A0(t) +A1(t)e\u0000i\ndt+A\u00001(t)ei\ndt)\n\u0002(A\u0003\n0(t0) +A\u0003\n1(t0)ei\ndt0+A\u0003\n\u00001(t0)e\u0000i\ndt0);(D9)\nwhere we have expressed this in terms of a Floquet expan-\nsion in the drive-frequency \n d, along with slowly varying\nenvelope functions A0;A\u00061, which vary over times of or-\nder\u001cd\u001d\n\u00001\nd.20\nThis gives, in the slowly varying envelope approximation for A's of\nN+\u0000(t;t0)=Neq\n+\u0000=\u0002\njA0(T)j2+A1(T)A\u0003\n\u00001(T)e\u00002i\ndT+A\u00001(T)A\u0003\n1(T)e2i\ndT\u0003\n2\u0019\u000e(!)\n+jA1(T)j22\u0019\u000e(!\u0000\nd) +jA\u00001(T)j22\u0019\u000e(!+ \nd)\n+\u0002\nA0(T)A\u0003\n1(T)ei\nDT+A\u0003\n0(T)A1(T)e\u0000i\nDT\u0003\n2\u0019\u000e(!\u0000\nd=2)\n+\u0002\nA0(T)A\u0003\n\u00001(T)e\u0000i\nDT+A\u0003\n0(T)A\u00001(T)ei\nDT\u0003\n2\u0019\u000e(!+ \nd=2):(D10)\nThis involves a number of terms, including some which\ncouple the slow-dynamics to the fast degrees of free-\ndom. These terms involve oscillatory couplings like ei\ndT.\nWhile these are important close to parametric resonance,\nor in the steady-state Floquet system, where the sepa-\nration of time scales completely disintegrates, or must\nbe treated non-perturbatively, we limit ourselves to the\nregime where the dynamics are still able to be disen-\ntangled. We therefore only keep in this expansion those\nterms which don't average out over long times T. This\nleaves only the terms\n\u0014\u0006(T;!) =jA0(T)j2\u0014\u0006(!)\n+jA1(T)j2\u0014\u0006(!\u0000\nd) +jA\u00001(T)j2\u0014\u0006(!+ \nd):(D11)In fact, the object we are interested in is the magnon\nKeldysh occupation function, whose equal-time value\nre\rects the time-dependence of the total number of\nmagnons. At the Gaussian level, one can \fnd that the\nthis Green's function is given by\nGK=GR\u000e\u0006K\u000eGA: (D12)\nIn order to proceed, we manipulate this to obtain an equation of motion of the form\n(GR)\u00001\u000eGK\u0000GK\u000e(GA)\u00001=\u0000(GR\u000e\u0006K\u0000\u0006K\u000eGA): (D13)\nWe now utilize the fact that this is diagonal in momentum space and take the Wigner transform of this equation. In\ngeneral, this will not yield a closed form since the Wigner transform is over convolutions of the functions. In the very\nlowest-order limit of a slowly-varying change in the self-energy, we get\n\u0014\ni@\n@T\u0000(\u0006R(T;!)\u0000\u0006A(T;!))\u0015\nGK\np(T;!) =\u0000\u0000\nGR\np(T;!)\u0000GA\np(T;!)\u0001\n\u0006K(T;!): (D14)\nThis is formulated in terms of the occupation and spectral functions as\n@GK\np(T;!)\n@T=\u0002\n\u0006R(T;!)\u0000\u0006A(T;!)\u0003\u0000\n\u0000iGK\np(T;!) + 2\u0019Amag(T;!;p)Forb(T;!)\u0001\n: (D15)\nThis is the simple frequency-dependent relaxation-time\napproximation. We \fnd a relaxation of the instantaneous\nmagnon occupation towards the bath temperature with\nthe relaxation rate given by the bath coupling.\nTo conclude, we implement the quasiparticle approx-\nimation, which assumes the linewidth of the magnon is\nmuch smaller than its central frequency. In this case we\ncan derive a simple equation solely for the total magnon\noccupation function\nfp(T) =iZd!\n2\u0019GK\np(T;!) (D16)as\n@tfp(t) =\u00001\n\u001cp(t)h\nfp(t)\u0000f(0)\np(t)i\n; (D17)\nwhere the instantaneous relaxation rate is given by\n1\n\u001cp(t)=\u00002Zd!\n2\u0019=\u0006R(!;t)Amag(!;p;t); (D18)\nand the instantaneous occupation function is\nf(0)\np(t) =Rd!\n2\u0019=\u0006R(!;t)Amag(!;p;t)Forb(!)Rd!\n2\u0019=\u0006R(!;t)Amag(!;p;t):(D19)\nWe approximate the spectral function as\nAmag(!;p;t) =\u000e(!\u0000\np(t)); (D20)21\nsince the quasiparticle decay rate is expected to be small."    },    {        "title": "2308.05955v2.Dynamical_Majorana_Ising_spin_response_in_a_topological_superconductor_magnet_hybrid_by_microwave_irradiation.pdf",        "content": "Dynamical Majorana Ising spin response in a topological superconductor-magnet\nhybrid by microwave irradiation\nYuya Ominato,1, 2Ai Yamakage,3and Mamoru Matsuo1, 4, 5, 6\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n2Waseda Institute for Advanced Study, Waseda University, Shinjuku, Tokyo 169-8050, Japan.\n3Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n4CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan\n6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: March 20, 2024)\nWe study a dynamical spin response of surface Majorana modes in a topological superconductor-\nmagnet hybrid under microwave irradiation. We find a method to toggle between dissipative and\nnon-dissipative Majorana Ising spin dynamics by adjusting the external magnetic field angle and\nthe microwave frequency. This reflects the topological nature of the Majorana modes, enhancing\nthe Gilbert damping of the magnet, thereby, providing a detection method for the Majorana Ising\nspins. Our findings illuminate a magnetic probe for Majorana modes, paving the path to innovative\nspin devices.\nIntroduction.— The quest for Majoranas within matter\nstands as one of the principal challenges in the study of\ncondensed matter physics, more so in the field of quan-\ntum many-body systems [1]. The self-conjugate nature\nof Majoranas leads to peculiar electrical characteristics\nthat have been the subject of intensive research, both\ntheoretical and experimental [2]. In contrast, the focus of\nthis paper lies on the magnetic properties of Majoranas,\nspecifically the Majorana Ising spin [3–8]. A distinctive\ncharacteristic of Majorana modes, appearing as a surface\nstate in topological superconductors (TSC), is its exceed-\ningly strong anisotropy, which makes it behave as an Ising\nspin. In particular, this paper proposes a method to ex-\nplore the dynamical response of the Majorana Ising spin\nthrough the exchange interaction at the magnetic inter-\nface, achieved by coupling the TSC to a ferromagnet with\nferromagnetic resonance (FMR) (as shown in Fig.1 (a)).\nFMR modulation in a magnetic hybrid system has at-\ntracted much attention as a method to analyze spin ex-\ncitations in thin-film materials attached to magnetic ma-\nterials [9, 10]. Irradiating a magnetic material with mi-\ncrowaves induces dynamics of localized spin in magnetic\nmaterials, which can excite spins in adjacent thin-film\nmaterials via the magnetic proximity effect. This setup\nis called spin pumping, and has been studied intensively\nin the field of spintronics as a method of injecting spins\nthrough interfaces [11, 12]. Recent studies have theoret-\nically proposed that spin excitation can be characterized\nby FMR in hybrid systems of superconducting thin films\nand magnetic materials [13–18]. Therefore, it is expected\nto be possible to analyze the dynamics of surface Majo-\nrana Ising spins using FMR in hybrid systems.\nIn this work, we consider a TSC-ferromagnetic insula-\ntor (FI) hybrid system as shown in Fig. 1 (a). The FMR\nis induced by microwave irradiation on the FI. At the\ninterface between the TSC and the FI, the surface Ma-\n(b)\n(c)(a)\nFI~~\n~~Microwave\nϑS\nY, yX\nxZhdcHex\nTSC\n(d)\nhdchdc+δhα+δα\nHz\nFIG. 1. (a) The TSC-FI hybrid schematic reveals how,\nunder resonance frequency microwave irradiation, localized\nspins commence precessional motion, consequently initiating\nthe dynamical Majorana Ising spin response at the TSC inter-\nface. (b) In the TSC context, the liaison between a spin-up\nelectron and a spin-down hole with the surrounding sea of\nspin-triplet Cooper pairs drastically modulate their proper-\nties; notably, a spin-down hole can engage with a spin-triplet\nCooper pair, thereby inheriting a negative charge. (c) No-\ntably, spin-triplet Cooper pairs amass around holes and scat-\nter around electrons, thereby eroding the rigid distinction be-\ntween the two. (d) The interplay between the Majorana mode\nand the localized spin manipulates the FMR spectrum, trig-\ngering a frequency shift and linewidth broadening.\njorana modes interact with the localized spins in the FI.\nAs a result, the localized spin dynamics leads to the dy-\nnamical Majorana Ising spin response (DMISR), which\nmeans the Majorana Ising spin density is dynamically in-\nduced, and it is possible to toggle between dissipative and\nnon-dissipative Majorana Ising spin dynamics by adjust-\ning the external magnetic field angle and the microwave\nfrequency. Furthermore, the modulation of the localizedarXiv:2308.05955v2  [cond-mat.mes-hall]  19 Mar 20242\nspin dynamics due to the interface interaction leads to a\nfrequency shift and a linewidth broadening, which reflect\nthe properties of the Majorana Ising spin dynamics. This\nwork proposes a setup for detecting Majorana modes and\npaves the way for the development of quantum comput-\ning and spin devices using Majoranas.\nModel.— We introduce a model Hamiltonian Hconsist-\ning of three terms\nH=HM+HFI+Hex. (1)\nThe first, second, and third terms respectively describe\nthe surface Majorana modes on the TSC surface, the bulk\nFI, and the proximity-induced exchange coupling. Our\nfocus is on energy regions significantly smaller than the\nbulk superconducting gap. This focus allows the spin ex-\ncitation in the TSC to be well described using the surface\nMajorana modes. The subsequent paragraphs provide\ndetailed explanations of each of these three terms.\nThe first terms HMdescribes the surface Majorana\nmodes,\nHM=1\n2Z\ndrψT(r)\u0010\nℏvˆkyσx−ℏvˆkxσy\u0011\nψ(r),(2)\nwhere r= (x, y),ˆk= (−i∂x,−i∂y),vis a constant\nvelocity, and σ= (σx, σy, σz) are the Pauli matrices.\nThe two component Majorana field operator is given by\nψ(r) = ( ψ→(r), ψ←(r))T, with the spin quantization\naxis along the xaxis. The Majorana field operators sat-\nisfy the Majorana condition ψσ(r) =ψ†\nσ(r) and the an-\nticommutation relation {ψσ(r), ψσ′(r)}=δσσ′δ(r−r′)\nwhere σ, σ′=→,←. We can derive HMby using surface-\nlocalized solutions of the BdG equation based on the bulk\nTSC Hamiltonian. The details of the derivation of HM\nare provided in the Supplemental Material [19].\nA notable feature of the surface Majorana modes is\nthat the spin density is Ising like, which we call the Majo-\nrana Ising spin [3–8]. The feature follows naturally from\nthe Majorana condition and the anticommutation rela-\ntion. The Majorana Ising spin density operator is given\nbys(r) := ψT(r)(σ/2)ψ(r) = (0 ,0,−iψ→(r)ψ←(r))\n(See the Supplemental Material for details [19]). The\nanisotropy of the Majorana Ising spin is the hallmark of\nthe surface Majorana modes on the TSC surface.\nThe second term HFIdescries the bulk FI and is given\nby the ferromagnetic Heisenberg model,\nHFI=− JX\n⟨n,m⟩Sn·Sm−ℏγhdcX\nnSZ\nn, (3)\nwhere J>0 is the exchange coupling constant, Snis the\nlocalized spin at site n,⟨n, m⟩means summation for near-\nest neighbors, γis the electron gyromagnetic ratio, and\nhdcis the static external magnetic field. We consider the\nspin dynamics of the localized spin under microwave irra-\ndiation, applying the spin-wave approximation. This al-\nlows the spin excitation to be described by a free bosonic\noperator, known as a magnon [20].The third term Hexrepresents the proximity exchange\ncoupling at the interface between the TSC and the FI,\nHex=−Z\ndrX\nnJ(r,rn)s(r)·Sn=HZ+HT,(4)\nHZ=−cosϑZ\ndrX\nnJ(r,rn)sz(r)SZ\nn, (5)\nHT=−sinϑZ\ndrX\nnJ(r,rn)sz(r)SX\nn, (6)\nwhere the angle ϑis shown in Fig. 1 (a). HZis the\ncoupling along the precession axis and HTis the coupling\nperpendicular to the precession axis. In our setup, HZ\nleads to gap opening of the energy spectrum of the surface\nMajorana modes and HTgives the DMISR under the\nmicrowave irradiation.\nDynamical Majorana Ising spin response.— We con-\nsider the microwave irradiation on the FI. The coupling\nbetween the localized spins and the microwave is given\nby\nV(t) =−ℏγhacX\nn\u0000\nSX\nncosωt−SY\nnsinωt\u0001\n,(7)\nwhere hacis the microwave amplitude, and ωis the mi-\ncrowave frequency. The microwave irradiation leads to\nthe precessional motion of the localized spin. When the\nfrequency of the precessional motion and the microwave\ncoincide, the FMR occurs. The FMR leads to the DMISR\nvia the exchange interaction. The DMISR is character-\nized by the dynamic spin susceptibility of the Majorana\nmodes, ˜ χzz(q, ω), defined as\n˜χzz(q, ω) :=Z\ndre−iq·rZ\ndtei(ω+i0)tχzz(r, t),(8)\nwhere χzz(r, t) := −(L2/iℏ)θ(t)⟨[sz(r, t), sz(0,0)]⟩\nwith the interface area L2and the spin den-\nsity operator in the interaction picture, sz(r, t) =\nei(HM+HZ)t/ℏsz(r)e−i(HM+HZ)t/ℏ. For the exchange cou-\npling, we consider configuration average and assume\n⟨P\nnJ(r,rn)⟩ave=J1, which means that HZis treated\nas a uniform Zeeman like interaction and the interface\nis specular [21]. Using eigenstates of Eq. (2) and after a\nstraightforward calculation, the uniform spin susceptibil-\nity is given by\n˜χzz(0, ω)\n=−X\nk,λ|⟨k, λ|σz|k,−λ⟩|2f(Ek,λ)−f(Ek,−λ)\n2Ek,λ+ℏω+i0,\n→ −Z\ndED (E)E2−M2\n2E2f(E)−f(−E)\n2E+ℏω+i0, (9)\nwhere |k, λ⟩is an eigenstate of HMwith eigenenergy\nEk,λ=λp\n(ℏvk)2+M2, (λ=±).M=J1Scosϑis\nthe Majorana gap, f(E) = 1 /(eE/kBT+ 1) is the Fermi3\ndistribution function, and D(E) is the density of states\ngiven by\nD(E) =L2\n2π(ℏv)2|E|θ(|E| − |M|), (10)\nwith the Heaviside step function θ(x). It is important to\nnote that the behavior of the uniform spin susceptibil-\nity is determined by the interband contribution, which is\nproportional to the Fermi distribution function, i.e., the\ncontribution of the occupied states. This mechanism is\nsimilar to the Van Vleck paramagnetism [22]. The con-\ntribution of the occupied states often plays a crucial role\nin topological responses [23].\nReplacing the localized spin operators with their statis-\ntical average values, we find the induced Majorana Ising\nspin density, to the first order of J1S, is given by\nZ\ndr⟨sz(r, t)⟩= ˜χzz\n0(0,0)J1Scosϑ\n+ Re[˜ χzz\n0(0, ω)]hac\nαhdcJ1Ssinϑsinωt, (11)\nwhere ˜ χzz\n0(0,0) is the spin susceptibility for M= 0. The\nfirst term originates from HZand gives a static spin den-\nsity, while the second term originates from HTand gives\na dynamic spin density. Figure 2 shows the induced Ising\nspin density as a function of time at several angles. As\nshown in Eq. (11), the Ising spin density consists of the\nstatic and dynamic components. The dynamic compo-\nnent is induced by the precessional motion of the local-\nized spin, which means one can induce the DMISR using\nthe dynamics of the localized spin.\nThe inset in Fig. 2 shows Im˜ χzz(0, ω) as a function of\nϑat a fixed frequency. When the frequency ℏωis smaller\nthan the Majorana gap, Im˜ χzz(0, ω) is zero. Once the\nfrequency overcomes the Majorana gap, Im˜ χzz(0, ω) be-\ncomes finite. The implications of these behaviors are that\nif the magnon energy is smaller than the Majorana gap,\nthere is no energy dissipation due to the DMISR. How-\never, once the magnon energy exceeds the Majorana gap,\nfinite energy dissipation associated with the DMISR oc-\ncurs at the surface of the TSC. Therefore, one can toggle\nbetween dissipative and non-dissipative Majorana Ising\nspin dynamics by adjusting the precession axis angle and\nthe microwave frequency.\nFMR modulation.— The retarded component of the\nmagnon Green’s function is given by GR(rn, t) =\n−(i/ℏ)θ(t)⟨[S+\nn(t), S−\n0(0)]⟩with the interaction picture\nS±\nn(t) =eiHFIt/ℏS±\nne−iHFIt/ℏ. The FMR signal is char-\nacterized by the spectral function defined as\nA(q, ω) :=−1\nπIm\"X\nne−iq·rnZ\ndtei(ω+i0)tGR(rn, t)#\n.\n(12)\nSSImχzz(0, ω) ˜⟨s   z⟩\n2\n1ωtϑ\nFInon-dissipativenon-dissipativedissipativedissipativeTSC\nFITSC000.00.51.0\nπ/4\nπ/2\n0 π/4 π/20\nϑ2π\nπFIG. 2. The induced Ising spin density, with a unit\n˜χzz\n0(0,0)J1S, is presented as a function of ωtandϑ. The\nfrequency and temperature are set to ℏω/J1S= 1.5 and\nkBT/J 1S= 0.1, respectively. The coefficient, hac/αhdc, is\nset to 0 .3. The static Majorana Ising spin density arises\nfrom HZ. When the precession axis deviates from the di-\nrection perpendicular to the interface, the precessional mo-\ntion of the localized spins results in the dynamical Majorana\nIsing spin response (DMISR). Energy dissipation due to the\nDMISR is zero for small angles ϑas the Majorana gap ex-\nceeds the magnon energy. However, once the magnon energy\novercomes the Majorana gap, the energy dissipation becomes\nfinite. Therefore, one can toggle between dissipative and non-\ndissipative DMISR by adjusting ϑ.\nFor uniform external force, the spectral function is given\nby\nA(0, ω) =2S\nℏ1\nπ(α+δα)ω\n[ω−γ(hdc+δh)]2+ [(α+δα)ω]2.\n(13)\nThe peak position and width of the FMR signal is given\nbyhdc+δhandα+δα, respectively. hdcandαcorre-\nspond to the peak position and the linewidth of the FMR\nsignal of the FI alone. δhandδαare the FMR modu-\nlations due to the exchange interaction HT. We treat\nHM+HFI+HZas an unperturbed Hamiltonian and HT\nas a perturbation. In this work, we assume the specular\ninterface, where the coupling J(r,rn) is approximated\nasDP\nn,n′J(r,rn)J(r′,rn′)E\nave=J2\n1. The dynamics\nof the localized spins in the FI is modulated due to the\ninteraction between the localized spins and the Majo-\nrana Ising spins. In our setup, the peak position and the\nlinewidth of the FMR signal are modulated and the FMR4\nmodulation is given by\nδh= sin2ϑSJ2\n1\n2NγℏRe˜χzz(0, ω), (14)\nδα= sin2ϑSJ2\n1\n2NℏωIm˜χzz(0, ω), (15)\nwhere Nis the total number of sites in the FI. These for-\nmulas were derived in the study of the FMR in magnetic\nmultilayer systems including superconductors. One can\nextract the spin property of the Majorana mode from the\ndata on δhandδα. Because of the Ising spin anisotropy,\nthe FMR modulation exhibits strong anisotropy, where\nthe FMR modulation is proportional to sin2ϑ.\nFigure 3 shows the FMR modulations (a) δαand (b)\nδh. The FMR modulation at a fixed frequency increases\nwith angle ϑand reaches a maximum at π/2, as can be\nread from Eqs. (14) and (15). When the angle ϑis fixed\nand the frequency ωis increased, δαbecomes finite above\na certain frequency at which the energy of the magnon\ncoincides with the Majorana gap. When ϑ < π/ 2 and\nℏω≈2M,δαlinearly increases as a function of ωjust\nabove the Majorana gap. The localized spin damping is\nenhanced when the magnon energy exceeds the Majorana\ngap. At ϑ=π/2 and ω≈0, the Majorana gap vanishes\nandδαis proportional to ω/T. In the high frequency\nregion ℏω/J 1S≫1,δαconverges to its upper threshold.\nThe frequency shift δhis almost independent of ωand\nhas a finite value even in the Majorana gap. This behav-\nior is analogous to the interband contribution to the spin\nsusceptibility in strongly spin-orbit coupled band insula-\ntors, and is due to the fact that the effective Hamiltonian\nof the Majorana modes includes spin operators. It is im-\nportant to emphasize that although the Majorana modes\nhave spin degrees of freedom, only the zcomponent of the\nspin density operator is well defined. This is a hallmark\nof Majorana modes, which differs significantly from elec-\ntrons in ordinary solids. Note that δhis proportional to\nthe energy cutoff, which is introduced to converge energy\nintegral for Re˜ χzz(0, ω). The energy cutoff corresponds\nto the bulk superconducting gap, which is estimated as\n∆∼0.1[meV] ( ∼1[K]). Therefore, our results are ap-\nplicable in the frequency region below ℏω∼0.1[meV]\n(∼30[GHz]). In addition, we assume that Majorana gap\nis estimated to be J1S∼0.01[meV] ( ∼0.1[K]).\nDiscussion.— Comparing the present results with spin\npumping (SP) in a conventional metal-ferromagnet hy-\nbrid, the qualitative behaviors are quite different. In con-\nventional metals, spin accumulation occurs due to FMR.\nIn contrast, in the present system, no corresponding spin\naccumulation occurs due to the Ising anisotropy. Also, in\nthe present calculations, the proximity-induced exchange\ncoupling is assumed to be an isotropic Heisenberg-like\ncoupling. However, in general, the interface interaction\ncan also be anisotropic. Even in such a case, it is no qual-\nitative change in the case of ordinary metals, although a\n0.00.5\n(a) (b)\nϑℏω/J1S 0\nπ/4\nπ/2024\nϑℏω/J1S 0\nπ/4\nπ/2024δ  α δ  h10\n0FIG. 3. The temperature is set to kBT/J 1S= 0.1. (a)\nThe damping modulation δαonly becomes finite when the\nmagnon energy exceeds the Majorana gap; otherwise, it van-\nishes. This behavior corresponds to the energy dissipation of\nthe Majorana Ising spin. (b) The peak shift is finite, except\nforϑ= 0, and is almost independent of ω. This behavior\nresembles the spin response observed in strongly spin-orbit\ncoupled band insulators, where the interband contribution to\nspin susceptibility results in a finite spin response, even within\nthe energy gap.\ncorrection term due to anisotropy is added [24]. There-\nfore, the Ising anisotropy discussed in the present work\nis a property unique to the Majorana modes and can\ncharacterize the Majorana excitations.\nLet us comment on the universal nature of the toggling\nbetween non-dissipative and dissipative dynamical spin\nresponses observed in our study. Indeed, such toggling\nbecomes universally feasible when the microwave fre-\nquency and the energy gap are comparable, and when the\nHamiltonian and spin operators are non-commutative,\nindicating that spin is not a conserved quantity. The\nnon-commutativity can be attributed to the presence of\nspin-orbit couplings [25–27], and spin-triplet pair corre-\nlations [28].\nMicrowave irradiation leads to heating within the FI,\nso that thermally excited magnons due to the heating\ncould influence the DMISR. Phenomena resulting from\nthe heating, which can affect interface spin dynamics, in-\nclude the spin Seebeck effect (SSE) [29], where a spin\ncurrent is generated at the interface due to a tempera-\nture difference. In hybrid systems of normal metal and\nFI, methods to separate the inverse spin Hall voltage due\nto SP from other signals caused by heating have been\nwell studied [30]. Especially, it has been theoretically\nproposed that SP and SSE signals can be separated us-\ning a spin current noise measurement [24]. Moreover, SP\ncoherently excites specific modes, which qualitatively dif-\nfers from SSE induced by thermally excited magnons [14].\nTherefore, even if heating occurs in the FI in our setup,\nthe properties of Majorana Ising spins are expected to\nbe captured. Details of the heating effect on the DMISR\nwill be examined in the near future.\nWe also mention the experimental feasibility of our the-\noretical proposals. As we have already explained, the\nFMR modulation is a very sensitive spin probe. Indeed,\nthe FMR modulation by surface states of 3D topological5\ninsulators [31] and graphene [32–36] has been reported\nexperimentally. Therefore, we expect that the enhanced\nGilbert damping due to Majorana Ising spin can be ob-\nservable in our setup when the thickness of the ferromag-\nnetic insulator is sufficiently thin.\nFinally, it is pertinent to mention the potential candi-\ndate materials where surface Majorana Ising spins could\nbe detectable. Notably, UTe 2[37], Cu xBi2Se3[38, 39],\nSrxBi2Se3and Nb xBi2Se3[40] are reported to be in a p-\nwave superconducting state and theoretically can host\nsurface Majorana Ising spins. Recent NMR measure-\nments indicate that UTe 2could be a bulk p-wave su-\nperconductor in the Balian-Werthamer state [41], which\nhosts the surface Majorana Ising spins with the per-\npendicular Ising anisotropy, as considered in this work.\nAxBi2Se3(A= Cu, Sr, Nb) is considered to possess in-\nplane Ising anisotropy [8], differing from the perpendic-\nular Ising anisotropy explored in this work. Therefore,\nwe expect that it exhibits anisotropy different from that\ndemonstrated in this work.\nConclusion.— We present herein a study of the spin\ndynamics in a topological superconductor (TSC)-magnet\nhybrid. Ferromagnetic resonance under microwave irra-\ndiation leads to the dynamically induced Majorana Ising\nspin density on the TSC surface. One can toggle between\ndissipative and non-dissipative Majorana Ising spin dy-\nnamics by adjusting the external magnetic field angle and\nthe microwave frequency. Therefore, our setup provides\na platform to detect and control Majorana excitations.\nWe expect that our results provide insights toward the\ndevelopment of future quantum computing and spintron-\nics devices using Majorana excitations.\nAcknowledgments.— The authors are grateful to R.\nShindou for valuable discussions. This work is partially\nsupported by the Priority Program of Chinese Academy\nof Sciences, Grant No. XDB28000000. We acknowl-\nedge JSPS KAKENHI for Grants (Nos. JP20K03835,\nJP21H01800, JP21H04565, and JP23H01839).\nSUPPLEMENTAL MATERIAL\nSurface Majorana modes\nIn this section, we describe the procedure for deriv-\ning the effective Hamiltonian of the surface Majorana\nmodes. We start with the bulk Hamiltonian of a three-\ndimensional topological superconductor. Based on the\nbulk Hamiltonian, we solve the BdG equation to demon-\nstrate the existence of a surface-localized solution. Us-\ning this solution, we expand the field operator and show\nthat it satisfies the Majorana condition when the bulk\nexcitations are neglected. As a result, on energy scales\nmuch smaller than the bulk superconducting gap, the\nlow-energy excitations are described by surface-localized\nMajorana modes. The above procedure is explained inmore detail in the following. Note that we use rfor three-\ndimensional coordinates and r∥for two-dimensional ones\nin the Supplemental Material.\nWe start with the mean-field Hamiltonian given by\nHSC=1\n2Z\ndrΨ†\nBdG(r)HBdGΨBdG(r), (16)\nwithr= (x, y, z ). We consider the Balian-Werthamer\n(BW) state, in which the pair potential is given by\n∆ˆk=∆\nkF\u0010\nˆk·σ\u0011\niσywith the bulk superconducting gap\n∆. Here, we do not discuss the microscopic origin of the\npair correlation leading to the BW state. As a result, the\nBdG Hamiltonian HBdGis given by\nHBdG=\nεˆk−EF 0 −∆\nkFˆk−∆\nkFˆkx\n0 εˆk−EF∆\nkFˆkx∆\nkFˆk+\n−∆\nkFˆk+∆\nkFˆkx−εˆk+EF 0\n∆\nkFˆkx∆\nkFˆk− 0 −εˆk+EF\n,\n(17)\nwith ˆk±=ˆky±iˆkz,ˆk=−i∇, and εˆk=ℏ2ˆk2\n2m. The four\ncomponent Nambu spinor ΨBdG(r) is given by\nΨBdG(r) :=\nΨ→(r)\nΨ←(r)\nΨ†\n→(r)\nΨ†\n←(r)\n, (18)\nwith the spin quantization axis along the xaxis. The\nmatrices of the spin operators are represented as\nσx=\u00121 0\n0−1\u0013\n, (19)\nσy=\u0012\n0 1\n1 0\u0013\n, (20)\nσz=\u00120−i\ni0\u0013\n. (21)\nThe fermion field operators satisfy the anticommutation\nrelations\n{Ψσ(r),Ψσ′(r′)}= 0, (22)\n{Ψσ(r),Ψ†\nσ′(r′)}=δσσ′δ(r−r′), (23)\nwith the spin indices σ, σ′=→,←.\nTo diagonalize the BdG Hamiltonian, we solve the BdG\nequation given by\nHBdGΦ(r) =EΦ(r). (24)\nWe assume that a solution is written as\nΦ(r) =eik∥·r∥f(z)\nu→\nu←\nv→\nv←\n, (25)6\nwithk∥= (kx, ky) and r∥= (x, y). If we set the four\ncomponents vector to satisfy the following equation (Ma-\njorana condition)\n\n0 0 1 0\n0 0 0 1\n1 0 0 0\n0 1 0 0\n\nu→\nu←\nv→\nv←\n=±\nu→\nu←\nv→\nv←\n, (26)\nwe can obtain a surface-localized solution. If we take a\npositive (negative) sign, we obtain a solution localized\non the top surface (bottom surface). As we will consider\nsolutions localized on the bottom surface below, we take\na negative sign. Finally, we obtain the normalized eigen-\nvectors of the BdG equation given by\nΦλ,k∥(r) =eik∥·r∥\n√\nL2fk∥(z)uλ,k∥, (27)\nwith\nfk∥(z) =Nk∥sin(k⊥z)e−κz, (28)\nNk∥=s\n4κ(k2\n⊥+κ2)\nk2\n⊥, (29)\nκ=m∆\nℏ2kF, (30)\nk⊥=q\nk2\nF−k2\n∥−κ2, (31)\nand\nu+,k∥=\nu+,→k∥\nu+,←k∥\nv+,→k∥\nv+,←k∥\n=1√\n2\nsinϕk∥+π/2\n2\n−cosϕk∥+π/2\n2\n−sinϕk∥+π/2\n2\ncosϕk∥+π/2\n2\n,(32)\nu−,k∥=\nu−,→k∥\nu−,←k∥\nv−,→k∥\nv−,←k∥\n=1√\n2\n−cosϕk∥+π/2\n2\n−sinϕk∥+π/2\n2\ncosϕk∥+π/2\n2\nsinϕk∥+π/2\n2\n.(33)\nThe eigenenergy is given by Eλ,k∥=λ∆k∥/kF. We can\nshow that the eigenvectors satisfy\nu−,−k∥=u+,k∥. (34)\nConsequently, the field operator is expanded as\nΨBdG(r) =X\nk∥\u0012\nγk∥eik∥·r∥\n√\nL2+γ†\nk∥e−ik∥·r∥\n√\nL2\u0013\n×fk∥(z)u+,k∥+ (bulk modes) ,(35)\nwhere γk∥(γ†\nk∥) is the quasiparticle creation (annihila-\ntion) operator with the eigenenergy E+,k∥. Substitutingthe above expression into Eq. (16) with omission of bulk\nmodes and performing the integration in the z-direction,\nwe obtain the effective Hamiltonian for the surface states\nHM=1\n2Z\ndr∥ψT(r∥)\u0010\nℏvˆkyσx−ℏvˆkxσy\u0011\nψ(r∥),(36)\nwhere v= ∆/ℏkFand we introduced the two component\nMajorana field operator\nψ(r∥) =\u0012ψ→(r∥)\nψ←(r∥)\u0013\n, (37)\nsatisfying the Majorana condition\nψσ(r∥) =ψ†\nσ(r∥), (38)\nand the anticommutation relation\nn\nψσ(r∥), ψσ′(r′\n∥)o\n=δσσ′δ(r∥−r′\n∥). (39)\nThe spin density operator of the Majorana mode is\ngiven by\ns(r∥) =ψ†(r∥)σ\n2ψ(r∥). (40)\nThexcomponent is given by\nsx(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00121/2 0\n0−1/2\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=1\n2\u0002\nψ†\n→(r∥)ψ→(r∥)−ψ†\n←(r∥)ψ←(r∥)\u0003\n=1\n2\u0002\nψ2\n→(r∥)−ψ2\n←(r∥)\u0003\n= 0. (41)\nIn a similar manner, the yandzcomponents are given\nby\nsy(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00120 1/2\n1/2 0\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=1\n2\u0002\nψ†\n→(r∥)ψ←(r∥) +ψ†\n←(r∥)ψ→(r∥)\u0003\n=1\n2\b\nψ→(r∥), ψ←(r∥)\t\n= 0, (42)\nand\nsz(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00120−i/2\ni/2 0\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=−i\n2\u0000\nψ†\n→(r∥)ψ←(r∥)−ψ†\n←(r∥)ψ→(r∥)\u0001\n=−iψ→(r∥)ψ←(r∥), (43)\nrespectively. As a result, the spin density operator is\ngiven by\ns(r∥) =\u0000\n0,0,−iψ→(r∥)ψ←(r∥)\u0001\n. (44)\nOne can see that the spin density of the Majorana mode\nis Ising like.7\nMajorana Ising spin dynamics\nIn this section, we calculate the Ising spin density in-\nduced on the TSC surface by the proximity coupling Hex.\nHexconsists of two terms, HZandHT.HZleads to the\nstatic spin density and HTleads to the dynamic spin\ndensity. First, we calculate the static spin density. Next,\nwe calculate the dynamic spin density.\nThe total spin density operator is given by\nsz\ntot=Z\ndr∥sz(r∥). (45)\nThe statistical average of the static spin density is calcu-\nlated as\n⟨sz\ntot⟩=−X\nk∥M\n2Ek∥\u0002\nf(Ek∥)−f(−Ek∥)\u0003\n→ −\u0012L\n2πℏv\u00132Z∆\nMEdEZ2π\n0dϕM\n2E[f(E)−f(−E)]\n=−Z∆\n0dED (E)f(E)−f(−E)\n2EM. (46)\nAt the zero temperature limit T→0, the static spin\ndensity is given by\n⟨sz\ntot⟩=1\n2L2\n2π(ℏv)2(∆−M)M≈˜χzz\n0(0,0)M, (47)\nwhere ˜ χzz\n0(0,0) = D(∆)/2 and we used ∆ ≫M.\nThe dynamic spin density is given by the perturbative\nforce\nHT(t) =Z\ndr∥sz(r∥)F(r∥, t), (48)\nwhere F(r∥, t) is given by\nF(r∥, t) =−sinϑX\nnJ(r∥,rn)\nSX\nn(t)\u000b\n≈ −sinϑJ1Sγhacp\n(ω−γhdc)2+α2ω2cosωt\n=:Fcosωt. (49)\nThe time dependent statistical average of the Ising spin\ndensity, to the first order of J1S, is given by\nZ\ndr∥\nsz(r∥, t)\u000b\n=Z\ndr∥Z\ndr′\n∥Z\ndt′χzz(r∥−r′\n∥, t′)F(r′\n∥, t−t′)\n= Re\u0002\n˜χzz(0, ω)Fe−iωt\u0003\n≈Re[˜χzz\n0(0, ω)]Fcosωt, (50)\nwhere we used Re˜ χzz\n0(0, ω)≫Im˜χzz\n0(0, ω). The real part\nof ˜χzz(0, ω) is given by\nRe˜χzz(0, ω) =−PZ\ndED (E)E2−M2\n2E2f(E)−f(−E)\n2E+ℏω,\n(51)where Pmeans the principal value. When the integrand\nis expanded with respect to ω, the lowest order correc-\ntion term becomes quadratic in ω. In the frequency range\nconsidered in this work, this correction term is signifi-\ncantly smaller compared to the static spin susceptibility\nRe˜χzz(0,0). Therefore, the spin susceptibility exhibits\nalmost no frequency dependence and remains constant\nas a function of ω. The imaginary part of ˜ χzz(0, ω) is\ngiven by\nIm˜χzz(0, ω)\n=πD(ℏω/2)(ℏω/2)2−M2\n2(ℏω/2)2[f(−ℏω/2)−f(ℏω/2)].\n(52)\nFMR modulation due to the proximity exchange\ncoupling\nIn this section, we provide a brief explanation for the\nderivation of the FMR modulations δhandδα. The FMR\nmodulations can be determined from the retarded com-\nponent of the magnon Green’s function, which is given\nby\n˜GR(k, ω) =2S/ℏ\nω−ωk+iαω−(2S/ℏ)ΣR(k, ω),(53)\nwhere we introduce the Gilbert damping constant αphe-\nnomenologically. In the second-order perturbation calcu-\nlation with respect to HT, the self-energy is given by\nΣR(k, ω) =−\u0012sinϑ\n2\u00132X\nq∥|˜J(q∥,k)|2˜χzz(q∥, ω),(54)\nwhere ˜J(q∥,0) is given by\n˜J(q∥,k) =1\nL2√\nNZ\ndr∥X\nnJ(r∥,rn)ei(q∥·r∥+k·rn)\n(55)\nThe pole of ˜GR(k, ω) signifies the FMR modulations,\nincluding both the frequency shift and the enhanced\nGilbert damping. These are given by\nδh=2S\nγℏReΣR(0, ω), δα =−2S\nℏωImΣR(0, ω).(56)\nFrom the above equations and Eq. (54), it is apparent\nthat FMR modulations provide information regarding\nboth the properties of the interface coupling and the dy-\nnamic spin susceptibility of the Majorana modes.\nThe form of matrix element ˜J(q∥,0) depends on the\ndetails of the interface. In this work, we assume the\nspecular interface. |˜J(q∥,0)|2is given by\n|˜J(q∥,0)|2=J2\n1\nNδq∥,0. (57)8\nUsing Eq. (57), the self-energy for the uniform magnon\nmode is given by\nΣR(0, ω) =−\u0012sinϑ\n2\u00132J2\n1\nN˜χzz(0, ω). (58)\n[1] F. Wilczek, Nat. Phys. 5, 614 (2009).\n[2] A. Yazdani, F. von Oppen, B. I. Halperin, and A. Ya-\ncoby, Science 380, eade0850 (2023).\n[3] S. B. Chung and S.-C. Zhang, Phys. Rev. Lett. 103,\n235301 (2009).\n[4] M. Sato and S. Fujimoto, Phys. Rev. B 79, 094504 (2009).\n[5] Y. Nagato, S. Higashitani, and K. Nagai, J. Phys. Soc.\nJpn.78, 123603 (2009).\n[6] R. Shindou, A. Furusaki, and N. Nagaosa, Phys. Rev. B\n82, 180505 (2010).\n[7] K. Shiozaki and M. Sato, Phys. Rev. B 90, 165114 (2014).\n[8] Y. Xiong, A. Yamakage, S. Kobayashi, M. Sato, and\nY. Tanaka, Crystals 7, 58 (2017).\n[9] Z. Qiu, J. Li, D. Hou, E. Arenholz, A. T. N’Diaye, A. Tan,\nK.-i. Uchida, K. Sato, S. Okamoto, Y. Tserkovnyak, Z. Q.\nQiu, and E. Saitoh, Nat. Commun. 7, 12670 (2016).\n[10] W. Han, S. Maekawa, and X.-C. Xie, Nat. Mater. 19,\n139 (2020).\n[11] Y. Tserkovnyak, A. Brataas, and G. E. Bauer, Phys.\nRev. Lett. 88, 117601 (2002).\n[12] Y. Tserkovnyak, A. Brataas, G. E. Bauer, and B. I.\nHalperin, Reviews of Modern Physics 77, 1375 (2005).\n[13] M. Inoue, M. Ichioka, and H. Adachi, Phys. Rev. B 96,\n024414 (2017).\n[14] T. Kato, Y. Ohnuma, M. Matsuo, J. Rech, T. Jonck-\nheere, and T. Martin, Phys. Rev. B 99, 144411 (2019).\n[15] M. A. Silaev, Phys. Rev. B 102, 144521 (2020).\n[16] M. A. Silaev, Phys. Rev. B 102, 180502 (2020).\n[17] Y. Ominato, A. Yamakage, T. Kato, and M. Matsuo,\nPhys. Rev. B 105, 205406 (2022).\n[18] Y. Ominato, A. Yamakage, and M. Matsuo, Phys. Rev.\nB106, L161406 (2022).\n[19] See Supplemental Material for the derivation of the effec-\ntive Hamiltonian of surface Majorana modes, the deriva-\ntion of the expression for Majorana Ising spin density,\nthe calculation of dynamic spin density and dynamic spin\nsusceptibility, and the formulation of ferromagnetic res-\nonance modulation due to proximity exchange coupling.\n[20] T. Holstein and H. Primakoff, Physical Review 58, 1098\n(1940).\n[21] In a realistic sample, correction terms would be added\ndue to interface roughness [17, 18]. Even with such cor-\nrection terms, the characteristic angular dependence is\nexpected to be preserved, since the Majorana Ising spinproperty of having spin density only in the perpendicular\ndirection is maintained.\n[22] A. Neil W. and M. N. David, Solid State Physics (Saun-\nders College Publishing, 1976).\n[23] S. Murakami, Phys. Rev. Lett. 97, 236805 (2006).\n[24] M. Matsuo, Y. Ohnuma, T. Kato, and S. Maekawa,\nPhys. Rev. Lett. 120, 037201 (2018).\n[25] A. Thakur, K. Sadhukhan, and A. Agarwal, Phys. Rev.\nB97, 035403 (2018).\n[26] J. Zhou and H.-R. Chang, Phys. Rev. B 97, 075202\n(2018).\n[27] Y. Ominato and K. Nomura, Phys. Rev. B Condens. Mat-\nter97, 245207 (2018).\n[28] M. Sigrist and K. Ueda, Reviews of Modern physics 63,\n239 (1991).\n[29] K. Uchida, S. Takahashi, K. Harii, J. Ieda, W. Koshibae,\nK. Ando, S. Maekawa, and E. Saitoh, Nature 455, 778\n(2008).\n[30] R. Iguchi and E. Saitoh, journal of the physical society\nof japan 86, 011003 (2017).\n[31] C. Tang, Q. Song, C.-Z. Chang, Y. Xu, Y. Ohnuma,\nM. Matsuo, Y. Liu, W. Yuan, Y. Yao, J. S. Moodera,\nS. Maekawa, W. Han, and J. Shi, Sci. Adv. 4, eaas8660\n(2018).\n[32] A. K. Patra, S. Singh, B. Barin, Y. Lee, J.-H. Ahn, E. del\nBarco, E. R. Mucciolo, and B. ¨Ozyilmaz, Appl. Phys.\nLett. 101, 162407 (2012).\n[33] Z. Tang, E. Shikoh, H. Ago, K. Kawahara, Y. Ando,\nT. Shinjo, and M. Shiraishi, Phys. Rev. B Condens. Mat-\nter87, 140401 (2013).\n[34] J. B. S. Mendes, O. Alves Santos, L. M. Meireles, R. G.\nLacerda, L. H. Vilela-Le˜ ao, F. L. A. Machado, R. L.\nRodr´ ıguez-Su´ arez, A. Azevedo, and S. M. Rezende,\nPhys. Rev. Lett. 115, 226601 (2015).\n[35] D. Indolese, S. Zihlmann, P. Makk, C. J¨ unger, K. Thod-\nkar, and C. Sch¨ onenberger, Phys. Rev. Appl. 10, 044053\n(2018).\n[36] J. B. S. Mendes, O. Alves Santos, T. Chagas, R. Mag-\nalh˜ aes Paniago, T. J. A. Mori, J. Holanda, L. M. Meireles,\nR. G. Lacerda, A. Azevedo, and S. M. Rezende, Phys.\nRev. B 99, 214446 (2019).\n[37] S. Ran, C. Eckberg, Q.-P. Ding, Y. Furukawa, T. Metz,\nS. R. Saha, I.-L. Liu, M. Zic, H. Kim, J. Paglione, and\nN. P. Butch, Science 365, 684 (2019).\n[38] K. Matano, M. Kriener, K. Segawa, Y. Ando, and G.-Q.\nZheng, Nat. Phys. 12, 852 (2016).\n[39] S. Yonezawa, K. Tajiri, S. Nakata, Y. Nagai, Z. Wang,\nK. Segawa, Y. Ando, and Y. Maeno, Nat. Phys. 13, 123\n(2017).\n[40] M. M. Sharma, P. Sharma, N. K. Karn, and V. P. S.\nAwana, Supercond. Sci. Technol. 35, 083003 (2022).\n[41] H. Matsumura, H. Fujibayashi, K. Kinjo, S. Kitagawa,\nK. Ishida, Y. Tokunaga, H. Sakai, S. Kambe, A. Naka-\nmura, Y. Shimizu, Y. Homma, D. Li, F. Honda, and\nD. Aoki, J. Phys. Soc. Jpn. 92, 063701 (2023)."    },    {        "title": "0807.1625v2.Magnetic_field_distribution_in_the_quiet_Sun__a_simplified_model_approach.pdf",        "content": "arXiv:0807.1625v2  [astro-ph]  3 Aug 2008Astronomy& Astrophysics manuscriptno.9683aph c/circleco√yrtESO 2018\nNovember30,2018\nMagneticfield distribution inthe quietSun:\na simplifiedmodel approach\nF.Berrilli, D.Del Moro, and B.Viticchi` e\nDipartimentodi Fisica,Universit` a degli Studidi Roma “To r Vergata”,Viadella Ricerca Scientifica1, I-00133 Roma, It aly\ne-mail:viticchie@roma2.infn.it\nPreprintonline version: November 30, 2018\nABSTRACT\nContext. The quiet Sun presents magnetized plasma whose field strengt hs vary from zero to about 2 kG. The probability density\nfunction of the magnetic fieldstrength Befficaciously describes the statisticalproperties of the quie t Sunmagnetic field.\nAims.We simulate the dynamics and the evolution of quiet Sun magne tic elements to produce a probability density function of th e\nfieldstrengths associated withsuch elements.\nMethods. The dynamics of the magnetic field are simulated by means of a n umerical model in which magnetic elements are driven\npassivelybyanadvection fieldcharacterized byspatio-tem poral correlations thatmimickthe granulationand mesogra nulation scales\nobservedonthesolarsurface.Thefieldstrengthcanincreas eduetoanamplificationprocess thatoccurswherethemagnet icelements\nconverge. Startingfrom a δ-like probability density function centered on B=30 G, we obtain magnetic field strengths of up to 2 kG\n(inabsolute value). Toderive the statisticalproperties o f the magnetic elements, several simulationruns are perfor med.\nResults. Our model is able to produce kG magnetic fields in a time interv al of the order of the granulation timescale. The mean\nunsigned flux density and the mean magnetic energy density of the synthetic quiet Sun reach values of /angbracketleftB/angbracketright≃100 G and/angbracketleftB2/angbracketright1/2≃\n350 G respectively inthe stationaryregime. The derived pro bability density function of the magnetic fieldstrength dec reases rapidly\nfromB=30 G toB∼100 G and has a secondary maximum at B=2 kG. From this result, it follows that magnetic fields ≥700 G\ndominatetheunsignedfluxdensityandmagneticenergydensi ty,althoughtheprobabilitydensityfunctionofthefieldst rengthreaches\na maximum at B∼10G.\nConclusions. Aphotospheric advectionfieldwithspatio-temporalcorrel ations,drivingthemagneticelements,andreducedmagneti c\namplificationrulesareabletocreatearealisticprobabili tydensityfunctionofthequietSunmagneticfield.Ithasbee nfoundthatthey\nnaturallyproduce anexcess of magnetic fieldsaround 2kG ifa nupper limitisimposed on the fieldstrength.\nKey words. Sun: magnetic fields–Sun: photosphere\n1. Introduction\nWhen observing the Sun in full disk magnetograms, magnetic\nconcentrations define the network pattern associated with\nthe supergranular motions of photospheric plasma, while th e\ninteriorofthenetworkpatternappearstobeunmagnetized.\nWe know that weak polarization signals are measur-\nable almost everywhere in the interior of network cells\n(Livingston&Harvey 1975; Smithson 1975). These signals\nare associated with the internetwork component of the solar\nmagnetic field, also called the quiet Sun magnetic field, whic h\ncovers more than 90% of the solar surface, independent of the\nsolar activity (Harvey-Angle 1993). Due to its large di ffusion\nonto the solar surface, the quiet Sun magnetic field may be\nuseful for understanding the physics of the photospheric ma g-\nnetism.Infact,itcouldholdalargefractionoftheunsigne dflux\ndensity and magnetic energy density of the solar photospher e\n(e.g. Unno 1959; Stenflo 1982; S´ anchezAlmeida 1998, 2004;\nSchrijver&Title 2003).\nThe weak polarization signals measured in quiet Sun regions\ncan be interpreted as the result of the linear combination\nof the polarization emerging from discrete magnetic flux\ntubes, which are typically smaller than the angular resolut ion,\nthat fill the solar atmosphere with a complex topology (e.g.\nEmonet&Cattaneo2001a; S´ anchezAlmeidaet al.2003).\nKnowledge of the quiet Sun has been enhanced sig-nificantly by improvements in instrumentation (e.g.\nSanchezAlmeidaet al.1996;Lin&Rimmele1999;Lites2002;\nDom´ ınguezCerde˜ naet al. 2003a,b; Khomenkoetal. 2003),\nin diagnostic techniques (e.g. Stenflo 1982; Faurobert-Sch oll\n1993; Faurobert-Scholletal. 1995; Landidegl’Innocenti\n1998; Rees etal. 2000; S´ anchezAlmeida&Lites 2000;\nSocas-Navarroet al. 2001; Socas-Navarro&S´ anchezAlmeid a\n2002; S´ anchezAlmeida 2004; TrujilloBuenoet al.\n2004; MansoSainzet al. 2004; S´ anchezAlmeida 2005;\nMart´ ınezGonz´ alezetal. 2006b), and numerical simula-\ntions (e.g. Cattaneo 1999; Emonet& Cattaneo 2001b;\nStein& Nordlund 2002; V¨ ogler 2003; V¨ ogleret al. 2005;\nStein& Nordlund2006; V¨ ogler&Sch¨ ussler2007).\nThese improvements were crucial in characterizing the quie t\nSun magnetic field and its di fferent observational parameters.\nOne parameter, the fraction of solar surface occupied by a\ncertainfieldstrength,isa verypowerfultool.Infact,this canbe\ntriviallytranslatedintotheprobabilitydensityfunctio n(PDF)of\nthe magnetic field strength, P(B), which is useful in describing\nthe statistical properties of a quiet Sun region. For exampl e,\ncalculatingthefirstandthesecondmomentumof P(B)provides\ninformationaboutthe mean unsignedflux density andthe mean\nmagnetic energy density (apart from a constant factor 1 /8π),\nrespectively. Using a PDF assumes implicitly a continuum fo r\nvalues of the magnetic field strength. The limit imposed by\nthe pressure equilibrium between the magnetic flux tubes and2 F.Berrilli,D.Del Moro andB.Viticchi´ e:Quiet SunPDF:as implifiedmodel approach\nphotospheric plasma also defines the variability range for q uiet\nSunmagneticfieldstobebetween0Gto about2kG.\nLin (1995) computed the PDF in active and quiet Sun regions\nusing Feiλ15648 and Feiλ15652 lines, while Lites (2002)\nobtained it from the analysis of network regions and their\nsurroundings using Fe ilines at 6300 Å. Both authors found a\nbimodal distribution with a dominant contribution at ≃150 G\nand a secondarypeak at ≃1.5kG. In contrast,Collados(2001),\nstudying quiet Sun regions using the same infrared lines as\nLin (1995), found an exponential decline in the magnetic flux\ndensity histogram from a peak at ≃200 G and no counts for\nmagneticfluxdensity >∼1kG.Dom´ ınguezCerde˜ naet al.(2006)\ndeveloped a set of PDFs for the quiet Sun magnetic fields\nthat was consistent with observations and magnetoconvecti on\nsimulations by using information from both Hanle and Zeeman\nobservations; their study created substantial interest be cause\nthe functions increased for strong fields, just before a cut- off\natB≃1.8 kG. This characteristic implies that strong fields are\nimportant because, even when present in a small fraction of\nthe photosphere, they dominate the unsigned flux density and\nmagnetic energy density. On the other hand, the magnetic fiel d\ndistributions proposed by Mart´ ınezGonz´ alezet al. (2006 a,b)\nand Mart´ ınezGonz´ alezetal. (2008), which were derived fr om\nsimultaneous infrared and visible observations of Fe ilines,\ndo not present any rise for strong fields. The inversion of the\nFeiλ6301 and Feiλ6302 profiles observed by the HINODE\nsatellite (OrozcoSu´ arezet al. 2007)confirmsthisresult.\nMagnetoconvection simulations provide a wide variety of\nquiet Sun magnetic field PDFs to be compared with those\nderived from observations. Simulations show, for exam-\nple, exponential decays (Cattaneo 1999; Steinet al. 2003;\nStein& Nordlund 2006; V¨ ogler& Sch¨ ussler 2007), bimodal\nbehaviors (V¨ ogleret al. 2003; V¨ ogler& Sch¨ ussler 2003), or\nan “exponential-like core and a modest peak” for B∼1 kG\n(Steiner2003).\nAn important step is to ascertain the presence or absence of a\nbump for magnetic fields ∼1 kG in quiet Sun magnetic field\nstrength PDFs; since the magnetic flux and magnetic energy\nboth scale as powers of the field strength, kG magnetic fields\nshould play a crucial role in defining the global properties\nof quiet Sun magnetic fields. The connectivity between the\nlower and the upper solar atmosphere may indeed be defined\nby the larger field strength concentrations in the photosphe re\n(Dom´ ınguezCerde˜ naet al.2006).\nS´ anchezAlmeida (2007) indicated that a bump in the PDF\nfor quiet Sun fields ∼kG is to be expected if “a magnetic\namplificationmechanismoperatesin thequiet Sun”.Theauth or\nderived an equation for the temporal evolution of the PDF,\nwhose shape can be modified by magnetic amplification and\nphotospheric convection. In the stationary state, the mode l is\ncharacterizedbya P(B)with abumpforstrongfields.\nIn our model, we implement the dynamics of a high number\nof photospheric magnetic elements, driven by a photospheri c\nvelocity field in which a large-scale organization (mesogra nu-\nlation) is naturally derived from the cooperation of small- scale\nflows (granulation). The model also takes into account simpl i-\nfied processes that are able to amplify and limit the magnetic\nfield strength. In§2, we describe the model, in §3, the results\nofthesimulationarepresented,whilein §4,theyarediscussed.\nFinally,in§5, theconclusionsareoutlined.\nFig.1.Illustrative representationof |B|in the computationaldo-\nmain. Black areas represent0 G regions;blue and sky-bluere p-\nresent∼10Gregions;greenrepresents ∼100Gregions;yellow\nandredrepresent∼kGmagneticconcentrations.Thepositionof\nconcentrationswith B∼kG correspondto the positionof stable\ndownwardplumesorganizedovermesogranularscales.\n2. Themodel\nThe model simulates the dynamics and evolution of photo-\nspheric magnetic elements driven by a photospheric velocit y\nfield characterized by spatio-temporal correlations. To re pro-\nduce these correlations, we simulated the photospheric vel oc-\nity field in a way similar to that proposed by Rast (2003); it\nwas computed by an n-body simulation that realizes the inter-\nactionbetween downflowplumesby consideringonlytheir mu-\ntual horizontal interaction, as in Rast (1998). The movemen t of\neach plume was determined by the advection produced by the\nother plumes. This model spontaneously creates stable down -\nward plumes,organizedona mesogranularpattern,fromthe i n-\nteraction of granular scale flows. The ratios between the mes o-\ngranular and granular spatio-temporal scales, over which t he\nmodelself-organizes,agreewiththeratiosmeasuredinthe solar\nphotosphere.Viticchi´ eet al.(2006)utilizedsuchaveloc ityfield\nto drive magnetic loop footpoints in a reconnection model th at\nreproducestheobservedstatistical propertiesnanoflares .\nIn the present work,the simulationsstart with the advectio n\nfield only, without any magnetic element in the domain. When\nthe velocityfield hasreacheda stationarybehavior,i.e.af ter the\nonset of mesogranular scale flows, the time tis set to zero and\nN0=500magneticelementsaredistributedatrandompositions\noverthe computationaldomain.Themagneticelementshavea n\nassociated field strength Bin=30 G with random orientation\n(±1) and size equal to the spatial resolution element. Hereaft er,\nBrepresents the absolute value of the magnetic field strength .\nWechoosesuchavaluefor Binsinceitisapproximatelyequalto\nthe minimum magnetic flux density measurable nowadays (e.g.\nTrujilloBuenoet al.2004;Bommieret al.2005;Carroll& Kop f\n2008). We allow the magnetic elements to be transported pas-\nsivelybythephotosphericvelocityfieldtowardsdi fferentplume\nsites.Ateachplumesite,acertainnumberofmagneticeleme nts\nconvergeundertheactionoftheadvectionfield.Whentwomag -F.Berrilli,D.Del Moro andB.Viticchi´ e:Quiet SunPDF:asi mplifiedmodel approach 3\nnetic elements, associated with B1andB2, overlap in the same\nresolutionelement,theycanproducethefollowingtworesu lts:\n1. If the two elements have opposite orientations, they are\nsubstituted by a new magnetic element at the same site with an\nassociated B=|B1−B2|andtheorientationofthestrongestele-\nment.Thetotalfluxisconservedsuchthatthespatialsizeof the\nnewelementremainsasoneresolutionelement.Inthepartic ular\ncase ofB1=B2, theelementscancelthee ffectofeachother.\n2. If the two elements have the same orientation, they are\nsubstituted by a new magnetic element in the same site with an\nassociated B=B1+B2and the same orientation.The total flux\nisagainconservedandthespatialsizeofthenewelementiso ne\nresolutionelement.\nThis process implicates an amplification of the magnetic fiel d\nstrength. Different mechanisms have been proposed to concen-\ntrate magnetic fields in quiet Sun regions up to the kG range\n(e.g. Weiss 1966; Parker 1978; Spruit 1979; S´ anchezAlmeid a\n2001). As reported by S´ anchezAlmeida (2007), all mecha-\nnisms are abruptly diminished at the maximum field strength\nimposed by the gas pressure of the quiet Sun photosphere.\nTherefore, we place the constraint that the amplification pr o-\ncess cannot create field strengths greater than Blim=2 kG.\nWe can interpret Blimas a representative value of the max-\nimum field strengths for quiet Sun regions (e.g. Parker\n1978; Spruit 1979; Grossmann-Doerthet al. 1998; Steinet al .\n2003;Dom´ ınguezCerde˜ naetal.2006;S´ anchezAlmeida200 7).\nWhenever the amplification process would produce B>Blim,\nthemagneticelementsdonotinteractandremainseparated.\nWe consider implicitly that the amplification mechanism op-\nerates on the same timescales as the granulation timescale\n(∼10 min). This hypothesis is consistent with the\namplification timescales reported by Parker (1978) and\nGrossmann-Doerthet al. (1998). Up to the equipartition val ue\nofabout500G,theamplificationcanbeascribedtothegranul ar\nflowsandfiveminutesaresu fficienttoproducekGfieldsfroma\nuniformdistributionof400G.\nAs proposed by Steiner (2003), starting from an initial PDF o f\nPt=0(B)=δ(B−Bin),the amplificationprocesswill broadenthe\nPDF andextendit to Blim.\nTosimulatetheemergenceofnewmagneticelementsontheso-\nlar surface, we continuously add randomly located elements of\nrandom orientation and Binmagnetic field strength. This injec-\ntion process maintains the total number of magnetic element s\n∼N0andcompensatesforthe reductionin the total numberdue\ntotheconcentrationprocess.\nFigure1showsanillustrativerepresentationofasnapshot ofthe\nsimulation in the stationary regime: we represent the |B|pattern\nemergingonthecomputationaldomain.\n3. PDFof thequiet Sunmagneticfield strengths\nTo derive a PDF of the quiet Sun magnetic field strengths, we\nperformedtensimulationrunswithdi fferentadvectionfieldsand\ndifferentinitialdistributionsofmagneticelements.Eachsim ula-\ntion reproduced the evolution of the quiet Sun for an equiva-\nlent time of about 2600 min. We considered the behavior of the\nmodelbyanalyzingitsaveragemaximumfieldstrength max(B),\nmeanunsignedfluxdensity /angbracketleftB/angbracketright,andmeanmagneticenergyden-\nsity/angbracketleftB2/angbracketright/8π,asfunctionsofthemeanevolutiontime t:\n<B>=/integraldisplayBlim\n0BP(B)dB,\n<B2>=/integraldisplayBlim\n0B2P(B)dB,\nFig.2.Averagemaximumfield strengthversusevolutiontime t.\nThe function is obtained by averaging the results from nRUN=\n10 distinct simulation runs and the error bars are calculate d to\nequalσ√nRUN.Thehorizontaldashedlinerepresentsthemaximum\nvalueallowedforthemagneticfluxdensity( Blim),whilethever-\nticaldot-dashedlineindicatestheinstantatwhichthemod elpro-\nducesthefirstkGelement.Theplotshowsonlyapartofthetot al\ntimedomain( t≃2600min).\nFig.3.Mean probability density function /angbracketleftP(B)/angbracketrightfor the mag-\nneticfieldstrengthinthestationaryregime. /angbracketleftP(B)/angbracketrighthasbeenob-\ntained averaging nTOT=40 non-correlated PDFs and the er-\nrorbarsare calculated to equalσ√nTOT. The histogrambin-sizeis\n30 G. The dashed line represents the upper limit for the mag-\nnetic amplification process Blim. The maximum at B=0 G is\nproduced by the fraction of the domain free of magnetic ele-\nments.\nina similarway toDom´ ınguezCerde˜ naet al. (2006).\nThe model creates magnetic concentrations with B=1 kG in\nabout30minand B≃Blimin about200min(Fig.2).\nThe quantity max(B) is unsuitable for describing the stationary\nregimeofthe magneticelement systembecause itsvalueis li m-\nited. As discussed in §2, we also expect to obtain a PDF span-\nningtheentiredomain B=0−Blim; thestationaryregimemust\ntherefore be identified referring to the integral propertie s of the\nPDF,suchasthemeanunsignedfluxdensityandthemeanmag-4 F.Berrilli,D.Del Moro andB.Viticchi´ e:Quiet SunPDF:as implifiedmodel approach\nFig.4.Meanprobabilitydensityfunction /angbracketleftP(B±)/angbracketrightforthe signed\nmagnetic field strength in the stationary regime. The histog ram\nbin-size is 30 G. The vertical dashed lines represent the upp er\nand lower limits of the magnetic amplification process ±Blim.\nThe maximum at B=0 G is produced by the fraction of the\ndomainfreeofmagneticelements.\nnetic energy density. /angbracketleftB/angbracketrightand/angbracketleftB2/angbracketright1/2saturate exponentially at\n/angbracketleftB/angbracketrightsat≃100 G and/angbracketleftB2/angbracketright1/2\nsat≃350 G, respectively. We inter-\npret this saturation as an indication that the entire system has\nreachedthestationaryregime.Inthisregime,wecompute /angbracketleftP(B)/angbracketright\nby averaging forty non-correlated PDFs. The computed /angbracketleftP(B)/angbracketright\nisshowninFig.3.\nAlternatively, we could compute the mean probability densi ty\nfunction/angbracketleftP(B±)/angbracketrightfor the signed magnetic field strength by con-\nsideringthedirectionofthemagneticfields(showninFig.4 ).As\nexpected,the/angbracketleftP(B±)/angbracketrightisquitesymmetricwithrespecttozero.\n4. Discussion\nInthe previoussectionwe presentedtheresultsofseverals imu-\nlations of our reducedmodel. We now focus on some particular\npointsindetail.\nThe maximum field strength, plotted in Fig. 2, as a function of\ntindicates that the timescale over which kG fields are produce d\nis approximately30 min.This is the same timescale overwhic h\nthe mesogranular pattern emerged in the numerical simulati on\nofViticchi´ eet al.(2006).\nDifferent descriptions of the magnetic amplification pro-\ncess (Grossmann-Doerthet al. 1998; S´ anchezAlmeida 2001)\nachieved similar results: weak magnetic-field structures, in a\ndownflow environment, develop field strengths ∼kG within a\nfew minutes. Berrilli et al. (2005) reportedthat the organi zation\nof structureson mesogranularboundariesis rapidwith a typ ical\ntimescaleof≃10min.\nThe analysis of the position of strong field magnetic element s\nin oursimulation revealedthat these are superimposedonto sta-\nble downward plumes on mesogranular scales (Fig. 1). Such a\ncorrelation between strong magnetic fields and the mesogran -\nular advection scale is in agreement with Dom´ ınguezCerde˜ na\n(2003), who found a correlation between mesogranular photo -\nsphericflowsandthepatternobservedin magnetograms.\nThe symmetry of the /angbracketleftP(B±)/angbracketright, reported in Fig. 4, confirms\nthe equal probability of having concentrations of positive and\nnegative direction. This originates in both the initial con di-tions and the new element addition criteria applied during\nthe evolution. Similar symmetries have been found in simu-\nlations (e.g., V¨ ogler&Sch¨ ussler 2007) and in observatio ns of\nMart´ ınezGonz´ alezetal. (2006a, 2008). In the latter work s the\nauthorspointedoutthat sucha symmetrymayimplya local dy-\nnamo origin for the internetwork magnetic field, that is a mag -\nneticfieldcontrolledbysolargranulation.\nThe sharp count decay found at B=Blimis obviously due to\nthe “interaction turn o ff” produced by Blim(§2). Other PDFs\nreported in the literature show a smooth decay of the peak for\nB≃1.8 kG (Dom´ ınguezCerde˜ naetal. 2006; S´ anchezAlmeida\n2007), as expected for natural PDFs. We could reproduce this\nsmoothdecayof/angbracketleftP(B)/angbracketrightforB/greaterorsimilarBlimbymodifyingthemagnetic\nelement interaction probability, but it is not within the sc ope of\nthisworkto reproducethese finedetailsofthe PDF.\nTo verify that the stationary regime has been reached, we com -\npared the/angbracketleftP(B)/angbracketrightwith four mean PDFs, Pt(B), representative\nof four different instants of the simulation during the station-\naryregime(Fig.5).Each Pt(B)representsthemeanoftenPDFs\nat a given time t. We consider the four Pt(B) to be uncorrelated\nbecause they are separated by more than 140 minutes, a time\ninterval that is longer than three times the mean mesogranul ar\nlifetime. As shown in Fig. 5, the deviation between the /angbracketleftP(B)/angbracketright\nandthefour Pt(B)showsnotimedependenceandtherootmean\nsquaredeviationsofallfour Pt(B)withrespecttothe /angbracketleftP(B)/angbracketrightare\napproximatelyonepartina thousand.\nThe values/angbracketleftB/angbracketrightsat≃100 G and/angbracketleftB2/angbracketright1/2\nsat≃350 G, reached in\nthe stationary regime, are in good agreement with the result s of\nDom´ ınguezCerde˜ naet al. (2006). The percentages of unsig ned\nflux density and magnetic energy density for B≥700 G are\n≃65% and≃97% respectively, indicating a dominant role\nfor kG fields. The percentages that we found are higher than\nthose reported by Dom´ ınguezCerde˜ naetal. (2006); these d if-\nferences can be explained by referring to the steeper decrea se\nin the/angbracketleftP(B)/angbracketrightfor the 0−700 G interval with respect to the log-\nnormaltrendfoundbytheseauthors.\nThe/angbracketleftP(B)/angbracketrightpresents a maximum at B=Binin agreement with\nDom´ ınguezCerde˜ naet al. (2006). This result originates i n the\ninjection value for the magnetic field strength Bin=30 G.\nAtB≃Blim, the PDF presents a secondary maximum, as\nfound by Steinet al. (2003), Dom´ ınguezCerde˜ naet al. (200 6),\nandS´ anchezAlmeida(2007).\nIn a similar way to S´ anchezAlmeida (2007), we may conclude\nthat this secondary maximum in the /angbracketleftP(B)/angbracketrightis due to the upper\nlimit placed on the amplification process at Blim. When a mag-\nneticamplificationprocessoperatesanda limitis placedon this\nprocess, an accumulation of field strength about the limit is ex-\npected. All assumptions of the model contribute in shaping t he\n/angbracketleftP(B)/angbracketright:thestrongconvergenceregionsofthegranularadvection\npattern create suitable conditions for amplifying the magn etic\nfield and,at the same time, supporttheamplificationprocess by\ngatheringnew magnetic elements to counteract the cancella tion\nprocess. The secondary maximum of the PDF is produced by\nboth the spatial and temporal correlations of the velocity fi eld,\nwhich advect the magnetic elements, and the upper limit of th e\namplificationprocessat Blim.\n5. Conclusions\nWe have presented a model that reproduces the dynamics and\nevolution of photospheric magnetic elements. These elemen ts\nare passively driven by an horizontal velocity field charact er-\nized by spatio-temporal correlations that agree with those ob-\nserved in the solar photosphere. The description of the velo cityF.Berrilli,D.Del Moro andB.Viticchi´ e:Quiet SunPDF:asi mplifiedmodel approach 5\nFig.5.Mean probability density function /angbracketleftP(B)/angbracketright(solid line,\nsame as Fig.3) compared with four Pt(B) (dotted lines) calcu-\nlated at different instants of the simulation. Each Pt(B) repre-\nsents the mean of ten PDFs from ten simulation runs at time\nt.Top left:t=2160 min, top right t=2308 min, bottom left\nt=2464 min and bottom right t=2593 min. The histogram\nbin-size is 30 G. The vertical dashed lines represent the upp er\nand lower limits of the magnetic amplification process ±Blim.\nThe maximum at B=0 G is produced by the fraction of the\ndomainfreeofmagneticelements.\nfield followsRast (1998, 2003). In their evolution,the magn etic\nelements interact and increase their magnetic field strengt hs\nthroughanamplificationprocessassumedtoworkonatimesca le\ncomparable with the granular timescale and able to recreate\nkG concentrations. The injection scale, constrained to be Bin=\n30 G, and the upper limit at Blim=2 kG are both derived from\nobservations and physical constraints (e.g. TrujilloBuen oet al.\n2004; Parker1978).\nWeinvestigatedthestatisticalpropertiesofthefieldstre ngthsas-\nsociated with the magneticelementsusing a seriesof numeri cal\nsimulations.Themainconclusionsare:\n1. The model produces kG fields in a time interval of the or-\nderofthemesogranulationtimescale.Thesestrongfieldsar e\norganizedinapatternsuperimposedonthemesogranulation\npatternoftheadvectionfield.\n2. In the stationary regime, the PDF of the magnetic field\nstrength has a stable shape and has a secondary maximum\nthat correspondstotheupperlimit set at2 kG.\n3. The associated mean unsigned flux density and mean mag-\nnetic energy density in the stationary regime are /angbracketleftB/angbracketrightsat≃\n100Gand/angbracketleftB2/angbracketright1/2\nsat≃350G, respectively.\n4. The magnetic element system described by this PDF is\nstrongly dominated by kG magnetic fields: the fractions of\nunsigned flux density and magnetic energy density for B≥\n700Gare≃65%and≃97%,respectively.\nThe quiet Sun is certainly more complex than the model pre-\nsented here. However, a more simplified approach, based on\nthe advectionfield proposedby Rast (2003), reproducedthe o b-\nserveddistributionsforthereleasedmagneticenergyandw aiting\ntimes of nanoflare events (Viticchi´ eet al. 2006). In the pre sent\nwork, we showed how the same approach can also be applied\nto reproduce some of the statistical properties of the quiet Sun\nmagneticfield.Acknowledgements. We are very grateful to the anonymous referee for invalu-\nable comments on the manuscript. Wethank Jorge S´ anchez Alm eida for fruitful\ndiscussionsaboutthetopic.Thispaperwaspartially suppo rtedbyRegioneLazio\nCVS (Centro per lo studio della variabilit` a del Sole) PhD gr ant and ASI/INAF\nI/015/07/0 ESSgrant.\nReferences\nBerrilli, F.,DelMoro,D.,Russo,S.,Consolini, G.,&Strau s, T.2005,ApJ,632,\n677\nBommier, V., Derouich, M., Landi degl’Innocenti, E., Molod ij, G., & Sahal-\nBr´ echot, S. 2005, A&A,432, 295\nCarroll, T.A.& Kopf,M.2008, A&A,481, L37\nCattaneo, F.1999, ApJ,515, L39\nCollados, M. 2001, in Astronomical Society of the Pacific Con ference\nSeries, Vol. 236, Advanced Solar Polarimetry – Theory, Obse rvation, and\nInstrumentation, ed. M.Sigwarth, 255– +\nDom´ ınguez Cerde˜ na, I.2003, A&A,412, L65\nDom´ ınguez Cerde˜ na, I.,Kneer,F.,&S´ anchez Almeida, J.2 003a, ApJ,582,L55\nDom´ ınguez Cerde˜ na, I., S´ anchez Almeida, J., & Kneer, F. 2 003b, A&A, 407,\n741\nDom´ ınguez Cerde˜ na, I.,S´ anchez Almeida, J.,& Kneer, F.2 006, ApJ,636, 496\nEmonet, T. & Cattaneo, F. 2001a, in Astronomical Society of t he Pacific\nConference Series, Vol. 236, Advanced Solar Polarimetry – T heory,\nObservation, and Instrumentation, ed.M. Sigwarth, 355– +\nEmonet, T.&Cattaneo, F.2001b, ApJ, 560,L197\nFaurobert-Scholl, M. 1993, A&A,268, 765\nFaurobert-Scholl, M., Feautrier, N., Machefert, F., Petro vay, K., & Spielfiedel,\nA.1995, A&A,298, 289\nGrossmann-Doerth, U.,Schuessler, M.,&Steiner, O.1998, A &A,337, 928\nHarvey-Angle, K.L.1993, PhD thesis, , Utrecht University, TheNetherlands\nKhomenko, E. V., Collados, M., Solanki, S. K., Lagg, A., & Tru jillo Bueno, J.\n2003, A&A,408, 1115\nLandi degl’Innocenti, E.1998, Nature, 392, 256\nLin,H. 1995, ApJ,446, 421\nLin,H. &Rimmele, T.1999, ApJ,514, 448\nLites, B. W.2002, ApJ, 573,431\nLivingston, W.C. & Harvey, J. 1975, in Bulletin of the Americ an Astronomical\nSociety, Vol. 7, Bulletin of the American Astronomical Soci ety, 346–+\nMansoSainz,R.,LandiDegl’Innocenti, E.,&Trujillo Bueno ,J.2004,ApJ,614,\nL89\nMart´ ınez Gonz´ alez, M. J., Collados, M., & Ruiz Cobo, B. 200 6a,\nin Astronomical Society of the Pacific Conference Series, Vo l. 358,\nAstronomical Society ofthePacificConference Series, ed.R .Casini&B.W.\nLites, 36–+\nMart´ ınez Gonz´ alez, M. J., Collados, M., & Ruiz Cobo, B. 200 6b, A&A, 456,\n1159\nMart´ ınez Gonz´ alez, M.J.,Collados, M.,RuizCobo,B.,&Be ck,C.2008,A&A,\n477, 953\nOrozco Su´ arez, D., Bellot Rubio, L. R., del Toro Iniesta, J. C., et al. 2007, ApJ,\n670, L61\nParker, E.N.1978, ApJ,221, 368\nRast, M. P.1998, Journal of Fluid Mechanics, 369, 125\nRast, M. P.2003, ApJ, 597,1200\nRees, D.E.,L´ opezAriste, A.,Thatcher, J.,&Semel, M.2000 , A&A,355, 759\nS´ anchez Almeida, J. 1998, in Astronomical Society of the Pa cific Conference\nSeries, Vol. 155, Three-Dimensional Structure of Solar Act ive Regions, ed.\nC. E.Alissandrakis & B.Schmieder, 54– +\nS´ anchez Almeida, J.2001, ApJ,556, 928\nS´ anchez Almeida, J. 2004, in Astronomical Society of the Pa cific Conference\nSeries, Vol. 325,TheSolar-B Mission and the Forefront ofSo lar Physics,ed.\nT.Sakurai & T.Sekii, 115– +\nS´ anchez Almeida, J.2005, A&A,438, 727\nS´ anchez Almeida, J.2007, ApJ,657, 1150\nS´ anchez Almeida, J.,Emonet, T.,&Cattaneo, F.2003, ApJ,5 85, 536\nSanchezAlmeida,J.,Landidegl’Innocenti, E.,MartinezPi llet,V.,&Lites,B.W.\n1996, ApJ,466, 537\nS´ anchez Almeida, J.&Lites, B. W.2000, ApJ,532, 1215\nSchrijver, C.J.& Title, A.M.2003, ApJ, 597,L165\nSmithson,R.C.1975,in Bulletin of theAmerican Astronomic al Society, Vol. 7,\nBulletin of theAmerican Astronomical Society, 346– +\nSocas-Navarro, H.,L´ opezAriste, A.,& Lites, B.W.2001, Ap J,553, 949\nSocas-Navarro, H.&S´ anchez Almeida, J.2002, ApJ,565, 132 3\nSpruit, H.C.1979, Sol. Phys.,61,363\nStein, R. F., Bercik, D., & Nordlund, Å. 2003, in Astronomica l Society of the\nPacific Conference Series, Vol. 286, Current Theoretical Mo dels and Future6 F.Berrilli,D.Del Moro andB.Viticchi´ e:Quiet SunPDF:as implifiedmodel approach\nHigh Resolution Solar Observations: Preparing for ATST, ed . A. A. Pevtsov\n&H.Uitenbroek, 121– +\nStein, R. F. & Nordlund, Å. 2002, in ESA Special Publication, Vol. 505,\nSOLMAG 2002. Proceedings of the Magnetic Coupling of the Sol ar\nAtmosphere Euroconference, ed. H.Sawaya-Lacoste, 83–89\nStein, R.F.&Nordlund, Å.2006, ApJ, 642,1246\nSteiner, O.2003, A&A,406, 1083\nStenflo, J.O. 1982, Sol.Phys.,80, 209\nTrujillo Bueno, J.,Shchukina, N.,&Asensio Ramos,A. 2004, Nature, 430,326\nUnno, W.1959, ApJ,129, 375\nViticchi´ e, B.,DelMoro, D.,&Berrilli, F.2006, ApJ,652, 1 734\nV¨ ogler, A.2003, PhDthesis, ,G¨ ottingen University, Germ any\nV¨ ogler, A.&Sch¨ ussler, M.2003, Astronomische Nachricht en, 324,399\nV¨ ogler, A.&Sch¨ ussler, M.2007, A&A,465, L43\nV¨ ogler,A.,Shelyag,S.,Sch¨ ussler,M.,etal.2003,inIAU Symposium,Vol.210,\nModelling of Stellar Atmospheres, ed. N. Piskunov, W. W. Wei ss, & D. F.\nGray, 157–+\nV¨ ogler, A.,Shelyag, S.,Sch¨ ussler, M.,et al. 2005, A&A,4 29, 335\nWeiss,N.O.1966, Royal Society of London Proceedings Serie s A,293, 310"    },    {        "title": "1406.5939v1.Magnetic_energy_dissipation_and_mean_magnetic_field_generation_in_planar_convection_driven_dynamos.pdf",        "content": "arXiv:1406.5939v1  [physics.flu-dyn]  23 Jun 2014APS/123-QED\nMagnetic energy dissipation and mean magnetic field generat ion in planar convection\ndriven dynamos\nA. Tilgner\nInstitute of Geophysics, University of G¨ ottingen,\nFriedrich-Hund-Platz 1, 37077 G¨ ottingen, Germany\n(Dated: October 10, 2018)\nA numerical study of dynamos in rotating convecting plane la yers is presented which focuses\non magnetic energies and dissipation rates, and the generat ion of mean fields (where the mean is\ntaken over horizontal planes). The scaling of the magnetic e nergy with the flux Rayleigh number is\ndifferent from the scaling proposed in spherical shells, whe reas the same dependence of the magnetic\ndissipation length on the magnetic Reynolds number is found for the two geometries. Dynamos both\nwith and without mean field exist in rapidly rotating convect ing plane layers.\nPACS numbers: 91.25.Cw, 47.65.-d\nIt is generally assumed that celestial bodies without a\nfossil magnetic field left over form the birth of the object\ncreate their magnetic fields through the dynamo effect,\nand that in most bodies, convectionis driving the motion\nofthefluidconductorwhosekineticenergyistransformed\ninto magnetic energy by the dynamo effect. Spherical\nshells and plane layers are two geometries in which con-\nvectiondrivendynamosareconvenientlystudiedwithnu-\nmerical simulations. More attention has been paid to the\nspherical shell because of its greater geo- and astrophys-\nical relevance and a larger data base exists for this ge-\nometry. A scaling for the magnetic field energy derived\nfrom these simulations [1] has matched observations well\n[2]and hasevenbeen invokedformechanicallydrivendy-\nnamos [3, 4], which raises the question of how universal\nthis scaling is. An obvious test is to compare convection\ndynamos in spherical shells with the most closely related\nstandard problem, which is convection driven dynamos\nin plane layers. Rotating convection in spherical shells\nis inhomogeneous in the sense that the region inside the\ncylinder tangent to the inner core and coaxial with the\nrotation axis behaves differently from the equatorial re-\ngion, and boundaries are curved. Convection in a plane\nlayer with its rotation axis perpendicular to the plane of\nthe layermaybe viewedasamodelforasmallregionsur-\nrounding the poles of a spherical shell. Are the physics in\nthis region representative for the rest? This question is\nthe motivation to look at the scaling of magnetic energy\nand energy dissipation in plane layer dynamos and to\ncompare the results with data from simulations in spher-\nical shells.\nField morphologies are more difficult to compare. Dy-\nnamos in spherical shells are frequently classified accord-\ning to whether they produce a magnetic field dominated\nby its dipole component (in which case they are a can-\ndidate for a model of the geodynamo) or not. A simi-\nlar distinction can be made among plane layer dynamos:\nThey either produce a mean field, obtained by averaging\nover horizontal planes, or not. Even though this issue is\nnot obviously analogous to the question of the dominat-\ning dipole field in the sphere, the end of this paper will\nbe devoted to showing that a transition in plane layerdynamos separates dynamos generating mean fields from\nthose who do not.\nThe model and the numerical method used here are\nthe same as in ref. 5 and are briefly reviewed here for\ncompleteness. The parameters of the numerical runs are\nthe same, too, except for some additional simulations at\nlarger aspect ratios. Consider a plane layer with bound-\naries perpendicular to the zaxis. Rotation Ω and grav-\nitational acceleration gare parallel and antiparallel to\nthis axis, respectively. The fluid in the layer has density\nρ, kinematic viscosity ν, thermal diffusivity κ, thermal\nexpansion coefficient α, and magnetic diffusivity λ. The\nboundaries are located in the planes z= 0 and z=d\nand periodic boundary conditions are applied in the lat-\neral directions imposing the periodicity lengths lxandly\nalong the xandydirections. In all simulations, lx=ly,\nand the aspect ratio Ais defined as A=lx/d. Four\nadditional control parameters govern magnetic rotating\nconvectionwithin the Boussinesqapproximation, namely\nthe Rayleigh number Ra, the Ekman number Ek, the\nPrandtl number Pr, and the magnetic Prandtl number\nPm. They are defined by\nRa =gα∆Td3\nκν,Ek =ν\nΩd2,Pr =ν\nκ,Pm =ν\nλ\n(1)\nwhere ∆Tis the temperature difference between bottom\nand top boundaries. With d,d2/κ,κ/d,ρκ2/d2, ∆Tand√µ0ρκ/dasunits oflength, time, velocity, pressure, tem-\nperature difference from the temperature at z=d, and\nmagnetic field, respectively, the nondimensional equa-\ntions for the velocity field v(r,t) as a function of po-\nsitionrand time t, the magnetic field B(r,t) and the\ntemperature field T(r,t), are given by:\n∂tρ+∇·v= 0 (2)\n∂tv+(v·∇)v+2Pr\nEkˆz×v=\n−c2∇ρ+Pr Ra θˆz+Pr∇2v+(∇×B)×B(3)\n∂tθ+v·∇θ−vz=∇2θ (4)2\n∂tB+∇×(B×v) =Pr\nPm∇2B (5)\n∇·B= 0 (6)\nwhereθis the deviationfromthe conductivetemperature\nprofile. Thenumericalcodeimplements anartificialcom-\npressibilitymethod [6] with the equationofstate p=c2ρ,\nwith pressure pand sound speed c. The standard Boussi-\nnesq equations, with eq. (2) replaced by ∇ ·v= 0 and\nwith the term −c2∇ρreplaced by ∇pin eq. (3), are\nrecovered in the limit of ctending to infinity. In all sim-\nulations, cwas chosen large enough to approximate well\nthe Boussinesq equations [5].\nEq. (2)isthefullcontinuityequationlinearizedaround\na density equal to 1, ρbeing the density perturbation.\nOnly∇ρenters the momentum equation so that we can\nset the unperturbed density to an arbitrary constant.\nThe system with the linearized continuity equation re-\nduces to the same Boussinesq limit for ctending to in-\nfinity as the full system, it also satisfies conservation of\nmass, and it is computationally more efficient because it\navoids round off errors in the term 1+ ρappearing in the\nfull continuity equation.\nOnemayalsowonderifitwouldnotbemoreefficientto\nsimulate the Boussinesq equations directly. Suppose we\nare content to approximate the Boussinesq solutions to\nanaccuracyof1%becauseweexpecterrorsduetolimited\ntimeaveragingoflargermagnitude. Theerrorintroduced\nby a finite sound speed is of the order ( U/c)2, whereUis\nthe typical flow velocity. We thus need c≈10Ufor the\ndesiredaccuracy. Withanexplicittimesteppingmethod,\nthe time step will need to be 10times smallerfor the arti-\nficial compressibility method than for the simulations of\nthe Boussinesq equations, assuming the advection CFL\ncriterion limits the size of the time step. However, ev-\nery time step solving the Boussinesq equations requires\nthe solution of a Poisson equation. One therefore has to\ncompare the execution time of 10 explicit time steps and\none Poisson inversion to decide which method is better\nsuited. The computations presented here solved eqs. (2-\n6)withafinitedifferencemethodimplementedongraphi-\ncalprocessingunits [5], whicharehighlyparallelwithrel-\nativelyslowcommunicationbetweensomecomponentsof\nthe board, so that the artificial compressibility method\nwas favored.\nThe boundary conditions implemented at the top and\nbottom boundaries were fixed temperature ( θ= 0), free\nslip (vz=∂zvy=∂zvx= 0), and a perfect conductor was\nassumedoutsidethefluidlayer( Bz=∂zBy=∂zBx= 0).\nSpatialresolutionwasupto2563points. Inallruns,Pr\nwas set to 0 .7, and Pm to either 1 or 3. For both Pm, the\nEkof2×10−4, 2×10−5,and2×10−6havebeensimulated.\nFor each of the six combinations of Pm and Ek, Ra was\nvariedfromits criticalvalueto upto100timescriticalfor\nEk = 2×10−4and three times critical for Ek = 2 ×10−6.\nThetypicallengthscaleofrotatingconvectionvarieswith\nEk as Ek1/3near the onset of convection and throughoutmuch of the range of Rayleigh numbers investigated here\n[7]. Accordingly, the aspect ratio Awas chosen to be\nA= 1, 1/2 and 1/4 for Ek = 2 ×10−4, 2×10−5and\n2×10−6, respectively. The aspect ratio dependence of\nthe mean magnetic field will be discussed towards the\nend of the paper.\nThedensitiesofkineticandmagneticenergies, ekinand\neB, are given by\nekin=1\nV/integraldisplay1\n2v2dV , e B=1\nV/integraldisplay1\n2B2dV,(7)\nwheretheintegrationextendsovertheentirefluidvolume\nV. Ifwedenotethetimeaveragebyangularbrackets,one\ncan compute average energy densities EkinandEBfrom\nEkin=∝angb∇acketleftekin∝angb∇acket∇ightandEB=∝angb∇acketlefteB∝angb∇acket∇ightas well as the Reynolds\nnumber Re and the magnetic Reynolds number Rm from\nRe =∝angb∇acketleft√\n2ekin∝angb∇acket∇ight/Pr,Rm = RePm .(8)\nIn the previous study of this model [5], it was found\nthat there is a transition at RmEk1/3= 13.5. The\ncombination RmEk1/3is proportional to the magnetic\nReynolds number based on the size of a columnar vortex\nnear the onset of convection. As the Rayleigh number is\nincreased starting from small values, the growth rate of\nkinematic dynamos first increases, then goes through a\nminimum at RmEk1/3= 13.5 and then increases again.\nThe growth rate is not a monotonic function of neither\nRm nor Ra at constant Ek. The amplitude of the sat-\nurated magnetic field obeys different scaling laws below\nand above this transition. These are given in ref. 5 in\nterms of Rm, Ek, and Pm. The Rm is not a control\nparameter of the problem, but it is more accessible to\nobservations than Ra, so that these scaling laws are of\ninterest even if they are not expressed in terms of control\nparameters only.\nAnotherparameterofgreaterrelevancetoobservations\nthan Ra is the flux Rayleigh number, Ra f, based on the\nheat flux. If the fluid is at rest, the heat flux across the\nlayer is purely diffusive and given by κρcp∆T/dwhere\ncpis the heat capacity at constant pressure. When\nconvection sets in, the heat flux may be written as\nκρcp∆T/d+Qadv, whereQadvis the difference between\nthe actual heat flux and the diffusive heat flux through\nthe fluid at rest. The Nusselt number Nu is defined as\nNu = 1+ Qadv/(κρcp∆T/d) (9)\nand\nRaf= Ra(Nu −1)Ek3/Pr = (gαQadv)/(ρcpΩ3d2).\n(10)\nThe flux Rayleigh number is independent of diffusivities,\nand the heat flux is better constrained by observations\nthan the temperature difference ∆ T.\nIt would be interesting to know a relation between the\nsaturation magnetic field strength and Ra f. From their\nsimulations in spherical shells, Christensen and Aubert\n[1] find (EB/fΩ)(Ek/Pr)2= (0.76Raf0.32Pm0.11)2where3\n10-710-610-510-410-310-210-1100\n10-910-810-710-610-510-410-310-210-1100EB (Ek/Pr)2 / fΩ\nRaf Pm1/3\n10-710-610-510-410-310-210-1100\n10-1110-1010-910-810-710-610-510-410-310-210-1EB (Ek/Pr)2 / fΩ\nRaf Ek1/3 Pm4/9\nFIG. 1. (Color online) ( EB/fΩ)(Ek/Pr)2as a function of\nRafPm1/3(top panel) and Ra fEk1/3Pm4/9(bottom panel).\nResults for Pm= 1 are shown in blue and those for Pm= 3\nare in red. For Pm= 1, the Ekman numbers of 2 ×10−4,\n2×10−5, and 2×10−6are indicated by the plus sign, trian-\ngle down, and circle, respectively, whereas for Pm= 3, the\nsame Ekman numbers are indicated by the x sign, triangle\nup, and square. The straight lines show power laws with the\nexponents 2/3 (top panel) and 3/5 (bottom panel).\nfΩis the ratio of ohmic to total dissipation, which in the\nunits used here is given by\nfΩ=ǫB/Pm\n(ǫv+ǫB/Pm)(11)\nwith\nǫB=1\nV/integraldisplay\n<(∇×B)2> dV (12)\nand\nǫv=1\nV/integraldisplay\n<(∂jvi)(∂jvi)> dV, (13)\nwhere summation over repeated indices is implied and\ntheintegrationextendsoverthewholecomputationalvol-\nume. The form of this scaling comes from an attempt to\ndetermine the magnetic field strength not from a balance\nof forces but from energy considerations. One can derive10-510-410-310-2\n105106107108109101010111012EB / εB Pm\n(Nu-1) Ra Pr-2 Pm-2/3\n10-510-410-310-2\n105106107108109101010111012EB / εB Pm\n(Nu-1) Ra Pr-2 Pm-2/3\nFIG. 2. (Color online) ( EB/ǫB)Pm as a function of Ra(Nu −\n1)Pr−2Pm−2/3with the same symbols as in fig. 1. The\nstraight lines indicate power laws with the exponents −2/5\n(solid line) and −1/3 (dashed line). The bottom panel con-\ntains the same data as the top panel but shows only points\nbelow the transition with RmEk1/3<13.5 and Pm = 3.\nfromthe equationsofevolution(2-6) (in the limit oflarge\nsound speed c, i.e. in the standard Boussinesq limit) the\nenergy budget\nǫv+ǫB\nPm= (Nu−1)Ra. (14)\nFor the spherical dynamo models with the radial varia-\ntionofgravityusuallysimulated, anexactequationofthe\nsame structure is not available, but a fit in ref. [1] shows\nthat the total dissipation is still approximately propor-\ntional to (Nu −1)Ra. The purely ohmic dissipation is\nrelated to the total dissipation by the factor fΩby def-\ninition, and EB/ǫBis the square of a magnetic length\nscale,lB, with\nlB=/radicalbig\nEB/ǫB. (15)\nThe magnetic dissipation time, defined as the ratio of\nmagnetic energy and ohmic dissipation, made nondimen-\nsional with the ohmic diffusion time, is also given by\nEB/ǫB. Ref. [1] finds an acceptable fit for lBas a func-\ntion of the control parameters of the flow, which together\nwith the fit for the total dissipation rate as a function4\nof (Nu−1)Ra leads to a relation between EB/ǫBand\nthe control parameters. A more elaborate fitting proce-\ndure [8] in which one searches directly a power law fit for\nEB/fΩas a function of Ra f, Ek and the Prandtl numbers\nleads to ( EB/fΩ)(Ek/Pr)2= (0.60Raf0.31Pm0.16)2. For\nthe purpose of the discussion below, we can round the\nexponents to\nEB\nfΩ/parenleftbiggEk\nPr/parenrightbigg2\n∝/parenleftBig\nRafPm1/3/parenrightBig2/3\n. (16)\nThe data available for the plane layer will not allow us to\ndetermine an exponent for Pm, and the analysis of the\nRafdependence will not depend on discrepancies of 0.01\nin the exponent. Note also that the factor Ek /Pr on the\nleft hand side is due to the different units of magnetic\nfield used here and in ref. 1. Eq. (16) has no predictive\npower for EBunless one guesses fΩ. However, an upper\nbound for fΩis 1, resulting in an upper bound for EBif\nfΩis set to 1 in Eq. (16).\nFigure 1 shows ( EB/fΩ)(Ek/Pr)2as a function of\nRafPm1/3for the plane layer dynamos and eq. (16)\nseems to provide a satisfying fit. Remarkably, there is\nno trace of a transition between different types of dy-\nnamos in this plot. However, one can simplify Eq. (16)\nby using the energy budget (14) in order to obtain\nEB\nǫBPm∝/parenleftBig\nRa(Nu−1)Pr−2Pm−2/3/parenrightBig−α\n.(17)\nwithα= 1/3. This equation is simpler than eq. (16)\nbecause common factorsRa(Nu −1) and Ekare removed.\nFigure 2 shows ( EB/ǫB)Pm as a function of Ra(Nu −\n1)Pr−2Pm−2/3. Because of the removal of the common\nfactors, the data points spread over fewer decades and\nit becomes apparent that α= 1/3 is not an acceptable\nexponent, neitherasafittothedatacloudasawhole,nor\nto the points below the transition at RmEk1/3= 13.5,\nnor to individual series of simulations at Ek, Pm and\nPr constant. Instead, the best fitting exponent is close\ntoα= 2/5. This exponent describes the dependence\non Ra(Nu −1). The dependence on Pm and Pr is not\nseriously tested by the data.\nWe can now reinflate eq. (17) for α= 2/5 with the\nhelp of the energy budget to obtain a relation analogous\nto eq. (16), which becomes\nEB\nfΩ/parenleftbiggEk\nPr/parenrightbigg2\n= 0.55/parenleftBig\nRafEk1/3Pm4/9/parenrightBig3/5\n.(18)\nwhere the prefactor is taken from fig. 1 which shows eq.\n(18) to be a satisfactory fit, again. An Ek dependence of\nthe right hand side therefore appears in eq. (18). In the\nspherical models on the contrary, the best fit does not\ncontain any Ek dependence (see table 4 of ref. 8).\nFor completeness, fig. 3 plots fΩas a function of\nRmEk1/3. It is plausible that fΩis small for dynamos\nclose to the onset in the case of a supercritical bifur-\ncation and that fΩapproaches 1 as the magnetic field00.20.40.60.81\n100101102103fΩ\nRm Ek1/3\nFIG. 3. (Color online) fΩas a function of RmEk1/3with the\nsame symbols as in fig. 1.\n10-210-1\n100101102103Ek-1/2 EB / εB\nRm Ek1/3\nFIG. 4. (Color online) Ek−1/2EB/ǫBas a function of\nRmEk1/3with the same symbols as in fig. 1. The straight\nlines indicate power laws with the exponents −5/6 (solid line)\nand−1 (dashed line).\ngrowsstronger. However, fΩisalready0.7atthesmallest\nRmEk1/3in fig. 3. This supports the scenario of a sub-\ncritical convection driven dynamo in plane layers [9, 10].\nAccording to [5], a second type of dynamo operates for\nRmEk1/3>13.5, and in this range, fΩis increasing as a\nfunction of RmEk1/3as expected. When scalings of EB\nare sought in terms of Rm and Ek, these two types of\ndynamos have to be considered separately [5], but they\ncan be fitted simultaneously in a graph of EB/fΩlike fig.\n1 because the complications of the transition are hidden\ninfΩ.\nThe magnetic length scale lBintroduced in eq. (15)\nis connected to the total dissipation Ra(Nu −1) in Eq.\n(17). It is more natural to seek a relation between lB\nand Rm. From spherical shell simulations, ref. 11 infers\nEB/ǫB∝1/Rm, whereas a more extended analysis [12]\nyieldedEB/ǫB∝Rm−5/6(Ek/Pm)1/6. There is no theo-\nretical basis for this relationship, it is at present a purely\nempirical finding. Fig. 4 shows Ek−1/2EB/ǫBas a func-5\n10-410-310-210-1100\n100101102103ÐE / (EB-ÐE)\nRm Ek1/3\nFIG. 5. ¯E/(EB−¯E) as a function of RmEk1/3forA= 0.5\n(circles), 1 (crosses) and 2 (stars). The dashed line follow s the\nprediction of first order smoothing and shows ¯E/(EB−¯E)∝\n(RmEk1/3)−2. The solid lines plot the functions 0 .09/√x,\n0.09/(4√x), and 0.09/(16√x).\ntion of RmEk1/3and confirms the dependence of lBon\nRm in Rm−5/6which is therefore identical in spherical\nand in planar geometry, and also confirms the Ek depen-\ndence found in ref. [12] to within a factor Ek1/18, which\nis too small to be discerned in the data.\nFig. 4 shows that the behavior of the magnetic dis-\nsipation length lBis not affected by the transition at\nRmEk1/3= 13.5 and that it behaves the same for the\ntwo types of dynamos, above and below the transition.\nThe variable RmEk1/3does on the other hand decide on\nwhether a mean field is generated. The energy in the\nmean field, ¯E, is computed as\n¯E=1\n21\nV</integraldisplay\ndz/parenleftbigg/integraldisplay\ndy/integraldisplay\ndxB/parenrightbigg2\n> .(19)\nIt is well known that close to the onset of dynamo action\nin rapidly rotating plane layer convection, the generated\nmagnetic field is dominated by its mean field component\n[9]. The dynamo is then accessible to the tools of mean\nfield magnetohydrodynamics and first order smoothing\n[13] which predict ¯E/(EB−¯E)∝(RmEk1/3)−2. In\nthe simulations presented here, the ratio ¯E/(EB−¯E)\nwas smaller than 0.01 at the highest RmEk1/3. The\nsimulations at Ek = 2 ×10−5and Pm = 3 have been\ncomplemented by simulations at different aspect ratios.\nMost points have been obtained at an aspect ratio of\n0.5, and a few points have been added for aspect ra-\ntios 1 and 2. The result is shown in fig. 5. If the as-\npect ratio is increased for points below the transition at\nRmEk1/3<13.5, one observes variations in both Rm\nand¯Ewhich increase as one approaches the transition.\nHowever, the variation in ¯E/(EB−¯E) is always less than\nby a factor of 2 even if the aspect ratio changes by a fac-\ntor of 4. Above the transition, on the other hand, an\nincrease of the aspect ratio Aby a factor of 2 always\nreduces ¯E/(EB−¯E) by a factor of 4. This behavior isreadily understood if one assumes that these dynamos do\nnot genuinely generate a mean field, but that the statis-\ntical fluctuations of the local field do not cancel exactly\nin a volume of finite size. Assume that the magnetic field\nhas a correlation length lc. The number of independent\ndegrees of freedom in a plane of cross section A×Ais\n(A/lc)2. The mean field computed in each plane is the\nsum of (A/lc)2random numbers drawn from a probabil-\nity distribution with a width proportional to√EB, so\nthat¯E/(EB−¯E)≈¯E/EB∝(lc/A)2. Doubling Athus\nreduces ¯E/(EB−¯E) by a factor of 4.\nThe evidence thus points at a dynamo without a mean\nmagnetic field above the transition, even though in any\nnumerical realization, the mean field is not exactly zero\nbut depends on the aspect ratio. Below the transition,\nthe dynamo does generate a mean field, but as its ampli-\ntude is small, the contribution from the statistical fluc-\ntuations of the mean field introduces some aspect ratio\ndependence in these dynamos as well.\nFavier and Bushby [14] also found in their simulations\nof dynamos in rotating compressible convection a mean\nfield which decreaseswith increasingaspect ratio, so that\nthe mean field detected in these simulations may well be\na statistical feature as described above. Cattaneo and\nHughes [15] simulate dynamos which produce magnetic\nenergy spectra which peak at small scales suggesting a\ndynamo process at small scales (similarly to [14]). They\nfor example present a case with RmEk1/3around 200\n(which is clearly above the transition) at the relatively\nlarge Ek of 2 .8×10−3and an aspect ratio of 10 and find\nasexpected asmallvaluefor EB/ǫBonthe orderof10−4.\nLarge mean fields were observed on the other hand in\nrefs. [9, 10, 16]. Stellmach and Hansen [9] used the ex-\nact same model as here, but simulated Rayleigh numbers\ncloserto onset than in the present study, so that the exis-\ntence of an important mean field is not surprising. Refs.\n[10, 16] used Rayleigh numbers a few times and up to\nten times critical, and Ekman numbers comparable to\nthese in the present study, so that these dynamos should\nbe examples of dynamos below the transition. The au-\nthors found mean fields about 2-3 times as large as here.\nOne may speculate that this is due to different boundary\nconditions: For the perfectly conducting boundaries used\nhere, the average over zof the mean field must be zero\n[10], but this constraint does not exist for the insulating\nboundaries used in refs. [10, 16].\nIn summary, convection in rotating plane layers sup-\nports dynamos both with and without a mean field. The\nscaling exponents for the energy and the magnetic dissi-\npationlength inferredfrom simulationsin sphericalshells\nat first glance fit perfectly well the data from the plane\nlayer. However, closer inspection reveals the field energy\nscaling proposed for the spherical shell to be unaccept-\nable for the plane layer data. Of course, more aspects\nof the model than the boundary geometry have been\nchanged in going from the usual spherical dynamo simu-\nlationtotheplanelayermodelpresentedhere,suchasthe\nspatial variation of gravity and the magnetic boundary6\nconditions, and it is not yetpossible to tell which ofthose\nfeatures is relevant for the magnetic field scalings. The\npresent work at any rate leads us to also expect differ-\nences between different spherical models, such as modelswith different ratios of outer and inner radii, with dif-\nferent radial dependencies of gravity, or with different\nboundary conditions.\n[1] U. Christensen and J. Aubert, Geophys. J. Int., 166, 97\n(2006).\n[2] U. Christensen, V. Holzwarth, and A. Reiners, Nature,\n457, 167 (2009).\n[3] C. Dwyer, D. Stevenson, and F. Nimmo, Nature, 479,\n212 (2011).\n[4] M. Le Bars, M. Wieczorek, O. Karatekin, D. C´ ebron,\nand M. Laneuville, Nature, 479, 215 (2011).\n[5] A. Tilgner, Phys. Rev. Lett., 109, 248501 (2012).\n[6] A. Chorin, J. Comp. Phys., 2, 12 (1967).\n[7] S. Schmitz and A. Tilgner, Geophys. Astrophys. Fluid\nDynam., 104, 481 (2010).[8] Z. Stelzer and A. Jackson, Geophys. J. Int., 193, 1265\n(2013).\n[9] S. Stellmach and U. Hansen, Phys. Rev. E, 70, 056312\n(2004).\n[10] C. Jones andP. Roberts, J. FluidMech., 404, 311(2000).\n[11] U. Christensen and A. Tilgner, Nature, 429, 169 (2004).\n[12] U. Christensen, Space Science Rev., 152, 565 (2004).\n[13] A. Soward, Phil. Trans. R.Soc. Lond.A, 275, 611(1974).\n[14] B. Favier and P. Bushby,J. Fluid Mech., 723, 529 (2013).\n[15] F. Cattaneo and D. Hughes, J. Fluid Mech., 553, 401\n(2006).\n[16] J. Rotvig and C. Jones, Phys. Rev. E, 66, 056308 (2002)."    },    {        "title": "2205.05379v1.Homotopy_transitions_and_3D_magnetic_solitons.pdf",        "content": "Homotopy transitions and 3D magnetic solitons\nV. M. Kuchkin1, 2and N. S. Kiselev1\n1)Peter Grünberg Institute and Institute for Advanced Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich,\nGermany\n2)Department of Physics, RWTH Aachen University, 52056 Aachen, Germany\n(*Electronic mail: v.kuchkin@fz-juelich.de)\n(Dated: 12 May 2022)\nThis work provides a concept for three-dimensional magnetic solitons based on mapping the homotopy path between\nvarious two-dimensional solutions onto the third spatial axis. The representative examples of statically stable configu-\nrations of that type in the model of an isotropic chiral magnet are provided. Various static and dynamic properties of\nsuch three-dimensional magnetic solitons are discussed in detail.\nI. INTRODUCTION\nChiral magnets are a distinct class of materials, where the\ncompetition between the Heisenberg exchange and the chi-\nral Dzyaloshinskii-Moria interaction1,2(DMI) gives rise to a\nwide diversity of magnetic solitons – localized magnetic tex-\ntures with particle-like properties. The most representative\nexample of magnetic solitons in chiral magnets are magnetic\nskyrmions which have been intensively studied both theoret-\nically3–5and experimentally6–12in the last decades. In thick\nfilms and bulky samples, the magnetization vector field of chi-\nral skyrmions resembles a filamentary structure composed of\nvortex-like tubes.\nMore recent studies have revealed a diversity of 3D solitons\nin chiral magnets beyond skyrmions. The most prominent ex-\namples are chiral bobbers13,14, skyrmion bags15–18, helikno-\ntons19, and skyrmion braids20. Because of distinct topology,\nthe above solitons possess different static, dynamic, and trans-\nport properties. In particular, the chiral bobbers are distinct\nby the presence of magnetic singularity – Bloch point, while\nother magnetic textures are characterized by smooth magnetic\nvector fields. Because of that, contrary to other solitons, the\nchiral bobber is not a topological soliton. The skyrmions and\nheliknotons, on the other hand, belong to different topological\ngroups characterized by distinct topological invariants.\nIn this work, we present another type of magnetic soli-\ntons which belong to a topological group of skyrmions and\nare stabilized in bulk crystals of chiral magnets. Because of\nthe unique properties of these solutions, it is reasonable to at-\ntribute them to a distinct class of magnetic solitons. Here we\nwill refer to them as hybrid skyrmion tubes. The crosssections\nof well-studied skyrmion tubes usually represent nearly iden-\ntical configurations with small additional modulations due to\nthe braiding effect20or the presence of the noncollinear back-\nground21and/or free surfaces22. We show that besides such\nnearly homogeneous tubes, there are also solutions where the\ncrosssections represent a continuous transformation between\nthe skyrmions of different configurations but the same topo-\nlogical index – skyrmions of the same homotopy group. In a\nrepresentative example, we illustrate the stability of such ex-\notic spin textures and discuss their unique dynamic properties.\nBesides that, we discuss an important consequence of apply-\ning such a homotopical concept to truly 3D localized magnetic\nspin textures. In particular, we present the solution represent-ing a topologically trivial but statically stable and truly three-\ndimensional soliton – 3D chiral droplet.\nII. MODEL\nThe model Hamiltonian for chiral magnet with bulk type\nDMI has the folowing form:\nE=Z\nVm(A(Ñn)2+Dn\u0001Ñn+U(n))dVm; (1)\nwhere n=M=Msis normalized magnetization jnj=1,Aand\nDare micromagnetic constants of the Heisenberg exchange\nand the DMI, respectively. Vmis the volume of the magnetic\nsample. The potential energy term U(n)includes an uniaxial\n(easy-axis or easy-plane) magnetic anisotropy, Ku, and the\ninteraction with the external magnetic field, Bextkez:\nU(n) =\u0000Kun2\nz\u0000MsBextnz: (2)\nFollowing the standard procedure3–5, the functional (1) can\nbe written in a more suited for analysis, dimensionless form\nE=Z\nV\u0012(Ñn)2\n2+2pn\u0001Ñn\u00004p2(un2\nz+hnz)\u0013\ndV;(3)\nwhereE=E=2Ais the reduced energy, u=Ku=MsBDand\nh=Bext=BDare the dimensionless values of anisotropy con-\nstant and the strength of the external magnetic field, respec-\ntively. The integration in (3) is over the reduced volume\nV=Vm=L3\nD. The characteristic parameters LD=4pA=Dand\nBD=D2=2MsAare the period of spin spiral and the satura-\ntion field for isotropic case, Ku=0.\nThe energy functional (3) remains valid for the 2D (or quasi\n2D) systems where the magnetization does not change along\nthez-axis and integration is carried out over d V=ldxdy,\nwhere lis the film thickness. The 2D model of the chi-\nral magnet, besides ordinary axially symmetric p-skyrmions,\nprovides a large variety of magnetic solitons e.g. skyrmion\nbags15,16and skyrmions with chiral kinks17. The homotopy\nclassification of localized solutions in 2D is provided by the\nfollowing invariant\nQ=1\n4pZ\nn\u0001(¶xn\u0002¶yn)dxdy: (4)arXiv:2205.05379v1  [cond-mat.mes-hall]  11 May 20222\nThe solutions with identical integer index Qare called the so-\nlutions of one homotopy class. The latter implies that one\ncan continuously transform the vector fields of such solu-\ntions into each other. For instance, let us assume there are\ntwo localized magnetic textures n1(x;y)andn2(x;y)of iden-\ntical charge Q. Then, there is a vector field transformation,\nn(x;y;s), where s2[0;1], such that for n(x;y;0) =n1(x;y)\nandn(x;y;1) =n2(x;y). When n(x;y;s)is differentiable\nwith respect to x,y, and sat any point such transformation\ncan be called a homotopy path. Since for any n1(x;y)and\nn2(x;y), there is an infinite number of homotopy paths, there\nis no unique method to construct such paths. To make the\nsearch more definite, we consider only those homotopy paths\nthat also satisfy the minimum energy path (MEP) criterion.\nIn particular, we use geodesic nudged elastic band (GNEB)\nmethod23,24implemented in Spirit code25. The details of the\nMEP calculation are provided in the Appendix A.\nReaction coordinate0 1.0E/E016\n141517\nd\nbc\nLD(a)\nbe\n2 0.5 1.5\n(b) (c)\n(d) (e)\nFIG. 1. aMEP between two stable skyrmion states with Q=1 de-\npicted in bandd. The spin configuration corresponding to the saddle\npoints are depicted in cande. The reaction coordinate is given in re-\nduced unit with respect to its value at intermediate state depicted in\nd. The calculations are performed at h=0:627 and u=0. The en-\nergy in ais given with respect to the energy of ferromagnetic state,\nE0.III. RESULTS\nIn Fig. 1 awe provide representative example of the MEP\nbetween two skyrmions with Q=1. We show only the spin\ntexture corresponding to local minima and saddle points. No-\nticeably, the textures corresponding to the central minimum\nstate ( d) and saddle points (c),(e)contain a chiral kink17while\nanother minimum state is a skyrmion bag free of kinks (b).\nThe MEP presented in Fig. 1 asatisfy the above criteria for\nthe homotopy path. The parameter scan be associated with\nthe reaction coordinate which has a meaning of the relative\ndistance between the images (snapshots of the vector field) in\nthe multidimensional parameter space.\nTo construct an initial configuration for the 3D skyrmion\ntube, one can take the stable 2D skyrmion configuration and\nplace it at each xy-plane of the 3D simulated domain. Stat-\nically stable configuration of such homogeneous skyrmion\ntube corresponding to the 2D skyrmion in Fig. 1 (b)is pro-\nvided in Fig. 2 (a). To construct the initial state for a nonho-\nmogeneous 3D skyrmion tube, we use a mapping from the\nhomotopy path to the third spatial axis, s!z. In other words,\nto create a 3D magnetic texture, we sequentially lay down the\nspin texture of the images from MEP on top of each other\nalong with the z-axis. The statically stable spin configuration\nobtained by the energy minimization of that initial state is pro-\nvided in Fig. 2 (b)and its crosssections are shown in (c)-(h).\nThe intermediate region resembling a knot on the isosurface of\nthe skyrmion string in Fig. 2 (b)is well localized and thus can\nbe thought of as a soliton that is hosted by a skyrmion string.\nThis nonhomogeneous 3D skyrmion tube is a representative\nexample of the solutions to which we refer as hybrid skyrmion\ntubes. For conciseness, the localized intermediate region, we\ncall a knot in the following. The choice of such terminology\nis justified by pure visual analogy and has nothing to do with\nthe topological knots. Whether one can continuously unwind\nthat knot without the appearance of Bloch points represents an\nintriguing question that, however, goes out of the scope of the\npresent work.\nAt the strong magnetic field, h\u00151\u00002u, the hybrid\nskyrmion tubes represent a metastable state embedded in the\nferromagnetic vacuum. For h<1\u00002u, the vacuum for such\nstates is the cone phase. The range of fields and anisotropies\nwhere hybrid skyrmion tubes remain stable depends on the\nparticular configuration – the type of 2D skyrmions in the tube\ncrosssections. For instance, the hybrid skyrmion tube depicted\nin Fig. 2 (b)atu=0 remain stable, at least in the range of\n0<h<0:35.\nIn general, the hybrid skyrmion tubes can be composed of\na few intermediate states representing stable 2D skyrmions.\nHere we consider the solution composed of two intermedi-\nate states only. Such configurations are easier to handle be-\ncause they satisfy periodical boundary conditions. It is worth\nnoting, the stabilization of hybrid skyrmion tubes does not re-\nquire the periodic boundary conditions along z-axis. The latter\nmakes the experimental observation of hybrid skyrmion tubes\nin thick films of chiral magnets quite promising. In the case\nof free boundaries along z-axis, the tube has an additional sur-\nface twist modulations. For thick enough films ( \u00186LD=4203\nxyz\n(a)\nBext\nKu\nz=0z=2LDLD(c) (d) (e) (f) (g)\nz=2.5LD z=3LD z=3.5LD z=4LD6LD(b)\n(h)\nFIG. 2. aandbshow isosurfaces nz=0 for ordinary and hybrid skyrmion tubes stabilized at h=0:45,u=0:3 in the box of size Lx=Ly=\nLz=6LD,LD=32awith periodic boundary conditions in all directions. b-gshow the cross-sections of the hybrid tube, while the ordinary\nskyrmion tube is characterized by identical magnetic texture (alike c) in every zsection.\nnm for FeGe) These modulations do not decrease the tube sta-\nbility. On the other hand the presence of the free boundaries\nthe hybrid skyrmion tube depicted in Fig. 2 (b)can continu-\nously unwind into the host skyrmion tube shown in Fig. 2 (a).\nThis process is illustrated by Supplementary Movie 1, where\neach frame was obtained by a manual shift of the entire spin\ntexture along the z-axis and following incomplete relaxation\nof the system. In the case of complete relaxation, the sys-\ntem converges to its initial state, which indicates the presence\nof the energy barrier for the knot to escape through the free\nboundary.\nTo demonstrate the unique dynamic properties of hybrid\nskyrmion tubes we perform micromagnetic simulations based\non the Landau-Lifshitz-Gilbert equation26:\n¶n\n¶t=\u0000gn\u0002Heff+an\u0002¶n\n¶t\u0000TZL; (5)\nwhere gis the gyromagnetic ratio, ais the Gilbert damping,\nHeff=\u00001\nMsdE\ndnis the effective field. The last term in (5) is\nthe Zhang-Li torque27because of the electric current:\nTZL=n\u0002[n\u0002(I\u0001Ñ)n]+xn\u0002(I\u0001Ñ)n; (6)\nwhere the vector I=jmBp(1+x2)\u00001(eMs)\u00001is proportionalto the current density j,xis the degree of non-adiabaticity, pis\nthe polarization of the spin current, mBis the Bohr magneton\nandeis the electron charge.\nWhen the 3D soliton is embedded in the FM state, the cur-\nrent direction can be chosen arbitrary. On the contrary, when\nthe soliton is embedded in non homogeneous vacuum, e.g.,\ncone phase, it is essential to chose the current direction per-\npendicular to the q-vector of the cone to prevent the excitation\nof the background. To simplify the following discussions we\nconsider the case of hybrid skyrmion tube in ferromagnetic\nvacuum.\nSupplementary Movie 2 illustrates the dynamics of the hy-\nbrid skyrmion tube when the current is along the z-axis. The\nsimulations were performed with Mumax28code. The exam-\nple of Mumax script and the initial state files are provided in\nSupplementary Data. We used periodical boundary conditions\nin all spatial directions to simulate a bulk crystal. Under the\nelectric current parallel to the skyrmion tube, the knot on the\nisosurfaces moves along the tube in the direction opposite to\nthe current. In this case, the position of the host skyrmion tube\nin the xy-plane remains fixed.\nIn Supplementary Movie 3, we show the dynamics of\nthe skyrmion tube when the current is perpendicular to the\nskyrmion tube, Ikex. In this case, both the host skyrmion4\ntube and the knot move. As a result, all three components of\nthe knot velocity are non-zero.\nTo quantify the knot velocity, we follow the approach used\nin our previous work on 2D skyrmion dynamics29. Extend-\ning this approach to 3D textures the position of the knot\nR=\u0000\nRx;Ry;Rz\u0001\ncan be defined as follows\nRk=Lk\n2ptan\u00001RNi j(rk)sin(2prk=Lk)drkRNi j(rk)cos(2prk=Lk)drk\u0006lkLk; (7)\nwhereNi j(rk) =Rnz(r)dridrjis magnon density averaged in\nrk-plane, indices i;j;k2fx;y;zg. Sigh\u0006accounts for the di-\nrection of solitons motion: along the basis vector ek(+) or\nin the opposite direction (\u0000). The integer lkis the number of\ntimes the soliton crossed the corresponding domain boundary.\nBy tracing the position Rat each moment in time, we can esti-\nmate the instant velocity of the knot, v=dR=dt, and compare\nit with the results of a semi-analytical approach based on the\nmethod of collective coordinates suggested by Thiele30. As-\nsuming the rigid motion of magnetic textures with velocity v,\ni.e.n(r;t) =n(r\u0000vt), from (5) and (6), one can derive the\nThiele equation:\nG\u0002(v+I)+ˆG(av+xI) =0; (8)\nwhere the gyro-vector, G, and the dissipation tensor, ˆG, have\ncomponents GiandGi jdefined as:\nGi=\u0000Z\nei jkei0j0k0ni0¶nj0\n¶rj¶nk0\n¶rkdV; (9)\nGi j=Z¶nk\n¶ri¶nk\n¶rjdV; (10)\nwhere ei jkis Levi-Civita symbol. The advantage of the Thiele\napproach is that the solution for the soliton velocity can be\nwritten explicitly. For instance, for I=Iex, the solution of (8)\nfor the uniform skyrmion tubes, Gxz=Gyz=Gzz=0 takes a\nform identical to the 2D case\nvx=\u0000IG2\nz+GzGxy(x\u0000a)+axdetˆG\nG2z+a2detˆG;\nvy=\u0000IGzGxx(x\u0000a)\nG2z+a2detˆG; (11)\nvz=0:\nFor the habridized skyrmion tube all the entries of the dissi-\npation tensor ˆGhave non-zero values and the solution of (8)\ntakes the form\nvx=\u0000IGzGxx+(GxyGzz\u0000GxzGyz)(x\u0000a)+axdetˆG\nG2zGzz+a2detˆG;\nvy=\u0000IGz(G2\nxz\u0000GxxGzz)(x\u0000a)\nG2zGzz+a2detˆG; (12)\nvz=\u0000IGz(GzGxz=a+GxxGyz\u0000GxzGxy)(x\u0000a)\nG2zGzz+a2detˆG:\nIt is worth noting two remarkable features of the solution\n(12). Firstly, there is no continuous transition between (12)\n(a)\n(c)\nLD(b)Reaction coordinateE/E06\n4\n2\n00 1.0 0.5 1.5 2.0bc c\nFM FMFIG. 3. aMEP between chiral droplet with Q=0 depicted in b\nand FM state. The spin configuration corresponding to the saddle\npoint is depicted in c. The reaction coordinate is given in reduced\nunit with respect to its value at intermediate state depicted in b. The\ncalculations are performed at h=0:65 and u=0. The energy in ais\ngiven with respect to the energy of ferromagnetic state, E0.\nand (11) even when Gxz,GyzGzztend to zero. Secondly, in\nthe most general case of 3D soliton, the components of gyro-\nvector Gcan be thought of as 2D topological charges in corre-\nsponding x,yandzcrosssections. For hybrid skyrmion tubes,\nhowever, similar to the 2D case, only the third component of\nthe gyro-vector is non zero and proportional to 2D topological\ncharge in xy-plane, G=4pQLzez.\nTo calculate the velocity components in (12), one has to find\nthe components of the gyro-vector (9) and the dissipation ten-\nsor (10) for a particular magnetization distribution. For that,\nwe use the magnetization vector field at the static equilibrium\nobtained by the numerical energy minimization of the func-\ntional (1). It is worth emphasizing that the soliton velocity\nobtained from the solutions of the Thiele equation (8) corre-\nsponds to the velocities in a steady-state only. Thus the veloc-\nities estimated from LLG simulations can be compared with\nthe analytical solutions only when the transient state is over,\nand all components of the vreached the saturation.\nIn the most general case of 3D soliton, when all three\ncomponents of the velocity are non-zero, one can intro-\nduce two deflection angles b1=arctan\u0000\nvy=vx\u0001\nandb2=\np=2\u0000arctan\u0010q\nv2x+v2y=vz\u0011\n. The angle b1is also known as\nskyrmion Hall angle. Because of that we refer to b2as the\nsecond skyrmion Hall angle. Table I demonstrates a good\nagreement between the values of b1andb2estimated from\nLLG simulations and calculated with the Thiele approach for\nparameters j=\u00005\u0001108exA/m2,a=0:01 and x=0:05. No-\nticeably, for particular tube depicted in Fig. 2(b), the angle b2\nis larger than b1. As follows from the Thiele equation (8),5\nwhen I=Iez, in agreement with the micromagnetic simula-\ntions, only the z-component of the knot velocity is non zero,\nvz=\u0000Ix=a.\nTABLE I. Two Hall angles for hybrid skyrmion tube estimated by\nthe LLG simulations and with the Thiele method of collective coor-\ndinates\nb1 b2\nLLG Thiele LLG Thiele\n\u000017:1\u000e\u000016:8\u000e57:3\u000e59:73\u000e\nThe homotopy concept used for constructing hybrid\nskyrmion tubes can be extended further for solitons localized\nin all three dimensions. For that one can consider the homo-\ntopy transformation of any 2D skyrmion with Q=0 into the\nFM phase. Here we examine such transformation on the ex-\nample of chiral droplet shown in Fig. 3. The MEP in Fig. 3 is\nquite similar to that shown in Fig. 1. Two stable states repre-\nsenting the isolated soliton and the saturated FM state are sep-\narated by the saddle point. In this transformation, the collapse\nof the droplet goes by its shrinking, which has been previously\ndiscussed in Refs.31,32.\nBy mapping the obtained images of the corresponding ho-\nmotopy path onto the spatial z-axis, we constructed the ini-\ntial state of the 3D soliton. However, it turned out that such\nan initial guess does not lead to a stable configuration in the\nFM phase but instead becomes stable in the cone phase vac-\nuum only. The isosurfaces and cross-sections of the stable\nconfiguration of the 3D chiral droplet obtained by numerical\nenergy minimization are shown in Fig. 4. As seen in ( a) the\nred color in the surface is missing, which means that by pro-\njecting the vector field of the 3D chiral droplet onto S2sphere,\none can not cover it entirely. The latter means that both topo-\nlogical invariant (4) and Hopf invariant are zero in this case.\nThe cross-sections in ( b) and ( c) show well-localization of the\nsoliton. We estimated the characteristic size of the 3D chiral\ndroplet equal to\u00181LDin all spatial directions.\nThe stability range for the 3D chiral droplet in terms of\nthe magnetic field, h, and anisotropy, u, is shown in Fig. 5.\nNote, the phase transition lines between the phases are repro-\nduced from Ref.33. As follows from the diagram, the 3D chi-\nral droplet stability region is in the range where the skyrmion\nlattice is the lowest energy state, while the cone phase is a\nmeta-stable state. The blurred edges at h.0:28 and u.0:15\ndenote that we did not succeed to identify the stability range\nbelow these values reliably. At these values of magnetic field\nand anisotropy, the helicoid competes in energy with the cone\nphase. A more precise estimation of the lower bound for the\n3D chiral droplet stability requires much larger sizes of the\nsimulated domain above the limit of our computational re-\nsources.\nThe 3D chiral droplet motion can be induced by apply-\ning different external stimuli. First, let us consider the mo-\ntion of the 3D chiral droplet induced by the external mag-\nnetic field slightly tilted and rotating about the z-axis, h=\n0:34(sinqhcos2pnt;sinqhsin2pnt;cosqh). We performed\nthe micromagnetic simulations assuming the tilt angle qh=\n0:1 and the small frequency of rotation n=5 MHz which isclose to a quasi-stationary process. Supplementary Movie 4\ndemonstrates such motion of the 3D chiral droplet consisting\nof two types of motion – the translation along the z-axis and\nthe rotation about the z-axis. Remarkably, similar dynamics\nhas been reported for heliknoton19and seems to be a com-\nmon property of all 3D localized solitons hosted by the conical\nphase. The motion of the 3D chiral droplet free of its rotation\ncan be induced by the cone phase rotation at the static exter-\nnal field. To demonstrate this, we performed the simulations\nwhere we manually rotated a small number of spins located\nfar from the droplet. As illustrated in Supplementary Movie\n5, the translation motion of the droplet along the z-axis, in this\ncase, occurs without its rotation.\nIn the case of Zhang-Li torque, the gyro vector of the 3D\ndroplet equals zero and the Thiele equation (8) provides a triv-\nial solution, v=xI=a. On the other hand, due to the presence\nof the cone phase, one has to choose I?qto suppress the\nexcitation of the cone phase itself. The analysis of the cone\nphase dynamics induced by Ikq, is provided in Appendix B.\nOne has to note that contrary to the Thiele equation, which\npredicts that the 3D chiral droplet should move without de-\nflection, vk\u0000I, in numerical simulations, we still observe a\nslight deflection in the soliton motion. Since, in the case of the\nhybrid skyrmion tube, we got pretty well agreement between\nnumerical simulations and the Thiele approach, we attribute\nthis effect to the numerical artifact caused by the excitation of\nthe cone phase, which in turn prevents reaching the regime of\na steady motion. Supplementary Movie 6 shows the dynamics\nof the droplet caused by the in-plane electric current Ikex. To\nprevent the excitation of the cone phase, in this simulation, we\npinned the spins on the bottom plane of the simulated domain.\nIV. CONCLUSIONS\nIn this work, we have discussed new types of 3D magnetic\nsolitons stabilized in chiral magnets – hybrid skyrmion tube\nand 3D chiral droplet. We show that the magnetic textures of\nthese solutions can be explained in terms of homotopy transi-\ntions between various 2D skyrmions.\nWe have studied the static and dynamic properties of new\nsolitons. The parameters of the external magnetic field and\nmagnetic anisotropy at which the presented solutions remain\nstabilized are different and characterized by the various states\nof the vacuum. hybrid skyrmion tubes can be stable in the FM\nstate and cone phase, while the 3D chiral droplet is stable only\nin the conical phase surroundings. It is shown that the dynam-\nics of the hybrid skyrmion tube can be induced by the electric\ncurrent modeled by the Zhang-Li torque in the LLG equation.\nThe electric current applied along the tube causes the motion\nof the knot on the skyrmion tube in the direction opposite to\nthe current. The electric current applied perpendicular to the\nskyrmion tube leads to more complicated dynamics, which\ncan be described in terms of two skyrmion Hall angles. The\nresults of LLG simulations agree well with the analysis based\non the Thiele equation.\nFor the case of the 3D chiral droplet, we show that its mo-\ntion besides the electric current can be induced by the rotating6\n(a)\nxyz xyzBext\nKu\n3LD\nz\nxyx(b)\n(c)\nFIG. 4. ashows isosurfaces nz=0 and nz=0:98 (black) for 3D chiral droplet stabilized at h=0:34,u=0:26 in the box of size Lx=Ly=\nLz=3LD,LD=64awith periodic boundary conditions in all directions. b,dshow the sections of the droplet by planes y=0 and z=0,\nrespectively.\n3D \ndroplet\nSkyrmion \n   lattice\nHelicoidCone\nAnisotropy, uMagnetic filed, h\n0 0.05 0.1 0.15 0.20.20.240.280.320.36\n0.25 0.3\nFIG. 5. The stability range of the 3D chiral droplet (blue) is shown.\nThe phase transition lines are taken from Ref.33.\nexternal magnetic field. Since the cone phase is easily excited\nwhen the electric current is not perpendicular to the cone wave\nvector, the analysis based on the Thiele approach is difficult in\nthis case. The analytic solution for the dynamics of the cone\nphase is provided. Supplementary Movies illustrate numerical\nsimulations for the motion of the 3D solitons.SUPPLEMENTARY MATERIAL\nSee supplementary material for additional movies which\nillustrate the dynamics of 3D solitons discussed in the main\ntext. Supplementary Movie 1 illustrates the escape of the knot\non the hybrid skyrmion tube through the free boundary of the\nplate. Supplementary Movie 2 and 3 show the dynamics of\nthe hybrid skyrmion tube induced by current IkezandIkex,\nrespectively. Supplementary Movie 4 illustrates the dynam-\nics of the 3D chiral droplet induced by the small precession\nof the external magnetic field about the z-axis with low fre-\nquency. Supplementary Movie 5 shows the dynamics of the\n3D chiral droplet under the cone spiral rotation induced by a\nweak electric current Ikezapplied in the bounded volume far\nfrom the droplet. Supplementary Movie 6 shows the dynam-\nics of the 3D chiral droplet induced by current Ikexwhen the\nspins in the plane z=0 are pinned.\nThe magnetization in the movies is presented by the iso-\nsurfaces nz=0 and the standard color code used in Mu-\nmax for visualization of the unit vector fields. The excep-\ntions are the Movies 4-6 where we use red-blue color code\nforny-component of magnetization: n= (0;\u00001;0)– red and\nn= (0;1;0)– blue.\nACKNOWLEDGMENTS\nWe are acknowledge Filipp Rybakov, Bernd Schroers, and\nBruno Barton-Singer for fruitful discussions. We also grate-\nful to Moritz Sallermann for technical support in minimum\nenergy path calculations with Spirit. We acknowledge fi-\nnancial support from the Deutsche Forschungsgemeinschaft\nthrough SPP 2137 “Skyrmionics\" Grant No. KI 2078/1-1 and7\nthe European Research Council under the European Union’s\nHorizon 2020 Research and Innovation Programme (Grant\nNo. 856538 - project “3D MAGiC”).\nAppendix A: Details of MEP calculation\nThe MEP in Fig. 1 (a)represents a homotopy path between\ntwo states (b)and(d). To get the MEP without singularities,\none has to make a reasonable assumption for the initial path.\nThe straightforward and often used approach based on the in-\nterpolation between two configurations by simple rotation of\nthe spins is not appropriate for this purpose. It is easy to check\nthat the topological index Qis often not conserved along the\nMEP with this approach. To avoid such a behavior of the\nGNEB solver, we have constructed the ansatz for the initial\npath using the energy minimization and drag-and-drop func-\ntion implemented in Spirit. In particular, as seen in Fig. 1( b)-\n(e) the intermedeate states represents the elongation and twist-\ning of the magnetic texture near the chiral kink. Starting with\nthe state depicted in Fig. 1( b) and using the drag-and-drop op-tion we enforce the chiral kink to elongate and bend to form\na state similar to that in ( c) and ( e). We intentionally stop the\nrelaxation process before the system converges to one of two\nlocal minima as in Fig. 1( b) or (d). Then, we collect such un-\nrelaxed snapshots of the spin texture with the different levels\nof elongation of the part containing the chiral kink and use\nthem as an initial guess for MEP.\nThe calculation were perfomed at the following parameters:\nLx=Ly=4LDwith LD=64a.\nAppendix B: Cone phase rotational dynamics\nIn terms of the spherical angles (Q;F)the magnetization\nwrites n= (sinQcosF;sinQsinF;cosQ)T. Assuming that at\nt=0, the magnetization profile is given by the cone phase\nQ=Qc,F=2pz+f0, we can find the analytic solution of\nthe LLG equation with the current I=Iezturned on at t>0.\nThe uniformity of the magnetic texture in the xy-plane leads\nto the following system of equations for (Q;F):\n8\n>><\n>>:¶2Q\n¶z2\u0000\u0012¶F\n¶z\u00132sin2Q\n2+2p¶F\n¶zsin2Q\u00004p2usin2Q\u00004p2hsinQ+ix¶Q\n¶z\u0000a¶Q\n¶t+\u0012¶F\n¶t\u0000i¶F\n¶z\u0013\nsinQ=0;\n\u00002¶Q\n¶z¶F\n¶zcosQ\u0000¶2F\n¶z2sinQ+4p¶Q\n¶zcosQ\u0000i¶Q\n¶z+¶Q\n¶t+\u0012\na¶F\n¶t\u0000ix¶F\n¶z\u0013\nsinQ=0:(B1)\nwhere t=gBDt=4p2is the dimensionless time and i=\n4p2I=gLDBDis the parameter of electric current. Although\nthe system (B1) represents two coupled non-linear partial dif-\nferential equations, its solution can be written for the case\nwhen Q=Q(t).\nWe found that strong currents can lead to the transition from\nthe cone phase to the FM state. The critical current at which\nthis happens is given by:\ni\u0006\nc=2pa(1\u00002u)\nx\u0000a(cosQc\u00061); (B2)\nThe formula (B2) has sense only at x6=a. In the case x=a,the cone phase can not be excited by the current. Below the\ncritical regime, i.e. i2(i\u0000\nc;i+\nc), the angle Qmonotonically\nchanges from Qc– the equilibrium cone angle without electric\ncurrent to QI– the equilibrium cone angle in dynamical steady\nstate in the presence of the current:\ncosQI=cosQc\u0000i(x\u0000a)\n2pa(1\u00002u): (B3)\nFori=2(i\u0000\nc;i+\nc)the angle QIequals to 0 or pdepending on the\nsign of the current, i. At i2(i\u0000\nc;i+\nc), the solution of (B1) for\nQcan be written as:\n8p2at\n1+a2=8\n>>><\n>>>:1\na11\ncosQc\u00071\u00001\na11\ncosQ\u00071+1\n2a1ln\u00121+z\n1+cosQc1\u0000cosQc\n1\u0000cosQ\u0013\n;a2=\u0006a1;a16=0;\n1\na2\n1\u0000a2\n2ln \u00121+cosQ\n1+cosQc\u0013a1\u0000a2\u00121\u0000cosQc\n1\u0000cosQ\u0013a1+a2\u0012a1\u0000a2cosQ\na1\u0000a2cosQc\u00132a2!\n;ja2j6=ja1j;(B4)\nwhere a1=h\u0000i(x\u0000a)=2pa,a2=1\u00002u. The solution for\nFfunction is given by:\nF(t;z) =2p\u0012\nz+ix\nat\u0013\n+f0+1\nalntan(Qc=2)\ntan(Q=2):(B5)As follows from (B4), at t!¥, one has cos Q=cosQI=\na1=a2. In this case, Eq. (B5) describes the rotation of the\ncone phase with constant velocity Ix=awhich agrees with the\nThiele prediction for topologically trivial configurations. No-8\n(a) (b) (c)\nTime, nsMagnetization0.00.51.0\n-0.5\n-1.00 2 4 6 8 10\n0 2 4 6 8 10\n0 0.5 1 1.5 2 3 2.5\nTime, ns Time, nsnxnynz\ncosΘI\nFIG. 6. a-cshow the functional dependencies of magnetization components on time at z=0 for current values i1=\u00000:15,i2=0:15 and\ni3=2. Solid lines are given by analytic solutions (B4), (B5) while points are obtained in numerical simulations with Mumax. Magenta solid\nline corresponds to the limit value for nzas follows from (B3).\nticeably, the dynamics described by the LLG equation reaches\nthe dynamical steady state exponentially fast. Note, the solu-\ntions (B4), (B5) remain valid not only for the bulk systems\nbut also for the films with free boundaries along z-axis. The\nlatter remain valid for any handuwhere the surface modu-\nlations13do not perturb the cone phase. This follows from\nthe fact that found solutions satisfy the boundary conditions:\n¶zQ=0,¶zF=2pon the free surface, z=const, for any t.\nTo verify the found solutions, we compare them with the re-\nsults of LLG simulations performed for different values of the\ncurrent but fixed values of h=0:34,u=0:26,a=0:01, and\nx=0:05. The critical current values (B2) for these parameters\nare:i\u0000\nc\u0019\u00000:22,i+\nc\u00191:29. We have performed simulations\nfor two current values within the critical range: i1=\u00000:15,\ni2=0:15, and for one out of this range i3=2. The results are\nprovided in Fig. 6. For all three cases we see a good agree-\nment between numerical and analytical solutions at least for\nthe chosen simulation time. At longer times, z-component\nof the magnetization tends to the limit value accordingly to\n(B3). For the cases shown in ( a) and ( b) these limit values\nare about 0 :51 and 0 :91, respectively. For the case shown in\n(c) corresponding to the current above the critical value, the\nmagnetization tends to \u00001. As one can deduce, applying the\ncurrent Ikqallows manipulating the cone phase angle and its\ndynamics in a controllable way.\nThe found analytical solutions besides the pure academic\ninterests can be used also for testing the accuracy of numeri-\ncal schemes for solving the LLG equation. In particular, the\nsolutions of (B1) in a special case of zero damping, a=0,\ncan be written in a more compact form:\nQ=2arctan\u0012\ntanQc\n2e2pixt\u0013\n;\nF=2p(z+it)+f0\u00004p2(1\u00002u\u0000h)t\n+2p\nixkln1+tan2Qc\n2e2pixt\n1+tan2Qc\n2: (B6)\nThe stability of LLG solvers typically requires the presence of\nnon-zero damping. Therefore, the solutions (B6) can be usefulfor testing new LLG solvers which are free of this limitation.\nBoth cases a=0 and a6=0 can be generalized for the case of\ntime-dependent currents I=I(t)easily. The detailed analysis\nof this case is out of the scope of the present study and will be\nprovided elsewhere.\n1I. Dzyaloshinsky, “A thermodynamic theory of “weak” ferromagnetism of\nantiferromagnetics,” Journal of Physics and Chemistry of Solids 4, 241 –\n255 (1958).\n2T. Moriya, “Anisotropic superexchange interaction and weak ferromag-\nnetism,” Physical review 120, 91 (1960).\n3A. N. Bogdanov and D. A. Yablonskii, “Thermodynamically stable “vor-\ntices” in magnetically ordered crystals. the mixed state of magnets,” Sov.\nPhys. JETP 68(1989).\n4A. Bogdanov and A. Hubert, “Thermodynamically stable magnetic vortex\nstates in magnetic crystals,” Journal of Magnetism and Magnetic Materials\n138, 255 – 269 (1994).\n5A. Bogdanov and A. Hubert, “The stability of vortex-like structures in uni-\naxial ferromagnets,” Journal of Magnetism and Magnetic Materials 195,\n182–192 (1999).\n6S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch,\nA. Neubauer, R. Georgii, and P. Böni, “Skyrmion lat-\ntice in a chiral magnet,” Science 323, 915–919 (2009),\nhttps://science.sciencemag.org/content/323/5916/915.full.pdf.\n7A. Tonomura, X. Yu, K. Yanagisawa, T. Matsuda, Y . Onose, N. Kanazawa,\nH. S. Park, and Y . Tokura, “Real-space observation of skyrmion lattice in\nhelimagnet mnsi thin samples,” Nano Letters 12, 1673–1677 (2012), pMID:\n22360155, https://doi.org/10.1021/nl300073m.\n8B. Lebech, J. Bernhard, and T. Freltoft, “Magnetic structures of cubic FeGe\nstudied by small-angle neutron scattering,” Journal of Physics: Condensed\nMatter 1, 6105–6122 (1989).\n9H. Wilhelm, M. Baenitz, M. Schmidt, U. K. Rößler, A. A. Leonov, and\nA. N. Bogdanov, “Precursor phenomena at the magnetic ordering of the\ncubic helimagnet fege,” Phys. Rev. Lett. 107, 127203 (2011).\n10X. Z. Yu, N. Kanazawa, Y . Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata,\nY . Matsui, and Y . Tokura, “Near room-temperature formation of a skyrmion\ncrystal in thin-films of the helimagnet fege,” Nature Materials 10, 106–109\n(2011).\n11X. Z. Yu, Y . Onose, N. Kanazawa, J. H. Park, J. H. Han, Y . Matsui,\nN. Nagaosa, and Y . Tokura, “Real-space observation of a two-dimensional\nskyrmion crystal,” Nature 465, 901–904 (2010).\n12Y . Onose, Y . Okamura, S. Seki, S. Ishiwata, and Y . Tokura, “Observa-\ntion of magnetic excitations of skyrmion crystal in a helimagnetic insulator\ncu2oseo 3,” Phys. Rev. Lett. 109, 037603 (2012).\n13F. N. Rybakov, A. B. Borisov, S. Blügel, and N. S. Kiselev, “New type of\nstable particlelike states in chiral magnets,” Phys. Rev. Lett. 115, 117201\n(2015).\n14F. Zheng, F. N. Rybakov, A. B. Borisov, D. Song, S. Wang, Z.-A. Li, H. Du,9\nN. S. Kiselev, J. Caron, A. Kovács, M. Tian, Y . Zhang, S. Blügel, and R. E.\nDunin-Borkowski, “Experimental observation of chiral magnetic bobbers\nin B20-type FeGe,” Nature nanotechnology 13, 451–455 (2018).\n15F. N. Rybakov and N. S. Kiselev, “Chiral magnetic skyrmions with arbitrary\ntopological charge,” Phys. Rev. B 99, 064437 (2019).\n16D. Foster, C. Kind, P. J. Ackerman, J.-S. B. Tai, M. R. Dennis, and I. I.\nSmalyukh, “Two-dimensional skyrmion bags in liquid crystals and ferro-\nmagnets,” Nature Physics 15(2019), 10.1038/s41567-019-0476-x.\n17V . M. Kuchkin, B. Barton-Singer, F. N. Rybakov, S. Blügel, B. J. Schroers,\nand N. S. Kiselev, “Magnetic skyrmions, chiral kinks, and holomorphic\nfunctions,” Phys. Rev. B 102, 144422 (2020).\n18J. Tang, Y . Wu, W. Wang, L. Kong, B. Lv, W. Wei, J. Zang, M. Tian, and\nH. Du, “Magnetic skyrmion bundles and their current-driven dynamics,”\narXiv:2108.02487 (2021), arXiv:2108.02487.\n19R. V oinescu, J.-S. B. Tai, and I. I. Smalyukh, “Hopf solitons in helical\nand conical backgrounds of chiral magnetic solids,” Phys. Rev. Lett. 125,\n057201 (2020).\n20F. Zheng, F. N. Rybakov, N. S. Kiselev, D. Song, A. Kovács, H. Du,\nS. Blügel, and R. E. Dunin-Borkowski, “Magnetic skyrmion braids,” Nat.\nComm. 12, 5316 (2021), arXiv:2104.01682.\n21H. Du, X. Zhao, F. N. Rybakov, A. B. Borisov, S. Wang, J. Tang, C. Jin,\nC. Wang, W. Wei, N. S. Kiselev, et al. , “Interaction of individual skyrmions\nin a nanostructured cubic chiral magnet,” Physical review letters 120,\n197203 (2018).\n22F. Rybakov, A. Borisov, and A. Bogdanov, “Three-dimensional skyrmion\nstates in thin films of cubic helimagnets,” Physical Review B 87, 094424\n(2013).\n23P. Bessarab, V . Uzdin, and H. Jónsson, “Method for finding mechanism\nand activation energy of magnetic transitions, applied to skyrmion and an-tivortex annihilation,” Computer Physics Communications 196, 335–347\n(2015).\n24P. F. Bessarab, “Comment on “path to collapse for an isolated néel\nskyrmion”,” Phys. Rev. B 95, 136401 (2017).\n25G. P. Müller, M. Hoffmann, C. Dißelkamp, D. Schürhoff, S. Mavros,\nM. Sallermann, N. S. Kiselev, H. Jónsson, and S. Blügel, “Spirit: Mul-\ntifunctional framework for atomistic spin simulations,” Phys. Rev. B 99,\n224414 (2019).\n26“18 - on the theory of the dispersion of magnetic permeability in ferromag-\nnetic bodies,” in Collected Papers of L.D. Landau , edited by D. TER HAAR\n(Pergamon, 1965) pp. 101 – 114.\n27S. Zhang and Z. Li, “Roles of nonequilibrium conduction electrons on\nthe magnetization dynamics of ferromagnets,” Phys. Rev. Lett. 93, 127204\n(2004).\n28A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F. Garcia-Sanchez,\nand B. Van Waeyenberge, “The design and verification of mumax3,” AIP\nAdvances 4, 107133 (2014), https://doi.org/10.1063/1.4899186.\n29V . M. Kuchkin, K. Chichay, B. Barton-Singer, F. N. Rybakov, S. Blügel,\nB. J. Schroers, and N. S. Kiselev, “Geometry and symmetry in skyrmion\ndynamics,” Phys. Rev. B 104, 165116 (2021).\n30A. A. Thiele, “Steady-state motion of magnetic domains,” Phys. Rev. Lett.\n30, 230–233 (1973).\n31F. Muckel, S. von Malottki, C. Holl, B. Pestka, M. Pratzer, P. F. Bessarab,\nS. Heinze, and M. Morgenstern, “Experimental identification of two dis-\ntinct skyrmion collapse mechanisms,” Nature Physics 17, 395–402 (2021).\n32P. F. B. Vladyslav M. Kuchkin and N. S. Kiselev, “Thermal generation of\ndroplet soliton in chiral magnet,” arXiv preprint arXiv:2111.12342 (2021).\n33M. N. Wilson, A. B. Butenko, A. N. Bogdanov, and T. L. Monchesky, “Chi-\nral skyrmions in cubic helimagnet films: The role of uniaxial anisotropy,”\nPhys. Rev. B 89, 094411 (2014)."    },    {        "title": "2402.02742v1.Bifurcation_to_complex_dynamics_in_largely_modulated_voltage_controlled_parametric_oscillator.pdf",        "content": "Bifurcation to complex dynamics in largely\nmodulated voltage-controlled parametric oscillator\nTomohiro Taniguchi1*\n1National Institute of Advanced Industrial Science and Technology (AIST), Research Center for Emerging\nComputing Technologies, Tsukuba, Ibaraki 305-8568, Japan\n*tomohiro-taniguchi@aist.go.jp\nABSTRACT\nAn experimental demonstration of a parametric oscillation of a magnetization in a ferromagnet was performed recently by\napplying a microwave voltage, indicating the potential to be applied in a switching method in non-volatile memories. In the\nprevious works, the modulation of a perpendicular magnetic anisotropy field produced by the microwave voltage was small\ncompared with an external magnetic field pointing in an in-plane direction. A recent trend is, however, opposite, where an\nefficiency of the voltage controlled magnetic anisotropy (VCMA) effect is increased significantly by material research and\nthus, the modulated magnetic anisotropy field can be larger than the external magnetic field. Here, we solved the Landau-\nLifshitz-Gilbert equation numerically and investigated the magnetization dynamics driven under a wide range of the microwave\nVCMA effect. We evaluated bifurcation diagrams, which summarize local maxima of the magnetization dynamics. For low\nmodulation amplitudes, the local maximum is a single point because the dynamics is the periodic parametric oscillation. The\nbifurcation diagrams show distributions of the local maxima when the microwave magnetic anisotropy field becomes larger than\nthe external magnetic field. The appearance of this broadened distribution indicates complex dynamics such as chaotic and\ntransient-chaotic behaviors, which were confirmed from an analysis of temporal dynamics.\nV oltage controlled magnetic anisotropy (VCMA) effect1modulates a perpendicular magnetic anisotropy at a ferromagnetic\nmetal/nonmagnetic insulating layer interface by modulating of electron states near the interface2–4and/or inducing magnetic\nmoments5. It enables us to manipulate the direction of the magnetization in a ferromagnet electrically without the Joule heating,\nand thus, is expected to be a new writing method in magnetoresistive random access memory (MRAM), whose writing scheme\ncurrently relies on spin-transfer torque6, 7. The material researches for highly efficient VCMA effect has been reported, where\nthe perpendicular magnetic anisotropy is largely modulated by a small voltage8–17. The efficiency recently achieved reaches to\nabout 300 fJ/(Vm)18, which corresponds to a magnetic anisotropy field on the order of kilo-Oersted for typical VCMA-based\nMRAM. At the same time, analyses on the magnetization dynamics driven by the VCMA effect have been investigated both\nexperimentally and numerically19–28. It has been revealed that the switching is unstable when the pulse width of the voltage is\nshort because the dynamics becomes very sensitive to the pulse shape in such condition24. For a long-pulse regime, however,\nthe switching also becomes unstable due to noise23.\nTo overcome the issue, a parametric oscillation of the magnetization by applying microwave voltage was proposed in\nRef.29. For the magnetization switching in the VCMA-based MRAM using a perpendicularly magnetized free layer, an\nexternal magnetic field Happlpointing in an in-plane direction is applied and induces the magnetization precession around Happl,\nwhich eventually relaxes to the direction of Happl. When the microwave voltage witn an oscillation frequency f=2fL, where\nfL=γHappl/(2π)(γis the gyromagnetic ratio) is the Larmor frequency, is applied, however, a sustainable oscillation of the\nmagnetization is excited. This oscillation is classified into the parametric oscillation, and for simplicity, we call the oscillator as\nvoltage-controlled parametric oscillator. Since this oscillation is stable, it can be used to manipulate the magnetization direction\neven in a long-pulse regime, which results in a reliable magnetization switching. Note that the previous work29focuses on a\nparameter region of HKa/Happl≪1, where HKais the amplitude of the modulated magnetic anisotropy field generated by the\nmicowave VCMA effect. The recent progress of the VCMA efficiency, however, makes the opposite limit, HKa/Happl≫1,\navailable because the value of HKais growing rapidly, as mentioned above. The dynamical behavior of the magnetization in\nthis limit has not been investigated yet.\nIn this work, we study the magnetization dynamics driven by the microwave VCMA effect by solving the Landau-Lifshitz-\nGilbert (LLG) equation numerically. We change the value of HKain a wide range including the limit of HKa/Happl≫1. To\nclarify the change of the dynamical behavior, bifurcation diagrams are evaluated, which summarize local maxima of the\noscillating magnetization. When the magnitude of HKais small, the parametric oscillation is induced, and the bifurcation\ndiagram becomes single points because the dynamics is periodic. When HKabecomes larger than Happl, however, the bifurcation\n1arXiv:2402.02742v1  [cond-mat.mes-hall]  5 Feb 2024Figure 1. (a) Schematic illustration of magnetization oscillation in a ferromagnetic/nonmagnetic trilayer. The unit vector m\npointing in the magnetization direction in free layer shows a sustainable oscillation around an external magnetic field Happlin\nthe in-plane direction when the frequency of a microwave voltage is twice the Larmor frequency fL. (b) Dynamical trajectory\nof the magnetization in a parametric oscillation state. Parameters are Happl=300 Oe and HKa=100 Oe. The black triangle\nindicates the direction of the magnetization motion. (c) Time evolution of mzin the parametric oscillation state. (d) Fourier\nspectrum of mz. The inset shows the spectrum around the main peak in a linear scale.\ndiagrams show broad distributions. It is revealed that the appearance of these complex, broadened structures indicated chaotic\nor transient-chaotic behavior, which are confirmed from evaluations of the Lyapunov exponent and analyses on temporal\ndynamics.\nSystem description\nIn Fig. 1(a), we show a schematic illustration of a ferromagnetic/nonmagnetic/ferromagnetic trilayer. The top and bottom\nferromagnets correspond to free and reference layer, respectively. We apply a macrospin assumption in the free layer, whose\nvalidity in dynamical state driven by the VCMA effect has been confirmed experimentally29. Thus, the unit vector mpointing\nin the magnetization direction in the free layer conserves its magnitude, i.e., |m|=1andd|m|/dt=0. An external magnetic\nfield Happlis applied to an in-plane direction. We assume that the shape of the free layer is a cylinder, and therefore, the free\nlayer does not have an in-plane magnetic anisotropy. For convenience, we use a Cartesian coordinate, where the xaxis is\nparallel to Happlwhile the zaxis is normal to the film plane. The magnetic field in the free layer H, in the absence of an applied\nvoltage, is given by\nH=Happlex+[HK−4πM(Nz−Nx)]mxez, (1)\nwhere ek(k=x,y,z) is the unit vector in the kdirection. The perpendicular magnetic anisotropy field includes the contribution\nfrom the interfacial magnetic anisotropy field HK30–32and the shape magnetic anisotropy field 4πM(Nz−Nx)with the\ndemagnetization coefficients Nk(Nx=Nydue to the cylindrical symmetry). The net perpendicular magnetic anisotropy field,\nHK−4πM(Nz−Nx), determines the retention time of MRAM. In the conventional scheme of the writing in the VCMA-based\nMRAM, the direct voltage modulates this net perpendicular magnetic anisotropy close to zero to excite the magnetization\nprecession around the external magnetic field33. If the perpendicular component largely remains finite, the magnetization just\nmoves its direction to the direction of the external field with the angle sin−1(Happl/H′\nK)and the precession cannot be excited,\nwhere H′\nKis the reduced perpendicular magnetic anisotropy field by the VCMA effect. In fact, the numerical simulation in\nRef.33assumes that the net perpendicular magnetic anisotropy field is completely canceled by the VCMA effect. Note that\nthis assumption is important not only for making the simulation simple but also for experiments. If the direct (or intrinsic)\ncomponent of the perpendicular magnetic anisotropy field remains finite during the precession, an instantaneous frequency\nbecomes nonuniform. Such a nonuniform frequency will increase the switching error because the pulse width of the voltage for\nwriting the bit is determined as a half of the Larmor precession period for VCMA-based MRAM. Therefore, it is preferable\nto make the direct component of the perpendicular magnetic anisotropy field zero during the switching for the conventional\nswitching scheme. In the parametric oscillation state, both direct and microwave voltages are applied to the trilayer, and the\ndirect voltage modulates the total perpendicular magnetic anisotropy so that it becomes close to zero29, while the microwave\nvoltage provides an oscillating magnetic anisotropy field. Accordingly, the magnetic field used in the following calculation\nbecomes,\nH=Happlex+HKasin(2πft)mzez, (2)\nwhere HKaand fare the magnitude and frequency of the magnetic anisotropy field due to the microwave voltage. The\nmagnetization dynamics driven by this magnetic field is described by the LLG equation,\ndm\ndt=−γm×H+αm×dm\ndt, (3)\n2/14Figure 2. (a) Dynamical trajectory of the magnetization in a parametric oscillation state. Parameters are Happl=300Oe and\nHKa=620 Oe. The black triangle indicates the direction of the magnetization motion. (b) Time evolution of mzin the\nparametric oscillation state. (c) Fourier spectrum of mz.\nwhere αis the damping constant. Throughout this paper, we use the values of γ=1.764×107rad/(Oe s) and α=0.005. The\npreparation of the initial state of mby investigating thermal equilibrium is explained in Methods34, 35. Recall that the LLG\nequation conserves the norm of mas|m|=1. Therefore, although mis a three-dimensional vector, its dynamical degree of\nfreedom is two due to this constraint. In fact, if we use a spherical coordinate, for example, the dynamics of mis described by\ntwo variables (zenith and azimuth angles). It should be noted that dynamical systems described by differential equations cannot\nshow chaotic behavior when the dynamical degree of freedom is less than or equal to two, according to the Poincaré-Bendixson\ntheorem36. The presence of the microwave voltage, however, makes the present system non-autonomous and provides a\npossibility to excite chaotic behavior, as shown below.\nResults\nHere, we study the change of the magnetization dynamics for various magnitude of HKa.\nParametric oscillation\nLet us first start by confirming the parametric oscillation studied previously29. It was shown in Ref.29that a sustainable\noscillation of the magnetization is excited when the frequency of the microwave voltage, f, is twice the Larmor frequency,\nfL=γHappl/(2π). Therefore, in the following, we fix the value of fto be f=2fL. Figure 1(b) shows the dynamical trajectory\nof the magnetization in a steady state, where Happl=300Oe while HKa=100Oe, i.e., HKa/Happl≪1, as in the case of the\nprevious work29. Since |m|=1is satisfied in the LLG equation, it is useful to draw the dynamical trajectory on a unit sphere,\nas shown in this figure. Time evolution of mzin a steady state is also shown in Fig. 1(c). These results indicate an appearance\nof the sustainable oscillation of the magnetization mentioned above. In Fig. 1(d), the Fourier spectrum of mzis shown, where\nthe inset shows it around the main peak in a linear scale. Its main peak appears at 0.84GHz, which is the same with fLwith\nHappl=300Oe. These results are consistent with the previous works29. Since the magnetization switches its direction between\nmz≃+1andmz≃ −1periodically with the period 1/(2fL), this parametric oscillation can be used as a switching scheme in\nVCMA-based MRAM29. Recall that this oscillation is sustained by the microwave modulation of the magnetic anisotropy; if\nthis time-dependent modulation is absent, the magnetization monotonically relaxes to the direction of the external magnetic\nfield. Spin-wave propagation through a parametric excitation is another example of the magnetization dynamics caused by\nmicrowave voltage, which has been studied previously37–40.\nAppearance of complex dynamics and bifurcation diagram\nWhen the value of HKafurther increases, the magnetization dynamics becomes complex. In Fig. 2(a), we show the dynamical\ntrajectory of the magnetization for HKa=620Oe, while Happl=300Oe is the same with that used in Fig. 1(b). We observe a\nclear change of the magnetization dynamics by comparing Figs. 1(b) and 2(a). The trajectory is not a simple circle in Fig. 2(a).\nThe time evolution of mzand its Fourier transformation are shown in Figs. 2(b) and 2(c), respectively. Figure 2(b) indicates that\nthe magnetization dynamics is still periodic, while Fig. 2(c) indicates the appearance of multipeak structure. These results\nindicate that the application of the microwave voltage is no longer applicable to the switching method for the VCMA-based\nMRAM when the modulation of the magnetic anisotropy field becomes larger than the external magnetic field due to the\nbreakdown of the simple parametric oscillation.\nA way to distinguish these dynamics, such as the difference between Figs. 1(b) and 2(a), qualitatively is to draw a bifurcation\ndiagram36. In Fig. 3(a), we summarize local maxima of mzfor various HKa, where Happl=300Oe. Recall that the parametric\n3/14Figure 3. (a) A bifurcation diagram summarizing local maxima of mz(t)and (b) Lyapunov exponent for various HKa, where\nHappl=300 Oe.\nFigure 4. (a) Dynamical trajectory of the magnetization in a parametric oscillation state. Parameters are Happl=300Oe and\nHKa=1200 Oe. (b) Time evolution of mzin the parametric oscillation state. (c) Fourier spectrum of mz.\noscillation is excited when HKais small. In this case, the dynamics is periodic and mzis similar to a simple trigonometric\nfunction, as shown in Fig. 1(c). The local maxima of mzfor this case, thus, saturate to a single point. When the dynamics\nbecome complex, the bifurcation diagram shows broadened structure. For example, there are three points at HKa=620 Oe in\nFig. 3(a), the validity of which is confirmed from Fig. 2(b). When the value of HKafurther increases, the bifurcation diagram\nshows largely broadened structure and finally shows simplified structure again. As discussed below, these correspond to chaotic\nand transient-chaotic behaviors41–43. Note that the appearance of complex but still periodic structure might weakly depend on\nthe initial state (see Methods) while the region of the broadened structure might depend on the measurement time, as will be\nmentioned below. It is difficult to analytically estimate the value of HKaat which the bifurcation from a simple parametric\noscillation to a complex oscillation occurs; see Methods. However, the numerical simulations for various parameters imply\nthat the complex oscillation appears when HKabecomes larger than Happl, as discussed below. We also evaluated Lyapunov\nexponent36by Shimada-Nagashima method44, as shown in Fig. 3(b). The Lyapunov exponent is an inverse of a time scale\ncharacterizing an expansion of a difference between two solutions of the LLG equation with an infinitesimally different initial\nconditions; see Methods explaining the evaluation method of the Lyapunov exponent. A negative Lyapunov exponent means\nthat the magnetization saturates to a fixed point. The Lyapunov exponent is zero when the magnetization dynamics are periodic,\nwhile it becomes positive when the dynamics are chaotic. We notice that the Lyapunov exponent in the present system is zero\nor positive, depending on the value of HKa. Thus, the Lyapunov exponent can be used as an indicator to distinguish between\nperiodic oscillation and chaotic dynamics. For example, we can conclude that the dynamics in Fig. 2(a) is periodic not only\nfrom the temporal dynamics shown in Fig. 2(b) but also from the fact that the Lyapunov exponent for HKa=620 Oe is zero.\nChaotic dynamics\nIn Fig. 4(a), we show the dynamical trajectory of the magnetization for HKa=1200 Oe. The dynamical trajectory covers\nalmost the whole region of the unit sphere. The time evolution of mzbecomes non-periodic, as shown in Fig. 4(b), and the\nFourier spectrum shows a broad structure having several peaks. These results imply chaotic dynamics of the magnetization41, 42.\n4/14Figure 5. (a) Dynamical trajectory of the magnetization in a parametric oscillation state. Parameters are Happl=300Oe and\nHKa=1600 Oe. The black triangle indicates the direction of the magnetization motion. (b) Time evolution of mzin the\nparametric oscillation state. (c) Fourier spectrum of mz.\nThe appearance of chaos for this parameter can also be concluded from the fact that the Lyapunov exponent shown in Fig. 3(b)\nis positive.\nAs mentioned above, chaotic dynamics in the present system are excited because of the presence of the microwave voltage.\nSimilar examples in spintronics devices have been found in spin-torque oscillators with time-dependent inputs45–47. The\ndifferences of the phenomena observed between the voltage-controlled parametric oscillator studied here and spin-torque\noscillator are as follows. In the spin-torque oscillators, a sustainable oscillation of the magnetization is driven by direct currents,\nand an injection of time-dependent inputs is not a necessary condition for the oscillation. Spin-torque oscillator cannot show\nchaotic behavior when only the direct current is injected due to the constraint by the Poincaré-Bendixson theorem. A way to\nexcite chaos in spin-torque oscillator is to inject time-dependent inputs such as alternating current and/or magnetic field45–47.\nWhen the magnitudes of these time-dependent inputs are relatively small, synchronization may occur. When their magnitudes\nare further increased, chaos might be induced. On the other hand, the sustainable oscillation of the magnetization in the present\nvoltage-controlled parametric oscillator is driven by microwave voltage. In other words, this time-dependent input is a necessary\ncondition for the oscillation. Chaotic dynamics appear when the magnitude of the microwave voltage becomes relatively large,\nas shown in Fig. 4.\nLet us briefly mention applicability of the chaotic dynamics for practical devices. Chaos in spintronics devices has been\nstudied both experimentally and theoretically using various methods48–57. An excitation of chaos in spintronics devices may\nevoke interest from a viewpoint of brain-inspired computing58, 59. For example, it was found that a computational capability of\nphysical reservoir computing is enhanced when spin-torque oscillators are near the edge of chaos58, where chaos was excited\nby adding another ferromagnet to the oscillator. An excitation of chaos in spin-torque oscillator by time-dependent inputs,\nhowever, seems to require large power consumption. For example, the amplitude of the alternating current density assumed\nin the numerical simulations in Refs.45–47is on the order of 107−108A/cm2. The value is larger than the switching current\ndensity of the state of the art spin-transfer torque driven MRAM60, and thus, is hardly desirable. The large current also causes\nlarge power consumption, which is also unsuitable for practical applications. The voltage-controlled parametric oscillator, on\nthe other hand, ideally reduces the power consumption significantly. In fact, this point has been a motivation for developing\nVCMA-based MRAM. However, these VCMA-based devices often require an external magnetic field for both the switching\nand parametric oscillation, which is not preferable in practical applications. A way to solve the issue might be to use an\neffective field, instead of applying external magnetic field, such as an interlayer exchange coupling field, as investigated in the\nstudy of spin-orbit torque driven MRAM61. Physical reservoir computing by the VCMA-based MRAM without an external\nmagnetic field was investigated recently62, however, switching nor parametric oscillation was used there. A development of the\nvoltage-controlled parametric oscillator without requiring an external magnetic field will be of interest as a future work for\napplying it to practical applications.\nTransient chaos\nIn Fig. 5(a), we show the dynamical trajectory of the magnetization for HKa=1600 Oe. The time evolution of mzand its\nFourier transformation are also shown in Figs. 5(b) and 5(c), respectively. We note that the dynamics shown here corresponds to\nmz(t)in a long-time limit, i.e., a steady state. The results look similar to those shown in Figs. 1 and 2. Also, the local maxima\nofmzin the bifurcation diagram in Fig. 3(a) concentrates on a single point. However, there is a critical difference between the\nmagnetization dynamics shown in Fig. 5 and those shown in Figs. 1 and 2.\nTo clarify their differences, it is necessary to focus on a process to reach the steady state. In Figs. 6(a) and 6(b), we show\nthe time evolution of mz(t)from t=0to the steady states for HKa=620Oe and 1600 Oe, respectively. When HKais relatively\n5/14Figure 6. Time evolution of mznear t=0 for (a) HKa=620 Oe and (b) HKa=1600 Oe.\nFigure 7. Time evolution of mzfor (a) HKa=1900 Oe, (b) HKa=1980 Oe, and (c) HKa=2000 Oe.\n6/14Figure 8. Bifurcation diagrams summarizing the local maxima of mzfor (a) Happl=200 Oe, (b) Happl=300 Oe, and (c)\nHappl=400Oe. Note that (b) is identical to Fig. 3(a) but is shown here for comparison. The value of the damping constant αis\n0.005. Those for α=0.020 are shown in (d)-(f).\nsmall [see Fig. 6(a)], the magnetization immediately reaches to the steady state shown in Fig. 2(b). When HKais relatively\nlarge [see Fig. 6(b)], on the other hand, chaotic behavior appears initially, and it suddenly changes to the steady state shown\nin Fig. 5(b). The phenomenon shown in Fig. 5(b) corresponds to a transient chaos43, where dynamical systems initially\nshow chaotic behavior but it suddenly disappears. Time necessary to move from chaos to a steady state is sensitive to various\nparameters and initial conditions of systems43. For example, in Figs. 7(a), 7(b), and 7(c), we show time evolution of mzfor\nHKa=1900 Oe, 1980 Oe, and 2000 Oe, respectively, where the initial conditions are common. The results indicate that the\ntime necessary to realize the steady state depends on the parameter HKaand it does not show, for example, a monotonic change\nwith respect to the change of the parameter. The transient chaos in spintronics devices was predicted in a spin-torque oscillator\nwith a delayed-feedback circuit54. The phenomenon has not been verified experimentally yet, although chaos was confirmed\nrecently57.\nWe note that the classification of chaos and transient chaos depends on a measurement time43. For example, in the present\nwork, we solve the LLG equation from t=0tot=5µs and classify the magnetization dynamics. If we change this maximum\ntime ( 5µs) to, for example 2µs, the dynamics for HKa=2000 Oe, shown in Fig. 7(c), will be classified as chaos. Another\nexample can be seen in the bifurcation diagram and the Lyapunov exponent shown in Fig. 3, where a broadened structure in\nthe bifurcation diagram and a positive Lyapunov for HKa=1920 Oe, indicating chaos for this parameter. If we measure the\ndynamics for this parameter longer, however, the dynamics might change to a steady state; in such a case, the dynamics will be\nclassified as transient chaos. As a general knowledge on transient chaos43, we should remember that the classification of chaos\nand transient chaos has such an arbitrary property.\nBifurcation diagrams for various applied fields\nAs mentioned above, a bifurcation diagram is useful to classify the magnetization dynamics, although, for example, the\ndifference between a simple steady oscillation and a transient chaos should be verified from temporal dynamics, as mentioned\nabove. We also notice that the whole structure of the bifurcation diagram does not change qualitatively even when the initial\ncondition is slightly changed. Recall that the present system includes only three parameters, HKa,Happl, andα. Therefore, in\nFigs. 8(a)-8(c), we show the bifurcation diagrams for (a) Happl=200Oe, (b) Happl=300Oe, and (c) Happl=400Oe. These\nfigures indicate that the local maxima of mzfor relatively small HKaconcentrate on single points, which indicate the excitation\nof the parametric oscillation. As HKaincreases, broadened structures appear, i.e., the local maxima of mzhave various values,\n7/14implying the appearance of complex oscillation and chaos. The figures also imply that this boundary between the parametric\noscillation and the complex dynamics, i.e., the boundary between the single and multiple points in the bifurcation diagram,\nlocates near HKa/Happl≃1.5−2.0. We examine similar calculations for different αbut the boundary seems to be unchanged;\nsee Figs. 8(d)-8(f), which are the bifurcation diagrams for α=0.020. Because of high nonlinearity in the LLG equation, it is\ndifficult to analytically verify these indications; however, it might be useful to design, for example, the modulation voltage for\nVCMA-based MRAM utilizing the parametric oscillation. We keep this question as a future work.\nWe note that the value of HKa/Happl≃1.5−2.0is available by current technology. The typical value of Happlis on the\norder of 100Oe; for example, it is 250Oe in Ref.23and720Oe in Ref.29. Although the value of Happlcan be further large\nexperimentally, a large Happlmight be unsuitable for practical applications; see Methods for analytical treatment of the LLG\nequation. On the other hand, the value of HKarelates to the VCMA efficiency η, the saturation magnetization M, the applied\nvoltage Vappl, and the thicknesses of the free and insulating layers, dFanddI, via HKa= (2ηVappl)/(MdFdI). Substituting their\ntypical values [ ηis about 300fJ/(Vm)18,Vapplis about 0.5V at maximum, Mis about 1000 emu/cm3,dFis about 1nm, and dI\nis about 2.5nm), HKacan be on the order of kilo Oersted at maximum, as written in the introduction. Therefore, we believe\nthat the value of HKa/Happl≃1.5−2.0 is experimentally achievable.\nConclusion\nIn summary, the magnetization dynamics in the voltage-controlled parametric oscillator for a large microwave voltage limit were\ninvestigated by numerical simulation of the LLG equation. As the modulation increases, the dynamical trajectory changes from\na simple parametric oscillation to complex oscillations, which are still periodic but have several local maxima in the amplitude.\nSuch dynamics will be of little preference in a switching scheme for VCMA-based MRAM. The evaluation of the bifurcation\ndiagrams for various values of the external magnetic field and the damping constant indicated that the simple parametric\noscillation is broken when the amplitude of the modulated magnetic anisotropy field becomes larger than the external magnetic\nfield, and this bifurcation point seems to be hardly sensitive to the value of the damping constant. A further enhancement of the\nmicrowave modulation leads to chaotic and transient-chaotic behaviors, which might make the voltage-controlled parametric\noscillator applicable to other usage in electric devices. These dynamics were classified from the bifurcation diagrams, Lyapunov\nexponent, and analyses on temporal dynamics.\nMethods\nPreparation of initial state\nWe prepare natural initial states by solving the LLG equation with thermal fluctuation34. For this purpose, we use Eq. (1) as the\nmagnetic field. We add a torque, −γm×h, due to thermal fluctuation to the right-hand side of Eq. (3). Here, the components of\nhsatisfy the fluctuation-dissipation theorem63,\n⟨hk(t)hℓ(t′)⟩=2αkBT\nγMVδkℓδ(t−t′). (4)\nIn the numerical simulation, the random field is given by\nhk(t) =s\n2αkBT\nγMV∆tξk(t), (5)\nwhere the time increment ∆tis 1 ps in this work. White noise ξkis defined from two random numbers, ζaandζb, in the range\nof0<ζa,ζb<1by the Box-Muller transformation as ξa=p\n−2lnζasin(2πζb)andξb=p\n−2lnζacos(2πζb). We added\nEq. (5) to the magnetic field and solved the LLG equation numerically using the Runge-Kutta method for the investigation\nof thermal equilibrium before applying microwave voltage. The value of the net perpendicular magnetic anisotropy field,\nHK−4πM(Nz−Nx), in the absence of microwave voltage is assumed to be 6.283kOe, while the saturation magnetization\nis set to be 955emu/cm329. The temperature Tis300K, and the volume is V=π×50×50×1.1nm3, which is typical for\nVCMA experiments. The thermal fluctuation excites a small-amplitude oscillation of the magnetization around the energetically\nminimum state with the ferromagnetic resonance frequency. We pick the temporal directions of the oscillating magnetization\nand use them as the natural initial states. It should be noted that the value of HKaat which the complex but still periodic\noscillation appears weakly depends on the choice of the initial state, despite the structure of the bifurcation diagram is not\nchanged qualitatively. It might relates to the presence of two stable phases of the parametric oscillation with respect to the\nmicrowave voltage35.\n8/14Analytical treatment of the LLG equation\nHere, we discuss an analytical treatment of the parametric oscillation, which provides a condition to excite the oscillation. It is,\nhowever, not applicable to investigate the bifurcation from the simple parametric oscillation to the complex but still periodic\noscillation shown in Figs. 1(b) and 2(a).\nSince we are interested in the oscillation around the xaxis, we introduce angles ΘandΦasm= (cosΘ,sinΘcosΦ,sinΘsinΦ).\nFor simplicity, we introduce notations k=γHKaandh=γHappl. The LLG equation for ΘandΦare given as dΘ/dt=\n−(1/sinΘ)(∂ε/∂Φ)−α(∂ε/∂Θ)and sin Θ(dΦ/dt) = (∂ε/∂Θ)−α(1/sinΘ)(∂ε/∂Φ), where\nε=−hcosΘ−k\n2sin(2πft)sin2Θsin2Φ, (6)\ni.e.,\ndΘ\ndt=ksin(2πft)sinΘsinΦcosΦ−α\u0002\nh−ksin(2πft)cosΘsin2Φ\u0003\nsinΘ, (7)\ndΦ\ndt=h−ksin(2πft)cosΘsin2Φ+αksin(2πft)sinΦcosΦ. (8)\nSince we are interested in an oscillation where myandmzoscillates with the frequency fL=f/2, we introduce Ψ=Φ−2πfLt,\nwhich obeys\ndΨ\ndt=h−2πfL−ksin(2πft)cosΘsin2(Ψ+2πfLt)+αksin(2πft)sin(Ψ+2πfLt)cos(Ψ+2πfLt). (9)\nIn the parametric oscillation states, the tilted angle Θof the magnetization from the xaxis and the phase difference Ψare\napproximately constants35. Therefore, after averaging Eqs. (7) and (9) over a period 1 /fL, we obtain,\ndΘ\ndt=k\n4sinΘcos2Ψ−α\u0012\nh−k\n4cosΘsin2Ψ\u0013\nsinΘ, (10)\ndΨ\ndt=σ−k\n4cosΘsin2Ψ+αk\n4cos2Ψ, (11)\nwhere σ=h−2πfL. Although the present work focuses on the case of σ=0only, we keep σas finite here, for generality.\nThe steady state solutions of ΘandΨafter averaging are\ncosΘ=±4(σ+α2h)p\n(1+α2)k2−[4α(h−σ)]2, (12)\ntan2Ψ=±p\n(1+α2)k2−[4α(h−σ)]2\n4α(h−σ). (13)\nThese solutions imply, for example, that (1+α2)k2>[4α(h−σ)]2for exciting the parametric oscillation, which can easily\nbe satisfied when α≪1. These solutions, however, cannot be used to discuss, for example, the bifurcation from the simple\nparametric oscillation to the complex oscillation because Φabove is assumed to oscillates with only a single frequency fL,\nwhile the complex oscillation is a superposition of multiple frequencies. Even for the parametric oscillation state, the above\nsolutions may not be quantitative due to the fact that, strictly speaking for example, Θis not constant.\nAlthough a reliability of the magnetization switching by the parametric oscillation is not the main scope of this work,\nlet us briefly provide a comment on the relationship between the switching reliability and the value of HKa/Happlmentioned\nin the introduction. First, the value of Happlshould be carefully chosen. For a large Happl, the precession frequency of the\nmagnetization becomes high. Such a fast precession makes the switching unstable because the dynamics becomes sensitive to\nthe pulse shape, as mentioned in the introduction. When Happlis small, however, the switching is also unstable because the\ndynamics is highly affected by thermal fluctuation, which is also written in the introduction. Summarizing them, the value of\nHapplshould be carefully determined, depending on the various factors, such as circuit quality controlling the pulse shape and\nthe volume of the ferromagnetic layer. Second, a large HKais considered to be preferable for a reliable switching due to the\n9/14following reasons. First, as mentioned below Eqs. (12) and (13), there is a threshold value of HKa[(1+α2)k2>[4α(h−σ)]2]\nto excite the parametric oscillation. Second, Eq. (12) indicates that the cone angle Θof the magnetization precession around\nHapplbecomes close to 90◦when HKais large. In other words, the cone angle decreases as HKadecreases. A precession with a\nsmall cone angle is unstable because such a small oscillation is easily disturbed by thermal fluctuation. Regarding these points,\nit will be preferable to increase HKa/Happl, orHKa, for a reliable switching because there is a threshold of HKaof the parametric\noscillation and it is necessary to make the oscillation robust against thermal fluctuation. However, as revealed in the main text,\nthe precession becomes complex when HKa/Happlbecomes greatly large, which is a main message in this work.\nEvaluation method of Lyapunov exponent\nThe evaluation method of the Lyapunov exponent is as follows34, 64. We denote the solution of the LLG equation at a certain time\ntasm(t). We introduce the zenith and azimuth angles, θandϕ, asm= (mx,my,mz) = ( sinθcosϕ,sinθsinϕ,cosθ). We also\nintroduce m(1)(t) = ( sinθ(1)cosϕ(1),sinθ(1)sinϕ(1),cosθ(1)), where θ(1)andϕ(1)satisfy ε=p\n[θ−θ(1)]2+[ϕ−ϕ(1)]2.\nNote that εcorresponds to the distance between m(t)andm(1)(t)attin the spherical space. Since the Lyapunov exponent\ncharacterizes the sensitivity of the dynamical system to the initial state, we study an expansion of εwith time. For this purpose,\nwe assume a small value of ε, which is ε=1.0×10−5in this work. For convenience, we introduce a notation,\nD[m(t),m(1)(t)] =q\u0002\nθ(t)−θ(1)(t)\u00032+\u0002\nϕ(t)−ϕ(1)(t)\u00032. (14)\nSolving the LLG equations of m(t)andm(1)(t), we obtain m(t+∆t)andm(1)(t+∆t), where ∆tis time increment. From them,\nwe evaluate the distance between m(t+∆t)andm(1)(t+∆t)as\nD[m(t+∆t),m(1)(t+∆t)] =q\u0002\nθ(t+∆t)−θ(1)(t+∆t)\u00032+\u0002\nϕ(t+∆t)−ϕ(1)(t+∆t)\u00032. (15)\nThen, a temporal Lyapunov exponent at t+∆tis given as\nΛ(1)=1\n∆tlnD(1)\nε, (16)\nwhereD(1)=D[m(t+∆t),m(1)(t+∆t)].\nNext, we introduce m(2)(t+∆t) = ( sinθ(2)cosϕ(2),sinθ(2)sinϕ(2),cosθ(2)), where θ(2)andϕ(2)are defined as\nθ(2)(t+∆t) =θ(t+∆t)+εθ(1)(t+∆t)−θ(t+∆t)\nD[m(t+∆t),m(1)(t+∆t)], (17)\nϕ(2)(t+∆t) =ϕ(t+∆t)+εϕ(1)(t+∆t)−ϕ(t+∆t)\nD[m(t+∆t),m(1)(t+∆t)]. (18)\nAccording to these definitions, m(t+∆t)andm(2)(t+∆t)satisfy\nD[m(t+∆t),m(2)(t+∆t)] =ε. (19)\nIt means that m(2)(t+∆t)is defined by moving m(t+∆t)to the direction of m(1)(t+∆t)with a distance εin the (θ,ϕ)phase\nspace. Solving the LLG equations for m(t+∆t)andm(2)(t+∆t), we obtain m(t+2∆t)andm(2)(t+2∆t). The temporal\nLyapunov exponent at t+2∆tis\nΛ(2)=1\n∆tlnD(2)\nε, (20)\nwhereD(2)=D[m(t+2∆t),m(2)(t+2∆t)].\nThese procedures are generalized. At t+n∆t, we have m(t+n∆t) = ( sinθ(t+n∆t)cosϕ(t+n∆),sinθ(t+n∆t)sinϕ(t+\nn∆t),cosθ(t+n∆t))andm(n)(t+n∆t) = ( sinθ(n)(t+n∆t)cosϕ(n)(t+n∆),sinθ(n)(t+n∆t)sinϕ(n)(t+n∆t),cosθ(n)(t+\nn∆t)). Then, we define m(n+1)(t+n∆t) = ( sinθ(n+1)(t+n∆t)cosϕ(n+1)(t+n∆),sinθ(n+1)(t+n∆t)sinϕ(n+1)(t+n∆t),cosθ(n+1)(t+\nn∆t))by moving m(t+n∆t)to the direction of m(n)(t+n∆t)with a distance εin the phase space as\nθ(n+1)(t+n∆t) =θ(t+n∆t)+εθ(n)(t+n∆t)−θ(t+n∆t)\nD[m(t+n∆t),m(n)(t+n∆t)], (21)\nϕ(n+1)(t+n∆t) =ϕ(t+n∆t)+εϕ(n)(t+n∆t)−ϕ(t+n∆t)\nD[m(t+n∆t),m(n)(t+n∆t)]. (22)\n10/14Note that m(t+n∆t)andm(n+1)(t+n∆t)satisfy D[m(t+n∆t),m(n+1)(t+n∆t)] =ε. Then, solving the LLG equations of\nm(t+n∆t)andm(n+1)(t+n∆t), we obtain m(t+ (n+1)∆t)andm(n+1)(t+ (n+1)∆t). A temporal Lyapunov exponent at\nt+(n+1)∆tis\nΛ(n+1)=1\n∆tlnD(n+1)\nε, (23)\nwhereD(n+1)=D[m(t+(n+1)∆t),m(n+1)(t+(n+1)∆t)]. The Lyapunov exponent is defined as a long-time average of the\ntemporal Lyapunov exponent as\nΛ=lim\nN→∞1\nNN\n∑\ni=1Λ(i). (24)\nIn the present study, we solve the LLG equation from t=0totmax=5µs with ∆t=1ps. Therefore, there are tmax/∆t=5×106\nsteps during the evaluation of the magnetization dynamics. We use the last 1×106steps for the evaluation of the Lyapunov\nexponent. Note that this method is an application of Shimada-Nagashima method44for the evaluation of the Lyapunov exponent\nfrom numerical simulation of an equation of motion to the LLG equation. Strictly speaking, the Lyapunov exponent evaluated\nhere corresponds to the conditional largest Lyapunov exponent in non-autonomous system34, 47.\nReferences\n1.Weisheit, M. et al. Electric Field-Induced Modification of Magnetism in Thin-Film Ferromagnets. Science 315, 349\n(2007).\n2.Duan, C.-G. et al. Surface Magnetoelectric Effect in Ferromagnetic Metal Films. Phys. Rev. Lett. 101, 137201 (2008).\n3.Nakamura, K. et al. Giant Modification of the Magnetocrystalline Anisotropy in Transition-Metal Monolayers by an\nExternal Electric Field. Phys. Rev. Lett. 102, 187201 (2009).\n4.Tsujikawa, M. & Oda, T. Finite Electric Field Effects in the Large Perpendicular Magnetic Anisotropy Surface Pt/Fe/Pt(001):\nA First-Principles Study. Phys. Rev. Lett. 102, 247203 (2009).\n5.Miwa, S. et al. V oltage controlled interfacial magnetism through platinum orbits. Nat. Commun. 8, 15848 (2017).\n6.Slonczewski, J. C. Current-driven excitation of magnetic multilayers. J. Magn. Magn. Mater. 159, L1 (1996).\n7.Berger, L. Emission of spin waves by a magnetic multilayer traversed by a current. Phys. Rev. B 54, 9353 (1996).\n8.Maruyama, T. et al. Large voltage-induced magnetic anisotropy change in a few atomic layers of iron. Nat. Nanotechnol.\n4, 158 (2009).\n9.Shiota, Y . et al. V oltage-Assisted Magnetization Switching in Ultrahin Fe 80Co20Alloy Layers. Appl. Phys. Express 2,\n063001 (2009).\n10.Nozaki, T., Shiota, Y ., Shiraishi, M., Shinjo, T. & Suzuki, Y . V oltage-induced perpendicular magnetic anisotropy change in\nmagnetic tunnel junctions. Appl. Phys. Lett. 96, 022506 (2010).\n11.Endo, M., Kanai, S., Ikeda, S., Matsukura, F. & Ohno, H. Electric-field effects on thickness dependent magnetic anisotropy\nof sputtered MgO/Co 40Fe40B20/Ta structures. Appl. Phys. Lett. 96, 212503 (2010).\n12.Wang, W.-G., Li, M., Hageman, S. & Chien, C. L. Electric-field-assisted switching in magnetic tunnel junctions. Nat.\nMater. 11, 64 (2011).\n13.Nozaki, T. et al. V oltage-Induced Magnetic Anisotropy Changes in an Ultrathin FeB Layer Sandwiched between Two\nMgO Layers. Appl. Phys. Express 6, 073005 (2013).\n14.Okada, A. et al. Electric-field effects on magnetic anisotropy and damping constant in Ta/CoFeB/MgO investigated by\nferromagnetic resonance. Appl. Phys. Lett. 105, 052415 (2014).\n15.Skowro ´nski, W. et al. Perpendicular magnetic anisotropy of Ir/CoFeB/MgO trilayer system tuned by electric fields. Appl.\nPhys. Express 8, 053003 (2015).\n16.Nozaki, T. et al. Highly effcient voltage control of spin and enhanced interfacial perpendicular magnetic anisotropy in\niridium-doped Fe/MgO magnetic tunnel junctions. NPG Asia Mater. 9, e451 (2017).\n17.Nozaki, T. et al. Enhancement in the interfacial perpendicular magnetic anisotropy and the voltage-controlled magnetic\nanisotropy by heavy metal doping at the Fe/MgO interface. APL Mater. 6, 026101 (2018).\n11/1418.Nozaki, T. et al. V oltage-cotrolled magnetic anisotropy in an ultrathin Ir-doped Fe layer with a CoFe termination layer.\nAPL Mater. 8, 011108 (2020).\n19.Shiota, Y . et al. Pulse voltage-induced dynamic magnetization switching in magnetic tunnel junctions with high resistance-\narea product. Appl. Phys. Lett. 101, 102406 (2012).\n20.Kanai, S. et al. Electric field-induced magnetization reversal in a perpendicular-anisotropy CoFeB-MgO magnetic tunnel\njunction. Appl. Phys. Lett. 101, 122403 (2012).\n21.Amiri, P., Upadhyaya, P., Alzate, J. G. & Wang, K. L. Electric-field-induced thermally assisted switching of monodomain\nmagnetic bits. J. Appl. Phys. 113, 013912 (2013).\n22.Grezes, C. et al. Ultra-low switching energy and scaling in electric-field-controlled nanoscale magnetic tunnel junctions\nwith high resistance-area product. Appl. Phys. Lett. 108, 012403 (2016).\n23.Shiota, Y . et al. Reduction in write error rate of voltage-driven dynamic magnetization switching by improving thermal\nstability factor. Appl. Phys. Lett. 111, 022408 (2017).\n24.Lee, H. et al. A word line pulse circuit technique for reliable magnetoelectric random access memory. IEEE Trans. Very\nLarge Scle Integ. Syst. 7, 3302 (2017).\n25.Deng, J., Liang, G. & Gupta, G. Ultrafast and lo-energy switching in voltage-controlled elliptical pMTJ. Sci. Rep. 25, 2027\n(2017).\n26.Song, J. et al. Evaluation of operating margin and switching probability of voltage-controlled magnetic anisotropy magnetic\ntunnel junctions. IEEE J. Explor. Solid-State Comput. Devices Circuits 4, 76 (2018).\n27.Wu, Y . C. et al. Study of precessional switching speed control in voltage-controlled perpendicular magnetic tunnel junction.\nAIP Adv. 10, 035123 (2020).\n28.Shao, Y . et al. Sub-volt switching of nanoscale voltage-controlled perpendicular magnetic tunnel junctions. Commun.\nMater. 3, 87 (2022).\n29.Yamamoto, T. et al. V oltage-Driven Magnetization Switching Controlled by Microwave Electric Field Pumping. Nano Lett.\n20, 6012 (2020).\n30.Yakata, S. et al. Influnence of perpendicular magnetic anisotropy on spin-transfer switching current in CoFeB/MgO/CoFeB\nmagnetic tunnel junctions. J. Appl. Phys. 105, 07D131 (2009).\n31.Ikeda, S. et al. A perpendicular-anisotropy CoFeB-MgO magnetic tunnel junction. Nat. Mater. 9, 721 (2010).\n32.Kubota, H. et al. Enhancement of perpendicular magnetic anisotropy in FeB free layers using a thin MgO cap layer. J.\nAppl. Phys. 111, 07C723 (2012).\n33.Shiota, Y . et al. Evaluation of write error rate for voltage-driven dynamic magnetization switching in magnetic tunnel\njunctions with perpendicular magnetization. Appl. Phys. Express 9, 013001 (2016).\n34.Taniguchi, T. Non-periodic input-driven magnetization dynamics in voltage-controlled parametric oscillator. J. Magn.\nMagn. Mater. 563, 170009 (2022).\n35.Taniguchi, T. Phase locking in voltage-controlled parametric oscillator. J. Magn. Magn. Mater. 578, 170806 (2023).\n36.Strogatz, S. H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering\n(Westview Press, 2001), first edn.\n37.Verba, R., Tiberkevich, V ., Krivorotov, I. & Slavin, A. Parametric Excitation of Spin Waves by V oltage-Controlled Magnetic\nAnisotropy. Phys. Rev. Appl. 1, 044006 (2014).\n38.Verba, R., Finocchio, M. C. G., Tiberkevich, V . & Slavin, A. Excitation of propagating spin waves in ferromagnetic\nnanowires by microwave voltage-controlled magnetic anisotropy. Sci. Rep. 6, 25018 (2016).\n39.Chen, Y .-J. et al. Parametric Resonance of Magnetization Excited by Electric Field. Nano Lett. 17, 572 (2017).\n40.Rana, B., Fukuma, Y ., Miura, K., Takahashi, H. & Otani, Y . Excitation of coherent propagating spin waves in ultrathin\nCoFeB film by voltage-controlled magnetic anisotropy. Appl. Phys. Lett. 111, 052404 (2017).\n41.Alligood, K. T., Sauer, T. D. & Yorke, J. A. Chaos. An Introduction to Dynamical Systems (Spinger (New York), 1997).\n42.Ott, E. Chaos in Dynamical Systems (Cambridge University Press (Cambridge), 2002), second edn.\n43.Lai, Y .-C. & Tél, T. Transient Chaos (Spinger (New York), 2011).\n12/1444.Shimada, I. & Nagashima, T. A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems. Prog. Theor.\nPhys. 61, 1605 (1979).\n45.Li, Z., Li, Y . C. & Zhang, S. Dynamic magnetization states of a spin valve in the presence of dc and ac currents:\nSynchronization, modification, and chaos. Phys. Rev. B 74, 054417 (2006).\n46.Yang, Z., Zhang, S. & Li, Y . C. Chaotic Dynamics of Spin-Valve Oscillators. Phys. Rev. Lett. 99, 134101 (2007).\n47.Yamaguchi, T. et al. Synchronization and chaos in a spin-torque oscillator with a perpendicularly magnetized free layer.\nPhys. Rev. B 100, 224422 (2019).\n48.Kudo, K., Sato, R. & Mizushima, K. Synchronized Magnetization Oscillations in F/N/F Nanopillars. Jpn. J. Appl. Phys.\n45, 3869 (2006).\n49.Petit-Watelot, S. et al. Commensurability and chaos in magnetic vortex oscillations. Nat. Phys. 8, 682–687 (2012).\n50.Devolder, T. et al. Chaos in magnetic nanocontact vortex oscillators. Phys. Rev. Lett. 123, 147701 (2019).\n51.Bondarenko, A. V ., Holmgren, E., Li, Z. W., Ivanov, B. A. & Korenivski, V . Chaotic dynamics in spin-vortex pairs. Phys.\nRev. B 99, 054402 (2019).\n52.Montoya, E. A. et al. Magnetization reversal driven by low dimensional chaos in a nanoscale ferromagnet. Nat. Commun.\n10, 543 (2019).\n53.Williame, J., Accoily, A. D., Rontani, D., Sciamanna, M. & Kim, J.-V . Chaotic dynamics in a macrospin spin-torque\nnano-oscillator with delayed feedback. Appl. Phys. Lett. 114, 232405 (2019).\n54.Taniguchi, T. et al. Chaos in nanomagnet via feedback current. Phys. Rev. B 100, 174425 (2019).\n55.Taniguchi, T. Synchronized, periodic, and chaotic dynamics in spin torque oscillator with two free layers. J. Magn. Magn.\nMater. 483, 281–292 (2019).\n56.Williame, J. & Kim, J.-V . Effects of delayed feedback on the power spectrum of spin-torque nano-oscillators. J. Phys. D:\nAppl. Phys. 53, 49501 (2020).\n57.Kamimaki, A. et al. Chaos in spin-torque oscillator with feedback circuit. Phys. Rev. Res. 3, 043216 (2021).\n58.Yamaguchi, T., Tsunegi, S., Nakajima, K. & Taniguchi, T. Computational capability for physical reservoir computing\nusing a spin-torque oscillator with two free layers. Phys. Rev. B 107, 054406 (2023).\n59.Tsunegi, S. et al. Information Processing Capacity of Spintronic Oscillator. Adv. Intell. Syst. 5, 2300175 (2023).\n60.Dieny, B., Goldfarb, R. B. & Lee, K.-J. (eds.) Introduction to Magnetic Random-Access Memory (Wiley-IEEE Press,\nHoboken, 2016).\n61.Lau, Y .-C., Betto, D., Rode, K., Coey, J. M. D. & Stamenov, P. Spin-orbit torque switching without an external field using\ninterlayer exchange coupling. Nat. Nanotechnol. 11, 758 (2016).\n62.Taniguchi, T., Ogihara, A., Utsumi, Y . & Tsunegi, S. Spintronic reservoir computing without driving current or magnetic\nfield. Sci. Rep. 12, 10627 (2022).\n63.Jr, W. F. Brown. Thermal Fluctuations of a Single-Domain Particle. Phys. Rev. 130, 1677 (1963).\n64.Taniguchi, T. Synchronization and chaos in spin torque oscillator with two free layers. AIP Adv. 10, 015112 (2020).\nAcknowledgements\nThe authors are grateful to Takayuki Nozaki for discussion. This work was supported by a JSPS KAKENHI Grant, Number\n20H05655.\nAuthor contributions statement\nT.T. designed the project, performed the numerical simulations, prepared figures, and wrote the manuscript.\nCompeting interests\nThe authors declare no competing interests.\nData availability\nThe datasets used and/or analyses during the current study available from the corresponding author on reasonable request.\n13/14Additional information\nCorrespondence and requests for materials should be addressed to T.T.\n14/14"    },    {        "title": "1911.09688v2.Coupled_spin_charge_dynamics_in_magnetic_van_der_Waals_heterostructures.pdf",        "content": "Coupled spin-charge dynamics in magnetic van der Waals heterostructures\nAvinash Rustagi,1,\u0003Abhishek Solanki,1Yaroslav Tserkovnyak,2and Pramey Upadhyaya1,y\n1School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907\n2Department of Physics and Astronomy, University of California, Los Angeles, CA 90095\n(Dated: December 4, 2019)\nWe present a phenomenological theory for coupled spin-charge dynamics in magnetic van der\nWaals heterostructures. The system studied consists of a layered antiferromagnet inserted into a\ncapacitive vdW heterostructure. It has been recently demonstrated that charge doping in such lay-\nered antiferromagnets can modulate the strength, and even the sign, of exchange coupling between\nthe layer magnetizations. This provides a mechanism for electrically generating magnetization dy-\nnamics. The central result we predict here is that the magnetization dynamics reciprocally results\nin inducing charge dynamics. Such dynamics makes magnetic van der Waals heterostructures inter-\nesting candidates for spintronics applications. To this end, we also show that these systems can be\nused to convert sub-THz radiation induced magnetization dynamics into electrical signals.\nIntroduction | Two-dimensional (2D) magnetism of-\nfers opportunities to investigate phenomena driven by\nenhanced \ructuations, reduced symmetries, and nontriv-\nial topology [1]. The recent discovery of stabilizing mag-\nnetic order in few layer systems held together by van der\nWaals forces (magnetic vdWs [2, 3]) have thus emerged\nas a promising avenue for fundamental research and tech-\nnological applications of 2D magnets. Furthermore, by\ntaking advantage of layer by layer assembly, magnetic\nvdWs can be interfaced with a wide range of materials\nto form heterostructures with desired functionality [4].\nA functionality of fundamental interest is the abil-\nity to interconvert between spin and charge degrees of\nfreedom. Such spin-charge conversion has been a key\nenabler of spintronics technology, which has attracted\nrigorous interest as an alternate to charge-based tech-\nnology [5, 6]. Consequently, a number of fundamen-\ntal phenomena allowing for conversion between spin and\ncharge has been uncovered in the recent past, which in-\ncludes reciprocal pairs such as spin torque-spin pumping\n[7{9], spin Hall-inverse spin Hall e\u000bect [10{13], Rashba\nEdelstein-inverse Rashba Edelstein e\u000bect [14{16], and\ndirect-converse magneto-electric e\u000bect [17, 18]. The\nsearch for alternate low-dissipation mechanisms and ma-\nterial platforms for e\u000ecient conversion between spin and\ncharge is an active area of research [6].\nBilayer Chromium Iodide (CrI 3)-based vdW het-\nerostructures have recently emerged as one such plat-\nform. CrI 3is a vdW magnet with ferromagnetically\nordered layers coupled via an antiferromagnetic inter-\nlayer coupling [19, 20]. The interlayer coupling is sen-\nsitive to charge residing on each layer. This has been\nutilized to switch the ground state spin con\fguration\nin gated bilayer CrI 3heterostructures between a layered\nantiferromagnetic and a ferromagnetic state electrically\n[21]. Reciprocity then dictates the existence of an inverse\nmechanism, wherein the charge distribution in the same\nheterostructure can be modulated by inducing magne-\ntization dynamics. Furthermore, the large out-of-plane\nanisotropy, in combination with the antiferromagneticcoupling, gives rise to sub THz ferromagnetic resonance\nmodes in the absence of external magnetic \felds [22, 23].\nThis suggests the existence of a new reciprocal pair in\ngated bilayer CrI 3heterostructures, which is capable of\ninterconverting between spin and charge up to sub tera-\nhertz frequencies. In this Letter, we thus present a phe-\nnomenological theory of dynamical spin-charge coupling\nin vdW hetrostructures involving bilayer CrI 3. In par-\nticular, we predict the phenomena of charge pumping by\nmagnetization dynamics, which can be utilized for electri-\ncal detection of antiferromagnetic resonance or convert-\ning GHz to sub THz radiation into an electrical signal.\nStructure and Model | Motivated by recent exper-\niments [21], here we consider a bi-layer CrI 3system\ninserted into a capacitor formed by hexagonal Boron-\nNitride (h-BN), which is contacted by metal layers as\nshown in the left panel of Fig. 1. The CrI 3layers are\nthemselves connected to ground (for example via contact-\ning them by graphene layers). The intralayer magnetic\nmoments in each layer of bi-layer CrI 3are ferromagneti-\ncally aligned. However, the interlayer magnetic moments\nare antiferromagnetically coupled [19, 22]. Moreover, the\nmagnetization of the layers have an out of plane easy\naxis, which arises due to anisotropic exchange interac-\ntion [23]. Thus, the free energy per unit area within the\nmacrospin approximation in its minimal form consists of\nan easy axis uniaxial anisotropy and an antiferromagnetic\ninterlayer exchange,\nF(~ m1;~ m2) =\u0000Km2\n1z\u0000Km2\n2z+J?~ m1\u0001~ m2: (1)\nThe charges on the CrI 3layers couple to the mag-\nnetic degrees of freedom giving rise to additional free en-\nergyF(~ mi;\u001bi) terms. The functional form of F(~ mi;\u001bi)\nis dictated by the time-reversal and structural symme-\ntry of our van der Waals heterostructure. In particular,\nour structure has inversion symmetry under which the\nsymmetric charge combination \u001b1+\u001b2remains invari-\nant, while the anti-symmetric charge combination \u001b1\u0000\u001b2\nchanges sign. Noting that the magnetization ~ mis not af-arXiv:1911.09688v2  [cond-mat.mes-hall]  3 Dec 20192\nFIG. 1. Schematic of the grounded (using graphene con-\ntacts) bilayer CrI 3inserted into a h-BN capacitor for coupled\nspin-charge dynamics. E\u000bective circuit model for the device\nstructure accounting for the magnetization dynamics induced\ncharge dynamics.\nfected by structural inversion, the coupling between the\nmagnetic and charge degrees of freedom to the lowest\norder in\u001bi, and obeying structural inversion and time-\nreversal, can be written as [24]:\nF(~ mi;\u001bi) =\u0000\u0015(\u001b1+\u001b2)~ m1\u0001~ m2: (2)\nThis can be identi\fed as doping-induced change in in-\nterlayer exchange coupling. We highlight that the ex-\nperimental observation that asymmetric charge does not\ninduce changes in the interlayer exchange [21] can be\ndirectly related to the structural symmetry of the van\nder Waals heterostructures. Additionally, the asymmet-\nric charge can also couple to the magnetic degrees of free-\ndom via a term of the form \u0018\u0010(\u001b1\u0000\u001b2)(~ m1\u0000~ m2)\u0001~H.\nThis \\magneto-electric coupling\" has recently been ob-\nserved in experiments [19]. However, the magnetoelectric\ne\u000bect is found to be less e\u000ecient for inducing magnetiza-\ntion dynamics when compared with the doping-induced\ninterlayer exchange [21]. Thus, here the spin-charge cou-\npling is restricted to the form given by Eq. (2).\nExperimentally, the coupling given by Eq. (2) has been\nutilized to electrically switch between a ferromagnetic\nand an antiferromagnetic con\fguration [21]. Recipro-\ncally, a change in the magnetic con\fguration should al-ter charge density on the CrI 3layer. We next derive an\nequivalent circuit for this coupled spin-charge dynamics.\nTo this end, we need to supplement Eqs. (1) and (2)\nwith the electrical energy solely due to charges, which is\ngiven by:\nF(\u001b1;\u001b2) =\u001b2\n1\n2Cg+\u001b2\n2\n2Cg: (3)\nHere,Cg=\u000f=dis the geometrical capacitance of the h-\nBN layer where \u000fanddare the permittivity and thickness\nof h-BN, respectively [25].\nCollecting the free energy terms involving charge den-\nsities, we thus have\nF\u001b=\u0000\u0015(\u001b1+\u001b2)~ m1\u0001~ m2+\u001b2\n1\n2Cg+\u001b2\n2\n2Cg: (4)\nAssumingRto be the series combination of the resis-\ntance of the external circuit and the resistance between\ngraphene contacts and the CrI 3layer, we can draw an\ne\u000bective circuit for the van der Waals heterostructure as\nshown in Fig. 1. The central result of this model is that\na dynamic magnetization gives rise to a time-dependent\nvoltage\nV=\u0000@F(~ mi;\u001b)=@\u001b=\u0015~ m1\u0001~ m2: (5)\nThis circuit is the \frst main result of this letter.\nCharge pumping | To demonstrate the e\u000bects of the\nreciprocal process of magnetization dynamics induced\nvoltage, we propose to utilize the antiferromagnetic reso-\nnance (AFMR) set up (see Fig. 2). In this case, the anti-\nferromagnet is subjected to an electromagnetic radiation\nin the presence of a dc magnetic \feld. A resonant exci-\ntation of the magnetic order parameter occurs when the\nfrequency of the electromagnetic radiation coincides with\nthe frequency of the inherent magnetic dynamical modes\nof the antiferromagnet (which can be tuned by varying\nthe strength of the dc magnetic \feld). The second main\nresult of this letter is that in the CrI 3-based vdW het-\nerostructures the absorbed radiation also gives rise to an\nelectrical signal. In particular, when the RC time as-\nsociated with vdW heterostructure is smaller than the\ntimescale associated with magnetization dynamics [26],\nthe induced voltage results in a \row of charge current in\nthe external circuit, which is given by:\njext(t) = _\u001b1+ _\u001b2= 2\u0015Cg@t~ m1\u0001~ m2: (6)\nWe next calculate analytically the charge current\npumped due to AFMR within the linear response, which\nis corroborated by numerical simulations.\nThe magnetization dynamics are governed by the\nLandau-Lifshitz-Gilbert (LLG) equation,\n_~ mi=\u0000\r ~ mi\u0002h\n~He\u000b;i+~Hext+~h\u0018i\n+\u000b~ mi\u0002_~ mi(7)3\nFIG. 2. Schematic of the proposed AFMR setup. The\nequilibrium magnetizations cant by an angle \u0012due to the\napplied dc magnetic \feld Hand the AFMR mode is excited\nby the radiation \feld hx(t).\nwhere index i= 1;2 correspond to the two layers, ~ miis\nthe unit magnetization, \r >0 is the gyromagnetic ratio,\n\u000bis the damping, ~He\u000b;i=\u0000@~ miF, and~h\u0018represents the\nac excitation \feld. We take the dc external \feld to be ori-\nented along the xaxis, that is ~Hext=H^x. This dc \feld\ncants the otherwise zaxis oriented ~ mito align with z0,\nwhich makes a polar angle \u0012with thezaxis (see Fig. 2).\nThe polar angle satis\fes sin \u0012=H=(Hk+ 2HJ) when\nH < Hk+ 2HJand\u0012=\u0019=2 forH > Hk+ 2HJ. Here\nHk= 2K=Msis the anisotropy \feld and HJ=J?=Ms\nis the interlayer exchange \feld. The eigenmodes, found\nby solving the LLG linearized about the tilted equilib-\nrium [27], correspond to the antiferromagnetic resonance\n(AFMR) frequencies given by\n!0;\u0006=q\n[!eq\u0006!J][!eq\u0007!Jcos 2\u0012\u0000!ksin2\u0012] (8)\nwhere!k=\rHk,!J=\rHJ, and!eq=\rHeqare\nthe frequencies corresponding to the anisotropy, inter-\nlayer exchange, and equilibrium \felds. The equilibrium\n\feld experienced by the magnetizations has contributions\nfrom external, anisotropy, and interlayer exchange \felds\nHeq=Hsin\u0012+Hkcos2\u0012+HJcos 2\u0012. The large easy axis\nout of plane anisotropy in few layer CrI 3, corresponding\nto a \feld of Hk= 3:75 T, results in a large spin wave gap\n\u0018110 GHz [23] for \u0012= 0. The presence of the in-plane\ndc \feldHlowers the AFMR frequency by e\u000bectively o\u000b-\nsetting the out of plane anisotropy \feld. Also, the two\ndegenerate Kittel modes in absence of the dc \feld, are\nnow non-degenerate [c.f. Fig. 3].\nTo calculate the charge current pumped by above\nmodes, we need to \fnd ~ m1\u0001~ m2[ c.f. Eq. (6)]. For this\npurpose, we also evaluate the deviations transverse to\nthe tilted equilibrium magnetizations ( z0andz00axis),\n(\u000em0\n1x;\u000em0\n1y) and (\u000em00\n2x;\u000em00\n2y), within the linearized\nLLG. A convenient choice for the solution is in termsof the collective coordinates X\u0006=\u000em0\n1x\u0006\u000em00\n2xand\nY\u0006=\u000em0\n1y\u0006\u000em00\n2y, which gives :\n~ m1\u0001~ m2=1\n4\u0014\n\u0000cos 2\u0012X2\n++Y2\n+\u00004 cos 2\u0012+\ncos 2\u0012X2\n\u0000\u0000Y2\n\u0000+ 2 sin 2\u0012X\u0000\u0015\n;(9)\nwhere\n\u0012\nX+\nY+\u0013\n= Re\"\n2\rhye\u0000i!t\n!2\u0000!2\n0;++i!\u0001!+\u0012\ni!\ni\u000b!\u0000!2+\u0013#\n\u0012X\u0000\nY\u0000\u0013\n= Re\"\n2\rhxcos\u0012e\u0000i!t\n!2\u0000!2\n0;\u0000+i!\u0001!\u0000\u0012i\u000b!\u0000!1\u0000\n\u0000i!\u0013#\n(10)\nassuming\u000b\u001c1. Here,hxandhyare thexandycompo-\nnents of the oscillating \feld ~h\u0018,!1\u0006=!eq\u0006!J,!2\u0006=\n!eq\u0007!Jcos 2\u0012\u0000!ksin2\u0012, and \u0001!\u0006=\u000b(!1\u0006+!2\u0006).\nAn important inference to make from Eq. (9) is that the\nmagnetization dot product has both quadratic and linear\nterms in the collective coordinates. This implies that the\ncharge dynamics can have response at both !and 2!,\nwhere!is the excitation \feld frequency. We next ana-\nlyze key experimental signatures of the proposed charge\npumping.\nCharge dynamics signatures | We begin by discussing\nthe charge pumping in the absence of the external dc \feld\nH= 0 when the canting angle \u0012= 0. In Fig. 3(a), we plot\nthe amplitude of the charge current pumped as a function\nof the excitation frequency of external radiation, which is\nobtained by substituting the numerical solution of Eq. (7)\ninto Eq. (6). Here, the AFMR are the well-known Kittel\nmodes!Kittel =\rp\nHk(Hk+ 2HJ)[28], which could be\nexcited via a circular or linearly polarized radiation. We\nemphasize \frst that when exposed to circularly polarized\noscillatory \felds, i.e. ~h\u0018=h(^x\u0006i^y)=p\n2, the eigenmodes\ndo not lead to any charge dynamics (i.e. @t~ m1\u0001~ m2= 0).\nThis is because in this case the magnetizations of the\ntwo layers precess keeping the angle between the magne-\ntizations constant. On the other hand, in the presence\nof linearly polarized excitation \felds ( xandy-polarized),\nthere is observable charge dynamics. Furthermore, from\nEq. (9) we see that in the absence of external \feld (i.e.\nwhen\u0012= 0), only quadratic terms in the magnetization\ndot product exists. Thus, the lowest order charge dynam-\nics arising from the magnetization dynamics for H= 0\nhas a response at twice the excitation frequency ( I2!\namp)\nwhich at resonance ( !=!0;\u0000) is approximately given by\nI2!\namp\u00192\u0015AC g\r2h2\nx(!2\n1\u0000+!2\n0;\u0000)\n!0;\u0000\u0001!2\n\u0000(11)\nwhich is in agreement with the amplitude evaluated from\nthe numerical solution of Eq. (7) plotted in Fig. 3(a).4\nFIG. 3. a) Electrical response ( I2!\namp) from Kittel modes at 2 !from magnetization dynamics induced by linearly (solid) and\ncircularly (dashed) polarized radiation \felds as a function of excitation \feld frequency in absence of dc \feld H= 0. The inset\ndisplays the precession of magnetization deviations transverse to the equilibrium orientation, of the top and bottom CrI 3layers\nfor both polarizations. b) Evaluated current amplitude at excitation \feld frequency I!\nampas a function of excitation magnetic\n\feld frequency and the in-plane dc magnetic \feld for x-polarized excitation \feld. Dashed lines show the two AFMR modes\n(Eq. (8)) as a function of in-plane dc \feld.\nA larger current response is generated in the presence\nof a DC canting \feld. This can be seen from Eq. (9),\nwhere due to non-zero \u0012we obtain a contribution to the\ncharge current of the form \u00182 sin 2\u0012@tX\u0000. This contri-\nbution, being linear in deviation, pumps a larger charge\ncurrent (when compared to the H= 0 case) oscillating\nat the frequency of incident radiation. The on-resonance\n(!=!0;\u0000) amplitude of charge current can be approxi-\nmated from Eq. (9) and Eq. (10), given by\nI!\namp\u00192\u0015AC gsin 2\u0012cos\u0012\rhxq\n!2\n1\u0000+\u000b2!2\n0;\u0000\n\u0001!\u0000:(12)\nIn Fig. 3(b), we plot the amplitude of this current\n(I!\namp) as a function of in-plane dc magnetic \feld and\nexcitation frequency for an x\u0000polarized oscillating mag-\nnetic \feld evaluated from substituting the numerical so-\nlution of Eq. (7) into Eq. (6). The linear response esti-\nmate in Eq. (12) agrees with the numerical result. The\nlinear term contribution sin 2 \u0012@tX\u0000\u0018!sin 2\u0012X\u0000peaks\nat an intermediate canting angle. This is because while\nthe AFMR frequency decreases with the in-plane DC\n\feld, the deviation in magnetization increases as the ef-\nfective \feld seen by the magnetization gets reduced (since\nthe DC \feld o\u000bsets the easy axis anisotropy). The com-\npetition between these two leads to an increase in the cur-\nrent amplitude at intermediate DC \feld strengths. For\ntypical experimentally realized values of the parameters\n(see Table. I), this peak current gives a value of \u00181.5\nnA per Oe of oscillating magnetic \feld, well within the\nreach of experiments.\nConclusions and outlook | In this letter, we con-\nstructed a phenomenological theory for coupled spin-TABLE I. CrI 3Parameters used in calculations.\nParameter Value\nSaturation Magnetization Ms[23] 1.37 \u000210\u00005emu/cm2\nEasy axis anisotropy K[23] 0.2557 erg/cm2\nInterlayer Exchange J?[29] 0.0354 erg/cm2\nSpin-Charge Coupling \u0015[21] 1 mV\nArea of bilayer CrI 3A 1\u0016m2\ncharge dynamics in symmetrically gated vdW het-\nerostructures of bilayer CrI 3. We \fnd that the spin-\ncharge coupling can be classi\fed into doping-induced\nmodi\fcations of magnetic properties and the so-called\nmagneto-electric coupling, which represents the interac-\ntion of electric \feld with the di\u000berence of magnetiza-\ntions in each layer. The structural symmetries restrict\nthe form of former (latter) to be directly proportional\nto the symmetric (antisymmetric) combination of charge\ndoping within each layer. Motivated by the experimental\nobservation of large doping-induced changes in interlayer\nexchange coupling, we speci\fcally construct an e\u000bective\ncircuit theory for coupled spin charge dynamics including\nthis term, which if needed can similarly be extended to\ninclude other spin charge couplings.\nA central \fnding of this theory is that the magnetiza-\ntion dynamics induces a voltage, which is proportional\nto the dot product of magnetizations within each layer.\nAs an experimental signature of this e\u000bect, we proposed\nand evaluated the charge current pumped by magnetiza-\ntion dynamics induced by the absorption of electromag-\nnetic radiation. This can be utilized for probing magnetic\nexcitations electrically, in addition to tunneling magne-5\ntoresistance [22], and harvesting electromagnetic radia-\ntion via conversion into an electrical signal with follow-\ning features. In the absence of any dc magnetic \feld,\nthe resulting charge dynamics have a response at twice\nthe excitation \feld frequency when exposed to linearly\npolarized time-dependent electromagnetic \felds. These\nhigh frequency AFMR modes can be softened by apply-\ning a dc magnetic \feld, which additionally allows a \frst\nharmonic response when time-dependent electromagnetic\n\feld is polarized along the dc \feld direction.\nFuture works should explore the implications and op-\nportunities o\u000bered by the reciprocity dictated pair of\n`voltage$magnetization dynamics' in addition to the\nproposed AFMR setup. For example, thanks to the easy\nintegration of various vdW materials, spin transistors\nhave recently been fabricated [30]. Here, a gate voltage-\ndependent magnetic con\fguration of bilayer CrI 3controls\nthe \row of tunneling charge current due to the tunnel-\ning magnetoresistance and through shift in the chemical\npotential of the vdW layers. Solving coupled spin-charge\ncircuits in such vdW magnet-based spin transistors, will\nbe addressed elsewhere.\nWe thank Vaibhav Ostwal, Se Kwon Kim, and Kin\nFai Mak for helpful discussions. A.R., A.S., and P.U. ac-\nknowledge support from the National Science Foundation\nthrough Grant No. DMR-1838513. Collaboration with\nY.T. was supported by the National Science Foundation\nthrough Grant No. ECCS-1810494.\n\u0003arustag@purdue.edu\nyprameyup@purdue.edu\n[1] K. S. Burch, D. Mandrus, and J.-G. Park, Nature 563,\n47 (2018).\n[2] M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S.\nNovoselov, Nature Nanotechnology 14, 408 (2019).\n[3] C. Gong, L. Li, Z. Li, H. Ji, A. Stern, Y. Xia, T. Cao,\nW. Bao, C. Wang, Y. Wang, et al. , Nature 546, 265\n(2017).\n[4] A. K. Geim and I. V. Grigorieva, Nature 499, 419 (2013).\n[5] I. \u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n[6] W. Han, Y. Otani, and S. Maekawa, NPJ Quantum Ma-\nterials 3, 27 (2018).\n[7] L. Berger, Phys. Rev. B 54, 9353 (1996).\n[8] J. C. Slonczewski, Journal of Magnetism and Magnetic\nMaterials 159, L1 (1996).\n[9] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. Phys. 77, 1375 (2005).\n[10] M. Dyakonov and V. Perel, Physics Letters A 35, 459\n(1971).\n[11] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Ap-\nplied physics letters 88, 182509 (2006).\n[12] T. Kimura, Y. Otani, T. Sato, S. Takahashi, and\nS. Maekawa, Phys. Rev. Lett. 98, 156601 (2007).\n[13] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back,\nand T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015).[14] V. M. Edelstein, Solid State Communications 73, 233\n(1990).\n[15] H. Zhang, S. Yamamoto, B. Gu, H. Li, M. Maekawa,\nY. Fukaya, and A. Kawasuso, Physical review letters\n114, 166602 (2015).\n[16] J. R. S\u0013 anchez, L. Vila, G. Desfonds, S. Gambarelli, J. At-\ntan\u0013 e, J. De Teresa, C. Mag\u0013 en, and A. Fert, Nature com-\nmunications 4, 2944 (2013).\n[17] M. Fiebig, Journal of physics D: applied physics 38, R123\n(2005).\n[18] J. Zhang, C. Fang, and G. J. Weng, Proceedings of the\nRoyal Society A 475, 20190002 (2019).\n[19] B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein,\nR. Cheng, K. L. Seyler, D. Zhong, E. Schmidgall, M. A.\nMcGuire, D. H. Cobden, et al. , Nature 546, 270 (2017).\n[20] N. Sivadas, S. Okamoto, X. Xu, C. J. Fennie, and\nD. Xiao, Nano letters 18, 7658 (2018).\n[21] S. Jiang, L. Li, Z. Wang, K. F. Mak, and J. Shan, Nature\nnanotechnology , 1 (2018).\n[22] D. R. Klein, D. MacNeill, J. L. Lado, D. Soriano,\nE. Navarro-Moratalla, K. Watanabe, T. Taniguchi,\nS. Manni, P. Can\feld, J. Fern\u0013 andez-Rossier, et al. , Sci-\nence 360, 1218 (2018).\n[23] J. L. Lado and J. Fern\u0013 andez-Rossier, 2D Materials 4,\n035002 (2017).\n[24] In addition, terms of the form \u0018(\u001b1+\u001b2)m2\ni;zi.e.\ndoping-induced change in uniaxial ansiotropy, are also\nallowed. However, motivated by the experimental obser-\nvation that the percentage change in anisotropy is typi-\ncally smaller than that in the exchange interaction, i.e.\n\u0001K=K\u001c\u0001J=J [21], we have not included those terms\nhere.\n[25] In addition to the energy stored in the h-BN capacitors,\ncharges also give rise to electrostatic energy stored in\nthe vdW gap between the graphene contact and the CrI 3\nlayer (parametrized by Cg;gc), as well as, the chemical\nenergy due to shifts in the Fermi levels of the graphene\ncontacts and the CrI 3layers (parameterized by the quan-\ntum capacitances CqgandCqc, respectively). However,\nfor typical experimental parameters we have Cg\u001cCqg\u0018\nCg;gc\u001cCqc. Thus, the electrical energy is dominated by\nthe one stored in the electrostatic \felds in the h-BN layer\n[21].\n[26] To estimate the contact resistance at the graphene-CrI 3\ninterface, we can consider typical contact resistance as-\nsociated with graphene-TMDC [31, 32] interfaces for mi-\ncron sized systems Rcontact Area\u00181000 \n\u0016m2. For typi-\ncal h-BN thicknesses, the geometric capacitance per unit\nareaCg\u001910\u00007F=cm2[33]. Neglecting the external cir-\ncuit resistance , the time constant associated with the\ncircuit isRcontact Area\u0002Cg\u00191ps.\n[27] S. V. Von Sovskii, Ferromagnetic resonance: the phe-\nnomenon of resonant absorption of a high-frequency mag-\nnetic \feld in ferromagnetic substances (Elsevier, 2016).\n[28] F. Ke\u000ber and C. Kittel, Phys. Rev. 85, 329 (1952).\n[29] D. Soriano, C. Cardoso, and J. Fern\u0013 andez-Rossier, Solid\nState Communications , 113662 (2019).\n[30] S. Jiang, L. Li, Z. Wang, J. Shan, and K. F. Mak, Nature\nElectronics 2, 159 (2019).\n[31] M. Houssa, K. Iordanidou, A. Dabral, A. Lu, R. Meng,\nG. Pourtois, V. Afanas' ev, and A. Stesmans, Applied\nPhysics Letters 114, 163101 (2019).\n[32] T. Yamaguchi, R. Moriya, Y. Inoue, S. Morikawa, S. Ma-\nsubuchi, K. Watanabe, T. Taniguchi, and T. Machida,6\nApplied Physics Letters 105, 223109 (2014). [33] Z. Wang, Y.-H. Chiu, K. Honz, K. F. Mak, and J. Shan,\nNano letters 18, 137 (2017)."    },    {        "title": "2206.05783v1.Exchange_enhancement_of_the_ultrafast_magnetic_order_dynamics_in_antiferromagnets.pdf",        "content": "Exchange-enhancement of the ultrafast magnetic order dynamics in antiferromagnets\nFlorian Jakobs and Unai Atxitia\u0003\nDahlem Center for Complex Quantum Systems and Fachbereich Physik,\nFreie Universität Berlin, 14195 Berlin, Germany\n(Dated: June 14, 2022)\nWe theoretically demonstrate that the ultrafast magnetic order dynamics in antiferromagnets\nis exchange-enhanced in comparison to their ferromagnetic counterparts. We provide an equation\nof motion for the magnetic order dynamics validated by computer simulations using atomistic spin\ndynamicsmethods. Theexchangeofangularmomentumbetweensublatticesspeedsupthedynamics\nin antiferromagnets, a process absent in ferromagnets.\nUltrafast optical control of the magnetization promises\nfaster data processing and storage [1–6]. Antiferro-\nmagnets (AFMs) show advantages over ferromagnets\n(FMs), such as faster magnetization dynamics [7–13].\nIn AFMs, the frequency of the magnetic oscillations\naround the anisotropy field in FMs ( !fm\u0018HA) are\nexchange-enhanced by the antiferromagnetic coupling\n(HE) between the spins at different sublattices, leading\nto a higher oscillation frequency in AFMs, ( !afm\u0018pHEHA), orders of magnitude higher than in FMs[14,\n15]. Femtosecond laser photo-excitation can induce\nsubpicosecond magnetic order quenching in both FMs\nand AFMs [1, 16, 17]. The speed of ultrafast quenching\nof the magnetic order is determined by the strength of\nthe exchange interaction ( 1=\u001cfm\u0018\u000bfmHE), and the FM\ndamping,\u000bfm[18]. This raises the fundamental question\nof whether the ultrafast magnetic order dynamics in\nAFMs is exchange-enhanced with respect to its FM\ncounterpart. We find that the AFM magnetic order\nrespondsfasterthantheFMonetoasuddentemperature\nchange due to the exchange-enhancement of the effective\nAFM damping, 1=\u001cafm\u0018\u000bafmHE. We show that,\ncontrary to FMs, the effective AFM damping depends on\nthe number of neighbours to which spins are exchange\ncoupled. Thus, low dimension magnets, such as 1D\nand 2D magnets[19], show a more pronounced speed\nup, while for lattices with higher coordination number\nthe exchange-enhancement reduces. As the system\napproaches the critical temperature, both the FM and\nAFM present a critical slow down of the relaxation\nprocess, however, the AFM critical exponent is smaller\nthan the FM one. In the temperature dominated\nregimeT\u001dTc, our model predicts intrinsically different\nrelaxation dynamics for AFMs and FMs. This scenario\ncorresponds to experiments using powerful femtosecond\nlaser pulses. For FMs, magnetic order quenching\nslows down as the magnetization reduces, while for\nAFMs speeds up. We demonstrate the validity of our\nmodel by direct comparison to computer simulations\nusing atomistic spin dynamics within an atomistic spin\ndynamics (ASD) model.\nEvidence of exchange-enhancement of the ultrafast\nmagnetization dynamics in AFMs is scarce due to the\ndifficulties to conduct a systematic comparison on thesame system presenting FM and AFM magnetic order.\nStudies in rare-earth Dy using femtosecond time-resolved\nresonant magnetic x-ray diffraction have measured the\ndynamics of its FM and AFM-spin-helix states. These\ninvestigations have shown that the dynamics of the\norder parameter in the AFM phase is faster than in the\nFM phase [17]. Laser induced ultrafast magnetization\ndynamics in FMs have been modeled using computer\nsimulations based on different approaches, from ASD\nmodels to macroscopic phenomenological approaches\n[1, 3, 18]. Within these approaches magnetization\ndynamics are explained on thermodynamic grounds.\nWhen the temperature of the heat-bath is rapidly\nmodified, the magnetic order changes according to _M\u0018\n\u000bfmH, driven by an effective field H=\u0000@F=@M\nat a rate \u000bfm, towards minimal free energy values,\nF(M)[20]. The thermodynamic argument explains\ndemagnetization and magnetization recovery when the\nsystemtemperatureincreasesanddecreases, respectively.\nIt can be expected to hold for other magnetic structures,\nsuch as antiferromagnets, as well. In AFMs however,\nan additional channel for angular momentum dissipation\nopens, by direct exchange of angular momentum between\nsublattices. For a two sublattice AFM, the dynamics of\nsublatticeacanbeexpressedas: _Ma\u0018\u000baHa+\u000bex(Ha\u0000\nHb), where\u000bexrepresents the rate of interatomic transfer\nof angular momentum between sublattices aandb. In\nthe simplest AFM case, both sublattices are equivalent,\nsuch thatHa=\u0000Hbis a valid approximation, leading to\n_Ma\u0018(\u000ba+2\u000bex)Ha=\u000bafmHa, where\u000baand\u000bexarethe\nOnsager coefficients describing exchange and relativistic\nrelaxations. For FM and AFM systems defined by\nthe same parameters, \u000bafm> \u000b fm, and consequently\nthe AFM is faster than the FM. In the present work,\nstarting from an ASD model, we derive an – so far\nunknown – expression for the exchange-enhancement of\nthe relaxation parameter, \u000bafmin AFMs.\nThe dynamics of the magnetic order parameter (in\nFMs and AFMs) are calculated within the framework of\na classical, atomistic spin model. The Hamiltonian reads\nH=\u0000J\n2X\nhi;jisisj\u0000dzX\ni(sz\ni)2: (1)\nThe unit vectors, si=\u0016i=\u0016at, represent the normalizedarXiv:2206.05783v1  [cond-mat.mtrl-sci]  12 Jun 20222\nmagnetic moment of the lattice site iwith magnetic\nmoment\u0016at=\u0016B. The first term describes nearest\nneighbors exchange coupling, with J=\u00063:450\u000210\u000021J\nfor the FM (+) and AFM(-). The second term represents\nthe uniaxial anisotropy, with dz= 1\u000210\u000022J for both\nAFM and FM. The dynamics at finite temperatures are\ndescribed by the stochastic Landau-Lifshitz-Gilbert (s-\nLLG) equation,\ndsi\ndt=\u0000j\rj\n(1 +\u00152)si\u0002[Hi\u0000\u0015(si\u0002Hi)]:(2)\nHere,\ris the gyromagnetic ratio. The first term\nrepresents a precession of the magnetic moments around\nan effective field Hi=\u0000(1=\u0016at)(@H=@si), while the\nsecond term represents the transverse relaxation. A\nphenomenological damping constant \u0015defines the rate\nof the relaxation. In order to include the effects of\nfinite temperature, we couple the spin system to a\nLangevin thermostat which adds an effective field-like\nstochastic term \u0010ito the effective field with white noise\nproperties [21].\nBy using ASD simulations, we first demonstrate\nthe existence of exchange-enhancement on the AFM\ndynamics in realistic conditions, similar to experiments.\nA high temperature regime, T\u001dTccan be accessed by\nsuddenlyheatingtheelectronsystemusingafemtosecond\nlaserpulse(Fig. 1(a))(seemoredetailsinsupplementary\nmaterialSec. S1). Onthetimescaleof100fs, theelectron\ntemperature will rise far above Tc.\nThe magnetic system responds to this temperature\nchange by reducing its magnetic order on similar time\nscales. The electron-phonon coupling allows energy\ntransfer from the hot electrons to the lattice in the\ntime scale of only a couple of picoseconds. This allows\nfor the investigation not only of the magnetic order\nquenching but also its recovery. Figure 1 (b) shows that\nthe demagnetization in the AFM is larger than in FM,\nowning to a faster response when excited by the same\ntemperature profile (Fig. 1 (a)). The magnetic order\nrecovery of the AFM is on the same time scale as the\nelectron-phonon temperature relaxation time, while the\nFM relaxes over longer time scales.\nThe exchange-enhancement of the AFM magnetic\ndamping is at the origin of this speed up as we shall\ndemonstrate.\nBuilding upon the described atomistic spin model, the\nnon-equilibrium macroscopic magnetization dynamics of\nthe sublattice ma=hsaican be described by [22]:\n1\n\rdma\ndt=\u000baHa+\u000bex(Ha\u0000Hb) (3)\nwherea6=b. The macroscopic damping parameter in\nEq. (3) is defined as, \u000ba= 2\u0015L(\u0018a)=\u0018a. Here,L(\u0018a)\nstands for the Langevin function, with the argument\n\u0018a=\f\u0016aHMFA\naand\f= 1=kBT[23]. In the exchange\n01T/T cPhonons\nElectrons\n0 1 2 3 4\nTime [ps]0.60.81.0mz(nz)a)\nb)\nFM AFMFIG. 1. (a) Electron and phonon temperature dynamics after\nan excitation by a 50 fs laser at t= 0. (b) The magnetic\norder dynamics of a FM, mz(red solid line), and an AFM, nz\n(blue dashed line) as a response to the electron temperature\ndynamics in (a).\napproximation, the MFA field acting on sublattice ais\n\u0016aHMFA\na =J0mb, hereJ0=zJ, wherezis the number\nof nearest neighbours of spins of type b. Moreover, in\nthe exchange approximation, one can fairly assume that\nma=mb=m, and therefore \u000ba=\u000bb. Under these\nassumptions, the exchange relaxation parameter can be\nwritten as \u000bex= 4\u000ba=(zm)[24]. One can recover the\nequation of motion for the FM case for \u000bex= 0(Eq. (3)),\nin that case, \u000bfm=\u000ba. The non-equilibrium effective\nfields are given by\nHa=(ma\u0000m0;a)\n\u0016a\fL0(\u0018a): (4)\nwhere,L0(\u0018) =dL=d\u0018andm0;a=L(\u0018a)[23, 25].\nFor the two sublattice AFM considered here, Ha=\n\u0000Hb=Hn, whereHn= (n\u0000n0)=\u0016a\fL0(\u0018). It follows\nthat the dynamics of the Néel order parameter is given\nby\n1\n\rdn\ndt=\u000bafmHn: (5)\nThis demonstrates that the origin of exchange-\nenhancement of the AFM dynamics can be traced back\nto the effective AFM damping parameter,\n\u000bafm=\u000bfm\u0012\n1 +4\nzjnj\u0013\n: (6)\nWe first address the differences and similarities between\nAFM and FM near thermal equilibrium, where the\nnon-equilibrium fields can be cast into Landau-like\nexpressions [22, 25], Hn= (\u0016at=2e\u001fk)\u000en2=n2\ne. Heree\u001fkis\nthe longitudinal susceptibility of the Néel vector at zero\nfield\ne\u001fk=\u0016at\nJ0\fJ0L0\n1\u0000\fjJ0jL0: (7)3\n0 50 100 150 200 250\nTime [fs]0.900.95mza)\nb)\nc)FM AFM\n101102103Relaxation time [fs]\nASDτ1,fm\nASDτ2,fm\nASDτafm(1−T/T c)ν\nτ1,fmMFA\nτafmMFA\n0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0\nTf/Tc101102Relaxation time [fs]\nτafm2D,z= 4\nτafmbcc,z= 8τafmsc,z= 6\nτafmfcc,z= 12\nFIG. 2. (a) Magnetic order dynamics of a FM and an\nAFM after a step-like temperature increase, Ti=Tc= 0:07\nand final temperature Tf=Tc= 0:33. (b) Magnetic order\nrelaxationtimeinFMsandAFMsasafunctionofthereduced\ntemperature Tf=Tc. Symbols correspond to ASD simulations,\nsolidlinestoMFA,anddashedlinesareafitofthescalinglaw,\n(1\u0000T=T N)\u0017with\u0017=\u00001:017(7)for\u001c1;fmand\u0017=\u00000:64(2)\nfor\u001cafm. The light red area around the FM data indicates the\nstatistical uncertainty coming from over 50 simulations. For\nAFMs is not shown since it is around 2-4% at maximum close\ntoTN. (c) The relaxation time for different lattice structures\n(sc, bcc, fcc and a 2D square), each with different number z\nof exchange coupled neighbouring spins. Symbols correspond\nto ASD simulations and lines to MFA.\nFor small deviations \u000eneof the order parameter n, from\nequilibrium ( \u000ene\u001cne), Eq. (5) can be expanded\naround the equilibrium state ne. The resulting linear\nequation in ncan be easily solved analytically as an\nexponential decay, with relaxation time given by \u001cafm=\ne\u001fk=\r\u000be\nafm, where\u000be\nafmis calculated for n=ne. At\nlow temperature, where ne\u00191the ratio between AFM\nand FM relaxation time is just given by \u000bafm=\u000bfm=\n1 + 4=zne, which equals to 1 + 4=z(5/3 for simple cubic)\nat strictly T= 0K. In striking contrast to the FMs,\nthe effective AFM damping depends on the number of\nnearest neighbours z. For the MFA limit, z!1, the\nexchange-enhancement vanishes ( \u000bafm=\u000bfm) whereasfor systems with low coordination number increases, for\nexample a spin chain with z= 2,\u000bafm= 3\u000bfm. Another\nrelevant example would be the metallic antiferromagnet\nMn2Au [26–28], the most promising candidate for future\nspintronic and memory applications, in which each Mn\nspin is antiferromagnetically exchange coupled to five\nneighbours, and thus a speed up of a factor 1+4=5 = 1:8\nis expected. At temperatures approaching Tc(N), the\norder parameter reduces, ne\u00190, and consequently,\n\u000bafm=\u000bfm\u00181=nediverges. In the MFA, close to\nTc(N), the order parameter at equilibrium scales with\ntemperature as ne\u0018(1\u0000T=T c(N))1=2. Hence the critical\nbehaviour of the AFM damping parameter is \u000bafm\u0018\n(1\u0000T=T N)\u00001=2. While the longitudinal susceptibility\nincreaseswithtemperatureuptothecriticaltemperature\nwhere it diverges, e\u001fk\u0019(1\u0000T=T c(N))\u00001. Thus,\nthe relaxation time in AFMs scales as \u001cafm\u0018(1\u0000\nT=T N)\u00001=2, which differs from the scaling for FMs \u001cfm\u0018\n(1\u0000T=T c)\u00001. Although both AFMs and FMs show\nthe so-called critical slow down near Tc(N), the effect\nof the exchange-enhancement of the AFM dynamics\nis to lower the critical exponent. Since, in general,\nthe MFA scaling laws are known to differ from the\nactual critical scaling exponents, \u001c\u0018(1\u0000T=T c)\u0000\u0017, in\nthe following we conduct ASD computer simulations to\nverify qualitatively these theoretical predictions, i) to\nfind quantitatively the critical exponents, \u0017, for AFMs\nand FMs, and ii) to demonstrate the dependence on\nthe number of exchange links between spins of the\nrelaxation time in AFMs. To do so, we compute the\nrelaxation time under the same conditions, namely, for\nsmall deviations from equilibrium, \u000ene=ne\u001c1. This\nis achieved by applying a step-like temperature increase,\n\u0001T=Tf\u0000Ti, such that\u000en=ne(Ti)\u0000ne(Tf) = 0:1ne(Ti)\nfor all initial/final temperatures Ti/Tf. We clarify for\ndirect comparison of the ASD simulations to analytical\nestimations, the parameters have to be calculated for\nthe final temperature, ne(Tf). An example of such\nmagnetization dynamics for the z\u0000component of the\norder parameter for FM and AFM orderings for a simple\ncubic lattice ( z= 6) are shown in Fig. 2 (a). We\nfind that for FMs the relaxation dynamics is defined\nby two characteristic times, \u001c1;fmand\u001c2;fm, associated\nto a fast and a slow relaxation process, respectively\nand for the AFM a single \u001cafmis enough to describe\nthe demagnetization process. Fig. 2 (b) shows the\nrelaxation times \u001c1;fm,\u001c2;fmand\u001cafmas function of\nthe reduced temperature Tf=Tcin comparison to the\nMFA analytical expression derived from Eq. (5). We\nnote that the values for the relaxation time in FMs\nin Fig. 2 (b) are obtained from averaging over 50\nindividual ASD simulations. Interestingly, we find that\nfor all temperatures the slow relaxation time \u001c2;fmis\nrelated to fast one \u001c1;fmas\u001c2;fm= 12\u001c1;fm. The\nfaster time decay is related to the relaxation of the4\nmagnetic order, while the slower one with the relaxation\nof short-wavelength spin waves[20]. Differently to this\ncharacteristic bi-exponential relaxation decay in the\nFMs, the relaxation process in AFMs is defined by only\none, fast characteristic time, \u001cafm. The relaxation time of\ntheAFMorderparameter \u001cafmisfasterthan \u001c1;fm, forthe\nsame microscopic magnetic parameters. In particular, at\nlowtemperatures, theratiobetweentherelaxationtimes,\n\u001c1;fm=\u001cafmis close to 5/3, like our predicted theory value\nfor a sc lattice. The absence of a second, slow relaxation\nprocessmakesthattheAFMsreachthefinal, equilibrium\nstate much faster than in FMs, indeed the characteristic\ntimes, in FMs and AFMs, are related as \u001c2;fm\u001912(1 +\n4=z)\u001cafm, which ranges from 12 for z! 1to 36 for\nz= 2. Asthefinalsystemtemperature Tfapproachesthe\ncritical temperature, TN, the magnetization dynamics\nslows down both for AFMs and FMs. Figure (2)(b)\nshows the good agreement between our model (MFA-\nsolid lines) and ASD (symbols) for both the AFMs and\nFMs. The critical behaviour of the relaxation time\ncan be also captured by a temperature scaling function,\n\u001ck\u0018(1\u0000T=T N)\u0000\u0017(dashed lines in Fig. 2(b)). By\nfitting our ASD simulation results, we find that for\nthe FM system, \u0017fm= 1:017(7), whereas for the AFM\nsystem,\u0017afm= 0:64(2), in qualitative agreement with the\nprediction of our theory, the critical exponent in AFMs is\nsmaller than in FMs. We note that for the AFM fit, the\ncriticalexponentisobtainedbytakingonlythedataclose\ntoTNinto account, where the second term in Eq. (6)\ndominates and therefore it coincides with our theoretical\nanalysis. Another fundamental difference between FMs\nand AFMs is the dependence of the relaxation time on\nthenumberofneighbours ztowhicheachspiniscoupled.\nFigure 2(c) shows the temperature dependence of the\n\u001cafmforthreedifferentlatticestructuresin3D,sc( z= 6),\nbcc (z= 8), and fcc ( z= 12), and in 2D, a square\nlattice (z= 4). Lines in Fig. 2(c) correspond to the\nanalytical estimation based in our model and symbols to\nthe ASD simulations. We stress that the lattice structure\ndependence of \u001cafmonly exists for AFMs. Relaxation\ntime in FMs is independent of the lattice structure.\nOne question remains, how could signatures of these\nexchange-enhanced dynamics in AFMs be found in\nexperiments? To address this problem, we first validate\nour model by comparing directly the dynamics of an\nAFM calculated via ASD simulations and Eq. (5) for\nthree different temperature profiles (see Fig. 3(a)). One\ntemperature profile corresponds to a step function and\nthe others to the TTM with two sets of parameters. The\nagreement between ASD simulations and our model is\nvery good. We note that for quantitative comparison\nbetween ASD simulations and MFA models, one needs\nto slightly rescale the exchange parameter, J(for more\ndetail we refer to the supplemental material Sec. S2).\nWe find that Eq. (5) describes ASD simulations as far\nas the microscopic spin configurations are homogeneous,\n0 1 2 3\nTime [ps]0.60.81.0nz\na)\nb)\nTTM 1\nTTM 2\nStep Function01Tel/TNTTM 1\nTTM 2Step FunctionFIG. 3. (a) Temperature step function ( Ti= 0:06TNand\nTf= 0:7TN) and two different Telprofiles for the same laser\nfluence and gep= 6\u00021017J/sKm3for two sets of parameters\n(TTM1:\r= 700J/K2m3,cph= 3\u0002106J/Km3, and TTM2:\n\r= 2000J/K2m3,cph= 5\u0002105J/Km3). (b) The magnetic\norder dynamics as a response to the temperature profiles in\npanel (a). The symbols correspond to ASD simulations and\nthe lines to the numerical solution of Eq. (5), ( \u0015= 0:01).\nas expected from the MFA grounds of our model (see\ndetails in supplemental materials, Sec. S3). By directly\ncomparing the dynamics of FMs and AFMs under\nlaser pulses, for instance see Figs. 1 and 3, one can\nbarely discern the effect of the exchange-enhancement\nin AFM dynamics. However, our model predicts striking\ndifferencesbetweenFMsandAFMsinthemagneticorder\ndynamics in the limit of high-temperatures, T\u001dTc,\nand small magnetic order parameter ( \u0018=\fJ0m!0).\nThis scenario corresponds to experiments using powerful\nfemtosecondlaserpulses. ForFMs, Eq. (3)approximates\nto a linear equation: (\u0016at=\r) _m= 2\u0015kBTm. Thus, the\ndynamicsisdescribedbyanexponentialdecay, namely, it\nslows down as the magnetization mreduces. In contrast,\nin the same limit, for AFMs, Eq. (3) approximates\nto(\u0016at=\r) _n\u00194(4=z)\u0015kBT, independent of n, which\nspeeds up the AFM dynamics. This different dynamic\ndirectly emerges by increasing the laser fluence so that\nthe electron temperature reaches very high temperatures\nand the magnetic reduces. In Fig. 4(a) one can observe\nthe diverse behaviour of the maximum demagnetization\n(\u0001maxm(n)) as a function of the reduced maximum\nelectron temperature Tel=Tcfor both AFMs and FMs.\nWe note that since the results depend on the chosen\nTTM parameters, the results are drawn as a function\nofTel=Tcinstead of laser intensity. Figure 4(a) shows\nthat for FMs the shape of \u0001maxm(Tel)is convex, while5\nLaser Power0.00.51.0Max ∆mz(nz)\nb)a)\nAFM\nFM\nAnalytical\n1.0 1.5 2.0 2.5 3.0\nMaxTel/Tc024Max Γ [1/ps]\nFIG. 4. (a) Maximum magnetic order quenching Max\n\u0001mz=Maxjm0\u0000mz(t)j(\u0001nz=Maxjn0\u0000nz(t)jfor AFM)\nas function of the reduced peak electron temperature Max\nTel=Tc. (b) Maximum demagnetization rate as function of the\npeak electron temperature. Dots, AFM (blue) and FM (red),\nrepresent ASD simulations and dashed lines the numerical\nsolution of Eq. (5). Results for a laser pulse of 50 fs duration\nand\u0015= 0:01in the ASD simulations.\nfor the AFM, \u0001maxn(Tel)is concave. These findings\nalign with an experimental work comparing the magnetic\norder dynamics of the AFM and FM phases in Dy [17],\nwhere for comparable laser powers, the maximum\ndemagnetization in AFMs was larger than in FMs [17].\nIt was also found that by increasing the laser intensity,\nthe maximum demagnetization rate \u0000maxincreased in\nAFMs is much stronger than in FMs. Figure 4 (b) shows\nhow the maximum demagnetization rate increases faster\nfor AFMs than for FMs, in qualitative agreement with\nexperimentsconductedinDy. Thedifferentslopeof \u0000max\nis directly related to the exchange-enhancement of the\neffective AFM damping (Eq. (6)), 1 + 4=z, for sc used\nhere, \u0000afm\nmax= (5=3)\u0000fm\nmax.\nSummary.– To summarize, we have shown that the\nultrafast magnetic order dynamics in antiferromagnets\nis exchange-enhanced in comparison to ferromagnets\nwith the same system parameters. The origin is the\nexchange-enhancement of the effective AFM damping.\nWe have provided an equation of motion for the AFM\nmagnetic order and predicted that AFMs have intrinsic\nfaster dynamics and distinct critical dynamics than FMs.\nNotably, we have found that the exchange-enhancement\nstrongly depends on the number of neighbours to which\nspins are exchange coupled, for instance in 2D magnets,\nthe speed up of the dynamics is larger. In the very\nhigh temperature regime, we have predicted a transition\nfromexponentialtolineardecaywhenthemagneticorder\nreduces. We propose a method to discern this effect in\nexperiments using powerful femtosecond laser pulses. We\nhave demonstrate the validity of our model by direct\ncomparison to atomistic spin dynamics simulations.\nAcknowledgments.– We gratefully acknowledge sup-port by the Deutsche Forschungsgemeinschaft through\nSFB/TRR 227 \"Ultrafast Spin Dynamics\", Project A08.\n\u0003unai.atxitia@fu-berlin.de.\n[1] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y.\nBigot, Ultrafast Spin Dynamics in Ferromagnetic Nickel,\nPhysical Review Letters 76, 4250 (1996).\n[2] A.Kirilyuk,A.V.Kimel,andT.Rasing,Ultrafastoptical\nmanipulation of magnetic order, Reviews of Modern\nPhysics 82, 2731 (2010).\n[3] B.Koopmans, G.Malinowski, F.DallaLonga, D.Steiauf,\nM. Fähnle, T. Roth, M. Cinchetti, and M. Aeschlimann,\nExplaining the paradoxical diversity of ultrafast laser-\ninduced demagnetization., Nature materials 9, 259\n(2009).\n[4] I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius,\nH.A.Dürr, T.A.Ostler, J.Barker, R.F.L.Evans, R.W.\nChantrell,A.Tsukamoto,A.Itoh,A.Kirilyuk,T.Rasing,\nand A. V. Kimel, Transient ferromagnetic-like state\nmediating ultrafast reversal of antiferromagnetically\ncoupled spins., Nature 472, 205 (2011).\n[5] T. A. Ostler, J. Barker, R. F. L. Evans, R. W. Chantrell,\nU. Atxitia, O. Chubykalo-Fesenko, S. El Moussaoui,\nL. Le Guyader, E. Mengotti, L. J. J. Heyderman,\nF. Nolting, A. Tsukamoto, A. Itoh, D. Afanasiev, B. A.\nIvanov, A. M. Kalashnikova, K. Vahaplar, J. Mentink,\nA. Kirilyuk, T. Rasing, and A. V. Kimel, Ultrafast\nheating as a sufficient stimulus for magnetization reversal\nin a ferrimagnet, Nature Communications 3, 666 (2012).\n[6] K.Carva,M.Battiato,D.Legut,andP.M.Oppeneer,Ab\ninitio theory of electron-phonon mediated ultrafast spin\nrelaxation of laser-excited hot electrons in transition-\nmetal ferromagnets, Physical Review B 87, 184425\n(2013).\n[7] A.V.Kimel, A.Kirilyuk, A.Tsvetkov, R.V.Pisarev,and\nT. Rasing, Laser-induced ultrafast spin reorientation in\nthe antiferromagnet TmFeO3, Nature 429, 850 (2004).\n[8] P. Wadley, B. Howells, J. Železný, C. Andrews, V. Hills,\nR. P. Campion, V. Novák, K. Olejník, F. Maccherozzi,\nS. S. Dhesi, S. Y. Martin, T. Wagner, J. Wunderlich,\nF. Freimuth, Y. Mokrousov, J. Kuneš, J. S. Chauhan,\nM. J. Grzybowski, A. W. Rushforth, K. W. Edmonds,\nB. L. Gallagher, and T. Jungwirth, Electrical switching\nof an antiferromagnet, Science 351, 587 LP (2016).\n[9] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono,\nand Y. Tserkovnyak, Antiferromagnetic spintronics,\nReviews of Modern Physics 90, 15005 (2018).\n[10] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich,\nAntiferromagnetic spintronics, Nature Nanotechnology\n11, 231 (2016).\n[11] K. Olejník, T. Seifert, Z. Kašpar, V. Novák, P. Wadley,\nR. P. Campion, M. Baumgartner, P. Gambardella,\nP. Němec, J. Wunderlich, J. Sinova, P. Kužel, M. Müller,\nT. Kampfrath, and T. Jungwirth, Terahertz electrical\nwriting speed in an antiferromagnetic memory, Science\nAdvances 4, eaar3566 (2018).\n[12] L. Rózsa, S. Selzer, T. Birk, U. Atxitia, and U. Nowak,\nReduced thermal stability of antiferromagnetic nanos-\ntructures, Phys. Rev. B 100, 064422 (2019).\n[13] Z. Kašpar, M. Surýnek, J. Zubáč, F. Krizek, V. Novák,6\nR. P. Campion, M. S. Wörnle, P. Gambardella,\nX. Marti, P. Němec, K. W. Edmonds, S. Reimers,\nO. J. Amin, F. Maccherozzi, S. S. Dhesi, P. Wadley,\nJ. Wunderlich, K. Olejník, and T. Jungwirth, Quenching\nof an antiferromagnet into high resistivity states using\nelectrical or ultrashort optical pulses, Nature Electronics\n10.1038/s41928-020-00506-4 (2020).\n[14] C.Kittel,Theoryofantiferromagneticresonance,Physial\nReview 82, 565 (1951).\n[15] T.Kampfrath, A.Sell, G.Klatt, A.Pashkin, S.Mährlein,\nT. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and\nR. Huber, Coherent terahertz control of antiferromag-\nnetic spin waves, Nature Photonics 5, 31 (2011).\n[16] M. Fiebig, N. P. Duong, T. Satoh, B. B. Van\nAken, K. Miyano, Y. Tomioka, and Y. Tokura,\nUltrafast magnetization dynamics of antiferromagnetic\ncompounds, Journal of Physics D: Applied Physics 41,\n164005 (2008).\n[17] N. Thielemann-Kühn, D. Schick, N. Pontius, C. Trabant,\nR. Mitzner, K. Holldack, H. Zabel, A. Föhlisch, and\nC. Schüßler-Langeheine, Ultrafast and Energy-Efficient\nQuenching of Spin Order: Antiferromagnetism Beats\nFerromagnetism, Physical Review Letters 119, 197202\n(2017).\n[18] U.AtxitiaandO.Chubykalo-Fesenko,Ultrafastmagneti-\nzation dynamics rates within the Landau-Lifshitz-Bloch\nmodel, Physical Review B 84, 144414 (2011).\n[19] Q. H. Wang, A. Bedoya-Pinto, M. Blei, A. H. Dismukes,\nA. Hamo, S. Jenkins, M. Koperski, Y. Liu, Q.-C. Sun,\nE. J. Telford, H. H. Kim, M. Augustin, U. Vool, J.-\nX. Yin, L. H. Li, A. Falin, C. R. Dean, F. Casanova,\nR. F. L. Evans, M. Chshiev, A. Mishchenko, C. Petrovic,\nR. He, L. Zhao, A. W. Tsen, B. D. Gerardot, M. Brotons-\nGisbert, Z. Guguchia, X. Roy, S. Tongay, Z. Wang, M. Z.\nHasan, J. Wrachtrup, A. Yacoby, A. Fert, S. Parkin,\nK. S. Novoselov, P. Dai, L. Balicas, and E. J. G. Santos,\nThe magnetic genome of two-dimensional van der waals\nmaterials, ACS Nano 10.1021/acsnano.1c09150 (2022).\n[20] V. G. Baryakhtar and A. G. Danilevich, The phenomeno-\nlogical theory of magnetization relaxation (Review\nArticle), Low Temperature Physics 39, 993 (2013).\n[21] U. Atxitia, O. Chubykalo-Fesenko, R. Chantrell,\nU. Nowak, and A. Rebei, Ultrafast spin dynamics: The\neffect of colored noise, Physical Review Letters 102,\n10.1103/PhysRevLett.102.057203 (2009).\n[22] J. H.Mentink, J.Hellsvik, D. V.Afanasiev, B.A. Ivanov,\nA. Kirilyuk, A. V. Kimel, O. Eriksson, M. I. Katsnelson,\nandT.Rasing,Ultrafastspindynamicsinmultisublattice\nmagnets, Physical Review Letters 108, 057202 (2012).\n[23] D. A. Garanin, Fokker-planck and landau-lifshitz-bloch\nequations for classical ferromagnets, Physical Review B\n55, 3050 (1997).\n[24] F. Jakobs and U. Atxitia, Universal criteria for single\nfemtosecond pulse ultrafast magnetization switching in\nferrimagnets (2021).\n[25] U. Atxitia, P. Nieves, and O. Chubykalo-Fesenko,\nLandau-Lifshitz-Bloch equation for ferrimagnetic mate-\nrials, Physical Review B 86, 104414 (2012).\n[26] V. Barthem, C. Colin, H. Mayaffre, M.-H. Julien, and\nD. Givord, Revealing the properties of mn 2 au for\nantiferromagnetic spintronics, Nat. Commun. 4, 2892\n(2013).\n[27] P. E. Roy, R. M. Otxoa, and J. Wunderlich, Robust\npicosecond writing of a layered antiferromagnet bystaggered spin-orbit fields, Phys. Rev. B 94, 014439\n(2016).\n[28] M. Meinert, D. Graulich, and T. Matalla-Wagner,\nElectrical switching of antiferromagnetic mn2au and the\nrole of thermal activation, Phys. Rev. Appl. 9, 064040\n(2018).\n[29] M. I. Kaganov, I. M. Lifshitz, and L. V. Tanatarov,\nRelaxation between electrons and crystalline lattices,\nJETP 173(1957).\n[30] J. K. Chen, D. Y. Tzou, and J. E. Beraun, A\nsemiclassical two-temperature model for ultrafast laser\nheating, International Journal of Heat and Mass Transfer\n49, 307 (2006).\n[31] R. Jedynak, Approximation of the inverse langevin\nfunction revisited, Rheologica Acta 54, 29 (2015).\n[32] A. N. Nguessong, T. Beda, and F. Peyraut, A new\nbasederrorapproachtoapproximatetheinverselangevin\nfunction, Rheologica Acta 53, 585 (2014).S1\nSupplementary Material to \"Exchange-enhancement of the ultrafast magnetic order dynamics in\nantiferromagnets\"\nS1. TWO-TEMPERATURE MODEL\nThe dynamics of the electron temperature Teland the phonon temperature Tphcan described via the two-\ntemperature model (TTM) [29, 30],\nCel@Tel\n@t=\u0000gep(Tel\u0000Tph) +Pl(t) (S1)\nCph@Tph\n@t= +gep(Tel\u0000Tph): (S2)\nwheregep= 6\u00021017J/sKm3istheelectron-phononcouplingconstant, Cph= 3\u0002106J/Km3andCel=\reTel(\re= 700\nJ/K2m3) represent the respective specific heats of the electron- and phonon system. Although we use standard values\nfor metals, these values are material-dependent. Pl(t)is Gaussian shaped and represents the absorbed energy of the\nelectron system coming from the laser.\nS2. RESCALING OF THE EXCHANGE CONSTANT FOR QUANTITATIVE COMPARISON BETWEEN\nMFA AND ASD SIMULATIONS\nIn the main text, our analytical model for the magnetic order dynamics is based in the mean field approximation\n(MFA). The equilibrium magnetization as a function of temperature calculated using the MFA slightly differs from\nthe ASD simulations. Fig. S1 shows the MFA results as a blue dashed line and the ASD simulations as red points for\na sc-lattice using J= 3:450\u000210\u000021J. For the MFA case, we have rescaled the exchange constant, Jmfa= 0:73Jasd,\nto obtainTMFA\nN =TASD\nN. We have estimated the ASD critical temperature by calculating the temperature at which\nthe magnetic specific heat diverge. The equilibrium magnetization as a function of temperature using the MFA\nFIG. S1. Equilibrium magnetization of a sc-lattice as a function of temperature from ASD simulations (red dots), mean field\napproximation (blue dashed line) and from the MFA model including a temperature dependent rescaled Heisenberg exchange\nconstantJ(T)(Eq. (S4)), (red line).\nstart to deviate from ASD simulations in the intermediate-to-high temperature region, TN=2< T < T N. In order\nto quantitatively compare our model to ASD simulations we resolve this discrepancy by introducing a temperature\ndependent Heisenberg exchange modulation J(T) =J0+J0(T), whereJ0describes the original MFA Heisenberg\nexchange constant, Jmfa= 0:73Jasd,J0(T)>0is a temperature dependent modulation that needs to be determined.\nWe determine it by forcing the equality the equilibrium magnetization calculated through ASD, me= (1\u0000T=TN)1=3\n(1=3for a sc-lattice), and the MFA, me=L(\fJ(T)me). Thus, the temperature dependent Heisenberg exchange J(T)S2\ncan be calculated from\n(1\u0000T=T c)1=3=L\u0012(J0+J0(T))m\nkBT\u0013\n(S3)\nwhich can be solves as\nJ0(T) =1\n\fmL\u00001((1\u0000T=T c)1=3)\u0000J0: (S4)\nL\u00001describes the inverse Langevin function for which no analytical expression is known. However there have been\nnumerous attempts at finding a simple and accurate approximation [31, 32]. In this work we have used the equation\nproposed by Nguessong et al. [32] to approximate the inverse Langevin function numerically.\nWenote, thatbyusing Eq.S4 J(T)becomes independentof the numericalvalueof J0and isinstead directly calculated\nfrom the magnetization curve m(T)via the inverse Langevin function. For a sc-lattice me(T) = (1\u0000T=Tc)1=3agrees\nwell with the atomistic results. However for other lattices (fcc, 2D or bcc), a different analytical expression for me(T)\nis needed to describe me(T).S3\nS3. BREAKDOWN OF THE MFA MODEL FOR HIGH FLUENCE LASER EXCITATION\nAs discussed in the main text, our model is based in the MFA. This means that better agreement between ASD\nand MFA would be expected when the microscopic spin configurations remain close to the MFA assumptions, when\neach atomic spin sees the same interactions from the neighbouring ones.\nFIG. S2. ASD simulations of AFM magnetic order dynamics for different laser powers ( \u0015= 0:01) (dots) in comparison to\nour analytical model (Eq. (5)) (lines). Higher laser powers yields larger demagnetization and the underlying MFA assumptions\nof Eq. (5) stops being a valid approximation. On the right different states of the spin system are shown, shortly after the\nexcitation with a laser pulse.\nWhen magnetic domains are be nucleated, our MFA macroscpin model no longer describes the spin state correctly.\nFigure S2. shows the magnetic order dynamics for a range of laser fluences, where symbols correspond to ASD\nsimulations and lines to the macrospin model. For higher fluences the agreement between the two models diminishes.\nThe right side shows snapshots of the microscopic spin configuration at different time delays corresponding to the a\ntime range where maximum demagnetization is achieved. When the laser fluence is only the 73 %of the maximum\nfluence simulated, the microscopic spin configuration is homogeneous. In that case, the agreement between theory\nand simulations is very good. As the laser fluence increases, magnetic domains start to nucleate and the theory and\nsimulations to deviate. For the maximum laser fluence that we simulate 100 %, large magnetic domains are nucleated\nand the MFA breaks down. The theory is not able to describe this situation. For those cases, a micromagnetic model\nshould be developed."    },    {        "title": "1103.4819v1.Spontaneous_demagnetization_of_a_dipolar_spinor_Bose_gas_at_ultra_low_magnetic_field.pdf",        "content": "arXiv:1103.4819v1  [cond-mat.quant-gas]  24 Mar 2011Spontaneous demagnetization of a dipolar spinor Bose gas at ultra-low magnetic field\nB. Pasquiou, E. Mar´ echal, G. Bismut, P. Pedri, L. Vernac, O. Gorc eix and B. Laburthe-Tolra\nLaboratoire de Physique des Lasers, CNRS UMR 7538,\nUniversit´ e Paris 13, 99 Avenue J.-B. Cl´ ement, 93430 Ville taneuse, France\n(Dated: November 13, 2018)\nQuantum degenerate Bose gases with an internal degree of fre edom, known as spinor condensates,\nare natural candidates to study the interplay between magne tism and superfluidity. In the spinor\ncondensates made of alkali atoms studied so far, the spinor p roperties are set by contact interac-\ntions, while magnetization is dynamically frozen, due to sm all magnetic dipole-dipole interactions.\nHere, we study the spinor properties of S=352Cr atoms, in which relatively strong dipole-dipole\ninteractions allow changes in magnetization. We observe a p hase transition between a ferromagnetic\nphase and an unpolarized phase when the magnetic field is quen ched to an extremely low value,\nbelow which interactions overwhelm the linear Zeeman effect . The BEC magnetization changes due\nto magnetic dipole-dipole interactions that set the dynami cs. Our work opens up the experimental\nstudy of quantum magnetism with free magnetization using ul tra-cold atoms.\nPACS numbers: 03.75.-b, 67.85.-d, 03.75.Mn\nWithin optical dipole traps, it is possible to trap all\nZeeman states of an atomic species in its electronic\nground state. This has enabled the field of multi-\ncomponent (spinor) Bose-Einstein condensates (BECs),\ni.e.BECs with internal degrees of freedom [1, 2]. With\nshort range interactions, collisions between atoms are\ndescribed by various scattering lengths, which leads to\nspin-dependent contact interactions. New phases arise,\nwhich have been investigated for F= 1 and F= 2\natoms, using Rb and Na, by studying miscibility [2], spin\ndynamics [3–6], and spin textures [7]. In these experi-\nments, spin-dependent contactinteractionsareextremely\nsmall compared to the linear Zeeman effect. In addition,\nthe gas magnetization remains constant for all practi-\ncal purposes because contact interactions are isotropic,\nand anisotropic dipole-dipole interactions between alkali\natomsarenegligible. Consequently, thetruegroundstate\nof the system, with free magnetization, at extremely low\nmagnetic fields has never been experimentally investi-\ngated.\nThe production of Cr BECs in optical dipole traps\n[8, 9] allows for the first time to study S= 3 spinor\nphysics, with a wealth of possible quantum nematic\nphases [10, 11], depending on the different scattering\nlengthsa0,2,4,6of the molecularpotentials Sm= 0,2,4,6.\nContrarily to the cases of Rb and Na where spin depen-\ndent interactions are very small due to similar scatter-\ning lengths in different molecular channels, Cr has large\nspin dependent contact interactions. One consequence is\nthat one can reach the regime where these interactions\noverwhelm the Zeeman effect at experimentally accessi-\nble magnetic fields (up to 1 mG, compared to typically\n10µG for Rb). While large magnetic fields favor a fer-\nromagnetic ground state, below a critical magnetic field\ndepending on the interactions the Cr spinor ceases to be\nferromagnetic [10, 11], which we here observe.\nAn important feature of Cr atoms, arising from\ntheir large electronic spin, is the strength of long-\n-3 -2 -1 0 1 2 3(a)\n(b)\n(c)\n(d)\nFIG. 1: Chromium BEC spin composition at low fields, revealed\nby Stern-Gerlach analysis (see Methods 1). The BEC spontane ously\ndepolarizes as the magnetic field is lowered. Absorption pic tures af-\nter 155 ms of hold time, in a field of: a) 1 mG; b) 0.5 mg c) 0.25\nmG and d) 0 mG. At the lowest magnetic field, the spin distribu-\ntion is{17.5±9,18±4,14±1.5,15±3,17±3,12.5±4,6±2}% cor-\nresponding to a magnetization of -0.5 gSµB.\nranged, anisotropic dipole-dipole interactions between\nthem. These interactions are too weak to greatly modify\nthe phase diagram of Cr at low magnetic fields [10, 12],\nbut they do crucially modify spinor properties. They\nintroduce magnetization-changing collisions [13], allow-\ning the study of spinor physics with free magnetization,\nwith an intriguing coupling between the spin degrees\nof freedom and mechanical rotation that resembles the\nEinstein-de-Haas effect [11, 14–16]. They are also ex-\npected to lead to spin textures, as investigated in [17],\nsimilar to domain formation in ferro-magnets. Here, we\nstudy how dipolar interactions induce magnetization dy-\nnamics, as the magnetic field is quenched below an ex-\ntremely low value, which corresponds to a phase transi-\ntion separating two spinor phases of different magnetiza-\ntion.\nIn this article, we show that a Cr BEC in which atoms\nare polarized in the lowest single particle energy state\nmS=−3 spontaneously depolarizes below a critical field\nof about 0.4 mG. We observe that the magnetic field be-\nlow which depolarization occurs is larger if the peak local2\ndensity of the BEC is increased by loading it into deep\n2D optical lattices. This is a consequence of an increased\nmeanfield energy, which raises the critical magnetic field\nup to 1.2 mG. We also analyze depolarization dynam-\nics with and without the lattice, and the role of dipole-\ndipole interactions; we find that depolarization is slower\nin the lattice although the peakdensity is then higher.\nThis counterintuitive behavior is a consequence of a re-\nduction of the averagedensity in the lattice and of the\nnon-locality of the mean-field due to long-ranged dipolar\ninteractions. Inaddition, withoutthe lattices, weobserve\nan increase of the thermal fraction, as expected from\nnon-interacting spinor thermodynamics theory; however\nthe observed depolarization of the BEC itself is a feature\nunique to the non trivial phase diagram at low magnetic\nfield due to spin-dependent contact interactions.\nWe produce Cr BECs by performing evaporative cool-\ning of atoms in the single particle lowest energy state\nmS=−3, in an optical dipole trap [9]. The magnetic\nfield during evaporationis sufficiently largethat any two-\nbody inelastic process is energetically forbidden. After a\nBEC has been obtained in mS=−3, with typically 20\n000 atoms, no discernible thermal fraction, and a peak\ndensity of 3 .5×1014cm−3in an harmonic trap of fre-\nquencies (320 ,400,550) Hz, we suddenly (within a 1 /e\ntime of 8 ms, set by the inductance of the coils and Eddy\ncurrents) reduce the magnetic field Bfrom 20 mG to an\nextremely low value, typically below 1 mG.\nThe magnetic field is controlled by three orthogonal\npairs of coils, and calibrated by rf spectroscopy. Ex-\ntremely low magnetic fields correspond to magnetic reso-\nnance frequencies of the order of 1 kHz or less. To reduce\ntechnical noise, and provide better long term field stabil-\nity, we have implemented an active stabilization of the\nambient magnetic field (see Methods 1). With active sta-\nbilization, typical shot to shot magnetic field fluctuations\nare 100µG.\nIf the final magnetic field is sufficiently small (below\n0.5 mG typically), we observe a spontaneous depolariza-\ntion of the BEC, while the total number of atoms re-\nmains constant. We measure the population in each of\nthe Zeeman sublevels by performing a Stern-Gerlach ex-\nperiment (see Methods 2). We finely tune the current\nsent to three orthogonal pairs of coils to maximize depo-\nlarization, which in practice precisely pinpoints B= 0.\nTypical depolarization results are displayed on Fig. 1.\nWe have repeated the same experiment for a BEC adi-\nabatically(within 15ms)loadedinthe groundstateband\nofa 2Doptical lattice [18], superimposed ontothe optical\ndipole trap. Our lattice configurationis described in [19].\nThe lattice depth in each direction is between 25 and 30\nER, whereER=h2\n2mλ2is the recoil energy, with λ= 532\nnm, and mthe atom mass. The vibrational trapping\nfrequency in a lattice site is 120 kHz, much larger than\nthe chemical potential ( µ/h=11 kHz); consequently, the\nmotion is frozen in two dimensions. We consider that the1.0\n0.8\n0.6\n0.4\n0.2\n0.0\n5 4 3 2 1 0\nMagnetic□field□(mG)BEC\nBEC□in□latticeFinal□m=-3□fraction\n-3.0-2.5-2.0-1.5-1.0-0.50.0\n4 3 2 1parallel□( ) /c113/c61/c48\nperpendicular ( ) /c113/c61/c112/c47/c50\nMagnetization□(/g□µ□)\nMagnetic□field□(mG)SB\nFIG. 2: Demagnetization curve for 3D BECs and for 1D quantum\ngases. The mS=−3 final population is plotted as a function of the\nmagnetic field. Full circles give the results for 3D-BECs and triangles\nfor 1D tubes. Inset: final gas magnetization as a function of t he mag-\nnetic field in the lattice, for two orthogonal orientations o f the field\nwith respect to axis of the lattice tubes. Lines are gaussian fits to the\ndata.\nBEC is split into an array of 1D quantum gases, with an\nestimated peak density of 2 ×1015cm−3.\nThe fraction of atoms remaining in mS=−3 after a\nhold time of 155 ms is displayed on Fig. 2 for the two\nexperimental configurations, as a function of magnetic\nfield. While we observe strong depolarization at zero\nmagnetic field for both configurations, the width of the\ndepolarization curve is significantly larger for the BEC\nloaded in the 2D optical lattice than without the lattice.\nA gaussian fit to the experimental data gives a 1 /ewidth\nof 0.4 mG for the BEC, and 1.2 mG for the BEC loaded\nin the lattice.\nWe interpret the increased width of the depolarization\ncurveinthelatticeasaneffectofanincreaseofthe mean-\nfield interactions. Indeed, the chemical potential of the\nBEC isµ/h= 3.8 kHz, and reaches 11 kHz when loaded\ninto the lattice. The widths of the depolarization curves\naretherefore roughly proportionalto the chemical poten-\ntial.\nAtT= 0, and at finite magnetic field B, sponta-\nneous depolarization from mS=−3 becomes energeti-\ncally possible when the energy cost to go from mS=−3\ntomS=−2 is compensated for by a reduction of in-\nteraction energy between the atoms. In the case of Cr,\nbecausea4< a6, this happens below a threshold Bc[10]:\ngSµBBc≈0.72π¯h2(a6−a4)n\nm(1)\nwithgS≈2 the Land´ e factor of Cr atoms, µBthe Bohr\nmagneton and nthe density. Bccorresponds to a critical\nmagnetic field separating the ferromagnetic phase to ei-\nther a polar phase (spin state ( α,0,0,0,0,0,β)), a cyclic\nphase (spin state ( α,0,0,0,0,β,0)), or a phase in which\nall Zeeman states are populated, depending on the un-\nknown value of a0[10, 11]. Phase separation is also pos-\nsible [20]. Let us compare the numerical values for Bcto\nthe values at which we do observe depolarization: for the\nBEC and the lattice respectively, Bccalculated from eq.3\n(1) is respectively 0.25 mG and 1.15 mG, which is com-\nparable to the observed 0.4 mG and 1.2 mG 1 /ewidths.\nThe slight discrepancy between experiment and theory\nis consistent with our magnetic field fluctuations. We\ninterpret the spontaneous depolarization as due to the\nphase transition at Bc. Our work therefore differs from\n[21], where depolarization of a thermal Cr gas occurred\nbecause the thermal energy was converted into Zeeman\nenergy through dipolar collisions between atoms.\nWe emphasize that technical noise can not mimic the\nincrease of the width of the depolarization curve as the\natoms are loaded in the lattice. Indeed, as the Zeeman\neffect is purely linear for Cr, rf spectroscopy, and more\ngenerally, the coupling of atoms to time-dependent mag-\nnetic fields, are independent of mean-field interactions:\nin a linear spin system, a magnetic field does not couple\ntwo different molecular potentials. We have also verified\nby rf spectroscopythat the quadraticeffect related to the\ntensorial light shift of Cr [22] in the lattice is not respon-\nsible for the increased width of the depolarization curve\n[23].\nWe have investigated the dynamics of depolarization,\nwith and without the lattice, as shown on Fig. 3. At\nt=0, the magnetic field control is suddenly switched to 0.\nThe magnetic field B(t) does not instantaneously reach\nits final value, and in practice, for the first 50 ms, the\natomsremaincompletelypolarizedas B(t <50ms)> Bc.\nThen, as B(t)< Bc, depolarization occurs. Without the\nlattice, depolarization occurs so suddenly ( ≤5 ms) that\nwe can barely resolve it; depolarization is slower in the\nlattice, although the peak density is then much larger.\nAs shown below, the typical timescale for depolarization\nis set by the magnetic field resulting from the surround-\ning dipolar atoms, i.e.by the non-local meanfield due to\ndipolar interactions [14]. For atoms loaded into an op-\ntical lattice, the large increase of the (repulsive) contact\nmean-field forces the cloud to swell in our experimental\nconfiguration; the overall volume of the cloud is then in-\ncreased by a factor of about three, hence reducing the\ndipolar mean-field. A slower depolarization dynamics in\nthe lattice is thus a consequence of the non local charac-\nter of dipole-dipole interactions, and indicates inter-site\ninelastic dipolar couplings in the lattice.\nAnother signature for inter-site coupling is provided in\ntheinsetofFig. 2, whereweshowthat thedepolarization\nisidenticalfortwodifferentorientations θofthemagnetic\nfield, either parallel or perpendicular to the tubes axis.\nIn a previous work, where we have studied dipolar relax-\nation in 2D optical lattices for atoms in mS= 3 at larger\nmagneticfields, the magnetizationchangingcollisionsoc-\ncur in a given lattice site, and they are suppressed for\nθ= 0 because of cylindrical symmetry [19]. The situa-\ntion is different herebecausethe experiment is performed\nat much smaller magnetic fields, and dipolar relaxation\noccurs at increasing inter-atomic distances for decreas-\ning magnetic field [13]. At 1 mG, the typical distance12\n8\n4\n0x103\n200 150 100 5012\n8\n4\n0x103\n200 150 100 50\nTime□(ms)Atom□left□in□m=-3Atom□left□in□m=-3\nFIG. 3: Time evolution of the mS=−3 population, as a function of\nthe time delay between the moment at which the magnetic field c ontrol\nis set to zero and the beginning of the Stern-Gerlach procedu re. a)\ndynamics for 3D BECs; b) dynamics for 1D quantum gases. Full l ines\nare guides to the eye. Depolarization starts after a delay, c orresponding\nto the time for B(t) to reach Bc(see text).\nat which dipolar relaxation occurs is 300 nm, exceeding\nthe lattice periodicity, which breaks cylindrical symme-\ntry. A description including inter-sites coupling is hence\nrequired.\nTo account for the depolarization dynamics, we\nconsider the time-dependent Gross-Pitaevskii equation,\nstarting with a polarized BEC in mS=−3, including\ncontact and dipole-dipole interactions. Dipole-dipole in-\nteractions provide the only mechanism to transfer the\natoms from the maximally polarized mS=−3 state to\nmS=−2. We calculate the dynamics at short times by\nassumingthat the population in mS=−2 remainssmall,\nand that population in mS>−2 is negligible. Then,\nthe coupling between the fields φj=(−3,−2)describing the\nmS= (−3,−2) components is given by (see also [24]):\nΓ(/vector r) =γ/integraldisplay\nd3r′[(x−x′)−i(y−y′)](z−z′)/vextendsingle/vextendsingle/vextendsingle/vector r−/vectorr′/vextendsingle/vextendsingle/vextendsingle5/vextendsingle/vextendsingle/vextendsingleφ−3(/vectorr′)/vextendsingle/vextendsingle/vextendsingle2\n(2)\nwhereγ=−3S3/2¯h2d2/√\n2, withd2=µ0(gSµB)2/4π,\nandµ0the magnetic constant. At short times, coupling\nbetween mS=−3 andmS=−2 atoms is only set by\ndipolar interactions via the term Γ( /vector r), and|Γ(/vector r)|−1is\nthe typical time-scale for depolarization at B= 0. This\ntimescale is about 3 ms for the BEC case and 10 ms for\nthe lattice case, in relatively good agreement with the\nobservations (Fig. 3) at the lowest achievable magnetic\nfield.\nWe have further monitored the state of the atoms at\nthe first stages of depolarization. For this, we have im-\nplemented an alternative Stern-Gerlach procedure (see\nMethods 2), allowing a simultaneous analysis of spin and\nmomentum distributions. As shown in Fig. 4, soon after\ndepolarization has started the momentum distributions\nof the Zeeman components remain extremely narrow, a\nsignature that the system is still Bose condensed. These4\nOptical□depth□(arb.□units)\nm =-3S -2 -1 0\nFIG. 4: Stern-Gerlach monitoring of the spin composition as the\ndepolarization develops, for the BEC case. These pictures u tilize our\nsecond Stern-Gerlach procedure (see Methods 2) which insur es simul-\ntaneously spin separation and a measurement of the momentum distri-\nbutions of each Zeeman components. The three curves, vertic ally offset\nfor better clarity, correspond to the three absorption pict ures shown in\ninsert, taken at three steps of depolarization, separated b y a few ms.\nSolid lines result from multiple gaussian fits. The width of t he nar-\nrow peaks shows that, during depolarization, atoms in differ ent spin\nstates remain Bose condensed. A broad additional pedestal r eveals the\npresence of a thermal component.\ndistributions are narrower than the one of the polarized\nBEC, presumably reflecting the decrease in mean-field\nenergy in the depolarized case, since a6is the largest of\nthe Cr scattering lengths. We could not detect vortices\nin themS=−2 component, although they arepredicted,\ndue to conservation of total angular momentum [11, 14].\nIt is not yet established whether the magnetic field is\nlow enough for such vortices to be observed, nor how\nlong they should survive in our non cylindrical trap in\nthe presence of thermal excitations. This will be further\ninvestigated in future experiments.\nBy measuring the momentum distribution using stan-\ndard time-of-flight techniques without separating the\nZeeman components, we also observe a decrease of the\ncondensatefractionwhen depolarizationisimportant, al-\nthough the temperature (deduced by fitting the thermal\nwings) does not change. Such condensate fraction reduc-\ntion at constant temperature is a direct consequence of\nthe release in the constraint of the spin degrees of free-\ndom, which lowers the critical degeneracy temperature\nTc[25]. Although the reduction of Tcmight be under-\nstood in the frameworkofnon-interactingspinor thermo-\ndynamics, it is worthwhile noting that depolarization of\nthe BEC can not be understood in such framework, as\na non-interacting BEC with free magnetization remains\nferromagnetic for all magnetic fields [26].\nIn conclusion, this work represents the first investiga-\ntion of quantum magnetism where magnetization is free\nand spin-dependent contact interactions set the many-\nbody mean-field ground state. We have observed the\nmagnetization dynamics ofa BECdue to dipolar interac-\ntions, as the magnetic field is quenched through a critical\nvalue correspondingto(at T=0)aquantum phasetransi-\ntion. We have established that demagnetization is set bythe dipolar mean-field. Demagnetization leads to a non-\npolarized BEC, although the spin state which is reached\ndoesnotnecessarilyrevealthenatureofthe groundstate,\ndue to diabaticity at the phase transition, and to ther-\nmal excitations: temperature ( ≈100 nK) is comparable\nto the difference of energy between the spinor phases at\nB= 0.\nWe will next try to observe the rotation necessarily\ninduced by demagnetization, in the spirit of the Einstein-\nde-Haas effect [11, 14–16]. We will also investigate how\nthis rotating system thermalizes at low magnetic fields,\nto possibly reach quantum nematic (mean-field) phases\n[10]. When the BEC is loaded into the 2D lattice, we\nproduce an array of mesoscopic samples (40 atoms per\nsite) at low magnetic field, high density, and with large\nspin-dependent interactions; such conditions are known\nto be favorable to the observation of non-classical spin\nstates [27] or fragmented BECs [28].\nThis research was supported by the Minist` ere de\nl’Enseignement Sup´ erieur et de la Recherche (within\nCPER) and by IFRAF.\n[1] T.-L. Ho, Phys. Rev. Lett. 81, 742 (1998); T. Ohmi and\nK. Machida, J. Phys. Soc. Jpn. 67, 1822 (1998).\n[2] Stenger, J. et al. Nature 396, 345 (1998).\n[3] M.-S. Chang, C. D. Hamley, M. D. Barrett, J. A. Sauer,\nK. M. Fortier, W. Zhang, L. You, and M. S. Chapman,\nPhys. Rev. Lett. 92, 140403 (2004)\n[4] H. Schmaljohann, M. Erhard, J. Kronjager, M. Kottke,\nS. van Staa, L. Cacciapuoti, J. J. Arlt, K. Bongs, and K.\nSengstock, Phys. Rev. Lett. 92, 040402 (2004)\n[5] J. Kronjager et al., Phys. Rev. Lett. 97, 110404 (2006).\n[6] A. T. Black, E. Gomez, L. D. Turner, S. Jung, and P. D.\nLett, Phys. Rev. Lett. 99, 070403 (2007)\n[7] L. E. Sadler et al., Nature (London), 443, 312 (2006)\n[8] Axel Griesmaier, Jorg Werner, Sven Hensler, Jurgen\nStuhler, and Tilman Pfau, Phys. Rev. Lett. 94, 160401\n(2005)\n[9] Q. Beaufils, R. Chicireanu, T. Zanon, B. Laburthe-Tolra,\nE. Marechal, L. Vernac, J.-C. Keller, and O. Gorceix,\nPhys. Rev. A 77, 061601 (2008) G. Bismut, B. Pasquiou,\nD. Ciampini, B. Laburthe-Tolra and E. Marechal, L.\nVernac, andO.Gorceix.AppliedPhysicsB, 102, 1(2011)\n[10] Roberto B. Diener and Tin-Lun Ho, Phys. Rev. Lett. 96,\n190405 (2006)\n[11] L. Santos and T. Pfau, Phys. Rev. Lett. 96, 190404\n(2006)\n[12] H. Makela and K.-A. Suominen, Phys. Rev. A 75, 033610\n(2007)\n[13] B. Pasquiou, G. Bismut, Q. Beaufils, A. Crubellier,\nE. Marechal, P. Pedri, L. Vernac, O. Gorceix, and B.\nLaburthe-Tolra, Phys. Rev. A 81, 042716 (2010)\n[14] YukiKawaguchi, Hiroki Saito, andMasahito Ueda, Phys.\nRev. Lett. 96, 080405 (2006)\n[15] Krzysztof Gawryluk, Miroslaw Brewczyk, Kai Bongs,\nand Mariusz Gajda, Phys. Rev. Lett. 99, 130401 (2007)\n[16] B. Sun and L. You, Phys. Rev. Lett. 99, 150402 (2007)5\n[17] M. Vengalattore, S. R. Leslie, J. Guzman, and D. M.\nStamper-Kurn, Phys. Rev. Lett. 100, 170403 (2008)\n[18] Denschlag JH, Simsarian JE, Haffner H, et al. JOSA B,\n35, 3095 (2002)\n[19] B. Pasquiou, G. Bismut, E. Marechal, P. Pedri, L.\nVernac, O. Gorceix, and B. Laburthe-Tolra, Phys. Rev.\nLett.106, 015301 (2011)\n[20] Liang He and Su Yi, Phys. Rev. A 80, 033618 (2009)\n[21] M. Fattori, T. Koch, S. Goetz, A. Griesmaier, S. Hensler ,\nJ. Stuhler, T. Pfau, Nature Physics 2, 765 (2006)\n[22] L. Santos, M. Fattori, J. Stuhler, andT. Pfau, Phys. Rev .\nA75, 053606 (2007), Chicireanu R, Beaufils Q, Pouder-\nous A, et al. Eur. Phys. J D 45189 (2007)\n[23] To rule out an impact of the quadratic effect, we have\nperformed rf spectroscopy in the optical lattice. We could\nnot observe any quadratic shift of the rf resonance fre-\nquency, which sets an upper bound for the tensorial light\nshift of 300 Hz ×m2\nS, insufficient to account for the 1.2\nmG width of the depolarization curve in the lattice. At\nthis level, though, the quadratic effect may impact the\nnature of the ground state at low magnetic fields, see L.\nSantos, M. Fattori, J. Stuhler, and T. Pfau, Phys. Rev.\nA75, 053606 (2007).\n[24] see T. Swislocki et al., arXiv:1102.1566 (2011).\n[25] T. Isoshima, T. Ohmi and K. Machida, J. Phys. Soc.\nJpn.,69, 3864 (2000)\n[26] M. V. Simkin and E. G. D. Cohen, Phys. Rev.A 59, 1528\n(1999)\n[27] C. K. Law, H. Pu, and N. P. Bigelow, Phys. Rev. Lett.\n81, 5257 (1998)\n[28] Tin-Lun Ho and Sung Kit Yip, Phys. Rev. Lett. 84, 4031\n(2000)\nMETHODS\nMagnetic field control\nThe three components of the magnetic field along or-\nthogonaldirections of space are stabilized using an active\ncancellation of the AC and DC magnetic field fluctua-\ntions: three large, 3-turn rectangular coils, with a typ-\nical size of 1 .5×1.7 m, and with their center located\nabout 1 m away from the BEC, along the 3 spatial direc-\ntions, are used as compensation coils. The magnetic field\nclose to the experimental chamber is measured using a 3\naxis fluxgate magnetic sensor (Bartington Mag 03MC).\nThe sensor is located outside the experimental chamber,\nabout 15 cm away from the BEC. The three components\nof the magnetic field are compared to a computer con-\ntrolled set-value, with a resolution of 50 µG. The error\nsignal is sent to a feedback loop which is a simple propor-\ntional controller, and the current in each coil is regulated\nwith a 1 Amp push-pull type voltage-controlled current\namplifier.\nWithout active compensation, the characteristics of\nthe magnetic noise are the following: the main noise con-\ntribution comes from the 50 Hz AC noise of magnitudes,\nupto4mGpeak/peakinoneofthedirections. Thisnoise\noriginates from the different equipment located aroundthe experiment. The DC magnetic field fluctuates up\nto 2 mG during the day time. This is much larger than\nthe natural fluctuations of the earth magnetic field which\nare of the order of 100 to 200 µG. These fluctuations are\nmainly caused by activities in the laboratory (positions\nofmetallicobjects)andoutside thelaboratory(elevators,\ncars).\nWhen the active stabilization is switched on, the AC\nnoise is decreased below 100 µG peak/peak and the DC\nmagnetic field value is stabilized to better than 20 µG at\nthe position of the sensor. We have also measured the\nresidual magnetic field fluctuations with a second inde-\npendent sensor located 20 cm away from the first one.\nWe observe that the drifts of the DC magnetic field stay\nbelow 100 µG over 1 hour. At this position, though,\nthe typical AC fluctuations are 500 µG peak/peak. This\ncomes from the fact that some of the AC noise sources\nare not very far from the BEC (around 1 meter) so that\nthe AC noise is not perfectly spatially homogeneous on\na 20 cm scale. We have nevertheless checked by rf spec-\ntroscopy that this AC noise is efficiently screened by the\nmetallic experimental chamber, so that the residual fluc-\ntuations at the BEC position are given by the 100 µG\nDC fluctuations. The experimental spectrum shown in\nFig. 2 corroborates this estimate.\nStern and Gerlach procedures\nWe have implemented two different Stern-Gerlach pro-\ncedures to monitor the spin populations at low magnetic\nfields. For both procedures, a small (10 mG) magnetic\nfieldBois first adiabatically applied to the atoms, with a\nrisetimeof10ms. Weinsurethat Boislargeenoughthat\na magnetic field gradient can also then be applied to the\natoms to separate the Zeeman components adiabatically,\ni.e.without changing the spin composition.\nIn the first Stern-Gerlach procedure, the applied mag-\nnetic field gradient is small (0.25 G/cm), and the mag-\nnetic field felt by each Zeeman component remains al-\nways small. The Zeeman components then separate very\nslowly. To prevent them from falling and keep them in\nthe imagingfield of view, their horizontalmotion is chan-\nneled in one direction by the horizontal laser beam used\nfor producing and trapping the BEC, and only the ver-\ntical beam is removed during the procedure. An absorp-\ntion image is taken after 45 ms of 1D expansion in the\nmagnetic field gradient, right after the horizontal opti-\ncal trap is switched off. Due to the very small magnetic\nfield employed, absorptionimagingoffers agood absolute\ndetermination of the number of atoms in each Zeeman\ncomponent, as the Zeeman shifts are small compared to\nthe transition linewidth for absorption imaging. We ver-\nify this by the fact that the sum of the populations in all\nZeeman components is constant while the gas depolar-\nizes. Unfortunately, the momentum distribution of each6\nZeeman component is lost when using this procedure, be-\ncause the atoms have then moved out of the focus of the\nimaging system. This first procedure was used for all\nresults in this paper, except the ones shown in Fig. 4.\nA second Stern-Gerlach procedure was used for Fig. 4.\nWe apply a stronger magnetic field gradient (1 G/cm).\nThe Zeeman components separate sufficiently rapidly\nthat the optical trap can be released almost simultane-ously. The Zeeman components are separated in 5 ms,\nand this faster procedure also gives access to the momen-\ntum distribution of each Zeeman component, although\nthe actual determination of their relative atom number\nis not as good as the one in the first procedure, because\nof the presence of a relatively strong magnetic field (a\nfew G) when absorption imaging is performed."    },    {        "title": "2107.02878v1.Static_and_dynamic_magnetic_properties_of_K3CrO4.pdf",        "content": "Static and dynamic magnetic properties of K3CrO4 \nLiliia D. Kulish, * Graeme  R. Blake  \n*l.kulish@rug.nl  \nZernike Institute for Advanced Materials, University of Groningen, Nijenborgh 4, 9747 AG, Groningen, \nthe Netherlands  \nABSTRACT : We report on the magnetic properties of geometrically  frustrated K3CrO 4, in which Cr5+ cations \nare arranged on a distorted pyrochlore lattice . The crystal structure, static and dynamic magnetic properties of the \ncompound are investigated  in detail . A combination of DC and AC magnetic susceptibility  measurements  together with  \nthermoremanent magnetization decay measurement s reveal several magnetic transitions: the onset of  glassy canted \nantiferromagnetic order occurs  at 36  K, followed by the appearance of  ferromagnetic/ferrim agnetic  cluster glass \nbehavior  below  the freezing temperature  of 20 K. Further field-induced , temperature -dependent transitions  are observed \nin the range  3-10 K. The frequency dependence of the freezing temperature  for the cluster glass state is analy zed on the \nbasis of dynamic scaling laws including  the critical slowing down formula and the  Vogel -Fulcher law . \n \nKeywords : geometrical frustration ; cluster glass ; AC susceptibility ; thermoremanent magnetization ; dynamic scaling .    \n1. INTRODUCTION  \nGeometrical ly induced  magnetic frustration can promote the existence of various exotic magnetic states in  a single  \ncompound. A manifold of ground states that are similar  in energy  is often manifested by  spin ice  / liquid behavior as \nwell as by the emergence of exotic  magnetic structure s such as helical  / cycloidal spirals and periodic states with non -\ntrivial topologies composed of skyrmions and antiskyrmions  [1-3]. Magnetic skyrmions are of particular interest because \nthey can be controlled by extremely small electric  currents, hence there are promising perspectives for application in \nnovel spintronic and information storage devices  [4,5] . The material science of skyrmions in frustrated magnets is \nexpected to be highly rich: skyrmions and antiskyrmions can co -exist in one material, which gives a new dimension to \nlogic operations that can be performed with these objects. Furthermore, in frustrated magnets , the skyrmion helicity \nbecomes a new collective degree of freedom coupled to the skyrmion motion.  Additionally, frustrated magnets are  \nexpected to exhibit smaller skyrmions  (a few nm in size) compared to  chiral magnets  which comprise most of the known \nskyrmion host materials; this has the potential to lead  to higher information density  [1,3,6] . \nA main  challenge  in this f ield is thus to find frustrated magnets that host skyrmions. A promising compound from \nthis point of view is K3CrO 4. It is a 3d1 system  (S = ½)  with the unusual 5+  oxidation state of chromium. K 3CrO 4 has a \ncubic structure with the same chiral space group P213 as the well -studied  skyrmionic materials  MnSi  [7] and Cu 2OSeO 3 \n[8]. \nIn the literature,  different  synthesis routes  have been described  [9,10] , including a  recent investigation  in which  in \nsitu high-temperatu re X -ray diffraction  measurements were performed  during the reduction  of the Cr6+ compound \nK2CrO 4, which led to  the formation of different K xCrO y compounds including K 3CrO 4 [11]. \nIn addition,  it has been  shown that single crystals of  a β-K3CrO 4 polymorph can be grown from the original cubic  \npolymorph  when heated  at 180  ˚C for 1 week.  β-K3CrO 4 adopts  a tetragonal  structure with  space group I4̅2m and show s \nparamagnetic behavior  to below 5 K  [9]. The magnetic behavior of cubic K 3CrO 4 has not been reported.  \nExamples of  compounds containing Cr5+ ions are scarce  in the literature. There are perchromate compounds \nM3CrO 8 (M is an alkali metal cation ) with ferroelectric properties , where  Cr is coordinated in eight -fold fashion by  \nperox ide (O22-) ions in a dodecahedral configuration  [12]. A combined EPR and magnetic susceptibility analysis  \ndemonstrated  the dynamics of the electron spin exchange and antiferromagnetic exchange coupling in K 3CrO 8 single \ncrystals  [13]. The i nfluence of the Cr5+ ions on the magnetic propertie s of YbCrO 4 was investigated  by both bulk \nmagnetic measurements and 170Yb Mössbauer spectroscopy.  Antiferromagnetic coupl ing between the Yb3+ and Cr5+ \nsublattices  leads to  ferrimagnetic ordering below 25  K, driven by exchange within the chromium sublattice  [14]. \nHere , we explore the magnetic behavior  of K3CrO 4 for the first time . We utilize a simple and fast reduction reaction \nto obtain the pure sample . Powder X-ray diffraction  show s that the cubic polymorph is obtained  with space group P213. \nWe use  DC and AC magnetic susceptibility studies together with thermoremanent magnetization decay measurement s \nto reveal that  K3CrO 4 undergoes a series of  magnetic phase transitions with temperature . A paramagnetic  to glassy \ncanted anti ferromagnetic transition takes place at ~ 36 K, and the sample becomes  glassy in nature below a freezing \ntemperature  of 20  K. The frequency dispersion of the temperature -dependent AC susceptibility is described by dynamic \nscaling theory and the Vogel -Fulcher law, which identifies  it as a ferromagnetic/ferrimagnetic  cluster glass  state. In \naddition, further field-induced temperature -dependent transitions are detected at lower temperatures of 3 -10 K.  2. MATERIALS AND METHOD S \nA polycrystalline  K3CrO 4 sample was prepared by the reduction  of potassium chromate  (K2CrO 4) by heating in \nflowing hydrogen , based on the process earlier described by Liang et al. [15]. Pre-ground K 2CrO 4 was placed in an \nalumina boat crucible with an alumina cap , which was inserted  in a tube furnace . A purified gas  mixture  (H2 + Ar, 15% \n+ 85%) was then introduced at a constant flow rate of  200 ml/min into the tube. The temperature was raised to 400  ˚C \nat 10 ˚C/min , and then to 450  ˚C at 1 ˚C/min. The sample was held at 450 ˚C for 30  min, and  then the furnace was  \nnaturally cooled to ambient temperature.  The sample was immediately transferred to a nitrogen -filled glove -box and \nkept in an inert atmosphere with O 2 and H 2O concentrations of less than 10  ppm due to the extreme air -sensitivity of the \nobtained compound.  \nThe reduction process that occurs on heating w as confirmed by means of simultaneous thermogravimetric analysis \n(TG) and differential scanning calorimetry (DSC) on a TG 2960 SDT instrument using a  hydrogen -argon flow  (H2 / Ar, \n15% / 85%) of 100  mL/min; the h eating rate was 5 °C/min over the temperature range 30  °C to 1000 °C (Fig. S 1). The \nobtained data are in good agreement  with a previous report  on the reduction of K2CrO 4 [16]. \nThe phase purity and c rystal structure of the product  were determined by X -ray powder diffraction (XRD ) using  a \nBruker D8 Advance d iffractometer operating with Cu  Kα radiation  in the 2θ range 10 -70 ˚. A rotating glass capillary \n(diameter  0.5 mm) containing the sealed sample was used in transmission geometry. The XRD data were fitted by \nRietveld refinement using  the GSAS software  [17]. Magnetic measurements were performed on a Quantum Design \nMPMS SQUID -based magnetometer. Magnetic susceptibility scans were performed on warming over the range 2 -\n400 K, and magnetization versus applied field curves were obtained between -7 T and 7  T at 3 -60 K. AC susceptibility \nmeasurements were per formed using a 3.8  Oe oscillating field superimposed on different DC fields: 0, 200 and 400 Oe. \nThermoremanent magnetization decay experiments were performed by applying a 1  T field, cooling the sample to 10  K \nat 10 K/min, then cooling to 3 or 7  K at 2  K/min (below the glass freezing temperature). After 30  sec, the field was \nremoved and the magnetization was measured as a function of time.  \n3. RESULTS  AND DISCUSSION  \n3.1.  Structu ral characterization  \nStructural analysis of the K3CrO 4 sample  using powder XRD  show s a single -phase  product  for which the peak \npositions are consistent with the previously reported cubic  space group P213 [9]. The refined lattice parameter  is \na = 8.3158(10)  Å and the unit cell volume is  V = 575.05(22)  Å3 (the fitted XRD data are shown in  Fig. S2 , the atomic \ncoordinates are  listed  in Table S1 ). In this structure  chromium atoms occupy  a single  crystallographic  position  and are \ntetrahedrally coordinated by oxygen  (Fig. 1(a)). The CrO 4 tetrahedra are not directly connected to each other , thus  Cr-\nCr magnetic exchange interaction s take place via two O atoms. In addition, it should be mentioned that the chromium \natoms  when envisaged alone  form a slightly distorted pyrochlore structure , which is geometrically frustrated and thus \nhas possible consequences for the magnetic properties.  The general chemical formula of a pyrochlo re is A 2B2X7, where \nthe A and B cations form  corner -sharing , interpenetrating  [A4] and [B 4] tetrahedra.  In the case of K3CrO 4, Cr occupies \nboth the A and B sites  giving rise to the configuration shown in  Fig. 1(b). \n \nFig. 1. (a) Crystal structure of K3CrO 4. The potassium, chromium , and oxygen atoms are represented by white, dark \ngreen, and red spheres respectively. The unit cell is outlined . (b) Schematic representation of the Cr-sublattice , which \nforms  a slightly distorted pyrochlore structure.  \n3.2.  DC magnetic susceptibility  \nThe magnetic properties of K3CrO 4 were  first investigated by performing  DC magnetization  versus temperature  \nmeasurements. Ze ro-field-cooled (ZFC) and field -cooled (FC) measurements were performed in applied magnetic field s \nof 200 Oe and 10  kOe on warming over the temperature range  2-400 K. A bifurcation  of the FC and ZFC curves  below \na charac teristic temperature ( Tirr = 38 K),  with a well -defined peak  in the ZFC branch ( Tg = 30 K) , is observed under  \n200 Oe DC field (Fig. 2(a)). Such splitting  shows a dependence on the thermal -magnetic history of the sample and can \narise from a variety of  phenomena such as glassy , spin  ice / liquid, superparamagnet ic, disordered antiferromagnet ic, \nand spin-spiral  states [18,19] . The FC magnetization shows a continuous increase with decreasing  temperature. At the \nsame time the ZFC magnetization crosses zero and becomes  negative below 14  K. Negative magnetization can appear \nin complex ferrimagnetic or canted antiferromagnetic systems [ 20]. There is  also a maximum  at 3 K in both the ZFC \nand FC curves  (Fig. 2(a), inset ; Fig. S3 ). However, measurement in a higher  applied DC field of 10 kOe leads to different \nmagnetic behavior of the sample below 60  K. Both ZFC and FC curves exhibit  a continuous increase with decreasing  \ntemperature and there is only a  tiny degree of ZFC-FC splitting below  8 K. No sign of the magnetic transition at 30  K \nis observed;  there are only two broad maxim a at 3 K and 38  K (Fig. S4 ).  \nThe inverse susceptibility of the sample  (Fig. 2(b)) has a linear part only in  10 kOe of applied DC field  and only \nin FC mode  (Fig. S5 ). The se data were  fitted according to the Curie -Weiss law, 𝜒𝑚𝑜𝑙 =𝐶/(𝑇−𝜃)+𝜒0, where C is \nthe Curie constant and θ is the Weiss constant. The temperature -independent term χ0 consists of the sum of the \ndiamagnetic contribution s of the core electrons  χdia (Cr5++3K++4O2-) [21] and the van Vleck paramagnetic contribution \nof the  Cr5+ ion [22,23]. Fitting was performed above 250  K; the deviation from linearity at lower temperature is most \nlikely due to short -range interactions. The extracted negative Weiss constant , θ = -172 K, implies  predominant \nantiferromagnetic interactions . From the extracted  Curie constant of 0. 796 emu· K·mol-1, an effective moment  μeff of \n2.52 µB per Cr atom i s determined using the  formula 𝐶=𝜇𝑒𝑓𝑓28⁄. The t heoretical spin-only μeff of Cr5+ is expected to \nbe 1.73 µB (μeff (Cr3+) = 3.87 µB; μeff (Cr2+) = 4.90 µB) [18]. \n \nFig. 2. (a) Temperature dependence of ZFC (open symbols) and FC (solid symbols) DC magnetic susceptibility of \nK3CrO 4 measured on warming in a field of 2 00 Oe. The inset shows the DC susceptibility of the samples in the 2 -40 K \nrange. ( b) FC inverse DC susceptibility of K3CrO 4 as a function of temperature measured under  10 kOe. The line  is a \nlinear fit to the experimental data above 250 K using the Curie -Weiss law. (c)  Magnetization vs applied DC field  curves  \nat 3-10 K  for K3CrO 4. The in sets show closer views of the low -field region.  (d) Magnetization decay as a function of \ntime measured after cooling the sample  under a 1  T field to  3 or 7  K and then removing  the field . The curves are fits to \nthe experimental data using the stretched exponential function (Eq. ( 3)).  \nThe possible presence of magnetic frustration can be inferred from the frustration parameter 𝑓=|𝜃𝐶𝑊|𝑇𝐶⁄ [24]. \nAlthough this parameter is strictly speaking only valid for a long -range -ordered state below a c ritical temperature TC, \nwhich does not seem to be achieved in the case of K 3CrO 4 (see discussion of AC susceptibility data below), it can \nnevertheless provide a useful indicator of frustration if TC is replaced by Tg. For K 3CrO 4 |θCW| is 4.8 times greater than \nTg (36 K) implying a moderate level of frustration.  \n3.3.  AC magnetic susceptibility  \nTo further investigate the origin of the features  observed in the ZFC curves (Fig. 2(a)), the temperature depe ndence \nof the AC susceptibility  χAC was measured over the temperature range 8-50 K at s even different frequencies: 10, 50, \n100, 250, 500, 750, 1000  Hz and in different  applied DC field s: zero, 200 or 400 Oe  (Fig. 3 ). AC measurements in the \nlower temperature range of  2.5-8 K (not shown)  do not show any peaks in either the  real or imaginary parts.   \nIn the case of zero applied DC field ( Fig. 3(a)), both the  real component χ'(T) (reversible magnetization process es) \nand the imaginary part  χ''(T) (losses due to irreversible processes ) [25] exhibit  frequency -dependent relaxation  (marked \nas 1 in Fig. 3( a)). There is a single peak in  χ'(T) which both decreases in height and shifts to higher tempera ture with \nincreasing frequency . The corresponding peak in  χ''(T) increases in height and shifts to higher  temperature with \nincreasing frequency.  The temperature of this maximum  (36 K at 10 Hz for χ'(T)), which we refer to as  Tf, is ~6 K higher  \nthan Tg at which the  peak in the  ZFC DC susceptibility  is observed.  Moreover,  Tf approximately corresponds to  the \ntemperature at which the inflection point  is observed in the  imaginary part χ''(T) . The peaks in χ'(T) and χ''(T)  are both \nasymmetric, with a n inflection point in χ'(T) at ~24  K. \nThe superposition  of a DC fie ld during the AC measurement leads to the appearance of another peak at a lower \ntemperature for both the real and the imaginary parts ( labelled as 2 in Fig. 3(b, c)), which might have developed from \nthe shoulder in χ'(T) in Fig. 3(a).  This transition at ~  20 K is dynamic  whereas  peak 1 at 38 K now has no frequency  \ndependen ce.  \nPeak 2 reflects  field-induced magnetic relaxation, which  is greatly enhanced with increasing  applied DC field  (Fig. \n3). In addition, peak  2 shifts to lower temperature while  peak 1 shifts to higher temperature with increasing  DC field. \nFurthermore, the AC susceptibility of χ'(T) and χ''(T)  for both transitions decreases  rapidly with increasing  DC field \n(Fig. S6 ).  \n \nFig. 3. Temperature dependence of the real and imaginary parts of the AC susceptibility in the temperature range 8 -\n50 K at 10-1000 Hz frequenc y, measured using a 3.8 Oe oscillating field and different applied DC field s: (a) without \nDC field; (b)  200 Oe DC field; (c)  400 Oe DC field.   \nAs mentioned above , the Cr5+ ions are arranged on  a distorted pyrochlore lattice . Therefore, it is possible that spin \nice behavior ty pical for pyrochlores is exhibited by K3CrO 4. Spin ices demonstrate a slowing down of the spin dynamics  \non cooling  in similar fashion to  spin-glass compounds , with a low -temperature peak in the DC susceptibility \naccompanied by a divergence between the ZFC and FC curves  [19]. However, there is  a striking difference between \nspin-glass and spin -ice freezing  with respect to  the distribution of relaxation times , which can be modelled  at a given \ntemperature using Cole -Cole analysis  [26]. The χ' and χ'' data for peak 1 of the AC susceptibility measurement without \nextra DC field , as well as the data for peak 2 with 200  Oe of DC field , were fitted within the Cole -Cole formalism, given \nby equation  [19]: \n \n𝜒′′(𝜒′)=−𝜒𝑇−𝜒𝑆\n2tan[(1−𝛼)𝜋\n2]+√(𝜒′−𝜒𝑆)(𝜒𝑇−𝜒′)+(𝜒𝑇−𝜒𝑆)2\n4𝑡𝑎𝑛2[(1−𝛼)𝜋\n2]   (1) \n \nHere χT is the isothermal susceptibility and χS is the  adiabatic susceptibility . The parameter α represents  the width \nof the distribution of relaxation times, where for a single spin relaxation  time α = 0. The values of  α obtained for peak  \n2 (α ≈ 0.7, 0 Oe DC field , Fig. S7 ) and peak  1 (α ≈ 0.56, 200 Oe DC , Fig. S8 ) lie in the expected range for glass -like \nbehavior, where the relaxation time typically  covers a broad range  of several orders  of magnitude  [19,2 7]. In contrast , \nspin ices exhibit either a  single spin relaxation or a narrow range of relaxation times with an extremely low value of α \n≈ 0.001  [28].  \nIn addition, a  quantitative measure of the frequency  dependence of  the maximum in χ'(T) can be  estimate d by the \nMydosh parameter  δ [29]:  \n \n𝛿=∆𝑇𝑓\n𝑇𝑓×∆(𝑙𝑛(𝜔))   (2) \n \nHere Tf is the freezing temperature, the frequenc y is ω = 2πf, and ∆Tf is the difference between the maximum and \nminimum values of Tf. The value of  δ distinguishes  a spin-glass state  [19,30-32] (0.00 1 < δ < 0.08) from  a non-\ninteracting ideal superparamagnet  with much larger δ values [33,34]. The values of the Mydosh parameter  for K 3CrO 4 \n(δ = 0.006, 0 Oe DC field; δ = 0.017, 200-400 Oe DC field) correspo nd to the intermediate situation  of a cluster glass \n(CG), also known  as a reentrant spin glass , for which  δ ~ 0.01-0.09 is expected  [35-38]. \nThese results suggest that the maxim a in χAC are associated with randomly arranged , interacting magnetic clusters \nwhich become frozen below  certain temperatures. The ir origin most probably relates to the relaxation of ferromagnetic \nor ferr imagnetic clusters, but not antiferromagnetic  clusters  because in all cases χ''(T)  ≠ 0 [25]. Moreover, it seems that \nthere is a paramagnetic to cluster glass transition at ~36  K from  the AC measurement without extra DC field, but with  \napplication of  a small additional DC field that transition is to a  ferromagnetic  or glassy canted antiferromagnetic state \nand is followed at ~20  K by another transition  to a cluster glass state. For such spin glass -like freezing occurring below \na ferromagnetic transition it has previously been  shown that t he DC magnetization first increase s with lowering \ntemperature due to the appearance of the ferroma gnetic state, but then decrease s at even lower temperatures due to \nfreezing  [39,40]. \n3.4.  Magneti zation versus applied field  \nThe nature of the magnetic states in K3CrO 4 was next probed by means of  magnetization (M) versus applied field \n(H) curve s. The M-H curve s exhibit an “S” shape at all temperatures below 60  K (Fig. 2(c); Fig. S9 ), indicating an \nuncompensated magnetic moment . However, t he magnetization does not reach saturation  (expected 1  μB / Cr5+ cation)  \nup to the highest applied field of 7  T at any temp erature , which exclud es a fully ferromagnetic state . There is  a hysteresis \nloop below 20  K (Fig. 2(c); Fig. S9 ). Glassy compounds can show these features: the absence of magnetic saturation \nand weak hysteresis in the frozen state due to competing ferromagnetic and antiferromagnetic exchange  interactions  \n[35,37,38]. Furthermore,  the M -H curves at 3 -4 K exhibit  a double pinched hysteresis loop  (a closer view is shown in \nthe left -hand panel of  Fig. 4(a)). This might be  associated with the magnetic transition that gives  a maximum at 3 K in \nboth the ZFC and FC DC magnetization measurements ( Fig. 2(a)). \nMoreover, signatures of  field-induced transitions  are observed in  the temperature  range 3 -10 K (Fig. 4(a)). These \ntransitions are temperature -dependent.  The small anomaly  at ±3.5  kOe at 3  K (marked with red triangles in the d M/dH \nplot of  Fig. 4(a)) shifts to higher magnetic field with increasing temperature  (±4.75  kOe at 4  K; ±6  kOe at 6  K). The \nmain peak i n the d M/dH plot at ±4  kOe at 3  K (marked with blue triangles in Fig. 4(a)) both shifts to lower field and \nbecomes narrow er with increasing temperature (±3.25 at 4  K; ±2  kOe at 6  K; ±1  kOe at 10  K). \n3.5.  Decay  of thermoremanent magnetization  \nTo investigate the mechanism by which the system decays back to equilibrium  after an external magnetic field is \napplied , time-dependen t thermoremanent magnetization measurements were carried out  at 3 and 7  K (Fig. 2(d)). At \nhigher temperatures, no significant relaxation was observed ; the magnetization decayed immediately to zero on  \nmeasure ment  time-scale s. The data collected at 3 and 7 K do not follow a power -law decay  or a simple logarithmic \ndependence . However , the curves  can be fitted well with a stretched exponential function :  \n \n𝑀(𝑡)=𝑀0+𝑀r 𝑒𝑥𝑝[−𝑡\n𝜏1−𝑛\n]   (3)  \nThis law has widely been used to describe the magnetic relaxation in different glass y systems  [41,42]. Here M0 is \nthe maximum magnetization at the start of the measurement  and other components are relate d to the observed relaxation \neffect , where  Mr is a glassy contribution; the time constant τ and parameter n are associated  with the relaxation rate . The \nparameter n = 0 corresponds to a single time -constant and there is no relaxation at n = 1. The fitted curve s match the \ndata well, with the following extracted  parameters  at 3 K: M0 = 49.61  emu/mol , Mr = 18.67  emu/mol, τ = 1776.12  s, \nn = 0.33. The corresponding parameters at 7 K are as follows : M0 = 15.04  emu/mol, Mr = 5.65 emu/mol, τ = 1508.85  s, \nn = 0.57.  \n3.6.  Dynamic scaling  \nMore detailed insight into the dynamics of the two temperature -induced magnetic transitions in K3CrO 4 can be \nobtained by further analysis of  the χAC measurements. The frequency dependen ce of χ'(T) can be described by the critical \nslowing down formula  [43] from dynamic scaling theory:  \n \n𝜏=𝜏0×[𝑇𝑓−𝑇𝑔\n𝑇𝑔]−𝑧𝜈\n   (4) \n \nHere Tf  is taken as the position of peak 2 at ~ 20 K (for measurements with 200 -400 Oe applied DC field) and \nfrom the position of peak  1 at ~ 36 K (for the measurement with zero DC field) in χ'(T) for a given frequency f. Tg is the \ntemperature at which the maximum in the ZFC DC susceptibility is observed  (29.7  K for 0  Oe DC field; 20  K for 200  Oe \nand 18  K for 400  Oe), because Tg can be regarded as the  value o f Tf for infinitely slow cooling (lim f→0 Tf) [35]. The \ncharacteristic relaxation time  of the dynamic fluctuations τ corresponds to the observation time  tobs = 1/ω = 1/(2πf) with \nthe attempt frequency  ω, and the shortest time  τ0 correspond s to the microscopic flipping t ime of the fluctuating entities. \nAccording to dynamic scaling theory, τ is related to the spin correlation length , τ ∝ ξz, and ξ diverges with temperature \nas ξ ∝ [Tf /(Tf −Tg)]ν with the dynamic exponent z and the critical exponent  ν [44].  \n \nFig. 4. (a) Magnetization vs ap plied DC field measurement at 3  K and field dependence of the susceptibility dM/dH \ncalculated  from M-H curves  at 3-10 K. The colors indicate the sequence in which data were collected (black,  red, blue, \ngreen, orange).  Triangles with dotted lines in the d M/dH plots show the shift s of field-induced  magnetic transitions at \ndifferent temperatures.  (b) Fits to maxima  in χ'(T) for different frequencies for peak 2 (measurements in 200 / 400 Oe \nDC field)  and peak  1 (measurement  in zero  DC field) , using the slowing down formula (Eq. ( 4)) and the Vogel -Fulcher \nlaw (Eq. ( 5)) for K3CrO 4. \nThe left part of Fig. 4(b) shows a linear fit of lnτ vs ln((T f-Tg)/Tg), allowing values of zν and τ0 to be obtained. These \nparameters are given in Table I. The zν values for peak  2 at 200 -400 Oe D C field lie in the characteristic range for glassy \nmagnetism 5  < zν < 13 [45-47]. Typical values of τ0 for a canonical SG  [47] lie in the range of ~ 10-12–10-14 s; for cluster \nglasses  [35,41,48]  with slower dynamics, τ0 is in the range ~ 10-9–10-11 s. The characteristic relaxation time of peak 2 in \n200-400 Oe D C field is of the order of  ~  10-9. These parameters strongly suggest that a magnetic cluster glass transition \noccurs at the temperature of  peak 2 when an additional  DC field  is applied . Peak 1 at ~ 36 K in 0 Oe DC field has \ndynamics that are too fast even for a canonical SG and the zν value is also too high . \nThe dynamic magnetic properties of a glassy system can also be described by  the Vogel -Fulcher law  [35,49], \nproposed for magnetically interacting clusters:  \n \n𝜏=𝜏∗×𝑒𝑥𝑝 [𝐸𝑎\n𝑘𝐵×(𝑇𝑓− 𝑇0)]   (5) \n \nHere T0 is a measure of  the inter-cluster interaction strength, and T0 is known as  the Vogel -Fulcher temperature  \n[29] and correspond s to the “ideal glass ” temperature. Close to T0, the Vogel -Fulcher law can be adjusted to match the \npower -law over a large frequency range  [47]: 𝑛⌊(40 𝑘𝐵𝑇𝑓)/𝐸𝑎⌋~ 25/𝑧𝜈 . This equation gives Ea/kB as listed  in Table I. \nThese  values allow the data to be fitted using  Eq. (5) (right -hand  panels  of Fig. 4(b)), yield ing the parameters τ*, T0 \n(Table  I). The extracted  τ* values  for peak  2 in 200-400 Oe D C field lie in the range of ~10-7–10-13 s anticipated for \nglassy bulk systems  [29,35,50]. The o btained τ* for peak  1 in zero DC field indicate s that the dynamics are  too fast , with \na probably unphysical value , similar to the analysis using  the critical slowing down formula.   \nTable I. Dynamic magnetic properties of K3CrO 4. \n Slowing down formula  Vogel -Fulcher law  \nτ0 (s) zν Ea/kB (K) τ* (s) T0 (K) \nPeak 2, 200  Oe  7.4×10-9 5.7 11 5.6×10-6 20.3 \nPeak 2, 400  Oe 1.1×10-9 6.9 22 4.37×10-7 17.6 \nPeak 1, 0 Oe 1.3×10-22 28 593 5.12×10-26 24.5 \n \n4. CONCLUSIONS  \nWe have synthesized phase -pure K3CrO 4 and for the first time investigated its magnetic properties. A Curie -Weiss \nfit to the DC magnetic  susceptibility  show s that antiferromagnetic  interactions dominate , although the S -shaped M -H \ncurves indicate a small uncompensated magnetic moment, likely from a canting of the spins . A bifurcation of the ZFC -\nFC magnetic susceptibility occurs below 38  K with a distinct peak in the ZFC branc h (30 K) that suggest s the presence \nof magnetic irreversibility. In addition, another magneti c transition  is observed with a maximum at 3  K for  the ZFC and \nFC branches .  \nAC magnetic susceptibility measurements indicate the presence of  two magnetic transitions. Moreover, depending \non whether an additional  DC field  is applied , the position s of the peaks associated with  both magnetic transitions become \nfrequency -dependent for both  the real and the imaginary components . The peak at ~  36 K seemingly corresponds to a \nparamagnetic  - glassy canted antiferromagnetic  transition, whereas the  peak at ~ 20 K likely originates from the \nformation of a ferromagnetic/ferrimagnetic cluster glass state.  Thermoremanent magnetization decay measurements  \nconfirm  that a cluster glass state forms below the glass freezing temperature  (20 K). Further evidence for a cluster glass \nstate is provided  by fits of the AC susceptibility data to  both the standard critical slowing -down formula and the Vogel -\nFulcher law.   \nIn addition, indications of field-induced temperature -dependent transitions at lower temperatures (3-10 K) are \nfound from  the magnetic field dependence of the susceptibility calculated from the M-H curves . Further investigation \nof the nature of the magnetic transitions in  K3CrO 4 will require mapping of the magnetic structure as a function of \ntemperature and applied field , for example from  neutron diffraction measurements . \n AUTHOR  CONTRIBUTIONS  \nLiliia D. Kulish: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Data Curation, \nWriting - Original Draft, Visualization, Project administration. Graeme R. B lake: Conceptualization, Methodology, \nValidation, Resources, Data Curation, Writing - Review & Editing, Supervision, Project administration.  \n \nACKNOWLEDGMENTS  \nThis work was supported by the European Union’s Horizon 2020 research and innovation program under  Marie \nSklodowska -Curie  Individual Fellowship,  grant agreement no. 833550.  We would like to thank Prof. Maxim Mostovoy \nand Joshua Levinsky for valuable discussion during the development of this research project as well as Ing. Jacob Baas \nfor technical supp ort. \n \nREFERENCES  \n[1] T. Okubo, S. Chung, H. Kawamura,  Multiple -q states and the skyrmion lat tice of the triangular -lattice H eisenberg \nantiferromagnet under magnetic fields , Phys. Rev. Lett. 108  (2012) 017206.  \nhttps://doi.org/10.1103/PhysRevLett.108.017206 .  \n[2] S. Hayami, S -Z. Lin, C.  D. Batista, Bubble and skyrmion crystals in frustrated magnets with easy-axis anisotropy , \nPhys. Rev. B 93 (2016)  184413. https://doi.org/10.1103/PhysRevB.93.184413 .  \n[3] A. O. Leonov, M. Mostovoy, Multiply periodic states and isolated skyrmions in an anisotropic fr ustrated magnet , \nNat. Commun. 6  (2015) 8275. https://doi.org/10.1038/ncomms9275 .  \n[4] S. Seki. M. Mochizuki, Skyrmions in Magnetic Materials, SpringerBriefs in Physics, 2015.  \n[5] M. Garst, Topological Stru ctures in Ferroic Materials, Springer , 2016.  \n[6] J. Zang, M . Mostovoy, J . H. Han, N . Nagaosa , Dynamics of skyrmion crystals in metallic thin films , Phys. Rev. Lett. \n107 (2011) 136804.  https://doi.org/10.1103/PhysRevLett.107.136804 .   \n[7] A. Bauer , C. Pfleiderer, Magnetic phase diagram of MnSi inferred from magnetization and ac susceptibility , Phys. \nRev. B 85  (2012) 214418.  https ://doi.org/10.1103/PhysRevB.85.214418 .  \n[8] F. Qian, H. Wilhelm, A. Aqeel, T. T. M. Palstra, A. J. E. Lefering, E. H. Brück, C. Pappas, Phase diagram and \nmagnetic relaxation phenomena in Cu 2OSeO 3, Phys. Rev. B 94  (2016) 064418.  \nhttps://doi.org/10.1103/PhysRevB.94.064418 .  \n[9] R. Olazcuaga, J. -M. Reau, G. LeFlem, P. Hacenmuller,  Préparation, proprietés cristallographiques et magnétiques \ndes p hases K 3XO 4 (X=V, Cr, Mn)  Z. anor g. allg. Chem. 412 (1975) 271 -280. \nhttps://doi.org/10.1002/zaac.19754120311 .  \n[10] L. H. Johnson, L. G. Hepler, C. E. Bamberger, D. M. Richardson, The enthaply of formation of potassium \nchromate(V), K3CrO 4 (c), Can. J. Chem. 56  (1978)  446-449. https://doi.org/10.1139/v78 -071.  \n[11] S.-t. Liang, H. -l. Zhang, M. -t. Luo, Y. -l. Bai, H. -b. Xu, Y. Zhang, In situ high -temperature X -ray diffraction studies \nof reduction of K 2CrO 4 and the formation of K xCrO y compounds , Powder Diffr., 32 (3)  (2017)  168-174. \nhttps://doi.org/10.1017/S0885715617000690 .  \n[12] R. Samantaray, R. J. Clark, E. S. Choi, H. Zhou, N. S. Dalal, M3−x(NH 4)xCrO 8 (M = Na, K, Rb, Cs): a new family \nof Cr5+-based magnetic ferroelectrics , J. Am. Chem. Soc. 133  (11) (2011) 3792 –3795.  \nhttps://doi.org/10.1021/ja1117683 .  \n[13] N. S. Dalal, J. M. Millar, M. S. Jagadeesh, M. S. Seehra, Paramagnetic resonance, magnetic susceptibility, and \nantiferromagnetic exchange in a Cr5+ paramagnet: p otassium perchromate (K 3CrO 8), J. Chem. Phys. 74 (1981)  1916. \nhttps://doi.org/10.1063/1.441284 .  \n[14] E. Jimenez, P. Bonville, J.  A. Hodges, P. C. M. Gubbens, J. Isasi, R. Saez -Puche,  Influence of the Cr5+ ion on the \nmagnetic properties of  YbCrO 4 oxide , J. Magn. Magn. Mater.  272–276 (1) (2004)  571–573. \nhttps://doi.org/10.1016/j.jmmm.2003.11.216 .  \n[15] S.-t. Liang, H. -l. Zhang, H. -b. Xu, Preparation of hexagonal and amorphous chromium oxyhydroxides by  facile \nhydrolysis of K xCrO y, Trans. Nonferrous Met. Soc. China 30 (2020)  1397−14053 . https://doi.org/10.1016/S1003 -\n6326(20)65305 -5.  \n[16] N. Naveen, S. C. Han, S. P. Singh, D. Ahn, K. -S. Sohn, M. Pyo,  Highly stable P′3 -K0.8CrO 2 cathode with limited \ndimensional changes for potassium ion batteries , J. Power Sources 430 (2019)  137–144. \nhttps://doi.org/10.1016/j.jpowsour.2019.05.017 .  \n[17] B. H. Toby , R. B. Von Dreele,  GSAS-II: the genesis of a modern open -source all purpose crystallography software \npackage , J. Appl. Cryst. 46  (2013)  544-549. https://doi.org/10.1107/S0021889813003531 .  \n[18] N. A. Spaldin, Magnetic Materials Fundamentals and Applications , Cambridge: Cambridge University Press, 2011.  [19] C. V. Topping , S. J. Blundell, A.C. susceptibility as a probe of low -frequency magnetic dynamics , J. Phys.: \nCondens. Matter 31  (2019)  013001 . https://doi.org/10.1088/1361 -648X/aaed96 .  \n[20] A. Kumar , S. M. Yusuf , The phenomenon of negative magnetization and its implications , Phys. Rep.  556 (2015) \n1-34, https://doi.org/10.1016/j.physrep.2014.10.003 . \n[21] G. A. Bain,  J. F. Berry , Diamagnetic corrections and Pascal's c onstants , J. Chem. Educ. 85 (4) (2008)  532-536. \nhttps://doi.org/10.1021/ed085p532 .  \n[22] T. A. Hewston , B. L. Chamberland, Magnetic and structural studies of Cr 2O5 and Cr 3O8, J. Magn. Magn. Mater. 43 \n(1984)  89-95. https://doi.org/10.1016/0304 -8853(84)90278 -6.  \n[23] J. Benkowitsch , H. Winter, Calculation of the orbital susceptibility of chromium , Physica Scripta , 31 (3) (1985)  \n222-224. https://doi.org/10.1088/0031 -8949/31/3/015 .  \n[24] A. P. Ramirez , Strongly geometrically frustrated magnets , Annu. Rev. Mater. Sci. 24 ( 1994 ) 453-480. \nhttps://doi.org/10.1146/annurev.ms.24.080194.002321 .  \n[25] M. Balanda, AC susceptibility studies of phase transitions  and magnetic relaxation:  conventional, molecular and \nlow-dimensional magnets , Acta Phys. Pol. A 124  (6) (2013) 964 -976. https://doi.org/10.12693/APhysPolA.124.964 .  \n[26] K. S. Cole, R. H. Cole, Dispersion and absorption in dielectrics I. alternating current characteristics , J. Chem. Phys. \n9 (1941)  341-351. https://doi.org/10.1063/1.1750906 .  \n[27] C. Dekker, A. F. M. Arts, H. W. de Wijn, A. J. van Duyneveldt, J. A. Mydosh, Activated dynamics in a two -\ndimensional Ising spin glass: Rb 2Cu1−xCoxF4, Phys. Rev. B 40  (1989) 11243 . \nhttps://doi.org/10.1103/PhysRevB.40.11243 .  \n[28] Sheetal, A. Ali, S. Rajput, Y. Singh, T. Maitra, C. S. Yadav,  Emergence of weak pyrochlore phase and signature \nof field induced spin ice groun d state in Dy 2−xLaxZr2O7; x = 0, 0.15, 0.3 , J. Phys.: Condens. Matter 32 (2020)  365804 . \nhttps://doi.org/10.1088/1361 -648X/ab8bf6 .  \n[29] J. A. Mydosh, Spin glasses: an experimental introduction , London: Taylor & Francis, 1993.  \n[30] R. Mahendiran, Y. Br éard, M. Hervieu, B. Raveau, P. Schiffer , Giant frequency dependence of dynamic freezing \nin nanocrystalline ferromagnetic LaCo 0.5Mn 0.5O3, Phys. Rev. B 68 (2003)  104402 . \nhttps://doi.org/10.1103/PhysRevB.68.104402 .  \n[31] J. A. Mydosh , Spin glasses: redux: an updated experimental/materials survey , Rep. Prog. Phys. 78  (5) (2015)  \n052501 . https://doi.org/10.1088/0034 -4885/78/5/052501 .  \n[32] S. Kundu, T. Dey, A. V. Mahajan, N. Büttgen, LiZn 2V3O8: a new geometrically frustrated cluster spin -glass , J. \nPhys.: Condens. Matter 32 (2020) 115601.  https://doi.org/10.1088/1361 -648X/ab58dc .  \n[33] L. Néel, Théorie du traînage magnétique des ferromagnétiques en grains fins avec application aux terres  cuites,  \nAnn. Géophys. 5  (1949)  99-136. \n[34] W. F. Brown, Thermal fluctuations of a single -domain particle , Phys. Rev. 130  (1963)  1677 -1686. \nhttps://doi.org/10.1103/PhysRev.130.1677 .   \n[35] A. Malinowski, V. L. Bezusyy, R. Minikayev, P. Dz iawa, Y. Syryanyy, M. Sawicki, Spin-glass behavior in Ni -\ndoped La 1.85Sr0.15CuO 4, Phys. Rev. B 84 (2011)  024409 . https://doi.org/10.1103/PhysRevB.84.024409 .  \n[36] U. Dutta, D. Ghosh, A. Haque, L. Kumar, T. K. Mandal, P. S. Walke, K. Pal, A. Gayen, A. K. Kundu, M. M. Seikh, \nA revisit to the effect of annealing temperature  on magnetic properties of LaFe 0.5Mn 0.5O3, J. Phys.: Condens. Matter \n31 (2019)  225801 . https://doi.org/10.1088/1361 -648X/ab0b98 .  \n[37] S. Pakhira, C. Mazumdar, M. Avdeev, R. N. Bhowmik, R. Ranganathan,  Spatially limited  antiferromagnetic order \nin a  cluster glass compound  Tb2Ni0.90Si2.94, J. Al loys Compd. 785  (2019)  72-79. \nhttps://doi.org/10.1016/j.jallcom.2019.01.123 .  \n[38] L. D. Kulish, R. Scholtens,  G. R. Blake,  Cluster glass behavior of the frustrated birnessites A xMnO 2·yH 2O (A = \nNa, K) , Phys. Rev. B 100 (2019) 214435 . https://doi.org/10.1103/PhysRevB.100.214435 .  \n[39] B. Aslibeiki, P. Kameli, H. Salamati, Reentrant spin glass behavior in La 0.8Sr0.2Mn 1−xTixO3 manganites , Solid State \nCommun. 149  (31-32) (2009)  1274 -1277. https://doi.org/10.1016/j.ssc.2009.05.012 .  \n[40] S. Abiko, S. Niidera, F. Matsubara, Reentrant spin -glass transition in a dilute magnet , Phys. Rev. Lett. 94  (2005)  \n227202 . https://doi.org/10.1103/PhysRevLett.94.227202 .  \n[41] C. A. Cardoso, F. M. Araujo -Moreira, V. P. S. Awana, E. Takayama -Muromachi, O. F. de Lima, H. Yamauchi, M. \nKarppinen, Spin glass behavior in RuSr 2Gd 1.5Ce0.5Cu2O10-δ, Phys. Rev. B  67 (2003)  020407(R) . \nhttps://doi.org/10.1103/PhysRevB.67.020407 .  \n[42] M. D. Mukadam, S. M. Yusuf, P.  Sharma, S. K. Kulshr eshtha,  G. K. Dey, Dynamics of spin clusers in amorphous \nFe2O3, Phys. Rev. B  72 (2005)  174408 . https://doi.org/10.1103/PhysRevB.72.174408 .  \n[43] P. C. Hohenberg , B. I. Halperin, Theory of dynamic critical phenomena , Rev. Mod. Phys. 49 (3) (1977)  435-479. \nhttps://doi.org/10.1103/RevModPhys.49.435 .  \n[44] A. T. Ogielski,  Dynamics of three -dimensional Ising spin glasses in thermal equilibrium , Phys. Rev. B  32 (1985)  \n7384 -7398. https://doi.org/10.1103/PhysRevB.32.7384 .  \n[45] C. Djurberg, P. Svedlindh, P. Nordblad, M. F. Hansen, F. Bodker, S. Morup,  Dynamics of an interacting particle \nsysem:  evidence of  critical slowing down , Phys. Rev. Lett. 79  (1997)  5154 -5157. \nhttps://doi.org/10.1103/PhysRevLett.79.5154 .  [46] K. Gunnarsson, P. Svedlindh, P. Nord blad, L. Lundgren, H. Aruga, A. Ito , Dynamics of an Ising spin-glass in the \nvicinity of the spin -glass temperature , Phys. Rev. Lett. 61  (1988)  754-757. \nhttps://doi.org/10.1103/PhysRevLett.61.754 .  \n[47] J. Souletie, J. L. Tholence, Critical slowing down in spin glasses and other glasses: Fulcher versus power law , Phys. \nRev. B 32 (1985)  516-519. https://doi.org/10.1103/PhysRevB.32.516 .  \n[48] R. Mathieu, A. Asamitsu, Y. Kaneko, J.  P. He, Y. Tokura, Eu0.5Sr1.5MnO 4: a three -dimensional XY spin glass , \nPhys. Rev. B 72 (2005)  014436 . https://doi.org/10.1103/PhysRevB.72.014436 .  \n[49] J. L. Tholence, On the frequency dependence of  the transition temperature in spin glasses , Solid State Commun . \n35 (2) (1980)  113-117. https://doi.org/10.1016/0038 -1098(80)90225 -2.  \n[50] M. H . Ehsani, P . Kameli, M. E . Ghazi, F. S. Razavi, An investigation on magnetic interacting La0.6Sr0.4MnO 3 \nnanoparticles , Adv. Mat. Res. 829  (2014)  712-716. https://doi.org/10.4028/www.scientific.net/AMR.829.712 . "    },    {        "title": "1411.2493v1.Induced_magnetic_moment_in_the_magnetic_catalysis_of_chiral_symmetry_breaking.pdf",        "content": "Nuclear Physics B Proceedings Supplement 00 (2022) 1–3Nuclear Physics B\nProceedings\nSupplement\nInduced magnetic moment in the magnetic catalysis of chiral symmetry breaking\nEfrain J. Ferrer and Vivian de la Incera\nDepartment of Physics, The University of Texas at El Paso 500 W. University Ave., EL Paso, TX 79968\nAbstract\nThe chiral symmetry breaking in a Nambu-Jona-Lasinio e \u000bective model of quarks in the presence of a magnetic\nfield is investigated. We show that new interaction tensor channels open up via Fierz identities due to the explicit\nbreaking of the rotational symmetry by the magnetic field. We demonstrate that the magnetic catalysis of chiral\nsymmetry breaking leads to the generation of two independent condensates, the conventional chiral condensate and a\nspin-one condensate. While the chiral condensate generates, as usual, a dynamical fermion mass, the new condensate\nenters as a dynamical anomalous magnetic moment in the dispersion of the quasiparticles. Since the pair, formed by\na quark and an antiquark with opposite spins, possesses a resultant magnetic moment, an external magnetic field can\nalign it giving rise to a net magnetic moment for the ground state. The two condensates contribute to the e \u000bective\nmass of the LLL quasiparticles in such a way that the critical temperature for chiral symmetry restoration becomes\nenhanced.\nKeywords: Magnetic catalysis of chiral symmetry breaking in NJL model.\n1. Introduction\nThe phases of matter under strong magnetic fields\nconstitute an active topic of interest in light of the exper-\nimental production of large magnetic fields in heavy-ion\ncollisions, and also because of the existence of strongly\nmagnetized astrophysical compact objects. On the other\nhand, from a theoretical point of view there exist con-\ntradictory results about the influence of a magnetic field\non the chiral and deconfinement transitions of QCD [1].\nOf particular interest for the present paper are some\nrecently obtained results [2] on the influence of a mag-\nnetic field on the condensate structures characterizing\nthe QCD chiral transition. A magnetic field is known\nto produce the catalysis of chiral symmetry breaking\n(MC\u001fSB) [3] in any system of fermions with arbitrarily\nweak attractive interaction. This e \u000bect has been actively\ninvestigated for the last two decades [4]. In the original\nstudies of the MC \u001fSB, the catalyzed chiral condensate\nwas assumed to generate only a dynamical mass for the\nfermion. Recently, however, it has been shown that in\nQED [5] the MC \u001fSB leads to a dynamical fermion massand inevitably also to a dynamical anomalous magnetic\nmoment (AMM). This is connected to the fact that the\nAMM does not break any symmetry that has not al-\nready been broken by the other condensate. The dy-\nnamical AMM in massless QED leads, in turn, to a non-\nperturbative Lande g-factor and Bohr magneton propor-\ntional to the inverse of the dynamical mass. The induc-\ntion of the AMM also gives rise to a non-perturbative\nZeeman e \u000bect [5]. An important aspect of the MC \u001fSB\nis its universal character for theories of charged mass-\nless fermions in a magnetic field. Therefore, it is nat-\nurally to expect that the dynamical generation of the\nAMM shall permeate all the models of interacting mass-\nless fermions in a magnetic field.\nAs follows, we consider the dynamical generation\nof a net magnetic moment in the ground state of a\none-flavor Nambu-Jona-Lasinio (NJL) model in a mag-\nnetic field and discuss its implications for the chiral\nphase transition at finite temperature. The AMM of the\nquark /antiquark in the pair points in the same direction,\nas the pair is formed by particles with opposite spins\nand opposite charges. Hence, the pair has a nonzeroarXiv:1411.2493v1  [nucl-th]  10 Nov 2014/Nuclear Physics B Proceedings Supplement 00 (2022) 1–3 2\nmagnetic moment that becomes aligned by the external\nmagnetic field and then producing the magnetization of\nthe ground state. This magnetization is reflected in the\nexistence of a second independent condensate. The two\ncondensates contribute to the e \u000bective dynamical mass,\nwhich is mainly determined by the quark /antiquark pair-\ning in the lowest Landau lever (LLL), resulting in a sig-\nnificant increase in the critical temperature for the chi-\nral restoration, as compared to the case where only the\nmagnetically catalyzed chiral condensate is considered.\n2. Model and Condensates\nLet us consider the following NJL model of massless\nquarks in the presence of a constant and uniform mag-\nnetic field. The new element of the proposed model is\nthe introduction of a four-fermion channel, with cou-\npling constant G0, that becomes relevant only in the\npresence of a magnetic field,\nL=¯ i\r\u0016D\u0016 +G\n2[(¯  )2+(¯ i\r5 )2]\n+G0\n2[(¯ \u00063 )2+(¯ i\r5\u00063 )2] (1)\nThe new interaction channel naturally emerges using the\nFierz identities in the one-gluon-exchange channels of\nQCD when the rotational symmetry is broken by the\nmagnetic field. Here, \u00063is the spin operator in the di-\nrection of the applied field.\nSolving the system gap equations in the LLL, we find\nthe condensate solutions\n\u001b=Gh¯  i=Aexp\u0000\"2\u00192\n(G+G0)NcqB#\n(2)\nand\n\u0018=G0h¯ i\r1\r2 i=A0exp\u0000\"2\u00192\n(G+G0)NcqB#\n(3)\nwith A=2G\u0003=(G+G0) and A0=2G0\u0003=(G+G0).\nThe condensate \u001bis associated with the dynamically\ngenerated mass and \u0018with the AMM.\nIt is worth to underline that the induced condensates\n(2)-(3) depend nonperturbatively on the coupling con-\nstants and the magnetic field. This behavior reflects the\nimportant fact that in a massless theory, chiral symme-\ntry can be only broken dynamically, that is, nonpertur-\nbatively. In this result, the LLL plays a special role due\nto the absence of a gap between it and the Dirac sea. The\nrest of the LLs are separated from the Dirac sea by en-\nergy gaps that are multiples ofp\n2qB, and hence do not\nsignificantly participate in the pairing mechanism at thesubcritical couplings where the magnetic catalysis phe-\nnomenon is relevant. Since the dynamical generation of\nthe AMM is produced mainly by the LLL pairing dy-\nnamics, one should not expect to obtain a linear-in-B\nAMM term, even at weak fields, in sharp contrast with\nthe AMM appearing in theories of massive fermions.\nIn the later case, not only the AMM is obtained pertur-\nbatively through radiative corrections, but considering\nthe weak-field approximation means first summing in\nall the LL’s, which contribute on the same footing, and\nthen taking the leading term in an expansion in powers\nof B [6, 7]. Notice that such a linear dependence does\nnot hold, even in the massive case, if the field is strong\nenough to put all the fermions in the LLL [8].\n3. Critical Temperature\nThe e \u000bect of the new condensate h¯ i\r1\r2 iis to in-\ncrease the e \u000bective dynamical mass of the quasiparticles\nin the LLL,\nM\u0018=\u001b+\u0018=2\u0003exp\u0000\"2\u00192\n(G+G0)NcqB#\n(4)\nIn QCD, for fields, qB\u0018\u00032, the dimensional re-\nduction of the LLL fermions would constraint the LLL\nquarks to couple with the gluons only through the longi-\ntudinal components. Thus, to consistently work in this\nregime within the NJL model, we should consider that\nG0=G(see Ref. [2] for details).\nBecause the e \u000bective coupling enters in the exponen-\ntial, the modification of the dynamical mass by the mag-\nnetic moment condensate can be significant. Thus, the\nquasiparticles become much heavier in our model than\nin previous studies that ignored the magnetic moment\ninteraction. Taking into account that for qB=\u00032\u00181,\n\u0011'1 inG0=\u0011G, and using the values G\u00032=1:835,\n\u0003 = 602:3 MeV [9], Nc=3 and q=jej=3'0:1, it was\nfound in [2] that due to the condensate \u0018the dynamical\nmass of the quasiparticles increases sixfold.\nStarting from the thermodynamic potential in the\ncondensate phase\n\nT\n0(\u001b;\u0018)=NcqBZ\u0003\n0dp3\n2\u00192\"\n\"0+2\n\fln\u0010\n1+e\u0000\f\"0\u0011#\n+\u001b2\n2G+\u00182\n2G0; \"2\n0=p2\n3+\u001b2+\u00182;(5)\nwe can analytically find the critical temperature TC\u001f\nfrom\n@2\nTC\u001f\n0\n@\u001b2j\u001b=\u0018=0=\u0000\"\nCZ\u0003\n0dp3\np3tanh \fC\u001fp3\n2!\n+1\nG#/Nuclear Physics B Proceedings Supplement 00 (2022) 1–3 3\n+\u001b2\n2G+\u00182\n2G0=0 (6)\nwhere C=NcqB(G+G0)=(2\u00192G).\nDoing in (6) the change p3!p3=TC\u001f, we have\nZ\u0003=TC\u001f\n0dp3\np3tanh\u0012p3\n2\u0013\n=2\u00192\n(G+G0)NcqB; (7)\nSolving this equation for TC\u001f, we get\nTC\u001f=1:16\u0003exp\u0000\"2\u00192\n(G+G0)NcqB#\n=0:58M\u0018(8)\nThe fact that the critical temperature is proportional to\nthe dynamical mass at zero temperature, is consistent\nwith what has been found in other models [10]. In the\npresent case, since the dynamical mass is increased by\nthe AMM, the critical temperature is proportionally in-\ncreased. Notice that, we would have arrived at the same\nresult by taking instead the derivative with respect to \u0018.\nThis is a consequence of the proportionality between \u001b\nand\u0018, (i.e.\u0018=(G0=G)\u001b), which follows from Eqs. (2)\nand (3). This implies that the two condensates evapo-\nrate at the same critical temperature. The fact that there\nexists a unique critical temperature for the evaporation\nof the two condensates indicates that the condensate \u0018\ndoes not break any new symmetry that was not already\nbroken by the condensate \u001band the magnetic field, as\npointed out above. The simultaneous evaporation of the\nchiral and magnetic moment condensates has been also\nreported in the context of lattice QCD [11].\n4. Conclusion and Discussion\nIn the presence of a magnetic field there is no mag-\nnetically catalyzed chiral condensate h  iwithout the\nsimultaneous generation of a second dynamical con-\ndensate of the form h \u00063 i. The genesis of this phe-\nnomenon lies in the fact that the chiral pairs possess net\nmagnetic moments that tend to align with the external\nmagnetic field. The collective e \u000bect of these magnetic\nmoments leads to the ground state magnetization and\nmanifests itself as a spin-one condensate h \u00063 iwhich\nenters in the quasiparticle spectrum as an AMM.\nAn important e \u000bect ofh \u00063 iis to increase the ef-\nfective dynamical mass of the LLL quarks, and conse-\nquently the critical temperature of the chiral phase tran-\nsition. Since the quasiparticles become heavier, com-\npared to the case when the spin-one condensate is ig-\nnored, and since they are charged, the electrical con-\nductivity in this case should be much smaller at strong\nfields.The characteristic increase of the critical tempera-\nture with an applied magnetic field in the MC \u001fSB phe-\nnomenon is in sharp contrast with the dropping of the\ntemperature with the field found in lattice QCD [1].\nA reconciliation between these apparently contradic-\ntory results can be worked out, once the running of the\ncoupling with the magnetic field is incorporated into\nthe analysis [12]. As discussed in [12], a strong mag-\nnetic field gives rise to an anisotropy of the strong cou-\npling constant. The main e \u000bect for the critical temper-\nature is then associated with the running of the paral-\nlel coupling \u000bk\ns, which characterizes the interactions in\nthe direction parallel to the field and can be connected\nto the conventional NJL couplings through the relation\nG+G0=4\u0019\u000bk\ns=qB. The behavior of this coupling with\nthe field is such that the critical temperature ends up\ndecreasing with the field, in agreement with the lattice\nresults. The physical mechanism behind this e \u000bect can\nbe traced back to the antiscreenning produced by the\nquarks confined to the LLL in a strong magnetic field\n[12].\nReferences\n[1] G. Bali, et. al , JHEP 1202, 044 (2012); G. S. Bali, et. al , Phys.\nRev. D 86, 071502 (2012).\n[2] E. J. Ferrer, V . de la Incera, I. Portillo and M. Quiroz, Phys. Rev.\nD 89, 085034 (2014).\n[3] K. G. Klimenko, Z. Phys. C 54, 323 (1992); V . P. Gusynin, V . A.\nMiransky and I. A. Shovkovy, Phys. Rev. Lett. 73, 3499 (1994).\n[4] C. N. Leung, Y . J. Ng and A. W. Ackley, Phys. Rev. D 54, 4181\n(1996); E. J. Ferrer and V . de la Incera, Phys. Rev. D 58, 065008\n(1998); V . P. Gusynin, V . A. Miransky and I. A. Shovkovy, Nucl.\nPhys. B 563, 361 (1999); E. J. Ferrer and V . de la Incera, Phys.\nLett. B 481, 287 (2000); Yu. I. Shilnov, and V . V . Chitov, Phys.\nAtom. Nucl. 64 (2001) 2051 [Yad. Fiz. 64, 2138 (2001)]; E. J.\nFerrer, V . P. Gusynin, and V . de la Incera, Eur. Phys. J. B 33, 397\n(2003); N. Sadooghi, A. Sodeiri Jalili, Phys. Rev. D 76, 065013\n(2007); E. Rojas, A. Ayala, A. Bashir, and A. Raya, Phys. Rev.\nD 77, 093004. (2008).\n[5] E. J. Ferrer and V . de la Incera, Phys. Rev. Lett. 102, 050402\n(2009); E. J. Ferrer and V . de la Incera, Nucl. Phys. B 824, 217\n(2010).\n[6] J. Schwinger, Phys. Rev. 73, 416 (1947).\n[7] B. L. Io \u000be and A. V . Smilga, Nucl. Phys. 232, 109 (1984).\n[8] B. Jancovici Phys. Rev. A 187, 2275 (1969); E. J. Ferrer, V\nde la Incera, D. Manreza Paret, A. Prez Martnez; ”Anomalous-\nMagnetic-Moment E \u000bects in a Strongly Magnetized and Dense\nMedium” in the electronic proceedings of the conference: Com-\npact Stars in the QCD Phase Diagram III (CSQCD III), Guaruj,\nSP, Brazil (arXiv:1307.5947 [nucl-th]).\n[9] P. Rehberg, S. P. Klevansky and J. H ¨ufner, Phys. Rev. C 53, 410\n(1996).\n[10] V . P. Gusynin and I. A. Shovkovy, Phys. Rev. D 56, 5251 (1997).\n[11] G. S. Bali, et. al , Phys. Rev. D 86, 094512 (2012).\n[12] E. J. Ferrer, V . de la Incera and X. J. Wen, Quark Antiscreen-\ning at Strong Magnetic Filed and Inverse Magnetic Catalysis ,\narXiv:1407.3503 [nucl-th]."    },    {        "title": "1909.13821v1.Integrable_magnetic_flows_on_the_two_torus__Zoll_examples_and_systolic_inequalities.pdf",        "content": "INTEGRABLE MAGNETIC FLOWS ON THE TWO-TORUS:\nZOLL EXAMPLES AND SYSTOLIC INEQUALITIES\nLUCA ASSELLE AND GABRIELE BENEDETTI\nAbstract. In this paper we study some aspects of integrable magnetic systems on the two-torus. On the\none hand, we construct the \frst non-trivial examples with the property that all magnetic geodesics with unit\nspeed are closed. On the other hand, we show that those integrable magnetic systems admitting a global\nsurface of section satisfy a sharp systolic inequality.\n1.Introduction\nAmagnetic system on a closed oriented surface \u0006 is a pair ( g;b), wheregis a Riemannian metric and\nb: \u0006!Ris a smooth function, which we refer to as the magnetic function . Every magnetic system yields a\n\row on S\u0006, the unit tangent bundle of \u0006 with respect to g, by\n\bt\ng;b(q;v) = (\r(t);_\r(t));8t2R;\nwhere\r:R!\u0006 is the unique ( g;b)-geodesic , that is, unit-speed curve satisfying the prescribed geodesic\ncurvature equation\n\u0014\r(t) =b(\r(t)); (1.1)\nwith initial conditions ( \r(0);_\r(0)) = (q;v). Here\u0014\ris the geodesic curvature of \rwith respect to gand\nthe given orientation of the surface. For every \u0015 >0, the solution of (1.1) for the pairs ( \u00152g;\u0015b) and (g;b)\ncoincide. Moreover, changing simultaneously the orientation and the sign of bdoes not change the solutions\nof (1.1), therefore we will assume throughout the paper that the average of b\nhbi:=1\narea(\u0006;g)Z\n\u0006b\u0016g;\nis non-negative. Here \u0016gis the area form given by gand the given orientation on \u0006, and area(\u0006 ;g) :=R\n\u0006\u0016g.\nWe readily see that, for every \u0015>0, (g;\u0015b)-geodesics correspond to \u0015\u00001-speed solutions of\nr_\r_\r= (b\u000e\r)\u0001_\r?; (1.2)\nwhererdenotes the Levi-Civita connection, and _ \r?is given by rotating _ \rby\u0019\n2in T\r\u0006.\nThis enables us to study ( g;b)-geodesics with symplectic methods. Indeed, solutions of (1.2) correspond,\nup to the isomorphism T\u0006 !T\u0003\u0006 given by the metric g, to the solutions of the Hamiltonian system on T\u0003\u0006\nwith kinetic Hamiltonian function\nH: T\u0003\u0006!R; H (q;p) =1\n2jpj2(1.3)\nand twisted symplectic form\n\ng;b:= dp^dq\u0000\u0019\u0003(b\u0016g); (1.4)\nwhere\u0019: T\u0003\u0006!\u0006 is the bundle projection. Such Hamiltonian systems are of physical interest since they\nmodel the motion of a charged particle under the e\u000bect of the magnetic \feld b\u0016g.\nThe easiest examples of magnetic systems are given by pairs ( gcon;bcon), wheregconhas constant Gaussian\ncurvatureKconandbconis constant. We see that, for\nb2\ncon+Kcon>0; (1.5)\n(gcon;bcon)-geodesics are given by geodesic spheres with radius r=r(Kcon;bcon)>0. In particular, the\ncorresponding \row \bt\ngcon;bconyields a free S1-action on S\u0006, whose quotient map associates to every geodesic\nsphere its center. We are prompted, therefore, to make the following general de\fnition.\nDe\fnition 1.1. A magnetic system ( g;b) is called Zoll, if \bt\ng;byields a free S1-action on S\u0006.\nDate : October 1, 2019.\n1arXiv:1909.13821v1  [math.DS]  30 Sep 20192 L. ASSELLE AND G. BENEDETTI\nRemark 1.2. If \u00066=S2, the classi\fcation of Seifert \fbrations yields that ( g;b) is Zoll if and only if all\n(g;b)-geodesics are closed. For \u0006 = S2there are instead examples where all ( g;b)-geodesics are closed but\nthe induced S1-action on S S2is only semi-free [Ben16b]. \u0004\nRemark 1.3. In the particular case b= 0 we retrieve the de\fnition of Zoll metric on \u0006, and in this case we\nnecessarily have \u0006 = S2. The space of Zoll metrics on S2has already been thoroughly investigated in the\npast century, and the local structure around gconis nowadays well understood: Every smooth one-parameter\nfamilys7!\u001as:S2!Rof functions with \u001a0= 0 yields a family of metrics s7!gs:=e\u001asgcon. Funk showed\nthat ifgsis Zoll, thend\u001as\nds(0) must be an odd function on S2[Fun13]. Conversely, using the Nash-Moser\nimplicit function theorem, Guillemin proved that for every odd function f:S2!Rthere is a deformation\nof Zoll metrics as above with f=d\u001as\nds(0) [Gui76]. In particular, the space of Zoll metrics on S2is in\fnite\ndimensional. We shall notice that already the space of Zoll metrics obtained from surfaces of revolutions in\nR3is in\fnite dimensional, as the very explicit examples constructed by Zoll in [Zol03] demonstrate. \u0004\nIf (g;b) is Zoll, then (1.5) generalizes to\nK(g;b) :=hbi2+2\u0019\u001f(\u0006)\narea(\u0006;g)>0\nand (g;b)-geodesics belong to the class of closed curves \u0003+(\u0006) on \u0006 whose lift ( \r;_\r) on S\u0006 is homotopic to\nan oriented \fber of \u0019: S\u0006!\u0006. Moreover, as shown in [BK18], Zoll magnetic systems are the protagonist of\na sharp local magnetic systolic inequality as we now recall.\nFor any magnetic system, ( g;b)-geodesics in the class \u0003+(\u0006) are critical points of the functional\n`(g;b): \u0003+(\u0006)!R; ` (g;b)(\r) :=`g(\r)\u0000Z\nD2\u0000\u0003(b\u0016g); (1.6)\nwhere`gis theg-length and \u0000 : D2!\u0006 is a capping disc for \robtained projecting a homotopy between\n(\r;_\r) and a\u0019-\fber on S\u0006. We denote by \u0003+(g;b) the (possibly empty) set of ( g;b)-geodesics in \u0003+(\u0006).\nTheorem 1.9 and Remark 1.12 in [BK18] show that if ( g;b) isC3-close to some ( g0;b0) Zoll, then\ninf\n\r2\u0003+(g;b)`(g;b)(\r)\u00142\u0019\nhbi+q\nK(g;b)\nwith equality if and only if ( g;b) is Zoll. As a special case, one recovers in a stronger topology the local\nsystolic inequality for Zoll metrics on S2proved in [ABHS17] (see also [ABHS18a]).\nThe local magnetic systolic inequality leaves the door open to a number of interesting questions. First, for\napplications of the local inequality it urges us to know more about the space of Zoll magnetic systems. As\nZoll did in the purely Riemannian case, one could start looking for non-trivial Zoll magnetic systems on S2\norT2which are rotationally symmetric, namely of the form\ng= dx2+a(x)2dy2; b =b(x); (1.7)\nwhereaandbare functions of the x-variable only.\nSecond, one would like to understand if the systolic inequality holds true also for magnetic systems which\nare not close to a Zoll one. Already in the purely Riemannian case this is known to be false, as the Calabi-\nCroke sphere demonstrates; see e.g. [Sab10]. Nevertheless, Abbondandolo, Bramham, Hryniewicz and Sa-\nlom~ ao did establish the global inequality in the space of rotationally symmetric metrics on S2[ABHS18b].\nThus, also for this question, we are prompted to look at rotationally symmetric magnetic systems \frst.\n2.Statement of results\nWe can now present our contribution to the two questions above.\n2.1.Zoll magnetic systems on \rat tori. Our \frst result constructs non-trivial 1-parameter families of\nrotationally symmetric Zoll systems on certain \rat tori. Throughout the paper, we will denote with T2the\ntorus obtained by quotienting R2by the lattice Z\u0002LZ\u001aR2for someL>0, and with gconthe corresponding\n\rat metric. Also, we de\fne the countable dense set of positive numbers\nB:=f2\u0019n=\u0018jn2N; \u00182(0;1); J1(\u0018) = 0g; (2.1)\nwhereJ1is the \frst Bessel function of the \frst kind.INTEGRABLE MAGNETIC FLOWS ON THE TWO-TORUS 3\nTheorem 2.1. Assume that bcon= 2\u0019n=\u00182B. Then there exists a family s7!bsof magnetic functions\nbs:T2!(0;1),s2(\u0000b\u00001\ncon;b\u00001\ncon), depending only on the x-variable such that b0=bcon,hbsi=bcon, and\n(gcon;bs)is Zoll for all s2(\u0000b\u00001\ncon;b\u00001\ncon). The deformation is non-trivial and unbounded:\ndbs\nds(0)6= 0; lim\ns7!\u0006b\u00001\nconmaxbs=1:\nIfbcon=2B, then the only magnetic function bdepending only on xwithhbi=bconsuch that (gcon;b)is Zoll\nis given by b=bcon.\nThe examples given by Theorem 2.1 are very explicit since the symmetry yields a multivalued \frst integral\nI: ST2!R=bconZthat can be used to solve the equations of motion by quadratures. The exact expressions\nare given in Remark 4.3. In Figures 1 and 2 we draw the ( gcon;bs)-geodesics for bcon= 2\u0019=\u00181andbcon= 2\u0019=\u00182\nrespectively, where \u00181and\u00182are the \frst two positive zeros of J1. In fact, as can be seen from the formulas\nforbsobtained in (4.1) and (4.2), for every m2Nthe systems for bconandb0\ncon=mbconare related by the\ncovering map ( x;y)7!(mx;my ) ofT2and so the corresponding geodesics have the same shape. Each plot in\nthe two \fgures below corresponds to s=k\n5b\u00001\nconfork= 0;:::; 5. In each plot we draw \fve ( gcon;bs)-geodesics\nwith di\u000berent colors representing di\u000berent values of the \frst integral I. Fork= 5, we have bs(0) = +1,\nwhich causes the displayed curves to have a sharp edge there.\n−101ys=0\n5b−1\ncon s=1\n5b−1\nconI=0\n4bcon\nI=1\n4bconI=2\n4bcon\nI=3\n4bconI=4\n4bcon\ns=2\n5b−1\ncon\n−1 0\nx−101ys=3\n5b−1\ncon\n−1 0\nxs=4\n5b−1\ncon\n−1 0\nxs=5\n5b−1\ncon\nFigure 1. (gcon;bs)-geodesics for the 1-parameter family corresponding to \u00181\nLet us now brie\ry comment on the proof of Theorem 2.1. The argument hinges on the fact that the \row\nof Zoll magnetic systems with symmetry has a global surfaces of section. The Zoll condition can then be\nreformulated by asking that the return map is the identity or, equivalently, that all the Fourier coe\u000ecients\nof the map vanish. Using in an essential fashion that the metric is \rat, we are able to compute such Fourier\ncoe\u000ecients and show that there exist non-constant functions bwithhbi=bconand a return map with zero4 L. ASSELLE AND G. BENEDETTI\n−101ys=0\n5b−1\ncon s=1\n5b−1\nconI=0\n4bcon\nI=1\n4bconI=2\n4bcon\nI=3\n4bconI=4\n4bcon\ns=2\n5b−1\ncon\n−2 0\nx−101ys=3\n5b−1\ncon\n−2 0\nxs=4\n5b−1\ncon\n−2 0\nxs=5\n5b−1\ncon\nFigure 2. (gcon;bs)-geodesics for the 1-parameter family corresponding to \u00182\nFourier coe\u000ecients if and only if bcon2B. In such a case, the functions bwith this property are parametrised\nby an open set of Rd(bcon), whered(bcon) is the number of zeros \u0018ofJ1such thatbcon= 2\u0019n=\u0018 . In particular,\nifd(bcon) = 1, then the examples of Theorem 2.1 are the only ones with \rat metric and hbi=bcon. We\nremark that d(bcon) = 1 for all bcon2Bif and only if the ratio of any two positive distinct zeros of J1were\nirrational. Unfortunately, this property does not seem to be known.\nTheorem 2.1 is only the \frst step in the investigation of Zoll magnetic systems. We do not know, for\nexample, if explicit rotationally symmetric Zoll magnetic systems can be found on S2or onT2for metrics g\ndi\u000berent from the \rat one considered above. Leaving symmetry behind, one could try to generalize Guillemin's\napproach to yield an in\fnite dimensional space of non-trivial Zoll deformations for ( gcon;bcon) on any surface.\nSuch systems, however, will not be explicit as will be obtained from a Nash-Moser implicit function theorem.\nHaving such a result will help us understand the local structure of Zoll magnetic systems at ( gcon;bcon).\nHowever, getting an intuition for what the global structure should be seems an even harder task. Is it true,\nfor instance, that for every gthere exists a magnetic function bmaking (g;b) into a Zoll magnetic system?\nWe conclude this subsection mentioning three results connected with Theorem 2.1. The \frst result is due\nto Tabachnikov, who constructs 1-homogeneous Lagrangians L: TR2!R, whose Euler-Lagrange solutions\nare Euclidean circles of given radius [Tab04]. Since the lift to R2of the \row of a magnetic system on T2\nis the Euler-Lagrange \row of a particular 1-homogeneous Lagrangian, we see that Tabachnikov's result is\ncomplementary to ours. In his case, the curves are \fxed to be Euclidean circles and one let the Lagrangian\nvary in a large class. In our case, we allow curves of di\u000berent shapes (as soon as they are all closed) but we\nlook only at Lagrangians coming from a magnetic system. We remark that the zeros of J1play an important\nrole also in Tabachnikov's work, which suggests that the two constructions might be related.INTEGRABLE MAGNETIC FLOWS ON THE TWO-TORUS 5\nThe second result is due to Burns and Matveev and asserts that if we require that ( g0;b0) and (g1;b1)\nhave the same solutions of (1.1) and one among b0andb1is non-zero, then ( g1;b1) = (\u00152g0;\u0015\u00001b0) for some\n\u0015>0 [BM06]. This means that the \rexibility of Theorem 2.1 and in Tabachnikov's result turns into rigidity,\nif we both want the curves to be \fxed and the Lagrangian to be of magnetic type.\nFor the third result, we \fx a subset C\u001a(0;1) and look for metrics gon \u0006 and functions b: \u0006!R\nsuch that ( g;\u0015b) is Zoll for all \u00152C. This condition means that the magnetic Hamiltonian system with\nHamiltonian (1.3) and symplectic form (1.4) is Zoll on every energy level in the set1\n2C2. Lange and the \frst\nauthor showed in [AL19] that, for \u0006 = T2andC= (0;1), this condition implies that ( g;b) = (gcon;bcon),\nthus showing that trivial Zoll magnetic systems on T2exhibit some rigidity properties. This provides an\ninteresting contrast with the \rexibility proved in Theorem 2.1.\n2.2.A systolic inequality for symmetric systems on the torus. Our second result establishes the\nsharp systolic inequality for rotationally symmetric magnetic systems on T2possessing a global torus-like\nsurface of section. The precise statement is as follows.\nTheorem 2.2. LetT2be a two-torus with coordinates (x;y), and consider a symmetric magnetic system\n(g;b)as in (1.7) , wherea;b:R=Z!Rare functions of the x-variable only. If the condition\n\f\f\fda\ndx\f\f\f<ab (2.2)\nis met, then the systolic inequality\ninf\n\r2\u0003+(g;b)`(g;b)(\r)\u0014\u0019\nhbi(2.3)\nholds. The equality case occurs if and only if (g;b)is Zoll.\nThe importance of condition (2.2) will be thoroughly discussed in the next section. Here we mention\nonly two properties which are equivalent to it. The \frst is that no curve with constant x-coordinate is a\n(g;b)-magnetic geodesic. In particular, all symmetric Zoll magnetic systems must satisfy (2.2) since in this\ncase all closed ( g;b)-geodesics are contractible. The second is that for each \u001202R=2\u0019Zthe embedding\n\u0013\u00120:T2!ST2;(x;y)7!cos\u00120@x+sin\u00120\na(x)@y (2.4)\nis a torus-like global surface of section for the \row \b (g;b)inside S T2, and the associated \frst-return map can\nbe written quite explicitly in terms of the \frst integral I, a fact that will be key in the proof of Theorem 2.2.\nA direction of future research will be to decide if (2.3) holds for symmetric systems not satisfying (2.2). In\nthis more general case one can still construct annulus-like surfaces of section which, however, are not global.\nFor this reason, already showing that \u0003+(g;b)6=?is an interesting problem.\n2.3.Structure of the paper. Section 3 is devoted to establishing basic properties of rotationally invariant\nmagnetic systems on T2and ends with the proof of Theorem 2.2. In Section 4 we prove Theorem 2.1.\n3.The first integral I\nAs above we denote with ( x;y) angular coordinates on T2= [0;1]\u0002[0;L]=\u0018, and consider a magnetic\nsystem (g;b) as in (1.7). We write a:R=Z!(0;1) andb:R=Z!Rfor the warping and magnetic\nfunctions respectively, which depend on the x-variable only. Up to rescaling the y-variable (x;y)7!(x;cy)\nwhich changes LtocLandatoa=c, we assume that the normalization\nZ1\n0a(x)dx= 1 (3.1)\nholds. This implies that\nhbi=Z1\n0a(x)b(x)dx: (3.2)\nWe de\fne the function B:R!Rby the properties\nB(0) = 0; B0(x) =a(x)b(x);8x2R\nso that\nB(x+ 1) =B(x) +hbi;8x2R:6 L. ASSELLE AND G. BENEDETTI\nIn view of this, we will also write B:R=Z!R=hbiZfor the corresponding quotient map. Here and below\nwe use a prime to denote derivatives with respect to the x-variable.\nFrom the formula for g, the embeddings \u0013\u00120in (2.4) are well-de\fned and we have an angular function\n\u0012: ST2!R=2\u0019Zyielding the oriented angle between an element v2ST2and@x. Ifv=vx@x+vy@y, then\nvx= cos\u0012; vy=sin\u0012\na(x): (3.3)\nOn the other hand, the \row of @ypreserves both the metric gand the magnetic function b. Therefore,\napplying Noether's Theorem, we get a multivalued conserved quantity along ( g;b)-geodesics.\nLemma 3.1. The function\nI: ST2!R=hbiZ; I (x;y;\u0012 ) =a(x) sin\u0012\u0000B(x):\nis constant along (g;b)-geodesics. The critical set of Ican be expressed in the coordinates (x;y;\u0012 )by\nCritI= CritI\u0000\u0002(R=LZ)\u0002f\u0000\u0019=2g t CritI+\u0002(R=LZ)\u0002f+\u0019=2g; (3.4)\nwhere\nI\u0006:R=Z!R=hbiZ; I\u0006(x) =\u0006a(x)\u0000B(x):\nFinally, a (g;b)-geodesic parametrizes the circle fx=x0gin the positive (respectively negative) y-direction if\nand only if x0belongs to CritI+(respectively to CritI\u0000).\nProof. The lifted \row of @yto the twisted cotangent bundle (T\u0003T2;\n(g;b)), where \n (g;b)is given by (1.4), is\ngenerated by the multivalued Hamiltonian py\u0000B(x). Indeed,\n\u0013@y\n(g;b)=\u0013@y\u0000\ndpx^dx+ dpy^dy\u0001\n\u0000\u0013@y\u0000\na(x)b(x)dx^dy\u0001\n= dpy\u0000a(x)b(x)dx\n= d\u0000\npy\u0000B(x)\u0001\n:\nSince the kinetic Hamiltonian Hfrom (1.3) is invariant under the \row of @y, Noether's theorem implies that\npy\u0000B(x) is invariant under the Hamiltonian \row of H. As \b (g;b)is the restriction of the Hamiltonian \row\nofHto ST2, after applying the metric isomorphism T T2!T\u0003T2, andpy=a(x) sin\u0012holds on S T2, we get\nthatIis a \frst integral for \b (g;b).\nWe compute\n@I\n@x=a0(x) sin\u0012\u0000a(x)b(x);@I\n@\u0012=a(x) cos\u0012: (3.5)\nThe second quantity is zero if and only if cos \u0012= 0. In this case, \u0012=\u0006\u0019=2 and the \frst quantity is zero if\nand only if xis a critical point of I\u0006.\nFinally, a (g;b)-geodesic parametrizes fx=x0gif and only if the corresponding trajectory of the Hamil-\ntonian \row of His a reparametrization of a trajectory of the Hamiltonian \row of py\u0000B(x). By the de\fnition\nof Hamiltonian vector \felds, this happens if and only if d( py\u0000B(x)) and dHhave the same kernel at this\norbit. Since S T2corresponds to H\u00001(1=2) and the restriction of py\u0000B(x) to this level set corresponds to I,\nwe see that the above conditions are equivalent to the fact that the orbit is contained in the critical set of I.\nBy (3.4) the assertion follows. \u0004\nCorollary 3.2. The equations of motion of a unit-speed (g;b)-geodesic (x;y) :R!T2, whose tangent vector\nmakes an angle \u0012:R!R=2\u0019Zwith@x, read\n8\n>>>>><\n>>>>>:_x= cos\u0012;\n_y=sin\u0012\na(x);\n_\u0012=b(x)\u0000a0(x)\na(x)sin\u0012:(3.6)\nProof. The \frst two equations follows immediately from (3.3). The third equation is equivalent to\n_\u0012=\u00001\na@I\n@x(3.7)INTEGRABLE MAGNETIC FLOWS ON THE TWO-TORUS 7\nLet us assume by contradiction that this equation does not hold. Since Iis a \frst integral, we compute using\nthe \frst equation in (3.6)\n0 =_I=@I\n@x_x+@I\n@\u0012_\u0012= cos\u0012\u0010\na_\u0012+@I\n@x\u0011\n: (3.8)\nThen cos\u0012\u00110 andx\u0011x02R=Z,\u0012\u0011\u0006\u0019\n2. According to Lemma 3.1, this means that I\u0006has a critical point\natx0. However, using the assumption that we want to contradict and the fact that _\u0012= 0, we get\ndI\u0006\ndx(x0) =@I\n@x(x0;\u0006\u0019=2)6= 0: \u0004\nAs observed in Section 2, Lemma 3.1 implies that if ( g;b) is Zoll, then Crit I=?. This prompts us to\nbetter investigate this condition.\nLemma 3.3. The following conditions are equivalent:\n(1)CritI=?;\n(2)ja0j<ab ;\n(3)@I\n@x<0;\n(4) the function \u0012is strictly monotone increasing along all (g;b)-geodesics;\n(5)\u0000d\u0012^!(g;b)is a volume form on ST2, where!(g;b)is the pull-back of \n(g;b)onST2;\n(6) for each \u001202R=2\u0019Zthe torus embedding \u0013\u00120from (2.4) is a global surface of section for \b(g;b).\nProof. (1),(2): The critical set of Iis empty if and only if the critical sets of I\u0000andI+are empty. This\nmeans that\u0006a0\u0000ab6= 0. By continuity, we either have \u0006a0\u0000ab> 0 or\u0006a0\u0000ab< 0. However,\nZ1\n0(\u0006a0\u0000ab)dx=\u0000Z1\n0abdx=\u0000hbi\u00140;\nwhich implies that only the second possibility can hold.\n(2),(3):ja0j<ab if and only if a0sin\u0012\u0000ab< 0 for all\u0012. By (3.5) the conclusion follows.\n(3),(4): It follows at once from (3.7).\n(3),(5): We compute\n!(g;b)= d(cos\u0012dx+asin\u0012dy)\u0000abdx^dy\n= (a0sin\u0012\u0000ab)dx^dy+ d\u0012^(\u0000sin\u0012dx+acos\u0012dy):\nTherefore,\u0000d\u0012^!(g;b)=@I\n@xd\u0012^dx^dyfrom which we see the desired equivalence.\n(4))(6): If\u0012is strictly monotone increasing along all \row lines of \b (g;b), then for every \u001202R=2\u0019Zthe\ntorusR=Z\u0002(R=LZ)\u0002f\u00120gis a global surface of section.\n(6))(1): Assume that \u0013\u00120is a global surface of section. Then,\n(i) no orbit of \b (g;b)can be entirely contained in \u0013\u00120;\n(ii) all orbits of \b (g;b)must intersect \u0013\u00120.\nIf\u001206=\u0006\u0019=2, then (ii) is su\u000ecient to see that Crit I\u0006=?. If\u00120=\u0019=2 (the case\u00120=\u0000\u0019=2 being analogous),\nthen from (i) we see that Crit I+=?and from (ii) that Crit I\u0000=?. \u0004\nRemark 3.4. Condition (e) implies that the odd-symplectic form !(g;b)is stable. Even if we will not make\nuse of this in the present paper, let us observe that this has the consequence that an arbitrary magnetic system\n(g1;b1) close to (g;b) satisfying (a) will also have an associated stable odd-symplectic form and therefore it\nwill possess a contractible ( g1;b1)-geodesic according to [AB16]. \u0004\nRemark 3.5. A su\u000ecient but not necessary condition for Crit I=?is the positivity of the magnetic\ncurvature of ( g;b). This phenomenon is analogous to the one ensuring that a convex Riemannian sphere of\nrevolution admits exactly one equator (see also [Ben16a, Proposition 4.6] for the case of magnetic spheres).\nLet us recall that the magnetic curvature, which in general can be obtained from Jacobi \felds along ( g;b)-\ngeodesics, in the symmetric setting takes the form\nK(g;b): ST2!R; K (g;b)(x;y;\u0012 ) =b2(x) +b0(x) sin'\u0000a00(x)\na(x):\nIf we assume that K(g;b)is everywhere positive, then we claim that all critical points of I\u0000andI+must be non-\ndegenerate maxima. This implies that the set of critical points of I\u0000andI+is empty. To prove the claim, we8 L. ASSELLE AND G. BENEDETTI\ntake a critical point x0forI+(the other case being completely analogous) such that a0(x0)\u0000a(x0)b(x0) = 0.\nUsing this equation, we see that the second derivative is negative\nd2\ndx2\f\f\f\nx=x0I+=a00(x0)\u0000a0(x0)b(x0)\u0000a(x0)b0(x0)\n=\u0000a(x0)\u0010\nf2(x0) +f0(x0)\u0000a00(x0)\na(x0)\u0011\n=\u0000a(x0)\u0001Kg;b(x0;y;\u0019= 2): \u0004\nFrom Lemma 3.3, we see that Crit I=?yields some strong dynamical consequences. First of all, thanks to\nCondition (c), we have a function x:R=hbiZ\u0002R=2\u0019Z!R=Zwhich is the unique solution of I=I(x(I;\u0012);\u0012)\nfor every (I;\u0012). Di\u000berentiating this relation with respect to \u0012we get\n@x\n@\u0012=\u0000a(x) cos\u0012@x\n@I: (3.9)\nUsing the function xwe can parametrize S T2by the coordinates ( I;y;\u0012 ). Moreover, by Condition (d), we can\nparametrize every ( g;b)-geodesic by the angle \u0012. If we do so, then the only non-trivial di\u000berential equation\nin the (I;y;\u0012 ) coordinates is the one determining y. Thanks to (3.6) and (3.7), it reads\ndy\nd\u0012=\u0000sin\u0012@x\n@I: (3.10)\nIf we parametrize the global surfaces of section \u0012\u00001(\u00120) appearing in Condition (f) using coordinates ( I;y),\nthen we have a \frst return map of the form ( I;y)7!(I;y+ \u0001(I)), where \u0001 : R=hbiZ!Ris the function\n\u0001(I) :=Z\nR\n2\u0019Zdy\nd\u0012d\u0012=\u0000Z\nR\n2\u0019Zsin\u0012@x\n@I(I;\u0012)d\u0012:\nLemma 3.6. Assume that CritI=?and let \u0001be de\fned as above. Then\ndS\ndI= \u0001;\nwhereS:R=hbiZ!Ris the function\nS(I) :=\u0000Z\nR\n2\u0019Za(x(I;\u0012)) cos2\u0012@x\n@I(I;\u0012)d\u0012: (3.11)\nProof. We de\fne ~S:R!Rby\n~S(I) =\u0000Z\nR\n2\u0019Zx(I;\u0012) sin\u0012d\u0012;\nwhere we identify xwith its lift to R. It is clear that ~Sis a primitive of the lift of \u0001. We are left to prove\nthat ~Sis the lift of S. Integrating by parts, we get\n~S(I) =Z\nR\n2\u0019Z@x\n@\u0012cos\u0012d\u0012=Z\nR\n2\u0019Z\u0000a(x) cos\u0012@x\n@Icos\u0012d\u0012=S(I);\nwhere we used (3.9) in the second equality. \u0004\nWe collect the main properties of Sin the next lemma.\nLemma 3.7. Assume that CritI=?and letSbe de\fned as above. Then\nZ\nR\nhbiZS(I)dI=\u0019:\nMoreover, an element I2R=hbiZis a critical point of Sif and only if the (g;b)-geodesics\rIwith integral I\nbelong to \u0003+(T2)(namely are contractible and with turning number +1). Moreover,\n`(g;b)(\rI) =S(I):\nIn particular, (g;b)is Zoll if and only if Sis constant (and equal to \u0019=hbi).INTEGRABLE MAGNETIC FLOWS ON THE TWO-TORUS 9\nProof. We compute\nZ\nR\nhbiZS(I)dI=Z\nR\n2\u0019Zcos2\u0012\u0010Z\nR\nhbiZ\u0000a(x(I;\u0012))@x\n@IdI\u0011\nd\u0012\n=Z\nR\n2\u0019Zcos2\u0012\u0010Z\nR\nZa(x)dx\u0011\nd\u0012\n=Z\nR\n2\u0019Zcos2\u0012d\u0012\n=\u0019;\nwhere we used (3.1) and the fact that for every \u0012the mapI7!x(I;\u0012) reverses the orientation.\nFrom the de\fnition of the \frst-return map we see that \u0001( I) = 0 if and only if the ( g;b)-geodesics with\nintegralIare contractible (and with turning number +1). By Lemma 3.6, the correspondence between critical\npoints ofSand elements in \u0003+(g;b) follows. It remains to compute\n`(g;b)(\rI) =`g(\rI)\u0000Z\nD2\u0000\u0003\nI(abdx^dy):\nFor the \frst term we use (3.7)\n`g(\rI) =Z\nR\n2\u0019Z1\n_\u0012d\u0012=\u0000Z\nR\n2\u0019Za(x)@x\n@Id\u0012:\nFor the second term we use Stokes Theorem, the de\fnition of the \frst integral and (3.10)\nZ\nD2\u0000\u0003\nI(abdx^dy) =Z\nR\n2\u0019Z\r\u0003\nI(B(x)dy)\n=Z\nR\n2\u0019Z(a(x) sin\u0012\u0000I)dy\n=Z\nR\n2\u0019Za(x) sin\u0012dy\nd\u0012d\u0012\u0000IZ\nR\n2\u0019Zdy\n=\u0000Z\nR\n2\u0019Za(x) sin\u0012sin\u0012@x\n@Id\u0012\u0000I\u00010:\nPutting the two pieces together yields the desired formula.\nFinally, the assertion about Zoll systems is a consequence of the fact thatR\nR\nhbiZdI=hbi. \u0004\nProof of Theorem 2.2. By the previous lemma, we have\ninf\n\r2\u0003+(g;b)`(g;b)(\r) = minS\u0014Rhbi\n0S(I)dI\nhbi=\u0019\nhbi:\nThe inequality is strict if and only if Sis not constant, which happens if and only if ( g;b) is not Zoll. \u0004\n4.Construction of symmetric Zoll magnetic systems on the flat torus\nFrom the previous section, we know that a rotationally symmetric ( g;b) on the two-torus is Zoll if and\nonly if the function S:R=hbiZ!Ris constant (and equal to \u0019=hbi). This condition is equivalent to asking\nthat the Fourier coe\u000ecients\n^S(m) =Zhbi\n0S(I)e\u00002\u0019im\nhbiIdI\nvanish for all m2Znf0g. These coe\u000ecients, although in general not computable, admit an explicit formula\nwhen the torus is \rat, namely when a\u00111. This formula involves the Bessel function (see [Wat44])\nJ1: (0;1)!R; J 1(\u0018) =\u0018Z2\u0019\n0cos\u0000\n\u0018sin\u0012\u0001\ncos2\u0012d\u0012:\nThe zeros of J1are all simple and form a strictly increasing, divergent sequence ( \u0018k)k2N.10 L. ASSELLE AND G. BENEDETTI\nLemma 4.1. Let(gcon;b)be a rotationally symmetric magnetic system with CritI=?. Then the function\nB:R=Z!R=hbiZis invertible and we have\nS(I) =Z2\u0019\n0f(sin\u0012\u0000I) cos2\u0012d\u0012; f :=dB\u00001\ndu:\nTherefore,\n^S(m) =hbi\n2\u0019jmjJ1\u00102\u0019jmj\nhbi\u0011\n^f(\u0000m);8m6= 0:\nProof. By Lemma 3.3, the condition Crit I=?is equivalent to the positivity of b. Thus,Bis invertible and\nfrom the de\fnition of I, we getx(I;\u0012) =B\u00001(sin\u0012\u0000I). Hence, (3.11) yields the desired formula for S. For\nthe Fourier coe\u000ecients we compute:\n^S(m) =Z2\u0019\n0\u0010Zhbi\n0f(sin\u0012\u0000I)e\u00002\u0019im\nhbiIdI\u0011\ncos2\u0012d\u0012\n=Z2\u0019\n0\u0010Zhbi\n0f(J)e\u00002\u0019im\nhbi(sin\u0012\u0000J)dJ\u0011\ncos2\u0012d\u0012\n=Z2\u0019\n0\u0010Zhbi\n0f(J)e\u00002\u0019i(\u0000m)\nhbiJdJ\u0011\ne\u00002\u0019im\nhbisin\u0012cos2\u0012d\u0012\n=^f(\u0000m)Z2\u0019\n0e\u00002\u0019im\nhbisin\u0012cos2\u0012d\u0012:\nThe desired formula follows using eiz= cosz+isinzand sin(\u0018sin(\u0000\u0012)) cos2(\u0000\u0012) =\u0000sin(\u0018sin\u0012) cos2\u0012.\u0004\nWe are now in position to prove Theorem 2.1.\nProof of Theorem 2.1. We assume \frst that hbidoes not belong to the set Bde\fned in (2.1). Then,\nJ1(2\u0019m=hbi)6= 0;8m6= 0:\nBy Lemma 4.1, it follows that ^f(\u0000m) = 0 for all m6= 0, which implies that f, henceb, is a constant.\nWe assume now that hbi2B . Then, there exists m02Nsuch thatJ1(2\u0019m0=hbi) = 0. Now, for all s2R\nsuch that the function\nfs(u) =1\nhbi+scos\u00102\u0019m0\nhbi(u\u0000u0)\u0011\nis positive, there exists an orientation preserving di\u000beomorphism\nFs:R=hbiZ!R=Z; Fs(u) =u\nhbi+shbi\n2\u0019m0sin\u00102\u0019m0\nhbi(u\u0000u0)\u0011\n(4.1)\nsuch thatdFs\ndu=fs, and the function\nbs=dF\u00001\ns\ndx(4.2)\nmakes the pair ( gcon;bs) into a Zoll system. Indeed, ^fs(m) = 0 ifm6= 0;m0andJ1(2\u0019m0=hbi) = 0 ensure\nthat ^S(m) = 0 for all m6= 0. It is now immediate to check that fsis positive if and only if jsj<hbi\u00001\nand that inf fstends to 0 asjsj!hbi\u00001. This last property implies that max bsdiverges asjsj!hbi\u00001.\nMoreover,\ndbs\nds(0) =\u0000dfs\nds(0) =\u0000cos\u00102\u0019m0\nhbi(u\u0000u0)\u0011\n6= 0: \u0004\nRemark 4.2. The proof of Theorem 2.1 shows that the solution fsconstructed above is the only one giving\nrise to a Zoll pair if d(hbi) = 1, namely if hbihas the property that there exists a unique m0such that\n2\u0019m0=hbiis a zero of J1(see the discussion after Figure 1). Moreover, shifting the parameter u0corresponds\nto shifting the xvariable in band so the associated ( gcon;b)-geodesics get also shifted in the x-direction.\nRemark 4.3. Using the formula for Fswithu0= 0 obtained above, we get an explicit expression for\n(gcon;b)-geodesics ( xI;yI) :R!T2parametrized by \u0012. Indeed,\nx(I;\u0012) =B\u00001(sin\u0012\u0000I) =Fs(sin\u0012\u0000I);@x\n@I=\u0000fs(sin\u0012\u0000I):INTEGRABLE MAGNETIC FLOWS ON THE TWO-TORUS 11\nThus, we get8\n>>><\n>>>:xI(\u0012) =sin\u0012\u0000I\nhbi+shbi\n2\u0019m0sin\u00102\u0019m0\nhbi(sin\u0012\u0000I)\u0011\n;\nyI(\u0012) =\u00001\nhbicos\u0012+sZ\u0012\n\u0000\u0019=2sin\u0012cos\u00102\u0019m0\nhbi(sin\u0012\u0000I)\u0011\nd\u0012;\nwhere we have used (3.10) to express yI. We used these formulas to produce Figure 1 and 2 using Python.\nAcknowledgements. We thank Alberto Abbondandolo and Serge Tabachnikov for fruitful discussions, and\nMattia Carlo Sormani for the precious help with the numerical integration. Luca Asselle is partially supported\nby the DFG-grant AS 546/1-1 \"Morse theoretical methods in Hamiltonian dynamics\". Gabriele Benedetti\nwas partially supported by the National Science Foundation under Grant No. DMS-1440140 while in residence\nat the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2018 semester for\nthe program \"Hamiltonian systems, from topology to applications through analysis\".\nReferences\n[AB16] L. Asselle and G. Benedetti, The Lusternik-Fet theorem for autonomous Tonelli Hamiltonian systems on twisted\ncotangent bundles , J. Topol. Anal. 8(2016), no. 3, 545{570.\n[ABHS17] A. Abbondandolo, B. Bramham, U.L. Hryniewicz, and P.A.S. Salom~ ao, A systolic inequality for geodesic \rows on\nthe two-sphere , Math. Ann. 367(2017), no. 1-2, 701{753.\n[ABHS18a] A. Abbondandolo, B. Bramham, U.L. Hryniewicz, and P.A.S. Salom~ ao, Sharp systolic inequalities for Reeb \rows\non the three-sphere , Invent. Math. 211(2018), no. 2, 687{778.\n[ABHS18b] A. Abbondandolo, B. Bramham, U.L. Hryniewicz, and P.A.S. Salom~ ao, Sharp systolic inequalities for Riemannian\nand Finsler spheres of revolution , preprint, arXiv:1808.06995, 2018.\n[AL19] L. Asselle and C. Lange, On the rigidity of Zoll magnetic systems on surfaces , in preparation, 2019.\n[Ben16a] G. Benedetti, The contact property for symplectic magnetic \felds on S2, Ergodic Theory Dynam. Systems 36\n(2016), no. 3, 682{713.\n[Ben16b] ,Magnetic Katok examples on the two-sphere , Bull. Lond. Math. Soc. 48(2016), no. 5, 855{865.\n[BK18] G. Benedetti and J. Kang, On a systolic inequality for closed magnetic geodesics on surfaces ,\nhttps://arxiv.org/abs/1902.01262, 2018, 2018.\n[BM06] K. Burns and V. S. Matveev, On the rigidity of magnetic systems with the same magnetic geodesics , Proc. Amer.\nMath. Soc. 134(2006), no. 2, 427{434.\n[Fun13] P. Funk, Uber Fl achen mit lauter geschlossenen geod atischen Linien , Math. Ann. 74(1913), no. 2, 278{300.\n[Gui76] V. Guillemin, The Radon transform on Zoll surfaces , Adv. Math. 22(1976), 85{119.\n[Sab10] S. Sabourau, Local extremality of the Calabi-Croke sphere for the length of the shortest closed geodesic , J. London\nMath. Soc. 82(2010), no. 3, 549{562.\n[Tab04] S. Tabachnikov, Remarks on magnetic \rows and magnetic billiards, Finsler metrics and a magnetic analog of\nHilbert's fourth problem , Modern dynamical systems and applications, Cambridge Univ. Press, Cambridge, 2004,\npp. 233{250.\n[Wat44] G.N. Watson, A treatise on the theory of Bessel functions (2nd edition) , Cambridge University Press, Cambridge,\n1944.\n[Zol03] O. Zoll, Ueber Fl achen mit Scharen geschlossener geod atischer Linien , Math. Ann. 57(1903), no. 1, 108{133.\nLuca Asselle\nJustus Liebig Universit at Gie\u0019en, Mathematisches Institut\nArndtstra\u0019e 2, 35392 Gie\u0019en, Germany\nE-mail address :luca.asselle@math.uni-giessen.de\nGabriele Benedetti\nUniversit at Heidelberg, Mathematisches Institut\nIm Neuenheimer Feld 205, 69120 Heidelberg, Germany\nE-mail address :gbenedetti@mathi.uni-heidelberg.de"    },    {        "title": "1808.08664v2.Dynamics_of_the_antiferromagnetic_skyrmion_induced_by_a_magnetic_anisotropy_gradient.pdf",        "content": "arXiv:1808.08664 [cond -mat.mes -hall] \nPage 1 of 19 Dynamics of  the antiferromagnetic skyrmion  induced by a \nmagnetic  anisotropy gradient  \n \nLaichuan Shen 1, *, Jing Xia 2, *, Guoping Zhao 1, †, Xichao Zhang 2, 3, Motohiko Ezawa 4, \nOleg A. Tretiakov 5, 6, Xiaoxi Liu 3, and Yan Zhou 2, † \n \n1College of Physics and Electronic Engineer ing, Sichuan Normal University, Chengdu \n610068, China \n2School of Science and Engineering, The Chinese Uni versity of Hong Kong, Shenzhen, \nGuangdong 518172, China \n3Department of Electrical and Computer Engineering, Shins hu University, 4 -17-1 Wakasato, \nNagano 380- 8553, Japan \n4Department of Applied Physics, The University of Tokyo, 7 -3-1 Hongo, Tokyo 113- 8656, \nJapan \n5Institute for Materials Research, Tohoku University, Sendai 980- 8577, Japan \n6School of Physics, The University  of New South Wales, Sydney  2052, Australia  \n \n* These authors contributed equally to this work.  \n† Authors to whom corr espondence should be addressed:  \nE-mail (G.Z.): zhaogp@uestc.edu.cn & E-mail (Y.Z.): zhouyan@cuhk.edu.cn  \n  arXiv:1808.08664 [cond -mat.mes -hall] \nPage 2 of 19 ABSTRACT  \nThe dynamics  of antiferromagnets is a  current  hot topic in  condensed matter physics and  \nspintronics. However, the dynamics of insulating antiferromagnets cannot be excited by an \nelectric current, which is a method usually used to manipulate ferromagnetic metals . Here , we \npropose  to use the  voltage -controlled magnetic  anisotropy gradient as an excitation  source to \nmanipulate  insulating antiferromagnetic texture s. We analytically and numerically  study the  \ndynamics of  an antiferromagnetic skyrmion driven  by a magnetic  anisotropy gradient. Our \nanalytical calculations demonstrate that such a  magnetic  anisotropy gradient can effectively \ndrive an antiferromagnetic skyrmion toward s the area of lower magnetic  anisotropy. The \nmicromagnetic simulations are in good agreement with our a nalytical solution . Furthermore , \nthe magnetic anisotropy gradient induced velocity  of an antiferromagnetic skyrmion is  \ncompared with that of a ferromagnetic skyrmion. Our results are useful for the  understanding \nof antiferromagnetic skyrmion dynamics and may open a new way for the  design of \nantiferromagnetic spintronic devices.  \n \nPACS: 75.50. Ee, 75.78.Fg, 75.78.- n, 75.30.Gw  \nKeywords: Skyrmion, Antiferromagnet, Perpendicular Magnetic Anisotropy, Spintronics  \n  arXiv:1808.08664 [cond -mat.mes -hall] \nPage 3 of 19 Introduction . – Skyrmions are topologically protected magnetic textures that  have been \nexperimentally observed in chiral materials (for examples, MnSi [1] , Fe 1−xCoxSi [2], FeGe  \n[3], Cu2OSeO 3 [4], and ultrathin Pt/Co/MgO nanostructure s [5]) without inversion symmetry. \nSkyrmion -based racetrack memory  [6-9] has attract ed great interest , where magnetic \nskyrmions are used as information carriers . Compared to domain walls, skyrmions have \nadvantages of nanoscale size and  topological  stability. Moreover , experiments have shown \nthat skyrmions require a  depinning current density of ~ 106 A/m2 [10, 11] , which is much \nsmaller than that of domain walls (~ 1010 − 1012 A/m2) [12, 13] . Various methods have been \nproposed to drive magnetic skyrmions, such as by emp loying electric current s [7-9, 14] , spin \nwaves  [15], magnetic field gradient s [16], and temperature gradient s [17]. Recently,  it has \nbeen experimentally  and theoretically  demonstrated  that voltage -controlled magnetic \nanisotropy  (VCMA)  gradient  can be used to drive and control the motion of ferro magnetic  \nskyrmions  [18-21]. Such  a VCMA effect has been experimentally proven in many systems, \nsuch as in Ir/CoFeB/MgO [22] , Ta/CoFeB/MgO [23] , Pt/Co [24, 25] , and Fe/Co/MgO [26]  \nsystems.  \nOn the other hand, a ntiferromagnets with compensated magnetic sublattices  have \nattracted great attention  [27-41] and are promising as candidate materials for advanced \nspintronic devices due to their zero stray fields and ultrafast magnetization dynamics.  \nTheoretical calculations demonstrate  the existence of stable skyrmions in antiferromagnets  \n[42-44]. Compared to ferrom agnetic (FM) skyrmion s, antiferromagnetic (AFM) skyrmion s \nare ideal information  carriers  because they have no skyrmion Hall effect  [42-45]. However, it \nis difficult to manipulate the AFM  skyrmion  using the magnetic field  due to the zero net \nmagnetic moments of a perfect antiferromagnet, and  also the dynamics of the  insulating  \nantiferromagnet cannot  be induced by an electric current  [32, 46- 48]. A vital question is how arXiv:1808.08664 [cond -mat.mes -hall] \nPage 4 of 19 to drive an insulating AFM skyrmion . Therefore, alternative methods are crucial and have \nbeen explored, for example, using temperature gradient s [30, 49]  and spin wave s [27] . \nIn this work, we analytically and numerically  study  the magnetization  dynamic s of an \nAFM skyrmion under a  voltage -controlled magnetic  anisotropy gradient . Such a  magnetic  \nanisotropy gradient is a new method to manipulate an AF M texture  and could be used in  the \ninsulating AF M materials . Our result s show that an  AFM skyrmion  will move toward s the \narea with low er anisotropy , and the  speed  of an AFM skyrmion  can reach 500 m/s  driven by a \nmagnetic anisotropy gradient  in principle . \n \nModel and simulation . – We consider an AFM  film with  two sublattices having  magnetic \nmoments M1(r, t) and M2(r, t), |M1| = |M2| = M S/2 with the saturation magnetization M S. The \ntotal magnetization is M (r, t) = M1(r, t) + M2(r, t) and the staggered magnetization is l(r, t) = \nM1(r, t) − M2(r, t), where the former is related to the canting of magnetic moments , and the \nlatter gives the unit Néel vector n (r, t) = l(r, t)/l (l = |l(r, t)|) that could be used to describe  \nconfigurations  of AFM skyrmion s. \nThe AF M free energy can be written as [27, 32, 46, 47, 50- 52] \n𝐸𝐸=∫𝑑𝑑𝑑𝑑[𝐴𝐴ℎ𝑴𝑴2+𝐴𝐴(∇𝒏𝒏)2−𝐾𝐾𝑛𝑛𝑧𝑧2+𝑤𝑤𝐷𝐷],                                    (1)  \nwhere Ah and A stand for the homogeneous and inhomogeneous exchange constant s, \nrespectively , K is the perpendicular magnetic anisotropy  (PMA) constant, which changes \nlinearly with the longitudinal spatial coordinate x [see Fig. 1 (a)], i.e., K (x) = K 0 − x·dK/dx, \nand wD represents  the energy density  arising from  the interfacial  Dzyaloshinskii -Moriya \ninteraction (DMI)  [53-55], whic h stabilize s the Néel -type skyrmion  [56] and is written as  [51, \n52, 57, 58]  \n𝑤𝑤𝐷𝐷=𝐷𝐷[𝑛𝑛𝑧𝑧∇∙𝒏𝒏−(𝒏𝒏∙∇)𝑛𝑛𝑧𝑧],                                               (2)  arXiv:1808.08664 [cond -mat.mes -hall] \nPage 5 of 19 where  D is the DMI constant  and n = sin θ(r)cosφex + sinθ(r)sinφey + cosθ(r)ez with the polar \ncoordinates ( r, φ). In this paper , the magnetic anisotropy  gradient  is assumed to be  induced by  \napplying a  voltage to the  sample with  a wedged  insulating layer  [see Fig. 1(a)]  [18, 59, 60]  \nand dK/dx  = 100 GJ/m4 unless otherwise addressed, which mean s that the  magnetic \nanisotropy decreases  by 10 kJ/m3 for every 100 nm  increase in the spa tial coordinate x . Such \nan anisotropy change of 10 kJ/m3 can be generated by a voltage  of 0.1 V , where the \nexperimental VCMA coefficient of 100 fJ/(V m)  [22, 23]  and the insulating layer  thickness of \n~ 1 nm are adopted.  \nEq. (1) is widely used for the AFM free energy  [51, 52, 57, 58] , from which more \nspecific forms can be obtained.  In particular, assuming that M  = 0 and an isolated  AFM  \nskyrmion  [Fig. 1(c)]  is in a thin film, Eq. (1)  can be simplified to a one -dimensional form , \nwhich is identical to  the free energy of a FM skyrmion  [Fig. 1(b)]  [61]. Further, using \nvariational calculus  yields  \n𝑑𝑑2𝜃𝜃\n𝑑𝑑𝑟𝑟2+1\n𝑟𝑟𝑑𝑑𝜃𝜃\n𝑑𝑑𝑟𝑟=�1\n𝑟𝑟2+𝐾𝐾\n𝐴𝐴�sin𝜃𝜃cos𝜃𝜃−𝐷𝐷sin2𝜃𝜃\n𝐴𝐴𝑟𝑟,                                    (3)  \nwhere θ is the angle between n  and the z -axis. Solving Eq. (3) , one can obtain the  profile  of a \nmetastable AFM skyrmion  [51] . \nThe variational derivatives of  Eq. (1) yield the effective fields  fM = − δE/μ0δM and fn = − \nδE/μ0δn with the vacuum permeability constant μ0, which are indispensable  in magnetization  \ndynamics of the  AFM system . The magnetization  dynamics  in ferromagnets is governed by \nLandau -Lifshitz -Gilbert (LLG) equation  [62], whereas i n AFM system,  the dynamics  of M \nand n  are controlled by the following two coupled equations  [27, 46] , \n𝒏𝒏̇=�𝛾𝛾𝒇𝒇𝑴𝑴−𝐺𝐺1𝑴𝑴̇�×𝒏𝒏,                                                 (4a)  \n𝑴𝑴̇=(𝛾𝛾𝒇𝒇𝒏𝒏−𝐺𝐺2𝒏𝒏̇)×𝒏𝒏+�𝛾𝛾𝒇𝒇𝑴𝑴−𝐺𝐺1𝑴𝑴̇�×𝑴𝑴,                              (4b ) \nwhere γ is the gyromagnetic ratio, and G 1 and G2 are the damping parameters.  Based on  Eq. \n(4), one can numerically  simulate the evolution  of staggered magnetization in arXiv:1808.08664 [cond -mat.mes -hall] \nPage 6 of 19 antiferromagnets  and also obtain a dynamic equation expressed by the unit Néel vector n  \n(see Ref.  [63]  for detail s) [46, 47] , \n(1+𝐺𝐺1𝐺𝐺2)𝒏𝒏̈/𝛾𝛾=2𝐴𝐴ℎ\n𝜇𝜇0(𝛾𝛾𝒇𝒇𝒏𝒏−𝐺𝐺2𝒏𝒏̇).                                         (5)  \nOur analytical solutions  for an AFM skyrmion driven by a magnetic anisotropy gradient  are \nobtained based on Eq. (5). \n \nAnalytical steady -motion  velocity of an AFM skyrmion  driven by a PMA gradient . – In the \nfollowing, only a voltage -controlled PMA  gradient along the x -direction is taken into account \nand other driving sources (e.g., currents) are not considered. Assuming that an AFM \nskyrmion  is rigid and will eventually move steadily  in the thin film, we take the scalar \nproduct of Eq. (5) with ∂ in and then integrate  over the space, it  gives the AFM skyrmion \nvelocity induced by a  PMA gradient  (see Ref.  [63]  for detail s), \n𝑣𝑣𝑥𝑥,AFM =𝛾𝛾𝛾𝛾\n𝜇𝜇0𝑀𝑀𝑆𝑆𝛼𝛼𝑑𝑑𝑑𝑑𝐾𝐾\n𝑑𝑑𝑥𝑥,                                                       (6a)  \n𝑣𝑣𝑦𝑦,AFM =0,                                                                    (6b)  \nwhere u and d are calculated  as, \n𝑑𝑑=∫𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑∂𝑥𝑥𝒏𝒏∙∂𝑥𝑥𝒏𝒏=∫𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑∂𝑦𝑦𝒏𝒏∙∂𝑦𝑦𝒏𝒏,                                     ( 7a) \n𝑢𝑢=∫𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (1−𝑛𝑛𝑧𝑧2),                                                      ( 7b) \nand α = G2/l. Equation (6a) shows that  the AFM  skyrmion moves towards  the area with low er \nmagnetic anisotropy  and the velocity  is proportional to 1/ α and dK/dx . It also means that \napplying an opposite voltage can drive a n AFM skyrmion towards  the inverse direction since \nan opposite voltage results  an inverse dK/dx  [25] . \nIt can be seen from Eq. ( 7) that the values of u and d are determined by the profile of \nAFM skyrmions. T he profile of the skyrmion can be approximated  by [61] , \nCdθ/dr  = − sinθ ,                                                           ( 8) arXiv:1808.08664 [cond -mat.mes -hall] \nPage 7 of 19 where C is a constant , which equals  the domain wall width  parameter  Δ  =  ( A/K)1/2 for the \nstraight domain wall . Then we can find  \nu ≈ 4π RsC, d ≈ 2π( Rs/C+C/Rs),                                               ( 9) \nwhere Rs is the skyrmion radius and sinθ ≠ 0 only for r ≈ Rs [61]. The skyrmion ra dius Rs is \ngiven by  [61]  \n𝑅𝑅𝑠𝑠≈∆\n�2−2𝐷𝐷/𝐷𝐷𝑐𝑐,                                                         ( 10) \nwhere Dc = 4(AK)1/2/π. In order to obtain the coefficient C , Eq. ( 3) is combined with Eq. ( 8) \nand integrated over  one period. Then, C  is obtained as  (see Ref.  [63]  for detail s) \n𝐶𝐶≈∆\n�∆2/𝑅𝑅𝑠𝑠2+1�1/2.                                                           ( 11) \nSubstituting it into Eq. ( 6a), the velocity of skyrmion takes the form  \n𝑣𝑣𝑥𝑥,AFM≈2𝛾𝛾\n𝜇𝜇0𝑀𝑀𝑆𝑆𝛼𝛼𝑑𝑑𝐾𝐾\n𝑑𝑑𝑥𝑥𝑅𝑅𝑠𝑠2\n𝑅𝑅𝑠𝑠2/∆2+2.                                                 ( 12) \nEq. (12) shows  that the velocity of an AFM skyrmion also increases for larger skyrmion s, \nwhich can be resulted from increasing D  or decreasing K . \n \nNumerical  velocity of an AFM skyrmion driven by a PMA gradient . – In order to verify the \nanalytical results, we now simulate the motion of the Néel- type AFM  skyrmion  based on Eq. \n(4). A metastable AFM skyrmion is relaxed initially and then a  PMA gradient is applied. The \nvelocity components are ( vx, vy) = ( Ṙx, Ṙy), where ( Rx, Ry) is the guiding center of skyrmion \nand defined as  \n𝑅𝑅𝑖𝑖=∫𝑑𝑑𝑥𝑥𝑑𝑑𝑦𝑦�𝑖𝑖𝒏𝒏∙�𝜕𝜕𝑥𝑥𝒏𝒏×𝜕𝜕𝑦𝑦𝒏𝒏��\n∫𝑑𝑑𝑥𝑥𝑑𝑑𝑦𝑦�𝒏𝒏∙�𝜕𝜕𝑥𝑥𝒏𝒏×𝜕𝜕𝑦𝑦𝒏𝒏��,   i = x, y.                                      (13)  \nFigure  2 shows vx and Rx of an AFM skyrm ion as functions of time for  dK/dx  = 100 \nGJ/m4 and α = 0.002. In Fig. 2(a), the AFM skyrmion is first accelerated to ~ 450 m/s in  0.2 \nns and then increases slowly  to 504 m/s by t = 0.5 ns. The velocity cannot reach a constant \nvalue due to the change of the skyrmion size  induced by the decreasing K. As shown in Fig. arXiv:1808.08664 [cond -mat.mes -hall] \nPage 8 of 19 2(c), the radius of the AFM skyrmion at t  = 0.5 ns is larger than  that of  the initial state . When \nthe AFM skyrmion moves in  the positive x -direction, the decreasing K  result s in the increase \nof skyrmion size, giving rise to  the slow increase of the speed. Such an effect also exists in \nthe case of FM skyrmion  driven by a magnetic anisotropy  gradient , as reported in Ref. [19]. \nFigure  2(b) show s the dependence of AFM skyrmion  velocity on magnetic anisotropy  K0. \nThe velocity  increases  with the decrease of K0 since a decreasing K 0 results in a larger \nskyrmion , as shown in Fig s. 2(c) and 2(d). T he analytical velocities given by  Eq. ( 12) are  also \nshown in Fig. 2(b) , where R s is adopted as the skyrmion radius at t = 0 ns. It can be  seen that \nthe analytical and numerical velocities are  in good agreement , especially for the large r K0 \nsince the change of skyrmion size is negligible when K 0 > 0.8 MJ/m3. \nFigure 3 shows the dependences of the velocity on the anisotropy gradient and damping \nconstant. To obtain the steady velocity, we adopt K 0 = 0.8 MJ/m3 and α = 0.02 ~ 0.2. The \nAFM skyrmion moves slowly and the change of skyrmion size is negligible. Thus, a steady \nvelocity can be reached at t = 0.2 ns ( see Ref. [63] for detail s). The velocity of an AFM \nskyrmion is proportional to the anisotropy gradient  dK/dx [see Fig. 3(a)] and 1/ α [see Fig. \n3(b)] , as expected from  Eq. (6). It can be seen that numerical results are in  a good agreement \nwith the analytical results. When dK/dx  = 100 GJ/m3 and α = 0.1, the velocity of an AFM \nskyrmion reaches 8.17 m/s. We also simulate the current -induced motion of an AFM \nskyrmion  for the purpose of comparison. It is found that the AFM skyrmion can gain the \nvelocity of 8.17 m/s when the current density j  = 9.1 × 109 A/m2. The simulation details of \ncurrent -induced motion are give n in Ref. [63]. It is worth mentioning that c ompared to the \ncurrent, the VCMA gradient  as a driving source has  some advantages in applications, f or \ninstance,  reducing  the Joule heating . The wedge heterostructures with finite length can be \nused to build the horizontal racetrack memory, as proposed in Ref. [64]. When these devices arXiv:1808.08664 [cond -mat.mes -hall] \nPage 9 of 19 are chained together, the  AFM or FM skyrmion will stop at the joint due to the i ncrease of \nmagnetic anisotropy, which can  also be used to build skyrmion- based diode  in principle  [65] . \nThe velocities  of a FM skyrmion driven by a PMA gradient  are also shown in Fig. 3.  \nThe simulation details are give n in Ref. [63]. It can be seen that t he velocity  of AFM \nskyrmion is much larger than that of FM skyrmion  under  the same PMA gradient. For a FM \nskyrmion, the velocity can be obtained by  solving the Thiele equation [66] (see Ref. [63] for \ndetail s), \n𝑣𝑣𝑥𝑥,FM=𝛼𝛼𝑑𝑑𝐹𝐹𝛾𝛾𝛾𝛾𝐹𝐹\n𝜇𝜇0𝑀𝑀𝑆𝑆[(4𝜋𝜋𝜋𝜋)2+(𝛼𝛼𝑑𝑑𝐹𝐹)2]𝑑𝑑𝐾𝐾𝐹𝐹\n𝑑𝑑𝑥𝑥, 𝑣𝑣𝑦𝑦,FM=4𝜋𝜋𝜋𝜋𝛾𝛾𝛾𝛾𝐹𝐹\n𝜇𝜇0𝑀𝑀𝑆𝑆[(4𝜋𝜋𝜋𝜋)2+(𝛼𝛼𝑑𝑑𝐹𝐹)2]𝑑𝑑𝐾𝐾𝐹𝐹\n𝑑𝑑𝑥𝑥,                  (14)  \nwhere Q is the  ferromagnetic topological number , 𝑢𝑢𝐹𝐹=∫𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 (1−𝑚𝑚𝑧𝑧2) and 𝑑𝑑𝐹𝐹=\n∫𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑∂𝑥𝑥𝒎𝒎∙∂𝑥𝑥𝒎𝒎=∫𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑∂𝑦𝑦𝒎𝒎∙∂𝑦𝑦𝒎𝒎. According  to Eq. (6), the AFM skyrmion speed is \nlarger than the FM sk yrmion speed   �𝑣𝑣𝑥𝑥,FM2+𝑣𝑣𝑦𝑦,FM2=𝛾𝛾𝛾𝛾𝐹𝐹\n𝜇𝜇0𝑀𝑀𝑆𝑆1\n�(4𝜋𝜋𝜋𝜋)2+(𝛼𝛼𝑑𝑑𝐹𝐹)2𝑑𝑑𝐾𝐾𝐹𝐹\n𝑑𝑑𝑥𝑥 with the same \nparameters. Moreover, the directions of AFM and FM skyrmion motion  are significantly \ndifferent. In particular, for a very small damping, a FM skyrmion moves almost along the y-\ndirection, i.e., perpendicular to the gradient direction, similar to the case where the magnetic \nfield gradient is adopted to drive a  FM skyrmion  [16, 67] . For small α , the velocity of a FM \nskyrmion  can be approximated  as 𝑣𝑣𝑥𝑥,FM≈0,𝑣𝑣𝑦𝑦,FM≈𝛾𝛾𝛾𝛾𝐹𝐹\n𝜇𝜇0𝑀𝑀𝑆𝑆4𝜋𝜋𝜋𝜋𝑑𝑑𝐾𝐾𝐹𝐹\n𝑑𝑑𝑥𝑥, which  shows that the FM \nskyrmion moves perpendicular to the gradient direction . The results are consistent with those  \nreported in Ref. [19] . Meanwhile, an AFM skyrmion moves along the  x-direction, i.e., \nparallel to the gradient direction.  \nThe velocities in Fig. 3  show that the AFM skyrmion has two obvious advantages \ncompared to the FM skyrmion. First, the AFM skyrmion has no velocity in the  y-direction ( vy \n= 0) and moves perfectly parall el to the racetrack , so that it will not be destroyed by touching \nsample edges  no matter how fast it moves. However, for the FM skyrmion, there is a \ntransverse dr ift, i.e., the skyrmion Hall effect [68-70], which may cause the skyrmion  to be arXiv:1808.08664 [cond -mat.mes -hall] \nPage 10 of 19 destroyed at the samp le edge [71, 72] . Various ways have been proposed to overcome or \nsuppress the skyrmion Hall effect, such as adopting an antiferromagnetically coupled bila yer \ngeometry  [45], using a curbed racetrack  [72], or applying high PMA  in the racetrack edge  \n[71]. Second, the speed of an AFM skyrmion is larger  than that of a FM skyrmion  under the \nsame driving force.  \nThese two advantages  of the AFM skyrmion are due to the cancellation  of the Magnus \nforce . Since the magnetization s in two sublattices are opposite, a n AFM skyrmion can be \nseen as the combination of  two skyrmions with the opposite ferromagnetic topological \nnumber Q , [1, 6, 42, 43, 73]  \n 𝑄𝑄=−1\n4π∫𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑�𝒎𝒎∙�∂𝑥𝑥𝒎𝒎×∂𝑦𝑦𝒎𝒎��,                                    ( 15) \nwhich equals  ± 1 for an isolated skyrmion. The opposite  Magnus force s (4πQez × v) acted on \nthe skyrmion  in each sublattice are counteracted perfectly, thus there is  no skyrmion Hall \neffect, as reported  in Refs.  [42, 43] . In addition, based on Thiele equation , the velocity of an \nAFM skyrmion is much la rger than that of a FM skyrmion. It should be mentioned that the \nsteady velociti es of AFM  skyrmion s and FM skyrmions  induced by a  magnetic anisotropy \ngradient  are given by the same formula (see Ref. [63] for detail s), Eq. (6), if the Magnus \nforce  acting on the FM skyrmion is balanced by the forces of the surrounding environment . \nHowever, the FM skyrmion will be destroyed by touching the edge when it  moves too fast  \ndue to large Magnus force . Our results can be extended to the antiferromagnetically  exchange \ncoupled bilayer system . As reported in Ref. [45], the Magnus forces acting on the skyrmions \nin the top and bottom FM layers are exactly cancelled when the interlayer AFM exchange \ncoupling is strong enough. Therefore, for this case of strong interlayer AFM exchange \ncoupling, the steady velocity  of the AFM bilayer skyrmion driven by a magnetic anisotropy \ngradient can also be calculated by Eq. (6).  arXiv:1808.08664 [cond -mat.mes -hall] \nPage 11 of 19 In summary, we demonstrate that the AFM skyrmion can be driven by  a magnetic \nanisotropy gradient. This motion has been investigated analytically and numerically. The  \nresults show  that its velocit y is proportional to 1/ α and dK/dx and can reach ~ 500 m/s . We \nalso give an expression which reveal s the effect of skyrmion size on the motion of  an AFM \nskyrmion, it shows  that an AFM  skyrmion  with larger size  obtains  higher velocity . The \nmotion  of AFM and FM skyrmion s driven by an anisotropy gradient are also compared . It is \nfound that the velocity of an AFM skyrmion is much larger than that of a FM skyrmion \nwithout showing skyrmion Hall effect. Our results provide a promising way to manipulate \nAFM textures for future spintronic applications. \n  arXiv:1808.08664 [cond -mat.mes -hall] \nPage 12 of 19 ACKNOWLEDGMENTS  \n \nG.Z. ack nowledges the support by the National Natural Science Foundation of China (Grant \nNos. 51771127, 51571126 and 51772004) of China, the Scientific Research Fund of Sichuan \nProvincial Education Department (Grant Nos. 18TD0010 and 16CZ0006). X.Z. was \nsupported by JSPS RONPAKU (Dissertation Ph.D.) Program. M.E. acknowledges the \nsupport by the Grants -in-Aid for Scientific Research from JSPS KAKENHI (Grant Nos. \nJP18H03676, JP17K05490 and JP15H05854), and also the support by CREST, JST (Grant \nNo. JPMJCR16F1). O.A.T. acknowledges support by the Grants -in-Aid for Scientific  \nResearch (Grant No s. 17K05511 and 17H05173) from MEXT, Japan and by JSPS and RFBR \nunder the Japan- Russian Research Cooperative Program . X.L. acknowledges the support by \nthe Grants -in-Aid for Scientific Research from JSPS KAKENHI (Grant Nos. 17K19074, \n26600041 and 22360122). Y.Z. acknowledges the support by the President's Fund of \nCUHKSZ, the National Natural Science Foundation of China (Grant No. 11574137), and \nShenzhen Fundamental Research Fu nd (Grant Nos. JCYJ20160331164412545 and \nJCYJ20170410171958839). \n  arXiv:1808.08664 [cond -mat.mes -hall] \nPage 13 of 19 REFERENCES  \n \n[1] S. Mühlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A. Neubauer, R. Georgii, \nand P. Böni, Science 323 , 915 (2009).  \n[2] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han, Y. Matsui, N. Nagaosa, and Y. \nTokura, Nature 465, 901 (2010).  \n[3] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z. Zhang, S. Ishiwata, Y. Matsui, \nand Y. Tokura, Nat. Mater. 10, 106 (2011).  \n[4] S. Sek i, X. Z. Yu, S. Ishiwata, and Y. Tokura, Science 336, 198 (2012).  \n[5] O. Boulle, J. Vogel, H. Yang, S. Pizzini, D. de Souza Chaves, A. Locatelli, T. O. \nMenteş, A. Sala, L. D. Buda -Prejbeanu, O. Klein, M. Belmeguenai, Y. Roussigné, A. \nStashkevich, S. M. Ché rif, L. Aballe, M. Foerster, M. Chshiev, S. Auffret, I. M. Miron, \nand G. Gaudin, Nat. Nanotech. 11, 449 (2016).  \n[6] N. Nagaosa and Y. Tokura, Nat. Nanotech. 8, 899 (2013).  \n[7] J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nat. Nanotech. 8, 742 (2013).  \n[8] A. Fert, V. Cros, and J. Sampaio, Nat. Nanotech. 8 , 152 (2013).  \n[9] X. Zhang, G. P. Zhao, H. Fangohr, J. P. Liu, W. X. Xia, J. Xia, and F. J. Morvan, Sci. \nRep. 5, 7643 (2015).  \n[10] F. Jonietz, S. Muhlbauer, C. Pfleiderer, A. Neubauer, W. Munzer, A. Bauer, T. A dams, \nR. Georgii, P. Boni, R. A. Duine, K. Everschor, M. Garst, and A. Rosch, Science 330, \n1648 (2010).  \n[11] X. Z. Yu, N. Kanazawa, W. Z. Zhang, T. Nagai, T. Hara, K. Kimoto, Y. Matsui, Y. \nOnose, and Y. Tokura, Nat. Commun. 3, 988 (2012). \n[12] I. M. Miron, T. Moore, H. Szambolics, L. D. Buda -Prejbeanu, S. Auffret, B. Rodmacq, \nS. Pizzini, J. Vogel, M. Bonfim, A. Schuhl, and G. Gaudin, Nat. Mater. 10, 419 (2011).  \n[13] S. H. Yang, K. S. Ryu, and S. Parkin, Nat. Nanotech. 10, 221 (2015).  \n[14] J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nat. Nanotech. 8 , 839 (2013).  \n[15] X. Zhang, M. Ezawa, D. Xiao, G. P. Zhao, Y. Liu, and Y. Zhou, Nanotechnology 26, \n225701 (2015).  \n[16] S. Komineas and N. Papanicolaou, Phys. Rev. B 92 , 064412 (2015).  \n[17] L. Kong a nd J. Zang, Phys. Rev. Lett. 111, 067203 (2013).  \n[18] C. Ma, X. Zhang, Y. Yamada, J. Xia, M. Ezawa, W. Jiang, Y. Zhou, A. Morisako, and \nX. Liu, arXiv preprint arXiv:1708.02023  (2017). arXiv:1808.08664 [cond -mat.mes -hall] \nPage 14 of 19 [19] R. Tomasello, S. Komineas, G. Siracusano, M. Carpentieri, and G. F inocchio, Phys. \nRev. B 98, 024421 (2018). \n[20] H. Xia, C. Song, C. Jin, J. Wang, J. Wang, and Q. Liu, J. Magn. Magn. Mater. 458, 57 \n(2018).  \n[21] X. Wang, W. L. Gan, J. C. Martinez, F. N. Tan, M. B. A. Jalil, and W. S. Lew, \nNanoscale 10, 733 (2018). \n[22] W. Skowroński, T. Nozaki, Y. Shiota, S. Tamaru, K. Yakushiji, H. Kubota, A. \nFukushima, S. Yuasa, and Y. Suzuki, Appl. Phys. Express 8, 053003 (2015). \n[23] X. Li, K. Fitzell, D. Wu, C. T. Karaba, A. Buditama, G. Yu, K. L. Wong, N. Altieri, C. \nGrezes, N. Kious sis, S. Tolbert, Z. Zhang, J. P. Chang, P. Khalili Amiri, and K. L. \nWang, Appl. Phys. Lett. 110, 052401 (2017).  \n[24] T. Hirai, T. Koyama, A. Obinata, Y. Hibino, K. Miwa, S. Ono, M. Kohda, and D. Chiba, \nAppl. Phys. Express 9, 063007 (2016).  \n[25] D. Chiba, M . Kawaguchi, S. Fukami, N. Ishiwata, K. Shimamura, K. Kobayashi, and T. \nOno, Nat. Commun. 3, 888 (2012).  \n[26] T. Kawabe, K. Yoshikawa, M. Tsujikawa, T. Tsukahara, K. Nawaoka, Y. Kotani, K. \nToyoki, M. Goto, M. Suzuki, T. Nakamura, M. Shirai, Y. Suzuki, and S. Miwa, Phys. \nRev. B 96, 220412(R) (2017). \n[27] E. G. Tveten, A. Qaiumzadeh, and A. Brataas, Phys. Rev. Lett. 112, 147204 (2014).  \n[28] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nat. Nanotech. 11, 231 (2016).  \n[29] C. Marrows, Science 351 , 558 (2016).  \n[30] S. Selzer, U. Atxitia, U. Ritzmann, D. Hinzke, and U. Nowak, Phys. Rev. Lett. 117, \n107201 (2016).  \n[31] O. Gomonay, T. Jungwirth, and J. Sinova, Phys. Rev. Lett. 117, 017202 (2016). \n[32] T. Shiino, S.- H. Oh, P. M. Haney, S.- W. Lee, G. Go, B.- G. Park, and K. -J. Lee, Phys. \nRev. Lett. 117, 087203 (2016). \n[33] K. D. Belashchenko, O. Tchernyshyov, A. A. Kovalev, and O. A. Tretiakov, Appl. \nPhys. Lett. 108, 132403 (2016). \n[34] A. K. Nayak, J. E. Fischer, Y. Sun, B. Yan, J. Karel, A. C. Komarek, C. Shekhar, N. \nKumar, W. Schnelle, J. Kübler, C. Felser, and S. S. P. Parkin, Sci. Adv. 2, e1501870 \n(2016).  \n[35] S. Fukami, C. Zhang, S. DuttaGupta, A. Kurenkov, and H. Ohno, Nat. Mater. 15, 535 \n(2016).  arXiv:1808.08664 [cond -mat.mes -hall] \nPage 15 of 19 [36] T. Kosub, M. Kopte, R. Hühne, P. Appel, B. Shields, P. Maletinsky, R. Hübner, M. O. \nLiedke, J. Fassbender, O. G. Schmidt, and D. Makarov, Nat. Commun. 8, 13985 (2017).  \n[37] K. Olejník, V. Schuler, X. Marti, V. Novak, Z. Kaspar, P. Wadley, R. P. Campion, K. \nW. Edmonds, B. L. Gallagher, J. Garces, M. Baumgartner , P. Gambardella, and T. \nJungwirth, Nat. Commun. 8, 15434 (2017).  \n[38] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Rev. Mod. \nPhys. 90, 015005 (2018).  \n[39] L. Šmejkal, Y. Mokrousov, B. Yan, and A. H. MacDonald, Nat. Phys. 14, 242 (2018).  \n[40] K. Olejník, T. Seifert, Z. Kašpar, V. Novák, P. Wadley, R. P. Campion, M. \nBaumgartner, P. Gambardella, P. Němec, J. Wunderlich, J. Sinova, P. Kužel, M. Müller, \nT. Kampfrath, and T. Jungwirth, Sci. Adv. 4, eaar3566 (2018).  \n[41] P. Wadley, S. R eimers, M. J. Grzybowski, C. Andrews, M. Wang, J. S. Chauhan, B. L. \nGallagher, R. P. Campion, K. W. Edmonds, S. S. Dhesi, F. Maccherozzi, V. Novak, J. \nWunderlich, and T. Jungwirth, Nat. Nanotech. 13, 362 (2018).  \n[42] X. Zhang, Y. Zhou, and M. Ezawa, Sci. R ep. 6, 24795 (2016).  \n[43] J. Barker and O. A. Tretiakov, Phys. Rev. Lett. 116, 147203 (2016).  \n[44] P. Bessarab, D. Yudin, D. Gulevich, P. Wadley, M. Titov, and O. A. Tretiakov, arXiv \npreprint arXiv:1709.04454  (2017).  \n[45] X. Zhang, Y. Zhou, and M. Ezawa, Nat. Commun. 7, 10293 (2016).  \n[46] K. M. D. Hals, Y. Tserkovnyak, and A. Brataas, Phys. Rev. Lett. 106, 107206 (2011).  \n[47] E. G. Tveten, A. Qaiumzadeh, O. A. Tretiakov, and A. Brataas, Phys. Rev. Lett. 110, \n127208 (2013).  \n[48] C. A. Akosa, O. Tretiakov, G . Tatara, and A. Manchon, arXiv preprint \narXiv:1709.02931  (2017). \n[49] S. K. Kim, O. Tchernyshyov, and Y. Tserkovnyak, Phys. Rev. B 92, 020402(R) (2015).  \n[50] S. A. Bender, H. Skarsvåg, A. Brataas, and R. A. Duine, Phys. Rev. Lett. 119, 056804 \n(2017).  \n[51] H. Velkov, O. Gomonay, M. Beens, G. Schwiete, A. Brataas, J. Sinova, and R. A. \nDuine, New J. Phys. 18, 075016 (2016).  \n[52] R. Zarzuela, S. K. Kim, and Y. Tserkovnyak, Phys. Rev. B 97, 014418 (2018). \n[53] I. E. Dzyaloshinskii, Sov. Phys. JETP 5, 1259 (1957).  \n[54] T. Moriya, Phys. Rev. 120, 91 (1960).  \n[55] K. Saiki, J. Phys. Soc. Japan 33, 1284 (1972).  arXiv:1808.08664 [cond -mat.mes -hall] \nPage 16 of 19 [56] M. C. T. Ma, Y. Xie, H. Sheng, S. J. Poon, and A. Ghosh, arXiv preprint \narXiv:1806.06334  (2018). \n[57] R. Zarzuela and Y. Tserkovnyak, Phys. Rev. B 95, 180402(R) (2017).  \n[58] A. N. Bogdanov, U. K. Rößler, M. Wolf, and K. H. Müller, Phys. Rev. B 66, 214410 \n(2002).  \n[59] T. Maruyama, Y. Shiota, T. Nozaki, K. Ohta, N. Toda, M. Mizuguchi, A. A. \nTulapurkar, T. Shinjo, M. Shiraishi, S. Mizukami, Y. Ando, and Y. Suzuki, Nat. \nNanotech. 4, 158 (2009).  \n[60] S. Yoichi, M. Takuto, N. Takayuki, S. Teruya, S. Masashi, and S. Yoshishige, Appl. \nPhys. Express 2, 063001 (2009).  \n[61] S. Rohart and A. Thiaville, Phys. Rev. B 88, 184422 (2013).  \n[62] T. L. Gilbert, IEEE Trans. M agn. 40, 3443 (2004). \n[63] See Supplemental Material at http://link.aps.org/supplemental/  for details on the \nanalytical velocity of an AFM skyrmion and a FM skyrmion driven by a PMA gradient, \nmicromagnetic simulations for the PMA gradient -induced motion a FM skyrmion and \ncurrent -induced motion of an AFM skyrmion.  \n[64] S. S. Parkin, M. Hayashi, and L. Thomas, Science 320, 190 (2008).  \n[65] J. Wang, J. Xia, X. Zhang, G. P. Zhao, L. Ye, J. Wu, Y. Xu, W. Zhao, Z.  Zou, and Y. \nZhou, J. Phys. D: Appl. Phys. 51, 205002 (2018).  \n[66] A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).  \n[67] S.-Z. Lin, Phy. Rev. B 96 , 014407 (2017).  \n[68] K. Litzius, I. Lemesh, B. Krüger, P. Bassirian, L. Caretta, K. Richter, F. Büttner, K. \nSato, O. A. Tretiakov, J. Förster, R. M. Reeve, M. Weigand, I. Bykova, H. Stoll, G. \nSchütz, G. S. D. Beach, and M. Kläui, Nat. Phys. 13, 170 (2017).  \n[69] W. Jiang, X. Zhang, G. Yu, W. Zhang, X. Wang, M. Benjamin Jungfleisch, J. E. \nPearson, X. Cheng, O. Heinonen, K. L. Wang, Y. Zhou, A. Hoffmann, and S. G. E. te \nVelthuis, Nat. Phys. 13, 162 (2017).  \n[70] I. A. Ado, O. A. Tretiakov, and M. Titov, Phys. Rev. B 95, 094401 (2017).  \n[71] P. Lai, G. P. Zhao, H. Tang, N. Ran, S. Q. Wu, J. Xia, X. Zhang, and Y. Zhou, Sci . Rep. \n7, 45330 (2017).  \n[72] I. Purnama, W. L. Gan, D. W. Wong, and W. S. Lew, Sci. Rep. 5, 10620 (2015).  \n[73] O. A. Tretiakov and O. Tchernyshyov, Phys. Rev. B 75, 012408 (2007).  \n  arXiv:1808.08664 [cond -mat.mes -hall] \nPage 17 of 19  \nFIG. 1. (a) A  sketch of the  voltage -controlled magnetic anisotropy device, where the voltage -\ncontrolled magnetic anisotropy K  linearly decreases  with the increase of the spacial  \ncoordinate x for K0 = 0.75 MJ/m3 and dK /dx = 100 GJ/m4. The spin textures of (b) a \nferromagnetic skyrmion and (c)  an antiferromagnetic skyrmion.  The magnetic parameters are \nadopted from Ref. [14] , that is,  MS = 580 kA/m , K0 = 0.75 MJ/m3 ~ 0.85  MJ/m3, A = 15 pJ/m,  \nD = 4 mJ/m2, α = 0.002 ~ 0.2, γ = 2.211 × 105 m/(A s) . In the simulation, we employed a \nsquare lattice model. The model size is 400 × 100 × 0.6 nm3 and mesh size is 1 × 1 × 0.6 nm3. \n  \narXiv:1808.08664 [cond -mat.mes -hall] \nPage 18 of 19  \nFIG. 2. (a) The evolution of velocity vx and position R x for an AFM skyrmion induced by a \nmagnetic anisotrop y gradient  for K0 = 0.75 MJ/m3. (b) The influence of magnetic anisotrop y \nK0 on velocity vx, where symbols stand for the numerical velocity at t = 0.5 ns, whereas  the \nline is calculated using Eq. ( 12). (c) -(d) The top -views of the AFM skyrmion motion  with K0 \n= 0.75 MJ/m3 and 0.8 MJ/m3, respectively.  In the simulation, anisotropy gradient  dK/dx  = \n100 GJ/m4 is used as the driving source, Ahl2 = 10 MJ/m3 is assumed , and the damping \nconstant  α (= G1l = G 2/l) = 0.002. \n  \narXiv:1808.08664 [cond -mat.mes -hall] \nPage 19 of 19  \nFIG. 3. The numerical (symbols) and analytical (lines) velocities of an AFM skyrmion and a \nFM skyrmion. (a) The velocities as functions of anisotropy gradient  dK/dx  for K0 = 0.8 \nMJ/m3 and α = 0.1. (b) The velocities as functions of 1/ α for K0 = 0.8 MJ/m3 and dK/dx  = 100 \nGJ/m4. Here, the numerical velocities are obtained at t = 0.2 ns , whereas the analytical results \nare based on Eq s. (6) and (14) . \n"    },    {        "title": "2312.02654v1.THz_Driven_Coherent_Magnetization_Dynamics_in_a_Labyrinth_Domain_State.pdf",        "content": "THz-Driven Coherent Magnetization Dynamics in a Labyrinth Domain State\nMatthias Riepp,1, 2,∗Andr´ e Philippi-Kobs,2, 3Leonard M¨ uller,2Wojciech Roseker,2Rustam Rysov,2\nRobert Fr¨ omter,4Kai Bagschik,2Marcel Hennes,5Deeksha Gupta,1Simon Marotzke,2, 3Michael Walther,2\nSaˇ sa Bajt,6, 7Rui Pan,2Torsten Golz,2Nikola Stojanovic,8Christine Boeglin,1and Gerhard Gr¨ ubel2,†\n1Institut de Physique et Chimie des Mat´ eriaux de Strasbourg,\nUMR 7504, Universit´ e de Strasbourg, CNRS, 67000 Strasbourg, France\n2Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany\n3Institut f¨ ur Experimentelle und Angewandte Physik,\nChristian-Albrechts-Universit¨ at zu Kiel, Leibnitzstraße 19, 24098 Kiel, Germany\n4Institute of Physics, Johannes Gutenberg-University Mainz, 55099 Mainz, Germany\n5Institut des NanoSciences de Paris, UMR7588 ,\nSorbonne Universit´ e, CNRS, 75005 Paris, France\n6Center for Free-Electron Laser Science CFEL, Deutsches Elektronen-Synchrotron DESY, 22607 Hamburg, Germany\n7The Hamburg Centre for Ultrafast Imaging, 22761 Hamburg, Germany\n8Institute for Optical Sensor Systems, Deutsches Zentrum f¨ ur Luft- und Raumfahrt, Rutherfordstraße 2, 12489 Berlin, Germany\nTerahertz (THz) light pulses can be used for an ultrafast coherent manipulation of the mag-\nnetization. Driving the magnetization at THz frequencies is currently the fastest way of writing\nmagnetic information in ferromagnets. Using time-resolved resonant magnetic scattering, we gain\nnew insights to the THz-driven coherent magnetization dynamics on nanometer length scales. We\nobserve ultrafast demagnetization and coherent magnetization oscillations that are governed by a\ntime-dependent damping. This damping is determined by the interplay of lattice heating and mag-\nnetic anisotropy reduction revealing an upper speed limit for THz-induced magnetization switching.\nWe show that in the presence of nanometer-sized magnetic domains, the ultrafast magnetization\noscillations are associated with a correlated beating of the domain walls. The overall domain struc-\nture thereby remains largely unaffected which highlights the applicability of THz-induced switching\non the nanoscale.\nI. INTRODUCTION\nUnderstanding the magnetization dynamics driven by\nultrashort light pulses is of key importance for developing\nfaster and more energy efficient opto-magnetic memory\ntechnologies. A promising way to a controlled manipu-\nlation of the magnetization in ferromagnetic thin films\non ultrafast time scales is the use of light pulses with\nfrequencies in the terahertz (THz) regime ( νTHz≈0.1–\n10·1012Hz) [1–3]. In contrast to incoherent ultrafast\ndemagnetization induced by femtosecond optical laser\npulses with frequencies in the infrared (IR) regime ( νIR≈\n1014Hz) [4], the electric field component ETHzis capable\nof driving a coherent ultrafast demagnetization with sig-\nnificantly lower energy transfer to the sample [5]. More-\nover, the magnetic field component HTHzmay induce\ncoherent oscillations [6–9] and, at high intensities, even\na switching of the magnetization [10–13]. The possibility\nof exciting coherent magnetization dynamics at THz fre-\nquencies in ferromagnets, i. e., far from the ferromag-\nnetic precession resonance, was explained by the iner-\ntia of the magnetization [14–17]. As a consequence, the\nmagnetization may undergo nutation dynamics, i. e., a\ntrembling of the magnetization vector at THz frequen-\ncies respectively femtosecond time scales. Ultrashort\n∗E-mail: matthias.riepp@ipcms.unistra.fr\n†Present address: European X-Ray Free-Electron Laser Facility\nGmbH, Holzkoppel 4, 22869 Schenefeld, GermanyTHz light pulses therefore promise high-speed and low-\npower-consumption information writing in ferromagnets.\nSo far, experiments mainly addressed THz-driven ul-\ntrafast magnetization dynamics in homogeneously mag-\nnetized thin films, i. e., in the single-domain state [7, 8,\n18–23]. Information on the THz-driven magnetization\ndynamics in non-uniformly magnetized states, such as\nthe nanoscale multi-domain states in Co/Pt multilay-\ners with perpendicular magnetic anisotropy (PMA), is\nstill lacking. A dependence of the THz-driven coher-\nent magnetization dynamics on the magnetic anisotropy\nenergy (MAE) was discovered recently by investigating\nCo thin films with fcc, bcc and hcp crystal structure [20].\nIn this article, we present THz-driven coherent mag-\nnetization dynamics in the labyrinth domain state of a\nCo/Pt multilayer with PMA. We employ time-resolved\nXUV resonant magnetic scattering (tr-XRMS) at the\nfree-electron laser (FEL) FLASH to resolve these co-\nherent dynamics with femtosecond time and nanometer\nspatial resolution [24–28]. The magnetization shows dif-\nferent responses depending on the used THz pump flu-\nence. For low-fluence excitation with a filtered THz spec-\ntrum ( ν <6.0 THz), the magnetization undergoes an ul-\ntrafast quenching and recovery within 1 ps. For high-\nfluence excitation with the full THz spectrum, a step-\nlike quenching within 2 ps occurs followed by oscillatory\ndynamics in resonance with the THz fundamental fre-\nquency ν0= 2.5 THz. The data are consistent with in-\ncoherent and coherent ultrafast magnetization dynamics\ndriven by the ETHz- and HTHz-field components. How-arXiv:2312.02654v1  [cond-mat.mtrl-sci]  5 Dec 20232\nTHz focusing parabolaXUV undulator THz undulatore−Bending \nmagnet Back reflection\nfocusing mirrorTHz beamline \nwith delay stage\nSamplee−\nCCDFrequency ν (THz)100\n10−1\n10−2\n10−3Inorm\nν0\n0246 8 10 12141618\nFigure 1. Schematics of tr-XRMS at the BL3 instrument of FLASH Relativistic electron bunches consecutively\ntraverse the XUV and THz undulator producing intrinsically synchronized pump and probe pulses. In a custom-made end\nstation, time-delayed THz-pump and XUV-probe pulses are focused quasi-collinearly onto the sample via a parabolic mirror\nand a back-reflection multilayer mirror, respectively. Included is the polychromatic THz pump spectrum with a fundamental\nfrequency ν0= 2.5 THz measured by electro-optical sampling (EOS).\never, a time-dependent damping has to be introduced\nwhich is modeled by the interplay of lattice heating and\nPMA reduction. The oscillatory magnetization dynam-\nics are associated with correlated dynamics of the domain\nstate’s form factor, interpreted as a successive broadening\nand narrowing of the domain walls, whereas the overall\ndomain structure is conserved.\nII. RESULTS AND DISCUSSION\nExperimental details The THz-driven magnetiza-\ntion dynamics were studied by tr-XRMS at the BL3 in-\nstrument of FLASH (see Methods). The schematics of\nthe experiment are shown in Fig. 1. The planar electro-\nmagnetic THz undulator at BL3 with nine full periods\nwas tuned to generate pump pulses with a fundamental\nfrequency ν0= 2.5 THz which results in a pump-pulse\nduration of 3 .6 ps. Importantly, the pump pulses con-\ntain a broad frequency spectrum, in particular, also high-\nfrequency components reaching up to the IR regime (see\nFig. 1). We call this the unfiltered THz radiation. From\na pump-pulse intensity of 23 µJ measured by a radiome-\nter [29] and a beam size of 200 ×160µm2measured by\na fluorescent screen at the sample position, the calcu-\nlated pump fluence is FTHz= 92 mJ cm−2. This corre-\nsponds to electric and magnetic field amplitudes ETHz=\n4 MV cm−1andµ0HTHz= 1.4 T, respectively. Alterna-\ntively, a longpass filter that blocks frequency components\nν>∼6.0 THz was inserted in the THz beamline. For this\nfiltered THz radiation, the pump fluence is reduced at\nleast by a factor of four [30].\nA typical labyrinth domain state mz(r) in Co/Pt mul-\ntilayers with PMA is shown in Fig. 2 a. Here, mz=\nMz/Msis the z-component of the magnetization normal-\nized to its value at saturation. Note that for THz pump\npulses incident normally on such an OOP domain state,\nthe Zeeman torque T=m×His maximized. The scat-tered intensity I(q, t=−1 ps), obtained by tr-XRMS\nfrom the labyrinth domain state of the [Co/Pt] 8multi-\nlayer used in this experiment (see Methods), is shown\nin Fig. 2 b. In the kinematical limit using linearly polar-\nized light incident normally on a thin film with PMA, the\nscattered intensity is given by pure charge and pure mag-\nnetic scattering contributions I(q) =Ic(q) +Im(q) [31].\nRecently it was shown that the charge scattering contri-\nbution is orders of magnitude smaller as compared to the\nfirst-order magnetic scattering contribution in the here\ninvestigated region of q-space [32]. Hence, we assume\nIc(q)≈0. The Im(q) were then corrected by dark im-\nages, normalized to the average FEL-pulse intensity and\nmasked from parasitic scattering. We take the azimuthal\naverage of Im(q) which reduces the 2D to a 1D inten-\nsity distribution (see Fig. 2 c). By that, we treat the 2D\nlabyrinth domain state as a 1D chain of up- and down-\nmagnetized domains with average domain characteris-\ntics. For analysis of the resulting magnetic scattering\nintensity Im(q), we employ a Lorentzian empirical fitting\nfunction [32]\nIm(q) =e−2q/qw|{z}\nF(q)2\nm0+m1\u0010\nq−q1\nw1\u00112\n+ 1\n2\n| {z }\nS(q)2. (1)\nThe first term outside of the square brackets is the form\nfactor F(q)2which is associated with the magnetic unit\ncell in real space. It is determined by the domain wall\nparameter qwand accounts for the asymmetric shape of\nIm(q). The term in the square brackets is the magnetic\nstructure factor S(q)2corresponding to the spatial ar-\nrangement of magnetic domains, i. e., the basic magnetic\nlattice in real space. It consists of a linear superposition\nof random uniform spatial fluctuations m0and the first-\norder Lorentzian diffraction peak with amplitude m1,3\na\nc\ndb\nNorm. Int. (arb. units)0.02 nm−1200 nmmz (arb. units)\nmask\n0 0.02 0.04 0.06 0.08\nq (nm−1)00.511.52Norm. Int. (arb. units)\nBloch domain wall Domain DomainUp- Down-Im(q)\nF(q)2S(q)2\nz\nx yETHz\nHTHz\nFigure 2. Processing tr-XRMS data a Fourier-\ntransform holography image of a typical labyrinth domain\nstate mz(r) in Co/Pt multilayers showing up- and down-\nmagnetized domains as dark and bright contrast. Arrows in-\ndicate the propagation directions of the electric and magnetic\nfield components ETHz(t) = (0 , Ey(t),0) and HTHz(t) =\n(Hx(t),0,0). bNormalized magnetic scattering image\nIm(q, t=−1 ps) obtained by tr-XRMS from the labyrinth\ndomain state of the [Co/Pt] 8multilayer used in this exper-\niment. cCorresponding azimuthal average of the magnetic\nscattering intensity Im(q). Included are a fit to the data using\neq. (1) and its individual contributions, i. e., the form factor\nF(q)2and the structure factor S(q)2. The Im(q) and S(q)2\nare normalized to the maximum of Im(q) for clarity. d1D\nillustration of the individual contributions to Im(q) in real\nspace: Bloch domain walls correspond to the magnetic unit\ncell, up- and down-magnetized domains to the magnetic lat-\ntice.\nposition q1and linewidth w1. Let us emphasize that\neq. (1) is purely phenomenological. The same function-\nality, however, has been used and shown to fit scattering\ndata from tr-XRMS up to the fifth diffraction order with\nexcellent accuracy by substituting S(q)2with a sum of\nLorentzian functions [32].\nA fit of eq. (1) to Im(q, t=−1 ps) is shown in\nFig. 2 ctogether with the individual contributions F(q)2andS(q)2. An illustration of the individual contribu-\ntions in real space is given in Fig. 2 d. The exponen-\ntial form factor contribution with qw(t=−1 ps) =\n0.1446±0.0118 nm−1is interpreted as the domain wall\nwidth δm= 2πq−1\nw= 43.4±3.5 nm. Labyrinth domain\nstates in Co/Pt multilayers with PMA exhibit strong\nBloch domain wall character [33] with a width defined\nbyδB=πp\nAex(|K1+K2|)−1[34]. Using the measured\nK1= 19.6 kJ m−3andK2=−159.1 kJ m−3(see Meth-\nods), as well as an exchange stiffness Aex= 23.3 pJ m−1\nin Co/Pt multilayers with PMA and an individual Co-\nlayer thickness of 0 .8 nm [35], we obtain a calculated\nδB= 40.6 nm in good agreement with the δmdetermined\nby XRMS. The magnetic structure of the labyrinth do-\nmain state is characterized by q1(t=−1 ps) = 0 .0466±\n0.0004 nm−1corresponding to an average domain period\nξm= 2π/q1= 135 .3±1.2 nm and w1(t=−1 ps) =\n0.0199±0.0011 nm−1corresponding to a lateral corre-\nlation length λm= 2π/w 1= 316 .1±17.5 nm.\nTHz-driven magnetization dynamics In the fol-\nlowing, we discuss the time evolution of the amplitude\nm1of the magnetic structure factor S(q)2which cor-\nresponds to the z-component of the magnetization mz.\nRelative changes ∆ m1(t)−1 are presented for the case\nof the filtered ( ν<∼6.0 THz) and the unfiltered THz ex-\ncitation in Fig. 3 aandb, respectively, together with\ntheHTHz-(ETHz-)field traces measured by electro-optical\nsampling (EOS) before the respective measurements.\nHere, ∆ m1(t) =m1(t)/⟨m1(t <0)⟩t.\nThe response of the z-component of the magnetization\nto the filtered THz-pump pulses is an ultrafast quench-\ning by 16% within τd≈400 fs followed by an equally fast\nand full recovery (Fig. 3 a). A maximum degree of de-\nmagnetization of 16% agrees well with the observations\nin a 15 nm Co thin film pumped with a comparable flu-\nence and can be explained by ETHz-field driven ultrafast\ndemagnetization [8]. According to time-dependent den-\nsity functional theory (TD-DFT), the ETHz-field drives\na coherent displacement of the electrons accompanied\nby a very efficient spin–orbit-coupling-(SOC-)mediated\ndemagnetization [5]. Thereafter, one could expect a\nstep-like reduction of mzwith each half-cycle of the\nETHzfield, i. e., within τd= 0.5/ν0= 200 fs per demag-\nnetization step for monochromatic THz radiation with\nν0= 2.5 THz. We speculate that the differences in the\nultrafast response originate from the polychromaticity of\nthe THz radiation (0 THz < ν <∼6.0 THz) that causes\na more incoherent demagnetization driven by the dif-\nferent ETHz-field components. Employing the tranfer\nmatrix method, we obtain an absorbed fluence of only\nabout 0 .7 mJ cm−2for the highest frequency component\nν= 6.0 THz. It is known that incoherent ultrafast de-\nmagnetization driven by low-fluence IR laser pulses is\ngoverned by an efficient energy equilibration with the\nlattice on sub-picosecond time scales. Here, the efficient\nenergy transfer among sub-systems could explain why no\nfurther demagnetization steps within the 3 .5 ps pump-\npulse duration but an ultrafast recovery is observed.4\nHTHz (arb. units)\n−0.20.00.2HTHz (arb. units)\n−0.20.00.2Filtered\n−2−1 0 1 2 3 4\nDelay time t (ps)\n−2−1 0 1 2 3 4\nDelay time t (ps)a\nb−0.4−0.20.0Δm1 − 1 Δm1 − 1\n−1.0−0.8−0.6−0.4−0.20.0Unfiltered\nFigure 3. THz-driven magnetization dynamics\naHTHz-field trace determined from electro-optical sam-\npling (EOS) and transient z-component of the magnetiza-\ntion ∆ m1(t)−1 using the filtered THz-pump pulses (inci-\ndent fluence FTHz<23 mJ cm−2).bThe same as abut\nfor the unfiltered THz-pump pulses (incident fluence FTHz≈\n92 mJ cm−2). Gray-shaded areas are fit errors. Vertical dot-\nted lines are guides to the eye.\nFurthermore, the weak magnetic field of the low-fluence\nTHz-pump pulses could explain the absence of HTHz-\nfield induced coherent oscillations of mz. We show in the\nnext paragraph that ∆ m1(t)−1 can be modeled by the\nconvolution of low-fluence incoherent ultrafast demagne-\ntization and strongly damped coherent oscillations due\nto the presence of PMA.\nThe situation completely changes when exciting\nthe Co/Pt multilayer with the unfiltered THz-pump\npulses (Fig. 3 b). Now, mzundergoes a 3-step demag-\nnetization reaching a maximum degree of 75% after 2 ps.\nThe recovery is governed by magnetization oscillationswith an amplitude of about ±20% in resonance with\nthe THz fundamental frequency ν0= 2.5 THz. The in-\ncrease of the maximum degree of demagnetization can\nbe explained by the additional frequency components\nν > 6.0 THz and the associated increase of the pump\nfluence. Employing the transfer matrix method as be-\nfore, we obtain a 10 times higher absorbed fluence for\nν= 20.0 THz which is the highest frequency component\nwith intensity Inorm(ν)>0.01·Inorm(ν0). We note that\nthe pump spectrum contains even higher-frequency com-\nponents up to the IR regime. The step-like demagne-\ntization qualitatively agrees with the ETHz-field driven\ncoherent displacement of electrons accompanied by SOC-\nmediated demagnetization predicted by TD-DFT [5].\nHowever, demagnetization steps with a duration of about\n0.8 ps = 2 /ν0are much longer than predicted, presum-\nably due to a combination of demagnetization processes\ndriven by the various ETHz-field components. The high-\nfrequency components thereby facilitate substantial en-\nergy transfer to the electron- and the spin-system reach-\ning thermal equilibrium with the lattice on picosecond\ntime scales. An onset of the oscillatory magnetization\ndynamics at t= 2 ps is rather surprising, as typically, an\ninstantaneous response ( t= 0) is observed in THz-pump–\nprobe experiments (see, e. g., [7, 8, 22]). In compari-\nson to these publications, we have to consider that the\nCo/Pt multilayer exhibits PMA, i. e., an energetic mini-\nmum of aligning the magnetization along the z-direction.\nWe show in the following paragraph that ∆ m1(t)−1 can\nbe modeled by the convolution of high-fluence incoherent\nultrafast demagnetization and delayed coherent oscilla-\ntions due to a heat-induced reduction of PMA.\nPhenomenological model We model ∆ m1(t) as\na convolution of incoherent ultrafast demagnetization\n∆mi(t) and coherent magnetization oscillations ∆ mc(t)\nconsistent with, e. g., [7, 22, 30]\n∆m1(t)−1 = [(∆ mi(t)−1) Θ( t)]∗∆mc(t).(2)\nHere, Θ( t) is the Heaviside function accounting for the\ndemagnetization onset at t= 0. We treat the incoherent\ncontribution ∆ mi(t) as pure thermal demagnetization in-\nduced by an IR pump pulse with λi= 800 nm. Obviously,\nthis is an oversimplification as the filtered THz spectrum\ndoes not contain any IR components and the unfiltered\nTHz spectrum contains a broad frequency spectrum. In\nour approach we cast all ETHz-field induced contribu-\ntions, may they be coherent or incoherent electronic ex-\ncitations, in one ∆ mi(t) that is comparable to what is\nknown from IR-induced ultrafast demagnetization.\nThe incoherent contribution is simulated within the\nudkm1Dsim toolbox [36] that contains the microscopic\nthree temperature model (M3TM) [37] with heat diffu-\nsion along the sample z-direction (see Methods). We sim-\nulated ∆ mi(t) for a number of fluences and selected the\ntransients that match the experimentally observed max-\nimum degrees of demagnetization. This is the case for\na fluence Fi= 4 mJ cm−2andFi= 18 mJ cm−2when\nusing the filtered and the unfiltered THz-pump pulses,5\na b\nd c\nDelay time t (ps)−2−1 0 1 2 3 4\nDelay time t (ps)−2−1 0 1 2 3 4Dnorm\n0.00.20.40.60.81.0Δmc\n−0.20.00.2−0.20.00.2Δmi− 1\n−1.0−0.8−0.6−0.4−0.20.0\nData24 mJ cm−2\n20 mJ cm−2\n16 mJ cm−2\n14 mJ cm−2\n10 mJ cm−2\n4 mJ cm−2Filtered 18 mJ cm−2Unfiltered \nHTHz(t)\nΔm1 − 1\n−0.8−0.6−0.4−0.20.00.2\nFigure 4. Phenomenological model a Incoherent ultrafast demagnetization ∆ mi(t)−1 determined from M3TM simu-\nlations using Fi= 4–24 mJ cm−2. Magnetization transients that were found to match the experimental data are shown in blue\nand red. bTime-dependent damping Dnorm(t) as given by eq. (3). Details in the text. cCoherent magnetization oscillations\n∆mc(t) =HTHz(t)Dnorm(t).dFinal model for the transient z-component of the magnetization ∆ m1(t)−1 given by a convo-\nlution of the incoherent ( a) and coherent ( c) contributions.\nrespectively. The results from simulating ∆ mi(t) via\nthe M3TM are presented in Fig. 4 a. The electron- and\nphonon-temperature transients are provided in the ex-\ntended data figures.\nThe coherent contribution ∆ mc(t) is modeled via the\nproduct of the HTHz-field trace and a time-dependent\ndamping\nD(t) =e−\u0000\n1−kBTp(t)\nK1(t)V\u0001\nt, (3)\nwhere V= 10×10×10 nm3is the volume of a mag-\nnetic grain (macrospin approximation). The phonon-\ntemperature transients Tp(t) are known from the M3TM\nsimulations and the anisotropy transients are calculated\naccording to [38]\nK1(Tp(t)) =K1m(Tp(t))10. (4)\nWe use m(τ) = [1 −sτ3/2−(1−s)τ5/2]1/3, with the\nreduced temperature τ(t) =Tp(t)/TC, and s= 0.11 for\nfcc Co [39]. The Curie temperature TC= 840 K was\ndetermined by vibrating sample magnetometry after the\nexperiment (see Methods). The calculated K1(Tp(t)) are\nprovided in the extended data figures.\nIn the limit of low fluences, kBTp(t)≪K1(t)Vat all\ntimes, i. e., the pump-induced heating of the lattice istoo weak to induce a substantial reduction of PMA. In\nthis case, D(t) =Dnorm(t) becomes an exponential de-\ncay (Fig. 4 b) and the coherent oscillations ∆ mc(t) are\nstrongly damped (Fig. 4 c). In the limit of high fluences,\nD(t) diverges, which corresponds to the unphysical case\nof strongly amplified oscillations. In case of D(t)>1,\nwe therefore normalize eq. (3) to its value of minimum\nmagnetic anisotropy K1,minfort < t (K1,min) and set\nDnorm(t) = 1 for t > t (K1,min). In other words, Dnorm\ndynamically changes as K1decreases and reaches the\nregime of the undamped coherent oscillations ( Dnorm =\n1) when K1=K1,min(Fig. 4 b). The anisotropy K1de-\ncreases by about 75% within the first 2 ps while the coher-\nent oscillations ∆ mc(t) develop in amplitude (Fig. 4 c).\nThe convolutions of ∆ mi(t) and ∆ mc(t) are presented for\nthe filtered and the unfiltered THz radiation in Fig. 4 d.\nThe model perfectly reproduces the features of both\nmagnetization transients along the entire measured time\nrange. For the unfiltered THz radiation, larger devia-\ntions that exceed the experimental noise at t≈2 ps might\nbe explained by the strong electromagnetic field that is\npredicted to lead to non-linearities in the magnetization\nresponse [19].\nNote that for the case of a sample with negligible MAE,\nthe criteria D(t)>1 holds from the start ( t= 0) and6\nEquilibrium state:a b c\nd\nz z\nDW DW DW DWExcited state:\nDW DW DW DW0.150.20\n0.060.080.100.120.140.160.18qw (nm−1)\n−2 −1 0 1 2 3 4\nDelay time t (ps)0.0450.047\n0.0460.0470.0480.0490.050q1 (nm−1)\n−2 −1 0 1 2 3 4\nDelay time t (ps)0.0200.025\n−2 −1 0 1 2 3 4\nDelay time t (ps)0.0100.0150.0200.025w1 (nm−1)\nUnfilteredFiltered\nFigure 5. THz-driven domain dynamics a andbTransient position q1(t) and width w1(t) of the domain state’s\nstructure factor. cTransient domain-wall parameter qw(t) of the domain state’s form factor. Filtered and unfiltered scenarios\nare shown as blue and red data, respectively. Gray-shaded areas are fit errors. d1D illustration of the equilibrium ( t <0) and\nmaximum excited domain state.\nour model predicts an instantaneous (undamped) coher-\nent response to the HTHzfield as it was observed, e. g.,\nin [7, 8, 22]. Furthermore, it is consistent with the ob-\nservation of an increasing delay of the coherent response\nwith increasing MAE from fcc, bcc to hcp Co [20]. Even\nthough our Dnorm(t) is purely phenomenological, it is\nqualitatively in agreement with a time-dependent nuta-\ntion damping factor derived from combining the time-\ndependent non-equilibrium Green function with the con-\nventional Landau-Lifshitz-Gilbert (LLG) formalism [40].\nTheir generalized LLG equation contains a memory ker-\nnel that describes time retardation effects and originates\nfrom the fact that electron-spin can not follow instanta-\nneously a change in the orientation of the local magnetic\nmoments. It was even suggested in [20] that the coherent\nmagnetization dynamics could be fully described by one\ntime-dependent damping parameter that is qualitatively\nlinked to a stronger electron–phonon scattering at sub-\npicosecond time scales and weaker spin–lattice relaxation\nat longer time scales.\nTHz-driven domain dynamics Finally, we investi-\ngate the effect of the THz-pump pulses on the lateral\ndomain configuration, determined by the position q1and\nwidth w1of the structure factor as well as the domain-\nwall parameter qwof the form factor (Fig. 5 a–c).\nWhen using the filtered THz-pump pulses, constant fit\nparameters q1,w1andqware obtained which is consistent\nwith the fluence threshold for ultrafast domain dynamics\nobserved when using IR pump pulses [41]. The parame-\ntersq1,w1andqweven remain constant for t <2 ps when\nusing high-fluence THz pump pulses (unfiltered) which\ndemonstrates that both the domain structure and thedomain walls maintain their equilibrium size-distribution\non ultrafast time scales. This is qualitatively different\nto the ultrafast q1-shift by 3–6% to smaller values when\nusing high-fluence IR-pump pulses [41]. Originally ex-\nplained by a broadening of the domain walls due to lateral\nsuperdiffusive spin transport, more recent experiments\nsuggest ultrafast domain reconfigurations as an explana-\ntion, with a larger effect in low-symmetry systems like\nlabyrinth domain states [32, 42]. However, no such ultra-\nfast domain reconfigurations can be observed here, even\nfor high-fluence THz pump pulses. The absence of such\nultrafast domain dynamics but rather the existence of a\nwaiting time, that is determined by the time needed to\ncompensate PMA, was reported for stripe domain states\nbefore [43, 44]. For a compensated PMA and in the pres-\nence of small IP magnetic fields, the stripes were found to\nundergo a reorientation along the external field direction.\nUpon compensation of PMA after t≈2 ps, here, the do-\nmain wall parameter qwundergoes oscillatory dynamics\nthat are highly correlated with the magnetization dy-\nnamics in Fig. 3 b. Assuming that qwinversely relates to\nthe Bloch-wall width, this could be interpreted as a suc-\ncessive broadening and narrowing of the Bloch domain\nwalls between 43 nm and 89 nm at maximum. A slight\nincrease of q1within the error of the fit in combination\nwith a sharp drop of w1to almost half its equilibrium\nvalue reveals an increased long-range order during these\ncoherent oscillations from O=q1/w1≈2.3 toO ≈ 3.0 at\nmaximum. A situation where the domain-wall width in-\ncreases while the average domain period remains largely\nthe same is illustrated in Fig. 5 d. A high correlation\nbetween m1(t) and qw(t) for t >2 ps is naturally con-7\nvincing as, for oscillatory dynamics of the magnetization\nvector, a reduction of the z-component of the magnetiza-\ntion has to be associated with an increase of the x- and\ny-components and thus an increase of the domain-wall\ncontribution in tr-XRMS.\nIII. CONCLUSIONS\nIn conclusion, the magnetization of a Co/Pt multilayer\nwith PMA undergoes fluence-dependent dynamics upon\nexcitation by polychromatic THz pump pulses. These\ndynamics can be explained by a convolution of ultra-\nfast demagnetization and coherent magnetization oscil-\nlations with time-dependent damping. For low pump\nfluences (filtered), PMA causes a rapid alignment of mz\nalong the z-direction, i. e., strongly damped coherent os-\ncillations of mz. For high pump fluences (unfiltered),\nPMA undergoes a substantial reduction which enables\nundamped coherent oscillations of mzupon lattice heat-\ning. Our results demonstrate the existence of an upper\nspeed limit for THz-driven magnetization switching in\nferromagnets with PMA, i. e., a limit that is determined\nby the time needed to overcome the anisotropy energy\nbarrier. It will be interesting to see if theoretical calcu-\nlations can confirm a time-dependent nutation damping\nas the one proposed here. A reduction of the mzcom-\nponent during these coherent oscillations is associated\nwith an increase in the mx,ycomponents which, in tr-\nXRMS from a labyrinth domain state, is directly seen\nvia highly correlated dynamics of the domain-wall pa-\nrameter. The overall domain structure thereby remains\nlargely unaffected, showing no signs of spin superdiffusion\nor ultrafast domain rearrangements, which highlights the\napplicability of THz driven magnetization switching on\nthe nanoscale. Our results thereby provide a guidelinefor controlling the THz-driven magnetization dynamics\nby tailoring PMA and changing the pump fluence.\nACKNOWLEDGMENTS\nWe acknowledge DESY (Hamburg, Germany), a mem-\nber of the Helmholtz Association HGF, for the provision\nof experimental facilities. Parts of this research were car-\nried out at FLASH. We thank S D¨ usterer, M Temme and\nthe whole experimental team at FLASH for assistance\nin using the BL3 instrument. Beamtime was allocated\nfor proposal F-20160531. We thank E Jal, N Bergeard\nand B Vodungbo for many fruitful discussions as well\nas D Hrabovsky at the MPBT platform of Sorbonne\nUniversit´ e for his support with the VSM measurements.\nWe acknowledge funding by the Deutsche Forschungsge-\nmeinschaft (DFG) – SFB-925 – project ID 170620586,\nthe Cluster of Excellence ‘Advanced Imaging of Mat-\nter’ of the DFG – EXC-2056 – project ID 390715994,\nthe European Union’s Horizon 2020 research and inno-\nvation programme under the Marie Sk lodowska-Curie\ngrant agreement number 847471 and ANR-20-CE42-\n0012-01(MEDYNA).\nAUTHOR CONTRIBUTIONS\nM. R., A. P.-K., L. M., W. R., R. R., R. F., K. B.,\nM. W., R. P., T. G. and N. S. performed the time-resolved\nexperiments at FLASH and exploited the data. M. R.,\nA. P.-K. and K. B. grew the samples. M. R., A. P.-\nK., S. M. and M. H. performed the MOKE and VSM\nmeasurements. M. R. conducted the simulations and\nwrote the paper. All authors discussed and improved\nthe manuscript.\n[1] T. Kampfrath, K. Tanaka, and K. A. Nelson, Nature Pho-\ntonics 7, 680 (2013).\n[2] J. Walowski and M. M¨ unzenberg, Journal of Applied\nPhysics 120, 140901 (2016).\n[3] A. Barman, G. Gubbiotti, S. Ladak, A. O. Adeyeye,\nM. Krawczyk, J. Gr¨ afe, C. Adelmann, S. Cotofana,\nA. Naeemi, V. I. Vasyuchka, et al. , Journal of Physics:\nCondensed Matter 33, 413001 (2021).\n[4] E. Beaurepaire, J.-C. Merle, A. Daunois, and J.-Y. Bigot,\nPhysical Review Letters 76, 4250 (1996).\n[5] C. Pellegrini, S. Sharma, J. K. Dewhurst, and A. Sanna,\nPhysical Review B 105, 134425 (2022).\n[6] T. Kampfrath, A. Sell, G. Klatt, A. Pashkin, S. M¨ ahrlein,\nT. Dekorsy, M. Wolf, M. Fiebig, A. Leitenstorfer, and\nR. Huber, Nature Photonics 5, 31 (2011).\n[7] S. Bonetti, M. Hoffmann, M.-J. Sher, Z. Chen, S.-H.\nYang, M. Samant, S. Parkin, and H. D¨ urr, Physical Re-\nview Letters 117, 087205 (2016).\n[8] M. Shalaby, A. Donges, K. Carva, R. Allenspach, P. M.\nOppeneer, U. Nowak, and C. P. Hauri, Physical ReviewB98, 014405 (2018).\n[9] K. Neeraj, N. Awari, S. Kovalev, D. Polley,\nN. Zhou Hagstr¨ om, S. S. P. K. Arekapudi, A. Semisalova,\nK. Lenz, B. Green, J.-C. Deinert, et al. , Nature Physics\n17, 245 (2021).\n[10] C. H. Back, D. Weller, J. Heidmann, D. Mauri, D. Guar-\nisco, E. L. Garwin, and H. C. Siegmann, Physical Review\nLetters 81, 3251 (1998).\n[11] I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C.\nSiegmann, J. St¨ ohr, G. Ju, B. Lu, and D. Weller, Nature\n428, 831 (2004).\n[12] D. Polley, M. Pancaldi, M. Hudl, P. Vavassori,\nS. Urazhdin, and S. Bonetti, Journal of Physics D: Ap-\nplied Physics 51, 084001 (2018).\n[13] K. Neeraj, M. Pancaldi, V. Scalera, S. Perna,\nM. d’Aquino, C. Serpico, and S. Bonetti, Physical Re-\nview B 105, 054415 (2022).\n[14] M.-C. Ciornei, J. M. Rub´ ı, and J.-E. Wegrowe, Physical\nReview B 83, 020410 (2011).8\n[15] S. Bhattacharjee, L. Nordstr¨ om, and J. Fransson, Physi-\ncal Review Letters 108, 057204 (2012).\n[16] E. Olive, Y. Lansac, M. Meyer, M. Hayoun, and J.-E.\nWegrowe, Journal of Applied Physics 117, 213904 (2015).\n[17] R. Mondal, M. Berritta, A. K. Nandy, and P. M. Oppe-\nneer, Physical Review B 96, 024425 (2017).\n[18] C. Vicario, C. Ruchert, F. Ardana-Lamas, P. M. Derlet,\nB. Tudu, J. Luning, and C. P. Hauri, Nature Photonics\n7, 720 (2013).\n[19] M. Hudl, M. d’Aquino, M. Pancaldi, S.-H. Yang, M. G.\nSamant, S. S. Parkin, H. A. D¨ urr, C. Serpico, M. C.\nHoffmann, and S. Bonetti, Physical Review Letters 123,\n197204 (2019).\n[20] V. Unikandanunni, R. Medapalli, M. Asa, E. Albisetti,\nD. Petti, R. Bertacco, E. E. Fullerton, and S. Bonetti,\nPhysical Review Letters 129, 237201 (2022).\n[21] R. Rouzegar, L. Brandt, L. N´ advorn´ ık, D. A. Reiss, A. L.\nChekhov, O. Gueckstock, C. In, M. Wolf, T. S. Seifert,\nP. W. Brouwer, et al. , Physical Review B 106, 144427\n(2022).\n[22] K. Neeraj, A. Sharma, M. Almeida, P. Matthes,\nF. Samad, G. Salvan, O. Hellwig, and S. Bonetti, Ap-\nplied Physics Letters 120, 102406 (2022).\n[23] R. Salikhov, I. Ilyakov, L. K¨ orber, A. K´ akay, R. A.\nGallardo, A. Ponomaryov, J.-C. Deinert, T. V. A. G.\nde Oliveira, K. Lenz, J. Fassbender, et al., Nature Physics\n19, 529 (2023).\n[24] J. St¨ ohr and H. C. Siegmann, Magnetism: from funda-\nmentals to nanoscale dynamics , Springer series in solid-\nstate sciences No. 152 (Springer, Berlin ; New York,\n2006).\n[25] C. Gutt, L.-M. Stadler, S. Streit-Nierobisch, A. P. Man-\ncuso, A. Schropp, B. Pfau, C. M. G¨ unther, R. K¨ onnecke,\nJ. Gulden, B. Reime, et al., Physical Review B 79, 212406\n(2009).\n[26] C. Gutt, S. Streit-Nierobisch, L.-M. Stadler, B. Pfau,\nC. M. G¨ unther, R. K¨ onnecke, R. Fr¨ omter, A. Kobs,\nD. Stickler, H. P. Oepen, et al. , Physical Review B 81,\n100401(R) (2010).\n[27] B. Vodungbo, J. Gautier, G. Lambert, A. B. Sardinha,\nM. Lozano, S. Sebban, M. Ducousso, W. Boutu, K. Li,\nB. Tudu, et al. , Nature Communications 3, 999 (2012).\n[28] T. Wang, D. Zhu, B. Wu, C. Graves, S. Schaffert, T. Ran-\nder, L. M¨ uller, B. Vodungbo, C. Baumier, D. P. Bern-\nstein, et al. , Physical Review Letters 108, 267403 (2012).\n[29] R. Pan, E. Zapolnova, T. Golz, A. J. Krmpot, M. D.\nRabasovic, J. Petrovic, V. Asgekar, B. Faatz, F. Tavella,\nA. Perucchi, et al. , Journal of Synchrotron Radiation 26,\n700 (2019).\n[30] X. Liu, E. Jal, R. Delaunay, R. Jarrier, G. S. Chiuzbaian,\nG. Malinowski, T. Golz, E. Zapolnova, R. Pan, N. Sto-\njanovic, et al. , Applied Sciences 12, 1323 (2022).\n[31] J. B. Kortright, Journal of Electron Spectroscopy and\nRelated Phenomena 189, 178 (2013).\n[32] D. Zusin, E. Iacocca, L. Le Guyader, A. H. Reid, W. F.\nSchlotter, T.-M. Liu, D. J. Higley, G. Coslovich, S. F.\nWandel, P. M. Tengdin, et al. , Physical Review B 106,\n144422 (2022).\n[33] M. Riepp, Multi-Scale Study of All-Optically Induced\nMagnetization Dynamics in Co/Pt Multilayers Utilizing\nTR-mSAXS at FEL Sources , Ph.D. thesis, Universit¨ at\nHamburg (2021).\n[34] H. Tr¨ auble, O. Boster, H. Kronm¨ uller, and A. Seeger,\nPhysica Status Solidi (b) 10, 283 (1965).[35] P. J. Metaxas, J. P. Jamet, A. Mougin, M. Cormier,\nJ. Ferr´ e, V. Baltz, B. Rodmacq, B. Dieny, and R. L.\nStamps, Physical Review Letters 99, 217208 (2007).\n[36] D. Schick, Computer Physics Communications 266,\n108031 (2021).\n[37] B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf,\nM. F¨ ahnle, T. Roth, M. Cinchetti, and M. Aeschlimann,\nNature Materials 9, 259 (2010).\n[38] J.-Y. Bigot, M. Vomir, L. Andrade, and E. Beaurepaire,\nChemical Physics 318, 137 (2005).\n[39] M. D. Kuz’min, Physical Review Letters 94, 107204\n(2005).\n[40] U. Bajpai and B. K. Nikoli´ c, Physical Review B 99,\n134409 (2019).\n[41] B. Pfau, S. Schaffert, L. M¨ uller, C. Gutt, A. Al-\nShemmary, F. B¨ uttner, R. Delaunay, S. D¨ usterer,\nS. Flewett, R. Fr¨ omter, et al. , Nature Communications\n3, 1100 (2012).\n[42] N. Zhou Hagstr¨ om, R. Jangid, M. Madhavi, D. Turenne,\nJ. A. Brock, E. S. Lamb, B. Stoychev, J. Schlappa,\nN. Gerasimova, B. Van Kuiken, et al. , Physical Review\nB106, 224424 (2022).\n[43] N. Bergeard, S. Schaffert, V. L´ opez-Flores, N. Jaouen,\nJ. Geilhufe, C. M. G¨ unther, M. Schneider, C. Graves,\nT. Wang, B. Wu, et al. , Physical Review B 91, 054416\n(2015).\n[44] V. L´ opez-Flores, M.-A. Mawass, J. Herrero-Albillos,\nA. A. Uenal, S. Valencia, F. Kronast, and C. Boeglin,\nJournal of Physics: Condensed Matter 32, 465801 (2020).\n[45] K. Tiedtke, A. Azima, N. von Bargen, L. Bittner, S. Bon-\nfigt, S. D¨ usterer, B. Faatz, U. Fr¨ uhling, M. Gensch,\nC. Gerth, et al. , New Journal of Physics 11, 023029\n(2009).\n[46] K. J. Kapcia, V. Tkachenko, F. Capotondi, A. Licht-\nenstein, S. Molodtsov, L. M¨ uller, A. Philippi-Kobs,\nP. Piekarz, and B. Ziaja, npj Computational Materials\n8, 212 (2022).\n[47] M. Riepp, L. M¨ uller, A. Philippi-Kobs, M. Walther,\nW. Roseker, R. Rysov, S. Marotzke, K. Bagschik,\nJ. Buck, R. Fr¨ omter, et al. , Journal of Physics: Con-\nference Series 2380 , 012134 (2022).\n[48] L. M¨ uller, M. Walther, M. Riepp, A. Philippi-Kobs,\nW. Jo, W. Roseker, K. Bagschik, R. Fr¨ omter, R. Pan,\nD. Lott, et al. , Journal of Physics: Conference Series\n2380 , 012081 (2022).\n[49] G. Winkler, A. Kobs, A. Chuvilin, D. Lott, A. Schreyer,\nand H. P. Oepen, Journal of Applied Physics 117, 105306\n(2015).\n[50] K. C. Kuiper, T. Roth, A. J. Schellekens, O. Schmitt,\nB. Koopmans, M. Cinchetti, and M. Aeschlimann, Ap-\nplied Physics Letters 105, 202402 (2014).\n[51] W. M. Haynes, ed., CRC Handbook of Chemistry and\nPhysics , 95th ed. (CRC Press, Boca Raton, 2014).\n[52] H. Ftouni, C. Blanc, D. Tainoff, A. D. Fefferman, M. De-\nfoort, K. J. Lulla, J. Richard, E. Collin, and O. Bourgeois,\nPhysical Review B 92, 125439 (2015).\n[53] W. S. M. Werner, K. Glantschnig, and C. Ambrosch-\nDraxl, Journal of Physical and Chemical Reference Data\n38, 1013 (2009).\n[54] D. Zahn, F. Jakobs, H. Seiler, T. A. Butcher, D. Engel,\nJ. Vorberger, U. Atxitia, Y. W. Windsor, and R. Ern-\nstorfer, Physical Review Research 4, 013104 (2022).9\nTIME-RESOLVED XUV RESONANT MAGNETIC\nSCATTERING (TR-XRMS)\nFor the tr-XRMS experiment, FLASH was operated in\nthe single-bunch mode providing 60 fs XUV probe pulses\nat a repetition rate of 10 Hz. The XUV undulator was\ntuned to generate XUV probe pulses with an average\nSASE spectrum centered around λXUV= 20.8 nm, i. e., a\nphoton energy EXUV = 59.6±0.6 eV in resonance with\nthe Co M2,3absorption edge [45]. Higher harmonics of\nthe FEL spectrum were blocked by a Si and Zr solid\nstate filter which, in combination with the back-reflection\nfocusing mirror, attenuate the probe-pulse intensity to\nabout 0 .037µJ. With a beam size of 52 ×40µm2, the\ncalculated probe fluence is 2 .2 mJ cm−2. As expected for\nsuch a moderate fluence, no XUV-induced demagnetiza-\ntion nor XUV-induced permanent domain modifications\nwere observed [46, 47]. A THz beam with an about four\ntimes larger diameter than the XUV beam ensured homo-\ngeneous excitation of the probed area. Diagnostic tools\non the sample holder allowed for measuring coarse tempo-\nral as well as spatial overlap of the two beams at the sam-\nple position [33]. The scattered intensity was recorded by\na CCD with 2048 ×2048 pixels and a pixel size of 13 .5µm.\nA beamstop-photodiode was installed centimeters from\nthe detector to block the intense direct FEL beam and,\nat the same time, monitor FEL-intensity fluctuations for\nnormalization of the data [48]. The scattering statistics\nwere improved by binning 4 ×4 pixels and accumulating\n50 FEL pulse exposures in one exposure of the CCD.\nSAMPLE PROPERTIES\nThe sample used in this study was a ferromagnetic\nPt(2.0)/[Co(0.8)/Pt(0.8)] 8/Pt(5.0) multilayer grown on\na Si3N4(50.0) multi-membrane substrate using sputtering\ntechniques (numbers in nanometer). Structural investi-\ngations of Co/Pt multilayers that were fabricated in the\nsame way revealed polychrystallinity with pronounced\n(1 1 1) texture and a grain size of about 10 nm [49].\nThe first and second-order magnetic anisotropy con-\nstants K1,2were determined by magneto-optical Kerr ef-\nfect (MOKE) in polar and longitudinal geometry. Polar\nMOKE measurements revealed magnetic easy-axis be-\nhavior along the OOP direction with a coercive field\nµ0Hc≈25 mT and a saturation field µ0Hs≈150 mT.\nLongitudinal MOKE revealed magnetic hard-axis behav-\nior along the IP direction. K1,2were determined by fit-\nting the (inverted) hard-axis hysteresis loop with\nµ0HIP(m∥) =2K1\nMsm∥+4K2\nMsm3\n∥, (5)\nwhere Ms= 1.4·106A m−1is the saturation magne-\ntization in bulk Co at T= 0 K and m∥is the re-\nduced magnetization component parallel to HIP. A fit\nof eq. (5) to the data yields K1= 19 .6±4.7 kJ m−3Table I. Material-specific parameters used for the M3TM sim-\nulations (∗assumtion)\nCo Pt Si 3N4\nCe(J kg−1K−1) 0.0734 Te[50] 0 .0335 Te[50] 0 .0100 T∗\ne\nCp(J kg−1K−1) 421 [51] 133 [51] 700 [51]\nκe(W m−1K−1) 20∗20∗20∗\nκp(W m−1K−1) 100 [51] 71 .6 [51] 2 .5 [52]\nρ(kg m−3) 8860 [51] 21500 [51] 3190 [51]\nn+ik 2.53 + 4 .88i[53] 0 .60 + 8 .38i[53] 2 .00 [51]\nandK2=−159.1±3.7 kJ m−3. The MOKE measure-\nments and fit to the data are provided in the extended\ndata figures. Prior to the FEL beamtime, the sample\nwas exposed to alternating OOP magnetic field cycles\nwith decreasing amplitude and µ0Hmax= 1 T to gener-\nate a labyrinth domain state mz(r) close to the magnetic\nground state.\nAfter the experiment, the temperature dependence of\nthe saturation magnetization Ms(T) was measured em-\nploying vibrating sample magnetometry (VSM) in an ex-\nternal magnetic field µ0HIP= 500 mT. The temperature\nwas increased from T= 300 K to T= 950 K at a rate\n∆T= 10 K min−1. The Curie temperature TC≈840 K\nwas determined by a linear extrapolation of Ms(T) at\nhigh temperatures. The VSM measurement and the fit\nto the data are provided in the extended data figures.\nM3TM SIMULATIONS\nIncoherent ultrafast demagnetization is simulated\nwithin the udkm1Dsim toolbox [36] that contains the mi-\ncroscopic three temperature model (M3TM) as proposed\nby B. Koopmans et al. [37], including heat diffusion along\nthe sample z-direction\nCeρ∂Te\n∂t=∂\n∂z\u0012\nκe∂Te\n∂z\u0013\n−Gep(Te−Tp) +S(z, t)\nCpρ∂Tp\n∂t=∂\n∂z\u0012\nκp∂Tp\n∂z\u0013\n+Gep(Te−Tp) (6)\n∂mi\n∂t=Rm iTp\nTC\u0012\n1−coth\u0012miTC\nTe\u0013\u0013\n.\nThe first two differentials describe the electron- and\nphonon-temperature transients, respectively, where Ce\nand Cpare the heat capacities, κeand κpare the\nthermal conductivities, Gepis the electron–phonon cou-\npling parameter and ρis the density. The initial\nheating of the electron system is given by the laser\nsource term S(z, t). Instead of a spin-temperature tran-\nsient, the M3TM considers a magnetization transient\nthat depends on TeandTp, with a shape defined by\nR= 8asfGepkBT2\nCVatµBµ−1\natE−2\nD. Here, asf= 0.15 is\nthe spin-flip probability, kBis the Boltzmann constant,\nTC= 840 K is the Curie temperature, Vat= 4πr3\nat/310\nis the atomic volume with atomic radius rat= 1.35˚A,\nµat/µB= 1.72 is the atomic magnetic moment in units\nof the Bohr magneton and ED= 0.0357 eV is the De-\nbye energy of Co [37]. For the electron–phonon cou-\npling parameter we take a constant value of Gep=\n1.5·1018W m−3K−1in Co [54]. The udkm1Dsim tool-\nbox yields a reflectivity of 85 .6% and a transmission of\n4.5% at λi= 800 nm, calculated by the transfer matrix\nmethod including multilayer absorption.\nWithin the udkm1Dsim toolbox, in a first step,\nthe Pt(2.0)/[Co(0.8)/Pt(0.8)] 8/Pt(6)/Si 3N4(50) sample\nstructure is generated as a 1D amorphous multilayer with\nmaterial-specific properties for each subsystem (see Ta-\nble I). In a second step, the laser source term S(z, t) is de-\nfined as a delta-like pulse of high frequency ( λi= 800 nm)with fluence Fi= 4–24 mJ cm−2. Note that the influence\nof the pump-pulse duration of 3 .6 ps is taken into account\nvia the coherent contribution ∆ mc(t). In the final step,\ntheudkm1Dsim toolbox calculates spatio-temporal heat-\nmaps of the electron temperature, phonon temperature\nand magnetization for a certain delay range by solving\neq. (6) with an ODE solver. The Te(t),Tp(t) and ∆ mi(t)\nare obtained by taking the spatial average along the z-\ndirection. The Te(t),Tp(t) are provided in the extended\ndata figures.\nEXTENDED DATA FIGURES11\na\nbc\nData\nFitData\nFit\n-0.40.00.4ε (mrad)\n-1000 0 1000\nμ0HIP (mT)\n-404Φ (mrad)\n-300 0 300\nμ0HOOP (mT)\n200 300 400 500 600 700 800 900 10000.00.20.40.60.81.01.21.4\nMs (106 A/m)\nT (K)\nFigure 6. Static magnetic properties of the [Co/Pt] 8multilayer a Polar and blongitudinal MOKE at room\ntemperature. The solid line in bis a fit to the inverted data µ0HIP(ε) for small ε(details in the main article). cTemperature\ndependence of the sponatneous magnetization measured by VSM in external magnetic field µ0HIP= 500 mT. The Curie\ntemperature TC≈840 K is determined by a linear extrapolation at high temperatures.\nDelay time t (ps)−2−1 0 1 2 3 4b\nTp (K)\n300400500600700a\nTe (K)\n300700110015001900\n24 mJ cm−2\n20 mJ cm−2\n18 mJ cm−2\n16 mJ cm−2\n14 mJ cm−2\n10 mJ cm−2\n4 mJ cm−2\nDelay time t (ps)−2−1 0 1 2 3 4c\n05101520K1 (kJ m−3)\nFigure 7. Results from M3TM simulations of the [Co/Pt] 8multilayer a Electron-temperature transient Te(t) and\nbphonon-temperature transient Tp(t) for fluences Fi= 4–24 mJ cm−2. The transients are extracted from spatio-temporal heat\nmaps averaged along the sample z-direction using the udkm1Dsim toolbox. cFirst-order magnetic anisotroy transient K1(t)\ncalculated as described in the main article."    }]